Handbook of Solvents - George Wypych - ChemTech - Ventech!

664 Mati Karelson
vector, containing the unknown surface charges. Ein is also a column vector collecting the
effective components of the solute electric field multiplied by the surface elements
�
0
E =−ΔS ∇Φ
s n [11.1.81]
( )
in, k k M, in k k
The D matrix depends only on the shape of the cavity and the dielectric constant of the
medium. Therefore, when the system of linear equations [11.1.80] has to be solved several
times with different E in, as in the case of the polarizable solute, it may be convenient to work
with the single inversion matrix D -1 .
The PCM model has been implemented for the calculation of the electronic excitation
energies of solvated molecules within the quantum-mechanical configuration interaction
method. 47,48 The respective final expression for the excitation energy from the ground state
(0) to the I-th state has the following form
1
ΔW = ΔE
− − + − +
( 0I) ( 0I)
CI [ J20( PI 2
P0) PT 0 0( PI P0)
]
[ J2∞( PI P0) P IT∞( PI − P0)
]
1
+ − +
2
[11.1.82]
where P I and P 0 are the electronic density matrices of the solute in the excited state and in
the ground state, respectively,
[ ( ) ( ) ( ) ]
( 0I)
1
ΔECI EI E0
I 0 0 0 I 0 10 20 I 0
2 2
= − + P − P T P + P − P J + J P −P
[11.1.83]
are the diagonal elements of CI matrix, and
−
T = C W C
t 1 ∂ � [11.1.84]
−1
J = CW ∂MZ � [11.1.85]
1
t t
J = Z M W
2
−1
−
B= Z M W MZ
∂C� [11.1.86]
t t 1 ∂ � [11.1.87]
In the last equations, C and ∂ � Care the matrices representing the electrostatic potential
and the electric field generated from the electron distribution in the solute molecule, respectively.
The matrices M and ∂ � Mare the matrices representing the electrostatic potential and
the electric field generated from the nuclear charges in the solute molecule, respectively.
The diagonal elements of the matrix W are defined as the following function of the dielectric
constant of the solvent
Wii = + ε 1
ε − 1
[11.1.88]
The subscripts 0 and ∞ in the J 1, J 2, and T matrices in equation [11.1.82] correspond to
the static and optical dielectric constant of the solvent. Equation [11.1.82] can be considered