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

This paper introduces the conformal model (an extension of the homogeneous coordinate system) for molecular geometry, where 3D space is represented within <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:semantics> <mml:mrow> <mml:msup><mml:mrow><mml:mi>ℝ</mml:mi></mml:mrow> <mml:mrow><mml:mn>5</mml:mn></mml:mrow> </mml:msup> </mml:mrow> <mml:annotation>$$ {\mathbb{R}}^5 $$</mml:annotation></mml:semantics> </mml:math> 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.