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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.