Geometry OptimisationModelling

1,2

1,4

1,6

1,8

2,0

2,21,0

1,5

2,0

2,5

3,0

-100

0

100

200

300

400

500

C--

--O

dis

tanc

e in

A

O---H distance in A

Energy in kJ / m

ol

OH + C2H4

*CH2-CH2-OH

CH3-CH2-O*

3D PES

What computational chemistry can do for you:

- structural properties (bond lengths, bond angles and dihedral)

-energetic properties (which isomer is more stable,

how fast a reaction should go: reactant and TS energies

- chemical reactivity (from electron distribution nucleophilic and electrophilic sites)

C O

Nuc

- spectral properties (IR, UV and NMR spectra)

- interaction properties (molecular fitting)

Lewars

Introduction

G-R

Structure1

ReagentStructure2

R - G1

ReagentR - G2

R - ClOH-

R - OH + Cl-

H3C

CH

H3C

CH2 Br

1-Bromo-2-methylpropane

1-bróm-2-metilpropán

izobutil-bromid

carbon skeletonfunctional group

the ultimate goal: interconversion of one structure to another one

architecture of the moleculestereochemistry

property=f(structure)

Physical properties

Chemical reactivities

Biological activities molecular structure

activity=f(structure)

reactivity=f(structure)

property=f(structure)

Optimization

Geometry

Geometrical distortion

Internalenergy

Stable structure

Geometrical distortion

Internalenergy

Multiple stable structures

the energy differences (DE) is a measure of relative stability.

stable structure 1 stable structure 2K

Stable structures and transition statesStable structures

and transition states TS

Typical reaction mech.

VARIABLES: 3 translational coordinates and 3 rotational coordinates of a general n-atomic molecule leave (3n – 6) internal coordinates.

Potential Energy Surface (PES) representation of chemical reaction

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)

Terminology

Geometry Optimization

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

Modeling Potential energy (1-d)

U(r) U(req ) dUdr rreq

(r req ) 12

d2Udr2

rreq

(r req )2

neq

rr

n

n

eq

rr

rrdr

Ud

nrr

dr

Ud

eqeq

)(!

1....)(

3

1 33

3

Modeling Potential energy (>1-d)

U(r a r ) U(ra ) dU

dr rra

ri

i

1

2ri

d2U

dridrj rreq

rj .....i, jij

c -b r +

1

2r T A

r

Hessian

Find Equilibrium Geometry for 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 for the Morse Oscillator

Re

RRe

aDiff

eeaDdR

dV

eDV

RRa

RRa

RRaRRa

RRaHH

,0 c)

,0)1( b)

02 a)

0 )1(2

)1(

)(

0)(

0

)()(0

2)(0

0

0

00

0

Bottlenecks

• No Functional Form• More than one variable• Coupling between variables

Geometry Optimization(No Functional Form)

• Bracketing (w/parabolic fitting)• Find energy (E1) for given value of coordinate xi

• Change coordinate (xi+1=xi- x) to give E2

• Change coordinate (xi+2=xi + x) 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)

Terminology

Potential Energy Surface (PES)

A force field defines for each molecule a unique PES.Each point on the PES represents a molecular conformation characterized by its structure and energy.Energy is a function of the coordinates.(Next) Coordinates are function of the energy.

ener

gy

coordinates

CH3

CH3

CH3

Goal of Energy Minimization

A system of N atoms is defined by 3N Cartesian coordinates or 3N-6 internal coordinates. These define a multi-dimensional potential energy surface (PES).

A PES is characterized by stationary points:

• Minima (stable conformations)• Maxima• Saddle points (transition states)

Goal of Energy Minimization• Finding the stable conformations

ener

gy

coordinates

Classification of Stationary Points

0.0

4.0

8.0

12.0

16.0

20.0

0 90 180 270 360

transition state

local minimum

global minimum

ener

gy

coordinate

TypeMinimum MaximumSaddle point

1st Derivative000

2nd Derivative*positivenegativenegative

* Refers to the eigenvalues of the second derivatives (Hessian) matrix

Minimization Definitions

0

ix

f0

2

2

ix

f

Given a function:

Find values for the variables for which f is a minimum:

),,( 3321 Nxxxxff

Functions• Quantum mechanics energy• Molecular mechanics energy

Variables• Cartesian (molecular mechanics)• Internal (quantum mechanics)

Minimization algorithms• Derivatives-based• Non derivatives-based

A Schematic Representation

Starting geometry

Ý Easy to implement; useful for well defined structuresß Depends strongly on starting geometry

Population of Minima

Most minimization method can only go downhill and so locate the closest (downhill sense) minimum.No minimization method can guarantee the location of the global energy minimum.No method has proven the best for all problems.

Global minimum

Most populated minimum

Active Structure

A General Minimization Scheme

Starting point x0

Minimum?

Calculatexk+1 = f(xk)

Stopyes

No

Two Questions

f(x,y)

Where to go (direction)?

How far to go (magnitude)?

This is where we want to go

How Far To Go? Until the Minimum

Real function

Cycle 1: 1, 2, 3

Cycle 2: 1, 2, 4

Line search in one dimension• Find 3 points that bracket the minimum

(e.g., by moving along the lines and recording function values).

• Fit a quadratic function to the points.• Find the function’s minimum through

differentiation.• Improved iteratively.

Arbitrary Step• xk+1 = xk + lksk, lk = step size.• Increasel as long as energy reduces.• Decrease l when energy increases. 4

3

21

5

Where to go?• Parallel to the force (straight downhill): Sk = -gk

How far to go?• Line search• Arbitrary Step

Steepest Descent

Steepest Descent: Example

-15 -10 -5 0 5 10 15

-15

-10

-5

0

5

10

15441

361289

169225

12181

4925

91

Starting point: (9, 9)

Cycle 1:Step direction: (-18, -36)Line search equation:Minimum: (4, -1)

Cycle 2:Step direction: (-8, 4)Line search equation:

Minimum: (2/3, 2/3)

92 xy

15.0 xy

22 2),( yxyxf

y

xg

4

2kk gS

Steepest Descent:Overshooting

SD is forced to make 90º turns between subsequent steps (the scalar product between the (-18,-36) and the (-8,4) vector is 0 indicating orthogonality) and so is slow to converge.

Ligand geometry

scoring: -11.2 kcal/mol scoring: -5.7 kcal/mol

Orientation - interactions

scoring: -11.2 kcal/mol scoring: -5.7 kcal/mol

KON

FORM

ÁCIÓ

S TÉ

R

.

Prot

ein

fold

ing

és

konf

orm

áció

s té

r

SZER

KEZE

TI

ÉS

Polim

er m

olek

ulák

sz

erke

zete

i és

reak

ciói

Kis

mol

ekul

ák

és re

akci

óik

Configuration and conformational space

C2H4O2

Energy landscape

R

RCT

TC

TS

3.ábra

●OH + + H2OC6H13N2O3

Summery I.

Summery II.