forte2.gradients.utils
======================

.. py:module:: forte2.gradients.utils




Module Contents
---------------

.. py:function:: flat_to_atom_gradient(gradient, natoms)

   
   Convert a flat atom-major Cartesian gradient to ``(natoms, 3)`` shape.


   :Parameters:

       **gradient** : array_like
           Flat gradient vector with shape ``(3 * natoms,)``.

       **natoms** : int
           Number of atoms.



   :Returns:

       NDArray
           Gradient array with shape ``(natoms, 3)``.











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.. py:function:: nuclear_repulsion_deriv(atoms)

   
   Compute point-charge nuclear repulsion derivatives.

   The derivative is returned in Hartree/Bohr for coordinates in Bohr:

   .. math::
       \frac{\partial E_\mathrm{nuc}}{\partial R_{A\alpha}}
       =
       -\sum_{B \ne A} Z_A Z_B
       \frac{R_{A\alpha} - R_{B\alpha}}{|\mathbf{R}_A-\mathbf{R}_B|^3}.

   :Parameters:

       **atoms** : list[tuple[float, Sequence[float]]]
           Nuclear charges and Cartesian centers.



   :Returns:

       NDArray
           Nuclear repulsion derivative with shape ``(natoms, 3)``.











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.. py:function:: compute_gradient(system, D1, W1, W2, W3)

   
   Compute the total gradient from the one-electron density matrix and two-electron derivative weights.

   The returned gradient is in Hartree/Bohr for coordinates in Bohr.

   :Parameters:

       **system** : System
           The system for which to compute the gradient.

       **D1** : NDArray
           The one-electron density matrix with shape ``(nbasis, nbasis)``.

       **W1** : NDArray
           The energy-weighted density matrix with shape ``(nbasis, nbasis)``.

       **W2** : NDArray
           The two-electron derivative weight for the metric with shape ``(naux, naux)``.

       **W3** : NDArray
           The two-electron derivative weight for the three-center integrals with shape ``(naux, nbasis, nbasis)``.



   :Returns:

       NDArray
           Total gradient with shape ``(natoms, 3)``.











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.. py:function:: build_metric_inverted_three_center(system)

   
   Computes the three-center integrals with the Coulomb metric inverse applied.

   Compute the quantity :math:`Z^{P}_{\mu\nu}` defined as:

   .. math::
       Z^{P}_{\mu\nu}
       =
       \sum_{Q} M^{-1}_{PQ} (Q|\mu\nu).

   :Parameters:

       **system** : System
           The system for which to compute the metric-inverted three-center integrals.



   :Returns:

       NDArray
           Metric-inverted three-center integrals with shape ``(naux, nbasis, nbasis)``.











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.. py:function:: apply_inverse_metric(system, M, J)

   
   Apply the density fitting metric inverse to a three-center tensor.
















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