2.5 Computational methods
2.5.2 Time-dependent density functional theory
Time-dependent density functional theory (TD-DFT) extends the idea behind DFT. The foundation of TD-DFT is the Runge-Gross-Theorem,[54] which proved (analogous to the Hohenberg-Kohn-Theorem for a static system) that all observables of a interacting many-body system, evolving from an initial (true) state / , can be extracted from the density ( , 1) of a non-interacting (Kohn-Sham) system of fermions starting in an initial state Φ . The proof of the Runge-Gross-Theorem is more involved than the Hohenberg- Kohn-Theorem, since the time-dependent effective potential at a given time is dependent on the densities at all previous times. Detailed explanation of the theory can be found in e.g. Ref. [55].
40
Suffice to say is that, analogous to DFT, the time-dependent density can be obtained according to
( , 1) = ( , 1) !
"#
(4)
with a set of N orthonormal ( , 1) orbitals obeying the time-dependent Kohn-Sham equation
3−12 ∇ + ( , 1)4 ( , 1) = 561 ( , 1)6 (5) Analogous to the potential of a Kohn-Sham system in the ground state, ( , 1) is split into three terms:
[ ; Φ ]( , 1) = $%[ ; Ψ ]( , 1) + & ( ,, 1)
| − ,| )*+,+ -.[ ; Ψ , Φ ]( , 1) (6) With the first term, $%[ ; Ψ ]( , 1), being the external time-dependent field, the second the time-dependent Hartree potential and the third the XC potential. Again, the XC potential has to be approximated as s functional of the density and, in contrast to DFT, also as a functional of the true initial state Ψ and the Kohn-Sham initial state Φ .
Within this work, geometry optimizations and vibrational frequency calculations were performed on the DFT level of theory, whereas for calculations of vertical transition energies TD-DFT was employed. Calculations were managed and monitored using mainly the Extensible Computational Chemistry Environment (ECCE)[56] interface running the Gaussian 09 program package.[57] Individual use of basis sets and functionals will be specified within each chapter. Calculations were gratefully performed on the computing clusters of the TU Kaiserslautern, which are under supervision of the theoretical research group of Prof. Dr. C. van Wüllen.
41
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45
protonated
fluorescein
studied
by
time-resolved
photofragmentation in an ion trap
Dimitri Imanbaew1), Maxim F. Gelin3), and Christoph Riehn1),2)
1Fachbereich Chemie, Technische Universität Kaiserslautern, Erwin-Schrödinger- Str. 52-54, 67663 Kaiserslautern, Germany
2Landesforschungszentrum OPTIMAS, Erwin-Schrödinger-Str. 46, 67663 Kaiserslautern, Germany
3Fakultät für Chemie, TU München, Lichtenbergstraße 4, 85747 Garching, Germany