Hyperspherical Coordinate Method Research Articles

Analytical Structure Matching and Very Precise Approach to the Coulombic Quantum Three-Body Problem

A powerful approach to solve the Coulombic quantum three-body problem is proposed. The approach is exponentially convergent and more efficient than the hyperspherical coordinate method and the correlation-function hyperspherical harmonic method. This approach is numerically competitive with the variational methods, such as that using the Hylleraas-type basis functions. Numerical comparisons are made to demonstrate the efficiency of this approach, by calculating the nonrelativistic and infinite-nuclear-mass limit of the ground state energy of the helium atom. The exponential convergency of this approach is due to the full matching between the analytical structure of the basis functions that are used in this paper and the true wavefunction. This full matching was not reached by most other methods. For example, the variational method using the Hylleraas-type basis does not reflects the logarithmic singularity of the true wavefunction at the origin as predicted by Bartlett and Fock. Two important approaches are proposed in this work to reach this full matching: the coordinate transformation method and the asymptotic series method. Besides these, this work makes use of the least square method to substitute complicated numerical integrations in solving the Schrödinger equation without much loss of accuracy, which is routinely used by people to fit a theoretical curve with discrete experimental data, but here is used to simplify the computation.

Electronically nonadiabatic transitions in a collinear H2+H+ system: Quantum mechanical understanding and comparison with a trajectory surface hopping method

Electronically nonadiabatic transitions in a collinear H2 + H+ system have been studied both quantum mechanically and with classical mechanics using a diatomics-in-molecules type potential energy surface fitted to recent ab initio data. The quantum dynamical calculations are carried out by employing a standard close-coupling method in hyperspherical coordinates and the quasiclassical trajectory calculations were carried out with a basic surface hopping method. Special emphasis is placed on qualitative analysis of the physical mechanisms of the electronically nonadiabatic transitions. To make such an analysis, the basic idea proposed by Nobusada et al. (K. Nobusada, O. I. Tolstikhin and H. Nakamura, J. Chem. Phys., 1998, 108, 8922) in the study of vibrationally nonadiabatic transitions is applied, and also the reaction dynamics is visualized in detail by using classical trajectories. It is found that the electronically nonadiabatic transition occurs in a rather narrow configuration space and its reactivity depends strongly on the initial vibrational state of H2.

ABC: a quantum reactive scattering program

This article describes a quantum mechanical reactive scattering program for atom–diatom chemical reactions that we have written during the past several years. The program uses a coupled-channel hyperspherical coordinate method to solve the Schrödinger equation for the motion of the three nuclei on a single Born–Oppenheimer potential energy surface. It has been tested for all possible deuterium-substituted isotopomers of the H+ H 2 , F+ H 2 , and Cl+ H 2 reactions, and tried and tested potential energy surfaces for these reactions are included within the program as Fortran subroutines.

Quantum scattering and quasi-classical trajectory calculations for the H2+OH ⇌ H2O+H reaction on a new potential surface

Six-dimensional (6D) quantum scattering calculations of reaction probabilities are reported for theOH+H2⇌H2O+H reaction. An arrangement channel hyperspherical coordinate method is used. A new potential energy surface due to Ochoa and Clary is employed. The results agree well with those calculated using the rotating bond approximation (RBA) and the quasi-classical trajectory (QCT) method. 6D quantum, RBA and QCT calculations of rate constants for the OH+H2 reaction agree well with experiment. In addition, RBA calculations of differential cross sections for the OH+D2→HOD+D reaction and the photodetachment spectrum for H3O− also agree well with experiment. These results suggest that the new potential surface is reliable for this reason.

3D wavepacket calculations of ozone photodissociation in the Hartley band: convergence of the autocorrelation function

3D wavepacket calculations on the O 3 molecule have been performed for a total angular momentum equal to zero. The split operator propagator and the fast Fourier transform method in hyperspherical coordinates are used in order to follow the quantum dynamics of photodissociation. The absorption spectrum for the transition from the ground X 1 A 1 to the excited D 1 B 2 state is obtained and compared with results from previous calculations. The effect of the finite lifetime of the excited state has been studied, in order to explain the disagreement between the theoretical calculations and experimental data.

Split operator method in hyperspherical coordinates: Application to CH2I2 and OClO

3D wave packet calculations on the CH2I2 and OClO molecules have been performed for a total angular momentum equal to zero. The initial wave function is found by solving the time independent Schrödinger equation in internal bond coordinates. The split operator method and the fast Fourier transform in hyperspherical coordinates are used in order to follow the quantum dynamics. An absorption spectrum of CH2I2 is obtained and compared with a previous 2D calculation. A Raman spectrum for the CH2I2 molecule at 355 nm is calculated and compared with experimental results. The absorption spectrum for the X2B1→A2 A2 transition of the OClO molecule is calculated using the same method as for CH2I2. Good agreement with experiment is obtained.

Four-dimensional quantum scattering calculations on the H+CH4→H2+CH3 reaction

Time-independent quantum scattering calculations have been performed to study the H+CH4→H2+CH3 reaction, using the analytic potential-energy surface developed by Jordan and Gilbert. A rotating bond umbrella (RBU) approximation with the implementation of a guided spectral transform subspace iteration technique has been applied together with a log-derivative method in hyperspherical coordinates. A single sector hyperspherical projection method was used to apply the boundary conditions to extract the S matrix at a large hyperradius. The results show that the H+CH4→H2+CH3 reaction occurs via a direct mechanism. The tunneling effect is pronounced, while there is little recrossing. Vibrational excitation of the C–H stretch and/or the H–CH3 bending modes of CH4 significantly enhance the reactivity. Exciting the umbrella mode of CH4 also enhance the reactivity, although less efficiently. The calculated thermal rate constants are larger than the experimental ones. However, good agreement has been obtained by including a barrier height correction of the potential function to make it agree with ab initio results. Finally, vibrational and rotational distributions of the reaction products are discussed in detail.

Hermite correction method in hyperspherical coordinates: Application to chemical reactions

The rotational product state distribution for the D+H 2(v=1, j=1) → HD( v′=1, j′ )+H reaction, has been calculated using the Hermite correction method in hyperspherical coordinates. We have also considered the geometric phase effect in our calculations either by introducing a vector potential in the system Hamiltonian or by multiplying the wavefunction by a phase factor. In the present application, we have investigated the performance of the simplest approach where only one time-dependent Gauss–Hermite basis function in the hyperradius is included.

Vibrational spectrum of ground state Li3 and statistical analysis of the energy levels

We report J = 0 calculations of all bound vibrational levels of ground-state Li3 using a realistic double many-body expansion potential energy surface, and a minimum-residual filter diagonalization technique. The action of the system Hamiltonian on the wavefunction is evaluated by the spectral transform method in hyperspherical coordinates, i.e. a fast Fourier transform for the ρ and ø variables and a discrete variable representation-finite basis representation transformation for θ. The spectrum shows significant changes when geometric phase effects are considered. Using random matrix theory, it is then shown from the neighbour spacing distributions of the vibrational levels that the spectra for the various symmetries are Brody-type while the full spectra are quasi-regular in short range and quasi-irregular in long range.

Vibrational spectrum of Li3 first-excited electronic doublet state: Geometric-phase effects and statistical analysis

We present J=0 calculations of all bound and pseudobound vibrational states of Li3 in its first-excited electronic doublet state by using a realistic double many-body expansion potential-energy surface and a minimum-residual filter diagonalization technique. The action of the system Hamiltonian on the wave function was evaluated by the spectral transform method in hyperspherical coordinates. Calculations of the vibrational spectra were carried out both without consideration and with consideration of geometric-phase effects. Dynamic Jahn–Teller and geometric-phase effects are found to play a significant role, while the calculated fundamental symmetric stretching frequency is larger by 8.3% than its reported experimental value of 326 cm−1. From the neighbor-spacing distributions of the levels, it is observed that the title vibrational spectrum is quasiregular in the short range and quasi-irregular in the long range. By the Δ2 standard defined in this article, it is found that the spectra are more nonuniform than those of the “trough” states for the ground electronic state. ©1999 John Wiley & Sons, Inc. Int J Quant Chem 75: 89–109, 1999

State-to-state three-atom reactive scattering using adiabatic rotation approximations

In this work we explore the applicability of the adiabatic rotation approximations to state-to-state quantum reactive scattering for three-body systems within the context of the coupled-channel hyperspherical coordinates method. A novel refinement to the adiabatic rotation approximation advocated by Bowman (Chem. Phys. Lett. 1994, 217, 36) has been obtained by the inclusion of the asymmetric rotational terms into the Hamiltonian. Within this approach, the total Hamiltonian is separated in an exact Hamiltonian for zero total angular momentum plus an effective rotational energy term that is added to the potential energy surface to form an effective potential on which the reactive dynamics evolves. Calculations of state-to-state reaction probabilities, integral and differential cross-sections for the prototypical reactions Cl+H2 and F+H2 are presented to test their accuracy and computational advantages in comparison with exact quantum mechanical calculations.

Semiclassical reactive scattering: the Hermite correction method in hyperspherical coordinates

We have used the classical path theory based on the Hermite correction to Gaussian wave packets by G.D. Billing [J. Chem. Phys. 107 (1997) 4286] to treat three center reactions using hyperspherical coordinates. With this approach the probabilities for the D + H 2 ( v=0, j=0) → HD ( v′, j′) + H, reaction have been calculated for the J=0 case. The results are compared with those of J.Z.H. Zhang and W.H. Miller [J. Chem. Phys. 91 (1989) 1528] obtained by time-independent scattering calculations.

Quantum theory of four-atom reactions using arrangement channel hyperspherical coordinates: Formulation and application to OH+H2↔H2O+H

An arrangement channel hyperspherical coordinate method for performing quantum scattering calculations on four-atom reactions is formulated. This method treats the vibrational and rotational states in different arrangement channels by a close-coupling expansion in nonorthogonal functions. The method is applied to the calculation of state-to-state probabilities for the OH+H2→H2O+H reaction. Good agreement is found with cumulative and state-selected reaction probabilities previously calculated by other methods. The major advantage of this general approach is that the whole S matrix can be obtained in a single calculation.

Criteria for the validity of the diabatic-by-sector expansion in the hyperspherical coordinate method

It is mathematically indicated that the widely used diabatic-by-sector recipe for solving coupled radial equations in the hyperspherical coordinate method causes non-negligible errors and these are revealed especially for extremely low-energy scattering. The intrinsic defects due to this method are illustrated for both bound state and the scattering states of $\mathrm{dt}\ensuremath{\mu}$. The calculated results are also compared with those obtained by the adiabatic expansion method.

Progress and applications of the hyperspherical coordinate method

The hyperspherical method as we know now stems from the pioneering work of Macek on doubly excited states of helium [J. Macek, J. Phys. B 1 (1968) 831]. It played an important role in classifying doubly excited Rydberg series according to the radial normal modes [U. Fano, Phys. Today 29 (1976) 32] of the two electrons. We review not only its application to the classification of doubly excited states but its related technical innovations that have turned the method into one of the most reliable numerical schemes for handling scattering problems for three-body systems. We take a glance at its wide applications from the low-energy electron impact ionization of H through the fast proton/antiproton impact excitation of He.

Coupled potential-energy surfaces and quantum reactive scattering for the Cl(2P)+HCl→ClH+Cl(2P) reaction

We construct global multiple-coupled potential-energy surfaces for the three-dimensional Cl( 2 P) + HCl → ClH + Cl( 2 P) reaction which we use to generate results from quantum reactive scattering calculations. The three potential surfaces (1 2 A′, 2 2 A′ and 1 2 A″) are derived from multi-configuration self-consistent field (MCSCF) electronic structure calculations for geometries where all atoms are strongly interacting. The MCSCF energies are used to construct diabatic potentials which are fitted with rotated Morse cubic spline functions. The diabatic surfaces are then combined with empirical long-range potentials due to Dubernet and Hutson, which are also used to determine empirical coupling surfaces. Spin–orbit interaction is added perturbatively to the resulting surfaces. Saddle-point properties on these fitted surfaces are scaled to reproduce experimental thermal rate coefficient data based on the quantum scattering calculations. The quantum scattering calculations are based on a coupled-channel hyperspherical coordinate (CCH) method. The basis expansion used in the scattering calculation contains diatomic vibration and rotation, and the atomic fine-structure states, including full angular momentum coupling of rotation with electronic spin and orbital angular momentum. All terms in the combined rovibronic Hamiltonian are evaluated explicitly, including spin–orbit coupling, Coriolis and electrostatic interactions. State-selected and total cumulative reaction probabilities and thermal rate coefficients are calculated using a J-shifting approximation, which starts from converged CCH results for total angular momentum quantum number J = 1/2. The calculated rate coefficients indicate that spin-state change collisions account for a few per cent of the thermal rate coefficient at 300 K. Transition-state ClHCl resonances are lower in energy and broader than those on earlier (single) surfaces, and there is a more extensive progression in the symmetric stretch mode.

Strong Induced-Dipole-Field Oscillations of the dt micro System above the t micro(n=2) Threshold.

Elastic, inelastic, and muon transfer processes of the $\mathrm{dt}\ensuremath{\mu}$ system above the $t\ensuremath{\mu}(n\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}2)$ threshold are studied theoretically using the hyperspherical coordinate method. Strong oscillation structures in the cross sections for these processes due to strong, attractive dipole potentials in this system are found. In addition to the expected Gailitis-Damburg Stark mixing oscillations, unexpected oscillations due to diabatic couplings of channels lacking attractive dipole potentials with those that have them are evident. The two types of oscillations interfere to produce the structure that appears in the computed cross sections.

HSTERM — A program to calculate potential curves and radial matrix elements for two-electron systems within the hyperspherical adiabatic approach

A FORTRAN 77 program is presented which calculates potential curves and matrix elements of radial coupling for two-electron systems using the hyperspherical coordinate method. The adiabatic and diabatic-by-sector close-coupling approaches are considered. The program calculates also the overlap matrices on borders of all sectors which are necessary for integration of close-coupling hyperradial equations within the sector-diabatic approach. It performs also the computation of the angular part of dipole amplitudes (in the length and acceleration forms) for dipole transitions between two given atomic states. The convergence and accuracy of the computational scheme elaborated are studied in details. Radial matrix elements computed by the HSTERM program can be used for the solution of the bound state and scattering problems for two-electron systems in both the adiabatic and diabatic-by-sector close-coupling approaches.

Double-electron excitation of H- by fast proton and antiproton impact.

A theoretical investigation is carried out for double-electron excitation processes of ${\mathrm{H}}^{\mathrm{\ensuremath{-}}}$ induced by proton and antiproton impact at the enegy of 1.5 MeV. Excitation cross sections to the 2${\mathit{s}}^{2}$ $^{1}$${\mathit{S}}^{\mathit{e}}$, 2s2p $^{1}$${\mathit{P}}^{\mathit{o}}$, and 2${\mathit{p}}^{2}$ $^{1}$${\mathit{D}}^{\mathit{e}}$ states are calculated by using the plane-wave Born approximation, the distorted-wave Born approximation, and the close-coupling method. Wave functions of ${\mathrm{H}}^{\mathrm{\ensuremath{-}}}$ are generated employing the hyperspherical coordinate method. It is shown that the low-lying continuum 1skp $^{1}$${\mathit{P}}^{\mathit{o}}$ is identified as an important intermediate state in the double-electron excitation process to the 2${\mathit{p}}^{2}$ $^{1}$${\mathit{D}}^{\mathit{e}}$ state. We have also evaluated the ejected-electron spectra from the 2s2p $^{1}$${\mathit{P}}^{\mathit{o}}$ shape-resonance state. It is found that its spectral shape is close to the observed line profile of the photodetachment of ${\mathrm{H}}^{\mathrm{\ensuremath{-}}}$ since the excitation mechanism to this state is mostly dominated by the optically allowed transition at the projectile energy concerned here.

A coupled channel hyperspherical scattering study of the Cl + HCl → ClH + Cl reaction: cumulative and state-selected probabilities, integral cross sections, and product rotational distributions

We present accurate quantum results for the Cl + HCl → ClH + Cl reaction using a coupled channel hyperspherical coordinate method (denoted CCH). A semiempirical extended London–Eyring–Polanyi–Sato potential energy surface is employed. In our calculations, the body-fixed z projection quantum number Ω in the rotational basis set has been truncated at Ωmax = 2, since this is sufficient to give converged results near to, and above, the reaction threshold (total energies E in the range 0.40–0.60 eV). We compare our results with earlier work using the centrifugal sudden hyperspherical (CSH) method, in which only Ω = 0 is included in the basis; other features of the calculations are identical. At E = 0.40 eV, where reactivity is low, the CCH(Ωmax = 2) and CSH(Ω = 0) degeneracy averaged reaction probabilities, integral cross sections and product rotational distributions are mostly in excellent agreement. This agreement deteriorates with increasing E, such that at E = 0.60 eV, the CCH(Ωmax = 2) cumulative reaction probability is approximately twice its CSH(Ω = 0) counterpart. Even at E = 0.40 eV there are some important differences between CCH(Ωmax = 2) and CSH(Ω = 0) concerning mj-resolved reaction probabilities. In addition, there are differences in the relative contributions of "tight bend" and "nearly free-rotor" mechanisms to the reaction probabilities and product rotational distributions, with the "nearly free-rotor" mechanism dominant in all cases.