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

Abstract

The response of the hydrogen molecular ion, H 2 + , 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 Schrodinger 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 highfrequency 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.

Density Functional Theory for Electronic Excited States

“…On the other hand, within the adiabatic approximation the initial-state dependence vanishes since the XC kernel is fully specified in terms of the instantaneous time-evolving density. 436 The topic of strong-field electron dynamics and how it can be described using TDKS calculations is not considered here, except to note that there have been successful TDKS simulations of strong-field photoionization, [437][438][439][440][441][442][443][444][445][446][447][448] and also of high harmonic generation, [449][450][451][452][453][454][455][456][457] both in Gaussian-orbital representations of the density. Unlike the gridbased treatments that are common in atomic physics, Gaussian-based methods are scalable to molecules.…”

Approximation Schemes to Include Nuclear Motion in Laser-Driven Ab Initio Electron Dynamics: Application to High Harmonic Generation

“…Here, E e , j ( R ) is the excess energy of the electron above the ionization potential, i.e., E e , j ( R ) = V j ( R ) – V nn ( R ) (with V nn ( R ) the nuclear repulsion), and d is an “escape length” beyond which the electron is considered as free. This model generalizes a heuristic ionization model originally developed for fixed-nuclei multistate models, to the case when the electronic states become potential energy surfaces or potential curves, giving for H 2 + . For H 2 , LiH, and also H 2 O, however, we neglected the coordinate dependence of Γ j and computed rates from virtual Hartree–Fock orbital energies, E e , j → ε r , evaluated at the ground-state equilibrium geometries, i.e., we used the heuristic ionization model for fixed-nuclei multistate cases of ref .…”

“…Regarding the electronic structure problem, for H 2 + , the same techniques as in ref were applied. That is, “exact” electronic potentials V j ( R ), dipole moments μ ij ( R ), and adiabatic electronic wavefunctions ψ j ( r ; R ) were calculated by solving eq () using the Gaussian-type orbital basis aug-cc-pVTZ, with additional first eight so-called Kaufmann shells with halved exponents (see ref for further details). This basis set is very diffuse and particularly suited for large-amplitude motion of the electron, as it occurs during HHG.…”

“…The quantum treatment of coupled electronic and nuclear motion has been accomplished for very small molecules, like H 2 + or related species, by numerically solving the molecular time-dependent Schrödinger equation on extended grids, among other things also in order to describe HHG. ,,− The general attention, however, has shifted from small molecules ,− to larger ones − recently. Chemically interesting molecules beyond H 2 + or alike are out of reach for the full solution of the coupled nuclear-electronic time-dependent Schrödinger equations.…”

“…We note that the coupled electron-nuclear problem can not only be solved on (extended) grids for all (three) degrees of freedom but also by using a standard Gaussian-type orbital basis, computing potential energy curves and describing only the nuclear wavepacket motion by grid methods. This approach was used in ref , where it was also shown that nuclear motion had a clear effect on HHG as just mentioned. Non-Born–Oppenheimer transitions between the many different potential energy surfaces via kinetic coupling terms in the coupled nuclear Schrödinger equation were also included, but found to be unimportant for the specific case studied in ref , namely, HHG following nonresonant (800 nm) laser-pulse excitation of H 2 + in its electronic ground state.…”