code: def sphere_analytical_gaussian(dr, a, alpha=0.2765):
      
      term1 = 0.5*(erf((dr+2*a)/(alpha*np.sqrt(2))) + erf(-dr/(alpha*np.sqrt(2))))
      term2 = np.sqrt(0.5/np.pi)*(alpha/(dr+a+1e-10)) * (
                  np.exp(-0.5*dr**2/alpha**2) - np.exp(-0.5*(dr+2*a)**2/alpha**2)
              )
      return term1 - term2
docs: Analytically calculate the sphere's functional form by convolving the
      Heavyside function with first order approximation to the sinc, a Gaussian.
      The alpha parameters controls the width of the approximation -- should be
      1, but is fit to be roughly 0.2765