Practical guide to GW calculations: Difference between revisions

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== General outline of a GW calculation ==
== General outline of a GW calculation ==
In the GW approximation following eigenvalue problem is solved
<math>
T+V_{ext}+V_h+\Sigma(\epsilon)|\phi\rangle = \epsilon|\phi\rangle
</math>
where <math>T</math> is the kinetic energy, <math>V_{ext}</math> the external potential of the nuclei and <math>V_h</math> the Hartree potential. In contrast to density functional theory, the exchange-correlation potential is replaced by the many-body (usually frequency dependent) self-energy <math>\Sigma</math> and approximated by <math>\Sigma=GW</math> in the GW approach with <math>G</math> being the Green's function and <math>W</math> the screened potential.
The main computational outline for GW calculations is, therefore, as follows:
*Determination of the polarizability <math>\chi=GG</math>
*Calculation of the screened potential <math>W=\frac{V}{1-\chi V}</math>
*Approximate Self-energy <math>\Sigma=GW</math>
Because all quantities, including the Green's function, are frequency dependent the eigenvalue equation should be solved self-consistently in principle. That is, one starts with a first guess for the energies <math>\epsilon^{(0)}</math> and one-electron states <math>\phi^{(0)}</math> (usually from a previous DFT calculation), calculates the self-energy <math>\Sigma^{(0)}</math>, solves the corresponding eigenvalue problem and obtains a new eigenset <math>{\epsilon^{(1)},\phi^{(1)}}</math> and iterates the procedure until self-consistency is reached.
However, the main complication arises due to the appearance of the appearance of the quasi-particle energies in the argument of the self-energy. In practice, one therefore, performs a Taylor expansion of the self-energy w.r.t. the frequency and solves following equation instead:
<math>
Z^{(i-1)}(T+V_{ext}+V_h+\Sigma^{(i-1)}-\Sigma^{'(i-1)}\epsilon^{(i-1)})|\phi^{(i)}\rangle = \epsilon^{(i)}|\phi^{(i)}\rangle
</math>
with the renormalization factor
<math>
Z^{(i-1)}=\left(1-\left.\frac{d}{dz}\right|_{z=\epsilon^{(i-1)}}\Sigma^{(i-1)}\right)^{-1}</math>.
This procedure can be repeated until a maximum of {{TAG|NELM}} self-consistency cycles and can be selected in VASP with the following lines in the {{TAG|INCAR}} file
{{TAGBL|ALGO}} = scGW ! self-consistent GW
{{TAGBL|NELM}} = 4    ! number of self-consistency cycles
Depending on which GW approximation is chosen, this equation is either exclusively solved for the quasi-particle energies <math>\epsilon_{nk}</math>  {{TAG|ALGO}}=G0W0,GW0, ...) or including the  corresponding eigenstates <math>|\phi_{nk}\rangle</math>.
VASP supports various kinds of GW calculations, which all differ in their details. However, the main procedure can be summarized as follows:


== Recipes ==
== Recipes ==

Revision as of 16:03, 21 September 2017

Available as of VASP.5.X. For details on the implementation and use of the GW routines we recommend the papers by Shishkin et al.[1][2][3] and Fuchs et al.[4]

General outline of a GW calculation

In the GW approximation following eigenvalue problem is solved

where is the kinetic energy, the external potential of the nuclei and the Hartree potential. In contrast to density functional theory, the exchange-correlation potential is replaced by the many-body (usually frequency dependent) self-energy and approximated by in the GW approach with being the Green's function and the screened potential.

The main computational outline for GW calculations is, therefore, as follows:

  • Determination of the polarizability
  • Calculation of the screened potential
  • Approximate Self-energy

Because all quantities, including the Green's function, are frequency dependent the eigenvalue equation should be solved self-consistently in principle. That is, one starts with a first guess for the energies and one-electron states (usually from a previous DFT calculation), calculates the self-energy , solves the corresponding eigenvalue problem and obtains a new eigenset and iterates the procedure until self-consistency is reached.

However, the main complication arises due to the appearance of the appearance of the quasi-particle energies in the argument of the self-energy. In practice, one therefore, performs a Taylor expansion of the self-energy w.r.t. the frequency and solves following equation instead:

with the renormalization factor

.

This procedure can be repeated until a maximum of NELM self-consistency cycles and can be selected in VASP with the following lines in the INCAR file

ALGO = scGW ! self-consistent GW
NELM = 4    ! number of self-consistency cycles

Depending on which GW approximation is chosen, this equation is either exclusively solved for the quasi-particle energies ALGO=G0W0,GW0, ...) or including the corresponding eigenstates .

VASP supports various kinds of GW calculations, which all differ in their details. However, the main procedure can be summarized as follows:

Recipes

Large systems

As of version 6, an additional 'R' can be added to the GW ALGO tags, i.e. ALGO=G0W0R, GW0R, scGW0R, GWR or scGWR to select the cubic scaling GW algorithms as described by Liu et. al.[5]

Related Tags and Sections

Examples that use this tag

References


Contents