One-particle reducibility

From the discussion so far, it is clear that the mapping to a system of noninteracting particles under the action of suitable effective potentials provides an efficient means for the calculation of the density and current density variables of the actual system of interacting electrons. The question that often arises is whether there are effective ways to obtain other properties of the interacting system from the calculation of the noninteracting model system. Examples of such properties are the one-particle reduced density matrix, response functions, etc. An excellent overview of response theory within TDDFT has been provided by Casida [15] and also more recently by van Leeuwen [17]. A recent formulation of density matrix-based TD density functional response theory has been provided by Furche [22]. 

Any arbitrary one-particle reduced Hamiltonian shifted by its A-particle ground-state energy must be expressible by the extreme HamUtonian elements in the convex set As we showed in Eq. (52), keeping the 1-RDM positive semide-finite is equivalent to applying the Al-representability constraints in Eq. (50) for the class of extreme positive semidefinite which may be parameterized by... 

D. A. Mazziotti, Geminal functional theory a synthesis of density and density matrix methods. J. Chem. Phys. 112, 10125 (2000). D. A. Mazziotti, Energy functional of the one-particle reduced density matrix a geminal approach, Chem. Phys. Lett. 338, 323 (2001). 

A direction for improving DPT lies in the development of a functional theory based on the one-particle reduced density matrix (1-RDM) D rather than on the one-electron density p. Like 2-RDM, the 1-RDM is a much simpler object than the A-particle wavefunction, but the ensemble A-representability conditions that have to be imposed on variations of are well known [1]. The existence [10] and properties [11] of the total energy functional of the 1-RDM are well established. Its development may be greatly aided by imposition of multiple constraints that are more strict and abundant than their DPT counterparts [12, 13]. 

D. A. Mazziotti, Energy functional of the one-particle reduced density matrix a geminal approach. 

Calculation of corrections generated by the gauge invariant set of diagrams with insertions of three radiative photons in the electron line in the skeleton diagrams in Fig. 9.2 is more complicated. Only insertions of the three-loop one-particle reducible diagrams with radiative photons in the electron line in Fig. 9.10 were considered thus far. Contribution of these diagrams in the Yennie gauge is equal to [29]... 

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... 

Reduced density matrices, introduced by Husimi [9], yield corresponding probability densities for the presence of n n < N) particles simultaneously in selected volume elements dxj,... x thus, for n = 1, the probability/unit volume of finding an electron (no matter which) at Xj will be obtained by integrating fV p (x x) over the positions of all fV — 1 volume elements dx2,...dxjv and dividing by [N — 1) (to avoid multiple counting). The quantity so defined is the one-particle reduced density matrix more explicitly, it becomes... 

In wavefunction form or in terms of first quantization, the one-particle reduced density matrix corresponding to an IV-electron state with wavefunction ( 1, 2, , n) is defined as... 

There exist such important properties of graphs as their one-particle reducibility (IPR) and, correspondingly, one-particle irreducibility (IPI). 

NATURAL ORBITALS AND THE ONE-PARTICLE REDUCED DENSITY MATRIX... 

Natural Orbitals and the One-Particle Reduced Density Matrix 252... 

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