The Wales Group in Context: Exploring Energy Landscapes Research Review by Ryan Babbush Applied Computation 298r February 8, 2013 Why potential energy surfaces? • Chemistry is the study of stationary points on the Born-Oppenheimer PES • Minima of this PES are molecules / conformers Caffeine in BO-Approximation • Saddle points are transition states between minima • Chemistry and biochemistry is all about structure-function relationships – think proteins • Global optimization is the goal Caffeine without BO-Approximation Example: water clusters • In liquid, water manifests as clusters, the complete structure of which is likely impossible to measure experimentally • For its 125th anniversary, Science released a special issue on the 125 most important open questions in science. Question #20: “What is the structure of water?” • Complex cluster structure may explain anomalies in thermodynamic properties of water and stabilities of many large molecules such as proteins • Proven to be NP Hard for atomic clusters, molecular structures are notoriously difficult to optimize* *LT Wille and J Vennik. Computational complexity of the ground-state determination of atomic clusters. 1985 J. Phys. A: Math. Gen. 18 L419 The TIP3P force-field Rules of the game: 1) Water has rigid bonds with fixed lengths, fixed angles, and fixed charges (see right) 2) Energy of system given by LennardJones potential and Coulomb potential only • How one specifies orientation of water molecule does not change “problem” but drastically changes PES • Entire research communities study this as an optimization problem 12 6ù é kq q æ ö æ ö s s Eab = ååê i j + e ç ÷ - e ç ÷ ú è rOO ø è rOO ø úû i j ê ë rij on a on b Disconnectivity graphs / trees ‘Martin Karplus’ • The tree idea belongs to Karplus but Wales has done a lot to popularize them and study their properties • The banyan tree on the right is for H20(20) cluster from Wales’ 1998 Nature paper - the structure of this tree shows that water is a “strong” liquid Optimizing molecular clusters • He is literally the record keeper: Cambridge Cluster Databse • Employs Monte Carlo and genetic algorithms with basin hopping • Wales is extremely good at this • I think (not sure) that he coined “basin hopping” Free energy surfaces • Free energy is the quantity which ensembles minimize at equilibrium (right is Helmholtz) • Entropy (the multiplicity of microstates in a macrostate) plays a role in free energy proportional to the temp • Local free energy does not really exist but is sometimes useful to think about • It is given by the local partition function • Ambiguity as to which coordinates to average Protein Folding • Made more difficult by high degrees of frustration in optimized structure • Probably hopeless in hardest case, possibly tractable in instances of proteins in nature • HP model can give insight into how choice of coordinate determines energy landscape Other Areas of Research • Atomic Lennard-Jones clusters • Polyhedra packing • Glass transition and disordered ground states • Classification of energy landscapes • Discrete path sampling, kinetic Monte Carlo • And more!