Handbook of Solvents - George Wypych - ChemTech - Ventech!

8.4 Two-body interaction energy 443
the potential of the dipole decreases faster than that of the monopole, the quadrupole faster
than that of the dipole, and so on.
To get the electrostatic interaction energy, the multipolar expansion of the potential
V B(r) is multiplied by a multipolar expansion of ρ A(r). The result is:
where
∞ ∞
≈ ∑∑
ll ′
l=
0 l′=
0
ES D R
m A
Dll = ∑ Cll MlmM − ( l+ l′+
1)
l
′ ′
B
lm ′
m=−l [8.49]
[8.50]
m
is the interaction energy of the permanent dipole l of A with the permanent dipole l� ofB(Cll′ is a numerical coefficient depending only on l, l�, and m).
The calculation of ES via eq. [8.49] is much faster than through eq. [8.44]: the integra-
X
tions are done once, to fix the M lm molecular multipole values and then used to define the
whole ES(R) surface.
For small size and almost spherical molecules, the convergence is fast and the expansion
may be interrupted at a low order: it is thus possible to use experimental values of the
net charge and of the dipole moment (better if supplemented by the quadrupole, if available)
to get a reasonable description of ES at low computational cost and without QM calculations.
We have, however, put a ≈symbol instead of = in eq. [8.49] to highlight a limitation of
this expansion. To analyze this problem, it is convenient to go back to the MEP.
Expansion [8.47] has the correct asymptotic behavior: when the number of terms of a
truncated expression is kept fixed, the description improves at large distances from the expansion
center. The expansion is also convergent at large values of R: this remark is not a
pleonasm, because for multipole expansions of other terms of the interaction energy (as for
example the dispersion and induction terms), the convergence is not ensured.
Convergence and asymptoticity are not sufficient, because the expansion theorem
holds (as we have already remarked) for points r lying outside a sphere containing all the elements
of the charge distribution. At the QM level this condition is never fulfilled because
eachρ(r) fades exponentially to zero when r →∞. This is not a serious problem for the use of
multipole expansions if the two molecules are far apart, and it simply reduces a little the
quality of the results if almost spherical molecules are at close contact. More important are
the expansion limitations when one, or both, molecules have a large and irregular shape. At
strict contact a part of one molecule may be inserted into a crevice of the partner, within its
nominal expansion sphere. For large molecules at close contact a systematic enlargement of
the truncated expression may lead to use high value multipoles (apparent indication of slow
convergence) with disastrously unphysical results (real demonstration of the lack of convergence).
The introduction of correction terms to the multipolar expansions of VB or of ES, acting
at short distances and called “penetration terms”, has been done for formal studies of
this problem but it is not used in practical applications.