Calibration-quality adiabatic potential energy surfaces for H3+ and its isotopologues

Trihydrogen Cation Helium Clusters: A New Potential Energy Surface

We present a new analytical potential energy surface (PES) for the interaction between the trihydrogen cation and a He atom,H 3 + - H e ${{H}_{3}^{+}-He}$ , in its electronic ground state. The proposed PES has been built as a sum of two contributions: a polarization energy term due to the electric field generated by the molecular cation at the position of the polarizable He atom, and an exchange-repulsion and dispersion interactions represented by a sum of "atom-bond" potentials between the three bonds ofH 3 + ${{H}_{3}^{+}}$ and the He atom. All parameters of this new PES have been chosen and fitted from data obtained from high-level ab-initio calculations. Using this new PES plus the Aziz-Slaman potential for the interaction between Helium atoms and assuming pair-wise interactions, we carry out classical Basin-Hopping (BH) global optimization, semiclassical BH with Zero Point Energy corrections, and quantum Diffusion Monte Carlo simulations. We have found the minimum energy configurations of small He clusters doped withH 3 + ${{H}_{3}^{+}}$ ,H 3 + H e N ${{H}_{3}^{+}{\left(He\right)}_{N}}$ , with N=1-16. The study of the energies of these clusters allows us to find a pronounced anomaly for N=12, in perfect agreement with previous experimental findings, which we relate to a greater relative stability of this aggregate.

Hitting the Trifecta: How to Simultaneously Push the Limits of Schrödinger Solution with Respect to System Size, Convergence Accuracy, and Number of Computed States

Methods for solving the Schrödinger equation without approximation are in high demand but are notoriously computationally expensive. In practical terms, there are just three primary factors that currently limit what can be achieved: 1) system size/dimensionality; 2) energy level excitation; and 3) numerical convergence accuracy. Broadly speaking, current methods can deliver on any two of these three goals, but achieving all three at once remains an enormous challenge. In this paper, we shall demonstrate how to "hit the trifecta" in the context of molecular vibrational spectroscopy calculations. In particular, we compute the lowest 1000 vibrational states for the six-atom acetonitrile molecule (CH₃CN), to a numerical convergence of accuracy 10^-2 cm^-1 or better. These calculations encompass all vibrational states throughout most of the dynamically relevant range (i.e., up to ∼4250 cm^-1 above the ground state), computed in full quantum dimensionality (12 dimensions), to near spectroscopic accuracy. To our knowledge, no such vibrational spectroscopy calculation has ever previously been performed.

Three states global fittings with improved long range: singlet and triplet states of H

Full dimensional analytical fits of the coupled potential energy surfaces for the three lower singlet and triplet adiabatic states of H are developed, providing analytic derivatives and non-adiabatic coupling matrix elements. The fits are highly accurate and include an improved description of the long range interactions, including new terms for the description of the long range in the diatomic fits and the atom-diatom dissociation channels. The fits are based on the DIM formalism including three body terms in Hamiltonian matrix elements, each of them obeying S₂ permutational symmetry, where the positive charge is placed in either of the three hydrogen atoms, but the full system obeys S₃ permutational symmetry, invariant under all permutations of the nuclei. The ab initio points used in the fitting are obtained from a complete basis set extrapolation, made for all electronic states. Total root mean square errors of the fits are 27 and 12 cm^-1, for the singlet and triplet states, respectively. The errors in the channels are lower than 2 cm^-1 and 6 cm^-1 for the H + H and H⁺ + H₂ channels respectively. The new fits have been used to calculate the rovibrational bound states of the lowest singlet and lowest triplet states showing very good agreement with previous calculations in the literature.

On neglecting Coriolis and related couplings in first-principles rovibrational spectroscopy: Considerations of symmetry, accuracy, and simplicity. II. Case studies for H2O isotopologues, H3+, O3, and NH3

For centuries, it has been known that vibrational and rotational degrees of freedom are in general not separable. Nevertheless, surprisingly little is known about the best strategies for approximately separating these degrees of freedom in practice-even in the case of semirigid molecules, where the separation is most meaningful. There is also some confusion in the literature about the proper way to quantify the magnitude of the Coriolis (i.e., rotation-vibration) coupling in rovibrational Hamiltonians or its effect on the rovibrational eigenenergies. In this study, a vibrational-coordinate-independent metric is proposed to quantify the magnitude of the Coriolis contribution to the rovibrational Hamiltonian. The impact of Coriolis coupling on the rovibrational eigenenergies is computed numerically exactly, using both full and various truncated Hamiltonians. The role played by the choice of the vibrational coordinate system-and especially by the choice of "embedding" or body-fixed frame-is examined extensively, both numerically and analytically. This investigation targets several molecular prototypes, all of which serve as important benchmarks for the high-resolution spectroscopic community. Most of these are triatomic molecules, including water (H₂¹⁶O), its deuterated isotopologues (D₂¹⁶O and HD¹⁶O), H₃⁺, and ozone (¹⁶O₃), but the tetratomic ammonia molecule (¹⁴NH₃) is also investigated. These studies provide important insight into the nature of Coriolis coupling under various circumstances. The findings of this study also have significant practical ramifications, vis-à-vis the use of simplifying numerical approximation techniques in nuclear-motion computations.

Near-resonant effects in the quantum dynamics of the H + H2 + → H2 + H+ charge transfer reaction and isotopic variants

The non-adiabatic quantum dynamics of the H + H₂ ⁺ → H₂ + H⁺ charge transfer reactions, and some isotopic variants, is studied with an accurate wave packet method. A recently developed 3 × 3 diabatic potential model is used, which is based on very accurate ab initio calculations and includes the long-range interactions for ground and excited states. It is found that for initial H₂ ⁺(v = 0), the quasi-degenerate H₂(v' = 4) non-reactive charge transfer product is enhanced, producing an increase in the reaction probability and cross section. It becomes the dominant channel from collision energies above 0.2 eV, producing a ratio between v' = 4 and the rest of v's, which that increase up to 1 eV. The H + H₂ ⁺ → H₂ ⁺ + H exchange reaction channel is nearly negligible, while the reactive and non-reactive charge transfer reaction channels are of the same order, except that corresponding to H₂(v' = 4), and the two charge transfer processes compete below 0.2 eV. This enhancement is expected to play an important vibrational and isotopic effect that needs to be evaluated. For the three proton case, the problem of the permutation symmetry is discussed when using reactant Jacobi coordinates.

Analysis of QED and non-adiabaticity effects on the rovibrational spectrum of H3 + using geometry-dependent effective nuclear masses

The influence of QED effects (including one- and two-electron Lamb-shift, Araki-Sucher term, one-loop self-energy, and finite nuclear size correction) together with non-adiabatic effects on the rovibrational bound states of H₃ ⁺ has been investigated. Non-adiabaticity is modeled by using geometry-dependent effective nuclear masses together with only one single potential energy surface. In conclusion, for rovibrational states below 20 000 cm^-1, QED and relativistic effects do nearly compensate, and a potential energy surface based on Born-Oppenheimer energies and diagonal adiabatic corrections has nearly the same quality as the one including relativity with QED; the deviations between the two approaches for individual rovibrational states are mostly below 0.02 cm^-1. The inclusion of non-adiabatic effects is important, and it reduces deviations from experiments mostly below 0.1 cm^-1.

General mathematical formulation of scattering processes in atom-diatomic collisions in the RmatReact methodology

Accurately modelling cold and ultracold reactive collisions occurring over deep potential wells, such as [Formula: see text], requires the development of new theoretical and computational methodologies. One potentially useful framework is the R-matrix method adopted widely for electron-molecule collisions which has more recently been applied to non-reactive heavy-particle collisions such as Ar-Ar. The existing treatment of non-reactive elastic and inelastic scattering needs to be substantially extended to enable modelling of reactive collisions: this is the subject of this paper. Herein, we develop the general mathematical formulation for non-reactive elastic and inelastic scattering, photoassociation, photodissociation, charge exchange and reactive scattering using the R-matrix method. Of particular note is that the inner region, of central importance to calculable R-matrix methodologies, must be finite in all scattering coordinates rather than a single scattering coordinate as for non-reactive scattering. This article is part of a discussion meeting issue 'Advances in hydrogen molecular ions: H₃⁺, H₅⁺ and beyond'.

Photodissociation as a probe of the H3+ avoided crossing seam

Experiments are conducted to investigate the role of the avoided crossing seam in the photodissociation of H₃⁺. Three-dimensional imaging of dissociation products is used to determine the kinetic energy release and branching ratio among the fragmentation channels. Vibrational distributions are measured by dissociative charge transfer of H₂⁺ products. It is found that the photodissociation of hot H₃⁺ in the near-ultraviolet produces cold H₂⁺, but hot H₂. Modelling the wavepacket dynamics along the repulsive potential energy surface accounts for the repopulation of the ground potential energy surface. The role of the avoided crossing seam is emphasized and its importance for the astrophysically relevant charge transfer reactions underlined. This article is part of a discussion meeting issue 'Advances in hydrogen molecular ions: H₃⁺, H₅⁺ and beyond'.

Non-adiabatic effects in the H3+ spectrum

The effect of non-adiabatic coupling on the computed rovibrational energy levels amounts to about 2 cm-1 for H3+ and must be included in high-accuracy calculations. Different strategies to obtain the corresponding energy shifts are reviewed in the article. A promising way is to introduce effective vibrational reduced masses that depend on the nuclear configuration. A new empirical method that uses the stockholder atoms-in-molecules approach to this effect is presented and applied to H3+. Furthermore, a highly accurate potential energy surface for the D3+ isotopologue, which includes relativistic and leading quantum electrodynamic terms, is constructed and used to analyse the observed rovibrational frequencies for this molecule. Accurate band origins are obtained that improve existing data. This article is part of a discussion meeting issue 'Advances in hydrogen molecular ions: H3+, H5+ and beyond'.

Hydrogen molecular ions: H3+, H5+ and beyond

Three decades after the spectroscopic detection of H₃⁺ in space, the inspiring developments in physics, chemistry and astronomy of H_n⁺ (n = 3, 5, 7) systems, which led to this Royal Society Discussion Meeting, are reviewed, the present state of the art as represented by the meeting surveyed and future lines of research considered. This article is part of a discussion meeting issue 'Advances in hydrogen molecular ions: H₃⁺, H₅⁺ and beyond'.

Highly accurate experimentally determined energy levels of H3. J Chem Phys 2019;150:214303. [PMID: 31176322 DOI: 10.1063/1.5099454]

A sub-Doppler rovibrational spectroscopic survey of H₃ ⁺ has been conducted which included 36 transitions in the ν₂ ← 0 fundamental band, 15 transitions in the 2ν₂ ²←ν₂ hot band, and 7 transitions in the 2ν₂ ²←0 overtone band, improving the uncertainties of most transitions by more than an order of magnitude to ∼4 MHz. Combination differences were used to determine relative energy levels and forbidden rotational transitions up to J = 6. A fit of the ground state to an effective Hamiltonian was used to connect ortho and para states, and to determine the absolute energy levels relative to the forbidden (0, 0) state. Ultimately, 62 rovibrational energy levels in the ground, ν₂, and 2ν₂ ² states were determined with ∼10 MHz uncertainty. Comparing the experimentally determined energy levels with ab initio calculations revealed an unexpected dependence of the residuals on the quantum number G.

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.

Explicitly correlated Gaussian functions with shifted-center and projection techniques in pre-Born-Oppenheimer calculations

Numerical projection methods are elaborated for the calculation of eigenstates of the non-relativistic many-particle Coulomb Hamiltonian with selected rotational and parity quantum numbers employing shifted explicitly correlated Gaussian functions, which are, in general, not eigenfunctions of the total angular momentum and parity operators. The increased computational cost of numerically projecting the basis functions onto the irreducible representations of the three dimensional rotation-inversion group is the price to pay for the increased flexibility of the basis functions. This increased flexibility allowed us to achieve a substantial improvement for the variational upper bound to the Pauli-allowed ground-state energy of the H 3 + = { p + , p + , p + , e - , e - } molecular ion treated as an explicit five-particle system. We compare our pre-Born-Oppenheimer result obtained for this molecular ion with rotational-vibrational calculations carried out on a potential energy surface.

Low-temperature chemistry using the R-matrix method

Techniques for producing cold and ultracold molecules are enabling the study of chemical reactions and scattering at the quantum scattering limit, with only a few partial waves contributing to the incident channel, leading to the observation and even full control of state-to-state collisions in this regime. A new R-matrix formalism is presented for tackling problems involving low- and ultra-low energy collisions. This general formalism is particularly appropriate for slow collisions occurring on potential energy surfaces with deep wells. The many resonance states make such systems hard to treat theoretically but offer the best prospects for novel physics: resonances are already being widely used to control diatomic systems and should provide the route to steering ultracold reactions. Our R-matrix-based formalism builds on the progress made in variational calculations of molecular spectra by using these methods to provide wavefunctions for the whole system at short internuclear distances, (a regime known as the inner region). These wavefunctions are used to construct collision energy-dependent R-matrices which can then be propagated to give cross sections at each collision energy. The method is formulated for ultracold collision systems with differing numbers of atoms.

Investigation of Nonadiabatic Effects for the Vibrational Spectrum of a Triatomic Molecule: The Use of a Single Potential Energy Surface with Distance-Dependent Masses for H3. J Phys Chem A 2017;121:7016-7030. [PMID: 28820589 DOI: 10.1021/acs.jpca.7b04703]

On the basis of first-principles, the influence of nonadiabatic effects on the vibrational bound states of H₃⁺ has been investigated using distance-dependent reduced masses and only one single potential energy surface. For these new vibrational calculations, potentials based on explicitly correlated wave functions are used where, in addition, adiabatic corrections and relativistic contributions are taken into account. For the first time, several different fully distance-dependent reduced mass surfaces in three dimensions have been incorporated in the vibrational calculations.

Orbit-orbit relativistic correction calculated with all-electron molecular explicitly correlated Gaussians

An algorithm for calculating the first-order electronic orbit-orbit magnetic interaction correction for an electronic wave function expanded in terms of all-electron explicitly correlated molecular Gaussian (ECG) functions with shifted centers is derived and implemented. The algorithm is tested in calculations concerning the H₂ molecule. It is also applied in calculations for LiH and H₃⁺ molecular systems. The implementation completes our work on the leading relativistic correction for ECGs and paves the way for very accurate ECG calculations of ground and excited potential energy surfaces (PESs) of small molecules with two and more nuclei and two and more electrons, such as HeH^-, H₃⁺, HeH₂⁺, and LiH₂⁺. The PESs will be used to determine rovibrational spectra of the systems.

Algorithms for calculating mass-velocity and Darwin relativistic corrections with n-electron explicitly correlated Gaussians with shifted centers

Algorithms for calculating the leading mass-velocity (MV) and Darwin (D) relativistic corrections are derived for electronic wave functions expanded in terms of n-electron explicitly correlated Gaussian functions with shifted centers and without pre-exponential angular factors. The algorithms are implemented and tested in calculations of MV and D corrections for several points on the ground-state potential energy curves of the H2 and LiH molecules. The algorithms are general and can be applied in calculations of systems with an arbitrary number of electrons.

Modelling non-adiabatic effects in H₃⁺: solution of the rovibrational Schrödinger equation with motion-dependent masses and mass surfaces

Introducing different rotational and vibrational masses in the nuclear-motion Hamiltonian is a simple phenomenological way to model rovibrational non-adiabaticity. It is shown on the example of the molecular ion H3(+), for which a global adiabatic potential energy surface accurate to better than 0.1 cm(-1) exists [M. Pavanello, L. Adamowicz, A. Alijah, N. F. Zobov, I. I. Mizus, O. L. Polyansky, J. Tennyson, T. Szidarovszky, A. G. Császár, M. Berg et al., Phys. Rev. Lett. 108, 023002 (2012)], that the motion-dependent mass concept yields much more accurate rovibrational energy levels but, unusually, the results are dependent upon the choice of the embedding of the molecule-fixed frame. Correct degeneracies and an improved agreement with experimental data are obtained if an Eckart embedding corresponding to a reference structure of D(3h) point-group symmetry is employed. The vibrational mass of the proton in H3(+) is optimized by minimizing the root-mean-square (rms) deviation between the computed and recent high-accuracy experimental transitions. The best vibrational mass obtained is larger than the nuclear mass of the proton by approximately one third of an electron mass, m(opt,p)((v))=m(nuc,p)+0.31224m(e). This optimized vibrational mass, along with a nuclear rotational mass, reduces the rms deviation of the experimental and computed rovibrational transitions by an order of magnitude. Finally, it is shown that an extension of the algorithm allowing the use of motion-dependent masses can deal with coordinate-dependent mass surfaces in the rovibrational Hamiltonian, as well.

Non-adiabatic effects in thermochemistry, spectroscopy and kinetics: the general importance of all three Born–Oppenheimer breakdown corrections

Analytical and numerical solutions describing Born–Oppenheimer breakdown in a simple, widely applicable, model depict shortcomings in modern computational methods.

Core-mass nonadiabatic corrections to molecules: H2, H2+, and isotopologues

For high-precision calculations of rovibrational states of light molecules, it is essential to include non-adiabatic corrections. In the absence of crossings of potential energy surfaces, they can be incorporated in a single surface picture through coordinate-dependent vibrational and rotational reduced masses. We present a compact method for their evaluation and relate in particular the vibrational mass to a well defined nuclear core mass derived from a Mulliken analysis of the electronic density. For the rotational mass we propose a simple, but very effective parametrization. The use of these masses in the nuclear Schrödinger equation yields numerical data for the corrections of a much higher quality than can be obtained with optimized constant masses, typically better than 0.1 cm(-1). We demonstrate the method for H(2), H(2)(+), and singly deuterated isotopologues. Isotopic asymmetry does not present any particular difficulty. Generalization to polyatomic molecules is straightforward.

State specific stabilization of H+ + H2(j) collision complexes

Stabilization of H3(+) collision complexes has been studied at nominal temperatures between 11 and 33 K using a 22-pole radio frequency (rf) ion trap. Apparent binary rate coefficients, k(*) = kr + k3[H2], have been measured for para- and normal-hydrogen at number densities between some 10(11) and 10(14) cm(-3). The state specific rate coefficients extracted for radiative stabilization, kr(T;j), are all below 2 × 10(-16) cm(3) s(-1). There is a slight tendency to decrease with increasing temperature. In contrast to simple expectations, kr(11 K;j) is for j = 0 a factor of 2 smaller than for j = 1. The ternary rate coefficients for p-H2 show a rather steep T-dependence; however, they are increasing with temperature. The state specific ternary rate coefficients, k3(T;j), measured for j = 0 and derived for j = 1 from measurements with n-H2, differ by an order of magnitude. Most of these surprising observations are in disagreement with predictions from standard association models, which are based on statistical assumptions and the separation of complex formation and competition between stabilization and decay. Most probably, the unexpected collision dynamics are due to the fact that, at the low translational energies of the present experiment, only a small number of partial waves participate. This should make exact quantum mechanical calculations of kr feasible. More complex is three-body stabilization, because it occurs on the H5(+) potential energy surface.

Spectroscopy of H3+ based on a new high-accuracy global potential energy surface

The molecular ion H(3)(+) is the simplest polyatomic and poly-electronic molecular system, and its spectrum constitutes an important benchmark for which precise answers can be obtained ab initio from the equations of quantum mechanics. Significant progress in the computation of the ro-vibrational spectrum of H(3)(+) is discussed. A new, global potential energy surface (PES) based on ab initio points computed with an average accuracy of 0.01 cm(-1) relative to the non-relativistic limit has recently been constructed. An analytical representation of these points is provided, exhibiting a standard deviation of 0.097 cm(-1). Problems with earlier fits are discussed. The new PES is used for the computation of transition frequencies. Recently measured lines at visible wavelengths combined with previously determined infrared ro-vibrational data show that an accuracy of the order of 0.1 cm(-1) is achieved by these computations. In order to achieve this degree of accuracy, relativistic, adiabatic and non-adiabatic effects must be properly accounted for. The accuracy of these calculations facilitates the reassignment of some measured lines, further reducing the standard deviation between experiment and theory.

Progress in calculating the potential energy surface of H3+

The most accurate electronic structure calculations are performed using wave function expansions in terms of basis functions explicitly dependent on the inter-electron distances. In our recent work, we use such basis functions to calculate a highly accurate potential energy surface (PES) for the H(3)(+) ion. The functions are explicitly correlated Gaussians, which include inter-electron distances in the exponent. Key to obtaining the high accuracy in the calculations has been the use of the analytical energy gradient determined with respect to the Gaussian exponential parameters in the minimization of the Rayleigh-Ritz variational energy functional. The effective elimination of linear dependences between the basis functions and the automatic adjustment of the positions of the Gaussian centres to the changing molecular geometry of the system are the keys to the success of the computational procedure. After adiabatic and relativistic corrections are added to the PES and with an effective accounting of the non-adiabatic effects in the calculation of the rotational/vibrational states, the experimental H(3)(+) rovibrational spectrum is reproduced at the 0.1 cm(-1) accuracy level up to 16,600 cm(-1) above the ground state.