Second-order reduced density matrix

Hamiltonians involving more than two electron interactions. I shall use this to illustrate the general case of arbitrary p. The second-order reduced density matrix (2-RDM) of a pure state ij/, a function of four particles, is defined as follows ... 

In 2001, Nakata and co-workers presented the results of realistic fermionic systems, like atoms and molecules, larger than previously reported for the variational calculation of the second-order reduced density matrix (2-RDM) [1]. 

C. Valdemoro, Approximating the second-order reduced density matrix in terms of the first-order one. Phys. Rev. A 45, 4462 (1992). 

C. Valdemoro, D. R. Alcoba, and L. M.Tel, Recent developments in the contracted Schrddinger equation method controlling the iV-representabiUty of the second-order reduced density matrix. Int. J. Quantum Chem 93, 212 (2003). 

This second-order reduced density matrix for SOAGP has a block diagonal form—one dense block for each geminal and one diagonal block for the mixing between geminals. 

P. W. Ayers and M. Levy, Generalized density-functional theory conquering the A-representability problem with exact functionals for the electron pair density and the second-order reduced density matrix. J. Chem. Set 117, 507-514 (2005). 

INEQUALITIES RELATING THE ELEMENTS OF THE SECOND-ORDER REDUCED DENSITY MATRIX... 

Abstract. The elements of the second-order reduced density matrix are pointed out to be written exactly as scalar products of specially defined vectors. Our considerations work in an arbitrarily large, but finite orthonormal basis, and the underlying wave function is a full-CI type wave function. Using basic rules of vector operations, inequalities are formulated without the use of wave function, including only elements of density matrix. 

The second-order reduced density matrix in geminal basis... 

The second-order reduced density matrix in geminal basis is expressed by the parameters of the wave function [6-9]. The second-order reduced density matrix (3) is the kernel of the second-order reduced density operator. Quantities 0 are matrix elements of the second-order reduced density operator in the basis of geminals. In spite of this, the expression element of density matrix is usual. In this sense, in the followings 0 is called as element of second-order reduced density matrix. 

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. 

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. 

By means of (3 20) and (3 21) we can write down the second order reduced density matrix with elements Piikl for the wave function (3 15) as... 

E. Brandas, C.A. Chatzidimitriou-Dreismann, On the Connection Between Certain Properties of the Second-Order Reduced Density Matrix and the Occurrence of Coherent-Dissipative Structures in Disordered Condensed Matter, Int. J. Quant. Chem. 40 (1991) 649. 

If the Hartree-Fock determinant dominates the wavefunction, some of the occupation numbers will be close to 2. The corresponding MOs are closely related to the canonical Hartree-Fock orbitals. The remaining natural orbitals have small occupation numbers. They can be analysed in terms of different types of correlation effects in the molecule . A relation between the first-order density matrix and correlation effects is not immediately justified, however. Correlation effects are determined from the properties of the second-order reduced density matrix. The most important terms in the second-order matrix can, however, be approximately defined from the occupation numbers of the natural orbitals. Electron correlation can be qualitatively understood using an independent electron-pair model . In such a model the correlation effects are treated for one pair of electrons at a time, and the problem is reduced to a set of two-electron systems. As has been shown by Lowdin and Shull the two-electron wavefunction is determined from the occupation numbers of the natural orbitals. Also the second-order density matrix can then be specified by means of the natural orbitals and their occupation numbers. Consider as an example the following simple two-configurational wavefunction for a two-electron system ... 

Methods using the second-order reduced density matrix as variable employ the exact energy functional in terms of 72, but must make approximations for enforcing the N-representability of 72. 

The exchange-correlation energy can thus be obtained by integrating the electron-electron interaction over the A variable and subtracting the Coulomb part. The right-hand side of eq. (B.18) can be written in terms of the second-order reduced density matrix eq. (6.14), and the definition of the exchange-correlation hole in eq. (6.21) allows the Coulomb energy to be separated out. 

Similarly, we can define the second-order reduced density matrix d whose matrix elements are determined from the expectation value of... 

We can write the first- and second-order reduced density matrix spatially resolved in terms of MOs (()J... 

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... 

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... 

In summary, this careful and systematic work has laid the groundwork for a topological force field (both inter and intra) drawn from ab initio electron densities and perhaps the second order reduced density matrix. Topological potentials are extractable and valid irrespective of computational... 

The exchange-correlation energy can be subject to the same type of multipole expansion. For any wavefunction we can write the second order reduced density matrix as a function of molecular orbitals via... 

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