Probabilistic inference applied to Quantum Mechanics

Although the unambiguous logic of scientific inference has been known for over two hundred years (cf. external link: T. Bayes), only modern computer power has made it possible to apply it rigourously to a wide variety of problems in science and engineering. I am interested in applying the computational tools of probability theory and machine learning to atomistic simulation.

The most significant difficulty in extending quantum mechanical simulation techniques to larger length and time scales is that all exact formulations of quantum mechanics are non-local. Indeed it seems that the fundamental difference between the simplest semi-empirical quantum mechanical model (e.g. Tight Binding) and the most complicated classical model (at least for insulators) is that the quantum model involves some global operation over the entire system, like diagonalizing the whole Hamiltonian. Great strides have been made in making this less and less painful, principally by taking advantage of the sparse nature of the Hamiltonian and the density matrix. I call this weak locality. However, it has long been known (indeed universally assumed and implicitly taken advantage of), that in most systems, there is a much stronger locality, namely that the forces, trajectories and general properties of an atom are not very dependent on the configuration of atoms far away. Every supercell calculation and every cluster calculation assumes this. But this strong locality implies that the properties of an atom (e.g. the force or charge on it) should only depend on the positions of nearby atoms. Therefore, predicting these properties can be cast as a learning task, where the input feature array is a set of descriptors of the neighbourhood of an atom and the output is the energy of the atom. Summing the atomic energies leads to the Born-Oppenheimer potential energy surface (for which the electronic system is in its ground state), and hence many material properties can be calculated cheaply and accurately, without further recourse to the expensive, non-local electronic system.

Of course many models of the Born-Oppenheimer surface exist, they are collectively called "interatomic potentials". But they are mostly made by trial and error, and guesswork. The paper below shows how interatomic potential models can be created automatically and rigourously using only the quantum mechanical data itself as input together with reasonable and universal assumptions about the smoothness of the potential energy surface.

Gaussian Approximation Potentials: the accuracy of quantum mechanics, without the electrons
external link: Physical Review Letters 104 136403 (2010)


The image above shows the potential energy landscape of a carbon atom in diamond, using a traditional model (left), the new GAP model with one datapoint (middle) and with 50 data points(right). The bluish surface represents the true quantum mechanical answer, the gray surface is the interatomic model.

Multiscale Materials Simulation

I develop computational schemes that allow materials simulation with simultaneous, concurrent use of multiple models, which address physical properties at different length scales. Most problems in the world are multiscale; however typical areas where direct multiscale simulation is unavoidable include e.g. plasticity, brittle fracture and enzyme catalysis.

multiscale methods There are many techniques that are each appropriate at a given scale and have a corresponding capability in terms of number of atoms handled or simulation time. Multiscale schemes use two or more of these techniques simultaneously.

Brittle fracture

crack kink The picture on the left shows the tip of a silicon crack just before the onset of crack propagation. It has a complex atomic structure, with a Stone-Wales rotation of a bond turning a pair of hexagons into a pentagon and a heptagon and a nanovoid opening up in front of the tip. This crack tip reconstruction phenomenon adds a new twist to our understanding of the atomic processes that underlie the catastrophic failure of brittle materials. In carefully controlled experiments on silicon, macroscopic consequences of tip reconstructions can be observed. The panels on the right show the grooved structure of slowly propagating cracks and the corresponding mesoscopic simulation based on the newly uncovered atomistic processes. Our simulations explain the counterintuitive observations that low speed cracks on the (111) surface of silicon (the supposedly "best" cleavage plane) are rough, whereas high speed cracks leave behind mirror smooth surfaces, because the crack tip does not have time to get trapped in the reconstruction: the atoms are dynamically steered towards rapid cracking in a cooperative manner.

On the one hand these simulations involve at least 200,000 atoms to describe the long range stress and strain fields of the crack tip, but on the other hand require not just quantum mechanics near the tip, but the use of a sophisticated and accurate model such as density functional theory (DFT). Using lower levels of theory the reconstruction does not always form. In our hybrid molecular dynamics simulation, the red atoms were described with DFT using external link: Castep while the rest of the system with a simple interatomic potential (Stillinger-Weber).

Low speed fracture instabilities of a brittle crystal
external link: Nature 455, 1224-1227 (30 October 2008)
A blurb for public consumption.

The original paper on the hybrid method:
"Learn on the fly": a hybrid classical and quantum-mechanical molecular dynamics simulation
Physical Review Letters 93 p. 175503 (2004) (external link: PDF [626 KB])

A more extended and detailed review of the same method:
Multiscale hybrid simulation methods for material systems
J. Phys. Cond. Mat. 17 R691-R703 Topical Review (2005) (external link: PDF [637 KB])


kink The image on the right shows the left kink of a 30 degree partial dislocation (again in silicon). The accepted equilibrium structure is in dashed black, but the true lowest free energy state at high temperatures is different, if we include quantum mechanics near the kink (at the level of tight binding in this case). Note the unusual appearance of a square arrangement of atoms. It is unusual for entropy to play such a decisive role in the structure of point defects in crystalline solids.

Molecular dynamics movies can be downloaded.

Superconductivity in Graphite

c6yb cmmm Following a recent discovery of superconductivity in Ca and Yb intercalated graphite (here in the Cavendish and UCL), we reexamined the class of materials known as GICs (Graphite Intercalation Compounds). A remarkable picture emerged: in those and only those compounds that superconduct, the donated electrons not only go into the pi* bands of graphite as expected, but also into an interlayer band, that is derived from the standard unoccupied interlayer band of graphite, and the metal free electron-like band. The microscopic pairing mechanism in these compounds is an open question, but the perfect correlation with the occupancy of this band opens up new possibilities.

The role of the interlayer state in the electronic structure of superconducting graphite intercalated compounds
Nature Physics 1 pp. 42-45 (2005) (external link: PDF [285 KB])

(This research featured in an external link: EPSRC newsletter)

Since the initial discovery of superconductivity in the rare earth intercalates, a singificant experimental research effort is underway to determine their properties. A particularly intriguing result is that the transition temperature goes up with increasing pressure (as opposed to most other materials), until some sort of phase transition when it drops abruptly. We carried out an extensive phase space search to determine the structure of C6Ca under high pressure, and found that a new (non-hexagonal) phase with Cmmm symmetry becomes much more favourable (see right).

Gábor Csányi, Chris J. Pickard, B. D. Simons, and R. J. Needs
Graphite intercalation compounds under pressure: A first-principles density functional theory study
Physical Review B 75 085432 (2007)

Fullerenes and Nanotubes

Computational materials simulation becomes really interesting when the system sizes that can be considered are large enough to be directly amenable to experimental study. This convergence has been happening just since the turn of the millenium, due to the advancement in experimental nanotechnology and the upkeep of the predictable pace in computer technology (Moore's "law"). One of the most exciting areas is that of carbon nanostructures.

Nature Materials CoverReinforcement of single-walled carbon nanotube bundles by intertube bridging
Nature Materials 3 pp. 153-157 (2004) (external link: PDF [343 KB])

Pristine nanotubes are extremely strong (Young's modulus is around 1 TPa), and this has spurred many people's imagination. However, the mass produced tubes from the CVD process (the only way one can obtain a large amount of tubes relatively cheaply) turn out to be much weaker for some reason. In addition to intrinsic weakness, bundles of nanotubes can slip and slide past each other, further weakening the macroscopic properties. In this work, we showed experimentally and theoretically how covalent intertube bridging can help this situation.

C59NC60Electron delocalization and dimerization in solid C59N-doped C60 fullerene
Physical Review Letters 94 066603 (2005)

For many years, people have been trying to make doped C60 fullerene. If successful, the electron doped crystal is expected to be a new semiconductor. The difficulty has been that most materials (e.g. the alkali doped fullerenes) are stoichiometric compounds, the doping level cannot be adjusted continuously. Recently, my collaborators in Hungary have successfully doped fullerene with azafullerene, C59N. In this paper, we use Electron Spin Resonance measurements and first principles calculations to elucidate the structure of this new material in a wide temperature range. In the equilibrium state, the C59N is covalently bound to a neighbouring C60 cage, and transfers its spin to it. At high temperatures, the electron delocalizes in the crystal.

Network Science

WiW Network A remarkable new field has emerged in the last 10 years, by the application of the methods of statistical physics to complex systems when modelled using graphs. A graph-based representation captures something very fundamental and robust in a complex system (e.g. economy, traffic, social relationships, world-wide-web etc.), and the resulting graph models turn out to have unexpected features that tell a unique story about the the organization and dynamics of these systems.

I am interested in social networks, which seem to be among the most difficult to model well using graphs. Beyond the inherent interest (it's about ourselves) and academic motivations, there are numerous applications to epidemiology, marketing and economics. We wrote some popular articles on these topics.