Handbook of Solvents - George Wypych - ChemTech - Ventech!

8.4 Two-body interaction energy 427
Figure 8.2. ES for two dipoles with collinear (left) and antilinear (right) orientation.
The electronic wave functions of A and B are separately computed, each with its own
Hamiltonian H A and H B. Ab initio methods give at every level of the formulation of the theory
antisymmetric electronic wave functions, satisfying the Pauli exclusion principle, on
which more details will be given later.
The two antisymmetric wave functions Ψ A and Ψ B are then used, without modifications,
in connection with the Hamiltonian of the whole system H AB to get the expectation
value of the energy; according to the standard notation of quantum chemistry we can write:
E I =ΨΨ A B HAB
ΨΨ A B
[8.13]
It simply means the integral over the whole space of the complex conjugate of the
function at the left (i.e., the function (Ψ AΨ B)* ) multiplied by the function H AB(Ψ AΨ B) (the
application of an operator such as H AB to a function always gives a function). We can neglect
complex conjugates (our functions are all real); the expression given above is a compact
and clear indication of a set of operations ending with an integral, and we shall use it
only to speed notations.
H AB differs from the sum of H A and H B according to the following expression
HAB = HA+ HB+ VAB
[8.14]
V AB collects terms describing electrostatic interactions between the nuclei and electrons of
A with electrons and nuclei of B. The wave function Ψ AB expressed as eigenfunction of H AB
is of course antisymmetric with respect to all the electrons, of A as well as of B, but this has
not yet been introduced in the model.
The energy obtained with this recipe (the calculations are a by-product of the calculation
of the dimer energy) may be so decomposed as:
() ()
E R = E + E + ES R
[8.15]
I A B