On the Floquet formulation of time-dependent density functional theory

The time-periodic density of a Floquet state of a time-periodic potential does not uniquely determine that potential. A simple example demonstrates this, and the implications are discussed. 2002 Published by Elsevier Science B.V. Ground-state density functional theory (DFT) [1] has been tremendously successful in predicting the electronic structure of atoms, molecules and solids relatively inexpensively [2]. DFT hugely simplifies calculations by introducing a fictitious system of non-interacting electrons with exactly the same density as the interacting many-electron system, the Kohn–Sham system [3]. In 1984, Runge and Gross generalized ground-state density functional theory to time-dependent problems (TDDFT) [4]. TDDFT has become popular for studying atoms and molecules in laser fields, calculating excitation spectra, polarizabilities, optical response of solids, etc. [5,6]. Most applications are in the linear response regime, where weak fields are applied, using adiabatic local-density and generalized gradient approximations (see in [6] for many references). But it is especially for intense laser fields that TDDFT would appear to be the only practical way of studying the dynamics of manyelectron systems, where correlation effects are important. This would prove very useful in quantum control problems [7]. In wavefunction methods, Floquet theory is an attractive approach for studies of species in intense laser fields. A time-periodic potential, such as in the case of laser fields, allows for a complete set of ‘steady-state’, or Floquet solutions, in which the problem reduces to a matrix diagonalization, similar to the finding of Bloch states for spatially periodic problems [8–11]. The system is assumed to reach a Floquet state by some adiabatic ramping of the time-dependent part of the potential (see e.g. [11]). Floquet theory is particularly useful because it is not limited to weak time-dependent fields. Floquet theory has been successful in describing a variety of phenomena, including multiphoton ionization and detachment problems [12,13], two-colour ionization [14,15], analysing microwave ionization experiments [16], high harmonic generation [17], selective excitation of molecular 20 June 2002 Chemical Physics Letters 359 (2002) 237–240 www.elsevier.com/locate/cplett * Corresponding author. Fax: +732-445-5312. E-mail address: [email protected] (N.T. Maitra). 0009-2614/02/$ see front matter 2002 Published by Elsevier Science B.V. PII: S0009-2614 (02 )00586-9 vibrational states using short laser pulses [18]. Most applications consider a one-electron picture, although recently an R-matrix Floquet theory [19– 21] has been developed to address multiphoton processes in many-electron systems. Time-dependent density functional theory transforms an interacting many-electron system into a fictitious non-interacting Kohn–Sham system with the same time-dependent density. The basis of any density functional theory is a demonstration of a one-to-one correspondence, for a certain class of problems, between densities and potentials, for a given interparticle statistics and interaction. Ground-state DFT is founded on a one-to-one mapping between densities of groundstates and their potentials, whereas TDDFT is based on a one-to-one mapping between time-dependent densities and potentials for a specified initial state. Consider problems in which the external potential is time-periodic, and the interacting system has reached a Floquet state, so that its density is time-periodic. It is natural to ask if the time-periodic density of a Floquet state uniquely determines the time-periodic potential. If it does, then all properties of the system are functionals of that density, which was the basis of [22,23]. However, we demonstrate here that two different Floquet states can be found that evolve with the same periodic density in different periodic potentials, so the mapping is not unique. Floquet states are steady-state solutions of the time-dependent Schr€ odinger equation when the Hamiltonian is time-periodic [8,9], Hðt þ T Þ 1⁄4 HðtÞ. There exists a complete set of Floquet solutions of the form [8,9] wnðtÞ 1⁄4 e i nunðtÞ; unðt þ T Þ 1⁄4 unðtÞ: ð1Þ The time-periodic functions unðtÞ are termed quasienergy eigenstates (QES), and n is termed the quasi-energy. The QESs satisfy fHðtÞ io=otgunðtÞ 1⁄4 nunðtÞ ð2Þ and play a role analogous to the stationary states of a time-independent Hamiltonian. We shall construct an example involving non-interacting electrons in the periodically driven one-dimensional harmonic oscillator H 1⁄4 1 2 d dx2 þ 1 2 x20x 2 þ kx sinðxtÞ: ð3Þ The QESs are known analytically [24]: unðxtÞ 1⁄4 /nð xðtÞÞ expfiðxAx cosðxtÞ þ aðtÞÞg; ð4Þ where /n are the eigenstates of the static harmonic oscillator (k 1⁄4 0 in Eq. (3)), xðtÞ 1⁄4 x A sinðxtÞ, the amplitude of the periodic shift A 1⁄4 k=ðx x20Þ; ð5Þ