Periodicity of the time-dependent Kohn-Sham equation and the Floquet theorem

Abstract

The Floquet theorem allows to reformulate periodic time-dependent problems such as the interaction of a many-body system with a laser field in terms of time-independent, field-dressed states, also known as Floquet states. If this was possible for density functional theory as well, one could reduce in such cases time-dependent density functional theory to a time-independent Floquet density functional theory. We analyze under which conditions the Floquet theorem is applicable in a density-functional framework. By employing numerical ab initio solutions of the interacting timedependent Schrödinger equation with time-periodic external potentials we show that the exact effective potential in the corresponding Kohn-Sham equation is not unconditionally periodic. Whenever several Floquet states in the interacting system are involved in a physical process the corresponding Hartree-exchange-correlation potential is not periodic with the external frequency only. Using an analytically solvable example we demonstrate that, in general, the periodicity of the time-dependent Kohn-Sham Hamiltonian cannot be restored by choosing a different initial state. Only if the external periodic potential is sufficiently weak such that the initial state of the interacting system evolves adiabatically to a single, field-dressed state, the resulting Kohn-Sham system admits the application of the Floquet theorem.

Defining a well-ordered Floquet basis by the average energy

“…These methods were constructed to calculate a poorly defined ground state, for which the Ritz variational principle is assumed. One of these methods, the Floquet Density Functional Theory [22], was shown to be fundamentally flawed due to the ambiguous definition of the ground state [24][25][26], an argument that can be extended to the other variational based ab-initio derivations as well. One method that remains valid is the Floquet Density Matrix Renormalization Group [27] and similar iterative methods.…”

Potential energy surfaces for electron dynamics modeled by floating and breathing Gaussian wave packets with valence-bond spin-coupling: An analysis of high-harmonic generation spectrum

“…Theoretical studies of HHG spectra with quantum dynamical calculations of realistic systems have been rather limited to small atoms and molecules such as H, He, H + 2 , H 2 , HeH + , D + 3 , and LiH [14][15][16][17][18][19][20]. Treating more electrons in larger and more complex molecules seems too demanding at present unless invoking the time-dependent mean-field approximations of various levels [21][22][23][24][25][26][27] or the density functional theory (DFT) [28][29][30]. The conventional MO and DFT calculations are based on atomic orbitals (AOs) that are clamped at nuclear centers, with the time-dependence carried by the coefficients of MO or the configuration-interaction.…”

Autoionization in time-dependent density-functional theory

“…Exact studies such as this have led to development of better exchange-correlation potentials in TDDFT for quite a few correlated process [13,14]. Such studies often reveal the essential features that any exchange-correlation potential should possess to be able to describe a correlated process correctly [13][14][15][16].…”