Static-exchange Approximation Research Articles

Elastic electron scattering by water molecules

Elastic differential and momentum transfer cross sections are reported for electron scattering by H2O at impact energies ranging from 4 to 50 eV. The iterative Schwinger variational method in the fixed-nuclei, static-exchange approximation is used to calculate the low partial wave scattering amplitudes and the higher partial wave contributions are included via closure using the Born approximation for a point-dipole. Comparison of our calculated cross sections with recent experimental and other theoretical results is encouraging.

Calculation of the differential two-photon ionization cross section of H2S

Differential cross section of nonresonant two-photon ionization of the outer valence shell of H2S is computed using ab initio methods for several choices of photon polarization and experimental setup. The theoretical model employed is based on the second-order perturbation theory in the dipole and vertical transition approximations and the calculations are performed adopting a large basis set of L2 integrable functions. The excitations to intermediate states are described in the random phase approximation, while the final states are described in the static-exchange approximation adopting for the continuum orbitals a K-matrix technique that allows a proper description of the angular distribution of the emitted electrons.

Calculation of low-energy elastic cross sections for electron-CF4 scattering

A new computational approach has been used to evaluate the rotationally summed, vibronically elastic integral cross sections from the scattering of slow electrons (energy ranging from 1.0 eV up to 40.0 eV) by tetrafluoromethane molecules in the gas phase. The various symmetry components have been analyzed using the exact static exchange approximation and also by including a nonempirical, model polarization potential employed before by our group. A comparison with earlier calculations and with existing experiments allows us to assign the symmetries of the shape resonances in the 5–30 eV energy region which are seen by experiments and are also shown by the present calculations.

Sturmian functions for nonlocal interactions

The problem of calculation of Sturmian functions (positive energy Weinberg states) for nonlocal (exchange) interactions is considered. It is shown that the method of continued fractions proposed by Horacek and Sasakawa makes the calculation of Sturmian eigenfunctions and eigenvalues feasible even for complicated nonlocal interactions. As an example Sturmian functions and Sturmian eigenvalues for the low energy electron-hydrogen scattering in the static exchange approximation are calculated. In addition a very general proof of convergence of the method of continued fractions is presented.

Low-energy elastic electron scattering by tetrafluoromethane (CF4)

We report cross sections for electronically elastic electron scattering by CF4 from 1 to 40 eV, calculated within the static-exchange approximation using the Schwinger multichannel method. Although the static-exchange approximation does not give results that are accurate in detail below 20 eV, it is useful in understanding resonant features in the elastic and vibrationally inelastic cross sections. Above 20 eV, where the static-exchange approximation is more reliable, we derive a dissociation cross section in fair agreement with experiment by subtracting our result from the measured total cross section. We compare our integral and differential cross sections with the results of recent elastic and vibrationally inelastic scattering experiments.

Studies of molecular Rydberg states by Schwinger variational-quantum defect methods: Application to molecular hydrogen

An ab initio electronic structure technique has been developed to study highly excited states of molecules by combining Schwinger variational methods of collision theory with generalized quantum defect theory. The technique exploits methods of scattering theory to study the region of highly excited Rydberg levels below and across ionization thresholds for molecules. The reaction matrix K, which describes the interaction of the Rydberg electron with the ionic core, is found at arbitrary negative electron energies by employing an unbounded Coulomb Green’s function in the Lippmann–Schwinger equation for the electronic wave function. Quantal conditions are imposed to obtain discrete molecular energy levels, associated Rydberg wave functions, and quantum defect functions, all as a function of the internuclear distance. Results within the static-exchange approximation for the 1,3Σ+u(1σgnσu) and 1,3Πu(1σgnπu) Rydberg states of H2, for n=2–20 and R=1.2–5.0 a0, are presented and discussed.

Low-energy electron scattering by C3H6 isomers

We report cross sections for the elastic scattering of electrons by the C3H6 isomers propene and cyclopropane from 5 to 40 eV. These results were calculated within the static-exchange approximation, using the Schwinger multichannel method as implemented on the Touchstone DELTA distributed-memory parallel computer. The integral elastic cross sections so obtained support recent observations of a substantial isomer effect in the total scattering cross section. Partial-wave analysis indicates the presence of an unusual A′2 shape resonance in cyclopropane.

Spurious singularities in the generalized Newton variational method.

The generalized Newton variational method is applied to the static-exchange approximation of the electron--hydrogen-atom scattering. Slater-type basis functions are employed to expand the amplitude density. Spurious singularities are encountered in both scattering processes. The width of the unphysical singularities is broader in the case of singlet scattering. Anomalous poles appear in narrow regions of the scale parameter and are in evident correlation with the zeros of the determinant of the free-particle Green's operator. As a by-product, simple least-squares extension of the generalized Newton variational method is developed in order to avoid spurious singularities and to recognize whether or not the convergence is of secondary nature.

Feshbach–Fano formalism in Hilbert space: Application to shape resonances in molecular photoionization

An explicit Hilbert-space formulation of Feshbach–Fano theory is described which is particularly well suited for treating the problem of shape resonances in molecular photoionization. The prepared states of Fano and Cooper are employed to resolve the irreducibly infinite degeneracy of molecular electronic continua that arises from the noncentral nature of molecular (body-frame) potentials. Previously defined L2 Stieltjes states generated employing appropriate test functions are shown to converge to these prepared states, avoiding prior construction of the associated degenerate channel functions. A novel choice of zeroth-order state particularly suitable for applications of the L2 Feshbach–Fano method to molecular shape resonances is introduced and shown to avoid many of the difficulties associated with more conventional treatments employing resonance scattering theory. Specifically, the energy of this zeroth-order state is seen to be a weighted average over the correct photoionization cross section, the associated background cross section is shown to vanish for all energies, and calculations of the associated ρ(E) and q(E) parameters are avoided entirely. Computational applications in the static-exchange approximation to selected valence-shell channels in N2, O2, and N2O illustrate the attributes of the method and provide accurate cross sections in each case. The L2 Feshbach–Fano formalism is seen to provide improved results from precisely the same information—pseudostate energies and oscillator strengths—as is required in the conventional L2 Stieltjes development, with no significant increase in computational effort required.

Studies of electron collisions with polyatomic molecules using distributed-memory parallel computers

Elastic electron scattering cross sections from 5–30 eV are reported for the molecules C2H4, C2H6, C3H8, Si2H6, and GeH4, obtained using an implementation of the Schwinger multichannel method for distributed-memory parallel computer architectures. These results, obtained within the static-exchange approximation, are in generally good agreement with the available experimental data. These calculations demonstrate the potential of highly parallel computation in the study of collisions between low-energy electrons and polyatomic gases. The computational methodology discussed is also directly applicable to the calculation of elastic cross sections at higher levels of approximation (target polarization) and of electronic excitation cross sections.

Fully L2 methods for multichannel scattering problems. Partial widths

We analyze the use of L2 discretization methods to calculate, in the framework of the Feshbach theory, partial and total widths of resonances lying in multichannel continua. When interchannel coupling is strong, partial widths cannot be obtained without solving the scattering equations and for this purpose a fully L2 method is proposed, starting from the static exchange approximation as a zeroth order result. For weak interchannel coupling, we study the validity of Born-type approximations and we justify why neglecting this coupling may be appropriate to evaluate total widths even for cases where it is totally inadequate for the corresponding partial ones. The methods proposed are illustrated for moderately strong (He) and weak (O6+) coupling in atomic resonances.

Theoretical studies of photoionization in C 2H 2, C 2H 4, and C 2H 6: correlation of prominent spectra features with CC and CH antibonding orbitals

Studies are reported of outer-valence-, inner-valence-, and K-shell photoionization cross sections in C 2H 2, C 2H 4, and C 2H 6 employing 2 methods particularly appropriate for descriptions of resonant partial channels. Calculations in the static-exchange approximation are found to give results in good agreement with corresponding measured values. Prominent spectral features in the cross sections correlate well with the positions and strengths of Mulliken N → V σ charge-transfer excitations involving CC and CH antibonding (σ*) virtual valence molecular orbitals. Accordingly, such excitations are seen to account for the origins and natures of the resonance features in these compounds, clarifying their spectral variations in terms of CC bond distances and the numbers of CH bonds. The presence of both CC and CH antibonding (σ * orbitals makes interpretation of the photoionizat cross sections of C 2H 2, C 2H 4, and C 2H 6 particularly interesting, complicating somewhat the 1/ l 2 variation in the spectral positions of resonance features with effective bond lengths that is common in linear compounds.

Low-energy electron scattering by silane (SiH4).

We have applied the Schwinger multichannel formulation [K. Takatsuka and V. McKoy, Phys. Rev. A 24, 2473 (1981)] to the elastic scattering of low-energy (1–30-eV) electrons by silane. Our results, obtained within the fixed-nuclei, static-exchange approximation, are in generally good agreement with recent differential, integral, and momentum-transfer cross-section measurements between 5 and 30 eV. At lower energies, our results reflect the known limitations of the static-exchange approximation. Near the shape resonance, the calculated differential cross section shows significant enhancement at high angles, suggesting an explanation for differences between integral cross sections determined from integrated differential data and those obtained from transmission measurements.

Electron-HF scattering in the static exchange approximation: treatment of the dipole potential

Low-energy electron scattering off an HF molecular target is investigated in the static exchange approximation. The scattering by the static potential is treated numerically exactly whereas a separable approximation is adopted for the exchange interaction. A perturbative correction of the asymptotic part of the wavefunction is introduced in order to accelerate the convergence with respect to the cut-off radius of the numerical integration of the Schrodinger equation. Finally the dependence of the separable approximation for the exchange potential on the size of the basis set is investigated.

Applications of the Schwinger multichannel method to electron-molecule collisions.

We discuss some recent developments in the implementation of the Schwinger multichannel method for electron-molecule collision calculations. The evaluation of matrix elements involving the operator ${\mathrm{VG}}_{P}^{(+)}$V, previously accomplished by insertion of a Gaussian basis on either side of ${G}_{P}^{(+)}$, is now done by direct numerical quadrature. This approach avoids the necessity of very large Gaussian basis sets, allowing the size of the basis to reflect only the dynamical requirements of the scattering wave function. We find that the reduction in the required basis size results in improved efficiency, in spite of the additional numerical effort of performing the quadrature. Trial applications to electron-${\mathrm{CH}}_{4}$ scattering in the static-exchange approximation and to electronic excitation of ${\mathrm{H}}_{2}$ illustrate the excellent convergence characteristics of the procedure.

Low-energy scattering of electrons from SiH4 molecules

The calculations are performed in the static-exchange formalism using a spherical approximate target wavefunction. The exchange interaction is treated in three different ways: exactly within the spherical model (EE), by means of a modified semiclassical exchange (MSCE) potential and by combining the MSCE potential with an orthogonalisation procedure (OMSCE). A parameter-free model correlation-polarisation potential is included to represent the interactions not considered in the pure static-exchange approximation. Elastic total and momentum transfer cross sections are reported for e-SiH4 scattering in the energy range 0.05-20 eV. The adequacy of the MSCE potential and the effects of the orthogonalisation on the final results are analysed by comparing the results obtained by the three methods. The error induced by the spherical approximation is estimated by comparing the present OMSCE results with those obtained without the sphericalisation.

Perturbative-polarization-propagator study of the photoionization cross section of the water molecule.

Perturbative-polarization-propagator methods of different orders employing large Gaussian basis sets are used to calculate discrete pseudostate representations of the oscillator strength distribution of the water molecule, from which continuous total and partial photoionization cross sections are then determined by moment-theoretical methods. The cross sections obtained in the first-order and second-order polarization-propagator approximation are in very good agreement with recent photoionization measurements. In particular, the present results agree much better with experiment than previous calculations that use the static-exchange approximation. The main reason for this is that first- and higher-order polarization-propagator approximations obey certain sum rules for the oscillator strength distribution, whereas the static-exchange approximation does not. This makes propagator methods particularly well suited for use in connection with moment theory. Disagreement of previously reported cross sections of the water molecule with experiment, such as the overestimation of the 1${b}_{2}$ partial cross section, have been partially attributed to the neglect of channel coupling in these calculations. Therefore, in the present work, the separated-channel approximation is avoided and all possible couplings between ionizations from all valence-shell molecular orbitals are allowed for.

Studies of elastic e-NH3 collisions.

We report differential and momentum-transfer cross sections for elastic scattering of electrons by ${\mathrm{NH}}_{3}$ for collision energies from 2.5 to 20 eV. These cross sections were obtained in the fixed-nuclei static-exchange approximation using the Schwinger variational principle. Although fixed-nuclei cross sections for electron--polar-molecule collisions are inherently divergent at small angles, such results can provide useful estimates of the differential cross sections at intermediate and larger angles. The agreement at intermediate and larger scattering angles between the calculated and relative experimental cross sections, which are available at 8.5 and 15 eV, is good. The differential cross sections also show evidence of a weak d-wave enhancement around 8 eV. A significant feature seen in these cross sections, particularly at 15 and 20 eV, is their substantial backward peaking beyond 120\ifmmode^\circ\else\textdegree\fi{}.

Two-photon ionization calculations. Results for H 2O

The two-photon first ionization cross section of H 2O was computed in the vertical transition approximation. The K-matrix method was employed, adopting a basis set of L 2 functions and an interpolation procedure, to obtain the final continuous orbitals needed to calculate the two-photon oscillator strength density. The results for the integral two-photon ionization cross section refer to the case of a single photon beam either linearly or circularly polarized. The static-exchange approximation was used to describe both intermediate and final states.

Two-photon ionization cross section calculation for H2O

The two-photon first ionization cross section was computed for H2O in the vertical transition approximation. A Stieltjes imaging technique was used to obtain from the results of a L2 basis set calculation, the two-photon oscillator strength density in the continuum. The calculations were performed in the independent particle static-exchange approximation for the final states. The ground-intermediate state transition matrix elements and excitation energies were computed by adopting the single channel static-exchange approximation or the multichannel random phase approximation; in the last case the intermediate-final state transition matrix elements were obtained by the equation of motion method. The calculated cross sections refer to the case of single beam of linearly or circularly polarized photons.