Finding the isometric embedding guaranteed by Alexandrov's theorem
in a robust way is presently an unsolved problem: the
state-of-the-art uses numerical force-field optimization
(Brinkmann et al. 2010,
Schwerdtfeger et al. 2013,
Wirz et al. 2015), which often
works, but it is prone to failure if the initial guess is far from
the correct shape, and it breaks down more often as fullerenes grow
in size.
This task will start from my recent work presented here.
The central idea is as follows: For fullerenes,
the positions of the pentagons uniquely determine the fullerene's
shape. The idea explored here is to thin out the graph in a certain
metric-preserving way by maintaining a Delaunay-triangulation, and to use
the surface metric to find the coarse polyhedron defined by the 12
pentagon vertices. The 12-vertex polyhedron can then be embedded
in a robust way (and uniquely, by Alexandrov) due to the fixed vertex count.
(a)
(b)
Neighbouring triangles share a coordinate system, but all three may not.
Computing the Delaunay triangulation is made difficult by the fact
that it isn't possible to make a single two-dimensional coordinate
system for the whole fullerene surface. However, it is possible
to make overlapping patches of flat coordinate systems. In particular,
any two adjacent triangles share a coordinate system, and this can be
extended to regions that contain no Gaussian curvature. Inside such
regions, angles and lengths can be worked out as in Cartesian
coordinates, but anything that goes outside of the ``safe'' region
potentially yields nonsense. The information used in the Delaunay
triangulation algorithm has to, in each step, remain within the
progressively growing flat regions.
The subtasks for Task E include:
Isometric coarsening of point-set to only cone points.
Unfolding/folding Delaunay triangulation to Eisenstein polygons.
3D embedding of coarse metric.
Preliminary experiments yielded a coarse shape, which
can be used for shape classification and search, in milliseconds.
Full 3D embedding derived from coarse embedding. Preliminary experiments
yielded 3D geometry very near to full DFT/B3LYP optimized geometry
in tens of seconds.
Preliminary Work:
In Intrinsic Geometries of Fullerenes, I presented a partial prototype of
this idea, but in which the Delaunay-step had to be hand annotated.
The figure above shows the steps of a test
calculation for a barrel-shaped $C_{480}$. The coarse and interpolated
shapes were calculated in miliseconds. Similar timing was obtained for
$C_{3000}$. However, more work is still necessary for creating a
robust, efficient solution.
MSc Thesis: Buster Pedersen, Shapes of Fullerenes: Fast, Parallel Molecular Geometry.
A master's student thesis is under way that builds
numerical codes for finding molecular 3D geometry from surface
metrics for entire isomer spaces at a time (millions of molecular
structures in parallel on GPU). The challenge is that optimization
either must always succeed, or always detect failure, as millions
of structures cannot be inspected by
hand.