Handbook of Solvents - George Wypych - ChemTech - Ventech!

428 Jacopo Tomasi, Benedetta Mennucci, Chiara Cappelli
and so:
() () ( )
ES R = E R − E + E
[8.16]
I A B
The energies of the two monomers are already known, so the calculation of ES is immediate.
Remark that in the right side of eq. [8.15] only ES depends on the geometry of the
dimer, according to cluster model we are using.
The induction term
The second term of the interaction energy, IND, is always negative. IND is related to the
mutual polarization of the electronic charge distributions of A and B (the nuclei are held
fixed), inducing additional stabilizing effects. This induction (or polarization) energy contribution
is defined in a different way in variational and perturbation theory approaches.
Perturbation theory approaches are compelled to compute at the first order of the perturbation
scheme only the effects due to the polarization of A with respect to the B distribution
kept fixed, and in parallel the effects due to the polarization of B with A kept fixed. Mutual
induction effects are introduced at higher order of the perturbation theory and have to be
separated in some way from dispersion effects computed at the same time.
In the variational approach use is made of an extension of the simple technique we
have used for ES.
The separation between electrons of A and B is maintained but the product ΨAΨB is
now subjected to a constrained variational optimization using the Hamiltonian HAB. The
two wave functions are so changed, allowing the effects of mutual polarization, because of
p p
the presence of the VAB term in the Hamiltonian: they will be so indicated as ΨA and ΨB .
The resulting expectation value of the energy:
()
p p
p
E R =ΨΨ H ΨΨ [8.17]
II A
may be so decomposed:
B
AB A
() () ()
p
B
E R = E R + IND R
[8.18]
II I
and so IND(R) = EII(R)-EI(R). In this way, IND(R) contains all the mutual polarization effects.
The exchange term
The next term, EX, is positive for all the molecular systems of interest for liquids. The name
makes reference to the exchange of electrons between A and B. This contribution to ΔE is
sometimes called repulsion (REP) to emphasize the main effect this contribution describes.
It is a true quantum mechanical effect, related to the antisymmetry of the electronic wave
function of the dimer, or, if one prefers, to the Pauli exclusion principle. Actually these are
two ways of expressing the same concept. Particles with a half integer value of the spin, like
electrons, are subjected to the Pauli exclusion principle, which states that two particles of
this type cannot be described by the same set of values of the characterizing parameters.
Such particles are subjected to a special quantum version of the statistics, the Fermi-Dirac
statistics, and they are called fermions. Identical fermions have to be described with an
antisymmetric wave function; the opposite also holds: identical particles described by an