Handbook of Solvents - George Wypych - ChemTech - Ventech!

11.1 Theoretical treatment of solvent effects 671
The calculation is performed, as usual, by comparing the coefficients of the
Schrödinger equation to successive orders.
The first order energy is the same as given by the usual Møller-Plesset treatment,
0
< 0| 0>
� H | and the first order electrostatic contribution to the free energy of solvation is
identical to the result obtained at the Hartree-Fock level of theory. The second order correction
to the free energy is given as
( 2) ( 2) m′
m m′
( 1) ( 1)
m′
m m′
ΔGs = 2∑ CS 0Ml′ S fl 0Ml′ 0 + ∑∑CDCD′
DMl′ D′
fl 0Ml′ 0 [11.1.120]
S≠0
D≠0
D′≠0
where |S� stands for the singly excited states, and |D� and |D�� for a pair of doubly excited
states different by just one orbital. Without excessive difficulty, it is possible to derive the
correction terms to the electrostatic free energy of solvation of higher orders.
A many-body perturbation theory (MBPT) approach has been combined with the
polarizable continuum model (PCM) of the electrostatic solvation. 64-66 The first approximation
called by authors the perturbation theory at energy level (PTE) consists of the solution
of the PCM problem at the Hartree-Fock level to find the solvent reaction potential and the
wavefunction for the calculation of the MBPT correction to the energy. In the second approximation,
called the perturbation theory at the density matrix level only (PTD), the calculation
of the reaction potential and electrostatic free energy is based on the MBPT
corrected wavefunction for the isolated molecule. At the next approximation (perturbation
theory at the energy and density matrix level, PTED), both the energy and the wave function
are solvent reaction field and MBPT corrected. The self-consistent reaction field model has
been also applied within the complete active space self-consistent field (CAS SCF)
theory 12,67 and the complete active space second-order perturbation theory. 12,67,68
Several groups 69-73 have also proposed the quantum mechanical density functional theory
(DFT) based methods for the calculation of the electrostatic solvation energy in dielectric
media. However, the application of this theory for excited states is not
straightforward. 74,75
11.1.4 THEORETICAL TREATMENT OF SOLVENT DISPERSION EFFECTS
ON ELECTRONIC-VIBRATIONAL SPECTRA OF MOLECULES
The dispersion interaction between two atomic or molecular systems can be theoretically
presented at different levels of theory. 76-78 The modelling of the dispersion interactions in
condensed media is more complicated and proceeds either from the discrete molecular description
of the liquid or from the continuum model. According to a contemporary classification,
4 the theoretical approaches to the dispersion effect in solutions can be divided into
following classes:
• pair-potential approaches
• reaction field based approaches
• cavity surface-dispersion energy relationship approaches
The pair-potential approach is based on the discrete representation of the pairs of solvent
and solute molecules or some fragments of them. The respective dispersion potentials
are expressed as truncated asymptotic expansions in powers of 1/r, the reciprocal of the distance
between the interacting entities 4
−k
ms( ) ∑ ms ms
k = 6810 , ,
k
U disp = d r
[11.1.121]