code: def expansion(la, lb, ra, rb, alpha, beta, t):
      r"""Compute Hermite Gaussian expansion coefficients recursively for two Gaussian functions.
  
      An overlap distribution, which defines the product of two Gaussians, can be written as a Hermite
      expansion as [`Helgaker (1995) p798 <https://www.worldscientific.com/doi/abs/10.1142/9789812832115_0001>`_]
  
      .. math::
  
          \Omega_{ij} = \sum_{t=0}^{i+j} E_t^{ij} \Lambda_t,
  
      where :math:`\Lambda` is a Hermite polynomial of degree :math:`t`, :math:`E` denotes the expansion
      coefficients, :math:`\Omega_{ij} = G_i G_j`, and :math:`G` is a Gaussian function. The overlap
      integral between two Gaussian functions can be simply computed by integrating over the overlap
      distribution which requires obtaining the expansion coefficients. This can be done recursively
      as [`Helgaker (1995) p799 <https://www.worldscientific.com/doi/abs/10.1142/9789812832115_0001>`_]
  
      .. math::
  
          E_t^{i+1,j} = \frac{1}{2p} E_{t-1}^{ij} - \frac{qr}{\alpha} E_{t}^{ij} + (t+1) E_{t+1}^{ij},
  
      and
  
      .. math::
  
          E_t^{i,j+1} = \frac{1}{2p} E_{t-1}^{ij} + \frac{qr}{\beta} E_{t}^{ij} + (t+1) E_{t+1}^{ij},
  
      where :math:`p = \alpha + \beta` and :math:`q = \alpha \beta / (\alpha + \beta)` are computed
      from the Gaussian exponents :math:`\alpha, \beta` and the position :math:`r` is computed as
      :math:`r = r_\alpha - r_\beta`. The starting coefficient is
  
      .. math::
  
          E_0^{00} = e^{-qr^2},
  
      and :math:`E_t^{ij} = 0` if :math:`t < 0` or :math:`t > (i+j)`.
  
      Args:
          la (integer): angular momentum component for the first Gaussian function
          lb (integer): angular momentum component for the second Gaussian function
          ra (float): position component of the first Gaussian function
          rb (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
          t (integer): number of nodes in the Hermite Gaussian
  
      Returns:
          array[float]: expansion coefficients for each Gaussian combination
  
      **Example**
  
      >>> la, lb = 0, 0
      >>> ra, rb = 0.0, 0.0
      >>> alpha = np.array([3.42525091])
      >>> beta =  np.array([3.42525091])
      >>> t = 0
      >>> c = expansion(la, lb, ra, rb, alpha, beta, t)
      >>> c
      array([1.])
      """
      p = alpha + beta
      q = qml.math.array(alpha * beta / p)
      r = ra - rb
  
      if la == lb == t == 0:
          return qml.math.exp(-q * r**2)
  
      if t < 0 or t > (la + lb):
          return 0.0
  
      if lb == 0:
          return (
              (1 / (2 * p))
              * expansion(la - 1, lb, ra, rb, alpha, beta, t - 1)
              - (q * r / alpha)
              * expansion(la - 1, lb, ra, rb, alpha, beta, t)
              + (t + 1)
              * expansion(la - 1, lb, ra, rb, alpha, beta, t + 1)
          )
  
      return (
          (1 / (2 * p))
          * expansion(la, lb - 1, ra, rb, alpha, beta, t - 1)
          + (q * r / beta)
          * expansion(la, lb - 1, ra, rb, alpha, beta, t)
          + (t + 1) * expansion(la, lb - 1, ra, rb, alpha, beta, t + 1)
      )
  
imports: import itertools
  import numpy
  from scipy.special import factorial2
  import pennylane
label_desc: Use image node_id 5 for calling a global function with example usage: expansion(la, lb, ra, rb, alpha, beta, t) and returns: unknown, qml, int, unknown