Handbook of Solvents - George Wypych - ChemTech - Ventech!

484 Jacopo Tomasi, Benedetta Mennucci, Chiara Cappelli
model one uses; thus, for example, in the original version of PCM it coincides with the normal
component of the electrostatic field to the cavity surface, while for both the COSMO
model 91-93 and the revised PCM version, known as IEF (integral equation formalism), 99 it is
represented by the electrostatic potential. In any case, however, this term induces a dependence
of the solvent reaction to the solute charge, and finally to its wave function; in a standard
SC procedure it has to be recomputed (and thus also the apparent charges q) at each
iteration.
Also, the form of M depends on the solvation model; contrary to f, it can be computed
once, and then stored, if the geometry of the system is not modified during the calculation.
Once the apparent charges have been defined in an analytical form through eq.
[8.122], it is possible to define the perturbation V R as
( ) ()
qsk
R
V = ∫ ρρ dr
[8.123]
r − sk
In particular, by distinguishing the source of ρ, i.e., the solute nuclei and electrons, we can
always define two equations [8.122], one for each source of f, and thus compute two sets of
apparent charges, q N and q e , depending on the solute nuclei and electrons, respectively. This
allows one to partition the reaction operators in two terms, one depending on the geometry
of the system and the solute nuclear charge, and the other still depending on the geometry
and on the solute wave function.
Such partition is usually introduced to get an effective Hamiltonian whose form resembles
that of the standard Hamiltonian for the isolated system. In fact, if we limit our exposition
to the Hartree-Fock approximation in which the molecular orbitals are expressed as
an expansion over a finite atomic orbital basis set, the Fock operators one defines for the two
systems, the isolated and the solvated one, become (over the atomic basis set)
F 0 = h + G(P) F = h + j + G(P) +X(P) [8.124]
where h and G(P) are the standard one and two-electron interaction matrices in vacuo (P
represents the one-electron density matrix). In [8.124] the solvent terms are expressed by
two matrices, j and X(P), whose form follows directly from the nuclei and electron-induced
equivalents of expression [8.122], respectively; for the electronic part, ρ has to be substituted
by P.
The parallel form of F 0 and F in [8.124] shows that the two calculations can be performed
exactly in the same way, i.e., that solvent does not introduce any complication or basic
modification to the procedure originally formulated for the isolated system. This is a
very important characteristic of continuum BE solvation methods which can be generalized
to almost any quantum mechanical level of theory. In other words, the definition of
‘pseudo’ one and two-electron solvent operators assures that all the theoretical bases and the
formal issues of the quantum mechanical problem remain unchanged, thus allowing a stated
solution of the new system exactly as in vacuo.
Above, we have limited the exposition to the basic features of single point HF calculations,
and to the thermodynamic functions (the solvation free energy is immediately given
by eqs. [8.121-8.124]). However, extensions to other QM procedures as well as to other
types of analysis (the evaluation of molecular response functions, the study of chemical re-