Construction of a nonlocal potential from the scattering amplitude in the born approximation

Within a framework of the Born approximation simplified by certain assumptions, a procedure is given which explicitly constructs a nonlocal potential that provides a fit to elastic scattering data. The data are assumed to consist of the scattering amplitude as a function of the momentum transfer over a finite range of energies. The constructed potential is unique only when the kernel of the nonlocal interaction is energy independent, or when its explicit energy dependence is known from information other than scattering data.     

Convergence problems in the solution of SCF equations

Some problems connected with the convergence of iterative solutions of the Hartree-Fock equations for the open shell systems are discussed. The nonuniqueness of the Hartree-Fock operator form is used. The method of iterating by the operator's power is developed for obtaining solutions of the Hartree-Fock equations. Some particular results for the molecules Li, Li are presented.     

Elastic scattering of electrons by H2 in the born approximation

This paper presents accurate computation of the differential cross sections for the elastic and vibrational excitation of ground state H₂ by fast electrons. The electronic matrix elements are evaluated at 20 internuclear separations and proper averaging over vibrational wavefunctions is carried out. The comparison to experiment yields excellent agreement. Different computational procedures appropriate for low and moderate values of momentum transfer are compared.     

An efficient and accurate approximation to double substitution coupled cluster wavefunctions

An approximate CCD theory in an efficient electron pair operator form has been applied to He₂ with very accurate results.     

A CSF-based multi-reference coupled pair approximation III. An application to F2, As2 and As+2

Using the newly developed multi-reference coupled pair approximation program code, the adiabatic potential curves of the ground states of F ₂, As ₂ and As ₂ ⁺ were calculated. Computed spectroscopic constants of these molecules were found to be in good agreement with experimental values. The resulting binding energy of As ₂ (3.86 eV) was compared with the experimental value of 3.99 eV [15] and the best multi-reference configuration interaction value (3.58 eV) reported previously by the present authors. The calculated first adiabatic ionization potential of As ₂ (9.67 eV) was found to be in good agreement with the experimental result. Received: 5 July 1997 / Accepted: 27 August 1997     

Hypervirial-perturbative approximation to the SCF eigenvalues of coupled oscillators

The hypervirial relations and perturbation theory are used in order to obtain analytical non-numerical expressions for the SCF eigenvalues of coupled oscillators.     

Time-averaging approximation in the interaction picture: absorption line shapes for coupled chromophores with application to liquid water

The time-averaging approximation (TAA), originally developed to calculate vibrational line shapes for coupled chromophores using mixed quantum/classical methods, is reformulated. In the original version of the theory, time averaging was performed for the full one-exciton Hamiltonian, while herein the time averaging is performed on the coupling (off-diagonal) Hamiltonian in the interaction picture. As a result, the influence of the dynamic fluctuations of the transition energies is more accurately described. We compare numerical results of the two versions of the TAA with numerically exact results for the vibrational absorption line shape of the OH stretching modes in neat water. It is shown that the TAA in the interaction picture yields theoretical line shapes that are in better agreement with exact results.     

Derivation from coupled channel equations of the resonance raman scattering amplitude for a diatomic molecule

The coupled equations of molecular collision theory can be formulated in such a way as to yield the amplitude for resonance Raman scattering of light by a diatomic molecule. Two variants of the method can be adopted, characterized by different choices for the potential matrices. All molecular potentials, either analytical or numerical, are treated on the same footing. Tests of accuracy are presented for the case of harmonic potentials intersected by linear potentials, for which an exact analytical solution can be given.     

A new technique for the early detection of stiffness in coupled differential equations and application to standard Runge-Kutta algorithms

Higher-order Runge-Kutta (RK) algorithms employing local truncation error (LTE) estimates have had very limited success in solving stiff differential equations. These LTEs do not recognize stiffness until the region of instability has been crossed after which no correction is possible. A new technique has been designed, using the local stiffness function (LSF), which can detect stiffness very early before instability occurs. The LSF is a normalized dimensionless ratio which is essentially based on the product of the step size and the geometric mean of all the slopes. It is exceedingly sensitive to the onset of stiffness. Together, the LSF and the LTE form a complementary pair which can cooperate to help solve some mildly stiff equations which were previously intractable to RK algorithms alone. Examples are given of implementation and LSF performance. Received: 18 September 1997 / Accepted: 13 February 1998 / Published online: 17 June 1998     

Corrections to the impulse approximation for use in electron compton scattering experiments

The breakdown of the impulse approximation for K-shell electrons in electron Compton scattering measurements on solids is parameterized in a form which makes it possible to assess the importance of this effect and where necessary to apply a suitable correction.     

A new method for solving the quantum hydrodynamic equations of motion: application to two-dimensional reactive scattering

The de Broglie-Bohm hydrodynamic equations of motion are solved using a meshless method based on a moving least squares approach and an arbitrary Lagrangian-Eulerian frame of reference. A regridding algorithm adds and deletes computational points as needed in order to maintain a uniform interparticle spacing, and unitary time evolution is obtained by propagating the wave packet using averaged fields. The numerical instabilities associated with the formation of nodes in the reflected portion of the wave packet are avoided by adding artificial viscosity to the equations of motion. The methodology is applied to a two-dimensional model collinear reaction with an activation barrier. Reaction probabilities are computed as a function of both time and energy, and are in excellent agreement with those based on the quantum trajectory method.     

An equation-free probabilistic steady-state approximation: dynamic application to the stochastic simulation of biochemical reaction networks

Stochastic chemical kinetics more accurately describes the dynamics of "small" chemical systems, such as biological cells. Many real systems contain dynamical stiffness, which causes the exact stochastic simulation algorithm or other kinetic Monte Carlo methods to spend the majority of their time executing frequently occurring reaction events. Previous methods have successfully applied a type of probabilistic steady-state approximation by deriving an evolution equation, such as the chemical master equation, for the relaxed fast dynamics and using the solution of that equation to determine the slow dynamics. However, because the solution of the chemical master equation is limited to small, carefully selected, or linear reaction networks, an alternate equation-free method would be highly useful. We present a probabilistic steady-state approximation that separates the time scales of an arbitrary reaction network, detects the convergence of a marginal distribution to a quasi-steady-state, directly samples the underlying distribution, and uses those samples to accurately predict the state of the system, including the effects of the slow dynamics, at future times. The numerical method produces an accurate solution of both the fast and slow reaction dynamics while, for stiff systems, reducing the computational time by orders of magnitude. The developed theory makes no approximations on the shape or form of the underlying steady-state distribution and only assumes that it is ergodic. We demonstrate the accuracy and efficiency of the method using multiple interesting examples, including a highly nonlinear protein-protein interaction network. The developed theory may be applied to any type of kinetic Monte Carlo simulation to more efficiently simulate dynamically stiff systems, including existing exact, approximate, or hybrid stochastic simulation techniques.     

The capture of inner-shell electrons in the strong-potential born (SPB) approximation

For the first time the capture of inner-shell electrons by protons is calculated by the SPB method without further approximation and using Hartree-Slater wavefunctions. The resulting cross-section has roughly the correct shape, but is much too large. The problem is traced to a loss of normalization caused by the SPB approximation. A suitably renormalized SPB gives close agreement with experiment.     

On the quantum momentum method for the exact solution of separale multiple well bound state and scattering problems

A simple method is suggested for the convenient application of Milne's method of solution of the one-dimensional Schrödinger equation to multiple well potentials. The method involves the introduction of “quantum momentum” functions which resemble the classical momentum over each “well”. The wavefunction is matched at intermediate point between the “wells”, and the quantization requirements are determined. The double oscillator and scattering by spherically symmetric potentials are cited as applications.     

Shift equations iteration solution to <Emphasis Type=Italic>n</Emphasis>-level close coupled equations,and the two-level nonadiabatic tunneling problem revisited

The shift equations iteration (SEI) solves the n-level quantum scattering problem in one dimension, i.e., the close-coupled equations, free from exponential instability arising from closed channels. SEI provides exponential-instability-free transmission and reflection coefficients, and is well suited to two-sided scattering problems such as conduction in molecular wires. Our most efficient implementation of SEI utilizes an adaptation of the log-derivative symplectic integrator described by Manolopoulos and Gray in (J Chem Phys 102:9214, 1995). The two-level nonadiabatic tunneling system is investigated—in the tunneling regime, above the barrier, and at resonance. Nonadiabatic components in the upper channel wavefunction (and lower channel wavefunction at resonance energies) are found to be non-adiabatic, i.e., not describable by WKB functions. Their behavior is characterized in terms of an empirical model relating these components to adiabatic components in the lower (upper) channel and the potential energy coupling.