Materials properties are very closely linked to the underlying crystal structure, the way how atoms or molecules are arranged in the crystal lattice. The knowledge of the atomic arrangement is the very first and most important information one needs to study and understand a material. Crystal structure prediction (CSP) methods are techniques to determine the crystal structure of condensed matter systems, like solids or clusters, from atomistic simulations. It has been the goal to predict the correct crystal structure of a material solely based on the chemical composition since the 1950s, but reliable CSP methods have only become available recently.

Sampling the energy landscape

A compound in thermodynamic equilibrium will crystallize in a structure that minimizes the Gibbs free energy at a given condition. Hence, the task in crystal structure prediction is equivalent to finding the global minimum on a high-dimensional free energy surface, G=E+pV-TS. The main difficulty in CSP is that this free energy landscape is very complex, and the number of possible atomic/molecular arrangements scales exponentially with the number of atoms in the system. In practice, only those structures are of importance which correspond to local minima on such an energy landscape, which are referred to as metastable structures. Therefore, many local minima need to be tested to screen for the lowest energy local minimum, the global minimum, which corresponds to the ground state structure at a given pressure and temperature.

The second challenge in CSP is how to represent the energy landscape using some method to model the atomic interactions. Depending on the method, the energetic ordering of the local minima can change, or in extreme cases completely vanish from the energy surface. Force fields are the computationally most affordable potentials, which completely neglect any quantum mechanical treatment of the atoms and rely on physically motivated analytical forms to represent the atomic interactions. The parameters of such force fields are fitted either to experimental data or highly accurate quantum mechanical calculations. Although such force fields work relatively well for simple ionic systems and can be used to treat millions of atoms, they are not very transferable and often fail for more complex materials.

Quantum mechanical methods on the other hand are much more accurate and don’t rely on any empirical data, but come at a higher computational cost. Thanks to the efforts of Walter Kohn, electronic structure calculations are now possible without explicitly operating on the many-body wave function, but on the 3-dimensional electron density (for which he was awarded the Nobel Prize in Chemistry in 1998). Density functional theory (DFT) has meanwhile become the main workhorse in physics, chemistry and materials science due to its good accuracy at a moderate computational demand, allowing the treatment of systems with thousands of atoms. Nevertheless, the approximations used in DFT lead to inaccurate results for some classes of materials, such as highly correlated systems or Van der Waals compounds. Quantum Monte Carlo methods are more accurate and can alleviate these issues, and are touted as the next generation tool in computational materials science.

The minima hopping method

Although there are meanwhile several available tools for CSP, they all come with their strengths and drawbacks. I developed my own CSP algorithm which is based on the minima hopping method, currently implemented in the Minhocao package (Minima Hopping for Crystal Optimization). Minhocao can treat various boundary conditions to investigate crystals, surfaces, nanowires, layered materials, molecules, etc, and can be coupled to any code that models an energy landscape. Some force fields are directly implemented into the code, such as a multi component Lennard-Jones potential, and several silicon/carbon potentials.

The algorithm employs efficient moves on the potential energy (or enthalpy) surface by performing consecutive molecular dynamics escape steps and geometry relaxations. The initial velocities for the dynamics are aligned preferably along soft-mode directions to exploit the Bell-Evans-Polanyi principle in order to favor the escape to low-energy structures. In contrast to many popular CSP methods like genetic algorithms, the minima hopping methods uses physical moves to sample the energy landscape, providing insight into the kinetics of a system. Minhocao has been used successfully for many applications, ranging from high-pressure materials like superconducting hydrides, to advanced energy materials like silicon allotropes for photovoltaics and thermoelectric materials. Currently, Minhocao works together with the following software packages: Abinit, Quantum Espresso, BigDFT, VASP, CP2K, mopac, lammps, Tinker, DFTB+, GULP and Siesta, but it can be coupled to any other code which evaluates energies, forces, and stresses. A network socket communication interface is implemented based on theĀ  i-Pi protocol, which makes the coupling to any external code very simple and straight forward.

The movie shows the evolution of a search for the ground state structure of the formaldehyde molecular crystal.

Polymorphism around us

A product that many people associate with Switzerland, my home country, is chocolate. Have you ever noticed that a white layer appears on chocolate if you keep it in the refrigerator for a long time? It is NOT chocolate gone bad or mold! This bloom appears due to a change in the crystal structure of the cocoa butter in chocolate. In fact, cocoa butter can form six different crystal structures at ambient pressure but different temperatures, and form V is the most desirable one for consumption due to its smooth texture and a melting point well below our body temperature. Other crystal forms, called polymorphs, are stable at lower temperatures. Now, if you refrigerate chocolate over a long time, a phase transition will occur to a low-temperature form which has a higher melting temperature and a brittle consistency, visible as white bloom on the surface. So don’t worry, the chocolate remains edible, but won’t taste as good. Always store chocolate in a cool, dry place, but never in a refrigerator.