:py:mod:`momentGW.energy`
=========================

.. py:module:: momentGW.energy

.. autoapi-nested-parse::

   Energy functionals.



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

.. py:function:: hartree_fock(rdm1, fock, h1e)

   Hartree--Fock energy functional.

   :param rdm1: One-particle reduced density matrix.
   :type rdm1: numpy.ndarray
   :param fock: Fock matrix.
   :type fock: numpy.ndarray
   :param h1e: One-electron Hamiltonian.
   :type h1e: numpy.ndarray

   :returns: **e_1b** -- Hartree--Fock energy.
   :rtype: float


.. py:function:: galitskii_migdal(gf, se, flip=False)

   Galitskii--Migdal energy functional.

   :param gf: Green's function object.
   :type gf: dyson.Lehmann
   :param se: Self-energy object.
   :type se: dyson.Lehmann
   :param flip: Default option is to use the occupied Green's function and the
                virtual self-energy. If `flip=True`, the virtual Green's
                function and the occupied self-energy are used instead. Default
                value is `False`.
   :type flip: bool, optional

   :returns: **e_2b** -- Galitskii--Migdal energy.
   :rtype: float

   .. rubric:: Notes

   This functional is the analytically integrated version of [1]_

   .. math::
       \frac{\pi}{4} \int d\omega \mathrm{Tr}[G(i\omega)
       \Sigma(i\omega)]

   in terms of the poles of the Green's function and the self-energy.
   This scales as :math:`\mathcal{O}(N^4)` with system size [2]_.

   .. rubric:: References

   .. [1] V. M. Galitskii and A. B. Migdal, Sov. Phys. JETP 7, 96,
       1958.
   .. [2] O. J. Backhouse, M. Nusspickel, and G. H. Booth, J. Chem.
       Theory Comput. 16, 2, 2020.


.. py:function:: galitskii_migdal_g0(mo_energy, mo_occ, se, flip=False)

   Galitskii--Migdal energy functional for the non-interacting Green's
   function.

   :param mo_energy: MO energies (poles of the Green's function).
   :type mo_energy: numpy.ndarray
   :param mo_occ: MO occupancies.
   :type mo_occ: numpy.ndarray
   :param se: Self-energy.
   :type se: dyson.Lehmann
   :param flip: Default option is to use the occupied Green's function and the
                virtual self-energy. If `flip=True`, the virtual Green's
                function and the occupied self-energy are used instead. Default
                value is `False`.
   :type flip: bool, optional

   :returns: **e_2b** -- Galitskii--Migdal energy.
   :rtype: float

   .. rubric:: Notes

   This functional is the analytically integrated version of [1]_

   .. math::
       \frac{\pi}{4} \int d\omega \\
           \mathrm{Tr}[G_{0}(i\omega) \Sigma(i\omega)]

   in terms of the poles of the mean-field Green's function and the
   self-energy. This scales as :math:`\mathcal{O}(N^3)` with system
   size [2]_.

   .. rubric:: References

   .. [1] V. M. Galitskii and A. B. Migdal, Sov. Phys. JETP 7, 96,
       1958.
   .. [2] O. J. Backhouse, M. Nusspickel, and G. H. Booth, J. Chem.
       Theory Comput. 16, 2, 2020.