Gabriel G. Balint-Kurti

The Fourier grid Hamiltonian method for bound state eigenvalues and eigenfunctions

A new method for the calculation of bound state eigenvalues and eigenfunctions of the Schrodinger equation is presented. The Fourier grid Hamiltonian method is derived from the discrete Fourier transform algorithm. Its implementation and use is extremely simple, requiring the evaluation of the potential only at certain grid points and yielding directly the amplitude of the eigenfunctions at the same grid points.

Quantum dynamics with real wave packets, including application to three-dimensional (J=0)D+H2→HD+H reactive scattering

We show how to extract S matrix elements for reactive scattering from just the real part of an evolving wave packet. A three-term recursion scheme allows the real part of a wave packet to be propagated without reference to its imaginary part, so S matrix elements can be calculated efficiently. Our approach can be applied not only to the usual time-dependent Schrodinger equation, but to a modified form with the Hamiltonian operator Ĥ replaced by f(Ĥ), where f is chosen for convenience. One particular choice for f, a cos−1 mapping, yields the Chebyshev iteration that has proved to be useful in several other recent studies. We show how reactive scattering can be studied by following time-dependent wave packets generated by this mapping. These ideas are illustrated through calculation of collinear H+H2→H2+H and three-dimensional (J=0)D+H2→HD+D reactive scattering probabilities on the Liu–Siegbahn–Truhlar–Horowitz (LSTH) potential energy surface.

Grid methods for solving the Schrödinger equation and time dependent quantum dynamics of molecular photofragmentation and reactive scattering processes

Abstract This review will concentrate on new theoretical methods for solving the quantum dynamics of molecular systems. The approach will be that of solving the time dependent Schrodinger equation, and extracting from it the measurable quantities of experimental interest. Applications to the modelling of molecular photodissociation processes and to the theory of reactive molecular collisions will be discussed.

Quantum mechanical three‐dimensional wavepacket study of the Li+HF→LiF+H reaction

A three‐dimensional time‐dependent quantum mechanical wavepacket method is used to calculate the state‐to‐state reaction probabilities at zero total angular momentum for the Li + HF → LiF +H reaction. Reaction probabilities starting from several different initial HF vibrational–rotational states (v=0,j=0,1,2) and going to all possible open channels are computed over a wide range of energies. A single computation of the wavepacket dynamics yields reaction probabilities from a specific initial quantum state of the reactants to all possible final states over a wide range of energies. The energy dependence of the reaction probabilities shows a broad background structure on which resonances of varying widths are superimposed. Sharp resonance features seem to dominate particularly at low product translational energies. There are marked changes in the energy dependence of the reaction probabilities for different initial or final diatom rotational quantum numbers, but it is noticeable that, for both reactants and...

Flux analysis for calculating reaction probabilities with real wave packets

Abstract A recent approach to obtain scattering probabilities involves propagating the real part of a wave packet and analysis in the asymptotic product channel. We show that real wave packet propagation can also be used with analysis methods based on flux through a surface. We also show the relation between asymptotic product analysis and flux-based methods. Flux analysis with real wave packets is illustrated with an application to D + H 2 → DH + H in three dimensions.

Reflection and transmission of waves by a complex potential - A semiclassical Jeffreys-Wentzel-Kramers-Brillouin treatment

In this paper, the reflection and transmission of plane waves are examined from a complex potential. Such potentials have the property of absorbing wave packets incident on them and are used widely in time‐dependent quantum scattering theory. The purpose of the study is to determine the optimal form of potential to be used for absorbing wave packets near the edges of finite grids in coordinate space. The best potentials for such purposes lead to the minimum possible transmission and reflection of the incident wave packet. The Jeffreys–Wentzel–Kramers–Brillouin (JWKB) theory is used to address this problem and a new form for the optimal complex potential is proposed. A scaled dimensionless form of the Schrodinger equation is also derived, so that the parameters of any optimized potential obtained for a particular collision energy and mass combination may be readily converted to apply to a new set of masses and energies.

Time-dependent quantum dynamics of molecular photofragmentation processes

The time-dependent quantum-mechanical description of molecular photodissociation processes is briefly reviewed. A new easily implementable method for the calculation of partial cross-sections to produce specific fragment quantum states is presented. The equivalence of the partial cross-sections calculated using these time-dependent quantum-mechanical methods to those calculated using standard time-independent quantum theory is explicitly demonstrated. Sample calculations using a model potential-energy surface for a system having physical parameters corresponding to the H2S molecule are presented. The power of the method is clearly demonstrated by explicitly showing, for this model system, how a single time-dependent calculation yields the partial photodissociation cross-sections for all photon energies. We furthermore point out the suitability of modern parallel computing techniques in connection with such methods.

Two computer programs for solving the Schrödinger equation for bound-state eigenvalues and eigenfunctions using the Fourier grid Hamiltonian method

Abstract Two computer programs (FGHEVEN and FGHFFT) for solving the one-dimensional Schrodinger equation for bound-state eigenvalues and eigenfunctions are presented. Both computer programs are based on the Fourier grid Hamiltonian method (J. Chem. Phys. 91 (1989) 3571). The method is exceptionally simple and robust. It relies on using the momentum representation for the kinetic energy operator and the coordinate representation for the potential energy. The first computer program (FGHEVEN) is based on an explicit (very simple) expression for the Hamiltonian matrix. The eigenvalues of this matrix give the required bound-state energies and the eigenvectors yield directly the eigenfunctions evaluated on the regularly spaced grid points. In this paper the theory has been slightly extended to encompass the situation where an even number of grid points is used. The second program (FGHFFT) is based on a complimentary theory which makes use of the discrete fast Fourier transform technique to evaluate the Hamiltonian matrix. The programs are self-contained and include subroutines to find the eigenvalues and eigenvectors needed.

Probing the effect of the H2 rotational state in O(1D)+H2→OH+H: Theoretical dynamics including nonadiabatic effects and a crossed molecular beam study

Theoretical estimates of reactive cross sections for O(1D)+H2(X,v=0,j)→OH(X)+H(2S), with H2 rotational quantum numbers j=0 and 1, are obtained for a range of collision energies, Ecol. Crossed molecular beam measurements are also used to infer the ratio, r1,0, of the j=1 and 0 cross sections at Ecol=0.056 eV. The theory indicates that the 1 1A′ potential surface is the most important one. However, the 2 1A′ and 1 1A″ surfaces can also contribute. Adiabatic dynamics on the 1 1A″ surface, particularly at Ecol above its 0.1 eV barrier to reaction plays a role. The 2 1A′ surface, while not correlating with ground electronic state products, can still lead to products via nonadiabatic interactions with the 1 1A′ surface. Many quantum dynamics and quasiclassical classical trajectory calculations are carried out. Accurate, ab initio based potential energy surfaces are employed. Quantum cross sections are based on helicity decoupled wave packet calculations for several values of total angular momentum. Nonadiabatic...

A new method for the exact calculation of vibrational–rotational energy levels of triatomic molecules

A new method is presented for the calculation of the vibrational–rotational energy levels of a triatomic molecule. The method provides a means of solving Schodinger’s equation for the bound energy levels of a triatomic system as exactly as desired. The relative motion of the atoms is governed by a potential energy surface, which can be of arbitrary form, but must, naturally, be specified. The method utilizes the well developed techniques of molecular scattering theory and is formulated in a body‐fixed reference frame. The bound state energies appear in the theory as first order poles of a specially constructed T matrix element. Because the analytic behavior of these poles, as a function of the energy, is well defined, the exact bound state energies are easily found. The theory has been applied to the water molecule and results are presented for the lowest five vibrational–rotational energy levels corresponding to zero total angular momentum. These results agree well with previously published values.