CARbon MAnifolds

Task W: Solving Wave Equations on Carbon Manifolds

 We will develop fast approximate solutions to electronic wave equations constrained to discrete non-Euclidean surfaces. A hierarchy of approximation methods will be tested and assessed, measuring tradeoffs between speed and accuracy. We will take advantage of specialized knowledge about fullerene chemistry and electron behavior on fullerene surfaces. Important subtasks:
  1. Wave equations restricted to bond-graph metric. This very first approximation is expected to be efficiently soluble using 1D finite-elements of high polynomial degree.
  2. Laplace-Beltrami operator for carbon manifolds.
  3. Jacobian and differential operators on the manifolds.
  4. Fullerene surface harmonics and momentum space solutions.
  5. Wave equations restricted to the manifold.
  6. Approximate solutions to the many-electron equations.
Subtasks 5-7 are expected to be the largest and most difficult tasks, and I expect that we will continue to work on improvements throughout the project.

Selected harmonics (1st, 3rd, 7th, etc.) computed by my prototype for a D$_{3h}$-symmetry $C_{2562}$ fullerene.}

Preliminary Work:

I have implemented a linear-element finite-volume representation of the fullerene Laplace-Beltrami operator, specializing the cotangent formula given, in e.g. Crane 2013. To test it, I have calculated fullerene harmonics, i.e. the eigenfunctions of the Laplace-Beltrami operator, for a selection of fullerenes. The figure above show a few of these. The harmonics of a manifold carry geometric information about its shape. The harmonics form a complete orthonormal basis for the Sobolev space of smooth functions on the surface, and are eigenfunctions to the kinetic energy operator. They can hence be used to build up solutions to the electronic wave equation. In addition, it may be possible to derive an analogous result to that of hyperspherical harmonics for the momentum-space form of the wave equation, but restricted to the fullerene surface. I have extended this work to implement a pilot finite-volume DFT prototype restricted to carbon manifolds: solving Poisson's equation with the non-Euclidean Laplace-Beltrami operator yields the Hartree-potential; the remaining exchange-correlation potential (LDA or GGA) is computed from electron density using libXC, and the system is iterated to self-consistency. Norm-preserving Troullier-Martins potentials are used to smooth out nuclear cusps for numerical efficiency.