# Expanding the periodic potentials

1. Apr 24, 2013

### hokhani

Could one always write the periodic potentials in the form:
v(r)=Ʃf(r-G)
where the sum is over G (reciprocal lattice vectors)?

2. Apr 24, 2013

### nasu

r-G does not make sense. They are vectors in different spaces. They have different units.

3. Apr 24, 2013

### hokhani

Excuse me. I wrote wrong.
Can we write v(r)=Ʃf(r-R) where the sum is over lattice vectors R?

4. Apr 24, 2013

### nasu

This looks like superposition. Adding the potentials of each atom, in the coordinate system with origin in the specific atom. Periodicity does not seem to be necessary for this.
f(r-R) will be the potential "produced" by the atom at location R.

5. Apr 25, 2013

### hokhani

Ok, thank you. But I like to know that could we earn this formula merely by having periodicity condition (without regarding atoms)?

6. Apr 25, 2013

### cgk

yes, if the potential v is periodic, then it can be written in this form. This is the most general form of a function v(r) which fulfills v(r+R) = v(r) for all lattice vectors R (i.e., which is periodic).

7. Apr 25, 2013

### hokhani

Could you please prove it or give me a reference which has proved it?

8. Apr 25, 2013

### cgk

Consider a large, finite lattice with periodic boundary conditions. Note that then v(r)=Ʃf(r-R) hold if you put in f(r) = v(r)/N_R itself, where N_R is the total number of lattice points. This also still holds if you set f(r) = v(r) if r is in one single unit cell, and 0 if not.