Practical guide to GW calculations

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. and Fuchs et al.

General outline of a GW calculation

In the GW approximation following eigenvalue problem is solved

[math]\displaystyle{ T+V_{ext}+V_h+\Sigma(\epsilon)|\phi\rangle = \epsilon|\phi\rangle }[/math]

where [math]\displaystyle{ T }[/math] is the kinetic energy, [math]\displaystyle{ V_{ext} }[/math] the external potential of the nuclei and [math]\displaystyle{ 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]\displaystyle{ \Sigma }[/math] and approximated by [math]\displaystyle{ \Sigma=GW }[/math] in the GW approach with [math]\displaystyle{ G }[/math] being the Green's function and [math]\displaystyle{ W }[/math] the screened potential.

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

• Determination of the polarizability [math]\displaystyle{ \chi=GG }[/math]
• Calculation of the screened potential [math]\displaystyle{ W=\frac{V}{1-\chi V} }[/math]
• Approximate Self-energy [math]\displaystyle{ \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]\displaystyle{ \epsilon^{(0)} }[/math] and one-electron states [math]\displaystyle{ \phi^{(0)} }[/math] (usually from a previous DFT calculation), calculates the self-energy [math]\displaystyle{ \Sigma^{(0)} }[/math], solves the corresponding eigenvalue problem and obtains a new eigenset [math]\displaystyle{ {\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]\displaystyle{ 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]\displaystyle{ 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 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

However, often simpler procedures yield also accurate quasi-particle energies.

Recipes

• G0W0 calculations: single-shot G₀W₀ calculations
• GW0, scGW0 calculations: partially selfconsistent, with respect to G (ALGO=GW0 or ALGO=scGW0)
• GW and scGW calculations: selfconsistent GW (ALGO=GW or ALGO=scGW)
• DM calculations with GW routines: Determination of the frequency dependent dielectric matrix (DM) using the GW routines

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.

Related Tags and Sections

• ALGO for response functions and GW calculations
• LMAXFOCKAE
• NOMEGA, NOMEGAR number of frequency points
• LSPECTRAL: use the spectral method for the polarizability
• LSPECTRALGW: use the spectral method for the self-energy
• OMEGAMAX, OMEGATL and CSHIFT
• ENCUTGW: energy cutoff for response function
• ENCUTGWSOFT: soft cutoff for Coulomb kernel
• ODDONLYGW and EVENONLYGW: reducing the k-grid for the response functions
• LSELFENERGY: the frequency dependent self energy
• LWAVE: selfconsistent GW
• NOMEGAPAR: frequency grid parallelization
• NTAUPAR: time grid parallelization

Examples that use this tag

References

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