Handbook of Solvents - George Wypych - ChemTech - Ventech!

11.1 Theoretical treatment of solvent effects 669
where C a are the CI expansion coefficients and D a are the basis functions (Slater determinants
or their linear combinations). In the case of orthogonal basis functions, the normalization
condition of the function Ψ CI is given as
N
2
∑ Ca a=
1
= 1
and the coefficients C a are determined from the following equation
� � ( rf) [11.1.106]
H 0 + V Ψ = EΨ
[11.1.107]
where � H 0 is the Hamiltonian for the molecule, unperturbed by the reaction field and � V rf is
the reaction field perturbation. The latter can be presented, for example, in the framework of
the boundary element method as follows (cf. Eq. [11.1.75])
() r′
V� rf d =
σ 2
∫ r
r − r′
[11.1.108]
where σ( r′ ) is the charge density on the surface of the cavity.
For the spectroscopic applications, it would be again instructive to separate the
noninertial and inertial components of the electrostatic polarization of the dielectric medium.
The first of them corresponds to the electrostatic polarization of the electron charge
distribution in the solvent that is supposedly instantaneous as compared to any electronic or
conformational transition of the solute. The second component arises from the orientational
polarization of the solvent molecules in the electrostatic field of the solute. The noninertial
polarization can be described by the optical dielectric permittivity of the solvent that corre-
2
sponds to the infinite frequency of external electromagnetic field (ε ∞ ≈ n D)
whereas the inertial
polarization represents the slow, orientational part of the total dielectric constant of
the solvent, ε. In order to separate the noninertial polarization, it is helpful to determine the
solute charge density as the sum of the respective nuclear and electronic parts
= + = ∑ ( r − r ) + ∑ ( |, r r′
) =∑
ρ ρ ρ δ Z C C ab C C ρ
n e A A a b a b ab
A
ab ,
ab ,
[11.1.109]
where δ(r - r A) is the Dirac’s delta function and (ab|r,r�) are the elements of the single-determinant
matrices of transitions between the configurations. Notably, the values of
( |, )
ρ = δ ρ − ρ rr ′
[11.1.110]
ab ab n ab
do not depend on the coefficients Ca and Cb. The noninertial component of the polarization
field, Φ∞ ( r ) , is always in equilibrium and thus it can be represented as follows
( ∞)
() r =∑CC
a b () r ab
Φ Φ
∞
ab ,
[11.1.111]
( ∞) where Φab ( r) is the solution of equation [11.1.107] for the basic charge distribution ρab.
( ∞) Since the latter does not depend on the coefficients Ca and Cb, the values of Φab ( r) can be