## Research Areas

Our research group deals with rather interdisciplinary questions that touch upon the fields of theoretical chemistry (solid state), theoretical physics (condensed matter, statistical mechanics, thermodynamics), crystallography (structure identification), materials science (properties of solids) and applied mathematics (mathematical modeling, optimization and efficient algorithms).

The individual projects are associated with five different research areas:

- Structure prediction of crystalline compounds
- Structure modeling of amorphous solids
- Analytical modeling and simulations of properties of solids
- General aspects of continuous and discrete energy landscapes
- Modern thermodynamics: calculation of free energies, finite time thermodynamics, geometry of thermodynamics

The red thread connecting all these topics is the investigation of energy landscapes of complex systems: the landscapes' structure, the thermodynamic and statistical mechanical properties inherent in the landscapes, and the physical and/or mathematical properties of the systems represented by the landscapes. This task requires the development of new methods and procedures for the study of multi-minima energy surfaces, followed by their application to interesting systems that range from toy-models over well-defined but more abstract mathematical and physical problems, to energy landscapes that reflect the real system (e.g. a solid compound) as much as possible.

These theoretical investigations are closely connected with the experimental work in the department, in particular with the efforts of the synthesis-via-atom-beam-deposition group and the high-pressure synthesis group.

## I. Structure prediction of crystalline compounds

A major task in solid state theory is the prediction of stable crystalline compounds in a chemical system: At which composition does a system form a kinetically and/or thermodynamically stable compound, what is its structure and chemical/physical properties, to what degree is this compound stable and what will happen, if such a metastable compound undergoes a phase transition?

To answer these questions, one investigates the energy landscape of the system, i.e. the (potential) energy as a function of the atomic positions in the system. At low temperatures local minima of the potential energy surrounded by sufficiently high energy barriers correspond to kinetically stable structures, while at higher temperatures, entropic barriers become increasingly important for the kinetic stability, and a "minimization" of the free energy should replace the simpler search for minima of the potential energy.

Due to the fact that the solution of the full Schrödinger equation for solid systems is very expensive computationally, we usually employ a sequence of approximations, in order to simplify the energy calculation: Born-Oppenheimer approximation, periodic boundary conditions, and effective potentials mostly based on empirical data. This simplified energy landscape is then analyzed using simulated annealing for the determination of the local minima. During the global optimization, the positions of the atoms, the vectors of the repeated simulations cell, the distribution of electrons on the atoms and the composition of the atoms in the cell are varied, depending on the specific problem under investigation. For the computation of energy barriers and local densities of states, the threshold algorithm has been developed. In addition, algorithms have been developed to calculate structural and physical properties.

Closely related to the question of structure prediction of not-yet- synthesized compounds is the issue of determining the structures of already synthesized compounds, where only powder diffraction data are available but no successful structural model has been found. Here, a combined optimization of the potential energy and the difference between calculated and observed powder diffractograms (Pareto-optimization) can serve to identify good structure candidates.

## II. Structure modeling of amorphous solids

An important class of solid compounds are (covalent) amorphous solids, both from a chemical and materials science point of view and because of their importance as classical disordered but nevertheless "static" systems. The actual structure of such compounds is very difficult to determine from experiment alone, and without reliable information about the geometry and topology of the solid, the interpretation of many curious and fascinating properties remains tentative. Theory in form of simple models and stochastic and deterministic simulations of such system, can help close this hole in our knowledge. Thus, we are currently investigating several amorphous materials, using MC/MD-simulations and the lid- and the threshold-algorithm. We study relaxation behavior, the properties of the energy landscape of realistic and model (network) systems, and calculate properties of such solids.

## III. Analytical modeling and simulations of properties of solids

Two important issues of theoretical work, especially if heavy numerical efforts are involved, are the comparison with experiment, and the connection with simple models that allow us to extract the essential physical and chemical aspects of the system under investigation (or a process for such a system), from the mass of numerical data. Therefore, we investigate simple, and more refined analytical mathematical models of systems and the processes present therein. In particular, rational planning of syntheses of chemical compounds (predicted or otherwise) requires the development of efficient models describing the various processes that occur during a chemical synthesis. Furthermore, we try to compute for real systems statistical and thermodynamic properties, using realistic potentials in the simulations.

Examples are the modeling of sintering processes and surface reconstruction, or the computation of vibrational properties of solids. Furthermore, we are modeling the low-temperature atom beam deposition synthesis starting from the evaporation of the constituents into the gas phase, over the deposition on the substrate to the final tempering at low temperatures. Finally, quantum mechanical methods are used to calculate the energy (and also for the determination of electronic properties), allowing e.g. the prediction of pressure-induced phase transitions.

## IV. General aspects of continuous and discrete energy landscapes

While the detailed investigation of a particular problem, with its specific energy landscape and properties, is highly rewarding, on a somewhat more abstract level several very interesting questions loom, e.g., what qualities of the energy landscape might be characteristic for what class of systems, and what general methods could and should be brought to bear on the efficient study of such landscapes.

Typically, complex systems possess multi-minima energy landscapes, with a more or less complicated structure of energetic and entropic barriers. Also e.g. the relaxation behavior of many systems can be modeled as a stochastic walk on such a landscape. We have therefore developed the lid-method, with its two implementations, the lid-algorithm for discrete landscapes, and the threshold algorithm for continuous landscapes. This method can be used to study in detail regions of the landscape, called pockets, whose properties might be representative for larger pieces of the landscape. An interesting observation is e.g. that for many complex systems the local density of states within such a pocket grows approximately exponential with energy, and that the reach of minima basins goes far beyond the nearest saddle points - a result that together with information about the energetic and entropic barriers could have profound implications for our understanding of the behavior of such systems.

## V. Modern thermodynamics: calculation of free energies, finite time thermodynamics, geometry of thermodynamics

With the increasing interest in multi-minima energy landscapes, which contain many metastable regions, the statistical mechanical and thermodynamic behavior of such systems moves into focus. The definition of the degree of metastability of such regions, the efficient calculation of their free energies, and the time-development of the individual metastable regions and of the system as a whole (especially, the question, whether a separation of time scales is appropriate), are major areas of research.

An important tool for the analysis of the efficiency of free energy calculations is finite time thermodynamics, which, of course, is also applicable to the question of optimality of real thermodynamic processes that are restricted to run for only a finite time. An important quantity that controls the efficiency of such a process is the Hessian matrix of the entropy or the internal energy as a function of the other thermodynamic variables. Thus, the geometry of thermodynamic surfaces needs to be investigated, leading to the field of geometric thermodynamics.

New developments in this regard deal with the optimal control of phase transitions and nucleation and growth processes. Clearly, the ultimate goal is the control of chemical syntheses using methods from finite time thermodynamics, both regarding the qualitative outcome (selection of modifications) and the quantitative aspect (speed of synthesis, energy efficiency, amount of product).

## Reviews

*J.C. Schön, M. Jansen:*Angew. Chem. Int. Ed.**35**(1996) 1286-1304*J.C. Schön, M. Jansen:*Z. Kristallographie Vol**216**(2001) 307-325, 361-383*M. Jansen:*Angew. Chem. Int. Ed.**41**(2002) 3746-3766*M. Jansen, J.C. Schön:*Angew. Chem. Int. Ed.**45**(2006) 3406-3412*M. Jansen:*"The Deductive Approach to Chemistry, a Paradigm Shift"

in "Turning points in Solid-State, Materials and Surface Chemistry"

K. Harris, P. Edwards (eds.), RSC Publishing, Cambridge UK (2008)*J.C. Schön, K. Doll, M. Jansen:*Phys. Status Solidi B Vol.**247**(2010) 23-39*M. Jansen, K. Doll, J. C. Schön:*Acta Cryst. A**66**(2010) 518-534