Calculations for the vibration-rotation levels of H+ 2 in its ground and first excited electronic states

Zero-Quantum-Defect Method and the Fundamental Vibrational Interval of H_{2}^{+}

The fundamental vibrational interval of H_{2}^{+} has been determined to be ΔG_{1/2}=2191.126 614(17)  cm^{-1} by continuous-wave laser spectroscopy of Stark manifolds of Rydberg states of H_{2} with the H_{2}^{+} ion core in the ground and first vibrationally excited states. Extrapolation of the Stark shifts to zero field yields the zero-quantum-defect positions -R_{H_{2}}/n^{2}, from which ionization energies can be determined. Our new result represents a 4-order-of-magnitude improvement compared to earlier measurements. It agrees, within the experimental uncertainty, with the value of 2191.126 626 344(17)(100)  cm^{-1} determined in nonrelativistic quantum electrodynamic calculations [V. Korobov, L. Hilico and J.-Ph. Karr, Phys. Rev. Lett. 118, 233001 (2017)PRLTAO0031-900710.1103/PhysRevLett.118.233001].

Deuteron-to-Proton Mass Ratio from Simultaneous Measurement of the Cyclotron Frequencies of H_{2}^{+} and D^{+}

By simultaneously measuring the cyclotron frequencies of an H_{2}^{+} ion and a deuteron in a coupled magnetron orbit we have made an extended series of measurements of their cyclotron frequency ratio. From the observed changes in H_{2}^{+} mass energy we have followed the decay of three H_{2}^{+} ions to the vibrational ground state. We are able to assign some of our measured ratios to specific rovibrational levels, hence reducing uncertainty due to H_{2}^{+} rotational energy. Assuming the most probable assignment, we obtain a deuteron-to-proton mass ratio, m_{d}/m_{p}=1.999 007 501 272(9). Combined with the atomic mass of the deuteron [S. Rau et al., Nature (London) 585, 43 (2020).NATUAS0028-083610.1038/s41586-020-2628-7] we also obtain a new value for the atomic mass of the proton, m_{p}=1.007 276 466 574(10)  u.

Deuteron-to-Proton Mass Ratio from the Cyclotron Frequency Ratio of H_{2}^{+} to D^{+} with H_{2}^{+} in a Resolved Vibrational State

We have measured cyclotron frequency ratios of H_{2}^{+} to D^{+} with sufficient precision to resolve the mass increase of H_{2}^{+} due to vibrational energy. Additional discrimination against excited vibrational levels was provided by increasing the rate of vibrational decay through Stark quenching. From our results we obtain a value for the deuteron-to-proton mass ratio, m_{d}/m_{p}=1.999 007 501 274(38), which has an uncertainty three times smaller than the current CODATA value.

Spectrum of Field-Ionized Triplet Gerade Rydberg States of H2 and Comparison to Multichannel Quantum-Defect Theory

A spectrum of field-ionized triplet Rydberg states of gerade symmetry H₂ has been measured, excited from the υ″ = 0, N″ = 1-3 rovibrational levels of the metastable c³Π_u^-2pπ state in a 6 keV fast molecular beam. The field-ionized spectrum is kinetic energy labeled in order to separate it from the well-studied υ⁺ ≥ 1 autoionization spectrum. The spectrum consists of both ns and nd Rydberg series with n between 10 and 28 converging to the υ⁺ = 0, N⁺ = 1-3 levels of the X^+ 2Σ_g⁺ ground state of H₂⁺. Transitions with changes in vibration and/or rotation are also identified. The spectral positions of 59 transitions in the field ionization spectrum are identified and assigned quantum numbers using the predictions of multichannel quantum-defect theory (MQDT). The transition energies and subsequent effective quantum defects are compared between the experiment and theory. Most of the observed transitions, within the experimental uncertainty of 0.2 cm^-1, agree with the energies predicted by MQDT.

Hyperfine-interaction-induced g/u mixing and its implication on the existence of the first excited vibrational level of the A + Σ u + 2 state of H 2 + and on the scattering length of the H + H+ collision

Ab initio calculations of the energy level structure of H 2 + that include relativistic and radiative corrections to nonrelativistic energies and the diagonal part of the hyperfine interaction have predicted the existence of four bound rovibrational levels [(v = 0, N = 0 - 2) and (v = 1, N = 0)] of the first electronically excited ( A +   Σ u + 2 ) state of H 2 + , the (v = 1, N = 0) level having a calculated binding energy of only E _b = 1.082 219 8(4)·10^-9 E_h and leading to an extremely large scattering length of 750(5) a₀ for the H⁺ + H collision [J. Carbonell et al., J. Phys. B: At., Mol. Opt. Phys. 37, 2997 (2004)]. We present an investigation of the nonadiabatic coupling between the first two electronic states ( X +   Σ g + 2 and A +   Σ u + 2 ) of H 2 + induced by the Fermi-contact term of the hyperfine-coupling Hamiltonian. This interaction term, which mixes states of total spin quantum number G = 1/2, is rigorously implemented in a close-coupling approach to solve the spin-rovibronic Schrödinger equation. We show that it mixes states of gerade and ungerade electronic symmetry, that it shifts the positions of all weakly bound rovibrational states of H 2 + , and that it affects both the positions and widths of its shape resonances. The calculations demonstrate that the G = 1/2 hyperfine component of the A⁺ (v = 1, N = 0) state does not exist and that, for G = 1/2, the s-wave scattering lengths of the H⁺ + H(1s) collision are -578(6) a₀ and -43(4) a₀ for the F = 0 and F = 1 hyperfine components of the H(1s) atom, respectively. The binding energy of the G = 3/2 hyperfine component of the A⁺ (v = 1, N = 0) state is not significantly affected by the hyperfine interaction and the corresponding scattering length for the H⁺ + H(1s, F = 1) collision is 757(7) a₀.

Non-adiabatic mass correction to the rovibrational states of molecules: Numerical application for the H 2 + molecular ion

General transformation expressions of the second-order non-adiabatic Hamiltonian of the atomic nuclei, including the kinetic-energy correction terms, are derived upon the change from laboratory-fixed Cartesian coordinates to general curvilinear coordinate systems commonly used in rovibrational computations. The kinetic-energy or so-called "mass-correction" tensor elements are computed with the stochastic variational method and floating explicitly correlated Gaussian functions for the H 2 + molecular ion in its ground electronic state. {Further numerical applications for the ⁴ He 2 + molecular ion are presented in the forthcoming paper, Paper II [E. Mátyus, J. Chem. Phys. 149, 194112 (2018)]}. The general, curvilinear non-adiabatic kinetic energy operator expressions are used in the examples, and non-adiabatic rovibrational energies and corrections are determined by solving the rovibrational Schrödinger equation including the diagonal Born-Oppenheimer as well as the mass-tensor corrections.

Observation and Calculation of the Quasibound Rovibrational Levels of the Electronic Ground State of H2+

Although the existence of quasibound rotational levels of the X^{+} ^{2}Σ_{g}^{+}  ground state of H_{2}^{+} was predicted a long time ago, these states have never been observed. Calculated positions and widths of quasibound rotational levels located close to the top of the centrifugal barriers have not been reported either. Given the role that such states play in the recombination of H(1s) and H^{+} to form H_{2}^{+}, this lack of data may be regarded as one of the largest unknown aspects of this otherwise accurately known fundamental molecular cation. We present measurements of the positions and widths of the lowest-lying quasibound rotational levels of H_{2}^{+} and compare the experimental results with the positions and widths we calculate using a potential model for the X^{+} state of H_{2}^{+} which includes adiabatic, nonadiabatic, relativistic, and radiative corrections to the Born-Oppenheimer approximation.

The He + H2+ → HeH+ + H reaction: ab initio studies of the potential energy surface, benchmark time-independent quantum dynamics in an extended energy range and comparison with experiments

In this work we critically revise several aspects of previous ab initio quantum chemistry studies [P. Palmieri et al., Mol. Phys. 98, 1835 (2000); C. N. Ramachandran et al., Chem. Phys. Lett. 469, 26 (2009)] of the HeH(2)(+) system. New diatomic curves for the H(2)(+) and HeH(+) molecular ions, which provide vibrational frequencies at a near spectroscopic level of accuracy, have been generated to test the quality of the diatomic terms employed in the previous analytical fittings. The reliability of the global potential energy surfaces has also been tested performing benchmark quantum scattering calculations within the time-independent approach in an extended interval of energies. In particular, the total integral cross sections have been calculated in the total collision energy range 0.955-2.400 eV for the scattering of the He atom by the ortho- and para-hydrogen molecular ion. The energy profiles of the total integral cross sections for selected vibro-rotational states of H(2)(+) (v = 0,...,5 and j = 1,...,7) show a strong rotational enhancement for the lower vibrational states which becomes weaker as the vibrational quantum number increases. Comparison with several available experimental data is presented and discussed.

Radiative charge transfer in He(+) + H2 collisions in the milli- to nano-electron-volt range: a theoretical study within state-to-state and optical potential approaches

The paper presents a theoretical study of the low-energy dynamics of the radiative charge transfer (RCT) reaction He(+)((2)S)+H2(X(1)Σg (+))→He((1)S)+H2 (+)(X(2)Σg (+))+hν extending our previous studies on radiative association of HeH2 (+) [F. Mrugała, V. Špirko, and W. P. Kraemer, J. Chem. Phys. 118, 10547 (2003); F. Mrugała and W. P. Kraemer, ibid. 122, 224321 (2005)]. The calculations account for the vibrational and rotational motions of the H2/H2 (+) diatomics and for the atom-diatom complex formation in the reactant and the product channels of the RCT reaction. Continuum states of He(+) + H2(v = 0, j = 0) in the collision energy range ~10(-7)-18.6 meV and all quasi-bound states of the He(+) - H2(para; v = 0) complex formed in this range are taken into account. Close-coupling calculations are performed to determine rates of radiative transitions from these states to the continuum and quasi-bound states of the He + H2 (+) system in the energy range extending up to ~0.16 eV above the opening of the HeH(+) + H arrangement channel. From the detailed state-to-state calculated characteristics global functions of the RCT reaction, such as cross-section σ(E), emission intensity I(ν, T), and rate constant k(T) are derived, and are presented together with their counterparts for the radiative association (RA) reaction He(+)((2)S) + H2(X(1)Σg (+))→ HeH2 (+)(X(2)A('))+hν. The rate constant k(RCT) is approximately 20 times larger than k(RA) at the considered temperatures, 0.1 μK-50 K. Formation of rotational Feshbach resonances in the reactant channel plays an important role in both reactions. Transitions mediated by these resonances contribute more than 70% to the respective rates. An extension of the one-dimensional optical potential model is developed to allow inclusion of all three vibrational modes in the atom-diatom system. This three-dimensional optical potential model is used to check to which extent the state-to-state RCT rate constant is influenced by the possibility to access ground state continuum levels well above the opening of the HeH(+)+ H arrangement channel. The results indicate that these transitions contribute about 30% to the "true" rate constant k(RCT) whereas their impact on the populations of the vibration-rotational states of the product H2 (+) ion is only minor. Present theoretical rate constant functions k(RCT)(T) obtained at different approximation levels are compared to experimental data: 1-1.1 × 10(-14) s(-1) cm(3) at T = 15-35 K and ∼7.5 × 10(-15) s(-1) cm(3) at 40 K [M. M. Schauer, S. R. Jefferts, S. E. Barlow, and G. H. Dunn, J. Chem. Phys. 91, 4593 (1989)]. The most reliable theoretical values of k(RCT), obtained by combining results from the state-to-state and the optical potential calculations, are between 2.5 and 3.5 times larger than these experimental numbers. Possible sources for discrepancies are discussed.

Spectroscopic observation and characterization of H(+)H(-) heavy Rydberg states

A series of discrete resonances was observed in the spectrum of H2, which can be unambiguously assigned to bound quantum states in the 1/R Coulombic potential of the H+H- ion-pair system. Two-step laser excitation was performed, using tunable extreme ultraviolet radiation at lambda = 94-96 nm in the first step, and tunable ultraviolet radiation in the range lambda = 310-350 nm in the second step. The resonances, detected via H+ and H2+ ions produced in the decay process, follow a sequence of principal quantum numbers (n = 140-230) associated with a Rydberg formula in which the Rydberg constant is mass scaled. The series converges upon the ionic H+H- dissociation threshold. This limit can be calculated without further assumptions from known ionization and dissociation energies in the hydrogen system and the electronegativity of the hydrogen atom. A possible excitation mechanism is discussed in terms of a complex resonance. Detailed measurements are performed to unravel and quantify the decay of the heavy Rydberg states into molecular H2+ ions, as well as into atomic fragments, both H(n = 2) and H(n = 3). Lifetimes are found to scale as n3.

High-resolution millimeter wave spectroscopy and multichannel quantum defect theory of the hyperfine structure in high Rydberg states of molecular hydrogen H2

Experimental and theoretical methodologies have been developed to determine the hyperfine structure of molecular ions from detailed studies of the Rydberg spectrum and have been tested on molecular hydrogen. The hyperfine structure in l=0-3 Rydberg states of H2 located below the X 2Sigmag+(v+=0,N+=1) ground state of ortho H2+ has been measured in the range of principal quantum number n=50-65 at sub-MHz resolution by millimeter wave spectroscopy following laser excitation to np and nd Rydberg states using a variety of single-photon and multiphoton excitation sequences. The np1(1), nd1(1), and the nf1(0-3) Rydberg states were found to be metastable and to have lifetimes of more than 5 micros beyond n=50. Members of other series, such as the nd1(2), nd1(3), and the np1(0) series, were found to have lifetimes of more than 1 mus. Local perturbations induced by low-n Rydberg states belonging to series converging on rovibrationally excited levels of H2+ reduce the lifetimes in narrow ranges of n values. The hyperfine structure is strongly dependent on the value of the orbital angular momentum l. In the penetrating s and p states at n approximately 50 the exchange interaction dominates over the hyperfine interaction and the levels can be labeled by the total electron spin angular momentum quantum number S (S=0 or 1). In the less penetrating d and f Rydberg states, the hyperfine interaction between the core nuclear and electron spins is larger than the exchange interaction and the Rydberg states are of mixed singlet and triplet character. A procedure based on the Stark effect and on the systematic analysis of selection rules and combination differences was developed to determine the orbital and the total angular momentum quantum numbers l and F and to construct an energy map of p and f Rydberg levels between n=54 and 64 with relative positions of an accuracy of better than 1 MHz. Multichannel quantum defect theory (MQDT) was extended to treat the hyperfine structure in molecular Rydberg states and was used to analyze the observed hyperfine structure of the p and f Rydberg states of H2. The frame transformation between the Born-Oppenheimer channels described by the angular momentum coupling scheme (abetaJ) and the asymptotic channels described by the (e[bbetaS+]) coupling scheme was derived and enables an elegant treatment of all intermediate coupling cases. Purely ab initio quantum defect theory reproduced the experimentally determined positions to within 40 MHz for the p levels and 13 MHz for the f levels. By slight adjustments of the quantum defect functions and their energy dependences and by consideration of the p-f interaction, of the singlet-triplet splittings of the f levels, and of the departure of the ionic levels from pure coupling case (bbetaS+), the agreement between theory and experiment could be improved to 600 kHz. By comparing the results of MQDT calculations of the hyperfine structure of f Rydberg levels with those of coupled equations calculations, the frame transformation approximation of MQDT was shown to be accurate to within 300 kHz. The extrapolated ionic hyperfine structure of the X 2Sigmag+(v+=0,N+=1) ionic level corresponds to the ab initio prediciton of Babb and Dalgarno [Phys. Rev. A 46, R5317 (1992)] within the experimental error.

Deflection and deceleration of hydrogen Rydberg molecules in inhomogeneous electric fields

Hydrogen molecules are excited in a molecular beam to Rydberg states around n=17-18 and are exposed to the inhomogeneous electric field of an electric dipole. The large dipole moment produced in the selected Stark eigenstates leads to strong forces on the H2 molecules in the inhomogeneous electric field. The trajectories of the molecules are monitored using ion-imaging and time of flight measurements. With the dipole rods mounted parallel to the beam direction, the high-field-seeking and low-field-seeking Stark states are deflected towards and away from the dipole, respectively. The magnitude of the deflection is measured as a function of the parabolic quantum number k and of the duration of the applied field. It is also shown that a large deflection is observed when populating the (17d2)1 state at zero field and switching the dipole field on after a delay. With the dipole mounted perpendicular to the beam direction, the molecules are either accelerated or decelerated as they move towards the dipole. The Rydberg states are found to survive for over 100 micros after the dipole field is switched off before being ionized at the detector and the time of flight is measured. A greater percentage change in kinetic energy is achieved by initial seeding of the beam in helium or neon followed by inhomogeneous field deceleration/acceleration. Molecular dynamics trajectory simulations are presented highlighting the extent to which the trajectories can be predicted based on the known Stark map. The spectroscopy of the populated states is discussed in detail and it is established that the N+=2, J=1, MJ=0 states populated here have a special stability with respect to decay by predissociation.

H+H2 thermal reaction: a convergence of theory and experiment

New experimental and theoretical rate constants for two isotopologs of the simplest chemical reaction, H+H2-->H2+H, are presented. The theoretical results are obtained using accurate quantum dynamics with a converged Born-Oppenheimer potential energy surface and include non-Born-Oppenheimer corrections. The new experiments are carried out using a shock tube and complement earlier investigations over a very large T range, 167 to 2112 K. Experiment and theory now agree perfectly, within experimental error, bringing this 75-year-old scientific problem to completion.

Microwave Spectroscopy at the Dissociation Limit

An ion beam technique has been developed that combines some of the methods of mass spectrometry and molecular spectroscopy and is designed for the study of molecular ions at energy levels lying very close to the lowest dissociation limit. Microwave radiation is used to drive spectroscopic transitions, and electric field dissociation of the weakly bound levels provides a high degree of state selection for sensitive detection of the spectra. The analysis of the spectra requires unconventional approaches to the description of the long-range levels and their spectroscopic study and provides stringent tests of ab initio theories.