Unitary group approach to the many-electron problem. III. Matrix elements of spin-dependent Hamiltonians

This is the final paper in a series of three directed toward the evaluation of spin-dependent Hamiltonians. In this paper we derive the reduced matrix elements of the U(2n) generators in a basis symmetry adapted to the subgroup U(n) × U(2) (i.e., spin-orbit basis), for the representations appropriate to many-electron systems. This enables a direct evaluation of the matrix elements of spin-dependent Hamiltonians in the spin-orbit basis. An alternative (indirect) method, which employs the use of U(2n) ↓ U(n) × U(2) subduction coefficients, is also discussed.     

Unitary group approach to the many-electron problem. II. Adjoint tensor operators for U(n)

In this paper we present a derivation of the U(n) adjoint coupling coefficients for the representations appropriate to many-electron systems. Since the states of a many-fermion system are to comprise the totally antisymmetric Nth rank tensor representation of U(2n), the work of this paper enables the matrix elements of the U(2n) generators to be evaluated directly in the U(n) × U(2) (i.e., spin orbit) basis using their transformation properties as adjoint tensor operators. A connection between the adjoint coupling coefficients, as derived in this paper, and the matrix elements of certain (spin independent) two-body operators is also presented. This indicates that in CI calculations, one may obtain the matrix elements of spin-dependent operators from the known matrix elements of certain spin-independent two-body operators. In particular this implies a segment-level formula for the matrix elements of the U(2n) generators in the spin-orbit basis.     

Bonded tableau unitary group approach to the many-electron correlation problem

A spin-free symmetry-adapted valence bond (VB ) state, named bonded tableau (BT ), is deduced from the classical bonded function and labeled by an at most two-column Weyl tableau. The complete set, which is composed of the BT basis or canonical bonded tableau (CBT ), can be constructed from an overcomplete set of BT states. CI CBT and VB CBT are two kinds of complete sets that are constructed in this paper. They can be used, respectively, in the CI and VB theory. It is shown that there is a one-to-one correspondence between the labeling scheme for CI CBT and the Gelfand–Tsetlin (GT ) basis. This relationship enables an efficient generation and compact representation of the BT basis if one desires to use the known global representation scheme for the GT basis. Effective algorithms for the matrix element evaluation of unitary group generators and products of generators between BT states are presented. In the formulation, the action of a generator on a BT state yields another BT state times a coefficient, so that the matrix elements of an arbitrary multiple product of generators are reduced to a calculation of the overlaps between BT states. The evaluation of the overlaps leads to a simple factorization into cycle contributions, whose values are given explicitly and only depend on the length parameters of the cycles. It is hoped that the presented formalism can facilitate the procedures for handling of the many-electron correlation problem.     

Unitary group tensor operator algebras for many-electron systems: I. Clebsch-Gordan and Racah coefficients

A basis for the Racah-Wigner algebra of irreducible representations of the unitary group U(n) that are pertinent to quantum chemical models of many-electron systems is developed. Standard Clebsch-Gordan coefficients and isoscalar factors (also called coupling factors or reduced Wigner coefficients) for both symmetric (S _N ) and unitary [U(n)] groups are extended to transformation coefficients and corresponding isoscalar factors relating canonical Young-Yamanouchi or Gel'fand-Tsetlin bases to simple partitioned bases. All these different types of isoscalar factors are interrelated using the well-known reciprocity between the S _N and U(n) tensor representations, and general expressions relating these different factors are given. For the two-column representations characterizing the many-electron theory, detailed explicit expressions are presented for both the above-mentioned relationships and for all relevant U(n) isoscalar factors. Finally, U(n) Racah coefficients are introduced and explicit expressions derived for certain special classes of these coefficients.Killam Research Fellow 1987–89.     

Unitary group approach to general system partitioning. II. U(n) matrix element evaluation in a composite basis

An explicit segment level formalism is derived for the matrix elements of the U(n) generators in an arbitrary (multishell) composite basis. The results of this paper, which contain the usual (spin-independent) unitary calculus approach as a limiting case, yield a more powerful and versatile algorithm than the traditional (spin-independent) unitary group formalism.     

A spin-dependent unitary group approach to many-electron systems

In this article we derive a segment-level formula for the matrix elements of the U(2n) generators in a basis symmetry adapted to the subgroup U(n) × U(2) (i.e., spin-orbit basis), for the representations appropriate to many-electron systems. This enables the direct evaluation of the matrix elements of spin-dependent Hamiltonians.     

Clifford algebra and unitary group formulations of the many-electron problem

An essential role of Clifford algebras for quantum-chemical finite-dimensional orbital models of many-electron systems is pointed out. The relationship between Clifford algebra matric units, the generators of the unitary group approach (UGA) and the higher order replacement or excitation operators, as well as between their first and second quantized realizations, is elucidated. The usefulness of higher order replacement operators in the spin-adaptation of various many-body theories is briefly outlined and illustrated on the orthogonally spin-adapted coupled-pair approach. A natural connection with the Clifford algebra UGA is explored and new possibilities for its exploitation in large scale configuration interaction calculations are suggested.Dedicated to Professor J. Koutecký on the occasion of his 65th birthdayKillam Research Fellow 1987–8     

Study of localized molecular orbitals using group theory methods and its approach to the many-electron correlation problem. III. Orthogonal bonded functions

By using the group symmetrical localized molecular orbitals (SLMOs) as configuration-generating orbitals (CGOs) of many-electron wave functions, the symmetry adaptation of many-electron spaces is greatly simplified, and novel orthogonal bonded functions (OBFs), as complete space- and spin-adapted antisymmetrized products, are introduced. The corresponding programs for the solutions of OBFs are developed. © 1993 John Wiley & Sons, Inc.     

Unitary group tensor operator algebras for many-electron systems: II. One- and two-body matrix elements

Relying on our earlier results in the unitary group Racah-Wigner algebra, specifically designed to facilitate quantum chemical calculations of molecular electronic structure, the tensor operator formalism required for an efficient evaluation of one- and two-body matrix elements of molecular electronic Hamiltonians within the spin-adapted Gel'fand-Tsetlin basis is developed. Introducing the second quantization-like creation and annihilation vector operators at the unitary group [U(n)] level, appropriate two-box symmetric and antisymmetric irreducible tensor operators as well as adjoint tensors are defined and their matrix elements evaluated in the electronic Gel'fand-Tsetlin basis as single products of segment values. Using these tensor operators, the matrix elements of one- and two-body components of a general electronic Hamiltonian are found. Explicit expressions for all relevant quantities pertaining to at most two-column irreducible representations that are required in molecular electronic structure calculations are given. Relationships with other approaches and possible future extensions of the formalism to partitioned bases or spin-dependent Hamiltonians are discussed.On leave from: Department of Chemistry, Xiamen University, Xiamen, Fujian, PR China.     

A graphical approach to permutation group representations for many-electron systems

A simple procedure is presented for obtaining the standard Young tableaux for the representation [(N/2) + S,(N/2) − S] of the permutation group ℒ_N for an N-electron system in spin state S directly from the spin branching diagram. We redefine the coordinate axes of the branching diagram to obtain a graph in terms of the partitions of the two-rowed Young diagram and define walks in this graph which yield directly the first rows of the allowed standard Young tableaux spanning a given representation when suitable weights have been assigned to the nodes in the graph. The allowed states are in a lexically ordered form and permit going easily from an index to an array and vice versa. © 1996 John Wiley & Sons, Inc.     

Unitary group approach to crystal field theory: Projection operator approach

A treatment for a strong crystal field of cubic symmetry is given using the unitary group approach. Projection operator techniques are utilized to achieve results equivalent to those given recently by Wen Zhenyi.     

Unitary group approach to spin-adapted open-shell coupled cluster theory

We show that the irreducible tensor operators of the unitary group provide a natural operator basis for the exponential Ansatz which preserves the spin symmetry of the reference state, requires a minimal number of independent cluster amplitudes for each substitution order, and guarantees the invariance of the correlation energy under unitary transformations of core, open-shell, and virtual orbitals. When acting on the closed-shell reference state with n_c doubly occupied and n_v unoccupied (virtual) orbitals, the irreducible tensor operators of the group U(n_c) ? U(n_V) generate all Gelfand-Tsetlin (GT) states corresponding to appropriate irreducible representation of U(n_c + n_v). The tensor operators generating the M-tuply excited states are easily constructed by symmetrizing products of M unitary group generators with the Wigner operators of the symmetric group S_M. This provides an alternative to the Nagel-Moshinsky construction of the GT basis. Since the corresponding cluster amplitudes, which are also U(n_c) ? U(n_s) tensors, can be shown to be connected, the irreducible tensor operators of U(n_c) ? U(n_v) represent a convenient basis for a spin-adapted full coupled cluster calculation for closed-shell systems. For a high-spin reference determinant with n, singly occupied open-shell orbitals, the corresponding representation of U(n), n=n_c + n_v + n_s is not simply reducible under the group U(n_c) ? U(n_s) ? U(n_v). The multiplicity problem is resolved using the group chain U(n) ? U(n_c + n_v) ? U(n_s) ? U(n_c) ?U(n_s)? U(n_v) ? U(n_v). The labeling of the resulting configuration-state functions (which, in general, are not GT states when n_c > 1) by the irreducible representations of the intermediate group U(n_c + n_v) ?U(n_s) turns out to be equivalent to the classification based on the order of interaction with the reference state. The irreducible tensor operators defined by the above chain and corresponding to single, double, and triple substitutions from the first-, second-, and third-order interacting spaces are explicitly constructed from the U(n) generators. The connectedness of the corresponding cluster amplitudes and, consequently, the size extensivity of the resulting spin-adapted open-shell coupled cluster theory are proved using group theoretical arguments. The perturbation expansion of the resulting coupled cluster equations leads to an explicitly connected form of the spin-restricted open-shell many-body perturbation theory. Approximation schemes leading to manageable computational procedures are proposed and their relation to perturbation theory is discussed. © 1995 John Wiley & Sons, Inc.     

Matrix element factorization in the unitary group approach for configuration interaction calculations

Techniques of diagrammatic spin algebra are employed to derive segment factorization formulas for spin-adapted matrix elements of one- and two-electron excitation operators. The spin-adapted basis is formed by the Yamanouchi–;Kotani geneological coupling method, and therefore constitutes an irreducible basis of the unitary group U(N), as prescribed by Gel'fand and Tsetlin. Several features distinguish this paper from similar work that has recently been published. First, intermediate steps in the derivation of each segment factor are fully documented. Comprehensive tables list the spin diagrams and phases that contribute to the possible segment factors. Second, a special effort has been made to distinguish between those parts of a segment factor that can be ascribed to a spin diagram and those parts which arise from the orbitals. The results of this paper should thus be useful for those who wish to extend diagrammatic spin algebra to evaluation of matrix elements for states built from nonorthogonal orbitals. Third, a novel graphical method has been introduced to keep track of phase changes that are induced by line up permutations of creation and annihilation operators. This technique may be useful for extension of our analysis to higher excitations. The necessary concepts of second quantization and diagrammatic spin algebra are developed in situ, so the present derivation should be accessible to those who have little prior knowledge of such methods.     

Freeon tensor product states and the unitary group formulation of the many-electron problem

The freeon tensor product basis provides a rapid method for the evaluation of matrix elements in the unitary group formulation of quantum chemistry. The method employs fast transformations between the Gel'fand and freeon tensor product basis.     

Theoretical approach to the evaluation of spin-only magnetic form factors for many-electron atomic systems

General expressions for evaluating spin-only magnetic form factors for many-electron atomic systems are derived using Racah algebra techniques. The formulas are derived in the |αLSM_LM_S〉 representation. The general formalism allows the evaluation of spin-only magnetic form factors beyond the Hartree-Fock approximation.     

Symmetric group approach to relativistic CI. I. General formalism

General formulas for matrix elements of spin-dependent operators in a basis of spin-adapted antisymmetrized products of orthonormal orbitals are derived. The resulting formalism may be applied to construction of the Hamiltonian matrices both for Pauli and for projected no-pair relativistic configuration interaction methods. From a formal point of view, it is a generalization of the symmetric group approach to the CI method for the case of spin-dependent Hamiltonians. © 1997 John Wiley & Sons, Inc.     

Study of localized molecular orbitals using group theory methods and its approach to the many-electron correlation problem. IV. The symmetry-adaptation of many-center integrals and hamiltonian matrix elements in MCSCF calculations

Group theoretic methods are presented for the transformations of integrals and the evaluation of matrix elements encountered in multiconfigurational self-consistent field (MCSCF) and configuration interaction (CI) calculations. The method has the advantages of needing only to deal with a symmetry unique set of atomic orbitals (AO) integrals and transformation from unique atomic integrals to unique molecular integrals rather than with all of them. Hamiltonian matrix element is expressed by a linear combination of product terms of many-center unique integrals and geometric factors. The group symmetry localized orbitals as atomic and molecular orbitals are a key feature of this algorithm. The method provides an alternative to traditional method that requires a table of coupling coefficients for products of the irreducible representations of the molecular point group. Geometric factors effectively eliminate these coupling coefficients. The saving of time and space in integral computations and transformations is analyzed. © 1994 by John Wiley & Sons, Inc.