code: def _hermite_coulomb(t, u, v, n, p, dr):
      """Evaluate the Hermite integral needed to compute the nuclear attraction and electron repulsion
      integrals.
  
      These integrals are computed recursively starting from the Boys function
      [`Helgaker (1995) p817 <https://www.worldscientific.com/doi/abs/10.1142/9789812832115_0001>`_]:
  
  
      .. math::
  
          R_{000}^n = (-2p)^n F_n(pR_{CP}^2),
  
      where :math:`F_n` is the Boys function, :math:`p` is computed from the exponents of the two
      Gaussian functions as :math:`p = \alpha + \beta`, and :math:`R_{CP}` is the distance between the
      center of the composite Gaussian centered at :math:`P` and the electrostatic potential generated
      by a nucleus at :math:`C`. The following recursive equations are used to compute the
      higher-order Hermite integrals
  
      .. math::
  
          R_{t+1, u, v}^n = t R_{t-1, u, v}^{n+1} + x R_{t, u, v}^{n+1},
  
          R_{t, u+1, v}^n = u R_{t, u-1, v}^{n+1} + y R_{t, u, v}^{n+1},
  
          R_{t, u, v+1}^n = v R_{t, u, v-1}^{n+1} + z R_{t, u, v}^{n+1},
  
      where :math:`x`, :math:`y` and :math:`z` are the Cartesian components of :math:`R_{CP}`.
  
      Args:
          t (integer): order of Hermite derivative in x
          u (integer): order of Hermite derivative in y
          v (float): order of Hermite derivative in z
          n (integer): order of the Boys function
          p (float): sum of the Gaussian exponents
          dr (array[float]): distance between the center of the composite Gaussian and the nucleus
  
      Returns:
          array[float]: value of the Hermite integral
      """
      x, y, z = dr[0:3]
      T = p * (dr**2).sum(axis=0)
      r = 0
  
      if t == u == v == 0:
          return ((-2 * p) ** n) * _boys(n, T)
  
      if t == u == 0:
          if v > 1:
              r = r + (v - 1) * _hermite_coulomb(
                  t, u, v - 2, n + 1, p, dr
              )
          r = r + z * _hermite_coulomb(t, u, v - 1, n + 1, p, dr)
          return r
  
      if t == 0:
          if u > 1:
              r = r + (u - 1) * _hermite_coulomb(
                  t, u - 2, v, n + 1, p, dr
              )
          r = r + y * _hermite_coulomb(t, u - 1, v, n + 1, p, dr)
          return r
  
      if t > 1:
          r = r + (t - 1) * _hermite_coulomb(t - 2, u, v, n + 1, p, dr)
      r = r + x * _hermite_coulomb(t - 1, u, v, n + 1, p, dr)
      return r
  
imports: import itertools
  import numpy
  from scipy.special import factorial2
  import pennylane
label_desc: Use image node_id 15 for calling a global function with example usage: _hermite_coulomb(t, u, v, n, p, dr) and returns: r, unknown, r, r