Handbook of Solvents - George Wypych - ChemTech - Ventech!

8.4 Two-body interaction energy 445
The charges obtained using Momamy’s idea of fitting the MEP with atomic charges
are sometimes called PDAC (potential derived atomic charges). It has been realized that a
fitting of MEP on the whole space was not convenient. It is better to reduce this fitting to the
portion of space of close contact between molecules, i.e., near the van der Waals surface.
This is the main technique now in use to define PDAC values.
The induction term
It is possible to define a molecular index PA for the induction term to be used in combination
with the MEP VA to get a detailed description of the spatial propensity of the molecule to develop
electrostatic interactions of classical type. Both functions are used under the form of
an interaction with a unit point charge q placed at position r. In the case of VA this means a
simple multiplication; in the case of PA there is the need of making additional calculations
(to polarize the charge distribution of A). There are fast methods to do it, both at the
variational level 25a and at the PT level. 25b The analysis of PA has not yet extensively been
used to model IND contributions to ΔE, and it shall not be used here. This remark has been
added to signal that when one needs to develop interactions potentials for molecular not yet
studied interactions including, e.g., complex solutes, the use of this approach could be of
considerable help.
The multipole expansion of IND may be expressed in the following way:
with
IND = IND( A ← B) + IND( B ← A)
[8.51]
∞ ∞
l<
k<
1
− ( l+ l′+ k+ k′+
2)
m m ′ A
B
IND( A ←B) ≈ − ∑R
∑ ∑C
ll′
Ckk ′ × πlk( m, m′ ) M l′
, −m
2
∑ ,
ll′=
1 kk ′= 1
m=− l<
m′=− k<
B
M k′ −m
[8.52]
and a similar expression for IND(B←A) with A and B labels exchanged.
X
We have here introduced the static polarizability tensor elements πlk ( m, m ′ ) : these
tensors are response properties of the molecule with respect to external electric fields F
(with components Fα), electric field gradients ∇F (with elements ∇Fαβ) and higher field derivatives,
such as ∇ 2 F (with elements ∇ 2 Fαβγ), etc. It is possible to compute these quantities
with the aid of PT techniques, but variational procedures are more powerful and more exact.
The polarizability tensors are defined as the coefficients of the expansion of one molecular
property (the energy, the dipole, etc.) with respect to an external field and its derivatives.
Among these response properties the most known are the dipole polarizability tensors
αβ , , and γ, that give the first three components of the expansion of a dipole moment μ subjected
to an external homogeneous field F:
μ
α
∂E
( 0)
=− = μ α + ααβFβ+ βαβγFβFγ+ γαβγFβFγFδ+
� [8.53]
∂F
α
the convention of summation over the repeated indexes (which are the Cartesian coordinates)
is applied here.