DFF-F2: CARbon MAnifolds

Understanding Low-dimensional Materials
Through Combinatorial Geometry

Click on each task for details.

Project Staff and Research Plan

Risk Assessment and Mitigation

Three tasks are high risk/high gain: W and B. B is a "terminal goal", in that other tasks do not depend on it. However, W is central the project's success, and at the same time involve PhD work. This makes it necessary to ensure (1) that fallback paths are built in to ensure the other project components can proceed if one or more high-risk task fail or stall, (2) that low-risk milestones are pursued in parallel, so that the two PhD are guaranteed successful theses with solid publications, and (3) that the project as a whole meaningfully contributes to science no matter the outcome of the high-risk tasks.

While success is not guaranteed, as the work has never been attempted before, I have already spent three years investigating feasibility of the most high risk tasks through my VILLUM Experiment grant Folding Carbon: A Calculus for Molecular Origami. Six 60ECTS MSc theses explored core functionality, and have laid out a much clearer view of how to make this difficult task succeed, much reducing the overall risk. The directions forward are laid out in the task details, found by clicking the tasks on the overview above.

Risk Mitigation for Task W:

The greatest risk of Task W, and indeed of the project, is the uncertainty of the degree to which the assumption holds that purely two-dimensional wave equations can approximate the electronic structure of fullerenes and similar surface molecules; i.e.~that the electron interaction through the surface is negligible compared to that along it. While the Coulomb repulsion is long range, the most important effects are local, whereby larger fullerene structures can more accurately be regarded as locally two-dimensional. However, even for large fullerenes, electron interaction through space can be important in certain regions, for example in high-curvature bends. Note, however, that we will at the very least be able to reach the same accuracy as tight-binding methods, which only interact along bond graph edges: the goal is to reach higher accuracy with at most the same cost. The exploratory VILLUM Experiment work investigated feasibility of sub-tasks that strengthen the approximation (1+2) and mitigate risk (3):

1. Recover 3D Electron interaction through Mixed-dimensional Geometry

The most common regions in which out-of-surface electron interaction is important for large fullerenes (or other surface molecules) is at high-curvature sites, for example "corners" of polyhedra. As the number of these are limited by combinatorial rules (e.g. never more than 12 for fullerenes), we can treat these in 3D while keeping the overall cost of analysis asymptotically the same as a full 2D treatment. Task M deals with this. The MSc thesis of D. Dedenbach showed that it is possible to cheaply reconstruct local molecular 3D geometry of these "pocket regions" without the need to find the global molecular 3D geometry. Thus, in these regions where it is important, we can accurately model full 3D electron interaction, while applying the 2D approximation for the rest of the surface molecule.

2. Recover 3D Electron density 3D through Holographic Encoding

Task D deals with encoding information about the 3D densities as surface fields: while numerically two-dimensional, the electronic structure calculations will not simply throw away the information about 3D electron densities, but model the 3D electron density around the surface through a low-parameter ansatz (e.g. exponential falloff away from surface), parameterized along the surface. This model is numerically represented as scalar fields along the surface, but approximately models the 3D electron density. Task ML can aid in fitting the model to best captures the physical densities, by applying machine learning on the carbon manifold surfaces (analogous to convolutional neural networks) trained on ab initio quantum chemical calculations.

3. Fallback path: Use Fast 3D Embedding

It is likely that there is a threshold size which a surface molecule must have before the surface electronic structure approximation becomes valid. For molecules below this threshold, or other cases where the carbon manifold DFT may be inaccurate, we will develop a simpler and more conventional method that requires approximate 3D molecular geometry. This can be calculated for fullerenes using a special fullerene geometry force-field (Schwerdtfeger, Wirz, Avery 2013, Wirz et al. 2015), developed further in B.Pedersen's MSc thesise and recently implemented on GPU by us to rapidly find geometries for full isomer spaces in parallel, using only a few microseconds per C100 isomer. This is designed to be used in Task P, but can also be used to build more conventional fall-back methods to complement the more risky methods in Task W, and let all tasks for the PhD proceed, ensuring that the work of PhD1 produces useful and publishable contributions to science regardless of whether the high-risk/high-gain components succeed.