Handbook of Solvents - George Wypych - ChemTech - Ventech!

656 Mati Karelson
media. The respective interaction energy is accounted for as a perturbation � V(a 0 , ε) of the
Hamiltonian of the isolated solute molecule, � H 0 .
� � 0
H= H + V�
a , ε [11.1.41]
( )
0
Within the approximation of electrostatic interaction between the solute dipole and the respective
reaction field, the perturbation term is simply
where
� 2
V( a , ε) =Γ μ� a
[11.1.42]
Γ=
0
21 ( −ε)
( ε+ ) a
3
2 1 0
[11.1.43]
In the case of ellipsoidal cavities, the last coefficient has to be substituted by the tensor
given in equation [11.1.35].
A self-consistent reaction field method (SCRF) has been developed at the level of
Hartree-Fock theory to solve the respective Schrödinger equation 25
�HΨ =E Ψ
[11.1.44]
Proceeding from the classical expression for the electrostatic solvation energy of a solute
molecule in a dielectric medium in the dipole-dipole interaction approximation, the total
energy of the solute is presented as follows 26
� � 2
( � � 2 � nuc nuc)
E E o 1
= − Γ ψμψ ψμψ + μ ψμψ + μ
2
[11.1.45]
where E o =, H ∧ 0 is the Hamiltonian for the reaction field unperturbed solute molecule
and ψ is the molecular electronic wave function. From the last equation, one can construct
the variational functional
� � 2
( � � 2 � nuc nuc)
( | 1)
o 1
L = E − Γ ψμψ ψμψ + μ ψμψ + μ −W ψ ψ −
2
[11.1.46]
where W is the Lagrange multiplier ensuring the normalization of the variational wave
function. The variation of the last equation with respect to the parameters of the wave function
yields
�
( nuc ) ( )
o
δL = δE − Γ δψμψ � ψμψ � + μ ψμψ � −Wδψ|
ψ
= δψ H�ψ− δψ μ� ψ μ − δψ| ψ + . . =
0
�
Γ
tot W c c 0
[11.1.47]
where μ� μ�
tot = nuc + is the total dipole moment of the solute molecule. The latter
is calculated during the SCRF procedure simultaneously with the total energy of the system.