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

Exact quantum dynamics developments for floppy molecular systems and complexes

Molecular rotation, vibration, internal rotation, isomerization, tunneling, intermolecular dynamics of weakly and strongly interacting systems, intra-to-inter-molecular energy transfer, hindered rotation and hindered translation over surfaces are important types of molecular motions. Their fundamentally correct and detailed description can be obtained by solving the nuclear Schrödinger equation on a potential energy surface. Many of the chemically interesting processes involve quantum nuclear motions which are 'delocalized' over multiple potential energy wells. These 'large-amplitude' motions in addition to the high dimensionality of the vibrational problem represent challenges to the current (ro)vibrational methodology. A review of the quantum nuclear motion methodology is provided, current bottlenecks of solving the nuclear Schrödinger equation are identified, and solution strategies are reviewed. Technical details, computational results, and analysis of these results in terms of limiting models and spectroscopically relevant concepts are highlighted for selected numerical examples.

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.

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 Hn+ (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'.

Non-adiabatic mass correction for excited states of molecular hydrogen: Improvement for the outer-well HH¯ 1Σg + term values

The mass-correction function is evaluated for selected excited states of the hydrogen molecule within a single-state nonadiabatic treatment. Its qualitative features are studied at the avoided crossing of the EF with the GK state and also for the outer well of the HH¯ state. For the HH¯ state, a negative mass correction is obtained for the vibrational motion near the outer minimum, which accounts for most of the deviation between experiment and earlier theoretical work.

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.

Toward breaking the curse of dimensionality in (ro)vibrational computations of molecular systems with multiple large-amplitude motions

Methodological progress is reported in the challenging direction of a black-box-type variational solution of the (ro)vibrational Schrödinger equation applicable to floppy, polyatomic systems with multiple large-amplitude motions. This progress is achieved through the combination of (i) the numerical kinetic-energy operator (KEO) approach of Mátyus et al. [J. Chem. Phys. 130, 134112 (2009)] and (ii) the Smolyak nonproduct grid method of Avila and Carrington, Jr. [J. Chem. Phys. 131, 174103 (2009)]. The numerical representation of the KEO makes it possible to choose internal coordinates and a body-fixed frame best suited for the molecular system. The Smolyak scheme reduces the size of the direct-product grid representation by orders of magnitude, while retaining some of the useful features of it. As a result, multidimensional (ro)vibrational states are computed with system-adapted coordinates, a compact basis- and grid-representation, and an iterative eigensolver. Details of the methodological developments and the first numerical applications are presented for the CH₄·Ar complex treated in full (12D) vibrational dimensionality.

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.

The kinetic energy operator for distance-dependent effective nuclear masses: Derivation for a triatomic molecule

The kinetic energy operator for triatomic molecules with coordinate or distance-dependent nuclear masses has been derived. By combination of the chain rule method and the analysis of infinitesimal variations of molecular coordinates, a simple and general technique for the construction of the kinetic energy operator has been proposed. The asymptotic properties of the Hamiltonian have been investigated with respect to the ratio of the electron and proton mass. We have demonstrated that an ad hoc introduction of distance (and direction) dependent nuclear masses in Cartesian coordinates preserves the total rotational invariance of the problem. With the help of Wigner rotation functions, an effective Hamiltonian for nuclear motion can be derived. In the derivation, we have focused on the effective trinuclear Hamiltonian. All necessary matrix elements are given in closed analytical form. Preliminary results for the influence of non-adiabaticity on vibrational band origins are presented for H₃⁺.

Core-valence stockholder AIM analysis and its connection to nonadiabatic effects in small molecules

A previous theory of separation of motions of core and valence fractions of electrons in a molecule [J. R. Mohallem et al., Chem. Phys. Lett. 501, 575 (2011)] is invoked as basis for the useful concept of Atoms-in-Molecules (AIM) in the stockholder scheme. The output is a new tool for the analysis of the chemical bond that identifies core and valence electron density fractions (core-valence stockholder AIM (CVSAIM)). One-electron effective potentials for each atom are developed, which allow the identification of the parts of the AIM which move along with the nuclei (cores). This procedure results in a general method for obtaining effective masses that yields accurate non-adiabatic corrections to vibrational energies, necessary to attain cm-1 accuracy in molecular spectroscopy. The clear-cut determination of the core masses is exemplified for either homonuclear (H2+, H2) or heteronuclear (HeH+, LiH) molecules. The connection of CVSAIM with independent physically meaningful quantities can resume the question of whether they are observable or not.

Rovibrational quantum dynamical computations for deuterated isotopologues of the methane–water dimer

Rovibrational states of methane–water isotopologues are computed in a variational procedure and the wave functions are analyzed in terms of the rigid-rotor and coupled-rotors models.