Molecular Modeling: Geometry Optimization C372 Introduction to Cheminformatics II Kelsey Forsythe
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!!
Why Extrema? Equilibrium structure/conformer MOST likely observed? Equilibrium structure/conformer MOST likely observed? Once geometrically optimum structure found can calculate energy, frequencies etc. to compare with experiment Once geometrically optimum structure found can calculate energy, frequencies etc. to compare with experiment Use in other simulations (e.g. dynamics calculation) Use in other simulations (e.g. dynamics calculation) Used in reaction rate calculations (e.g. saddle reaction time ) Used in reaction rate calculations (e.g. saddle reaction time ) Characteristics of transition state Characteristics of transition state PES interpolation (Collins et al) PES interpolation (Collins et al)
Nomenclature PES equivalent to Born-Oppenheimer surface PES equivalent to Born-Oppenheimer surface Point on surface corresponds to position of nuclei Point on surface corresponds to position of nuclei Minimum and Maximum Minimum and Maximum Local Local Global Global Saddle point (min and max) Saddle point (min and max)
Cyclohexane Local maxima Global minimum Global maxima Local minima
Ex. PES Saddle point Local minimum Global minimum
Recall glycine? Global Local
Methods Steepest Descent Steepest Descent Conjugate Gradient Conjugate Gradient Fletcher Powell Fletcher Powell Simplex Simplex Geometric Direct Inversion of Iterative Subspace Geometric Direct Inversion of Iterative Subspace Newton-Raphson Newton-Raphson Minimize w.r. each individual coordinate No gradients required
Methods (1-d) No Functional Form No Functional Form Bracketing Bracketing Parabolic Interpolation (Brent’s method) Parabolic Interpolation (Brent’s method)
Methods (1-d)(w/ gradients) Steepest Descent Steepest Descent
Methods (n-d)(w/o gradients) Line Search Line Search Simplex Simplex Fletcher-Powell Fletcher-Powell
Methods (n-d)(w/ gradients) Conjugate Gradient (space N) Conjugate Gradient (space N) Fletcher-Reeves Fletcher-Reeves Polak-Ribiere Polak-Ribiere Quasi-Newton/Variable Metric (space N 2 ) Quasi-Newton/Variable Metric (space N 2 ) Davidon-Fletcher-Powell Davidon-Fletcher-Powell Broyden-Fletcher-Goldfarb-Shanno Broyden-Fletcher-Goldfarb-Shanno
Multidimensional Methods Stochastic Tunneling Stochastic Tunneling Monte Carlo Monte Carlo Simulated Annealing Simulated Annealing Genetic Algorithm Genetic Algorithm Surface smoothing: proteins Multi-dimensional Global (uphill jumps allowed)
Bottleneck Typically many function evaluations are required in order to estimate derivatives and interpolate/extrapolate along PES Typically many function evaluations are required in order to estimate derivatives and interpolate/extrapolate along PES Want simple analytic form for energy ! Want simple analytic form for energy ! q1q2q3..qnq1q2q3..qn E(q 1,q 2..) Molecular mechanics Semi- Empirical Ab Initio Analytic?
What is the optimum point? ? ? ? At extremum
Local vs. Global? Conformational Analysis (Equilibrium Conformer) Equilibrium Geometry A conformational analysis is global geometry optimization which yields multiple structurally stable conformational geometries (i.e. equilibrium geometries) 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 where the potential gradient is zero) E local greater than or equal to E global
Geometry Optimization Basic Scheme Basic Scheme Find first derivative (gradient) of potential energy Find first derivative (gradient) of potential energy Set equal to zero Set equal to zero Find value of coordinate(s) which satisfy equation Find value of coordinate(s) which satisfy equation
Modeling Potential energy (1-d)
Modeling Potential energy (>1-d) Hessian
Find Equilibrium Geometry for the Morse Oscillator
Bottlenecks No Functional Form No Functional Form More than one variable More than one variable Coupling between variables Coupling between variables
Geometry Optimization (No Functional Form) Bracketing (w/parabolic fitting) Bracketing (w/parabolic fitting) Find energy (E 1 ) for given value of coordinate x i Find energy (E 1 ) for given value of coordinate x i Change coordinate (x i+1 =x i - x) to give E 2 Change coordinate (x i+1 =x i - x) to give E 2 Change coordinate (x i+2 =x i + x) to give E 3 Change coordinate (x i+2 =x i + x) to give E 3 If (E 2 >E 1 and E 3 >E 1 ) then x i+1 > x min >x i+2 If (E 2 >E 1 and E 3 >E 1 ) then x i+1 > x min >x i+2 Fit to parabola and find parabolic minimum Fit to parabola and find parabolic minimum Use value of coordinate at minimum as starting point for next iteration Use value of coordinate at minimum as starting point for next iteration Repeat to satisfaction (Minimum Energy error tolerance) Repeat to satisfaction (Minimum Energy error tolerance)
What is the optimum point? HO-trivial case 434
Line Search For given point V(r a ) choose u vector For given point V(r a ) choose u vector u chosen in direction opposite to gradient (I.e. steepest descent) u chosen in direction opposite to gradient (I.e. steepest descent) Approaches Approaches Constant Constant Steepest descent Steepest descent Minimize V(x i + u) Minimize V(x i + u) Want s.t. vectors f and u perpendicular Want s.t. vectors f and u perpendicular Repeat to minimum Repeat to minimum
Line Search(1-d) Steepest Descent (Gradient Descent Method) Steepest Descent (Gradient Descent Method)
Conjugate Methods No “Spoiling” No “Spoiling” Reduces #iterations Reduces #iterations Numerical Gradient Numerical Gradient Powell Method ( speed n 2 ) Powell Method ( speed n 2 ) Analytic Gradient Analytic Gradient Conjugate Gradient (speed n) Conjugate Gradient (speed n)