Handbook of Solvents - George Wypych - ChemTech - Ventech!

8.5 Three- and many-body interactions 451
In the MM approach, the energy of the system is decomposed as follows:
E tot = [E str +E ben +E tor +E other] +E elec +E vdW
[8.63]
the terms within square brackets regard contributions due to the bond stretching, to the angle
bending, and to torsional interactions, supplemented by other contributions due to more
specific deformations and by couplings among different internal coordinates. The E elec
terms regard Coulomb and inductions terms between fragments not chemically bound:
among them there are the interactions between solute and solvent molecules. The same
holds for the van der Waals interactions (i.e., repulsion and dispersion) collected in the last
term of the equation.
It must be remarked that E tot is actually a difference of energy with respect to a state of
the system in which the internal geometry of each bonded component of the system is at
equilibrium.
Table 8.4
STRETCH E str =k s(1-l 0) 2
BEND E ben =k b(θ-θ 0) 2
TORSION E tor =k t[1 ± cos(nω)]
ks kb kt l0 θ0 stretch force constant
bend force constant
torsion force constant
reference bond length
reference bond angle
The versions of MM potentials (generally
called force fields) for solvent molecules
may be simpler, and in fact limited libraries of
MM parameters are used to describe internal
geometry change effects in liquid systems. We
report in Table 8.4 the definitions of the basic,
and simpler, expressions used for liquids.
To describe internal geometry effects in
the dimeric interaction, these MM parameters
are in general used in combination with a partitioning
of the interaction potential into
atomic sites. The coupling terms are neglected
and the numerical values of the parameters of the site (e.g., the local charges) are left unchanged
when there is a change of internal geometry produced by these local deformations.
Only the relative position of the sites changes. We stress that interaction potential only regards
interaction among sites of different molecules, while the local deformation affects the
internal energy of the molecule.
8.5 THREE- AND MANY-BODY INTERACTIONS
The tree-body component of the interaction energy of a trimer ABC is defined as:
1
ΔE( ABC; R ) = E ( R ) − ∑∑E(
R ) −∑E
2
ABC ABC ABC AB AB K
A B
K
[8.64]
This function may be computed, point-by-point, over the appropriate R ABC space, either
with variational and PT methods, as the dimeric interactions. The results are again affected
by BSS errors, and they can be corrected with the appropriate extension of the CP procedure.
A complete span of the surface is, of course, by far more demanding than for a dimer,
and actually extensive scans of the decomposition of the E ABC potential energy surface have
thus far been done for a very limited number of systems.
Analogous remarks hold for the four-body component ΔE(ABCD; R ABCD) of the cluster
expansion energy, as well as for the five- and six- body components, the definition of