Handbook of Solvents - George Wypych - ChemTech - Ventech!

8.4 Two-body interaction energy 429
antisymmetric wave function are fermions and satisfy the Pauli exclusion principle. Introducing
these concepts in the machinery of the quantum mechanical calculations, it turns out
that at each coulomb interaction between two electrons described by MOs φ µ(1) and φυ (2)
(the standard expression of this integral, see eq.[8.13], is: ) one
has to add a second term, in which there is an exchange of the two electrons in the conjugate
function: with a minus sign (the exchange in the label of the
two electrons is a permutation of order two, bearing a sign minus in the antisymmetric case).
There are other particles, called bosons, which satisfy other quantum statistics, the
Bose-Einstein statistics, and that are described by wave functions symmetric with respect to
the exchange, for which the Pauli principle is not valid. We may dispense with a further consideration
of bosons in this chapter.
It is clear that to consider exchange contributions to the interaction energy means to introduce
the proper antisymmetrization among all the electrons of the dimer. Each monomer
is independently antisymmetrized, so we only need to apply to the simple product wave
functions an antisymmetrizer restricted to permutations regarding electrons of A and B at
the same time: it will be called AAB. p p
By applying this operator to ΨΨ A Bwithout
other changes and computing the expectation
value, one obtains:
with
and
p p p p
EIII(R) =�AABΨ Ψ|HAB|AABΨ
Ψ�
[8.19]
() () ()
A
B
A
B
E R = E R + EX R
[8.20]
III II
() () ()
EX R = E R − E R
[8.21]
III II
Morokuma has done a somewhat different definition of EX: it is widely used, being inserted
into the popular Kitaura-Morokuma decomposition scheme. 4 In the Morokuma definition
EIII(R) is computed as in eq. [8.19] using the original Ψm monomer wave functions
p
instead of the mutually polarized Ψm ones. This means to lose, in the Morokuma definition,
the coupling between polarization and antisymmetrization effects that have to be recovered
later in the decomposition scheme. In addition, EX can be no longer computed as in eq.
[8.20], but using EI energy: (i.e., EX� =EIII(R) - EI(R)). The problem of this coupling also
appears in the perturbation theory schemes that are naturally inclined to use unperturbed
monomeric wave functions, not including exchange of electrons between A and B: we shall
come back to this subject considering the perturbation theory approach.
The charge transfer term
The charge transfer contribution CT may play an important role in some chemical processes.
Intuitively, this term corresponds to the shift of some electronic charge from the occupied
orbitals of a monomer to the empty orbitals of the other. In the variational
decomposition schemes this effect can be separately computed by repeating the calculations
on the dimer with deletion of some blocks in the Hamiltonian matrix of the system and tak-