Approximate excited-state potential energy surfaces for defects in solids

A description of electron-phonon coupling at a defect or impurity is essential to characterizing and harnessing its functionality for a particular application. Electron-phonon coupling limits the amount of useful light produced by a single-photon emitter and can destroy the efficiency of optoelectronic devices by enabling defects to act as recombination centers. Information on atomic relaxations in the excited state of the center is needed to assess electron-phonon coupling but may be inaccessible due to failed convergence or computational expense. Here we develop an approximation technique to quantify electron-phonon coupling using only the forces of the excited state evaluated in the equilibrium geometry of the ground state. The approximations are benchmarked on well-studied defect systems, namely, ${\mathrm{C}}_{\mathrm{N}}$ in GaN, the nitrogen-vacancy center in diamond, and the carbon dimer in h-BN. We demonstrate that the zero-phonon line energy can be approximated with just a single mode, while the Huang-Rhys factor converges by including displacements up to the second-nearest neighbors. This work also provides important insight into the success of the widely utilized one-dimensional accepting-mode approximation, specifically demonstrating that the accepting-mode Huang-Rhys factor is a strict upper bound on the full multidimensional Huang-Rhys factor.