This module offers two strategies for automagically calculating the gradients (and optionally force constants) of a potential energy function (or any other function of vectors, for that matter). The more convenient strategy is to create an object of the class PotentialWithGradients. It takes a regular Python function object defining the potential energy and is itself a callable object returning the energy and its gradients with respect to all arguments that are vectors.
Example:
def _harmonic(k,r1,r2):
dr = r2-r1
return k*dr*dr
harmonic = PotentialWithGradients(_harmonic)
energy, gradients = harmonic(1., Vector(0,3,1), Vector(1,2,0))
print energy, gradients
prints
3.0 [Vector(-2.0,2.0,2.0), Vector(2.0,-2.0,-2.0)]The disadvantage of this procedure is that if one of the arguments is a vector parameter, rather than a position, an unnecessary gradient will be calculated. A more flexible method is to insert calls to two function from this module into the definition of the energy function. The first, DerivVectors(), is called to indicate which vectors correspond to gradients, and the second, EnergyGradients(), extracts energy and gradients from the result of the calculation. The above example is therefore equivalent to
def harmonic(k, r1, r2):
r1, r2 = DerivVectors(r1, r2)
dr = r2-r1
e = k*dr*dr
return EnergyGradients(e,2)
To include the force constant matrix, the above example has to be
modified as follows:
def _harmonic(k,r1,r2):
dr = r2-r1
return k*dr*dr
harmonic = PotentialWithGradientsAndForceConstants(_harmonic)
energy, gradients, force_constants = harmonic(1.,Vector(0,3,1),Vector(1,2,0))
print energy
print gradients
print force_constants
The force constants are returned as a nested list representing a matrix. This can easily be converted to an array for further processing if the numerical extensions to Python are available.