Wave Equations Without Coordinates I: Fullerenes

TY - JOUR

T1 - Wave Equations Without Coordinates I

T2 - Fullerenes

AU - Avery, James Emil

PY - 2018/9/1

Y1 - 2018/9/1

N2 - Can we solve electronic wave equations absent a coordinate system? The question arises from the wish to treat polyhedral molecules such as fullerenes as two-dimensional closed surfaces. This would allow us to study electronic structure on intrinsic surface manifolds, which can be derived directly from the bond structure. The wave equation restricted to the (non-Euclidean) surface could then be solved without reference to any three-dimensional geometry of the molecule, and hence without the need for quantum chemical geometry optimization. The resulting 2D system can potentially be solved several orders of magnitude faster than the full wave equation. However, because these curved surfaces do not admit any simple coordinate system, we must devise methods that can do without. In this paper, I describe how surface geometries can be derived from fullerene bond graphs as combinatorial objects, and how electronic structure may be studied by solving wave equations directly on these intrinsic surface manifolds, without needing to find three-dimensional geometries. The goal is approximation methods that are rapid enough to systematically analyze entire isomer spaces consisting of millions of molecules, so as to identify structures with desired properties.

AB - Can we solve electronic wave equations absent a coordinate system? The question arises from the wish to treat polyhedral molecules such as fullerenes as two-dimensional closed surfaces. This would allow us to study electronic structure on intrinsic surface manifolds, which can be derived directly from the bond structure. The wave equation restricted to the (non-Euclidean) surface could then be solved without reference to any three-dimensional geometry of the molecule, and hence without the need for quantum chemical geometry optimization. The resulting 2D system can potentially be solved several orders of magnitude faster than the full wave equation. However, because these curved surfaces do not admit any simple coordinate system, we must devise methods that can do without. In this paper, I describe how surface geometries can be derived from fullerene bond graphs as combinatorial objects, and how electronic structure may be studied by solving wave equations directly on these intrinsic surface manifolds, without needing to find three-dimensional geometries. The goal is approximation methods that are rapid enough to systematically analyze entire isomer spaces consisting of millions of molecules, so as to identify structures with desired properties.

U2 - 10.1007/s12210-018-0717-4

DO - 10.1007/s12210-018-0717-4

M3 - Journal article

SN - 2037-4631

VL - 29

SP - 609

EP - 621

JO - Rendiconti Lincei. Scienze Fisiche e Naturali

JF - Rendiconti Lincei. Scienze Fisiche e Naturali

IS - 3

ER -