Length vs Velocity Gauge Formulation of the Frequency-Dependent Polarizability for 1D Periodic Systems at Coupled Cluster with Single and Double Excitations Level

This work presents the implementation of the linear response function for 1D periodic systems at coupled cluster with single and double excitations with periodic boundary conditions level (LR-CCSD-PBC) for the calculation of the frequency-dependent electric dipole-electric dipole polarizability tensor, α(ω), with the modified velocity gauge (MVG) formalism. We compare this approach with the length gauge (LG) formalism that we presented in a previous work [APL Comp. Phys. 1, 026107 (2025)] in terms of theoretical analysis and computational results. Calculations on molecular systems with large Dunning basis sets including diffuse functions show a smooth convergence towards the complete basis set limit (CBS) with both formalisms, but LG and MVG converge to different values when CCSD provides an incomplete treatment of electron correlation. With the small basis sets used in the PBC calculations, the difference between the gauge formalisms increases due to basis set incompleteness. For the PBC calculations on 1D periodic systems, the MVG formalism shows a faster convergence towards the thermodynamic limit with k sampling compared to LG. Furthermore, the MVG formalism provides perfect agreement between PBC and long molecular cluster models. This is not the case for LG due to the so-called missing integer issue. However, a comparison with the molecular data suggests that the LG-PBC results with the small basis set affordable in this work are closer to the CBS results than those with MVG. This work represents a further step forward for the simulation of optical response properties for solid-state materials with systematically improvable quantum mechanical methods.