Essentials of Computational Chemistry

396 11 IMPLICIT MODELS FOR CONDENSED PHASES
where ε is the exterior dielectric constant. Taking r on the surface of the sphere, i.e., |r| =a,
Eq. (11.3) becomes
G =− 1
q
2 4πa2
− q
ds =
εa
q2
2εa
(11.11)
As the square of the charge, the dielectric constant, and the ionic radius must all be positive,
work must be expended to charge the sphere, but the work is less for higher exterior dielectric
constants, as expected. Recalling that the polarization energy is the difference in the required
work in the gas phase and solution, we may write
GP =− 1
1 −
2
1
2 q
(11.12)
ε a
which is the so-called Born equation for the polarization energy of a monatomic ion in
atomic units.
If instead of carrying a charge, our sphere appears to be characterized by a perfectly
dipolar distribution having dipole moment µ, an analogous analysis provides
GP =− 1
2
2
2 (ε − 1) µ
(2ε + 1) a3 (11.13)
which is the so-called Kirkwood–Onsager equation in atomic units.
An important difference between the Born and Kirkwood–Onsager formulae is that the
former depends on the charge, which is a property of the system restricted to integral values,
while the latter depends on the dipole moment, which can potentially vary in different
environments. In the context of quantum mechanical calculations, let us define the Kirkwood–Onsager
polarization energy operator by invoking µ as the dipole moment operator
in Eq. (11.13). In that case, the Schrödinger equation in solution becomes
H − 1
2
2(ε − 1) 〈|µ|〉
(2ε + 1) a3
µ = E (11.14)
where H is the usual gas-phase Hamiltonian. Written in this fashion, the components of the
second term on the l.h.s. that precede the final dipole moment operator may be regarded as
the reaction field.
Equation (11.14) is an example of a non-linear Schrödinger equation. It can be solved in
the usual HF fashion by construction of a Slater determinant formed from MOs ψ that are
optimized using a modified Fock operator according to
2(ε − 1) 1
Fi −
〈|µ|〉2 ψi = eiψi
(11.15)
(2ε + 1) a3 where Fi is the usual gas-phase Fock operator for MO i (Ángyán 1992). A critical feature
of Eq. (11.15) is that it involves an additional level of iteration compared to the standard
HF approach. Not only must the final wave function render the density matrix and Fock