One-particle reduced density matrix

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]. 

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. 

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... 

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

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

Knowledge of the 2-particle reduced density matrix (2-RDM) allows one to calculate the energy and other observables for atomic and molecular systems with... 

In the last decade, a series of functionals has been developed [1, 2] using a reconstruction proposed by Piris [3] of the two-particle reduced density matrix (2-RDM) in terms of the one-particle RDM (1-RDM). In particular, the Piris natural orbital functional 5 (PNOF5) [4, 5] has proved to... 

In general, the equations for the density operator should be solved to describe the kinetics of the process. However, if the nondiagonal matrix elements of the density operator (with respect to electron states) do not play an essential role (or if they may be expressed through the diagonal matrix elements), the problem is reduced to the solution of the master equations for the diagonal matrix elements. Equations of two types may be considered. One of them is the equation for the reduced density matrix which is obtained after the calculation of the trace over the states of the nuclear subsystem. We will consider the other type of equation, which describes the change with time of the densities of the probability to find the system in a given electron state as a function of the coordinates of heavy particles Pt(R, q, Q, s,...) and Pf(R, q, ( , s,... ).74,77 80... 

In Section 8.4.2, we considered the problem of the reduced dynamics from a standard DFT approach, i.e., in terms of single-particle wave functions from which the (single-particle) probability density is obtained. However, one could also use an alternative description which arises from the field of decoherence. Here, in order to extract useful information about the system of interest, one usually computes its associated reduced density matrix by tracing the total density matrix p, (the subscript t here indicates time-dependence), over the environment degrees of freedom. In the configuration representation and for an environment constituted by N particles, the system reduced density matrix is obtained after integrating pt = T) (( over the 3N environment degrees of freedom, rk Nk, ... 

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. 

We can word the results up to this point for an N-particle fermion system, using M-dimensional one-particle function basis, the elements of the second-order reduced density matrix in geminal basis are scalar products of ( ) piece of ( 2) dimensional vectors. 

We now introduce a set of preferred coordinates ), cf Refs. [Kiibler 1973 Zeh 1973 Zurek 1981 Zurek 1982 Zurek 2003], These are the relevant degrees of freedom coupled to the neutron probe. The density matrix needed in (13) is then the reduced one in the space spanned by these states, and it is obtained by tracing out the (huge number of the) remaining degrees of freedom belonging to the "environment" of the microscopic scattering system (e.g., a proton and its adjacent particles). To simplify notations, we denote this reduced density matrix by p too. 

He goes on to show that for the description of the energy it is sufficient to know the second-order reduced density matrix F(jc, jc2 i 2)- This was a favorite subject when he was lecturing and led to speculations about the possibility to compute the second-order density matrix directly, and discussions of the so called A/ -represent-ability problem. In spite of several attempts, this way of attacking the quantum chemical many-particle problem has so far been unsuccessful. Of special interest was the first-order reduced density matrix, 7( 1 jC ), which when expanded in a complete one-electron basis, ij/, is obtained as... 

As shown in [3], this cannot explain the large rates because the decreased interchromophoric distance is counteracted by a reduced orientation factor due to the deviation from the optimal collinear arrangement. As a further point, the vahdity of the dipole approximation, Eq. (7), was investigated [3], To this end, the electronic coupling matrix element in Eq. (6) was calculated from the Coulomb integral over the one-particle transition densities of the donor yofr.r ) and the acceptor yA(<">> ) ... 

Note that the term open system refers here to exchange of energy and phase with the environment, as the number of particles is conserved throughout. The reduced density matrix p t) evolves coherently under the influence of the nuclear Hamiltonian, Hnuc, and the non-adiahatic effects enter the equation via the dissipative Liouvillian superoperator jSfn- The latter is also termed memory kernel , as it contains information about the entire history of the environmental evolution and its interaction with the system. The definition of the memory kernel is by no means unique nor straightforward. One possible solution is to start from the microscopic Hamiltonian of the total system, eqn (1). Using the projector formalism, it is possible to separate the evolution of the system, i.e., the... 

Here we will present the formulae needed for calculating the reduced one-particle density matrices from the floating correlated Gaussians used in this work. The first-order density matrix for wave function T (ri,r2,..., r ) for particle 1 is defined as... 

Direct minimization of the energy as a functional of the p-RDM may be achieved if the p-particle density matrix is restricted to the set of Al-represen-table p-matrices, that is, p-matrices that derive from the contraction of at least one A-particle density matrix. The collection of ensemble Al-representable p-RDMs forms a convex set, which we denote as P. To define P, we first consider the convex set of p-particle reduced Hamiltonians, which are... 

A quantum system of N particles may also be interpreted as a system of (r — N) holes, where r is the rank of the one-particle basis set. The complementary nature of these two perspectives is known as the particle-hole duality [13, 44, 45]. Even though we treated only the iV-representability for the particles in the formal solution, any p-hole RDM must also be derivable from an (r — A)-hole density matrix. While the development of the formal solution in the literature only considers the particle reduced Hamiltonian, both the particle and the hole representations for the reduced Hamiltonian are critical in the practical solution of N-representability problem for the 1-RDM [6, 7]. The hole definitions for the sets and are analogous to the definitions for particles except that the number (r — N) of holes is substituted for the number of particles. In defining the hole RDMs, we assume that the rank r of the one-particle basis set is finite, which is reasonable for practical calculations, but the case of infinite r may be considered through the limiting process as r —> oo. 

In this section we review the known theorems that relate entanglement to the ranks of density matrices [52]. The rank of a matrix p, denoted as rank(p), is the maximal number of linearly independent row vectors (also column vectors) in the matrix p. Based on the ranks of reduced density matrices, one can derive necessary conditions for the separability of multiparticle arbitrary-dimensional mixed states, which are equivalent to sufficient conditions for entanglement [53]. For convenience, let us introduce the following definitions [54—56]. A pure state p of N particles Ai, A2,..., is called entangled when it cannot be written... 

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