Non-adiabatic mass-correction functions and rovibrational states of 4 He 2 + ( X 2 Σ u + )

A nearly complete treatment of the effect of non-adiabaticity on rovibrational energies of H3+ (Part III)

In this article, significant contributions of non-adiabaticity for the rovibrational bound states up to 25 000 cm-1 and total angular momentum J = 0-20 of H3+ are investigated. A coupled-perturbed full configuration interaction (CP-FCI) treatment is applied to calculate all couplings between electronic states caused by the nuclear motion. These derivative couplings were evaluated up to the second order by means of a perturbation treatment and include all nuclear Cartesian first and second derivatives of the electronic wavefunctions. In particular, the coupling of special derivatives with respect to r and R in the Jacobi coordinate representation is more significant than thought. The perturbation approach is especially optimal for the treatment of weak non-adiabaticity in case of rovibrational energies in H3+ and had not been available before for H3+ or other triatomics. Using exclusively Gaussian basis functions for CP-FCI appears to be sufficient, because explicit correlated wavefunctions are already used for all other potential energy contributions. Our work is an extension of earlier non-adiabatic investigations based on first derivative couplings of electronic states that led to the concept of geometry-dependent effective nuclear masses and which needs only a single potential energy surface for the dynamics. The implementation allows us to include all non-adiabatic effects up to the order of O(μ-2), μ being the reduced nuclear mass. Our treatment works for any isotopologue and for the whole potential energy curve or surface. By this treatment, a further reduction in deviations to experimental data for most rovibrational levels to less than 0.1 cm-1 is possible. For the related transition frequencies, 1366 of 1720 known rovibrational transitions in H3+ have deviations less than 0.1 cm-1 without using any empirically adjustable parameters or optimizing the nuclear mass for a specific transition. For many questionable assignments (deviations >0.3 cm-1) of observed transitions in H3+, a new labeling is proposed.

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.

Nonadiabatic, Relativistic, and Leading-Order QED Corrections for Rovibrational Intervals of ^{4}He_{2}^{+} (X ^{2}Σ_{u}^{+})

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].

Precision Measurements in Few-Electron Molecules: The Ionization Energy of Metastable ^{4}He_{2} and the First Rotational Interval of ^{4}He_{2}^{+}

Molecular helium represents a benchmark system for testing ab initio calculations on few-electron molecules. We report on the determination of the adiabatic ionization energy of the a ^{3}Σ_{u}^{+} state of He_{2}, corresponding to the energy interval between the a ^{3}Σ_{u}^{+} (v^{''}=0, N^{''}=1) state of He_{2} and the X^{+} ^{2}Σ_{u}^{+} (v^{+}=0, N^{+}=1) state of He_{2}^{+}, and of the lowest rotational interval of He_{2}^{+}. These measurements rely on the excitation of metastable He_{2} molecules to high Rydberg states using frequency-comb-calibrated continuous-wave UV radiation in a counterpropagating laser-beam setup. The observed Rydberg states were extrapolated to their series limit using multichannel quantum-defect theory. The ionization energy of He_{2} (a ^{3}Σ_{u}^{+}) and the lowest rotational interval of He_{2}^{+} (X^{+} ^{2}Σ_{u}^{+}) are 34 301.207 002(23)±0.000 037_{syst}  cm^{-1} and 70.937 589(23)±0.000 060_{syst}  cm^{-1}, respectively.

H3 + as a five-body problem described with explicitly correlated Gaussian basis sets

Various explicitly correlated Gaussian (ECG) basis sets are considered for the solution of the molecular Schrödinger equation with particular attention to the simplest polyatomic system, H₃ ⁺. 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 H₂ and H₃ ⁺.

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'.

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.

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.