Kinetic Energy Operator Research Articles

Lévy processes and Schrödinger equation

We analyze the extension of the well known relation between Brownian motion and the Schrödinger equation to the family of the Lévy processes. We consider a Lévy–Schrödinger equation where the usual kinetic energy operator–the Laplacian–is generalized by means of a selfadjoint, pseudodifferential operator whose symbol is the logarithmic characteristic of an infinitely divisible law. The Lévy–Khintchin formula shows then how to write down this operator in an integro-differential form. When the underlying Lévy process is stable we recover as a particular case the fractional Schrödinger equation. A few examples are finally given and we find that there are physically relevant models–such as a form of the relativistic Schrödinger equation–that are in the domain of the non stable Lévy–Schrödinger equations.

A concise method for kinetic energy quantisation

We present a straightforward method for obtaining exact classical and quantum molecular Hamiltonians in terms of arbitrary coordinates. As compared to other approaches the resulting expressions are rather compact, the physical meaning of each quantity is quite transparent, and in some cases the calculation effort will be greatly reduced. We also investigate systems with constraints to find the suggested method to be applicable in contrast to most conventional approaches to kinetic energy operators which cannot directly be applied to constrained systems. Two examples are discussed in detail, the monohydrated hydroxide anion and the protonated ammonia dimer.

Exact and constrained kinetic energy operators for polyatomic molecules: The polyspherical approach

Abstract A review of the polyspherical approach to the kinetic operators for polyatomic molecules is given. This approach provides general and correct forms of the kinetic energy operator (KEO) expressed in terms of curvilinear coordinates. These forms are well adapted to the physical description of molecular systems and to the numerical methods used to solve the Schrodinger equation. The approach derives its name, polyspherical , from the fact that the operators are expressed in terms of spherical coordinates eventually. These kinetic energy operators can be exploited to treat a large variety of problems such as the calculation of infrared or photo-absorption spectra or the study of reactive scattering systems. Special emphasis is placed on concrete examples.

Level distributions, partition functions, and rates of chirality changing processes for the torsional mode around O–O bonds

In view of the particular attention recently devoted to hindered rotations, we have tested reduced kinetic energy operators to study the torsional mode around the O-O bond for H(2)O(2) and for a series of its derivatives (HOOCl, HOOCN, HOOF, HOONO, HOOMe, HOOEt, MeOOMe, ClOOCl, FOOCl, FOOF, and FOONO), for which we had previously determined potential energy profiles along the dihedral ROOR(') angle [R,R(')=H,F,Cl,CN,NO,Me (=CH(3)), Et (=C(2)H(5))]. We have calculated level distributions as a function of temperature and partition functions for all systems. Specifically, for the H(2)O(2) system we have used two procedures for the reduction in the kinetic energy operator to that of a rigid-rotor-like one and the calculated partition functions are compared with previous work. Quantum partition functions are evaluated both by quantum level state sums and by simple classical approximations. A semiclassical approach, using a linear approximation of the classical path and a quadratic Feynman-Hibbs approximation of Feynman path integral, introduced in previous work and here applied to the torsional mode, is shown to greatly improve the classical approximations. Further improvement is obtained by the explicit introduction of the dependence of the moment of inertia from the torsional angle. These results permit one to discuss the characteristic time for chirality changes for the investigated molecules either by quantum mechanical tunneling (dominating at low temperatures) or by transition state theory (expected to provide an estimate of racemization rates in the high energy limit).

The kinetic energy operator in the subspaces of wavelet analysis

At any resolution level of wavelet expansions the physical observable of the kinetic energy is represented by an infinite matrix which is “canonically” chosen as the projection of the operator − Δ/2 onto the subspace of the given resolution. It is shown, that this canonical choice is not optimal, as the regular grid of the basis set introduces an artificial consequence of its periodicity, and it is only a particular member of possible operator representations. We present an explicit method of preparing a near optimal kinetic energy matrix which leads to more appropriate results in numerical wavelet based calculations. This construction works even in those cases, where the usual definition is unusable (i.e., the derivative of the basis functions does not exist). It is also shown, that building an effective kinetic energy matrix is equivalent to the renormalization of the kinetic energy by a momentum dependent effective mass compensating for artificial periodicity effects.

Time-dependent V-representability on lattice systems

We study the mapping between time-dependent densities and potentials for noninteracting electronic systems on lattices. As discovered recently by Baer [J. Chem. Phys. 128, 044103 (2008)], there exist well-behaved time-dependent density functions on lattices which cannot be associated with any real time-dependent potential. This breakdown of time-dependent V-representability can be tracked down to problems with the continuity equation which arise from discretization of the kinetic-energy operator. Examples are given for lattices with two points and with N points, and implications for practical numerical applications of time-dependent density-functional theory are discussed. In the continuum limit, time-dependent noninteracting V-representability is restored.

Calculation of the O−H Stretching Vibrational Overtone Spectrum of the Water Dimer

The O-H stretching vibrational overtone spectrum of the water dimer has been calculated with the dimer modeled as two individually vibrating monomer units. Vibrational term values and absorption intensities have been obtained variationally with a computed dipole moment surface and an internal coordinate Hamiltonian, which consists of exact kinetic energy operators within the Born-Oppenheimer approximation of the monomer units. Three-dimensional ab initio potential energy and dipole moment surfaces have been calculated using the internal coordinates of the monomer units using the coupled cluster method including single, double, and perturbative triple excitations [CCSD(T)] with the augmented correlation consistent valence triple zeta basis set (aug-cc-pVTZ). The augmented correlation consistent valence quadruple zeta basis set (aug-cc-pVQZ), counterpoise correction, basis set extrapolation to the complete basis set limit, relativistic corrections, and core and valence electron correlations effects have been included in one-dimensional potential energy surface cuts. The aim is both to investigate the level of ab initio and vibrational calculations necessary to produce accurate results when compared with experiment and to aid the detection of the water dimer under atmospheric conditions.

Cluster sum rules for three-body systems with angular-momentum dependent interactions

We derive general expressions for non-energy-weighted and energy-weighted cluster sum rules for systems of three charged particles. The interferences between pairs of particles are found to play a substantial role. The energy-weighted sum rule is usually determined by the kinetic energy operator, but we demonstrate that it has similar additional contributions from the angular momentum and parity dependence of two- and three-body potentials frequently used in three-body calculations. The importance of the different contributions is illustrated with the dipole excitations in $^{6}\mathrm{He}$. The results are compared with the available experimental data.

Kinetic energy operators in linearized internal coordinates

It is customary to describe molecular vibrations using as exact kinetic energy operators and as accurate potentials as possible. It has become a standard approach to express Hamiltonians in curvilinear internal displacement coordinates, because they offer a simple and physical picture of vibrational motions, including large amplitude changes in the shape. In the older normal mode model of molecular vibrations, the nuclei are thought to vibrate infinitesimally about the reference configuration, and the shape of the molecule is described using linearized approximations of the true geometrically defined internal displacement coordinates. It is natural to ask how the two approaches are related. In this work, I present a general yet practical way to obtain curvilinear displacement coordinates as closed function of their linearized counterparts, and vice versa. In contrast to the conventional power series approach, the body-frame dependency is explicitly taken into account, and the relations are valid for any value of the coordinates. The present approach also allows one to obtain easily exact kinetic energy operators in linearized shape coordinates.

THE APPLICATION OF BESSEL DISCRETE VARIABLE REPRESENTATION TO ATOMIC HYDROGEN IN INTENSE LASER FIELDS

The Bessel discrete variable representation (DVR) method is tested to describe the interaction of atomic hydrogen with intense laser fields by numerically solving the time-dependent Schrödinger equation. Using the Bessel functions of the first kind, the singular terms, r-2 or r-1 at the origin, in the kinetic energy operators are analytically solved. As an illustration example, the high-order harmonic generation (HOHG) spectra in atomic hydrogen is calculated in length and acceleration forms. From the numerical results, it is concluded that this simple Bessel DVR may be a useful method for describing the interaction of atomic hydrogen with intense laser fields.

Full-dimensional (15-dimensional) quantum-dynamical simulation of the protonated water dimer. I. Hamiltonian setup and analysis of the ground vibrational state

Quantum-dynamical full-dimensional (15D) calculations are reported for the protonated water dimer (H5O2+) using the multiconfiguration time-dependent Hartree (MCTDH) method. The dynamics is described by curvilinear coordinates. The expression of the kinetic energy operator in this set of coordinates is given and its derivation, following the polyspherical method, is discussed. The potential-energy surface (PES) employed is that of Huang et al. [J. Chem. Phys. 122, 044308 (2005)]. A scheme for the representation of the PES is discussed which is based on a high-dimensional model representation scheme, but modified to take advantage of the mode-combination representation of the vibrational wave function used in MCTDH. The convergence of the PES expansion used is quantified and evidence is provided that it correctly reproduces the reference PES at least for the range of energies of interest. The reported zero point energy of the system is converged with respect to the MCTDH expansion and in excellent agreement (16.7 cm(-1) below) with the diffusion Monte Carlo result on the PES of Huang et al. The highly fluxional nature of the cation is accounted for through use of curvilinear coordinates. The system is found to interconvert between equivalent minima through wagging and internal rotation motions already when in the ground vibrational state, i.e., T=0. It is shown that a converged quantum-dynamical description of such a flexible, multiminima system is possible.

Photoionization-induced dynamics of the ammonia cation studied by wave packet calculations using curvilinear coordinates

The determination of the photoelectron spectrum of NH 3 and of the internal conversion dynamics of NH 3 + recently published [A. Viel, W. Eisfeld, S. Neumann, W. Domcke, U. Manthe, J. Chem. Phys. 124 (2006) 214306] is complemented by the investigation of the effect of the vibrational angular momenta couplings on the dynamics. The multi-configurational time-dependent Hartree method is used to propagate a wave packet on the analytical anharmonic six-dimensional three-sheeted potential energy surface for the ground and first excited states of the ammonia cation. Curvilinear coordinates and the associated quasi-exact kinetic energy operator suitable for the multi-configurational time-dependent Hartree scheme are employed. A non-negligible effect of the use of Cartesian normal modes instead of the curvilinear coordinates is observed on the low energy part of the photoelectron spectrum. However, the three different time scales found in the dynamical calculations for the second absorption band are very similar regardless of the use of either normal modes or curvilinear coordinates.

Few-body problem in terms of correlated Gaussians

In their textbook, Suzuki and Varga [ (Springer, Berlin, 1998)] present the stochastic variational method with the correlated Gaussian basis in a very exhaustive way. However, the Fourier transform of these functions and their application to the management of a relativistic kinetic energy operator are missing and cannot be found in the literature. In this paper we present these interesting formulas. We also give a derivation for formulations concerning central potentials.

Quantum dynamics of the CH3 fragment: A curvilinear coordinate system and kinetic energy operators

A curvilinear coordinate system for AB(3) fragments is given. The corresponding exact kinetic energy operator is derived and a series of simpler, progressively more approximate kinetic energy operators are suggested. The operators are tailored for quantum dynamics simulations using the multiconfigurational time-dependent Hartree approach. It is outlined how these fragment coordinates can be utilized to set up coordinate systems for larger systems such as AB(3)C or AB(3)CD. Calculations of the vibrational levels of CH(3) and quantum dynamics studies investigate the accuracy of the different kinetic energy operators suggested.

The vibration–rotation kinetic energy operator for sequentially bonded tetra-atomic molecules in internal coordinates

An exact vibration–rotation kinetic energy operator for polyatomic molecules has been obtained. Using this Hamiltonian for sequentially bonded tetra-atomic molecules, all rovibrational terms have been derived with internal coordinates as the vibrational variables. The present approach is greatly simplified with less algebra compared with conventional methods. Also, simple and explicit expressions for the vibration–rotation coupling terms in internal coordinates are presented.

Vibrational energy levels with arbitrary potentials using the Eckart-Watson Hamiltonians and the discrete variable representation

An effective and general algorithm is suggested for variational vibrational calculations of N-atomic molecules using orthogonal, rectilinear internal coordinates. The protocol has three essential parts. First, it advocates the use of the Eckart-Watson Hamiltonians of nonlinear or linear reference configuration. Second, with the help of an exact expression of curvilinear internal coordinates (e.g., valence coordinates) in terms of orthogonal, rectilinear internal coordinates (e.g., normal coordinates), any high-accuracy potential or force field expressed in curvilinear internal coordinates can be used in the calculations. Third, the matrix representation of the appropriate Eckart-Watson Hamiltonian is constructed in a discrete variable representation, in which the matrix of the potential energy operator is always diagonal, whatever complicated form the potential function assumes, and the matrix of the kinetic energy operator is a sparse matrix of special structure. Details of the suggested algorithm as well as results obtained for linear and nonlinear test cases including H(2)O, H(3) (+), CO(2), HCNHNC, and CH(4) are presented.

Theoretical ROVibrational Energies (TROVE): A robust numerical approach to the calculation of rovibrational energies for polyatomic molecules

We present a new computational method with associated computer program TROVE (Theoretical ROVibrational Energies) to perform variational calculations of rovibrational energies for general polyatomic molecules of arbitrary structure in isolated electronic states. The (approximate) nuclear kinetic energy operator is represented as an expansion in terms of internal coordinates. The main feature of the computational scheme is a numerical construction of the kinetic energy operator, which is an integral part of the computation process. Thus the scheme is self-contained, i.e., it requires no analytical pre-derivation of the kinetic energy operator. It is also general, since it can be used in connection with any internal coordinates. The method represents an extension of our model for pyramidal XY 3 molecules reported previously [S.N. Yurchenko, M. Carvajal, P. Jensen, H. Lin, J.J. Zheng, W. Thiel, Mol. Phys. 103 (2005) 359]. Non-rigid molecules are treated in the Hougen–Bunker–Johns approach [J.T. Hougen, P.R. Bunker, J.W.C. Johns, J. Mol. Spectrosc. 34 (1970) 136]. In this case, the variational calculations employ a numerical finite basis representation for the large-amplitude motion using basis functions that are generated by Numerov-Cooley integration of the appropriate one-dimensional Schrödinger equation.

Valence-band mixing in first-principles envelope-function theory

This paper presents a numerical implementation of a first-principles envelope-function theory derived recently by the author [B. A. Foreman, Phys. Rev. B 72, 165345 (2005)]. The examples studied deal with the valence subband structure of $\mathrm{Ga}\mathrm{As}∕\mathrm{Al}\mathrm{As}$, $\mathrm{Ga}\mathrm{As}∕{\mathrm{Al}}_{0.2}{\mathrm{Ga}}_{0.8}\mathrm{As}$, and ${\mathrm{In}}_{0.53}{\mathrm{Ga}}_{0.47}\mathrm{As}∕\mathrm{In}\mathrm{P}$ (001) superlattices calculated using the local-density approximation to density-functional theory and norm-conserving pseudopotentials without spin-orbit coupling. The heterostructure Hamiltonian is approximated using quadratic-response theory, with the heterostructure treated as a perturbation of a bulk reference crystal. The valence subband structure is reproduced accurately over a wide energy range by a multiband envelope-function Hamiltonian with linear renormalization of the momentum and mass parameters. Good results are also obtained over a more limited energy range from a single-band model with quadratic renormalization. The effective kinetic-energy operator ordering derived here is more complicated than in many previous studies, consisting in general of a linear combination of all possible operator orderings. In some cases, the valence-band Rashba coupling differs significantly from the bulk magnetic Luttinger parameter. The splitting of the quasidegenerate ground state of no-common-atom superlattices has non-negligible contributions from both short-range interface mixing and long-range dipole terms in the quadratic density response.

A simple and efficient evolution operator for time-dependent Hamiltonians: the Taylor expansion

No compact expression of the evolution operator is known when the Hamiltonian operator is time dependent, like when Hamiltonian operators describe, in a semiclassical limit, the interaction of a molecule with an electric field. It is well known that Magnus [N. Magnus, Commun. Pure Appl. Math. 7, 649 (1954)] has derived a formal expression where the evolution operator is expressed as an exponential of an operator defined as a series. In spite of its formal simplicity, it turns out to be difficult to use at high orders. For numerical purposes, approximate methods such as "Runge-Kutta" or "split operator" are often used usually, however, to a small order (<5), so that only small time steps, about one-tenth or one-hundredth of the field cycle, are acceptable. Moreover, concerning the latter method, split operator, it is only very efficient when a diagonal representation of the kinetic energy operator is known. The Taylor expansion of the evolution operator or the wave function about the initial time provides an alternative approach, which is very simple to implement and, unlike split operator, without restrictions on the Hamiltonian. In addition, relatively large time steps (up to the field cycle) can be used. A two-level model and a propagation of a Gaussian wave packet in a harmonic potential illustrate the efficiency of the Taylor expansion. Finally, the calculation of the time-averaged absorbed energy in fluoroproprene provides a realistic application of our method.

Parallelizable split-operator propagator for treating Coriolis-coupled quantum dynamics

We show that the split-operator propagator of Feit and Fleck can be converted into a form that is easy to parallelize for a quantum system with Coriolis-coupled quantum dynamics. The angular kinetic energy part of the propagator is replaced by the Crank–Nicholson approximation, which makes the entire propagator tridiagonal in the quantum number Ω (the projection of the total angular momentum quantum number J onto the intermolecular axis). We demonstrate that the modified propagator is numerically stable and accurate, provided that the Crank–Nicholson approximation is applied to the entire angular kinetic energy operator.