Essentials of Computational Chemistry

11.2 ELECTROSTATIC INTERACTIONS WITH A CONTINUUM 397
operator stationary, but it must also lead to a stationary dipole moment. Solution of the HF
equations (or the equivalent Kohn–Sham DFT equations) in such a fashion, where accounting
for solvation leads to a non-linear Schrödinger equation, is referred to as a self-consistent
reaction field (SCRF) calculation.
Inspection of Eq. (11.14) should make it clear that the manner in which the dipoledependence
enters into the equations will lead to an increase in dipole moments in increasingly
polar solvents. As noted in Section 11.1, the increase in the dipole moment in such
an SCRF formalism provides an energy lowering that is counterbalanced by an increase in
the energy computed from the ‘usual’ Hamiltonian H (the first operator on the l.h.s.) so that
a stationary solution is reached when additional distortion costs associated with H exactly
balance additional energy lowering associated with further increasing the dipole moment.
In describing the results from SCRF calculations, it is useful to keep careful track of the
various components of the energy. The electrostatic component of the solvation free energy
is the difference between the energy in the gas phase and the energy in solution. This may
be written
GENP = 〈 (sol) |H | (sol) 〉+〈 (sol) |GP| (sol) 〉 −〈 (gas) |H | (gas) 〉
= EEN + GP
(11.16)
where the difference between the first and third expectation values on the r.h.s. in the first
line of Eq. 11.16 defines the distortion energy EEN, which must be positive since (gas)
minimizes H . The ‘EN’ subscript on this term emphasizes it is associated with the electronic
and nuclear components of the total energy; in the absence of any geometry reoptimization,
the N subscript is superfluous. As written, Eq. (11.16) mixes potential and free energies, but
we will ignore this issue for now.
The Kirkwood–Onsager equations can be generalized to include multipole moments higher
than the dipole, leading to the expression
GP =− 1
2
L
l
L
l=0 m=−l l ′ l
=0
′
m ′ =−l ′
M m l
mm′
fll ′ Mm′ l ′
(11.17)
where each component m of every molecular multipole M of order l interacts with the
reaction field, which is itself expressed as a multipole expansion equal and opposite to the
molecular multipoles, through the reaction field factors f that carry the dependence on
dielectric constant and cavity radius. In principle, the multipole expansion may be carried
out to infinite order, but in practice, some judicious choice of l is made in Eq. (11.17) to
keep things tractable. A fairly typical choice is l = 6 (note that l = 0andl = 1definethe
Born and Born–Kirkwood–Onsager (BKO) approaches, respectively).
The simplicity of the BKO approach to computing polarization free energies led to its
widespread use for the qualitative analysis of solvation effects on various properties for
many years (including in the absence of any explicit theoretical calculations). For quantitative
purposes, however, it suffers from a number of undesirable features. One such feature is the
slow nature of the convergence of Eq. (11.17) with respect to l. Table 11.2 lists GEP