Dr. Yukuan Hu, École nationale des ponts et chaussées, Institut Polytechnique de Paris

Weiying Zheng, Professor

Mathematical Theories and Numerical Methods for Computing Electronic Excitation Energies of Molecular Systems

15:00-16:00 June 15(Monday), N202

Electronic excitation play a central role in photochemical and photophysical processes. In principle, electronically excited states are defined as higher-energy eigenstates of the electronic Hamiltonian in the many-body Schrödinger equation; excitation energies are computed as differences between excited-states and ground-state energies. However, the curse of dimensionality renders direct numerical solutions computationally prohibitive. To address this challenge, a plethora of more tractable yet nonlinear variational approximations have emerged, and many of them naturally possess Riemannian manifold structures. 

Within these variational approximate theories, methods for computing excitation energies fall broadly into two categories: linear response theory (LR) and critical point search (CP). LR-based methods essentially boil down to solving eigenvalue problems (i.e., LR equations), whose derivations often rely on ad-hoc algebraic manipulations and lack a clear geometric interpretation. CP-based methods approximate excited states as saddle points on the underlying Riemannian manifolds. They typically encounter convergence difficulties caused by the interplay of intrinsic instability of saddle points and complicated manifold structures. 

This talk presents our recent progress on both fronts. For LR-based methods, we develop a universal and conceptually straightforward framework for deriving LR equations, grounded in the abstract Kähler formalism. This framework unveils the geometric nature of LR-based methods: excitation energies are approximated as vibration frequencies of linearized Hamiltonian dynamics on tangent spaces. For the CP viewpoint, we propose a constrained saddle dynamics on Riemannian manifolds, which is formulated compactly on the Grassmann bundle of the tangent bundle. We rigorously establish the theoretical properties of both the continuous dynamics and the resulted discretized algorithm. As a byproduct, our analysis highlights the importance of non-redundant parametrizations for locating saddle points.

We have implemented the proposed saddle search algorithm with interfaces to the PySCF and CFOUR programs, and conducted tests on standard molecular systems, described within the Hartree-Fock (HF), Complete Active-Space Self-Consistent Field (CASSCF), and Full Configuration Interaction (FCI) theories. Numerical results demonstrate the effectiveness and robustness of the proposed algorithm, and provide comparisons between LR- and CP-based methods. 

This talk is based on joint work with Éric Cancès (ENPC-IPP & Inria), Laura Grazioli (ENPC-IPP & Inria), Tommaso Nottoli (UniPi), and Filippo Lipparini (UniPi).