Harmonic oscillator tensors. III: Efficient algorithms for evaluating matrix elements of vibrational operators

Succinct expressions for the matrix elements of various vibrational operators have been derived in the basis of the nondegenerate harmonic oscillator. Among these are the matrix elements of and , which are found to be dependent upon two quantities and their derivatives. Furthermore, the derivative property of the commutator is used to obtain an explicit expression for the derivatives of an operator in terms of its nested commutator with the conjugate momentum. It may be applied to any of the above cases to obtain the matrix representatives of expressions such as the mixed products , for example. In addition, a simple expression for 1/q is given and its derivatives may be evaluated by this commutator technique. Also the matrix elements of a Gaussian-type operator has been evaluated. This revised version was published online in July 2006 with corrections to the Cover Date.     

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. 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 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.     

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     

Convergence of vibrational angular momentum terms within the Watson Hamiltonian

Vibrational angular momentum terms within the Watson Hamiltonian are often considered negligible or are approximated by the zeroth order term of an expansion of the inverse of the effective moment of inertia tensor. A multimode expansion of this tensor up to second order has been used to study the impact of first and second order terms on the vibrational transitions of N(2)H(2) and HBeH(2)BeH. Comparison with experimental data is provided. The expansion of the tensor can be exploited to introduce efficient prescreening techniques.     

Generating functions for oscillator matrix elements

Two general harmonic oscillator elements \documentclass{article}\pagestyle{empty}\begin{document}$$ \left\langle m \right|\hat x^s \hat p\left| n \right\rangle $$\end{document} and \documentclass{article}\pagestyle{empty}\begin{document}$$ \left\langle m \right|\exp \left[ { - \alpha \left( {\hbar /m*\omega } \right)\hat x^2 } \right]\exp \left( {\beta \hat x} \right)\left| n \right\rangle $$\end{document} are derived by a generating function method using operator techniques which contain practically all one- and two-centre integrals with equal frequencies of chemical physics.     

Analytical evaluation of pseudopotential matrix elements with Gaussian‐type solid harmonics of arbitrary angular momentum

An efficient formalism for evaluating pseudopotential matrix elements with Gaussian‐type solid harmonics of arbitrary angular momentum is presented. It is based on the tensor coupling technique, which is especially well suited for treating Gaussian‐type solid harmonics of arbitrary angular momentum. Closed analytical expressions are derived for the matrix elements as well as for their nuclear displacement derivatives. The efficiency of the implementation into our new parallel density functional program PARA GAUSS and the quality of the pseudopotential approach is tested for a set of representative molecules and cluster models. To this end the results of pseudopotential calculations are compared to those of nonrelativistic and scalar‐relativistic all‐electron calculations. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 79: 209–221, 2000     

Harmonic analysis and discrete polynomials. From semiclassical angular momentum theory to the hyperquantization algorithm

Orthogonal polynomials of a discrete variable have been widely investigated as fundamental tools of numerical analysis. This work aims to propose the extension of their use to quantum mechanical problems. By exploiting both their connection with coupling and recoupling coefficients of angular momentum theory and their asymptotic relationships (semiclassical limit) with spherical and hyperspherical harmonics, a discretization procedure, the hyperquantization algorithm, has been developed and applied to the study of anisotropic interactions and of reactive scattering. One of the most appealing features of this method turns out to be a drastic reduction of memory requirements and computing time for extensive dynamical calculations. Examples of the application of this technique to stereodirected dynamics via an exact representation for the S matrix as well as to the characterization of molecular beam polarization are also illustrated. Received: 17 September 1999 / Accepted: 3 February 2000 / Published online: 5 June 2000     

Representing and selecting vibrational angular momentum states for quasiclassical trajectory chemical dynamics simulations

Linear molecules with degenerate bending modes have states, which may be represented by the quantum numbers N and L. The former gives the total energy for these modes and the latter identifies their vibrational angular momentum jz. In this work, the classical mechanical analog of the N,L-quantum states is reviewed, and an algorithm is presented for selecting initial conditions for these states in quasiclassical trajectory chemical dynamics simulations. The algorithm is illustrated by choosing initial conditions for the N = 3 and L = 3 and 1 states of CO2. Applications of this algorithm are considered for initial conditions without and with zero-point energy (zpe) included in the vibrational angular momentum states and the C-O stretching modes. The O-atom motions in the x,y-plane are determined for these states from classical trajectories in Cartesian coordinates and are compared with the motion predicted by the normal-mode model. They are only in agreement for the N = L = 3 state without vibrational angular momentum zpe. For the remaining states, the Cartesian O-atom motions are considerably different from the elliptical motion predicted by the normal-mode model. This arises from bend-stretch coupling, including centrifugal distortion, in the Cartesian trajectories, which results in tubular instead of elliptical motion. Including zpe in the C-O stretch modes introduces considerable complexity into the O-atom motions for the vibrational angular momentum states. The short-time O-atom motions for these trajectories are highly irregular and do not appear to have any identifiable characteristics. However, the O-atom motions for trajectories integrated for substantially longer period of times acquire unique properties. With C-O stretch zpe included, the long-time O-atom motion becomes tubular for trajectories integrated to approximately 14 ps for the L = 3 states and to approximately 44 ps for the L = 1 states.     

Improved treatment for matrix elements of spin-dependent operators

In this paper a general method for the evaluation of the matrix elements of spin-dependent operators is proposed to improve the treatment primitively suggesteed by Cooper and Musher. This approach is largely based on the recent results which the present authors have achieved in the representation theory for the inner- and outer-product reduction of the symmetric group. It is shown that the so-called outer-product coupling coefficients (OPCC ) can be used to generalize the method for constructing the irreducible tensor operators of group S_n. Together with the use of inner-product coupling coefficients (IPCC ), an expression for the matrix elements of spin-dependent operators is presented as the product of a Racah coefficient for S_n and a reduced matrix element which can be expressed in terms of IPCC, OPCC , and the related integrals. The treatment for one- and two-electron spin-dependent operators is discussed in detail.     

On the computation of the dipole change arising from vibrational angular momentum and some inter-relationships between the elements of the atomic polar tensor

A programme is developed for the computation of the contribution of the rotation of the dipole moment to vibrational transition moments. It is argued that this return to fundamentals has much in its favour compared with earlier methods based on pseudo isotope calculations. The programme is applied to H₂O. Some discrepancies are noted with the results of the matrix method originally proposed by Biarge, Herranz and Morcillo. The existence of internal relationships between the elements of the APT is demonstrated. Such relationships are of importance in the refinement of the APT to fit experimental transition moments.     

Uniform approximate Franck-Condon matrix elements for bound-continuum vibrational transitions

Franck-Condon matrix elements are calculated approximately for vibrational transitions of a diatomic molecule from a bound electronic potential curve to a purely repulsive curve. The bound states are approached by exactly normalized Miller-Good wavefunctions uniform in both turning points. For the continuum wavefunction a single turning point uniform Airy approximation is taken. The resulting Franck-Condon matrix element is approximately done in closed form with the help of a new canonical integral for a product of harmonic oscillator wavefunctions and Airy functions. The degree of agreement with a closed form exact result is qualitatively discussed for transitions from the ground state of a Morse curve to the continuum of a particular repulse exponential curve.Dedicated to Professor Hermann Hartmann on the occasion of his 65th birthday.     

Addition theorem and matrix elements of radiative multipole operators

The addition theorem for radiative multipole operators, i.e., electric-dipole, electric-quadropole, or magnetic-dipole, etc., is derived through a translational transformation. The addition theorem of μth component of the angular momentum operator, L _μ (r), is also derived as a simple expression that represents a general translation of the angular momentum operator along an arbitrary orientation of a displacement vector and when this displacement is along the Z-axis. The addition theorem of the multipole operators is then used to analytically evaluate the matrix elements of the electric and magnetic multipole operators over the basis functions, the spherical Laguerre Gaussian-type function (LGTF), . The explicit and simple formulas obtained for the matrix elements of these operators are in terms of vector-coupling coefficients and LGTFs of the internuclear coordinates. The matrix element of the magnetic multipole operator is shown to be a linear combination of the matrix element of the electric multipole operator     

Generating functions and recurrence relations for harmonic oscillator matrix elements

A simple and straightforward way of obtaining generating functions and recurrence relations for harmonic oscillator integrals is presented. The method, which is based on the properties of unnormalized coherent states and canonical transformations, allows a unified treatment of several different physical problems. Matrix elements of Gaussian and exponential functions, Franck-Condon overlaps, and transition probabilities for time-dependent quadratic Hamiltonians are discussed as illustrative examples.     

The relationship between matrix elements of position and momentum in vibronic coupling

A general expression is derived that relates the matrix elements of position and momentum allowing for the possibility of a mixed basis. It is shown that when Born-Oppenheimer wavefunctions for two different electronic states are used as basis functions, the use of the usual expression relating matrix elements of position and momentum can lead to results that may not be of the correct order of magnitude.     

Calculation of j and jm symbols for arbitrary compact groups. II. An alternate procedure for angular momentum

The various orthogonality and sum rules which the 6j and 3jm symbols satisfy are sufficient to obtain the algebraic formulas for these symbols for SO₃ ? SO₂. Character theory enters in that the j's and m's occurring in the various sums are given by the triangle rule together with information on the symmetrized product and the branching, SO₃ ? SO₂, The resulting calculation is somewhat simpler, algebraically speaking, than previous calculations and has the pedagogical advantage that only the concept of an irreducible representation of a group is required, instead of the more elaborate concept of ladder operators.     

Relation between momentum and angular momentum correlation times. analysis of the uncorrelated successive binary-collision approximation

A relation between the correlation times τ_P and τ_J for a system of hard spheres of arbitrary roughness has been found in the uncorr