422 Hirata et al.

The second term of the right-hand side of Eq. (13) corresponds to the change of the solute– solvent distribution function due to the modification of the intramolecular correlation function ω. Other notations used here have the usual meanings. It has been well recognized that the energy gradient technique in the ab initio electronic structure theory is a powerful tool for investigating the mechanism of chemical reactions of polyatomic systems, and it opens up a variety of applications to the actual chemical processes in solution: carrying out the geometric optimization of reactant, transition state, and product in the solvated molecular system; constructing the free energy surfaces along the proper reaction coordinates; computing the vibrational frequencies and modes; and so on.

In analyzing the computational results, the following quantities are very important:

where Eisolate is the total energy of the solute molecule in an isolated condition and Esolute is the energy of the solute molecule defined above. The quantity Ereorg represents the reorganization energy associated with the relaxation or distortion of the electronic cloud and molecular geometry in solution.

Now we have the tools in hand to tackle various problems in solvated molecules. In the following sections, we present our recent efforts to explore such phenomena by means of the RISM-SCF/MCSCF method.

III.SOLVATION EFFECT ON A VARIETY OF CHEMICAL PROCESSES IN SOLUTION

A. Molecular Polarization in Neat Water*

The molecular and liquid properties of water have been subjects of intensive research in the field of molecular science. Most theoretical approaches, including molecular simulation and integral equation methods, have relied on the effective potential, which was determined empirically or semiempirically with the aid of ab initio MO calculations for isolated molecules. The potential parameters so determined from the ab initio MO in vacuum should have been readjusted so as to reproduce experimental observables in solutions. An obvious problem in such a way of determining molecular parameters is that it requires the reevaluation of the parameters whenever the thermodynamic conditions such as temperature and pressure are changed, because the effective potentials are state properties.

One of the most efficient ways to treat this problem is to combine the ab initio MO method and the RISM theory, and this has been achieved by a slight modification of the original RISM-SCF method. Effective atomic charges in liquid water are determined such that the electronic structure and the liquid properties become self-consistent, and along the route of convergence the polarization effect can be naturally incorporated.

The temperature dependence of the effective charges and dipole moment of water

* This discussion is based on Ref. 15.

T / K

Figure 1 Temperature dependence of the reorganization energy (Ereorg) and effective charges on oxygen atom based on ( , ■) SPC and ( , ) TIP3P models.

are plotted in Figure 1. The parameters associated with the short-range part of the interaction and geometry are borrowed from two typical models of water, SPC and TIP3P. In both models, the magnitudes of the effective charges and dipole moment monotonically decrease with increasing temperature. The results can be explained in terms of the increase in molecular motion, especially rotational motion, with increasing temperature. As the motion of a molecule (molecule A) increases, the average electrostatic field produced by the surrounding water molecules becomes less anisotropic, which decreases the polarity of molecule A. Conversely, the reaction field from the water (molecule A) become more isotropic, which decreases the polarity of other molecules.

The pair correlation function of water has a marked feature that distinguishes water from other liquids (Fig. 2). One of the important features characterizing the liquid water

˚

structure is a peak around r 1.8 A observed in the oxygen–hydrogen (O–H) pair, which is a direct manifestation of the hydrogen bond between a pair of water molecules. Another feature is the position of the second peak in the oxygen–oxygen (O–O) PCF, which is caused by the tetrahedral icelike coordination. Since the icelike structure becomes less pronounced as temperature increases because of the thermal disruption of the hydrogenbonded network, those features in PCF become less prominent.

B. Autoionization of Water*

A water molecule has amphoteric character. This means it can act as both an acid and a base. The autoionization equilibrium process in water,

* This discussion is based on Ref. 16.

The free energy change associated with the reaction in Eq. (15) can be written in terms of the energy change associated with the reaction in vacuo (∆Gvac) and the free energy change of the reacting species due to solvation as

where δG(H3O ), δG(OH ), and δG(H2O) are, respectively, the free energy changes of H3O , HO , and H2O upon solvation. It is also possible to decompose ∆Gaq into intraand intermolecular contributions as

where ∆Eelecvac , ∆δGkin, and ∆δGreorg are electronic energy in vacuo, kinetic free energy, and electronic reorganization energy, respectively, which are intramolecular contributions. ∆δµ is the solvation free energy change. (We use ∆ for changes of quantities associated with the chemical reaction and δ for changes due to solvation.)

The value of pKw at temperature T relative to that at T 273.15 K, given by

is further decomposed into four contributions corresponding to the free energy components:

The resultant ∆T pKw(T) values and their components are plotted in Figure 3. As shown

Figure 3 Temperature dependence of calculated pKw. Dashed line indicates experimental values.