
Optimization Methods
Geometry Optimization Methods
Types of algorithm
Simple Energy Minimization
1st Derivative Methods
2nd Derivative Methods
Energy Minimizations
 Simplest
 Consistently moves downhill
 Rapid at first, can oscillate near minimum
 No test for convergence
 Simplex method
 Sequential Univariate
 Change one coordinate at a time
 Fit parabola to initial point plus 2 displacements
 Adjust individual coordinate to minimum of parabola
 Problem with long narrow valley
1 + 3N x m evaluations
1st Derivative/Gradient Methods
 Order of magnitude faster
 Convergence to stationary points (minima, saddle, maxima)
 Steepest Descent
 Consistently moves downhill
 Can oscillate near minimum
 In effect approximates Hessian w/ unit matrix
 Conjugate Gradient/FletcherReeves
 Faster than Steepest Descent (& BDNR for PP)
 Uses gradients as if updating Hessian
 Fast convergence when far from stationary point
 Slow to converge for floppy
 May oscillate near minimum
 NLLSQ (NonLinear Least Squares)
 Nearest stationary point
 Not on shoulder
 Modified FletcherPowell
 Two displacements along each coordinate
 Fit quadratic surface w/ interactions (in effect calculating g & elements of H numerically)
 Two displacements along downhill direction
 Fit parabola to downhill direction
 Displace atoms
2N + 4 + m x (N + 3) evaluations
Second Derivative/Hessian Methods
 Assume quadratic gradient surface: f(E, g, H)
NewtonRaphson
 Calculate Hessian matrix directly at each step
 Augmented Hessian for transition state
QuasiNewton
 Estimate Hessian (Inverse Hessian/Green’s Function)
 Calculate E, g
 Update Hessian
 Move to stationary point of model surface
N+1 steps, but each one longer
 Block Diagonal NewtonRaphson
 Guesses Hessian
 Unit matrix for cartesian, better for internal coordinate
 Better convergence near minimum
 DFP (DavidsonFletcherPowell)
 Better for transition states
 MurtaghSargent
 BFGS (BroydenFletcherGoldfarbShanno)
 Faster than NR
 Better for optimizations
 EF (Eigenvector Following)
 Faster than NLLSQ, Sigma, BFGS
 May fail with dummy atoms
 Analytical Gradients
 Analytical rather than finite differences
 More accurate
Single Hessian Calculation
 Hessian not recomputed after 1st step
 Rapid for nearest stationary point
 Fails if initial gradient large
 Sigma
 Gradient Norm Squared/Powell
 Refine NR geometry
 Stationary point including shoulder
Back to Modeling Reference Page
Link to .
4/7/99 Ernie Chamot / Chamot Labs / (630) 6371559 / echamot@chamotlabs.com
