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WIREs Comput Mol Sci
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Geometry optimization

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Abstract Geometry optimization is an important part of most quantum chemical calculations. This article surveys methods for optimizing equilibrium geometries, locating transition structures, and following reaction paths. The emphasis is on optimizations using quasi‐Newton methods that rely on energy gradients, and the discussion includes Hessian updating, line searches, trust radius, and rational function optimization techniques. Single‐ended and double‐ended methods are discussed for transition state searches. Single‐ended techniques include quasi‐Newton, reduced gradient following and eigenvector following methods. Double‐ended methods include nudged elastic band, string, and growing string methods. The discussions conclude with methods for validating transition states and following steepest descent reaction paths. © 2011 John Wiley & Sons, Ltd. WIREs Comput Mol Sci 2011 1 790–809 DOI: 10.1002/wcms.34 This article is categorized under: Electronic Structure Theory > Ab Initio Electronic Structure Methods

Model potential energy surface showing minima, transition structures, second‐order saddle points, reaction paths, and a valley ridge inflection point (Reprinted with permission from Ref 5. Copyright 1998 John Wiley & Sons.)

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An example of a double‐ended reaction path optimization on the Muller–Brown surface.154 The path optimization with 11 points starts with the linear synchronous transit path (black); the first two iterations are shown in blue and green, respectively. The path after 12 steps is in red and can be compared with the steepest descent path in light blue.

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Various classes of reaction channels near the transition structure on reactive potential energy surfaces: (a) I‐shaped valley, (b) L‐ or V‐shaped valley, (c) T‐shaped valley, and (d) H‐ or X‐shaped valley. (Reprinted with permission from Ref 113. Copyright 2009 ACS Publications.)

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Plot of the displacement squared, |Δ x|2 = |(H − λ I)−1 g|2, as a function of the Hessian shift parameter λ (Reprinted with permission from Ref 102. Copyright 1983 ACS Publications.). The singularties occur at the eigenvalues of the Hessian, hi. (a) For displacement to a minimum with a trust radius of τ1, shift parameter λ1 must be less than the lowest eigenvalue of the Hessian, h1. (b) For a displacement to a transition structure, the shift parameter must be between h1 and ½ h2. For τ1, the lower of the two solutions is chosen. For a smaller trust radius τ2, there is no solution; λ2 is chosen as the minimum in the curve [or λ2 = ½ (h1 + ½ h2) if λ2 > ½ h2] and the displacement is scaled back to the trust radius (see Refs 101–104 for more details).

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