A Comparison of Relativistic Coupled Cluster and Equation of Motion Coupled Cluster Quadratic Response Theory

TL;DR

This work implements relativistic quadratic response theory (QR-CC) within the ExaCorr/DIRAC framework and systematically compares it with QR-EOMCC across static and frequency-dependent properties. By evaluating $\beta_{||}$, $\beta_{zzz}$, $\beta_{zxx}$ for hydrogen halides and Verdet constants for Xe–Og, the study dissects scalar-relativistic and spin-orbit contributions and assesses basis-set, core-valence, and $(n-1)d$-electron effects. The findings show that QR-EOMCC and QR-CC agree closely for dipole polarizabilities and Verdet constants (differences typically around 1%), but large discrepancies emerge for hyperpolarizabilities down the periodic table (up to tens of percent), driven by state-to-state transition elements. These results underscore the practical utility of QR-EOMCC as a cost-effective alternative while highlighting the need for careful interpretation in heavy-element systems and motivating future work on excited-state properties.

Abstract

We present the implementation of relativistic coupled cluster quadratic response theory (QR-CC), following our development of relativistic equation of motion coupled cluster quadratic response theory (QR-EOMCC) [X. Yuan et al., J. Chem. Theory Comput. 2023, 19, 9248]. These codes, which can be used in combination with relativistic (2- and 4-component based) as well as non-relativistic Hamiltonians, are capable of treating both static and dynamic perturbations for electric and magnetic operators. We have employed this new implementation to revisit the calculation of static and frequency-dependent first hyperpolarizabilities of hydrogen halides (HX, X=F-Ts) and the Verdet constant of heavy noble gas atoms (Xe, Rn, Og) and of selected hydrogen halides (HF to HI), in order to investigate the differences and similarities of QR-CC and the more approximate QR-EOMCC. Furthermore, we have determined the relative importance of scalar relativistic effects and spin-orbit coupling to these properties, through a comparison of different Hamiltonians, and extended our calculations to superheavy element species (HTs for hyperpolarizabilities, Og for the Verdet constant). Our results show that as one moves towards the bottom of the periodic table, QR-EOMCC can yield rather different results (hyperpolarizabilities) or perform rather similarly (Verdet constant) to QR-CC. These results underscore the importance of further characterizing the performance of QR-EOMCC for heavy element systems.

Figures (4)

• Figure 1: QR-CC and QR-EOMCC dipole hyperpolarizabilities components ($\beta_{zxx}$ and $\beta_{zzz}$, in a.u.) and the respective differences ($\Delta\beta_{zxx}$ and $\Delta\beta_{zzz}$, in a.u.) calculated with the $^{2}$DC$^M$ Hamiltonians for the HX series, upon correlating the $n$s$n$p shells (and with virtual threshold at 5 a.u.). [Adapted from zenodo.17215085, available under the Creative Commons Attribution 4.0 International license. Copyright X. Yuan, L. Halbert, L. Visscher, A. S. P. Gomes, 2025].
• Figure 2: QR-CC and QR-EOMCC $^2$DC$^M$ dispersion curves for the $\beta_{zzz}, \beta_{zxx}$ components (top: general view; bottom: detailed view of the low-frequency region) of the hyperpolarizability of the HI molecule for second harmonic generation (SHG) calculations ($\omega = \omega_B = \omega_C$, in a.u.). We note the pole at around 0.10 a.u. corresponding to half of the excitation energy towards the $a^{3}\Pi_{0+}$ state Yuan_Xiang_QR. [Adapted from zenodo.17215085, available under the Creative Commons Attribution 4.0 International license. Copyright X. Yuan, L. Halbert, L. Visscher, A. S. P. Gomes, 2025].
• Figure 3: Mean dipole polarizabilities (left, $\bar{\alpha} = 1/3(\alpha_{zz}+2\alpha_{xx})$ in a.u.) and anisotropies as a fraction of mean polarizability (right, $\Delta\alpha / \bar{\alpha}=\left( \alpha_{zz}-\alpha_{xx}\right)/\bar{\alpha}$ in % ) calculated with $^{2}$DC$^M$ Hamiltonian for the HX series with LR-CC and LR-EOMCC, upon correlating the $n$s$n$p shells (and with virtual threshold at 5 a.u.). The absolute differences between the two approaches ($\Delta$ : LR-CC - LR-EOMCC) are also presented. [Adapted from zenodo.17215085, available under the Creative Commons Attribution 4.0 International license. Copyright X. Yuan, L. Halbert, L. Visscher, A. S. P. Gomes, 2025].
• Figure 4: Left: frequency dependence of the QR-CC Verdet constants calculated with $^{2}$DC$^M$ Hamiltonian for Xe, Rn and Og, upon correlating the $n$s$n$p shells (and with virtual threshold at 5 a.u.). ${}^{\dagger}$ Experimental data from Ingersoll:56 and cadene_circular_2015. Left: frequency dependence of the difference between QR-CC and QR-EOMCC ($\Delta V = (V_{CC}-V_{EOM})/V_{CC}$, in %). [Adapted from zenodo.17215085, available under the Creative Commons Attribution 4.0 International license. Copyright X. Yuan, L. Halbert, L. Visscher, A. S. P. Gomes, 2025].