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2 years ago

Ewald summation 

# share by jovian
import numpy
from numpy.linalg import norm 
from scipy.special import erfc
from math import cos, sin, exp, pi, sqrt
from itertools import product
import jovian
jovian.commit()
[jovian] Updating notebook "sabg0430/ewaldsum-f4c50" on https://jovian.ai/ [jovian] Committed successfully! https://jovian.ai/sabg0430/ewaldsum-f4c50 
'https://jovian.ai/sabg0430/ewaldsum-f4c50'

Define

  • Aim : Calculate potential energy of ion(ii) within unit cell. (periodic boundary condition)

    ϕ(i)=∑l∈I∑j=1nqjl∣ri−(rj+la)∣\phi(i) = \sum_{l \in I} \sum^{n}_{j=1} \frac{q^{l}_{j}}{\left| r_i -\left(r_j + la \right) \right|}
  • charge density
    qjl=(qjl−ρ(i,l))+ρ(i,l)q^l_j = \left( q^l_j -\rho(i,l) \right) +\rho(i,l)

    ϕ(i)=∑l∈I∑j=1n(qjl−ρ(i,l))+ρ(i,l)∣ri−(rj+la)∣\phi(i) = \sum_{l \in I} \sum^{n}_{j=1} \frac{\left( q^l_j -\rho(i,l) \right) +\rho(i,l)}{\left| r_i -\left(r_j + la \right) \right|}

    →ϕ(i)=ϕ1(i)+ϕ2(i)\to \phi(i) = \phi_1(i) + \phi_2(i)

The total potential are separated into two different but related potentials.