Methane dimer rovibrational states and Raman transition moments.

Benchmark-quality rovibrational data are reported for the methane dimer from variational nuclear motion computations using an ab initio intermolecular potential energy surface reported by [M. P. Metz et al., Phys. Chem. Chem. Phys., 2019, 21, 13504-13525]. A simple polarizability model is used to compute Raman transition moments that may be relevant for future direct observation of the intermolecular dynamics. Non-negligible ΔK ≠ 0 transition moments arise in this symmetric top system due to strong rovibrational couplings.

QED corrections to the correlated relativistic energy: One-photon processes.

This work is a collection of initial calculations and formal considerations within the Salpeter-Sucher exact equal-time relativistic quantum electrodynamics framework. The calculations are carried out as preparation for the computation of pair, retardation, and radiative corrections to the relativistic energy of correlated two-spin-1/2-fermion systems. In this work, particular attention is paid to the retardation and the "one-loop" self-energy corrections, which are known to be among the largest corrections to the correlated relativistic energy. The theoretical development is supplemented with identifying formal connections to the non-relativistic quantum electrodynamics framework, which is based on a correlated but non-relativistic reference, as well as to the "1/Z approach," which is built on a relativistic but independent-particle zeroth order. The two complementary directions currently provide the theoretical framework for light atomic-molecular precision spectroscopy and heavy-atom phenomena. The present theoretical efforts pave the way for relativistic QED corrections to (explicitly) correlated relativistic computations.

Regularized relativistic corrections for polyelectronic and polyatomic systems with explicitly correlated Gaussians.

Drachmann's regularization approach is implemented for floating explicitly correlated Gaussians (fECGs) and molecular systems. Earlier applications of drachmannized relativistic corrections for molecular systems were hindered due to the unknown analytic matrix elements of 1/rix1/rjy-type operators with fECGs. In the present work, one of the 1/r factors is approximated by a linear combination of Gaussians, which results in calculable integrals. The numerical approach is found to be precise and robust over a range of molecular systems and nuclear configurations, and thus, it opens the route toward an automated evaluation of high-precision relativistic corrections over potential energy surfaces of polyatomic systems. Furthermore, the newly developed integration approach makes it possible to construct the matrix representation of the square of the electronic Hamiltonian relevant for energy lower-bound as well as time-dependent computations of molecular systems with a flexible and high-precision fECG basis representation.

Vibrational infrared and Raman spectra of HCOOH from variational computations.

All vibrational energies of the formic acid molecule in different forms (trans-, cis-, delocalized-) were converged up to 4500 cm-1 beyond the zero-point vibrational energy with the GENIUSH-Smolyak variational approach and using an ab initio potential energy surface [D. P. Tew and W. Mizukami, J. Phys. Chem. A, 120, 9815-9828 (2016)]. The full-dimensional dipole and polarizability surfaces were fitted to points computed at the CCSD/aug-cc-pVTZ level of theory. Then, body-fixed vibrational dipole and polarizability transition moments were evaluated and used to simulate jet-cooled infrared and Raman spectra of HCOOH. The benchmark-quality vibrational energy, transition moment, and wave function data will be used in further work for comparison with vibrational experiments, and in further rovibrational computations.

Evaluation of the Bethe Logarithm: From Atom to Chemical Reaction.

A general computational scheme for the (nonrelativistic) Bethe logarithm is developed, opening the route to "routine" evaluation of the leading-order quantum electrodynamics correction (QED) relevant for spectroscopic applications for small polyatomic and polyelectronic molecular systems. The implementation relies on the Schwartz method and minimization of a Hylleraas functional. In relation with electronically excited states, a projection technique is considered, which ensures positive definiteness of the functional over the entire parameter (photon momentum) range. Using this implementation, the Bethe logarithm is converged to a relative precision better than 1:103 for selected electronic states of the two-electron H2 and H3+, and the three-electron He2+ and H+H2 molecular systems. The present work focuses on nuclear configurations near the local minimum of the potential energy surface, but the computations can be repeated also for other structures.

Relativistic two-electron atomic and molecular energies using LS coupling and double groups: Role of the triplet contributions to singlet states.

The triplet contribution is computed to the 1 and 2 S0e1 states of the He atom, to the 1S0e1 state of the Li+ and Be2+ ions, and to the X1Σg + ground state of the H2 molecule by extensive use of double-group symmetry (equivalent to LS coupling for the atomic systems) during the course of the variational solution of the no-pair Dirac-Coulomb-Breit (DCB) wave equation. The no-pair DCB energies are converged within sub-parts-per-billion relative precision, using an explicitly correlated Gaussian basis optimized to the non-relativistic energies. The α fine-structure constant dependence of the triplet sector contribution to the variational energy is α4Eh at leading order, in agreement with the formal perturbation theory result available from the literature.

High-dimensional neural network potentials for accurate vibrational frequencies: the formic acid dimer benchmark.

In recent years, machine learning potentials (MLP) for atomistic simulations have attracted a lot of attention in chemistry and materials science. Many new approaches have been developed with the primary aim to transfer the accuracy of electronic structure calculations to large condensed systems containing thousands of atoms. In spite of these advances, the reliability of modern MLPs in reproducing the subtle details of the multi-dimensional potential-energy surface is still difficult to assess for such systems. On the other hand, moderately sized systems enabling the application of tools for thorough and systematic quality-control are nowadays rarely investigated. In this work we use benchmark-quality harmonic and anharmonic vibrational frequencies as a sensitive probe for the validation of high-dimensional neural network potentials. For the case of the formic acid dimer, a frequently studied model system for which stringent spectroscopic data became recently available, we show that high-quality frequencies can be obtained from state-of-the-art calculations in excellent agreement with coupled cluster theory and experimental data.

Variational vs perturbative relativistic energies for small and light atomic and molecular systems.

Variational and perturbative relativistic energies are computed and compared for two-electron atoms and molecules with low nuclear charge numbers. In general, good agreement of the two approaches is observed. Remaining deviations can be attributed to higher-order relativistic, also called non-radiative quantum electrodynamics (QED), corrections of the perturbative approach that are automatically included in the variational solution of the no-pair Dirac-Coulomb-Breit (DCB) equation to all orders of the α fine-structure constant. The analysis of the polynomial α dependence of the DCB energy makes it possible to determine the leading-order relativistic correction to the non-relativistic energy to high precision without regularization. Contributions from the Breit-Pauli Hamiltonian, for which expectation values converge slowly due the singular terms, are implicitly included in the variational procedure. The α dependence of the no-pair DCB energy shows that the higher-order (α4Eh) non-radiative QED correction is 5% of the leading-order (α3Eh) non-radiative QED correction for Z = 2 (He), but it is 40% already for Z = 4 (Be2+), which indicates that resummation provided by the variational procedure is important already for intermediate nuclear charge numbers.

On the Breit interaction in an explicitly correlated variational Dirac-Coulomb framework.

The Breit interaction is implemented in the no-pair variational Dirac-Coulomb (DC) framework using an explicitly correlated Gaussian basis reported in the previous paper [P. Jeszenszki, D. Ferenc, and E. Mátyus, J. Chem. Phys. 156, 084111 (2022)]. Both a perturbative and a fully variational inclusion of the Breit term are considered. The no-pair DC plus perturbative Breit and the no-pair DC-Breit energies are compared with perturbation theory results including the Breit-Pauli Hamiltonian and leading-order non-radiative quantum electrodynamics corrections for low Z values. Possible reasons for the observed deviations are discussed.

Variational Dirac-Coulomb explicitly correlated computations for atoms and molecules.

The Dirac-Coulomb equation with positive-energy projection is solved using explicitly correlated Gaussian functions. The algorithm and computational procedure aims for a parts-per-billion convergence of the energy to provide a starting point for further comparison and further developments in relation with high-resolution atomic and molecular spectroscopy. Besides a detailed discussion of the implementation of the fundamental spinor structure, permutation, and point-group symmetries, various options for the positive-energy projection procedure are presented. The no-pair Dirac-Coulomb energy converged to a parts-per-billion precision is compared with perturbative results for atomic and molecular systems with small nuclear charge numbers. Paper II [D. Ferenc, P. Jeszenszki, and E. Mátyus, J. Chem. Phys. 156, 084110 (2022).] describes the implementation of the Breit interaction in this framework.

Fingerprint region of the formic acid dimer: variational vibrational computations in curvilinear coordinates.

Curvilinear kinetic energy models are developed for variational nuclear motion computations including the inter- and the low-frequency intra-molecular degrees of freedom of the formic acid dimer. The coupling of the inter- and intra-molecular modes is studied by solving the vibrational Schrödinger equation for a series of vibrational models, from two up to ten active vibrational degrees of freedom by selecting various combinations of active modes and constrained coordinate values. Vibrational states, nodal assignment, and infrared vibrational intensity information is computed using the full-dimensional potential energy surface (PES) and electric dipole moment surface developed by Qu and Bowman [Phys. Chem. Chem. Phys., 2016, 18, 24835; J. Chem. Phys., 2018, 148, 241713]. Good results are obtained for several fundamental and combination bands in comparison with jet-cooled vibrational spectroscopy experiments, but the description of the ν8 and ν9 fundamental vibrations, which are close in energy and have the same symmetry, appears to be problematic. For further progress in comparison with experiment, the potential energy surface, and in particular, its multi-dimensional couplings representation, requires further improvement.

Orientational decoherence within molecules and emergence of the molecular shape.

The question of classicality is addressed in relation with the shape of the nuclear skeleton of molecular systems. As the most natural environment, the electrons of the molecule are considered as continuously monitoring agents for the nuclei. For this picture, an elementary formalism of decoherence theory is developed and numerical results are presented for few-particle systems. The numerical examples suggest that the electron-nucleus Coulomb interaction is sufficient for inducing a blurred shape with strong quantum coherences in compounds of the lightest elements, H2, D2, T2, and HeH+.

Performance of a black-box-type rovibrational method in comparison with a tailor-made approach: Case study for the methane-water dimer.

The present work intends to join and respond to the excellent and thoroughly documented rovibrational study of X. G. Wang and T. Carrington, Jr. [J. Chem. Phys. 154, 124112 (2021)] that used an approach tailored for floppy dimers with an analytic dimer Hamiltonian and a non-product basis set including Wigner D functions. It is shown in the present work that the GENIUSH black-box-type rovibrational method can approach the performance of the tailor-made computation for the example of the floppy methane-water dimer. Rovibrational transition energies and intensities are obtained in the black-box-type computation with a twice as large basis set and in excellent numerical agreement in comparison with the more efficient tailor-made approach.

All-order explicitly correlated relativistic computations for atoms and molecules.

A variational solution procedure is reported for the many-particle no-pair Dirac-Coulomb and Dirac-Coulomb-Breit Hamiltonians aiming at a parts-per-billion (ppb) convergence of the atomic and molecular energies, described within the fixed nuclei approximation. The procedure is tested for nuclear charge numbers from Z = 1 (hydrogen) to 28 (iron). Already for the lowest Z values, a significant difference is observed from leading-order Foldy-Woythusen perturbation theory, but the observed deviations are smaller than the estimated self-energy and vacuum polarization corrections.

Nonadiabatic, Relativistic, and Leading-Order QED Corrections for Rovibrational Intervals of

The rovibrational intervals of the ^{4}He_{2}^{+} molecular ion in its X ^{2}Σ_{u}^{+} ground electronic state are computed by including the nonadiabatic, relativistic, and leading-order quantum-electrodynamics corrections. Good agreement of theory and experiment is observed for the rotational excitation series of the vibrational ground state and the fundamental vibration. The lowest-energy rotational interval is computed to be 70.937 69(10) cm^{-1} in agreement with the most recently reported experimental value, 70.937 589(23)(60)_{sys} cm^{-1} [L. Semeria et al., Phys. Rev. Lett. 124, 213001 (2020)PRLTAO0031-900710.1103/PhysRevLett.124.213001].

Exact quantum dynamics background of dispersion interactions: case study for CH4·Ar in full (12) dimensions.

A full-dimensional ab initio potential energy surface of spectroscopic quality is developed for the van-der-Waals complex of a methane molecule and an argon atom. Variational vibrational states are computed on this surface including all twelve (12) vibrational degrees of freedom of the methane-argon complex using the GENIUSH computer program and the Smolyak sparse grid method. The full-dimensional computations make it possible to study the fine details of the interaction and distortion effects and to make a direct assessment of the reduced-dimensionality models often used in the quantum dynamics study of weakly-bound complexes. A 12-dimensional (12D) vibrational computation including only a single harmonic oscillator basis function (9D) to describe the methane fragment (for which we use the ground-state effective structure as the reference structure) has a 0.40 cm-1 root-mean-square error (rms) with respect to the converged 12D bound-state excitation energies, which is less than half of the rms of the 3D model set up with the 〈r〉0 methane structure. Allowing 10 basis functions for the methane fragment in a 12D computation performs much better than the 3D models by reducing the rms of the bound state vibrational energies to 0.07 cm-1. The full-dimensional potential energy surface correctly describes the dissociation of the system, which together with further development of the variational (ro)vibrational methodology opens a route to the study of the role of dispersion forces in the excited methane vibrations and the energy transfer from the intra- to the intermolecular vibrational modes.

Molecular dimers of methane clathrates: ab initio potential energy surfaces and variational vibrational states.

Motivated by the energetic and environmental relevance of methane clathrates, highly accurate ab initio potential energy surfaces (PESs) have been developed for the three possible dimers of the methane and water molecules: (H2O)2, CH4·H2O, and (CH4)2. While only a single monomer geometry was used for each monomer in the ab initio calculations, the PES parameterization makes it possible to produce distinct surfaces for all isotopologues within the rigid-monomer approximation. The PESs were fitted to computations at the frozen-core coupled-cluster level with single, double, and non-iterative triple excitations, employing basis sets of augmented triple- and quadruple-zeta quality plus bond functions, followed by extrapolations to the complete basis set limit. The long-range parts of the PESs are computed using the asymptotic version of symmetry-adapted perturbation theory based on a density-functional description of the monomers. All PESs are polarizable, i.e., in cluster or condensed-phase applications they approximate many-body effects by the induced dipole polarization model. The PESs were developed in a fully automated procedure applying the autoPES method, which is used for the first time to generate near-spectroscopic quality surfaces. The stationary points (SPs) on the PESs have been determined and compared with literature data. For CH4·H2O, previously unknown SPs have been identified and the first detailed study of the (CH4)2 potential energy landscape has been carried out. The PESs were used in variational quantum nuclear motion computations. For the water dimer, the resulting vibrational transitions are in excellent agreement with available high-resolution spectroscopic data. For (CH4)2, the intermonomer vibrational states are reported for the first time.

Full-dimensional (12D) variational vibrational states of CH4·F-: Interplay of anharmonicity and tunneling.

The complex of a methane molecule and a fluoride anion represents a 12-dimensional (12D), four-well vibrational problem with multiple large-amplitude motions, which has challenged the quantum dynamics community for years. The present work reports vibrational band origins and tunneling splittings obtained in a full-dimensional variational vibrational computation using the GENIUSH program and the Smolyak quadrature scheme. The converged 12D vibrational band origins and tunneling splittings confirm complementary aspects of the earlier full- and reduced-dimensionality studies: (1) the tunneling splittings are smaller than 0.02 cm-1; (2) a single-well treatment is not sufficient (except perhaps the zero-point vibration) due to a significant anharmonicity over the wells; and thus, (3) a full-dimensional treatment appears to be necessary. The present computations extend to a higher energy range than earlier work, show that the tunneling splittings increase upon vibrational excitation of the complex, and indicate non-negligible "heavy-atom" tunneling.

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.

Effective non-adiabatic Hamiltonians for the quantum nuclear motion over coupled electronic states.

The quantum mechanical motion of the atomic nuclei is considered over a single- or a multidimensional subspace of electronic states which is separated by a gap from the rest of the electronic spectrum over the relevant range of nuclear configurations. The electron-nucleus Hamiltonian is block-diagonalized up to O(εn+1) through a unitary transformation of the electronic subspace, and the corresponding nth-order effective Hamiltonian is derived for the quantum nuclear motion. Explicit but general formulas are given for the second- and the third-order corrections. As a special case, the second-order Hamiltonian corresponding to an isolated electronic state is recovered which contains the coordinate-dependent mass-correction terms in the nuclear kinetic energy operator. For a multidimensional, explicitly coupled electronic band, the second-order Hamiltonian contains the usual Born-Oppenheimer terms and nonadiabatic corrections, but generalized mass-correction terms appear as well. These, earlier neglected terms, perturbatively account for the outlying (discrete and continuous) electronic states not included in the explicitly coupled electronic subspace.