Local energy and chemical potential equations and the exchange-correlation potential

Local energy and chemical potential equations are considered in some detail in relation to low-order density matrices. Some asymptotic properties can be extracted in exact form. The spatial derivative of the chemical potential equation referred to above yields the external force, defined as the (negative of the) gradient of the potential energy of the nuclear framework. This quantity, by utilizing the differential virial theorem, can be expressed as a sum of three terms: (i) a Laplacian contribution known explicitly in terms of the ground-state electron density; (ii) a kinetic part derivable from the ''near-diagonal'' behaviour of the first-order density matrix; and (iii) a term from electron-electron interactions, that involves the electronic pair correlation function. Following the work of Holas and March, this allows the exchange-correlation potential of density functional theory to be expressed in terms of low-order density matrices. Finally, scaling of electron-electron interactions is briefly considered, as well as the adiabatic connection formula in density functional theory. Such scaling arguments lead to a kinetic correction to the Harbola-Sahni form of the exchange-only potential.