Many Body Perturbation Theory for Non-Uniform Electronic Systems

Abstract

In the now famous papers by P. Hohenberg and W. Kohn [1] and in a sequel by W. Kohn and L.J. Sham [2] use was made of the fact that the total energy of an interacting electron system is a functional of the electronic charge density:

$$E[n(r)]=\int{\upsilon (r)n(r)dr+F[n(r)]}$$

(1)

where υ(r) is the external potential and F is a universal functional of the density. For the exact n(r), E[n(r)] is a minimum relative to variations of n(r) consistent with a variation of υ(r). Unfortunately, for an approximate F = F ^a the approximate energy functional E ^a obtained from equation (1) by replacing F[n] with F ^a[n] does not have to have any relation in particular to E. That is, E ^a may be greater or smaller or equal to E. An example is the Slater X _α scheme [3] in which one can vary α to obtain any E ^a that one wants. Another fundamental objection is that given an F ^a there may or may not exist a solution to the Kohn and Sham equations in a region of physical interest. Even when the approximate energy functional has the appropriate properties leading to a n ^a(r) (there exists a point ñ ^a(r) such that E ^a[n] = min), the constraint used by Kohn and Sham that

$$\tilde{n}(r)=\sum\limits_{i}{{{\left| {{{\tilde{\phi }}}_{i}}(r) \right|}^{2}}}N=\int{\tilde{n}(r)}dr$$

(2)

(using the Lagrange multiplier metyhod this constraint leads to the K-S equations upon minimizing the functional [E ^a[n(r)] − λ^x∫n(r)dr] with respect to variations in n compatible with being produced by a variation in the external potential, supposing, of course, that the functional derivative exists) effectively raises the calculated energy even though it does not necessarily ensure N-representability. In principle other N-conserving constraints than (2) could be used.

NAS-NRC Post-Doctoral Associate