Density Matrix Theory in Many-Electron Quantum Mechanics I. General Properties of Density Matrices and Their Eigenfunctions

János G. Ángyán

N. H. March

Many-electron theory is treated here with much attention focused on the synthesis of the density functional and density matrix theory. This synthesis shows that the exchange–correlation potential, normally obtained as the functional derivative with respect to the electron density of an assumed approximate exchange–correlation energy, can be written exactly in terms of the “near-diagonal” behavior of the fully interacting first-order density matrix and the diagonal of the second-order density matrix, thereby bypassing the functional derivative. Also involved, since the exchange–correlation potential is an “artifact” of reducing the many-electron Schrödinger equation to one-body Slater–Kohn–Sham equations, are the one-body orbitals derived from these latter equations, not just the density thereby obtained. This situation has then motivated a discussion of what constraints are placed on the low-order density matrices of atoms and molecules, described by an assumed spin-independent Hamiltonian, by the requirement that these matrices are constructed from an N-electron wave function which is an eigenfunction of the spin operators S2 and Sz. Some briefer discussion is added on the optimally convergent determinantal expansion of such an N-electron wave function, the relation between Brueckner and Löwdin natural orbitals being commented on as a by-product. In this latter context, it is noted that the exchange-only potential of density functional theory can be represented as a sum of two line integrals, each of which is path-dependent, the sum being, of course, path-independent. But in addition to knowledge of Slater–Kohn–Sham orbitals appearing as in the Harbola–Sahni exchange-only potential (which is one of the path-dependent terms), there is a kinetic correction. This involves Löwdin natural orbitals and their occupation numbers, but only now in the weak coupling limit of the electron–electron interaction, as emphasized in the kinetic “correction” to the Harbola–Sahni formula exhibited by Levy and the writer. Some directions for further work are briefly set out. ©1999 John Wiley & Sons, Inc. Int J Quant Chem 74: 163–176, 1999

Eugene S. Kryachko, Eduardo V. Ludeña

Werner Kutzelnigg

The density operator (density matrix) of a quantum mechenical system can be decomposed into operators which transform as irreducible representations of the symmetry group in coordinate and spin space. Each of these components has a physical meaning connected with the expectation values of certain operators. The reduced density matrices can be decomposed in a completely analogous way. The symmetry properties of the total wave function give rise to degeneracies of the eigenvalues of the reduced density matrices. These degeneracies can be removed by requiring that the natural spin orbitals (NSO, defined as the eigenfunctions of the first-order density matrix), as well as the natural spin geminals (NSG, the eigenfunctions of the second-order density matrix) and their spinless counterparts transform as irreducible representations of the symmetry group and are eigenfunctions of S/sup 2/ and S/sub z/. In many cases this requirement is compatible with the original definition of the NSO, the NSG, etc., e.g., when there is no spatial degeneracy of the total wave function and when the Z- component of the total spin vanishes. When these conditions are not fulfilled an alternative definition of the NSO and the NSG is proposed. (auth)

Nicholas J. Cox

A summary is presented of the properties of the coefficient matrices formed by expanding the two-body reduced density matrix in a complete set of two-electron wave functions. Calculating the relationship between the many electron wave function and these so-called two-body density matrices enables the derivation of necessary $N$-representability conditions that ensure a given matrix obeys the postulates of many-body quantum mechanics.