Handbook of Solvents - George Wypych - ChemTech - Ventech!

8.7 Theoretical and computing modeling 483
to be added to the Hamiltonian H 0 so to obtain an effective Hamiltonian H which will satisfy
a different Schrödinger equation with respect to [8.119], namely, we have
HΨ = EΨ
[8.120]
Both the wave function and the eigenvalue E will be modified with respect to [8.119], due to
the presence of the solvent perturbation.
An important aspect, until now not introduced, is that the solvent apparent charges,
and consequently the reaction operator V R , depend on the solute charge, i.e., in the present
QM framework, on the wave function they contribute to define. This mutual interactions between
Ψ and V R induces a complexity in the problem which can be solved through the standard
iterative procedures characterizing the self-consistent (SC) methods. Only for an
aspect the calculation in solution has to be distinguished from that in vacuo way; the energy
functional to be minimized in a variational solution of [8.120] is not the standard functional
E but the new free energy functional G
G H V R
0
= Ψ + 1/ 2 Ψ
[8.121]
The 1/2 factor in front of V R accounts for the linear dependence of the operator on the
solute charge (i.e., the quadratic dependence onΨ). In a more physical description, the same
factor is introduced when one considers that half of the interaction energy has been spent in
polarizing the solvent and it has not been included in G.
In the standard original model the perturbation V R is limited to the electrostatic effects
(i.e., the electrostatic interaction between the apparent point charges and the solute charge
distribution); however, extensions to include dispersion and repulsion effects have been
formulated. In this more general context the operator V R can be thus partitioned in three
terms (electrostatic, repulsive and dispersive), which all together contribute to modify the
solute wave function.
For clarity’s sake in the following we shall limit the analysis to the electrostatic part
only, referring the reader to refs. [31,83], and to the original papers quoted therein, for details
on the other terms. For a more complete report we recall that nonelectrostatic effects
can be also included a posteriori, i.e., independently of the QM calculation, through approximated
or semiempirical models still exploiting the definition of the molecular cavity.
In PCM and related methods, the term corresponding to the energy spent in forming the solute
cavity in the bulk liquid, is usually computed using the SPT integral equation method
(see Section 8.7.1.1).
Until now no reference to the specific form of the operator V R has been given, but here
it becomes compulsory. Trying to remain at the most general level possible, we start to define
the apparent charges q(s k) through a matrix equation, i.e.,
()
−1
q =−M
f ρ [8.122]
where M is a square matrix depending only on the geometry of the system (e.g., the cavity
and its partition in tesserae) and the solvent dielectric constant and f(ρ) is a column vector
containing the values of a function f, which depends on the solute distribution charge ρ,
computed on the cavity tesserae. The form of this function is given by the specific solvation