Calculating state-to-state transition probabilities within time-dependent density-functional theory

Abstract

The determination of the elements of the S-matrix within the framework of time-dependent density-functional theory (TDDFT) has remained a widely open question.We explore two different methods to calculate state-to-state transition probabilities. The first method closely follows the extraction of the S-matrix from the time-dependent Hartree-Fock approximation. This method suffers from cross-channel correlations resulting in oscillating transition probabilities in the asymptotic channels. An alternative method is proposed which corresponds to an implicit functional in the time-dependent density. It gives rise to stable and accurate transition probabilities. An exactly solvable two-electron system serves as benchmark for a quantitative test.As a matter of principle, time-dependent density functional theory [1] provides a highly efficient method to solve the time-dependent quantum many-body problem. It yields directly the time-dependent one-particle density n( r, t) of the many-body system. All physical observables of the quantum system can, in principle, be determined from the density. In practice there a two essential ingredients to a TDDFT calculation. First an approximation to the time-dependent exchange-correlation potential V xc [n]( r, t) has to be found which via the non-interacting Kohn-Sham system determines the evolution of the density. The second ingredient are functionals that allow the extraction of physical observables from the density.For some of the observables such as the ground-state energy extraction is straight forward within ground-state density functional theory [2]. Excited-state spectra have been obtained from linear-response functionals [3,4]. Beyond linear response, the time-dependent dipole moment which governs the emission of high-harmonic radiation can be directly determined from n( r, t). Ionization probabilities can be approximately extracted by identifying the integrated density beyond a certain critical distance from the bound system with the flux of ionized particles [5,6]. However, in general, on the most fundamental level, state-to-state transition probabilities contain the full information on the response of a many-body system

An effective method for state population within time-dependent density functional theory

“…In fact, the challenge of extracting excited-state densities via a suitable time propagation was raised. 3 Furthermore, many more read-out methods can be developed based on the density distribution of eigenstates, which are not readily accessible within the TDDFT previously, 14 therefore, our proposed FT method is important for its potential applications. Figures 4 and 5 show the characteristic signature of detuned Rabi oscillations of a Na atom, Na 2 dimer, and Na 4 cluster, like in character and differ in their amplitudes.…”

“…14 In the non-degenerate case of ω m = ω n for m = n, the time-averaged density may be expanded in the form In case of degenerate states, Eq. (4) is then modified in the form…”

Time-Dependent Two-Particle Reduced Density Matrix Theory: Application to High-Harmonic Generation

“…The validity of this approximation has remained an open question [23,24]. The TD-2RDM method, on the other hand, has the advantage that if threeparticle ionization is negligible it allows to accurately extract this information from the twoparticle density as…”

First principles multielectron mixed quantum/classical simulations in the condensed phase. I. An efficient Fourier-grid method for solving the many-electron problem

“…[46][47][48][49][50][51][52][53] This is because Eq. Harmonium has received considerable attention as a model by which to test density functionals.…”