Spin reduced density matrices

G. Gidofalvi and D. A. Mazziotti, Spin and symmetry adaptation of the variational two-electron reduced-density-matrix method. Phys. Rev. A 72, 052505 (2005). 

The matrix elements of the reduced density matrix needed to calculate the entanglement can be written in terms of the spin-spin correlation functions and the average magnetization per spin. The spin-spin correlation functions for the ground state are dehned as [62]... 

If electronic spin is not a focus of attention, then the spin-traced versions of these density matrices can be used. The r-space, spin-traced, first-order, reduced-density matrix is... 

The two-electron reduced density matrix is a considerably simpler quantity than the N-electron wavefunction and again, if the A -representability problem could be solved in a simple and systematic manner the two-matrix would offer possibilities for accurate treatment of very large systems. The natural expansion may be compared in form to the expansion of the electron density in terms of Kohn-Sham spin orbitals and it raises the question of the connection between the spin orbital space and the -electron space when working with reduced quantities, such as density matrices and the electron density. 

Reduced density matrix treatment of spin-orbit interaction terms in niany-electron systems 63... 

The reduced density-matrix element for LS coupling is found similarly, treating the orbital and spin factors independently. 

The quantum mechanical equation of motion for an ensemble of nuclei may be written in terms of the reduced density matrix, o-,37 which is defined as, where V>k(0) is a possible wave function for the ensemble of nuclear spins and py is the probability of the nuclei being in that state. The equation of motion is,... 

Explicit account of the electron interaction within a self-consistent approach modifies the interpretation of the parameters. Slater s notion of the average of configurations and fractional occupation will be consistently applied in the grand canonical ensemble form. The one-particle reduced density matrix retains the symmetry of the crystal field and spin-orbit matrices, thus... 

Since the bath is made up of harmonic oscillators, it can be eliminated analytically (provided that one is only interested in the reduced density matrix of the reaction coordinate). After performing this elimination, the bath-plus-coupling part H of the Hamiltonian leads to an influence functional 0[cr] in terms of the spins cr, alone [38,39], and the correlation function takes the form... 

In order to obtain a bond order formula for open-shell systems that can be applied for both the indep)endent-partide model and correlated wave functions and which simultaneously yields unique bond orders for all spin multiplet components (in the absence of a magnetic field), Alcoba et al. [151, 152] derived a general expression (in the Hilbert space partitioning scheme) from a second-order reduced density matrix. Furthermore, as the first- and second-order reduced density matrices are invariant with respect to the spin projection, they are only a function of the total spin or similarly of the maximum projection S = and the bond order can be evaluated for the highest spin-projected state = S. They arrived at the following expression for the bond order... 

The orbitals that diagonalize the first-order reduced density matrix are the natural spin orbitals x-... 

First, we provide the main EUE definitions using the conventional reduced density matrix (RDM) methods. In singlet states, the first-order RDM (1-RDM) can be defined as a spin-free matrix which is also termed the charge density matrix. Throughout the paper, the capital letter D will be denoted the charge density matrix. In Dirac s bra-ket notation, D conveniently takes a compact form of a spectral resolution, that is the following diagonal form ... 

We begin with the spin-free reduced density matrix p (r, r t) is just the electron density. In chemical applications, it is often expanded in atomic orbitals XaA (r) as... 

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