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

Nina Rohringer, Simone Peter, and Joachim Burgdörfer

Phys. Rev. A 74, 042512 – Published 24 October 2006

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 of the time-dependent density. Evaluated with the exact time-dependent density it gives rise to stable and accurate transition probabilities. However, the functional shows an extreme sensitivity with respect to errors in the time-dependent density when evaluated using an approximate density from an actual TDDFT calculation. Two exactly solvable two-electron systems serve as a benchmark for a quantitative test.

Received 8 August 2005

DOI:https://doi.org/10.1103/PhysRevA.74.042512

Authors & Affiliations

Nina Rohringer^*, Simone Peter, and Joachim Burgdörfer

Institute for Theoretical Physics, Vienna University of Technology, A-1040 Vienna, Austria

^*Present address: Argonne National Laboratory, Argonne, IL 60439, USA. Electronic address: nrohringer@anl.gov

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Vol. 74, Iss. 4 — October 2006

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Images

Figure 1
Ground- and excited-state transition probabilities for the laser-driven harmonic 1D two-electron quantum dot of a confining frequency $ω = 0.25$ in the dependence of time: Comparison of occupation probabilities of the ground state (a), first excited state $∣ N_{c . m .} = 1, n_{r e l} = 0 ⟩$ (b), and second excited state $∣ N_{c . m .} = 2, n_{r e l} = 0 ⟩$ (c) of the exact calculation (full line) and the TDDFT calculation (dashed line). The TDDFT occupation probabilities are obtained using the approximate $S$ matrix Eq. (9); final channel states are Slater determinants of Kohn-Sham orbitals $∣ 0, 0 ⟩$ , $∣ 0, 1 ⟩$ , and $∣ 0, 2 ⟩$ . Parameters of the laser pulse: $F_{0} = 0.07$ , $ω_{L} = 0.1839$ , $τ = 168$ .Reuse & Permissions
Figure 2
Ground- and excited-state transition probabilities for the laser-driven harmonic 1D two-electron quantum dot for a confining frequency $ω = 0.25$ in the dependence of time: Comparison of occupation probabilities of the ground state (a), first excited state $∣ N_{c . m .} = 1, n_{r e l} = 0 ⟩$ (b), and second excited state $∣ N_{c . m .} = 2, n_{r e l} = 0 ⟩$ (c) of exact calculation (full line) and the TDDFT calculation (dashed line). The TDDFT transition probabilities are obtained by the inversion of Eq. (16). Parameters of the laser pulse: $F_{0} = 0.07$ , $ω_{L} = 0.1839$ , $τ = 168$ .Reuse & Permissions
Figure 3
Comparison of the time-dependent dipole moment of the laser-driven 1D helium of the exact solution (solid line) vs the TDDFT with $V_{x c}$ of Eq. (21) (dotted line) and the ALSDA-SIC exchange only (dashed line). The figure inset is an enlargement of the region after the turnoff of the laser pulse. Parameters of the laser pulse: driving frequency $ω_{L} = 0.0615$ , field amplitude $A_{0} = 0.16$ , pulse duration of $τ = 361$ .Reuse & Permissions
Figure 4
(Color online) Comparison of ground-state occupation probabilities calculated for the laser-driven 1D He model for several approaches: exact occupation probability calculated by projection of the time-dependent many-body wave function (solid black line starting at $t = 0$ ) compared to the results of Eq. (16) evaluated with the exact time-dependent density and exact final-state densities [solid bold black line starting after the turnoff of the laser pulse, (online: blue)]. The curves are almost indistinguishable within the graphical resolution. The TDDFT ground-state occupation according to the projection approach of Eq. (9) [projection onto the exact many-body ground state (black dashed line)]. The results of the read-out functional of Eq. (16) evaluated for the TDDFT density with the exact final-state densities $ρ_{f, f}^{(1) e x}$ for $N_{f} = 7$ and $N_{f} = 5$ [upper and lower gray solid lines, respectively (online: red lines)]. The results of the read-out functional of Eq. (16) evaluated for the TDDFT density with DFT final-state densities $ρ_{f, f}^{(1) D F T}$ (dotted line).Reuse & Permissions

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