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

• Published in: Journal of chemical theory and computation Vol. 16; no. 3; pp. 1578 - 1585
• Main Author: Cioslowski, Jerzy
• Format: Journal Article
• Language: English
• Published: United States American Chemical Society 10.03.2020
• Subjects: Density functional theory; Eigenvalues; Eigenvectors; Electron density; Electron states; Mathematical analysis; Matrix methods; Operators (mathematics); Polarization (spin alignment); Tensors; Wave functions
• ISSN: 1549-9618; 1549-9626; 1549-9626
• DOI: 10.1021/acs.jctc.9b01155

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) G 2 and the tensor of two-electron integrals g is critically analyzed. The dependence of G 2 on both the vector of occupation numbers n and g derived from the corresponding vector ϕ ­(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, G 2 (and thus U) is found to be a function of not only n and g , but also of the vector s ≡ { ⟨ ϕ p ( 1 ) | ŝ z ( 1 ) | ϕ p ( 1 ) ⟩ } and the matrix S ≡ { ⟨ ϕ p ( 1 ) | Ŝ ( 1 ) | ϕ q ( 1 ) ⟩ } , where Ŝ ( 1 ) is the spin-flip operator. The presence of spin polarization imposes additional constraints upon G 2 , including a simple condition that, when satisfied, assures the underlying wavefunction being an eigenstate of both the Ŝ 2 and Ŝ z operators with the eigenvalues S ( S + 1 ) and ± S , 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 G 2 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” G 2 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.