The Sternheimer Approach to Linear Response Time-Dependent Density Functional Theory

As natural light-harvesting mechanisms are more efficient and robust than artificial solar technology, a deeper understanding of the energy absorption and conversion processes in plants and bacteria is at the center of a lot of current research. The theoretical prediction and interpretation of these phenomena requires methods that facilitate a quantum mechanical description of systems made from several thousands of electrons. Density functional theory in its Kohn-Sham formulation is by far the most popular method for the study of molecules, clusters and solids due to its beneficial ratio of accuracy to computational cost, and its time-dependent extension (TDDFT) is commonly used for the description of dynamical properties of molecular and nanostructures. The reliability and efficiency of Kohn-Sham density functional theory is determined by the approximation to the exchange-correlation energy (or potential) employed in practical applications. For the study of electronic excitations one usually resorts to TDDFT in the linear regime. Linear response calculations in the Casida formulation are routinely done with most quantum chemistry codes. While the Casida approach is technically highly developed and quite efficient for medium-sized systems, it involves virtual orbitals, which leads to an unfavorable scaling with the system size. Additionally, as an eigenvalue problem with a dense matrix, it is not suited for a high degree of parallelization. Therefore, the Casida scheme is not optimal for the study of the larger systems occurring in natural light-harvesting complexes, which prompts the development of alternative linear response methods. The investigation of the relevant processes in these systems through TDDFT is further complicated by the fact that they involve certain types of electronic excitations that most commonly used exchange-correlation approximations cannot predict reliably. This includes, among others, charge-transfer excitations, which play an important role in, e. g., photosynthetic reaction centers. More involved approximations that give a qualitatively correct description of charge-transfer exist, but are too expensive for applications in larger systems. This has motivated the development of various cheaper approximations that aim at mimicking the decisive features of, e. g., expensive exact-exchange based range-separated hybrid functionals. Meta-generalized gradient approximations seem to particularly well-suited for this task, but like most other approximations that might improve the description of charge transfer and other difficult excitations, they are orbital dependent approximations, which makes their application in time-dependent Kohn-Sham calculations highly nontrivial. In order to address these problems, this thesis is focused on advancing an alternative, lesser-known linear response scheme, the Sternheimer method. While only relatively few applications of the scheme have been reported so far, it is particularly promising for the study of large systems since it only involves occupied orbitals, scales favorably with the system size, and can be parallelized massively, most notably because different frequencies can be treated completely independently. In the first part of the thesis the scheme is developed further, regarding both formal and technical aspects. Among other things, a new derivation is presented, it is extended to the treatment of triplet excitations, and novel strategies for the efficient evaluation of excitation energies are put forward. Then the scheme is employed to study an orbital independent exchange-correlation approximation designed to mimic properties of exact exchange, the Armiento-Kümmel generalized gradient approximation. To be able to study more flexible approximations, a new and efficient way of treating orbital dependent exchange-correlation potentials in the Sternheimer approach is developed, which suggests that the Sternheimer method might be better suited for the application for orbital functionals then linear response schemes based on Casida’s equations or on real-time propagation. Finally, this method is applied to the recently developed TASK meta-generalized gradient approximation, and TASK’s performance in the description of charge transfer in a donor-acceptor-donor system of experimentally relevant size is studied.