Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physical Review A

Two-body-operator matrix element factorization technique in U(n)↓ SN unitary bases. II. Raising-lowering operators

R. D. Kent, M. Schlesinger, Pardu S. Ponnapalli

Published Apr 1, 1989• 9 citations
Publisher

Abstract

A detailed exposition of explicit formulas used in the evaluation of raising-lowering forms of two-body-operator matrix elements is presented. The methods are based on the use of the unitary group distinct row table and graphical representation of the many-particle basis. All matrix elements are expressible in terms of scalar and simple matrix factors. In order to facilitate the derivations of simple, computationally efficient forms for the matrix factors, we utilize a recently developed calculus based on elementary graphs. The methods are applicable to systems of particles involving spins greater than (1/2 and reduce to previously known results for the case of spin (1/2.

DOI
10.1103/physreva.39.3260

Related papers

Two-body-operator matrix-element factorization technique in U(n)↓ SN unitary bases: Raising-raising operators

R. D. Kent, M. Schlesinger, Pardu S. Ponnapalli

A detailed exposition of explicit formulas used in the evaluation of raising-raising forms of two-body-operator matrix elements is presented. The methods are based on the use of the unitary-group distinct-row tabular-graphical representation of the many-particle basis. All matrix elements are expressible in the form of scalar and 2\ifmmode\times\else\texttimes\fi{}2 matrix factors. In order to facilitate the derivations of simple, computationally efficient forms for the matrix factors we develop a calculus based on elementary graphs. The methods are applicable to systems of particles involving spins greater than (1/2 and reduce to previously known results for the case of spin (1/2.

Physical Review A•Jan 1, 1989•15 citations

Two-body-operator matrix elements in SU(n) for application to complex spectroscopy

R. D. Kent, M. Schlesinger

We present a derivation of closed-form algebraic expressions for two-body operators in the context of the special-unitary-group approach to complex spectroscopy. The method involves the use of a graphical representation of the permutational symmetry-adapted irreducible-representation basis which facilitates the decomposition of the matrix elements into simple factors. The results afford significant reductions of computational complexity in cases pertaining to higher-than-spin-1/2 N-particle systems.

Physical Review A•Nov 1, 1987•5 citations

Particle-hole formulation of the unitary group approach to the many-electron correlation problem. II. Matrix element evaluation

M. J. Boyle, Josef Paldus

Graphical methods of spin algebras are used to derive the expressions for the matrix elements of the total particle-number-conserving operators in the basis of hole-particle states, adapted to the chain $\mathrm{U}({n}^{\ensuremath{'}}+{n}^{\ensuremath{'}\ensuremath{'}})\ensuremath{\supset}\mathrm{U}({n}^{\ensuremath{'}})\ensuremath{\bigotimes}\mathrm{U}({n}^{\ensuremath{'}\ensuremath{'}})$, where ${n}^{\ensuremath{'}}$ and ${n}^{\ensuremath{'}\ensuremath{'}}$ designate dimensions of particle and hole subspaces, respectively. The matrix elements are expressed as a product of segment values, each associated with one orbital level as in the particle formalism, and of an additional segment value, representing a linkage of the hole and particle subspaces. It is shown that the particle formalism segment values can be used throughout except for the link segment, whose possible values are derived. An example of hole-particle bases and of their graphical representations is given and the advantages of the hole-particle formalism in shell-model calculations are outlined. An extension of this formalism to particle-number-nonconserving operators, needed in applications involving the $m\mathrm{p}\ensuremath{-}n\mathrm{h}$ propagators with $m\ensuremath{\ne}n$, $m$, $n=0$, 1, and 2, is discussed.

Physical Review A•Dec 1, 1980•49 citations

Evaluation of two-body operator matrix elements in the unitary group approach

Hai‐Lun Lin, Yu-Fang Cao

The authors extend the Weyl graphical method to the two-body operator matrix elements in the many-particle system. The evaluation is accomplished by a simple and efficient recursion scheme.

Journal of Physics A Mathematical and General•Feb 21, 1990•0 citations

Unitary-group approach to spin-dependent operators

R. D. Kent, M. Schlesinger

General, closed-form expressions are presented for the matrix elements of ${\mathit{ssS}}_{\mathit{K}}$(${\mathbf{s}}_{\mathit{i}}$,${\mathbf{s}}_{\mathit{j}}$) (rank K=0,1,2), tensors in the single-particle spin operators ${\mathbf{s}}_{\mathit{i}}$, in the unitary-group approach. Factorization of the matrix elements in terms of 6-j and 9-j coefficients is achieved through methods of spin-graphical analysis. Results are also obtained for the rank-2 spin-spin operator.

Physical Review A•Aug 1, 1990•31 citations

Unitary group approach to reduced density matrices

M. D. Gould, Josef Paldus, Graham S. Chandler

A fully spin-adapted approach to many-electron density matrices is developed in the context of the unitary group approach to many-electron systems. An explicit expression for the single-electron spin-density operator, as a polynomial of degree two in the orbital U(n) generators, is derived for the case of spin-independent systems. Extensions to spin-dependent systems are also considered, leading to the appearance of total-spin transition densities, whose general properties are investigated. A corresponding formalism for the two-electron density matrix, which is capable of further generalization, is also developed. The results of this paper, together with recent developments on the matrix elements of the U(2n) generators in the electronic Gel’fand basis, afford a versatile method for the direct calculation of one- and two-body density matrices in the unitary group approach framework.

The Journal of Chemical Physics•Sep 15, 1990•33 citations
A detailed exposition of explicit formulas used in the evaluation of raising-lowering forms of two-body-operator matrix elements is presented. The methods are based on the use of the unitary group distinct row table and graphical representation of the many-particle basis. All matrix elements are expressible in terms of scalar and simple matrix factors. In order to facilitate the derivations of simple, computationally efficient forms for the matrix factors, we utilize a recently developed calculus based on elementary graphs. The methods are applicable to systems of particles involving spins greater than (1/2 and reduce to previously known results for the case of spin (1/2.
A detailed exposition of explicit formulas used in the evaluation of raising-raising forms of two-body-operator matrix elements is presented. The methods are based on the use of the unitary-group distinct-row tabular-graphical representation of the many-particle basis. All matrix elements are expressible in the form of scalar and 2\ifmmode\times\else\texttimes\fi{}2 matrix factors. In order to facilitate the derivations of simple, computationally efficient forms for the matrix factors we develop a calculus based on elementary graphs. The methods are applicable to systems of particles involving spins greater than (1/2 and reduce to previously known results for the case of spin (1/2.
We present a derivation of closed-form algebraic expressions for two-body operators in the context of the special-unitary-group approach to complex spectroscopy. The method involves the use of a graphical representation of the permutational symmetry-adapted irreducible-representation basis which facilitates the decomposition of the matrix elements into simple factors. The results afford significant reductions of computational complexity in cases pertaining to higher-than-spin-1/2 N-particle systems.
Graphical methods of spin algebras are used to derive the expressions for the matrix elements of the total particle-number-conserving operators in the basis of hole-particle states, adapted to the chain $\mathrm{U}({n}^{\ensuremath{'}}+{n}^{\ensuremath{'}\ensuremath{'}})\ensuremath{\supset}\mathrm{U}({n}^{\ensuremath{'}})\ensuremath{\bigotimes}\mathrm{U}({n}^{\ensuremath{'}\ensuremath{'}})$, where ${n}^{\ensuremath{'}}$ and ${n}^{\ensuremath{'}\ensuremath{'}}$ designate dimensions of particle and hole subspaces, respectively. The matrix elements are expressed as a product of segment values, each associated with one orbital level as in the particle formalism, and of an additional segment value, representing a linkage of the hole and particle subspaces. It is shown that the particle formalism segment values can be used throughout except for the link segment, whose possible values are derived. An example of hole-particle bases and of their graphical representations is given and the advantages of the hole-particle formalism in shell-model calculations are outlined. An extension of this formalism to particle-number-nonconserving operators, needed in applications involving the $m\mathrm{p}\ensuremath{-}n\mathrm{h}$ propagators with $m\ensuremath{\ne}n$, mmm, n=0n=0n=0, 1, and 2, is discussed.
General, closed-form expressions are presented for the matrix elements of ssSK{\mathit{ssS}}_{\mathit{K}}ssSK​(si{\mathbf{s}}_{\mathit{i}}si​,sj{\mathbf{s}}_{\mathit{j}}sj​) (rank K=0,1,2), tensors in the single-particle spin operators si{\mathbf{s}}_{\mathit{i}}si​, in the unitary-group approach. Factorization of the matrix elements in terms of 6-j and 9-j coefficients is achieved through methods of spin-graphical analysis. Results are also obtained for the rank-2 spin-spin operator.
A fully spin-adapted approach to many-electron density matrices is developed in the context of the unitary group approach to many-electron systems. An explicit expression for the single-electron spin-density operator, as a polynomial of degree two in the orbital U(n) generators, is derived for the case of spin-independent systems. Extensions to spin-dependent systems are also considered, leading to the appearance of total-spin transition densities, whose general properties are investigated. A corresponding formalism for the two-electron density matrix, which is capable of further generalization, is also developed. The results of this paper, together with recent developments on the matrix elements of the U(2n) generators in the electronic Gel’fand basis, afford a versatile method for the direct calculation of one- and two-body density matrices in the unitary group approach framework.