Geometry Optimization for Nonlocal Excited State Using the Divide-and-Conquer Method

We present an analytic gradient method for divide-and-conquer (DC)-based nonlocal excited-state calculations, enabling efficient geometry optimization of large molecular systems exhibiting delocalized or charge-transfer excitations. Transition density matrices are extracted from response densities near polarizability poles, and subsystem Z-vector equations are solved independently, reducing the dominant computational scaling from O(N^3.55) in standard time-dependent Hartree-Fock/density functional theory to O(N^1.60). Applications to push-pull polyenes and [9]cycloparaphenylene ([9]CPP) demonstrate that the DC-based gradients systematically converge toward standard results with increasing buffer size and accurately reproduce excited-state structural relaxation, including the pronounced quinoid distortion and large Stokes shift in [9]CPP. These results establish the present method as a practical and scalable approach for excited-state geometry optimization in systems beyond the feasible range of conventional excited-state techniques.