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Abstract

Nowadays, X-ray absorption spectroscopy (XAS) techniques are important tools to investigate the electronic structure of molecules. Mostly, these methods are applied in the field of organic electronics to study unoccupied molecular levels, which provide information about charge generation and transport properties. With the help of modern synchrotron soft beam sources, molecules can absorb high-energy X-ray photons, thereby promoting an electron from the core level, e.g. K-shell 1s orbitals, to the unoccupied molecular level. As a result, meta-stable bound core-excited states are generated. Since core orbitals are energetically well-separated from the remaining occupied and virtual orbital space, they are strongly contracted and the corresponding core-excited states are very localized. As a consequence, the generated core-hole interaction induces a rearrangement of the valence electrons, because the effective shielding of the nucleus is reduced. This effect leads to a lowering of the core-excitation energy of the final state. This rearrangement of the electrons can be understood as an orbital relaxation effect. To fully understand and interpret experimental spectra, an accurate knowledge about core-excitation energies, transition moments, the character of the core-excited states as well as their corresponding properties is necessary. Such information can be obtained with quantum chemical (QC) methods. They help to analyze and interpret experimental spectra, thereby providing a deep insight into the nature of core-excited states. Generally, a plethora of methods is available to calculate excited states and simulate absorption spectra. The larger the system, the more expensive are the computations. Hence, certain levels of approximation have to be introduced to lower the computational cost. This leads to a loss of reliability and accuracy of the results. The time-dependent density functional theory (TD-DFT), for example, currently is the prevalently used excited-state method for the calculation of large molecules up to 300 atoms. However, TD-DFT has several disadvantages like the self-interaction error (SIE), which leads to wrong descriptions of certain kinds of excited states, e.g. charge-transfer states or core-excited states. The excitation energies of these types of states are strongly underestimated, but if these issues are kept in mind, TD-DFT is a useful tool, providing proper spectral features. The algebraic diagrammatic construction scheme (ADC) is a prominent QC method for the calculation of excited states, which is known to provide accurate valence-excited states of small- and medium-sized molecules in an adequate computational time. The ADC approach is based on a Green’s function formalism in combination with partitioning the Hamiltonian using perturbation theory. Due to its size-consistency and Hermitian ADC secular matrix structure, the level of approximation can be improved systematically and properties can be computed straightforwardly. It is possible to calculate one-particle state properties in combination with the intermediate state representation (ISR) approach, e.g. static dipole moments and state densities, which altogether provide enhanced information about absorption spectra. A further advantage of ADC is the indirect inclusion of orbital relaxation effects via couplings to higher-excited configurations, which are important to describe core-excited states properly. However, the calculation of core excitations is tedious using the unmodified ADC approach, because, in order to solve the ADC eigenvalue problem, numerical iterative eigenvalue solvers are employed usually only providing the energetically lowest eigenstates. Core-excited states, however, are located in the high energy X-ray region of the optical spectrum and in order to calculate them directly, one has to compute all energetically underlying valence excitations as well. This is computationally very expensive and not feasible for medium-sized systems. The direct calculation of the core excitations is prevented by couplings between the valence and core-excited states. A solution to this issue is the application of the core-valence separation (CVS) approximation to the ADC approach, which results in the CVS-ADC method. This approximation is based on the fact that core orbitals are energetically well-separated from the remaining orbital spaces and as a consequence, the couplings between core- and valence-excited states are small and can be neglected. In other words, the CVS approximation decouples the core and valence excitation spaces from each other and allows for a direct computation of core-excited states. In former work, it was proven that a very good agreement with experiments can be obtained at the extended second order level CVS-ADC(2)-x.

My PhD project mainly consists of two important parts. One was to enhance and develop variants of the CVS-ADC method and implement all approaches efficiently in the adcman program, which is part of the Q-chem program package. Secondly, I benchmarked these implementations and simulated X-ray absorption spectra of small- and medium-sized molecules from different fields. In this thesis, I present my implementations, as well as the results and applications obtained with the CVS-ADC methods and give a general introduction into quantum chemical methods. At first, I implemented the CVS-ADC approach up to the extended second in an efficient way. The program is able to deal with systems up to 500 basis functions in an adequate computational time, which allows for accurate calculations of medium-sized closed-shell molecules, e.g. acenaphthenequinone (ANQ). Afterwards, the CVS-ADC implementation was extended for the first time to deal with open-shell systems, i.e. ions and radicals, which implies a treatment of unrestricted wave functions and spin-orbitals. The resulting method is denoted as CVS-UADC(2)-x. For the first time, I applied the CVS approximation to the the third order ADC scheme, derived the working equations, and implemented the CVS-ADC(3) method in adcman. As the last step, I applied the CVS formalism for the first time to the ISR approach to enable calculations of core-excited state properties and densities. This provides the basis for subsequent evaluations of transition- and density matrices, which give access to exciton sizes, e.g. hole sizes or distances between hole and electron densities. All implementations are presented and discussed in the scope of my thesis.

To benchmark all restricted and unrestricted CVS-ADC/CVS-ISR methods up to third order in perturbation theory, I chose a set of small molecules, e.g. carbon monoxide (CO). The calculated values of core-excitation energies, transition moments and static dipole moments are compared with experimental data or other approaches, thereby estimating complete basis set (CBS) limits. Furthermore, a comprehensive study of different basis sets is performed. As it turns out, the CVS-ADC(2)-x method provides the best agreement with experiments, while CVS-ADC(3) overestimates the core excitation energies. In combination with the CBS limit of the aug-cc-series, a mean error of -0.23%±0.12% for core-excitation energies can be identified at the CVS-ADC(2)-x level for carbon, nitrogen and oxygen K-edge excitations, whereas CVS-ADC(3) exhibits errors of 0.61%±0.32%. This is due to fortuitous error compensation of basis set truncation, electron correlation, orbital relaxation and neglect of relativistic effects at the CVS-ADC(2)-x level. I show that this error compensation is broken at the third order level, because the ratio between terms describing relaxation and polarization effects is shifted in a way that the excitation energy increases. However, transition moments and spectral features, as well as static dipole moments, are excellently described with both CVS-ADC(2)-x and CVS-ADC(3). Overall, considering the detailed investigation of the basis set influence on the results, I conclude that the use of restricted or unrestricted CVS-ADC(2)-x in combination with a diffuse triple-ζ basis set in its Cartesian version can be seen as a black-box method for the calculation of core-excited states of organic molecules. Especially the 6-311++G** basis set provides an excellent ratio of accuracy to computational time. Another important topic is the description of orbital relaxation effects. In the scope of this thesis, I show, how these effects are included indirectly within the CVS-ADC approaches. For this purpose, two different descriptors are used, i.e. electron promotion numbers and the amount of doubly excited amplitudes. Furthermore, with the help of detachment/attachment (D/A) densities, which can be constructed via the CVS-ISR approach, relaxation effects can be visualized. For this purpose, the (D/A) densities are compared with hole/electron (h/e) densities based on the transition density matrix. With this knowledge, the X-ray absorption spectra of medium-sized molecules and radicals from the fields of organic electronics and biology are investigated and analyzed. On the basis of these studies, the restricted and unrestricted versions of CVS-ADC(2)-x in combination with the 6-311++G** basis set exhibit mean errors of core-excitation energies around 0.1%, compared to experimental values. Additionally, core-excited state characters are analyzed with the help of state densities obtained via the CVS-ISR approach or the transition density matrix. To demonstrate that the CVS-ADC(2)-x approach can be employed as a benchmark black-box method, TD-DFT results are compared directly with the ones at the CVSADC(2)-x level. As expected, TD-DFT underestimates core-excitation energies up to 4% due to the SIE, which is about 10 eV in the case of carbon 1s excitations. Since the CVS approximation leads to both a simplification of the ADC working equations, as well as a restriction of the excitation space to correspond only to core excitations, the computational cost is reduced compared to the general ADC approach. To demonstrate the computational savings as a function of the size of the core space, several systems are investigated. CVS-ADC(3) calculations take about 8 – 10 times longer than CVS-ADC(2)-x calculations and since the results are generally more accurate with the latter method, the use of CVS ADC(3) is not justified. Compared to general ADC(2)-x, the speed-up at the CVS-ADC(2)-x level is about a factor of 4.0, but this factor strongly depends on the size of the system and the size of the core space. Next, I present applications from the field of organic electronics. The remarkable agreement with experimental data at the CVS-ADC(2)-x level justifies the prediction of yet non-recorded experimental X-ray absorption spectra. Therefore, I chose the anthracene cation, which can be seen as a model system of pentacene and its derivatives, which are commonly used as hole conductors (p-type). X-ray absorption spectra of the pentacene cation could provide deeper insight into its charge carrier properties, but measurements of experimental spectra of ionized species are usually very challenging. With the help of CVS-UADC(2)-x calculations, I show that the anthracene cation exhibits additional peaks due to the half-filled single-occupied molecular orbital. They are located approximately 3.5 eV – 1.5 eV below the first peak of neutral anthracene, which may help to distinguish a cation from the neutral species. Furthermore, the cationic spectrum exhibits peak broadening, compared to the two first peaks of neutral anthracene. Other applications concentrate on the trends of core-excited state properties along important potential energy surfaces (PES) of ANQ, phenol and bithiophene. Therefore, static dipole moments, energies, and exciton sizes are analyzed as a function of the C–O distances of ANQ and phenol, as well as the torsion around the central dihedral angle of bithiophene. Finally, another aspect of the CVS-ISR method is the accessibility of transition moments between two states, which can be used to calculate oscillator strengths for core-excited state absorption (CESA) spectra. To the best of my knowledge, no experimental data of CESA processes between two core-excited states have been recorded yet. However, such spectroscopic data could exhibit new insights and the calculation of CESA transition moments using the CVS-ADC/CVS-ISR approach is straightforward. Hence, first results of CESA processes were calculated and are presented in this thesis. In the case of ANQ, particularly bright transitions can be identified from the lowest oxygen 1s excited-state to higher ones.