First order reduced density operator

Factoring the Energy Functional through the First Order Reduced Density Operator. 

An appealing way to apply the constraint expressed in Eq. (3.14) is to make connection with Natural Orbitals (31), in particular, to express p as a functional of the occupation numbers, n, and Natural General Spin Ckbitals (NGSO s), yr,, of the First Order Reduced Density Operator (FORDO) associated with the N-particle state appearing in the energy expression Eq. (3.8). In order to introduce the variables n and yr, in a well-defined manner, the... 

The first-order reduced density operator y can be defined in terms of its kernel function37... 

If we are interested only in properties that can be expressed in terms of one-electron operators, then it is sufficient to work with the first-order reduced density matrix rather than the A-electron wave function [23-27]. 

Natural resonance theory (NRT) is an optimal ab initio realization of the resonance weighting concepts expressed in Equation 7.10 and Equation 7.11. The necessary and sufficient condition that Equation 7.10 be satisfied for all possible density-related (one-electron) properties p is that the first-order reduced density operator be expressible as such a weighted average of localized density operators t... 

Alternative approaches to the many-electron problem, working in real space rather than in Hilbert space and with the electron density playing the major role, are provided by Bader s atoms in molecule [11, 12], which partitions the molecular space into basins associated with each atom and density-functional methods [3,13]. These latter are based on a modified Kohn-Sham form of the one-electron effective Hamiltonian, differing from the Hartree-Fock operator for the inclusion of a correlation potential. In these methods, it is possible to mimic correlated natural orbitals, as eigenvectors of the first-order reduced density operator, directly... 

Density matrices, in particular, the so-called first- and second-order reduced density matrices, are important quantities in the theoretical description of electronic structures because they contain all the essential information of the system under study. Given a set of orthonormal MOs, we define the first-order reduced density matrix D with matrix elements as the expectation value of the excitation operator E = aLa, -I- with respect to some electronic wave function Fgi,... 

To facilitate the discussion, we couch DFT in the language of p, the first-order reduced density operator of the noninteracting reference system. Consider an N electron system in a spin-compensated state and in an external potential Wext(r) (extension to spin-polarized state is trivial). The real space representation of p is the density matrix... 

Reduced density operators were first introduced by K. Husimi (Proc. Phys. Math. Soc. Japan 22 [1940], 264) to describe subsets of the IV-electron distribution (first-order for one-particle distributions, second-order for pair distributions, etc.) and are obtained from the full Mh-order (von Neumann) density operator electronic coordinates see, e.g., E. R. Davidson, Reduced Density Matrices in Quantum Chemistry (New York, Academic Press, 1976) and note 31. 

Elements of second order reduced density matrix of a fermion system are written in geminal basis. Matrix elements are pointed out to be scalar product of special vectors. Based on elementary vector operations inequalities are formulated relating the density matrix elements. While the inequalities are based only on the features of scalar product, not the full information is exploited carried by the vectors D. Recently there are two object of research. The first is theoretical investigation of inequalities, reducibility of the large system of them. Further work may have the chance for reaching deeper insight of the so-called N-representability problem. The second object is a practical one examine the possibility of computational applications, associate conditions above with known methods and conditions for calculating density matrices. 

The formulas above give the gradient and the Hessian in terms of matrix elements of the excitation operators. They can be evaluated in terms of one-and two-electron integrals, and first and second order reduced density matrices, by inserting the Hamiltonian (3 24) into equations (4 9), (4 11), and (4 13)-(4 15). Note that transition density matrices and are needed for the evaluation of the Cl coupling matrix (4 15). 

In Bohmian mechanics, the way the full problem is tackled in order to obtain operational formulas can determine dramatically the final solution due to the context-dependence of this theory. More specifically, developing a Bohmian description within the many-body framework and then focusing on a particle is not equivalent to directly starting from the reduced density matrix or from the one-particle TD-DFT equation. Being well aware of the severe computational problems coming from the first and second approaches, we are still tempted to claim that those are the most natural ways to deal with a many-body problem in a Bohmian context. 

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