A geometric picture of excited states in variational theories. – (Laura Grazioli / LCPQ / Seminar). – 2/02/2026, 14H

Séminaire du LCPQ

Laura Grazioli CERMICS, Paris

Seminar LCPQ, 2/02/2026, 14H

Summary :
In exact theory, excited states correspond to higher-energy solutions of the Schrödinger equation. For an exact wave function, these states appear as saddle points of the electronic energy functional, and for a Morse function they can be classified by the number of negative eigenvalues of the Hessian matrix—the nth excited state having Morse index n. When the linear Schrödinger equation is solved within a nonlinear wave-function parameterization, however, spurious critical points may emerge.

To address this, we develop manifold-constrained saddle-point search algorithms defined on the manifold of admissible electronic states. These methods target saddle points of fixed index k. A global exploration of the energy landscape is first performed using stochastic algorithms adapted to Riemannian manifolds, to identify regions likely to contain saddle points. Within these regions, local critical-point algorithms are then employed, relying on the Riemannian gradient and selected information from the Riemannian Hessian. A careful treatment of the underlying manifold geometry is essential in both stages, enabling the construction of stable algorithms for locating saddle points, which are naturally unstable.

In quantum chemistry, excited states are also commonly computed through linear response theory, which analyses the linearized dynamics around a stable ground state. Although linear response formulations exist for many variational theories, their derivations are often technically involved and rely on ad-hoc constructions. We provide a unified derivation for variational theories by exploiting the geometric structure of Kähler manifolds.

Both excited-state characterisations—saddle-point theory and linear response theory—are developed and applied first within Hartree–Fock theory, represented by a Grassmann manifold, and subsequently within Complete Active Space Self-Consistent Field (CASSCF) theory, represented by a flag manifold.