Molecular Modeling:Molecular Modeling:Geometry OptimizationGeometry Optimization

C372C372

Introduction to Introduction to Cheminformatics IICheminformatics II

Kelsey ForsytheKelsey Forsythe

Geometry OptimizationGeometry Optimization

Le Chatliers’ PrincipleLe Chatliers’ PrincipleThe optimum geometry is the The optimum geometry is the

geometry which minimizes the strain geometry which minimizes the strain on a given system. Any perturbation on a given system. Any perturbation from this geometry will induce the from this geometry will induce the

system to change so as to reduce this system to change so as to reduce this perturbation unless prevented by perturbation unless prevented by

external forcesexternal forcesMathematical Surface Reflects This Principle!!

Why Extrema?Why Extrema? Equilibrium structure/conformer MOST Equilibrium structure/conformer MOST

likely observed? likely observed? Once geometrically optimum structure found Once geometrically optimum structure found

can calculate energy, frequencies etc. to can calculate energy, frequencies etc. to compare with experimentcompare with experiment

Use in other simulations (e.g. dynamics Use in other simulations (e.g. dynamics calculation)calculation)

Used in reaction rate calculations (e.g. Used in reaction rate calculations (e.g. saddlesaddlereaction timereaction time ))

Characteristics of transition stateCharacteristics of transition state PES interpolation (Collins et al)PES interpolation (Collins et al)

NomenclatureNomenclature

PES equivalent to Born-PES equivalent to Born-Oppenheimer surfaceOppenheimer surface

Point on surface corresponds to Point on surface corresponds to position of nuclei position of nuclei

Minimum and MaximumMinimum and Maximum LocalLocal Global Global Saddle point (min and max)Saddle point (min and max)

CyclohexaneCyclohexane

Local maxima

Global minimum

Global maxima

Local minima

Ex. PESEx. PES

Saddle point

Local minimumGlobal minimum

Recall glycine?Recall glycine?

Global

Local

MethodsMethods

Steepest DescentSteepest Descent Conjugate GradientConjugate Gradient Fletcher Powell Fletcher Powell SimplexSimplex Geometric Direct Geometric Direct

Inversion of Inversion of Iterative SubspaceIterative Subspace

Newton-RaphsonNewton-Raphson

Minimize w.r. each Minimize w.r. each

individualindividual coordinate coordinate No gradients requiredNo gradients required No gradients requiredNo gradients required

Methods (1-d)Methods (1-d)

No Functional FormNo Functional Form Bracketing Bracketing Parabolic Interpolation (Brent’s method) Parabolic Interpolation (Brent’s method)

Methods (1-d)(w/ gradients)Methods (1-d)(w/ gradients)

Steepest DescentSteepest Descent

Methods (n-d)(w/o Methods (n-d)(w/o gradients)gradients)

Line SearchLine Search SimplexSimplex Fletcher-PowellFletcher-Powell

Methods (n-d)(w/ gradients)Methods (n-d)(w/ gradients)

Conjugate Gradient (space Conjugate Gradient (space N) N) Fletcher-Reeves Fletcher-Reeves Polak-RibierePolak-Ribiere

Quasi-Newton/Variable Metric (space Quasi-Newton/Variable Metric (space NN22)) Davidon-Fletcher-PowellDavidon-Fletcher-Powell Broyden-Fletcher-Goldfarb-ShannoBroyden-Fletcher-Goldfarb-Shanno

Multidimensional MethodsMultidimensional Methods

Stochastic TunnelingStochastic Tunneling

Monte CarloMonte Carlo

Simulated AnnealingSimulated Annealing

Genetic AlgorithmGenetic Algorithm

Surface Surface smoothing: smoothing: proteinsproteins

Multi-dimensionalMulti-dimensional

Global (uphill Global (uphill jumps allowed)jumps allowed)

BottleneckBottleneck Typically many function evaluations are required Typically many function evaluations are required

in order to estimate derivatives and in order to estimate derivatives and interpolate/extrapolate along PESinterpolate/extrapolate along PES

Want simple analytic form for energy !Want simple analytic form for energy !

q1

q2

q3

.

.qn

E(q1,q2..)Molecular mechanics

Semi-Empirical

Ab Initio

Analytic?

8.35E-28 8.77567E+14 20568787140 2.03098E-18 1.05374E-188.35E-28 8.77567E+14 20568787140 1.77569E-18 9.66155E-198.35E-28 8.77567E+14 20568787140 1.54682E-18 8.82365E-198.35E-28 8.77567E+14 20568787140 1.34201E-18 8.02375E-198.35E-28 8.77567E+14 20568787140 1.15913E-18 7.26185E-198.35E-28 8.77567E+14 20568787140 9.96207E-19 6.53795E-198.35E-28 8.77567E+14 20568787140 8.51451E-19 5.85205E-198.35E-28 8.77567E+14 20568787140 7.23209E-19 5.20415E-198.35E-28 8.77567E+14 20568787140 6.09973E-19 4.59425E-198.35E-28 8.77567E+14 20568787140 5.10362E-19 4.02235E-198.35E-28 8.77567E+14 20568787140 4.2311E-19 3.48845E-198.35E-28 8.77567E+14 20568787140 3.47061E-19 2.99255E-198.35E-28 8.77567E+14 20568787140 2.81155E-19 2.53465E-198.35E-28 8.77567E+14 20568787140 2.24426E-19 2.11475E-198.35E-28 8.77567E+14 20568787140 1.75987E-19 1.73285E-198.35E-28 8.77567E+14 20568787140 1.35031E-19 1.38895E-198.35E-28 8.77567E+14 20568787140 1.0082E-19 1.08305E-198.35E-28 8.77567E+14 20568787140 7.26787E-20 8.15147E-208.35E-28 8.77567E+14 20568787140 4.99924E-20 5.85247E-208.35E-28 8.77567E+14 20568787140 3.22001E-20 3.93347E-208.35E-28 8.77567E+14 20568787140 1.87901E-20 2.39447E-208.35E-28 8.77567E+14 20568787140 9.29638E-21 1.23547E-208.35E-28 8.77567E+14 20568787140 3.29443E-21 4.56475E-21

Empirical Potential for Hydrogen Molecule

0

2E-19

4E-19

6E-19

8E-19

1E-18

1.2E-18

1.4E-18

0 0.5 1 1.5 2 2.5 3 3.5 4

What is the optimum point?What is the optimum point?

?

?

?

0)(=

rdrdV

vv

At extremum

Local vs. Global?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 wherethe potential gradient is zero)•Elocal greater than or equal to Eglobal

Geometry OptimizationGeometry Optimization

Basic Scheme Basic Scheme Find first derivative (gradient) of Find first derivative (gradient) of

potential energypotential energy Set equal to zeroSet equal to zero Find value of coordinate(s) which Find value of coordinate(s) which

satisfy equationsatisfy equation

Modeling Potential energy Modeling Potential energy (1-d)(1-d)

€

U(r) U(req ) −dU

dr r= req

(r − req ) +1

2

d2U

dr2

r= req

(r − req )2

€

−1

3

d3U

drr= req

(r − req )3 ....+1

n!

dnU

drn

r= req

(r − req )n€

=

€

≈

Modeling Potential energy Modeling Potential energy (>1-d)(>1-d)

€

U(v r a + δ

v r ) = U(ra ) −

dU

dr r= ra

δri

i

∑ +1

2δri

d2U

dridrj r= req

δrj + .....i, ji≤ j

∑

≈ c -v b δ

v r +

1

2δ

v r T A

≈δ

v r

Hessian

Find Equilibrium Geometry Find Equilibrium Geometry for the Morse Oscillatorfor the Morse Oscillator

)()(0

)()(0

)()(0

2)(0

00

00

00

0

)1(2

) )1(2

))(0( )1(2

)1(

RRaRRa

RRaRRa

RRaRRa

RRaHH

eeaD

aeeD

eaeDdRdV

eDV

−−−−

−−−−

−−−−

−−

−=

−=

−−×−=

−=

Find Equilibrium Geometry Find Equilibrium Geometry for the Morse Oscillatorfor the Morse Oscillator

BottlenecksBottlenecks

No Functional FormNo Functional Form More than one variableMore than one variable Coupling between variablesCoupling between variables

Geometry OptimizationGeometry Optimization(No Functional Form) (No Functional Form)

Bracketing (w/parabolic fitting)Bracketing (w/parabolic fitting) Find energy (EFind energy (E11) for given value of coordinate x) for given value of coordinate xii

Change coordinate (xChange coordinate (xi+1i+1=x=xii--x) to give Ex) to give E22

Change coordinate (xChange coordinate (xi+2i+2=x=xii + +x) to give Ex) to give E33

If (EIf (E22>E>E11 and E and E33>E>E11) then x) then xi+1i+1> x> xminmin >x >xi+2i+2

Fit to parabola and find parabolic minimumFit to parabola and find parabolic minimum Use value of coordinate at minimum as starting Use value of coordinate at minimum as starting

point for next iterationpoint for next iteration Repeat to satisfaction (Minimum Energy error Repeat to satisfaction (Minimum Energy error

tolerance)tolerance)

8.35E-28 8.77567E+14 20568787140 2.03098E-18 1.05374E-188.35E-28 8.77567E+14 20568787140 1.77569E-18 9.66155E-198.35E-28 8.77567E+14 20568787140 1.54682E-18 8.82365E-198.35E-28 8.77567E+14 20568787140 1.34201E-18 8.02375E-198.35E-28 8.77567E+14 20568787140 1.15913E-18 7.26185E-198.35E-28 8.77567E+14 20568787140 9.96207E-19 6.53795E-198.35E-28 8.77567E+14 20568787140 8.51451E-19 5.85205E-198.35E-28 8.77567E+14 20568787140 7.23209E-19 5.20415E-198.35E-28 8.77567E+14 20568787140 6.09973E-19 4.59425E-198.35E-28 8.77567E+14 20568787140 5.10362E-19 4.02235E-198.35E-28 8.77567E+14 20568787140 4.2311E-19 3.48845E-198.35E-28 8.77567E+14 20568787140 3.47061E-19 2.99255E-198.35E-28 8.77567E+14 20568787140 2.81155E-19 2.53465E-198.35E-28 8.77567E+14 20568787140 2.24426E-19 2.11475E-198.35E-28 8.77567E+14 20568787140 1.75987E-19 1.73285E-198.35E-28 8.77567E+14 20568787140 1.35031E-19 1.38895E-198.35E-28 8.77567E+14 20568787140 1.0082E-19 1.08305E-198.35E-28 8.77567E+14 20568787140 7.26787E-20 8.15147E-208.35E-28 8.77567E+14 20568787140 4.99924E-20 5.85247E-208.35E-28 8.77567E+14 20568787140 3.22001E-20 3.93347E-208.35E-28 8.77567E+14 20568787140 1.87901E-20 2.39447E-208.35E-28 8.77567E+14 20568787140 9.29638E-21 1.23547E-208.35E-28 8.77567E+14 20568787140 3.29443E-21 4.56475E-21

Empirical Potential for Hydrogen Molecule

0

2E-19

4E-19

6E-19

8E-19

1E-18

1.2E-18

1.4E-18

0 0.5 1 1.5 2 2.5 3 3.5 4

What is the optimum point?What is the optimum point?

HO-trivial case

1

42

3 34

Line SearchLine Search

For given point V(rFor given point V(raa) choose u vector) choose u vector u chosen in direction opposite to gradient u chosen in direction opposite to gradient

(I.e. steepest descent)(I.e. steepest descent) ApproachesApproaches

Constant Constant Steepest descentSteepest descent

Minimize V(xMinimize V(xii++ u)u)

Want Want s.t. vectors f and u perpendiculars.t. vectors f and u perpendicular

Repeat to minimumRepeat to minimum

€

vu = −

dV

dv x i

uxx ii

vvv +=+

€

dV (v x i+1)

dλ=

dV

dv x i+1

⎛

⎝ ⎜

⎞

⎠ ⎟

f1 2 3

Td

v x i+1

dλu

{= 0

Line Search(1-d)Line Search(1-d)

Steepest Descent (Gradient Descent Steepest Descent (Gradient Descent Method)Method)

ix

ii

dx

dfu

uxx

=

+==

+

.022)( 23 +−= xxxf

Conjugate MethodsConjugate Methods

No “Spoiling”No “Spoiling” Reduces #iterationsReduces #iterations Numerical GradientNumerical Gradient

Powell Method (Powell Method (speedspeednn22)) Analytic GradientAnalytic Gradient

Conjugate Gradient (speed Conjugate Gradient (speed n)n)