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 Saddle points are transition states between minima Chemistry and biochemistry is all about structure-function relationships – think proteins Global optimization is the goal Caffeine in BO-Approximation 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 125 th 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 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 Lennard- Jones 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

Disconnectivity graphs / trees 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 ‘Martin Karplus’

Optimizing molecular clusters He is literally the record keeper: Cambridge Cluster Databse 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!