This talk presents a geometric and optimization-based perspective on computing electronic excited states. Starting from the many-body Schrödinger equation, we discuss variational wavefunction approximations and the nonlinearity introduced by restricting the electronic energy to parameter manifolds. We review linear-response, state-averaged, and state-specific approaches, emphasizing their distinct formulations of approximate excited states.  

Focusing on state-specific calculations, we introduce a constrained saddle dynamic for locating saddle points of prescribed Morse index on Riemannian manifolds. The Grassmann bundle geometry provides a framework for analyzing the dynamic and constructing discretizations using retractions and vector transport, with linear stability and local linear convergence guarantees at nondegenerate saddle points. Numerical experiments with Hartree–Fock and complete active space self-consistent field models illustrate the approach. We conclude with challenges in developing faster saddle-search methods and navigating excited-state potential energy surfaces.