INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY, VOL. XXVII, 293-301 (1985)

On the SCF Theory of Continuum Solvent Effects Representation: Introduction of

Local Dielectric Effects RENATO CONTRERAS AND ARIE AIZMAN

Departamento de Quimica, Faculrad de Ciencias Basicas y Farinuceuticas, Universidad de Chile, Casilla 653, Santiago, Chile

Abstract

The introduction of local dielectric effects within the SCF theory of continuum solvent effects representa- tion is examined at a semiempirical level. The formalism is developed in the frame of the reaction field theory within the effective charge approximation. The solvation free energies of Li', Na', F-, and C1- ions in water were calculated in order to illustrate the reliability of the proposed model. The extension to molecules and molecular ions was performed including a desolvation corrective term related to the specific neighborhood of each atomic center. The results show a qualitative agreement with experimental data. A comment on the solvatonlike models for incorporating the solvent effect into the Hamiltonian is also given.

1. Introduction

Solvation in aqueous and nonaqueous media has been the object of extensive re- search [l-31. Within the continuum models, the Born formula [4] has been the basis for most calculations of solvation free energies [ 1,2].

Quantum chemical schemes based on Born's formula have been proposed in the literature [5-71. They are known as effective charge models [8] and were developed in order to extend the calculation of solvation energies to molecules and molecular ions. In this approach, these energies are expressed as a sum of Born-like terms each one corresponding to the partially charged atomic centers of the solute. Within a cm0/2-like approximation, we have

which is known as the generalized Born formula (GBF). In this expression, QA(P) = ZA - xpdPpp are the net charges associated to the atomic A centers of the solute, yAB are the two-electron Coulomb integrals representing the solute-solvent interaction and E~ is the bulk dielectric constant of the solvent. The GBF has been used within a perturbation theory scheme by Kloprnan [5] (the solvaton model of solvent effects representation).

0 1985 John Wiley & Sons, Inc. CCC 0200-7608/85/030293-09$04.00

294 CONTRERAS AND AIZMAN

Several extensions of this approach, implemented within the SCF scheme of calcu- lation have been proposed up to now [6-91. The main problem in these models is the definition of an appropriate effective Fock operator compatible with Eq. (1). For instance, Miertus and Kysel [lo] have commented Germer’s method in which the effective Hamiltonian of the solute in the field of the solvent is written as

[Fori definitions of the quantities involved in Eq. (2) , see Ref. 10.1 They have noted that the application of Germer’s model leads to positive hydration energies for Lit and F- ions. By using the same Hamiltonian but with a minus sign preceding the solute-solvent term, they have restored the stabilizing character of these energies.

Nevertheless, Miertus and Kysel’s Hamiltonian is not satisfactory because the polarization energy of the solvent (or solvaton self-energy) has been neglected ac- cording to the original formulation of Klopman’s model [5]. In the frame of the CND0/2 approximation, Miertus and Kysel’s effective Fock operator can be written as

where F(1,P) is the Fock operator for the isolated solute. (The off-diagonal matrix elements of the Fock operator do not suffer any modification in this approach.)

The solvent polarization energy was considered previously by Constanciel and Tapia in the virtual charge model of solvent effects, but the reaction potential ob- tained in this last approach was incompatible with the GBF [7].

This problem was solved recently by Tapia [9] and further discussed in the frame of the reaction field theory (RF model hereafter) by Constanciel and Contreras [ 111. The effective Fock operator compatible with the GBF including the solvent polariza- tion energy is given in both approaches by

Comparing expressions (4) and (3) it may be observed that a sporious factor 1. 2 appears in Miertus and Kysel’s formulation as a consequence of neglecting the solvent polar- ization energy [9, 111.

An extension of the RF model that takes into account the specific influence of the neighborhood of each atomic center of solvated molecules has been proposed by Constanciel and Contreras 1121. In this approach, the solvent is represented by a set of “polarization charges” characteristic of a particular atomic A center as follows:

wherefA is a parameter measuring the specific steric hindrance effect to solvation of the atomic A center. fA has been represented empirically by

SCF THEORY OF SOLVENT EFFECTS 295

where SAB is the overlap integral between the 2s atomic orbitals (1s for hydrogen atom) [13].

Several attempts to extend the Born formula in order to include local dielectric ef- fects without abandoning the simplicity of this approach have been worked out classi- cally [ 14-16]. These local effects are produced by the strong interaction between the ion and the primary solvation layer. Accordingly, the value of the dielectric constant E in the vicinity of the ion may be significatively lower than the bulk value consid- ered in the original Born formula.

To take into account this drop in E it has been proposed to replace in the Born expression E~ by an effective E value (eeff) [ 161, or by a dielectric constant continu- ously varying with the distance r from the ion, E(r) [17]. An alternative to this procedure implies the partition of the continuum into concentric shells characterized by different dielectric constants [ 181. This last approach simulates within a continuum framework the so-called “chemical models,” where at least the first layer of solvent is explicitly considered.

In this article we present a quantum chemical formalism within the SCF approxima- tion that includes local dielectric effects (LRF model hereafter), via a partition of the continuum based on the classical model recently proposed by Beveridge and Schnnuelle [MI. In order to test the reliability of this method, calculation of hydra- tion energies for Li’, Na’, F-, and C1- ions were performed with both RF and LRF models. The extension to molecules and molecular ions is illustrated by performing RF, LRF, and LRF calculations that include the overlap correction (LRF/O model) on molecular systems. The results are compared with experimental data and to the re- sults reported by Miertus and Kysel [ 101.

2. Introduction of Local Dielectric Effects

A . Derivation of the Reaction Potential Expression: Classical Aspects

solute represented by a discrete charge distribution, {qk}k = l ,N, is given by In the frame of the reaction field theory, the electrostatic solvation free energy of a

The solute is supposed to be enclosed in a spherical cavity S(0, a) imbedded in a con- tinuum polarizable medium characterized by a dielectric constant E ~ . VR(rk, E ~ ) is the reaction potential induced by the dissolved charges and acting back on themselves.

According to Beveridge and Schnnuelle model, the continuum is now partitioned into concentric shells characterized by a region where a local dielectric constant cL is defined and a region characterized by a bulk dielectric constant eB (see Fig. 1). The solvation free energy is given in this case by

(n + 1 ) ( 1 - E,,) Qn (n + l ) ~ , , + n a’“+

(n + 1)(1 - &b) 1 - n ( l - E,,) Qn + n = O

(8)

(we have conserved the same notation used in Ref. 18). In EQ. (8), the Qn are the electric moments of order n: a and b are the radii of the internal and external cavities,

296 CONTRERAS AND AIZMAN

Figure 1. Definition of parameters for the partition of the dielectric continuum.

respectively (see Fig. 1 ) . The quantities 8,. and &b are related to the local and bulk dielectric constants by

and

respectively.

Eq. (8) reduces to In the frame of the effective charge approximation, (i.e., for n = 0 and Qo = 4 4 ,

where V, (E , , , & b ) is the reaction potential acting on the charge distribution whose net charge is qo. By using expressions (9) and (10) and putting c0 = 1 , VR(&,,, & b ) may be expressed in terms of the local and bulk dielectric constants as follows:

We may then consider that the polarization of the concentric dielectric continua is re- duced to the creation at the surfaces S(0, a) and S(0, b) of two sets of polarization charges:

4Y' = -(I - ;)qo

and

SCF THEORY OF SOLVENT EFFECTS 297

respectively. With expressions (13) and (14) in mind and putting b = a + R, we get, after a little algebra,

Equation (15) shows that the reaction potential may be separated in two contribu- tions: one coming from the “internal” polarization charge distribution in the local dielectric region and the other coming from the “external” polarization charge distri- bution, corrected by the screening effect of q?’, induced in the bulk region. Using Eq. (15), the solvation free energy AA(eL, E ~ ) may be rewritten as

B . The Effective Fock Operator Including Local Dielectric Effects

into account overlap effects [12,13], Remaining within the GBF framework, we may write, based on Eq. (16) and taking

and

representing the internal and external reaction potentials, respectively. The effec- tive Fock operator is then derived by seeking the minimum of the total solute-solvent free energy

A(&L,&B,P) = E(1,P) + AA(&L,&B,P) (20)

through the variational method. The quantity E ( 1 , P) in Eq. (20) represents the total energy of the isolated solute. By using a methodology proposed recently [11], we obtain for the effective Fock operator the expression

[FLW’O(&L, EB, PI,, = [F(1, P)Ipp

+ - - cQB(P)[l - (f4 +fB) + 2fAfB)I(yAB - TEA) ( ‘ J B

298 CONTRERAS AND AIZMAN

where the F A A solute-solvent interaction integrals are related to the respective 3/AA

integrals by

and the T A B integrals are calculated as

r,= R + - - + - - 1 . [ AB r 2 l If local dielectric and overlap effects are neglected we have

and the RF expression (4) of the effective Fock operator is then restored

3. Discussion

A. Monoatomic Ions

The calculated and experimental free energies of Li', Na', F-, and C1- ions in water will be used to discuss some points introduced in previous sections.

It is well known that Born's formula overestimates the free-energy changes for hy- dration of ions when the bulk value for the dielectric constant and the crystallo- graphic radii of the ions are used [ 151. The implementation of a quantum-mechanical analog of this continuum approach should give then results that are lower bounds to the experimental free energies of hydration. For monoatomic ions the comparison of the classical and quantum expression of the Born formula allows the identification of YAA with 1/ri, where r; is the ionic radius. If the original CND0/2 3/AA integrals are used, incorrect qualitative results in a calculation of solvation free energies are to be expected because these integrals do not reproduce the trend in the experimental radii Li' < F- - Na' < C1- [3], Accordingly, the solute-solvent integrals YAA have been parametrized as 1/ri. In the simple case of solvation of monoatomic ions, this choice allows a numerical testing of a correct quantum-mechanical implementation of both the original GBF and the version including local effects (classical and quantum results must be identical).

Recently, Miertus and Kysel have reported theoretical calculations for hydration energies for Li' and F- which are shown in the first column in Table I. These results were obtained using the original CND0/2 3/AA integrals (OMK method). In order to make significative comparisons, we have recalculated, using Miertus and Kysel's Hamiltonian within the reaction field framework, the hydration energies for these ions plus Na' and C1- cases with YAA parametrized according to the present work (MMK method). The results are displayed in the second column of Table I. It may be seen that the MMK values are above the experimental results shown in column five of Table I. The correct qualitative trend has been restored using the RF model as shown in the third column of Table I, where all the theoretical results are below the experimental values.

SCF THEORY OF SOLVENT EFFECTS 299

TABLE I. Hydration free energies of Li', Na', F-, and C1- ions."

Solute ion Calculated hydration free energies AA expLc

O M K ~ MMK RF LRFd Li+ -306.5 -368.6 -737.3 -488.2 -516.7

Na' . . . -293.0 -586.0 -406.0 -411.3

F- - 1226.0 -295.5 -591.1 -408.8 -434.3

-209.0 -418.1 ... -317.2 c1-

-651.4

-523.9

-528.0 ...

"All values are given in kJ mol-'. bFrom Ref. 10. Values corresponding to hydration enthalpies. 'From Ref. 2. dFirst value corresponds to E~ = eq and the second to E~ = E,.

In a model where the radii of the ions is supposed to be known, it could be argued that the differences between the theoretical RF results and the experimental free- energy values show the extent of local dielectric effects. From this perspective our RF results are qualitatively correct. They show local effects to be less important in the order C1- < Nat - F- < Li' (our model does not discriminate between cations and anions having the same crystallographic radii).

Results using the LRF model are displayed in fourth column in Table I. Two choices of cL were used: E~ = cop = 1.8 [19,20] and E~ = E~ = 5.0 [16]. The inter- action integral r A A has been calculated according to Eq. (22) with Rs equal to the diameter of Q water molecule (i.e., 2 . 8 0 A).

For small cations like Lit and Na' and for F- ion, local dielectric effects are ex- pected to be important as a consequence of their strong interaction with the first solvation layer. Accordingly, it may be considered that the polarization induced in this region of the solvent is essentially electronic. The LRF results are consistent with the above argument: the choice eL = cop gives a better description of the solva- tion free energies of these ions in agreement with empirical results obtained by Glueckauf [14]. Since for large halide ions, microwave studies do not show a signif- icative depression of the bulk dielectric constant of water in the vicinity of the ions [ 161, we are not reporting LRF calculations on C1-.

B. Molecules and Molecular Ions

Results of calculations on HF, NH3, NHf , and CH3NH: systems are shown in Table 11. The different models of solvation (RF, LRF, and LRF~O) are illustrated in Figure 2 for the HF case. The values of Rs and cL are the same to those used for monoatomic ions. The solute-solvent interaction integrals y A B have not been parame- trized but the original CND0/2 integrals were used. The r A B integrals have been cal- culated according to Eq. (23).

For a neutral molecule as compared to small monoatomic ion, the interaction with the first solvation is expected to be weaker. For a large ion as NH: and CH3NH: the same behavior is to be expected (remember that this effect is essentially governed

300 CONTRERAS AND AIZMAN

TABLE 11. Hydration free energies for some molecules and molecular ions.a

I I1 111 Solute RF LRFC LRF/Od Expt.b

HF -283.6 - 169.4 -30.2 -61.5 -70.4

-43.9

-355.1

-324.8

NH3 -123.1 -66.1 -18.7 -30.5

NH: -581.9 -513.7 -209.7 -347.5

CH,NH: -564.9 -412.3 -208.2 -316.4

“All values are given in W/mol. bValue for HF from Ref. 10, the remaining values from Ref. 21. ‘Corresponds to eL = E+

dFirst value corresponds to eL = ew and the second to eL = ei.

by the ratio q / r ) . The above physical considerations should be reflected in our dielectric model by a value of cL > cop. Table I1 shows that the best description is accomplished within the LRF/O model with cL = cir > cop in agreement with the above argument.

It is interesting to note that for all the systems studied in this work, upper and lower bounds to the experimental values of the solvation free energies are obtained by using EL = .sir and E~ = cop, respectively.

4. Concluding Remarks

We have shown how to include local dielectric effects via a partition of the con- tinuum within a SCF formalism. Our interest was to discuss the reliability of a continuum approach which simulates the “chemical model” of solvent effects repre- sentation, remaining within the simplicity of the Born formula. These preliminary results suggest the feasibility of applying this kind of procedure to those cases where the application of a supermolecule model becomes prohibitive. We are currently ap-

I 11 Figure 2. Solvation models for HF.

III

SCF THEORY OF SOLVENT EFFECTS 301

plying this model to an extensive array of molecules and molecular ions. Complete calculations will be soon submitted for publication.

Acknowledgments

We thank Dr. J. E. Sanhueza and Dr. R. Constanciel for helpful discussions. We are also indebted to Professor R. Daudel and Professor G. Klopman for the reading of the manuscript. This work was supported by the DDI office, University of Chile, Project No. 4-1759-8312.

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Received May 15, 1984 Accepted for publication May 23, 1984