GW approximation of Hedin's equations

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Green's functions

The GW method can be understood in terms of the following eigenvalue equation[1]

Here is the kinetic energy, the external potential of the nuclei, the Hartree potential and the quasiparticle energies with orbitals . In contrast to DFT, the exchange-correlation potential is replaced by the many-body self-energy and should be obtained together with the Green's function , the irreducible polarizability , the screened Coulomb interaction and the irreducible vertex function in a self-consistent procedure. For completeness, these equations are[2]

Here the common notation was adopted and denotes the bare Coulomb interaction. Note, that these equations are exact and provide an alternative to the Schrödinger equation for the many-body problem. Nevertheless, approximations are necessary for realistic systems. The most popular one is the GW approximation and is obtained by neglecting the equation for the vertex function and using the bare vertex instead:

This means that the equations for the polarizability and self-energy reduce to

while the equations for the Green's function and the screened potential remain the same.

However, in practice, these equations are usually solved in reciprocal space in the frequency domain

In principle Hedin's equations have to be solved self-consistently, where in the first iteration is the non-interacting Green's function

with being a set of one-electron orbitals and the corresponding energies. Afterwards the polarizability is determined, followed by the screened potential and the self-energy . This means that GW calculations require a first guess for the one-electron eigensystem, which is usually taken from a preceding DFT step.

In principle, one has to repeat all steps by the updating the Green's function with the Dyson equation given above in each iteration cycle until self-consistency is reached. In practice, this is hardly ever done due to computational complexity on the one hand (in fact fully self-consistent GW calculations are available as of VASP 6 only).

On the other hand, one observes that by keeping the screened potential in the first iteration to the DFT level one benefits from error cancelling,[3] which is the reason why often the screening is kept on the DFT level and one aims at self-consistency in Green's function only.

Following possible approaches are applied in practice.

Single Shot: G0W0

Partially self-consistent: GW0 or EVGW0

Self-consistent Quasi-particle approximation: QPGW0

Low-scaling GW: The Space-time Formalism

The scaling with system size (number of electrons) of GW calculations can be reduced[4] by performing a so-called Wick-rotation to imaginary time .

References