# The Weighted Gaussian Curvature Derivative of a Space-Filling Diagram

## TL;DR

This paper derives a formula for the derivative of the weighted Gaussian curvature within the morphometric approach to solvation free energy, enabling comprehensive differentiation of the energy expression based on geometric measures.

## Contribution

It provides the first explicit formula for the derivative of weighted Gaussian curvature, completing the set of derivatives needed for the morphometric solvation energy model.

## Key findings

- Derived the formula for the derivative of weighted Gaussian curvature.
- Enabled full differentiation of the morphometric solvation energy expression.
- Facilitated more accurate and efficient calculations in solvation modeling.

## Abstract

The morphometric approach [HRC13,RHK06] writes the solvation free energy as a linear combination of weighted versions of the volume, area, mean curvature, and Gaussian curvature of the space-filling diagram. We give a formula for the derivative of the weighted Gaussian curvature. Together with the derivatives of the weighted volume in [EdKo03], the weighted area in [BEKL04], and the weighted mean curvature in [AkEd19], this yields the derivative of the morphometric expression of solvation free energy.