Higher-order relativistic corrections to the vibration–rotation levels of H2S

https://doi.org/10.1016/S0009-2614(02)00926-0Get rights and content

Abstract

Relativistic corrections beyond the simple one-electron mass–velocity–Darwin (MVD1) approximation to the ground-state electronic energy of H2S are determined at over 250 geometries. The corrections considered include the two-electron Darwin, the Gaunt and Breit corrections, and the one-electron Lamb shift. Fitted correction surfaces are constructed and used with an accurate ab initio nonrelativistic Born–Oppenheimer potential, determined previously (J. Chem. Phys. 115 (2001) 1229), to calculate vibrational and rotational levels for H232S. The calculations suggest that one- and two-electron relativistic corrections have a noticable influence on the levels of H2S. As for water, the effects considered have markedly different characteristics for the stretching and bending states.

Introduction

The major factor determining the accuracy of variationally computed rovibrational energy levels is the potential energy hypersurface (PES) employed for such calculations. State-of-the-art ab initio electronic structure techniques are now capable [1], [2], [3], [4], [5], [6], perhaps after well-defined and systematic adjustments including extrapolations [3], [7], [8], to predict vibrational band origins (VBOs) and other spectroscopic properties with an accuracy of better than 0.1%. To achieve this accuracy in the case of the ground-state PES of H2O it was necessary to consider not only core–valence electron correlation [1] and coupling between electronic and nuclear motion [9], [10], but also effects originating from special relativity [11], [12], [13], [14]. It became clear from these concerted studies on water that the relativistic effects are sizeable both for the stretching and bending degrees of freedom (Table 3, vide infra). Since previously it was not expected that geometry dependence of relativistic effects can be so large for light molecules, more experience is needed to understand the chemical significance of such small corrections on rovibrational spectra of small light molecules. In this letter we extend our investigation of electronic relativistic effects [4] to the ground-state PES and the rovibrational levels of H2S.

Starting from the most sophisticated molecular theory, quantum electrodynamics (QED), the effects not considered in simple non-relativistic electronic structure theory can be separated into three distinct contributions: (a) one-body effects, arising directly and indirectly from the high velocity of (some of) the electrons; (b) two-body effects, arising through the exchange of virtual photons between electrons; and (c) radiative corrections, involving emission of virtual photons by an electron and subsequent reabsorption of the photon by the same particle (the leading terms are the so-called one- and two-electron Lamb shifts) [11], [13], [14], [15], [16], [17], [18]. All these effects have been investigated for the ground-state PES of H2O, but not for other molecules, with the result that their importance decreases in the order given above. Although almost all of the one-electron relativistic energy correction is associated with the fast-moving core electrons, the differential (geometry-dependent) one-electron relativistic energy corrections are clearly [19] associated with changes in the valence shell; namely, it seems [18], [19], [20] that rehybridisation in the valence shell results in significant changes in the relativistic energies.

One-electron relativistic corrections have been considered for the ground-state PES of H2S [4], [18] and the resulting shifts in the rovibrational states proved to be rather sizeable, of the same order as the corrections found for water. After the dominant one-electron mass–velocity (MV) and Darwin (D1) corrections are covered, one needs to consider the two-electron Darwin (D2) term. It is important for several reasons: (a) it appears to be the most important two-electron relativistic correction; (b) the sum MVD2=MVD1+D2 defines the Coulomb–Pauli approximation as spin–orbit interactions can be neglected for light closed-shell molecules [18] and it is generally assumed that the Coulomb–Pauli Hamiltonian [15] yields good approximations to results obtained from variational four-component solutions of the many-electron relativistic Dirac–Coulomb equation [15], [18]; (c) it is closely related to the spin–spin interaction term; and (d) it allows [13] an estimation of the two-electron Lamb-shift effect. In order to go one step beyond MVD2, the Dirac–Coulomb–Gaunt/Breit Hamiltonian includes magnetic interactions between pairs of electronic currents, neglecting certain O(()2) contributions [17], [21], [11]. We have already probed the Gaunt and Breit energy corrections for water [11], [14].

The relativistic studies mentioned above and simple physical arguments suggest that the inclusion of the geometry dependence of one-body relativistic effects in the ground-state PES of H2S has a noticable effect on the calculated VBOs and rotational term values. On the other hand, two-body effects are expected to be rather insensitive to changes in the geometry not affecting the volume of the molecule; this is especially the case for changes in bond angle. The third and possibly the smallest correction is due to the leading QED effect requiring renormalisation of divergences, the one-electron Lamb-shift effect (self-energy and vacuum polarisation [21]). The effect of Lamb-shift correction energies on the ground-state PES of water and on the related rovibrational states has been investigated by some of us [13], yielding significant corrections up to 1 cm−1 in magnitude for higher-lying vibrational states of water. In that paper it was proposed that for molecular calculations the one-electron Lamb shift can be estimated from the standard D1 term(s) by using simple atomic scale factors. Inclusion of the two-electron Lamb-shift effect in the PES has had, on the other hand, negligible influence on the rovibrational states of water [13]; therefore, it is not considered further in this study. Similarly, a recent calculation has shown that higher-order corrections such as spin–orbit interactions make a negligible contribution to the shape of the water ground state potential energy surface [18].

Section snippets

Computational techniques

The energy corrections due to the two-electron Darwin term, D2, have been computed with aug-cc-pCVTZ CCSD(T) [22], [23] wave functions, at the same level as the previous calculations of MVD1 correction energies [4].

Relativistic energy corrections due to the Gaunt and Breit interactions were obtained in first order of perturbation theory using the four-component Dirac–Hartree–Fock (DHF) wave function [17], the recommended exponent factors for the Gaussian nuclear charge distribution [24], and

Discussion

Table 1, Table 2 summarise calculations for selected vibrational and rotational term values of H2S, respectively. These calculations were all performed with PESs being a sum of non-relativistic and relativistic correction surfaces, where the nonrelativistic surface is the CBS FCI  + C ab initio Born—Oppenheimer (BO) surface of Tarczay et al. [4]. Table 3 compares relativistic energy corrections on the stretching and bending VBOs of H2S to those of water.

The MVD1 relativistic correction [4] for

Conclusions

We have calculated ab initio the contribution of various two-electron relativistic correction terms and the one-electron Lamb shift to the potential energy surface of H2S and investigated their consequence on the vibration–rotation energy levels. Using this information it is possible to quantify the contributions of various terms which are neglected in a standard non-relativistic Born–Oppenheimer Schrödinger treatment of the electronic structure problem. For H2S the largest relativistic

Acknowledgements

The work of P.B. and J.T. was supported by the UK Engineering and Physical Science Research Council under Grant GR/K47702. The work of A.G.C. has been supported by the Scientific Research Foundation of Hungary (OTKA T033074). H.M.Q. acknowledges the support of the Victorian Partnership for Advanced Computing. Scientific exchanges between London and Budapest were supported by the Hungarian–British Joint Academic and Research Programme (Project No. 076). This work was partially carried out on the

References (34)

  • N.F. Zobov et al.

    Chem. Phys. Lett.

    (1996)
  • H.M. Quiney et al.

    Chem. Phys. Lett.

    (1998)
  • A.G. Császár et al.

    Chem. Phys. Lett.

    (1998)
    A.G. Császár et al.

    Chem. Phys. Lett.

    (1999)
  • K. Balasubramanian

    Relativistic Effects in Chemistry, Part A: Theory and Techniques and Part B: Applications

    (1997)
  • Relativistic Effects in Heavy Element Chemistry (REHE, A program of the European Science Foundation) Newsletter No. 13,...
  • G. Tarczay et al.

    Chem. Phys. Lett.

    (2000)
  • I. Kozin et al.

    J. Mol. Spectrosc.

    (1994)
  • J.-M. Flaud et al.

    J. Mol. Spectrosc.

    (1998)
  • A.D. Bykov et al.

    Can. J. Phys.

    (1994)
  • H. Partridge et al.

    J. Chem. Phys.

    (1997)
  • A.G. Császár et al.
  • A.G. Császár et al.
  • G. Tarczay et al.

    J. Chem. Phys.

    (2001)
  • T. van Mourik et al.

    J. Chem. Phys.

    (2001)
  • O.L. Polyansky, A.G. Császár, S.V. Shirin, N.F. Zobov, P. Barletta, J. Tennyson, D.W. Schwenke, P.J. Knowles,...
  • A.G. Császár et al.

    J. Chem. Phys.

    (2001)
  • T. Helgaker et al.

    J. Chem. Phys.

    (1997)
  • Cited by (10)

    • The dipole moment surface for hydrogen sulfide H<inf>2</inf>S

      2015, Journal of Quantitative Spectroscopy and Radiative Transfer
      Citation Excerpt :

      As discussed below, it transpires that our best ab initio results are in much better agreement with directly-measured line intensities in HITRAN than with those line intensities calculated from effective dipole moment models. Core-correlation and relativistic corrections are known to be essential for the calculation of accurate ab initio potential energy surfaces [35,36]. Their effect on DMSs was shown to be small but relevant for the water molecule [16,17].

    • Calculation of the two-electron Darwin term using explicitly correlated wave functions

      2012, Chemical Physics
      Citation Excerpt :

      Quiney and co-workers [4] have shown that the D2 correction needs to be taken into account for accurate computations of vibration-rotation levels of light molecules such as H2O. Moreover, the D2 term is the most-important two-electron relativistic correction: it completes the Coulomb–Pauli approximation for closed-shell systems when added to the D1 and MV terms and it allows for the computation (or estimation) of the two-electron spin–spin term and Lamb shift [5]. In case of the two-electron Darwin (D2) correction, the two major obstacles are the calculation of time-consuming two-electron four-center integrals and the slow convergence to the basis-set limit with X−1, where X is the cardinal number of the orbital basis set [6] or with (L + 1)−1, where L is the maximum angular momentum quantum number contained in the orbital basis set.

    • ExoMol: Molecular line lists for exoplanet and other atmospheres

      2012, Monthly Notices of the Royal Astronomical Society
    View all citing articles on Scopus
    View full text