code: def _diff2(i, j, ri, rj, alpha, beta):
      r"""Compute the second order differentiated integral needed for evaluating a kinetic integral.
  
      The second-order integral :math:`D_{ij}^2`, where :math:`i` and :math:`j` denote angular
      momentum components of Gaussian functions, is computed from overlap integrals :math:`S` and the
      Gaussian exponent :math:`\beta` as
      [`Helgaker (1995) p804 <https://www.worldscientific.com/doi/abs/10.1142/9789812832115_0001>`_]:
  
      .. math::
  
          D_{ij}^2 = j(j-1)S_{i,j-2}^0 - 2\beta(2j+1)S_{i,j}^0 + 4\beta^2 S_{i,j+2}^0.
  
      Args:
          i (integer): angular momentum component for the first Gaussian function
          j (integer): angular momentum component for the second Gaussian function
          ri (array[float]): position component of the first Gaussian function
          rj (array[float]): position component of the second Gaussian function
          alpha (array[float]): exponent of the first Gaussian function
          beta (array[float]): exponent of the second Gaussian function
  
      Returns:
          array[float]: second-order differentiated integral between two Gaussian functions
      """
      p = alpha + beta
  
      d1 = (
          j
          * (j - 1)
          * qml.math.sqrt(np.pi / p)
          * expansion(i, j - 2, ri, rj, alpha, beta, 0)
      )
      d2 = (
          -2
          * beta
          * (2 * j + 1)
          * qml.math.sqrt(np.pi / p)
          * expansion(i, j, ri, rj, alpha, beta, 0)
      )
      d3 = (
          4
          * beta**2
          * qml.math.sqrt(np.pi / p)
          * expansion(i, j + 2, ri, rj, alpha, beta, 0)
      )
  
      return d1 + d2 + d3
  
imports: import itertools
  import numpy
  from scipy.special import factorial2
  import pennylane
label_desc: Use image node_id 11 for calling a global function with example usage: _diff2(i, j, ri, rj, alpha, beta) and returns: unknown