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Molecular Modeling: Geometry Optimization

Molecular Modeling: Geometry Optimization. C372 Introduction to Cheminformatics II Kelsey Forsythe. Geometry Optimization. Le Chatliers’ Principle

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Molecular Modeling: Geometry Optimization

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  1. Molecular Modeling:Geometry Optimization C372 Introduction to Cheminformatics II Kelsey Forsythe

  2. Geometry Optimization Le Chatliers’ Principle The optimum geometry is the geometry which minimizes the strain on a given system. Any perturbation from this geometry will induce the system to change so as to reduce this perturbation unless prevented by external forces Mathematical Surface Reflects This Principle!!

  3. Why Extrema? • Equilibrium structure/conformer MOST likely observed? • Once geometrically optimum structure found can calculate energy, frequencies etc. to compare with experiment • Use in other simulations (e.g. dynamics calculation) • Used in reaction rate calculations (e.g. 1/nsaddle a reaction time) • Characteristics of transition state • PES interpolation (Collins et al)

  4. Nomenclature • PES equivalent to Born-Oppenheimer surface • Point on surface corresponds to position of nuclei • Minimum and Maximum • Local • Global • Saddle point (min and max)

  5. Cyclohexane Global maxima Local maxima Local minima Global minimum

  6. Ex. PES Local minimum Saddle point Global minimum

  7. Recall glycine? Global Local

  8. Steepest Descent Conjugate Gradient Fletcher Powell Simplex Geometric Direct Inversion of Iterative Subspace Newton-Raphson Minimize w.r. each individual coordinate No gradients required No gradients required Methods

  9. No Functional Form Bracketing Parabolic Interpolation (Brent’s method) Methods (1-d)

  10. Steepest Descent Methods (1-d)(w/ gradients)

  11. Line Search Simplex Fletcher-Powell Methods (n-d)(w/o gradients)

  12. Conjugate Gradient (space a N) Fletcher-Reeves Polak-Ribiere Quasi-Newton/Variable Metric (space a N2) Davidon-Fletcher-Powell Broyden-Fletcher-Goldfarb-Shanno Methods (n-d)(w/ gradients)

  13. Stochastic Tunneling Monte Carlo Simulated Annealing Genetic Algorithm Surface smoothing: proteins Multi-dimensional Global (uphill jumps allowed) Multidimensional Methods

  14. Typically many function evaluations are required in order to estimate derivatives and interpolate/extrapolate along PES Want simple analytic form for energy ! Bottleneck Analytic? q1q2 q3 . . qn Molecular mechanics Semi-Empirical Ab Initio E(q1,q2..)

  15. What is the optimum point? At extremum ? ? ?

  16. Local vs. Global? Conformational Analysis (Equilibrium Conformer) A conformational analysis is global geometry optimization which yields multiple structurally stable conformational geometries (i.e. equilibrium geometries) Equilibrium Geometry An equilibrium geometry may be a local geometry optimization which finds the closest minimum for a given structure (conformer)or an equilibrium conformer • BOTH are geometry optimizations (i.e. finding wherethe potential gradient is zero) • Elocal greater than or equal to Eglobal

  17. Geometry Optimization • Basic Scheme • Find first derivative (gradient) of potential energy • Set equal to zero • Find value of coordinate(s) which satisfy equation

  18. Modeling Potential energy (1-d)

  19. Modeling Potential energy (>1-d) Hessian

  20. Find Equilibrium Geometry for the Morse Oscillator

  21. Find Equilibrium Geometry for the Morse Oscillator

  22. Bottlenecks • No Functional Form • More than one variable • Coupling between variables

  23. Geometry Optimization(No Functional Form) • Bracketing (w/parabolic fitting) • Find energy (E1) for given value of coordinate xi • Change coordinate (xi+1=xi-Dx) to give E2 • Change coordinate (xi+2=xi +Dx) to give E3 • If (E2>E1 and E3>E1) then xi+1> xmin >xi+2 • Fit to parabola and find parabolic minimum • Use value of coordinate at minimum as starting point for next iteration • Repeat to satisfaction (Minimum Energy error tolerance)

  24. What is the optimum point? HO-trivial case 3 1 34 2 4

  25. Line Search • For given point V(ra) choose u vector • u chosen in direction opposite to gradient (I.e. steepest descent) • Approaches • Constant l • Steepest descent • Minimize V(xi+lu) • Want l s.t. vectors f and u perpendicular • Repeat to minimum

  26. Line Search(1-d) • Steepest Descent (Gradient Descent Method)

  27. Conjugate Methods • No “Spoiling” • Reduces #iterations • Numerical Gradient • Powell Method (speed a n2) • Analytic Gradient • Conjugate Gradient (speed a n)

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