Electron Dynamics: MCTDHF
- The Multi-Configuration Time-Dependent Hartree-Fock method is a very powerful approach to correlated many
electron dynamics. The high-dimensional wave function is expanded in a sum of Slater determinants of spin orbitals. Both, the orbitals
and the coefficients of the determinants are variationally optimized and time-dependent. The method can therefore also be seen as
TD-CASSCF and TD-MCSCF.
The picture below shows the time-dependent electron density for a one-dimensional model system, where an electron scatters from a thin metal film. The metal is modelled by five electrons in a jellium potential. The colour coding is logarithmic.
The electron excites coherent oscillations inside the metal, which are then dephasing on a timescale of several fs. Also, in the upper half of the picture one can identify non-classical reflection, and that some part of the incident electron is trapped in long lived resonances, from which it is periodically emitted.
This time-dependent approach can also be used to calculate eigenstates of the system by propagation in imaginary time. The following graph shows the evolution of energy expectation value for different sizes of the active space (number of space orbitals), for the relaxation of a 6 electron wave function in a jellium system. Note the high correlation energy, and the convergence once sixfold excitations are included.
Nuclear Dynamics: MCTDH
- The Multi-Configuration Time-Dependent Hartree method is a well established tool for quantum dynamical
simulations for distinguishable particles (atoms). Ample documentation can be found on the
Homepage of the Heidelberg MCTDH group.
On the same site benchmark calculations can be found for the dynamics of high dimensional anharmonic oscillators (up to 61D), and Henon-Heiles (up to 18D) systems. These calculation provide high accuracy data for comparison with approximative (like semiclassical or reduced dynamics).
The following figure compares the sticking probability for a particle hitting a non-rigid surface (inelastic scattering), as a function of its initial incident kinetic energy. The agreement between the 61-dim wave packet calculation (red) and the reduced density matrix calculation (green) with generalized raising/lowering operators as Lindblad operators is surprisingly good.
Reduced Quantum Dynamics/Chemical Physics
- I am also working on the simulation of chemical reactions using several quantum dynamical methods, like wave packet- or density matrix propagation.
A typical scenario is the light induced desorption of small molecules from metal surfaces:
The following systems were treated:
- Ar@Cu
- O2@Pt
- NO@Pt
The effect of the surfaces was modelled by a dissipative functional. In order to reproduce the correct distance dependence of the dissipation strength (no dissipation at large distances, therefore no bilinear coupling) we developed generalized raising/lowering operators (RLOs) for anharmonic systems. The problem to be solved was, that any operator for which O|n> \propto |n+1> holds in anharmonic systems, depends explicitly on n: O => O_n. Therefore we used the idea to go "sideways" in a (supersymmetric) hierarchy of shape invariant Hamiltonians, and use these new operators as approximate RLOs:
