27: Silicon — Selected columns of density matrix algorithm for automated MLWFs¶

Note: This tutorial requires a recent version of the pw2wannier90.x post-processing code of Quantum ESPRESSO (v6.4 or above).

Outline: For bulk crystalline Silicon, generate the \(A_{mn}\) matrices via the selected columns of density matrix (SCDM) algorithm and the corresponding MLWFs for 1) Valence bands 2) Valence bands and 4 low-lying conduction bands 3) Conduction bands only. To better understand the input files and the results of these calculations, it is crucial that the Reader has familiarized with the concepts and methods explained in Ref. ¹. More info on the keywords related to the SCDM method may be found in the user_guide.
Directory: tutorials/tutorial27/ Files can be downloaded from here
Input Files: input_files, and in the three subfolders isolated, erfc and gaussian. The input_files folder contains:
- si.scf The pwscf input file for the ground state calculation
- si_4bands.nscf The pwscf input file to obtain Bloch states on a uniform grid for 4 bands.
- si_12bands.nscf The pwscf input file to obtain Bloch states on a uniform grid for 12 bands.
Whereas the three subfolders isolated, erfc and gaussian contain the si.win wannier90 input files and si.pw2wan pw2wannier90 input files each corresponding to one of the scenarios listed in the outline.

Valence bands¶

In this case we will compute 4 localized WFs corresponding to the 4 valence bands of Silicon. These 4 bands constitute a manifold that is separated in energy from other bands. In this case the columns of the density matrix are already localized in real space and no extra parameter is required.

Copy the input files si.scf and si_4bands.nscf from the input_files directory into the isolated folder
Run pwscf to obtain the ground state charge of bulk Silicon.
Terminal
```
pw.x < si.scf > scf.out
```
Run pwscf to obtain the Bloch states on a uniform k-point grid of 4x4x4 for 4 bands.
Terminal
```
pw.x < si_4bands.nscf > nscf.out
```
Inspect the si.win input file and make sure that the auto_projections flag is set to .true.. Also, make sure that no projections block is present.
Run wannier90 to generate a list of the required overlaps and also info on the SCDM method (written into the si.nnkp file).
Terminal
```
wannier90.x -pp si
```
Inspect the si.nnkp file and make sure you find the auto_projections block and that no projections have been written in the projections block.
Inspect the .pw2wan input file. You will find two new keywords, i.e. scdm_proj and scdm_entanglement. The former, will instruct pw2wannier90.x to use the SCDM method when generating the \(A_{mn}\) matrix. The latter, defines which formula to adopt for the function \(f(\varepsilon_{n\mathbf{k}})\) (see ¹ and point below).
Run pw2wannier90 to compute the overlap between Bloch states and the projections via the SCDM method (written in the si.mmn and si.amn respectively).
Terminal
```
pw2wannier90.x < si.pw2wan > pw2wan.out
```
Run wannier90 to compute the MLWFs.
Terminal
```
wannier90.x si
```

At this point, you should have obtained 4 Wannier functions and the interpolated valence bands for Silicon. Inspect the output file si.wout. In particular, look at the geometric centres of each WF, do they lie at the centre of the Si-Si bond as for the MLWFs computed from user-defined initial \(s\)-like projections (see Tutorial 11)? Plot these WFs using Vesta. Do they show the \(\sigma\) character one would expect from chemical arguments?

Valence bands + conduction bands¶

In this case we will compute 8 localized WFs corresponding to the 4 valence bands and 4 low-lying conduction bands. Here, we don't have a separate manifold, since the conduction bands are entangled with other high-energy bands and the columns of the density matrix are not exponentially localized by construction. A modified density matrix is required in this case¹, and it is defined as:

\[ P(\mathbf{r},\mathbf{r}') = \sum_{n,\mathbf{k}} \psi_{n\mathbf{k}}(\mathbf{r})f(\varepsilon_{n,\mathbf{k}})\psi_{n\mathbf{k}}^\ast(\mathbf{r}'), \]

where \(\psi_{n\mathbf{k}}\) and \(\varepsilon_{n,\mathbf{k}}\) are the energy eigestates and eigenvalues from the first-principle calculation respectively. The function \(f(\varepsilon_{n,\mathbf{k}})\) contains two free parameters \(\mu\) and \(\sigma\) and is defined as a complementary error function:

\[ f(\varepsilon_{n,\mathbf{k}}) = \frac{1}{2}\mathrm{erfc} \left(\frac{\varepsilon_{n,\mathbf{k}} - \mu}{\sigma}\right). \]

Copy the input files si.scf and si_12bands.nscf from the input_files folder into the erfc folder
Run pwscf to obtain the ground state charge of bulk Silicon.
Terminal
```
pw.x < si.scf > scf.out
```
Run pwscf to obtain the Bloch states on a uniform k-point grid of 4x4x4 for 12 bands this time.
Terminal
```
pw.x < si_12bands.nscf > nscf.out
```
Inspect the si.win input file and make sure that the auto_projections flag is set to .true.. Also, make sure that no projection block is present.
Run wannier90 to generate a list of the required overlaps and also info on the SCDM method (written into the si.nnkp file).
Terminal
```
wannier90.x -pp si
```
Inspect the si.nnkp file and make sure you find the auto_projections block and that no projections have been written in the projections block.
Inspect the .pw2wan input file. You will find other two new keywords, i.e. scdm_mu and scdm_sigma. These are the values in eV of \(\mu\) and \(\sigma\) in \(f(\varepsilon_{n,\mathbf{k}})\), respectively.
Run pw2wannier90 to compute the overlap between Bloch states and the projections via the SCDM method (written in the si.mmn and si.amn respectively).
Terminal
```
pw2wannier90.x < si.pw2wan > pw2wan.out
```
Run wannier90 to compute the MLWFs.
Terminal
```
wannier90.x si
```

At this point, you should have obtained 8 localized Wannier functions and the interpolated valence and conduction bands for Silicon. Again, compare the results for the geometric centres and the individual spreads with the ones from Tutorial 11. Is the final value of total spread bigger or smaller than the one from Tutorial 11? Look at the WFs with Vesta. Can you explain what you see? Where do the major lobes of the \(sp3\)-like WFs point in this case?

Conduction bands only¶

In this case we will compute 4 localized WFs corresponding to the 4 low-lying conduction bands only. As for the previous point, we need to define a modified density matrix¹. Since we are only interested in a subset of the conduction states, within a bounded energy region, a good choice for \(f(\varepsilon_{n,\mathbf{k}})\) is:

\[ f(\varepsilon_{n,\mathbf{k}}) = \exp\left(-\frac{(\varepsilon_{n,\mathbf{k}} - \mu)^2}{\sigma^2}\right). \]

Copy the input files si.scf and si_12bands.nscf from the input_files directory into the gaussian folder
Run pwscf to obtain the ground state charge of bulk Silicon.
Terminal
```
pw.x < si.scf > scf.out
```
Run pwscf to obtain the Bloch states on a uniform k-point grid of 4x4x4 for 12 bands this time.
Terminal
```
pw.x < si_12bands.nscf > nscf.out
```
Inspect the si.win input file and make sure that the auto_projections flag is set to .true.. Also, make sure that no projections block is present.
Run wannier90 to generate a list of the required overlaps and also info on the SCDM method (written into the si.nnkp file).
Terminal
```
wannier90.x -pp si
```
Inspect the si.nnkp file and make sure you find the auto_projections block and that no projections have been written in the projections block.
Run pw2wannier90 to compute the overlap between Bloch states, the projections for the starting guess via the SCDM method (written in the si.mmn and si.amn respectively).
Terminal
```
pw2wannier90.x < si.pw2wan > pw2wan.out
```
Run wannier90 to compute the MLWFs.
Terminal
```
wannier90.x si
```

At this point, you should have obtained 4 localized Wannier functions and the interpolated conduction bands for Silicon. From chemical intuition, we would expect these functions to be similar to anti-bonding orbitals of molecules with tetrahedral symmetry. Plot the WFs and check if this is confirmed.

A. Damle and L. Lin. Disentanglement via entanglement: A unified method for Wannier localization. ArXiv e-prints, March 2017. URL: http://adsabs.harvard.edu/abs/2017arXiv170306958D, arXiv:1703.06958. ↩↩↩↩