Relativistic LCAO with Minimax Principle and New Balanced Basis Sets

Relativistic density functional theory is widely applied in molecular calculations with heavy atoms, where relativistic and correlation effects are on the same footing. Variational stability of the Dirac Hamiltonian is a very important field of research from the beginning of relativistic molecular calculations on, among efforts for accuracy, efficiency, and density functional formulation, etc. Approximations of one- or two-component methods and searching for suitable basis sets are two major means for good projection power against the negative continuum. The minimax two-component spinor linear combination of atomic orbitals (LCAO) is applied in the present work for both light and super-heavy one-electron systems, providing good approximations in the whole energy spectrum, being close to the benchmark minimax finite element method (FEM) values and without spurious and contaminated states, in contrast to the presence of these artifacts in the traditional four-component spinor LCAO. The variational stability assures that minimax LCAO is bounded from below. New balanced basis sets, kinetic and potential defect balanced (TVDB), following the minimax idea, are applied with the Dirac Hamiltonian. Its performance in the same super-heavy one-electron quasi-molecules shows also very good projection capability against variational collapse, as the minimax LCAO is taken as the best projection to compare with. The TVDB method has twice as many basis coefficients as four-component spinor LCAO, which becomes now linear and overcomes the disadvantage of great time-consumption in the minimax method. The calculation with both the TVDB method and the traditional LCAO method for the dimers with elements in group 11 of the periodic table investigates their difference. New bigger basis sets are constructed than in previous research, achieving high accuracy within the functionals involved. Their difference in total energy is much smaller than the basis incompleteness error, showing that the traditional four-spinor LCAO keeps enough projection power from the numerical atomic orbitals and is suitable in research on relativistic quantum chemistry. In scattering investigations for the same comparison purpose, the failure of the traditional LCAO method of providing a stable spectrum with increasing size of basis sets is contrasted to the TVDB method, which contains no spurious states already without pre-orthogonalization of basis sets. Keeping the same conditions including the accuracy of matrix elements shows that the variational instability prevails over the linear dependence of the basis sets. The success of the TVDB method manifests its capability not only in relativistic quantum chemistry but also for scattering and under the influence of strong external electronic and magnetic fields. The good accuracy in total energy with large basis sets and the good projection property encourage wider research on different molecules, with better functionals, and on small effects.

On Solution Sets of Information Inequalities

We investigate solution sets of a special kind of linear inequality systems. In particular, we derive characterizations of these sets in terms of minimal solution sets. The studied inequalities emerge as information inequalities in the context of Bayesian networks. This allows to deduce important properties of Bayesian networks, which is important within causal inference.

Photoperiod-dependent plasticity of circadian pacemaker center in the brain of the Madeira cockroach Rhyparobia maderae

To various degrees, insects in nature adapt to and live with two fundamental environmental rhythms around them: (1) the daily rhythm of light and dark, and (2) the yearly seasonal rhythm of the changing photoperiod (length of light per day). It is hypothesized that two biological clocks evolved in organisms on earth which allow them to harmonize successfully with the two environmental rhythms: (1) the circadian clock, which orchestrates circadian rhythms in physiology and behavior, and (2) the photoperiodic clock, which allows for physiological adaptations to changes in photoperiod during the course of the year (insect photoperiodism). The circadian rhythm is endogenous and continues in constant conditions, while photoperiodism requires specific light inputs of a minimal duration. Output pathways from both clocks control neurosecretory cells which regulate growth and reproduction. This dissertation focuses on the question whether different photoperiods change the network and physiology of the circadian clock of an originally equatorial cockroach species. It is assumed that photoperiod-dependent plasticity of the cockroach circadian clock allows for adaptations in physiology and behavior without the need for a separate photoperiodic clock circuit. The Madeira cockroach Rhyparobia maderae is a well established circadian clock model system. Lesion and transplantation studies identified the accessory medulla (aMe), a small neuropil with about 250 neurons, as the cockroach circadian pacemaker. Among them, the pigment-dispersing factor immunoreactive (PDF-ir) neurons anterior to the aMe (aPDFMes) play a key role as inputs to and outputs of the circadian clock system. The aim of my doctoral thesis was to examine whether and how different photoperiods modify the circadian clock system. With immunocytochemical studies, three-dimensional (3D) reconstruction, standardization and Ca2+-imaging technique, my studies revealed that raising cockroaches in different photoperiods changed the neuronal network of the circadian clock (Wei and Stengl, 2011). In addition, different photoperiods affected the physiology of single, isolated circadian pacemaker neurons. This thesis provides new evidence for the involvement of the circadian clock in insect photoperiodism. The data suggest that the circadian pacemaker system of the Madeira cockroach has the plasticity and potential to allow for physiological adaptations to different photoperiods. Therefore, it may express also properties of a photoperiodic clock.