Linear response and resonances in adiabatic time-dependent density functional theory

TL;DR

This paper analyzes the mathematical properties of adiabatic time-dependent density functional theory, proving well-posedness of the time evolution and the emergence of resonances at specific frequencies.

Contribution

It establishes the well-posedness of the linear response in adiabatic TDDFT and introduces a novel reformulation as a real Hamiltonian system to handle complex linearity issues.

Findings

Proved well-posedness of time evolution near a non-degenerate ground state.

Demonstrated the appearance of resonances at relevant frequencies.

Reformulated the linearized equations as a real Hamiltonian system for stability.

Abstract

We consider the electrons of a molecule in the adiabatic time-dependent density functional theory approximation. We establish the well-posedness of the time evolution and its linear response close to a non-degenerate ground state, and prove the appearance of resonances at relevant frequencies. The main mathematical difficulty is due to the structure of the linearized equations, which are not complex-linear. We bypass this difficulty by reformulating the linearized problem as a real Hamiltonian system, whose stability is ensured by the second-order optimality conditions on the energy.