Valence bond approach exploiting Clifford algebra realization of Rumer–Weyl basis

A detailed algorithm is described that enables an implementation of a general valence bond (VB ) method using the Clifford algebra unitary group approach (CAUGA ). In particular, a convenient scheme for the generation and labeling of classical Rumer–Weyl basis (up to a phase) is formulated, and simple rules are given for the evaluation of matrix elements of unitary group generators, and thus of any spin-independent operator, in this basis. The case of both orthogonal and nonrothogonal atomic orbital bases is considered, so that the proposed algorithm can also be exploited in molecular orbital configuration interaction calculations, if desired, enabling a greater flexibility for N-electron basis-set truncation than is possible with the standard Gel'fand–Tsetlin basis. Finally, an exploitation of this formalism for the VB method, based on semiempirical Pariser–Parr–Pople (PPP )-type Hamiltonian and nonorthogonal overlap-enhanced atomic orbital basis, and its computer implementation, enabling us to carry out arbitrarily truncated or full VB calculations, is described in detail.     

Unitary group tensor operator algebras for many-electron systems. III. Matrix elements in U(n 1 +n 2) ⊃ U(n 1) × U(n 2) partitioned basis

Exploiting our earlier results [J. Math. Chem. 4 (1990) 295–353 and 13 (1993) 273–316] on the unitary group U(n) Racah-Wigner algebra, specifically designed for quantum chemical calculations of molecular electronic structure, and the related tensor operator formalism that enabled us to introduce spin-free orbital equivalents of the second quantization-like creation and annihilation operators as well as higher rank symmetric, antisymmetric and adjoint tensors, we consider the problem of U(n) basis partitioning that is required for group-function type approaches to the many-electron problem. Using the U(n) U(n ₁) × U(n ₂),n =n ₁ +n ₂ adapted basis, we evaluate all required matrix elements of U(n) generators and their products that arise in one- and two-body components of non-relativistic electronic Hamiltonians. The formalism employed naturally leads to a segmented form of these matrix elements, with many of the segments being identical to those of the standard unitary group approach. Relationship with similar approaches described earlier is briefly pointed out.     

A unitary group formulation of many-body theory: Diagram systematics and use of the spin shifts

This paper shows that the spin-shift formalism developed in B. T. Pickup and A. Mukhopadhyay [Int. J. Quantum Chem. 26 , 101 (1984)] supports a one-component diagrammatics which has a systematics akin to that in the spin-orbital many-body theory. The diagrams are neither Goldstone nor Yutsis type, and characterize the chain U(2R) ? U(R)?SU(2) on which the spin-shift formalism is based. Accordingly, while the lines in such diagrams are labeled by the orbital indices, the diagram structure adequately reflects the irreducible representation of the group U(R). In this sense the paper presents a unitary group approach to the natural generalization of the usual many-body theory for the spin-adapted cases. A set of very simple rules is derived; their similarity with the corresponding rules in the ordinary many-body theory and practical utility are discussed in connection with (a) matrix elements over many-electron spin states and (b) closed- and open-shell many-body perturbation theory. A possibility of integral-driven many-body perturbation theory for open-shells is indicated. Connections of this formalism with others are also discussed.     

Application of the unitary group approach to evaluate spin density for configuration interaction calculations in a basis of S2 eigenfunctions

We present an implementation of the spin‐dependent unitary group approach to calculate spin densities for configuration interaction calculations in a basis of spin symmetry‐adapted functions. Using S² eigenfunctions helps to reduce the size of configuration space and is beneficial in studies of the systems where selection of states of specific spin symmetry is crucial. To achieve this, we combine the method to calculate U(n) generator matrix elements developed by Downward and Robb (Theor. Chim. Acta 1977, 46, 129) with the approach of Battle and Gould to calculate U(2n) generator matrix elements (Chem. Phys. Lett. 1993, 201, 284). We also compare and contrast the spin density formulated in terms of the spin‐independent unitary generators arising from the group theory formalism and equivalent formulation of the spin density representation in terms of the one‐ and two‐electron charge densities.     

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.     

Spin-free quantum theory. XXV. The unitary-group formulation of fermion many-body theory

The fermion unitary group formulation (UGF ) of many-body theory is based on the unitary group U(2n) where n is the number of freeon orbitals. This formulation, which conserves particle-number but not spin, is isomorphic to the particle-number-conserving, second-quantized formulation (SQF ). In UGF we derive the familiar diagrammatic algorithm for matrix elements, M(Y) = (?1)^H+L where H and L denote the numbers of hole lines and loops in the diagram D(Y) of M(Y). The unitary group derivation is considerably simpler than is the conventional, second-quantized derivation that employs time-dependence, Wick's theorem, normal-order, and contractions. In neither fermion UGF nor SQF is spin conserved. We carry out in UGF the spin-projection (symmetry adaptation to SU (2)) of the fermion vectors and obtain with a spin-free Hamiltonian the same matrix elements as with the freeon UGF (part 24 of this series). The fermion unitary group formulation for a spin-free Hamiltonian should be regarded as an alternate path to spin-free quantum chemistry.     

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.     

Spin-independent three-body effective valence-shell operators: Application to molecular oxygen

A mathematical construction is presented that uniquely defines a set of spin-independent effective valence-shell Hamiltonian (H^v) three-body matrix elements. These spin-independent H^v matrix elements separate direct and exchange portions of the three-body H^v matrix elements and therefore provide the most natural form for comparisons with parameterization schemes of semiempirical electronic structure methods in which the three-body matrix elements are incorporated into semiempirical one- and two-body Hamiltonian matrix elements in an averaged manner. Ab initio H^v three-body matrix elements of O₂ are computed through third order of quasidegenerate perturbation theory and are analyzed as a function of internuclear distance and atomic orbital overlap to aid in understanding how these three-body matrix elements may be averaged into semiempirical one- and two-body matrix elements. © 1992 John Wiley & Sons, Inc.     

The massey parameter expansion for the semiclassical scattering matrix

The Massey parameter expansion method for the semiclassical scattering matrix is proposed, using the Magnus formalism. The first Massey parameter, a unitary member of this expansion, is found for the interaction hamiltonian, depending on time, as exp(-τ) and τ^?n. Other possible applications of the proposed method are discussed.     

Method of recurrent construction of Löuwdin spin-adapted wave functions. II. Local representation of creation–annihilation operators

Some compositions of the addition and subtraction operators and recurrence relations for the Sanibel-type coefficients c_{u, v} (n, s, M) generated by these compositions are studied. A local representation of the fermion creation–annihilation operators via the addition and subtraction operators is obtained. Operators of single excitations, coupling, and decoupling operators, in terms of which the unitary group generators can be expressed are defined. The resulting representation of the nonelementary unitary group generators is much more simple than in the Gelfand–Tzetlin basis and in the most general case contains only six logically different terms, each of them possessing quite transparent physical significance.     

Configuration interaction matrix elements. II. Graphical approach to the relationship between unitary group generators and permutations

The explicit expressions for the matrix elements of unitary group generators between geminally antisymmetric spin-adapted N-electron configurations in terms of the orbital occupancies and spin factors, given as spin function matrix elements of appropriate orbital permutations, are derived using the many-body time-independent diagrammatic techniques. It is also shown how this approach can be conveniently combined with graphical methods of spin algebras to obtain explicit expressions for the spin factors, once a definite coupling scheme is chosen. This method yields explicit expressions for the orbital permutations defining the spin factors. However, if desired, the explicit determination of line-up permutations can be avoided in this approach, since they are implicitly contained in the orbital diagrams. It also clearly indicates why the geminally antisymmetric spin functions have to be used when a simple formalism is desired.     

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.     

Direct minimization of the energy functional in LCAO-MO density matrix formalism I. Closed and open shell systems

A direct method of minimization of the energy expression for closed and open shell systems in LCAO-MO density matrix formalism is presented. The method makes use of a unitary transformation acting directly on the density matrices. Expressions of the gradient and second energy derivatives are worked out. Some preliminary calculations to test the rate of minimization using a variable metric method have been made on H₂S and SO molecules and have given satisfactory results.[/p]     

量子化学酉群方法(UGA)的新算法——Wey1基图解法

本文是作者在近二年中创立的酉群计算新方法——weyl基图解法的一个系统介绍和总结。具体介绍了处理量子化学中多电子体系组态相互作用问题(包括与自旋无关和与自旋有关的哈密顿体系)的方法和公式。     

Calculating spin density in the quantum-chemical unitary formalism

Spin density derivation is considered within the framework of the unitary quantum-chemical formalism without spin. A relationship previously established between the spin and charge densities enables one to write the spin-density matrix Q explicitly in terms of the standard generators E of the unitary group U_N. This method of calculating the spin density includes only those operations that have already been devised in the unitary formalism of spin-free quantum chemistry.Translated from Teoreticheskaya i Éksperimental'naya Khimiya, No. 3, pp. 344–346, May–June, 1985.     

Treatment of hydrogen bonding within CNDO/2 and MINDO/3: CNDO/2H and MINDO/3H

A formalism has been developed to treat hydrogen-bonded A—H…?B systems within the CNDO /2 and the MINDO /3 methodologies. In this formalism the interactions are divided into three distinct classes; those between (a) two hydrogen-bonded atoms, (b) one hydrogen-bonded and non-hydrogen-bonded atom, and (c) two non-hydrogen-bonded atoms. The last class of interactions is treated solely by the existing CNDO /2 or MINDO /3 method. For A –H…?B systems, the core resonance integrals are individually parametrized depending upon the class of the interaction. Three types of A—H…?B systems have been thus far parametrized. Nine hydrogen-bonded dimers have been studied using the new formalism and the current CNDO /2 and the MINDO /3 methods. MINDO /3 predicts very large interatomic (A –B) distances for the equilibrium geometry, and relatively small stabilization values for the hydrogen-bond energies. CNDO/2 predicts the reverse. The new formalism for both CNDO /2 and MINDO /3 predicts accurate geometries as well as energies for all nine dimers. The new formalisms are called CNDO /2H and MINDO /3H. A general discussion of the nature of hydrogen bonding as exhibited by CNDO /2H and MINDO /3H is presented.     

Spin-free approach for evaluation of electronic matrix elements using character operators of ℒN

A spin-free method is presented for evaluating electronic matrix elements over a spin-independent many-electron Hamiltonian. The spin-adapted basis of configuration state functions is obtained using a nonorthogonal spin basis consisting of projected spin eigenfunctions. The general expressions for the matrix elements are given explicitly, and it is demonstrated how the matrix elements may be obtained simply from the knowledge of the irreducible characters of the permutation group ℒ_N. The presented formulas are very general and may be applied in connection with both spin-coupled valence bond studies and in conventional configuration interaction (CI) methods based on an orthonormal orbital basis. © 1996 John Wiley & Sons, Inc.     

Spin-free quantum chemistry. XXIII. The generator-state approach

In the unitary-group formulation of quantum chemistry, the spin-projected, configuration-state spaces of quantum chemistry are realized by the irreducible representation spaces (IRS ) of the freeon unitary group U_(n), where n is the number of freeon orbitals. The Pauli-allowed IRS are labeled by the partitions [λ] = [2^(N/2)?s, 1^2S], where N and S are the particle number and the spin, respectively. The generator-state approach (GSA ) to the unitary-group formulation consists of (1) the construction of the overcomplete, nonorthonormal generator basis for each IRS ; (2) the Lie-algebraic computation of matrix elements over generator states; (3) the Moshinsky–Nagel construction of the complete, orthonormal Gel'fand basis in terms of the generator basis; and (4) the computation of matrix elements over Gel'fand states in terms of matrix elements over generator states.     

Magnetosensitive reactions in ionic crystals during their plastic deformation

A new method for investigating spin-independent chemical reactions between radical centers at point defects and dislocations in solids is proposed. The method is based on studying the mobility of dislocations in a field of mechanical stresses. The application of a magnetic field with inductionB=1 T results in a change in the state of the defects of the crystal lattice in ionic crystals, which can be observed a long time (up to 10³ s) after switching off the field. Radical reactions were analyzed in which the kinetics of changes in the magnetic field can cause the effect of magnetic “memory” in distorted microregions of the lattice. Translated fromIzvestiya-Akademii Nauk Seriya Khimicheskaya, No. 4 pp. 739–744, April. 1997.     

Usage de l'Algèbre de Lie su (n) dans l'Etude des Systèmes Quantiques à n Etats. V. Application aux Systèmes composés

We adapt the use of the Lie algebra su (n) we proposed in former papers to a system made up of two subsystems (respectively, n_A and n_B states). We show how reduced density operators, correlation problems, and overall and reduced evolution equations, may be made precise with this formalism.