Velocity Formulations for Hyper-Rayleigh Scattering Optical Activity Spectroscopy: Addressing the Origin-dependence Problem

TL;DR

This paper introduces a velocity form formulation for hyper-Rayleigh scattering optical activity spectroscopy that ensures origin-independence and is suitable for approximate wavefunction calculations.

Contribution

It provides an alternative velocity-based formulation of hyperpolarizabilities that addresses origin-dependence issues present in the length formulation.

Findings

Velocity formulation is origin-independent by design.

Relations between length and velocity formulations ensure consistent gauge-origin shifts.

Velocity formulation is more basis set dependent but suitable for approximate wavefunctions.

Abstract

The theory of hyper-Rayleigh scattering optical activity (HRS-OA) spectroscopy has previously been described within the length formulation of the pure electric-dipole and mixed (electric-dipole/magnetic-dipole and electric-dipole/electric-quadrupole) first hyperpolarizabilities required for the description of this process. In this work, we provide an alternative formulation of these pure and mixed hyperpolarizabilities. This new formulation made use of the velocity form of the electric-dipole and electric-quadrupole moment operators that enter in the quadratic response functions. A one-to-one correspondence is found for the gauge-origin shifts obtained in the two formulations. These relations ensure the origin-independence of the theory also for the velocity formulation. Furthermore, even though the basis set dependence of the velocity formulation is more significant compared to the length one, the former is origin-independent by design. This property makes it particularly suitable for calculations of HRS-OA invariants using approximated (variational or not) wavefunctions.