Electric field gradient in accurate quantum chemical calculations

TL;DR

This work systematically analyzes how to compute electric field gradients (EFGs) in both molecular and solid-state environments, benchmarking HF, DFT, and multiconfigurational methods against experimental data and across periodic and embedded-cluster models. It tackles sign-convention disparities among popular quantum-chemistry codes, providing a self-contained derivation to standardize EFG conventions and ensure consistent interpretation of $V_{zz}$ and the asymmetry parameter $oldsymbol{eta}$. The study demonstrates that embedding can faithfully reproduce symmetry and yield accurate $V_{zz}$ when the active space and basis sets are carefully chosen, with CASPT2 often delivering the best agreement with experiments. It also highlights the sensitivity of EFGs to geometry and active-space selection, offering practical guidelines for computing, interpreting, and exploiting EFGs as quantitative descriptors of electronic structure and local chemical environments.

Abstract

The electric field gradients (EFGs) at the (non-spherical) nucleus contribute to atomic and molecular hyperfine structure and govern Nuclear Quadrupole Resonance (NQR) and Mössbauer spectra. EFGs provide a highly sensitive probe of local bonding, symmetry, and crystal defect geometry and electronic structure. The EFGs can be obtained from electronic structure calculations and can also be extracted from spectroscopic measurements, thus linking electronic structure theory and spectroscopic observables. In this work, we present a methodological study of EFGs for a range of molecules and crystalline materials, using both periodic boundary conditions and embedded cluster models, and compare the results with reported experimental data. We analyze the sensitivity of EFG values to details of the calculations, such as the selection of the model Hamiltonians, basis sets, and the geometries of molecules and crystals. We also address persistent differences in EFG sign conventions and tensor definitions employed in the literature and in widely used quantum chemistry codes. While the EFG sign does not affect zero B-field NQR spectra, they can become critical in Mossbauer spectroscopy or when the quadrupolar interactions are combined with other interactions of the nucleus with the environment. Together, our systematic study results provide practical guidelines for computing, interpreting, and exploiting EFGs as quantitative descriptors of electronic structure and chemical environment.

Figures (3)

• Figure 1: Different symmetry patterns of EFG for octahedrons, shown from left to right: distorted along the $C_{3v}$ direction (rhombohedral), non-distorted (cubic) and distorted along the $z$-direction (tetragonal).
• Figure 2: EFG at the oxygen in a water molecule as a function of the H–O–H bond angle, $\theta$. Panel (a) shows the principal EFG component $V_{zz}$ (in a.u.), with solid black circles for $V_{zz}$ and open red circles for $|V_{zz}|$. Panel (b) shows the corresponding asymmetry parameter $\eta$. Calculations were performed using Molcas at the Hartree-Fock theory level with the ANO-L-VTZP basis set.
• Figure 3: The effect of distortions in the Co--O bond lengths along the $z$ axis (expressed as a relative change in bond length $Z$) on the principal component $V_{zz}$ of the electric-field–gradient tensor. This plot is for the symmetrized cubic structure of LaCoO3. In all cases the asymmetry parameter vanishes ($\eta = 0$). The calculations were performed at the density-functional-theory level using the PBE functional and the ANO-RCC-VQZP basis set.