In this paper we will address the question of how to obtain energies from functionals when only the functional derivative is given. It is shown that one can obtain explicit expressions for the exchange-correlation energy from approximate exchange-correlation potentials using line integrals along paths within the space of densities. The path dependence of the results is discussed and criteria for path independence are given. Derivations are given of upper and lower bounds to the exchange-correlation energy in terms of the exchange-correlation potential at the beginning and the end point of a certain path. We further express the kinetic part ${\mathit{T}}_{\mathrm{xc}}$ of the exchange-correlation energy in terms of a line integral and derive a constraint on approximate correlation potentials. We show how to use the line-integral formalism to derive the requirements that exchange-correlation potentials must fulfill in order to make the exchange-correlation functional satisfy some symmetry property such as rotational and translational invariance and scaling properties. Finally, we will discuss the use of line integrals along a path in density space to obtain energy differences, notably, the bonding energies of molecules, from exchange-correlation potentials. These last results generalize the transition-state formulations of Slater and Ziegler.

• Received 17 March 1994

DOI:https://doi.org/10.1103/PhysRevA.51.170