XVIII. On the Relation Between the Spatial Electron Densityand the Conformational Properties of Molecular Systems

A. T.HAGLERChemical Physics Department, The Weizmann Institute of Science, Rehovot, Israel

Abstract. The increasing power of computer simulation experiments for molecularsystems is discussed. These include such apparently disparate calculations as MonteCarlo simulation of solvent structure in biological systems, conformational analysis andfolding of oligopeptide chains, and lattice-energy calculations and the effect of latticeforces on molecular geometry. All such calculations have in common the requirementof an analytic expression of the energy of the system as a function of the inter- andintramolecular coordinates of interest.

The nature and derivation of these force fields and their relation to the spatialelectron densities in relevant molecular systems is reviewed. The method of thederivation of intermolecular force fields from crystal data of amides is outlined. It isshown that information which may be obtained from ab-inirio molecular orbital orX-ray diffraction studies, population analysis, and total and difference electron densitymaps, may be used in conjunction with the crystal data both to verify (or test)assumptions made in the derivation and to help in the formulation of the models onwhich the analytical expressions for the energy are based. Finally, the relation betweenintramolecular forces and the electron distribution as represented in differenceelectron density maps is discussed. Several examples involving carbonyl compoundsare presented, in which changes in the molecular geometry, intramolecular forces andvibrational frequencies on protonation and substitution are discussed both in terms of"valence or Urey-Bradley" type interactions and the changes which occur in thedifference electron density maps.

The intramolecular energy has long been known to beexpressible approximately, as the sum of terms which arecommon to whole families of molecules. These termsdepend on the types of interacting atoms or groups, andon the molecular internal coordinates and interatomicdistances. Similarly, the energy of condensed molecularsystems is given approximately by the sum of inter­molecular interactions composed of the same atom-atompotentials which affect intramolecular nonbonded in­teractions. Theoretical as well as empirical estimates ofthese functions have made their application possible inmany areas of physical chemistry, organic chemistry andmolecular biology. Due to the availability of fast compu­ters, it has been possible in recent years to calculate alarge number of molecular and crystalline properties of awide variety of systems. Such calculations serve, on theone hand, as a means to analyze or predict the behaviourof molecular systems of interest, and, as such, they maybe considered as a kind of "computer simulation experi­ment". On the other hand they may lead to a better

Table I. Current Computer Applications of Molecular EnergyFunctions

Crystals1. Crystal packing analysis2. Reactive sites in crystals3. Effect of crystal forces on molecular conformation4. Spontaneous resolution of chiral moleculesS. Crystal dislocations

Conformations of single moleculesI. Conformations of small organic molecules and conforma­

tional equilibria, including strained rings and over-crowdedstructures

2. Conformational reaction paths3. Structural determination with gas-phase electron diffraction

Energy1. Strain energies2. Lattice energies3. Rotational barriers4. Conformational barriers

Biological molecules1. Conformation of biological molecules2. Enzyme-substrate interactions3. Protein refinement4. Protein foldingS. Protein-solvent interactions6. Protein-denaturant interactions7. Protein-nucleic acid interactions

Monte Carlo and molecular dynamic simulations1. Liquid structure and properties - especially water and

organic liquids2. Hydrophobic bonding3. Solvent effect on conformation4. Chiral recognition - gas liquid chromatography of peptides

Vibrational phenomena1. Vibrational frequencies and normal modes2. Effect of vibrational entropy on conformational equilibria3. Effect of amplitudes of vibration on electron diffraction and

crystal structure4. Lattice dynamicsS. Non-bonded effects on vibrational frequencies - especially

in strained molecules

Polymers1. Chain statistics, including Monte Carlo calculation of charac­

teristic ratio, distribution of end-to-end distances, etc.

IsraelJoumal ofChemistry VoL 16 1977 pp.202-212

xr

a

b

203

understanding of the nature of molecular forces, and maybe used as a powerful tool of theoretical chemistry. Thescope of this field may be appreciated from Table 1 whichlists some of the applications of "computer simulationexperiments" which are of current interest.

Examples of the results obtained from "computersimulation experiments" on three rather different: appli­cations with which we have been concerned are given inFigs. 1-3. Figure 1 represents a simulation of the foldingof a polypeptide chain from an extended structure into astable globular structure. In Fig. 2 we show an exampleof the results of a lattice energy minimization in whichthe effect of crystal forces on molecular conformation("conformational polymorphism") was studied. Finally,Fig. 3 shows some energy distributions of water indifferent environments in the triclinic lysozyme crystaland in bulk water, obtained from a Monte Carlo simula­tion. These three examples, as well as all the applicationslisted in Table 1, have in common the requirement of aforce field which adequately represents the energy sur­face of the system of interest as a function of itsconfigurational coordinates.

In this paper we discuss some aspects of the nature ofthese force fields, their relation to the spatial electrondensity in molecules, and applications to several crystaland molecular systems. In the first part of the paper wediscuss the nature of intermolecular force fields currentlybeing used in applications to dense systems, the approxi­mations implicit in these force fields and how they areobtained. Since the energetics of molecular interactionsare intimately related to the electron densities in theinteracting molecules, the approximations made in theseanalytical representations are discussed in terms of theelectronic structure of model compounds.

In the second part of the paper we turn our attentionto intramolecular properties. Here we focus on therelation between the electron distribution as representedin electron difference-density maps and intramolecularforces. Several examples are presented of carbonyl com­pounds relating changes in the electron density on sub­stitution and protonation, to changes in intramolecularforces, molecular geometry and vibrational frequencies.

Fig. 1. Three stages in the simulation of the folding of apolypeptide chain from an extended to a native like conforma­tion (A. T. Hagler and B. Honig, Proc. Nat. Acad. Sci. (USA),in press).

~ <Lat \~~).<:. .~-, .' f'7"}l.;.' ()~t</ ~/~_ {){h "~ '-, ~n ,)<{

-'\,..{~ " I' , ....

-' 'J . I',I' ,\_, \. I

, " \.,0 '{ <t..1.1"'\ .-{)_'-.)7'.vc{ ~7'..v.-,(.7 r<ir-I' "

Fig. 2. "Conformational Polymorphism". The effect of crystal forces on molecular confor­mation. (a) A stereo view of the triclinic crystal structure of N-(p-chlorobenzylidine)-p­chloroaniline in which the molecule exhibits a planar conformation. (b) A stereo view of theorthorhombic form of the same molecule in which the exocyclic torsion angles are twisted by- 25°in opposite directions (J. Bernstein and A. T. Hagler, J. Am. Chern. Soc., in press).

Hagler I Electron Density and Conformational Properties

-30.0 -20.0 -10.0 0.0 10.0

ENERGY KCALIMOL

Fig. 3a. Energy distribution of water in the interstices of aprotein crystal (A. T. Hagler and J. Moult, Nature (London), inpress).

204

~a

~

~

....~z

::!~

::>;!!a: 'I~

<I>

Cl

>-c:>a: ......z ......

:;

- TDTIL EI£IllIT• • WATEJHjflTEfl E1EAOT

-- WATEfI-l'RlTEIN EI£IllIY

Various sources have been used for deriving potentialfunctions for non-bonded interactions;"? however, prob­lems in these methods" have led to the recognition thatone of the best techniques for deriving and testingintermolecular forces is to use the information inherentin the crystal structures and lattice energies of modelcompounds. We have recently used these propertiesalong with molecular dipole moments to derive anintermolecular force field of the form given in (1) foramides.'" Nine parameters were optimized over 81 ob­servables including lattice constants, energies and dipolemoments. The amide crystals used in the optimizationalong with the hydrogen bond lengths observed in thesecrystals are given in Table 2.

Table 2. The Hydrogen Bond Geometries in the AmideCrystals Used in the Optimization"

180°-NH· .. O H· .. O=C

H· .. O(A) (deg) (deg)

2.02 29 1551.94 4 1191.90 2 1211.97 13 1221.95 23 1211.96 5 1161.97 20 1202.13 29 1441.99 21 1232.31 41 1491.94 3 1381.95 4 1172.02 28 1511.98 11 1211.98 18 1272.06 35 1462.07 12 1062.09 28 1481.91 19 1251.95 13 1211.85 4 1231.91 11 1201.93 13 123

1.92 10 1381.98 10 1152.03 18 122

Urea

Formamide

The results were further tested by minimization overall crystalline degrees of freedom of the crystals used inthe optimization as wen as additional crystals not in­cluded in the original optimization," The variability in thepacking arrangements of the crystals studied may be seenby considering a few examples. In Fig. 4 the three­dimensional structure of oxamide" (1) is given. Oxamidepacks in space group PI. The structure consists of

a. Based on hydrogen positions refined by minimization.'

DKPLL-DMDKP

Oxamide

Glutaramide

Adipamide

Malonamide

Succinamide

Cyclopropane­carboxamide

(1)

0.0-s.o

-1lLll EI£IllIy

•• IE"" EI£IllIY

-20.0

....~ .

b

-25.0

'1c-'I.... ..

-15.0 -10.0

ENERGY KCALIMOLFig. 3b. Comparison of the energy distribution for watermolecules in contact with a protein molecule with that of bulkwater (A. T. Hagler andJ. Moult, Nature (London), in press).

where , is the interatomic distance and A, C., q, and qjare constants. The first term in this equation representsthe atom-atom repulsion. The most common value of nis 12, and the combination of the first two terms corres­ponds to the wen known Lennard-Jones "6-12" poten­tial! The second term is attractive and corresponds to thedispersion energy. The third term represents the elec­trostatic forces between polar groups and is usuallyrepresented by Coulombic interaction between partialatomic charges placed on the atoms. Finally a fourthterm is sometimes added, to explicitly represent thehydrogen-bond contribution.'

INTERMOLECULAR POTENTIAL FUNCTIONSMuch of the work done in this field has been inspired

to a large degree by the need for a force field to studypeptide and protein conformation."! It should be notedin this connection that in treating large molecules, suchas oligopeptides and proteins, it is common to fix thebond lengths and angles and allow only torsion aboutsingle bonds. Thus, aside from an intrinsic torsion poten­tial, the crucial interactions which must be accounted forare of the atom-atom type between non-bonded atoms.The most usual form of the potential used in thesecalculations, and in other applications given in Table I, is

Israel Journal of Chemistry 16 1977

205

Fig. 4. A stereo view of the crystal packing and hydrogen bonding in oxamide. (See text for adescription of the packing.)

Fig. 5. A stereo view of the packing arrangement and hydrogen bonding in adipamide.(See text for a description of the packing.)

translation related layers of hydrogen-bonded molecules.The layers may be thought of as being built from chainsof centrosymmetric hydrogen-bonded dimers along c(formed across centres of symmetry at each end of themolecule), which then hydrogen bond by translationalong b. The three-dimensional packing arrangement ofadipamide (2)10 is shown in Fig. 5. Adipamide packs in

(2) Adipamide

space group P2 1/c. The unusual feature of adipamide isthe lack of centrosymmetric cyclic dimers. Adipamide isone of the very few primary amides which do not formcyclic dimers." The final example we present is the ureacrystal structure given in Fig. 6.12 Urea (3) is unique

oII

/C,H-N N-H

\ /H H

(3) Urea

among the crystals studied in that it has four "amide"protons per carbonyl oxygen rather than the usual two.The "flexibility" of the hydrogen bond, as well as theimportance of forming all possible hydrogen bonds, isreflected in this structure by the fact that all four protons"hydrogen bond" to the same oxygen. Two of thehydrogen bonds are formed between pairs of moleculeswhich are essentially coplanar, while two more N-Hbonds hydrogen bond to the carbonyl by "coming in"along a direction perpendicular to the plane of the ureamolecule.

Several of the conclusions which emerged from thisstudy concerning the nature of the intermolecular forcesin amides are: (1) No explicit hydrogen-bonding poten-

Hagler / Electron Density and Conformational Properties

206

Fig. 6. A stereo view of the crystal packing and hydrogen bonding in urea. (See text for adescription of the packing arrangement.)

tial term is needed to account for this interaction inamide crystals. Hydrogen-bonding interactions in thesecrystals may be accounted for simply by electrostaticinteractions. (2) One of the main features of the amidehydrogen bond is the negligible repulsion of the amidehydrogen, which allows a short N ... 0 distance. (3) Thepotential functions derived in this study give as good ­or better - fits to crystal structures as any used hitherto(including those with explicit hydrogen-bonding terms).(4) Analysis of the remaining deviations of the minimizedversus experimental structures, especially in the case offormamide, indicate that the explicit inclusion of thelone-pair orbitals on the oxygen is necessary in order tofurther improve the energy surface. (5) The electrostaticenergy in these crystals accounts for - 50-75% in pri­mary amides and - 33% in secondary amides.

ELECTRON DENSITY STUDIES OF AMIDES AND CARBOXYLICACIDS

Ultimately all chemical and conformational propertiesof molecules depend on the electron distribution withinthese molecules. A study of the electron density inamides, peptides and carboxylic acids was undertakenwith several objectives in mind." The first objective ofthis study was simply the elucidation of the nature of theelectron density in these important biological com­pounds, in particular the determination of the propertiesof the difference electron density in peptide and car­boxyl groups. In addition, it was of interest to investigateseveral of the standard approximations made in confor­mational analysis, such as the transferability of potentialconstants, and the use of partial atomic charges, from thepoint of view of the electron density in these compounds.

As discussed above, the electrostatic interaction be­tween polar molecules is usually represented by theinteraction of partial atomic charges (the third term inEq. (1)). This term is extremely important in determiningthe conformational and crystal properties of moleculescontaining very polar functional groups such as amide,peptide, or carboxyl groups.!" One of the commonmethods of obtaining partial charges for use in Eq. (1),has been to use the results of population analysis ofmolecular-orbital calculations.v":" The problem in as­signing physical significance to these charges has beenpointed out previously,":" and the representation of the

Israel Journal of Chemistry 16 1977

molecular potential field in terms of atomic contributionsdiscussed at length in Chapter XVII.

The net atomic charges in a series of primary andsecondary amides as obtained from population analysisof wave functions obtained from three different basis setsis given in Fig. 7." The difficulty in assigning partialcharges from population analysis is emphasized from thelarge differences obtained in the three basis sets. On theother hand, it was suggested on the basis of the resultsobtained in this study" that population analysis offamilies of molecules may yield useful information aboutthe transferability of parameters for atoms in differentfunctional groups and the "patterns of charge" in suchgroups. The latter may be useful in the derivation ofsimple, transferable sets of partial charges to representthe charge distribution in various functional groups."

The Lennard-Jones potential represented in Eq. (1),and indeed all central potentials, implicitly assume thatthe molecule consists of spherical atoms. The totalelectron density of N-methylacetamide (NMA) in theplane of the amide group as calculated from the splitvalence 6-31G basis set".•7 is presented in Fig. Sa. Forcomparison, the corresponding electron density obtainedfrom the superposition of the electron densities of spheri­cal atoms is given in Fig. 8b. Aside from a generalwithdrawal of electrons from the peripheral regions ofthe molecule, the general features associated withmolecular formation, such as the bonding density andcarbonyl oxygen lone-pair orbitals, are not evident inFig. 8a. In fact the overall shape of the molecule is verysimilar to that of the density produced by superposedspherical atoms as given in Fig. 8b. The size and shape ofa molecule are related to the volume excluded to othermolecules or parts of the same molecule. This factor,which is represented in Eq. (1) by the Lennard-Jonesterm, is one of the important factors determining theconformation of large molecules"?" and the packingmodes of crystals.':" The similarity between the actual(calculated) molecular density and that obtained fromsuperposed spherical atoms in Fig. 8a is due to the factthat molecular formation involves a relatively smallredistribution of electrons.f'" These features indicatethat the use of spherical atoms to represent the excluded­volume effect in Eq. (1) would seem to be a reasonableapproximation.

207

CIlDOI1-aJ

-o.Ol' ~,o• -c 0."5

" O.ltl.O"i'. .-.0.17.

0-<1·610.1' ~.- C o.n' .. _.0...

-<I."z.,-O.Jl

o0.0' "

0.0': O.Jl 0.12•'J /C, /.

C •/ -0.19 J -0.45

• • 0.10O.Of

o-O.u

0.11 0.11 "0.71 0.40.-J /C, / •c Ni O•51 J -0.'1

• •0.11 O.J'

-0 ••1

,•

O.lt

-0.1' 0.:, 0.0'

o \'

/ ~ ,29/ C~0.05N • -o.J7 • 0.07

0.05 IN 0.11

O.lt• O.U

o "II O.U ~.Jl

/C, / \,• • -0'1 0.160.1' ,.

•O.J'

-0.J9 O~OJ O.OJ

0.:5 il \~.0.05 , CO.. C 0.05

.'-1 / './ <,c -o.J, O.OJ~ -0.17 I

• •0.05 0.1.

0.07 -o~Jl 0.:, 0.0'

II-, ~.

0.07 H ..".

N~ .« /,"-:5C -0.1' 1-o·J7 0.07, .

D.:' 0.11

-.~, 0.;, 0.1'

0.10 I \;.0.10 .,

.. .0.7J C~ / "<, / ~.10l1

• •(-0.51 ,-0." 0.1'

• •0.1f O.J'

fbi

0.1.

-0.40 H, .-i..44 H

o / D.l'

" 0."'" 0.47c-/ ~

...... -0.'4 0-0.40

0.1' 'H0.11

H,-0.10 II

O"-CO,u !0.12 / "-

II _" -0.44 0

'Ho.lJ

-H

-O.u0, 0,71

C -

/H_ II -0.92

0.'0 'H0.4J

H-,,,- H

/C'0

-0.10 0.12

\ .".H-0.10 C_N -0.450,,- O.Jl / -,

c- C -O.lS H

0.22 / ~ 0.2111-_ -0.45 ~ "0.07

"" 0.070.11

N

<,

/-C

'0

"N~-0. JO C

'O.Jl /C-C-O.ll

/ ~H_H -0.4' I ..

0.11 .......H :'07 0.07

D.IJ

Fig. 7. Partial atomic charges in amides as obtained from population analysis of three basis sets(taken from Ref. 13).

Hagler / Electron Density and Conformational Properties

208

Fig. 8. Contour map of the electrondensityin the amideplaneof N-methylacetamide as obtained with the split valence 6-31Gbasisset." (a) Total density as calculated from molecular wavefunction. (b) Density obtained from superposition of sphericalatoms."

The electron density contours given in Fig. 8 are arepresentation of the electron density in the molecularplane. A better picture of the molecular shape may beobtained by drawing a contour of constant electrondensity in three dimenions. Such a representation of theamide group in NMA is given in Figs. 9a and 9b.13

.%2 InFig. 9a, a contour level of 0.75 electrons/A' is used, fromwhich the atomic features can be clearly distinguished. InFig. 9b, the contour level is chosen as 0.027 electrons/A",which corresponds roughly to the "van der Waals"radius. These plots also show the roughly spherical shapeof the atom, at both contour levels. Furthermore, theshape plot of the amide group contoured at the "van derWaals" radius shows qualitatively why it was found from

Israel Journal of Chemistry 16 1977

Fig. 9. The "shape" of the amidegroup in NMAas obtainedbya three-dimensional plot of a surface of constant electrondensity. The carbonyl oxygen is in the upper right-hand cornerof the figures with the amide hydrogen being in the foregroundat the left. (a) Surface of constant electron density of 0.75electron/A", (b) Surface of constant electron density of 0.027electron!A'. 13

the crystal studies that the repulsion due to the amidehydrogen was negligible (conclusion 2 above). Thus atthis level the electron density surrounding the amidenitrogen "envelops" the hydrogen, and the N-H groupappears to be one large atom, whose excluded volumemay be represented by a single central repulsion. (Thiswas reflected in the least-squares analysis of the energyparameters by the indeterminancy of the amide hy­drogen parameters, which had standard deviations sev­eral times larger than the parameters themselves.')

The density maps considered above involved the totalmolecular density, which displays few of the features ofmost interest to the chemist - namely the electronmigration on molecular formation. The lack of suchfeatures as bonding density and lone-pair orbitals is dueto the fact, as discussed above, that the electron migra­tion accompanying molecule formation is small whencompared to the total number of electrons in themolecule." The features of interest may be obtained byconsidering a difference density map." These maps arecalculated by subtracting the superposed spherical atomsfrom the total molecular density (both calculated in thesame basis set). The difference map for NMA obtained in

(3)

-------

H

Fig. 10. Difference electron density map of NMA in the planeof the amide group, as calculated with the 6-310 basis set. Solid,dashed and dotted lines represent positve, zero and negativedifference densities respectively. 13

Fig. 11. A "shape" plot of the positve difference density of theamide group in NMA. The 0.1 electron/A' contour is plotted.The lone pair orbitals are denoted A, the C-N bonding densityB, the N-H density C, and the C'-C density 0. 13

this way with the 6-31G basis is given in Fig. 10. Suchfeatures as the bonding densities, lone-pair orbitals, anddifferent polarization of the amide and C-H hydrogenatoms become apparent in such a map. Once again it isimportant to recognize that the contours represented insuch a map are in the plane of the amide group, while theactual features extend in three dimensions. The shape ofthese features may be seen by again plotting a surface ofconstant density in three dimenisons. The shape plot ofthe positive difference density of NMA is shown in Fig.11,where the 0.1 electron/A' contour is plotted. It shouldbe noted that in many amide crystals the planes of theamide group involved in the hydrogen bond are oftendisplaced by as much as 1 A.II From the shape plots it isseen that the lone-pair density extends considerablyabove the plane, and this may account somewhat for theapparent "shallowness" of the hydrogen bond poten-

209

tial.,··,I1·23 Finally it should be noted that although thecharge migration on molecular formation is small, it is thesole contribution to the polarity of the molecule. Whereasthe excluded volume depends on all the electrons (or atleast all valence electrons), the polarity of the moleculedepends only on the fraction of electrons which havebeen "displaced". (A molecule composed of perfectlyspherical atoms would have no dipole moment.) It wouldthen appear that the use of partial atomic charges torepresent the charge distribution is a more serious ap­proximation than the use of an atom-atom representa­tion of the excluded volume. Thus, the first step inattempts to further refine intermolecular force fieldswould appear to be to introduce a better representationof the charge separation as, for example, observed inthese difference maps. Work is in progress in this direc­tion in our laboratory.

INTRAMOLECULAR POTENTIAL FUNCTIONS AND MOLECULARPROPERTIES

We now turn to a discussion of the difference electrondensity in relation to the intramolecular properties ofmolecules. For this purpose we will discuss two exam­ples, taken from recent work on the properties of neutraland protonated carbonyl compounds.t':"

There are essentially two types of analytical expres­sions used to represent the energy of a molecule as afunction of its internal coordinates, the so-calledUrey-Bradley force field and the, Valence Force Field. 26

In both force fields, the total strain energy, E" of themolecule is represented by a sum of terms such as

E, = E bond + E.nl l• + E t _ on + E<t~'''''' + E nan-bond' (2)

The first three terms represent the strain due to bondstretching, angle bending and torsional deformations,and are common to both force fields. They are commonlygiven by expressions such as

E bond = ~ 2: Kb(b - bo)2,

E th... = ~ 2: K.(6 - 80 ) "

1E.onion = 22: K.(1 + cos nq,),

respectively, where the sums go over the appropriateinternal coordinates in the molecule. The difference inthe two types of force fields arises in the representationof the cross terms which in the Urey-Bradley force fieldare given by !~ F';j(rij - r~)2, where the sum runs over allatoms in the molecule connected to a common atom("1-3" interactions), and rij is the distance between theseatoms. In the valence-type force field the cross terms areof the type ~Kbb(bl - blO)(bj - bjo), ~KbO(b - bo)(8 - 60)

etc., which represent a coupling between bond and angledeformations (which have at least one atom in common).

The second difference between the representations ofthe force field involves the last term, E 110ft-bond, and ingeneral is independent of whether the UBFF or VFF isused but rather depends upon the application. In spec­troscopic analysis the non-bonded term is usually omit­ted, and thus repulsion (or attractions) of atoms notbonded to a given atom are simply ignored." In "molecu­lar mechanics", where the object is usually restricted tothe minimization of the molecular energy in order toobtain the molecular geometry and/or strain energy, this

Hagler / Electron Density and Conformational Properties

210

term is usually included (in a form similar to Eq. (1)).26(The result of this practice is that the force constantsobtained from one type of analysis are usually incompati­ble with the second.) An exception to these latterremarks is the work of Lifson and coworkers in which theapproach has been to develop a "consistent force field"which can consistently treat both vibrational and confor­mational properties."

The force fields derived from the molecular propertiesdescribed above contain a wealth of information aboutthe nature of the molecular forces operating in thesemolecules. It would seem worthwhile to combine thesetechniques with other theoretical approaches in order toobtain better model force fields, and to help interpret the

results obtained from the different methods. RecentlyBartell et al." have investigated the importance of vari­ous cross terms in valence force fields for alkanes byab-initio methods. Below we present two examples ofab-initio calculations in which the results are discussed interms of the nature of the intramolecular forces underly­ing these expressions.

The first example concerns formaldehyde and itshalogenated derivatives, carbonyl fluoride and phosgene.The experimental geometries.P:" along with the fullyoptimized geometries of these compounds as obtainedwith the split valence 4-31G basis set," are given inTable 3.

'"8

1.15

r'

0.97 1.300.510.18

".---,

-1.01 ~.61 ~.221.-1""'._40_~_~_~_~__~_~_~ ---r'

k k

--' -;

? ' )2 .,.'

I

I

Fig. 12. Difference electron density map of formaldehyde ascalculated in the molecular plane with the 4-310 basis set.

Fig. 13. Difference electron density map of carbonyl fluoride ascalculated in the molecular plane with the 4-310 basis set.

Table 3. Experimental and Optimized Geometries of Substituted Carbonyls

Compound Experimentalgeometry

Optimizedgeometry"

Formaldehyde

Carbonyl fluoride

Phosgene

H 121.4

,.'O~~ 0H 1.204

125.7

1~~OF/ 1.174

124.5

C~1.746""C-O

CL/ 1.166

(0 ) H 121.8

I.O~OH/ 1.206

J25.8

1.3~~OF/ 1.170

C~1.810yC- O

CL/ 1.167

Protonatedformaldehyde

a. Ref. 29. b. Ref. 30. c. Ref. 31.

Israel Journal of Chemistry 16 1977

The difference electron density maps of formaldehydeand car~onyl fl~orid~ ct?rresponding to the optimizedgeo~etn.es are given In Figs. 12 and 13, respectively. ThesUbstltu~lOn.of hydrogen by an electronegative atom suchas ftuonne IS known to result in a electron withdrawalfrom the carbonyl bond and some 71' back donation."Des~ite ~h~ a withdrawing effect of the electronegativeftuonne It IS seen from the difference electron densitymaps that the carbonyl bonding density in the molecularplane of carbonyl fluoride increases relative to that informaldehyde. This is borne out by the overlap popula­t~ons .as well, which indicate that the o overlap popula­non In the carbonyl bond is greater in F2Ca than inH2Ca, confirming the trend observed in the map andconsistent with the shorter carbonyl bond length inF2Ca .

Although the shortening of the carbonyl bond in F2Ca

relative to H2Ca may be explained in terms of thesUbstit~ent effects on the electron density, the story is notas straightforward when phosgene is considered. Thetotal overlap populations in the carbonyl bonds in H2Ca,

F2Ca and cico are 0.83, 0.95, 0.75, respectively." Thusalthough as expected, the carbonyl bond is shorter inF2<?0 than in H2Ca, consi~erationof the overlap popu­lations leads to the conclusion that in phosgene this bondshould be "weaker" and longer than in either H2CO orF2CO: However, as observed in Table 3, both theexpenmental and calculated results indicate that thecarbonyl bond is shorter in phosgene than in either of theother two compounds.

An explanation for these results may be formulateda?d te~ted in terms of the expression for the strain energygiven In Eq. (2) and the following discussion. Accordingto the overlap population the C=O bond in ChCO is"weaker" than in F2Ca, yet contrary to the usualcorrelation between bond length and strength (Badger'sRule") it is not shorter. It was postulated that thisapparent discrepancy was due to the difference instrength of the CI· .. a and F· .. 0, "1-3" interactions."Thus although the c=a bond in CI2Ca is weaker it wassuggested that there is a smaller "1, .. 3" repulsion inphosgene than in F2CO, resulting in essentially equalbond lengths. It follows from this analysis that thecarbonyl force constant should be significantly larger inF2CO than in CI2Ca, again contrary to what would beexpected from consideration of the bond lengths. Thisindeed is the case as evidenced from both the calculatedand "experimental" force constants.":" Furthermore thevibrational analysis indicates that the Urey-Bradley("I ... 3") force constant is larger in F2CO than inCI2CO" again in agreement with the proposed analysis.

The last example we present is related to the largechanges !n electron density and geometry accompanyingprotonanon of the carbonyl oxygen. The optimizedgeometry of protonated formaldehyde as obtained in the4-310 basis set is given in Table 3.",3> Comparison withthe structure of the neutral molecule shows significantchanges on protonation. The carbonyl bond length islengthened by approximately 0.04 A, while the HCOangle trans to the proton decreases by - 6°. The protonmakes an .angle of - 125° with the carbonyl bond,corresponding roughly to the lone-pair direction.

The difference electron density map of protonatedformaldehyde is given in Fig. 14. The large migration ofelectron density on protonation may be readily observedby comparing t~is m~p with the corresponding map forthe neutral species (FIg. 12). Protonat ion is accompanied

211

Fig. 14., Di1ference electron density map of protonated formal.dehyde In the molecular plane as calculated with the 4-310 basisset.

by a large polarization of the molecule toward theproton. The. electron dens~ty in the lone pair envelopingthe proton Increases considerably, while the volume ofthe ~nproton.at~ l?n~ .pair decreases. The carbonylbonding density IS diminished and polarized toward theproton. Slight changes are also observed in the C-Hbonds.

The sUbst~ntial changes in molecular geometry may beunderstood In terms of these large shifts in electrondensity. The increase in the carbonyl bond length followsnatu~ally from. the decrease in carbonyl bondingdensity (there IS also a contribution from the "1-3"proton-carbon interaction). The striking decrease in thetrans angle is es~ecially interesting, since it would appearthat the proton IS too far from the trans substituent tointeract with it directly. Rather it would appear that thedecrease in this angle is related to the "bond-bond" and"bond-orbital" interactions between the C-H bond andthe carbonyl bonding and lone-pair orbital density."~us the contr~cti~n in the volume of the lone pair alongWith the polarization of the carbonyl bonding densitytowards the proton both lead to a decrease in therepulsion between these features and the C-H bond. Thedecrease in these repulsions allows the C-H bond to"swing in" closer to the carbonyl bond, leading to theobserved decrease in the HCa angle.

CONCLUSION

In this paper we have tried to give some indication ofthe breadth of subjects which can be investigated by"computer simulation experiments". We have attemptedto indicate ho~ a combination of electron density maps,molecular orbital calculations, and empirical energy sur­faces may be used together with experimental observa­tions to give more insight into molecular properties thanany of these techniques used separately, The role ofelectron density maps, which up to now have beenemployed less often than the other techniques wasemphasized. '

Acknowledgement. The work presented here is the result ofcooperation with many people whose collaboration it is apleasure to acknowlege. Among them are Professor ShneiorLifson, Drs. A. Lapiccirella, L. Leiserowitz, Z. Karpas and Mr.F. Sussman. I am also grateful to Ms. Pnina Dauber and RuthSharon for able and enthusiastic assistance with computationalaspects of the work,

Hagler I Electron Density and Conformational Properties

212

REFERENCES AND NOTES1. H. A. Scheraga, Adv. Phys. Org. Chem., 6,103 (1968); H.

A. Scheraga, Chern. Rev., 71, 195 (1971).2. G. N. Ramachandran and V. Sasisekharan, Adv. Protein

Chem., 23, 283 (1968).3. A. T. Hagler and S. Lifson in H. Neurath and R. Hill, eds.,

The Proteins, in press.4. The "Buckingham" potential in which the reciprocal nth

power repulsion is replaced by an exponential (A exp-6r) isalso commonly used.

5. A. T. Hagler, S. Lifson and E. Huler in E. R. Blout, F. A.Bovey, M. Goodman and N. Lotan, eds., Peptides, Polypep­tides and Proteins, John Wiley and Sons, New York, 1974, p.35.

6. A. T. Hagler and S. Lifson, Acta Crystal/ogr., 830, 1336(1974).

7. A. T. Hagler, E. Huler and S. Lifson, J. Am. Chem. Soc., 96,5319 (1974).

8. A. T. Hagler and S. Lifson, J. Am. Chem. Soc., 96, 5327(1974).

9. E. M. Ayerst and J. R. C. Duke, Acta Crystal/ogr., 7, 588(1954).

10. M. Hospital and J. Housty, Acta Crystallogr., 20, 626 (1966).11. L. Leiserowitz and G. M. J. Schmidt, J. Chem. Soc., A, 2372

(1969).12. J. E. Worsham, Jr., H. A. Levy and S. W. Peterson, Acta

Crystallogr., 10, 319 (1957).13. A. T. Hagler and A. Lapiccirella, Biopolymers, 15, 1167

(1976).14. A. Pullman, Proc. Jerusalem Symp. Quantum Chem.

Biochem., 1, 9 (1970).15. J. E. Gabenstener and M. A. Whitehead, J. Chem. Soc.

Faraday Trans. 2, 69, 962 (1973).16. R. Ditchfield, W. J. Hehre and J. A. Pople, J. Chem. Phys.,

54, 724 (1971).17. W. J. Hehre, R. Ditchfield and J. A. Pople, J. Chem. Phys.,

56, 2257 (1972).18. P. J. Flory, Statistical Mechanics of Chain Molecules, Inter­

science, 1969.19. A. I. Kitaigorodskii, Molecular Crystals and Molecules,

Israel Journal of Chemistry 16 1977

Academic Press, New York, 1973.20. C. A. Coulson in The Shape and Structure of Molecules,

Oxford Chem. Soc., Clarendon Press, Oxford, 1974.21. See Ref. 13 and references therein for further discussion of

this point.22. See V. Smith (Chapter XVI) for a discussion of the various

representations of electron density. It should be noted thatthe shape plots presented in these studies differ slightly inthat our representation is drawn relative to a base plane,whereas the shape plots in the work of Smith et al. representthe entire surface.

23. J. Donohue in A. Rich and W. Davidson, eds., StructuralChemistry and Molecular Biology, W. H. Freeman, SanFrancisco, 1968.

24. A. T. Hagler, Z. Karpas and F. Sussman, submitted forpublication.

25. Z. Karpas, F. Sussman and A. T. Hagler, submitted forpublication.

26. See for example a. C. Altona and D. H. Faber in DyamicChemistry, Topics in Current Chemistry, 45, Springer­Verlag, Berlin, 1974, p. 1; b. O. Ermer, in Structure andBonding, 17, Bonding Forces,Springer-Verlag, 1976, p. 161.

27. S. Lifson and A. Warshel, J. Chem. Phys., 49, 5116 (1968);A. Warshel and S. Lifson, J. Chem. Phys., 53, 582 (1970); A.Warshel, M. Levitt and S. Lifson, J. Mol. Spectrosc., 33, 84(1970); O. Ermer and S. Lifson, J. Am. Chem. Soc., 95, 4121(1973).

28. L. S. Bartell, S. Fitzwater and W. J. Hehre, J. Chem. Phys.,63, 4750 (1975).

29. T. Oka, J. Phys. Soc. Jpn., 15, 2274 (1960).30. V. W. Laurie, D. T. Pence and R. H. Jackson, J. Chem.

Phys., 37, 2995 (1962).31. G. W. Robinson, J. Chem. Phys., 11, 1741 (1953).32. See e.g. J. E. Del Bene, G. T. Worth, F. T. Marchase and M.

E. Conrad, Theor. Chim. Acta, 36, 195 (1975).33. R. M. Badger, J. Chem. Ph)'s., 3, 710 (1935).34. J. Overend and J. R. Scherer, Spectrochim. Acta, 16, 773

(1960).35. F. Bernardi, I. Csizmadia, H. B. Schlegel and S. Wolfe, Can.

J. Chem., 53, 1114 (1975).