Handbook of Solvents - George Wypych - ChemTech - Ventech!

430 Jacopo Tomasi, Benedetta Mennucci, Chiara Cappelli
ing then a difference of the energies (in other words: the same strategy adopted for the preceding
terms, but changing the matrix blocks). In the Kitaura-Morokuma scheme CT also
contains the couplings between induction and exchange effects.
We consider unnecessary to summarize the technical details; they can be found in the
source paper 4 as well as in ref. [3] for the version we are resuming here.
In standard perturbation theory (PT) methods, the CT term is not considered; there are
now PT methods able to evaluate it but they are rarely used in the modeling of interaction
potentials for liquids.
The dispersion term
The dispersion energy contribution DIS intuitively corresponds to electrostatic interactions
involving instantaneous fluctuations in the electron charge distributions of the partners:
these fluctuations cancel out on the average, but their contribution to the energy is different
from zero and negative for all the cases of interest for liquids. The complete theory of these
stabilizing forces (by tradition, the emphasis is put on forces and not on energies, but the
two quantities are related) is rather complex and based on quantum electrodynamics concepts.
There is no need of using it here.
The concept of dispersion was introduced by London (1930), 5 using by far simpler arguments
based on the application of the perturbation theory, as will be shown in the following
subsection. A different but related interpretation puts the emphasis on the correlation in
the motions of electrons.
It is worth spending some words on electron correlation.
Interactions among electrons are governed by the Coulomb law: two electrons repel
each other with an energy depending on the inverse of the mutual distance: e 2 /rij, where e is
the charge of the electron. This means that there is correlation in the motion of electrons,
each trying to be as distant as possible from the others. Using QM language, where the electron
distribution is described in terms of probability functions, this means that when one
electron is at position rk in the physical space, there will be a decrease in the probability of
finding a second electron near rk, or in other words, its probability function presents a hole
centered at rk. We have already considered similar concepts in discussing the Pauli exclusion principle
and the antisymmetry of the electronic wave functions. Actually, the Pauli principle
holds for particles bearing the same set of values for the characterizing quantum numbers,
including spin. It says nothing about two electrons with different spin.
This fact has important consequences on the structure of the Hartree-Fock (HF) description
of electrons in a molecule or in a dimer. The HF wave function and the corresponding
electron distribution function take into account the correlation of motions of electrons
with the same spin (there is a description of a hole in the probability, called a Fermi hole),
but do not correlate motions of electrons of different spin (there is no the second component
of the electron probability hole, called a Coulomb hole).
This remark is important because almost all the calculations thus far performed to get
molecular interaction energies have been based on the HF procedure, which still remains
the basic starting approach for all the ab initio calculations. The HF procedure gives the best
definition of the molecular wave function in terms of a single antisymmetrized product of
molecular orbitals (MO). To improve the HF description, one has to introduce in the calculations
other antisymmetrized products obtained from the basic one by replacing one or
more MOs with others (replacement of occupied MOs with virtual MOs). This is a proce-