Handbook of Solvents - George Wypych - ChemTech - Ventech!

11.1 Theoretical treatment of solvent effects 659
ΔEia = a − i − Jia + Kia a i Jia Kia
⎧ ⎫
⎨ ⎬ = − − +
⎩ ⎭
⎧ 2
2
0 0 ⎫
ε ε ε ε ⎨ ⎬ −
0
⎩0⎭
−Γψμψ � a
ψ μψ � a
− ψ μψ �
[ i i ]
0 0 0 0
[11.1.62]
where Jia and Kia are the respective Coulomb’ and exchange matrix elements and Γ is the re-
0
action field tensor at the dipole level. The terms ε i are the eigenvalues of the Fock operator
for the k-th electron in the isolated solute molecule. The off-diagonal CI matrix elements are
given by
� Γ � � � Γ � � [11.1.63]
0 0
HIJ = ψI H + ψμψ μψJ = ψI H ψJ − ψ0μψ0 ψI μψJ
Equations [11.1.60] - [11.1.63] demonstrate that some part of the solvent effect is already
included in the ordinary CI treatment when proceeding from the SCRF Fock matrix. It
has to be noticed that the terms �ψ 0 |�μ|ψ 0 � should represent the ground-state dipole moment
after CI, and therefore, an iterative procedure would be required to obtain a proper solution.
However, at the CIS (CI single excitations) level, commonly used for the spectroscopic calculations,
this is no concern because of Brillouin’s theorem, which implies that the CI does
not change the dipole moment of the molecule. Even at higher levels of excitation in CI, this
effect should not be large and might be estimated from the respective perturbation operator.
27
There are two approaches to address the instantaneous electronic polarization of the
solvent during the excitation of the solute molecule. In the first case, the following correction
term has to be added to the CI excitation energy
1
2
ΔE I = Γ(
ε ) ⎡
I I − ⎤
∞ ψ0μψ � 0 ψ μψ � ψI μψ � I
[11.1.64]
2 ⎣⎢
⎦⎥
where Γ( ε∞ ) is the reaction field tensor for the optical relative dielectric permittivity of the
solvent, ε ∞ . In the last equation, the first term removes the incorrect term arising from the
SCRF orbitals and energies in forming the CI matrix whereas the second term adds the response
of the electronic polarization of the solvent to the dipole of the excited state. Equation
[11.1.64] is first order in electron relaxation. Higher orders can be examined by
defining the perturbation
X I ()
( )
= λΓ ε
∞
[ ψ � �
0μψ0 − ψI μψI
]
2
[11.1.65]
which is clearly different for each excited state and would, if pursued, lead to a set of excited
states that were nonorthogonal. In principle, these corrections need not to be small.
Depending on the Fock operator used (equation [11.1.58] or [11.1.59]), the excitation
energy from the ground state |ψ 0 � to the excited state |ψ I � of a solute molecule in a dielectric
medium is given as follows
[ ]
A A
1
WI − W0
= ψ I H� ψ I − ψ 0 H� ψ 0 + Γ ψ 0μ� ψ ψ Iμ� 0 ψ I − ψ 0μ� ψ 0 −
2