Handbook of Solvents - George Wypych - ChemTech - Ventech!

444 Jacopo Tomasi, Benedetta Mennucci, Chiara Cappelli
For larger molecules it is necessary to pass to many-center multipole expansions. The
formalism is the same as above, but now the expansion regards a portion of the molecule,
for which the radius of the encircling sphere is smaller than that of the whole molecule. Here
it is no longer possible to use experimental multipoles. One has to pass to QM calculations
supplemented by a suitable procedure of partition of the molecule into fragments (this operation
is necessary to define the fragmental definition of multipoles with an analog of eq.
[8.42]). There are many approaches to define such partitions of the molecular charge distribution
and the ensuing multipole expansions: for a review see, e.g., Tomasi et al. 20
It is important to remark here that such local expansions may have a charge term even
if the molecule has no net charge (with these expansions the sum of the local charges must
be equal to the total charge of the molecule).
There is freedom in selecting the centers of these local expansions as well as their
number.
A formal solution to this problem is available for the molecular wave functions expressed
in terms of Gaussian functions (as is the general rule). Each elementary electron distribution
entering in the definition of Ψ, is described by a couple of basic functions
*
χsχtcentered at positions rs and rt. This distribution may be replaced by a single Gaussian
function centered at the well defined position of the overlap center: χu at ru. The new Gaussian
function may be exactly decomposed into a finite local multipole expansion. It is possible
to decompose the spherical function Y into a finite (but large) number of local multipole
expansions each with a limited number of components. This method works (the penetration
terms have been shown to be reasonably small) but the number of expansion centers is exceedingly
large.
Some expedient approximations may be devised, introducing a balance between the
number of expansion centers and the level of truncation of each expansion. This work has
been done for years on empirical bases. One strategy is to keep each expansion at the lowest
possible order (i.e., local charges) and to optimize number and location of such charges.
The modern use of this approach has been pioneered by Alagona et al., 21 using a number
of sites larger than the number of atoms, with values of the charges selected by
minimization of the difference of the potential they generate with respect to the MEP function
VB(r) (eq. [8.40]). This is the second application of the MEP function we have mentioned.
Alagona’s approach has been reformulated by Momany in a simpler way, by reducing
the number of sites to that of the nuclei present in the molecule. 22 This strategy has gained
wide popularity: almost all the potentials in use for relative large molecules reduce the electrostatic
contributions to Coulomb contributions between atomic charges. A relatively
larger number of sites are in use for the intermolecular potentials of some simple molecules,
as, for example, water, for which more accuracy is sought.
Momany’s idea has led to a new definition of atomic charges. It would be possible to
write volumes about atomic charges (AC), a concept that has no a precise definition in QM
formalism, but is of extreme utility in practical applications. Many definitions of AC are
based on manipulations of the molecular wave function, as, for example, the famous
Mulliken charges. 23 Other definitions are based on different analyses of the QM definition
of the charge density, as for example Bader’s charges. 24 There are also charges derived from
other theoretical approaches, such as the electronegativity equalization, or from experimental
values, such as from the vibrational polar tensors.