Analytic $G_0W_0$ gradients based on a double-similarity transformation equation-of-motion coupled-cluster treatment

TL;DR

This work develops analytic gradients for $G_0W_0$ energies by recasting the non-variational GW problem into an IP/EA-EOM-$\lambda$-drCCD framework, enabling standard CC Lagrangian derivatives. Central to the approach are formal links between RPA and direct-ring CCD, block-diagonalization of the RPA matrix, and a double similarity transformation that introduces the $\lambda$-drCCD operator, capturing missing correlation effects. The resulting IP/EA energies satisfy $E_{p}^{IP/EA}=E_0^{G_0W_0}\pm \epsilon_p^{G_0W_0}$ with a diagonal approximation yielding $\epsilon_p^{IP/EA}=-\epsilon_p^{G_0W_0}$, while the Lagrangian formalism provides analytic derivatives with respect to nuclear coordinates. Benchmarking on the $GW$20 molecular set shows that the new $G_0W_0$ gradients yield IPs with MAEs around $0.3$–$0.4$ eV, outperforming second-order methods and aligning closely with high-level CC benchmarks, thereby enabling accurate adiabatic-IP calculations and geometry optimizations for larger systems.

Abstract

The accurate prediction of ionization potentials (IPs) is central to understanding molecular reactivity, redox behavior, and spectroscopic properties. While vertical IPs can be accessed directly from electronic excitations at fixed nuclear geometries, the computation of adiabatic IPs requires nuclear gradients of the ionized states, posing a major theoretical and computational challenge, especially within correlated frameworks. Among the most promising approaches for IP calculations is the many-body Green's function $GW$ method, which provides a balanced compromise between accuracy and computational efficiency. Furthermore, it is applicable to both finite and extended systems. Recent work has established formal connections between $GW$ and coupled-cluster doubles (CCD) theory, leading to the first derivation of analytic $GW$ nuclear gradients via a unitary CCD framework. In this work, we present an alternative, fully analytic formulation of $GW$ nuclear gradients based on a modified version of the traditional equation-of-motion CCD formalism, enabling the inclusion of missing correlation effects in the traditional CCD methods.