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

The GW approximation is a many-body method from quantum field theory and goes beyond density functional theory (DFT). Since the paper of Hedin many GW schemes with increasing complexity levels were proposed. In the following we discuss the details of these GW methods and how they are selected in VASP.

The GW approach gained popularity in the community after Hybertsen and Louie showed that the GW band gaps and spetral properties yield very good agreement with measurements for semiconductors and insulators. Today, this method is known as quasi-particle GW (QPGW) and can be understood in terms of the following eigenvalue equation

$(T+V_{ext}+V_h)\phi_{n{\bf k}}({\bf r})+\int d{\bf r}\Sigma({\bf r},{\bf r}',\omega=E_{n{\bf k}})\phi_{n{\bf k}}({\bf r}') = E_{n{\bf k}}\phi_{n{\bf k}}({\bf r})$

Here $T$ is the kinetic energy, $V_{ext}$ the external potential of the nuclei, $V_h$ the Hartree potential and $E_{n{\bf k}}$ the quasiparticle energies with orbitals $\phi_{n{\bf k}}$. In contrast to DFT, the exchange-correlation potential is replaced by the many-body self-energy $\Sigma$ and should be obtained together with the Green's function $G$, the irreducible polarizability $\chi$, the screened Coulomb interaction $W$ and the irreducible vertex function $\Gamma$ in a self-consistent procedure. For completeness, these equations are

$G(1,2)=G_0(1,2)+\int d(3,4) G_0(1,3)\Sigma(3,4)G(4,2)$

$\chi(1,2)=\int d(3,4) G(1,3)G(4,1)\Gamma(3,4;2)$

$W(1,2)=V(1,2)+\int d(3,4) V(1,3)\chi(3,4)W(4,2)$

$\Sigma(1,2)=\int d(3,4) G(1,3)\Gamma(3,2;4)W(4,1)$

$\Gamma(1,2;3)=\delta(1,2)\delta(1,3)+\int d(4,5,6,7)\frac{\delta(\Sigma(1,2))}{\delta G(4,5)}G(4,6)G(7,5)\Gamma(6,7;3)$

where the common notation $1=({\bf r}_1,t_1)$ was adopted and $V$ denotes the bare Coulomb interaction.

Due to the complexity, approximations to this coupled set of integro-differential equations are made. The most important one, the GW approximation, is obtained by neglecting the equation for the vertex function and using the bare vertex $\Gamma(1,2;3)=\delta(1,2)\delta(1,3)$ instead. Then the polarizability and self-energy reduce to

$\chi(1,2)=G(1,2)G(2,1)$

$\Sigma(1,2)=G(1,2)W(2,1)$

The form of the self-energy give this approximation its name.

Quasi particle GW Method

In the QPGW method the equation for the Green's function is never solved and the spetral form

$G({\bf r},{\bf r}',z)=\sum_{n{\bf k}}\frac{\phi_{n{\bf k}}^* ({\bf r})\phi_{n{\bf k}} ({\bf r}')}{z-\epsilon_{n{\bf k}}}$

is used instead. Furthermore the self-energy in the eigenvalue equation is lineraized with a Taylor expansion around the quasi-particle energies from the previous step $(T+V_{ext}+V_h)\phi^{(i+1)}_{n{\bf k}}({\bf r})+\int d{\bf r}\left[ \Sigma({\bf r},{\bf r}',E^{(i)}_{n{\bf k}}) - E^{(i)}_{n{\bf k}}\Sigma'({\bf r},{\bf r}',E^{(i)}_{n{\bf k}}) \right]\phi^{(i+1)}_{n{\bf k}}({\bf r}') = E_{n{\bf k}} \int d{\bf r}\left[ \delta({\bf r}-{\bf r}')-\Sigma'({\bf r},{\bf r}',E^{(n)}_{n{\bf k}})\right] \phi^{(n+1)}_{n{\bf k}}({\bf r}')$

Consequently, the QPGW procedure is as follows

• Determine the Green's function using the spectral form
• Evaluate the polarizability $\chi=GG$
• Calculate the screened potential $W=\frac{V}{1-\chi V}$
• Compute the self-energy $\Sigma=GW$
• Solve linearized eigenvalue problem
• Repeat until self-consistency reached

This method is selected by following lines in the INCAR

ALGO = QPGW ! self-consistent quasi-particle GW
NELM = 4    ! number of self-consistency cycles
LOPTICS = .TRUE. ; LPEAD = .TRUE. ! long-wave limit of chi with finite differences

Note that as of VASP.6 the GW ALGO tags have been renamed, see here for VASP.5.X tags.

After each iteration cycle the quasi-particle energies are updated and the polarizability and screened interaction are recalculated. This approach overestimates band gaps of solids, as shown by Shishkin et al. and a simpler approach, where the screened interaction is not updated, often yields better agreement with experiment. This method is named QPGW0 and is selected via

ALGO = QPGW0 ! partial self-consistent GW
NELM = 4     ! number of self-consistency cycles

VASP supports a third QPGW method, where only the diagonal matrix elements of the eigenvalue equation are updated. This implies that the one-electron orbitals $\phi_{n{\bf k}}$ remain unchanged. This scheme used with following INCAR tags

ALGO = EVGW0 ! partial self-consistent GW
NELM = 4     ! number of self-consistency cycles

Using this option, VASP will write the quasi-particle energies into the OUTCAR file for a set of NBANDSGW bands for every k-point in the Brillouin zone. Look for lines similar to

 band No.  KS-energies  QP-energies   sigma(KS)   V_xc(KS)     V^pw_x(r,r')   Z         occupation Imag(sigma) 
     
     1      -9.8139     -10.2490     -11.6986     -11.0666     -17.7440       0.6884       2.0000       1.1996
     2       2.9032       2.3351     -12.1377     -11.3852     -13.3214       0.7550       2.0000       0.0987
     3       2.9032       2.3351     -12.1377     -11.3852     -13.3214       0.7550       2.0000       0.0987
     4       2.9032       2.3351     -12.1377     -11.3852     -13.3214       0.7550       2.0000       0.0987
     5       3.4884       4.3792      -9.8672     -11.0531      -5.9700       0.7511       0.0000      -0.1035
     6       6.5738       6.9295      -8.9534      -9.4305      -4.7050       0.7455       0.0000      -0.1463
     7       6.5738       6.9295      -8.9534      -9.4305      -4.7050       0.7455       0.0000      -0.1463
     8       6.5738       6.9295      -8.9534      -9.4305      -4.7050       0.7455       0.0000      -0.1463

The first column is the band index, the second column are the quasi-particle energies from the previous (current) self-consistency step. If the self-consistent QPGW algorithms are chosen the OUTCAR will contain additional information of the form

 band No. DFT-energies  QP-energies  QP-e(diag)   sigma(DFT)    Z            occupation
 
     1      -9.8139     -10.2500     -10.2490     -10.4460       0.6884       2.0000
     2       2.9032       2.3350       2.3352       2.1508       0.7550       2.0000
     3       2.9032       2.3350       2.3350       2.1506       0.7550       2.0000
     4       2.9032       2.3352       2.3350       2.1506       0.7550       2.0000
     5       3.4884       4.3792       4.3792       4.6743       0.7511       0.0000
     6       6.5738       6.9297       6.9295       7.0510       0.7455       0.0000
     7       6.5738       6.9297       6.9295       7.0509       0.7455       0.0000
     8       6.5738       6.9297       6.9295       7.0509       0.7455       0.0000

Here the forth column should be identical to the third column of the EVGW0 results shown above, whereas the third column reports the quasi-particle energies obtained from including the off-diagonal matrix elements in the eigenvalue equation.

Self-consistent GW

The GW implementations in VASP described in the papers of Shishkin et al. avoids storage of the Green's function $G$ as well as Fourier transformations between time and frequency domain entirely. That is, all calculations are performed solely on the real frequency axis using Kramers-Kronig transformations for convolutions in the equation of $\chi$ and $\Sigma$ in reciprocal space.

As of VASP.6 a new cubic scaling GW algorithm (called space-time implementation in the following) can be selected. This approach follows the idea of Rojas et al. and allows to solve the Dyson equation for the Green's function self-consistently providing access for true self-consistent GW calculations. The main procedure for a self-consistent GW calculation consists of four steps

• Determine the Green's function $G=\frac{G_0}{1-\Sigma G_0}$
• Evaluate the polarizability $\chi=GG$
• Calculate the screened potential $W=\frac{V}{1-\chi V}$
• Compute the self-energy $\Sigma=GW$
• Repeat until self-consistency reached
• Solve the eigenvalue problem

This algorithm can be selected via

ALGO = scGWR ! self-consistent GW 
NELM = 4     ! number of self-consistentcy steps 

and updates the Green's function as well as the screened potential in every self-consistency step. Replacing scGWR by scGW0R updates the Green's function only, similar to the QPGW and EVGW options. After NELM iterations the self-energy is analytically continued on the real frequency axis using a continued fraction and the same linearized eigenvalue equation of the QPGW routines is solved. The resulting quasi-particle energies can be found in the OUTCAR file as discussed above.

Furthermore, the space-time implementation can be forced to perform QPGW and EVGW calcualtions as well by adding a R to the ALGO GW options described above.

Large systems

Note, that the cubic scaling GW algorithm works on the imaginary time and frequency axis and makes extensive use of Fast Fourier Transformations (FFT). The frequent usage of FFTs gives a relatively large overall scaling pre-factor, so that for small unit cells the QPGW and EVGW ALGO options are often more efficient than the QPGWR and EVWGR options. Only for larger unit cells (more than ~30 atoms) we recommend to use the space-time implementation.

Here similar settings for PRECFOCK,NOMEGAPAR, NTAUPAR and NOMEGA should be chosen as for ACFDTR/RPAR calculations, with one difference. OMEGAMIN(OMEGAMAX) should be smaller(larger) than the HOMO(LUMO) w.r.t. the Fermi energy.

The reason for this is the additional Fourier transform (FT) has to capture all possible one-particle energies in the Green's function

$G(i\tau)\propto e^{-(\epsilon_{n{\bf k}}-\mu)\tau}(\Theta(\tau)(1-n_{n{\bf k}})-\Theta(-\tau)n_{n{\bf k}})$

Only then, the FT $G(i\tau)W(i\tau)=\Sigma(i\tau)\to\Sigma(i\omega)$ is carried out with small error.

Recipes

• G0W0 calculations: single-shot EVGW0 (G₀W₀) calculations
• EVGW0, QPGW0 calculations: partially self-consistent, with respect to G (ALGO=EVGW0 or ALGO=QPGW0)
• EVGW and QPGW calculations: self-consistent QPGW (ALGO=EVGW or ALGO=QPGW)
• DM calculations with GW routines: Determination of the frequency dependent dielectric matrix (DM) using the GW routines

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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