10 Density matrix and spin-polarization phenomena

B. Z. Kopeliovich

В. И. Стражев, P. L. Shkolnikov

V. L. Jacobs

The theory of multiphoton ionization for an atomic system of arbitrary complexity is developed using a density matrix formalism. An expression is obtained which determines the differential N-photon ionization cross section as a function of the polarization states of the target atom and the incident radiation. The parameters which characterize the photo-electron angular distribution are related to the general reduced matrix elements for the N-photon transition. Two-photon ionization of unpolarized atoms is treated as an illustration of the use of the theory. The dependence of the multiphoton ionization cross section on the polarization state of the incident radiation, which has been observed in two- and three-photon ionization of Cs, is accounted for by the theory. Finally, the photoelectron spin polarization produced by the multiphoton ionization of unpolarized atoms, like the analogous polarization resulting from single-photon ionization, is found to depend on the circular polarization of the incident radiation.

Jerzy Ciosłowski

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) and the tensor of two-electron integrals g is critically analyzed. The dependence of on both the vector of occupation numbers n and g derived from the corresponding vector (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, (and thus U ) is found to be a function of not only n and g, but also of the vector and the matrix, where is the spin-flip operator. The presence of spin polarization imposes additional constraints upon, including a simple condition that, when satisfied, assures the underlying wavefunction being an eigenstate of both the and operators with the eigenvalues and, 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 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” 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.