Non-adiabatic theory in terms of a single potential energy surface. The vibration–rotation levels of and

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.

Numerical Equivalence of Diabatic and Adiabatic Representations in Diatomic Molecules

The (time-independent) Schrödinger equation for atomistic systems is solved by using the adiabatic potential energy curves (PECs) and the associated adiabatic approximation. In cases where interactions between electronic states become important, the associated nonadiabatic effects are taken into account via derivative couplings (DDRs), also known as nonadiabatic couplings (NACs). For diatomic molecules, the corresponding PECs in the adiabatic representation are characterized by avoided crossings. The alternative to the adiabatic approach is the diabatic representation obtained via a unitary transformation of the adiabatic states by minimizing the DDRs. For diatomics, the diabatic representation has zero DDR and nondiagonal diabatic couplings ensue. The two representations are fully equivalent and so should be the rovibronic energies and wave functions, which result from the solution of the corresponding Schrödinger equations. We demonstrate (for the first time) the numerical equivalence between the adiabatic and diabatic rovibronic calculations of diatomic molecules using the ab initio curves of yttrium oxide (YO) and carbon monohydride (CH) as examples of two-state systems, where YO is characterized by a strong NAC, while CH has a strong diabatic coupling. Rovibronic energies and wave functions are computed using a new diabatic module implemented in the variational rovibronic code Duo. We show that it is important to include both the diagonal Born-Oppenheimer correction and nondiagonal DDRs. We also show that the convergence of the vibronic energy calculations can strongly depend on the representation of nuclear motion used and that no one representation is best in all cases.

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'.

Hyperfine-interaction-induced g/u mixing and its implication on the existence of the first excited vibrational level of the A + Σ u + 2 state of H 2 + and on the scattering length of the H + H+ collision

Ab initio calculations of the energy level structure of H 2 + that include relativistic and radiative corrections to nonrelativistic energies and the diagonal part of the hyperfine interaction have predicted the existence of four bound rovibrational levels [(v = 0, N = 0 - 2) and (v = 1, N = 0)] of the first electronically excited ( A +   Σ u + 2 ) state of H 2 + , the (v = 1, N = 0) level having a calculated binding energy of only E _b = 1.082 219 8(4)·10^-9 E_h and leading to an extremely large scattering length of 750(5) a₀ for the H⁺ + H collision [J. Carbonell et al., J. Phys. B: At., Mol. Opt. Phys. 37, 2997 (2004)]. We present an investigation of the nonadiabatic coupling between the first two electronic states ( X +   Σ g + 2 and A +   Σ u + 2 ) of H 2 + induced by the Fermi-contact term of the hyperfine-coupling Hamiltonian. This interaction term, which mixes states of total spin quantum number G = 1/2, is rigorously implemented in a close-coupling approach to solve the spin-rovibronic Schrödinger equation. We show that it mixes states of gerade and ungerade electronic symmetry, that it shifts the positions of all weakly bound rovibrational states of H 2 + , and that it affects both the positions and widths of its shape resonances. The calculations demonstrate that the G = 1/2 hyperfine component of the A⁺ (v = 1, N = 0) state does not exist and that, for G = 1/2, the s-wave scattering lengths of the H⁺ + H(1s) collision are -578(6) a₀ and -43(4) a₀ for the F = 0 and F = 1 hyperfine components of the H(1s) atom, respectively. The binding energy of the G = 3/2 hyperfine component of the A⁺ (v = 1, N = 0) state is not significantly affected by the hyperfine interaction and the corresponding scattering length for the H⁺ + H(1s, F = 1) collision is 757(7) a₀.

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.

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⁺.

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.

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.

Observation and Calculation of the Quasibound Rovibrational Levels of the Electronic Ground State of H2+

Although the existence of quasibound rotational levels of the X^{+} ^{2}Σ_{g}^{+}  ground state of H_{2}^{+} was predicted a long time ago, these states have never been observed. Calculated positions and widths of quasibound rotational levels located close to the top of the centrifugal barriers have not been reported either. Given the role that such states play in the recombination of H(1s) and H^{+} to form H_{2}^{+}, this lack of data may be regarded as one of the largest unknown aspects of this otherwise accurately known fundamental molecular cation. We present measurements of the positions and widths of the lowest-lying quasibound rotational levels of H_{2}^{+} and compare the experimental results with the positions and widths we calculate using a potential model for the X^{+} state of H_{2}^{+} which includes adiabatic, nonadiabatic, relativistic, and radiative corrections to the Born-Oppenheimer approximation.

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.

A systematic investigation of the ground state potential energy surface of H3+

Based on different ab initio electronic structure calculations (CI-R12 and Gaussian Geminals) of the Born-Oppenheimer electronic energy E(BO) of H(3)(+) from high to highest quality, we build up a potential energy surface which represents a highly reliable form of the topology of the whole potential region, locally and globally. We use the CI-R12 method in order to get within reasonable CPU-time a relatively dense grid of energy points. We demonstrate that CI-R12 is good enough to give an accurate surface, i.e., Gaussian Geminals are not absolutely necessary. For different types of potential energy surface fits, we performed variational calculations of all bound vibrational states, including resonances above the dissociation limit, for total angular momentum J = 0. We clarify the differences between different fits of the energy to various functional forms of the potential surface. Small rms-values (<1 cm(-1)) of the fit do not provide precise information about the interpolatory behaviour of the fit functions.

Very accurate potential energy curve of the He2+ ion

A very accurate ground-state potential energy curve (PEC) of the He(2)(+) molecule is calculated with 1200 explicitly correlated Gaussian functions with shifted centers in the range between 0.9 and 100 a(0). The calculations include the adiabatic corrections determined for the (3)He(4)He(+), (3)He(2)(+), and (4)He(2)(+) isotopologues. The absolute accuracy of the PEC is better than 0.05 cm(-1) and that of the adiabatic corrections is around 0.01 cm(-1). The depths of the PECs augmented with the adiabatic corrections for the three isotopologues are: 19 956.708 cm(-1) for (4)He(2)(+), 19 957.054 cm(-1) for (3)He(4)He(+), and 19 957.401 cm(-1) for (3)He(2)(+). The rovibrational energies are also determined. For (3)He(4)He(+) the computed rovibrational transitions corresponding to the ν = 1-0 band differ from the experiment by less than 0.005 cm(-1). For the rovibrational transitions corresponding to the ν = 23-22 band the difference is around 0.012 cm(-1). Presently, this represents the best agreement between theory and experiment for He(2)(+).

Rovibrational energy levels of H3(+) with energies above the barrier to linearity

The H(3)(+) potential energy surface is sampled at 5900 geometries with the emphasis on the nonequilibrium and asymptotic points. Apart from the Born-Oppenheimer energy converged to the accuracy better than 0.02 cm(-1), the adiabatic and the leading relativistic corrections are computed at each geometry. To represent analytically the potential energy surface, the parameters of a power series are determined by fitting to the computed energy points. Possible choice of nuclear masses simulating the nonadiabatic effects in solving the nuclear Schrodinger equation is analyzed. A set of theoretically predicted rovibrational transitions is confronted with the experimental data in the 10,700-13,700 cm(-1) window of the spectra.