Harmonic oscillator tensors. I. The nondegenerate case

It is shown that the Heisenberg Lie algebra of the nondegenerate harmonic oscillator leads to a basis {J₊, J₀, J_?} of LASU (2). The Hamiltonian of the system is proportional to J₀, and the basis elements give rise to irreducible tensors in the associative enveloping algebra of the Heisenberg Lie algebra. The construction of these irreducible tensors is studied with special attention being paid to the case in which they act upon a single vector space spanned by the harmonic oscillator basis functions. A tensor coupling rule is developed, and useful application is made of it in the calculation of general expressions for vibrational operators and their matrix elements. Throughout, the value of the additional algebraic quantum numbers (l, m) is emphasized.     

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.     

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     

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.     

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.     

Analogy between real irreducible tensor operator and molecular two-particle operator

. Molecular matrix elements of a physical operator are expanded in terms of polycentric matrix elements in the atomic basis by multiplying each by a geometrical factor. The number of terms in the expansion can be minimized by using molecular symmetry. We have shown that irreducible tensor operators can be used to imitate the actual physical operators. The matrix elements of irreducible tensor operators are easily computed by choosing rational irreducible tensor operators and irreducible bases. A set of geometrical factors generated from the expansion of the matrix elements of irreducible tensor operator can be transferred to the expansion of the matrix elements of the physical operator to compute the molecular matrix elements of the physical operator. Two scalar product operators are employed to simulate molecular two-particle operators. Thus two equivalent approaches to generating the geometrical factors are provided, where real irreducible tensor sets with real bases are used. Received: 3 September 1996 / Accepted: 19 December 1996     

Icosahedral irreducible tensors and their applications

After a discussion of different concepts of anisotropy with reference to the icosahedron, the unusual problems surrounding the construction of icosahedral irreducible tensors are discussed. Tables of these tensors are then presented and applied to orbital problems, molecular polarization phenomena and to the construction of projection operators.     

Alternative equivalent icosahedral irreducible tensors

Some sets of icosahedral irreducible tensors can only be defined with respect to one of two seemingly equivalent choices of axes. The problem is explained and the cases when such an alternative choice may cause mixing of sets belonging to different irreducible representations are determined. It is found that the amount of mixing is a property of the spherical basis sets and is independent of the rank and of the permutation symmetry of the tensorial sets. It is further found that certain operators which change the handedness of the coordinate axes can similarly affect these sets, notwithstanding the centrosymmetry of the icosahedron.     

Irreducible representation and projection operator application to understanding nonlinear optical phenomena: hyper-Raman, sum frequency generation, and four-wave mixing spectroscopy

Symmetry plays an essential role in understanding optical activities of a molecule in infrared and Raman vibrational spectroscopy as well as in nonlinear optical vibrational spectroscopy. Each vibrational mode belongs to an irreducible representation of the underlying symmetry group. In this paper, using the alpha-helical polypeptide symmetry as an example, we calculate all the third rank nonzero hyper-Raman tensors as well as the infrared and Raman tensors by applying the projection operators to each irreducible species. We demonstrate that the projection operator method provides selection rules for the infrared, Raman, and hyper-Raman vibrational transitions and also other nonlinear optical spectroscopy such as sum frequency generation and the four-, five-, and six-wave mixing coherent vibrational transitions. Specific expressions for all nonzero elements of the corresponding nonlinear susceptibility tensors in a laboratory-fixed coordinate frame are also deduced.     

Isospin basis for atoms in relativistic approximation

The isospin basis is put into operation for investigation of atomic configurations, having two shells of equivalent electrons, characterized by the same orbital (LS coupling) or total (jj coupling) angular momenta of each electron. Tensorial properties of both the operators and the wave functions are studied in this basis. The two-particle operator is expressed in terms of the tensors irreducible in the isospin space. The problem of the additional classification of the levels is considered. The accuracy of the quantum numbers of the isospin basis in jj coupling scheme is discussed.     

Geometry of density matrices. VI. Superoperators and unitary invariance

We examine and compare ways of dividing into subspaces the space whose elements are density matrices or other operators for the class of model problems defined by a finite one-particle basis set. One method of decomposition makes the significance of the subspaces apparent. We show that this decomposition is also complete, in the group-theoretic sense, for the group of unitary transformations of the set of one-electron basis functions. The irreducible subspaces are labeled by particle number and by an additional integer we call the reduction index. For spaces of particle-number-conserving operators, all subspaces with the same reduction index are isomorphic, and an analogous isomorphism exists for non-particle-number-conserving cases. The general linear group also plays a key role, and we introduce the term “canonical superoperators” to characterize those superoperators which commute with this group. When an appropriate basis set is chosen for the matrix spaces, the supermatrices corresponding to these superoperators have a particularly simple form: a block structure with the only nonzero blocks being multiples of unit matrices. The superoperators of interest can be constructed in terms of two operators, , and these two have been expressed simply in terms of creation and annihilation operators. When only real orthogonal transformations of the basis are considered, a further decomposition is possible. We have introduced superoperators associated with this decomposition.     

Theory of two shells of atomic electrons using non-orthogonal radial orbitals

A theory for handling non-orthogonal radial orbitals of two shells of atomic electrons based on the mathematical apparatus of irreducible tensor operators is presented. The general expressions for one- and two-electron operator matrix elements are given.     

The isotropic invariants of fifth-rank cartesian tensors

An orthonormal set of irreducible fifth-rank tensors having the required permutation symmetry is constructed. Various problems not encountered in the analogous problem for tensors of ranks two, three, four and six are discussed.     

Analytic quantum mechanics of the morse oscillator

The approximate eigenfunctions of the Morse oscillator, expressed in terms of Laguerre polynomials, are shown to form an approximately orthogonal basis. Analytic expressions for the matrix elements of common operators are obtained within this representation. With such matrix elements in closed form, the Morse oscillator becomes, as the harmonic oscillator has been, a practical building block in molecular theory.     

Spin relaxation in cubic liquid crystals. The role of symmetry

Abstract

Cubic liquid-crystalline phases are usually regarded as isotropic systems. This view is justified for physical properties that transform as second rank tensors. However, the time correlation functions describing spin relaxation in cubic phases include components that transform as fourth rank tensors, which distinguish between cubic and spherical symmetry. In this work we explore the consequences of this fact for spin relaxation behaviour in cubic phases using group theoretical methods. We identify the two irreducible crystal frame time correlation functions of a cubic phase, derive the orientation dependence of the laboratory frame time correlation functions for single crystal samples, and discuss the relation of the cubic (fourth rank) order parameter to the microstructure of the phase. Finally, as an illustration of the general results, we derive the time correlation functions for a specific model of a micellar cubic phase.     

Anisotropic oscillator strength of 1-hydronaphthyl radical in naphthalene single crystal

The absolute value of the optical absorption strength of the 539 nm intra-radical transition of the 1-hydronaphthyl radical in a naphthalene crystal was measured with light polarized along the a, b and c′ axes. The long and short axial components of the oscillator strength were determined by using the local field tensors obtained with several methods. It was found that the local field tensors obtained recently by Munn, assuming each molecule to be two polarizable points, yield good agreement between the molecular axial components of the oscillator strengths obtained experimentally and theoretically. The local field tensors obtained previously with the single point-dipole approximation were found to result in inconsistency.     

Usage de l'Algèbre de Lie su (n) dans l'Etude des Systèmes Quantiques à n Etats. II. Transformation de l'Espace des Observables,Problèmes d'Evolution

In the previous paper we examined, for a quantum system, the relation between its n-dimensional state space and the su (n) Lie algebra. The present paper is devoted to relations between unitary transformations in the state space and orthogonal transformations in Lie's algebra. Two cases can happen. First, the transformations are independently chosen in the two spaces; this amounts to changing the former relation. On the other hand, the relation is maintained and the unitary operators are then related to some of the orthogonal operators. This second case is used to study the evolution operators.     

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.     

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.     

General harmonic oscillator integrals by the operator method

The operator technique with a minimum of commutator algebra is employed to calculate matrix elements of any number of operators between distorted, displaced harmonic oscillator wavefunctions. The results are valid for multidimensional integrals, and regardless of the extent of the Duschinsky effect. General recursion relations useful in machine calculations are given. The formalism is illustrated for the well-known one-dimensional Franck–Condon integrals.