Density Matrices for Electrons With Strong Interactions

Eugene S. Kryachko, Eduardo V. Ludeña

Norman Henry March

Forms of canonical (Bloch) and Dirac density matrices for free electrons with and without external applied fields are first considered. The basic tool employed is the Bloch equation with a one-electron Hamiltonian. Such an approach is used to obtain a perturbation theory to all orders for the idempotent Dirac density matrix when a common potential energy V(r) is switched on to originally free electrons. The relation to density functional theory is then considered and the exchange–correlation contribution Vxc(r) to V(r) is expressed in terms of first- and second-order density matrices following Holas and March. These latter density matrices are now for the fully interacting system and, in particular, the first-order density matrix is no longer idempotent, though it must still satisfy generalized Pauli Principle conditions. Reference is also made to a localized Wigner electron in a strong magnetic field.

Francesc Malet, André Mirtschink, Klaas J. H. Giesbertz +1 more

Jerzy Ciosłowski

Explicit asymptotic expressions are derived for the reduced one-electron density matrices (the 1-matrices) of strongly correlated two- and three-electron harmonium atoms in the ground and first excited states. These expressions, which are valid at the limit of small confinement strength ω, yield electron densities and kinetic energies in agreement with the published values. In addition, they reveal the ω(5/6) asymptotic scaling of the exchange components of the electron-electron repulsion energies that differs from the ω(2/3) scaling of their Coulomb and correlation counterparts. The natural orbitals of the totally symmetric ground state of the two-electron harmonium atom are found to possess collective occupancies that follow a mixed power/Gaussian dependence on the angular momentum in variance with the simple power-law prediction of Hill's asymptotics. Providing rigorous constraints on energies as functionals of 1-matrices, these results are expected to facilitate development of approximate implementations of the density matrix functional theory and ensure their proper description of strongly correlated systems.

Katarzyna Pernal

Abstract Random phase approximation (RPA) electron correlation methods have gained in popularity in the recent years. A number of RPA correlation energy variants emerged in the Kohn–Sham DFT framework and in the theory of strongly orthogonal geminals. Foundations of most RPA approaches trace back to an exact expression for two‐electron reduced density matrix written in terms of one‐electron density matrix and dynamic one‐electron response functions, originally presented in the seminal paper of McLachlan and Ball (Rev. Mod. Phys. 1964, 36, 844). The aim of this article is to give a pedagogical introduction to possible approaches for describing electron correlation energy based on the McLachlan and Ball formula. The focus of the presentation is to formulate electron‐interaction energy expressions as functions of reduced density matrices. On one hand, it provides a common umbrella for RPA approximations proposed for uncorrelated (Hartree–Fock or Kohn–Sham) as well as partially correlated (strongly orthogonal geminals) references. On the other hand, such presentation may stimulate new developments in density matrix functional theory.