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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 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 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.

Potential energy surfaces of the hydrogen molecular ion H$_2^+$ in the Born-Oppenheimer approximation are computed by means of the Riccati-Pad\'e method (RPM). The convergence properties of the method are analyzed for different states. The equilibrium internuclear distance, as well as the corresponding electronic plus nuclear energy, and the associated separation constants, are computed to 40 digits of accuracy for several bound states. For the ground state the same parameters are computed with more than 100 digits of accuracy. Additional benchmark values of the electronic energy at different internuclear distances are given for several additional states. The software implementation of the RPM is given under a free software license. The results obtained in the present work are the most accurate available so far, and further additional benchmarks are made available through the software provided.

Generally, driven by a specific laser waveform, a single-order harmonic from the atomic harmonic spectrum can be enhanced, caused by the multiple acceleration-recombination processes. However, in t...

In the hydrogen molecular ion, the kinetic energy lowering of the electron is associated with its delocalization due to electron exchange between the two protons of the molecule. This decrease in the kinetic energy of the exchanged electron in the hydrogen molecular ion and the decrease in the dynamical mass of the two exchanged pions in the nucleon-nucleon interaction are at the origin of the attraction mechanism in the molecular covalent bonding and in the nuclear interaction. Based on this unitary approach of the attraction mechanism, the formulas of molecular potential and central nucleon-nucleon potential were derived. The decrease in the mass of the exchanged pions in the nucleon-nucleon bound state, actually means the decrease in the mass of the nucleons. This nucleon mass decrease could be a manifestation of the partial chiral symmetry restoration in nuclear matter.

As the simplest molecules with only one electron and two nuclei in nature, H2+ and its isotopes are of special importance and serve as benchmark systems for ultrafast science and technologies. Their dissociation in strong laser fields has attracted a lot of interests in past decades because it directly relates to the production of chemical reactions, and a series of great achievements have already been made successively under the cooperative efforts between the theoretical and experimental physicists. With the rapid technological development of lasers as well as the advanced experimental techniques in recent years, some new trends in this topic are presented. In this paper, we focus on recent research progress of experimental and theoretical works on hydrogen molecular ions dissociation. We first give a background information about past achievements, and then introduce general numerical methods to deal with dissociation, i.e., the numerical simulation of the time-dependent Schrodinger equation (TDSE), and the combination of the strong field approximation and TDSE simulation. The latter was developed to investigate dissociative ionization of hydrogen molecules. Next three related hot research topics are reviewed sequentially, including interference between dissociation channels, photon energy sharing between electron and nuclei, and molecular rotation in dissociation. In detail, some strategies have been raised to control the unique electron localization after the molecular dissociation, which is due to the interference between the lowest two electronic states with opposite parities. In addition, the pump-probe technique was also utilized to observe electron localization dynamics in real time. However, interference of nuclear wave packets carrying different angular momenta in the same electronic state and ending with the same momentum contributes novel spiral momentum distribution under a circularly polarized laser pulse. The electron-nuclear correlation for different ionization mechanisms is mainly indicated in the joint energy spectra, which was extracted and explained by coincident measurement and various theoretical methods. The molecular rotation is generally assumed frozen in dissociation process, which is the common axial recoil approximation. However, recent theoretical and experimental researches show that the ultrafast electron dynamics may play roles in the molecular rotation though their timescales are significantly different. In conclusion, the studies of hydrogen molecular ions not only further deepen our understanding of ultrafast dynamics in small molecules, but also shed light on observing, understanding, and ultimately controlling complex chemical reactions.

Through controlling the charge resonance enhanced ionization process of the isotope molecules of hydrogen molecular ion, deuterium molecular ion and tritium molecular ion, the intense single harmon...

The occurrence of Hamiltonian chaos in the system of hydrogen molecular ion placed in external magnetic field has been investigated by using electronic trajectory plot, Poincare surface of section and maximal Lyapunov exponent. Our analysis reveals that the underlying dynamics of this system is very much sensitive to the magnetic field. The numerical calculations show that the regular dynamics in the field free case begins to deviate, with the addition of the external magnetic field, and shows chaotic character for certain field strengths. Moreover, a gradual increase in the strength of the magnetic field makes the system to posses chaos–order–chaos type of regions. When the magnetic field is very large, we have observed that the entire phase space has been occupied by chaotic trajectories. drafts

We present a critical analysis of the mechanism of reversible hydrogen evolution reaction based on thermodynamics of hydrogen processes considering atomic and ionic species as intermediates. Clear distinction between molecular hydrogen evolution/oxidation (H2ER and H2OR) and atomic hydrogen evolution/oxidation (HER and HOR) reactions is made. It is suggested that the main reaction describing reversible H2ER and H2OR in acidic and basic solutions is: H3O++2e−⇌(H2+)adH2+OH− and its standard potential is E0 = −0.413 V (vs. standard hydrogen electrode, SHE). We analyse experimentally reported data with models which provide a quantitative match (R.J.Kriek et al., Electrochem. Sci. Adv. e2100041 (2021)). Presented analysis implies that reversible H2 evolution is a two-electron transfer process which proceeds via the stage of adsorbed hydrogen molecular ion H2+ as intermediate, rather than Had as postulated in the Volmer-Heyrovsky-Tafel mechanism. We demonstrate that in theory, two slopes of potential vs. lg(current) plots are feasible in the discussed reversible region of H2 evolution: 2.3RT/F≈60 mV and 2.3RT/2F≈30 mV, which is corroborated by the results of electrocatalytic hydrogen evolution studies reported in the literature. Upon transition to irreversible H2ER, slowdown of H2+ formation in the first electron transfer stage manifests, and the slope increases to 2.3RT/0.5F≈120 mV, R,F,T are the universal gas, Faraday constants and absolute temperature, respectively.

A study of information-theoretic properties of molecular bond on the example of the ground (bound) 1 s σ g and the lowest excited (unbound) 2 p σ u states of H 2 + was conducted. Data about general information-theoretic measures, information products (Cramer-Rao, Fisher-Shannon information, and LMC complexity) and their inequalities in position and momentum spaces as a function of internuclear distance R = 0.2 –20 a.u. were obtained by an exact numerical calculation of the wave function. A systematic correlation between these information products was established. It was found that LMC complexity is most sensitive and detailed among these information products. A clear correlation between accuracy of determination of energy and information products in the simplest LCAO model compared to results of the exact calculation was observed.

Ultrafast photoelectron imaging allows to measure information about coherent electron dynamic processes in materials or chemical compounds on femtosecond to attosecond time scales. We show that molecular time-resolved photoelectron diffraction produced by a time-delayed soft X-ray attosecond pulse can be used to monitor the ultrafast coherent excitation induced by a resonant UV pump pulse with variable carrier-envelope phases. Asymmetric diffraction angular patterns illustrate coherent electron dynamics of charge migration with spatiotemporal resolution on the attosecond and ångström scale. This allows the temporal reconstruction of phases and amplitudes of electronic states and geometry of molecules as a function of time delay of the probe pulse and carrier-envelope phases of the pump pulse. Results are obtained from solutions of time-dependent Schrödinger equations of the hydrogen molecular ion, and analyzed by ultrafast photoelectron diffraction models for coherent superposition of electronic states. The present demonstration provides a guiding principle for monitoring ultrafast spatiotemporal coherent electron dynamics and imaging molecular electronic structure in complex systems by ultrafast pump-probe experiments and their dependence on carrier-envelope phases and time delays.

We solve the time-dependent Schrodinger equation using the coherent states as basis sets for computing high harmonic generation (HHG) in a full-dimensional single-electron "realistic" system. We apply the static coherent states (SCS) method to investigate HHG in the hydrogen molecular ion induced by a linearly polarized laser field. We show that SCS gives reasonable agreement compared to the three dimensional unitary split-operator approach. Next, we study isolated attosecond pulse generation in H2+ . To do so, we employ the well-known polarization gating technique, which combines two delayed counter-rotating circular laser pulses, and opens up a gate at the central portion of the superposed pulse. Our results suggest that the SCS method can be used for full-dimensional quantum simulation of higher dimensional systems such as the hydrogen molecule in the presence of an external laser field.

We numerically simulate the photoionization of a hydrogen molecular ion exposed to a monochromatic extreme ultraviolet laser field beyond the dipole approximation. In such a diatomic system, the laser field reaches the two nuclei at different instants, adding extra phase on the ionization events from two nuclei and resulting in the shift of the angular maxima in the photoelectron momentum distribution. In the limit of the infinite internuclear distance, the double-slit interference scenario can be adopted to explain the simulated photoelectron momentum distribution. However, when the internuclear distance is as short as 1 \AA{}, the electron distributes more in the center of the molecule, and the photoelectron momentum distribution carries the single-slit diffraction character. The extracted birth time delay of photoemission from two nuclei is about a few hundred of zeptoseconds, which is similar to the experimental measurement in ${\mathrm{H}}_{2}$ [Grundmann et al., Science 370, 339 (2020)].

In this paper we study the ionization rate and the Stark shift of a one-dimensional model of the ${\mathrm{H}}_{2}^{+}$ ion. The finding of these two quantities is reduced to the solution of a complex eigenvalue problem. We solve this problem both numerically and analytically. In the latter case we consider the regime of small external electrostatic fields and small internuclear distances. We find an excellent agreement between the ionization rate computed with the two approaches, even when the approximate result is pushed beyond its expected validity. The ionization rate is very sensitive to small changes of the external electrostatic field, spanning many orders of magnitude for small changes of the intensity of the external field. The dependence of the ionization on the internuclear distance is also studied, as this has a direct connection with experimental methods in molecular physics. It is shown that for large distances the ionization rate saturates, which is a direct consequence of the behavior of the energy eigenvalue with the internuclear distance. The Stark shift is computed and from it we extract the static polarizability of ${\mathrm{H}}_{2}^{+}$ and compare our results with those found by other authors using more sophisticated methods. This work provides an extension of what is usually covered in a course of quantum mechanics where the double delta-potential is studied, while also establishing the connection to a concrete physics problem., NMRP acknowledges support from the European Commission through the project 'Graphene-Driven Revolutions in ICT and Beyond' (Ref. No. 881603-core 3), and the Portuguese Foundation for Science and Technology (FCT) in the framework of the Strategic Financing UID/FIS/04650/2019. JCGH acknowledges FCT for a scholarship in the context of the Summer school 'Quantum Matter: Materials and Concepts'.

The largest hyperfine interaction coefficients in the hydrogen molecular ion ${\mathrm{HD}}^{+}$, i.e., the electron-proton and electron-deuteron spin-spin scalar interactions, are calculated with estimated uncertainties slightly below 1 ppm. The ${(Z\ensuremath{\alpha})}^{2}{E}_{F}$ relativistic correction for which a detailed derivation is presented, QED corrections up to the order ${\ensuremath{\alpha}}^{3}{ln}^{2}(\ensuremath{\alpha})$ along with an estimate of higher-order terms, and nuclear structure corrections are taken into account. Improved results are also given for the electron-proton interaction coefficient in ${\mathrm{H}}_{2}^{+}$, in excellent agreement with rf spectroscopy experiments. In ${\mathrm{HD}}^{+}$, a $4\ensuremath{\sigma}$ difference is found in the hyperfine splitting of the $(v,L)=(0,3)\ensuremath{\rightarrow}(9,3)$ two-photon transition that was recently measured with high precision. The origin of this discrepancy is unknown.

We report on the production of cold, state-selected ${\mathrm{H}}_{2}^{+}$ molecular ions in a linear rf trap. The ions are produced by ($3+1$) resonance-enhanced multiphoton ionization (REMPI) of ${\mathrm{H}}_{2}$, and sympathetically cooled by laser-cooled ${\mathrm{Be}}^{+}$ ions. After demonstrating and characterizing the REMPI process, we use photodissociation by a deep UV laser at 213 nm to verify the high vibrational purity of the produced ${\mathrm{H}}_{2}^{+}$ ion samples. Moreover, the large difference between the photodissociation efficiencies of ions created in the $v=0$ and $v=1$ levels provides a way to detect a $v=0\ensuremath{\rightarrow}1$ transition. These results pave the way towards high-resolution vibrational spectroscopy of ${\mathrm{H}}_{2}^{+}$ for fundamental metrology applications.

In the framework of a quasi-molecular approach, the formation of hydrogen atom in the pre-recombination period of evolution of the universe is analysed quantitatively. Calculations in an adiabatic multi-level representation enable estimates of probabilities of radiative transitions. The quasi-molecular mechanism of recombination allows the formation of hydrogen molecular ion, $H_2^+$, in its ground state. The probability of this process is comparable with the probability of the creation of atomic hydrogen. The participation of a second proton in the recombination increases the binding energy of an electron and decreases the rate of recombination of hydrogen., 6 pages, 1 figure

We present results on the accurate one-dimensional (1D) modeling of simple atomic and molecular systems excited by strong laser fields. We use atomic model potentials that we derive from the corrections proposed earlier using the reduced ground-state density of a three-dimensional (3D) single-active electron atom. The correction involves a change of the asymptotics of the 1D Coulomb model potentials while maintaining the correct ground-state energy. We present three different applications of this method: we construct correct 1D models of the hydrogen molecular ion, the helium atom, and the hydrogen molecule using improved parameters of existing soft-core Coulomb potential forms. We test these 1D models by comparing the corresponding numerical simulation results with their 3D counterparts in typical strong-field physics scenarios with near- and mid-infrared laser pulses, having peak intensities in the ${10}^{14}\ensuremath{-}{10}^{15}\phantom{\rule{4pt}{0ex}}\mathrm{W}/{\mathrm{cm}}^{2}$ range, and we find an impressively increased accuracy in the dynamics of the most important atomic quantities on the time scale of the excitation. We also present the high-order harmonic spectra of the He atom, computed using our 1D atomic model potentials. They show a very good match with the structure and phase obtained from the 3D simulations in an experimentally important range of excitation amplitudes.

HD$^+$ hydrogen molecular ion is of metrological interest due to possibility of precise theoretical evaluation of its spectrum and of external-field-induced shifts. In present work, the rates of laser-induced electric quadrupole transitions are calculated for a wide range of rovibrational transitionsin particular for L0-L’2, L1-L’3, L2-L’4 (v=(0-10)-v’=(0-3)). Results are presented with six significant digits.

The trajectory-based method for calculating the probabilities of transitions in the quantum system developed in our previous works and tested for atoms is applied to calculating the photoionization probability for the simplest molecule – hydrogen molecular ion. In a weak field it is established a good agreement between our photoionization cross section and the data obtained by other theoretical methods for photon energy in the range from one-photon ionization threshold up to 25 a.u. Photoionization cross section in the range 25 ω ≤ 100 a.u. was calculated for the first time judging by the literature known to us. It is also confirmed that the trajectory method works in a wide range of the field magnitudes including superatomic values up to relativistic intensity.

Isolated attosecond pulse (IAP) generation is theoretically investigated by using a few-cycle laser pulse from a two-dimensional model of hydrogen molecular ion. The harmonic spectra from two nuclei of hydrogen molecular ion lead to interference. We investigate the spatial distribution in harmonic generation and propose a scheme to restrain the harmonic generation from the nucleus along the positive-x direction, and thus the interference is weaken in spatial. By using a few-cycle 800 nm chirped laser pulse, the harmonics are mainly generated from the nucleus along negative-x direction in the region of 130th to 230th order. The harmonic spectra are smooth and are mainly contributed by the short quantum path near the cutoff region and IAP with a duration of 97 as is generated. The semiclassical of three-step model is also used to illustrate the physical mechanism.

Charge transfer from the hydrogen molecular ion by positron impact is studied using Fourier transform techniques. The interaction amplitude and the differential cross section for the reaction are obtained within the first born approximation. The interference patterns and the effect of the orientation angle of the molecular axis are analyzed, and compared with the available data in the literature.

The nonrelativistic variational calculation of a complete set of ro-vibrational states in the H$_2^+$ molecular ion supported by the ground $1s\sigma$ adiabatic potential is presented. It includes both bound states and resonances located above the $n=1$ threshold. In the latter case we also evaluate a predissociation width of a state wherever it is significant. Relativistic and radiative corrections are discussed and effective adiabatic potentials of these corrections are included as supplementary files., Comment: 7 pages, 1 figure, 2 tables, 5 ancillary files

Using the Born – Oppenheimer approximation, and considering the nuclei fixed at the foci, a study of the hydrogen molecular ion (H2+), confined strongly in two dimensions by ellipses of different size ξ0, is done. The Schrodinger equation is solved numerically in elliptic coordinates (ξ, η), applying the separation of variables method. The equations for ξ and η are solved following an iterative process, until the energy and separation constant become consistent with the size of the confining ellipse characterized by the parameter ξ0 and the internuclear distance R. The energies for the ion at its ground state and the equilibrium distance between the nuclei are obtained, for different values of ξ0, as well as the polarizability of the molecule.

We investigate the attosecond streaking spectrum of ${\mathrm{H}}_{2}{}^{+}$ by numerically solving the time-dependent Schr\"odinger equation. The results show that the streaking spectrum of ${\mathrm{H}}_{2}{}^{+}$ depends sensitively on the internuclear distance $R$. In addition to the regular streaking structure, some streaking spectra of ${\mathrm{H}}_{2}{}^{+}$ with the splitting and distorted structures are also observed as $R$ changes. The splitting and distorted streaking spectra of ${\mathrm{H}}_{2}{}^{+}$ are demonstrated to be related to the destructive two-center interference and resonant transition triggered by the IR field, respectively. Meanwhile, the mixture structure of split and distortion on the streaking spectrum of ${\mathrm{H}}_{2}{}^{+}$ is attributed to the coupling of the destructive two-center interference and resonant transition. The $R$-dependent streaking spectra might pave an accessible route toward monitoring the ultrafast nuclear dynamics of molecules.

Strong-field photoelectron momentum distributions (PEMDs) generated in the ionization of molecules by circularly polarized laser pulses contain valuable information on the structure of the ionizing molecular orbital. In this paper, we investigate using the adiabatic theory how this information is imprinted in and can be extracted from such PEMDs. The PEMDs have a donutlike shape. The variation of a PEMD along the donut reproduces the orientation dependence of the ionization rate of the ionizing orbital, and hence reflects its nodal structure and shape. The transverse structure of the PEMD in cross sections of the donut corresponding to orientations of the molecule at which the ionizing orbital has a node in the direction opposite to that of the instantaneous laser field reflects the interplay of two competing ionization channels, which provides additional information on the orbital structure. We illustrate these features by the analysis of PEMDs for three states of different symmetries of the hydrogen molecular ion ${\mathrm{H}}_{2}{}^{+}$.

We apply the extended multiconfiguration time-dependent Hartree-Fock method to the simulation of a hydrogen molecular ion exposed to an intense laser pulse. By comparing the results obtained by this method with the results obtained by a method in which the time-dependent Schr\"odinger equation is solved directly on a three-dimensional grid, we find that the results obtained by these two methods are in good agreement with each other when the number of time-dependent expansion terms exceeds 8. We further compare the results with those obtained by the conventional two-state Born-Oppenheimer approximation. In order to interpret the resultant time-dependent wave functions, we decompose the total wave function into the natural electronic and protonic orbitals that diagonalize the single-particle density matrices and find that the pair of electronic and protonic natural orbitals carrying the largest population describes the vibrational excitation, while the pair carrying the second largest population describes the dissociation into $\text{H}+{\mathrm{H}}^{+}$. We also examine the time-dependent motion of the protons in ${\mathrm{H}}_{2}^{+}$ in terms of the time-dependent adiabatic potential-energy curves, which are defined as the instantaneous eigenvalues of the Hamiltonian matrix governing the time-dependent motion of the protonic orbitals. We show that two potential minima are formed on the lowest-energy adiabatic potential-energy curve and that the nuclear wave packet localized in the inner minimum corresponds to the bound vibrational motion, while the nuclear wave packet localized in the outer minimum corresponds to the dissociation.

We use the exponential explicitly correlated variational basis set of the type exp(-αnR - βnr1 - γnr2) to calculate systematically the nonrelativistic bound-state energies for the hydrogen molecular ion H2+ and HD+. We perform calculations for the states of the total angular momentum L=0-4 with the complete set of vibrational numbers v=0-10.

The high harmonic generation of heteronuclear hydrogen molecular ions in few-cycle infrared strong laser fields is studied by numerically simulating the time-dependent Schr\"odinger equation. By reversing the carrier envelope phase of the few-cycle laser pulse, the electron dynamics is reversed as well. However, due to the different nuclear masses, two nuclei move with different repulsive speeds, which makes the rescattering processes not identical when the carrier envelope phase shifts by $\ensuremath{\pi}$. The different rescattering processes lead to distinct harmonic spectra in the cutoff region. The mass-dependent harmonic spectrum offers a view angle to diagnose the isotope dynamics in molecules with unprecedent high time and spatial resolutions.

In this reply we clarify questions point out in the Comment on: “The Effect of Confinement on the Electronic Energy and Polarizability of a Hydrogen Molecular Ion” by J. F. da Silva, F. R. Silva and E. Drigo Filho, Int. J. Quantum Chem.2016, 116, 497–503 written by S. A. Cruz and H. Olivares-Pilon. In particular, we show how we made the calculations of ground state energy from the confined hydrogen molecule ion for cavities of different volumes. The internuclear distances to the excited state 2pσu are also presented.

The control of the electronic states of a hydrogen molecular ion by photoexcitation is considerably difficult because it requires multiple sub-10 fs light pulses in the extreme ultraviolet (XUV) wavelength region with a sufficiently high intensity. Here, we demonstrate the control of the dissociation pathway originating from the 2pσu electronic state against that originating from the 2pπu electronic state in a hydrogen molecular ion by using a pair of attosecond pulse trains in the XUV wavelength region with a train-envelope duration of ∼4 fs. The switching time from the peak to the valley in the oscillation caused by the vibrational wavepacket motion in the 1sσg ground electronic state is only 8 fs. This result can be classified as the fastest control, to the best of our knowledge, of a molecular reaction in the simplest molecule on the basis of the XUV-pump and XUV-probe scheme., Attosecond laser pulses allow to explore the ultrafast atomic or molecular processes occurring in chemical reactions. Here the authors utilize a sequence of two XUV pump-probe pulses to control the dissociation dynamics of the hydrogen molecular ion.

The interference minimum in the molecular high-order harmonic gener- ation is studied by numerically solving the non-Born-Oppenheimer approximation time-dependent Schrodinger equation. The results show that two kinds of interference minima appear in the high-order harmonic spectrum when the hydrogen molecular ion is exposed to the laser pulse. Furthermore, by adjusting wavelength of the laser as well as initial condition of nucleus, it is found that two kinds of interference minima are ascribed to the molecular structure and the electronic dynamics behavior induced by the laser field, respectively. Besides, the time-frequency distributions and the elec- tron wave packet distributions are calculated to better understand the mechanism of the minimum.

Quantum path control on the harmonic emission from hydrogen molecular ion is investigated via combining an intense few-cycle laser and a half-cycle pulse (HCP). The results show that the quantum paths of returning electrons can be effectively modified due to the introduction of the HCP which extremely dominates the acceleration process of ionized electron. With the favorable conditions, the long quantum path is completely eliminated and an isolated 61 as pulse can be obtained by superposing proper harmonics. Moreover, the electron wave packet distributions and the double-well model are presented to explore the underlying physical mechanism.

H2+ exposed to strong mid-infrared laser pulses undertakes the tunneling ionization, followed by the rescattering of the photoelectron with the parent ion. The two nuclei exclude each other during the tunneling and rescattering, which modifies the Coulomb potential experienced by the rescattering electron. We numerically study the role of the nuclear movement in the tunneling ionization of hydrogen molecular ions. The classical trajectory Monte-Carlo simulations give intuitive explanations of photoelectron momentum distributions driven by laser pulses having different wavelengths and different intensities. The competition of the Coulomb attraction and the Lorentz push determines the photoelectron momentum distribution along the laser propagation direction. The nuclear movement modifies the Coulomb focusing and thus shifts the peaks of the photoelectron momentum distribution along the laser propagation direction.

Atom in a coherent superposition state reveals an advantage in the enhancement conversion efficiency of high-order harmonic generation (HHG), which is meaningful in producing attosecond pulses. In this study, we expand to investigate a more complicated system, H\(_2^+\) molecule in the superposition of the ground and second excited states, exposed to an ultrashort intense laser pulse by numerically solving the time-dependent Schrödinger equation. Firstly, we examine the enhancement of HHG from this system. Then, we study the depletion effect on the cutoff energy of HHG spectra with the coherent superposition state. We found that these effects on the HHG from molecules are similar to those from atoms. Finally, we study the signature of the interesting effect, which is absent for atoms -- two-center interference effect in the HHG from H\(_2^+\) in the coherent superposition state. We recognize that the minimum positions in HHG from molecules in the superposition state, and in the pure ground state are the same. Especially, for weak laser intensity, in the HHG with the superposition state, the minimum due to the interference effect is apparent, while it is invisible in the HHG from pure ground state. As a result, in comparison with the ground-state molecule, the coherent molecule can be used as a more accurate tool to determine the internuclear distance of molecule.

Positron scattering from the vibrational ground state of the hydrogen molecular ion has been investigated using the single-center convergent close-coupling (CCC) method. Cross sections for dissociative excitation, dissociative ionization, total inelastic scattering, and proton production have been calculated at incident positron energies from 10 to 500 eV. For this energy range the mass stopping power and mean excitation energy have also been calculated for both the positron and electron cases. Since there are no experiments or other calculations for positron scattering on the hydrogen molecular ion, the obtained results have been compared with corresponding CCC results for positron collisions with He+ and electron collisions with H2+. Comparative analysis allows to estimate the quality of the approximations used in the present calculations.

Generation of high-order harmonics (HOHs) and attosecond pulses in the barrier suppression ionization regime for a target of hydrogen molecular ion, $${\text{H}}_{2}^{+}$$ , with large inter nuclear distance is introduced in this work. Some broadband femtosecon laser pulses are designed based on multicolor beam superposition method in order to interact with $${\text{H}}_{2}^{+}$$ . Behavioral study of harmonics cutoff, attosecond pulse widths and intensities, along with the maximum ionization intensity with respect to the spectral width of driving pulses was carried out. Molecular orbitals in this study are estimated via the linear combination of atomic orbital approximation, and temporal variations of these orbitals are studied in the various ionization regimes. It is seen in this work that although the cutoff harmonic is increased by the driving pulse intensity, but it is insensitive to the spectral driving pulsewidth. This study showed that intensities of the generated attosecond pulses gradually decreased by increasing the bandwidth of the driving laser pulse, but the duration of them is shown to be insensitive in this respect.

This contribution presents a dual-frequency molecular clock with trapped HD+ ions based on the two-photon rotational transition (v,L)=(0,1)->(0,3) at 3.268 THz and the two-photon rovibrational transition (v,L)=(0,3)->(9,3) at 207.634 THz which are detected by photodissociation of the (v,L)=(9,3) state. The two-photon transition rates between hyperfine components of HD+ energy levels and the lightshifts are calculated with the two-photon operator formalism. Temporal dependences of the populations of trapped HD+ ions are described by a set of coupled rate equations. The two-photon transitions may be detected efficiently with resolutions at the 10-13 level. The comparison, with an accuracy estimated at the 10-12 level, between experimental frequencies of two-photon rotational and rovibrational transitions of selected HD+ and H2+ ions and the values derived from quantum electrodynamics calculations may be exploited to determine the Rydberg constant, the proton-to-electron and deuteron-to-proton mass ratios, the proton and the deuteron radii independently on previous results. Depending on possible issues of the proton radius puzzle, measurements of hydrogen molecular ion two-photon transitions may improve the determination of the proton-to-electron and deuteron-to-proton mass ratios beyond the 10-11 level.

International audience; We present a new derivation of the proton-electron mass ratio from the hydrogen molecular ion, HD$^+$. The derivation entails the adjustment of the mass ratio in highly precise theory so as to reproduce accurately measured ro-vibrational frequencies. This work is motivated by recent improvements of the theory, as well as the more accurate value of the electron mass in the recently published CODATA-14 set of fundamental constants, which justifies using it as input data in the adjustment, rather than the proton mass value as done in previous works. This leads to significantly different sensitivity coefficients and, consequently, a different value and larger uncertainty margin of the proton-electron mass ratio as obtained from HD$^+$.

Before entering the wide field of molecule-laser interaction in this chapter, some basics of molecular theory will be repeated by discussing the solution of the time-independent Schrodinger equation of the hydrogen molecular ion. The concept of electronic potential energy surfaces, on which the nuclear dynamics occurs, will thereby be introduced and the analytical Morse potential will be reviewed in some detail. The solution of the time-dependent Schrodinger equation for the hydrogen molecular ion then serves as an example for the increasing numerical effort that has been taken in the course of time, starting with frozen nuclei and then taking the nuclear dynamics into account.

We multiply the anisotropic Gaussian atomic orbital by a field-dependent gauge phase to describe the wave function for the hydrogen molecular ion in non-aligned magnetic fields. With the kind of basis set, the convergence of the total energy at the equilibrium distance for the 1u state is much improved compared to the same atomic orbital without the gauge phase. For 2.35 × 104 ≤ B ≤ 107 T, better total energies of the 1u state at the corresponding equilibrium are obtained for the deviations 15°–90° of the magnetic field relative to the molecular axis. The result also shows that, there is a transition of the equilibrium configuration from the vertical orientation to the parallel orientation with increasing field strength.

High-order harmonic generation from hydrogen molecular ion in the inhomogeneous laser field is studied theoretically by numerically solving the non-Born–Oppenheimer time-dependent Schrodinger equation. The results show that ionized electrons can acquire more accelerating energy in inhomogeneous laser field than in homogeneous field, which is beneficial to generate a broader supercontinuum. Furthermore, by optimizing spatial non-uniformity of the inhomogeneous field, the long quantum path is suppressed obviously, and an isolated 60-as pulse is achieved by superposing harmonics from 110th to 150th orders. Besides, combined Coulomb and laser field potential as well as time–frequency distributions are applied to illustrate the physical mechanism.

In TPD-SheetIV, a simulation experiment on a long leg divertor configuration has been conducted and hydrogen molecular ion formation in detached plasma was observed. Improvements to the simulation geometry used in DEGAS 2 allowed us to study the effect of leg length on neutral transport. This paper also describes a simple ion balance equation and the ion density ratios obtained with it. From these data, it is shown that a long leg divertor enhances molecular ion density in the divertor, which might enhance molecular assisted recombination.

We present systematic calculations of the leading order relativistic corrections including the hyperfine splitting (HFS) for a wide range of rotational and vibrational states of the HT+ molecular ion. We also calculate the DC and AC Stark effects for the molecular hydrogen ions \(H_{2}^{+}\) and HD+ in the non-relativistic approximation. The influence of the DC Stark polarizability effect on the hyperfine substates of a ro-vibrational state is carefully analyzed. Our results enable the detailed evaluation of certain systematic shifts of the transition frequencies for the purpose of ultra-high-precision optical, microwave, or radio-frequency spectroscopy of the hydrogen molecular ions in a trap.

We present new theoretical results for the transition energies of the hydrogen isotope molecular ions and antiprotonic helium atoms. Our consideration includes corrections at the mα7 order in the nonrecoil limit such as the one-loop self-energy, one-loop vacuum polarization, Wichman-Kroll, and complete two-loop contributions. That allowed to get transition energies for the fundamental transition \((v = 0,L = 0)\rightarrow (1,0)\) in the hydrogen molecular ion with the relative theoretical uncertainty of ∼7⋅10−12 that corresponds to a fractional precision of 1.5⋅10−11 in determination of the electron-to-proton mass ratio, mp/me. Correspondingly, for the two-photon transitions in the antiprotonic helium we have 4.7⋅10−11 as a relative uncertainty for the \((33,32)\rightarrow (31,30)\) transition frequency and a fractional precision of 3.6⋅10−11 for an inferred antiproton-to-electron mass ratio.

The Effective Hamiltonian of the hydrogen molecular ions is a convenient tool for the evaluation of the shift of the energy levels of the ro-vibrational states and the frequencies of the transitions between them, due to external electric and magnetic fields. Using the Effective Hamiltonian, composite frequencies of suppressed susceptibility to external fields are constructed.

Various conceptual and numerical errors reported in the paper cited in the title of this contribution are pointed out and clarified. Our main concern is the evident violation of the variational principle, which is used in that paper for the study of the confined hydrogen molecular ion to obtain both energies and properties.

Theoretical analysis of laser-assisted electron impact ionization of a hydrogen molecular ion H+ 2 at high impact energy and large momentum transfer is carried out. The laser-field effects on the incoming and outgoing electrons are taken into account using the Volkov functions. The field-dressing of the target electron is treated with a quasistatic state approach. Calculations for laser radiation with frequency ω = 1.55 eV and intensity I = 5 × 1011 W/cm2 exhibit strong laser influence on the molecular bond oscillation in laser-assisted electron momentum distributions.