Essentials of Computational Chemistry

2.2 POTENTIAL ENERGY FUNCTIONAL FORMS 33
a plausible choice might be
⎧
⎪⎨
∞ if A and B are 1,2- or 1,3-related
εAB = 3.0
⎪⎩
1.5
if A and B are 1,4-related
otherwise
(2.24)
which dictates that electrostatic interactions between bonded atoms or between atoms sharing
a common bonded atom are not evaluated, interactions between torsionally related atoms are
evaluated, but are reduced in magnitude by a factor of 2 relative to all other interactions,
which are evaluated with a dielectric constant of 1.5. Dielectric constants can also be defined
so as to have a continuous dependence on the distance between the atoms. Although one
might expect the use of high dielectric constants to mimic to some extent the influence
of a surrounding medium characterized by that dielectric (e.g., a solvent), this is rarely
successful – more accurate approaches for including condensed-phase effects are discussed
in Chapters 3, 11, and 12.
Bonds between heteroatoms and hydrogen atoms are amongst the most polar found in
non-ionic systems. This polarity is largely responsible for the well-known phenomenon of
hydrogen bonding, which is a favorable interaction (usually ranging from 3 to 10 kcal/mol)
between a hydrogen and a heteroatom to which it is not formally bonded. Most force
fields account for hydrogen bonding implicitly in the non-bonded terms, van der Waals and
electrostatic. In some instances an additional non-bonded interaction term, in the form of a
10–12 potential, is added
U(rXH) = a′ XH
r 12
XH
− b′ XH
r 10
XH
(2.25)
where X is a heteroatom to which H is not bound. This term is analogous to a Lennard–Jones
potential, but has a much more rapid decay of the attractive region with increasing bond
length. Indeed, the potential well is so steep and narrow that one may regard this term as
effectively forcing a hydrogen bond to deviate only very slightly from its equilibrium value.
Up to now, we have considered the interactions of static electric moments, but actual
molecules have their electric moments perturbed under the influence of an electrical field
(such as that deriving from the electrical moments of another molecule). That is to say,
molecules are polarizable. To extend a force field to include polarizability is conceptually
straightforward. Each atom is assigned a polarizability tensor. In the presence of the permanent
electric field of the molecule (i.e., the field derived from the atomic charges or the
bond–dipole moments), a dipole moment will be induced on each atom. Following this,
however, the total electric field is the sum of the permanent electric field and that created
by the induced dipoles, so the determination of the ‘final’ induced dipoles is an iterative
process that must be carried out to convergence (which may be difficult to achieve). The total
electrostatic energy can then be determined from the pairwise interaction of all moments and
moment potentials (although the energy is determined in a pairwise fashion, note that manybody
effects are incorporated by the iterative determination of the induced dipole moments).
As a rough rule, computing the electrostatic interaction energy for a polarizable force field is
about an order of magnitude more costly than it is for a static force field. Moreover, except for