Algebraic approach to closed formulation of Kratzer potential integrals

In this work, closed form expressions for the calculation of the Kratzer potential integrals are obtained by means of a procedure based on the algebraic representation of the Kratzer eigenfunctions along with the usual ladder properties and commutation relations. For that, the exact formulae for matrix elements are achieved with the aid of the raising operator applied repeatedly over the ket and with the lowering operator acting reiteratively over the bra for the symmetric closed form expression. Comparatively, the formulae algebraically obtained in this work are quite similar to the ones derived from usual methods involving the evaluation of integrals. Besides, when considering some particular cases the results show that the closed formulae that comes from the algebraic procedure are an improvement to the closed form expressions already published. This revised version was published online in July 2006 with corrections to the Cover Date.     

The algebraic approach for the derivation of ladder operators and coherent states for the Goldman and Krivchenkov oscillator by the use of supersymmetric quantum mechanics

The ladder operators for the Goldman and Krivchenkov anharmonic potential have been derived within the algebraic approach. The method is extended to include the rotating oscillator. The coherent states for the Goldman and Krivchenkov oscillator, which are the eigenstates of the annihilation operator and minimize the generalized position-momentum uncertainty relation, are constructed within the framework of supersymmetric quantum mechanics. The constructed ladder operators can be a useful tool in quantum chemistry computations of non-trivial matrix elements. In particular, they can be employed in molecular vibrational–rotational spectroscopy of diatomic molecules to compute transition energies and dipole matrix elements.     

Improvement to the hypervirial theorem and Kramer's rule for the calculation of central potential matrix elements

Ameliorated recurrence relations for the calculation of matrix elements of central potential wavefunctions are presented. These were obtained by using the hypervirial theorem with V(r) three-dimensional potentials and f (r) arbitrary functions. This procedure leads to a generalization of the usual l = l′ diagonal three-dimensional and l == 0 one-dimensional hypervirial relations of first and second class. The use of this kind of generalization to the calculation of r^k integrals, allows one to obtain all off-diagonal recursion formulas for any V(r). Besides, for hydrogenic wavefunctions one gets to equations that reduce to the usual Kramer's rule as a particular diagonal case. The proposed approach can be straightforwardly extended to obtain recurrence relations for the calculation of two center integrals.     

Oscillators in one and two dimensions and ladder operators for the Morse and Coulomb problems

We exhibit that the radial eigenfunctions of a 2D-harmonic oscillator (2DHO) may be regarded as 1D-harmonic oscillator (1DHO) matrix elements. From this simple fact and using as a starting point the ladder operators â^± for 1DHO, we obtain ladder operators for 2DHO. Furthermore, by using the relationship between the Coulomb and Morse problems with a 2DHO, we are able to obtain the ladder operators for the former problems without explicitly recurring to the factorization method. Some uses of the technique presented are suggested. © 1997 John Wiley & Sons, Inc.     

Ladder operators for the Morse potential

A realization of the raising and lowering operators for the Morse potential is presented. It is shown that these operators satisfy the commutation relations for the SU(2) group. Closed analytical expressions are obtained for the matrix elements of different operators such as 1/y and d/dy. The harmonic limit of the SU(2) operators is also studied and an approach previously proposed to calculate the Franck–Condon factors is discussed. © 2001 John Wiley & Sons, Inc. Int J Quantum Chem, 2001     

Method of computer algebraic calculation of the matrix elements in the second quantization language

An automated method by the algebraic programming language REDUCE3 for specifying the matrix elements expressed in second quantization language is presented and then applied to the case of the matrix elements in the TDHF theory. This program works in a very straightforward way by commuting the electron creation and annihilation operators (a^? and a) until these operators have completely vanished from the expression of the matrix element under the appropriate elimination conditions. An improved method using singlet generators of unitary transformations in the place of the electron creation and annihilation operators is also presented. This improvement reduces the time and memory required for the calculation. These methods will make programming in the field of quantum chemistry much easier. © 1995 John Wiley & Sons, Inc.     

Matrix elements of the linear Boltzmann collision operator for systems of two components at different temperatures

Matrix elements of the linearized collision operators that arise in the linearization of the Boltzmann equations for a binary gas system are calculated. The collision operators employed here differ from those usually considered in that the Maxwell—Boltzmann distribution functions which appear are parametrized by two different temperatures, one for each component. The matrix representations of the isotropic portion of the collision operators are calculated with the Sonine polynomials as basis functions, and for the hard sphere cross section, recursion relations for the matrix elements are derived which permit their efficient numerical calculation. The dependene of a few matrix elements on the mass and temperature ratios of the two components is considered. In particular, the disparate mass limit is investigated and the range of validity of the Fokker—Planck operator as an approximation to the collision operator in this limit is briefly discussed.     

Correlated wave-function theory for many-center many-electron problems

The correlated electronic wave-function theory developed by S. Obara and K. Hirao [Bull. Chem. Soc. Jpn. 66 , 3300 (1993)], as applied to two-electron molecular systems, is generalized to many-center many-electron problems. The exact formulas for effective Hamiltonian operators are given. The rules for the calculation of matrix elements with three-electron operators over Slater determinants are formulated. From the energy-minimum principle, the system of master equations is derived for variational coefficients of a trial wave function for the molecules with closed electronic shells. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 69: 639–648, 1998     

The construction of the Gilmore–Perelomov coherent states for the Kratzer–Fues anharmonic oscillator with the use of the algebraic approach

By applying the algebraic approach and the displacement operator to the ground state, the unknown Gilmore–Perelomov coherent states for the rotating anharmonic Kratzer–Fues oscillator are constructed. In order to obtain the displacement operator the ladder operators have been applied. The deduced SU(1, 1) dynamical symmetry group associated with these operators enables us to construct this important class of the coherent states. Several important properties of these states are discussed. It is shown that the coherent states introduced are not orthogonal and form complete basis set in the Hilbert space. We have found that any vector of Hilbert space of the oscillator studied can be expressed in the coherent states basis set. It has been established that the coherent states satisfy the completeness relation. Also, we have proved that these coherent states do not possess temporal stability. The approach presented can be used to construct the coherent states for other anharmonic oscillators. The coherent states proposed can find applications in laser-matter interactions, in particular with regards to laser chemical processing, laser techniques, in micro-machinning and the patterning, coating and modification of chemical material surfaces.     

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 the 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.     

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     

A general diagrammatic algorithm for contraction and subsequent simplification of second-quantized expressions

We present a general computer algorithm to contract an arbitrary number of second-quantized expressions and simplify the obtained analytical result. The functions that perform these operations are a part of the program Nostromo which facilitates the handling and analysis of the complicated mathematical formulas which are often encountered in modern quantum-chemical models. In contrast to existing codes of this kind, Nostromo is based solely on the Goldstone-diagrammatic representation of algebraic expressions in Fock space and has capabilities to work with operators as well as scalars. Each Goldstone diagram is internally represented by a line of text which is easy to interpret and transform. The calculation of matrix elements does not exploit Wick's theorem in a direct way, but uses diagrammatic techniques to produce only nonzero terms. The identification of equivalent expressions and their subsequent factorization in the final result is performed easily by analyzing the topological structure of the diagrammatic expressions.     

Matrix elements in configuration interaction calculations

A method is proposed for the calculation of matrix elements among various states of atoms. A set of tensor operators is the only entity in the formalism, and all formulas involve merely the vacuum expectation values of these tensor operators and the recoupling transformation coefficients. Some numerical examples are given for the Coulomb interaction matrix elements.     

Generalized spherical functions in the theory of many-electron atoms

A new method for finding non-relativistic and relativistic wave-functions of an electron moving in the field of a nuclear charge in the jj coupling scheme is proposed. It is based on the usage of generalized spherical functions. The mathematical apparatus necessary to find the expressions for matrix elements of the non-relativistic and relativistic energy or electron transition operators is developed. The formulas obtained for these matrix elements are more convenient than those usually used in jj coupling scheme; only their radial integrals and some phase multipliers depend on orbital quantum numbers.     

Integrals and derivatives for correlated Gaussian functions using matrix differential calculus

The matrix differential calculus is applied for the first time to a quantum chemical problem via new matrix derivations of integral formulas and gradients for Hamiltonian matrix elements in a basis of correlated Gaussian functions. Requisite mathematical background material on Kronecker products, Hadamard products, the vec and vech operators, linear structures, and matrix differential calculus is presented. New matrix forms for the kinetic and potential energy operators are presented. Integrals for overlap, kinetic energy, and potential energy matrix elements are derived in matrix form using matrix calculus. The gradient of the energy functional with respect to the correlated Gaussian exponent matrices is derived. Burdensome summation notation is entirely replaced with a compact matrix notation that is both theoretically and computationally insightful. © 1996 John Wiley & Sons, Inc.     

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.     

Recoupling coefficients of group SO(4) and their application to linear tetratomic molecules

The general expressions for the recoupling coefficients of group SO(4) are obtained by employing the subgroup chains (2) and (11). When considering a special case, we give results that can be used for tetratomic molecules. Using these recoupling coefficients, the matrix elements of C and M operators that are not diagonal can be given. Within the dynamic algebra framework, an effective algebraic Hamiltonian of linear tetratomic molecules is obtained. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem 94: 293–301, 2003     

Discrete and continuum quantum states for the Kratzer oscillator

Kratzer oscillator is a realistic zero‐order model for describing the anharmonic ro‐vibrational motion in diatomic molecules. Kratzer oscillator has an energy spectrum containing both discrete and continuum parts. Wavefunctions belonging to the continuum would be useful in the study of transitions to the continuum in molecular dissociation processes. In this article, bound and scattering wavefunctions of the Kratzer oscillator are reviewed and the bound–bound and the bound–free matrix elements are obtained. © 2002 Wiley Periodicals, Inc. Int J Quantum Chem, 2002