Essentials of Computational Chemistry

270 8 DENSITY FUNCTIONAL THEORY
where, again, the pairwise terms can be computed once for every pair of atoms and then
either fit to analytic functions or tabulated for future reference.
With these further simplifications, enormously large systems may be handled fairly easily
to include geometry optimization. This non-self-consistent protocol defines DFTB.
The critical assumption of DFTB, however, is that the charge density of a composite
system is well represented by the sum of the charge densities of its unperturbed constituent
atoms. Clearly such a situation is inconsistent with the polarization that occurs in bonds
between elements having significantly different electronegativities. To address such polarized
systems, we consider a generalization of Eq. (8.36) that is valid to second order in the density
fluctuation δρ(r) about the fixed density ρ0(r), namely
E[ρ0(r) + δρ(r)] = E[ρ0(r)]
+ 1
2
1
|r1 − r2| +
δ2
Exc
δρ(r1)δρ(r2) δρ(r1)δρ(r2) dr1dr2
ρ0
(8.39)
In the spirit of DFTB, one may consider δρ(r) to be decomposable into atomic contributions
according to
δρ(r) =
atoms
A
qA
(8.40)
where q is used for the atomic contribution to emphasize the analogy to partial atomic
charge. The second-order term in Eq. (8.39) may then be written as
1
2
1
|r1 − r2| +
δ2
Exc
δρ(r1)δρ(r2) δρ(r1)δρ(r2) dr1dr2 =
ρ0
1
atoms
qAqBγAB
2
A,B
(8.41)
If the atomic charge distributions are assumed to be spherically symmetric, the effective
inverse distance γ is computed as
2ηA, A = B
γAB =
(8.42)
(aa|bb), A = B
where η is an atomic hardness (formally the second derivative of the atomic energy with
respect to a change from neutrality in charge; η is well approximated as (IP − EA)/2, where
IP and EA are the atomic ionization potential and electron affinity, respectively) and a and b
are Slater-type s orbitals on atoms A and B, respectively, in which case the electron-repulsion
integral has an analytic solution. Once again this function, for which the off-diagonal terms
depend on the distance between atoms A and B, may be approximated with a simpler analytic
form or tabulated for all relevant atomic pairs.
The presence of the partial atomic charges in Eq. (8.41), however, poses the question of
how they are to be computed. A popular choice is to compute them from Mulliken population
analysis (see Section 9.1.3.2), in which case the partial atomic charges depend on the KS