Seeking a Direct Optimization Method Using Distance Geometry Space for Molecular Orientation

Abstract

We report a direct method to determine the relative configuration of two molecules using distance geometry, in which the structure is expressed as a set of distances between atoms. In the calculation, the gradient and bordered Hessian with inequality and equality constraints, based on the Cayley-Menger determinant for embedding the structure in three-dimensional space, are evaluated by introducing slack variables. The developed method is then applied to three molecular systems: $H_2$O with ${\mathrm{Li}}^{+}$, HF dimer, and ${\mathrm{NH}}_3$ with ${\mathrm{Na}}^{+}$. The present approach could be related to a straightforward route to explore the entire potential-energy landscape for a small molecular system described with simple analytical functions.