Derivative of SVD V and U matrices

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SUMMARY

The discussion centers on deriving the rotation matrix R, defined as R=V diag(1, 1, |VU^T|)U^T, where V and U are obtained from the Singular Value Decomposition (SVD) of matrix K. Matrix K is constructed by summing the outer products of atomic coordinate vectors from two molecular geometries. The user seeks the derivative of each element of R with respect to atomic coordinates for use in energy approximations and analytic forces. A relevant paper providing insights into this derivation is referenced.

PREREQUISITES
  • Understanding of Singular Value Decomposition (SVD)
  • Familiarity with rotation matrices in 3D geometry
  • Knowledge of molecular geometry and atomic coordinates
  • Basic calculus for derivatives and Jacobians
NEXT STEPS
  • Study the derivation of the Jacobian matrix in the context of SVD
  • Explore the application of rotation matrices in molecular dynamics simulations
  • Research analytic force calculations in energy approximations
  • Read the referenced paper on SVD Jacobians for deeper insights
USEFUL FOR

Researchers in computational chemistry, molecular dynamics specialists, and anyone involved in the mathematical modeling of molecular geometries and energy calculations.

saulg
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sorry I am new and posted instead of previewing...im currently writing the post
 
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I find a rotation matrix which best describes how to get from one set of atomic coordinates (molecular geometry) to another by just a pure rotation.

The rotation matrix R is defined

[tex]R=V \left( \begin{array}{ccc}<br /> 1 & & \\<br /> & 1 & \\<br /> & & \left|VU^T\right| \end{array} \right)U^T[/tex]

where V and U are from the SVD of matrix K:

[tex]K=V\Lambda U^T[\tex]<br /> <br /> K is formed by summing over i the outer products of the coordinate vectors of atom i in the first and second geometry. (i follow the method described at http://homepages.inf.ed.ac.uk/rbf/CVonline/LOCAL_COPIES/MARBLE/high/pose/least.htm" )<br /> <br /> <br /> I.e K depends on atomic coordinates, and so do U and V.<br /> <br /> I require the derivative of each element of R with respect to atomic coordinate. (The rotation matrix is used in an energy approximation and I need analytic forces)<br /> <br /> <br /> Any answers or hints much appreciated.[/tex]
 
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