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Development and Application of Hermitian Methods for Molecular Properties and Excited Electronic States

Hodecker, Manuel

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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 light-induced 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 non-Hermitian 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 algebraic-diagrammatic construction (ADC) scheme, as well as the related unitary coupled-cluster (UCC) method, are presented. Within these methods, one-electron properties such as dipole moments are available via the so-called intermediate state representation (ISR) approach, which corresponds to an expectation value of the respective one-electron operator with the wave function. The ISR formalism is also used to derive explicit working equations for the second-order 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 first-order MP wave function is required, which contains only doubly-excited determinants for a Hartree–Fock reference. Due to the form of the first-order doubles amplitudes, several cancellations occur in the singles block of the ADC(2) matrix. In order to remedy the breakdown of MP2, the first-order doubles amplitudes from MP are replaced by the ones obtained from a coupled-cluster (CC) calculation, which are formally correct through infinite order. The resulting schemes, referred to as CC-ADC(2), are applied to several sets of small to medium-sized 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 quantum-chemical methods, the experimental first excitation energy is 1.6 eV; standard ADC(2) is far off with 2.14 eV, and CCD-ADC(2) yields 1.59 eV. Excited-state 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 CCD-ADC(2) curves remain reasonable up to about 3.5 Å. The CC-ADC(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 CC-ADC schemes compared to standard ADC, particularly for aromatic systems like benzene or pyridine, which had proven difficult cases for standard ADC. Specifically, the CC-ADC(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 first-order MP ones, the aforementioned cancellations in the second-order singles block do not occur. Hence, the final matrix elements differ between CCD-ADC(2) and this scheme referred to as CCD-ISR(2). As the expansion of the UCC similarity-transformed 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 similarity-transformed Hamiltonian is treated. By employing the Baker–Campbell–Hausdorff expansion, the second-order singles block is equivalent to CCD-ISR(2), but by employing the Bernoulli expansion, the matrix elements are equivalent to CCD-ADC(2), with differences only in the correlation amplitudes. In a strict perturbation-theoretical framework, all methods turn out to be identical. All different Hermitian second-order 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 medium-sized organic molecules as well as ground- and excited-state 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 IP-UCC2 and IP-UCC3 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 quantum-chemical methods, in particular MP and ADC. It is shown that for MP2 the expectation value is very close to the orbital-relaxed 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 singly-excited 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
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