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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. A comparison of the results obtained by using manifold-constrained saddle-point search algorithm with those given by already established quantum chemical methods is also shown for test systems.

We introduce an interacting particle system originating from a nucleation process and investigate the nucleation time as a function of the interaction strength, ranging from weak to strong. Using (uniform) propagation of chaos, we study the non-linear mean-field limit. A standard analysis yields a Yaglom limit conditionned on non-nucleation and its associated tails for the distribution of the nucleation time. The most surprising result arises in the strong interaction regime: the tails follow a decay, where denotes the nucleus size. This result is obtained through an application of the centre manifold theorem. This is a joint work with Frédéric Paccaut.

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