Free energies and structures of hydrated cations, based on effective pair potentials

ELSEVIER Chemical Physics 195 (1995) 207-220

Chemical Physics

Free energies and structures of hydrated cations, based on effective pair potentials

Franca Floris, Maurizio Persico, Alessandro Tani, Jacopo Tomasi Dipartimento di Chimica e Chim. Ind., Universitgt di Pisa,

v. Risorgimento 35, 56126 Pisa, Italy

Received 22 November 1994

Abstract

We present a method, based on a continuum representation of the solvent, to compute ab initio effective interaction potentials for solvated pairs. Such potentials take into account many-body effects, thus overcoming the non-additivity errors affecting uncorrected pair potentials. We apply the method to cation-water interactions, for a variety of cations: Li +, Be 2+, Mg 2+, Ca 2., Ni 2+, Zn 2+ and A13+. The potentials thus obtained are suitable for simulations of ionic solutions or clusters of water molecules surrounding a cation. We exploit them to compute hydration free energies AGhyd of cations, with the constraint that the first solvation shell contains a given number of water molecules. This enables us to find the thermodynamically most stable solvation number. The effective potential results compare well with experimental values of AGhyd and with full ab initio calculations on the [M(H20)n] q+ complexes.

1. In t roduc t ion

Thermodynamics, structure and reactivity in condensed phase systems are successfully investigated by means of computer simulations. The most widely employed simulation techniques rely on classical force- field representations of inter- and intramolecular interactions. The potential energy functions can be fitted to the results of ab initio calculations for dimers and other small clusters. Other potentials are evaluated by calibration against experimental data. Many of such potentials, however obtained, can be regarded as effective potentials, in the sense that they incorporate some effects of the condensed phase environment on the interacting molecules.

For instance, a typical water-water potential contains coulombic terms, parametrised in the form of point charges, located on the nuclei or at related po-

sitions. The dipole moment associated with the point charge distribution for several such potentials [ 1-8] is larger than that of an isolated H20 molecule ( 1.85 D [9] ), but close to the average dipole of a polarised molecule in liquid water (about 2.4 D [ 10] ). As a con- sequence, the long-range part of the two-body potential does not represent correctly a gas phase situation, but it is suitable for the liquid state. Exceptions are Rowlinson's [ l 1 ], Stillinger's CF [ 12,13] and Watts' [14-16] potentials. We note in passing that dipole moments larger than 2 D are obtained in HF calculations for H20, with medium quality basis sets (double- zeta or triple-zeta plus polarisation functions), such as those often employed in computing water-other molecule interaction potentials. The effect of the liquid environment can be well represented by the intro- duction of many-body terms, for instance by means of polarisable water molecule models [8 ,17-21] : in

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208 E Floris et al./Chemical Physics 195 (1995) 207-220

such cases, the dipole associated with the zero-field charge distribution is close to that of H20 in vacuo.

Coming to the topic of this paper, the failure of pairwise additivity for cation-water potentials has been recognised since longtime, especially for doubly and triply charged cations [22-40]. There are two aspects of the problem. The first one concerns the long-range behaviour of the ground state potential of [M(H20) ] q+ systems, with q /> 2. In most cases (exceptions are M q+ = Ca 2+, Sr 2+, Ba 2+) the electron affinity of M q+, i.e. the qth IP of M, is larger than the first IP of water (12.615 eV): therefore, at large M- (H20) distances, the ground state of the system is represented by the charge-transfer (CT) configura- tion [M(q-J)+(H20) + ]. The long-range potential is therefore repulsive, but at shorter distances the nature of the ground state changes to [Mq+(H20)] , with only partial CT character (electron donation from water to metal) [37,41,42]. The presence of an avoided crossing between CT and non-CT states can be important in vacuo for the dynamics of [M(H20)]q+ systems [42], but is quite irrelevant in solution. The +q charge of the cation is strongly stabilised by solvation, even with only one or two water molecules: therefore, when a water molecule is extracted from a [ M(H20), , ] q+ complex, it is substantially neutral at all distances, with a long-range potential dominated by a charge-dipole term. The presence of the other n - 1 water molecules thus changes qualitatively the shape of the potential, in such a way that many-body terms would be in principle unavoidable. This problem is easily bypassed, and often not even mentioned, when constructing ab initio potentials for simulations. In fact, the restricted HF cation-water wavefunctions yield necessarily the Mq+-(H20) dissociation, at least for closed shell cations: this is qualitatively adequate for the study of the liquid phase, where other water molecules always solvate the cation, although not correct in vacuo. In this sense, most ab initio potentials should be considered as effective potentials. Notice that this simple approach may fail when correlated wavefunctions are introduced, unless ad hoc procedures are devised to limit the charge-transfer effects [37,43].

The second, more subtle aspect of the non-additivity problem, concerns the strength of the binding between a cation and the second, third ..... nth water molecule in a complex. Two important interaction energy terms, in-

duction and charge-transfer [44,45], are clearly non- additive [ 37 ]. Unlike the crossing with charge-transfer states, this second non-additivity problem is important only at short distances. Simulations based on ab initio pair potentials, neglecting many-body effects, have consistently overestimated the binding properties of cation-water complexes. Both structural and energetic results are affected. Hydration numbers nh (number of water molecules in the first hydration shell) can be estracted from Monte Carlo (MC) or Molecular Dynamics (MD) results, by integrating the cation- oxygen radial correlation functions gMo(R). Uncor- rected pair-potentials often yield hydration numbers in excess of the diffraction or EXAFS results [46]: for instance, nh = 6 has been obtained for Be 2+ [47] instead of the experimentally determined nh = 4; nh = 8 instead of 6 has been obtained for Fe 2+ [27,48], Fe 3+ [27], Ni 2+ [49] and Cu 2+ [35]. In some cases the agreement between simulation and experiment (nh = 6 for Zn 2+ [50] and Mg 2+ [51-53]) may be due to a sort of error compensation: in fact, the binding energies obtained with small basis sets may be underestimated, thus making up for the non-additivity error.

We also observe that enthalpies and free energies of hydration, obtained by MC calculations with uncorrected pair potentials, are clearly overestimated, for monocations [54] as well as for dications [40,50].

The problem of non-additivity can be dealt with in several ways. The most obvious and rigorous approach is to supplement the potential energy functions, to be employed in simulations, with many-body terms. In several cases three-body potentials have been found adequate to reproduce correctly hydration numbers (Be 2+ [32], Cu 2+ [35,40]) and free energies (Li + [30] ). The determination of three-body potentials by ab initio methods and their use in simulations increase the cost and the complexity of the calculations. The short range character of the many-body corrections can be exploited to improve the computational effi- ciency [35,36]. As the non-additivity problem is substantially limited to the first solvation shell, a quite adequate, though costly, strategy, consists in computing the interaction pair potential between a water molecule and the [M(H20)n]q+ complex as a whole [55,56].

Yet another approach is to define effective pair potentials, representing the average interaction of a water molecule with a cation already solvated by other water molecules. Effective potentials can be derived

E Floris et al./Chemical Physics 195 (1995) 207-220 2 0 9

from empirical data [27,57,58] or from calculations on cation-water clusters [40]. We presented in a previous letter [39] the definition of an effective pair potential based on the Polarisable Continuum Model (PCM) of the solvent [59,60]. MD simulations per- formed with our effective potential were successful in predicting the correct hydration number (nh = 6) for Fe 2+ and Fe 3+.

In this paper the effective pair potentials (EPP) for several cations (Li +, Be 2+, Mg 2+, Ca 2+, Ni 2+, Zn 2q , AI 3+) are presented and applied to the evaluation of hydration free energies, AGhya. AGhyd is computed by a thermodynamic cycle, involving evaporation of n water molecules, gas phase reaction to form the complex [M(H20)n] q+, and solvation of the complex. Several values of n can be tried, each corresponding to a different first solvation shell structure: the lowest AGhyd identifies nh, the normal or dominant hydration number in dilute solutions. This approach is comple- mentary to MC or MD simulations, which normally yield information on the structural, energetic and dy- namical properties of the most stable structure only. Once AGhyd is known as a function of n, one can assess the possibility that changes in the theoretical model (interaction potentials, electrostatic, dispersion and cavitation energies) or in the physical conditions (temperature, pressure, composition) may lead to a different stability ordering. Comparisons with experimental free energies of hydration [61 ] and with full ab initio calculations on [M(H20)n]q+ clusters are presented as tests for the accuracy of the effective pair potentials.

2. Effective pair potentials

The gist of our proposal is to evaluate the interaction between two partners A and B, starting from the w avefunctions ,It of the solvated molecules. A (or B) can be a neutral or ionic solute or a solvent molecule. We stress that the effective potential UAB must not contain pair interactions between the AB system and other solvent molecules: these will be taken explicitly into account in the simulations; therefore the Hamilto- nian defining UAB is that of the isolated pair or supermolecule AB, ~(0). On the other hand, the wavefunction • should represent, at least in average, the situation of the AB system embedded in the solvent, with

charge distribution and other properties modified by the interaction with the surrounding molecules. There- fore, the A-B potential is defined as

= -

- ( 1 )

where 7~ (°) is the molecular Hamiltonian. The labels A, B and AB denote Hamiltonians, wavefunctions and matrix elements for the A and B monomers and for the AB supermolecule, respectively, q~ is a wavefunction perturbed by the presence of the solvent, therefore not an eigenfunction of 7~ (°) . We shall compare/JAB with the uncorrected potential H (°) computed in vacuo: ~ A B '

u ( 0 ) _- u _ A B

- ( ,I ,(°) I,r (0 ) )B . ( 2 )

Here ~(0) is an (approximate) eigenfunction of T~ (°). In order to evaluate q~ one may include a number

of solvent molecules in the calculations for A, B and AB. Such a supermolecule approach can be compu- tationally very demanding, and requires an a priori knowledge of the structure of the first solvation shell. Moreover, there would be the problem of separating the solute and solvent wavefunctions. A classical representation of the solvent in this case offers consider- able advantages. In our procedure ~P is determined in the framework of the Polarisable Continuum Model (PCM), in which the solvent is represented as a lin- ear isotropic dielectric [ 59,60]. The solute is accomo- dated in a cavity made of interlocking spheres, centred on the nuclei, with additional spheres to smooth out the most concave portions of the surface [62]. The solvent reaction field determines the one-electron interaction operator ~('~) which is added to ~(0) to give the effective Hamiltonian of the solvated system:

= + ( 3 )

q~ is ideally an eigenfunction of 7~, to be approximated by any suitable ab initio technique. The difference between • and the wavefunction of an isolated molecule, xpC0), depends on the dielectric constant of the solvent and on the shape of the cavity. The radii of the spheres constituting the cavity are somewhat arbitrary, although a standard choice has been found adequate for neutral solutes: p = 1.2Rvaw, where Rvaw is the van der Waals radius of the atom [63-65]. A

210 F. Floris et al./Chemical Physics 195 (1995) 207-220

variation in the radii pl of the spheres will then affect the computed effective potential.

We can take advantage of the degrees of freedom represented by the sphere radii, to fine tune the additivity of the effective pair potential. The uncorrected potential computed in vacuo is not additive, in the sense that for a trimer ABC:

u(o) = (~(O) I~(O)I~(O))ABC _ (~(0)i~(0)1.(0>>~

_ % _

H (°) H ( 0 ) H(0) VAB ~ ' -vAC "[- ~ B C " (4)

In contrast, we expect at least an approximate additivity to hold for the effective potential:

UAB C = - -

---- UAB "}- UAC "}- UBC • ( 5 )

Here of course the geometries of A, B, C, AB, AC, and BC are those of the corresponding subsystems of ABC. By adjusting the radii of the spheres, we can impose the validity of Eq. (5) at least for a few representative geometries. In this way the only arbitrary parameters of the procedure are determined by a cri- terium of internal consistency: that is, the perturbation induced by a continuum dielectric solvent on the AB, AC and BC pairs is directly equated to the effect of adding explicitly a third body in the ab initio calculation.

3. Cation-water potentials

In this section we specialise the procedure outlined above to the determination of cation-water potentials: the A-B pair is composed of a metal cation M n+ and a water molecule W:

- (6)

The U(M°~W uncorrected potential is defined analo- gously, according to Eq. (2). We shall not explicitly address the problem of non-additivity of water-water potentials: we shall make use of the empirical effective potential elaborated by Berendsen and coworkers

(SPC/E [5,7] ). The water-water interaction is actually modified in the proximity of ions, but we shall see that this effect is included in the cation-water effective potential. Inductive and charge transfer interactions are quite important in cation-water binding [45] and both are non-additive, in the sense that they are made weaker when the cation is complexed by more water molecules [26,31,33,37]. In fact, the electric field felt by an individual water molecule is primarily gen- erated by the cation, plus a contribution, of opposite sign, by the other ligands: in this way, the presence of the latter weakens the induction and CT effects.

In the PCM calculations with a cation and one or two water molecules, the role of the rest of the solvent is taken up by the polarisation charges, which generate the solvent reaction field. The surface charge on the sphere containing a cation is negative, therefore its effect is to contrast the cation electric field. Qualitatively, this is correct: the induction and CT interactions are decreased. The extent of this effect depends mainly on the radius of the sphere centred on the cation, PM: with smaller radii, the effect is more pronounced, therefore the binding energy is weaker. For oxygen and hydrogen we assume the standard radii po = 1.68 ,& and pn = 1.44 /~. To determine the radii of cations we apply Eq. (5), where C is a second water molecule:

U w M w = -

- 2(q,l~¢°)l,I,)w = 2UMw + Uww, (7)

As in this case the water-water potential Uww, computed ab initio in vacuo, does not contain the effect of the electric field of the cation, this is implicitely taken into account by UMW. Both UWMW and 2UMw are functions of PM, with negative slopes: however, as the PM dependence is stronger for 2UMW than for UwMw, the solution of Eq. (7) is unique (see also Ref. [39] ). Because we adjust only one parameter, we must chose a single representative geometry of the [ M (H20) 2 ] u+ system to implement Eq. (7). The water internal geometry is the experimental one [66]. The metal oxygen distances RMO are set to values close to those found for the first solvation shell, i.e. slightly larger than the optimal distances for the [M(H20) ]q+ complexes: RLiO = 2.0, RBeO = 1.7, RMgO = 2.0, Rcao = 2.4, RNiO = 2 . 0 5 , Rzno = 2 .0 , RA10 = 1.9 ]k. The two water molecules are face-to-face, with the cation at the crossing of the two HOH bisectors. The overall

E Floris et al./Chemical Physics 195 (1995) 207-220 211

symmetry is C2v (see Fig. 1). The non-additivity of the water-cation-water potential depends on the OMO angle. We have computed the three-body term which is missing in the right-hand side of Eq. (4) :

l :(o) ~.,(o) i1(o) U3 = ~ wM w - ~ UMw -- ~ww. ( 8 )

The basis sets were the same as in the calculation of two-body potentials (see next section). Fig. 1 shows /_13 as a function of ZOMO: the behaviour is qualitatively similar for all cations, with steeper curves and higher values for smaller or more charged cations. The somewhat irregular behaviour of Ni 2+ may be due to the partial filling of the 3d shell. We have computed the weighted average of/-/3 for ZOMO = 70.5 °, 90 °, 109.5 °, 180 °, taking into account the recurrence of these angles in tetrahedral, octahedral and cubic complexes: for all cations, the average corresponds ap- proximately to the value obtained for / O M O = 90 °. Moreover, AOMO = 90 ° is the most representative ~alue on statistical grounds. Therefore, we assume the [M(H20)2]q+ geometry with / O M O = 90 ° to implement Eq. (7).

The resulting sphere radii are shown in Table 2. They are somewhat larger than the ionic radii in most cases, especially for Be 2+ and A13+; PLi is practically equal to the ionic radius. Although this means that the radii here employed are of the same order of magnitude as those recommended for direct PCM evalu- ations of AGhyd [67,68], we stress that our aim and numerical values are different.

4. C o m p u t a t i o n a l d e t a i l s a n d f i t t i n g

We computed the energies of [M(H20) ]q+ complexes at the HF level with two kinds of basis sets, A and B, as detailed in Table 1. The basis sets A are of triple-zeta quality. B denotes split-valence basis sets, employed in conjunction with the Effective Core Potential (ECP) approximation when appropri- ate. When not already present in the original basis sets, we added polarisation d functions on the oxygen and metal atoms, because a few tests demonstrated their importance.

Table 2 shows the RHF results of calculations on [ M (H20) ] q+ systems. Only the RMo distance was optimised; the H20 geometry was fixed at Roll = 0.9575 /~ and / H O H = 104.51 ° [66]. The binding energies

50

45

40

35

f ,, I I I I [

0

AI 3+

30 ". "',, F3+

_ . . . . . - ~ . -- 25

20

15

10

5

0

. . . . . .

F2-~. Z~+. " ' - . . "" 'T.---:.: .....................

, \ , " . . B e ~

~. . . - - - . . . . . . ~> " . > ,

t 7 i ""- . . . . :"=: .... - ~}Z_ - _ " 2 ~ ? Y - - . . . . . . . . .

M? . . . . . . - . . . . . . >-=2~--: -)£ c2+-- ...->>.~

I I I I [

70 c~() l(19 I ~0 155 ] ~1(}

o-M (3 angle, degree

Fig. 1. Three-body potential term for [ M(H20)2 ] q+ complexes, as a function of the O - M - O angle. Pairwise interactions are subtracted from the full three-body interaction energy.

D e obtained with the smaller basis sets, B, are in good agreement with those of basis A: the largest relative error (3.8%, 5.6 kcal /mol) , concerns [Be(H20) ]2+. Therefore we adopted basis set B for the calculation of effective potentials. About 80 to 110 energies were computed for different positions of each cation with respect to a rigid water molecule. The cation was located on the C 2 axis of H20, or off-axis by at most 45 ° (angle between the C2 axis and the O - M line).

Both uncorrected pair potentials U~°) w and effective potentials UMW were fitted with analytic functions of the same form:

U -- UMO -+- UMH 1 -{- UMH2, ( 9 )

where the three terms on the r.h.s, are cation-oxygen and cation-hydrogen potentials, depending only on the respective distances:

qqK A B C D UMK(R) = ~ + ~-g -+- ~ - Jr- ~ -+- ~ - b E e -rR •

(10)

Here q is the cation charge; qH = 0,4238 and qo = -0 .8476 are the charges characterising the SPC/E model of water [5,7] we adopt in a parallel work

212 F. Floris et al./Chemical Physics 195 (1995) 207-220

Table I Basis sets

Atom A B

contracted exp. of Refs. contracted exp. of Refs. basis set pol. fun. basis set pol. fun.

0 15s3pld] 1.280 1691 a 13s2pld] 0.850 1701 a H 13slp] 1.000 [691 b 12sl 1701 Li 14s3pldl 0.200 [71] c 13s2pld] 0.500 1701 d Be [4s3pld] 0.255 [71] c 13s2pldl 0.400 17010 Mg [6s5pld] 0.234 [691 a 13s2pldl 0.175 1721 c Ca I10s8pldl 0.103 [73,74] f [4s4pldl 0.103 1751 g Ni I10s8p3d] [73,76] h [4s4p2d] 1751 i Zn [10s8p3d] 173,76] h 12s2p2d] 1771 i AI 16s5pld] 0.311 1691 a 13s2pld] 0.175 1721 c

a 6 Cartesian d functions, exponent as in the HONDO7 program [78[. b p functions from Ref. 170[. c 6-31G* [711. J 6 Cartesian d functions, exponent optimised for [Li(H20)]+. c 6-6-31G* 1721. f 5111111111/411111 contraction of Wachters' basis set [ 73 ], plus 2 sets of p functions (exp. = 0.122,0.048) and 6 Cartesian d functions from Ortega-Blake et al. [74].

ECP calculation, ls and 2sp electrons in core [751; 6 Cartesian d functions from Ortega-Blake et al. [74]. h HONDO7 [78] TZP basis set, modified from Wachters [73] and Rappe et al. [76]. i ECP calculation, ls and 2sp electrons in core [75]. J ECP calculation, ls 2sp and 3sp electrons in core [77].

Table 2 HF results for [M(H20) ]q+. Distances in /~, energies in kcal/mol

Cation Calculations in vacuo

basis set A basis set B

Effective potential

basis set B

RMO De RMO De pa RMO De o "b

Li + 1.83 38.1 1.87 37.7 0.57 1.89 33.3 0.8 Be 2+ 1.49 146.7 1.52 141.1 0.62 1.47 116.0 2.0 Mg 2+ 1.92 82.8 1.94 81.7 0.88 1.94 68.3 1.2 Ca 2+ 2.37 51.7 2.37 53.2 1.30 2.37 47.7 0.8 Ni 2+ 1.92 88.2 1.93 86.7 1.11 1.93 71.9 2.1 Zn 2+ 1.89 91.5 1.91 88.9 1.04 1.91 70.0 2.0 A13+ 1.73 193.1 1.75 189.6 0.93 1.70 151.0 2.7

a Sphere radius of the cation in the PCM calculations. b Standard deviation for the fitting of ab initio energies.


on MD simulations [79,80] and in the evaluation of free energies (see next section). The standard devi- ations o- for the fitting of the effective potentials are given in Table 2. Similar results were obtained also for the uncorrected potentials 1r(°~ with one excep- vMW, tion. In fact, for [ N i ( H 2 0 ) ] 2+ in vacuo, at large Ni- O distances the lowest energy RHF triplet wavefunction represents the CT state [ N i + ( H 2 0 ) + ] . There- fi)re, no results concerning the uncorrected potential of [ N i ( H 2 0 ) ] 2+ will be reported in the following. The PCM calculations converged instead to the non- CT state at all distances, because of the stabilising effect of the solvent reaction field on the Ni 2+ charge.

As anticipated, the effective potentials UMw are characterised by smaller binding energies with respect to tr (°) (see Table 2) The difference in De between vMW UMw and H(°l is clearly correlated with the size and ~MW charge of the cation, i.e. with the electric field felt by the water molecules in the first solvation shell. Although the UMw potentials are less attractive than u ( o ) ~. MW, me optimal RMO distances are almost the same. This feature has a substantial effect on the structure of the first solvation shell, whether determined by simulation of water solutions or by a cluster approach, as described in the next section. In fact, a displace- ment of the minimum in UMW to longer RMO distances would allow to accomodate a larger number of water molecules in positions where the binding with the cation is optimal: this would compensate, at least partially, the effect of smaller De associated with UMw. This is not the case, as shown in Table 2; actually the UMW and H(°) ~MW curves are almost parallel in the region of the minimum and up to the somewhat hmger distances characterising [M(H20)n] q+ complexes: therefore the differences in the [M(H20)] q+ and [ M ( H 2 0 ) n ] q+ binding energies obtained with the two potentials are well correlated.

5. The evaluation of hydration free energies

In this section we show how the effective pair potentials UMw can be applied to the evaluation of binding energies AE of [M(H20)n]q+ complexes and of the hydration free energies AGhyd for the corresponding cations. No ab initio calculations of the complexes with n water molecules are needed, although we per- formed them to compare with the effective potential

results. In order to compute AGhyd, we separate the solva-

tion process into three steps: (A) evaporation of n water molecules:

n H20( l iq ) , n H20(gas ) , ( 11 )

(B) complexation of the cation by N water molecules in the gas phase:

q--. q+ ~ [M(H20)n](gas ) , n H20(gas) + M(gas ) (12)

(C) solvation of the complex:

q+ [M(H20)n](gas ) ----+ [M(H20)n]{a~ ) . (13)

As far as we are able to optimise the geometry and compute the binding energy of a complex with n water molecules in the first solvation shell, we can evaluate the free energy of hydration associated with the hydration number n. In this way, the most stable structure under given thermodynamic conditions can be identified.

The complexation reaction, step B, implies a free energy change AGcom, which can be separated into electronic, vibrational, rotational and translational contributions:

AGcom = A E + AGvib -1- AGrot q- AGtra. (14)

AE, the binding energy without zero point energy (ZPE) corrections, was evaluated by means of the rigid SPC/E model of water, plus the effective cation- water potential, or, for comparison, the uncorrected potential. In principle, the effective potential UMw is computed for a cation-water pair surrounded by the bulk liquid. However, it has been noted that the non- additivity effects are saturated with a very small number of water molecules [22,26,33,37,39]. Although the importance of four-body or higher terms has been advocated in some case [31], three-body potentials usually give satisfactory results [30,32,35,40]. There- fore, the same effective potential which has been devised for the simulation of solutions, should be adequate also for clusters, provided that a few water molecules do complexate the cation.

For the geometry optimisations and the normal co- ordinate analyses we made use of a modified version of the program VENUS [81 ]. We obtained four kinds of structures:


[M(H20) 4] q+, with the oxygen atoms at the vertices of a slightly distorted tetrahedron; the point group is $4, and the 4 water molecules are perfectly equivalent; [M(H20)6] q+, octahedral complexes, point group

Th; [M(H20)8]q+; the oxygen atoms are at the corners of two squares with a differential rotation of 45 ° around the common C4 axis; the point group is $8, with 8 equivalent water molecules; [M(H20)9]q+; the oxygen atoms are at the corners of three equilateral triangles with a common C3 axis: two equivalent triangles, upper and lower, and a larger one in the middle; the point group is D3; there are two groups of equivalent water molecules (6+3) . In all complexes the axis of each water molecule points towards the cation, either exactly or only ap- proximately; exactly symmetric water molecules, with two equivalent hydrogen atoms, are only those of the octahedral complexes and the 3 equatorial ones of [M(H20)9 ] q+. Further details concerning the geometrical parameters will be given in a forth- coming paper [80], together with the results of MD simulations (see also Ref. [79] ).

The AGvib term contains the ZPE and thermal contributions (at 25°C), computed assuming rigid water molecules. In practice, we imposed very large force constants for the O-H stretching and H - O - H bend- ing coordinates, and we disregarded the contributions of the internal modes of water to AGvib. All other vibrational coordinates of the complex were treated as harmonic oscillators. This approximation may not be warranted for large amplitude slow modes, and we obtained indeed frequencies as low as 34 cm -1, for the [Ca(H20)9] 2+ complex. In some cases the lowest frequencies could be clearly associated with tor- sional motions of some kind, but other deformations involving the simultaneous change of several O - M - O angles were also present. We compared the harmonic oscillator and hindered rotor [82] approximations for frequencies between 30 and 150 cm -1, and for a wide interval of reduced moments of inertia: the latter ranged from that of a single water molecule, to that of two rigid squares with O atoms in the corners, rotating coaxially one with respect to the other, as in [Ca(H20)8] 2+. The A G values computed in the two approximations never differed by more than 0.08 kcal/mol, therefore the simpler harmonic model was adopted.

Not all of the geometry optimisations we tried were successful in placing n water molecules in the first solvation shell, as desired. For too large n, as in [Ca(H20)10] 2+, one or more water molecules pre- fer to occupy positions corresponding to the second solvation shell. This can be taken as an indication that the most stable hydration number is smaller than n: however, one must consider that the presence of a complete second hydration shell and of the bulk liquid are disregarded at this stage of the calculation; in our approach, all interactions involving the solvent outside the first solvation shell are included in the AG associated with step C. Four [M(H20)8 ] 2. complexes (M q+ = Mg 2+, Ni 2+, Zn 2+ , A13+) yielded degenerate pairs of imaginary frequencies, because of their tendency to expel two water molecules from the first solvation shell; only by imposing an $8 symmetry we succeded in optimising their cubic antiprism structures. None of these complexes turned out to be thermodynamically more stable than the corresponding hexacoordinated one, when all contributions to AGhyd were taken into account.

The free energy change associated with process A is nAGvap, where AGvap is well known experimentally. However, for a sake of consistency, the free energy changes of both processes A and C where evaluated by a simplified version of the PCM method. In this framework the solvation free energy of a given molecule is given as

AGsol = AGel + AGcav + AGdis-rep . (15)

The cavitation energy, AGcav, was evaluated by Pierotti's formula [83,84], computing the radius of the solute from its van der Waals surface. For the dispersion-repulsion term, AGdis-rep, we applied the pair-potential approach of Floris and Tomasi [ 85-87 ] in the uniform approximation; O-O, O-H and H-H atom-atom potentials were taken from Ref. [ 88]; no M - O or M - H potential was included in the calculation, because the distance between the cation and the second solvation shell is large with respect to the range of dispersion interactions.

We have computed the electrostatic contribution, AGe1, by a rigid classical model, in which the charge distribution of the solute is approximated by point charges: the q charge of the cation and the SPC/E charges of the water molecules. This simplified model

F. Floris et al./Chemical Physics 195 (1995) 20~220

Table 3 Test calculations of AGd with different approximations. Free energies in kcal/mol

215

Quantal PCM Point charges, Point charges, Single charge, calculation molecular cavity spherical cavity spherical cavity

[ Mg( H20)6 ] 2+ - 182.0 - 182.9 -235 .0 192.7 [ Be(H20)412+ -212.1 -219.3 -203.7 -224.1 [ Ca(H20) 812+ - 162.2 - 161.4 - 189.9 - 170.0 H20 -6 .930 -6 .919 -8 .324

permits to avoid the ab initio calculation of the wavefunction, as required in the full PCM procedure: therefore, it complements ideally the evaluation of the binding energy of [M(H20) , ] q+ based on the effective potential. For a few test cases we have compared the results of classical and quantal PCM calculations. As shown in Table 3, we can expect a very good agree- merit, except probably for the smallest complexes, such as [ Be(H20) 4 ] 2+. The almost spherical symmetry probably makes the polarisation of the solute in the solvent reaction field quite unimportant. The shape of the cavity was the standard one, made of interlocking spheres; here the radius of the cation, buried among the water molecules, is irrelevant. Adopting a spherical cavity of the same volume in connection with the point charge model defined above, one obtains much larger errors (third column of Table 3). The simple Born formula [89], treating only the cation charge in the centre of the sphere, is somewhat more reliable (lburth column of Table 3).

For a single water molecule, AGvap is related to AG~ol by

AGvap = -AGsol - RT ln( MliqRT/P) , (16)

where Mliq is the molarity of liquid water (in the ar- gument of the logarithm function, RT/P is given in 1/mol). The second term in the r.h.s. (4.27 kcal/mol) accounts for the different concentrations of gas and liquid phase in the standard states. With the same approximations applied for the complexes, we obtain AGel = -6 .92, AGcav = 4.34 and AGdis_rep = -4.01 kcal/mol. The overall result, AGvap = 2.32 kcal/mol, is in good agreement with the experimental value of 2.06 kcal/mol.

6. Results and discussion

The total AGhyd for a cation with assumed hydration number n is

AGhyd = nAGvap + AGcom + AGsol + RT ln(RT/P) ,

(17)

where, again, the term RT In (RT/P) = 1.89 kcal/mol is due to the different standard concentrations of the gas phase (RT/P) and of the solution (unit molarity). The various contributions to AGhya for several complexes, computed with the effective potential UMW or with the uncorrected potential U (°) MW, are given in Ta- bles 4 and 5, respectively. To facilitate the interpreta- tion of the results, we have grouped together all the terms which are multiplied by the factor n (number of water molecules) : AGvap = 2.32 kcal/mol, plus the rotational and translational free energies of H20 ( -2 .23 and -8 .84 kcal/mol, respectively), which contribute to AGcom with a negative sign. Therefore, the entry AGtra in Tables 4 and 5 only contains the difference in the translational free energies of the complex and of the bare cation, while AGrot belongs to the complex alone.

Altogether, for each water molecule to be included in the first shell, we have a contribution of 13.38 kcal/mol to the hydration free energy. The only other components of AGhyd showing a large and monotone change with n are the binding energy and the electrostatic contribution to AGsol. The latter, AGel, becomes less negative when increasing the size of the complex, while the smaller changes in AGcav and AGdis rep almost cancel each other. Therefore, an expansion of the first shell of solvation is thermodynamically con- venient only if the binding energy AE increases in absolute value by at least 15-25 kcal/mol for each added water molecule. This can be assumed as a rule


Table 4

Solvation free energies ( k c a l / m o l ) based on effective pair potentials

Cation n AE AGtra AGrot AGvib AGel AGcav AGdis-rep AGhyd

calc. a exp.

Li + 4 - 1 1 5 . 6 - 2 . 2 - 6 . 5 2.7 - 5 1 . 7 13.4 - 1 1 . 5 - 1 1 5 . 9 - 1 1 5 . 0

Li + 6 - 1 3 5 . 0 - 2 . 5 - 6 . 3 1.1 - 4 9 . 0 16.8 - 1 4 . 5 - 1 0 7 . 2

Be 2+ 4 - 3 8 8 . 1 - 2 . 0 - 6 . 1 13.5 - 2 1 9 . 3 12.1 - 1 1 . 1 - 5 4 5 . 7 - 5 7 4 . 6

Be 2+ 6 - 4 2 6 . 7 - 2 . 3 - 6 . 1 14.2 - 1 8 9 . 9 15.5 - 1 4 . 4 - 5 2 7 . 4 -

M g 2+ 4 - 2 5 9 . 5 - 1.2 - 6 . 5 6.5 - 2 2 0 . 2 13.4 - 11.5 - 4 2 3 . 6 M g 2+ 6 - 3 4 5 . 9 - 1 . 5 - 6 . 3 12.3 - 1 8 2 . 9 16.9 - 1 4 . 4 - 4 3 9 . 6 - 4 3 9 . 3

M g 2+ 8 - 3 6 5 . 9 - 1 . 7 - 6 . 9 12.4 - 1 6 8 . 0 19.8 - 1 7 . 6 - 4 1 9 . 1 -

Ca 2+ 6 - 2 5 8 . 3 - 1 . 2 - 6 . 5 3.4 - 1 7 4 . 2 19.0 - 1 5 . 0 - 3 5 0 . 6 -

Ca 2+ 8 - 3 0 8 . 5 - 1 . 4 - 7 . 1 6.6 - 1 6 1 . 4 21.5 - 1 7 . 7 - 3 5 9 . 0 - 3 6 2 . 1

Ca 2+ 9 - 3 2 4 . 2 - 1 . 4 - 7 . 4 5.9 - 1 5 7 . 1 22.9 - 1 9 . 2 - 3 5 8 . 2 - 3 6 2 . 1

Ni 2+ 4 - 2 7 3 . 7 - 0 . 7 - 6 . 5 5.8 - 2 1 9 . 2 13.3 - 1 1 . 5 - 4 3 7 . 0 - Ni 2+ 6 - 3 6 5 . 4 - 0 . 9 - 6 . 2 12.1 - 1 8 3 . 4 16.8 - 1 4 . 4 - 4 5 9 . 4 - 4 7 7 . 5

Ni 2+ 8 - 3 8 0 . 9 - 1 . 1 - 6 . 9 12.9 - 1 6 9 . 0 19.5 - 1 7 . 6 - 4 3 4 . 1 -

Zn 2+ 4 - 2 6 8 . 3 - 0 . 7 - 6 . 4 6.9 - 2 1 7 . 5 13.2 - 1 1 . 5 - 4 2 8 . 9 -

Zn 2+ 6 - 3 5 4 . 4 - 0 . 9 - 6 . 2 13.7 - 183.9 16.7 - 14.4 - 4 4 7 . 1 - 4 6 9 . 2

Zn 2+ 8 - 3 6 0 . 2 - 1 . 0 - 6 . 9 14.3 - 1 6 9 . 2 19.3 - 1 7 . 7 - 4 1 2 . 5 -

AI 3+ 4 - 5 8 1 . 6 - 1 . 2 - 6 . 3 14.9 - 5 1 9 . 8 12.7 - 1 1 . 3 - 1 0 3 7 . 1 -

AI 3+ 6 - 7 7 2 . 5 - 1 . 4 - 6 . 1 25.1 - 4 4 0 . 9 15.6 - 1 4 . 3 - 1 1 1 2 . 4 - 1 0 8 2 . 9

AI 3+ 8 - 7 9 2 . 6 - 1.6 - 6 . 8 28.9 - 3 9 9 . 5 18.0 - 17.4 - 1062.0 -

a F rom Eq. ( 1 7 ) , AGhy d = 13.38n + AE + AGtra -l- AGmt -I- AGvib -1- AGel + AGcav + AGdis_rep.

Table 5 Solvation free energies ( k c a l / m o l ) based on uncorrec ted pair potentials

Cat ion n AE AGtra AGmt AGvib AGel AGcav AGdis-rep AGhyd

calc. a exp.

Li + 4 - 1 3 0 . 9 - 2 . 2 - 6 . 4 2.8 - 5 1 . 6 13.4 - 1 1 . 5 - 1 3 1 . 0 - 1 1 5 . 0

Li + 6 - 1 5 5 . 3 - 2 . 5 - 6 . 3 1.8 - 4 9 . 1 16.6 - 1 4 . 5 - 1 2 7 . 0 -

Be 2+ 4 - 5 0 1 . 4 - 2 . 0 - 6 . 2 12.7 - 2 1 8 . 7 12.2 - 1 1 . 1 - 6 5 9 . 0 - 5 7 4 . 6

Be 2+ 6 - 6 1 2 . 1 - 2 . 3 - 6 . 1 15.7 - 1 8 9 . 8 15.5 - 1 4 . 4 - 7 1 1 . 3 -

M g 2+ 4 - 3 1 2 . 7 - 1 . 2 - 6 . 5 6.5 - 2 2 0 . 6 13.4 - 1 1 . 5 - 4 7 7 . 2 -

M g 2+ 6 - 4 2 4 . 6 - 1 . 5 - 6 . 2 12.8 - 1 8 3 . 3 16.9 - 1 4 . 4 - 5 1 8 . 2 - 4 3 9 . 3

Mg 2+ 8 - 4 6 2 . 5 - 1 . 7 - 6 . 9 13.2 - 1 6 9 . 0 19.6 - 1 7 . 6 - 5 1 5 . 9 -

Ca 2+ 6 - 2 8 7 . 9 - 1 . 2 - 6 . 5 3.6 - 1 7 4 . 4 18.9 - 1 5 . 0 - 3 8 0 . 4 -

Ca 2+ 8 - 3 4 6 . 3 - 1 . 4 - 7 . 1 6.9 - 1 6 2 . 0 21.2 - 1 7 . 8 - 3 9 7 . 5 - 3 6 2 . 1 Ca 2+ 9 - 365.3 - 1.4 - 7 . 4 6.0 - 157.7 22.6 - 19.1 - 4 0 0 . 0 - 362.1

Zn 2+ 4 - 3 4 3 . 1 - 0 . 7 - 6 . 4 6.7 - 2 1 7 . 7 13.2 - 1 1 . 4 - 5 0 4 . 0 - Zn 2+ 6 - 4 6 5 . 3 - 0 . 9 - 6 . 2 13.6 - 1 8 4 . 1 16.6 - 1 4 . 4 - 5 5 8 . 5 - 4 6 9 . 2

Zn z+ 8 - 5 0 1 . 3 - 1 . 0 - 6 . 9 11.5 - 1 6 9 . 9 19.0 - 1 7 . 6 - 5 5 7 . 2 - A13+ 6 - 1 0 3 0 . 3 - 1 . 4 - 6 . 1 23.0 - 4 3 8 . 5 15.8 - 1 4 . 0 - 1 3 6 9 . 3 - 1 0 8 2 . 9

A13+ 8 - -1173 .7 - -1 .6 --6.8 28.3 - -397.2 18.2 - 1 7 . 4 - 1 4 4 1 . 3 -

a From Eq. ( 1 7 ) , AGhyd = 13.38n + AE + AGtra -I- AGrot -I- AGvib "k- Aael -t- AGcav + AGdis_re p.

E FIoris et al./Chemical Physics 195 (1995) 207-220 217

of thumb to judge of the thermodynamic stability of complexes when only the binding energy is computed.

For Be 2+, Mg 2+, Ni 2+, Zn 2+ and AI 3+ we find hydration numbers in agreement with most experimental determinations [46] : the most stable structure has nh = 4 for Be 2+ and nh = 6 for the other cations. The differences in AGhya with respect to other structures are rather large, more than 15 kcal/mol. The uncorrected potential gives, for Be 2+ and A13+, larger nh; for Mg 2+ and Zn 2+, the structures with n = 6 and n = 8 have almost the same AGhyd.

The cases of Li + and Ca 2+ are somewhat more problematic. Experimentally, the most frequent coor- dination number for Li + is 4, but a tendency to larger numbers for low concentrations has been observed. We find the tetrahedral complex more stable than the octahedral one, by 8.7 kcal/mol. The Ca 2+ results, with almost equal AGhya for n = 8 and n = 9, perfectly agree with MD simulations run by us with the same potential, only computed for a larger number of geometries and fitted to a more flexible function [79]. The simulations actually show that the two structures alternate in time, with a slight prevalence of the [Ca(H20)9] 2+ complex. It is clear that any small change in the physico-chemical conditions (ionic strength, pH, temperature, pressure) may shift the balance towards one or the other side. [ Ca(H20) 6 ] 2+ lies about 8 kcal/mol higher than the two other complexes. Most X-ray diffraction experiments gave nh = 6, but nh = 9 was also found, while neutron diffraction yielded higher values, up to nh = 10 [46]. Computer simulations by other authors predict nh values slightly higher than 9, and also our uncorrected potential favours the [Ca(H20) 9 ] 2+ complex.

The computed AGhyd values are in excellent to good agreement with experiment [ 61 ]. The largest relative errors, about 5%, are obtained for Be 2+ and Zn 2+. In these two cases AGhyd is underestimated (in absolute ~alue), while for A13+ it is slightly overestimated. The very good result obtained for Ca 2+ makes us confi- dent about the structural predictions proposed above and in Ref. [79]. The AGhyd values computed with the uncorrected potential are much larger, with errors ranging from 10%, for [Ca(H20)9] 2+, to 33%, for [ A1 (H20) 8 ] 2+. Previous ab initio-MC determinations of AGhyd for Li + gave the following results: AGhyd ---=

141.5 :t_ 1.5 kcal/mol with an uncorrected pair potential [54], and AGhyd = --114 4-8 kcal/mol with a

three-body potential [ 30]. The latter value is in good agreement with experiment and with the present work, although the rather large standard deviation reflects the slow convergency of three-body MC simulations.

These results show that the effective potential is definitely superior to the uncorrected one; still there may be lesser deficiencies both in the model we set up to evaluate the free energies (reaction in the gas phase followed by solvation of the complex), and in the approximations made computing the potentials. As to the model we may list the following sources of error:

(A1) treatment of the vibrational contribution AGvib in the harmonic approximation, neglecting the coupling with rotation;

(A2) calculation of AG~ol within a continuum representation of the solvent (PCM) ;

(A3) neglect of changes in the geometry of the complex, in AGvib and in AGrot, upon solvation;

(A4) rigid model of water. Some of these points have already been discussed in the previous section; more details about the perfor- mance of continuum methods may be found in Ref. [63]. An overall validation of the model will be partially supplied by the results of MD simulations, to be published separately [80].

Easily identified sources of error related to the limited accuracy of the potentials are:

(BI) the basic assumption of additivity of pair potentials, even in the effective potential version we have presented;

(B2) the SPC/E (or any other) empirical potential for water-water interactions;

(B3) the Hartree-Fock approximation for cation- water interactions;

(B4) the basis set truncation, and in particular the basis set superposition error (BSSE) ;

(B5) the fitting of the ab initio energies with analytic functions.

On the basis of previous experience (see Refs. [45,90,37] and Section 4) we know that errors B3, B4 and B5 are of the same order of magnitude (although not of the same sign): ~1-2 kcal/mol per water molecule. In a previous work, S~inchez Marcos et al. [91 ] apply a model substantially similar to ours, but without resorting to pair potentials: the first shell binding energies are computed ab initio, at the HF 3-21G* level. Another difference with respect to the present work is the full optimisation of geometries

218

Table 6 Solvation free energies

E Floris et al./Chemical Physics 195 (1995) 207-220

(kcal /mol) based on ab initio calculations on the [ M(H20)n ]q+ complexes

HF/basis B HF/basis A MP2/basis A

AE AGrot AGvib AGhyd a AE AGhyd a AE AGhyd a

[L i (H20)4 ]+ -115.1 -7 .3 1.2 -117 .8 -113 .2 -115 .9 -116 .7 -119 .4 [Li (H20)6] + -130 .8 6.3 0.7 -103 .5 -127 .6 -100.3 -138 .3 - I l i a I Be(H20) 4 ] 2+ -398 .5 - 7 . 0 9.5 -561 .0 -399 .5 -562 .0 -399 .4 -561 .9 I Mg(H20)6 12+ -332 .8 -6 .3 8.5 -430 .6 -329 .8 -427 .6 -337 .6 -435 .4 I Ca(H20) ~ 12+ 285.2 -7 .5 3.3 -339 .7 276.4 330.8 -287 .2 -341 .6 I Ca( H20)9 12`- -295 .3 -7 .8 2.5 -333 .2 -285 .6 323.5 -300 . I -338 .0 I Ni ( H20 ) 612+ - 343.0 -6 .3 7.2 -442 .0 320.3 -419 .4 - 343.0 -442 . I I Zn (H20) 612+ -334.7 -6 .3 6.5 -434 .9 -332.1 -432.3 -354 .3 -454 .5 [ AI (H20) 6 ] 3+ -703.5 -6.1 14.2 - 1054.5 -709 .0 - 1060.0 -719 .6 - 1070.6

a From Eq. (17) , AGhyd = 13.36n + AE + AGtra + AG,-,,t + AGvib + AGel

and the determination of vibrational frequencies with bulk solvent effects taken into account. Therefore, in their treatment the inaccuracies related to points A3, A4, B 1, B2 and B5 are totally or partially eliminated. S~inchez Marcos et al. present results only for hexahy- drated cations (Be 2+, Mg 2+, Ca 2+, Zn 2+, A13+) and

lbr [Be(H20)4] 2+. Their AGhyd values are not more accurate than ours, with a maximum relative error of 9% (lk)r Zn 2+, as in our study); adding our estimate of AGdis rep, which they neglected, slightly improves the results (maximum error 6%).

We decided to investigate further the basis set and the correlation energy dependence of the computed AGhyct, by running full ab initio calculations on the most stable [M(H20) , , ] q+ complexes. In this way, we also have an independent test of the validity of the effective pair potential approach, because AE is computed at three ab initio levels: one is the same applied for UMW and lr~°) HF with the basis sets B; the sec- vMW, ond is HF with the basis sets A and the third is MP2 with the basis sets A. The AE and AGhyd results are shown in Table 6. Geometries have been fully opti- raised and normal mode frequencies computed at the HF/B level. The AGvib and AGrot entries in Table 6 are then slightly different from those in Table 4; all other contributions have been left unchanged. The AE computed at the HF/B level differ from those based on the effective cation-water potential and SPC/E model of water, by up to 10%: in most cases, the ab initio AE are slightly smaller. This fact indicates that the non additivity correction contained in UMW is not exag- gerated, on the contrary it is probably underestimated.

+ AGcav + AGdis_re p (AGcl, AGcav and AGdi s rep from Table 4).

However, some cancellation of errors apparently oc- curs in the AGhyd calculations based on the effective potentials, as it yields AGhya values more accurate than the HF/B ones. After improving the quality of the basis set and taking into account the electron correlation (MP2 /A level) we obtain results comparable in accuracy with those derived from the effective potentials. Correlation effects are generally more important than switching from DZ to TZ basis sets, and tend to increase the binding energies.

7. Conclusions

We have presented a simple and inexpensive procedure to obtain ab initio effective pair potentials tot solvated pairs. The application to cation-water interactions is successful in predicting the correct hydration numbers (when experimentally ascertained) and yields rather accurate values of the hydration free energies. The use of uncorrected pair potentials should be avoided, not only for highly charged cations. To go beyond the present level of approximation, it is advis- able to improve first the accuracy of ab initio calculations, in particular by taking into account the electron correlation: this is straightforward within our computational scheme. More sophisticated approaches, such as full ab initio calculations of [M(H20) , , ] q+ complexes or simulations based on many-body potentials, may be useful only if the ab initio level is correspond- ingly upgraded.


Acknowledgement

This work was partly supported by the EEC grant CI 1-0629.

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Free energies and structures of hydrated cations, based on effective pair potentials

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