Interactions in solvents and solutions

8

Interactions in Solvents andSolutions

Jacopo Tomasi, Benedetta Mennucci, Chiara CappelliDipartimento di Chimica e Chimica Industriale, Università di Pisa, It aly

8.1 SOLVENTS AND SOLUTIONS AS ASSEMBLIES OF INTERACTING MOLECULESA con ve nient start ing point for the ex po si tion of the top ics col lected in this chap ter is givenby a naïve rep re sen ta tion of liq uid sys tems.

Ac cord ing to this rep re sen ta tion a liq uid at equi lib rium is con sid ered as a large as sem -bly of mol e cules un der go ing in ces sant col li sions and ex chang ing en ergy among col lid ingpart ners and among in ter nal de grees of free dom. The par ti cles are dis or dered at large scale,but of ten there is a lo cal or der that fades away. So lu tions may be en closed within this rep re -sen ta tion.

The col lec tion of par ti cles con tains at least two types of mol e cules − those hav ing ahigher mo lar frac tion are called the sol vent − the oth ers the sol ute. Col li sions and ex changeof en ergy pro ceed as in pure liq uids, lo cal or der ing may be dif fer ent, be ing ac tu ally de pend -ent on the prop er ties of the mol e cule on which at ten tion is fo cused, and on those of thenearby mol e cules.

Liq uids at a bound ary sur face re quire more spec i fi ca tions to be en closed in the rep re -sen ta tion. There is an other phase to con sider which can be ei ther a solid, an other liq uid, or agas. Fur ther spec i fi ca tions must be added to char ac ter ize a spe cific bound ary sys tem (seesec tion 8.8), but here it is suf fi cient to stress that the dy namic col li sion pic ture and the oc -cur rence of lo cal or der ing is ac cept able even for the liq uid por tion of a bound ary.

This naïve de scrip tion of liq uid sys tems ac tu ally rep re sents a model − the ba sic modelto de scribe liq uids at a lo cal scale. As it has been here for mu lated, it is a clas si cal model: usehas been made of phys i cal clas si cal con cepts, as en ergy, col li sions (and, im plic itly, clas si cal mo ments), spa tial or der ing (i.e., dis tri bu tion of el e ments in the space).

It is clear that the model, as for mu lated here, is se verely in com plete. Noth ing has beensaid about an other as pect that surely has a re mark able im por tance even at the level of naïverep re sen ta tions: mol e cules ex ert mu tual in ter ac tions that strongly de pend on their chem i calcom po si tion.

To say some thing more about mo lec u lar in ter ac tions, one has to pass to a quan tum de -scrip tion.

Quan tum me chan ics (QM) is uni ver sally ac knowl edged as the ap pro pri ate the ory totreat ma te rial sys tems at the level of phe nom ena of in ter est to chem is try. There fore quan -tum me chan ics is the le git i mate the ory level at which to treat liq uids.

We re call here the open ing state ment of a fa mous book on quan tum chem is try:1 “In sofar as quan tum me chan ics is cor rect, chem i cal ques tions are prob lems in ap plied math e mat -ics.” Actually, the math e mat ics to ap ply to liq uid sys tems is a hard nut to crack. For tu natelya se quence of many, but rea son able, ap prox i ma tions can be in tro duced. We shall con siderand ex ploit them in the fol low ing sec tion of this chap ter.

It is im por tant to stress one point be fore be gin ning. The naïve pic ture we have sum ma -rized can be re cov ered with out much dif fi culty in the quan tum for mu la tion. This will be asemiclassical model: the iden tity of the con sti tut ing par ti cle is pre served; their mo tions, aswell as those of nu clei within each mol e cule, are treated as in clas si cal me chan ics: the quan -tum meth ods add the nec es sary de tails to de scribe in ter ac tions. This semiclassical quan tumde scrip tion can be ex tended to treat prob lems go ing be yond the pos si bil i ties of purely clas -si cal mod els, as, for ex am ple, to de scribe chem i cal re ac tions and chem i cal equi lib ria in so -lu tion. The lim its be tween the clas si cal and quan tum parts of the model are quite flex i ble,and one may shift them in fa vor of the quan tum part of the model to treat some spe cific phe -nom e non, or in fa vor of the clas si cal part, to make the de scrip tion of larger classes of phe -nom ena faster.

We shall en ter into more de tails later. It is suf fi cient here to un der line that the choice of us ing a quan tum ap proach as ref er ence is not a ca price of the o re ti cians: it makes de scrip -tions (and pre dic tions) safer and, even tu ally, sim pler.

We shall start with the in tro duc tion of some ba sic sim pli fi ca tions in the quan tummodel.

8.2 BASIC SIMPLIFICATIONS OF THE QUANTUM MODELThe quan tum me chan i cal de scrip tion of a ma te rial sys tem is ob tained as so lu tion of the per -ti nent Schrödinger equa tions.

The first Schrödinger equa tion is the fa mous equa tion ev ery body knows:

( ) ( ) ( )H x x E xΨ Ψ= [8.1]

The func tion Ψ(x) de scribes the “state of the sys tem” (there are many states for each sys -tem). It ex plic itly de pends on a set of vari ables, col lec tively called x, that are the co or di nates of the par ti cles con sti tut ing the sys tem (elec trons and nu clei). E is a num ber, ob tained bysolv ing the equa tion, which cor re sponds to the en ergy of the sys tem in that state. H(x),called Hamiltonian, tech ni cally is an op er a tor (i.e., a math e mat i cal con struct act ing on thefunc tion placed at its right to give an other func tion). Eq. [8.1] is an eigenvalue func tion:among the in fi nite num ber of pos si ble func tions de pend ing on the vari ables x, only a fewhave the no ta ble prop erty of giv ing, when H(x) is ap plied to them, ex actly the same func -tion, mul ti plied by a num ber. To solve eq. [8.1] means to find such func tions.

The first step in the se quence of op er a tions nec es sary to reach a de scrip tion of prop er -ties of the sys tem is to give an ex plicit for mu la tion of the Hamiltonian. This is not a dif fi culttask; of ten its for mu la tion is im me di ate. The prob lems of “ap plied math e mat ics” are re latedto the so lu tion of the equa tion, not to the for mu la tion of H.

The sec ond Schrödinger equa tion adds more de tails. It reads:

388 Jacopo Tomasi, Benedetta Mennucci, Chiara Cappelli

i£ ( ) ( )∂∂t

x t H x tΨ Ψ, ,= [8.2]

In prac tice, it ex presses how the state Ψ(x) of the sys tem evolves in time (i.e., it gives ∂Ψ(x)/∂t when Ψ(x) is known).

Note that in eq. [8.1] the time t was not in cluded among the pa ram e ters de fin ing H(and Ψ): with eq. [8.1] we are look ing at sta tion ary states, not de pend ing on time. In spite ofthis, eq. [8.1] may be ap plied to liq uids, which are char ac ter ized by a con tin u ous dy namicex change of en ergy through col li sions, and by a con tin u ous dis place ment of the con sti tut ing mol e cules. This is not a prob lem for the use of the time in de pend ent for mu la tion of theSchrödinger equa tion for liq uids: eq. [8.1] ac tu ally in cludes ki netic en ergy and all re lateddy namic as pects. A di rect use of eq. [8.1] for liq uids means to treat liq uids at equi lib rium.We shall not make ex plicit use of the sec ond equa tion, even when the nonequilibrium prob -lem is con sid ered; the use of the semiclassical ap prox i ma tion per mits us to treat time de -pend ent phe nom ena at the clas si cal level, sim pler to use.

Let us come back to the ex pres sion of H(x). The num ber of pa ram e ters within (x) is ex -ceed ingly large if the sys tem is a liq uid. For tu nately, things may be sim pli fied by us ingfactorization tech niques.

When a sys tem con tains two sub sys tems (say A and B) that do not in ter act, theHamiltonian can be par ti tioned so:

( ) ( ) ( )H x H x H xA A B B= + [8.3]

This rig or ously leads to a factorization of Ψ:

( ) ( ) ( )Ψ Ψ Ψx x xA A B B= + [8.4]

and to a par ti tion of the en ergy:

E E EA B= + [8.5]

Equa tion [8.1] is in con se quence trans formed, with out sim pli fi ca tions, into two sim -pler equa tions:

( ) ( ) ( )H x x E xA A A A A A AΨ Ψ= [8.6a]

( ) ( ) ( )H x x E xB B B B B B BΨ Ψ= [8.6b]

In prac tice, there are al ways in ter ac tions be tween A and B. The Hamiltonian may al -ways be writ ten as:

( ) ( ) ( ) ( )H x H x H x H xA A B B AB= + + [8.7]

The rel a tive mag ni tude of the cou pling term per mits us to dis tin guish among three no -ta ble cases.

1) If the cou pling term is very small, it may be ne glected, com ing back to equa tions [8.6].

8.2 Basic simplifications of quantum model 389

2) If it is small but not neg li gi ble, it may be used to cor rect the so lu tions given by equa tions [8.6] with the use of ap pro pri ate math e mat i cal tools: the fac tor ized for mu la tion with cor rec tions con tin ues to be sim pler than the orig i nal one.

3) There are cases in which this factorization is prof it ably used, in spite of the fact that in ter ac tions be tween par ti cles of A and B are of the same mag ni tude of in ter ac tions within both A and B.

We shall now briefly in tro duce two factorizations of the last type that are of par a mount im por tance in mo lec u lar quan tum chem is try, then con sider other factorizations of di rect in -ter est for the de scrip tion of liq uid sys tems.

The first factorization we are in tro duc ing re gards the elec tronic and nu clear co or di -nates (called r and R re spec tively) of the same sys tem: H(x) = He(r) + Hn(R). It is clear thatthe elec tro static in ter ac tion be tween two elec trons (and be tween two nu clei) within a givenmol e cule is of the same mag ni tude as that con cern ing a cou ple elec tron-nu cleus. In spite ofthis, the factorization is per formed, with a tre men dous ef fect on the evo lu tion of quan tumchem is try. Its phys i cal jus ti fi ca tion rests on the very dif fer ent masses nu clei and elec tronshave, and there fore on their ve loc i ties. It is of ten called Born-Oppenheimer (BO) ap prox i -ma tion and in the fol low ing we shall use this ac ro nym. The BO ap prox i ma tion makes pos si -ble the in tro duc tion of the con cept of po ten tial en ergy sur face (PES), a very use ful model tode scribe the mo tion of nu clei (of ten in the semiclassical ap prox i ma tion) tak ing into ac countthe in ter ac tions with the quan tum de scrip tion of the elec trons.

Some more de tails can be use ful in the fol low ing. To ap ply this factorization it is im -per a tive to fol low a given or der. First, to solve the elec tronic equa tion with He(R) at a fixedge om e try of nu clei: the out put is a wave func tion and an en ergy both parametrically de pend -ent on the nu clear ge om e try R given as in put: Ψ e(r;R) and Ee(R) (ac tu ally there will be a setof elec tronic states, each with its en ergy). Sec ond, to re peat the same cal cu la tions at dif fer -ent nu clear ge om e tries R’, many times, un til a suf fi ciently de tailed de scrip tion of the func -tion Ee(R) is reached. Ee(R) is the PES we have men tioned: it may be used to de fine thepo ten tial op er a tor within the nu clear Hamiltonian Hn(R) and then to com pute vi bra tionaland ro ta tional states (an other factorization is here in tro duced) or used in a semiclassical way to study the ef fect of nu clear mo tions.

The sec ond factorization we are in tro duc ing re gards the elec tronic part of the sys tem(af ter the BO factorization). The main pro ce dure in use leads to factorization into many sep -a rate parts, each re gard ing one elec tron only:

( ) ( )H r h rei i

i

elec

= ∑ .

Here again the cou pling terms are of the same or der of mag ni tude as the in ter ac tionsleft within each one-elec tron Hamiltonian (that ex plic itly re gards the elec tro static in ter ac -tion of the elec tron with all the nu clei of the sys tems, placed at fixed po si tions). The trick al -low ing this factorization con sists in in tro duc ing within each one-elec tron Hamiltonian (thesym me try of elec trons makes them all equal) an av er aged de scrip tion of the cou plings,based on the yet un known wave func tions of the other elec trons. The cal cu la tion pro ceedsiteratively: start ing from a first guess of the av er aged in ter ac tion, the de scrip tion is pro gres -sively re fined us ing in ter me di ate val ues of the one-elec tron wave func tions. The fi nal out -


put is a Ψe(r;R) ex pressed as an (anti)symmetrized prod uct of the cor rect num ber ofone-elec tron wave func tions. Each elec tronic state will be de scribed by its spe cific col lec -tion of such one-elec tron wave func tions, called mo lec u lar orbitals (MO). The whole pro ce -dure is called SCF (self-con sis tent field), a name that re minds us of the tech nique in use, orHF (Hartree-Fock) to honor sci en tists work ing in its elab o ra tion.

This par ti tion is the sec ond cor ner stone of mo lec u lar quan tum chem is try. Al most allQM mo lec u lar cal cu la tions use the SCF the ory (pass ing then, when nec es sary, to a higherlevel of the mo lec u lar QM the ory). The con cepts of MOs and of or bital en er gies de rive from this one-elec tron factorization.

In the fol low ing we shall make lit tle ex plicit use of the two factorizations we have here ex am ined, be cause we shall not en ter into too tech ni cal de tails about how to ob tain ac cu ratemo lec u lar in ter ac tion po ten tials. They will be al ways in the back ground, how ever, andsome con cepts ex posed here will be re called when nec es sary.

The last factorization we have an tic i pated in the in tro duc tion re gards sys tems com -posed by many mol e cules, as mo lec u lar crys tals, clus ters and liq uids.

A factorization of a liq uid into mo lec u lar sub units was im plicit in the naïve model wehave used in the in tro duc tion. The in ter ac tions within a mol e cule surely are larger thanthose among mol e cules; how ever, we can not ne glect these cou plings, which are es sen tial tode scribe a liq uid. This is the main sub ject of this chap ter and will be treated with due at ten -tion in the fol low ing sec tions.

To com plete this pre lim i nary over view, we stress that the factorization tech niques arequite flex i ble and that they may be ap plied at dif fer ent lev els. One among them de servesmen tion, be cause it will be used in the fol low ing.

In the study of liq uid sys tems there are sev eral rea sons to put more at ten tion to a lim -ited por tion of the liq uid, with the re main der of the liq uid sys tems treated at a lower level ofac cu racy. We shall call the de scrip tions in which there is a por tion of the whole sys tem con -sid ered as the main com po nent (called M) “fo cused mod els” while the re main der (called S)plays a sup ple men tary or as sist ing role. In such an ap prox i ma tion, the Hamiltonian is par ti -tioned in the fol low ing way:

( ) ( ) ( ) ( )H x H x H x H xM M S S MS= + + [8.8]

Fo cused mod els are used to study lo cal prop er ties in pure liq uids, so lu tions, and in ter -faces. The larg est use is to study sol va tion ef fects and re ac tions in so lu tions. In these casesM is com posed of a sol ute sup ple mented by a sol va tion clus ter (also a sin gle sol ute mol e -cule may be used). In these mod els the more de tailed de scrip tion of M is en sured by us ingthe BO ap prox i ma tion, fol lowed by a MO-based de scrip tion of the elec tronic struc ture(there are also meth ods that re place the QM de scrip tion of the elec tronic struc ture withsome sim pler semiclassical model). The S com po nents are gen er ally de scribed with the aidof the intermolecular po ten tials we shall ex am ine in the fol low ing sec tions. The de scrip tionof the cou pling takes into ac count the na ture of the de scrip tion cho sen for both M and S.

Fo cused mod els may also be used to get more de tailed in for ma tion on the struc ture ofliq uids, be ing in prin ci ple more ac cu rate than de scrip tions solely based on intermolecularpo ten tials. The com pu ta tional cost is higher, of course, and this ap proach is now used onlyat the fi nal stage of the as sess ment of mod els of the mo lec u lar in ter ac tion po ten tial.

8.2 Basic simplifications of quantum model 391

8.3 CLUSTER EXPANSIONThe pure ab in itio ap proach we have sum ma rized has never been used to de ter mine the mo -lec u lar in ter ac tion po ten tials. Fur ther sim pli fi ca tions are gen er ally used.

We will start con sid er ing a model in which only the main com po nent HM(xM) of theHamiltonian [8.8] is con sid ered. This means to pass from a liq uid to a clus ter of mol e cules.The pres ence of the mol e cules within M is ex plic itly ac knowl edged. Adding now the fur -ther con straint of keep ing each mol e cule at a fixed ge om e try, the dimensionality of the Rspace is se verely re duced. In fact, six co or di nates for each mol e cule will be suf fi cient to de -fine the rel a tive po si tion of its cen ter of mass an its ori en ta tion with re spect to an ar bi traryfixed ref er ence frame. If M is com posed of n mol e cules, the BO co or di nate space will have3n-6 co or di nates.

It is con ve nient to con sider now the whole PES EM(Rm), not lim it ing our at ten tion tothe min ima (that cor re spond to equi lib rium po si tions). For each point Rm of the hyperspacewe in tro duce the fol low ing clus ter ex pan sion of the en ergy

( ) ( ) ( )E R E E R E RM m A AB AB ABC ABCCBABAA

= + + +∑∑∑∑∑∑ 12

16

K [8.9]

In this rather ar ti fi cial (but ex act) de com po si tion the whole en ergy is de com posed intothe sum of the en er gies of the sep a rate mol e cules, each at the ge om e try they have in the gasphase, fol lowed by the sums of two-, three-, many-body terms. For each term of eq. [8.9] we have put in pa ren the ses the in di ca tion of the per ti nent nu clear co or di nate subspace to em -pha size that each dimeric in ter ac tion is de fined in a 6-di men sional space (RAB), eachtrimeric in ter ac tion over a 9-di men sional space, etc., while EM is de fined over a 3n-6 space.

The to tal in ter ac tion en ergy is de fined as the dif fer ence be tween [8.9] and the sum ofthe mono mers’s en er gies:

( ) ( ) ( )∆E m R E R E Rm AB AB ABC ABCCBABA

1212

16

, , ;K K= + +∑∑∑∑∑ [8.10]

The con ver gence of this ex pan sion is rel a tively fast for clus ters com posed by neu tralmol e cules, less fast when there are charged spe cies. In any case, it is not pos si ble to in ter rupt the ex pan sion to the two-body terms. This con tri bu tion gives the ad di tive terms of the in ter -ac tion en ergy; the other terms de scribe non-ad di tive ef fects that in prin ci ple can not be ne -glected.

It is al most com pul sory to pro ceed step by step and to study two-body in ter ac tionsfirst. Each cou ple can be con sid ered sep a rately.

8.4 TWO-BODY INTERACTION ENERGY: THE DIMERThe def i ni tion of a two-body po ten tial is given by:

( ) ( ) [ ]∆E R E R E EAB AB A B= − + [8.11]

We have kept the pa ram e ter R=RAB to un der line that this po ten tial de pends both on the rel a tive po si tion of the two mol e cules and on their mu tual ori en ta tion: it is a 6-di men sionalfunc tion de pend ing on these two sets of 3 pa ram e ters each, in the fol low ing in di cated,


where nec es sary, with rab and Ωab. ∆EAB(R)has the sta tus of a PES, with a shift in theref er ence en ergy, given here by the sum ofthe en er gies of the two mono mers.

A func tion de fined in a 6-di men sional space is hard to vi su al ize. Many de viceshave been in tro duced to ren der in graphicform se lected as pects of this func tion.Some will be used in the fol low ing: here we shall use the sim plest graph i cal ren der ing,con sist ing in fix ing an ori en ta tion (Ωab) andtwo co or di nates in the rab set in such a waythat the re main ing co or di nate cor re spondsto the mu tual ap proach be tween mol e culeA and B, along a given straight tra jec tory

and a fixed mu tual ori en ta tion. A typ i cal ex am ple is re ported in Fig ure 8.1. The en ergy curve may be roughly di vided into three re gions. Re gion I cor re sponds to

large sep a ra tion be tween in ter ac tion part ners; the in ter ac tion is fee ble and the curve is rel a -tively flat. Re gion II cor re sponds to in ter me di ate dis tances; the in ter ac tions are stron gercompared to re gion I, and in the case shown in the fig ure, the en ergy (neg a tive) reaches amin i mum. This fact in di cates that the in ter ac tion is bind ing the two mol e cules: we have here a dimer with sta bi li za tion en ergy given, in first ap prox i ma tion, by the min i mum value of the curve. Pass ing at shorter dis tances we reach re gion III; here the in ter ac tion rap idly in creases and there it gives or i gin to re pul sion be tween the two part ners. Be fore mak ing more com -ments, a re mark must be added. The shape of the in ter ac tion en ergy func tion in mo lec u larsys tems is quite com plex: by se lect ing an other path of ap proach and/or an other ori en ta tion,a com pletely dif fer ent shape of the curve could be ob tained (for ex am ple, a com pletely re -pul sive curve). This is quite easy to ac cept: an ex am ple will suf fice. The curve of Fig ure 8.1could cor re spond to the mu tual ap proach of two wa ter mol e cules, along a path lead ing to the for ma tion of a hy dro gen bond when their ori en ta tion is ap pro pri ate: by chang ing the ori en -ta tion bring ing the ox y gen at oms point ing against each other, the same path will cor re spondto a con tin u ously re pul sive curve (see Fig ure 8.2 be low for an even sim pler ex am ple). Wehave to con sider paths of dif fer ent shape, all the paths ac tu ally, and within each path wehave to con sider all the three re gions. To de scribe a liq uid, we need to know weaklong-range in ter ac tions as well as strong short-range re pul sion at the same de gree of ac cu -racy as for the in ter me di ate re gion. Studies lim ited to the sta bi li za tion en er gies of the dimers are of in ter est in other fields of chem i cal in ter est, such as the mod el ing of drugs.

The ∆EAB(R) func tion nu mer i cally cor re sponds to a small frac tion of the whole QMen ergy of the dimeric sys tem. It would ap pear to be computationally safer to com pute ∆E di -rectly in stead of ob tain ing it as a dif fer ence, as done in the for mal def i ni tion [8.11]. This canbe done, and in deed in some cases it is done, but ex pe ri ence teaches us that al go rithms start -ing from the en er gies of the dimer (or of the clus ter) and of the mono mers are sim pler andeven tu ally more ac cu rate. The ap proaches mak ing use of this dif fer ence can be de signed asvariational ap proaches, the oth ers di rectly aim ing at ∆E are called per tur ba tion ap proaches,be cause use is made of the QM per tur ba tion the ory. We shall pay more at ten tion, here be -

8.4 Two-body interaction energy 393

Fig ure 8.1. In ter ac tion en ergy for a dimer with re spect tothe mu tual ap proach dis tance at a fixed ori en ta tion.

low on the variational ap proach, then add ing the ba sic el e ments of the per tur ba tion the oryap proach.

8.4.1 DECOMPOSITION OF THE INTERACTION ENERGY OF A DIMER: VARIATIONAL APPROACH

Com ing back to Figure 8.1, its shape ex hib it ing a min i mum makes it man i fest that there aresev eral types of in ter ac tions at work, with dif fer ent signs and with dif fer ent dis tance de cays(at in fin ity all in ter ac tions go to zero). This sub ject has been am ply stud ied and the maincon clu sions (there are de tails dif fer ing in the var i ous schemes of in ter ac tion en ergy de com -po si tion) are so widely known and in tu itive that we feel au tho rized to in tro duce them herebe fore giv ing a for mal def i ni tion (that will be con sid ered later). In Ta ble 8.1 we re port thenames of these com po nents of the in ter ac tion en ergy.

Table 8.1. The main components of the bimolecular interactions ∆EAB(R)

Com po nent name Ac ro nym Phys i cal Mean ing

Elec tro static ES Cou lomb in ter ac tions be tween rigid charge dis tri bu tions

In duc tion IND Mu tual elec tro static de for ma tion of the two charge dis tri bu tions

Ex change EX Quan tum ef fect due to the Pauli ex clu sion prin ci ple

Dis per sion DISIn ter ac tions among fluc tu a tions in the charge dis tri bu tions ofthe two part ners

Charge trans fer CT Trans fer of elec trons be tween part ners

All com po nents of ∆EAB(R) are pres ent in all points of the RAB 6-di men sional spacewe have in tro duced. It is con ve nient to ex am ine them sep a rately. We shall make ref er encesto a sin gle de com po si tion scheme2 that we con sider more con ve nient. Ref er ence to otherschemes will be done when nec es sary. An other view of meth ods may be found in Tomasi etal.3

The electrostatic termThe ES term may be pos i tive or neg a tive: the shape of the ES(R) func tion strongly de pendson the elec tric char ac ter is tics of the part ner. If both mol e cules are re duced to di poles, wehave the two ex treme sit u a tions (see Fig ure 8.2) which give rise to re pul sive (i.e., pos i tive)and at trac tive (i.e., neg a tive) elec tro static con tri bu tions to ∆EAB, re spec tively. With dif fer -ent ori en ta tions of the two di poles there will be dif fer ent val ues of ES that in this sim ple case can be ob tained with an an a lyt i cal ex pres sion:

( )ES R R, cos /θ µ µ θ= −2 1 23 [8.12]

In more com plex mol e cules, the ES(R) func tion has a com plex shape that of ten de ter -mines the sa lient fea tures of the whole ∆EAB(R) func tion. This is the rea son why in the stud -ies of mo lec u lar rec og ni tion and mo lec u lar dock ing, great at ten tion is paid to a properrep re sen ta tion of ES(R).

In the var i ous meth ods for the de com po si tion of ∆EAB(R) there are no dif fer ences inthe def i ni tion of ES. This quan tity can be eas ily com puted with ab in itio meth ods, and thefol low ing rec ipe is the speed i est and more used way of do ing it.


The elec tronic wave func tions of A and B are sep a rately com puted, each with its ownHamiltonian HA and HB. Ab in itio meth ods give at ev ery level of the for mu la tion of the the -ory antisymmetric elec tronic wave func tions, sat is fy ing the Pauli ex clu sion prin ci ple, onwhich more de tails will be given later.

The two antisymmetric wave func tions ΨA and ΨB are then used, with out mod i fi ca -tions, in con nec tion with the Hamiltonian of the whole sys tem HAB to get the ex pec ta tionvalue of the en ergy; ac cord ing to the stan dard no ta tion of quan tum chem is try we can write:

E HI A B AB A B= Ψ Ψ Ψ Ψ [8.13]

It sim ply means the in te gral over the whole space of the com plex con ju gate of thefunc tion at the left (i.e., the func tion (ΨAΨB)* ) mul ti plied by the func tion HAB(ΨAΨB) (theap pli ca tion of an op er a tor such as HAB to a func tion al ways gives a func tion). We can ne -glect com plex con ju gates (our func tions are all real); the ex pres sion given above is a com -pact and clear in di ca tion of a set of op er a tions end ing with an in te gral, and we shall use itonly to speed no ta tions.

HAB dif fers from the sum of HA and HB ac cord ing to the fol low ing ex pres sion

H H H VAB A B AB= + + [8.14]

VAB col lects terms de scrib ing elec tro static in ter ac tions be tween the nu clei and elec trons ofA with elec trons and nu clei of B. The wave func tion ΨAB ex pressed as eigenfunction of HAB

is of course antisymmetric with re spect to all the elec trons, of A as well as of B, but this hasnot yet been in tro duced in the model.

The en ergy ob tained with this rec ipe (the cal cu la tions are a by-prod uct of the cal cu la -tion of the dimer en ergy) may be so de com posed as:

( ) ( )E R E E ES RI A B= + + [8.15]


Fig ure 8.2. ES for two di poles with col lin ear (left) and antilinear (right) ori en ta tion.

and so:

( ) ( ) ( )ES R E R E EI A B= − + [8.16]

The en er gies of the two mono mers are al ready known, so the cal cu la tion of ES is im -me di ate. Re mark that in the right side of eq. [8.15] only ES de pends on the ge om e try of thedimer, ac cord ing to clus ter model we are us ing.

The induction termThe sec ond term of the in ter ac tion en ergy, IND, is al ways neg a tive. IND is re lated to themu tual po lar iza tion of the elec tronic charge dis tri bu tions of A and B (the nu clei are heldfixed), in duc ing ad di tional sta bi liz ing ef fects. This in duc tion (or po lar iza tion) en ergy con -tri bu tion is de fined in a dif fer ent way in variational and per tur ba tion the ory ap proaches.Per tur ba tion the ory ap proaches are com pelled to com pute at the first or der of the per tur ba -tion scheme only the ef fects due to the po lar iza tion of A with re spect to the B dis tri bu tionkept fixed, and in par al lel the ef fects due to the po lar iza tion of B with A kept fixed. Mu tualin duc tion ef fects are in tro duced at higher or der of the per tur ba tion the ory and have to besep a rated in some way from dis per sion ef fects com puted at the same time.

In the variational ap proach use is made of an ex ten sion of the sim ple tech nique wehave used for ES.

The sep a ra tion be tween elec trons of A and B is main tained but the prod uct ΨAΨB isnow sub jected to a con strained variational op ti mi za tion us ing the Hamiltonian HAB. Thetwo wave func tions are so changed, al low ing the ef fects of mu tual po lar iza tion, be cause ofthe pres ence of the VAB term in the Hamiltonian: they will be so in di cated as ΨA

p and ΨBp .

The re sult ing ex pec ta tion value of the en ergy:

( )E R HII Ap

Bp

AB Ap

Bp= Ψ Ψ Ψ Ψ [8.17]

may be so de com posed:

( ) ( ) ( )E R E R IND RII I= + [8.18]

and so IND(R) = EII(R) - EI(R). In this way, IND(R) con tains all the mu tual po lar iza tion ef -fects.

The exchange termThe next term, EX, is pos i tive for all the mo lec u lar sys tems of in ter est for liq uids. The namemakes ref er ence to the ex change of elec trons be tween A and B. This con tri bu tion to ∆E issome times called re pul sion (REP) to em pha size the main ef fect this con tri bu tion de scribes.It is a true quan tum me chan i cal ef fect, re lated to the antisymmetry of the elec tronic wavefunc tion of the dimer, or, if one pre fers, to the Pauli ex clu sion prin ci ple. Actually these aretwo ways of ex press ing the same con cept. Par ti cles with a half in te ger value of the spin, likeelec trons, are sub jected to the Pauli ex clu sion prin ci ple, which states that two par ti cles ofthis type can not be de scribed by the same set of val ues of the char ac ter iz ing pa ram e ters.Such par ti cles are sub jected to a spe cial quan tum ver sion of the sta tis tics, the Fermi-Diracsta tis tics, and they are called fer mions. Iden ti cal fer mions have to be de scribed with anantisymmetric wave func tion; the op po site also holds: iden ti cal par ti cles de scribed by an


antisymmetric wave func tion are fer mions and sat isfy the Pauli ex clu sion prin ci ple. In tro -ducing these con cepts in the ma chin ery of the quan tum me chan i cal cal cu la tions, it turns outthat at each cou lomb in ter ac tion be tween two elec trons de scribed by MOs φ µ(1) and φυ (2)(the stan dard ex pres sion of this in te gral, see eq.[8.13], is: <φ µ(1)φυ (2)|1/r12|φ µ(1)φυ (2)>) one has to add a sec ond term, in which there is an ex change of the two elec trons in the con ju gatefunc tion: <φυ (1)φ µ(2)|1/r12|φ µ(1)φυ (2)> with a mi nus sign (the ex change in the la bel of thetwo elec trons is a per mu ta tion of or der two, bear ing a sign mi nus in the antisymmetric case).

There are other par ti cles, called bos ons, which sat isfy other quan tum sta tis tics, theBose-Ein stein sta tis tics, and that are de scribed by wave func tions sym met ric with re spect to the ex change, for which the Pauli prin ci ple is not valid. We may dis pense with a fur ther con -sid er ation of bos ons in this chap ter.

It is clear that to con sider ex change con tri bu tions to the in ter ac tion en ergy means to in -tro duce the proper antisymmetrization among all the elec trons of the dimer. Each mono meris in de pend ently antisymmetrized, so we only need to ap ply to the sim ple prod uct wavefunc tions an antisymmetrizer re stricted to per mu ta tions re gard ing elec trons of A and B atthe same time: it will be called AAB.

By ap ply ing this op er a tor to Ψ ΨAp

Bp with out other changes and com put ing the ex pec ta -

tion value, one ob tains:

EIII(R) = £AABΨ ΨAp

Bp |HAB|AABΨ ΨA

pBp § [8.19]

with

( ) ( ) ( )E R E R EX RIII II= + [8.20]

and

( ) ( ) ( )EX R E R E RIII II= − [8.21]

Morokuma has done a some what dif fer ent def i ni tion of EX: it is widely used, be ing in -serted into the pop u lar Kitaura-Morokuma de com po si tion scheme.4 In the Morokuma def i -ni tion EIII(R) is com puted as in eq. [8.19] us ing the orig i nal Ψm mono mer wave func tionsin stead of the mu tu ally po lar ized Ψm

p ones. This means to lose, in the Morokuma def i ni tion,the cou pling be tween po lar iza tion and antisymmetrization ef fects that have to be re cov eredlater in the de com po si tion scheme. In ad di tion, EX can be no lon ger com puted as in eq.[8.20], but us ing EI en ergy: (i.e., EX' = EIII(R) - EI(R)). The prob lem of this cou pling alsoap pears in the per tur ba tion the ory schemes that are nat u rally in clined to use un per turbedmonomeric wave func tions, not in clud ing ex change of elec trons be tween A and B: we shallcome back to this sub ject con sid er ing the per tur ba tion the ory ap proach.

The charge transfer termThe charge trans fer con tri bu tion CT may play an im por tant role in some chem i cal pro -cesses. In tu itively, this term cor re sponds to the shift of some elec tronic charge from the oc -cu pied orbitals of a mono mer to the empty orbitals of the other. In the variationalde com po si tion schemes this ef fect can be sep a rately com puted by re peat ing the cal cu la tions on the dimer with de le tion of some blocks in the Hamiltonian ma trix of the sys tem and tak -


ing then a dif fer ence of the en er gies (in other words: the same strat egy adopted for the pre -ced ing terms, but chang ing the ma trix blocks). In the Kitaura-Morokuma scheme CT alsocon tains the cou plings be tween in duc tion and ex change ef fects.

We con sider unnec es sary to sum ma rize the tech ni cal de tails; they can be found in thesource paper4 as well as in ref.3 for the ver sion we are re sum ing here.

In stan dard per tur ba tion the ory (PT) meth ods, the CT term is not con sid ered; there arenow PT meth ods able to eval u ate it but they are rarely used in the mod el ing of in ter ac tionpo ten tials for liq uids.

The dispersion termThe dis per sion en ergy con tri bu tion DIS in tu itively cor re sponds to elec tro static in ter ac tionsin volv ing in stan ta neous fluc tu a tions in the elec tron charge dis tri bu tions of the part ners:these fluc tu a tions can cel out on the av er age, but their con tri bu tion to the en ergy is dif fer entfrom zero and neg a tive for all the cases of in ter est for liq uids. The com plete the ory of thesesta bi liz ing forces (by tra di tion, the em pha sis is put on forces and not on en er gies, but thetwo quan ti ties are re lated) is rather com plex and based on quan tum elec tro dy nam ics con -cepts. There is no need of us ing it here.

The con cept of dis per sion was in tro duced by Lon don (1930),5 us ing by far sim pler ar -gu ments based on the ap pli ca tion of the per tur ba tion the ory, as will be shown in the fol low -ing sub sec tion. A dif fer ent but re lated in ter pre ta tion puts the em pha sis on the cor re la tion inthe mo tions of elec trons.

It is worth spend ing some words on elec tron cor re la tion. In ter ac tions among elec trons are gov erned by the Cou lomb law: two elec trons re pel

each other with an en ergy de pend ing on the in verse of the mu tual dis tance: e2/rij, where e isthe charge of the elec tron. This means that there is cor re la tion in the mo tion of elec trons,each try ing to be as dis tant as pos si ble from the oth ers. Using QM lan guage, where the elec -tron dis tri bu tion is de scribed in terms of prob a bil ity func tions, this means that when oneelec tron is at po si tion rk in the phys i cal space, there will be a de crease in the prob a bil ity offind ing a sec ond elec tron near rk, or in other words, its prob a bil ity func tion pres ents a holecen tered at rk.

We have al ready con sid ered sim i lar con cepts in dis cuss ing the Pauli ex clu sion prin ci -ple and the antisymmetry of the elec tronic wave func tions. Actually, the Pauli prin ci pleholds for par ti cles bear ing the same set of val ues for the char ac ter iz ing quan tum num bers,in clud ing spin. It says noth ing about two elec trons with dif fer ent spin.

This fact has im por tant con se quences on the struc ture of the Hartree-Fock (HF) de -scrip tion of elec trons in a mol e cule or in a dimer. The HF wave func tion and the cor re spond -ing elec tron dis tri bu tion func tion take into ac count the cor re la tion of mo tions of elec tronswith the same spin (there is a de scrip tion of a hole in the prob a bil ity, called a Fermi hole),but do not cor re late mo tions of elec trons of dif fer ent spin (there is no the sec ond com po nentof the elec tron prob a bil ity hole, called a Cou lomb hole).

This re mark is im por tant be cause al most all the cal cu la tions thus far per formed to getmo lec u lar in ter ac tion en er gies have been based on the HF pro ce dure, which still re mainsthe ba sic start ing ap proach for all the ab in itio cal cu la tions. The HF pro ce dure gives the bestdef i ni tion of the mo lec u lar wave func tion in terms of a sin gle antisymmetrized prod uct ofmo lec u lar orbitals (MO). To im prove the HF de scrip tion, one has to in tro duce in the cal cu -la tions other antisymmetrized prod ucts ob tained from the ba sic one by re plac ing one ormore MOs with oth ers (re place ment of oc cu pied MOs with vir tual MOs). This is a pro ce -


dure we shall see in ac tion also in the con text of the per tur ba tion the ory. At the variationallevel con sid ered here, the pro ce dure is called Con fig u ra tion In ter ac tion (CI): each con fig u -ra tion cor re sponds to one of the antisymmetrized prod ucts of MO we have in tro duced, in -clud ing the HF one, and the co ef fi cients in front of each com po nent of this lin ear ex pan sionof the ex act (in prin ci ple) wave func tion are de ter mined by ap ply ing the variational prin ci -ple.

There are nu mer ous al ter na tive meth ods that in tro duce elec tron cor re la tion in the mo -lec u lar cal cu la tions at a more pre cise level that can be prof it ably used. We men tion here theMC-SCF ap proach (the ac ro nym means that this is a vari ant of HF (or SCF) pro ce dure start -ing from the op ti mi za tion no more of a sin gle antisymmetric or bital prod uct, but of manydif fer ent prod ucts, or con fig u ra tions), the Cou pled-Clus ter the ory, etc., all meth ods basedon a MO de scrip tion of sin gle-elec tron func tions.

For read ers wish ing to reach a better ap pre ci a tion of pa pers re gard ing the for mu la tionof in ter ac tion po ten tials, we add that there is an other way of in tro duc ing elec tron cor re la tion ef fects in the cal cu la tions. It is based on the den sity func tional the ory (DFT).6 There is a va -ri ety of DFT meth ods (de tailed in for ma tion can be found in the quoted monograph6); a fam -ily of these meth ods again makes use of MOs: they are called hy brid func tional meth ods and give, on av er age, better re sults than other cor re lated meth ods at a lower com pu ta tional cost.

The in tro duc tion of elec tron cor re la tion in the de scrip tion of the mono mers pro duceschanges in their charge dis tri bu tion and in their pro pen sity to be po lar ized. For this rea sonES, IND, and EX com puted with cor re lated wave func tions are some what dif fer ent with re -spect to the val ues ob tained with the cor re spond ing HF wave func tions. The pro ce duresketched with equa tions [8.13]-[8.21] can be adapted, with some mod i fi ca tions, to cor re -lated wave func tion us ing a MO ba sis.

The reader must be warned that here, as well as in other points of this chap ter, we havesim pli fied the dis cus sion by omit ting many de tails nec es sary for a proper han dling and afuller un der stand ing the prob lem, but not es sen tial to grasp the ba sic points.

What is of more prac ti cal in ter est here is that HF de scrip tions of the dimer can not givea DIS term. This may be re cov ered by in tro duc ing CI de scrip tions of the sys tem. The sim -pler CI de scrip tion, now largely used in rou tine cal cu la tions on mol e cules and mo lec u larag gre gates, is called MP2. This ac ro nym means that use has been made of a spe cial ized ver -sion of the per tur ba tion the ory (called Møller-Plesset) lim ited to sec ond or der to de ter minethe ex pan sion co ef fi cients.

MP2 wave func tions con tain el e ments able to give an ap pre ci a tion of DIS, even if oflim ited ac cu racy. It is in fact pos si ble to de com pose MP2 val ues of ∆EAB(R) us ing the strat -egy we have out lined for the HF case, add ing to each term the ap pro pri ate MP2 cor rec tion.Each term of the de com po si tion is some what mod i fied, be cause of the MP2 cor rec tions: there main der of the MP2 con tri bu tion can be taken as a first ap prox i ma tion to DIS.

There are other meth ods to get DIS val ues start ing from HF wave func tions, whichmay be more pre cise. Among them we quote the meth ods based on the re sponse the ory andon the use of dy namic polarizabilities. This pow er ful method has been de vel oped dur ing the years, and the out stand ing con tri bu tions are due to McWeeny (1984) and Claverie (1986),to which ref er ence is made for more de tails.7


The decomposition of the interaction energy through a variationalapproach: a summary

We have so far given a de scrip tion of the el e ments in which ∆EAB(R) may be de com -posed us ing variational ap proaches. We re port here the fi nal ex pres sion:

∆E ES PL EX CT COUP DISAB = + + + + + [8.22]

At the HF level, all terms (but one: DIS) can be ob tained with al most zero com pu ta -tional cost with re spect to the nu mer i cal de ter mi na tion of ∆EAB; in the HF de com po si tion ofthe in ter ac tion en ergy there is a small ad di tional term (called COUP) de scrib ing fur ther cou -plings be tween the other com po nents, which is of ten left as it is ob tained (i.e., as the dif fer -ence be tween the orig i nal ∆EAB value and the sum of the other four com po nents) or

sub jected to fur ther de com po si -tions.8 The last term, DIS, may becal cu lated at this level us ingspecialistic time-de pend ent for -mu la tions of the HF pro ce dure.Going be yond the HF level, thede com po si tion can be ob tained atthe MP2 level es sen tially us ingthe same tech niques (DIS may beap pre ci ated by sep a rat ing some ofthe MP2 con tri bu tions that arestrictly ad di tive).

The variational ap proacheswe have con sid ered are able to de -scribe, and to de com pose, the in -ter ac tion en ergy at the level ofac cu racy one wishes, once thenec es sary com pu ta tional re -sources are avail able. In fact, aswe shall see later, there is no need

of reach ing ex treme pre ci sion in the pre lim i nary cal cu la tions to model in ter ac tion po ten tials for liq uids, be cause other ap prox i ma tions must be in tro duced that will dras ti cally re duce the ac cu racy of the de scrip tion.

We re port here as an ex am ple the de com po si tion of the in ter ac tion en ergy of the wa terdimer, in the same ori en ta tion Ω used in Fig ure 8.1.

At large dis tances the in ter ac tion is dom i nated by ES; this con tri bu tion also gives area son able ap prox i ma tion to ∆E at the equi lib rium dis tance. The IND de cays with the dis -tance more rap idly than ES; this con tri bu tion is par tic u larly sen si tive to the qual ity of theex pan sion ba sis set χ. Old cal cu la tions of IND us ing re stricted ba sis sets are not re li able.EX is a short ranged term: its con tri bu tion is how ever es sen tial to fix the po si tion of the min -i mum (and to de scribe por tion III of the PES). CT and DIS terms have both a short-rangechar ac ter.

This sche matic anal y sis is valid for al most all the dimeric in ter ac tions where at leastone part ner has a dipolar char ac ter. The pres ence of a hy dro gen bond (as is the case for the


Fig ure 8.3. De com po si tion of the in ter ac tion en ergy ∆E inH2OLHOH.

ex am ple given in the fig ure) only in tro duces quan ti ta tive mod i fi ca tions, suf fi cient how everto show by sim ple vi sual in spec tion if there is a hy dro gen bond or not.

Anal o gous trends in the de com po si tion of ∆E are pres ent in the in ter ac tions in volv ingcharged-neu tral spe cies (also in the case of apolar mol e cules). A spe cial case is given by thein ter ac tion of a mol e cule with the bare pro ton: in this case there is no EX con tri bu tion.

If the part ners have no per ma nent charge, or di pole, the in ter ac tion at large-me diumdis tances is dom i nated by DIS, and by EX at small val ues of R.

In com par ing graphs like Fig ure 8.4 for a set of dimers of dif fer ent chem i cal com po si -tion, it is easy to draw em pir i cal rules con nect ing the chem i cal com po si tion to some chem i -cal con cept, as electronegativity, hy brid iza tion, etc. A stricter con nec tion with thesecon cepts can be found when the ∆E de com po si tions have been sub jected to the coun ter poise cor rec tions in tro duced here be low.

8.4.2 BASIS SET SUPERPOSITION ERROR AND COUNTERPOISE CORRECTIONS

Cal cu la tions of the in ter ac tion en er gies are af fected by a for mal er ror that may have im por -tant con se quences on the fi nal value of the en ergy and on its de com po si tion. We shall con -sider here the case of variational cal cu la tions for dimers, but the ba sic con sid er ations can beex tended to larger clus ters and to other com pu ta tional meth ods. The or i gin of the er ror is anon-per fect bal ance in the qual ity of the cal cu la tion of dimer en ergy, EAB and of en er gies ofthe two mono mers, EA and EB. In fact, there are more com pu ta tional de grees of free domavail able for the dimer than for each mono mer sep a rately. The num ber of de gree of free dom cor re sponds to the num ber of ba sis func tions avail able for the op ti mi za tion of the elec tronicstruc ture of the mol e cule, and hence for the minimization of the en ergy. Let us con sider, toclar ify the con cept, the case of two wa ter mol e cules giv ing or i gin to a dimer; each wa termol e cule has ten elec trons, while the qual ity and num ber of ex pan sion func tions is se lectedat the be gin ning of the cal cu la tion. This is called the ex pan sion ba sis set (just ba sis set, orBS, for brev ity) and it will be in di cated for the mol e cule A with χA. The num ber of theseba sis func tions is fixed, for ex am ple, 30 func tions. The sec ond mol e cule will be de scribedby a sim i lar ba sis set χB con tain ing in this ex am ple ex pan sion func tions of the same qual -ity and num ber as for mol e cule A (the two mol e cules are in this ex am ple of the same chem i -cal na ture). The wave func tion of the dimer AB and its en ergy will be de ter mined in terms of the un ion of the two ba sis sets, namely χAB= χA⊕ χB, com posed of 60 func tions. It is ev -i dent that it is eas ier to de scribe 20 elec trons with 60 pa ram e ters than 10 elec trons with 30only. The con clu sion is that the dimer is better de scribed than the two mono mers, and so thedimer en ergy is rel a tively lower than the sum of the en er gies of the two mono mers.

This is called the ba sis set su per po si tion (BSS) er ror. Why su per po si tion er ror? Whenthe two com po nents of the dimer are at large dis tance, χA and χB are well sep a rated, i.e., they have small su per po si tion (or over lap), and so the rel a tive er ror we are con sid er ing ismod est, zero at in fin ity. When the two mono mers are at shorter dis tances the su per po si tionof the two ba sis sets in creases (the ba sis func tions are al ways cen tered on the per ti nent nu -clei) as well as the er ror.

There is a sim ple rec ipe to cor rect this er ror: it con sists of per form ing all the nec es sarycal cu la tions with the same ba sis set, the dimeric ba sis χAB which de pends on the ge om e try of the dimer.9 This means that the en ergy of the mono mers must be re peated for each po si -tion in the R6 con fig u ra tion space (see Section 8.4 for its def i ni tion). We add a su per script


CP (coun ter poise) to de note quan ti ties mod i fied in such a way and we also add the spec i fi -ca tion of the ba sis set. We re place eq. [8.11] with the fol low ing one:

( ) ( ) ( ) ( )[ ]∆E R E R E R E RABCP

AB AB AB ACP

AB BCP

ABχ χ χ χ; ; ; ;= − + [8.23]

∆ECP is by far more cor re spond ing to the ex act po ten tial en ergy func tions in cal cu la -tions per formed at the HF level with a ba sis set of a small-me dium size. Pass ing to cal cu la -tions with larger ba sis sets, the BSS er ror is ob vi ously smaller, but the CP cor rec tion isal ways ben e fi cial.

We give in Fig ure 8.4 a graph i cal view of how CP cor rec tions mod ify the equi lib riumpo si tion of the dimer and, at the same time, its sta bi li za tion en ergy.

The same holds for cal cu la tions per formed at higher lev els of the QM the ory, be yondthe HF ap prox i ma tion, with de creas ing ef fects of the CP cor rec tions, how ever. We mayleave this last sub ject to the at ten tion of the spe cial ists be cause for the de ter mi na tion of in -ter ac tion po ten tial for liq uids, the HF ap prox i ma tion is in gen eral suf fi cient; in some cases it may be sup ple mented by ap ply ing sim ple lev els of de scrip tion for elec tron cor re la tion, aswe have al ready said.

The CP cor rected in ter ac tion en ergy may be de com posed into terms each hav ing a def -i nite phys i cal mean ing, in anal ogy with what we have ex posed in the pre ced ing sub sec tionfor ∆E with out CP cor rec tions. There are slightly dif fer ent ways of do ing it. We sum ma rizehere the strat egy that more closely fol lows the phys ics of the prob lem.10 When this cor rec -tion is ap plied, the var i ous terms better sat isfy chem i cal in tu ition in pass ing from one dimerto an other of dif fer ent chem i cal com po si tion. Each term of the ∆E de com po si tion is cor -


Fig ure 8.4. Lo ca tion of the min i mum en ergy for H2OLHF dimer ac cord ing to the var i ous ba sis sets and the var i ous meth ods. Crosses (×) re fer to SCF cal cu la tions with out CP cor rec tions, full cir cles (l) to CP cor rected cal cu la -tions. The ovoidal area re fer to an es ti mate of the lo ca tion of the min i mum at the HF limit (in clud ing an es ti mate ofthe er ror). Left side: min i mal ba sis sets; right side: dou ble va lence-shell ba sis sets.

rected with an ad di tive term ∆X (X stays for one of the com po nents of the in ter ac tion en ergy) which is ex pressed as a dif fer ence in the mono mers’ en er gies com puted with the op por tuneba sis set.

Let us con sider again the de com po si tion we have done in eq. [8.19].The phys i cal def i ni tion of ES (elec tro static in ter ac tions among rigid part ners) clearly

in di cates that there is no room here for CP cor rec tions, which would in volve some shift ofthe mono mer’s charge on the ghost ba sis func tions of the part ner. This phys i cal ob ser va tioncor re sponds to the struc ture of the blocks of the Hamiltonian ma trix used to com pute ES: noel e ments re gard ing the BS of the part ner are pres ent in con clu sion and thus no cor rec tions to ES are pos si ble.

In anal ogy, IND must be left un mod i fied. Phys i cal anal y sis of the con tri bu tion and thestruc ture of the blocks used for the cal cu la tion agree in sug gest ing it.

CP cor rec tions must be per formed on the other el e ments of ∆E, but here again phys i cal con sid er ations and the for mal struc ture of the block par ti tion sug gest us ing dif fer ent CP cor -rec tions for each term.

Let us in tro duce a par ti tion into the BS space of each mono mer and of the dimer. Thispar ti tion can be in tro duced af ter the cal cu la tion of the wave func tion of the two mono mers.At this point we know, for each mono mer M (M stays for A or for B), how the com pletemono mer’s BS is par ti tioned into oc cu pied and vir tual orbitals φ M:

χ φ φΜ = ⊕M Mv0 [8.24]

For the ex change term we pro ceed in the fol low ing way. The CP cor rected term is ex -pressed as the sum of the EX con tri bu tion de ter mined as de tailed above, plus a CP cor rec -tion term called ∆EX:

EX EXCP EX= + ∆ [8.25]

∆EX in turn is de com posed into two con tri bu tions:

∆ ∆ ∆ΑEX EX

BEX= + [8.26]

with

( ) ( )[ ]∆AEX

A A A AEXE E= −χ χ [8.27]

and a sim i lar ex pres sion for the other part ner. The CP cor rec tion to EX is so re lated to thecal cu la tion of an other en ergy for the mono mers, per formed on a ba sis set con tain ing oc cu -pied MOs of A as well as of B:

χ φ φAEX

A B= ⊕0 0 [8.28]

We have used here as ghost ba sis the oc cu pied orbitals of the sec ond mono mer, fol -low ing the sug ges tions given by the phys ics of the in ter ac tion.


The other com po nents of the in ter ac tion en ergy are changed in a sim i lar way. The ∆X

cor rec tions are all pos i tive and com puted with dif fer ent ex ten sions of the BS for the mono -mers, as de tailed in Ta ble 8.2.

The cor rec tion ∆TOT per mits to re cover the full CP cor rec tion. It may be used to de finethe ∆COUP cor rec tion:

Table 8.2 Explicit expressions of the ∆X counterpoise corrections to the interactionenergy components

( ) ( )[ ]∆AEX

A A A A0

B0E E= − ⊕χ φ φ

( ) ( )[ ]∆BEX

B B B A0

B0E E= − ⊕χ φ φ

( ) ( )[ ]∆ACT

A A A A0

BE E= − ⊕χ φ φ

( ) ( )[ ]∆BCT

B B B B0

AE E= − ⊕χ φ φ

( ) ( )[ ]∆ATOT

A A A A BE E= − ⊕χ χ χ

( ) ( )[ ]∆BTOT

B B B A BE E= − ⊕χ χ χ

( )∆ ∆ ∆ ∆COUP TOT EX CT= − + [8.29]

or the cor rec tion to the re main der, if COUP is fur ther de com posed.8b This last step is of lit tle util ity for sol vent-sol vent in ter ac tions, but use ful for stron ger chem i cal in ter ac tions. Thereis no need of an a lyz ing here these re fine ments of the method.

The per for mances of the CP cor rec tions to the in ter ac tion en ergy de com po si tion canbe ap pre ci ated in a sys tem atic study on rep re sen ta tive hy dro gen-bond dimers.11 A gen eralre view on the CP the ory has been done by van Dujneveldt et al.12 (who pre fer to use a dif fer -ent de com po si tion).

8.4.3 PERTURBATION THEORY APPROACH

The per tur ba tion the ory (PT) ap proach aims at ex ploit ing a con sid er ation we have al readyex pressed, namely, that ∆E is by far smaller than the sum of the en er gies of the sep a ratemono mers. It would be safer (and hope fully eas ier) to com pute ∆E di rectly in stead of get -ting it as dif fer ence be tween two large num bers. The for mu la tion of the per tur ba tion the oryfor this prob lem that we shall give here be low has all the ba sic pre mises to sat isfy this pro -gram. In prac tice, things are dif fer ent: the in tro duc tion of other as pects in this for mu la tionof PT, made nec es sary by the phys ics of the prob lem, and the ex am i na tion and cor rec tion offiner de tails, put in ev i dence by the anal y sis of the re sults, make the PT ap proach morecostly than variational cal cu la tions of com pa ra ble ac cu racy. Mod ern PT meth ods are com -pet i tive in ac cu racy with variational pro ce dures but are rarely used at ac cu rate lev els tomodel po ten tials for liq uids, the main rea son be ing the com pu ta tional cost.

In spite of this we ded i cate a sub sec tion to PT meth ods, be cause this the o ret i cal ap -proach shows here, as in many other prob lems, its unique ca pa bil ity of giv ing an in ter pre ta -tion of the prob lem. To model in ter ac tions, we need, in fact, to have a clear vi sion of the


phys i cal el e ments giv ing or i gin to the in ter ac tion, and some in for ma tion about the ba sicmath e mat i cal be hav ior of such el e ments (be hav ior at large dis tances, etc.).

The for mal per tur ba tion the ory is sim ple to sum ma rize.The the ory ad dresses the prob lem of giv ing an ap prox i mate so lu tion for a tar get sys -

tem hard to solve di rectly, by ex ploit ing the knowl edge of a sim pler (but sim i lar) sys tem.The tar get sys tem is rep re sented by its Hamiltonian H and by the cor re spond ing wave func -tion Ψ de fined as so lu tion of the cor re spond ing Schrödinger equa tion (see eq. [8.1])

H EΨ Ψ= [8.30]

The sim pler un per turbed sys tem is de scribed in terms of a sim i lar equa tion

H E00 0 0Φ Φ= [8.31]

The fol low ing par ti tion of H is then in tro duced

H H V= +0 λ [8.32]

as well as the fol low ing ex pan sion of the un known wave func tion and en ergy as pow ers ofthe pa ram e ter λ:

Ψ Φ Φ Φ Φ= + + + +01 2 2 3 3λ λ λ( ) ( ) ( ) K [8.33]

E E E E E= + + + +01 2 2 3 3λ λ λ( ) ( ) ( ) L [8.34]

The cor rec tions to the wave func tion and to the en ergy are ob tained in tro duc ing thefor mal ex pres sions [8.32]-[8.34] in the equa tion [8.30] and sep a rat ing the terms ac cord ingto their or der in the power of λ. In this way one ob tains a set of integro-dif fer en tial equa tions to be sep a rately solved. The first equa tion is merely the Schrödinger equa tion [8.31] of thesim ple sys tem, sup posed to be com pletely known. The oth ers give, or der by or der, the cor -rec tions Φ (n) to the wave func tion, and E(n) to the en ergy. These equa tions may be solved byex ploit ing the other so lu tions, Φ Φ Φ Φ1 2 3, , , ,K K , of the sim pler prob lem [8.31], that con -sti tute a com plete ba sis set and are sup posed to be com pletely known. With this ap proachev ery cor rec tion to Ψ is given as lin ear com bi na tion

Φ Φ( )nKn

kK

C= ∑ [8.35]

The co ef fi cients are im me di ately de fined in terms of the integrals VLK = <ΦL|V|ΦK> wherethe in dexes L and K span the whole set of the eigenfunctions of the un per turbed sys tem (in -clud ing Φ0 where nec es sary). The cor rec tions to the en ergy fol low im me di ately, or der byor der. They only de pend on the VLK integrals and on the en er gies E1, E2, ..., EK, ... of the sim -ple (un per turbed) sys tem. The prob lem is so re duced to a sim ple sum ma tion of el e ments, allde rived from the sim pler sys tem with the ad di tion of a ma trix con tain ing the VLK integrals.

This for mu la tion ex actly cor re sponds to the orig i nal prob lem, pro vided that the ex pan -sions [8.33] and [8.34] of Ψ and E con verge and that these ex pan sions are com puted un tilcon ver gence.


We are not in ter ested here in examining other as pects of this the ory, such as the con -ver gence cri te ria, the def i ni tions to in tro duce in the case of in ter rupted (and so ap prox i mate) ex pan sions, or the prob lems of prac ti cal im ple men ta tion of the method.

The MP2 wave func tions we have in tro duced in a pre ced ing sub sec tion are just the ap -pli ca tion of this method to an other prob lem, that of the elec tronic cor re la tion. In this case,the sim pler un per turbed sys tem is the HF ap prox i ma tion, the cor rec tions are lim ited to thesec ond or der, and the cor re spond ing con tri bu tions to the en ergy are ex pressed as a sim plesum ma tion of el e ments. The MP2 method is cur rently used in the PT de scrip tion of theintermolecular po ten tial:13 in such cases, two dif fer ent ap pli ca tions of PT are used at thesame time.

We pass now to ap ply PT to the cal cu la tion of ∆EAB(R). The most rea son able choicecon sists of de fin ing the un per turbed sys tem as the sum of the two non-in ter act ing mono -mers. We thus have:

H H HA B0 = + [8.36]

Φ Ψ Ψ0 0 0= >| A B [8.37]

E E EA B0 0 0= + [8.38]

Φ0 is the sim ple prod uct of the two mono mers’ wave func tions. The per tur ba tion op er -a tor is the dif fer ence of the two Hamiltonians, that of the dimer and H0. The per tur ba tion pa -ram e ter λ may be set equal to 1:

V H H VAB= − =0 [8.39]

The set Φ Φ Φ Φ1 2 3, , , ,K K of the so lu tions of the un per turbed sys tem can be ob tainedby re plac ing within each mono mer wave func tion, one, two or more oc cu pied MO with va -cant MOs be long ing to the same mono mer. The per tur ba tion op er a tor V only con tains one-and two-body in ter ac tions, and so, be ing the MOs orthonormal, the only VLK integrals dif -fer ent from zero are those in which L and K dif fer at the max i mum by two MOs.

The for mu la tion is quite ap peal ing: there is no need for re peated cal cu la tions, the de -com po si tion of the in ter ac tion en ergy can be im me di ately ob tained by sep a rately col lect ingcon tri bu tions cor re spond ing to dif fer ent ways of re plac ing oc cu pied with vir tual orbitals.

In ter rupting the ex pan sion at the sec ond or der, one ob tains the fol low ing re sult: ∆E E E(1) (2)≈ + =

Ψ Ψ Ψ ΨA B A BV| + I or der: elec tro static term

−−

∑| | |Ψ Ψ Ψ Ψ0 0 0

2

0

A BKA B

KA A

K

V

E E + II or der: po lar iza tion of A

−−

∑| | |Ψ Ψ Ψ Ψ0 0 0

2

0

A B AKB

KB B

K

V

E E + II or der: po lar iza tion of B [8.40]

( ) ( )−+ − +

∑∑| | |Ψ Ψ Ψ Ψ0 0

2

0 0

A BKA

LB

KA

LB A B

lk

V

E E E E + II or der: dis per sion A-B + higher or der terms


We have here ex plic itly writ ten the mono mer wave func tions, with a ge neric in di ca -tion of their elec tronic def i ni tion: Ψ0

A is the start ing wave func tion of A, al ready in di catedwith ΨA, ΨK

A is an other con fig u ra tion for A with one or two oc cu pied one-elec tron orbitalsre placed by vir tual MOs.

The first or der con tri bu tion ex actly cor re sponds to the def i ni tion we have done of ESwith equa tion [8.16]: it is the Cou lomb in ter ac tion be tween the charge dis tri bu tions of Aand B.

The sec ond term of the ex pan sion, i.e., the first el e ment of the sec ond or der con tri bu -tions, cor re sponds to the po lar iza tion of A, due to the fixed un per turbed charge dis tri bu tionof B; the next term gives the po lar iza tion of B, due to the fixed charge dis tri bu tion of A. Thetwo terms, summed to gether, ap prox i mate IND. We have al ready com mented that per tur ba -tion the ory in a stan dard for mu la tion can not give IND with a unique term: fur ther re fine -ments re gard ing mu tual po lar iza tion ef fects have to be searched at higher or der of the PTex pan sion.

The last term of the sec ond or der con tri bu tion is in ter preted as a dis per sion en ergycon tri bu tion. The man i fold of ex cited mono mer states is lim ited here at sin gle MO re place -ments within each mono mer; the re sult ing en ergy con tri bu tion should cor re spond to a pre -lim i nary eval u a tion of DIS, with re fine ments com ing from higher or ders in the PTex pan sion.

Re mark that in PT meth ods, the fi nal value of ∆E is not avail able. It is not pos si ble here to get a nu mer i cal ap praisal of cor rec tion to the val ues ob tained at a low level of the ex pan -sion. All con tri bu tions are com puted sep a rately and added to gether to give ∆E.

In con clu sion, this PT for mu la tion, which has dif fer ent names, among which “stan -dard” PT and RS (Ray leigh-Schrödinger) PT, gives us the same ES as in the variationalmeth ods, a un com pleted value of IND and a un com pleted ap praisal of DIS: higher or der PTcon tri bu tions should re fine both terms. One ad van tage with re spect to the variational ap -proach is ev i dent: DIS ap pears in PT as one of the lead ing terms, while in variational treat -ments one has to do ad hoc ad di tional cal cu la tions.

Con versely, in the RS-PT for mu la tion CT and EX terms are not pres ent. The ab senceof CT con tri bu tions is quickly ex plained: RS-PT works on sep a rated mono mers and CTcon tri bu tions should be de scribed by re place ments of an oc cu pied MO of A with an emptyMO of B (or by an oc cu pied MO of B with an empty MO of A), and these elec tronic con fig -u ra tions do not be long to the set of state on which the the ory is based. For many years thelack of CT terms has not been con sid ered im por tant. The at ten tion fo cused on the ex am i na -tion of the in ter ac tion en ergy of very sim ple sys tems, such as two rare gas at oms, in whichCT ef fects are in fact of very lim ited im por tance.

The ab sence of EX terms, on the con trary, in di cates a se ri ous de fi ciency of the RS for -mu la tion, to which we have to pay more at ten tion.

The wave func tion ΦK used in the RS for mu la tion does not fully re flect the elec tronpermutational sym me try of the dimer: the per mu ta tions among elec trons of A and B are ne -glected. This leads to se vere in con sis ten cies and large er rors when RS PT is ap plied over the whole range of dis tances. One has to re work the per tur ba tion the ory in the search of otherap proaches. The sim plest way would sim ply re place Φ Ψ Ψ0

0= | A B0 > with |AABΦ0 > where

AAB is the ad di tional antisymmetry op er a tor we have al ready in tro duced. Un for tu nately|AABΦ0 > is not an eigenfunction of H0 as asked by the PT. Two ways of over com ing this dif -fi culty are pos si ble. One could aban don the nat u ral par ti tion ing [8.36] and de fine an other


un per turbed Hamiltonian hav ing |AAB Φ0 > as eigenfunction. Changes in the def i ni tion ofthe un per turbed Hamiltonian are no rare events in the field of ap pli ca tion of the per tur ba tion the ory and there are also other rea sons sug gest ing a change in the def i ni tion of it in the study of dimeric in ter ac tions. This strat egy has been ex plored with mixed suc cess: some im prove -ments are ac com pa nied by the lack of well-de fined mean ing of the en ergy com po nentswhen the ex pan sion ba sis set grows to ward com plete ness. A sec ond way con sists in keep ing the orig i nal, and nat u ral, def i ni tion of H0 and in chang ing the PT. This is the way cur rentlyused at pres ent: the var i ous ver sions of the the ory can be col lected un der the ac ro nym SAPT (sym me try adapted per tur ba tion the ory)

It is worth re mark ing that in the first the o ret i cal pa per on the in ter ac tion be tween twoat oms (or mol e cules (Eisenschitz and Lon don, 193014) use was made of a SAPT; the prob -lem was for got ten for about 40 years (rich, how ever, in ac tiv i ties for the PT study of weakin ter ac tions us ing the stan dard RS for mu la tion). A re newed in tense ac tiv ity started at theend of the six ties, which even tu ally led to a uni fy ing view of the dif fer ent ways in whichSAPT may be for mu lated. The first com plete mono graph is due to Arrighini15 (1981), andnow the the ory can be found in many other mono graphs or re view ar ti cles. We quote hereour fa vor ites: Claverie16 (1976), a mon u men tal mono graph not yet giv ing a com plete for mal elab o ra tion but rich in sug ges tions; Jeziorski and Kolos17 (1982), short, clear and crit i cal;Jeziorski et al.13 (1994), clear and cen tered on the most used ver sions of SAPT.

There are a score of SAPT the o ries that con tinue to be re fined and ex tended. Readersof this chap ter surely are not in ter ested to find here a syn op sis of a very in tri cate sub ject thatcould be con densed into com pact and el e gant for mu la tions, hard to de code, or ex pandedinto long and com plex se quences of for mu las. This sub ject can be left to spe cial ists, or tocu ri ous peo ple, for which the above given ref er ences rep re sent a good start ing point.

The es sen tial points can be sum ma rized as fol lows. The in tro duc tion of theintra-mono mer antisymmetry can be done at dif fer ent lev els of the the ory. The sim plest for -mu la tion is just to use |AABΦ0 > as un per turbed wave func tion, in tro duc ing a trun cated ex -pan sion of the antisymmetry op er a tor. This means to pass from rig or ous to ap prox i matefor mu la tions.

One ad van tage is that the per tur ba tion en er gies (see eq. [8.30]), at each or der, may bewrit ten as the sum of the orig i nal RS value and of a sec ond term re lated to the in tro duc tionof the ex change:

E E EnRSn

excn( ) ( ) ( )= + [8.41]

The re sults at the lower or ders can be so sum ma rized:

∆E E E E ERS exc RS exc= + + + +( ) ( ) ( ) ( )1 1 2 2 L [8.42]

Eexc(1) does not fully cor re spond to EX ob tained with the variational ap proach. The main rea -

son is that use has been made of an ap prox i ma tion of the antisymmetrizer: other con tri bu -tions are shifted to higher or der of the PT se ries. A sec ond rea son is that use has been heremade of the orig i nal MOs ϕ ito be con trasted with the po lar ized ones ϕ i

p , see Sec tion 8.4.1:for this rea son there will be in the next or ders con tri bu tions mix ing ex change and po lar iza -tion ef fects.

At the sec ond or der we have:


E E Eexc exc dis exc ind( ) ( ) ( )2 2 2= +− − [8.43]

The par ti tion of this sec ond or der con tri bu tion into two terms is based on the na ture ofthe MO re place ments oc cur ring in each con fig u ra tion ΨK

m ap pear ing in the sums of eq.[8.40]. These two terms give a mix ing of ex change and dis per sion (or in duc tion) con tri bu -tions.

Pass ing to higher or ders E(n) the for mu las are more com plex. In the RS part it is pos si -ble to de fine pure terms, as ERSdis

( )3 , but in gen eral they are of mixed na ture. The same hap pens for the Eexc

(n) con tri bu tions.The ex am i na tion of these high or der con tri bu tions is ad dressed in the stud ies of the

math e mat i cal be hav ior of the sep a rate com po nents of the PT se ries. Lit tle use has so farbeen made of them in the ac tual de ter mi na tion of mo lec u lar in ter ac tion po ten tials.

8.4.4 MODELING OF THE SEPARATE COMPONENTS OF ∆E

The nu mer i cal out put of variational de com po si tions of ∆E (sup ple mented by some PT de -com po si tions) now a days rep re sents the main source of in for ma tion to model mo lec u lar in -ter ac tion po ten tials. In the past, this mod el ing was largely based on ex per i men tal data(sup ported by PT ar gu ments), but the dif fi culty of add ing new ex per i men tal data, com binedwith the dif fi culty of giv ing an in ter pre ta tion and a de coup ling of them in the cases of com -plex mol e cules, has shifted the em pha sis to the o ret i cally com puted val ues.

The rec i pes for the de com po si tion we have done in the pre ced ing sec tions are too com -plex to be used to study liq uid sys tems, where there is the need of re peat ing the cal cu la tionof ∆E(r,Ω) for a very large set of the six vari ables and for a large num ber of dimers. There isthus the need of ex tract ing sim pler math e mat i cal ex pres sions from the data on model sys -tems. We shall ex am ine sep a rately the dif fer ent con tri bu tions to the dimer in ter ac tion en -ergy. As will be shown in the fol low ing pages, the ba sic el e ments for the mod el ing are to agood ex tent drawn from the PT ap proach.

The electrostatic termES may be writ ten in the fol low ing form, com pletely equiv a lent to eq. [8.16]

( ) ( )ES rr

r dr drAT

BT= ∫∫ ρ ρ1

122 1 2

1[8.44]

we have here in tro duced the to tal charge den sity func tion for the two sep a rate mono mers.The den sity func tion ρ M

T r)( is a one-elec tron func tion, which de scribes the dis tri bu tion in the space of both elec trons and nu clei. For mula [8.44] is sym met ric both in A and B, as well asin r1 and r2.

It is of ten con ve nient to de com pose the dou ble in te gra tion given in eq. [8.44] in thefol low ing way

( ) ( )ES r V r drAT

B= ∫ ρ 1 1 1 [8.45]

where

( )V r rr

drB BT( )1 2

122

1= ∫ ρ [8.46]


VB(r1) is the elec tro static po ten tial of mol e cule B (of ten called MEP or MESP, ac cord -ing to the authors3,18,19). This a true mo lec u lar quan tity, not de pend ing on in ter ac tions, and itis of ten used to look at lo cal de tails of the elec tro static in ter ac tions between mol e cules as re -quired, for ex am ple, in chem i cal re ac tiv ity and mo lec u lar dock ing prob lems.

To model sim pli fied ex pres sions of the intermolecular po ten tial the di rect use of eq.[8.45] does not in tro duce sig nif i cant im prove ments. The MEP, how ever, may be used intwo ways. We con sider here the first, con sist ing of de fin ing, and us ing, a mul ti po lar ex pan -sion of it. The the ory of mul ti po lar ex pan sion is re ported in all the text books onelectrostatics and on mo lec u lar in ter ac tions, as well as in many pa pers, us ing widely dif fer -ent formalisms. It can be used to sep a rately ex pand ρA, ρB, and 1/r12 of eq. [8.44] or VB and ρA

of eq. [8.45]. We adopt here the sec ond choice. The mul ti po lar ex pan sion of VB(r1) may beso ex pressed:

( ) ( )V r M R YB l mB l

lm

m l

l

l1

1

0

= − +

=−=

∞

∑∑ ,( ) ,θ ϕ [8.47]

It is a Tay lor ex pan sion in pow ers of the dis tance R from the ex pan sion cen ter. Thereare neg a tive pow ers of R only, be cause this ex pan sion is con ceived for points ly ing out sidea sphere con tain ing all the el e ments of the charge dis tri bu tion.

The other el e ments of [8.47] are the har monic spher i cal func tions Ylm and the

multipole el e ments M l,mB which have val ues spe cific for the mol e cule:

( ) ( )Ml

r Y r d r Ml mB l

lm

B B lm

B, , |=+

=∫

42 1

3πθ ϕ ρ Ψ Ψ [8.48]

The spher i cal har mon ics are quite ap pro pri ate to ex press the ex plicit orientational de -pend ence of the in ter ac tion, but in the chem i cal prac tice it is cus tom ary to in tro duce a lin eartrans for ma tion of the com plex spher i cal func tions Yl

m into real func tions ex pressed overCar te sian co or di nates, which are eas ier to vi su al ize. In Ta ble 8.3 we re port the ex pres sionsof the multipole mo ments.

Table 8.3. Multipole moments expressed with the aid of real Cartesian harmonics

M0 1 el e ment Charge Q

M1m 3 el e ments Di pole µx, µy, µz

M 2m 5 el e ments Quadrupole θxy, θxz, θyz, θx y2 − 2 , θz 2

M 3m 7 el e ments Octupole ωxxy, etc.

The lead ing pa ram e ter is l, which de fines the 2l poles. They are, in or der, the monopole (l=0, a sin gle el e ment cor re spond ing to the net mo lec u lar charge), the di pole (l=1, three el e -ments, cor re spond ing to the 3 com po nents of this vec tor), the quadrupole (l=2, five com po -nents, cor re spond ing to the 5 dis tinct el e ments of this first rank ten sor), the octupole, etc.We have re placed the po ten tial given by the dif fuse and de tailed charge dis tri bu tion ρB withthat of a point charge, plus a point di pole, plus a point quadrupole, etc., placed all at the ex -pan sion cen ter. Note that the po ten tial of the 2l pole is pro por tional to r-(l+1). This means that


the po ten tial of the di pole de creases faster than that of the monopole, the quadrupole fasterthan that of the di pole, and so on.

To get the elec tro static in ter ac tion en ergy, the mul ti po lar ex pan sion of the po ten tialVB(r) is mul ti plied by a mul ti po lar ex pan sion of ρA(r). The re sult is:

ES D Rlll l

ll

≈ ′− + ′+

′=

∞

=

∞

∑∑ ( )1

00

[8.49]

where

D C M Mll llm

lmA

l mB

m l

l

′ ′ ′=−

= ∑ [8.50]

is the in ter ac tion en ergy of the per ma nent di pole l of A with the per ma nent di pole l' of B (Cllm

′is a nu mer i cal co ef fi cient de pend ing only on l, l', and m).

The cal cu la tion of ES via eq. [8.49] is much faster than through eq. [8.44]: the in te gra -tions are done once, to fix the M lm

X mo lec u lar multipole val ues and then used to de fine thewhole ES(R) sur face.

For small size and al most spher i cal mol e cules, the con ver gence is fast and the ex pan -sion may be in ter rupted at a low or der: it is thus pos si ble to use ex per i men tal val ues of thenet charge and of the di pole mo ment (better if sup ple mented by the quadrupole, if avail able) to get a rea son able de scrip tion of ES at low com pu ta tional cost and with out QM cal cu la -tions.

We have, how ever, put a ≈ sym bol in stead of = in eq. [8.49] to high light a lim i ta tion ofthis ex pan sion. To an a lyze this prob lem, it is con ve nient to go back to the MEP.

Ex pan sion [8.47] has the cor rect as ymp totic be hav ior: when the num ber of terms of atrun cated ex pres sion is kept fixed, the de scrip tion im proves at large dis tances from the ex -pan sion cen ter. The ex pan sion is also con ver gent at large val ues of R: this re mark is not aple o nasm, be cause for multipole ex pan sions of other terms of the in ter ac tion en ergy (as forex am ple the dis per sion and in duc tion terms), the con ver gence is not en sured.

Con ver gence and asymptoticity are not suf fi cient, be cause the ex pan sion the o remholds (as we have al ready re marked) for points r ly ing out side a sphere con tain ing all the el -e ments of the charge dis tri bu tion. At the QM level this con di tion is never ful filled be causeeach ρ(r) fades ex po nen tially to zero when r → ∞. This is not a se ri ous prob lem for the use of multipole ex pan sions if the two mol e cules are far apart, and it sim ply re duces a lit tle thequal ity of the re sults if al most spher i cal mol e cules are at close con tact. More im por tant arethe ex pan sion lim i ta tions when one, or both, mol e cules have a large and ir reg u lar shape. Atstrict con tact a part of one mol e cule may be in serted into a crev ice of the part ner, within itsnom i nal ex pan sion sphere. For large mol e cules at close con tact a sys tem atic en large ment of the trun cated ex pres sion may lead to use high value multipoles (ap par ent in di ca tion of slowcon ver gence) with di sas trously unphysical re sults (real dem on stra tion of the lack of con ver -gence).

The in tro duc tion of cor rec tion terms to the mul ti po lar ex pan sions of VB or of ES, act -ing at short dis tances and called “pen e tra tion terms”, has been done for for mal stud ies ofthis prob lem but it is not used in prac ti cal ap pli ca tions.


For larger mol e cules it is nec es sary to pass to many-cen ter multipole ex pan sions. Thefor mal ism is the same as above, but now the ex pan sion re gards a por tion of the mol e cule,for which the ra dius of the en cir cling sphere is smaller than that of the whole mol e cule. Here it is no longer pos si ble to use ex per i men tal multipoles. One has to pass to QM cal cu la tionssup ple mented by a suit able pro ce dure of par ti tion of the mol e cule into frag ments (this op er -a tion is nec es sary to de fine the fragmental def i ni tion of multipoles with an an a log of eq.[8.42]). There are many ap proaches to de fine such par ti tions of the mo lec u lar charge dis tri -bu tion and the en su ing multipole ex pan sions: for a re view see, e.g., Tomasi et al.20

It is im por tant to re mark here that such lo cal ex pan sions may have a charge term evenif the mol e cule has no net charge (with these ex pan sions the sum of the lo cal charges mustbe equal to the to tal charge of the mol e cule).

There is free dom in se lect ing the cen ters of these lo cal ex pan sions as well as theirnum ber.

A for mal so lu tion to this prob lem is avail able for the mo lec u lar wave func tions ex -pressed in terms of Gaussi an func tions (as is the gen eral rule). Each el e men tary elec tron dis -tri bu tion en ter ing in the def i ni tion of Ψ, is de scribed by a cou ple of ba sic func tionsχ χs

*t cen tered at po si tions rs and rt. This dis tri bu tion may be re placed by a sin gle Gaussi an

func tion cen tered at the well de fined po si tion of the over lap cen ter: χu at ru. The new Gaussi -an func tion may be ex actly de com posed into a fi nite lo cal multipole ex pan sion. It is pos si -ble to de com pose Y into a fi nite (but large) num ber of lo cal multipole ex pan sions each witha lim ited num ber of com po nents. This method works (the pen e tra tion terms have beenshown to be rea son ably small) but the num ber of ex pan sion cen ters is ex ceed ingly large.

Some ex pe di ent ap prox i ma tions may be de vised, in tro duc ing a bal ance be tween thenum ber of ex pan sion cen ters and the level of trun ca tion of each ex pan sion. This work hasbeen done for years on em pir i cal bases. One strat egy is to keep each ex pan sion at the low estpos si ble or der (i.e., lo cal charges) and to op ti mize num ber and lo ca tion of such charges.

The mod ern use of this ap proach has been pi o neered by Alagona et al.,21 us ing a num -ber of sites larger than the num ber of at oms, with val ues of the charges se lected byminimization of the dif fer ence of the po ten tial they gen er ate with re spect to the MEP func -tion VB(r) (eq. [8.40]). This is the sec ond ap pli ca tion of the MEP func tion we have men -tioned.

Alagona’s ap proach has been re for mu lated by Momany in a sim pler way, by re duc ingthe num ber of sites to that of the nu clei pres ent in the mol e cule.22 This strat egy has gainedwide pop u lar ity: al most all the po ten tials in use for rel a tive large mol e cules re duce the elec -tro static con tri bu tions to Cou lomb con tri bu tions be tween atomic charges. A rel a tivelylarger num ber of sites are in use for the intermolecular po ten tials of some sim ple mol e cules,as, for ex am ple, wa ter, for which more ac cu racy is sought.

Momany’s idea has led to a new def i ni tion of atomic charges. It would be pos si ble towrite vol umes about atomic charges (AC), a con cept that has no a pre cise def i ni tion in QMfor mal ism, but is of ex treme util ity in prac ti cal ap pli ca tions. Many def i ni tions of AC arebased on ma nip u la tions of the mo lec u lar wave func tion, as, for ex am ple, the fa mousMulliken charges.23 Other def i ni tions are based on dif fer ent anal y ses of the QM def i ni tionof the charge den sity, as for ex am ple Bader’s charges.24 There are also charges de rived from other the o ret i cal ap proaches, such as the electronegativity equal iza tion, or from ex per i men -tal val ues, such as from the vi bra tional po lar ten sors.


The charges ob tained us ing Momamy’s idea of fit ting the MEP with atomic chargesare some times called PDAC (po ten tial de rived atomic charges). It has been re al ized that afit ting of MEP on the whole space was not con ve nient. It is better to re duce this fit ting to thepor tion of space of close con tact be tween mol e cules, i.e., near the van der Waals sur face.This is the main tech nique now in use to de fine PDAC val ues.

The induction termIt is pos si ble to de fine a mo lec u lar in dex PA for the in duc tion term to be used in com bi na tionwith the MEP VA to get a de tailed de scrip tion of the spa tial pro pen sity of the mol e cule to de -velop elec tro static in ter ac tions of clas si cal type. Both func tions are used un der the form ofan in ter ac tion with a unit point charge q placed at po si tion r. In the case of VA this means asim ple mul ti pli ca tion; in the case of PA there is the need of mak ing ad di tional cal cu la tions(to po lar ize the charge dis tri bu tion of A). There are fast meth ods to do it, both at thevariational level25a and at the PT level.25b The anal y sis of PA has not yet ex ten sively beenused to model IND con tri bu tions to ∆E, and it shall not be used here. This re mark has beenadded to sig nal that when one needs to de velop in ter ac tions po ten tials for mo lec u lar not yetstud ied in ter ac tions in clud ing, e.g., com plex sol utes, the use of this ap proach could be ofcon sid er able help.

The multipole ex pan sion of IND may be ex pressed in the fol low ing way:

( ) ( )IND IND A B IND B A= ← + ← [8.51]

with

( ) ( )IND A B R C C m m Ml l k kllm

kkm

lkA

l← ≈ − × ′− + ′+ + ′+′ ′

′′ −

12

2( ),,π m

Bk mB

m k

k

m l

l

kkll

M ′ −< ′+ − <

<

= −

<

′=

∞

′=

∞

∑∑∑∑ ,11

[8.52]

and a sim i lar ex pres sion for IND(B←A) with A and B la bels ex changed.We have here in tro duced the static polarizability ten sor el e ments π lk

X m,m )( ′ : theseten sors are re sponse prop er ties of the mol e cule with re spect to ex ter nal elec tric fields F(with com po nents Fa), elec tric field gra di ents ∇F (with el e ments ∇Fαβ) and higher field de -riv a tives, such as ∇2F (with el e ments ∇2Fαβγ), etc. It is pos si ble to com pute these quan ti tieswith the aid of PT tech niques, but variational pro ce dures are more pow er ful and more ex act.

The polarizability ten sors are de fined as the co ef fi cients of the ex pan sion of one mo -lec u lar prop erty (the en ergy, the di pole, etc.) with re spect to an ex ter nal field and its de riv a -tives.

Among these re sponse prop er ties the most known are the di pole polarizability ten sors α β, , and γ, that give the first three com po nents of the ex pan sion of a di pole mo ment µ sub -jected to an ex ter nal ho mo ge neous field F:

µ∂∂

µ α β γαα

α αβ β αβγ β γ αβγ β γ δ= − = + + + +EF

F F F F F F( )0 K [8.53]

the con ven tion of sum ma tion over the re peated in dexes (which are the Car te sian co or di -nates) is ap plied here.


Other polarizabilities that may have prac ti cal im por tance are the quadrupolepolarizability ten sors A, B, etc. whose el e ments can be de fined in terms of the ex pan sion ofa mo lec u lar quadrupole θ sub jected to a uni form elec tric field:

θ∂

∂θαβ

αβαβ γ αβ γ γδ αβ γ δ= − = + + +3

12

0EF

A F B F F( ), , K [8.54]

The A, B, ... terms are also pres ent in the ex pan sion se ries of the di pole when the mol e -cule is sub jected to higher de riv a tives of F. For ex am ple, the ten sor A de ter mines both thequadrupole in duced by a uni form field and the di pole in duced by a field gra di ent.

The elec tri cal in flu ence of a mol e cule on a sec ond mol e cule can not be re duced to acon stant field F or to the com bi na tion of it with a gra di ent ∇F. So an ap pro pri ate han dling ofthe for mal ex pan sion [8.52] may turn out to be a del i cate task. In ad di tion, the for mal anal y -sis of con ver gence of the se ries gives neg a tive an swers: there are only dem on stra tions thatin spe cial sim ple cases there is di ver gence. It is ad vis able not to push the higher limit oftrun cated ex pan sions much with the hope of mak ing the re sult better.

The use of sev eral ex pan sion cen ters surely im proves the sit u a tion. The av er agedvalue of di pole first polarizability α can be sat is fac to rily ex pressed as a sum of trans fer ablegroup con tri bu tions, but the cal cu la tions of IND with dis trib uted polarizabilities are ratherun sta ble, prob a bly di ver gent, and it is con ve nient to limit the cal cu la tion to the first termalone, as is ac tu ally done in al most all the prac ti cal im ple men ta tions. In sta bil ity and lack ofcon ver gence are fac tors sug gest ing an ac cu rate ex am i na tion of the func tion to fit the op por -tune val ues for these dis trib uted α polarizabilities.

The dispersion termThe dis per sion term DIS can be for mally treated as IND. Asymptotically (at large R), DIScan be ex pressed in terms of the dy namic multipole polarizabilities of the mono mers.

( )DIS R C C m m il l k kllm

kkm

lkA≈ − × ′− + ′+ + ′+

′ ′′

∞

<∫

12

2

0ππ ω( ) , ; ( )

′=− <

<

=−

<

′=

∞

′=

∞

′ ′∑∑∑∑ − − ′m k

k

m l

l

k kl ll kB m m i d

,,

, ;11

π ω ω [8.55]

The dy namic polarizabilities are sim i lar to the static ones, but they are de pend ent on the fre -quency of the ap plied field, and they have to be com puted at the imag i nary fre quency iω.

Here again the stan dard PT tech niques are not ac cu rate enough, and one has to employoth ers tech niques. It is worth re mark ing that at short or in ter me di ate dis tances the dy namicmultipole polarizabilities give an ap pre ci a tion of DIS of poor qual ity, un less com puted onthe dimer ba sis set. This is a con se quence of the BSS er rors: the PT the ory has thus to aban -don the ob jec tive of com put ing ev ery thing solely re ly ing on mono mers prop er ties.

At the variational level there are no for mal prob lems to use the dy namicpolarizabilities, for which there are ef fi cient variational pro ce dures.

The ex pan sion is of ten trun cated to the first term (di pole con tri bu tions only), giv ingor i gin to a sin gle term with dis tance de pend ence equal to R-6. From eq. [8.49] it turns outthat all the mem bers of this ex pan sion have an even neg a tive de pend ence on R. For ac cu ratestud ies, es pe cially for sim ple sys tems in the gas phase, the next terms, C8R

-8, C10R-10, ... are

of ten con sid ered. To have an odd term, one has to con sider third or der el e ments in the PTthe ory, which gen er ally are rarely used.


Lim i ta tions due to the non-con ver gence of these ex pan sion se ries are re duced by theuse of multicenter ex pan sions: in such a way it is pos si ble to reach a sat is fac tory rep re sen ta -tion even for polyatomic mol e cules at short dis tances us ing the first ex pan sion term only.

In gen eral, the ex pan sion sites are the (heavy) at oms of the dimer, us ing a sim pler for -mu la tion due to Lon don. The Lon don for mu la tion uses the av er aged (iso tro pic) value of thestatic di pole polarizabilities, α j , of the at oms and their first ion iza tion po ten tials Ii and Ij:

( ) ( )DIS C AB R C i j Rijj

B

i

A

( ) ,6 66

66≈ =− −∑∑ [8.56]

with

( )C i jI I

I I

i j i j

i j6

32

, =+

α α[8.57]

This for mula de rives from the stan dard RS-PT with the ex pan sion over the ex cited statetrun cated at the first term. For iso lated at oms i and j, the co ef fi cients C6(i,j) can be drawnfrom ex per i men tal data, while for atomic frag ments of mol e cules only from com pu ta tions.

The exchange (or repulsion) termThe ex change terms are re lated to the in tro duc tion of the intermonomer antisymmetrizerAAB in the ex pres sion giv ing the elec tro static con tri bu tion. The cor rec tion fac tors in tro -duced in the SAPT methods15 may be re lated to a renormalization fac tor that may be writ tenin the fol low ing form when AAB is re placed with 1+P for sim plic ity:

( )1 1 1

10 0Φ Φ Ψ Ψ Ψ ΨΑ| |P P PB A B

= =+

[8.58]

This fac tor is dif fer ent from 1, be cause con tains all the mul ti ple over lap val ues be tweenMOs of A and B. <P> may be then ex panded into terms con tain ing an in creas ing num ber ofmul ti ple over laps (or ex changes of orbitals):

P P P P P= + + + +1 2 3 4 K [8.59]

The ex change en ergy may be ex panded into in creas ing pow ers of MO over lapintegrals S:

( ) ( ) ( )EX EX S EX S EX S= + + +2 3 4 K [8.60]

with

( )EX S VP P2 2 2= − [8.61]

This first term is suf fi cient for mod el ing intermolecular po ten tials. The over laps de -pend on the mono mer sep a ra tion roughly as exp(-αR) so the fol low ing terms give smallcon tri bu tion at large-me dium dis tances. At short dis tances the pos i tive ex change term rap -idly in creases. Rea sons of uni for mity with the other terms of the in ter ac tion po ten tial sug -


gest re plac ing the exp(-αR) de pend ence with a highly neg a tive power of R: R-12 in mostcases. The ex change terms de scribe what is of ten called the steric re pul sion be tween mol e -cules (ac tu ally there are other short range re pul sive con tri bu tions, as the elec tro static pen e -tra tion com po nents not de scribed by the multipole ex pan sion).

The other termsCharge trans fer con tri bu tions (CT) have been rarely in tro duced in the mod el ing of in ter ac -tion po ten tials for liq uids. Some at tempts at mod el ing have been done for the study of in ter -ac tions lead ing to chem i cal re ac tions, but in such cases, di rect cal cu la tions of the in ter ac tion term are usu ally em ployed.

For cu ri ous read ers, we add that the ten ta tive mod el ing was based on an al ter na tivefor mu la tion of the PT (rarely used for com plete stud ies on the intermolecular in ter ac tion,be cause it pres ents sev eral prob lems) in which the pro mo tion of elec trons of a part ner on the vir tual or bital of the sec ond is ad mit ted. The for mal ex pres sions are sim i lar to those shownin eq. [8.40] for the II or der con tri bu tions. We have

−−

∑Ψ Ψ Ψ ΨΚ0 0 0

2

0

a b a b

Ka a

K

V

E E

|

The sum over K is strongly re duced (of ten to just one term, the HOMO-LUMO in ter ac tion), with K cor re spond ing to the re place ment of the oc cu pied MO φr

0 of A with the vir tual MO φt

v of B (A is the do nor, B the ac cep tor). The ex pres sion is then sim pli fied: the nu mer a tor isre duced to a com bi na tion of two-elec tron Cou lomb integrals mul ti plied by the op por tuneover lap. The CT con tri bu tion rap idly de cays with in creas ing R.

COUP con tri bu tions, ob tained as a re main der in the variational de com po si tion of ∆E,are not mod eled. The con tri bu tions are small and of short-range char ac ter.

A conclusive viewIn this long anal y sis of the mod el ing of the sep a rate com po nents we have learned the fol -low ing:

• All terms of the decomposition may be partitioned into short- and long-rangecontributions.

• The long-range contributions present problems of convergence, but these problemsmay be reduced by resorting to multicenter distributions based on suitable partitionsof the molecular systems.

• The short-range contributions are in general of repulsive character and aredominated by the EX terms. The effect of the other short-range contributions isstrongly reduced when multicenter expansions are used.

Adapting these re marks, one may write a ten ta tive an a lyt i cal ex pres sion for ∆E(R):

∆E C Rrtn

rtn

ntr

≈ −∑∑∑ ( ) [8.62]

The ex pan sion cen ters (called “sites”) are in di cated by the in dexes r and t for the mol e -cule A and B re spec tively. The in dex n in di cates the be hav ior of the spe cific term with re -spect to the intersite dis tance Rrt; each cou ple r-t of sites has a spe cific set of al lowed nval ues.


In such a way it is pos si ble to com bine sites cor re spond ing only to a point charge withsites hav ing more terms in the ex pan sion. For two sites r-t both de scribed by a point chargethe only in ter ac tion term is C Rrt

(1)rt-1, while for a cou ple r-t de scribed by a point charge and a

di pole, the con tri bu tion to the in ter ac tion is lim ited to the R-2 term, and, if it is de scribed bytwo di poles, to the R-3 term. The last ex am ple shows that eq. [8.62] is cryp tic or not well de -vel oped. In fact, if we ex am ine eq. [8.12], which re ports the in ter ac tion en ergy be tween twodi poles, it turns out that the de pend ence on the ori en ta tion an gle θ ap par ently is miss ing ineq. [8.62]. The co ef fi cients Crt

(n) must be spec i fied in more de tail as given in equa tions [8.50]and [8.52] by add ing the op por tune l, l' and m in dexes (or the cor re spond ing com bi na tion ofCar te sian co or di nates), or must be re duced to the iso tro pic form. This means to re place thethree com po nents of a di pole, and the five com po nents of a quadrupole, and so on, by an av -er age over all the ori en ta tions (i.e., over the m val ues). There is a loss of ac cu racy, very large in the case of two di poles, and of de creas ing im por tance in pass ing to higher multipoles.The loss in ac cu racy is greatly de creased when a multi-site de vel op ment is em ployed. Many an a lyt i cal ex pres sions of in ter ac tion po ten tials use this choice. In such cases the Crt

(n) co ef fi -cients are just num bers.

The co ef fi cients Crt(n) can be drawn from ex per i men tal val ues (but this is lim ited to the

cases in which there is one site only for both A and B), from ad hoc cal cu la tions, or by a fit -ting of ∆E val ues. In the past, large use has been made of “ex per i men tal” ∆E val ues, de rived, e.g., from crys tal pack ing en er gies, but now the main sources are the variationally com puted QM val ues, fol lowed by a fit.

The use of ex pres sion [8.62] for the fit ting of QM val ues rep re sents a re mark able im -prove ment with re spect to the past strat egy of reach ing a good fit ting with com plete free -dom of the an a lyt i cal form of the ex pres sion. There are in the lit er a ture in ter ac tionpo ten tials us ing, e.g., non-in te ger val ues of n and/or other an a lyt i cal ex pres sions with outphys i cal mean ing. This strat egy leads to po ten tials that can not be ex tended to sim i lar sys -tems, and that can not be com pared with the po ten tial ob tained by oth ers for the same sys -tem.

Ex pres sion [8.62] has sev eral mer its, but also sev eral de fects. Among the latter wenote that given pow ers of R may col lect terms of dif fer ent or i gin. For ex am ple, the term R-6

de scribes both the in duc tion di pole-in duced di pole in ter ac tion and the dis per sion di pole-di -pole in ter ac tion: two con tri bu tions with a dif fer ent sen si tiv ity with re spect to changes in themo lec u lar sys tem. The ory and com pu ta tional meth ods both per mit get ting two sep a rate Crt

(6)

co ef fi cients for the two con tri bu tions. This is done in a lim ited num ber of po ten tials (themost im por tant cases are the NEMOn26 and the ASP-Wn27 fam i lies of mod els). The mostpop u lar choice of a unique term for con tri bu tions of dif fer ent or i gin is mo ti vated by theneed of re duc ing com pu ta tional ef forts, both in the der i va tion and in the use of the po ten tial. To fit a unique Crt

(6) co ef fi cient means to use ∆E val ues only: for two co ef fi cients there is theneed of sep a rately fit ting IND and DIS con tri bu tions.

8.4.5 THE RELAXATION OF THE RIGID MONOMER CONSTRAINT

We have so far ex am ined de com po si tions of two-body in ter ac tion po ten tials, keep ing fixedthe in ter nal ge om e try of both part ners. This con straint is clearly unphysical, and does notcor re spond to the naïve model we have con sid ered in the in tro duc tion, be cause mol e culesal ways ex hibit in ter nal mo tions, even when iso lated, and be cause mo lec u lar col li sions in aliq uid (as well as in a clus ter) lead to ex changes of en ergy be tween in ter nal as well as ex ter -nal de grees of free dom.


It may be con ve nient to in tro duce a loose clas si fi ca tion of the or i gin of changes in thein ter nal ge om e try of mol e cules in clus ters and liq uids, and to use it for the dimer case we are con sid er ing here:

1) per ma nent or semipermanent mo lec u lar in ter ac tions;2) in ter nal dy na mism of the mol e cule;3) mo lec u lar col li sions.In the va ri ety of liq uid sys tems there are quite abun dant cases in which rel a tively

strong in ter ac tions among part ners in duce changes in the in ter nal ge om e try. The for ma tionof hy dro gen-bonded ad ducts is an ex am ple of gen eral oc cur rence (e.g., in wa ter so lu tions).The in ter ac tions of a metal cat ions and their first sol va tion shells is a sec ond out stand ing ex -am ple but the va ri ety of cases is very large. It is not easy to give gen eral rules on the dis tinc -tion be tween per ma nent and semipermanent in ter ac tions of this type. In gen eral it de pendson the time scale of the phe nom e non be ing stud ied, but it is clear that the long res i dencetime of wa ter mol e cule (in some cases, on the or der of years28) in the first sol va tion shellaround a cat ion leads us to con sider this ef fect as per ma nent.

In such cases, it is con ve nient to re con sider the def i ni tion of the clus ter ex pan sion andto in tro duce some ex tra vari ables in the nu clear co or di nate subspace to span for the anal y sis. For ex am ple, in the case of a dimer Mn+·H2O, it is con ve nient to add three co or di nates cor re -spond ing to the in ter nal co or di nates of the wa ter mol e cule (the to tal num ber of de grees offree dom in such case is again 6, be cause of the spher i cal sym me try of the metal cat ion).There is no need of re peat ing, for this model, the variational de com po si tion of the en ergy(the PT ap proaches have more dif fi cul ties to treat changes in the in ter nal ge om e try). Thecon clu sions do not change qual i ta tively, but the quan ti ta tive re sults can be sen si bly mod i -fied.

An other im por tant case of per ma nent in ter ac tions is re lated to chem i cal equi lib riawith mo lec u lar com po nents of the liq uid. The out stand ing ex am ple is the prototropic equi -lib rium, es pe cially the case AH + B → A- + HB+. There are wa ter-wa ter po ten tials, in clud -ing the pos si bil ity of de scrib ing the ionic dis so ci a tion of H2O.

The semipermanent in ter ac tions are gen er ally ne glected in the mod el ing of the po ten -tials for liq uids. Things are dif fer ent when one looks at prob lems re quir ing a de tailed lo calde scrip tion of the in ter ac tions. Mo lec u lar dock ing prob lems are a typ i cal ex am ple. In suchcases, use is made of variational QM cal cu la tions, or es pe cially for dock ing, where one mol -e cule at least has a large size, use is made of mo lec u lar me chan ics (MM) al go rithms, al low -ing lo cal mod i fi ca tion of the sys tems.

The sec ond cat e gory cor re sponds to mo lec u lar vi bra tions; ro ta tions of a mol e cule as awhole have been al ready con sid ered in, or def i ni tion of, the ∆E(R) po ten tial. The third cat e -gory, mo lec u lar col li sions, gives rise to ex change of en ergy among mol e cules that can beex pressed as changes in the traslational and ro ta tional en er gies of the rigid mol e cule andchanges in the in ter nal vi bra tional en ergy dis tri bu tions. In con clu sion, all the cases ofnon-ri gid ity we have to con sider here can be lim ited to mo lec u lar vi bra tions con sid ered in abroad sense (cou plings among vi bra tions and ro ta tions are gen er ally ne glected inintermolecular po ten tials for liq uids).

The vi bra tions are gen er ally treated at the clas si cal level, in terms of lo cal de for ma tion co or di nates. The lo cal de for ma tion func tions are of the same type of those used in mo lec u -lar me chan ics (MM) meth ods. Now a days, MM treats ge om e try changes for mol e cules ofany di men sion and chem i cal com po si tion.


In the MM ap proach, the en ergy of the sys tem is de com posed as fol lows:

Etot = [Estr + Eben + Etor + Eother] + Eelec + EvdW [8.63]

the terms within square brack ets re gard con tri bu tions due to the bond stretch ing, to the an -gle bend ing, and to tor sional in ter ac tions, sup ple mented by other con tri bu tions due to morespe cific de for ma tions and by cou plings among dif fer ent in ter nal co or di nates. The Eelec

terms re gard Cou lomb and in duc tions terms be tween frag ments not chem i cally bound:among them there are the in ter ac tions be tween sol ute and sol vent mol e cules. The sameholds for the van der Waals in ter ac tions (i.e., re pul sion and dis per sion) col lected in the lastterm of the equa tion.

It must be re marked that Etot is ac tu ally a dif fer ence of en ergy with re spect to a state ofthe sys tem in which the in ter nal ge om e try of each bonded com po nent of the sys tem is atequi lib rium.

The ver sions of MM po ten tials (gen er -ally called force fields) for sol vent mol e culesmay be sim pler, and in fact lim ited li brar ies ofMM pa ram e ters are used to de scribe in ter nalge om e try change ef fects in liq uid sys tems. We re port in Ta ble 8.4 the def i ni tions of the ba sic,and sim pler, ex pres sions used for liq uids.

To de scribe in ter nal ge om e try ef fects inthe dimeric in ter ac tion, these MM pa ram e tersare in gen eral used in com bi na tion with a par -ti tion ing of the in ter ac tion po ten tial intoatomic sites. The cou pling terms are ne glected

and the nu mer i cal val ues of the pa ram e ters of the site (e.g., the lo cal charges) are left un -changed when there is a change of in ter nal ge om e try pro duced by these lo cal de for ma tions.Only the rel a tive po si tion of the sites changes. We stress that in ter ac tion po ten tial only re -gards in ter ac tion among sites of dif fer ent mol e cules, while the lo cal de for ma tion af fects thein ter nal en ergy of the mol e cule.

8.5 THREE- AND MANY-BODY INTERACTIONSThe tree-body com po nent of the in ter ac tion en ergy of a trimer ABC is de fined as:

( ) ( ) ( )∆E ABC R E R E R EABC ABC ABC AB AB KKBA

; = − − ∑∑∑12

[8.64]

This func tion may be com puted, point-by-point, over the ap pro pri ate RABC space, ei therwith variational and PT meth ods, as the dimeric in ter ac tions. The re sults are again af fectedby BSS er rors, and they can be cor rected with the ap pro pri ate ex ten sion of the CP pro ce -dure. A com plete span of the sur face is, of course, by far more de mand ing than for a dimer,and ac tu ally ex ten sive scans of the de com po si tion of the EABC po ten tial en ergy sur face havethus far been done for a very lim ited num ber of sys tems.

Anal o gous re marks hold for the four-body com po nent ∆E(ABCD; RABCD) of the clus -ter ex pan sion en ergy, as well as for the five- and six- body com po nents, the def i ni tion of

8.5 Three- and many-body interactions 419

Table 8.4

STRETCH Estr =ks(1 - l0)2

BEND Eben = kb(θ - θ0)2

TORSION Etor = kt[1 ± cos(nω )]

ks stretch force con stantkb bend force con stantkt tor sion force con stantl0 ref er ence bond lengthθ0 ref er ence bond an gle

which can be eas ily ex tracted from the eq. [8.9]. The com pu ta tional task be comes more andmore ex act ing in in creas ing the num ber of bod ies, as well as the di men sions which rise to 18 with fixed ge om e tries of the com po nents, and there are no rea son able per spec tives of hav -ing in the fu ture such a sys tem atic scan as that avail able for dimers.

The stud ies have been so far cen tered on four types of sys tems that rep re sent four dif -fer ent (and typ i cal) sit u a tions:

1) clus ters com posed of rare gas at oms or by small al most spher i cal mol e cules (e.g., meth ane);

2) clus ters com posed of po lar and protic mol e cules (es pe cially wa ter);3) clus ters com posed of a sin gle non-po lar mol e cule (e.g., an alkane) and

a num ber of wa ter mol e cules;4) clus ters com posed of a sin gle charged spe cies (typ i cally atomic ions) and

a num ber of wa ter mol e cules.In side the four cat e go ries, at ten tion has mostly been paid to clus ters with a uni form

chem i cal com po si tion (e.g., tri mers AAA more than A2B or ABC for cases 1 and 2). Clearly the types of many-body anal y ses are not suf fi cient to cover all cases of chem i -

cal rel e vance for liq uids; for ex am ple, the mixed po lar sol vents (be long ing to type 2) and the wa ter so lu tions con tain ing a cat ion in pres ence of some ligand (type 4) are poorly con sid -ered.

In spite of these de fi cien cies the avail able data are suf fi cient to draw some gen eraltrends that in gen eral con firm what phys i cal in tu ition sug gests. We com bine here be lowcon clu sions drawn from the for mal anal y sis and from the ex am i na tion of nu mer i cal cases.

ES con tri bu tions are strictly ad di tive: there are no three- or many body terms for ES.The term “many-body cor rec tion” to ES in tro duced in some re views ac tu ally re gards twoother ef fects. The first is the elec tron cor re la tion ef fects which come out when the start ingpoint is the HF de scrip tion of the mono mer. We have al ready con sid ered this topic that doesnot be long, strictly speak ing, to the many-body ef fects re lated to the clus ter ex pan sion [8.9]. The sec ond re gards a screen ing ef fect that we shall dis cuss later.

The other con tri bu tions are all non-ad di tive. The clus ter ex pan sion [8.9] ap plies tothese terms too:

( ) ( ) ( )TERM ABCDE TERM R TERM RAB AB ABC ABCCBABA

K K= + +∑∑∑∑∑12

16

[8.65]

The most im por tant are the IND and DIS terms. CT is ac tive for some spe cial sys tems(long range elec tron trans fer) but of scarce im por tance for nor mal liq uids. EX many-bodycon tri bu tions rap idly fade away with the num ber of bod ies: the three-body terms have beenstud ied with some de tails,29 but they are not yet used ex ten sively to model in ter ac tion po ten -tials for liq uids.

The IND many-body con tri bu tions are dom i nated by the first di pole polarizability αcon tri bu tions. The field F pro duced by the charge dis tri bu tion of the other mol e cules andact ing on the units for which α is de fined, at a first (and good) ap prox i ma tion, as ad di tive.The non-lin ear ity de rives from the fact that the con tri bu tion to the in ter ac tion en ergy is re -lated to the square of the elec tric field. So in mod el ing many body ef fects for IND the at ten -tion is lim ited to mo lec u lar con tri bu tions of the type


( ) ( )E M F rind mind

mm

M

= −=

∑12 1

µ [8.66]

the in dex m runs over all the sites of the mol e cule M. The di pole mo ment of site m is de com -posed into a static and an in duc tion di pole:

µ µ µm m mind= +0 [8.67]

with the in duc tion term αm de pend ing on the polarizability and on the to tal elec tric field Ftot.The lat ter co mes from the other per ma nent charges and di poles, as well as from the in duceddi poles pres ent in the sys tem:

( )µ α αmind

mtot

m s ms

S

F F r= ==

∑1

[8.68]

Gen erally, the cal cu la tion of F is not lim ited to three-body con tri bu tions but in cludeshigher or der terms; for ex am ple, in sim u la tion meth ods the con tri bu tions com ing from allthe mol e cules in cluded in the sim u la tion box, which may be of the or der of hun dreds. Thesecon tri bu tions are not com puted sep a rately, but just used to have a col lec tive value of F.

The sum of three- and higher body con tri bu tions to IND thus may be writ ten and com -puted in the fol low ing way:

( ) ( )IND many body E m Findtot

m

− ≈ ∑ ; [8.69]

When the sys tem con tains mol e cules of large size it is ad vis able to use a many-site de -scrip tion of the mo lec u lar polarizability α. This of course means to in crease the com pu ta -tional ef fort, which is not neg li gi ble, es pe cially in com puter sim u la tions.

The many-body con tri bu tion to IND may be neg a tive as well as pos i tive. In gen eral, itre in forces with a neg a tive con tri bu tion min ima on the PES.

The non-ad di tive DIS con tri bu tions are de scribed in terms of the first di pole dy namicpolarizability or by the cor re spond ing ap prox i mate ex pres sion we have in tro duced for thedimeric case.

DIS con tri bu tions are al ready pres ent and are quite im por tant for the liq uid ag gre ga -tion of spher i cal sys tems. Even spher i cal mono mers ex hibit dy nam i cal di poles and the re -lated two- and many-body con tri bu tions to DIS.

The most im por tant term is known as the Axilrod-Teller term and de scribes the in ter -ac tion of three in stan ta neous di poles. The sign of the Axilrod-Teller term strongly de pendson the ge om e try. It is neg a tive for al most lin ear ge om e tries and pos i tive in tri an gu lar ar -range ments. Four body con tri bu tions of the same na ture are gen er ally of op po site sign andthus they re duce the ef fect of these terms on the PES shape.

The po ten tials for po lar liq uids gen er ally ne glect many-body dis per sion con tri bu tions.

Screening many-body effectsWhen the ma te rial con densed sys tem con tains mo bile charges, the elec tro static in ter ac -tions, which are strong and with the slow est de cay with the dis tance, are se verely damped.Ev ery charged com po nent of the sys tem tends to at tract mo bile com po nents bear ing the op -


po site charge. Typ i cal ex am ples are the counterion charge cloud sur round ing each ion insalt so lu tions, and the counterion con den sa tion ef fects act ing on im mo bile charged spe ciesas DNA or other polyelectrolytes in salt so lu tions. In both cases, at large dis tance the elec -tro static ef fect of the charge is screened by the sim i lar elec tro static ef fect, but with an op po -site sign, of the counterion dis tri bu tion.

A sim i lar screen ing ef fect is pres ent for dipolar liq uids: each mol e cule bear ing a per -ma nent di pole tends to or ga nize around it self other dipolar mol e cules with the ori en ta tionwhich optimizes sta bi li za tion en ergy and screens the field pro duced by the sin gled out mol -e cule.

The or ga ni za tion of the ori en ta tion of mo lec u lar di poles around a sin gled-out chargedcom po nent is even more ef fec tive. The in ter ac tion at long dis tance be tween two pointcharges screened by the sol vent di poles is Q1Q2/εR12 where ε is the di elec tric con stant of theliq uid. When the two charges are placed in a salt so lu tion the value of the elec tro static en -ergy is even lower, be ing screened by counterions as well as by ori ented di poles.

For the dis per sion con tri bu tions there is no screen ing. We ar rive to the ap par ently par -a dox i cal sit u a tion that for rel a tively large bod ies, even when bear ing net charges, the in ter -ac tions at large dis tances are gov erned by the dis per sion forces, weak in com par i son withoth ers, and having a more rapid de crease with the dis tance.

Effective interaction potentialsIn the last sen tences we have changed view with re spect to the be gin ning of Section 8.4,where we in tro duced the clus ter ex pan sion of the in ter ac tion po ten tials. Still keep ing a mi -cro scopic view, we passed from the ex am i na tion of the prop er ties of PES of in creas ing di -men sions to a view that takes into con sid er ation very lim ited por tions of such sur facesse lected on the ba sis of en er getic ef fects, or, in other words, on the ba sis of proba bil is ticcon sid er ations. The screen ing ef fects we have men tioned are due to an av er aged dis tri bu -tion of the other com po nents around a sin gle part ner of the sys tem, and its av er age is basedon the rel a tive en er gies of all the con for ma tions of the sys tem, span ning the whole per ti nentPES.

This view based on av er aged dis tri bu tions is re lated to the “fo cused” model we in tro -duced in Section 8.2.

If we sup pose that the as sist ing part S of the sys tem de scribed by the Hamiltonian [8.8] with co or di nates XS is in equi lib rium with the main part M at ev ery co or di nate XM, we mayre peat the clus ter de com po si tion anal y ses given in Sec tion 8.3, start ing again from thedimer and pro ceed ing to the tri mers, etc. The po ten tial func tion ∆EAB(R) we have called po -ten tial en ergy sur face (PES) now as sumes a dif fer ent name, po ten tial of mean force (MFP),em pha siz ing the fact that it re gards no more in ter ac tions be tween A and B but also the meanef fects of the other bod ies pres ent in the sys tem. It must be re marked that the ap proach is not lim ited to equi lib rium dis tri bu tions of S: it may be ex tended to dis tri bu tions of S dis placedfrom the equi lib rium. The non-equi lib rium de scrip tion has an im por tant role, and the im ple -men ta tion of non-equi lib rium meth ods to day rep re sents one of the fron tiers of the the ory ofliq uids.

An im por tant as pect of this re vis ited anal y sis of the dis sec tion of ∆EAB into sep a ratecom po nents is that all terms, in clud ing ES, are now de scribed in terms of mono mer dis tri bu -tions which feel the ef fects of S. The anal y ses we have done sug gest that these changes canbe ex pressed in terms of a mono mer’s MO ba sis φi

p “po lar ized” in a way sim i lar to that wehave used to de scribe IND. Actually, this “po lar iza tion” or de for ma tion of the mono mer is


not due to clas si cal po lar iza tion ef fects only, but to all the com po nents of the in ter ac tion of S with A-B, namely dis per sion, ex change, etc.

The re sults of the de com po si tion of the intermolecular in ter ac tion en ergy of clus ter Mare not qual i ta tively dif fer ent from those found for the same clus ter M in vacuo. There arequan ti ta tive changes, of ten of not neg li gi ble en tity and that cor re spond, in gen eral, to adamp ing of the ef fects.

The use of this ap proach re quires us to con sider the avail abil ity of ef fi cient and ac cu -rate mod els and pro ce dures to eval u ate the ef fects of S on M, and at the same time, of M onS. The ba sic pre mises of this ap proach have been laid many years ago, es sen tially with thein tro duc tion of the con cept of sol vent re ac tion field made by Onsager in 1936,30 but only re -cently have they been sat is fac tory for mu lated. There are good rea sons to ex pect that theiruse on the for mu la tion of “ef fec tive” intermolecular po ten tials will in crease in the next fewyears.

Actually, the de com po si tion of the ∆EM(R) in ter ac tion en ergy is pre ceded by a step not pres ent for iso lated sys tems.

There is the need of de fin ing and an a lyz ing the in ter ac tions of each mono mer A, B,etc. with the sol vent alone. The fo cused model is re duced to a sin gle mol e cule alone, let ussay A, which may be a sol ute mol e cule but also a com po nent of the dom i nant mole frac tion,the sol vent, or a mo lec u lar com po nent of the pure liq uid.

This sub ject is treated in de tail in other chap ters of this book and some thing more willbe added in the remainder of this chap ter. Here, we shall be con cise.

There are two main ap proaches, the first based on dis crete (i.e., mo lec u lar) de scrip -tions of S, the sec ond on a con tin u ous de scrip tion of the as sist ing por tion of the me dium, via ap pro pri ate in te gral equa tions based on the den sity dis tri bu tion of the me dium and on ap -pro pri ate in te gral ker nels de scrib ing the var i ous in ter ac tions (clas si cal elec tro static, dis per -sion, ex change-re pul sion). Both ap proaches aim at an equil i bra tion of S with M: there is theneed of re peat ing cal cu la tions un til the de sired con ver gence is reached. More de tails onboth ap proaches will be given in the spe cific sec tions; here we shall limit our selves toquoting some as pects re lated to in ter ac tion po ten tials, which con sti tute the main topic ofthis sec tion.

The ap proach based on dis crete de scrip tions of the sol vent makes ex plicit use of themo lec u lar in ter ac tion po ten tials which may be those de fined with out con sid er ation of S; thecal cu la tions are rather de mand ing of com pu ta tion time, and this is the main rea son ex plain -ing why much ef fort has been spent to have sim ple an a lyt i cal ex pres sions of such po ten tials.

The sec ond con tin u ous ap proach is of ten called Ef fec tive Hamiltonian Ap proach(EHA) and more re cently Im plicit Sol va tion Method (ISM).31 It is based on the use of con -tin u ous re sponse func tions not re quir ing ex plicit sol vent-sol vent and sol ute-sol vent in ter ac -tion po ten tials, and it is by far less com puter-de mand ing than the sim u la tion ap proach.

As we shall show in the sec tion de voted to these meth ods, there are sev eral pos si blever sions; here it is worth quot ing the Polarizable Con tin uum Model (PCM) for cal cu la tionsat the ab in itio QM level.32

PCM it is the only method used so far to get ef fec tive two body A-B po ten tials over the whole range of dis tances R (three-body cor rec tions have been in tro duced in Ref. [33]).

Un til now, the EHA ap proach has been mostly used to study sol vent ef fects on a sol ute mol e cule. In such stud ies, M is com posed of just a sin gle sol ute mol e cule (M = A) or of asol ute mol e cule ac com pa nied by a few sol vent mol e cules (M = ASn). These al ter na tive


choices make the re sults in di rectly rel e vant to the ques tion of the dif fer ence be tween tra di -tional and ef fec tive intermolecular po ten tial en er gies. How ever, some hints can be ex -tracted from the avail able data. The ex pe ri ence thus far col lected shows that when both Mand S have a po lar na ture, the clas si cal elec tro static con tri bu tions are dom i nant with an ex -change-re pul sion term, giv ing an im por tant con tri bu tion to the en ergy but a small ef fect onthe charge dis tri bu tion of M and, as a con se quence, on the mo lec u lar prop er ties on whichthe in ter ac tion po ten tial de pends (e.g., multipole dis tri bu tion of the charge den sity, staticand dy namic polarizabilities). Dis per sion ef fects play a role when M is not po lar and theybe come the dom i nant com po nent of the sol va tion ef fects when both M and S are not po lar.

The cal cu la tion of ef fec tive PCM in ter ac tion po ten tials has been so far lim ited to thecases in which the sol vent ef fects are more size able, namely the in ter ac tions of metal cat -ions with wa ter. They have been used in com puter sim u la tions of mod els of very di luteionic so lu tions (a sin gle ion in wa ter), no tice ably im prov ing the re sults with re spect to sim i -lar stud ies us ing tra di tional po ten tials.34

8.6 THE VARIETY OF INTERACTION POTENTIALSIn the pre ced ing sec tions we have ex am ined the def i ni tion of the com po nents of theintermolecular po ten tial, pay ing at ten tion mostly to rig or ous meth ods and to some ap prox i -ma tions based on these meth ods.

The be gin ning of qual i ta tive and semi-quan ti ta tive com pu ta tional stud ies on the prop -er ties of liq uids dates back at least to the 1950s, and in the early years of these stud ies, thecom pu ta tional re sources were not so pow er ful as they are to day. For this rea son, for manyyears, the po ten tials used have been sim ple, dis card ing what our un der stand ing of thedimeric in ter ac tions sug gested. There is no rea son, how ever, to ne glect here these sim pli -fied ver sions of the po ten tial, be cause very sim ple and naïve ex pres sions have given ex cel -lent re sults and are still in use. The strug gle for better and more de tailed po ten tials isjus ti fied, of course, be cause chem is try al ways tends to have more and more ac cu rate es ti -mates of the prop er ties of in ter est, and there are many prob lems in which a semi-quan ti ta -tive as sess ment is not suf fi cient.

We shall try to give a cur sory view of the va ri ety of intermolecular po ten tials in use for liq uid sys tems, pay ing more at ten tion to the sim plest ones. We shall al most com pletely ne -glect po ten tials used in other fields, such as the scat ter ing of two iso lated mol e cules, the de -ter mi na tion of spec tro scopic prop er ties of dimers and tri mers, and the ac cu rate study oflo cal chem i cal in ter ac tions, which all re quire more so phis ti cated po ten tials.

The main cri te rion to judge the qual ity of a po ten tial is a pos te ri ori, namely, based onthe ex am i na tion of the per for mance of that po ten tial in de scrib ing prop er ties of the liq uidsys tem. In fact, the per for mances of a po ten tial are to a good ex tent based on sub tle fac torsex press ing the equi lib rium reached in the def i ni tion of its var i ous com po nents. The ex pe ri -ence gained in us ing this cri te rion has un for tu nately shown that there is no def i nite an swer.Typically, a po ten tial, or a fam ily of po ten tials, better de scribe some prop er ties of the liq uid, while a sec ond one is more proper for other prop er ties. For this rea son there are many po ten -tials in use for the same liq uid and there are in the lit er a ture con tin u ous com par i sons amongdif fer ent po ten tials. A sec ond cri te rion is based on the com pu ta tional cost, an as pect of di -rect in ter est for peo ple plan ning to make nu mer i cal stud ies but of lit tle in ter est for peo pleonly in ter ested in the re sults.


Our ex po si tion will not try to clas sify po ten tials ac cord ing to such cri te ria, but sim plyshow the va ri ety of po ten tials in use. We shall pay at ten tion to the de scrip tion adopted for asin gle mol e cule, that must then be com bined with that adopted for the in ter ac tion part ners.In par tic u lar, we shall con sider the num ber of sites and the shape of the mol e cule used ineach de scrip tion.

The num ber of sites re flects the pos si bil ity we have ex am ined, and ad vo cated, of us ing many-cen ter ex pan sions to im prove the rep re sen ta tion. Each ex pan sion cen ter will be a site. There are mod els with one, two, and more sites. This se quence of in creas ing com plex ityreaches the num ber of heavy at oms of the mol e cule and then the whole num ber of at oms, in -clud ing hy dro gens. It is not lim ited to the nu clei as ex pan sion sites. There are po ten tials in -tro duc ing other lo ca tions of sites, in sub sti tu tion or in ad di tion to the nu clei. For ex am plepo ten tials widely used in sim u la tions adopt for wa ter a four-site model; other po ten tials(rarely used in sim u la tions) pre fer to use the mid dle of the bonds in stead of (or in ad di tionto) nu clei. Each site of the mol e cule must be com bined with the sites of the sec ond (andother) mol e cule to give the po ten tial.

The shape of the mol e cule re flects the ef fect of the ex change-re pul sion in ter ac tion.For al most all many-site mod els the shape is not given, but it im plic itly re sults to be that ofthe un ion of the spheres cen tered on the ex pan sion sites pro vided by a source of ex -change-re pul sion po ten tial. There are some sim ple mod els in which the shape is ex plic itlystated. There will be spheres, el lip soids, cyl in ders and more com plex shapes, as fusedspheres, spherocylinders, etc. Some typ i cal ex am ples are re ported in Ta ble 8.5.

Table 8.5. Single site-based potentials

# Shape In ter ac tion Name

1 Sphere hard HS

2 Sphere soft SS

3 Sphere hard with charge CHS

4 Sphere soft with charge CSS

5 Sphere hard with rigid di pole DHS

6 Sphere soft with di pole DSS

7 Disc hard HD

8 Disc hard with di pole SD

9 El lip soid hard

10 El lip soid Gay-Berne35

11 Quadrupolar shape Zewdie36

12 Sphere re pul sion dis per sion Lennard Jones (LJ)

13 Sphere re pul sion dis per sion Buckingham

14 Sphere LJ + charge LJ+q

15 Sphere LJ + di pole Stockmayer

16 Sphere LJ + di pole+ soft sticky SSD37

8.6 The variety of interaction potentials 425

Po ten tial 1 is ex tremely sim ple: the only pa ram e ter is the ra dius of the sphere. In spiteof this sim plic ity, an im pres sive num ber of phys i cal re sults have been ob tained us ing the HS po ten tial on the whole range of den si ties and ag gre ga tions. Also, mix tures of liq uids havebeen suc cess fully treated, in tro duc ing in the com pu ta tional ma chin ery the de sired num berof spheres with ap pro pri ate ra dii. The HS model is at the ba sis of the Scaled Par ti cle The ory(SPT),38 which still con sti tutes a ba sic el e ment of mod ern sol va tion meth ods (see later formore de tails).

Of course, HS can not give many de tails. A step to ward re al ism is given by po ten tial 2,in which the hard po ten tial is re placed by a steep but smoothly re pul sive po ten tial. It isworth re mind ing read ers that the shift from HS to SS in com puter sim u la tions re quired theavail abil ity of a new gen er a tion of com put ers. Sim u la tions had in the past, and still have atpres ent, to face the prob lem that an in crease in the com plex ity of the po ten tial leads to alarge in cre ment of the com pu ta tional de mand.

Po ten tials 3 and 4 have been in tro duced to study ionic so lu tions and sim i lar flu ids con -tain ing mo bile elec tric charges (as, for ex am ple, mol ten salts). In ionic so lu tions the ap pro -pri ate mix ture of charged and un charged spheres is used. These po ten tials are the firstex am ples of po ten tials in our list, in which the two-body char ac ter is tics be come ex plicit.The in ter ac tion be tween two charged spheres is in fact de scribed in terms of the charge val -ues of a cou ple of spheres: QkQl/rlk (or QkQl/εrlk).

Po ten tials 5 and 6 are the first ex am ples in our lists of anisotropic po ten tials. The in ter -ac tion here de pends on the mu tual ori en ta tions of the two di poles. Other ver sions of the di -pole-into-a-sphere po ten tials (not re ported in Ta ble 8.5) in clude in duced di poles andac tu ally be long to a higher level of com plex ity in the mod els, be cause for their use there isthe need of an it er a tive loop to fix the lo cal value of F. There are also other sim i lar mod elssim u lat ing liq uids in which the lo ca tion of the di poles is held fixed at nodes of a reg u lar 3Dgrid, the hard sphere po ten tial is dis carded, and the op ti mi za tion only re gard ori en ta tion and strength of the lo cal di poles. This last type of model is used in com bi na tion with sol utes Mde scribed in an other, more de tailed way.

Po ten tials 7-11 add more re al ism in the de scrip tion of mo lec u lar shape. Among them,the Gay-Berne35 po ten tial (10) has gained a large pop u lar ity in the de scrip tion of liq uid crys -tals. The quite re cent po ten tial (11)36 aims at re plac ing Gay-Berne po ten tial. It has beenadded here to show that even in the field of sim ple po ten tials there is space for in no va tion.

Po ten tials 12 and 13 are very im por tant for chem i cal stud ies on liq uids and so lu tions.The Lennard-Jones po ten tial (12) in cludes dis per sion and re pul sion in ter ac tions in the

form:

LJ: ER RAB

LJ =

−

εσ σ12 6

[8.70]

The de pend ence of the po ten tial on the cou ple of mol e cules is here ex plicit, be causethe pa ram e ters ε and σ de pend on the cou ple. In the orig i nal ver sion, ε is de fined as the depthof the at trac tive well and σ as the dis tance at which the steep re pul sive wall be gins.

More re cently, the LJ ex pres sion has been adopted as an an a lyt i cal tem plate to fit nu -mer i cal val ues of the in ter ac tion, us ing in de pend ently the two pa ram e ters.:


LJ with 2 pa ram e ters: ( )E R AR

CRAB

LJAB AB=

−

1 112 6

[8.71]

LJ ex pres sions are of ex ten sive use in MM to model non-co va lent in ter ac tions amongmo lec u lar sites, of ten with the ad di tion of lo cal charges (po ten tial 14)

LJ with charge: ( )E r Ar

Cr

q q

rms ms msms

msms

m s

ms

=

−

+

1 112 6

[8.72]

Some po ten tials add an ef fec tive di elec tric con stant in the de nom i na tor of the Cou -lomb term to mimic po lar iza tion ef fects.

The Buckhingam po ten tial is sim i lar to LJ with a more phys i cal de scrip tion of the re -pul sion term given in terms of a de cay ing ex po nen tial func tion:

Er

rr

rABBU( ) exp

* *=

−−

−

−

−

−

εα

αα

α6

61

6

6

[8.73]

where:ε depth of the at trac tive wellr RAB

r* dis tance of the min i mumα nu mer i cal pa ram e ter

The dif fer ence in the com pu ta tional costs of LJ and Buckingham po ten tials is re latedto the dif fer ence be tween com put ing the square of a value al ready avail able (R-12 from R-6)and that of com put ing an ex po nen tial. Here again the in cre ment in com pu ta tional costs ofsim u la tions due to small changes in the po ten tial plays a sig nif i cant role. Also, theBuckingham + charge po ten tial is used to de scribe liq uids.

The last po ten tial of Ta ble 8.5 is spe cial ized for wa ter. It has been en closed in the ta bleto doc u ment the prog ress in one-site po ten tials; in this case, the LJ + di pole is sup ple mentedby a short-range “sticky” tet ra he dral in ter ac tion.

The list ing of one-cen ter po ten tials is not ex hausted by the ex am ples given in Ta ble8.5. We have, for ex am ple, ne glected all the po ten tials in use for rare gases sys tems in which much at ten tion is paid to using higher terms to de scribe the dis per sion con tri bu tion. Our aim was just to show with a few ex am ples how it is pos si ble to de fine a large va ri ety of po ten tials re main ing within the con straints of us ing a sin gle cen ter.

The prob lem of giv ing a cur sory but sig nif i cant enough view of po ten tials be comesharder when one passes to many-site po ten tials. In Ta ble 8.6 we re port some an a logues ofHS with a more com plex shape.

They con sist of fused reg u lar forms (spheres or com bi na tion of spheres with other reg -u lar sol ids). The hard ver sion of these po ten tials is ac com pa nied by soft mod i fi ca tions andby ver sions in clud ing di pole, charges, as in Ta ble 8.5.

A step fur ther along this way is given by a po ten tial com posed by spheres linked by‘spac ers’ with a con stant length but al low ing changes of con for ma tion with ap pro pri ate tor -sion po ten tials. These po ten tials are used for poly mers or for mol e cules hav ing long hy dro -car bon chains. This is not, how ever, the main trend in the evo lu tion of po ten tials.

8.6 The variety of interaction potentials 427

Table 8.6. Many-site simple potentials

# Shape In ter ac tion Name

1 three fused spheres hard hard dumb bell HD

2 two half spheres + cyl in der hard hard spherocylinder HSC

3 gen eral con vex shape hard hard con vex core HCC

The larg est num ber of sol vents are com posed by polyatomic mol e cules of small-me -dium size, ex hib it ing a vari able (but in gen eral not ex ces sive) de gree of flex i bil ity. In so lu -tions, the in ter ac tions in volve these sol vents and mol e cules hav ing the same char ac ter is ticsas sol vent or with more com plex chem i cal com po si tion. In all cases (pure liq uids, so lu tionswith sol ute of vari able com plex ity), one has to take into ac count in ter ac tions hav ing a re -mark able de gree of spec i fic ity. The chem i cal ap proach to the prob lem ad dresses this spec i -fic ity on which the whole chem is try is based. This is the real field of ap pli ca tion of thedef i ni tion and anal y sis of mo lec u lar in ter ac tions on which we have spent the first sec tionsof this chap ter.

It is clear that we are en ter ing here into a very com plex realm, hard to sum ma rize. Awhole book would be nec es sary.

It is pos si ble to make a rough dis tinc tion be tween po ten tial of gen eral ap pli ca bil ity and po ten tials con ceived for spe cific cou ples (or col lec tions) of mol e cules.

In both cases, the many-site ex pan sions are used, but for gen eral po ten tials, in whichtrans fer abil ity is asked, the de scrip tion is ob vi ously less de tailed. The LJ+charge (po ten tial14 in Ta ble 8.5) is a pop u lar choice for these po ten tials; the sites are in gen eral lim ited to theheavy at oms of the mol e cule. There are, how ever, many other ver sions with a va ri ety ofchanges to this stan dard set ting.

The po ten tials elab o rated for spe cific cases are of ten more de tailed and they adapt alarger num ber of de vices to in crease the ac cu racy. Many have been de vel oped for pure liq -uids, but the num ber of po ten tials re gard ing spe cific com bi na tions of sol ute and sol vent isnot neg li gi ble.

The num ber of sol vents in chem is try is quite large. The num ber of avail able spe cificpo ten tials is by far more lim ited, but too large to be summarized here. In ad di tion, a sim plelist of names and ref er ences, not ac com pa nied by crit i cal re marks about the per for mances of such po ten tials in de scrib ing liq uid sys tems would have lit tle util ity.

The nu mer ous text books on intermolecular po ten tials are of lit tle help, be cause out -dated or pay ing lit tle at ten tion to the an a lyt i cal ex po si tion of com pu ta tional mod els. Moreuse ful are mono graphs and re views re gard ing com puter sim u la tions of liquids39 or orig i nalpa pers mak ing com par i sons among dif fer ent po ten tials.

These last are more abun dant for wa ter. This sol vent, be cause of its im por tance and itsmo lec u lar sim plic ity, has been the bench mark for many ideas about the de scrip tion andmod el ing of intermolecular in ter ac tions. Al most all po ten tials that have been thus far pro -posed for not ex tremely flex i ble mo lec u lar com po nent of liq uid sys tems may be found inthe lit er a ture re gard ing wa ter. Al ter na tive strat e gies in the de fin ing num ber and po si tion ofsites, use dif fuse in stead of point charges, dif fer ent ap proaches to model the many-bodycor rec tions, flex i ble po ten tials, dis so ci a tion of the mol e cule are some fea tures that can beex am ined by look ing at the wa ter po ten tial lit er a ture.


There is a large crit i cal lit er a ture, but two re cent re views (Floris and Tani40 andWallqvist and Moun tain,41 both pub lished in 1999) are here rec om mended as an ex cel lentguide to this sub ject. The two re views con sider and an a lyze about 100 po ten tials for purewa ter. WM re view starts from his tor i cal mod els , FM re view pays at ten tion to re cent mod -els (about 70 mod els, sup ple mented by ion-wa ter po ten tials). The two re views partly over -lap, but they are to a good ex tent com ple men tary, es pe cially in the anal y sis of theper for mances of such mod els.

There are, to the best of our knowl edge, no re views of com pa ra ble ac cu racy for the po -ten tials re gard ing other liq uid sys tems.

8.7 THEORETICAL AND COMPUTING MODELING OF PURE LIQUIDS AND SOLUTIONS8.7.1 PHYSICAL MODELS

In this chap ter we shall pres ent a nec es sarily par tial re view of the main the o ret i cal ap -proaches so far de vel oped to treat liq uid sys tems in terms of phys i cal func tions. We shall re -strict our selves to two ba sic the o ries, in te gral equa tion and per tur ba tion the o ries, to keep the chap ter within rea son able bounds. In ad di tion, only the ba sic the o ret i cal prin ci ples un der ly -ing the orig i nal meth ods will be dis cussed, be cause the prog ress has been less rapid for the -ory than for nu mer i cal ap pli ca tions. The lat ter are in fact de vel op ing so fast that it is anim pos si ble task to try to give an ex haus tive view in a few pages.

A fun da men tal ap proach to liq uids is pro vided by the in te gral equa tion meth ods42-44

(some times called dis tri bu tion func tion meth ods), ini ti ated by Kirkwood and Yvon in the1930s. As we shall show be low, one starts by writ ing down an ex act equa tion for the mo lec -u lar dis tri bu tion func tion of in ter est, usu ally the pair func tion, and then in tro duces one ormore ap prox i ma tions to solve the prob lem. These ap prox i ma tions are of ten mo ti vated bycon sid er ations of math e mat i cal sim plic ity, so that their va lid ity de pends on a pos te ri oriagree ment with com puter sim u la tion or ex per i ment. The the o ries in ques tion, called YBG(Yvon-Born-Green), PY (Percus-Yevick), and the HNC (hypernetted chain) ap prox i ma -tion, pro vide the dis tri bu tion func tions di rectly, and are thus ap pli ca ble to a wide va ri ety ofprop er ties.

An al ter na tive, and par tic u larly suc cess ful, ap proach to liq uids is pro vided by the ther -mo dy namic per tur ba tion the o ries.42,44 In this ap proach, the prop er ties of the fluid of in ter estare re lated to those of a ref er ence fluid through a suit able ex pan sion. One at tempts to choose a ref er ence sys tem that is in some sense close to the real sys tem, and whose prop er ties arewell known (e.g., through com puter sim u la tion stud ies or an in te gral equa tion the ory).

As a last phys i cal ap proach we men tion, but do not fur ther con sider, the scaled-par ti -cle-the ory (SPT)38,45 which was de vel oped about the same time as the Percus-Yevick the -ory. It gives good re sults for the ther mo dy namic prop er ties of hard mol e cules (spheres orcon vex mol e cules). It is not a com plete the ory (in con trast to the in te gral equa tion and per -tur ba tion the o ries) since it does not yield the mo lec u lar dis tri bu tion func tions (al thoughthey can be ob tained for some fi nite range of intermolecular sep a ra tions).

Early work on the the ory of dense flu ids dealt al most ex clu sively with sim ple atomicflu ids, in which the intermolecular forces are be tween the cen ters of spher i cal mol e culesand de pend only on the sep a ra tion dis tance r. How ever, in real flu ids the intermolecularforces de pend on the mo lec u lar ori en ta tions, vi bra tional co or di nates, etc., in ad di tion to r.

8.7 Theoretical and computing modeling 429

The com plex ity of the prob lem thus re quires the as sump tion of some ap prox i ma tions;most of the for mu lated the o ries, for ex am ple, as sume

(a) a rigid mol e cule ap prox i ma tion, (b) a clas si cal treat ment of the traslational and ro ta tional mo tions, and (c) a pairwise additivity of the intermolecular forces.In the rigid mol e cule ap prox i ma tion it is as sumed that the intermolecular po ten tial en -

ergy V(rN,ωN) de pends only on the po si tions of the cen ters of mass rN=r1r2...rN for the N mol -e cules and on their ori en ta tions ω ω ω ωN

N= 1 2K . This im plies that the vi bra tionalco or di nates of the mol e cules are dy nam i cally and sta tis ti cally in de pend ent on the cen ter ofmass and ori en ta tion co or di nates, and that the in ter nal ro ta tions are ei ther ab sent, or in de -pend ent of the rN and ωN co or di nates. The mol e cules are also as sumed to be in their groundelec tronic states.

At the bases of the sec ond ba sic as sump tion made, e.g., that the flu ids be have clas si -cally, there is the knowl edge that the quan tum ef fects in the ther mo dy namic prop er ties areusu ally small, and can be cal cu lated readily to the first ap prox i ma tion. For the struc turalprop er ties (e.g., pair cor re la tion func tion, struc ture fac tors), no de tailed es ti mates are avail -able for mo lec u lar liq uids, while for atomic liq uids the rel e vant the o ret i cal ex pres sions forthe quan tum cor rec tions are avail able in the lit er a ture.

The third ba sic ap prox i ma tion is usu ally in tro duced that the to tal intermolecular po -ten tial en ergy V(rN,ωN) is sim ply the sum of the intermolecular po ten tials for iso lated pairs ij of mol e cules, i.e.,

( ) ( )V V rN Nij i j

i j

r , , ,ω ω ω=<∑ [8.74]

In the sum i is kept less than j in or der to avoid count ing any pair in ter ac tion twice. Eq. [8.74] is ex act in the low-den sity gas limit, since in ter ac tions in volv ing three or

more mol e cules can be ig nored. It is not ex act for dense flu ids or sol ids, how ever, be causethe pres ence of ad di tional mol e cules nearby dis torts the elec tron charge dis tri bu tions inmol e cules i and j, and thus changes the intermolecular in ter ac tion be tween this pair from the iso lated pair value. In or der to get a more re li able de scrip tion, three-body (and highermulti-body) cor rec tion terms should be in tro duced.

The in flu ence of three-body terms on the phys i cal prop er ties has been stud ied in de tailfor atomic flu ids,46,47 while much less is known about mo lec u lar flu ids. In the lat ter case, theac cu rate po ten tials are few and sta tis ti cal me chan i cal cal cu la tions are usu ally done withmodel po ten tials. A par tic u lar model may be purely em pir i cal (e.g., atom-atom), orsemiempirical (e.g., gen er al ized Stockmayer), where some of the terms have a the o ret i calba sis. The so-called gen er al ized Stockmayer model con sists of cen tral and non-cen tralterms. For the cen tral part, one as sumes a two-pa ram e ter cen tral form (the clas si cal ex am ple is the Lennard-Jones, LJ, form in tro duced in the pre vi ous sec tions). The long-range,non-cen tral part in gen eral con tains a trun cated sum of mul ti po lar, in duc tion and dis per sionterms. In ad di tion, a short-range, an gle-de pend ent over lap part, rep re sent ing the shape orcore of the po ten tial, is usu ally in tro duced. The mul ti po lar in ter ac tions are pair-wise ad di -tive, but the in duc tion, dis per sion and over lap in ter ac tions con tain three-body (and highermulti-body) terms. Hence, three-body in ter ac tions are strongly sus pected to be of large im -por tance also for mo lec u lar liq uids.


This re view will be mainly re stricted to a dis cus sion of small rigid mol e cules in theirground elec tronic states. How ever, one can gen er al ize in a num ber of di rec tions by ex tend -ing the pair po ten tial v(rij,ω i,ω j) of eq. [8.74] so as to treat:

(a) non-rigid mol e cules, (b) mol e cules with in ter nal ro ta tion, (c) large (e.g., long chain) mol e cules which may be flex i ble, (d) elec tronic ex cited state mol e cules.

In ad di tion, the in clu sion of co va lence and charge trans fer ef fects are also pos si ble. The first two gen er al iza tions are straight for ward in prin ci ple; one lets v de pend on co or di nates de -scrib ing the ad di tional de grees of free dom in volved. As for (c), for very large mol e cules thesite-site plus charge-charge model is the only vi a ble one. Un der (d) we can quote theso-called long-range ‘res o nance’ in ter ac tions.

As the last note, we an tic i pate that in the fol low ing only equi lib rium prop er ties will becon sid ered; how ever, it is fun da men tal to re call from the be gin ning that dy nam i cal (e.g.,non-equi lib rium) anal y ses are an im por tant and ac tive field of re search in the the ory ofphys i cal mod els.42,48,49

Be fore en ter ing more de tails into each the ory, it is worth in tro duc ing some ba sic def i -ni tions of sta tis ti cal me chan ics. All equi lib rium prop er ties of a sys tem can be cal cu lated ifboth the intermolecular po ten tial en ergy and the dis tri bu tion func tions are known. In con -sid er ing flu ids in equi lib rium, we can dis tin guish three prin ci pal cases:

(a) iso tro pic, ho mo ge neous flu ids, (e.g., liq uid or com pressed gas in the ab sence of an ex ter nal field),

(b) anisotropic, ho mo ge neous flu ids (e.g., a polyatomic fluid in the pres ence of a uni form elec tric field, ne matic liq uid crys tals), and

(c) inhomogeneous flu ids (e.g., the in ter fa cial re gion). These fluid states have been listed in or der of in creas ing com plex ity; thus, more in de pend -ent vari ables are in volved in cases (b) and (c), and con se quently the eval u a tion of the nec es -sary dis tri bu tion func tions is more dif fi cult.

For mo lec u lar flu ids, it is con ve nient to de fine dif fer ent types of dis tri bu tion func -tions, cor re la tion func tions and re lated quan ti ties. In par tic u lar, in the pair-wise ad di tivethe ory of ho mo ge neous flu ids (see eq. [8.74]), a cen tral role is played by the an gu lar paircor re la tion func tion g(r12ω 1ω 2) pro por tional to the prob a bil ity den sity of find ing two mol e -cules with po si tion r1 and r2 and ori en ta tions ω 1 and ω 2 (a sche matic rep re sen ta tion of suchfunc tion is re ported in Fig ure 8.5).

In fact, one is fre quently in ter ested in some ob serv able prop erty <B>, which is (ex per -i men tally) a time av er age of a func tion of the phase vari ables B(rN,ω N); the lat ter is of ten asum of pair terms b(rij,ω i,ω j) so that <B> is given by

( ) ( ) ( ) ( )B d d P B N dr g bN N N N N N= = ∫∫ r r r r rω ω ω ρ ω ω ω ωω ω

12 1 2 1 2

1 2

[8.75]

i.e., in terms of the pair cor re la tion func tion g(12). Ex am ples of such prop er ties are the con -fig u ra tional con tri bu tion to en ergy, pres sure, mean squared torque, and the mean squaredforce. In eq.[8.75] g b( ) ( )r r 1ω ω ω ω

ω ω1 2 21 2

means the un weight ed av er age over ori en ta -tions.

In ad di tion to g(12), it is also use ful to de fine:


a) the site-site cor re la tion func tion gαβ(rαβ) pro por tional to the prob a bil ity den sity that sites α and β on dif fer ent mol e cules are sep a rated by dis tance r, re gard less of mo lec u lar ori en ta tions;

b) the to tal cor re la tion func tion h(r12ω 1ω 2)= g(r12ω 1ω 2) - 1;c) the di rect cor re la tion func tion c(r12ω 1ω 2).Sim ple ex am ples of h and c func tions are re ported in Figure 8.6.In the list above, item (a) im plies de finite sites within mol e cules; these sites may be the

nu clei them selves or sites at ar bi trary lo ca tions within the mol e cules. In ad di tion, the to tal cor re la tion, h, be tween mol e cules 1 and 2 can be sep a rated into

two parts: (i) a di rect ef fect of 1 on 2 which is short-ranged and is char ac ter ized by c, and (ii)and in di rect ef fect in which 1 in flu ences other mol e cules, 3, 4, etc., which in turn af fect 2.The in di rect ef fect is the sum of all con tri bu tions from other mol e cules av er aged over theircon fig u ra tions. For an iso tro pic and ho mo ge neous fluid formed by non-spher i cal mol e -cules, we have

( ) ( ) ( ) ( )h c d c hr r r r r12 1 2 12 1 2 3 13 1 3 32 3 2ω ω ω ω ρ ω ω ω ωω3

= + ∫ [8.76]

which is the gen er al iza tion of the Ornstein-Zernike (OZ) equation50 to non-spher i cal mol e -cules. The OZ equa tion is the start ing point for many the o ries of the pair cor re la tion func -tion (PY, HNC, etc.); how ever, nu mer i cal so lu tions start ing from [8.76] are com pli cated bythe large num ber of vari ables in volved. By ex pand ing the di rect and to tal cor re la tion func -tions in spher i cal har mon ics, one ob tains a set of al ge bra ically cou pled equa tions re lat ingthe har monic co ef fi cients of h and c. These equa tions in volve only one vari able in the placeof many in the orig i nal OZ equa tion. In ad di tion, the the o ries we shall de scribe be low trun -cate the in fi nite set of cou pled equa tions into a fi nite set, thereby en abling a rea son ably sim -ple so lu tion to be car ried out.


Fig ure 8.5. Ex am ple of the ra dial dis tri bu tion func tiong(r) of a typ i cal monoatomic liq uid. Fig ure 8.6. Ex am ples of h(r) and c(r) func tions for a typ i -

cal monoatomic liq uid.

8.7.1.1 Integral Equation MethodsThe struc ture of the in te gral equa tion ap proach for cal cu lat ing the an gu lar pair cor re la tionfunc tion g(r12ω 1ω 2) starts with the OZ in te gral equa tion [8.76] be tween the to tal (h) and thedi rect (c) cor re la tion func tion, which is here sche mat i cally re writ ten as h=h[c] where h[c]de notes a func tional of c. Cou pled to that a sec ond re la tion, the so-called clo sure re la tionc=c[h], is in tro duced. While the for mer is ex act, the lat ter re la tion is ap prox i mated; the form of this ap prox i ma tion is the main dis tinc tion among the var i ous in te gral equa tion the o ries to be de scribed be low.

In the OZ equa tion, h de pends on c, and in the clo sure re la tion, c de pends on h; thus the un known h de pends on it self and must be de ter mined self-con sis tently. This (self-con sis -tency re quir ing, or in te gral equa tion) struc ture is char ac ter is tic of all many body prob lems.

Two of the clas sic in te gral equa tion ap prox i ma tions for atomic liq uids are the PY(Percus-Yevick)51 and the HNC (hypernetted chain)52 ap prox i ma tions that use the fol low ing clo sures

h c y− = −1 (PY) [8.77]

h c y− = ln (HNC) [8.78]

where y is the di rect cor re la tion func tion de fined by g(12)=exp(-β v(12))y(12) with v(12)the pair po ten tial and β=1/kT.

The clo sures [8.77-8.78] can be also writ ten in the form

( )c g e v= −1 12β ( ) (PY) [8.79]

( )c h v g= − −β 12 ln (HNC) [8.80]

For atomic liq uids the PY the ory is better for steep re pul sive pair po ten tials, e.g., hardspheres, whereas the HNC the ory is better when at trac tive forces are pres ent, e.g.,Lennard-Jones, and Cou lomb po ten tials. No stated tests are avail able for mo lec u lar liq uids;more de tails are given be low.

As said be fore, in prac tice the in te gral equa tions for mo lec u lar liq uids are al most al -ways solved us ing spher i cal har monic ex pan sions. This is be cause the ba sic form [8.76] ofthe OZ re la tion con tains too many vari ables to be han dled ef fi ciently. In ad di tion, har monicex pan sions are nec es sar ily trun cated af ter a fi nite num ber of terms. The va lid ity of the trun -ca tions rests on the rate of con ver gence of the har monic se ries that de pends in turn on thede gree of ani so tropy in the intermolecular po ten tial.

We re call that the so lu tion to the PY ap prox i ma tion for the hard sphere atomic fluid isan a lyt i cal and it also forms a ba sis for other the o ries, e.g., in mo lec u lar flu ids the MSA andRISM the o ries to be dis cussed be low.

The mean spher i cal ap prox i ma tion (MSA)52 the ory for flu ids orig i nated as the ex ten -sion to con tin uum flu ids of the spher i cal model for lat tice gases. In prac tice it is usu ally ap -plied to po ten tials with spher i cal hard cores, al though ex ten sions to soft core andnon-spher i cal core po ten tials have been dis cussed.

The MSA is based on the OZ re la tion to gether with the clo sure


h = −1 r < σ [8.81]

c v= −β ( )12 r < σ [8.82]

where σ is the di am e ter of the spher i cal hard core part of the pair po ten tial v(12).The con di tion [8.81] is ex act for hard core po ten tials since g(12)=0 for r < σ, while the

con di tion [8.82] is the ap prox i ma tion, be ing cor rect only as ymp tot i cally (large r). We thusex pect the ap prox i ma tion to be worst near the hard core; we also note that the the ory is notex act at low den sity (ex cept for a pure hard sphere po ten tial, where MSA=PY), but this isnot too im por tant since good the o ries ex ist for low den si ties, and we are there fore mainly in -ter ested in high den si ties. Once again the so lu tion of the MSA prob lem is most eas ily ac -com plished us ing the spher i cal har monic com po nent of h and c.

In the list of the var i ous cor re la tion func tions we have in tro duced at item (b) theso-called site-site pair cor re la tion func tions gαβ(rαβ); since they de pend only on the ra dialvari ables rαβ (be tween sites), they are nat u rally sim pler than g(r12;ω ω1 2) but, at the sametime, they con tain less in for ma tion. The the o ries for gαβ fall into two cat e go ries based onsite-site or par ti cle-par ti cle OZ equa tions, re spec tively. Var i ous clo sures can be used withei ther cat e gory.

As con cerns the site-site ap proach the most im por tant the ory is the ‘ref er ence in ter ac -tion site model’, or RISM.53,54 This method ap plies to an intermolecular pair po ten tial mod -eled by a site-site form, i.e., V(r v rω ω αβ αβ αβ1 2 ) ( )= Σ and its orig i nal in tu itive der i va tion isbased on ex plor ing the pos si bil ity of de com pos ing g(r;ω ω1 2) also in the same form, i.e., as asum of site-site gαβ(r)s.

The RISM the ory con sists of (i) an OZ-like re la tion be tween the set hαβ (wherehαβ(r)=gαβ(r)-1) and cor re spond ing set cαβ of di rect cor re la tion func tions, and (ii) aPY-type or other type of clo sure. The OZ-like re la tion is here a ma trix one.

Al though suc cess ful in many ap pli ca tions, the RISM ap prox i ma tion suf fers from anum ber of ma jor de fects. First, it is not the best choice to cal cu late the equa tion of state, andthe re sults that are ob tained are ther mo dy nam i cally in con sis tent. Sec ondly, cal cu latedstruc tural prop er ties show an unphysical de pend ence on the pres ence of ‘aux il iary’ sites,i.e., on sites that la bel points in a mol e cule but con trib ute noth ing to the intermolecular po -ten tial. Thirdly, use of the RISM ap prox i ma tion leads to triv ial and in cor rect re sults for cer -tain quan ti ties de scrip tive of an gu lar cor re la tions in the fluid.48

At tempts to de velop a more sat is fac tory the ory within the in ter ac tion-site pic ture havede vel oped along two dif fer ent lines. The first re lies on treat ing the RISM-OZ re la tion aspro vid ing the def i ni tion of the site-site di rect cor re la tion func tions, cαβ. In this way the usual RISM clo sures can be mod i fied by add ing terms to cαβ such that the as ymp totic re sult is sat -is fied.

In the al ter na tive ap proach, the ba sic ob ser va tion is that the RISM-OZ re la tion, though plau si ble, does not pro vide an ad e quate ba sis for a com plete the ory of the struc ture of mo -lec u lar flu ids. Ac cord ingly, it is there rather than in the clo sure re la tion that im prove mentmust be sought.55

De spite the de fects we have un der lined, in the last years the RISM method hasreceived much more at ten tion than other in te gral equa tion meth ods; prom is ing de vel op -ments in this area are rep re sented by the in clu sion of polarizable me dia in RISM56 and thecou pling of the RISM equa tions to a quan tum me chan i cal de scrip tion.57


8.7.1.2 Perturbation TheoriesIn per tur ba tion theories44,58 one re lates the prop er ties (e.g., the dis tri bu tion func tions or freeen ergy) of the real sys tem, for which the intermolecular po ten tial en ergy is E, to those of aref er ence sys tem where the po ten tial is E0, usu ally by an ex pan sion in pow ers of the per tur -ba tion po ten tial E1=E - E0.

The meth ods used are con ve niently clas si fied ac cord ing to whether the ref er ence sys -tem po ten tial is spher i cally sym met ric or anisotropic. The ories of the first type are most ap -pro pri ate when the ani so tropy in the full po ten tial is weak and long ranged; those of thesec ond type have a greater phys i cal ap peal and a wider range of pos si ble ap pli ca tion, butthey are more dif fi cult to im ple ment be cause the cal cu la tion of the ref er ence-sys tem prop er -ties poses greater prob lems.

In con sid er ing per tur ba tion the ory for liq uids it is con ve nient first to dis cuss the his tor -i cal de vel op ment for atomic liq uids.

The intermolecular pair po ten tial can in many cases be sep a rated in a nat u ral way intoa sharp, short-range re pul sion and a smoothly vary ing, long range at trac tion. A sep a ra tionof this type is an ex plicit in gre di ent of many em pir i cal rep re sen ta tions of the intermolecularforces in clud ing, for ex am ple, the Lennard-Jones po ten tial. It is now gen er ally ac cepted that the struc ture of sim ple liq uids, at least of high den sity, is largely de ter mined by geo met ricfac tors as so ci ated with the pack ing of the mo lec u lar hard cores. By con trast, the at trac tivein ter ac tions may, in the first ap prox i ma tion, be re garded as giv ing rise to a uni form back -ground po ten tial that pro vides the co he sive en ergy of the liq uid but has lit tle ef fect on itsstruc ture. A fur ther plau si ble ap prox i ma tion con sists of mod el ing the short-range forces bythe in fi nitely steep re pul sion of the hard-sphere po ten tial. In this way, the prop er ties of agiven liq uid can be re lated to those of a hard-sphere ref er ence sys tem, the at trac tive part ofthe po ten tial be ing treated as a per tur ba tion. The choice of the hard-sphere fluid as a ref er -ence sys tem is an ob vi ous one, since its ther mo dy namic and struc tural prop er ties are wellknown.

The idea of rep re sent ing a liq uid as a sys tem of hard spheres mov ing in a uni form, at -trac tive po ten tial well is an old one; suf fice here to re call the van der Waals equa tion. Roughly one can thus re gard per tur ba tion meth ods as at tempts to im prove the the ory of vander Waals in a sys tem atic fash ion.

The ba sis of all the per tur ba tion the o ries we shall con sider is a di vi sion of the pair po -ten tial of the form

( ) ( ) ( )v v w12 12 120= + [8.83]

where v0 is the pair po ten tial of the ref er ence sys tem and w(12) is the per tur ba tion. The fol -low ing step is to com pute the ef fect of the per tur ba tion on the ther mo dy namic prop er tiesand pair dis tri bu tion func tion of the ref er ence sys tem. This can be done sys tem at i cally viaan ex pan sion in pow ers ei ther of in verse tem per a ture (the “λ ex pan sion”) or of a pa ram e terthat mea sures the range of the per tur ba tion (the “γ ex pan sion”).

In spite of the fact that hard spheres are a nat u ral choice of the ref er ence sys tem, for the rea sons dis cussed above, re al is tic intermolecular po ten tials do not have an in fi nitely steephard core, and there is no nat u ral sep a ra tion into a hard-sphere part and a weak per tur ba tion.A pos si ble im prove ment is that to take proper ac count of the “soft ness”’ of the re pul sivepart of the intermolecular po ten tial. A method of do ing this was pro posed by Rowlinson,59


who used an ex pan sion in pow ers of a soft ness pa ram e ter, n-1, about n-1=0 (cor re spond ing toa hard sphere fluid). This method gives good re sults for the re pul sive, but not for the at trac -tive, part of the po ten tial. Sub se quently sev eral re search ers at tempted to com bine the ad -van tages of both ap proaches. The first suc cess ful method was that of Barker and Henderson(BH),60 who showed that a sec ond-or der the ory gave quan ti ta tive re sults for the ther mo dy -namic prop er ties of a Lennard-Jones liq uid. Even more rap idly con ver gent re sults werelater ob tained by Weeks, Chan dler and Andersen (WCA)61 us ing a some what dif fer ent ref -er ence sys tem.

Given the great suc cess of per tur ba tion the o ries in treat ing the prop er ties of atomic liq -uids, a large ef fort has been de voted to ex tend ing the meth ods to deal with mo lec u lar sys -tems. The first rig or ous ap pli ca tion of these the o ries to mo lec u lar flu ids seems to have beenmade in 1951 by Barker,62 who ex panded the par ti tion func tion for a po lar fluid about thatfor a fluid of iso tro pic mol e cules.

Roughly, the ba sic prob lem is the same as in the atomic case, but the prac ti cal dif fi cul -ties are much more se vere. A pos si ble ap proach is to choose a ref er ence-sys tem po ten tialv0(r) spher i cally sym met ric so that the in te gra tions over the ori en ta tions (ab sent in atomicflu ids) in volve only the per tur ba tion. The real sys tem can be thus stud ied by a straight for -ward gen er al iza tion of the λ-ex pan sion de vel oped for atomic flu ids.

Per tur ba tion the o ries based on spher i cally sym met ric ref er ence po ten tials, how ever,can not be ex pected to work well when (as in the most real mol e cules) the short-range re pul -sive forces are strongly anisotropic. The nat u ral ap proach in such cases is to in clude thestrongly vary ing in ter ac tions in the spec i fi ca tion of an iso tro pic ref er ence po ten tial to re latethe prop er ties of the ref er ence sys tem to those of hard mol e cules hav ing the same shape.63

Cal cu la tion of this type have been made by Tildesley,64 ex ploit ing a gen er al iza tion of theWBA ap proach quoted above. In a sim i lar way also, the BH per tur ba tion the ory has beengen er al ized with good re sults.65 The main dis ad van tage of these meth ods is the large com -pu ta tional ef fort that their im ple men ta tion re quires.

8.7.2 COMPUTER SIMULATIONS

In this sec tion we will give a brief re view of meth od ol o gies to per form com puter sim u la -tions. These ap proaches now a days have a ma jor role in the study of liq uid-state phys ics:they are de vel op ing so fast that it is im pos si ble to give a com plete view of all the dif fer entmeth od ol o gies used and of the over whelm ingly large num ber of ap pli ca tions. There fore wewill only give a brief ac count of the ba sic prin ci ples of such meth od ol o gies, of how a com -puter sim u la tion can be car ried out, and we will briefly dis cuss lim i ta tions and ad van tagesof the meth ods. The lit er a ture in the field is enor mous: here we will re fer in ter ested read ersto some of the ba sic text books on this sub ject.42,66,67

The mi cro scopic state of a sys tem may be spec i fied in terms of the po si tions andmomenta of the set of par ti cles (at oms or mol e cules). Making the ap prox i ma tion that a clas -si cal de scrip tion is ad e quate, the Hamiltonian H of a sys tem of N par ti cles can be writ ten asa sum of a ki netic K and a po ten tial V en ergy func tions of the set of co or di nates qi andmomenta pi of each par ti cle i, i.e.:

( ) ( ) ( )H q p K p V q, = + [8.84]

In the equa tion the co or di nates q can sim ply be the set of Car te sian co or di nates of each atom in the sys tem, but in this case we treat the mol e cules as rigid bod ies (as is usu ally done


in most sim u la tions); q will con sist of the Car te sian co or di nates of each mo lec u lar cen ter ofmass to gether with a set of other pa ram e ters spec i fy ing the mo lec u lar ori en ta tion. In anycase p is the set of con ju gate momenta.

Usually the ki netic en ergy K takes the form:

Kp

mi

ii

N

= ∑∑=

µ

µ

2

1 2[8.85]

where the in dex µ runs over the com po nents (x, y, z) of the mo men tum of mol e cule i and mi

is the mo lec u lar mass.The po ten tial en ergy V con tains the in ter est ing in for ma tion re gard ing intermolecular

in ter ac tions. A de tailed anal y sis on this sub ject has been given in the pre vi ous sec tions; here we shall re call only some ba sic as pects.

It is pos si ble to con struct from H an equa tion of mo tion that gov erns the time-evo lu -tion of the sys tem, as well as to state the equi lib rium dis tri bu tion func tion for po si tions andmomenta. Thus H (or better V) is the ba sic in put to a com puter sim u la tion pro gram. Al mostuni ver sally, in com puter sim u la tion the po ten tial en ergy is bro ken up into terms in volv ingpairs, trip lets, etc. of mol e cules, i.e.:

( ) ( ) ( )V v r v r r v r r ri i j i j kk j ij iij iii

= + + +> >>>∑∑∑∑∑∑ 1 2 3, , , K [8.86]

The no ta tion in di cates that the sum ma tion runs over all dis tinct pairs i and j or trip letsi, j and k, with out count ing any pair or trip let twice. In eq. [8.86], the first term rep re sents the ef fect of an ex ter nal field on the sys tem, while the re main ing terms rep re sent in ter ac tionsbe tween the par ti cles of the sys tem. Among them, the sec ond one, the pair po ten tial, is themost im por tant. As said in the pre vi ous sec tion, in the case of mo lec u lar sys tems the in ter ac -tion be tween nu clei and elec tronic charge clouds of a pair of mol e cules i and j is gen er ally acom pli cated func tion of the rel a tive po si tions ri and rj, and ori en ta tions Ωi and Ωj (for atomicsys tems this term de pends only on the pair sep a ra tion, so that it may writ ten as v2(rij)).

In the sim plest ap prox i ma tion the to tal in ter ac tion is viewed as a sum of pair-wise con -tri bu tions from dis tinct sites a in mol e cule i, at po si tion ria, and b in mol e cule j, at po si tion rjb, i.e.:

( ) ( )v r v rij i j ab ab, ,Ω Ω = ∑∑ [8.87]

In the equa tion, vab is the po ten tial act ing be tween sites a and b, whose inter-sep a ra tion is rab.The pair po ten tial shows the typ i cal fea tures of intermolecular in ter ac tions as shown

in Fig ure 8.1: there is an at trac tive tail at large sep a ra tion, due to cor re la tion be tween theelec tron clouds sur round ing the at oms (‘van der Waals’ or ‘Lon don’ dis per sion), a neg a tivewell, re spon si ble for co he sion in con densed phases, and a steeply ris ing re pul sive wall atshort dis tances, due to over lap be tween elec tron clouds.

Turn ing the at ten tion to terms in eq. [8.86] in volv ing trip lets, they are usu ally sig nif i -cant at liq uid den si ties, while four-body and higher are ex pected to be small in com par i sonwith v2 and v3. De spite of their sig nif i cance, only rarely trip let terms are in cluded in com -puter sim u la tions: that is due to the fact that the cal cu la tion of quan ti ties re lated to a sumover trip lets of mol e cules are very time-con sum ing. On the other hand, the av er age


three-body ef fects can be par tially in cluded in a pair-wise ap prox i ma tion, lead ing it to givea good de scrip tion of liq uid prop er ties. This can be achieved by de fin ing an “ef fec tive” pairpo ten tial, able to rep re sent all the many-body ef fects. To do this, eq. [8.86] can be re-writ ten as:

( ) ( )V v r v rieff

ijj iii

≈ +>

∑∑∑ 1 2 [8.88]

The most widely used ef fec tive po ten tial in com puter sim u la tions is the sim pleLennard-Jones 12-6 po ten tial (see eq. [8.70]), but other pos si bil i ties are also avail able; seeSec tion 8.6 for dis cus sion on these choices.

8.7.2.1 Car-Parrinello Direct QM SimulationUn til now we have fo cused the at ten tion to the most usual way of de ter min ing a po ten tial in -ter ac tion to be used in sim u la tions. It is worth men tion ing a dif fer ent ap proach to the prob -lem, in which the dis tri bu tion of elec trons is not treated by means of an “ef fec tive”in ter ac tion po ten tial, but is treated ab in itio by den sity func tional the ory (DFT). The mostpop u lar method is the Car-Parrinello (CP) ap proach,68,69 in which the elec tronic de grees offree dom are ex plic itly in cluded in the de scrip tion and the elec trons (to which a fic ti tiousmass is as signed) are al lowed to re lax dur ing the course of the sim u la tion by a pro cess called “sim u lated-an neal ing”. In that way, any di vi sion of V into pair-wise and higher terms isavoided and the in ter atomic forces are gen er ated in a con sis tent and ac cu rate way as thesim u la tion pro ceeds. This point con sti tutes the main dif fer ence be tween a CP sim u la tionand a con ven tional MD sim u la tion, which is pre ceded by the de ter mi na tion of the po ten tialand in which the pro cess lead ing to the po ten tial is com pletely sep a rated from the ac tualsim u la tion. The forces are com puted us ing elec tronic struc ture cal cu la tions based on DFT,so that the in ter atomic po ten tial is pa ram e ter-free and de rived from first prin ci ples, with noex per i men tal in put.

Let us con sider a sys tem for which the BO ap prox i ma tion holds and for which the mo -tion of the nu clei can be de scribed by clas si cal me chan ics. The in ter ac tion po ten tial is givenby:

( )V R H= ψ ψ0 0| $| [8.89]

where H is the Hamiltonian of the sys tem at fixed R po si tions and ψ0 is the cor re spond ingin stan ta neous ground state. Eq. [8.89] per mits to de fine the in ter ac tion po ten tial from firstprin ci ples.

In or der to use eq. [8.89] in a MD sim u la tion, cal cu la tions of ψ0 for a num ber of con -fig u ra tions of the or der of 104 are needed. Ob vi ously this is computationally very de mand -ing, so that the use of cer tain very ac cu rate QM meth ods (for ex am ple, the con fig u ra tionin ter ac tion (CI)) is pre cluded. A prac ti cal al ter na tive is the use of DFT. Fol low ing Kohn and Sham,70 the elec tron den sity ρ(r) can be writ ten in terms of oc cu pied sin gle-par ti cleorthonormal orbitals:

( ) ( )ρ ψr rii

= ∑ 2[8.90]


A point on the BO po ten tial en ergy sur face (PES) is then given by the min i mum with re -spect to the ψi of the en ergy func tional:

[ ] ( ) ( ) ( ) [ ]V R d r rhm

r U r Ri I v ii

i I vψ α ψ ψ ρ α, , , ,*= − ∇

+∫∑ 3

22

2Ω[8.91]

where RI are the nu clear co or di nates, αv are all the pos si ble ex ter nal con straints im -posed on the sys tem (e.g., the vol ume Ω). The func tional U con tains the inter-nu clear Cou -lomb re pul sion and the ef fec tive elec tronic po ten tial en ergy, in clud ing ex ter nal nu clear,Hartree, and ex change and cor re la tion con tri bu tions.

In the con ven tional ap proach, the minimization of the en ergy func tional (eq. [8.91])with re spect to the orbitals ψi sub ject to the orthonormalization con straint leads to a set ofself-con sis tent equa tions (the Kohn-Sham equa tions), i.e.:

( )( ) ( )− ∇ +

=hm

Ur

r ri i i

22

2∂

∂ρψ ε ψ [8.92]

whose so lu tion in volves re peated ma trix diagonalizations (and rap idly grow ing com pu ta -tional ef fort as the size of the sys tem in creases).

It is pos si ble to use an al ter na tive ap proach, re gard ing the minimization of the func -tional as an op ti mi za tion prob lem, which can be solved by means of the sim u lated an neal ing pro ce dure.71 A sim u lated an neal ing tech nique based on MD can be ef fi ciently ap plied tomin i mize the KS func tional: the re sult ing tech nique, called “dy nam i cal sim u lated an neal -ing” al lows the study of fi nite tem per a ture prop er ties.

In the “dy nam i cal sim u lated an neal ing,” the Ri, αv and ψi pa ram e ters can becon sid ered as de pend ent on time; then the Lagrangian is in tro duced, i.e.:

[ ]L d r M R V Ri I I v v i I vvIi

= + + −∑∑∫∑ 12

12

12

3 2 2µ ψ µ α ψ α& & & , ,Ω

[8.93]

with:

( ) ( )d r r t r ti j ij3 ψ ψ δ* , , =∫

Ω

[8.94]

In eq.[8.94] the dot in di cates time de riv a tive, MI are the nu clear masses and µ are ar bi -trary pa ram e ters hav ing the di men sion of mass.

Using eq. [8.94] it is pos si ble to gen er ate a dy nam ics for Ri, αv and ψi throughthe fol low ing equa tions:

M R VI I R I

&& = −∇ [8.95]

µ α∂∂αv v

v

V&& = −

[8.96]


( )( )

( )µψ∂

∂ψψ&& ,

,,

*r t

V

r tr t

i

ik kk

= − + ∑ Λ [8.97]

In eq. [8.97] the Λ are the Lagrange mul ti pli ers in tro duced to sat isfy eq. [8.94]. It is worthno tic ing that, while the nu clear dy nam ics [8.95] can have a phys i cal mean ing, that is nottrue for the dy nam ics as so ci ated with the αv and ψi: this dy nam ics is fic ti tious, like theas so ci ated “masses” µ.

If µ and the ini tial con di tions ψi0 and dψi/dt0 are cho sen such that the two clas si calset of de grees of free doms (nu clear and elec tronic) are only weakly cou pled, the trans fer ofen ergy be tween them is small enough to al low the elec trons to adi a bat i cally fol low the nu -clear mo tion, then re main ing close to their in stan ta neous BO sur face. In such a metastablesit u a tion, mean ing ful tem po ral av er ages can be com puted. The men tioned dy nam ics ismeant to re pro duce what ac tu ally oc curs in real mat ter, that is, elec trons adi a bat i cally fol -low ing the nu clear mo tion.

QM po ten tials have been widely used in mo lec u lar dy nam ics sim u la tion of liq uid wa -ter us ing the CP DFT al go rithm. See, for ex am ple, refs. [72,73].

8.7.2.2 Semi-classical SimulationsCom puter sim u la tions are meth ods ad dressed to per form “com puter ex per i men ta tion”. Theim por tance of com puter sim u la tions rests on the fact that they pro vide quasi-ex per i men taldata on well-de fined mod els. As there is no un cer tainty about the form of the in ter ac tion po -ten tial, the o ret i cal re sults can be tested in a way that is usu ally im pos si ble with re sults ob -tained by ex per i ments on real liq uids. In ad di tion, it is pos si ble to get in for ma tion onquan ti ties of no di rect ac cess to ex per i men tal mea sures.

There are ba si cally two ways of sim u lat ing a many-body sys tem: through a sto chas ticpro cess, such as the Monte Carlo (MC) sim u la tion,74 or through a de ter min is tic pro cess,such as a Mo lec u lar Dy nam ics (MD) sim u la tion.75,76 Nu mer i cal sim u la tions are also per -formed in a hy brid ized form, like the Langevin dynamics42 which is sim i lar to MD ex ceptfor the pres ence of a ran dom dissipative force, or the Brownian dy nam ics,42 which is basedon the con di tion that the ac cel er a tion is bal anced out by drift ing and ran dom dissipativeforces.

Both the MC and the MD meth od ol o gies are used to ob tain in for ma tion on the sys temvia a clas si cal sta tis ti cal anal y sis but, whereas MC is lim ited to the treat ment of static prop -er ties, MD is more gen eral and can be used to take into ac count the time de pend ence of thesys tem states, al low ing one to cal cu late time fluc tu a tions and dy namic prop er ties.

In the fol low ing, we shall briefly de scribe the main fea tures of MD and MC meth od ol -o gies, fo cus ing the at ten tion to their use in the treat ment of liq uid sys tems.

Molecular DynamicsMo lec u lar Dy nam ics is the term used to re fer to a tech nique based on the so lu tion of theclas si cal equa tion of mo tion for a clas si cal many-body sys tem de scribed by a many-bodyHamiltonian H.

In a MD sim u la tion, the sys tem is placed within a cell of fixed vol ume, usu ally of cu -bic shape. A set of ve loc i ties is as signed, usu ally drawn from a Maxwell-Boltzmann dis tri -bu tion suit able for the tem per a ture of in ter est and se lected to make the lin ear mo men tumequal to zero. Then the tra jec to ries of the par ti cles are cal cu lated by in te gra tion of the clas si -cal equa tion of mo tion. It is also as sumed that the par ti cles in ter act through some forces,


whose cal cu la tion at each step of the sim u la tion con sti tutes the bulk of the com pu ta tionalde mand of the cal cu la tion. The first for mu la tion of the method, due to Al der,75 re ferred to asys tem of hard spheres: in this case, the par ti cles move at con stant ve loc ity be tween per -fectly elas tic col li sions, so that it is pos si ble to solve the prob lem with out mak ing any ap -prox i ma tions. More dif fi cult is the so lu tion of the equa tions of mo tion for a set ofLennard-Jones par ti cles: in fact, in this case an ap prox i mated step-by-step pro ce dure isneeded, since the forces be tween the par ti cles change con tin u ously as they move.

Let us con sider a point Γ in the phase space and sup pose that it is pos si ble to write thein stan ta neous value of some prop erty A as a func tion A(Γ). As the sys tem evolves in time, Γand hence A(Γ) will change. It is rea son able to as sume that the ex per i men tally ob serv able“mac ro scopic” prop erty Aobs is re ally the time av er age of A(Γ) over a long time in ter val:

( )( ) ( )( )A A A tt

A t dtobs time time t

t

= = =→∞ ∫Γ Γlim

1

0

[8.98]

The time evo lu tion is gov erned by the well-known New ton equa tions, a sys tem of dif -fer en tial equa tions whose so lu tion is prac ti cal. Ob vi ously it is not pos si ble to ex tend the in -te gra tion to in fi nite time, but the av er age can be reached by in te grat ing over a long fi nitetime, at least as long as pos si ble as de ter mined by the avail able com puter re sources. This isex actly what is done in a MD sim u la tion, in which the equa tions of mo tion are solvedstep-by-step, tak ing a large fi nite num ber τ of steps, so that:

( )( )A Aobs = ∑1τ

ττ

Γ [8.99]

It is worth stress ing that a dif fer ent choice in the time step is gen er ally re quired to de -scribe dif fer ent prop er ties, as mo lec u lar vi bra tions (when flex i ble po ten tials are used),trans la tions and ro ta tions.

A prob lem aris ing from the meth od ol ogy out lined above is whether or not a suit ablere gion of the phase space is ex plored by the tra jec tory to yield good time av er ages (in a rel a -tively short com put ing time) and whether con sis tency can be ob tained with sim u la tionswith iden ti cal mac ro scopic pa ram e ters but dif fer ent ini tial con di tions. Gen erally, ther mo -dy nam i cally con sis tent re sults for liq uids can be ob tained pro vided that care ful at ten tion ispaid to the se lec tion of ini tial con di tions.

Monte CarloAs we have seen, apart from the choice of the ini tial con di tions, a MD sim u la tion is en tirelyde ter min is tic. By con trast, a proba bil is tic (sto chas tic) el e ment is an es sen tial part of anyMonte Carlo (MC) sim u la tion.

The time-av er age ap proach out lined above is not the only pos si ble: it is in fact prac ti -cal to re place the time av er age by the en sem ble av er age, be ing the en sem ble a col lec tion ofpoints Γ dis trib uted ac cord ing to a prob a bil ity den sity ρ(Γ) in the phase space. The den sity is de ter mined by the mac ro scopic pa ram e ters, NPT, NVT, etc., and gen er ally will evolve intime. Making the as sump tion that the sys tem is “ergodic”, the time av er age in eq. [8.98] canbe re placed by an av er age taken over all the mem bers of the en sem ble at a par tic u lar time:

( ) ( )A A Aobs ensemble ensemble= = ∑ Γ ΓΓ

ρ [8.100]


Some times, in stead of us ing the ρ(Γ), a “weight” func tion w(Γ) is used:

( ) ( )ρ Γ

Γ=

w

Q[8.101]

( )Q w= ∑ ΓΓ

[8.102]

( ) ( )

( )A

w A

wensemble=

∑∑

Γ Γ

ΓΓ

Γ

[8.103]

Q is the par ti tion func tion of the sys tem.From eq. [8.102], it is pos si ble to de rive an ap proach to the cal cu la tion of ther mo dy -

nam ics prop er ties by di rect eval u a tion of Q for a par tic u lar en sem ble. Q is not di rectly es ti -mated, but the idea of gen er at ing a set of states in phase space sam pled in ac cor dance withthe prob a bil ity den sity ρ(Γ) is the cen tral idea of MC tech nique. Pro ceeding ex actly as donefor MD, re plac ing an en sem ble av er age as in Eq.[8.103] with a tra jec tory av er age as in Eq.[8.99], a suc ces sion of states is gen er ated in ac cor dance with the dis tri bu tion func tion ρNVE

for the microcanonical NVE en sem ble. The ba sic aim of the MC method (so-called be causeof the role that ran dom num bers play in the method), which is ba si cally a tech nique for per -form ing nu mer i cal in te gra tion, is to gen er ate a tra jec tory in phase space that sam ples from acho sen sta tis ti cal en sem ble. It is pos si ble to use en sem bles dif fer ent from themicrocanonical: the only re quest is to have a way (phys i cal, en tirely de ter min is tic or sto -chas tic) of gen er at ing from a state Γ(τ) a next state Γ(τ + 1). The im por tant point to bestressed is that some con di tions have to be ful filled: these are that the prob a bil ity den sity ρ( )Γ for the en sem ble should not change as the sys tem evolves, any start ing dis tri bu tion ρ( )Γ should tend to a sta tion ary so lu tion as the sim u la tion pro ceeds, and the ergodicity ofthe sys tems should hold. With these rec om men da tions, we should be able to gen er ate froman ini tial state a suc ces sion of points that in the long term are sam pled with the de sired prob -a bil ity den sity ρ( )Γ . In this case, the en sem ble av er age will be the same as a “time av er age”(see eq. [8.99]). In a prac ti cal sim u la tion, τ runs over the suc ces sion of states gen er ated fol -low ing the pre vi ously men tioned rules, and is a large fi nite num ber. This ap proach is ex -actly what is done in an MC sim u la tion, in which a tra jec tory is gen er ated through phasespace with dif fer ent rec i pes for the dif fer ent en sem bles. In other words, in a MC sim u la tiona sys tem of par ti cles in ter act ing through some known po ten tial is as signed a set of ini tial co -or di nates (ar bi trarily cho sen) and a se quence of con fig u ra tions of the par ti cles is then gen er -ated by suc ces sive ran dom dis place ments (also called ‘moves’).

If f(R) is an ar bi trary func tion of all the co or di nates of the mol e cule, its av er age valuein the ca non i cal en sem ble is:

( )f

f r e dR

e dR

v R

V R= ∫

∫

−

−

β

β

( )

( )[8.104]

where β = 1/kT and V(R) is the po ten tial en ergy.


In the MC meth ods the in te gra tion is done by re-writ ing <f> in terms of the prob a bil i -ties Pk of find ing the con fig u ra tion Rk:

( )f f R Pk kk

= ∑ [8.105]

Pe

ek

V R

V R

k

k

k=

−

−∑β

β

( )

( )[8.106]

The search for the most prob a ble con fig u ra tion Rk is done us ing Markov’s chain the -ory, so that af ter set ting the sys tem at a given con fig u ra tion Ri, an other con fig u ra tion Rj isse lected, mak ing a ran dom move in the co or di nates. Not all con fig u ra tions are ac cepted; thede ci sion on whether or not to ac cept a par tic u lar con fig u ra tion is made in such a way as toen sure that as ymp tot i cally the con fig u ra tion space is sam pled ac cord ing to the prob a bil ityden sity ap pro pri ate to a cho sen en sem ble. In par tic u lar:

if ( ) ( )V R V Rj i<

the move is ac cepted in the chain. Otherwise,

if ( ) ( )V R V Rj i>

the move is sub jected to a sec ond screen ing and ac cepted

if e V Vj i− − <β γ( )

where γ is a ran dom num ber be tween 0 and 1.The orig i nal mod els used in MC were highly ide al ized rep re sen ta tions of mol e cules,

such as hard spheres and disks, but now a days MC sim u la tions are car ried out on the ba sis ofmore re li able in ter ac tion po ten tial. The use of re al is tic mol e cule-mol e cule in ter ac tion po -ten tial makes it pos si ble to com pare data ob tained from ex per i ments with the com puter gen -er ated ther mo dy namic data de rived from a model. The par ti cle momenta do not en ter thecal cu la tion, there is no scale time in volved, and the or der in which the con fig u ra tions oc curhas no spe cial sig nif i cance.

It is worth no tic ing that, be cause only a fi nite num ber of states can be gen er ated in asim u la tion, the re sults ob tained from a MC sim u la tion are af fected by the choice of ini tialcon di tions, ex actly as pre vi ously said for the MD re sults.

As we have al ready said, it is pos si ble to ex tend the MC method, orig i nally for mu lated in the microcanonical en sem ble, to other en sem bles. Par tic u larly im por tant in the study ofliq uid sys tems is the ex ten sion of the Monte Carlo method to the grand ca non i cal µVT en -sem ble:76 that is due to the fact that such an ex ten sion per mits to cal cu late the free en ergy ofthe sys tem, a quan tity of par tic u lar sig nif i cance in the chem is try and phys ics of liq uids. Thegrand ca non i cal MC in volves a two-stage pro cess: the first stage is ex actly iden ti cal to whatis done in the con ven tional MC, with the par ti cles moving and the move ac cepted or re -jected as pre vi ously said. The sec ond stage in volves an at tempt ei ther to in sert a par ti cle at aran domly cho sen point in the cell or to re move a par ti cle that is al ready pres ent. The de ci -


sion whether to make an in ser tion or a re moval is made ran domly with equal prob a bil i ties.The trial con fig u ra tion is ac cepted if the pseudo-Boltzmann fac tor:

[ ]W eN

N N V rNN

=− −βµ βlog ! ( )

[8.107]

in creases. If it de creases, the change is ac cepted when equal to:

[ ]W

W NeN

N

V VN N+ − −=+

+1 11

1βµ β ( [8.108]

for the in ser tion, or to:

[ ]W

WNeN

N

V VN N− − − −= +1 1βµ β( [8.109]

for the re moval.This method works very well at low and in ter me di ate den si ties. As the den sity in -

creases, it be comes dif fi cult to ap ply, be cause the prob a bil ity of in sert ing or re mov ing a par -ti cle is very small.

As we have al ready said, the grand ca non i cal Monte Carlo pro vides a mean to de ter -min ing the chem i cal po ten tial, and hence, the free en ergy of the sys tem. In other MC andMD cal cu la tions a nu mer i cal value for the free en ergy can al ways be ob tained by means ofan in te gra tion of ther mo dy namic re la tions along a path which links the state of in ter est toone for which the free en ergy is al ready known, for ex am ple, the di lute gas or the low-tem -per a ture solid. Such a pro ce dure re quires con sid er able com pu ta tional ef fort, and it has a low nu mer i cal sta bil ity. Sev eral meth ods have been pro posed and tested.

The dif fi cul ties in us ing MC and MD arise from the heavy com pu ta tional cost, due tothe need of ex am in ing a large num ber of con fig u ra tions of the sys tem usu ally con sist ing of a large num ber of par ti cles. The size of the sys tem one can study is lim ited by the avail ablestor age on the com puter and by the speed of ex e cu tion of the pro gram. The time taken toeval u ate the forces or the po ten tial en ergy is pro por tional to N2.

Other dif fi cul ties in mo lec u lar sim u la tions arise from the so-called quasi-ergodicprob lem,77 i.e., the pos si bil ity that the sys tem be comes trapped in a small re gion of the phase space. To avoid it, what ever the ini tial con di tions, the sys tem should be al lowed toequilibrate be fore start ing the sim u la tion, and dur ing the cal cu la tion, the bulk prop er tiesshould be care fully mon i tored to de tect any long-time drift.

As al ready said, apart from the ini tial con di tions, the only in put in for ma tion in a com -puter sim u la tion are the de tails of the inter-par ti cle po ten tial, al most al ways as sumed to bepair-wise ad di tive. Usually in prac ti cal sim u la tions, in or der to econ o mize the com put ingtime, the in ter ac tion po ten tial is trun cated at a sep a ra tions rc (the cut-off ra dius), typ i cally ofthe or der of three mo lec u lar di am e ters. Ob vi ously, the use of a cut-off sphere of small ra dius is not ac cept able when the inter-par ti cle forces are very long ranged.

The trun ca tion of the po ten tial dif fer ently af fects the cal cu la tion of bulk prop er ties,but the ef fect can be re cov ered by us ing ap pro pri ate “tail cor rec tions”. For en ergy and pres -sure for monoatomic flu ids, for ex am ple, these “tail cor rec tions” are ob tained by eval u at ing


the fol low ing integrals [8.110] and [8.111] be tween the lim its r=rc and r=∞, and as sum ingthat g(r)=1 for r>rc:

( ) ( )EN

v r g r r dr= ∫2 2πρ [8.110]

( ) ( )βρ

πβρPv r g r r dr= − ∫1

2

33 [8.111]

As al ready men tioned above, the ex tent of “tail cor rec tions” strongly de pends on theprop erty un der study. In the case of pres sure, this cor rec tion can be very large; for ex am ple,in the case of a Lennard-Jones po ten tial trun cated at rc=2.5σ, at con di tions close to the tri plepoint, E/Nε=-6.12, to which the tail cor rec tion con trib utes -0.48, but βP/ρ=0.22, of which-1.24 co mes to the tail. The ne glect ing of the tail cor rec tion would lead to a strongly pos i tive value for pres sure.42

Usually, when a cut-off ra dius is ex ploited, for the cal cu la tion of the in ter ac tion be -tween a par ti cle and the oth ers, the “near est-neigh bor” con ven tion is used. It means that, as -sum ing the par ti cles of in ter est lie in a cell and that this ba sic unit is sur rounded bype ri od i cally re peated im ages of it self, a par ti cle i ly ing on the cen tral cell will in ter act onlywith the near est im age of any other par ti cle j. The in ter ac tion is then set to zero if the dis -tance be tween i and the near est im age is greater than rc. The usual as sump tion rc<L/2, with L the cell length, means it is as sumed that at most one im age of j lies in any sphere of ra diusr<L/2 cen tered on the par ti cle i.

For the cal cu la tions of prop er ties of a small liq uid sys tem, like a very small liq uiddrop, the co he sive forces be tween mol e cules may be suf fi cient to hold the sys tem to getherdur ing the course of a sim u la tions: oth er wise the sys tem of mol e cules may be con fined by apo ten tial act ing as a con tainer. Such an ar range ment is, how ever, not sat is fac tory for thesim u la tion of bulk liq uids, be ing the main ob sta cle the large frac tion of mol e cules which lieon the sur face. Mol e cules on the sur face will ex pe ri ence quite dif fer ent forces from mol e -cules in the bulk.

To over come the prob lem, in the sim u la tion of bulk liq uids a pos si ble so lu tion is to im -pose pe ri odic bound ary con di tions. The sys tem is in a cu bic box, which is rep li catedthrough out space to form an in fi nite lat tice. Dur ing the sim u la tion, as a mol e cule moves inthe orig i nal box, its im age in each of the neigh bor ing boxes moves ex actly in the same way.In par tic u lar as a mol e cule leaves the cen tral box, one of its im ages will en ter the cen tral box through the op po site face. A com mon ques tion to ask about the use of pe ri odic bound arycon di tions is if the prop er ties of a small, in fi nitely pe ri odic sys tem and the mac ro scopic sys -tem that it rep re sents are the same. This will usu ally de pend both on the range of theintermolecular po ten tial and the phe nom e non un der in ves ti ga tion. In some cases, pe ri odicbound ary con di tions have lit tle ef fect on the equi lib rium ther mo dy namic prop er ties andstruc ture of flu ids away from phase tran si tions and where the in ter ac tions are short-ranged,but in other cases this is not true.

It is worth men tion ing an al ter na tive method to the stan dard pe ri odic bound ary con di -tions for sim u lat ing bulk liq uids. A three di men sional sys tem can be em bed ded in the sur -face of an hypersphere: the hypersphere is a fi nite sys tem and can not be con sid ered as partof an in fi nitely re peat ing pe ri odic sys tem. Ef fects com ing from the curved ge om e try


(non-Eu clid ean ge om e try has to be con sid ered) are ex pected to de crease as the sys tem in -creases, so that in the case of sol utes of large size, “spher i cal bound ary con di tions” are ex -pected to be a no tice able method for sim u lat ing bulk liq uids. In this case, bound arycon di tions can be ob tained by in sert ing the spher i cal cell into a large sphere and then by per -form ing a sim u la tion of a lower com pu ta tional level in the ex tended re gion. How ever, re -cent ver sions of the method tend to re place the sec ond por tion of the liq uid, de scribed at amo lec u lar level, with a de scrip tion by means of con tin uum mod els (see Sec tion 8.7.3),which can be eas ily ex tended up to r=∞.

Turn ing at ten tion to the ac cu racy gained in the cal cu la tion of liq uids prop er ties by mo -lec u lar sim u la tions, it turns out that cer tain prop er ties can be cal cu lated more ac cu ratelythan oth ers. For ex am ple, the mean po ten tial en ergy can be ob tained with an un cer tainty less than 1%; larger er rors are as so ci ated with the cal cu la tion of ther mo dy namic prop er ties,such as, the spe cific heat, that are linked to fluc tu a tions in a mi cro scopic vari able. Typically, an un cer tainty of the or der of 10% is ex pected.42

The dy nam i cal prop er ties of liq uids can be gen er ally com puted by cal cu lat ing thetime-cor re la tion func tions. They pro vide a quan ti ta tive de scrip tion of the mi cro scopic dy -nam ics in liq uids. Here, com puter sim u la tions play a key role, since they give ac cess to alarge va ri ety of cor re la tion func tions, many of which are not mea sur able by lab o ra tory ex -per i ments. In a MD sim u la tion, the value of the cor re la tion func tion CAB(t) at time t is cal cu -lated by av er ag ing the prod uct A(t+s)B(s) of two dy nam i cal vari ables, A(t) and B(t) overmany choices of the time or i gin s. It has been shown that the un cer tainty in time cor re la tionsbe tween events sep a rated by an in ter val τ in creases as τ1/2. In ad di tion, the cor re lated mo tionof large num bers of par ti cles de ter mines many dy nam i cal prop er ties. These col lec tive prop -er ties are usu ally sub jected to larger er rors than in the case of sin gle par ti cle prop er ties, andtheir cal cu la tion is very de mand ing from a com pu ta tional point of view.42

QM/MMAs fi nal re mark of this sec tion, we would like to briefly mention the so-called QM/MMmethod.78,79

In this ap proach, a part of the sys tem (the sol ute M) is treated ex plic itly by a quan tumme chan i cal (QM) method, while an other part (the bulk sol vent S) is ap prox i mated by a stan -dard Mo lec u lar Me chanics (MM) force field. It is clear that such a method takes ad van tageof the ac cu racy and gen er al ity of the QM treat ment, and of the com pu ta tional ef fi ciency ofthe MM cal cu la tion.

Using the BO ap prox i ma tion and as sum ing that no charge trans fer oc curs be tween Sand M, the Hamiltonian of the sys tem can be sep a rated into three terms:

$ $ $ $H H H HM MS SS= + + [8.112]

where the first term is the Hamiltonian of the sol ute in vacuo, the sec ond one rep re sents theQM/MM sol ute-sol vent in ter ac tion and cou ples sol vent ef fects into QM cal cu la tions. It canbe sep a rated in an elec tro static term, a van der Waals terms and a po lar iza tion term, i.e.:

$ $ $ $H H H HMS MSel

MSvdW

MSpol= + + [8.113]


The third term in eq. [8.112] is the in ter ac tion en ergy be tween sol vent mol e cules. It isusu ally ap prox i mated by the mo lec u lar me chan ics force field, which in gen eral con tainsbond stretch ing, an gle bend ing, di hed ral tor sion and non-bonded terms (see Sec tion 8.4.5).

Once the Hamiltonian of the sys tem is de fined, the to tal en ergy of an in stan ta neouscon fig u ra tion sam pled dur ing an MC or MD sim u la tion is de ter mined as:

E H E E EM MS SS= = + +Φ Φ$ [8.114]

More de tails on the pro ce dure can be found in lit er a ture.78

The forces used to in te grate New ton’s equa tion of mo tion in MD sim u la tions are de -ter mined by dif fer en ti at ing eq. [8.114] with re spect to nu clear co or di nates.

In the prac ti cal QM/MM cal cu la tion, the sol vent S is equil i brated first us ing sim u la -tions (or a RISM ver sion of the in te gral equa tion ap proach, see Sec tion 8.7.1.1), and thenthe elec tronic struc ture of the sol ute M is mod i fied via an it er a tive QM pro ce dure in thepres ence of a fixed po ten tial of the S com po nent. The pro ce dure is re peated iteratively; atthe next step the S dis tri bu tions is de ter mined again, now tak ing into ac count the mod i fiedde scrip tion of M. This se quence of steps is re peated un til con ver gence. A prob lem of such ameth od ol ogy is that changes in the in ter nal ge om e try of M must be treated apart, scan ningpoint-by point the rel e vant PES, not be ing avail able an a lyt i cal ex pres sions for first and sec -ond de riv a tives of the en ergy with re spect to nu clear co or di nates, which would be nec es sary to take into ac count any mod i fi ca tion in the sol ute ge om e try.

To over come prob lems aris ing from the fi nite sys tem size used in MC or MD sim u la -tion, bound ary con di tions are im posed us ing pe ri odic-sto chas tic ap prox i ma tions or con tin -uum mod els.75 In par tic u lar, in sto chas tic bound ary con di tions the fi nite sys tem is notdu pli cated but a bound ary force is ap plied to in ter act with at oms of the sys tem. This force isset as to re pro duce the sol vent re gions that have been ne glected. Any way, in gen eral any ofthe meth ods used to im pose bound ary con di tions in MC or MD can be used in the QM/MMap proach.

8.7.3 CONTINUUM MODELS

Or i ginally, con tin uum mod els of sol vent were for mu lated as di elec tric mod els for elec tro -static ef fects. In a di elec tric model the sol vent is mod eled as a con tin u ous me dium, usu allyas sumed ho mo ge neous and iso tro pic, char ac ter ized by a sca lar, static di elec tric con stant ε.This model of the sol vent, that can be re ferred to the orig i nal work by Born, Onsager andKirkwood 60-80 years ago, as sumes lin ear re sponse of the sol vent to a per turb ing elec tricfield due to the pres ence of the mo lec u lar sol ute.

This sim ple def i ni tion then has been largely ex tended to treat more com plex phe nom -ena, in clud ing not only elec tro static ef fects; and now a days con tin uum sol va tion mod elsrep re sent very ar tic u late meth od ol o gies able to de scribe dif fer ent sys tems of in creas ingcom plex ity.

The his tory of, and the the ory be hind, con tin uum sol va tion mod els have been de -scribed exhaustively in many reviews31a,80,81 and articles82-84 in the past, so we pre fer not tore peat them here. In ad di tion, so large and con tin u ously in creas ing is the amount of ex am -ples of the o ret i cal de vel op ments on one hand, and of nu mer i cal ap pli ca tions on the other,that we shall limit our at ten tion to a brief re view of the ba sic char ac ter is tics of these mod elswhich have gained wide ac cep tance and are in use by var i ous re search groups.


Trying to give a more the o ret i cally-stated def i ni tion of con tin uum mod els, but stillkeep ing the same con cise ness of the orig i nal one re ported above, we have to in tro duce some ba sic no tions of clas si cal electrostatics.

The elec tro static prob lem of a charge dis tri bu tion ρM (rep re sent ing the mo lec u lar sol -ute) con tained in a fi nite vol ume (the mo lec u lar cav ity) within a con tin uum di elec tric can be ex pressed in terms of the Pois son equa tion:

( ) ( )[ ] ( )r r r r r∇ ∇ = −→

ε φ πρr r rM4 [8.115]

where φ is the to tal elec tro static po ten tial.Eq. [8.115] is sub ject to the con straint that the di elec tric con stant in side the cav ity is

(εin=1, while at the as ymp totic bound ary we have:

( )limr

r r→∞

=φ αr ( )lim

rr r

→∞=2 φ β

r[8.116]

with α and β fi nite quan ti ties.At the cav ity bound ary the fol low ing con di tions hold:

( ) ( )φ φin outr rr r

= ( )∂φ

∂ε

∂φ∂

inout

out

nr

n∧=

r

$[8.117]

where sub scripts in and out in di cate func tions de fined in side and out side the cav ity, and $n isthe unit vec tor per pen dic u lar to the cav ity sur face and point ing out wards.

Fur ther con di tions can be added to the prob lem, and/or those shown above (e.g., thelast one) can be mod i fied; we shall con sider some of these spe cial cases later.

As said above, in sev eral mod els an im por tant sim pli fi ca tion is usu ally in tro duced,i.e., the func tion ε out (r)

r is re placed by a con stant ε (from now on we skip the re dun dant sub -

script out). With this sim pli fi ca tion we may re write the elec tro static prob lem by the fol low -ing equa tions:

in side the cav ity: ( ) ( )∇ = −2 4φ πρr rr rM , out side the cav ity: ( )∇ =2 0φ

rr [8.118]

Many al ter na tive ap proaches have been for mu lated to solve this prob lem. In bio chem is try, meth ods that discretize the Pois son dif fer en tial op er a tor over fi nite

el e ments (FE) have long been in use. In gen eral, FEM ap proaches do not di rectly use themo lec u lar cav ity sur face. Nev er the less, as the whole space filled by the con tin u ous me diumis par ti tioned into lo cally ho mo ge neous re gions, a care ful con sid er ation of the por tion ofspace oc cu pied by the mo lec u lar sol ute has still to be per formed.

There is an other fam ily of meth ods, known as FD (fi nite dif fer ence) meth ods,85 whichex ploit point grids cov er ing the whole space, but con versely to the FEM point, in FD meth -ods the points are used to re place dif fer en tial equa tions by al ge braic ones.

These meth ods can be ap plied both to the “sim pli fied” model with con stant ε, and tomod els with space de pend ent ε( )

rr or real charges dis persed in the whole di elec tric me dium.

They aim at solv ing the Pois son equa tion (1) ex pressed as a set of fi nite dif fer ence equa tions for each point of the grid. The lin ear sys tem to be solved has el e ments de pend ing both on


ε

ε and ρM, and its so lu tion is rep re sented by a set of φ val ues in the grid points. These val ueshave to be reached iteratively.

Many tech ni cal de tails of the meth ods have been changed in the years. We sig nal,among oth ers, the strat egy used to de fine the por tion of the grid, es pe cially near the mo lec u -lar sur face where there is a sharp change in ε, and the math e mat i cal al go rithms adopted toim prove con ver gence in the it er a tive pro cesses. Al most all FD meth ods have been con -ceived for, and ap plied to, clas si cal de scrip tions of sol utes, gen er ally rep re sented by atomiccharges. The lat ter are usu ally drawn from pre vi ous QM cal cu la tions in vacuo or from Mo -lec u lar Me chanics ef fec tive po ten tials, as in the study of large mol e cules, usu ally pro teins.The de fi cien cies due to the use of rigid charges are par tially al le vi ated by in tro duc ing a di -elec tric con stant greater than one for the in ner space (ε be tween 2 and 4), and/or by in clud -ing atomic po lar iza tion func tions (akin to atomic di pole polarizabilities).

The DelPhi program85a is one of these pro pos als. This method, which has been ex ten -sively im proved dur ing the years, ex ists now in a QM version86 ex ploit ing a re cent highlevel QM method, called PS-GVB,87 based on a syn the sis of grid point in te gra tion tech -niques and stan dard Gaussi an or bital meth ods. From a PS-GVB cal cu la tion in vacuo, a setof atomic charges, the po ten tial de rived (PD) charges, is ob tained by a fit ting of the mo lec u -lar elec tro static po ten tial (MEP). These charges are used in DelPhi to solve the Pois sonequa tion in the me dium, and then to get, with the aid of a sec ond ap pli ca tion of DelPhi invacuo, the re ac tion field (RF). The lat ter is rep re sented by a set of point charges on the cav -ity sur face, which de fine the sol ute-sol vent in ter ac tion po ten tial to be in serted in thePS-GVB Hamiltonian. Then a new com pu ta tional cy cle starts, and the whole pro cess is re -peated un til con ver gence is reached.

In par al lel, the clas si cal FD pro ce dure of Bashford and Karplus88 has been cou pledwith the Am ster dam den sity fuctional the ory (DFT)89 code to give an other ver sion ofQM-FD meth ods. Starting from a DFT cal cu la tion in vacuo, the RF po ten tial is ob tained asthe dif fer ence be tween the so lu tions of the Pois son equa tion ob tained with a FD method inthe me dium and in vacuo (in both cases, the elec tro static po ten tial φM is ex pressed in termsof PD atomic charges). The RF po ten tial is then re-com puted to solve a mod i fiedKohn-Sham equa tion. The cal cu la tions are re peated un til con ver gence is achieved.

An al ter na tive to the use of fi nite dif fer ences or fi nite el e ments to discretize the dif fer -en tial op er a tor is to use bound ary el e ment meth ods (BEM).90 One of the most pop u lar ofthese is the polarizable con tin uum model (PCM) de vel oped orig i nally by the Pisa group ofTomasi and co-work ers.32 The main as pect of PCM is to re duce the elec tro static Pois sonequa tion (1) into a bound ary el e ment prob lem with ap par ent charges (ASCs) on the sol utecav ity sur face.

A method that is very sim i lar to PCM is the con duc tor-like screen ing model(COSMO) de vel oped by Klamt and coworker91 and used later by Truong and co work ers,92

and by Barone and Cossi.93 This model as sumes that the sur round ing me dium is well mod -eled as a con duc tor (thus sim pli fy ing the electrostatics com pu ta tions) and cor rec tions aremade a pos te ri ori for di elec tric be hav ior.

Elec tro static BE meth ods can be sup ported by nonelectrostatic terms, such as dis per -sion and ex change, in dif fer ent ways, and their ba sic the ory can be ex tended to treat bothclas si cal and quan tum me chan i cal sol utes; in ad di tion, many fea tures, in clud ing an a lyt i calgra di ents with re spect to var i ous pa ram e ters, have been added to the orig i nal mod els so as to


make BE meth ods one of the most pow er ful tools to de scribe solvated sys tems of in creas ing com plex ity. More de tails on their main as pects will be given in the fol low ing.

Fur ther al ter na tive meth ods for solv ing the Pois son equa tion are to rep re sent the sol ute and the sol vent re ac tion field by sin gle-cen ter multipole ex pan sions (MPE), as first pi o -neered by the Nancy group of Rivail and co work ers.94 Or i ginally, this for mal ism wasadopted be cause it sim pli fied com pu ta tion of sol va tion free en er gies and an a lytic de riv a -tives for sol ute cav i ties hav ing cer tain ideal shapes, e.g., spheres and el lip soids. The for mal -ism has been ex tended to ar bi trary cav ity shapes and multipole ex pan sions at mul ti plecen ters. An al ter na tive method still adopt ing the same MPE ap proach is that of Mikkelsenand co work ers;95 this method has been im ple mented in the Dal ton com pu ta tional pack age.96

The idea of a dis trib uted multipole ex pan sion to rep re sent the charge dis tri bu tion isalso em ployed by the so-called gen er al ized Born (GB) ap proach to con tin uum sol va tion. Inthis case, how ever, only monopoles (i.e., atomic par tial charges) are em ployed and in steadof solv ing the Pois son equa tion with this charge dis tri bu tion, one uses the gen er al ized Bornap prox i ma tion. The most widely used sol va tion mod els adopt ing this pro ce dure are theorig i nal gen er al ized Born/sur face area (GBSA) ap proach of Still and co-work ers,97 in clud -ing the sub se quent ex ten sions and ad ap ta tions of sev eral groups in which the GBelectrostatics are com bined with a clas si cal MM treat ment of the sol ute, and the SMx mod -els of Cramer, Truhlar and coworkers98 in which GB electrostatics are em ployed in a quan -tum me chan i cal treat ment of the sol ute. In both of these ap proaches, semiempirical atomicsur face ten sions are used to ac count for nonelectrostatic sol va tion ef fects. For the SMxmod els, the value of x ranges from 1 to 5.42, cor re spond ing to suc ces sively im proved ver -sions of the pre scrip tions for atomic charges and ra dii, the di elec tric descreening al go rithm,and the func tional forms for the atomic sur face ten sions.

8.7.3.1 QM-BE methods: the Effective HamiltonianAs an tic i pated above, in BE meth ods for sol va tion the ba sic fea ture is to solve the Pois sonequa tion [8.115] with the re lated bound ary con di tions [8.117] (or in a sim pler iso tro pic case the sys tem [8.118]) by de fin ing an ap par ent charge dis tri bu tion σ on the sol ute cav ity sur -face (such charge is usu ally in di cated as ASC).31a This con tin u ous charge which de pends onboth the sol vent di elec tric con stant and the sol ute charge and ge om e try, is then discretizedin terms of point charges each placed on a dif fer ent por tion of sur face iden ti fied by a col lo -ca tion point sk and an area ak. This par ti tion of the cav ity sur face into patches, or tesserae, rep re sents one of the main fea tures of the method as well as the shape of the cav ity it self,which here has not been lim ited to an a lyt i cal sim ple shapes (as in the orig i nal MPE meth -ods), but it can be well mod eled ac cord ing to the ac tual sol ute ge om e try.

The par tic u lar as pect we shall treat here is the gen er al iza tion of the ba sic the ory ofthese meth ods to QM de scrip tions of the sol ute. To do that, one has to in tro duce the con ceptof the Ef fec tive Hamiltonian (EH); for the iso lated sys tem, the fun da men tal equa tion tosolve is the stan dard Schrödinger equa tion

H E0 0 0 0Ψ Ψ= [8.119]

where the su per script 0 in di cates the ab sence of any in ter ac tions with other mol e cules.In the pres ence of a sol vent treated in the ASC-BE frame work, an ex ter nal per tur ba -

tion has to be taken into ac count: the re ac tion field due to the ap par ent charges placed on thecav ity sur face. Such per tur ba tion can be rep re sented in terms of an elec tro static po ten tial VR


to be added to the Hamiltonian H0 so to ob tain an ef fec tive Hamiltonian H which will sat isfy a dif fer ent Schrödinger equa tion with re spect to [8.119], namely, we have

H EΨ Ψ= [8.120]

Both the wave func tion and the eigenvalue E will be mod i fied with re spect to [8.119], due to the pres ence of the sol vent per tur ba tion.

An im por tant as pect, un til now not in tro duced, is that the sol vent ap par ent charges,and con se quently the re ac tion op er a tor VR, de pend on the sol ute charge, i.e., in the pres entQM frame work, on the wave func tion they con trib ute to de fine. This mu tual in ter ac tions be -tween Ψ and VR in duces a com plex ity in the prob lem which can be solved through the stan -dard it er a tive pro ce dures char ac ter iz ing the self-con sis tent (SC) meth ods. Only for anas pect the cal cu la tion in so lu tion has to be dis tin guished from that in vacuo way; the en ergyfunc tional to be min i mized in a variational so lu tion of [8.120] is not the stan dard func tionalE but the new free en ergy func tional G

G H V R= +Ψ Ψ0 1 2/ [8.121]

The 1/2 fac tor in front of VR ac counts for the lin ear de pend ence of the op er a tor on thesol ute charge (i.e., the qua dratic de pend ence on Ψ). In a more phys i cal de scrip tion, the same fac tor is in tro duced when one con sid ers that half of the in ter ac tion en ergy has been spent inpo lar iz ing the sol vent and it has not been in cluded in G.

In the stan dard orig i nal model the per tur ba tion VR is lim ited to the elec tro static ef fects(i.e., the elec tro static in ter ac tion be tween the ap par ent point charges and the sol ute chargedis tri bu tion); how ever, ex ten sions to in clude dis per sion and re pul sion ef fects have beenfor mu lated. In this more gen eral con text the op er a tor VR can be thus par ti tioned in threeterms (elec tro static, re pul sive and dispersive), which all to gether con trib ute to mod ify thesol ute wave func tion.

For clar ity’s sake in the fol low ing we shall limit the anal y sis to the elec tro static partonly, re fer ring the reader to refs. [31,83], and to the orig i nal pa pers quoted therein, for de -tails on the other terms. For a more com plete re port we re call that nonelectrostatic ef fectscan be also in cluded a pos te ri ori, i.e., in de pend ently of the QM cal cu la tion, through ap -prox i mated or semiempirical mod els still ex ploit ing the def i ni tion of the mo lec u lar cav ity.In PCM and re lated meth ods, the term cor re spond ing to the en ergy spent in form ing the sol -ute cav ity in the bulk liq uid, is usu ally com puted us ing the SPT in te gral equa tion method(see Section 8.7.1.1).

Un til now no ref er ence to the spe cific form of the op er a tor VR has been given, but hereit be comes com pul sory. Trying to re main at the most gen eral level pos si ble, we start to de -fine the ap par ent charges q(sk) through a ma trix equa tion, i.e.,

( )q M f= − −1 ρ [8.122]

where M is a square ma trix de pend ing only on the ge om e try of the sys tem (e.g., the cav ityand its par ti tion in tesserae) and the sol vent di elec tric con stant and f(ρ) is a col umn vec torcon tain ing the val ues of a func tion f, which de pends on the sol ute dis tri bu tion charge ρ,com puted on the cav ity tesserae. The form of this func tion is given by the spe cific sol va tion


model one uses; thus, for ex am ple, in the orig i nal ver sion of PCM it co in cides with the nor -mal com po nent of the elec tro static field to the cav ity sur face, while for both the COSMOmodel91-93 and the re vised PCM ver sion, known as IEF (in te gral equa tion for mal ism),99 it isrep re sented by the elec tro static po ten tial. In any case, how ever, this term in duces a de pend -ence of the sol vent re ac tion to the sol ute charge, and fi nally to its wave func tion; in a stan -dard SC pro ce dure it has to be re com put ed (and thus also the ap par ent charges q) at eachit er a tion.

Also, the form of M de pends on the sol va tion model; con trary to f, it can be com putedonce, and then stored, if the ge om e try of the sys tem is not mod i fied dur ing the cal cu la tion.

Once the ap par ent charges have been de fined in an an a lyt i cal form through eq.[8.122], it is possible to de fine the per tur ba tion VR as

( ) ( )Vq sk

r skdrR =

−∫ ρ ρ [8.123]

In par tic u lar, by dis tin guish ing the source of ρ, i.e., the sol ute nu clei and elec trons, we canal ways de fine two equa tions [8.122], one for each source of f, and thus com pute two sets ofap par ent charges, qN and qe, de pend ing on the sol ute nu clei and elec trons, re spec tively. This al lows one to par ti tion the re ac tion op er a tors in two terms, one de pend ing on the ge om e tryof the sys tem and the sol ute nu clear charge, and the other still de pend ing on the ge om e tryand on the sol ute wave func tion.

Such par ti tion is usu ally in tro duced to get an ef fec tive Hamiltonian whose form re -sem bles that of the stan dard Hamiltonian for the iso lated sys tem. In fact, if we limit our ex -po si tion to the Hartree-Fock ap prox i ma tion in which the mo lec u lar orbitals are ex pressed as an ex pan sion over a fi nite atomic or bital ba sis set, the Fock op er a tors one de fines for the two sys tems, the iso lated and the solvated one, be come (over the atomic ba sis set)

F0 = h + G(P) F = h + j + G(P) +X(P) [8.124]

where h and G(P) are the stan dard one and two-elec tron in ter ac tion ma tri ces in vacuo (Prep re sents the one-elec tron den sity ma trix). In [8.124] the sol vent terms are ex pressed bytwo ma tri ces, j and X(P), whose form fol lows di rectly from the nu clei and elec tron-in ducedequiv a lents of ex pres sion [8.122], re spec tively; for the elec tronic part, ρ has to be sub sti -tuted by P.

The par al lel form of F0 and F in [8.124] shows that the two cal cu la tions can be per -formed ex actly in the same way, i.e., that sol vent does not in tro duce any com pli ca tion or ba -sic mod i fi ca tion to the pro ce dure orig i nally for mu lated for the iso lated sys tem. This is avery im por tant char ac ter is tic of con tin uum BE sol va tion meth ods which can be gen er al izedto al most any quan tum me chan i cal level of the ory. In other words, the def i ni tion of‘pseudo’ one and two-elec tron sol vent op er a tors as sures that all the the o ret i cal bases and the for mal is sues of the quan tum me chan i cal prob lem re main un changed, thus al low ing a stated so lu tion of the new sys tem ex actly as in vacuo.

Above, we have lim ited the ex po si tion to the ba sic fea tures of sin gle point HF cal cu la -tions, and to the ther mo dy namic func tions (the sol va tion free en ergy is im me di ately givenby eqs. [8.121-8.124]). How ever, ex ten sions to other QM pro ce dures as well as to othertypes of anal y sis (the eval u a tion of mo lec u lar re sponse func tions, the study of chem i cal re -


ac tiv ity, the re pro duc tion of spec tra of var i ous na ture, just to quote the most com mon ap pli -ca tions) have been al ready suc cess fully per formed.83 Many of these ex ten sions have beenmade pos si ble by the for mu la tion and the fol low ing im ple men ta tion of ef fi cient pro ce duresto get an a lyt i cal de riv a tives of the free en ergy in so lu tion with re spect to var i ous pa ram e -ters.100 To un der stand the role of this meth od olog i cal ex ten sion in the ac tual de vel op ment of the sol va tion mod els we just re call the im por tance of an ef fi cient com pu ta tion of an a lyt i calgra di ents with re spect to nu clear co or di nates to get re li able ge om e try optimizations and dy -nam ics.101

It is here im por tant to re call that such im prove ments are not lim ited to BE sol va tionmeth ods; for ex am ple, Rivail and the Nancy group102 have re cently ex tended theirmultipole-ex pan sion for mal ism to per mit the an a lytic com pu ta tion of first and sec ond de -riv a tives of the sol va tion free en ergy for ar bi trary cav ity shapes, thereby fa cil i tat ing the as -sign ment of sta tion ary points on a solvated po ten tial en ergy sur face. An a lytic gra di ents forSMx mod els at ab in itio the ory have been re cently described103 (even if they have beenavail able lon ger at the semiempirical level104), and they have been pre sented also for fi nitedif fer ence so lu tions of the Pois son equa tion and for fi nite el e ment so lu tions.

Many other ex ten sions of dif fer ent im por tance have been added to the orig i nal BEmod els in the last few years. Here it may be worth re call ing a spe cific gen er al iza tion whichalso has made BE mod els pref er a ble in par tic u lar ap pli ca tions un til now al most com pletelyre stricted to FE and FD meth ods.

At the be gin ning of the pres ent sec tion we in di cated the sim pli fied sys tem (4), validonly in the limit of an ho mo ge neous and iso tro pic di elec tric, the com monly stud ied prob -lem; ac tu ally, this has been the only af ford able sys tem un til re cently, with some ex cep tionsrep re sented by sys tems con sti tuted by two iso tro pic sys tems with a def i nite in ter face. Thesit u a tion changed only a few years ago (1997-1998) when BE-ASC meth ods were ex tendedto treat mac ro scop i cally anisotropic di elec trics (e.g., liq uid crys tals),99 ionic so lu tions (e.g.,iso tro pic so lu tions with non zero ionic strength),99 and super criti cal liq uids.105

An other im por tant fea ture in cluded in BE meth ods, as well in other con tin uum meth -ods, are the dy nam i cal ef fects.

In the pre vi ous sec tion de voted to Phys i cal Models, we have re called that an im por tant is sue of all the sol va tion meth ods is their abil ity to treat nonequilibrium, or dy nam i cal, as -pects. There, for clar ity’s sake, we pre ferred to skip the anal y sis of this fea ture, lim it ing theex po si tion to static, or equi lib rium prob lems; here, on the con trary, some com ments will beadded.

In con tin uum mod els no ex plicit ref er ence to the dis crete mi cro scop i cal struc ture ofthe sol vent is in tro duced; on one hand, this largely sim pli fies the prob lem, but on the otherhand it rep re sents a limit which can ef fect the qual ity of the cal cu la tions and their ex ten si bil -ity to more com plex prob lems such as those in volv ing dy nam i cal phe nom ena. How ever,since the first for mu la tions of con tin uum mod els, dif fer ent at tempts to over come this limithave been pro posed.

One of the the o ries which has gained more at ten tion iden ti fies the sol vent re sponsefunc tion with a po lar iza tion vec tor de pend ing on time, P(t), which var ies ac cord ing to thevari a tions of the field from which it is orig i nated (i.e., a sol ute and/or an ex ter nal field). This vec tor ac counts for many phe nom ena re lated to dif fer ent phys i cal pro cesses tak ing place in -side and among the sol vent mol e cules. In prac tice, strong ap prox i ma tions are nec es sary inor der to for mu late a fea si ble model to deal with such com plex quan tity. A very sim ple but


also ef fec tive way of see ing this prob lem is to par ti tion the po lar iza tion vec tor in terms ac -count ing for the main de grees of free dom of the sol vent mol e cules, each one de ter mined bya proper re lax ation time (or by the re lated fre quency).

In most ap pli ca tions, this spec tral de com po si tion is lim ited to two terms rep re sent ing“fast” and “slow” phe nom ena, respectively106,107

( )P t P Pfast slow= + [8.125]

In par tic u lar, the fast term is eas ily as so ci ated with the po lar iza tion due to the boundelec trons of the sol vent mol e cules which can in stan ta neously ad just them selves to anychange of the in duc ing field. The slow term, even if it is of ten re ferred to as a gen eralorientational po lar iza tion, has a less def i nite na ture. Roughly speak ing, it col lects many dif -fer ent nu clear and mo lec u lar mo tions (vi bra tional re lax ations, ro ta tional and traslationaldif fu sion, etc.) re lated to gen er ally much lon ger times. Only when the fast and the slowterms are ad justed to the ac tual de scrip tion of the sol ute and/or the ex ter nal field, on onehand, and to each other on the other hand, do we have sol va tion sys tems cor re spond ing tofull equi lib rium. Nonequilibrium sol va tion sys tems, on the con trary, are char ac ter ized byonly par tial re sponse of the sol vent. A very ex pli ca tive ex am ple of this con di tion, but notthe only one pos si ble, is rep re sented by a ver ti cal elec tronic ex ci ta tion in the sol ute mol e -cule. In this case, in fact, the im me di ate sol vent re sponse will be lim ited to its elec tronic po -lar iza tion only, as the slower terms will not able to fol low such fast change but will re mainfrozen in the equi lib rium sta tus ex ist ing im me di ately be fore the tran si tion.

This kind of anal y sis is eas ily shifted to BE-ASC mod els in which the po lar iza tionvec tor is sub sti tuted by the ap par ent sur face charge in the rep re sen ta tion of the sol vent re ac -tion field.107 In this frame work, the pre vi ous par ti tion of the po lar iza tion vec tor into fast andslow com po nents leads to two cor re spond ing sur face ap par ent charges, σf and σs, the sum ofwhich gives the to tal ap par ent charge σ. The def i ni tion of these charges is in turn re lated tothe static di elec tric con stant ε(0) (for the full equi lib rium to tal charge σ), to the fre quencyde pend ent di elec tric con stant ε(ω) (for the fast com po nent σf), and to a com bi na tion of them(for the re main ing slow com po nent σs).

The an a lyt i cal form of the fre quency de pend ence of ε(ω) in gen eral is not known, butdif fer ent re li able ap prox i ma tions can be ex ploited. For ex am ple, if we as sume that the sol -vent is po lar and fol lows a Debye-like be hav ior, we have the gen eral re la tion:

( ) ( ) ( ) ( )ε ω ε

ε ε

ωτ= ∞ +

− ∞

−

0

1 i D

[8.126]

with ε(0)=78.5, ε(∞)=1,7756, and τD=0.85×10-11s in the spe cific case of wa ter.It is clear that at the op ti cal fre quen cies usu ally in volved in the main phys i cal pro -

cesses, the value of ε(ω) is prac ti cally equal to ε(∞); vari a tions with re spect to this, in fact,be come im por tant only at ωτD <<1, where ε(ω ) is close to the static value. In prac tice, thismeans that for a po lar sol vent like wa ter, where ε(∞) is much smaller than the static value,the dy nam i cal sol vent ef fects are small with re spect to the static ones. Nu merically, thismeans that all pro cesses de scribed in terms of a static or a dy namic sol vent ap proach, as well as sol ute re sponse prop er ties com puted in the full sol ute-sol vent equi lib rium or, in stead in acon di tion of nonequilibrium, will pres ent quite dif fer ent val ues.


As a last note we un der line that, while it is easy to see that nonequilibrium ef fects haveto be taken into ac count in the pres ence of fast changes in the elec tronic dis tri bu tion of thesol ute, or of an os cil lat ing ex ter nal field (as that ex ploited to mea sure mo lec u lar op ti calprop er ties), nonequilibrium ap proaches for nu clear vi bra tional anal y ses are still open.

8.8 PRACTICAL APPLICATIONS OF MODELINGIn this sec tion we will pres ent some ex am ples of how to de ter mine prop er ties of pure liq uids by means of the pre vi ously-shown com pu ta tional pro ce dures. In spite of the fact that mostlit er a ture in this field is con cerned with the study of the prop er ties of wa ter, we will ratherfo cus the at ten tion on the de ter mi na tion of the prop er ties of a few or ganic sol vents of com -mon use (for a re view of stud ies on wa ter, we ad dress in ter ested read ers to the al readyquoted pa per of Floris and Tani).40 Ob vi ously, the fol low ing dis cus sion can not be con sid -ered as a com plete re view of all the lit er a ture in the field, but it is rather in tended to give thereader a few sug ges tions on how the pre vi ously-shown com pu ta tional meth od ol o gies canbe used to ob tain data about the most usual prop er ties of pure liq uids.

From what we said in the pre vi ous sec tion, it should be clear to the reader that con tin -uum mod els can not prop erly be used in or der to de ter mine liq uid prop er ties: they are in factcon cerned in the treat ment of a sol ute in a sol vent, and so not of di rect use for the study ofthe sol vent prop er ties. For this rea son, we will pres ent in the fol low ing some ex am ples con -cern ing the ap pli ca tion of in te gral equa tions and com puter sim u la tions.

The anal y sis will be lim ited to the ex am i na tion of the most usual static and dy namicprop er ties of liq uids.

Dielectric ConstantLet us be gin the dis cus sion by con sid er ing a cou ple of ex am ples re gard ing the cal cu la tion of the di elec tric con stant of pure liq uids.

The crit i cal role of the di elec tric con stant in de ter min ing the prop er ties of a liq uid andin par tic u lar in mod er at ing the intermolecular in ter ac tions can not be over em pha sized. It isthen ex tremely im por tant that po ten tial en ergy func tions used, for ex am ple, in com putersim u la tions are able to re pro duce the ex per i men tal value of the di elec tric con stant and thatsuch a cal cu la tion can be used to es ti mate the va lid ity of any em pir i cal po ten tial to be usedin sim u la tions. The prob lem of the cal cu la tion of the di elec tric re sponse is com plex:108

many the o ries and meth ods are avail able, giv ing some times dif fer ent and con trast ing re -sults.

It is ac cepted that for the cal cu la tion of the di elec tric con stant (as well as other prop er -ties de pend ent on intermolecular an gu lar cor re la tions) via com puter sim u la tions, it is com -pul sory to in clude long-range elec tro static in ter ac tions in the treat ment. Such in ter ac tionscan be taken into ac count by us ing ei ther the Ewald sum ma tion ap proach or us ing the re ac -tion field ap proach. With out go ing into de tails, we sim ply re call here that in the Ewald ap -proach, elec tro static in ter ac tions are eval u ated us ing an in fi nite lat tice sum over all pe ri odicim ages, i.e., by im pos ing a pseudo-crys tal line or der on the liq uid. In the re ac tion field ap -proach, how ever, elec tro static in ter ac tion be yond the spher i cal cut off (see Sec tion 8.7.2 forde tails) are ap prox i mated by treat ing the part of the sys tem out side the cut off ra dius as apolarizable con tin uum of a given di elec tric con stant εrf.

Sev eral meth ods ex ist for the cal cu la tion of the di elec tric con stant by means of com -puter sim u la tions. The most widely used is the cal cu la tion of the av er age of the square of the to tal di pole mo ment of the sys tem, <M2> (fluc tu a tion method). Using re ac tion field bound -

8.8 Practical applications of modeling 455

ary con di tions, the di elec tric con stant ε of the liq uid within the cut off ra dius is cal cu lated byap ply ing the fol low ing ex pres sion, where V is the vol ume of the sys tem, k the Boltzmanncon stant and T the tem per a ture:

49

13

2 1

2

2π ε ε

ε ε

M

kVTrf

rf

=−

++

[8.127]

In the ap pli ca tion of eq. [8.127] care ful at ten tion has to be paid to the choice of εrf, thedi elec tric con stant to use for the con tin uum part of the sys tem. Usually, rapid con ver gencein cal cu la tions is ob tained by choos ing εrf ≈ ε.

A dif fer ent ap proach to the cal cu la tion of the di elec tric con stant via com puter sim u la -tions is given by the po lar iza tion re sponse method, i.e., to de ter mine the po lar iza tion re -sponse of a liq uid to an ap plied elec tric field E0. If <P> is the av er age sys tem di pole mo mentper unit vol ume along the di rec tion of E0, ε can be cal cu lated us ing the fol low ing ex pres -sion:

49

13

2 1

20

π ε εε ε

P

Erf

rf

=−

++

[8.128]

Even if this sec ond ap proach is more ef fi cient than the fluc tu a tion method, it re quiresthe per tur ba tion of the liq uid struc ture fol low ing the ap pli ca tion of the elec tric field.

As an ex am ple, such a meth od ol ogy has been ap plied by Essex and Jorgensen109 to thecal cu la tion of the di elec tric con stant of formamide and dimethylformamide us ing MonteCarlo sta tis ti cal me chan ics sim u la tions (see Sec tion 8.7.2 for de tails). The sim u la tion re sultfor dimethylformamide, 32±2, is rea son ably in agree ment with re spect to the ex per i men talvalue, 37. How ever, in the case of formamide, the ob tained value, 56±2, un der es ti mates theex per i men tal value of 109.3. The poor per for mance here addresses the fact that force fieldmod els with fixed charges un der es ti mate the di elec tric con stant for hy dro gen-bonded liq -uids.

Other meth od ol o gies ex ist for the cal cu la tion of the static di elec tric con stant of pureliq uids by means of com puter sim u la tions. We would like to re call here the use of ion-ionpo ten tials of mean force,110 and an um brella sam pling ap proach whereby the com plete prob -a bil ity dis tri bu tion of the net di pole mo ment is cal cu lated.111

Com puter sim u la tions are not the only meth ods which can be used to cal cu late the di -elec tric con stant of pure liq uids. Other ap proaches are given by the use of in te gral equa -tions, in par tic u lar, the hypernetted chain (HNC) mo lec u lar in te gral equa tion and themo lec u lar Ornstein-Zernike (OZ) the ory (see Section 8.7.1 for de tails on such meth od ol o -gies).

The mo lec u lar HNC the ory gives the mo lec u lar pair dis tri bu tion func tion g(12) fromwhich all the equi lib rium prop er ties of a liq uid can be eval u ated. It is worth re call ing herethat the HNC the ory con sists for a pure liq uid in solv ing a set of two equa tions, namely, theOZ equa tion [8.76] and the HNC clo sure re la tion [8.78] and [8.80].

In or der to solve the HNC equa tions, the cor re la tion func tions are writ ten in terms of ase ries of ro ta tional invariants Φµv

mnl ( )12 , which are de fined in terms of co sines and sines ofthe an gles de fin ing the rel a tive ori en ta tion of two in ter act ing par ti cles 1 and 2:


( ) ( ) ( )c c Rvmnl

vmnl

mnlv

12 12= ∑ µ µ

µ

Φ [8.129]

where the co ef fi cients cmnlµν ( )12 de pend only on the dis tance be tween the mass cen ters of the

par ti cles, and the in dexes m, n, l, µ, and v are in te gers sat is fy ing the fol low ing in equal i ties:

m n l m n m m n v n− ≤ ≤ + − ≤ ≤ − ≤ ≤, ,µ [8.130]

The di elec tric con stant can be ob tained from the ro ta tional in vari ant co ef fi cients of the pair cor re la tion func tion. For ex am ple, fol low ing Fries et al.,112 it can be eval u ated us ing theKirkwood g fac tor:

( )g R h R dR= −∞

∫14

32

00110

0

πρ [8.131]

( )( )49

1 2 1

92π

βρε ε

εm ge =

− +[8.132]

In the pre vi ous re la tions, ρ is the num ber den sity of mol e cules, β=1/kT, and me is theef fec tive di pole of mol e cules along their sym me try axes.

Such an ap proach has been used to eval u ate the di elec tric con stant of liq uidacetonitrile, ac e tone and chlo ro form by means of the gen er al ized SCMF-HNC (self-con sis -tent mean field hypernetted chain), and the values ob tained are 30.8 for acetonitrile, 19.9 for ac e tone, and 5.66 for chlo ro form. The agree ment with the ex per i men tal data, 35.9 foracetonitrile, 20.7 for ac e tone, and 4.8 for chlo ro form (un der the same con di tions), is good.

An other way of eval u at ing the di elec tric con stant is that used by Richardi et al.113 whoap plied mo lec u lar Ornstein-Zernike the ory and eq. [8.131] and [8.132]. The ob tained val -ues for both ac e tone and chlo ro form are low with re spect to ex per i men tal data, but in ex cel -lent agree ment with re spect to Monte Carlo sim u la tion cal cu la tions. In this case, thedi elec tric con stant is eval u ated by us ing the fol low ing ex pres sions:

( )( )( )

ε ε

ε ερµ

ε

µ µ

µ

− +

+=

==∑∑

1 2 1

3 2 9

2

0

11

2

rf

rf

i jj

N

i

N

kT N[8.133]

where εrf is the di elec tric con stant of the con tin uum part of the sys tem, i.e., the part out sidethe cut off sphere, N the num ber of mol e cules of the sys tem and µ the di pole mo ment.

A fur ther im prove ment in the men tioned ap proach has been achieved by Richardi etal.114 by cou pling the mo lec u lar Ornstein-Zernike the ory with a self-con sis tent mean-fieldap prox i ma tion in or der to take the polarizability into ac count. For the pre vi ously-men tioned sol vents (ac e tone, acetonitrile and chlo ro form), the cal cu lated val ues are in ex cel lent agree -ment with re spect to ex per i men tal data, show ing the cru cial role of tak ing into ac countpolarizability con tri bu tions for po lar polarizable aprotic sol vents.


Thermodynamical propertiesPaying at ten tion again to cal cu la tions for ac e tone and chlo ro form, we would like to men tion two pa pers by the Regensburg group113,114 ex ploit ing mo lec u lar Ornstein-Zernike (MOZ)the ory and site-site Ornstein-Zernike (SSOZ) the ory for the cal cu la tion of ex cess in ter nalen er gies. In the MOZ frame work, the in ter nal ex cess en ergy per mol e cule is com puted by acon fig u ra tional av er age of the pair po ten tial V act ing be tween mol e cules:

( ) ( )E V R g R dR d dex = ∫∫∫ρ

2 2 12 1 2 12 1 2 12 1 2ΩΩ Ω Ω Ω Ω Ω, , , , [8.134]

with ρ the den sity of mol e cules.Using in stead SSOZ, Eex can be di rectly cal cu lated from the site-site dis tri bu tion func -

tion, gαβ, as:

( ) ( )E g r V r r drexkk

===

∞

∑∑∫2110

2πρ αββα

αβ [8.135]

where k is the num ber of sites per mol e cule.In both cases the cal cu lated val ues are in good agree ment with ex per i men tal data and

with cal cu la tions done ex ploit ing the Monte Carlo com puter sim u la tion. In this case, Eex can be ob tained as:

( )EN

V rz z e r

rex

ijrf

rf

ij

cuto

= +−+

1 1

2 1 402 2

0

αβ αβα β αβε

ε πε,

,

ff

kk

j

N

i

N

31111 βα ==>=

∑∑∑∑ [8.136]

where k is the num ber of sites per mol e cule, rαβ,ij is the dis tance be tween site α of mol e cule iand site β of mol e cule j. Only pairs i,j of mol e cules for which rij<rcutoff are con sid ered.

Fur ther re fine ments of the men tioned ap proach have been pro posed in the al readyquoted pa per by Richardi et al.114 by cou pling the mo lec u lar Ornstein-Zernike the ory with aself-con sis tent mean-field ap prox i ma tion. Here again, com puted data are in ex cel lentagree ment with ex per i ments.

As al ready said, the use of in te gral equa tion meth od ol o gies is not the only pos si bleway of ob tain ing thermodynamical prop er ties for liq uids. Com puter sim u la tions are alsowidely used, and it is then pos si ble to ob tain equi lib rium prop er ties by mean of both MonteCarlo and Mo lec u lar Dy nam ics sim u la tions.

As an ex am ple, we would like to men tion some cal cu la tions on equi lib rium prop er tiesfor liq uid eth a nol at var i ous tem per a tures.115 The cal cu lated val ues for the heat of va por iza -tion are in over all agree ment with ex per i ments, show ing an er ror of the or der of 1-1.5%.

CompressibilitiesAs an ex am ple of how to de ter mine compressibilities by mean of in te gral equa tions, wewould like to quote here the al ready cited pa per of Richardi et al.113 in which mo lec u larOrnstein-Zernike (MOZ) the ory and site-site Ornstein-Zernike (SSOZ) the ory have beenex ploited. In par tic u lar, us ing the MOZ, the com press ibil ity κT can be eval u ated using theequa tion:


( )( )ρ κ πρkT g R R dRT = + −∞

∫1 4 100000

12 122

0

12 [8.137]

and ex ploit ing SSOZ can be cal cu lated from any site-site dis tri bu tion func tion gαβ as:

( )( )ρ κ πρ αβkT g r r drT = + −∞

∫1 4 1 2

0

[8.138]

For ac e tone and chlo ro form, the ob tained re sults are too large.

Relaxation times and diffusion coefficientsLet us now con sider an ex am ple of how to cal cu late dy namic prop er ties of pure liq uids bycom pu ta tional meth od ol o gies. As al ready said in Section 8.7.2, mo lec u lar dy nam ics sim u -la tions are able to take into ac count the time-de pend ence in the cal cu la tion of liq uid prop er -ties.

Paying at ten tion to pure ac e tone, Brodka and Zerda116 have cal cu lated ro ta tional re lax -ation times and translational dif fu sion co ef fi cients by mo lec u lar dy nam ics sim u la tions. Inpar tic u lar, the cal cu lated ro ta tional times of the di pole mo ment can be com pared with a mo -lec u lar re lax ation time τM ob tained from the ex per i men tally de ter mined τD by us ing the fol -low ing ex pres sion, which con sid ers a lo cal field fac tor:

τε ε

ετM

s

sD=

+ ∞2

3[8.139]

where εs and ε∞ are the static and op ti cal di elec tric con stants, re spec tively. The cal cu lateddata are in good agree ment with ex per i men tal values.

Dif fu sion co ef fi cients can be cal cu lated di rectly from the ve loc ity cor re la tion func -tions or from mean square dis place ments, as:

( ) ( )D v t v dtvcf =∞

∫13

00

[8.140]

( ) ( )Dt

r t rmsd

i=

→∞lim

16

02

[8.141]

The val ues ob tained by us ing the [8.140] or the [8.141] are al most the same, and prop -erly de scribe the ex per i men tal tem per a ture and den sity de pend en cies of the dif fu sion co ef -fi cients, even if about 30% smaller than ob tained us ing NMR spin-echo tech niques.

Among the pure sol vents we have treated so far, other avail able data re gard the cal cu -la tion of the self dif fu sion co ef fi cient (D) for liq uid eth a nol at dif fer ent tem per a tures.115d

The D pa ram e ter was ob tained from the long-time slope of the mean-square dis place mentsof the cen ter of mass: ex per i men tal changes of D over the 285-320K range of tem per a tureswere ac cept ably re pro duced by mo lec u lar dy nam ics sim u la tions.


Shear viscosityTo end this sec tion, we would like to re port an ex am ple of cal cu la tion of the shear vis cos ityagain tak ing as an ex am ple pure ac e tone.

The vis cos ity can be com puted by us ing mo lec u lar dy nam ics sim u la tions from thevirial form of the mo lec u lar pres sure ten sor P, which can be rep re sented as a sum of fourcon tri bu tions:

PVp p

mP V P V P Vi i

iS

i

N

R C= + + +=∑

1

[8.142]

where V is the vol ume of the cell, N the num ber of mol e cules, pi and mi the mo men tum andmass of mol e cule i.

The first term in eq. [8.142] is the ki netic con tri bu tion, while PS is the short-range po -ten tial in ter ac tion con tri bu tion, PR the re cip ro cal-space por tion of the pres sure ten sor, andPC a cor rec tion ten sor term.

From the xy el e ments of the time-av er aged mo lec u lar pres sure ten sor [8.142] and theap plied shear (or strain) rate γ ∂ ∂= v yx / , the vis cos ity η can be com puted us ing the con sti tu -tive re la tion:

ηγ

= −−P Pxy yx

2[8.143]

the cal cu lated val ues for ac e tone (ex trap o lated data) are in rea son able agree ment with ex -per i men tal data, as well as re sults for ac e tone/meth a nol and ac e tone/wa ter mix tures.117

8.9 LIQUID SURFACESUn til now our task has been rel a tively easy, as we have only con sid ered the gen eral meth od -olog i cal as pects of the com pu ta tional de scrip tion of liq uids.

To pro ceed fur ther, we have to be more spe cific, en ter ing into de tails of spe cific prob -lems, all re quir ing the in tro duc tion and the char ac ter iza tion of dif fer ent prop er ties, and ofthe op por tune meth od olog i cal (com pu ta tional) tools to treat them. We also note that, as ad -di tional com pli ca tion, it can hap pen that things change con sid er ably when the full range ofmac ro scopic vari ables (tem per a ture, pres sure, vol ume) is con sid ered, and when other com -po nents are added to the liq uid sys tem.

Not much ef fort is need to con vince read ers of the over whelm ing com plex ity of thetask we are here con sid er ing. Let us take a sim ple pure liq uid: to span the range of thethermodynamical vari ables means to con sider sys tems in which the liq uid is in pres ence ofthe solid, of the va por, in clud ing super criti cal as well as super-cooled states, and all the re -lated phe nom ena, as eb ul li tion, va por iza tion, con den sa tion and freez ing.

All states and all dy nam i cal pro cesses con nect ing states are of in ter est for chem is try,chem i cal en gi neer ing, and phys i cal chem is try. Some prop er ties can be de fined for all thestates (such as the ther mo dy namic ba sic quan ti ties); oth ers are spe cific for some sit u a tions,such as the sur face ten sion. But also for prop er ties of gen eral def i ni tion, the tech niques touse have of ten to be very dif fer ent. It is not pos si ble, for ex am ple, to use the same tech nique


to study the tem per a ture de pend ence of the Helmholtz free en ergy in a bulk liq uid or in thesame liq uid un der the form of a fog near con den sa tion.

Things be come more com plex when a sol ute is added to the liq uid, and more and morecom plex when the sys tem con tains three or more com po nents.

We are not sure whether any case oc cur ring in this field has been sub jected to the o ret i -cal anal y sis and mod el ing; how ever, the num ber of prob lems con sid ered by the o ret i cal re -search is very large, and there are mod els, gen er ally based on the use of mo lec u larpo ten tials, that can give hints, and of ten use ful com pu ta tional rec i pes.

These re marks, which are triv ial for ev ery reader, with even a min i mal ex pe ri ence inwork ing with sol vents, have been added here to jus tify our choice of not try ing to give asche matic over view, and also to jus tify the ex tremely sche matic pre sen ta tion of the uniquespe cific more ad vanced sub ject we shall here con sider, the in ter fa cial prop er ties of liq uids.

8.9.1 THE BASIC TYPES OF LIQUID SURFACES

The sur face of a liq uid is of ten rep re sented as a very small frac tion of the liq uid spec i men,and it may be ne glected. This state ment is not of gen eral va lid ity, how ever, and of ten thesur face plays a role; in some cases, the dom i nant role. To be more spe cific, it is con ve nientto con sider a clas si fi ca tion of liq uid sur faces, start ing from a crude clas si fi ca tion, and in tro -duc ing more de tails later.

The ba sic clas si fi ca tion can be ex pressed as fol lows:• Liquid/gas surfaces;• Liquid/liquid surfaces;• Liquid/solid surfaces.Liq uid/gas sur faces may re gard a sin gle sub stance pres ent in both phases, liq uid and

va por at equi lib rium or out of equi lib rium, but they may also re gard a liq uid in the pres enceof a dif fer ent gas, for ex am ple, air (in prin ci ple, there is al ways a given amount of sol vent inthe gas phase). In the pres ence of a dif fer ent gas, one has to con sider it in the de scrip tion ofthe liq uid, both in the bulk and in the in ter face. In many cases, how ever, the role of the gas ismar ginal, and this is the rea son why in many the o ret i cal stud ies on this in ter face the gas isre placed by the vac uum. This is a very un re al is tic de scrip tion for liq uids at equi lib rium, butit sim ply re flects the fact that in sim pli fy ing the model, sol vent-sol ute in ter ac tions may besac ri ficed with out los ing too much in the de scrip tion of the in ter fa cial re gion.

The po ten tials used to de scribe these in ter faces are those used to de scribe bulk liq uids. The dif fer ences stay in the choice of the thermodynamical en sem ble (grand ca non i cal en -sem bles are of ten nec es sary), in the bound ary con di tions to be used in cal cu la tions, and inthe ex plicit in tro duc tion in the model of some prop er ties and con cepts not used for bulk liq -uids, like the sur face ten sion. Much could be said in this pre lim i nary pre sen ta tion of liq -uid/gas in ter faces, but we post pone the few as pects we have de cided to men tion, be causethey may be treated in com par i son with the other kind of sur faces.

Liq uid/liq uid sur faces re gard, in gen eral, a cou ple of liq uids with a low mis ci bil ity co -ef fi cient, but there are also cases of one-com po nent sys tems pre sent ing two dis tinct liq uidphases, for which there could be in ter est in ex am in ing the in ter fa cial re gion at the co ex is -tence point, or for sit u a tions out of the thermodynamical equi lib rium.

If the sec ond com po nent of the liq uid sys tem has a low mis ci bil ity with the first, andits mo lar ra tio is low, it is easy to find sit u a tions in which this sec ond com po nent is ar rangedas a thin layer on the sur face of the main com po nent with a depth re duc ing, in the op por tunecon di tions, to the level of a sin gle mol e cule. This is a sit u a tion of great prac ti cal im por tance

8.9 Liquid Surfaces 461

to what we experience in our ev ery day life (petrol on the sur face of wa ter); how ever, it hasto be re lated to a finer clas si fi ca tion of liq uid in ter faces we shall con sider later.

Once again the po ten tials used for liq uid/liq uid sur faces are in gen eral those used forbulk liq uids even tu ally with the in tro duc tion of a change in the nu mer i cal val ues of the pa -ram e ters.118 The dif fer ence from the bulk again re gards the thermodynamical en sem ble, thebound ary con di tions for the cal cu la tions and the use of some ad di tional con cepts.

The liq uid/solid sur faces may be of very dif fer ent types. They can be clas si fied ac -cord ing to the elec tri cal na ture of the solid as:

• Liquid/conductor• Liquid/dielectricThe pres ence of mo bile elec trons in met als and other solid sub stances (to which some

liq uids may be added, like mercury119), in tro duces in the mod el ling of the liq uid in ter facenew fea tures not pres ent in the pre vi ously-quoted in ter faces nor in the bulk liq uids. Formany phe nom ena oc cur ring at such in ter faces there is the need for tak ing into ex plicit ac -count elec trons flow ing from one phase to the other, and of re lated elec tron trans fers oc cur -ring via re dox mech a nisms. The in ter ac tion po ten tials have to be mod i fied and ex tendedac cord ingly.120

The good elec tri cal con duc tiv ity of the solid makes more size able and ev i dent the oc -cur rence of phe nom ena re lated to the pres ence of elec tric po ten tials at the in ter face (sim i larphe nom ena also oc cur at in ter faces of dif fer ent type, however121). A well-known ex am ple is the dou ble layer at the liq uid side of an elec tro lyte/elec trode sur face. For the dou ble layer,ac tu ally there is no need of in ter ac tion po ten tials of spe cial type: the changes in themodelling mainly re gard the bound ary con di tions in the sim u la tion or in the ap pli ca tion ofother mod els, of con tin uum or in te gral equa tion type.

It is clear that at ten tion must be paid to the elec tro static part of the in ter ac tion. Modelsfor the solid part sim i lar to those used for bulk liq uids play a mar ginal role. There are mod els of LJ type, but they are mainly used to in tro duce in the solid com po nent the coun ter part ofthe mo lec u lar mo tions of mol e cules, al ready pres ent in the de scrip tion of the liq uid (vi bra -tions of the nu clei: phon ons, etc.).

The elec tro static prob lem may be treated with tech niques sim i lar to those we haveshown for the con tin uum elec tro static model for so lu tions but with a new bound ary con di -tion on the sur face, e.g., V=con stant. In this case, a larger use is made (es pe cially for pla narsur faces) of the im age method,122 but more gen eral and pow er ful meth ods (like the BEM)are gain ing im por tance.

An al ter na tive de scrip tion of the metal that can be prof it ably cou pled to the con tin uumelec tro static ap proach is given by the jellium model. In this model, the va lence elec trons ofthe con duc tor are treated as an in ter act ing elec tron gas in the neu tral iz ing back ground of theav er aged dis tri bu tion of the pos i tive cores (some dis crete ness in this core dis tri bu tion maybe in tro duced to de scribe atomic mo tions). The jellium is de scribed at the quan tum level bythe Den sity Func tional The ory (DFT) and it rep re sents a solid bridge be tween clas si cal andfull QM de scrip tions of such in ter faces:123 fur ther im prove ments of the jellium which in tro -duce a dis crete na ture of the lattice124 are now of ten used.125

The liq uid/di elec tric sur faces ex hibit an even larger va ri ety of types. The di elec tricmay be a reg u lar crys tal, a microcrystalline solid, a glass, a poly mer, an as sem bly of mol e -cules held to gether by forces of dif fer ent type.


The po ten tials in use for crys tals and those used for co va lently bound sys tems are cog -nate with those used for liq uids, so it is not com pul sory to open here a di gres sion on thistheme. We have al ready quoted the MM meth ods gen er ally used for sol ids of mo lec u lartype; for more gen eral classes of com pounds, a va ri ety of ‘atom-atom’ po ten tials have beengiven.126 The dif fer ence with re spect to the bulk is mainly due, again, to dif fer ent ways ofhan dling the op por tune bound ary con di tions. It is ev i dent that the tech niques used to de -scribe a wa ter-pro tein in ter face greatly dif fer, in mi cro scopic de tail, time scale, etc., to those used to de scribe the sur faces be tween a liq uid and a reg u lar crys tal sur face.

For many prob lems, the solid sur face in con tact with the liq uid may be held fixed: thischar ac ter is tic has been am ply ex ploited in sim u la tion stud ies. We quote here two prob lemsfor which this sim pli fi ca tion is not pos si ble: prob lems in which the struc ture of the solid (orsolid-like) com po nent is sub jected to conformational changes, and prob lems in which a pro -gres sive ex pan sion (or re duc tion) of the solid phase plays the main role.

To the first case be long prob lems ad dress ing the struc ture of the in ter fa cial zone in thecase of biopolymers (e.g., pro teins) or sim i lar co va lently bound mas sive mol e cules, and or -dered as sem blies of mol e cules held to gether by strong non-co va lent in ter ac tions (as mem -branes or sim i lar struc tures). In both cases, the liq uid and the solid com po nent of thesys tems are gen er ally treated on the same foot ing: in ter ac tion force fields are used for bothcom po nents, in tro duc ing, where ap pro pri ate, dif fer ent time scales in the mo lec u lar dy nam -ics cal cu la tions.

The sec ond case mainly re gards the prob lem of crys tal growth or dis so lu tion in a liq -uid sys tem (pure liq uid or so lu tion). A mo bile sur face of this type is not easy to treat withstan dard meth ods: it is better to pass to more specialistic ap proaches, for which there is theneed of in tro duc ing a dif fer ent math e mat i cal treat ment of the (mo bile) bound ary sur facecon di tions. This is a sub ject that we can not prop erly treat here.

8.9.2 SYSTEMS WITH A LARGE SURFACE/BULK RATIO

To com plete this quick pre sen ta tion of the three main types of liq uid sur faces, it is con ve -nient to con sider sys tems in which the in ter fa cial por tion of the liq uid is no tice ably largerthan in nor mal liq uids.

Some among them are de noted in lit er a ture as con strained liq uids, oth ers as dis persedsys tems. This clas si fi ca tion is not universal, and it does not take in ac count other cases.

A typ i cal ex am ple of con strained liq uid is that pres ent in pores or cap il lar ies. The sur -face is of liq uid/solid type, and the same re marks about in ter ac tion po ten tials and com pu ta -tional meth ods we have al ready offered may be ap plied here. More at ten tion must be paid incom pu ta tions of in ter ac tions among sep a rate por tions of the liq uid in ter face. In the slit porecase (i.e., in a sys tem com posed of two par al lel plane solid sur faces, with a thin amount ofliq uid be tween them), there may be an in ter fer ence be tween the two dis tinct sur faces, moreev i dent when the thick ness of the liq uid is small, or when there is an elec tri cal po ten tial be -tween the two faces. The slit pore model is ex ten sively used to de scribe capillarity prob -lems: in the most usual cy lin dri cal cap il lary types there is a ra dial in ter ac tion sim i lar insome sense to the in ter ac tion with a sin gle solid sur face.

Pores of dif fer ent shape are pres ent in many solid ma te ri als: zeolites and otheraluminosilicates are out stand ing ex am ples. When the pore is very small there is no moredis tinc tion be tween su per fi cial and bulk mol e cules of the liq uid. Na ture (and hu man in ge -nu ity) of fers ex am ples over the com plete range of sit u a tions. For all the sys tems we havehere in tro duced (quite im por tant in prin ci ple and for prac ti cal rea sons), there is no need of


in tro duc ing im por tant changes in the stan dard the o ret i cal ap proach with re spect to that usedfor other liq uid sys tems. Com puter sim u la tions are the meth ods more largely used to de -scribe pores and mi cro-pores in mo lec u lar sieves of var i ous na ture.

A dif fer ent type of con fined liq uid is given by a thin layer spread on a sur face. The dif -fer ence with re spect to the mi cro-pore is that this is a three-com po nent sys tem: the sur facemay be a solid or a liq uid, while the sec ond bound ary of the layer may re gard an other liq uidor a gas. An ex am ple is a thin layer liq uid on a metal, in the pres ence of air (the prodrome ofmany phe nom ena of cor ro sion); a sec ond ex am ple is a layer on a sec ond liq uid (for ex am -ple, oil on wa ter). The stan dard ap proach may be used again, but other phe nom ena oc cur -ring in these sys tems re quire the in tro duc tion of other con cepts and tools: we quote, forex am ple, the wet ting phe nom ena, which re quire a more ac cu rate study of sur face ten sion.

The thin layer in some case ex hib its a con sid er able self-or ga ni za tion, giv ing rise to or -ga nized films, like the Blodgett-Langmuir films, which may be com posed by sev eral lay ers, and a va ri ety of more com plex struc tures end ing with cel lu lar mem branes. We are here atthe bor der line be tween liq uid films and co va lently held lam i nar struc tures. Ac cord ing to the de gree of ri gid ity, and to the inter-unit per me ation in ter ac tion, the com pu ta tional modelmay tune dif fer ent po ten tials and dif fer ent ways of per form ing the sim u la tion.

An other type of con fined liq uid re gards drops or mi cro-drops dis persed in a me dium.We are here pass ing from con fined liq uids to the large realm of dis persed sys tems, someamong which do not re gard liq uids.

For peo ple in ter ested in liq uids, the next im por tant cases are those of a liq uid dis persed into a sec ond liq uid (emul sions), or a liq uid dis persed in a gas (fogs), and the op po site casesof a solid dis persed in a liq uid (colloids) or of a gas dis persed in a liq uid (bub bles).

Sur face ef fects on those dis persed sys tems can be de scribed, us ing again the stan dardap proach de vel oped in the pre ced ing sec tions: intermolecular po ten tials, com puter sim u la -tions (ac com pa nied where con ve nient by in te gral equa tion or con tin uum ap proaches).There is not much dif fer ence with re spect to the other cases.

A re mark of gen eral va lid ity but par tic u lar for these more com plex liq uid sys temsmust be ex pressed, and strongly un der lined.

We have ex am ined here a given line of re search on liq uids and of cal cu la tions of per ti -nent physico-chem i cal prop er ties. This line starts from QM for mal de scrip tion of the sys -tems, then spe cial izes the ap proach by de fin ing intermolecular po ten tials (two- ormany-body), and then it uses such po ten tials to de scribe the sys tem (we have paid more at -ten tion to sys tems at equi lib rium, with a mod er ate ex ten sion to non-equi lib rium prob lems).

This is not the only re search line con ceiv able and used, and, in ad di tion, it is not suf fi -cient to de scribe some pro cesses, among them, those re gard ing the dis persed sys tems wehave just con sid ered.

We ded i cate here a lim ited space to these as pects of the o ret i cal and com pu ta tional de -scrip tion of liq uids be cause this chap ter spe cif i cally ad dresses in ter ac tion po ten tials and be -cause other ap proaches will be used and de scribed in other chap ters of the Hand book.Sev eral other ap proaches have the QM for mu la tion more in the back ground, of ten nevermen tioned. Such mod els are of a more clas si cal na ture, with a larger phenomenologicalchar ac ter. We quote as ex am ples the mod els to de scribe light dif frac tion in dis or dered sys -tems, the clas si cal mod els for evap o ra tion, con den sa tion and dis so lu tion, the trans port ofthe mat ter in the liq uid. The num ber is fairly large, es pe cially in pass ing to dy nam i cal and


out of equi lib rium prop er ties. Some mod els are old (dat ing be fore the in tro duc tion of QM),but still in use.

A more de tailed de scrip tion of mo lec u lar in ter ac tions may lead to re fine ments of suchmod els (this is cur rently be ing done), but there is no rea son able pros pect of re plac ing themwith what we have here called the stan dard ap proach based on po ten tials. The stan dard ap -proach, even when used, may be not suf fi cient to sat is fac torily de scribe some phe nom ena.We quote two ex am ples drawn from the pre sen ta tion of liq uid sur faces in dis persed sys -tems: bub bles and solid small par ti cles.

The de scrip tion of the energetics of a sin gle bub ble, and of the den sity in side andaround it, is not suf fi cient to de scribe col lec tive phe nom ena, such as eb ul li tion or sep a ra tion of a bub ble layer. The de scrip tion of a sin gle bub ble is at a low level in a lad der of mod els,each in tro duc ing new con cepts and en larg ing the space (and time) scale of the model.

The same holds for small solid par ti cles in a liq uid. There is a whole sec tion of chem is -try, col loi dal chem is try, in which our stan dard ap proach gives more ba sic el e ments, butthere is here, again, a lad der of mod els, as in the pre ced ing ex am ple.

Our stan dard model ex tends its range of ap pli ca bil ity to wards these more com plexlev els of the ory. This is a gen eral trend in phys ics and chem is try: mi cro scopic ap proaches,start ing from the study of the in ter ac tion of few el e men tary par ti cles, are pro gres sivelygain ing more and more im por tance over the whole range of sci ences, from bi ol ogy to cos -mol ogy, pass ing through en gi neer ing and, in par tic u lar, chem is try. In the case of liq uids,this ex ten sion of mi cro scop i cal mod els is far from cov er ing all the phe nom ena of in ter est.

8.9.3 STUDIES ON INTERFACES USING INTERACTION POTENTIALS

To close this sec tion on liq uid sur faces, lim ited to a rapid ex am i na tion of the sev eral types of sur face of more fre quent oc cur rence, we re port some gen eral com ments about the use of thestan dard ap proach on liq uid sur faces.

The first in for ma tion com ing from the ap pli ca tion of the method re gards the den sitypro file across the in ter face. Den sity may be as sumed to be con stant in bulk liq uids at theequi lib rium, with lo cal de vi a tions around some sol utes (these de vi a tions be long to the fam -ily of cybotactic ef fects, on which some thing will be said later). At each type of liq uid sur -face there will be some de vi a tions in the den sity, of ex tent and na ture de pend ing on thesys tem.

A par tic u lar case is that of mo bile sur faces, i.e., liq uid/gas and liq uid/liq uid sur faces.Hav ing liq uids a mo lec u larly grained struc ture, there will al ways be at a high level of spa tial de vi a tion a lo cal de vi a tion of the sur face from pla nar ity. If there are no other ef fects, this de -vi a tion (some times called cor ru ga tion, but the term is more con ve nient for other cases, likeliq uid/solid sur faces, where cor ru ga tion has a more per ma nent sta tus) av er ages to zero.

Entropic forces are re spon si ble for other ef fects on the den sity pro file, giv ing or i gin ata large scale to a smooth be hav ior of the den sity pro file, con nect ing the con stant val ues ofthe two bulk den si ties. The most nat u ral length unit in liq uids is the bulk cor re la tion length, ξ , i.e., the dis tance at which the pair cor re la tion gAB(r) has de cayed to 1. For bulk wa ter ξ isaround 5-6 Å. This large scale be hav ior hides dif fer ent lo cal be hav iors, that range amongtwo ex tremes, from cap il lary waves to van der Waals re gime.127 Cap il lary waves cor re spond to a sharp bound ary (cor ru ga tion apart) with “fin gers” of a liq uid pro trud ing out of the bulk.Cap il lary waves may hap pen both at liq uid and gas sur faces, in the for mer, in a more sta blesta tus (i.e., with a large mean life) than in the lat ter.


The van der Waals re gime lo cally cor re sponds to the large scale en ergy pro file, with asmooth shape.

The be hav ior of the den sity pro file largely de pends on the chem i cal na ture of the com -pos ite sys tem. Some phenomenological pa ram e ters, like mu tual sol u bil ity, may give a good guess about the prob a ble en ergy pro file, but “sur prises” are not rare events. It may be addedthat the den sity pro file is sen si tive to the qual ity of the po ten tial and the method us ing thepo ten tials. It is not clear, in our opin ion, if some claims of the oc cur rence of a thin void (i.e.,at very low den sity) layer sep a rat ing two liq uid sur faces is a real find ing or an ar ti fact due tothe po ten tials. Old re sults must be con sid ered with cau tion, be cause com pu ta tions donewith old and less pow er ful com put ers tend to re duce the size of the spec i men, thus re duc ingthe pos si bil ity of find ing cap il lary waves. The di men sion of the sim u la tion box, or othersim i lar lim it ing con straints, is a crit i cal pa ram e ter for all phe nom ena with a large lengthscale (this is true in all con densed sys tems, not only for sur faces).

The den sity pro file for liq uid/solid sur faces has been firstly stud ied for hard spheres on hard walls, and then us ing sim i lar but a bit more de tailed mod els. It is easy to mod ify the po -ten tial pa ram e ters to have an in cre ment of the liq uid den sity near the wall (ad he sion) or adec re ment. Both cases are phys i cally pos si ble, and this ef fect plays an im por tant role incapillarity stud ies.

A cor ru ga tion at the solid sur face can be achieved by us ing po ten tials for the solidphase based on the atomic and mo lec u lar con sti tu tion. This is al most com pul sory for stud ies on sys tems hav ing a large in ter nal mo bil ity (biomolecules are the out stand ing ex am ple), but it is also used for more com pact solid sur faces, such as met als. In the last quoted ex am plethe dis crete ness of the po ten tial is not ex ploited to ad dress cor ru ga tion ef fects but rather toin tro duce mo bil ity (i.e., vi bra tions. or phon ons) in the solid part of the sys tem, in par al lel tothe mo bil ity ex plic itly con sid ered by the mod els for the liq uid phase.

An other type of mo bil ity is that of elec trons in the solid phase: this is partly de scribedby the con tin uum elec tro static ap proach for di elec trics and con duc tors, but for met als (themost im por tant con duc tors) it is better to re sort to the jellium model. A QM treat ment ofjellium mod els per mits us to de scribe elec tric po lar iza tion waves (polarons, solitons) andtheir mu tual in ter play with the liq uid across the sur face.

Jellium, and the other con tin uum de scrip tions of the solid, have the prob lem of ex actly de fin ing the bound ary sur face. This is the an a log of the prob lem of the def i ni tion of the cav -ity bound ary for sol utes in bulk sol vents, oc cur ring in con tin uum sol va tion meth ods (seeSec tion 8.7.3). The only dif fer ence is that there are more ex per i men tal data for sol utes thanfor liq uid/solid sur faces to have hints about the most con ve nient mod el ing. The few ac cu -rate ab-in itio cal cu la tions on liq uid/metal sys tems are of lit tle help, be cause in or der to reach an ac cept able ac cu racy, one is com pelled to re duce the solid to a small clus ter, too small tode scribe ef fects with a large length scale.

The dis tur bances to the bulk den sity pro file are of ten lim ited to a few sol vent di am e -ters (typ i cally 2-3), but the ef fect may be larger, es pe cially when ex ter nal elec tric fields areap plied.

The sec ond in for ma tion de rived from the ap pli ca tion of our meth ods to sur faces re -gards the pre ferred ori en ta tion of liq uid mol e cules near the sur face. To treat this point thereis the need of a short di gres sion.

In bulk ho mo ge neous liq uids at the equi lib rium ori en ta tion, the lo ca tion of a spe cificmol e cule is im ma te rial. We need to know the orientational modes (of ten re duced to li bra -


tions) of the com po nents of the liq uid sys tem to write the ap pro pri ate sta tis ti cal par ti tionfunc tion. In the case of sol va tion en er gies, the ro ta tional com po nent of the fac tor ized sol utepar ti tion func tion gives a not neg li gi ble con tri bu tion to ∆Gsolv . The translational com po nentof the sol ute par ti tion func tion can be fac tor ized apart, as first sug gested by Ben-Naim,128

and in cluded in the cratic com po nent of ∆Gsolv with a small cor rec tion (the cratic com po nent is the nu mer i cal fac tor tak ing into ac count the re la tion ships be tween stan dard states in achange of phase).

In the ho mo ge neous bulk liq uid we have to ex plic itly con sider ori en ta tions for dy nam -i cal re lax ation ef fects. The same holds for trans la tion in other dy nam i cal prob lems (dif fu -sion, trans port, etc.).

Com ing back to the static prob lems, it turns out that near the lim it ing sur face, the freeen ergy of the com po nents of the liq uid also de pends on po si tions and ori en ta tions. A quan -tity like ∆Gsolv for the trans fer of M in the sol vent S in the bulk must be re placed by ∆Gsolv (r, Ω) spec i fy ing po si tion and ori en ta tion of M with re spect to the sur face of S.

This ef fect must not be con fused with the cybotactic ef fects we have men tioned, norwith the hole in the sol ute-sol vent cor re la tion func tion gMS(r) (see Fig ure 8.5). The hole inthe ra dial cor re la tion func tion is a con se quence of its def i ni tion, cor re spond ing to a con di -tional prop erty, namely that it gives the ra dial prob a bil ity dis tri bu tion of the sol vent S, when the sol ute M is kept at the or i gin of the co or di nate sys tem. Cybotactic ef fects are re lated tochanges in the cor re la tion func tion gMS(r) (or better gMS(r, Ω)) with re spect to a ref er ence sit -u a tion. Sur face prox im ity ef fects can be de rived by the anal y sis of the gMS(r, Ω) func tions, or di rectly com puted with con tin uum sol va tion meth ods. It must be re marked that theobtention of gMS(r) func tions near the sur face is more dif fi cult than for bulk ho mo ge neousliq uids. Re li able de scrip tions of gMS(r, Ω) are even harder to reach.

Orientational pref er ences are the or i gin of many phe nom ena, es pe cially when at thesur face there are elec tric charges, mo bile or fixed. Out stand ing ex am ples oc cur in bi o log i -cal sys tems, but there is a large lit er a ture cov er ing other fields. A rel a tively sim ple case thathas been abun dantly stud ied is the elec tric po ten tial dif fer ence across the sur face.129

We have not con sid ered yet the be hav ior of sol utes near the in ter face. The spe cial, butquite im por tant, case of the charged com po nents of a salt so lu tion near a charged bound aryis widely known, and it is sim ply men tioned here.

More gen er ally, the be hav ior of sol utes largely de pends on the chem i cal com po si tionof the sys tem. Small ions are re pelled at the air/gas in ter face, hy dro pho bic mol e cules are at -tracted to the in ter face. Even the pres ence of a sim ple CH3 group has some ef fect: the air in -ter face of di lute so lu tions of meth a nol is sat u rated in CH3OH over a size able mo lar frac tionrange.130 At the liq uid/liq uid in ter face be tween wa ter and hy dro car bon sol vents, sol uteswith a po lar head and an alkylic chain tend to find an equi lib rium po si tion, with the po larhead in wa ter, at a dis tance from the sur face de pend ing on the chain length.131

All the quoted re sults have been ob tained at the static level, em ploy ing a mean forcepo ten tial, ob tained in some cases with in ex pen sive con tin uum mod els; in other cases, withcom puter de mand ing sim u la tions of the free en ergy pro file.

Sim u la tions can give more de tails than con tin uum meth ods, which ex ploit a sim pli -fied and av er aged model of the me dium. Among the sim u la tion re sults we quote the rec og -ni tion that small cat ions, ex hib it ing strong in ter ac tions with wa ter mol e cules, tend to keepbound the first sol va tion layer when forced to pass from wa ter to an other sol vent. Anal o -gous ef fects are un der ex am i na tion for neu tral po lar sol utes at wa ter/liq uid in ter faces in case


the bulk sol u bil ity of the two part ners is not sym met ric (a typ i cal ex am ple is the wa -ter/octanol sys tem).132

We have here men tioned the pro cess of trans port of a sol ute from a liq uid phase to an -other. This is a typ i cal dy namic pro cess for which static de scrip tions based on mean forcessim ply rep re sent the start ing-point, to be com pleted with an ex plicit dy nam i cal treat ment.Mo lec u lar dy nam ics is the cho sen tool for these stud ies, which are quite ac tive and promiseil lu mi nat ing re sults. MD sim u la tions also are the main ap proach to study an other im por tanttopic: chem i cal re ac tions at the sur face.133

The study of the var i ous as pects of chem i cal re ac tions in bulk liq uids, from thethermodynamical bal ance of the re ac tion, to the mech a nis tic as pects, to the re ac tion rates,rep re sents now a days a well ex plored field, in which sev eral ap proaches may be used. Thecom plex ity of chem i cal sys tems does not per mit us to con sider this prob lem com pletely set -tled un der the meth od olog i cal and com pu ta tional point of view, but im por tant ad vanceshave been achieved.

Re ac tions at the sur face are a fron tier field, and the same re marks hold for re ac tions indis persed liq uid sys tems. In ter ac tion po ten tials, sim u la tions and other meth ods will findhere re newed stim uli for more pre cise and pow er ful for mu la tions.

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Interactions in solvents and solutions

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