Download energyminimization

Presented by
Mr. Pradeep V. Kore
M. Pharm. ( II Sem )
Guided by
Mr.
M. Pharm.
Department of Pharmaceutical Chemistry
JSPM’s Charak College of Pharmacy & Research,
Gat No. 720(1/2), Wagholi, Pune-Nagar road,
Wagholi-412 207.
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CONTENTS:
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INTRODUCTION
 In computational chemistry energy minimization (also called
energy optimization or geometry optimization) methods are
used to compute the equilibrium configuration of molecules
and solids.
 Energy minimizatiom methods can precisely locate minimum
energy confirmation by mathematically “homing in” on the
energy function minima (one at a time).
 The goal of energy minimization is to find a route (consisting
of variation of the intramolecular degrees of freedom) from an
initial confirmation to nearest minimum energy confirmation
using the smallest number of calculations possible.
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 In molecular modeling we are interested in minimum points on
the energy surface.
 Minimum energy arrangments of the atoms corresponds to
stable states of the system:
 Any movement away from a minimum gives a configuration
with higher energy.
 There may be a very large number of minima on energy
surface. The minimum with very lowest energy is known as the
Global Energy Minimum.
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Fig: One dimentional energy surface
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What can energy minimization do?
 It can repair distorted geometries by moving atoms to release
internal constraints, as shown below:
 In this example, the CZ of a phenylanalnine ring was artificially
stretched out, which lead to bonds much too long.
Fig: Phenylanalnine
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 By first invoking an energy computation, the C-terminal Oxygen
(OXT) is added to the residue. Note that the residue must also be
protonated, and in this case an N-terminal blocking group (HHT)
is added. Then the energy computation can be done:
The direction in which atoms should be displaced in order to
reach a lowe energy state are shown by dotted lines. A
minimal deplacement appears in dark blue, while a big
deplacement appear in red (blue-green-red gradient).
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 After
an energy minimization (200 cycles of Steepest
Descent), the geometry is repaired, and all the force vectors are
dark blue, which means a minimum has been reached.
Fig: Repaired geometry structure
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Energy minimising procedures
1) Conformational energy searching
2) Energy minimisation
3) Minimisation algorithms
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Conformational energy searching
 Energy is a function of the degrees of freedom in a molecule:
bonds, angles, dihedrals.
 Conformational energy searching is used to find all of the
energetically preferred conformations of a molecule.
 This is mathematically equivalent to locating all of the minima of
the energy function of the molecule.
 The possible conformations for a molecule lie on an n-dim.
Lattice, with n being the number of degrees of freedom.
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 Systematic energy sampling is thus technically impossible for
almost all molecules in question, due to the high large number
of required sampling points.
 Need for methods to speed up energy minima localisation.
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Energy minimization:
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 Minimisation algorithms are designed to head down-hill towards
the nearest minimum.
 Remote minima are not detected, because this would require some
period of up-hill movement.
 Minimisation algorithms monitor the energy surface along a series
of incremental steps to determine a down-hill direction.
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•The local shape of the energy surface around a given conformation
en route to a minimum is often assumed to be quadratic so as to
simplify the mathematics.
•An energy minimum can be characterised by a small change in
energy between steps and/or by a zero gradient of the energy
function
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Approximation of the quadratic energy function
 Approximation of the quadratic energy function is given by
aTaylor series:
f(x)= f(P) – bx + 1/2Ax2
P- is the current point
x -an arbitrary point on the energy surface
b- is the gradient at
P,
A- is the
Hessian matrix
(the second partial derivatives) at P. A and b can be viewed
as parameters that fit the idealised quadratic form to the actual
energy surface.
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Minimisation algorithms
Simplex algorithm
- Not a gradient minimization method.
- Used mainly for very crude, high energy starting structures.
Steepest descent minimiser
- Follows the gradient of the energy function (b) at each step.
This results in successive steps that are always mutually
perpendicular, which can lead to backtracking.
- Works best when the gradient is large (far from a minimum).
- Tends to have poor convergence because the gradient becomes
smaller as a minimum is approached.
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Conjugate gradient and Powell minimiser
 Remembers the gradients calculated from previous steps to help
reduce backtracking.
 Generally finds a minimum in fewer steps than Steepest
Descent.
 May encounter problems when the initial conformation is far
from a minimum.
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Newton-Raphson and BFGS minimiser
- Predicts the location of a minimum, and heads in that direction.
- Calculates (Newton-Raphson) or approximates (BFGS) the second
derivatives in A.
- Storage of the A term can require substantial amounts of
computer memory
- May find a minimum in fewer steps than the gradient-only
methods.
- May encounter serious problems when the initial conformation is
far from a minimum.
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Types of minima:
f(x)
strong
local
minimum
weak
local
minimum
strong
global
minimum
strong
local
minimum
feasible region
x
which of the minima is found depends on the starting point
such minima often occur in real applications
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