code: def scf(mol, n_steps=50, tol=1e-8):
      r"""Return a function that performs the self-consistent-field calculations.
  
      In the Hartree-Fock method, molecular orbitals are typically constructed as a linear combination
      of atomic orbitals
  
      .. math::
  
          \phi_i(r) = \sum_{\mu} C_{\mu i} \chi_{\mu}(r),
  
      with coefficients :math:`C_{\mu i}` that are initially unknown. The self-consistent-field
      iterations are performed to find a converged set of molecular orbital coefficients that minimize
      the total energy of the molecular system. This optimization problem can be reduced to solving a
      linear system of equations which are usually written as
  
      .. math::
  
          FC = SCE,
  
      where :math:`E` is a diagonal matrix of eigenvalues, representing the molecular orbital
      energies, :math:`C` is the matrix of molecular orbital coefficients, :math:`S` is the overlap
      matrix and :math:`F` is the Fock matrix, which also depends on the coefficients. Fixing an
      initial guess :math:`C_0`, the corresponding :math:`F_0` is built and the system
      :math:`F_0C_0 = SC_0E` is solved to obtain a solution :math:`C_1`. This process is iteratively
      repeated until the coefficients are converged.
  
      The key step in in this process is constructing the Fock matrix which is defined as
  
      .. math::
  
          F = H + \frac{1}{2} J - K,
  
      where :math:`H`, :math:`J` and :math:`K` are the core Hamiltonian matrix, Coulomb matrix and
      exchange matrix, respectively. The entries of :math:`H` are computed from the electronic kinetic
      energy and the electron-nuclear attraction integrals, which are integrals over atomic basis
      functions. The elements of the :math:`J` and :math:`K` matrices are obtained from the Coulomb
      and exchange integrals over the basis functions.
  
      Following the procedure in
      [`Lehtola et al. Molecules 2020, 25, 1218 <https://www.mdpi.com/1420-3049/25/5/1218>`_], we
      express the molecular orbital coefficients in terms of a matrix :math:`X` as
      :math:`C = X \tilde{C}` which gives the following transformed equation
  
      .. math::
  
           \tilde{F} \tilde{C} = \tilde{S} \tilde{C} E,
  
      where :math:`\tilde{F} = X^T F X`, :math:`\tilde{S} = X^T S X` and :math:`S` is the overlap
      matrix. We chose :math:`X` such that :math:`\tilde{S} = 1` as
  
      .. math::
  
          X = V \Lambda^{-1/2} V^T,
  
      where :math:`V` and :math:`\Lambda` are the eigenvectors and eigenvalues of :math:`S`,
      respectively. This gives the eigenvalue equation
  
      .. math::
  
           \tilde{F}\tilde{C} = \tilde{C}E,
  
      which is solved with conventional methods iteratively.
  
      Args:
          mol (~qchem.molecule.Molecule): the molecule object
          n_steps (int): the number of iterations
          tol (float): convergence tolerance
  
      Returns:
          function: function that performs the self-consistent-field calculations
  
      **Example**
  
      >>> symbols  = ['H', 'H']
      >>> geometry = np.array([[0.0, 0.0, 0.0], [0.0, 0.0, 1.0]], requires_grad = False)
      >>> alpha = np.array([[3.42525091, 0.62391373, 0.1688554],
      >>>                   [3.42525091, 0.62391373, 0.1688554]], requires_grad=True)
      >>> mol = qml.qchem.Molecule(symbols, geometry, alpha=alpha)
      >>> args = [alpha]
      >>> v_fock, coeffs, fock_matrix, h_core, rep_tensor = scf(mol)(*args)
      >>> v_fock
      array([-0.67578019,  0.94181155])
      """
  
      def _scf(*args):
          r"""Perform the self-consistent-field iterations.
  
          Args:
              *args (array[array[float]]): initial values of the differentiable parameters
  
          Returns:
              tuple(array[float]): eigenvalues of the Fock matrix, molecular orbital coefficients,
              Fock matrix, core matrix
          """
          basis_functions = mol.basis_set
          charges = mol.nuclear_charges
          r = mol.coordinates
          n_electron = mol.n_electrons
  
          if r.requires_grad:
              args_r = [
                  [args[0][i]] * mol.n_basis[i]
                  for i in range(len(mol.n_basis))
              ]
              args_ = [*args] + [
                  qml.math.vstack(list(itertools.chain(*args_r)))
              ]
              rep_tensor = repulsion_tensor(basis_functions)(*args_[1:])
              s = overlap_matrix(basis_functions)(*args_[1:])
              h_core = core_matrix(basis_functions, charges, r)(*args_)
          else:
              rep_tensor = repulsion_tensor(basis_functions)(*args)
              s = overlap_matrix(basis_functions)(*args)
              h_core = core_matrix(basis_functions, charges, r)(*args)
  
          rng = qml.math.random.default_rng(2030)
          s = s + qml.math.diag(rng.random(len(s)) * 1.0e-12)
  
          w, v = qml.math.linalg.eigh(s)
          x = v @ qml.math.diag(1.0 / qml.math.sqrt(w)) @ v.T
  
          eigvals, w_fock = qml.math.linalg.eigh(
              x.T @ h_core @ x
          )  # initial guess for the scf problem
          coeffs = x @ w_fock
  
          p = mol_density_matrix(n_electron, coeffs)
  
          for _ in range(n_steps):
              j = qml.math.einsum("pqrs,rs->pq", rep_tensor, p)
              k = qml.math.einsum("psqr,rs->pq", rep_tensor, p)
  
              fock_matrix = h_core + 2 * j - k
  
              eigvals, w_fock = qml.math.linalg.eigh(
                  x.T @ fock_matrix @ x
              )
  
              coeffs = x @ w_fock
  
              p_update = mol_density_matrix(n_electron, coeffs)
  
              if qml.math.linalg.norm(p_update - p) <= tol:
                  break
  
              p = p_update
  
          mol.mo_coefficients = coeffs
  
          return eigvals, coeffs, fock_matrix, h_core, rep_tensor
  
      return _scf
  
imports: import itertools
  import pennylane
  from .matrices import core_matrix, mol_density_matrix, overlap_matrix, repulsion_tensor
label_desc: Use image node_id 1 for calling a global function with example usage: scf(mol, n_steps, tol) and returns: _scf, eigvals,  coeffs,  fock_matrix,  h_core,  rep_tensor