Deriving forces from an energy potential

[pholvey]

Homework Statement

This is not a homework problem. I'm trying to do this for research purposes. I have an energy potential (the Keating potential) of the following form (attached). In this form, A, B, and C are force constants, and r_ab is the vector along the bond between atoms a and b. For my purposes, subscripts i and i' represent silicon atoms while j and k represent oxygen atoms. r_0 is the equilibrium Si-O bond length and theta_0 is the average oxygen bond angle.

From this form, I need to derive an equation describing the forces so that I can speed up a conjugate gradient optimization so I don't have to keep estimating the gradient from multiple calls to the energy function.

Homework Equations

r_ij = r_i - r_j (where r's are vectors)

cos theta_ijk = (r_ij dot r_ik)/(abs(r_ij)*abs(r_ik))

The Attempt at a Solution

Honestly, I'm at a loss as to how to even attempt a solution to this problem. Any help/guidance on the subject would be appreciated.

Many thanks,

pholvey

 

[pholvey]

Does anyone have any idea on this? Even showing me how to take the derivative of a dot product summation would be greatly appreciated. Thanks.