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Kluwer Academic Publishers eBooks

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

Gergely Dezső, Imre Bálint, I.K. Gyémánt

Published May 4, 2006• 0 citations
Publisher

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.

DOI
10.1007/1-4020-4528-x_6

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The density operator (density matrix) of a quantum mechenical system can be decomposed into operators which transform as irreducible representations of the symmetry group in coordinate and spin space. Each of these components has a physical meaning connected with the expectation values of certain operators. The reduced density matrices can be decomposed in a completely analogous way. The symmetry properties of the total wave function give rise to degeneracies of the eigenvalues of the reduced density matrices. These degeneracies can be removed by requiring that the natural spin orbitals (NSO, defined as the eigenfunctions of the first-order density matrix), as well as the natural spin geminals (NSG, the eigenfunctions of the second-order density matrix) and their spinless counterparts transform as irreducible representations of the symmetry group and are eigenfunctions of S/sup 2/ and S/sub z/. In many cases this requirement is compatible with the original definition of the NSO, the NSG, etc., e.g., when there is no spatial degeneracy of the total wave function and when the Z- component of the total spin vanishes. When these conditions are not fulfilled an alternative definition of the NSO and the NSG is proposed. (auth)

•Oct 1, 1963•0 citations
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 structure of second-order reduced density matrix (2RDM) of N-electron systems is studied. The wave function corresponding to the 2RDM is assumed to be linear combination of Slater-deteminants (a FCI-type wave function) constructed over a finite dimensional orthonormal one-particle basis set. Our considerations are independent from the size and the quality of the one-particle basis. Using the definition of second-order reduced density matrix, and properties of Slater-determinants a set of inequalities are derived for the 2RDM. Inequalities can be formulated without the wave function, and without higher order density matrices, involving only the density matrix, and can be considered as necessary N-representability conditions.
The reduction of density matrices defined with respect to a finite basis set is considered. A basis set can be introduced into the vector space of Hermitian matrices acting on functions of the coordinates of ppp particles for each value of ppp. An analogy with the construction of spin eigenfunctions is used to obtain a particular basis set called the reducing basis. Reduction of one of these basis elements is either one to one or maps into the origin. Any element in the preimage of a density matrix can thus be resolved into two components, one uniquely determined by the density matrix and the other arbitrary within a certin subset. A restatement of the NNN-representability problem is given, and two sufficient conditions and one necessary condition for NNN representability are given in terms of distances and norms.
Part I. CHAPTER 1: N-REPRESENTABILITY (A. John Coleman). CHAPTER 2: HISTORICAL INTRODUCTION (Mitja Rosina). Part II. CHAPTER 3: VARIATIONAL TWO-ELECTRON REDUCED-DENSITY-MATRIX THEORY (David A. Mazziotti). CHAPTER 4: THE LOWER BOUND METHOD FOR DENSITY MATRICES AND SEMIDEFINITE PROGRAMMING (Robert M. Erdahl). CHAPTER 5: THE T1 AND T2 REPRESENTABILITY CONDITIONS (Bastiaan J. Braams, Jerome K. Percus, and Zhengji Zhao). CHAPTER 6: SEMIDEFINITE PROGRAMMING: FORMULATIONS AND PRIMAL-DUAL INTERIOR-POINT METHODS (Mituhiro Fukuda, Maho Nakata, and Makoto Yamashita). Part III. CHAPTER 7: THEORY AND METHODOLOGY OF THE CONTRACTED SCHRO DINGER EQUATION (By C. Valdemoro) CHAPTER 8: CONTRACTED SCHRO DINGER EQUATION (By David A. Mazziotti). CHAPTER 9: PURIFICATION OF CORRELATED REDUCED DENSITY MATRICES: REVIEW AND APPLICATIONS (By D. R. Alcoba). CHAPTER 10: CUMULANTS, EXTENSIVITY, AND THE CONNECTED FORMULATION OF THE CONTRACTED SCHRO DINGER EQUATION (By John M. Herbert and John E. Harriman). CHAPTER 11: GENERALIZED NORMAL ORDERING, IRREDUCIBLE BRILLOUIN CONDITIONS, AND CONTRACTED SCHRO DINGER EQUATIONS (By Werner Kutzelnigg and Debashis Mukherjee). CHAPTER 12: ANTI-HERMITIAN FORMULATION OF THE CONTRACTED SCHRO DINGER THEORY (By David A. Mazziotti). CHAPTER 13: CANONICAL TRANSFORMATION THEORY FOR DYNAMIC CORRELATIONS IN MULTIREFERENCE PROBLEMS (By Garnet Kin-Lic Chan and Takeshi Yanai). Part IV. CHAPTER 14: NATURAL ORBITAL FUNCTIONAL THEORY (By Mario Piris). CHAPTER 15: GEMINAL FUNCTIONAL THEORY (By B. C. Rinderspacher). CHAPTER 16: LINEAR INEQUALITIES FOR DIAGONAL ELEMENTS OF DENSITY MATRICES (By Paul W. Ayers and Ernest R. Davidson). Part V. CHAPTER 17: PARAMETERIZATION OF THE 2-RDM (A. John Coleman). CHAPTER 18: ENTANGLEMENT, ELECTRON CORRELATION, AND DENSITY MATRICES (By Sabre Kais). AUTHOR INDEX. SUBJECT INDEX.
The density operator (density matrix) of a quantum mechenical system can be decomposed into operators which transform as irreducible representations of the symmetry group in coordinate and spin space. Each of these components has a physical meaning connected with the expectation values of certain operators. The reduced density matrices can be decomposed in a completely analogous way. The symmetry properties of the total wave function give rise to degeneracies of the eigenvalues of the reduced density matrices. These degeneracies can be removed by requiring that the natural spin orbitals (NSO, defined as the eigenfunctions of the first-order density matrix), as well as the natural spin geminals (NSG, the eigenfunctions of the second-order density matrix) and their spinless counterparts transform as irreducible representations of the symmetry group and are eigenfunctions of S/sup 2/ and S/sub z/. In many cases this requirement is compatible with the original definition of the NSO, the NSG, etc., e.g., when there is no spatial degeneracy of the total wave function and when the Z- component of the total spin vanishes. When these conditions are not fulfilled an alternative definition of the NSO and the NSG is proposed. (auth)