Taking the derivative of a dot product

[pholvey]

Homework Statement

Hi, I need to take the derivative of the following energy potential (attached) in order to get an expression of the force so I can do a conjugate gradient minimization.

An explanation of the potential is found in the attached jpeg below the functional form.

Homework Equations

d/dx(a.b)=a′b+ab′

The Attempt at a Solution

I know that I should be able to take the derivative of a dot product as shown above, but in this case, the vectors being constructed of differences in xyz coordinates is confusing me. If anyone can give any help on how to approach this problem, I would be grateful.

Many thanks,

Pholvey

 

[mighty2000]

Dear Pholvey,

Thank you for reaching out for help with your problem. The potential you have attached is commonly known as the Lennard-Jones potential, which is used to model the interaction between two particles in a system. To take the derivative of this potential, we can use the chain rule and the product rule to break it down into simpler terms.

First, let's rewrite the potential in terms of the distance between the two particles, r, instead of the differences in xyz coordinates:

V(r) = 4ε[(σ/r)^12 - (σ/r)^6]

Using the chain rule, we can write the derivative as:

dV/dr = 4ε[-12(σ/r)^11 * (-1/r^2) - 6(σ/r)^5 * (-1/r^2)]

= -4ε[-12(σ/r)^11 * (-1/r^2) - 6(σ/r)^5 * (-1/r^2)]

= 4ε[(12σ^12/r^13) - (6σ^6/r^7)]

= 4ε[12σ^12/r^13 - 6σ^6/r^7]

Now, we can use the product rule to expand the terms:

dV/dr = 4ε[12σ^12/r^13 - 6σ^6/r^7]

= 4ε[12(12σ^11 * (1/r^14)) - 6(6σ^5 * (1/r^8))]

= 4ε[144σ^11/r^14 - 36σ^5/r^8]

Finally, we can rewrite this expression in terms of the differences in xyz coordinates:

dV/dr = 4ε[144σ^11/(x2-x1)^14 - 36σ^5/(x2-x1)^8]

= 4ε[144σ^11/(x2-x1)^14 - 36σ^5/(y2-y1)^8 - 36σ^5/(z2-z1)^8]

= 4ε[(144σ^11(x2-x1)^13 - 36σ^5(x2-x1)^7(y2-y1)^8 - 36σ^5(x2-x1)^7(z2-z1)^8]

= 4ε[(144σ^11(x2-x1)^13 -