Handbook of Solvents - George Wypych - ChemTech - Ventech!

8.4 Two-body interaction energy 439
We have here explicitly written the monomer wave functions, with a generic indica-
A
tion of their electronic definition: Ψ0 is the starting wave function of A, already indicated
A
with ΨA, ΨK is another configuration for A with one or two occupied one-electron orbitals
replaced by virtual MOs.
The first order contribution exactly corresponds to the definition we have done of ES
with equation [8.16]: it is the Coulomb interaction between the charge distributions of A
and B.
The second term of the expansion, i.e., the first element of the second order contributions,
corresponds to the polarization of A, due to the fixed unperturbed charge distribution
of B; the next term gives the polarization of B, due to the fixed charge distribution of A. The
two terms, summed together, approximate IND. We have already commented that perturbation
theory in a standard formulation cannot give IND with a unique term: further refinements
regarding mutual polarization effects have to be searched at higher order of the PT
expansion.
The last term of the second order contribution is interpreted as a dispersion energy
contribution. The manifold of excited monomer states is limited here at single MO replacements
within each monomer; the resulting energy contribution should correspond to a preliminary
evaluation of DIS, with refinements coming from higher orders in the PT
expansion.
Remark that in PT methods, the final value of ΔE is not available. It is not possible here
to get a numerical appraisal of correction to the values obtained at a low level of the expansion.
All contributions are computed separately and added together to give ΔE.
In conclusion, this PT formulation, which has different names, among which “standard”
PT and RS (Rayleigh-Schrödinger) PT, gives us the same ES as in the variational
methods, a uncompleted value of IND and a uncompleted appraisal of DIS: higher order PT
contributions should refine both terms. One advantage with respect to the variational approach
is evident: DIS appears in PT as one of the leading terms, while in variational treatments
one has to do ad hoc additional calculations.
Conversely, in the RS-PT formulation CT and EX terms are not present. The absence
of CT contributions is quickly explained: RS-PT works on separated monomers and CT
contributions should be described by replacements of an occupied MO of A with an empty
MO of B (or by an occupied MO of B with an empty MO of A), and these electronic configurations
do not belong to the set of state on which the theory is based. For many years the
lack of CT terms has not been considered important. The attention focused on the examination
of the interaction energy of very simple systems, such as two rare gas atoms, in which
CT effects are in fact of very limited importance.
The absence of EX terms, on the contrary, indicates a serious deficiency of the RS formulation,
to which we have to pay more attention.
The wave function ΦK used in the RS formulation does not fully reflect the electron
permutational symmetry of the dimer: the permutations among electrons of A and B are neglected.
This leads to severe inconsistencies and large errors when RS PT is applied over the
whole range of distances. One has to rework the perturbation theory in the search of other
0 0
approaches. The simplest way would simply replace Φ0 =| ΨAΨB > with |AABΦ0 > where
AAB is the additional antisymmetry operator we have already introduced. Unfortunately
|AABΦ0 > is not an eigenfunction of H 0 as asked by the PT. Two ways of overcoming this difficulty
are possible. One could abandon the natural partitioning [8.36] and define another