Electron Translation Factors Research Articles

Practical phase-space electronic Hamiltonians for abinitio dynamics.

Modern electronic structure theory is built around the Born-Oppenheimer approximation and the construction of an electronic Hamiltonian Ĥel(X) that depends on the nuclear position X (and not the nuclear momentum P). In this article, using the well-known theory of electron translation (Γ') and rotational (Γ″) factors to couple electronic transitions to nuclear motion, we construct a practical phase-space electronic Hamiltonian that depends on both nuclear position and momentum, ĤPS(X,P). While classical Born-Oppenheimer dynamics that run along the eigensurfaces of the operator Ĥel(X) can recover many nuclear properties correctly, we present some evidence that motion along the eigensurfaces of ĤPS(X,P) can better capture both nuclear and electronic properties (including the elusive electronic momentum studied by Nafie). Moreover, only the latter (as opposed to the former) conserves the total linear and angular momentum in general.

Surface hopping, electron translation factors, electron rotation factors, momentum conservation, and size consistency.

For a system without spin-orbit coupling, the (i) nuclear plus electronic linear momentum and (ii) nuclear plus orbital electronic angular momentum are good quantum numbers. Thus, when a molecular system undergoes a nonadiabatic transition, there should be no change in the total linear or angular momentum. Now, the standard surface hopping algorithm ignores the electronic momentum and indirectly equates the momentum of the nuclear degrees of freedom to the total momentum. However, even with this simplification, the algorithm still does not conserve either the nuclear linear or the nuclear angular momenta. Here, we show that one way to address these failures is to dress the derivative couplings (i.e., the hopping directions) in two ways: (i) we disallow changes in the nuclear linear momentum by working in a translating basis (which is well known and leads to electron translation factors) and (ii) we disallow changes in the nuclear angular momentum by working in a basis that rotates around the center of mass [which is not well-known and leads to a novel, rotationally removable component of the derivative coupling that we will call electron rotation factors below, cf. Eq. (96)]. The present findings should be helpful in the short term as far as interpreting surface hopping calculations for singlet systems (without spin) and then developing the new surface hopping algorithm in the long term for systems where one cannot ignore the electronic orbital and/or spin angular momentum.

Ab Initio Study of Single- and Double-Electron Capture Processes in Collisions of He2+ Ions and Ne AtomsSupported by the National Natural Science Foundation of China (Grant Nos. 11774344 and 11474033), and the National Key Research and Development Program of China (Grant No. 2017YFA0402300).

The single- and double-electron capture (SEC, DEC) processes of He2+ ions colliding with Ne atoms are studied by utilizing the full quantum-mechanical molecular-orbital close-coupling method. Total and state-selective SEC and DEC cross sections are presented in the energy region of 2 eV/u to 20 keV/u. Results show that the dominant reaction channel is Ne+(2s2p 6 2 S) + He+(1s) in the considered energy region due to strong couplings with the initial state Ne(2s 22p 6 1 S) + He2+ around the internuclear distance of 4.6 a.u. In our calculations, the SEC cross sections decrease initially and then increase whereby, the minimum point is around 0.38 keV/u with the increase of collision energies. After considering the effects of the electron translation factor (ETF), the SEC cross sections are increased by 15%–25% nearby the energy region of keV/u and agree better with the available results. The DEC cross sections are smaller than those of SEC because of the larger energy gaps and no strong couplings with the initial state. Due to the Demkov-type couplings between DEC channel Ne2+(2s22p 4 1 S) + He(1s 2) and the dominating SEC channel Ne+(2s2p 6 2 S) + He+(1s), the DEC cross sections increase with increasing impact energies. Good consistency can also be found between the present DEC and the experimental measurements in the overlapping energy region.

Ground-to-excited derivative couplings for the density functional-based tight-binding method: semi-local and long-range corrected formulations

A derivation of non-adiabatic coupling vectors for the density functional-based tight-binding method (DFTB) between ground and excited states is presented. The analytical result is valid both for semi-local and long-range corrected DFTB and includes all required Pulay terms. Electron-translation factors lead to a conceptual simplification of the Slater-Koster scheme for precomputed integrals. Compared to scalar couplings obtained from numerical derivatives, the present approach is computationally more efficient and can be applied to systems with hundreds of atoms. The accuracy of DFTB derivative couplings is assessed by comparison to full density functional theory (DFT) calculations using semi-local and hybrid exchange-correlation functionals with promising results. As exemplified by a case study of furan, DFTB provides non-adiabatic coupling vectors that are close to DFT counterparts in size and direction also in the vicinity of conical intersections.

Single- and Double-Electron Capture Processes in Low-Energy Collisions of N4+ Ions with He**Supported by the National Natural Science Foundation of China under Grant Nos. 11774344, 11474033 and 11574326, and the National Key Research and Development Program of China under Grant No. 2017YFA0402300.

We investigate the electron capture processes of N4+(1s22s) colliding with He(1s2) in the energy range of 10–1700 eV/amu using the quantum-mechanical molecular-orbital close-coupling (QMOCC) method. Total and state-selective single-electron capture and double-electron capture (SEC and DEC) cross sections are obtained and compared with other available studies. The results agree better with the experimental data in both trend and magnitude when the electron translation factor (ETF) effects are included. Our results indicate that both the SEC and DEC processes play important roles in the considered energy region. For the SEC processes, the N3+(1s22p2) + He+(1s) states are the dominant capture states, and the N2+(1s22s2p2) + He2+ states are the main DEC states.

The influence of pseudo states on the single-electron capture processes in low-energy collisions of N5+ with He

The nonradiative single-electron capture (SEC) processes for N5+(1s2) colliding with He atoms are investigated by using the quantum-mechanical molecular-orbital close-coupling method. Total and state-selective SEC cross sections are obtained in the energy range of 3–1700 eV amu−1, for which the ab initio potential curves and nonadiabatic coupling matrix elements (involving physical and pseudo states) are computed by using the multireference single- and double-excitation configuration-interaction method. It is found that the pseudo states are also important to obtain the convergent results, especially in the low-energy region. Moreover, our results indicate that the n = 3 states are the dominant states in the considered energy region, and that the electron translation factor effects become important when the energy is above 1 keV amu−1. In addition, it is observed that the total and state-selective cross sections for SEC are in general agreement with all available experimental data.

Time-dependent spin-density-functional-theory description of He+ -He collisions

Theoretical total cross-section results for all ionization and capture processes in the ${\mathrm{He}}^{+}$-He collision system are presented in the approximate impact energy range of 10--1000 keV/amu. Calculations were performed within the framework of time-dependent spin-density functional theory. The Krieger-Li-Iafrate approximation was used to determine an accurate exchange-correlation potential in the exchange-only limit. The results of two models, one where electron translation factors in the orbitals used to calculate the potential are ignored and another where partial electron translation factors are included, are compared with available experimental data as well as a selection of previous theoretical calculations.

Spin-resolved state-selective electron capture in C5+-H collisions

The electron capture processes in the C5+(1s)+H(1s) collision system are investigated by the quantum-mechanical molecular orbital close-coupling (QMOCC) method in the energy range of 10−5–10 keV u−1. Accurate molecular structure calculations are performed by the ab initio multireference single- and double-excitation configuration interaction method. The electron translational effects are included in the calculations. The total and spin-resolved state-selective cross sections are presented and compared with the available experimental and theoretical data. The present results have a good agreement with the experimental measurements. Our calculations show that the electron translation factors play a very important role for energies above 0.1 keV u−1 leading to significant differences between the present and the previous QMOOC cross section results of Nolte et al 2012 (J. Phys. B: At. Mol. Opt. Phys. 45 245202). The effects of the core electron also cannot be ignored below 2 keV u−1. Model potential calculations, in which the core electron is treated as frozen, cannot give accurate spin-resolved cross sections.

Charge exchange collisions of slow C6+ with atomic and molecular H

Charge exchange in collisions of C6+ ions with H and H2 is investigated theoretically at projectile energies 0.1 < E < 10 keV/amu, using electron nuclear dynamics (END) − a semi-classical approximation which not only includes electron translation factors for avoiding spurious couplings but also employs full dynamical trajectories to treat nuclear motions. Both the total and partial cross sections are reported for the collision of C6+ ions with atomic and molecular hydrogen. A comparison with other theoretical and experimental results shows, in general good agreement except at very low energy, considered here. For H2, the one- and two-electron charge exchange cross sections are calculated and compared with other theoretical and experimental results. Small but non-negligible isotope effects are found at the lowest energy studied in the charge transfer of C6+ with H. In low energy region, it is observed that H2 has larger isotope effects than H atom due to the polarizability effect which is larger than the mass effect.

Derivative Couplings with Built-In Electron-Translation Factors: Application to Benzene

Derivative couplings are the essential quantities at the interface between electronic-structure calculations and nonadiabatic dynamics. Unfortunately, standard approaches for calculating these couplings usually neglect electronic motion, which can lead to spurious electronic transitions. Here we provide a general framework for correcting these anomalies by incorporating perturbative electron-translation factors (ETFs) into the atomic-orbital basis. For a range of representative organic molecules, we find that our ETF correction is often small but can be qualitatively important, especially for few-atom systems or highly symmetric molecules. Our method entails no additional computational cost, such that ETFs are “built-in,” and it is equivalent to a simple rule of thumb: We should set the antisymmetrized version of the nuclear overlap-matrix derivative to zero wherever it appears. Thus, we expect that built-in ETFs will be regularly incorporated into future studies of nonadiabatic dynamics.

Analytic derivative couplings between configuration-interaction-singles states with built-in electron-translation factors for translational invariance

We present a method for analytically calculating the derivative couplings between a pair of configuration-interaction-singles (CIS) excited states obtained in an atom-centered basis. Our theory is exact and has been derived using two completely independent approaches: one inspired by the Hellmann-Feynman theorem and the other following from direct differentiation. (The former is new, while the latter is in the spirit of existing approaches in the literature.) Our expression for the derivative couplings incorporates all Pulay effects associated with the use of an atom-centered basis, and the computational cost is minimal, roughly comparable to that of a single CIS energy gradient. We have validated our method against CIS finite-difference results and have applied it to the lowest lying excited states of naphthalene; we find that naphthalene derivative couplings include Pulay contributions sufficient to have a qualitative effect. Going beyond standard problems in analytic gradient theory, we have also constructed a correction, based on perturbative electron-translation factors, for including electronic momentum and eliminating spurious components of the derivative couplings that break translational symmetry. This correction is general and can be applied to any level of electronic structure theory.

Charge exchange in Be4+–H(n = 1, 2) collisions studied systematically by atomic-orbital close-coupling calculations

Charge exchange in Be4+–H(n = 1, 2) collisions was studied using the atomic-orbital close-coupling formalism in its impact parameter description and applying electron translational factors. Calculations were carried out in the energy range of 1–750 keV amu−1 with fully stripped beryllium ions impacting on atomic hydrogen in its ground state (1s) and in the excited n = 2 states. An optimized description of both collision centres was determined by analysing the convergence of the total cross sections and also by comparison with other theoretical approaches. This optimized description consists of up to 170 basis states involving hydrogen-like states and pseudostates. We present total, n-resolved as well as nℓ-resolved cross sections. The latter are needed to evaluate the emission cross sections in two limiting collisional plasma environments, i.e. single-collisional and multi-collisional. The calculated emission cross sections are compared with experimental data from the fusion experiment JET. Taking the experimental environment and certain characteristics of the theoretical methods into account, the agreement can be considered to be quite good.

Ionic-covalent collision interactions

Several classes of collision problems are mediated by interactions between ionic and covalent potential energy curves. Two prominent cases are ion-ion recombination, A+ + B− A + B, and electron capture in multiply charged ion, atom collisions, Aq+ + B A(q – 1)+ + B+. Professor Bates and his students provided the theoretical framework upon which many successful theories are based for the above reactions. Advances in computational methods have allowed the calculation of cross sections with accuracy unrealizable only a decade ago. Important physical effects, such as electron translation factors that were pointed out by Bates and McCarroll in 1958, are routinely incorporated in present calculations. Pseudopotential and model-potential molecular structure methods now provide the input of accurate eigenenergies and eigenfunctions for the scattering calculations. However, important problems are still unsolved. One of these problems is the molecular and collisional description of discrete states embedded in a continuum as it arises in collisions of highly charged, multiply charged ions with many electron atomic targets.

The molecular model of electronic transitions in atom-atom collisions: Successes and shortcomings

The application of the molecular model to describe inelastic electronic transitions in atom-atom collisions is reviewed. The removal of defects of the model by the introduction of electron translation factors is discussed with reference to some specinc examples. A brief survey is given of the difficulties associated with the transformation from an adiabatic to a diabatic representation for complex systems.

Quantum theory of slow atomic collisions

Quantum-mechanical and semiclassical theories of slow atomic collisions are reviewed, with attention to electron-translation factors and their effects.

He2+-He charge transfer collisions using a 27-state close-coupled calculation with a diabatic molecular basis

A comparative study of two earlier three-state and fifteen-state [Chaudhuri et al., Pramana- J. Phys., 43, 175 (1994); ibid., Phys. Rev. A, 52, 1137 (1995)] close-coupled treatments of He2+-He single and double charge transfer collisions is made in this paper with a larger, 27-state close-coupled calculation. The calculations have been done using the diabatic molecular basis set used in the earlier work extended by adding excited orbitals leading to higher excitation channels up to 3s and 3p. For such molecular basis functions that go to the correct separated-atom limits used in this work, the present results show that without the inclusion of the electron translation factors (ETFs) the quantitative cross-section calculations up to velocity ~1.1 a.u. (~30 keV/amu) are in good agreement with both experiment and other calculations. This suggests that if ETFs are properly incorporated into the charge transfer collision studies at low energies this diabatic molecular basis can be used for benchmark calculations. With the aid of the Wannier's picture of the ground state correlations, a combined detailed analysis and comparison has been carried out to find a connection with the dynamic two-electron correlation picture in charge transfer collision processes which involve sequential/simultaneous two-electron exchange. If ETFs are included, the model approach of the present work may open up opportunities to investigate dynamic two-electron correlation effect in charge transfer ion-atom collision processes with benchmark accuracy. PACS codes: 34.70.+e, 34.20.-b

Nonresonant formation ofH−near unreconstructedSi(100)surfaces

We calculate ab initio the fraction of outgoing negative hydrogen ions that are normally incident on an unreconstructed Si(100) surface with kinetic energies between 50 and 150 eV. The ground-state electronic structure of the surface is derived from a self-consistent screened Thomas-Fermi-von Weizsaecker pseudopotential including Wang-Teter shell structure corrections. Orbitals and energies of the electronic states in this potential are obtained by solving Kohn-Sham equations. The dynamics of the transfer of a single electron during the ion-surface collision is represented within the Newns-Anderson model, including image-charge interactions and electron translation factor. We show that the outgoing H{sup -} fraction evolves at large distances from the surface due to nonresonant transitions from the valence band levels of the substrate into the affinity level of H{sup -}. In particular, we show that electron capture from dangling-bond surface-state resonances determines the final negative-ion fraction. We find good qualitative agreement with the experimental results of Maazouz et al. [Surf. Sci. 398, 49 (1998)] for the scattering of hydrogen atoms and ions on silicon surfaces, even though our calculations do not include the effects of reconstruction and projectile motion parallel to the surface.

Total, Partial, and Differential Ionization Cross Sections in Proton–Hydrogen Atom Collisions in the Energy Region of 0.1–10 keV/u

Single-differential, partial, and total ionization cross sections for the proton–hydrogen atom collision system in the energy region of 0.1–10 keV/u are determined by using the molecular-orbital close-coupling method within a semiclassical formalism. The present cross sections are in an excellent agreement with the recent experiments of Shah et al. [J. Phys. B. 31, L757 (1998)], but decrease more rapidly than the cross sections measured by Pieksma et al. [Phys. Rev. Lett. 73, 46 (1994)] with decreasing energy. The numerical data for all calculated cross sections are included in this paper. A critical evaluation of the existing data for the ionization process in the keV energy range is performed both for the experiment and theory. The recommended data are obtained from a converged close-coupling expansion, which in total includes 362 bound and continuum channels with their wave functions augmented by the electron translation factor in order to insure the correct scattering boundary condition.

Hyperspherical close-coupling calculations for charge-transfer cross sections inHe2++H(1s)collisions at low energies

A theory for ion-atom collisions at low energies based on the hyperspherical close-coupling (HSCC) method is presented. In hyperspherical coordinates the wave function is expanded in analogy to the Born-Oppenheimer approximation where the adiabatic channel functions are calculated with B-spline basis functions while the coupled hyperradial equations are solved by a combination of R-matrix propagation and the slow/smooth variable discretization method. The HSCC method is applied to calculate charge-transfer cross sections for He{sup 2+}+H(1s){yields}He{sup +}(n=2)+H{sup +} reactions at center-of-mass energies from 10 eV to 4 keV. The results are shown to be in general good agreement with calculations based on the molecular orbital (MO) expansion method where electron translation factors (ETF's) or switching functions have been incorporated in each MO. However, discrepancies were found at very low energies. It is shown that the HSCC method can be used to study low-energy ion-atom collisions without the need to introduce the ad hoc ETF's, and the results are free from ambiguities associated with the traditional MO expansion approach.

Total and differential cross-section calculations for proton-impact ionization of hydrogen at low energies

We have computed the single-differential and total ionization cross sections for the proton-hydrogen collision system at low-energy range (0.1--10 keV/amu), using the electron translation factor corrected close-coupling method. Full convergence of ionization cross sections as a function of ${\mathrm{H}}_{2}^{+}$ molecular basis size was achieved by including up to ten bound states and 11 continuum partial waves. The present results are compared with the available experimental data and various theoretical models. Our calculated cross sections are in an excellent agreement with the recent experiments of Shah et al. [J. Phys. B. 31, L757 (1998)], but decrease more rapidly than the cross sections measured by Pieksma et al. [Phys. Rev. Lett. 73, 46 (1994)] with decreasing energy. We have found that the $1s{\ensuremath{\sigma}}_{g}$ electron ionization is a rather higher-level ladder climbing process than a direct mechanism; the $2p{\ensuremath{\sigma}}_{u}$ electron, on the contrary, is ionized directly and the higher levels act as a temporary trap.