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Energy Minimisation Cheat Sheet (DRAFT) by

Structural Bioinformatics – Slide 7

This is a draft cheat sheet. It is a work in progress and is not finished yet.

Defini­tions

Minimi­sation algorithm
Identifies geometries corres­ponding to minimum points on the energy surface
Saddle point
Highest points on the path between two minima­/maxima i.e. a transition structure
At a minimum point, first deriva­tives are zero, and second deriva­tives are positive
Parameter coordi­nates
Molecular mechanics – Cartesian (3N)
Quantum mechanics – Internal (3N-6)
Categories of min algo
1. Derivative
2. Non-de­riv­ative
Derivative methods
- Obtained analyt­ically or numeri­cally
- Analytical preferred
- If only numerical, non-de­riv­ative may be more effective
Numerical derivative
Change in energy divided by change in coordi­nates
 

Non-de­riv­ative methods

Simplex method
- Non derivative (zeroth order)
- Locates minimum on energy surface by moving around like an amoeba
Simplex
M cartesian coord => M+1 vertices
M internal coord => M-5 vertices
- Direction of first derivative => Minima location
- Magnitude of deriv. => Steepness of local slope
Movements
Reflection -
Reflection and Expansion -
Contra­ction -
 

Derivative methods

- Direction of first derivative => Minima location
- Magnitude of deriv. => Steepness of local slope
- Second derivative => curvature of function
Force = -dV(r)/dr
First order algos
- steepest descent
- conjugate gradient
STEEPEST DESCENT
- moves in dir. || net force (walking straight downhill
- both gradient and direction orthogonal
1) line search (2) arbitrary step (3) lanrange multip­liers
- robust when starting point is far from minimum
- relieves higest energy features
1D Line search
- bracket search
- comput­ati­onally expensive
Arbitrary step
- random step size
- if lower energy, step size increased by multip­lic­ation factor
- higher energy, step size reduced
- more steps but less function evalua­tions
Cons
- forced to make right angles
- path oscill­ates, overco­rrects, and reintr­oduces errors
CONJUGATE GRADIENT
- no oscill­ation
- gradient orthogonal but direction conjugate
- for quadratic function of M variables, min reached in M steps
- can be used from 2nd step (1st step SD)