Introduction 2.Potential energy surface 3.Basics of energy minimization Energy Minimization Algorithms 1.Steepest Descent 2.Conjugate Gradient

1. Energy Minimization
Aishwarya M. Rane
M.Sc. Bioinformatics(part-2)
G. N. Khalsa CollegeJul 24, 2018 1

2. Contents
Energy Minimization
1. Introduction
2. Potential energy surface
3. Basicsof energy minimization
Energy Minimization Algorithms
1. Steepest Descent
2. ConjugateGradient
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3. Introduction
• energy minimization is the process of finding an arrangement
in space of a collection of atoms where, according to some
computational model of chemical bonding, the net inter-
atomic force on each atom is acceptably close to zero and the
position on the potential energy surface (PES) is a stationary
point.
single molecule, an ion,
a transition state or even a
collection of any of these
Quantum mechanics
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4. General Terms
• A potential energy surface (PES) describes the
energy of a system, especially a collection of atoms, in
terms of certain parameters, normally the positions of
theatoms.
Explores properties of structures
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5. Forexample
•When optimizing the geometry
of a water molecule, our aim is
to obtain thehydrogen-oxygen
bond lengths and thehydrogen-
oxygen-hydrogen bond angle
which minimizetheforcesthat
would otherwisebepulling atoms
together or pushing them apart.
Fig1: PES for water molecule: Shows the energy minimum
corresponding to optimized molecular structure for water- O-H bond
length of 0.0958nm and H-O-H bond angle of 104.5°
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6. Significance of geometry
optimization
• Physically significant structures.
• Optimized structures often correspond to a substance as it is
found in nature.
• Optimized structurescan beused in avariety of
experimental and theoretical investigationsin thefieldsof
chemical structure, thermodynamics,chemicalkinetics.
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7. Molecular Geometry and
Mathematical Interpretation
Thegeometry of aset of atoms
Vector of theatoms' positions
Described byDescribed by
Single
atom
Cartesian
Coordinates
Cartesian
Coordinates
Molecule
Internal
Coordinates
Internal
Coordinates
Bond lengths , Bond
Angles , Dihedral Angles
Bond lengths , Bond
Angles , Dihedral Angles
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8. Mathematical Interpretation
• r– A vectorthe describes the atoms' positions
• E(r) – Energy as a function of position
• Mathematically, our aim is to to find the
value of r for which E(r) is at a local
minimum.
Smallest value of a function
Obtained by calculating
∂E/∂r
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9. Mathematical Interpretation
• Second aim is to find the curvature of the
PES at r.
Obtained by calculating
∂∂E/∂ri∂rj
Hessian matrixHessian matrixJul 24, 2018 9

10.  Thecomputational model that providesan approximate E(r) could bebased
on
1. Quantum mechanics
2. Forcefields
3. Or acombination of thosein caseof QM/MM.
 Using thiscomputational model and an initial information of thecorrect
geometry, an iterativeoptimization procedureisfollowed, for example:
 Calculate the force on each atom (that is, -∂ E/∂ r).
 If the force is less than some threshold, finish.
 Otherwise, move the atoms by some computed step .
 Repeat from the start.
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11. Energy Optimization
Algorithms
• An optimization algorithm can use some or all
of E(r) , ∂E/∂r and ∂∂E/∂ri∂rj to try to minimize the
forces.
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12. Steepest Descent
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13. Steepest Descent
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Fig2 : Steepest descent
• Slow near minimum.
• Used for structures away
from minimum.
• Used asrough and
introductory method
followed by moreadvanced
methods.

14. Conjugate Gradient
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Valuefrom
previousstep

15. Conjugate Gradient
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Fig3: ConjugateGradient
• Used for larger system.
• Morecomputational power
and requirements.
• Moreefficient
convergence.

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