Handbook of Solvents - George Wypych - ChemTech - Ventech!

670 Mati Karelson
determined before the calculation of the wavefunction Ψ CI [11.1.105]. The coefficients C a
can be calculated, as usual, from the matrix equation
H|C = E|C [11.1.112]
where |C denotes the column-vector of the coefficients and the elements of the matrix H are
given as follows 61
∑
ab a
b a b∫ ab
( )
ab ∫ ab
ab ,
0 3 ∞ 3
H = D H D + C C d rρ Φ + d rρ
Φ [11.1.113]
where Φ0 is inertial (nuclear) part of the polarization field. For a given Φ0, the set of equations
[11.1.98) can be solved iteratively. Implicitly, the last equations describe both the
electron subsystems of the solute and solvent, the latter being taken into account as the field
( )
of noninertial (electron) polarization of the solvent, Φ ∞ .
The electron correlation effects on the solvation energy of a solute have been also accounted
for within the framework of the perturbation theory. 62,63 By starting from the
Hamiltonian of the solute molecule as follows
H = H
0 m m m′
− MlflψMlψ [11.1.114]
′
where ψ denotes the exact (correlated) wavefunction, the Hartree-Fock operator may be
written as
0 m m
m′
F= F − Mlflψ0Mlψ ′ 0
[11.1.115]
where ψ 0 denotes the electronic wavefunction at the Hartree-Fock level. The Hamiltonian
may be then written as a perturbed expression of the Hartree-Fock operator
0 0 m m
m′
m′
( ) Mlfl( 0Ml′ 0 Ml′
)
H = F+ H − F + ψ ψ − ψ ψ [11.1.116]
The perturbation has two contributions, the standard Møller-Plesset perturbation and
(i)
the non-linear perturbation due to the solute-solvent interaction. If C j denotes the coefficient
of the eigenstate |J in the corrections of ψ to the i-th order, then the perturbation operators
H (i) of the i-th order are given by the following formulae
() 1
H H 0 0
= − F
[11.1.117]
( 2) H =− 2
( 1)
I
m m
l l 0
m′
l′
I≠0
∑C M f M I
( 3) ( 2)
m m m′
( 1) ( 1)
m m
H =−2 0 −
J l l l′
∑∑ J P l l
J≠0
J≠0
P≠0
∑C M f M J C C M f IM
0
m′
l′
P
[11.1.118]
[11.1.119]