Speaker: Laura Grazioli, École nationale des ponts et chaussées, INRIA Paris

In exact theory, excited states are defined as higher-energy solutions of the Schrödinger equation. From a mathematical point of view, we can interpret excited states as saddle points on the electronic energy functional of a molecular system. For a Morse function, saddle points can be classified on the basis of the number of negative eigenvalues of the Hessian matrix. The nth excited state can be seen as a saddle point with n negative eigenvalue (Morse-index n). However, when applying a nonlinear parameterization of the wave function to the linear Schrödinger equation, spurious critical points may appear. Therefore, a careful analysis of the saddle points is needed to identify, among the critical points, those having the physical interpretation of excited states. We will develop manifold constrained saddle-point search algorithms on the manifold of admissible electronic states to locate all index-1 saddle points. A first global exploration of the energy can be performed through stochastic algorithms, to identify the area in which the saddle points are located, extending ideas from [1] to Riemannian manifolds. In the identified area, local critical-point search algorithms can be applied to find the saddle point, using the Riemannian gradient of the energy functional and part of the information contained in the Riemannian Hessian. For both steps, it is key to carefully exploit the geometry of the manifold of admissible electronic states. We will start from Hartree-Fock theory, which can be identified as models on Grassmann manifolds, and then turn to MCSCF and CASSCF theory. This definition of the excited states will be compared to the one obtained through linear-response theory, for which a new derivation scheme, based on the structure of Kähler manifolds, has been developed [2].

[1] Lelièvre, T.; Parpas, P. J. Sci. Comput., 2024, 46-2, A770.
[2] Grazioli, L.; Hu, Y.; Cancès, E., arXiv:2506.16420.