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Abstract
In many parts of physics, chemistry, biology, or material science, excited electronic states, accessible via the interaction of atoms or molecules with electromagnetic radiation, play an essential role. Experimental spectra, however, generally provide only indirect information on molecular structure and dynamics. Thus, a theoretical description of excitation energies and transition strengths is crucial for a comprehensive understanding of lightinduced processes. In this dissertation, the theory, implementation, and application of several Hermitian methods to calculate the properties mentioned above are described. If excitation energies are obtained by diagonalization of a nonHermitian secular matrix, both left and right eigenvectors need to be calculated to obtain spectral intensities and other properties. In this case, the eigenvectors are not orthogonal to each other, and the energy may become complex. Hermiticity is thus a very desirable property since none of the aforementioned problems occurs. Thus, several approaches based on the algebraicdiagrammatic construction (ADC) scheme, as well as the related unitary coupledcluster (UCC) method, are presented. Within these methods, oneelectron properties such as dipole moments are available via the socalled intermediate state representation (ISR) approach, which corresponds to an expectation value of the respective oneelectron operator with the wave function. The ISR formalism is also used to derive explicit working equations for the secondorder ADC scheme, which is based on a ground state described by Møller–Plesset (MP) perturbation theory. This implies that ADC inherits all weaknesses from the underlying MP model. For the ADC(2) scheme, merely the firstorder MP wave function is required, which contains only doublyexcited determinants for a Hartree–Fock reference. Due to the form of the firstorder doubles amplitudes, several cancellations occur in the singles block of the ADC(2) matrix. In order to remedy the breakdown of MP2, the firstorder doubles amplitudes from MP are replaced by the ones obtained from a coupledcluster (CC) calculation, which are formally correct through infinite order. The resulting schemes, referred to as CCADC(2), are applied to several sets of small to mediumsized molecular systems, where generally minor improvements in excitation energies compared to the standard ADC(2) scheme can be observed. For the ozone molecule, which is known to be a difficult test case for quantumchemical methods, the experimental first excitation energy is 1.6 eV; standard ADC(2) is far off with 2.14 eV, and CCDADC(2) yields 1.59 eV. Excitedstate potential energy curves along the dissociation of the nitrogen molecule calculated with ADC(2) break down at around 2 Å due to the failure of MP2. The CCDADC(2) curves remain reasonable up to about 3.5 Å. The CCADC(2) methods are successively extended to the calculation of static dipole polarizabilities. It is shown that the correlation amplitudes play a more important role in the modified transition moments than in the ADC secular matrix itself, and consistent improvement is obtained for static polarizabilities with the CCADC schemes compared to standard ADC, particularly for aromatic systems like benzene or pyridine, which had proven difficult cases for standard ADC. Specifically, the CCADC(2) schemes yield significantly better results than the ADC(3/2) scheme, at a computational cost amounting to only 1% of the latter. The ISR derivation can also be carried out with a CC wave function correct through first order instead of the MP one. However, having converged CCD amplitudes instead of the firstorder MP ones, the aforementioned cancellations in the secondorder singles block do not occur. Hence, the final matrix elements differ between CCDADC(2) and this scheme referred to as CCDISR(2). As the expansion of the UCC similaritytransformed Hamiltonian does not truncate naturally, it needs to be truncated manually, usually by using arguments from MP perturbation theory. The UCC2 doubles amplitudes correspond to those from LCCD, but the secular matrix elements depend on the treatment of the similaritytransformed Hamiltonian is treated. By employing the Baker–Campbell–Hausdorff expansion, the secondorder singles block is equivalent to CCDISR(2), but by employing the Bernoulli expansion, the matrix elements are equivalent to CCDADC(2), with differences only in the correlation amplitudes. In a strict perturbationtheoretical framework, all methods turn out to be identical. All different Hermitian secondorder methods have been implemented and tested on a set of small molecules, where it turned out that the differences in excitation energies between the methods are small whenever the systems are well described by means of perturbation theory. The Bernoulli UCC scheme is further extended to third order, where excitation energies and oscillator strengths on mediumsized organic molecules as well as ground and excitedstate dipole moments are reported for the first time. While vertical excitation energies of the UCC3 scheme are similar to those obtained with ADC(3), significant improvements can be observed for the dipole moments in the ground and excited states. Furthermore, this UCC scheme is applied to the electron propagator, and ionization potentials of the IPUCC2 and IPUCC3 schemes of selected amino acids are reported for the first time. Apart from expectation values, molecular properties can be calculated as derivatives of the energy with respect to a perturbation connected to the observable. The two approaches are only equivalent if the Hellmann–Feynman theorem is fulfilled. By using explicit working equations, the relationship between the two approaches is investigated with a focus on orbital relaxation for all standard quantumchemical methods, in particular MP and ADC. It is shown that for MP2 the expectation value is very close to the orbitalrelaxed property. In contrast, for ADC(1) the expectation value includes no orbital relaxation and for ADC(2) only a small fraction. With ADC(3) eigenvectors, on the other hand, the ISR gets closer to the relaxed values, but only for singlyexcited states. Numerical investigations underline all the theoretical predictions.
Item Type:  Dissertation 

Supervisor:  Dreuw, Prof. Dr. Andreas 
Place of Publication:  Heidelberg 
Date of thesis defense:  29 April 2020 
Date Deposited:  12 May 2020 07:15 
Date:  2020 
Faculties / Institutes:  Fakultät für Chemie und Geowissenschaften > Institute of Physical Chemistry Service facilities > Interdisciplinary Center for Scientific Computing 
Subjects:  500 Natural sciences and mathematics 530 Physics 540 Chemistry and allied sciences 
Controlled Keywords:  quantum chemistry, electronic structure theory, excitation energies, molecular properties, coupled cluster 