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energy minimization
1. Presented by Guided by
Mr. Pradeep V. Kore Mr.
M. Pharm. ( II Sem ) 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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3. 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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4. 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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6. 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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7. 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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8. 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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9. Energy minimising procedures
1) Conformational energy searching
2) Energy minimisation
3) Minimisation algorithms
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10. 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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11. 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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13. 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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14. •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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15. 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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16. 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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17. 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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18. 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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19. Types of minima:
weak strong
strong local local
local minimum minimum
f(x) minimum strong
global
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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