One-Electron Reduced Density Matrix Functional Theory of Spin-Polarized Systems.

Abstract

The formulation of the density matrix functional theory (DMFT) in which the correlation component U of the electron-electron repulsion energy is expressed in terms of a model two-electron density cumulant matrix (a.k.a. the 2-cumulant) [Formula: see text] and the tensor of two-electron integrals g is critically analyzed. The dependence of [Formula: see text] on both the vector of occupation numbers n and g derived from the corresponding vector [Formula: see text](1) of natural spinorbitals, and its concomitant invariance to the nature of the spin-independent interparticle interaction potential that enters the definition of g are emphasized. In the case of spin-polarized systems, [Formula: see text] (and thus U) is found to be a function of not only n and g, but also of the vector [Formula: see text] and the matrix [Formula: see text], where [Formula: see text] is the spin-flip operator. The presence of spin polarization imposes additional constraints upon [Formula: see text], including a simple condition that, when satisfied, assures the underlying wavefunction being an eigenstate of both the [Formula: see text] and [Formula: see text] operators with the eigenvalues [Formula: see text] and [Formula: see text], respectively. This feature allows targeting electronic states with definite multiplicities, which is virtually impossible in the case of the Kohn-Sham implementation of density functional theory. Among the four possible pairing schemes for the natural spinorbitals that give rise to approximations employing [Formula: see text] with only two independent indices, three are found to result in unphysical constraints even for spin-unpolarized systems, whereas the failure of the fourth one turns out to be precipitated by the presence of spin polarization. Consequently, any implementation of DMFT based upon "two-index" [Formula: see text] is shown to be generally unsuitable for spin-polarized systems (and incapable of yielding the spin-parallel components of U for the spin-unpolarized ones). A clear distinction is made between the genuine 1-matrix functionals that are defined for arbitrary N-representable 1-matrices and general energy expressions that depend on auxiliary quantities playing the role of fictitious 1-matrices subject to additional (often unphysical) constraints.