# Unitary group approach to reduced density matrices

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.

## Summary (2 min read)

### I. INTRODUCTION

- Since then there have appeared many applications and extensions which are discussed in several books and review articles.
- I 4--16 The experiments and the interpretation of such neutron-diffraction data are, however, difficult and theoretical information on the distribution of unpaired spins, as afforded by the one-and two-electron spin densities, can be of crucial importance and help.
- 38 Thus, as indicated above, the UGA affords a convenient method for the calculation of CI wave functions and hence provides an ideal framework in which to develop a spin-and charge-density matrix formalism.
- The authors then consider extensions to two-body density matrices which, for spinindependent systems, are shown to be completely determined by six density operators, which in turn are expressible in terms of the one-body charge and spin-density operators.

### II. ONE-PARTICLE DENSITY MATRICES

- The authors shall consider density matrices arising from wave functions constructed from 2n spin orbitals "'il' (x). they introduce corresponding fermion creation and annihilation operators.
- The generators (2) and ( 3) collectively constitute the generators of the subgroup U(n) XU(2) (outer direct product) of U(2n), referred to herein as the spin-orbit (SO) subgroup.
- 3 .4 This tensor form for reduced density matrices is useful, particularly for discussing twoand higher-electron density operators, which will be considered in Sec. IV and V.
- For one-body densities, it isjust as convenient to work with the matrix formulation ofEq. ( 12), which the authors adopt throughout most of the paper.

### III. SPIN DENSITIES AND TOTAL-SPIN TRANSITION DENSITIES

- In this section the authors consider the partitioning of the singleelectron density matrix into components according to their shifts on the total-spin quantum number S.
- This leads to the appearance of total-spin transition densities, which increase or decrease the total-spin quantum number S, as well as a density-matrix component which commutes with the total spin S.
- The resulting SO basis states may be written as The VGA was originally designed 20 . 21 for spin-independent problems, in which case the electronic Hamiltonian is expressible solely in terms of the orbital V(n) generators.
- This latter problem, which has been addressed by Gould and Chandler 41 .
- 45 and more recently by Gould and PaId us, 39 is also essential for the evaluation of the single-electron density matrix, Eq. (6'), in the SO basis, Eq. ( 14), to which the authors now tum.

### IV. TWO-BODY DENSITY OPERATORS

- In this section the authors consider extensions of their previous results to two-body density matrices.
- The operators (30) constitute a convenient basis for the universal enveloping algebra of U (2n) which, as shall be discussed in Sec. V, is useful for investigating polynomial identities satisfied by the group generators.
- This leads to the number density operator p~, Eq. ( 38), which is an SU (2) scalar, together with the two vector densities [cf.

### a l =HlTl®lTs(S2) -1T I (S2)®I-I®lT,(S2)],

- The authors note that the vector densities can only increase or decrease the total spin S by one unit, while the rank-two tensor densities may also increase or decrease S by two units,.
- The authors have thus demonstrated that the full two-electron density matrix is completely determined by the six scalar densities, Eqs. ( 38), ( 42), (45), and ( 49), together with the ten spin transition density operators, Eq, (53).
- For wave functions with well-defined total spin S, these latter transition densities do not contribute and the density matrix is fully determined by the scalar densities, as noted earlier.
- Finally, it is clear that the spin transition density operators may be conveniently expressed using the off-diagonal operator equivalent method, which has been extensively investigated 55 -59 in the hyperbolic operator formalism ofSchwinger 60.

### CASE

- It follows that on the antisymmetric tensor irrep of U(2n), pertinent to the many-electron problem, these operators must satisfy the relations 5 \ EQUATION For the orbital U(n) generators, such relations have been already discussed by Paldus and Jeziorski.
- Thus, the relations of Eq. ( 59) imply the U(2n) identities, Eq. ( 8), but also contain extra information which the authors utilize below.
- The reduced identities satisfied by the unitary group generators on a given k-column irrep may similarly be derived and will be investigated elsewhere.

### VI. CONCLUSIONS

- A fully spin-adapted approach to one-and two-electron density matrices has been presented in the framework of the UGA.
- It follows that the U (n) matrix .6., Eq. ( 22), plays a fundamental role in the determination of the spin density of a molecule and hence also of two-(and higher-order) electron densities.
- The authors have investigated this algebra for one-electron density operators and will present these results elsewhere.
- In particular, it is of interest to determine the k-electron density matrix, for wave functions with well-defined total spin S, in terms of the onebody charge-and spin-density operators, as required when dealing with spin-independent systems.
- As noted earlier, this problem has already been solved for the one-and twoelectron case in Secs. III and V.

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