4.5 Discussion
In the harmonic potential a spontaneous refilling of the core hole is observed. Si-multaneously charge is redistributed into higher orbitals. This process resembles what is expected during an Auger process. The carbon atom and the carbon monoxide molecule immediately respond to the ionization. The charge oscillates between different orbitals. In the carbon monoxide molecule this results in an oscillating dipole moment.
For the carbon atom a transition is observed. During this transition the hole is filled. While electrons of the opposite spin occupy higher energy basis functions.
However, the energy is not conserved as it should.
We can see three possible reasons for the violation of the energy conservation.
The time evolution operator (see equation (2.41)) is unitary. This ensures norm conservation during the time propagation. Since the Fock operator is time-dependent, the norm conservation does not imply the conservation of the ex-pectation value of the Fock operator. The exex-pectation value is only conserved, if the Fock operators at different times commute. Similarly the energy, which is not the energy of the Fock operator, is only conserved if the Hamiltonian commutes with the Fock operator.
A numerical solution of the time-dependent Hartree-Fock equations will only yield norm and energy conservation with a finite accuracy imposed by the integrator deployed. Thus a suitable integrator has to be used, to keep the errors caused by the approximations within a reasonable range. Both the second-order differencing and the time evolution operator scheme exhibit deficiencies, so further researcj in this direction is clearly required.
The second-order differencing scheme clearly fails in the application to the car-bon atom. It is not strictly unitary, which causes the norm to grow rapidly, as can be seen from figures 3.10 and 3.11. In the case of the harmonic oscillator the
CHAPTER 4. RESULTS
norm and energy drifts are within an acceptable range. This is due to the weaker interaction between the particles. The Coulomb repulsion between electrons is not bounded, whereas the model potential (see equation (3.2)) stays finite for all particle-particle distances.
The time evolution operator scheme is unitary. Accordingly, in all cases seen so far the norm is conserved. However, it violates the energy conservation in the transition observed for both the harmonic oscillator and the carbon atom. The effect is more pronounced for the carbon. This effect may be due to the stronger Coulomb interaction.
Unfortunately we observed that the life time of the decay is proportional to the step size used for the integration and therefore is likely caused by the first order approximation of the integral in the time evolution operator (see equation (2.41)) in equation (3.1).
It is of course, finally, possible, that - despite extensive testing - the problems seen for both the second-order differencing and the time evolution operator scheme are caused by programming errors.
64
5
Conclusion & Outlook
We have shown that time-dependent Hartree-Fock theory is capable of describing the decay of a core ionized state in a harmonic model system. The ionized state was found to decay in an Auger-like process. The application to more realistic systems could not be judged so far because the analysis has to be limited to the first femtosecond after the ionization due to numerical violation of the en-ergy conservation. This violation is most likely caused by still not sufficiently accurate integration schemes or mistakes in the implementation. The response to the ionization of both the carbon atom and the carbon monoxide in the first femtosecond was rationalized.
Instead of directly going from closed shell to directly open inner shell systems, situation should be considered where the initial state is perturbed less. The time evolution of excited states is a starting point for further analysis.
To study larger systems, the performance of the implementation has to be im-proved considerably. The most time consuming step for larger systems is the formation of the Fock matrix. To increase the performance of the formation of the Fock matrix, the integration schemes will be implemented into a quantum chemistry program. So it is possible to use the optimized routines for this task.
Furthermore, it is then possible to use direct SCF schemes. Thus the two electron
CHAPTER 5. CONCLUSION & OUTLOOK
integrals do not have to be stored in memory anymore and the system size is not limited by the memory anymore like in the current implementation. This also of-fers the possibility to use correlated methods, if the time-dependent Hartree-Fock theory turns out to be insufficient for the three-dimensional systems. However, this would again strongly limit the achievable system size.
What is missing in the currently used basis sets, is the description of an unbound electron. When studying the Auger effect, the wave function of the escaping elec-tron needs to be modeled with a finite basis set. But the basis sets used so far are designed to describe only bound states. It is therefore necessary to use additional basis functions. Two different approaches have been used to construct a basis set for an unbound electron. Cederbaum et al. [4] use a grid of gaussian basis functions. The grid was constructed to resemble a large s-type wave function.
This approach has the advantage of using a pure gaussian basis. The second pos-sibility are hybrid basis sets in which the unbound electron is described by plane waves. This approach has, for instance, been used to calculate photoelectron spectra [17]. In future work we would like to test which of these two approaches performs better for the systems at hand.
66
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A
Appendix
A.1 Calculation
Here the calculations of the relations (2.36), (2.37) and (2.38), which were skipped, in 2.3 are shown.
hδΨ|∂t|Ψi=
APPENDIX A. APPENDIX
Here the group property of SN has been used to eliminate the first sum overSN.
=
follows from an analogous calculation.
hδΨ| 1
A.1. CALCULATION
The term proportional to (1−δlj)(1−δlk) in the third line has been eliminated using equation (2.33).
= Z
dxjdxk
½
δχ∗j(xj)χ∗k(xk) 1
rjkχj(xj)χk(xk)−
δχ∗j(xj)χ∗k(xk) 1
rjkχj(xj)χk(xj)+
χ∗j(xj)δχ∗k(xk) 1
rjkχj(xj)χk(xk)−
χ∗j(xj)δχ∗k(xk) 1
rjkχj(xk)χk(xj)
¾
=hδχjχk| 1
r12|χjχki+hχjδχk| 1
r12|χjχki−
hδχjχk| 1 r12
|χkχji − hχjδχk| 1 r12
|χkχji (A.3)
APPENDIX A. APPENDIX