How to write this in most efficient way

Question

I have an array (N = 10^4) and I need to find a difference between each two of the entries( calculating a potential given the coordinates of the atoms) Here is the code I am writing in pure python, but its really not effective, can anyone tell me how to speed it up? (using numpy or weave). Here x,y are arrays of coordinates of atoms(just simple 1D array)

def potential(r):
   U = 4.*(np.power(r,-12) - np.power(r,-6))
   return U
def total_energy(x):
   E = 0.
   #need to speed up this part
   for i in range(N-1):
     for j in range(i):
         E += potential(np.sqrt((x[i]-x[j])**2))  
return E

Answer 1

first you can use array arithmetics:

def potential(r):
    return 4.*(r**(-12) - r**(-6))

def total_energy(x):
    E = 0.
    for i in range(N-1):
        E += potential(np.sqrt((x[i]-x[:i])**2)).sum()
    return E

or you can test the fully vectorized version:

def total_energy(x):
    b=np.diag(x).cumsum(1)-x
    return potential(abs(b[np.triu_indices_from(b,1)])).sum()