The method of complex coordinate rotation and its applications to atomic collision processes

One of the most significant developments in computational atomic and molecular physics in recent years has been the introduction of B-spline basis sets in calculations of atomic and molecular structure and dynamics. B-splines were introduced in applied mathematics more than 50 years ago, but it has been in the 1990s, with the advent of powerful computers, that the number of applications has grown exponentially. In this review we present the main properties of B-splines and discuss why they are useful to solve different problems in atomic and molecular physics. We provide an extensive reference list of theoretical works that have made use of B-spline basis sets up to 2000. Among these, we have focused on those applications that have led to the discovery of new interesting phenomena and pointed out the reasons behind the success of the approach.

https://iopscience.iop.org/article/10.1088/0034-4885/64/12/205/pdf

Applications of B-splines in atomic and molecular physics

As part of a systematic study of approximations commonly made in solutions of the Boltzmann equation for electrons in molecular gases, we have investigated the effects of anisotropic scattering on electron transport coefficients in ${\mathrm{N}}_{2}$ and have extended our study of the multiterm expansion technique to higher E/n. A critical survey of published data yields a set of differential and integral cross sections for electron energies from 0.003 to ${10}^{4}$ eV. For electric-field--to--gas-density ratios E/n between 10 and 500 Td (1 Td${=10}^{\mathrm{\ensuremath{-}}21}$ V ${\mathrm{m}}^{2}$), the changes in the commonly measured transport and reaction coefficients resulting from the introduction of anisotropic elastic and inelastic scattering, while keeping the elastic momentum-transfer cross section constant, are less than 1%.These calculations were made with use of the multiterm spherical-harmonic expansion solution of the Boltzmann equation. For 500lE/nl1500 Td the changes in scattering anisotropy cause changes in transport and reaction coefficients which increase with E/n to about 10%. The errors in drift velocity, mean energy, and the reaction coefficients resulting from the use of the two-term spherical-harmonic expansion rather than a six-term expansion are less than 3% at 1500 Td.However, the errors in the diffusion coefficients become large (g25%) at our highest E/n. The calculated transport coefficients are in generally good agreement with experiment for E/n less than 300 Td, but the differences increase at higher E/n. The importance of proper interpretation of ionization and excitation experiments at high E/n is illustrated by calculations which model either an exponential growth of density in time or an exponential growth with position. The calculated ionization coefficients are low compared to most experiments for E/n less than 200 Td. At E/ng600 Td the agreement is good for the spatial growth experiments, but the calculated values are below experiment from the temporal growth experiments. The calculated excitation coefficients are generally higher than experiment at low and high E/n but in agreement with experiment at E/n near 150 Td.

Anisotropic scattering of electrons by N2 and its effect on electron transport.

The current state of the theory and its application to low-energy electron-molecule collisions is reviewed. The emphasis is on elastic scattering and vibrational and rotational excitation of small diatomic and polyatomic molecules. New and traditional theoretical approaches are described, and the results of calculations are compared with existing experimental measurements.

The theory of electron-molecule collisions

Electron–molecule collision calculations using the R-matrix method

Application of the Close-Coupling Method to Electron-Molecule Scattering.- The Coupled-Channels Integral-Equations Method in the Theory of Low-Energy Electron-Molecule Scattering.- Roundtable on Numerical Methods.- Contribution of the Variable Phase Method to the Frame Transformation Theory of Rotational Excitation of Molecules by Electron Impact.- The R-Matrix Method for Electron-Molecule Scattering: Theory and Computation.- The T-Matrix Method in Electron-Molecule Scattering.- Roundtable on L2-Methods.- The Separable Approximation in Multichannel Electron-Molecule Collisions.- Nonempirical Polarization in Low-Energy Electron-Molecule Scattering Theory.- R-Matrix Calculations for Electron Scattering by Diatomic Molecules.- Discussion on Electron-Molecule Scattering.- Polarization Potentials for Electron Scattering.- Vibrational Excitations of Low-Energy e-CO Scattering.- Improved Hybrid Theory Calculation of e-N2 Vibrational Excitation.- Stieltjes-Tchebycheff Moment-Theory Approach to Molecular Photoionization Studies.- The Continuum Multiple-Scattering Approach to Electron-Molecule Scattering and Molecular Photo ionization.- Molecular Resonance Phenomena.- Stieltjes-Moment-Theory Technique for Calculating Resonance Widths.- Progress Toward the Application of Complex Coordinte and Complex Basis Function Techniques to Molecular Resonance Calculations.- Discussion on Photoionization and Molecular Resonances.- A Modification of the Langhoff Imaging Technique.- Workshop on L2-Methods.- Workshop on Single-Center Techniques.

Electron-Molecule and Photon-Molecule Collisions

We present a rigorous technique, which should be applicable to both atoms and molecules, for calculating photoabsorption cross sections using square-integrable basis functions. The technique is based on the method of complex coordinates as developed by Nuttall and co-workers. In contrast to some other L2 methods, the calculations converge directly at real energies. The method is illustrated by application to the case of atomic hydrogen.

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Rigorous method for computing photoabsorption cross sections from a basis-set expansion

We present the results of ab initio calculations for elasticelectron scattering by tetrahydrofuran (THF) using the complex Kohnvariational method. We carried out fixed-nuclei calculations at theequilibrium geometry of the target molecule for incident electronenergies up to 20 eV. The calculated momentum transfer cross sectionsclearly reveal the presence of broad shape resonance behavior in the 8-10eV energy range, in agreement with recent experiments. The calculateddifferential cross sections at 20 eV, which include the effects of thelong-range electron-dipole interaction, are alsofound to be in agreementwith the most recent experimental findings.

Elastic Scattering of Low-Energy Electrons byTetrahydrofuran

We present cross sections for the excitation of the two lowest (b3Σu+ and a3Σg+) triplet states of molecular hydrogen by electron impact for incident electron energies ≤ 20 eV. The cross sections are calculated using the distorted-wave approximation with the inelastic transition density obtained in the random-phase approximation. An efficient expansion technique using Gaussian basis functions allows us to avoid numerical integrations and to treat accurately the noncentral nature of the scattering process with full allowance for electron exchange. The sum of the two triplet cross sections is found to be in good agreement with the experimental cross section for dissociation of H2 into 2H.

See Also:

Erratum: T. N. Rescigno, C. W. McCurdy, V. McKoy, and C. F. Bender, Erratum: Low-energy electron-impact excitation of the hydrogen molecule, Phys. Rev. A 15, 2569 (1977).

/pdf/low-energy-electron-impact-excitation-of-the-hydrogen-xvzd52viqx.pdf

Low-energy electron-impact excitation of the hydrogen molecule

A method is described for computing scattering amplitudes for a cylindrically symmetric potential using square-integrable basis functions. The addition of variational corrections using the Kato formula is also outlined. Results for a two-center Gaussian potential are presented.

/pdf/discrete-basis-set-approach-to-nonspherical-scattering-ii-3mx639yx64.pdf

Thomas N. Rescigno

Papers

Electron-Molecule and Photon-Molecule Collisions

Rigorous method for computing photoabsorption cross sections from a basis-set expansion

Elastic Scattering of Low-Energy Electrons byTetrahydrofuran

Low-energy electron-impact excitation of the hydrogen molecule

Discrete-basis-set approach to nonspherical scattering. II.