Exchange Energy In Density Functional Theory Research Articles

Relation between single-particle kinetic energy and exchange energy in DFT for the inhomogeneous electron liquid in the Be atom

In a recent study, the authors have used the semi empirical fine-tuned Hartree–Fock ground-state electron density n(r) of Cordero et al. [Phys. Rev. A 75, 052502 (2007)] for the Be atom to calculate the phase θ(r) from a non-linear pendulum-like equation. Since the density amplitude n(r)1/2 plus θ(r) determine, in turn, the idempotent Dirac density matrix γ(r, r′), we use n(r) and θ(r) first of all to calculate the exchange energy density e X (r) of the density functional theory (DFT). This enables us to obtain the Slater (Sl) approximation to the exchange-only potential. A comparison can then be made, by integrating the earlier predicted exchange-correlation force −∂V XC (r)/∂r, of V XC (r) with . Relationship to the Becke semiempirical density gradient approximation for exchange is also established. Some brief discussion of the Perdew–Burke–Ernzerhof density functional is added.

TREATMENTS OF THE EXCHANGE ENERGY IN DENSITY-FUNCTIONAL THEORY

Following a recent work [Gál, Phys. Rev. A64, 062503 (2001)], a simple derivation of the density-functional correction of the Hartree–Fock equations, the Hartree–Fock–Kohn–Sham equations, is presented, completing an integrated view of quantum mechanical theories, in which the Kohn–Sham equations, the Hartree–Fock–Kohn–Sham equations and the ground-state Schrödinger equation formally stem from a common ground: density-functional theory, through its Euler equation for the ground-state density. Along similar lines, the Kohn–Sham formulation of the Hartree–Fock approach is also considered. Further, it is pointed out that the exchange energy of density-functional theory built from the Kohn–Sham orbitals can be given by degree-two homogeneous N-particle density functionals (N = 1, 2, …), forming a sequence of degree-two homogeneous exchange-energy density functionals, the first element of which is minus the classical Coulomb-repulsion energy functional.

Self-consistent multiplicative constant method for the exchange energy in density-functional theory

We advance a self-consistent multiplicative constant (SCMC) method based on a local exchange potential that yields total, exchange, and atomization energies approaching quite closely, but at a much lower computational cost, those of the exact orbital-dependent exchange treatment [S. Ivanov, S. Hirata, and R. J. Bartlett, Phys. Rev. Lett. 83, 5455 (1999); A. G\orling, ibid. 83, 5459 (1999)]. Application of the SCMC method to any approximate exchange energy functional permits us to remove the self-interaction correction from the latter as well as to restore its variational character. The SCMC method is implemented here using the following exchange energy functionals: local-density approximation, the Perdew-Wang 1991, Becke 1988, and the one-parameter hybrid Becke Lee-Yang-Parr functional. Calculations for atoms and diatomic molecules are presented. In addition, we prove that the exchange energy evaluated from the Kohn-Sham exchange-only determinant is an upper bound to the Hartree-Fock exchange energy.