from math import exp,sqrt

# Constants
sigma = 1.0             # Value of sigma in nm
accuracy = 1e-6         # Required accuracy in nm
z = (1+sqrt(5))/2       # Golden ratio

# Function to calculate the Buckingham potential
def f(r):
    return (sigma/r)**6 - exp(-r/sigma)

# Initial positions of the four points
x1 = sigma/10
x4 = sigma*10
x2 = x4 - (x4-x1)/z
x3 = x1 + (x4-x1)/z

# Initial values of the function at the four points
f1 = f(x1)
f2 = f(x2)
f3 = f(x3)
f4 = f(x4)

# Main loop of the search process
while x4-x1>accuracy:
    if f2<f3:
        x4,f4 = x3,f3
        x3,f3 = x2,f2
        x2 = x4 - (x4-x1)/z
        f2 = f(x2)
    else:
        x1,f1 = x2,f2
        x2,f2 = x3,f3
        x3 = x1 + (x4-x1)/z
        f3 = f(x3)

# Print the result
print("The minimum falls at",0.5*(x1+x4),"nm")