Exchange potential via functional differentiation of the Dirac idempotent density matrix

Abstract

Following two illustrative examples, a quite general form for the functional derivative of the Dirac density matrix $\ensuremath{\gamma}({\mathbf{r}}_{1},{\mathbf{r}}_{2})$ with respect to the electron density $\ensuremath{\rho}(\mathbf{r})$ is proposed. In turn, this functional derivative is related to a further three-point object, which reduces to the linear response function ${\ensuremath{\chi}}_{0}({\mathbf{r}}_{1},\mathbf{r})$ when ${\mathbf{r}}_{2}$ tends to ${\mathbf{r}}_{1}$. The main application is then to derive the exchange-only potential of density functional theory. To avoid heavy numerical work in molecules and clusters, an approximation is suggested, with major calculational simplifications as a probable consequence.