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
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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!