Handbook of Solvents - George Wypych - ChemTech - Ventech!

440 Jacopo Tomasi, Benedetta Mennucci, Chiara Cappelli
unperturbed Hamiltonian having |A AB Φ 0 > as eigenfunction. Changes in the definition of
the unperturbed Hamiltonian are no rare events in the field of application of the perturbation
theory and there are also other reasons suggesting a change in the definition of it in the study
of dimeric interactions. This strategy has been explored with mixed success: some improvements
are accompanied by the lack of well-defined meaning of the energy components
when the expansion basis set grows toward completeness. A second way consists in keeping
the original, and natural, definition of H 0 and in changing the PT. This is the way currently
used at present: the various versions of the theory can be collected under the acronym SAPT
(symmetry adapted perturbation theory)
It is worth remarking that in the first theoretical paper on the interaction between two
atoms (or molecules) (Eisenschitz and London, 1930 14 ) use was made of a SAPT; the problem
was forgotten for about 40 years (rich, however, in activities for the PT study of weak
interactions using the standard RS formulation). A renewed intense activity started at the
end of the sixties, which eventually led to a unifying view of the different ways in which
SAPT may be formulated. The first complete monograph is due to Arrighini 15 (1981), and
now the theory can be found in many other monographs or review articles. We quote here
our favorites: Claverie 16 (1976), a monumental monograph not yet giving a complete formal
elaboration but rich in suggestions; Jeziorski and Kolos 17 (1982), short, clear and critical;
Jeziorski et al. 13 (1994), clear and centered on the most used versions of SAPT.
SAPT theories are continuously refined and extended. Readers of this chapter surely
are not interested to find here a synopsis of a very intricate subject that could be condensed
into compact and elegant formulations, hard to decode, or expanded into long and complex
sequences of formulas. This subject can be left to specialists, or to curious people, for which
the above given references represent a good starting point.
The essential points can be summarized as follows. The introduction of the
intra-monomer antisymmetry can be done at different levels of the theory. The simplest formulation
is just to use |A ABΦ 0 > as unperturbed wave function, introducing a truncated expansion
of the antisymmetry operator. This means to pass from rigorous to approximate
formulations.
One advantage is that the perturbation energies (see eq. [8.30]), at each order, may be
written as the sum of the original RS value and of a second term related to the introduction
of the exchange:
( n)
( n)
( n)
E = E + E
[8.41]
RS
exc
The results at the lower orders can be so summarized:
() () ( ) ( )
{ RS exc} { RS exc}
1 1 2 2
ΔE = E + E + E + E + � [8.42]
(1)
Eexc does not fully correspond to EX obtained with the variational approach. The main reason
is that use has been made of an approximation of the antisymmetrizer: other contributions
are shifted to higher order of the PT series. A second reason is that use has been here
p
made of the original MOs ϕito be contrasted with the polarized ones ϕi , see Section 8.4.1:
for this reason there will be in the next orders contributions mixing exchange and polarization
effects.
At the second order we have: