Practical guide to GW calculations: Difference between revisions
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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.''<ref name="shishkin-PRB74"/><ref name="shishkin-PRB75"/><ref name="shishkin:prl:07"/> and Fuchs ''et al.''<ref name="fuchs-PRB76"/> | 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.''<ref name="shishkin-PRB74"/><ref name="shishkin-PRB75"/><ref name="shishkin:prl:07"/> and Fuchs ''et al.''<ref name="fuchs-PRB76"/> | ||
As of VASP.6 all GW approximations can be selected directly via the {{TAG|ALGO}} tag (omitting {{TAG|NBANDS}}) without a preceding DFT calculation. | = Single step procedure: GW in one go = | ||
The actual GW calculation is performed in the second step. | |||
As of VASP.6.3.0 all GW approximations can be selected directly via the {{TAG|ALGO}} tag (omitting {{TAG|NBANDS}}) without a preceding DFT calculation. | |||
{{TAGBL|System}} = SiC | |||
{{TAGBL|ALGO}} = EVGW0, QPGW0, EVGW, QPGW, GW0R or GWR # use an algorithgm described below | |||
{{TAGBL|NELMGW}} = 1,2,.. # number of self-consistency cycles | |||
{{TAGBL|ISMEAR}} = 0 ; {{TAGBL|SIGMA}} = 0.05 ! small sigma is required to avoid partial occupancies | |||
{{TAGBL|LOPTICS}} = .TRUE. # for insulators, omit for metals | |||
Note, {{TAGBL|NBANDS}} must not be present in the INCAR. | |||
Older versions require a two step procedure, where the first step is always a DFT calculation. | |||
The actual GW calculation is performed in the second step. The two-step procedure is described below. | |||
<span id="DFT_Step_for_GW"> | <span id="DFT_Step_for_GW"> | ||
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However, one ''caveat'' applies to this case: when the orbitals are updated, the derivatives of the orbitals with respect to <math>k</math> (stored in the {{FILE|WAVEDER}} file) will become incompatible with the orbitals. | However, one ''caveat'' applies to this case: when the orbitals are updated, the derivatives of the orbitals with respect to <math>k</math> (stored in the {{FILE|WAVEDER}} file) will become incompatible with the orbitals. | ||
This can cause severe problems and convergence to the incorrect solution. For metals, we recommend to | This can cause severe problems and convergence to the incorrect solution. | ||
{{NB|warning|For metals, in general, we recommend to omit the {{TAG|LOPTICS}} tag and remove the {{FILE|WAVEDER}} file from the directory.}} | |||
For insulators, VASP (version 5.3.2 or higher) can update the {{FILE|WAVEDER}} file in each electronic iteration if the finite difference method is used to calculate the first derivative of the orbitals with respect to <math> k </math>: | For insulators, VASP (version 5.3.2 or higher) can update the {{FILE|WAVEDER}} file in each electronic iteration if the finite difference method is used to calculate the first derivative of the orbitals with respect to <math> k </math>: | ||
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<span id="scGW0R"> | <span id="scGW0R"> | ||
== Partially self-consistent GW caluclations: GW0R | == Partially self-consistent GW caluclations: GW0R == | ||
The space-time implementation allows for true self-consistent GW calculations. That is, the solution of the Dyson equation for the Green's function can be obtained with a modest computational effort. The main procedure of a self-consistent GW calculation consists of four main steps | The space-time implementation allows for true self-consistent GW calculations. That is, the solution of the Dyson equation for the Green's function can be obtained with a modest computational effort. The main procedure of a self-consistent GW calculation consists of four main steps | ||
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<span id="scGWR"> | <span id="scGWR"> | ||
== Fully self-consistent GW caluclations: GWR | == Fully self-consistent GW caluclations: GWR == | ||
If the screened potential should be updated during the self-consistency circle<ref name="grumet"/> the following {{TAG|INCAR}} file can be used | If the screened potential should be updated during the self-consistency circle<ref name="grumet"/> the following {{TAG|INCAR}} file can be used | ||
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{{sc|GW calculations|Examples|Examples that use this tag}} | {{sc|GW calculations|Examples|Examples that use this tag}} | ||
= References = | |||
<references> | <references> | ||
<ref name="shishkin-PRB74">[http://link.aps.org/doi/10.1103/PhysRevB.74.035101 M. Shishkin and G. Kresse, Phys. Rev. B 74, 035101 (2006).]</ref> | <ref name="shishkin-PRB74">[http://link.aps.org/doi/10.1103/PhysRevB.74.035101 M. Shishkin and G. Kresse, Phys. Rev. B 74, 035101 (2006).]</ref> |
Revision as of 10:57, 22 November 2021
A brief summary of the theoretical background can be found here.
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]
Single step procedure: GW in one go
As of VASP.6.3.0 all GW approximations can be selected directly via the ALGO tag (omitting NBANDS) without a preceding DFT calculation.
System = SiC ALGO = EVGW0, QPGW0, EVGW, QPGW, GW0R or GWR # use an algorithgm described below NELMGW = 1,2,.. # number of self-consistency cycles ISMEAR = 0 ; SIGMA = 0.05 ! small sigma is required to avoid partial occupancies LOPTICS = .TRUE. # for insulators, omit for metals
Note, NBANDS must not be present in the INCAR.
Older versions require a two step procedure, where the first step is always a DFT calculation. The actual GW calculation is performed in the second step. The two-step procedure is described below.
First step: DFT calculation
GW calculations always require a one-electron basis set. Usually this set is obtained from a standard DFT calculation and written into the WAVECAR file and can be calculated for instance the following INCAR file:
System = SiC NBANDS = 512 ISMEAR = 0 ; SIGMA = 0.05 ! small sigma is required to avoid partial occupancies LOPTICS = .TRUE.
Note, that a significant number of empty bands is required for GW calculations, so that it might be better to perform the calculations in two steps: first a standard grounstate calculation with few unoccupied orbitals only,
System = SiC groundstate occupied orbitals ISMEAR = 0 ; SIGMA = 0.05 ! small sigma is required to avoid partial occupancies EDIFF = 1E-8 ! required tight tolerance for groundstate orbitals
and, second, a calculation of a large number of unoccupied orbitals
System = SiC unoccupied orbitals ALGO = Exact ! use exact diagonalization of the Hamiltonian NELM = 1 ! since we are already converged stop after one step NBANDS = 512 ! maybe even larger ISMEAR = 0 ; SIGMA = 0.05 ! small sigma is required to avoid partial occupancies LOPTICS = .TRUE.
Furthermore note that the flag LOPTICS=.TRUE. is required in order to write the file WAVEDER, which contains the derivative of the orbitals with respect to k. This derivative is used to construct the head and wings of the dielectric matrix employing k·p perturbation theory and is important to accelerate k-point convergence for insulators and semiconductors.
Warning: For metals, in general, we recommend to omit the LOPTICS tag and remove the WAVEDER file from the directory. |
Optional: Use Hybrid functionals
Optionally, one can start a GW calculation from a hybrid functional, such as HSE. For hybrid functionals, the two step procedure will accordingly involve the following INCAR files. In the first step, converged HSE03 orbitals are determined (see here for a selection of available hybrid functionals):
System = SiC groundstate occupied orbitals ISMEAR = 0 ; SIGMA = 0.05 ALGO = Damped ; TIME = 0.5 ! or ALGO = Conjugate LHFCALC = .TRUE. ; AEXX = 0.25 ; HFSCREEN = 0.3 EDIFF = 1E-6 ! required tight tolerance for groundstate orbitals
Secondly, determine the HSE03 orbitals for unoccupied states:
System = SiC unoccupied orbitals NBANDS = 512 ! maybe even larger ALGO = Exact NELM = 1 ! since we are already converged stop after one step ISMEAR = 0 ; SIGMA = 0.05 LHFCALC = .TRUE. ; AEXX = 0.25 ; HFSCREEN = 0.3 LOPTICS = .TRUE.
Step 2: GW step
The actual GW calculation is done in a second step. Here different GW flavors are possible and are selected with the ALGO tag.
Note that as of VASP.6 the GW ALGO tags have been renamed, see here for VASP.5.X tags.
Single shot quasi-particle energies: G0W0
This is the simplest GW calculation and computationally the most efficient one. A single-shot calculation is often referred to as G0W0 and calculates the quasi-particle energies from a single GW iteration by neglecting all off-diagonal matrix elements of the self-energy and employing a Taylor expansion of the self-energy around the DFT energies Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): E_{{n{{\bf {q}}}}}^{{(0)}} . The corresponding equation becomes
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): E_{{n{{\bf {q}}}}}=E_{{n{{\bf {q}}}}}^{{(0)}}+Z_{{n{{\bf {q}}}}}\langle \phi _{{n{{\bf {q}}}}}^{{(0)}}|\Sigma (E_{{n{{\bf {q}}}}}^{{(0)}})-V_{{xc}}|\phi _{{n{{\bf {q}}}}}^{{(0)}}\rangle
with the renormalization factor
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): Z_{{n{{\bf {q}}}}}={\frac {1}{1-{{\rm {Re}}}\langle \phi _{{n{{\bf {q}}}}}^{{(0)}}|\Sigma '(E_{{n{{\bf {q}}}}}^{{(0)}})|\phi _{{n{{\bf {q}}}}}^{{(0)}}\rangle }}
In VASP, G0W0 calculations are selected using an INCAR file such as
System = SiC NBANDS = 512 ISMEAR = 0 ; SIGMA = 0.05 NELM = 1 ; ALGO = EVGW0 ! use "GW0" for VASP.5.X NOMEGA = 50
Convergence with respect to the number of empty bands NBANDS and with respect to the number of frequencies NOMEGA must be checked carefully.
To avoid complicated inter-nested tests, we recommend to calculate all orbitals that the plane wave basis set allows to calculate (except for simple tests). For further reading please consult the section on ENCUTGW.
After a successful G0W0 run, 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
QP shifts <psi_nk| G(iteration)W_0 |psi_nk>: iteration 1 for sc-GW calculations column KS-energies equals QP-energies in previous step and V_xc(KS)= KS-energies - (<T + V_ion + V_H > + <T+V_H+V_ion>^1 + <V_x>^1) k-point 1 : 0.0000 0.0000 0.0000 band No. KS-energies QP-energies sigma(KS) V_xc(KS) V^pw_x(r,r') Z occupation Imag(sigma) 1 -7.1627 -8.3040 -14.5626 -12.7276 -21.6682 0.6219 2.0000 1.2037 2 -2.0901 -3.4347 -15.7660 -14.2799 -21.7439 0.9048 2.0000 0.6914 3 -2.0901 -3.4347 -15.7660 -14.2799 -21.7439 0.9048 2.0000 0.6914 4 -2.0901 -3.4347 -15.7660 -14.2799 -21.7439 0.9048 2.0000 0.6914 5 0.4603 -0.4663 -13.7603 -12.5200 -18.1532 0.7471 2.0000 0.2167 6 0.4603 -0.4663 -13.7603 -12.5200 -18.1532 0.7471 2.0000 0.2167
The first column is the band index and the thrid column denotes the quasi-particle energies Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): E_{{n{{\bf {q}}}}} . Column two, four, five and seven refer to the DFT energies Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): E_{{n{{\bf {q}}}}}^{{(0)}} , diagonal matrix elements of the self-energy Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): \langle \phi _{{n{{\bf {q}}}}}^{{(0)}}|\Sigma (\omega =E_{{n{{\bf {q}}}}}^{{(0)}})|\phi _{{n{{\bf {q}}}}}^{{(0)}}\rangle , the exchange-correlation potential Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): \langle \phi _{{n{{\bf {q}}}}}^{{(0)}}|V_{{xc}}|\phi _{{n{{\bf {q}}}}}^{{(0)}}\rangle and the renormalization factor Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): Z_{{n{{\bf {q}}}}} defined above, respectively.
Partially self-consistent calculations: EVGW0
In most cases, the best results (i.e., closest to experiment) are obtained by iterating only Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): G via the spectral representation
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): G^{{(i)}}({{\bf {r}}},{{\bf {r}}}',\omega )=\sum _{{n{{\bf {k}}}}}{\frac {\phi _{{n{{\bf {k}}}}}^{{*(0)}}({{\bf {r}}})\phi _{{n{{\bf {k}}}}}^{{(0)}}({{\bf {r}}}')}{\omega -E_{{n{{\bf {k}}}}}^{{(i)}}}}
but keeping Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): W and the orbitals Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): \phi _{{n{{\bf {q}}}}}^{{(0)}} fixed to the initial DFT level. This method goes back to Hybertsen and Louie[5] and can be achieved in two ways.
If the spectral method is not selected (LSPECTRAL=.FALSE., requiring much more compute time), the quasi-particle (QP) shifts are iterated automatically four times, and one finds four sets of QP shifts in the OUTCAR file. The first one corresponds to the G0W0 case. The INCAR file is simply:
System = SiC NBANDS = 512 ISMEAR = 0 ; SIGMA = 0.05 ALGO = EVGW0 ! use "GW0" in VASP.5.X LSPECTRAL =.FALSE.
For technical reasons, it is not possible to iterate Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): G in this manner if LSPECTRAL=.TRUE. is set in the INCAR file (this is the default). In this case, an iteration number must be supplied in the INCAR file using the NELM-tag. Usually three to four iterations are sufficient to obtain accurate QP shifts.
System = SiC NBANDS = 512 ISMEAR = 0 ; SIGMA = 0.05 ALGO = EVGW0 ! use "GW0" in VASP.5.X NELM = 4
The results are found again in the OUTCAR file
QP shifts <psi_nk| G(iteration)W_0 |psi_nk>: iteration 4 k-point 1 : 0.0000 0.0000 0.0000 band No. old QP-enery QP-energies sigma(KS) T+V_ion+V_H V^pw_x(r,r') Z occupation Imag(sigma) 1 -8.6924 -8.7107 -14.2871 5.5647 -21.6681 0.6076 2.0000 1.1648 2 -3.4692 -3.4806 -15.6742 12.1894 -21.7437 0.7304 2.0000 0.6351 3 -3.4692 -3.4806 -15.6742 12.1894 -21.7437 0.7304 2.0000 0.6351 4 -3.4692 -3.4806 -15.6742 12.1894 -21.7437 0.7304 2.0000 0.6351 5 -0.6957 -0.7006 -13.6827 12.9802 -18.1531 0.7264 2.0000 0.2769 6 -0.6957 -0.7006 -13.6827 12.9802 -18.1531 0.7264 2.0000 0.2769 7 -0.6957 -0.7006 -13.6827 12.9802 -18.1531 0.7264 2.0000 0.2769
In contrast to single shot GW calculations, the second column represent now the QP-energies form previous iteration.
Partially self-consistent quasi-particle calculations: QPGW0
If non diagonal components of the self-energy (in the orbital basis) should be included use ALGO=QPGW0. The following setting can be used:
System = SiC NBANDS = 512 ISMEAR = 0 ; SIGMA = 0.05 ALGO = QPGW0 ! or "scGW0" for VASP.5.2.11 and older LOPTICS = .TRUE. ; LPEAD = .TRUE. ! ommit this lines for metals NELM = 4
In this case, the orbitals are updated as well by constructing a hermitian (energy independent) approximation to the self-energy.[3] The "static" COHSEX approximation can be selected by setting NOMEGA = 1.[6] To improve convergence to the groundstate, the charge density (and the charge density only) is mixed using a Kerker type mixing in VASP.5.3.2 and more recent versions (see IMIX). The mixing parameters AMIX, BMIX, AMIX_MAG, BMIX_MAG, AMIN can be adjusted, if convergence problems are encountered.
We strongly urge the user to monitor convergence by inspecting the lines
charge density residual
in the OUTCAR files.
Alternatively the mixing may be switched off by setting IMIX=0 and controlling the step width for the orbitals using the parameter TIME (which defaults to 0.4). This selects a fairly sophisticated damped MD algorithm, that is also used for DFT methods when ALGO=Damped. In general, this method is more reliable for metals and materials with strong charge sloshing.
After every iteration VASP writes following lines into the OUTCAR file
QP shifts <psi_nk| G(iteration)W_0 |psi_nk>: iteration 1 GWSYM: cpu time 15.8978: real time 15.9528 k-point 1 : 0.0000 0.0000 0.0000 band No. DFT-energies QP-energies QP-e(diag) sigma(DFT) Z occupation 1 -7.1626 -8.4217 -8.3038 -8.9978 0.6219 2.0000 2 -2.0899 -3.4394 -3.4347 -3.5765 0.9047 2.0000 3 -2.0899 -3.4394 -3.4347 -3.5765 0.9047 2.0000 4 -2.0899 -3.4394 -3.4347 -3.5765 0.9047 2.0000 5 0.4604 -0.4787 -0.4663 -0.7800 0.7471 2.0000 6 0.4604 -0.4787 -0.4663 -0.7800 0.7471 2.0000 7 0.4604 -0.4787 -0.4663 -0.7800 0.7471 2.0000 8 5.1013 4.1883 4.2149 3.9518 0.7711 2.0000
For the first iteration, here the forth column should be identical to the third column of the G0W0 results discussed above, while the third column reports the quasi-particle energies obtained from including the off-diagonal matrix elements in the eigenvalue equation.
Caveats
The QPGW0 (or scGW0 in VASP.5.2.11 and older) must be used with great caution, in particular, in combination with symmetry. Symmetry is handled in a rather sophisticated manner, specifically, only the minimal number of required combination of q and k points is considered. In this case, symmetry must be applied to restore the full star of q. This is done by determining degenerate eigenvalue/eigenvector pairs and restoring their symmetry according to their irreducible representation. Although the procedure is generally rather reliable, it fails to work properly if the degenerate states do not posses eigenvalues that are sufficiently close, due to insufficient convergence in the preceding DFT calculations. States are treated as degenerate if, and only if, their eigenenergies are within 0.01 eV.
For large supercells with low symmetry, we strongly recommend to switch off symmetry.
Self-consistent EVGW and QPGW calculations
Self-consistent QPGW calculations are only supported in a QP picture. As for QPGW0, it is possible to update the eigenvalues only (ALGO=EVGW or GW for VASP.5.X), or the eigenvalues and one-electron orbitals (ALGO=QPGW or scGW in VASP.5.2.11 and older). In all cases, a QP picture is maintained, i.e., satellite peaks (shake ups and shake downs) can not be accounted for in the self-consistency cycle. Self-consistent QPGW calculations can be performed by simply repeatedly calling VASP using:
System = SiC NBANDS = 512 ISMEAR = 0 ; SIGMA = 0.05 ALGO = EVGW ! "GW" in VASP.5.X, eigenvalues only or alternatively ALGO = QPGW ! "scGW" in VASP.5.2.11 and older, eigenvalues and one electron orbitals
For QPGW0 or QPGW non diagonal terms in the Hamiltonian are accounted for, e.g. the linearized QP equation is diagonalized, and the one electron orbitals are updated.[3] Alternatively (and preferably), the user can specify an electronic iteration counter using NELM:
System = SiC NBANDS = 512 ISMEAR = 0 ; SIGMA = 0.05 NELM = 3 ALGO = EVGW ! "GW" in VASP.5.X # or ALGO = QPGW ! "scGW" in VASP.5.2.11 and older
In this case, the one-electron energies (=QP energies) are updated 3 times (starting from the DFT eigenvalues) in both G and W. For ALGO=QPGW (or ALGO=scGW in VASP.5.2.11 and older), the one electron energies and one electron orbitals are updated 3 times.[3] As for ALGO = QPGW0 (or scGW0 in vasp.5.2.11 and older), the "static" COHSEX approximation can be selected by setting NOMEGA=1.[6]
To improve convergence to the groundstate, the charge density is mixed using a Kerker type mixing starting with VASP.5.3.2 (see IMIX). The mixing parameters AMIX, BMIX, AMIX_MAG, BMIX_MAG, AMIN can be adjusted, if convergence problems are encountered. Alternatively the mixing may be switched off by setting IMIX=0 and controlling the step width for the orbitals using the parameter TIME (which defaults to 0.4). This selects a fairly sophisticated damped MD algorithm, that is also used for DFT methods when ALGO=Damped. In general, this method is more reliable for metals and materials with strong charge sloshing.
Additional information about this method is found here.
Caveats
Fully self-consistent QPGW calculations with an update of the orbitals in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): G and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): W [3] require significant care and are prone to diverge (QPGW0 calculations are usually less critical). As discussed, above, one can select this mode using:
System = SiC NBANDS = 512 ISMEAR = 0 ; SIGMA = 0.05 ALGO = QPGW ! or "scGW" in VASP.5.2.11 and older, eigenvalues and one-electron orbitals NELM = number of steps
However, one caveat applies to this case: when the orbitals are updated, the derivatives of the orbitals with respect to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): k (stored in the WAVEDER file) will become incompatible with the orbitals. This can cause severe problems and convergence to the incorrect solution.
Warning: For metals, in general, we recommend to omit the LOPTICS tag and remove the WAVEDER file from the directory. |
For insulators, VASP (version 5.3.2 or higher) can update the WAVEDER file in each electronic iteration if the finite difference method is used to calculate the first derivative of the orbitals with respect to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): k :
System = SiC NBANDS = 512 ISMEAR = 0 ; SIGMA = 0.05 ALGO = QPGW ! "scGW" in VASP.5.2.11 and older, eigenvalues and one-electron orbitals NELM = 10 LOPTICS = .TRUE. ; LPEAD = .TRUE.
The combination LOPTICS=.TRUE.; LPEAD=.TRUE. is required since Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\frac {\delta H-\epsilon _{{n{{\bf {k}}}}}S}{\delta {k}_{i}}} is not available for GW like methods. LPEAD=.TRUE. circumvents this problems by calculating the derivatives of the orbitals using numerical differentiation on the finite k-point grid (this option is presently limited to insulators).
Vertex corrections are presently not documented. This is a feature still under construction, and we recommend to collaborate with the Vienna group if you are desperately in need of that feature.
Low scaling GW algorithms
The GW implementations in VASP described in the papers of Shishkin et al.[1][2] avoids storage of the Green's function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): \chi and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): \Sigma in reciprocal space.
As of VASP.6 a new cubic scaling GW algorithm[7] (called space-time implementation in the following) can be selected. This approach follows the idea of Rojas et al.[8] and performs the GW self-consistency cycle on imaginary time Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): t\to i\tau and imaginary frequency axes Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): \omega \to i\omega .
Low scaling, single shot GW calculations: G0W0R
The low-scaling analogue of G0W0 is selected with ALGO=G0W0R. In contrast to the single-shot GW calculations on the real-axes, here the self-energy Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\displaystyle \Sigma =G_{0}W_{0}} is determined on the imaginary frequency axis. To this end, the overall scaling is reduced by one order of magnitude and is cubic with respect to the system size, because a small value for NOMEGA can be used (usually <20).
This algorithm evaluates:
- Single-shot GW quasi-particle energies (from an analytical continuation of the self-energy to the real axis)[7]
- Natural orbitals from the first order change of the density matrix (i.e. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\displaystyle G_{0}\Sigma G_{0}} ), see the NATURALO tag for more information.[9]
N.B. This selection ignores NELM.
Following INCAR file selects the low-scaling GW algorithm:
System = SiC ISMEAR = 0 ; SIGMA = 0.05 LOPTICS = .TRUE. ALGO = G0W0R NOMEGA = 12 ! small number of frequencies necessary
Search the OUTCAR file for following lines
QP shifts evaluated in KS or natural orbital/ Bruckner basis k-point 1 : 0.0000 0.0000 0.0000 band No. KS-energies sigma(KS) QP-e(linear) Z QP-e(zeros) Z occupation Imag(E_QP) QP_DIFF TAG 1 -7.1627 -8.6732 -8.2451 0.7166 -8.2346 0.7026 2.0000 -1.3101 0.0000 2 2 -2.0901 -3.4155 -3.0350 0.7129 -3.0272 0.7011 2.0000 -0.5582 -0.0000 2 3 -2.0901 -3.4155 -3.0350 0.7129 -3.0272 0.7011 2.0000 -0.5582 0.0000 2 4 -2.0901 -3.4155 -3.0350 0.7129 -3.0272 0.7011 2.0000 -0.5582 -0.0000 2 5 0.4603 -0.8219 -0.4904 0.7414 -0.4814 0.7273 2.0000 -0.1902 0.0000 2 6 0.4603 -0.8219 -0.4904 0.7414 -0.4814 0.7273 2.0000 -0.1902 -0.0000 2
Here column four is obtained by a linearization of the self-energy around the Kohn-Sham energies (second column) and can be compared to the third column of single-shot GW calculations on the real axis. Column six represents an other set of QP-energies that is obtained from the roots of following equation
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\displaystyle \langle \phi _{n{\bf {q}}}^{(0)}|T+V_{ext}+V_{h}+\Sigma (\omega )|\phi _{n{\bf {q}}}^{(0)}\rangle -\omega =0}
These roots represent the poles of the Green's function in the spectral representation.
Partially self-consistent GW caluclations: GW0R
The space-time implementation allows for true self-consistent GW calculations. That is, the solution of the Dyson equation for the Green's function can be obtained with a modest computational effort. The main procedure of a self-consistent GW calculation consists of four main steps
- Obtain Green's function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): G_{{{\bf {GG'}}}}^{{(j)}}({{\bf {q}}},i\omega _{n})=\left[\delta _{{{\bf {GG'}}}}-\Sigma _{{{\bf {GG'}}}}^{{(j-1)}}({{\bf {q}}},i\omega _{n})G_{{{\bf {GG'}}}}^{{(j-1)}}({{\bf {q}}},i\omega _{n})\right]^{{-1}}G_{{{\bf {GG'}}}}^{{(j-1)}}({{\bf {q}}},i\omega _{n})
- Compute irreducible polarizability Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): \chi ^{{(j)}}({{\bf {r}}},{{\bf {r}}}',i\tau _{m})=-G^{{(j)}}({{\bf {r}}},{{\bf {r}}}',i\tau _{m})G^{{(j)}}({{\bf {r}}}',{{\bf {r}}},-i\tau _{m})
- Determine screened potential Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): W_{{{{\bf {G}}}{{\bf {G}}}'}}^{{(j)}}({{\bf {q}}},\omega _{m})=\left[\delta _{{{{\bf {G}}}{{\bf {G}}}'}}-\chi _{{{{\bf {G}}}{{\bf {G}}}'}}^{{(j)}}({{\bf {q}}},\omega _{m})V_{{{{\bf {G}}}{{\bf {G}}}'}}({{\bf {q}}})\right]^{{-1}}V_{{{{\bf {G}}}{{\bf {G}}}'}}({{\bf {q}}})
- Calculate GW self-energy Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): \Sigma ^{{(j)}}({{\bf {r}}},{{\bf {r}}}',i\tau _{m})=-G^{{(j)}}({{\bf {r}}},{{\bf {r}}}',i\tau _{m})W^{{(j)}}({{\bf {r}}}',{{\bf {r}}},i\tau _{m})
This procedure can be selected with the following INCAR settings
System = SiC ISMEAR = 0 ; SIGMA = 0.05 LOPTICS = .TRUE. ; LPEAD = .TRUE. NELM = number of iterations wanted ALGO = GW0R ! ALGO = scGW0R has the same effect here, that is self-consistency in G, no update in W
The number of self-consistency steps can be set with the NELM tag.
Due to efficiency, VASP performs each step in the Hartree-Fock basis. This is the reason why there are two sets of QP-energies found after the first iteration (one for the QP-energies in the KS-basis and one for the QP energies in the HF basis) After the second iteration only the QP energies obtained in the HF basis are printed and a similar output as follows is found in the OUTCAR file
QP shifts evaluated in HF basis k-point 1 : 0.0000 0.0000 0.0000 band No. KS-energies sigma(KS) QP-e(linear) Z QP-e(zeros) Z occupation Imag(E_QP) QP_DIFF TAG 1 -7.1626 -8.6510 -8.2275 0.7154 -8.2173 0.7017 2.0000 -1.3177 0.0000 2 2 -2.0899 -3.4157 -3.0348 0.7127 -3.0269 0.7008 2.0000 -0.5614 0.0000 2 3 -2.0899 -3.4157 -3.0348 0.7127 -3.0269 0.7008 2.0000 -0.5614 -0.0000 2 4 -2.0899 -3.4157 -3.0348 0.7127 -3.0269 0.7008 2.0000 -0.5614 0.0000 2 5 0.4604 -0.8170 -0.4857 0.7407 -0.4768 0.7266 2.0000 -0.1945 0.0000 2 6 0.4604 -0.8170 -0.4857 0.7407 -0.4768 0.7266 2.0000 -0.1945 -0.0000 2 7 0.4604 -0.8170 -0.4857 0.7407 -0.4768 0.7266 2.0000 -0.1945 0.0000 2 8 5.1013 4.0069 4.2594 0.7693 4.2645 0.7598 2.0000 -0.0602 0.0000 2
Here the meaning of each column is the same as for the other low-scaling GW algorithms.
Fully self-consistent GW caluclations: GWR
If the screened potential should be updated during the self-consistency circle[10] the following INCAR file can be used
System = SiC ISMEAR = 0 ; SIGMA = 0.05 LOPTICS = .TRUE. ; LPEAD = .TRUE. NELM = number of iterations wanted ALGO = GWR ! ALGO = scGWR has the same effect here, that is self-consistency in G and W
Caveats
Using this option, similar caveats can be expected as for ALGO=EVGW and QPGW calculations and we recommend to leave out the LOPTICS and LPEAD line for metals. Memory requirements of low-scaling GW and RPA algorithms
Related Tags and Sections
- ALGO for response functions and GW calculations
- LOPTICS, derivative of wavefunction w.r.t. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): k
- LPEAD, derivative of wavefunction with finite differences
- LMAXFOCKAE overlap densities and multipoles
- MAXMEM, memory available to one mpi rank on each node
- 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
- NATURALO, natural orbitals
References
- ↑ a b M. Shishkin and G. Kresse, Phys. Rev. B 74, 035101 (2006).
- ↑ a b M. Shishkin and G. Kresse, Phys. Rev. B 75, 235102 (2007).
- ↑ a b c d e M. Shishkin, M. Marsman, and G. Kresse, Phys. Rev. Lett. 99, 246403 (2007).
- ↑ F. Fuchs, J. Furthmüller, F. Bechstedt, M. Shishkin, and G. Kresse, Phys. Rev. B 76, 115109 (2007).
- ↑ M. S. Hybertsen, S. G. Louie Phys. Ref. B 34, 5390 (1986)
- ↑ a b F. Bruneval, N. Vast, and L. Reining, Phys. Rev. B 74, 45102 (2006).
- ↑ a b P. Liu, M. Kaltak, J. Klimes and G. Kresse, Phys. Rev. B 94, 165109 (2016).
- ↑ H. N. Rojas, R. W. Godby, R. J. Needs, Phys. Rev. Lett. 74, 1827 (1995)
- ↑ B. Ramberger, Z. Surkuma, T. Schäfer, G. Kresse, J. Chem. Phys. 151, 214106 (2019).
- ↑ M. Grumet, P. Liu, M. Kaltak, J. Klimeš and Georg Kresse, Phys. Ref. B 98, 155143 (2018).