momentGW.energy

Energy functionals.

Module Contents

momentGW.energy.hartree_fock(rdm1, fock, h1e)

Hartree–Fock energy functional.

Parameters:

  • rdm1 (numpy.ndarray) – One-particle reduced density matrix.

  • fock (numpy.ndarray) – Fock matrix.

  • h1e (numpy.ndarray) – One-electron Hamiltonian.

Returns:

e_1b – Hartree–Fock energy.

Return type:

float

momentGW.energy.galitskii_migdal(gf, se, flip=False)

Galitskii–Migdal energy functional.

Parameters:

  • gf (dyson.Lehmann) – Green’s function object.

  • se (dyson.Lehmann) – Self-energy object.

  • flip (bool, optional) – 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.

Returns:

e_2b – Galitskii–Migdal energy.

Return type:

float

Notes

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

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

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

References

momentGW.energy.galitskii_migdal_g0(mo_energy, mo_occ, se, flip=False)

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

Parameters:

  • mo_energy (numpy.ndarray) – MO energies (poles of the Green’s function).

  • mo_occ (numpy.ndarray) – MO occupancies.

  • se (dyson.Lehmann) – Self-energy.

  • flip (bool, optional) – 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.

Returns:

e_2b – Galitskii–Migdal energy.

Return type:

float

Notes

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

\[\begin{split}\frac{\pi}{4} \int d\omega \\ \mathrm{Tr}[G_{0}(i\omega) \Sigma(i\omega)]\end{split}\]

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

References