Conformal Coordinates for Molecular Geometry: from 3D to 5D

TL;DR

The paper tackles the inefficiency of distance computations in molecular geometry when using traditional homogeneous coordinates. It proposes a conformal model that embeds 3D space into $\mathbb{R}^5$ via $\hat{x}=x+e_0+\tfrac{1}{2}\|x\|^2 e_{\infty}$ and defines the Conformal Coordinate Matrix (C-matrix) to encode isometries, yielding an orthogonal representation under a non-positive inner product. A key result is the distance formula $r_{i,j}^2=2 e_{\infty}^t B_{[i+1,j]} e_0$, enabled by the orthogonality of the C-matrix, and the operation count for computing a distance drops to $28(j-i)-45$ compared to the Euclidean or homogeneous models. This approach offers a computationally efficient framework for molecular geometry and has potential cross-domain applications in robotics and graphics where conformal models are used.

Abstract

This paper introduces the conformal model (an extension of the homogeneous coordinate system) for molecular geometry, where 3D space is represented within R^5 with an inner product different from the usual one. This model enables efficient computation of interatomic distances using what we call the Conformal Coordinate Matrix (C-matrix). The C-matrix not only simplifies the mathematical framework but also reduces the number of operations required for distance calculations compared to traditional methods.

Figures (1)

• Figure 1: Cartesian and internal coordinates