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On coordinate transformations in steepest descent path and stationary point locations
Author(s) -
Banerjee Ajit,
Adams Noah P.
Publication year - 1992
Publication title -
international journal of quantum chemistry
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.484
H-Index - 105
eISSN - 1097-461X
pISSN - 0020-7608
DOI - 10.1002/qua.560430610
Subject(s) - hessian matrix , gradient descent , mathematics , stationary point , eigenvalues and eigenvectors , saddle point , method of steepest descent , decoupling (probability) , invariant (physics) , path (computing) , directional derivative , transformation (genetics) , mathematical analysis , minification , mathematical optimization , geometry , computer science , physics , control engineering , machine learning , biochemistry , chemistry , quantum mechanics , artificial neural network , engineering , mathematical physics , gene , programming language
Two aspects of the problems of calculating steepest descent paths and locating stationary points on surfaces E ( X ), which are sources of some confusion in the literature, are addressed. These include writing proper expressions for the gradient and Hessian, and their transformation properties relative to coordinate transformations, based on the invariance of the surface E ( X ). The appropriate transformation is derived, based on a constrained energy minimization condition, to achieve what we call the Hessian eigenvalue representation. This not only allows decoupling of the variables, but also points to the minimization direction and preserves the eigenvalues of the Hessian. These results allow one to use the steepest descent path and stationary point location algorithms in any coordinate system and obtain invariant results. The validity of these considerations are also confirmed through numerical examples. The stationary condition with constrained kinematic path length is also shown to yield a Hessian eigenvalue representation for the normal modes for small vibrations. Lastly, we have constructed a mathematically consistent definition of mass‐weighted Cartesians where the intrinsic reaction path of Fukui is a steepest descent path. © 1992 John Wiley & Sons, Inc.