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The Journal of Chemical Physics

Unitary group approach to reduced density matrices

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

Published Sep 15, 1990• 33 citations
Publisher

Abstract

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.

DOI
10.1063/1.458747

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Physical Review A•Apr 1, 1989•9 citations
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.
A new spin-dependent unitary group approach to the many-electron correlation problem is investigated. It is demonstrated that the matrix elements of the U(2n) generators, in the U(n)×U(2) adapted electronic Gel’fand basis, are determined by the matrix elements of a single U(n) adjoint tensor operator, herein denoted by Δij(1≤i, j≤n), where Δ is a polynomial of degree two in the U(n) matrix E=[Eij]. The method is then applied, in the second paper of the series, to derive a simple segment level formula for the matrix elements of all U(2n) generators. The advantages of this new procedure for computer implementation are discussed and the central role played by the matrix Δ for the determination of molecular spin densities is highlighted.
A formalism based on the unitary- and symmetric-group approaches to configuration-interaction methods has been applied to an evaluation of matrix elements of a spin-adapted reduced Hamiltonian (SRH). It has been shown that there is a simple closed-form expression for the matrix elements of the SRH matrices. This formula has been expressed as a linear combination of two-electron integrals with coefficients given as explicit functions of the parameters defining the finite-dimensional Hilbert space (number of orbitals, number of electrons, and the total spin quantum number).
Abstract The spin-independent properties of an electronic system are completely determined by spinless density functions, P1(1; 1') and P 2( 1,2; 1',2 '), derived from the 1-and 2-electron density matrices, p1(1; 1') and p2(1,2 ; 1',2'). These functions, and their physical inter­-pretation, have been dealt with in part I: the present paper contains the generalizations necessary when the Hamiltonian contains spin terms (as in discussions of electron and nuclear spin resonance). The density matrices, and more generally the transition density matrices p1(kλ|1; 1') and p2(kλ|l,2; 1',2') associated with states and Φkand Φλ are resolved into spinless components associated with different spin situations: p1 has four such components (one giving the density of a-spin electrons, a second the density of β-spin electrons, and two with less direct physical significance), while p2 has sixteen components. It is shown that all components, for states k=(S, M),λ= (S',M') of given S, S', are determined by three 2-electron functions.For k=λ these reduce to p2, the spinless pair function, from which the electron density p1 may be derived by integration; Q2a ‘conditional' spin density, from which the (1-electron) spin density Q1 may be derived (as in the case of p1 and P2); Q c. a., a ‘coupling anisotropy function', which describes the anisotropy of spin-spin coupling associated with quantization of the spin z-component. It is also useful to introduce (though it is not an independent function). Qc, a ‘coupling density function', which measures the coupling between the spins of two electrons at different points in space. All spin-dependent effects, including the usual spin-spin and spin-orbit couplings and also (indirect) nuclear and electron-nuclear spin-spin couplings, may be related to the many- electron wave function through the above density functions. All the densities have a physical interpretation, independent of the precise nature of the wave functions (which may, for example, be the exact eigenfunctions of a spinless Hamiltonian or merely approximations of orbital form). The derivation of the density functions from approximate wave functions is considered in some detail. For functions describing weakly coupled electron groups (e. g. inner-shell and valence electrons) the densities may be expressed in terms of those of the separate groups. This reduction may be followed by an orbital description of each group; and subsequently, by a linear-combination-of-atomic-orbitals (l. c. a. o.) approximation. In the simpler approxi­mations, all the coupling effects discussed are determined by electron and spin ‘populations'. Preliminary calculations suggest that such approximations may in many cases be adequate.
Abstract The representation of an N‐electron Schrödinger Hamiltonian on an orthonormal, spin‐free (freon) orbital product space is exactly modeled by a second‐degree polynomial in the infinitesimal generators of the unitary group. Symmetry adaptation of the orbital product space with respect to the symmetric group yields Gel'fand states which provide base for irreducible represntations of the unitary group. These exist in closed form as, in consequence, does the representation of the model Hamiltonian in this same basis. We retain as physical only those states characterized by tableaux with no more than two columns for which the spin labelling is one‐half the difference in the lengths of the two columns. The unitary group formulation is equivalent to standard, number‐conserving, second‐quantized, many‐body theory.
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.