Geometric theory of constrained Schr\"odinger dynamics with application to time-dependent density-functional theory on a finite lattice

TL;DR

This paper develops a geometric framework for constrained Schr"odinger dynamics, offering new insights into the mathematical foundations of TDDFT and proposing novel Kohn--Sham schemes with numerical demonstrations.

Contribution

It introduces a geometric approach to constrained Schr"odinger dynamics, revealing multiple formulations and proposing new, potentially more robust TDDFT methods.

Findings

Multiple natural definitions of constrained dynamics arise from geometry.

A new geometric constrained Schr"odinger evolution is proposed.

Numerical results on the Hubbard dimer illustrate the new approaches.

Abstract

Time-dependent density-functional theory (TDDFT) is a central tool for studying the dynamical electronic structure of molecules and solids, yet aspects of its mathematical foundations remain insufficiently understood. In this work, we revisit the foundations of TDDFT within a finite-dimensional setting by developing a general geometric framework for Schr\"odinger dynamics subject to prescribed expectation values of selected observables. We show that multiple natural definitions of such constrained dynamics arise from the underlying geometry of the state manifold. The conventional TDDFT formulation emerges from demanding stationarity of the action functional, while an alternative, purely geometric construction leads to a distinct form of constrained Schr\"odinger evolution that has not been previously explored. This alternative dynamics may provide a more mathematically robust route to TDDFT and may suggest new strategies for constructing nonadiabatic approximations. Applying the theory to interacting fermions on finite lattices, we derive novel Kohn--Sham schemes in which the density constraint is enforced via an imaginary potential or, equivalently, a nonlocal Hermitian operator. Numerical illustrations for the Hubbard dimer demonstrate the behavior of these new approaches.