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Using quaternions to calculate RMSD.

Evangelos A Coutsias·Chaok Seok·Ken A Dill
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Abstract

A widely used way to compare the structures of biomolecules or solid bodies is to translate and rotate one structure with respect to the other to minimize the root-mean-square deviation (RMSD). We present a simple derivation, based on quaternions, for the optimal solid body transformation (rotation-translation) that minimizes the RMSD between two sets of vectors. We prove that the quaternion method is equivalent to the well-known formula due to Kabsch. We analyze the various cases that may arise, and give a complete enumeration of the special cases in terms of the arrangement of the eigenvalues of a traceless, 4 x 4 symmetric matrix. A key result here is an expression for the gradient of the RMSD as a function of model parameters. This can be useful, for example, in finding the minimum energy path of a reaction using the elastic band methods or in optimizing model parameters to best fit a target structure.

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Evangelos A Coutsias, Chaok Seok, Ken A Dill. 2004-11-30. Using quaternions to calculate RMSD.. https://doi.org/10.1002/jcc.20110

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