Improving the dielectric function

Overview > Dielectric properties of Si using BSE > Improving the dielectric function > Plotting the BSE fatband structure of Si > List of tutorials

Task

Calculate the dielectric function of Si using an averaging over multiple grids or a model-BSE to improve k-sampling in BSE calculations.

Input

Si
 5.4300
0.5 0.5 0.0
0.0 0.5 0.5
0.5 0.0 0.5
2
cart
0.00 0.00 0.00 
0.25 0.25 0.25

INCAR

This is the INCAR file for the basic DFT calculation:

System  = Si
PREC = Normal ; ENCUT = 250.0
ISMEAR = 0 ; SIGMA = 0.01
KPAR = 2
EDIFF = 1.E-8

KPOINTS

Automatic
 0
Gamma
 6 6 6 
 0 0 0

Calculation

The calculated spectra can be improved in two ways:

Averaging over multiple grids:

Compute N independent dielectric functions using BSE (or any other method), using shifted grids of k-points, and take the average over the results.

Model-BSE:

Use a parametrized model^[1] for the dielectric screening, and DFT eigenenergies moved with a scissor operator, instead of RPA screening and GW quasiparticle energies.

Averaging over multiple grids

Construct shifted k-point grid with the same density.
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): n\times n\times n k-point grid Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): \rightarrow Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): X_{{n}} irreducible k-points Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {K_{{n}}} with weights Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): W_{{n}} . We do Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): x_{{n}} calculations on a Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): m\times m\times m grid, shifted of Gamma by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): K_{{n}} .
Extract the dielectric function of each calculation and average over them with respect to the weights Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): W_{{n}} :

We have now effectively constructed the result for a Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): (n\times m)\times (n\times m)\times (n\times m) grid. But interactions of range longer than Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): m times the supercell size have been ignored.

In our example we use Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): n=4 and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): m=4 : Effectively we use Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): 16\times 16\times 16 k-points.

In the script doall-average.sh the scheme is written for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): n=4 and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): n={\textrm {\$NKPT}} . At the end the dielectric functions are extracted and averaged accordingly. You can choose up to which level of theory (DFT, RPA, BSE) the dielectric function is computed by commenting out the corresponding lines in the script (default is all the way up to BSE).
Because of the shifted grids we have to use density functional perturbation theory to calculate the derivatives of the wave functions 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://wikimedia.org/api/rest_v1/":): {\mathbf {k}} and not the finite difference scheme. We also have to switch off all k-points symmetry in all INCAR files. These two important parameters look like the following in the INCAR file:

PREC = Normal ; ENCUT = 250.0
 
ALGO = EXACT ; NELM = 1
ISMEAR = 0 ; SIGMA = 0.01
KPAR = 2

NBANDS = 128  
LOPTICS = .TRUE.; LPEAD = .FALSE.
ISYM = -1
OMEGAMAX = 40

Finally the averaging over multiple grids should should give spectra that are in much closer agreement than the calculations using Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): 4\times 4\times 4 k-points:

Model-BSE

The dielectric function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): \epsilon _{{{\mathbf {G}},{\mathbf {G'}}}}^{{-1}}({\mathbf {k}}) is replaced by the local model function:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): \epsilon ^{{-1}}({\mathbf {k}}+{\mathbf {G}})=1-(1-\epsilon _{{\infty }}^{{-1}})\exp \left({\frac {-(2\pi |{\mathbf {k}}+{\mathbf {G}}|)^{{2}}}{4\lambda ^{{2}}}}\right) .

This makes the screened Coulomb kernel diagonal Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): ({\mathbf {G}}={\mathbf {G'}}) in the screened Coulomb potential:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): W_{{c'v'{\mathbf {k}}}}^{{cv{\mathbf {k}}}}={\frac {4\pi e^{{2}}}{\Omega }}\sum \limits _{{{\mathbf {G}}}}B_{{c'{\mathbf {k}}}}^{{c{\mathbf {k}}}}({\mathbf {G}}){\frac {\epsilon ^{{-1}}({\mathbf {k}}+{\mathbf {G}})}{|{\mathbf {k}}+{\mathbf {G}}|^{{2}}}}B_{{v'{\mathbf {k}}}}^{{v{\mathbf {k}}}}({\mathbf {G}}) ,

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): B_{{n'{\mathbf {k}}}}^{{n{\mathbf {k}}}}({\mathbf {G}}) denote Bloch integrals of the cell-periodic part of the Bloch waves.

In addition to a model dielectric function we need approximate quasiparticle energies and wave functions.

Approximation:

Use DFT single particle eigenvalues + SCISSOR (SCISSOR=GW band gap - DFT band gap).
Use DFT single particle orbitals.

Extract Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\mathbf {G}}={\mathbf {G'}} dielectric function from the vasprun.xml file from the previous GW calculation using the script extract_optics.sh or view the file dieG_g6x6x6-GW0.dat. Use AEXX=0.088 for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): \epsilon _{{\infty }}^{{-1}} and HFSCREEN=1.26 for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): \lambda .Then fit the model to get:

Check the GW+BSE and DFT+mBSE calculations for constistency:

The sequence of calculations as given in the script doall-model.sh consists of two steps:

Step 1: standard DFT calculation. The INCAR file (INCAR.DFT) for this step looks as follows:

PREC = Normal ; ENCUT = 250.0
ISMEAR = 0 ; SIGMA = 0.01
EDIFF = 1.E-8
NBANDS = 16
PRECFOCK = Normal
 
#WAVEDER file must be made:
LOPTICS = .TRUE.
LPEAD = .TRUE.
OMEGAMAX = 40

Step2: model BSE calculation. The INCAR file (INCAR.mBSE) for this step looks as follows:

PREC = Normal ; ENCUT = 250.0
 
ALGO = TDHF
ANTIRES = 0 ; SIGMA = 0.01
ENCUTGW = 150
 
EDIFF = 1.E-8
NBANDS = 16
NBANDSO = 4
NBANDSV = 8
OMEGAMAX = 20

PRECFOCK = Normal

LMODELHF = .TRUE.
HFSCREEN = 1.26
AEXX = 0.088
SCISSOR = 0.69