FinitedifferenceOEPESCEPESEpESCdiadiO"EPES-haqi2h-hhOEPESCEPESaqiadi-A12
Finite difference 12
Finitedifferenceh must be small enough to stay in the harmonicregion, but big enough to avoid numerical noiseswamping the gradients;For a molecule with N atoms, to construct Hessian(3N X 3N,diagonalize),have to evaluate gradients2*3Ntimes fortwo-sided differencing.Obviouslytendsto be quite expensive;To get better precision and accuracy, could calculateevenmorethantwodisplacements,and couldevenfit to amore complicatedfunction thana harmonicpotential13
Finite difference 13 • h must be small enough to stay in the harmonic region, but big enough to avoid numerical noise swamping the gradients; • For a molecule with N atoms, to construct Hessian (3N × 3N , diagonalize), have to evaluate gradients 2*3N times for two-sided differencing. Obviously tends to be quite expensive; • To get better precision and accuracy, could calculate even more than two displacements, and could even fit to a more complicated function than a harmonic potential
OptimizationalgorithmsGeometry optimization is one of the most common tasks;Finding a minimum point on Epes;"Energy-only" algorithms search for minima without anygradient/force information;Effectively infer gradients and Hessian from lots ofdisplacements;Veryslowto convergeand areonly usedin specializedsituations;It is computationally costly to calculate gradientsanalytically, but they can vastly increase the speed ofoptimizations;Algorithms apply to both geometry optimization andelectronic structure calculationJensenChp13Schlegel, J.Comp.Chem.2003, 24, 151414
Optimization algorithms 14 • Geometry optimization is one of the most common tasks; • Finding a minimum point on EPES; • “Energy-only” algorithms search for minima without any gradient/force information; • Effectively infer gradients and Hessian from lots of displacements; • Very slow to converge and are only used in specialized situations; • It is computationally costly to calculate gradients analytically, but they can vastly increase the speed of optimizations; • Algorithms apply to both geometry optimization and electronic structure calculation Jensen Chp 13 Schlegel, J. Comp. Chem. 2003, 24, 1514
SteepestdescentConsider some set of coordinates R for which we have Epesand g;Most straightforward approach would be to move atomsin direction of -g, i.e. find 入 such that coordinates R' = R-入g have mimimum energy; Called a line-search;Repeat starting from updated R' until △Epes or g issufficiently small;While every step lowers the energy,this algorithm tendsto convergeveryslowly near a minimum.https://en.wikipedia.org/wiki/Gradientdescent15
Steepest descent 15 • Consider some set of coordinates R for which we have EPES and g; • Most straightforward approach would be to move atoms in direction of –g, i.e. find λ such that coordinates R′ = R −λg have mimimum energy; • Called a line-search; • Repeat starting from updated R′ until ΔEPES or g is sufficiently small; • While every step lowers the energy, this algorithm tends to converge very slowly near a minimum. https://en.wikipedia.org/wiki/Gradient_descent
ConjugategradientDisadvantage of steepest descent is that itloses information about previous steps.Notion of "conjugate gradient" search is toaugment steepest descent with requirementtokeep each step orthogonalto some numberofprevious steps.Generallyperformsmuchbetterthansteepestdescent,andisoftenthefirstchoiceiffarfrom anequilibriumgeometryhttps://en.wikipedia.org/wiki/Conjugategradientmethod16
Conjugate gradient 16 • Disadvantage of steepest descent is that it loses information about previous steps. • Notion of “conjugate gradient” search is to augment steepest descent with requirement to keep each step orthogonal to some number of previous steps. • Generally performs much better than steepest descent, and is often the first choice if far from an equilibrium geometry. https://en.wikipedia.org/wiki/Conjugate_gradient_method