Kinetic Energy Matrix Elements Research Articles

Variational rovibrational calculations for tetra atomic linear molecules using Watson’s isomorphic Hamiltonian, I: The C8v4 approach

The implementation of an algorithm for the variational determination of rovibrational state energies and wave functions for linear tetra atomic molecules is presented. The approach, termed C8v4, is based on Watson’s isomorphic Hamiltonian formulated in normal coordinates. Products of harmonic oscillator and rigid-rotor functions are used to expand the rovibrational wave function. Intricacies related to the coupling of the bending–rotation basis and the vibrational angular momentum in linear molecules require a careful study of symmetry properties to set up a symmetry adapted basis set. Kinetic energy matrix elements can be evaluated in a fast mixed numerical/analytical fashion. The main computational bottlenecks are the integration of the potential energy matrix and the diagonalization of the Hamiltonian. These can be overcome by exploiting the block structure of the Hamiltonian which is discussed in some detail. Test calculations on acetylene (HCCH) and boranimine (HBNH) are presented and the C8v4 results reproduce previous variational calculations based on a Hamiltonian formulated in internal coordinates.

General formulation of rovibrational kinetic energy operators and matrix elements in internal bond-angle coordinates using factorized Jacobians.

We show how inverse metric tensors and rovibrational kinetic energy operators in terms of internal bond-angle coordinates can be obtained analytically following a factorization of the Jacobian worked out by Frederick and Woywod. The structure of these Jacobians is exploited in two ways: On one hand, the elements of the metric tensor as well as its determinant all have the form ∑rmsin(αn)cos(βo). This form can be preserved by working with the adjugate metric tensor that can be obtained without divisions. On the other hand, the adjugate can be obtained with less effort by exploiting the lower triangular structure of the Jacobians. Together with a suitable choice of the wavefunction, we avoid singularities and show how to obtain analytical expressions for the rovibrational kinetic energy matrix elements.

Perspective on the Lagrange–Jacobi mesh

This paper presents a unified treatment of the kinetic energy matrix elements related to a number of Lagrange functions associated with the Lagrange–Jacobi mesh. The matrix elements can be readily modified for application to problems requiring eigenfunction expansion with Lagrange–Legendre, Lagrange–Chebyshev, Lagrange–Gegenbauer, as well as the Lagrange–Jacobi functions. The applicability of and the accuracy attainable with the matrix elements is demonstrated with the solution to the Schrödinger equation for confining trigonometric Pöschl–Teller potentials. The results obtained are within machine accuracy when appropriate choices of the basis functions are used.

Truncated spherical-wave basis set for first-principles pseudopotential calculations

Analytic results for two- and three-centre integrals are derived for the truncated spherical-wave basis set designed for first-principles pseudopotential calculations within density-functional theory. These allow the overlap, kinetic energy and non-local pseudopotential matrix elements to be calculated efficiently and accurately. In particular, the scaling of the computational effort with maximum angular momentum component is dramatically improved and the projection method takes full account of the discontinuities in the basis functions arising from their localization within spherical regions.

A fast method for solving both the time-dependent Schrödinger equation in angular coordinates and its associated “m-mixing” problem

An efficient split-operator technique for solving the time-dependent Schrödinger equation in an angular coordinate system is presented, where a fast spherical harmonics transform accelerates the conversions between angle and angular momentum representations. Unlike previous techniques, this method features facile inclusion of azimuthal asymmetries (solving the "m-mixing" problem), adaptive time stepping, and favorable scaling, while simultaneously avoiding the need for both kinetic and potential energy matrix elements. Several examples are presented.

Treating thebquark distribution function with reliable uncertainties

The parton distribution function for a $b$ quark in the $B$ meson (called the shape function) plays an important role in the analysis of the $B\ensuremath{\rightarrow}{X}_{s}\ensuremath{\gamma}$ and $B\ensuremath{\rightarrow}{X}_{u}\ensuremath{\ell}\overline{\ensuremath{\nu}}$ data, and gives one of the dominant uncertainties in the determination of $|{V}_{ub}|$. We introduce a new framework to treat the shape function, which consistently incorporates its renormalization group evolution and all constraints on its shape and moments in any short distance mass scheme. At the same time it allows a reliable treatment of the uncertainties. We develop an expansion in a suitable complete set of orthonormal basis functions, which provides a procedure for systematically controlling the uncertainties due to the unknown functional form of the shape function. This is a significant improvement over fits to model functions. Given any model for the shape function, our construction gives an orthonormal basis in which the model occurs as the first term, and corrections to it can be studied. We introduce a new short distance scheme, the ``invisible scheme,'' for the kinetic energy matrix element ${\ensuremath{\lambda}}_{1}$. We obtain closed form results for the differential rates that incorporate perturbative corrections and a summation of logarithms at any order in perturbation theory, and present results using known next-to-next-to-leading order expressions. The experimental implementation of our framework is straightforward.

Energy and energy gradient matrix elements with N-particle explicitly correlated complex Gaussian basis functions with L=1

In this work we consider explicitly correlated complex Gaussian basis functions for expanding the wave function of an N-particle system with the L=1 total orbital angular momentum. We derive analytical expressions for various matrix elements with these basis functions including the overlap, kinetic energy, and potential energy (Coulomb interaction) matrix elements, as well as matrix elements of other quantities. The derivatives of the overlap, kinetic, and potential energy integrals with respect to the Gaussian exponential parameters are also derived and used to calculate the energy gradient. All the derivations are performed using the formalism of the matrix differential calculus that facilitates a way of expressing the integrals in an elegant matrix form, which is convenient for the theoretical analysis and the computer implementation. The new method is tested in calculations of two systems: the lowest P state of the beryllium atom and the bound P state of the positronium molecule (with the negative parity). Both calculations yielded new, lowest-to-date, variational upper bounds, while the number of basis functions used was significantly smaller than in previous studies. It was possible to accomplish this due to the use of the analytic energy gradient in the minimization of the variational energy.

Self-consistent-field – Hartree–Fock method with finite nuclear mass corrections

We have upgraded a Self-consistent-field – Hartree–Fock routine to include a finite nuclear mass correction for molecules developed in our laboratory. The new routine can handle isotopomers without calculating any nuclear kinetic energy matrix element. Tests on H2, LiH, HF, F2, and H2O isotopomers indicate the equivalence of our correction to the standard diagonal adiabatic correction. A further original application to C2H6 illustrates the usefulness of the method for polyatomic molecules. The resulting molecular orbitals carry the nuclear mass signature, exemplified with Koopmans’ ionization potentials.

Flexible Transition State Theory for a Variable Reaction Coordinate: Analytical Expressions and an Application

Completely general canonical and microcanonical (energy-resolved) flexible transition state theory (FTST) rate constant expressions for an arbitrary choice of reaction coordinate have recently been derived [Robertson et al. J. Chem. Phys. 2000, 113, 2648.] by the present authors. The rate expressions apply to any definition of the separation distance between fragments in a barrierless recombination (or dissociation) that is held fixed during hindered rotations at the transition state, and to any combination of fragment structure (atom, linear top, nonlinear top). The minimization of the rate constant with respect to this definition can be regarded as optimizing the reaction coordinate within a canonical or microcanonical framework. The expressions are analytic, with the exception of a configuration integral whose evaluation generally requires numerical integration over an integrand which depends on internal angles (from one to five depending on the fragment structures). The primary component of the integrand is the determinant of the inverse G-matrix associated with the external rotations and the relative internal rotation of the fragments. In this paper, we derive closed-form, analytic expressions for the inverse G-matrix determinant for all combinations of fragment top types for an arbitrary reaction coordinate definition entirely in terms of kinetic energy matrix elements for a centers-of-mass reaction coordinate. For a model potential for CFH2 + H, the effect of optimizing the reaction coordinate definition is displayed, and the optimized coordinate is compared to the traditional center-of-mass definition at the canonical level. The associated rate constant is about a factor of 20% to 45% lower than that obtained using a centers-of-mass reaction coordinate.

Quasirandom distributed Gaussian bases for bound problems

We introduce quasirandom distributed Gaussian bases (QDGB) that are well suited for bound problems. The positions of the basis functions are chosen quasirandomly while their widths and density are functions of the potential. The basis function overlap and kinetic energy matrix elements are analytical. The potential energy matrix elements are accurately evaluated using few-point quadratures, since the Gaussian basis functions are localized. The resulting QDGB can be easily constructed and is shown to be accurate and efficient for eigenvalue calculation for several multidimensional model vibrational problems. As more demanding examples, we used a 2D QDGB-DVR basis to calculate the lowest 400 or so energy levels of the water molecule for zero total angular momentum to sub-wave-number precision. Finally, the lower levels of Ar3 and Ne3 were calculated using a symmetrized QDGB. The QDGB was shown to be accurate with a small basis.

Rovibrational Hamiltonians for general polyatomic molecules in spherical polar parametrization. III. Global vs local axis system and angular coordinates

We study different parametrizations of the angular space of polyatomic molecules for an orthogonal description of the molecular geometry. Kinetic energy operators for pentatomic molecules, given by compact and computationally useful forms in a global and a local formulation of the axis system, are compared and discussed. A new decomposition of T̂ for sequentially bonded pentatomic molecules in conjunction with a basis of Wigner and associated Legendre functions provides kinetic energy matrix elements which are free of singularities. Practical problems caused by an unusual volume element for a description involving only intervector (bending) angles are addressed. The corresponding rovibrational kinetic energy operators are derived for the two-vector body-fixed formulation.

Rovibrational Hamiltonians for general polyatomic molecules in spherical polar parametrization. II. Nonorthogonal descriptions of internal molecular geometry

A rovibrational kinetic energy operator for a two-vector embedded reference frame is derived for general polyatomic molecules of arbitrary structure. New compact expressions for T̂ in terms of angular momentum and linear momentum vector operators are explicitly given for tri-, tetra-, and pentatomic molecules. The necessary actions for the evaluation of kinetic energy matrix elements in a standard rotation-angular basis are summarized. The structure of kinetic energy operators in the body-fixed formulation is explained with the help of Cartesian components of angular momentum and linear momentum vector operators. Kinetic energy operators for tetratomic molecules in a bisector embedded body-fixed formulation and for sequentially bonded pentatomic molecules are derived to show how to use our general result for diverse coordinate systems.

Long Range Binding in Alkali-Helium Pairs

Variational calculations are performed to search for bound rovibrational states of diatomic molecules formed from alkali atoms and helium in the very shallow $^{2}\ensuremath{\Sigma}$ electronic ground state. Examination of a recent set of potential surfaces and several older potentials indicates that all pairs possess a single very diffuse bound state with $J\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0$. Such marginally bound states will have profound effects on low energy collisions between alkali atoms and helium atoms. The sensitivity of these states with respect to retardation effects has been studied. The variational calculations employ a basis set of generalized Laguerre functions and new analytical expressions for kinetic energy matrix elements.

Effective three-body Born-Oppenheimer Hamiltonian including adiabatic corrections

The concept of effective, position-dependent reduced mass is introduced for three-body molecules. It varies continuously between the asymptotic limits and generates a scaled Born-Oppenheimer-type Hamiltonian that contains a substantial amount of the adiabatic correction. A close analytical expression for this correction is obtained, without the explicit calculation of the nuclear kinetic energy matrix element. The approach accounts for the isotope effect and allows the first successful Born-Oppenheimer-type calculation on e-e+e-.

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.

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.

A CLOSER LOOK AT PERTURBATIVE CORRECTIONS IN THE b→c SEMILEPTONIC TRANSITIONS

We comment on two recent calculations of the second order perturbative corrections in the heavy flavor semileptonic transitions within the Brodsky-Lepage-Mackenzie approach. It is pointed out that the results do not show significant enhancement either in the inclusive b→c decays or in the exclusive amplitudes at zero recoil provided that the expansion parameter is chosen in a way appropriate to the kinematics at hand. The values of the second order coefficients inferred from the BLM type calculations appear to be of order unity in both cases. Thus, in both cases no significant uncertainty in extracting Vcb can be attributed to perturbative effects. The theoretical accuracy is mostly determined by the existing uncertainty in [Formula: see text] nonperturbative corrections in the exclusive B→D* amplitude, and, to a lesser extent, by the uncertainty in the estimated value of the kinetic energy matrix element [Formula: see text] in the case of Γ sl (b→c). The theoretical accuracy of the inclusive method of determining Vcb seemingly competes with and even exceeds the experimental accuracy.

Above-threshold multiphoton detachment from the H- ion by 10.6- microm radiation: Angular distributions and partial widths.

We present a general procedure for accurate nonperturbative treatment of the angular distribution and partial widths for multiphoton above-threshold detachment (ATD) of atoms or negative ions in intense laser fields. The procedure consists of the following two steps: (1) The resonance wave function is determined by means of the non-Hermitian Floquet Hamiltonian method. The Floquet Hamiltonian is discretized by the complex scaling generalized pseudospectral method recently developed by Yao and Chu [Chem. Phys. Lett. 204, 381 (1993)]. No computation of potential matrix elements is required, and the kinetic-energy matrix elements can be evaluated analytically. (2) The angular distribution and partial rates are calculated, based on an exact differential expression, and a procedure is developed for the backrotation of the total complex resonance wave function to the real axis. The method is applied to the study of multiphoton ATD of ${\mathrm{H}}^{\mathrm{\ensuremath{-}}}$ in strong fields at 10.6 \ensuremath{\mu}m. An accurate one-electron model potential [Laughlin and Chu, Phys. Rev. A 48, 4654 (1993)], which reproduces the known ${\mathrm{H}}^{\mathrm{\ensuremath{-}}}$ binding energy and the low-energy e-H(1s) elastic scattering phase shifts, is employed. At this low frequency, the resonance wave functions can be obtained efficiently and rather accurately by means of a nonperturbative adiabatic approach recently developed by Telnov [J. Phys. B 24, 2967 (1991)]. This adiabatic theory is also valid in the limit of weak fields, and its validation is justified by its agreement with the exact perturbation calculations for the seven- and eight-photon detachment of ${\mathrm{H}}^{\mathrm{\ensuremath{-}}}$. Detailed results for the angular distribution and partial widths for multiphoton ATD of ${\mathrm{H}}^{\mathrm{\ensuremath{-}}}$ are presented for the moderately strong laser intensity regime (${10}^{10}$--${10}^{11}$ W/${\mathrm{cm}}^{2}$) at 10.6 \ensuremath{\mu}m.

New stretching vibrational basis functions for vibrational variational calculations in curvilinear internal coordinates

New basis functions for the stretching degrees of freedom for vibrational variational calculations in curvilinear internal coordinates are proposed. Analytical kinetic energy matrix elements for the curvilinear internal coordinate vibrational Hamiltonian for triatomic molecules are derived. Use of the basis functions in practical calculations is demonstrated by calculating vibrational term values for H 2 16O variationally up to 30000 cm −1.

Optimal treatment of diffraction coordinates in wave packet scattering from surfaces

In the context of wave packet methodology we show how to take advantage of the diffractive scattering symmetry arising when the incident beam is normal to the surface or to a surface principal axis. This may lead to a reduction in dimensionality being up to a factor of 8. The Fourier transformation is applied to evaluate the translational kinetic energy operator. Two alternative treatments are possible depending on whether the transformation is utilized to calculate the kinetic energy matrix elements in coordinate space, or whether it is applied to the wave function itself to switch between coordinate and momentum representations. The first approach is similar to the discrete variable representation treatment in the spirit of Light and co-workers whereas the second one enables the use of the fast Fourier transform (FFT) scheme of Kosloff and Kosloff. We provide a detailed comparison between the two approaches as a function of the size of the grid, with and without the presence of symmetry in the diffractive scattering.