Reviews in Computational Chemistry Volume 18

104 Polarizability in Computer Simulations
dipoles, there are iterative, adiabatic techniques as well as fully dynamic methods.
In the adiabatic methods, the correspondence between the shell charge
and the effective electronic degrees of freedom is invoked, along with the
Born–Oppenheimer approximation. In this case, the slow-moving nuclei and
core charges are said to move adiabatically in the field generated by the shell
charges. In other words, the positions of the shell charges are assumed to
update instantaneously in response to the motion of the nuclei, and thus
always occupy the positions in which they feel no net force (i.e., the positions
that minimize the total energy of the system). The forces on the core charges
are then used to propagate the dynamics, using standard numerical integration
methods. The other alternative is to treat the charges fully dynamically, allowing
them to occupy positions away from the minimum-energy position dictated
by the nuclei, and thus experience nonzero forces.
When the charges are treated adiabatically, a self-consistent method
must be used to solve for the shell displacements, fdig (just as with the dipoles
fl ig in the previous section). Combining Eqs. [26], [28], and [29], we can
write the total energy of the shell model system as
U indðfrig; fdigÞ ¼ XN
i ¼ 1
þ 1
2
1
2 kid 2 i þ qi½ri E 0 i ðri þ diÞ E 00
i Š
X n
X
i ¼ 1 j 6¼ i
1
rij
1
jrij djj
1
jrij þ dij þ
1
jrij di þ djj
which is the equivalent of Eq. [18] for a model with polarizable point dipoles,
but with one important difference: Eq. [18] is a quadratic function of the fl ig,
guaranteeing that its derivative (Eq. [19]) is linear and that a standard matrix
method can be used to solve for the fl ig. Equation [30] is not a quadratic function
of the fdig. Moreover, the dependence of the short-range interactions on
the displacements of the shell particles further complicates the matter. Consequently,
matrix methods are typically not used to find the shell displacements
that minimize the energy.
Iterative methods are used instead. In one such approach, 101 the nuclear
(core) positions are updated, and the shell displacements from the previous
step are used as the initial guess for the new shell displacements. The net force,
Fi, on the shell charge is calculated from the gradient of Eq. [30], together with
any short-range interactions. Because the harmonic spring interaction is, by
far, the fastest varying component of the potential felt by the shell charge,
the incremental shell displacement ddi ¼ Fi=ki represents a very good estimate
of the equilibrium (energy minimizing) position of the shells. The forces are
recalculated at this position, and the procedure is iterated until a (nearly)
force-free configuration is obtained. Alternatively, this steepest descent style
½30Š