Harmonic oscillator tensors. V. The doubly degenerate harmonic oscillator

The step operators of the two-dimensional isotropic harmonic oscillator are shown to be separable into the basis elements of two disjoint Heisenberg Lie algebras. This separability leads to two sets of irreducible tensors, each of which is based upon its associated underlying Heisenberg Lie algebra. The matrix elements of these tensors are evaluated, along with those of some vibrational operators of physical interest. The possibility of other irreducible tensors are discussed and their usefulness is compared with that of those found here. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 67: 343–357, 1998     

Harmonic oscillator tensors. II. angular momentum expressions of matrix elements of vibrational operators

The spatial symmetries of the harmonic oscillator and the recently found irreducible tensors constructed from the associated annihilation and creation operators are exploited to obtain new expressions for the elements of the matrix representatives of several examples of vibrational operators. Since all vibrational operators are expressible in terms of the irreducible tensors, their matrix elements reflect the angular momentum symmetry inherent in them, for the results derived here are in terms of the Clebsch–Gordan coefficients and the isoscalar factors that arise from the couplinig rule of the irreducible tensors. Familiarity with the mathematical properties of these quantities derived from the elementary theory of angular momentum facilitates the evaluation of many vibrational operators that may be of importance in the study of potentials in this basis. In particular, it is shown that the nonvanishing of matrix elements is governed by a law of conservation of angular momentum along the axis of quantization of the nondegenerate harmonic oscillator. © 1993 John Wiley & Sons, Inc.     

A general expression for matrix elements of reciprocal powers of the displacement coordinateq in the harmonic oscillator basis

A formula is derived that allows one to determine the matrix elements of an arbitrary integral reciprocal power of the dimensionless displacement coordinate q of the harmonic oscillator from those ofq ^–1 in an exact manner. This relation is obtained from the use of the chain rule and irreducible tensors expressed in terms of the creation and annihilation operators of the harmonic oscillator.     

Approach of the associated Laguerre functions to the su(1,1) coherent states for some quantum solvable models

Using second‐order differential operators as a realization of the su(1,1) Lie algebra by the associated Laguerre functions, it is shown that the quantum states of the Calogero‐Sutherland, half‐oscillator and radial part of a 3D harmonic oscillator constitute the unitary representations for the same algebra. This su(1,1) Lie algebra symmetry leads to derivation of the Barut‐Girardello and Klauder‐Perelomov coherent states for those models. The explicit compact forms of these coherent states are calculated. Also, to realize the resolution of the identity, their corresponding positive definite measures on the complex plane are obtained in terms of the known functions. © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2009     

Harmonic oscillator tensors in contact transformation theory

The application of contact transformation theory to the perturbed harmonic oscillator is reexamined in the light of the harmonic oscillator tensors previously presented. It is found that the recasting of the formalism of this problem in terms of harmonic oscillator tensors results in great simplifications, most of which stem from the introduction of the additional algebraic quantum numbers (l, m). The order of magnitude of each fragment of the Hamiltonian is easily recognizable, and the diagonal and nondiagonal parts contained therein are readily identifiable. The determination of the contact transformation operator is reduced to a simple formula. First, an analysis is made for a single mode of vibration, and it is subsequently extended to a multimode case. The perturbed diatomic vibrator is presented as an example.     

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.     

Sequence-adapted molecular tensors: Algebraic methods and application to crystal field theory

Tensorial sets adapted to sequences of finite subgroups are applied to the crystal field problem, and a general method for generating sequence-adapted molecular tensors using finite group algebra is formulated. All subgroup sequences of the abstract finite group G(24), isomorphic to the octahedral, O, tetrahedral, T_d, and symmetric, S(4), groups are tabulated with explicit isomorphisms provided. The sequences fall into eight equivalence classes. A catalog of irreducible representations of G(24) adapted to a member of each of the eight sequence classes is given together with the transformations which generate representations adapted to all other sequences. With this data it is possible to systematically generate tensorial sets adapted to any sequence of a realization of G(24). Unitary transformations which adapt conventional forms of first- and second-rank irreducible tensorial sets of the rotation group to the eight sequences of the octahedral group are provided. Forms suitable for use with magnetic fields are included. The problem of a d¹ ion in a trigonal crystal field is treated with sequence-adapted molecular tensors, and the utility of different sequences for descent in symmetry is discussed.     

Local density functional theory applied to spin coupling in Fe6S supercluster

Using density functional theory (DFT), the multiplicity of the ground state was determined for Fe₆S₆Cl ions as well as the order of the excited spin states. A method to determine the exchange integrals J of the Fe₆S cluster is presented based on these results and a spin‐coupling algebra. The following order of the spin states was established with respect to the total spin 5/2, 1/2, 7/2, 3/2, 9/2, 11/2, 15/2, 13/2, and 17/2. We also calculated the Heisenberg coupling parameters J₁, J₂, J₃, and J₄ as 22, 146, −130, and 81 cm⁻¹, respectively. The possibility of the ground state of high multiplicity as well as the necessary conditions for such state are discussed. ©1999 John Wiley & Sons, Inc. Int J Quant Chem 72: 39–51, 1999     

Harmonic oscillator tensors. IV. A tensorial approach to internal rotations in molecules

A mapping of 2×2 matrices into the space of single boson operators is shown to lead to the angular momentum operators that give rise to irreducible tensors for the harmonic oscillator. The mapping may also be used to define an axis of quantization. A rotation about this axis induces a wave function and Hamiltonian that may be applied to the study of internal rotations in molecules. The example of a molecule containing two coaxial symmetric tops is presented as a case in point. The case of a potential with a high barrier leads to the approximation of an internal rotation as a torsional oscillator and, consequently, to torsional oscillator tensors whose properties are the same as those of the harmonic oscillator. The possibility of studying more complex potentials is discussed. © 1997 John Wiley & Sons, Inc. Int J Quant Chem 65 : 305–315, 1997     

On the time‐dependent solutions of the Schrödinger's equation. II. The one‐mode field perturbed harmonic oscillator

The solution of the time‐dependent Schrödinger's equation for a perturbed harmonic oscillator is obtained using a solvable Lie algebra. We choose a harmonic oscillator interacting with a one‐mode field, where the perturbation happens to be periodic in time. This leads to one of the simplest Floquet problems. Using the Wei–Norman theorem, the Floquet wave function is obtained as well as the semiclassical Floquet shift in the energy. © 2011 Wiley Periodicals, Inc. Int J Quantum Chem, 2011     

The heisenberg exchange integral in a non-orthogonal basis for antiferromagnetic and ferromagnetic systems

The Dirac-Van Vleck-Serber permutation degeneracy method is used to demonstrate that the Heisenberg spin exchange Hamiltonian, –2J ₁₂s₁·s₂, is a good approximate Hamiltonian for the theoretical interpretation of antiferromagnetic and ferromagnetic systems. The approach does not neglect double or higher-order permutations and covers the general case of a singleN-electron configuration as well as that of configuration interaction. An analogy between antiferromagnetic and hydrogen-molecule-like systems is established, and a formula for the estimation of the Heisenberg exchange integral is derived.     

Intramolecular Antiferromagnetism in μ-Oxo-bis[(5,15-dimethyl-2,3,7,8,12,13,17,18-octaethylporphyrinato)iron(III)]

The magnetism of μ-oxo-bis[(5,15-dimethyl-2,3,7,8,12,13,17,18-octaethylporphyrinato)iron(III)] with bridge geometry d(Fe? O) = 1.752 Å and ?(Fe? O? Fe) = 178.6° can be explained in terms of antiferromagnetically exchange coupled iron(III)-3d⁵ pairs. The magnetochemical analysis in the temperature range 6K–295K on the basis of the isotropic Heisenberg model (spin Hamiltonian: ? = ?2J?₁ · ?₂ S₁ = S₂ = 5/2) leads to the exchange parameter J = ?125 cm^?1. With regard to the Fe? O bond length the J value corresponds to the series of data observed for other μ-oxodiiron-porphyrins and -porphycenes. Compared to the spin-spin coupling in [Fe₂Cl₆O]^2?, |J| is enhanced by ≈ 10%.     

A method for the construction of orthogonal spin eigenfunctions

A method for the construction of the essentially idempotent and Hermitian diagonal elements of the matric algebra of the permutation group S_n is proposed. For the irreducible representation [λ] = [λ₁, λ₂] characterising a spin state S of an n-electron system, it is found that this method generates the complete set of spin projections from the appropriate primitive spin functions. The method is applied to a 7-electron system in the spin state S = M_S = 1/2 and the results are listed in the Appendix.     

Effects of deuteration on the structure and magnetic properties of bis-quinolinium tetrabromidocuprate(II) dihydrate

Abstract

The partially [d₆, 2] and fully [d₂₀, 3] deuterated analogues of (QuinH)₂CuBr₄·2H₂O (d₀, 1) were prepared and their crystal structures were determined [Quin = quinoline]. In both compounds, there is a clear disorder in the positions of the bromide ions which was resolved. This led to a reexamination of the structure of the parent, fully protonated compound (1) where a small percentage of previously unrecognized disorder was also observed and the structure rerefined. Variable temperature magnetization measurements over the range 1.8–310?K indicate that all three materials behave as magnetically well-isolated layers that can be evaluated using the 2D-quantum Heisenberg antiferromagnetic model. Final fitting results for the partially (J = ?5.96(5) K) and fully (J = ?5.77(2) K) deuterated compounds indicate slightly weaker exchange compared to the protonated compound (J = ?6.17(3) K), likely as a result of the increased disorder in the deuterated phases.     

Permutation representations in molecular symmetry

The reducible representations of the point groups are generally studied because of their relevance to molecular orbital and vibration theory. Triple correlations within the polyhedra are described by group-theoretical invariants that are related to the permutation representations and termed polyhedral isoscalar factors. These invariants are applied in theorems on matrix elements referring to the symmetry-adapted bases at different centres. Further invariants or geometrical weight factors inter-relate different types of reduced matrix elements of irreducible tensors (generalization of the Wigner-Eckart theorem to the polycentric case). As a demonstration a complete tabulation is given for the point group C ₄.     

Two approaches to fractional supersymmetrical quantum mechanics

Two complementary approaches of ?? = 2 fractional supersymmetrical quantum mechanics of order k are studied in this article. The first, based on a generalized Weyl–Heisenberg algebra W_k (which comprises the affine quantum algebra U_q(sl₂) with q^k = 1 as a special case), apparently contains solely one bosonic degree of freedom. The second uses generalized bosonic and k‐fermionic degrees of freedom. As an illustration, a particular emphasis is put on the fractional supersymmetrical oscillator of order k. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem, 2003     

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.     

On quantum groups and their potential use in mathematical chemistry

The quantum algebrasu _q (2) is introduced as a deformation of the ordinary Lie algebrasu(2). This is achieved in a simple way by making use ofq-bosons. In connection with the quantum algebrasu _q (2) we discuss theq-analogues of the harmonic oscillator and the angular momentum. We also introduceq-analogues of the hydrogen atom by means of aq-deformation of the Pauli equations and of the so-called Kustaanheimo Stiefel transformation.     

Transferable integrals in a deformation density approach to crystal calculations. I. Crystal harmonics and their properties

The term “crystal harmonic” is introduced to denote a symmetrized plane wave in the special case where the wave vector is a reciprocal lattice vector. Crystal harmonics, thus defined, have the translational symmetry of the lattice, and they also have the transformation properties of the irreducible representations of the crystal's point group. An expansion is derived expressing crystal harmonics in terms of spherical Bessel functions and in terms of the functions ??_??,ξ (eigenfunctions of L² which are also basis functions for IRS of the crystal's point group). A sum rule for the functions ??_??,ξ is derived. Methods are given for expanding periodic functions of special symmetry in terms of crystal harmonics. Methods are also presented for calculating matrix elements of the potential in a crystal using crystal harmonics as a basis and for transforming to a STO basis. It is shown that the invariant component of the product of two crystal harmonics can be expressed as a sum of a few invariant crystal harmonics, and expressions for the coefficients in the sum are derived. Orthogonality with respect to summation over networks of points and normalization are also discussed. The properties mentioned above are illustrated in detail in the case of cubic crystals with point group O_h.