Fixed-nuclei Approximation Research Articles

Positronium collisions with polar molecules

We calculate elastic and positronium (Ps) break-up cross sections for collisions of Ps with the polar molecules CO, HCl, and LiF in the fixed-nuclei approximation. We incorporate electron exchange and correlation for these processes by using the free-electron-gas model developed earlier for Ps scattering by rare-gas atoms, ${\mathrm{N}}_{2}$, ${\mathrm{O}}_{2}$, and ${\mathrm{CO}}_{2}$ molecules. The present target molecules provide a range of dipole moments from the weakly polar CO to the strongly polar LiF. We find that Ps scattering is similar to electron scattering when the cross sections are plotted as a function of projectile velocity for the targets with smaller dipole moments (CO, HCl). However, we do not see such a similarity for LiF which has a large dipole moment. Below the Ps break-up threshold we observe resonance structures similar to those obtained earlier for the other molecular targets that we have studied.

Non-Born-Oppenheimer confined variational calculation of low-energy Ps- H2 scattering

Using the confined variational method, the low-energy $S$-wave elastic scattering of positronium from molecular hydrogen is studied nonadiabatically at incident energy up to 0.13 eV. Accurate $S$-wave phase shifts, scattering lengths, and cross sections are calculated and compared with other theoretical and experimental values. Using the modified effective-range formula, the $S$-wave scattering length is determined to be 1.97 Bohr radii, which is about $2.5%$ smaller than that of the confined variational calculation under the fixed-nucleus approximation. The consistency of the present cross sections with those from the angular correlation of annihilation radiation experiment indicates that the results of Doppler broadening spectroscopy might be problematic. We also discuss the distortion effects of positronium and molecular hydrogen.

Elastic and Inelastic Cross Sections for Low-Energy Electron Collisions with ClF Molecule Using the R-Matrix Method

The ClF molecule belongs to an interhalogen family and is important in laser physics and condensed phase molecular dynamics. The elastic and excitation scattering cross sections are obtained in a fixed nuclei approximation using the UKRmol+ codes based on R-matrix formalism. The scattering calculations were performed in the static-exchange (SE), static-exchange-plus-polarisation (SEP), and close-coupling (CC) models. Three CC models with different target states were employed, namely, the 1-state, 5-states, and 12-states. In the CC model, the target states were represented by configuration interaction (CI) wavefunctions. A good agreement of dipole and quadrupole moments of the ground state was obtained with the experimental values, which indicates a good representation of the target modelling. The study predicted the existence of a shape resonance in the SE, SEP, and 5-states CC models. This resonance vanished in the 12-states CC model. The excitation cross sections from ground to the lowest two excited states were also reported. The elastic differential and momentum transfer cross sections were obtained in the 12-states CC models. The contribution of long-range interactions to elastic scattering was included via Born closure approach. The quantities like collision frequencies and rate coefficients were also presented over a wide range of electron temperatures. The ionization cross sections were computed using the binary-encounter-Bethe (BEB) model. The results were reported in C2v point group representation.

Search for long-lasting electronic coherence using on-the-fly ab initio semiclassical dynamics.

Using a combination of high-level ab initio electronic structure methods with efficient on-the-fly semiclassical evaluation of nuclear dynamics, we performed a massive scan of small polyatomic molecules searching for a long-lasting oscillatory dynamics of the electron density triggered by the outer-valence ionization. We observed that in most of the studied molecules, either the sudden removal of an electron from the system does not lead to the appearance of the electronic coherence or the created coherences become damped by the nuclear rearrangement on a time scale of a few femtoseconds. However, we report several so far unexplored molecules with the electronic coherences lasting up to 10fs, which can be good candidates for experimental studies. In addition, we present the full-dimensional simulations of the electronic coherences coupled to nuclear motion in several molecules which were studied previously only in the fixed nuclei approximation.

Gaussian-Type Orbital Calculations for High Harmonic Generation in Vibrating Molecules: Benchmarks for H2.

The response of the hydrogen molecular ion, H2+, to few-cycle laser pulses of different intensities is simulated. To treat the coupled electron-nuclear motion, we use adiabatic potentials computed with Gaussian-type basis sets together with a heuristic ionization model for the electron and a grid representation for the nuclei. Using this mixed-basis approach, the time-dependent Schrödinger equation is solved, either within the Born-Oppenheimer approximation or with nonadiabatic couplings included. The dipole response spectra are compared to all-grid-based solutions for the three-body problem, which we take as a reference to benchmark the Gaussian-type basis set approaches. Also, calculations employing the fixed-nuclei approximation are performed, to quantify effects due to nuclear motion. For low intensities and small ionization probabilities, we get excellent agreement of the dynamics using Gaussian-type basis sets with the all-grid solutions. Our investigations suggest that high harmonic generation (HHG) and high-frequency response, in general, can be reliably modeled using Gaussian-type basis sets for the electrons for not too high harmonics. Further, nuclear motion destroys electronic coherences in the response spectra even on the time scale of about 30 fs and affects HHG intensities, which reflect the electron dynamics occurring on the attosecond time scale. For the present system, non-Born-Oppenheimer effects are small. The Gaussian-based, nonadiabatically coupled, time-dependent multisurface approach to treat quantum electron-nuclear motion beyond the non-Born-Oppenheimer approximation can be easily extended to approximate wavefunction methods, such as time-dependent configuration interaction singles (TD-CIS), for systems where no benchmarks are available.

Time-dependent variational dynamics for nonadiabatically coupled nuclear and electronic quantum wavepackets in molecules

We propose a methodology to unify electronic and nuclear quantum wavepacket dynamics in molecular processes including nonadiabatic chemical reactions. The canonical and traditional approach in the full quantum treatment both for electrons and nuclei rests on the Born–Oppenheimer fixed nuclei strategy, the total wavefunction of which is described in terms of the Born–Huang expansion. This approach is already realized numerically but only for small molecules with several number of coupled electronic states for extremely hard technical reasons. Besides, the stationary-state view of the relevant electronic states based on the Born–Oppenheimer approximation is not always realistic in tracking real-time electron dynamics in attosecond scale. We therefore incorporate nuclear wavepacket dynamics into the scheme of nonadiabatic electron wavepacket theory, which we have been studying for a long time. In this scheme thus far, electron wavepackets are quantum mechanically propagated in time along nuclear paths that can naturally bifurcate due to nonadiabatic interactions. The nuclear paths are in turn generated simultaneously by the so-called matrix force given by the electronic states involved, the off-diagonal elements of which represent the force arising from nonadiabatic interactions. Here we advance so that the nuclear wavepackets are directly taken into account in place of path (trajectory) approximation. The nuclear wavefunctions are represented in terms of the Cartesian Gaussians multiplied by plane waves, which allows for feasible calculations of atomic and molecular integrals together with the electronic counterparts in a unified manner. The Schrödinger dynamics of the simultaneous electronic and nuclear wavepackets are to be integrated by means of the dual least action principle of quantum mechanics [K. Takatsuka, J. Phys. Commun. 4, 035007 (2020)], which is a time-dependent variational principle. Great contributions of Vincent McKoy in the electron dynamics in the fixed nuclei approximation and development in time-resolved photoelectron spectroscopy are briefly outlined as a guide to the present work.

All-order explicitly correlated relativistic computations for atoms and molecules.

A variational solution procedure is reported for the many-particle no-pair Dirac-Coulomb and Dirac-Coulomb-Breit Hamiltonians aiming at a parts-per-billion (ppb) convergence of the atomic and molecular energies, described within the fixed nuclei approximation. The procedure is tested for nuclear charge numbers from Z = 1 (hydrogen) to 28 (iron). Already for the lowest Z values, a significant difference is observed from leading-order Foldy-Woythusen perturbation theory, but the observed deviations are smaller than the estimated self-energy and vacuum polarization corrections.

Temporal coherent control of resonant two-photon double ionization of the hydrogen molecule via doubly excited states

We use time-delayed, counter-rotating, circularly polarized few-cycle attosecond nonoverlapping pulses to study the temporal coherent control of the resonant process of two-photon double ionization (TPDI) of hydrogen molecule via doubly excited states for pulse propagation direction along $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathbf{k}}$ either parallel or perpendicular to the molecular axis $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathbf{R}}$. For $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathbf{k}}\ensuremath{\parallel}\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathbf{R}}$ and a pulse carrier frequency of 36 eV resonantly populating the ${Q}_{2}^{1}\mathrm{\ensuremath{\Pi}}_{u}^{+}(1)$ doubly excited state as well as other $^{1}\mathrm{\ensuremath{\Pi}}_{u}^{+}$ doubly excited states, we find that the indirect ionization pathway through these doubly excited states changes the character of the kinematical vortex-shaped momentum distribution produced by the two direct ionization pathways from fourfold to twofold rotational symmetry. This result is similar to what found in TPDI of the He atom involving $^{1}P_{\ifmmode\pm\else\textpm\fi{}1}^{o}$ doubly excited states [Ngoko Djiokap and Starace, J. Opt. 19, 124003 (2017)]; however, angular distributions exhibiting a quantum beat effect between the ground state and a doubly excited state seen for the He atom are observed here for its molecular counterpart with an anomaly in shape and magnitude, not in frequency. The sixfold differential probability integrated over the azimuthal angle of the photoelectron pair shows that this anomaly is due to autoionization decays and quantum beats between doubly excited states. For $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathbf{k}}\ensuremath{\perp}\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathbf{R}}$ and a broadband pulse carrier frequency of 30 eV populating the ${Q}_{1}^{1}\mathrm{\ensuremath{\Pi}}_{u}^{+}(1)$, ${Q}_{1}^{1}\mathrm{\ensuremath{\Sigma}}_{u}^{+}(1)$, ${Q}_{2}^{1}\mathrm{\ensuremath{\Pi}}_{u}^{+}(1)$, and ${Q}_{1}^{1}\mathrm{\ensuremath{\Sigma}}_{u}^{+}(2)$ doubly excited states, the momentum distribution is shown to exhibit dynamical electron vortices with four spiral arms, which originates from the interplay between the $^{1}\mathrm{\ensuremath{\Delta}}_{g}^{+}$, $^{1}\mathrm{\ensuremath{\Pi}}_{g}^{+}$, and $^{1}\mathrm{\ensuremath{\Sigma}}_{g}^{+}$ dynamical ionization amplitudes. Our treatment within either the adiabatic-nuclei approximation or fixed-nuclei approximation shows that the latter provides a very good account for this correlated process.

Convergent close-coupling calculations of electrons scattering on electronically excited molecular hydrogen

We use the adiabatic-nuclei molecular convergent close-coupling method to perform calculations of 0.01--1000 eV electrons scattering on the $c^{3}\mathrm{\ensuremath{\Pi}}_{u}$, $a^{3}\mathrm{\ensuremath{\Sigma}}_{g}^{+}$, $B^{1}\mathrm{\ensuremath{\Sigma}}_{u}^{+}$, $C^{1}\mathrm{\ensuremath{\Pi}}_{u}$, and $EF^{1}\mathrm{\ensuremath{\Sigma}}_{g}^{+}$ states of ${\mathrm{H}}_{2}$ in the $v=0$ vibrational level. Elastic, superelastic, ionization, and grand-total cross sections are presented, as well as cross sections for excitation of the $n=2$--3 singlet and triplet states of ${\mathrm{H}}_{2}$ (where $n$ is the atomic-limit principle quantum number). Comparison with available theoretical results is made. Good agreement is found with the recent $R$-matrix results [J. Phys. B 53, 245203 (2020)] for most of the exchange and dipole-forbidden transitions, but not for the dipole-allowed transitions. The sources of disagreement were found to be an unconverged partial-wave expansion and the utilization of the fixed-nuclei approximation (as opposed to adiabatic-nuclei) in the $R$-matrix calculations.

Positron scattering by the Ar2 and Xe2 dimers

Positron scattering by Ar2 and Xe2 dimers are studied for low incident energies. The Zero Range Potential approximation is used and the dimer is modeled as two independent atoms under an internuclear distance constraint. The scattering calculations are performed within the fixed nuclei and the rigid rotor formulations. For the fixed nuclei approximation, analytical expressions are obtained for the positron-dimer scattering length and for the elastic cross section. In the rigid rotor treatment, an analytical expression is obtained for the rotational excitation cross sections. The main results obtained in this investigation are that the short-range interaction between the positron and the molecular dimer is the dominant mechanism for rotational excitation in this energy range. Such effect is explained by the small values of the quadrupole moments of the Ar2 and Xe2 dimers.

Electron-scattering on molecular hydrogen: convergent close-coupling approach

Calculations of electron-impact excitation cross sections for molecular hydrogen have been performed using spherical- and spheroidal-coordinate formulations of the molecular convergent close-coupling method. We present a comparison and find good agreement between the results of these two techniques performed within the fixed-nuclei approximation for excitation from the ground X1Σg+ (v = 0) state of H2 to the B1Σu+ , C 1Πu, B′1Σu+ , D 1Πu, EF1Σg+ , b3Σu+ , c3Πu+ , a3Σu+ , e3Σu+ , h3Σg+ , and d 3Πu states. For the spheroidal-coordinate approach the adiabatic-nuclei method has been applied, allowing for a more reliable estimate of cross sections at near-threshold energies. Comparison of the adiabatic-nuclei cross sections with the corresponding fixed-nuclei cross sections is also presented.

RMT: R-matrix with time-dependence. Solving the semi-relativistic, time-dependent Schrödinger equation for general, multielectron atoms and molecules in intense, ultrashort, arbitrarily polarized laser pulses

RMT is a programme which solves the time-dependent Schrödinger equation for general, multielectron atoms, ions and molecules interacting with laser light. As such it can be used to model ionization (single-photon, multiphoton and strong-field), recollision (high-harmonic generation, strong-field rescattering) and, more generally, absorption or scattering processes with a full account of the multielectron correlation effects in a time-dependent manner. Calculations can be performed for targets interacting with ultrashort, intense laser pulses of long wavelength and arbitrary polarization. Calculations for atoms can optionally include the Breit–Pauli correction terms for the description of relativistic (in particular, spin–orbit) effects. Program summaryProgram Title: (RMT) R-matrix with time-dependenceProgram Files doi:http://dx.doi.org/10.17632/3ptyfg2bmx.1Licensing provisions: GPLv3Programming language: FortranNature of problem: The interaction of laser light with matter can be modelled with the time-dependent Schrödinger equation (TDSE). The solution of the TDSE for general, multielectron atomic and molecular systems is computationally demanding, and has previously been limited either to particular laser wavelengths and intensities, or to simple, few-electron cases. RMT overcomes this limitation by using a general approach to modelling dynamics in atoms and molecules which is applicable to multielectron systems and a wide range of perturbative and non-perturbative phenomena.Solution method: We use the R-matrix paradigm, partitioning the interaction region into an ‘inner’ and an ‘outer’ region. In the inner region (within some small radius of the nucleus/nuclei), full account is taken of all multielectron interactions including electron exchange and correlation. In the outer region, far from the nucleus/nuclei, these are neglected and a single, ionized electron moves in the long-range potential of the residual ionic system and the laser field. The key computational aspect of the RMT approach is the use of a different numerical scheme in each region, facilitating efficient parallelization without sacrificing accuracy. Given an initial wavefunction and the electric field of the driving laser pulse, the wavefunction for all subsequent times and the associated observables are computed using an explicit, Arnoldi propagator method.Additional comments including restrictions and unusual features: The description of the atomic/molecular structure is provided from other, time-independent R-matrix codes [1], [2], [3], and the capabilities (in terms of structure) are, in some sense, inherited therefrom. Thus, the atomic calculations can optionally include Breit–Pauli relativistic corrections to the Hamiltonian, in order to account for the spin–orbit effect. However, no such capability exists for the molecular case. Furthermore, the fixed-nuclei approximation is adopted in the molecular calculations (so nuclear motion is neglected). Similarly, all calculations are restricted to the description of a single electron in the outer region, and consequently the study of double-ionization phenomena is not yet within the capabilities of the method. Finally, the parallel strategy employed necessitates the use of at least two (and usually many more) computer cores. As a result, there is no option for serial calculations and, for most realistic cases, a massively parallel architecture (several hundred cores) will be required.Program repository available at: https://gitlab.com/Uk-amor/RMTReferences[1] C. P. Ballance Parallel R-matrix codes, http://connorb.freeshell.org.[2] R-matrix II codes, http://gitlab.com/uk-amor/rmt/rmatrixii.[3] Z. Mašín et al UKRmol+: a suite for modelling of electronic processes in molecules interacting with electrons, positrons and photons using the R-matrix method, Comput. Phys. Commun., accepted, http://dx.doi.org/10.1016/j.cpc.2019.107092.

Hartree-Fock implementation using a Laguerre-based wave function for the ground state and correlation energies of two-electron atoms.

An implementation of the Hartree-Fock (HF) method using a Laguerre-based wave function is described and used to accurately study the ground state of two-electron atoms in the fixed nucleus approximation, and by comparison with fully correlated (FC) energies, used to determine accurate electron correlation energies. A variational parameter A is included in the wave function and is shown to rapidly increase the convergence of the energy. The one-electron integrals are solved by series solution and an analytical form is found for the two-electron integrals. This methodology is used to produce accurate wave functions, energies and expectation values for the helium isoelectronic sequence, including at low nuclear charge just prior to electron detachment. Additionally, the critical nuclear charge for binding two electrons within the HF approach is calculated and determined to be ZHFC=1.031 177 528.This article is part of the theme issue 'Modern theoretical chemistry'.

Resonance-modulated wavelength scaling of high-order-harmonic generation from H2+

Wavelength scaling of high-order harmonic generation (HHG) in a non-Born-Oppenheimer treatment of ${\mathrm{H}}_{2}{}^{+}$ is investigated by numerical simulations of the time-dependent Schr\odinger equation. The results show that the decrease in the wavelength-dependent HHG yield is reduced compared to that in the fixed-nucleus approximation. This slower wavelength scaling is related to the charge-resonance-enhanced ionization effect, which considerably increases the ionization rate at longer driving laser wavelengths due to the relatively larger nuclear separation. In addition, we find an oscillation structure in the wavelength scaling of HHG from ${\mathrm{H}}_{2}^{+}$. Upon decreasing the laser intensity or increasing the nuclear mass, the oscillation structure will shift towards a longer wavelength of the laser pulse. These results permit the generation of an efficient harmonic spectrum in the midinfrared regime by manipulating the nuclear dynamics of molecules.

Investigation of low energy electron interaction with silane molecules

SynopsisTotal and momentum transfer cross sections are calculated for the scattering of electrons from SiH4 molecules in electron energy region of 0.1 − 20 eV. A parameter–free spherical optical potential (SOP) approach in the fixed nuclei approximation is applied. The present calculated results are compared with the available other calculations and the experimental measurements. The quantitative features of the scattering parameters (such as Ramsauer - Townsend minimum and shape resonance structure) as observed in experiments, are well reproduced in this study.

Electron-impact excitation of molecular hydrogen

We report the electron impact integrated and differential cross sections for excitation to the $b{\phantom{\rule{0.16em}{0ex}}}^{3}{\mathrm{\ensuremath{\Sigma}}}_{u}^{+}, a{\phantom{\rule{0.16em}{0ex}}}^{3}{\mathrm{\ensuremath{\Sigma}}}_{g}^{+}, c{\phantom{\rule{0.16em}{0ex}}}^{3}{\mathrm{\ensuremath{\Pi}}}_{u}, B{\phantom{\rule{0.16em}{0ex}}}^{1}{\mathrm{\ensuremath{\Sigma}}}_{u}^{+}, E,F{\phantom{\rule{0.16em}{0ex}}}^{1}{\mathrm{\ensuremath{\Sigma}}}_{g}^{+}, C{\phantom{\rule{0.16em}{0ex}}}^{1}{\mathrm{\ensuremath{\Pi}}}_{u}, e{\phantom{\rule{0.16em}{0ex}}}^{3}{\mathrm{\ensuremath{\Sigma}}}_{u}^{+}, h{\phantom{\rule{0.16em}{0ex}}}^{3}{\mathrm{\ensuremath{\Sigma}}}_{g}^{+}, d{\phantom{\rule{0.16em}{0ex}}}^{3}{\mathrm{\ensuremath{\Pi}}}_{u}, {B}^{\ensuremath{'}}{\phantom{\rule{0.16em}{0ex}}}^{1}{\mathrm{\ensuremath{\Sigma}}}_{u}^{+}, D{\phantom{\rule{0.16em}{0ex}}}^{1}{\mathrm{\ensuremath{\Pi}}}_{u}, {B}^{\ensuremath{'}}{}^{\ensuremath{'}}{\phantom{\rule{0.16em}{0ex}}}^{1}{\mathrm{\ensuremath{\Sigma}}}_{u}^{+}$, and ${D}^{\ensuremath{'}}{\phantom{\rule{0.16em}{0ex}}}^{1}{\mathrm{\ensuremath{\Pi}}}_{u}$ states of molecular hydrogen in the energy range from 10 to 300 eV. Total scattering and total ionization cross sections are also presented. The calculations have been performed by using the convergent close-coupling method within the fixed-nuclei approximation. Detailed convergence studies have been performed with respect to the size of the close-coupling expansion and a set of recommended cross sections has been produced. Significant differences with previous calculations are found. Agreement with experiment is mixed, ranging from excellent to poor depending on the transition and incident energies.

Alignment dependence of photoelectron momentum distributions of atomic and molecular targets probed by few-cycle circularly polarized laser pulses

We present theoretical photoelectron momentum distributions (PMDs) for ionization from Ar($3p$) and ${\mathrm{H}}_{2}{}^{+}$(${\ensuremath{\sigma}}_{g}$) orbitals by few-cycle, high-intensity, near-infrared laser fields circularly polarized in the $xy$ plane. The three-dimensional time-dependent Schr\odinger equation is solved numerically within the single-active-electron approximation for Ar and within the fixed nuclei approximation for ${\mathrm{H}}_{2}{}^{+}$. The PMDs are investigated for alignment of the probed target orbitals relative to the polarization plane of the laser field. In the atomic case, the PMDs in the polarization plane for aligned $3p$ Ar orbitals are, up to an overall scaling factor, insensitive to alignment of the probed orbital, while the lateral PMDs show a signature of the orbital node when that node is sufficiently close to the polarization plane. For the molecular case of ${\mathrm{H}}_{2}{}^{+}$(${\ensuremath{\sigma}}_{g}$), our results show a significant impact of alignment on the PMDs due to the anisotropic molecular potential and the alignment-dependent coupling between the ground state and excited states.

Complete Solution of Electronic Excitation and Ionization in Electron-Hydrogen Molecule Scattering.

The convergent close-coupling method has been used to solve the electron-hydrogen molecule scattering problem in the fixed-nuclei approximation. Excellent agreement with experiment is found for the grand total, elastic, electronic-excitation, and total ionization cross sections from the very low to the very high energies. This shows that for the electronic degrees of freedom the method provides a complete treatment of electron scattering on molecules as it does for atoms.

Controlling the high frequency response of H2 by ultra-short tailored laser pulses: A time-dependent configuration interaction study.

We combine the stochastic pulse optimization (SPO) scheme with the time-dependent configuration interaction singles method in order to control the high frequency response of a simple molecular model system to a tailored femtosecond laser pulse. For this purpose, we use H2 treated in the fixed nuclei approximation. The SPO scheme, as similar genetic algorithms, is especially suited to control highly non-linear processes, which we consider here in the context of high harmonic generation. Here, we will demonstrate that SPO can be used to realize a "non-harmonic" response of H2 to a laser pulse. Specifically, we will show how adding low intensity side frequencies to the dominant carrier frequency of the laser pulse and stochastically optimizing their contribution can create a high-frequency spectral signal of significant intensity, not harmonic to the carrier frequency. At the same time, it is possible to suppress the harmonic signals in the same spectral region, although the carrier frequency is kept dominant during the optimization.

Differential cross sections for the ionization of by proton impact without using the two-effective center approximation

The ionization of through the impact of fast protons is treated within the fixed nuclei approximation by using the Born and CDW-EIS models, in which accurate electronic wavefunctions are applied to the ion. The differential cross sections are compared to those calculated by an alternative CDW-EIS treatment via use of the two-effective center (TEC) approximation. Generally, the calculations with and without the TEC approximation give similar results, however, discrepancies can be observed in the domains of differential cross sections with relatively small values.