Introduction to GeometryIntroduction to Geometry Optimizationp

Computational Chemistry labComputational Chemistry lab2014

Determination of the molecule configuratione e a o o e o ecu e co gu a o

H H • Diatomic molecule – determine the interatomic distance

H HO • Triatomic molecule – determine distances and the angle

• Four-atomic molecule – determine angles and distances

H H

HN

H H

3N 6 t f N t i l l (N>2)3N- 6 parameters for an N-atomic molecule (N>2)(3N atomic coordinates-3 c.o.m coordinates -3 angles)

Optimal configurations correspond to minima of h f i f hthe energy as a function of these parameters

Solution of the Schrödingerequation in the Born-Oppenheimer

approximation:approximation:Htot = (Tn + Vn ) + Te + Vne + Ve= (Hn) + He

Electronic SE: He Ψ (r,R)=Ee(R) Ψ (r,R)Ab-initio or semi-empirical solutionAb initio or semi empirical solution Ee(R) – electronic energyR l di tR – nuclear coordinatesVn(R) – nuclei-nuclei interaction

E(R)=Vn(R)+ Ee(R) = potential energy surface (PES)

Molecular mechanics: PES ←molecular force field

Potential Energy Surface (PES)o e e gy Su ce ( S)A hypersurface in 3N-6 -dimensional hyperspace, where Nis the number of atomsis the number of atoms.

Minima– equilibrium structures of one orstructures of one or several molecules

S ddl i t i iSaddle points - transition structures connecting equilibrium structuresequilibrium structures.

Global Minimum - that point that is the lowest value in the PES

H2 H2O – 3N-6=3

H H

H

H

O

H

• One configuration is known for H2

• One configuration is known for H2O

g 2

• The planar and pyramidal configurations are known for NH3

The number of local minima typically goes exponentially with the number of variables (degrees of freedom). Combinatorial Explosion Problem

Possible Conformations (3n) for linear alkanes CH3(CH2)n+1CH3

n = 1 3n 2 9n = 2 9n = 5 243n = 10 59,049n 10 59,049n = 15 14,348,907

Global minimum - Mariana Trenchin the Pacific Oceanin the Pacific Ocean

Optimization/GoalsOptimization/Goals

• Find the Local Minimum Structure

• Find the Global Minimum Structure

• Find the Transition State Structure

We would like to Find the Reaction Pathway Connecting Two MinimaPathway Connecting Two Minima and Passing through the Transition State Structure

Geometry optimization is the name for the procedure that attempts to find the configuration of minimum energy of the p g gymolecule.

One-dimensional optimization

• f’(x)=0 – stationary i

3

point• f’’(x)<0 – maximum

2 10

f

maximum

• f’’(x)>0 – minimum0

1

f

0

5minimum

-1

0

-5

0

f'

f''

-0.5 0.0 0.5 1.0 1.5 2.0 2.5x

f''

x

Multidimensional Optimization

- gradient

The coordinates can be Cartesian (x,y,z..),Cartesian (x,y,z..), or internal, such as bondlength and angle

Force: F=-E (Potential energy)displacements)

f=0 - Stationary Point (minimum, maximum, or saddle point)

Hessian matrix

A matrix of second-order derivatives ofA matrix of second order derivatives of the energy with respect to atomic coordinates (e.g., Cartesian or internal coordinates)coordinates)Sometimes called force matrix –matrix size of (3N-6)x(3N-6)

ji xxf

ij fH

2

)(Quadratic approximation

Approximate the complex energy landscape by harmonic potentials

n n

stststst xxxxEHxxExxE )()(1)()( ))()(()()(

]0),...([ )()(1 st

nst xxE),...( )()(

1st

nst xxaround a stationary point

i j

jjiiijnn xxxxEHxxExxE1 1

211 ))()((),...(),...(

n n

stjj

stiiij

stn

stn xxxxEHxxExxE )()(

21)()(

11 ))()((),...(),...(

n

i

kiij

kjk lHl

1

)()(Hessian matrix diagonalization (eigenproblem)

i j

jjiiijnn1 1

211 ))()(()()(

i 1Eigenvalues 2

kk m k - vibrational frequenciesEigenvectors )(k

jl give normal coordinates n

stii

kik xxlq )()( )(

ji 1

n

kkk

stn

stn qxxExxE

1

221)()(

11 ),...(),...(

)real(0 All kk minimum

)imaginary(0 All kk maximum

)realimaginary/(0 kk saddle pointp

q1

q2

Steepest Descentp

)(

Steepest descent direction g=-

, g - (3N-6) -dimensional vectors

||/1 iiiii ggrr

),...( 1 nxxr||1 iiiii gg

Advantage:•local minimization is guaranteed•local minimization is guaranteed•fast minimization far from the minimum Disadvantage:g• slow descent along narrow valleys• slow convergence near minima

iiiiii EE 5.0 else 2.1 then)()( if 111 rrIn HYPERCHEM

History gradient information is not kept

Conjugate Gradient MethodsSearch direction is chosen using history gradient informationInitial direction – the steepest descent direction, h0 = g0 =-

hrr

is defined by a one-dimensional minimization along the search direction

iii hrr 1

minimization along the search direction. iiii hgh 111

Fl h R h d2

1 || ig jFletcher-Reeves method: 21

1 ||||

i

ii g

g

Polak Ribiere method: 11 )( iii ggg

steepest conjugateg1Polak-Ribiere method: 21 || i

i g

Polak-Ribiere may be superior for non-quadratic functions.

Gradient history is equivalent to implicit use of the Hessian matrix.

Newton-Raphson MethodsExplicit use of Hessian matrix. nExplicit use of Hessian matrix.

Quadratic approximation : )()()( 221

10 kk

n

kkk qqFEE

rr

n

n

qk- normal coordinates,

k

kkq1

0 lrr

k

kkFE1

0 )( lr

εk and lk - eigenvalues and eigenvectors of the Hessian matrix at r0

One-step optimization f d ti f ti

kFkq

k k g g 0

0)( rEof quadratic functions kkq

For arbitrary functionsconjugateNewton-

0)( rE

)(/)()(1

1 ik

n

kikikii F rrrlrr

0)(fdi iD

conjugateNewtonRaphson

i d h l i i ( i h

0)(for directionDescent ik r

Finds the closest stationary point (either minimum, maximum, or saddle point).

Obtaining, storing, and diagonalaizing the Hessian

Numerical calculation – (3N)2 calculations of energy

diagonalaizing the Hessian

Diagonalization - (3N)3 operationsMemory - (3N)2

E t l i f l l lExtremely expensive for large molecules.

Block Diagonal Newton-Raphson

22211

2

11

2

21

2

000

000

EEEzx

Eyx

ExE

Calculation - 9N

222

21

2

11

2

11

211

2111

000

000

EEEzE

zyE

zxE

zyyyx Diagonalization - 27N

Memory – 9N

22

2

22

2

22

222

2

22

2

22

2

000

000

zyE

yE

yxE

zxE

yxE

xE Memory 9N

22

2

22

2

22

222222

000zE

zyE

zxE

yyyx

Gaussian The Hessian matrix is calculated and processed on the first step onlyIt is updated using the computed energies and gradients

Rational functionBerny algorithm

n

ikF )()( rl

Rational function optimization (RFO) step

k ik

ikikii

11 )(

)()( r

rlrrRFO

r0εk and lk - eigenvalues and eigenvectors

<min( ) (<0 for <0)Newton-Raphson

k k g gof the Hessian matrix at ri

<min(k) (<0 for k<0) - step always toward to a minimum.

A linear search between the latestA linear search between the latest point and the best previous point.

Convergence criteriaGradient or force F=-g=- Gradient

Convergence criteriaGradient or force F=-g=- Gradient

Convergence criteriaGradient or force F=-g=-

Predicted displacement r r

Gradient

Predicted displacement ri+1-ri

Maximum force: max(F)=max(F F )<thresholdmax(F)=max(F1,…Fn)<threshold

Root-mean-square (RMS) force: threshold/)(|| 22 nFFF

Maximum displacement:

threshold/)(|| 1 nFF nF

max(ri+1-ri) <thresholdRMS displacement:

| |/ <th h ld|ri+1-ri|/n<thresholdDisplacement

• One configuration is known for H2

• One configuration is known for H2O

g 2

• The planar and pyramidal configurations are known for NH3

The number of local minima typically goes exponentially with the number of variables (degrees of freedom). Combinatorial Explosion Problem

Possible Conformations (3n) for linear alkanes CH3(CH2)n+1CH3

n = 1 3n 2 9n = 2 9n = 5 243n = 10 59,049n 10 59,049n = 15 14,348,907

Global minimization

• "building up" the structure: combining a large l l f i i d f ( imolecule from preoptimized fragments (protein

from preoptimized aminoacids)• conformational sampling: take various starting

points for local minimization (simulation of nature)

• None of these are guaranteed to find the global g gminimum!

Global Minimum- conformational sampling

• Molecular Dynamics• Monte CarloMonte Carlo• Simulated Annealing• Tabu searchTabu search• Genetic Algorithms• Ant Colony y

Optimizations• Diffusion Methods • Distance Geometry

Methods