How Do I Calculate the Dynamical Matrix for a Crystal with a Pair Potential?

[aihaike]

Dear all,

I'm trying to compute the dynamical matrix of a crystal with a pair potential.
One way would be to use the second derivatives of the potential, and an other is to compute it numerically.
Well, in both cases, I get troubles where atom i equal atom j.
Is there something I haven't get yet?
I go through each atoms (i=1,N) and for each pairs I go through each coordinates (x,y,z). But in the analytical expressions of the derevatives there is always 1/r which diverge when r=0.
Numerically I face to the same problem ...
Any advices?
Thanks in advance,

Eric.

 

[kanato]

I assume your pair potential has the form:

V(r_1,...,r_N) = \frac12 \sum_{i \neq j} \phi(|r_i - r_j|)

That has N*(N-1) terms in it. When you take the first derivative wrt to a single coordinate, than you end up with N-1 terms. Then you have two different outcomes depending on whether your second derivative is taken with respect to the same coordinate as the first one or not. If it is, then you still have N-1 terms, like:

\frac{\partial^2 V}{\partial r_i^2} = \sum_{j} \left. \frac{\partial^2 \phi(r)}{\partial r^2} \right|_{|r_i-r_j|}

where the sum over j excludes the j=i term.

 

[aihaike]

Hi Kanato,

thanks it's clear.
I think I have to sum over 2 different cells to not exclude terms i=j.


Eric.

 

[nsundar]

Dear All,
Can some one help me derive a general form of dynamical matrix for lattice more than one atom?
Many thanks in advance,
Sundar