Dependence of the multipole moments, static polarizabilities, and static hyperpolarizabilities of the hydrogen molecule on the H–H separation in the ground singlet state

Abstract

In this work, we provide values for the quadrupole moment Θ, the hexadecapole moment Φ, the dipole polarizability α, the quadrupole polarizability C, the dipole-octopole polarizability E, the second dipole hyperpolarizability γ, and the dipole-dipole-quadrupole hyperpolarizability B for the hydrogen molecule in the ground singlet state, evaluated by finite-field configuration interaction singles and doubles (CISD) and coupled-cluster singles and doubles (CCSD) methods for 26 different H–H separations r, ranging from 0.567 a.u. to 10.0 a.u. Results obtained with various large correlation-consistent basis sets are compared at the vibrationally averaged bond length r0 in the ground state. Results over the full range of r values are presented at the CISD/d-aug-cc-pV6Z level for all of the independent components of the property tensors. In general, our values agree well with previous ab initio results of high accuracy for the ranges of H–H distances that have been treated in common. To our knowledge, for H2 in the ground state, our results are the first to be reported in the literature for Φ for r > 7.0 a.u., γ and B for r > 6.0 a.u., and C and E for any H–H separation outside a narrow range around the potential minimum. Quantum Monte Carlo values of Θ have been given previously for H–H distances out to 10.0 a.u., but the statistical error is relatively large for r > 7.0 a.u. At the larger r values in this work, αxx and αzz show the expected functional forms, to leading order in r−1. As r increases further, Θ and Φ vanish, while α, γ, and the components of B converge to twice the isolated-atom values. Components of C and E diverge as r increases. Vibrationally averaged values of the properties are reported for all of the bound states (vibrational quantum numbers υ = 0–14) with rotational quantum numbers J = 0–3.

Absorption spectra of electrified hydrogen molecules

“…Good numerical accuracy was demonstrated by Miliordos & Hunt (2018), with test calculations for the case of a ground state hydrogen atom yielding electrical re-sponse tensors that differ from the known (exact) values by only ∼ 0.4%. Although that is a small error it is large compared to the systematic errors in the eigenfunctions we have constructed ( §2.1), so we do not expect the latter to limit the accuracy of the matrix elements we have evaluated.…”

“…Recent ab initio calculations by Miliordos & Hunt (2018) have substantially improved the characterisation of the electrical response of the H 2 molecule. In addition to the second-rank response tensors -the permanent quadrupole moment, Θ, and the dipole polarizability, α -Miliordos & Hunt (2018) evaluated the permanent hexadecapole moment, Φ, the quadrupole polarizability, C, the dipole-octupole polarizability, E, the dipole-dipolequadrupole hyperpolarizability, B, and the second dipole hyperpolarizability, γ.…”

“…Because of the symmetries of the H 2 molecule (point group D ∞h ) only 16 independent components are required to fully describe the seven response tensors under consideration (McLean & Yoshimine 1967;Buckingham 1967;Miliordos & Hunt 2018). For the convenience of readers, the behaviour of those components as a function of internuclear separation is presented in Appendix A; they are specified in a cartesian coordinate system (X, Y, Z) in which the Z-axis is aligned with the internuclear separation vector.…”

“…For the convenience of readers, the behaviour of those components as a function of internuclear separation is presented in Appendix A; they are specified in a cartesian coordinate system (X, Y, Z) in which the Z-axis is aligned with the internuclear separation vector. The response tensors in the laboratory coordinate system, (x, y, z), depend also on the orientation of the internuclear axis and can be related to those in the molecular basis using the expressions given by Buckingham (1967), for Θ, α, C, B, and γ, and by Bohr & Hunt (1987) for Φ and E. We are thus in a position to evaluate the perturbation of the energy of the molecule (Miliordos & Hunt 2018)…”

Raman signal from a hindered hydrogen rotor

“…is the polarizability in the reference frame of the hydrogen molecule. α is a known parameter, taken to have a value of 1.43 from previous experimental work [23,24] and considered to be pressure and temperature independent here. The rotation matrix R transforms the E fields into the frame of the molecule, before they interact with the polarizability ellipsoid.…”