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Silicon isotopic ratios measured in meteoritic presolar grains can provide useful information about the nucleosynthesis origin of these isotopes if the rates of nuclear reactions responsible for their production are known. One of the key reactions determining the Si isotopic abundances is 29Si(p,gamma)30P. Its reaction rate is not known with sufficient precision due in part to some ambiguous resonance strength values. In the present work, the strength of the E_p = 416.9 keV resonance has been measured with high precision using the activation technique. The new strength of omega_gamma = 219 +- 16 meV can be used in updated reaction rate estimations and astrophysical models., Comment: Accepted for publication in European Physical Journal A

Rovibrational energies, wave functions, and Raman transition moments are reported for the lowest-energy states of the H$_3^+$ molecular ion including the magnetic couplings of the proton spins and molecular rotation in the presence of a weak external magnetic field. The rovibrational-hyperfine-Zeeman Hamiltonian matrix is constructed and diagonalized using the rovibrational eigenstates and the proton spin functions. The developed methodology can be used to compute hyperfine-Zeeman effects also for higher-energy rovibrational excitations of H$_3^+$ and other polyatomic molecules. These developments will guide future experiments extending quantum logic spectroscopy to polyatomic systems.

Full-dimensional (12D) vibrational states of the methanol molecule (CH$_3$OH) have been computed using the GENIUSH-Smolyak approach and the potential energy surface from Qu and Bowman (2013). All vibrational energies are converged better than 0.5 cm$^{-1}$ with respect to the basis and grid size up to the first overtone of the CO stretch, ca. 2000 cm$^{-1}$ beyond the zero-point vibrational energy. About seventy torsion-vibration states are reported and assigned. The computed vibrational energies agree with the available experimental data within less than a few cm$^{-1}$ in most cases, which confirms the good accuracy of the potential energy surface. The computations are carried out using curvilinear normal coordinates with the option of path-following coefficients which minimize the coupling of the small- and large-amplitude motions. It is important to ensure tight numerical fulfilment of the $C_{3\mathrm{v}}$(M) molecular symmetry for every geometry and coefficient set used to define the curvilinear normal coordinates along the torsional coordinate to obtain a faithful description of degeneracy in this floppy system. The reported values may provide a computational reference for fundamental spectroscopy, astrochemistry, and for the search of the proton-to-electron mass ratio variation using the methanol molecule.

This work is concerned with two-spin-1/2-fermion relativistic quantum mechanics, and it is about the construction of one-particle projectors using an inherently two(many)-particle, `explicitly correlated' basis representation, necessary for good numerical convergence of the interaction energy. It is demonstrated that a faithful representation of the one-particle operators, which appear in intermediate but essential computational steps, can be constructed over a many-particle basis set by accounting for the full Hilbert space beyond the physically relevant anti-symmetric subspace. Applications of this development can be foreseen for the computation of quantum-electrodynamics corrections for a correlated relativistic reference state and high-precision relativistic computations of medium-to-high-$Z$ helium-like systems, for which other two-particle projection techniques are unreliable.

Precision physics aims to use atoms and molecules to test and develop the fundamental theory of matter, possibly beyond the Standard Model. Most of the atomic and molecular phenomena are described by the QED (quantum electrodynamics) sector of the Standard Model. Do we have the computational tools, algorithms, and practical equations for the most possible complete computation of atoms and molecules within the QED sector? What is the fundamental equation to start with? Is it still Schr\"odinger's wave equation for molecular matter, or is there anything beyond that? This paper provides a concise overview of the relativistic QED framework and recent numerical developments targeting precision physics and spectroscopy applications with common features with the robust and successful relativistic quantum chemistry methodology.

In several processes of stellar nucleosynthesis, like the astrophysical gamma-process, nuclear reactions involving alpha particles play an important role. The description of these reactions necessitates the knowledge of the alpha-nucleus optical model potential (AOMP) which is highly ambiguous at low, astrophysical energies. This ambiguity introduces a substantial uncertainty in the stellar models for predicting elemental and isotopic abundances. The experimental study of the AOMP is thus necessary which can be implemented by measuring the cross section of alpha-induced nuclear reactions. At low energies, (a,n) reactions are suitable for such a purpose. Therefore, in the present work, the (a,n) cross sections of four Te isotopes have been measured, mostly for the first time, and compared with theoretical predictions. The (a,n) cross sections of 120,122,124,130Te have been measured in the energy range between about 10 and 17 MeV using the activation method. The detection of the gamma radiation following the decay of the radioactive reaction products were used to determine the cross sections. The measured cross sections are compared with statistical model calculations obtained from the widely used TALYS nuclear reaction simulation code. Predictions using various available AOMPs are investigated. It is found that the recently developed Atomki-V2 AOMP provides the best description for all studied reactions and this potential also reproduces well the total reaction cross sections from elastic scattering experiments, when they are available in literature. We recommend therefore to use the astrophysical reaction rates based on this potential for nucleosynthesis models of heavy elements., Comment: Accepted for publication in Phys. Rev. C

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 $1/r_{ix}1/r_{jy}$-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 towards 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.

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 $\Delta K\neq 0$ transition moments arise in this symmetric top system due to strong rovibrational couplings.

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.

Nuclear reactions involving alpha particles play an important role in various astrophysical processes such as the gamma-process of heavy element nucleosynthesis. The poorly known low-energy alpha-nucleus optical (AOMP) potential is a key parameter to estimate the rates of these reactions. The AOMP can be tested by measuring the cross section of alpha-scattering as well as alpha-induced reactions. Low energy elastic alpha-scattering on 144Sm has recently been measured with high precision. The aim of the present work was to complement that work by measuring the (a,n) cross sections on 144Sm at low energies. The experimental data shall be used to constrain the AOMP. From this potential the 144Sm(a,g)148Gd reaction rate can be derived with reduced uncertainties. The 144Sm(a,n)147Gd reaction was studied by bombarding Sm targets with alpha-beams provided by the cyclotron accelerator of Atomki. The cross section was determined using the activation method. The gamma-radiation following the decay of the 147Gd reaction product was measured with a HPGe detector. The experimental data are analyzed within the statistical model. The cross section was measured in the alpha-energy range between 13 and 20 MeV in 1 MeV steps, i.e., from close above the (a,n) threshold. The results were compared with statistical model calculations using various approaches and parametrizations for the AOMP, and excellent agreement was obtained for two recent potentials. However, these potentials cannot reproduce literature data for the 144Sm(a,g)148Gd reaction with the same accuracy. Constraints for the AOMP were derived from an analysis of the new 144Sm(a,n)147Gd data and literature data for 144Sm(a,g)148Gd. These constraints enable a determination of the reaction rate of the 144Sm(a,g)148Gd reaction with significantly reduced uncertainties of less than a factor of two., Comment: Accepted for publication in Phys. Rev. C

All vibrational energies of the (trans-, cis-, delocalized-) formic acid molecule are 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)]. Full-dimensional dipole and polarizability surfaces are fitted to points computed at the CCSD/aug-cc-pVTZ level of theory. Then, body-fixed vibrational dipole and polarizability transition moments are evaluated and used to simulate jet-cooled infrared and Raman spectra of HCOOH. The benchmark-quality vibrational energy, transition moment, and wave function list will be used in further work in comparison with vibrational experiments, and in further rovibrational computations.

The sixteen-component, no-pair Dirac--Coulomb--Breit equation, derived from the Bethe--Salpeter equation, is solved in a variational procedure using Gaussian-type basis functions for the example of positronium, muonium, hydrogen atom, and muonic hydrogen. The $\alpha$ fine-structure-constant dependence of the variational energies, through fitting a function of $\alpha^n$ and $\alpha^n\text{ln}\alpha$ terms, shows excellent agreement with the relevant energy expressions of the (perturbative) non-relativistic QED framework, and thereby, establishes a solid reference for the development of a computational relativistic QED approach.

The development of computational resources has made it possible to determine upper bounds for atomic and molecular energies with high precision. Yet, error bounds to the computed energies have been available only as estimates. In this paper, the Pollak-Martinazzo lower-bound theory, in conjunction with correlated Gaussian basis sets, is elaborated and implemented to provide sub-parts-per-million convergence of the ground- and excited-state energies for the He, Li, and Be atoms. The quality of the lower bounds is comparable to that of the upper bounds obtained from the Ritz method. These results exemplify the power of lower bounds to provide tight estimates of atomic energies.

The triplet contribution is computed to the 1 and 2 $^1S^\text{e}_0$ states of the He atom, to the $1\ ^1S^\text{e}_0$ state of the Li$^+$ and Be${^{2+}}$ ions, and to the $X\ ^1\Sigma_\text{g}^+$ ground state of the H$_2$ 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 wave equation. The no-pair Dirac-Coulomb-Breit energies are converged within a sub-parts-per-billion relative precision using an explicitly correlated Gaussian basis optimized to the non-relativistic energies. The $\alpha$ fine-structure constant dependence of the triplet sector contribution to the variational energy is $\alpha^4E_\text{h}$ at leading order, in agreement with the formal perturbation theory result available from the literature.

Interactions in atomic and molecular systems are dominated by electromagnetic forces and the theoretical framework must be in the quantum regime. The physical theory for the combination of quantum mechanics and electromagnetism, quantum electrodynamics has been established by the mid-twentieth century, primarily as a scattering theory. To describe atoms and molecules, it is important to consider bound states. In the non-relativistic quantum mechanics framework, bound states can be efficiently computed using robust and general methodologies with systematic approximations developed for solving wave equations. With the sight of the development of a computational quantum electrodynamics framework for atomic and molecular matter, the field theoretic Bethe-Salpeter wave equation expressed in space-time coordinates, its exact equal-time variant and emergence of a relativistic wave equation is reviewed. A computational framework, with initial applications and future challenges in relation with precision spectroscopy, is also highlighted.

For rovibronic states corresponding to the $B$ and $B'\ ^1\Sigma_\text{u}^+$ electronic states of the hydrogen molecule, the pre-Born--Oppenheimer (four-particle) non-relativistic energy is converged to a 1-3 parts-per-billion relative precision. The four-particle non-relativistic energy is appended with leading-order relativistic, leading- and estimated higher-order quantum-electrodynamics corrections. The resulting term values referenced to the rovibronic ground state are obtained in an excellent agreement with the experimental results. Further results are reported and discussed for other rovibronic states assignable to the $C\ ^1\Pi_\text{u}$ and the $EF,GK,$ and $HH\ ^1\Sigma_\text{g}^+$ electronic states.

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 Schr\"odinger 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 Schr\"odinger 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.

The automated development of a new ab initio full-dimensional potential energy surface (PES) is reported for the CH$_4\cdot$F$^-$ complex using the ROBOSURFER program package. The new potential provides a near-spectroscopic quality description over a broad configuration range including the methane-ion dissociation, as well as isolated methane vibrations. In particular, it improves upon the earlier [Czak\'o, Braams, Bowman (2008)] PES over intermediate methane-fluoride distances. Full-dimensional (12D) variational vibrational computations using the new PES and the GENIUSH-Smolyak algorithm show that tunneling splittings larger than 0.1 cm$^{-1}$ appear below the top of the interconversion barrier of the four equivalent minima of the complex.

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.

A general computational scheme for the (non-relativistic) 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 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 1:10$^3$ for selected electronic states of the two-electron H$_2$ and H$_3^+$, and the three-electron He$_2^+$ and H+H$_2$ molecular systems. The present work focuses at nuclear configurations near the local minimum of the potential energy surface, but the computations can be repeated also for other structures.

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 $\alpha$ fine-structure constant. The analysis of the polynomial $\alpha$ 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 $\alpha$ dependence of the no-pair DCB energy shows that the higher-order ($\alpha^4 E_\mathrm{h}$) non-radiative QED correction is 5 % of the leading-order ($\alpha^3 E_\mathrm{h}$) non-radiative QED correction for $Z=2$ (He), but it is 40 % already for $Z=4$ (Be$^{2+}$), which indicates that resummation provided by the variational procedure is important already for intermediate nuclear charge numbers.

A benchmark-quality potential energy curve is reported for the H$_3$ system in collinear nuclear configurations. The electronic Schr\"odinger equation is solved using explicitly correlated Gaussian (ECG) basis functions using an optimized fragment initialization technique that significantly reduces the computational cost. As a result, the computed energies improve upon recent orbital-based and ECG computations. Starting from a well-converged basis set, a potential energy curve with an estimated sub-parts-per-billion precision is generated for a series of nuclear configurations using an efficient ECG rescaling approach.

Vibrational states of the formic acid molecule are converged using the GENIUSH-Smolyak approach and the potential energy surface taken from [D. Tew and W. Mizukami, J. Phys. Chem. A 120, 9815 (2016)]. The quantum nuclear motion is described by using the $cis$-$trans$ torsional coordinate and eight curvilinear normal coordinates defined with respect to an instantaneous reference configuration changing as a function of the torsional degree of freedom. Harmonic oscillator basis functions are used for the curvilinear normal coordinates, a Fourier basis for the torsional coordinate, and a simple basis pruning condition is combined with a Smolyak integration grid. $Trans$, $cis$, and $delocalized$ vibrational states are reported up to and slightly beyond the isomerization barrier.

A variational procedure is described for the computation of the non-adiabatic mass-correction tensor applicable for multi-dimensional electronic manifolds. The 30-year-old computations of Wolniewicz, Dressler, and their co-workers are appended with the computed vibronic mass-correction functions corresponding to the $EF$-$GK$-$H\bar{H}$-$S5$-$S6$ $^1\Sigma_\text{g}^+$ manifold of the hydrogen molecule. Initial results are reported for the vibronic energies. Necessary further improvements and further developments are discussed.

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 [Paper I: P. Jeszenszki, D. Ferenc, and E. M\'atyus (2022)]. Both a perturbative and a fully variational inclusion of the Breit term is considered. The no-pair DC plus perturbative Breit as well as the no-pair Dirac-Coulomb-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.

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. The subsequent paper [Paper II: D. Ferenc, P. Jeszenszki, and E. M\'atyus (2022)] describes the implementation of the Breit interaction in this framework.

This paper elaborates the integral transformation technique of [K. Pachucki, W. Cencek, and J. Komasa, J. Chem. Phys. 122, 184101 (2005)] and uses it for the case of the non-relativistic kinetic and Coulomb potential energy operators, as well as for the relativistic mass-velocity and Darwin terms. The techniques are tested for the ground electronic state of the helium atom and new results are reported for the ground electronic state of the H$_3^+$ molecular ion near its equilibrium structure.

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 `only' 2-3 times larger basis set sizes and in an excellent numerical agreement in comparison with the more efficient tailor-made approach.

A variational solution procedure is reported for the many-particle no-pair Dirac-Coulomb-Breit Hamiltonian 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.

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 Schr\"odinger 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 the full-dimensional potential energy surface (PES) and electric dipole moment surface developed by Qu and Bowman [Phys. Chem. Chem. Phys. 18, 24835 (2016); J. Chem. Phys. 148, 241713 (2018)]. Good results are obtained for several fundamental and combination bands in comparison with with jet-cooled vibrational spectroscopy experiments, but the description of the $\nu_8$ and $\nu_9$ 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.

Molecular structure is often considered as emerging from the decoherence effect of the environment. Electrons are part of the environment of the nuclei in a molecule. In this work, their contribution to the classical-like geometrical relationships often observed between nuclei in molecular systems is investigated. Reduced density matrix (RDM) elements are evaluated from electron-nucleus wave functions. The computational results show that the electrons play a role in the localization of the nuclei around specific geometries. Although the electronic environment alone cannot explain molecular symmetry-broken isomers, it can contribute to their dynamical stability by reducing off-diagonal RDM elements., Comment: Theoretical Chemistry Accounts: Theory, Computation, and Modeling, Springer Verlag, In press

The rovibrational intervals of the $^4$He$_2^+$ molecular ion in its $X\ ^2\Sigma_\text{u}^+$ ground electronic state are computed by including the non-adiabatic, 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, $70.937\ 589(23)(60)_\text{sys}$ cm$^{-1}$ [L. Semeria, P. Jansen, G.-M. Camenisch, F. Mellini, H. Schmutz, and F. Merkt, Phys. Rev. Lett. 124, 213001 (2020)].

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, H$_2$, D$_2$, T$_2$, and HeH$^+$.

A full-dimensional \emph{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 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 $\langle r \rangle_0$ methane structure. Allowing 10 basis functions for the methane fragment, the rms of the bound state vibrational energies is reduced to 0.07 cm$^{-1}$, which is much better than the 3D models. 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 the route for the study of the role of dispersion forces on the excited methane vibrations and the energy transfer from the intra- to the intermolecular vibrational modes.

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.

Various explicitly correlated Gaussian (ECG) basis sets are considered for the solution of the molecular Schr\"odinger equation with particular attention to the simplest polyatomic system, H$_3^+$. Shortcomings and advantages are discussed for plain ECGs, ECGs with the global vector representation, floating ECGs and their numerical projection, and ECGs with complex parameters. The discussion is accompanied with particle density plots to visualize the observations. In order to be able to use large complex ECG basis sets in molecular calculations, a numerically stable algorithm is developed, the efficiency of which is demonstrated for the lowest rotationally and vibrationally excited states of H$_2$ and H$_3^+$., Comment: 22 pages, 5 figures

The mass-correction function is evaluated for selected excited states of the hydrogen molecule within a single-state non-adiabatic treatment. Its qualitative features are studied under the avoided crossing of the $EF$ with the $GK$ state and also for the outer well of the $H\bar{H}$ state. For the $H\bar{H}$ 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.

Selected states of the $EF\ ^1\Sigma_\mathrm{g}^+$ electronic manifold of the hydrogen molecule are computed as resonances of the four-body problem. Systematic improvement of the basis representation for the variational treatment is achieved through an energy-tracking optimization procedure. The resulting non-relativistic energy is converged within a few nano Hartree, while the predissociative width is found to be negligible at this level of accuracy. The four-particle non-relativistic energies are appended with relativistic and quantum electrodynamics corrections which close the gap between the experimental observations and earlier theoretical work.

Methodological progress is reported in the challenging direction of a black-box-type variational solution of the (ro)vibrational Schr\"odinger 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 [E. M\'atyus, G. Czak\'o, and A. G. Cs\'asz\'ar, J. Chem. Phys. 130, 134112 (2009)] and (ii) the Smolyak non-product grid method of [G. Avila and T. 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, multi-dimensional (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 CH$_4\cdot$Ar complex treated in full (12D) vibrational dimensionality.

The quantum mechanical motion of the atomic nuclei is considered over a single- or a multi-dimensional 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 $\mathcal{O}(\varepsilon^{n+1})$ through a unitary transformation of the electronic subspace and the corresponding $n$th-order effective Hamiltonian is derived for the quantum nuclear motion. Explicit but general formulae 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 multi-dimensional, explicitly coupled electronic band, the second-order Hamiltonian contains the usual BO terms and non-adiabatic 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.

Peculiarities of the intermolecular rovibrational quantum dynamics of the methane-argon complex are studied using a new, ab initio potential energy surface [Y. N. Kalugina, S. E. Lokshtanov, V. N. Cherepanov, and A. A. Vigasin, J. Chem. Phys. 144, 054304 (2016)], variational rovibrational computations, and detailed symmetry considerations within the molecular symmetry group of this floppy complex as well as within the point groups corresponding to the local minimum structures. The computed (ro)vibrational states up to and beyond the dissociation asymptote are characterized using two limiting models: the rigidly rotating molecule's model and the coupled-rotor model of the rigidly rotating methane and an argon atom orbiting around it.

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.) 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 Schr\"odinger equation including the diagonal Born--Oppenheimer as well as the mass-tensor corrections.

The main motivation for this work is the exploration of rotational-vibrational states corresponding to electronic excitations in a pre-Born-Oppenheimer quantum theory of molecules. These states are often embedded in the continuum of the lower-lying dissociation channel of the same symmetry, and thus are thought to be resonances. In order to calculate rovibronic resonances, the pre-Born-Oppenheimer variational approach of [J. Chem. Phys. 137, 024104 (2012)], based on the usage of explicitly correlated Gaussian functions and the global vector representation, is extended with the complex coordinate rotation method. The developed computer program is used to calculate resonance energies and widths for the three-particle positronium anion, Ps$^-$, and the four-particle positronium molecule, Ps$_2$. Furthermore, the excited bound and resonance rovibronic states of the four-particle H$_2$ molecule are also considered. Resonance energies and widths are estimated for the lowest-energy resonances of H$_2$ beyond the $b\ ^3\Sigma_\mathrm{u}^+$ continuum.

The mass-correction functions in the second-order non-adiabatic Hamiltonian are computed for the $^4$He$^+_2$ molecular ion using the variational method, floating explicitly correlated Gaussian functions, and a general coordinate-transformation formalism. When non-adiabatic rovibrational energy levels are computed using these (coordinate-dependent) mass-correction functions and a highly accurate potential energy and diagonal Born-Oppenheimer correction curve, significantly improved theoretical results are obtained for the nine rotational and two rovibrational intervals known from high-resolution spectroscopy experiments.

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 for this molecular ion with rovibrational results including non-adiabatic corrections., Comment: 29 pages, 3 figures, 4 tables

In the present work we introduce a computational approach to the absolute rovibrational quantum partition function using the path-integral formalism of quantum mechanics in combination with the nested sampling technique. The numerical applicability of path-integral nested sampling is demonstrated for small molecules of spectroscopic interest. The computational cost of the method is determined by the evaluation time of a point on the potential-energy surface (PES). For efficient PES implementations, the path-integral nested-sampling method can be a viable alternative to the direct Boltzmann summation technique of variationally computed rovibrational energies, especially for medium-sized molecules and at elevated temperatures.

In pre-Born-Oppenheimer (pre-BO) theory a molecule is considered as a quantum system as a whole, including the electrons and the atomic nuclei on the same footing. This approach is fundamentally different from the traditional quantum chemistry treatment, which relies on the separation of the motion of the electrons and the atomic nuclei. A fully quantum mechanical treatment of molecules has a great promise for future developments and applications. Its most accurate versions may contribute to the definition of new schemes for metrology and testing fundamental physical theories; its approximate versions can provide an efficient theoretical description for molecule-positron interactions and, in general, it would circumvent the tedious computation and fitting of potential energy surfaces and non-adiabatic coupling vectors. To achieve these goals, the review points out important technical and fundamental open questions. Most interestingly, the reconciliation of pre-BO theory with the classical chemistry knowledge touches upon fundamental problems related to the measurement problem of quantum mechanics.