Mel Levy

Hybrid schemes combining the Hartree–Fock method and density-functional theory: Underlying formalism and properties of correlation functionals

Hybrid schemes that combine elements of the Hartree–Fock and the Kohn–Sham procedures are shown here to have a rigorous formal basis within exact density-functional theory. Properties of the exact correlation energy and its functional derivative, corresponding to each hybrid scheme, are introduced and the correlation energy is expressed by a coupling constant integral. The coupling constant expansions of hybrid correlation energies are considered.

Generalized density-functional theory : Conquering the N-representability problem with exact functionals for the electron pair density and the second-order reduced density matrix

Using the constrained search and Legendre-transform formalisms, one can derive “generalized” density-functional theories, in which the fundamental variable is either the electron pair density or the second-order reduced density matrix. In both approaches, theN-representability problem is solved by the functional, and the variational principle is with respect to all pair densities (density matrices) that are nonnegative and appropriately normalized. The Legendre-transform formulation provides a lower bound on the constrained-search functional. Noting that experience in density-functional and density-matrix theories suggests that it is easier to approximate functionals than it is to approximate the set ofN-representable densities sheds some light on the significance of this work.

Generalizations of the Hohenberg-Kohn theorem: I. Legendre Transform Constructions of Variational Principles for Density Matrices and Electron Distribution Functions

Given a general, N-particle Hamiltonian operator, analogs of the Hohenberg-Kohn theorem are derived for functions that are more general than the particle density, including density matrices and the diagonal elements thereof. The generalization of Liebs Legendre transform ansatz to the generalized Hohenberg-Kohn functional not only solves the upsilon-representability problem for these entities, but, more importantly, also solves the N-representability problem. Restricting the range of operators explored by the Legendre transform leads to a lower bound on the true functional. If all the operators of interest are incorporated in the restricted maximization, however, the variational principle dictates that exact results are obtained for the systems of interest. This might have important implications for practical work not only for density matrices but also for density functionals. A follow-up paper will present a useful alternative approach to the upsilon- and N-representability problems based on the constrained search formalism.

On the adiabatic connection method, and scaling of electron–electron interactions in the Thomas–Fermi limit

In this paper we examine three aspects of electron–electron scaling: (i) the electron–electron repulsions are only scaled in Thomas–Fermi theory; (ii) the electron–electron repulsions are scaled, and the one electron potential is adjusted to give a prescribed density, in Thomas–Fermi–Dirac theory; and (iii) new approaches to the adiabatic connection formulas are presented to help improve the exchange–correlation functional. A new generalized two‐point expression is presented. Models (i) and (ii) are solved exactly.

The pair density functional of the kinetic energy and its simple scaling property

For electronic systems, a simple property of the recently introduced kinetic energy T as a functional of the pair density n(r1,r2)is derived. Approximate explicit expressions for T[n] are presented.

Exact high-density limit of correlation potential for two-electron density

Present approximations to the correlation energy, Ec[n], in density functional theory yield poor results for the corresponding correlation potential, vc([n];r)=δEc[n]δ/n(r). Improvements in vc([n];r), are especially needed for high-quality Kohn–Sham calculations. For a two-electron density, the exact form of vc([n];r) in its high-density limit is derived in terms of the density of the system and the first-order wave function from the adiabatic perturbation theory. Our expression leads to a formula for the difference 2Ec[n]−∫vc([n];r)n(r)dr, valid for any two-electron density in the high-density limit, thus generalizes previous results. Numerical results (both exact and approximate) are presented for both Ec[n] and ∫vc([n];r)n(r)dr in this limit for two electrons in a harmonic oscillator external potential (Hooke’s atom).

The Constrained Search Formulation of Density Functional Theory

Consider N interacting electrons in a local spin-independent external potential v. The Hamiltonian is

Sum rules for exchange and correlation potentials

{\text{H = T + Vee + }}\sum\limits_{i = 1}^N {v(\vec r_i )} ,