A Density-Matrix Formulation of Vibrational Circular Dichroism Spectra for Møller-Plesset and Configuration Interaction Methods
Abstract
We reformulate the overlap of wave-function derivatives that defines the atomic axial tensors (AATs) and velocity-gauge atomic polar tensors (VG-APTs) used to compute vibrational circular dichroism (VCD) rotatory strengths. In place of our earlier determinant-level derivatives, the new approach expresses these tensors as contractions involving molecular-orbital derivative integrals, perturbed and unperturbed one- and two-electron density matrices, and dot products of the perturbed correlated wave-function amplitudes. Following the elegant treatment of the diagonal Born-Oppenheimer correction by Gauss and co-workers, the formulation holds for any real, normalized CI-like wave function. The two-electron density contribution vanishes exactly, because one perturbation is real (nuclear) and the other imaginary (magnetic field or momentum); this removes multiple terms from our earlier AAT and VG-APT expressions for second-order Møller-Plesset perturbation theory (MP2) and configuration interaction singles and doubles (CISD). We also present a simplified form of the origin-independent length-gauge [LG(OI)] rotatory strength that avoids the singular-value decomposition of the outer product of the length-gauge APT and VG-APT vectors for each mode, and that clarifies how the LG(OI) and velocity-gauge rotatory strengths are related. Applying the density-based AATs and VG-APTs to (P)-hydrogen peroxide, (\textit{S})-methyloxirane, and (\textit{S})-3-chloro-1-butyne, we find that discrepancies among the gauges arise chiefly from basis-set incompleteness in the VG-APT. The charge-sum rule serves as a useful measure of that incompleteness, independent of the correlation method.
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BibTeX
@misc{krishnan2026densitymatrix,
title = {{A Density-Matrix Formulation of Vibrational Circular Dichroism Spectra for Møller-Plesset and Configuration Interaction Methods}},
author = {Aparna Krishnan and Brendan M. Shumberger and T. Daniel Crawford},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.35392},
eprint = {2610.35392},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.35392},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A Density-Matrix Formulation of Vibrational Circular Dichroism Spectra for Møller-Plesset and Configuration Interaction Methods
AU - Aparna Krishnan
AU - Brendan M. Shumberger
AU - T. Daniel Crawford
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2610.35392
UR - https://arxiv.org/abs/2610.35392
AB - We reformulate the overlap of wave-function derivatives that defines the atomic axial tensors (AATs) and velocity-gauge atomic polar tensors (VG-APTs) used to compute vibrational circular dichroism (VCD) rotatory strengths. In place of our earlier determinant-level derivatives, the new approach expresses these tensors as contractions involving molecular-orbital derivative integrals, perturbed and unperturbed one- and two-electron density matrices, and dot products of the perturbed correlated wave-function amplitudes. Following the elegant treatment of the diagonal Born-Oppenheimer correction by Gauss and co-workers, the formulation holds for any real, normalized CI-like wave function. The two-electron density contribution vanishes exactly, because one perturbation is real (nuclear) and the other imaginary (magnetic field or momentum); this removes multiple terms from our earlier AAT and VG-APT expressions for second-order Møller-Plesset perturbation theory (MP2) and configuration interaction singles and doubles (CISD). We also present a simplified form of the origin-independent length-gauge [LG(OI)] rotatory strength that avoids the singular-value decomposition of the outer product of the length-gauge APT and VG-APT vectors for each mode, and that clarifies how the LG(OI) and velocity-gauge rotatory strengths are related. Applying the density-based AATs and VG-APTs to (P)-hydrogen peroxide, (\textit{S})-methyloxirane, and (\textit{S})-3-chloro-1-butyne, we find that discrepancies among the gauges arise chiefly from basis-set incompleteness in the VG-APT. The charge-sum rule serves as a useful measure of that incompleteness, independent of the correlation method.
ER -