Analytical gradients for excitation energies from frozen-density embedding

The formulation of analytical excitation-energy gradients from time-dependent density functional theory within the frozen-density embedding framework is presented. In addition to a comprehensive mathematical derivation, we discuss details of the numerical implementation in the Slater-function based Amsterdam Density Functional (ADF) program. Particular emphasis is put on the consistency in the use of approximations for the evaluation of second-and third-order non-additive kinetic-energy and exchange-correlation functional derivatives appearing in the final expression for the excitation-energy gradient. We test the implementation for different chemical systems in which molecular excited-state potential-energy curves are affected by another subsystem. It is demonstrated that the analytical implementation for the evaluation of excitation-energy gradients yields results in close agreement with data from numerical differentiation. In addition, we show that our analytical results are numerically more stable and thus preferable over the numerical ones.