# QSGW Tutorial

This tutorial begins with an LDA calculation for Si, starting from an init file. Following this is a demonstration of a quasi-particle self-consistent GW (QSGW) calculation. An example of the 1-shot GW code is provided in a separate tutorial. Click on the dropdown menu below for a brief description of the QSGW scheme. A complete summary of the commands used throughout is provided in a separate dropdown menu. Theory for GW and QSGW, and its implementation in the Questaal suite, can be found in Phys. Rev. B76, 165106 (2007).

Each iteration of a QSGW calculation has two main parts: a section that uses effective one-body hamiltonians to make the density *n* (as in DFT), and the *GW* code that makes the self-energy Σ(*ω*) of an interacting hamiltonian. For quaisparticle self-consistency, the *GW* code makes a “quasiparticlized” self-energy Σ^{0}, which is used to construct the effective one-body hamiltonian for the next cycle. The process is iterated until the change in Σ^{0} becomes small.

The one-body executable is **lmf**. The script **lmfgwd** is similar to **lmf**, but it is a driver whose purpose is to set up inputs for the *GW* code. Σ^{0} is made by a shell script **lmgw**. The entire cycle is managed by a shell script **lmgwsc**.

Before any self-energy Σ^{0} exists, it is assumed to be zero. Thus the one-body hamiltonian is usually the LDA, though it can be something else, e.g. LDA+U.

*Note:* in some circumstances, e.g. to break time reversal symmetry inherent in the LDA, you need to start with LDA+U.

Thus, there are two self-energies and two corresponding Green’s functions: the interacting *G*[Σ(*ω*)] and non-interacting *G*^{0}[Σ^{0}]. At self-consistency the poles of *G* and *G*^{0} coincide: this is a unique and very advantageous feature of QSGW. It means that there is no “mass renormalization” of the bandwidth — at least at the *GW* level.

Usually the interacting Σ(*ω*) isn’t made explicitly, but you can do so, as explained in this tutorial.

In short, a QSGW calculation consists of the following steps. The starting point is a self-consistent DFT calculation (usually LDA). The DFT eigenfunctions and eigenvalues are used by the GW code to construct a self-energy Σ^{0}. This is called the “0^{th} iteration.” If only the diagonal parts of Σ^{0} are kept, the “0^{th}” iteration corresponds to what is sometimes called *GW*, or as *G*^{LDA}W^{LDA}.

In the next iteration, Σ^{0}−*V*_{xc}^{LDA} is added to the LDA hamiltonian. The density is made self-consistent, and control is handed over to the *GW* part. (Note that for a fixed density *V*_{xc}^{LDA} cancels the exchange-correlation potential from the LDA hamiltonian.) This process is repeated until the RMS change in Σ^{0} falls below a certain tolerance value. The final self-energy (QSGW potential) can be thought of as an effective exchange-correlation functional that is tailored to the system. This is very convenient as it can now be used in an analogous way to standard DFT to calculate properties such as the band structure.

```
$ nano init.si #create init file using lines from box below
$ blm init.si --express --gmax=5 --nk=4 --nit=20 --gw #use blm tool to create actrl and site files
$ cp actrl.si ctrl.si #copy actrl to recognised ctrl prefix
$ lmfa ctrl.si; cp basp0.si basp.si #run lmfa and copy basp file
$ lmf ctrl.si > out.lmfsc #make self-consistent
$ echo -1 | lmfgwd si #make GWinput file
$ vim GWinput #change GW k mesh to 3x3x3
$ lmgwsc --wt --insul=4 --tol=2e-5 --maxit=5 si #self-consistent GW calculation
$ vim ctrl.si #change number of iterations to 1
$ lmf si --rs=1,0 #lmf with QSGW potential to get QSGW band gap
$ lmgwclear #clean up directory
```

### LDA calculation

The starting point is a self-consistent LDA density, you may want to review the DFT tutorial for silicon. Copy the following lines to a file called *init.si*:

```
LATTICE
ALAT=10.26
PLAT= 0.00000000 0.50000000 0.50000000
0.50000000 0.00000000 0.50000000
0.50000000 0.50000000 0.00000000
# pos means cartesian coordinates, units of alat
SITE
ATOM=Si POS= 0.00000000 0.00000000 0.00000000
ATOM=Si POS= 0.25000000 0.25000000 0.25000000
```

Run the following commands to obtain a self-consistent density:

```
$ blm init.si --express --gmax=5 --nk=4 --nit=20 --gw
$ cp actrl.si ctrl.si && lmfa ctrl.si && cp basp0.si basp.si
$ lmf ctrl.si > out.lmfsc
```

Note that we have included an extra --gw switch, which tailors the *ctrl* file for a *GW* calculation. To see how it affects the *ctrl* file, try running blm without --gw. The switch modifies the basis set section (see the **autobas** line) to increase the size of the basis, which is necessary for *GW* calculations.

Two new blocks of text, the **HAM** and **GW** categories, are also added towards the end of the *ctrl* file. The extra parameters in the **HAM** category handle the inclusion of a self-energy, actually a GW potential (see theory notes above), in a DFT calculation. The **GW** category provides some default values for parameters that are required in the *GW* calculation. The *GW* code has its own input file and the DFT *ctrl* file influences what defaults are set in it, we will come back to this later. One thing to note is the **NKABC=** token, which defines the *GW* k-point mesh. It is specified in the same way as the lower case **nkabc** for the LDA calculation.

Now check the output file *out.lmfsc*. The self-consistent gap is reported to be around 0.58 eV as can be seen by searching for the last occurence of the word ‘**gap**’. Note that this result differs slightly to that from the LDA tutorial because the **gw** switch increases the size of the basis set.

Now that we have the input eigenfunctions and eigenvalues, the next step is to carry out a *GW* calculation. For this, we need an input file for the *GW* code.

### Making GWinput

The *GW* package (both one-shot and QSGW) uses one main user-supplied input file, *GWinput*. The script lmfgwd can create a template *GWinput* file for you by running the following command:

```
$ echo -1 | lmfgwd si #make GWinput file
```

The **lmfgwd** script has multiple options and is designed to run interactively. Using ‘**echo -1**’ automatically passes it the ‘**-1**’ option that specifies making a template input file. You can try running it interactively by just using the command ‘**lmfgwd si**’ and then entering ‘**-1**’. Take a look at *GWinput*, it is a rather complicated input file but we will only consider the *GW* k-point mesh for now (further information can be found on the *GWinput* page). The k mesh is specified by **n1n2n3** in the *GWinput* file, look for the following line:

```
$ n1n2n3 4 4 4 ! for GW BZ mesh
```

When creating the *GWinput* file, **lmfgwd** checks the GW section of the *ctrl* file for default values. The ‘**NKABC= 4**’ part of the DFT input file (*ctrl.si*) is read by **lmfgwd** and used for **n1n2n3** in the GW input file. Remember if only one number is supplied in **NKABC** then that number is used as the division in each direction of the reciprocal lattice vectors, so 4 alone means a $4\times4\times4$ k mesh. To make things run a bit quicker, change the k mesh to $3\times3\times3$ by editing the *GWinput* file line:

```
$ n1n2n3 3 3 3 ! for GW BZ mesh
```

The k mesh of 3 x 3 x 3 divisions is rough, but it makes the calculation fast and for Si the results are reasonable.

### Running QSGW

We are now ready for a QSGW calculation, this is run using the shell script **lmgwsc**:

```
$ lmgwsc --wt --insul=4 --tol=2e-5 --maxit=0 si #zeroth iteration of QSGW calculation
```

The switch ‘**–wt**’ includes additional timing information in the printed output, **insul** refers to the number of occupied bands (normally spin degenerate so half the number of electrons), **tol** is the tolerance for the RMS change in the self-energy between iterations and **maxit** is the maximum number of QSGW iterations. Note that **maxit** is zero, this specifies that a single iteration is to be carried out starting from DFT with no self-energy (zeroth iteration).

Take a look at the line containing the file name *llmf*:

```
lmgw 15:26:47 : invoking mpix -np=8 /h/ms4/bin/lmf-MPIK --no-iactive cspi >llmf
```

Each QSGW iteration begins with a self-consistent DFT calculation by calling the program **lmf** and writing the output to the file *llmf*. We are starting from a self-consitent LDA density (we already ran **lmf** above) so this step is not actually necessary here. The next few lines are preparatory steps. The main *GW* calculation begins on the line containing the file name ‘**lbasC**’:

```
lmgw 16:27:55 : invoking /h/ms4/bin/code2/hbasfp0 --job=3 >lbasC
lmgw 16:27:55 : invoking /h/ms4/bin/code2/hvccfp0 --job=0 >lvccC ... 0.0m (0.0h)
lmgw 16:27:58 : invoking /h/ms4/bin/code2/hsfp0_sc --job=3 >lsxC ... 0.0m (0.0h)
lmgw 16:27:59 : invoking /h/ms4/bin/code2/hbasfp0 --job=0 >lbas
lmgw 16:27:59 : invoking /h/ms4/bin/code2/hvccfp0 --job=0 >lvcc ... 0.0m (0.0h)
lmgw 16:28:02 : invoking /h/ms4/bin/code2/hsfp0_sc --job=1 >lsx ... 0.0m (0.0h)
lmgw 16:28:02 : invoking /h/ms4/bin/code2/hx0fp0_sc --job=11 >lx0 ... 0.1m (0.0h)
lmgw 16:28:07 : invoking /h/ms4/bin/code2/hsfp0_sc --job=2 >lsc ... 0.1m (0.0h)
lmgw 16:28:13 : invoking /h/ms4/bin/code2/hqpe_sc 4 >lqpe
```

The three lines with **lbasC**, **lvccC** and **lsxC** are the steps that calculate the core contributions to the self-energy and the following lines up to the one with **lsc** are for the valence contribution to the self-energy. The **lsc** step, calculating the correlation part of the self-energy, is usually the most expensive step. The self-energy is converted into an effective exchange-correlation potential in the **lqpe** step and the final few lines are to do with its handling. A full account of the *GW* steps can be found here in the annotated output file.

Run the command again but this time set the number of iterations (**maxit**) to something like 5:

```
$ lmgwsc --wt --insul=4 --tol=2e-5 --maxit=5 si #self-consistent GW calculation
```

This time the iteration count starts from 1 since we are now starting with a self-energy from the zeroth iteration. Again, the iteration starts with a self-consistent DFT calculation but this time the zeroth iteration *GW* potential is used. The following line in the *llmf* file specifies that the *GW* potential is being used:

```
RDSIGM: read file sigm and create COMPLEX sigma(R) by FT ...
```

The *GW* potential is contained in the file *sigm*, **lmgwsc** also makes a soft link *sigm.si* so **lmf** can read it. The *GW* potential is automatically used if present, this is specified by the RDSIG variable in the *ctrl* file. Take a look at the *GW* output again and you can see that the rest of the steps are the same as before. After 3 iterations the RMS change in the self-energy is below the tolerance - the calculation is converged.

```
lmgwsc : iter 3 of 5 RMS change in sigma = 1.19E-05 Tolerance = 2e-5
```

Now that we have a converged self-energy (sigm) we can go back to using **lmf** to calculate additional properties. We only want to run a single iteration so change the number of iterations (nit) to 1 in the *ctrl* file. Run the following command:

```
$ lmf si --rs=1,0 #lmf with QSGW potential to get QSGW band gap
```

Inspect the **lmf** output and you can find that the gap is now around 1.41 eV. The **--rs** option is explained here.

Check your directory and you will see that a large number of files were created. The following command removes many redundant files:

```
$ lmgwclear #clean up directory
```

Further details can be found in the Additional exercises below.

### Additional exercises

- Correct gap: This is actually the Γ-X gap; the true gap is around 1.31 eV as can be seen by running
**lmf**with a fine k mesh. - Changing k mesh: Test the convergence with respect to the
*GW**k*mesh by increasing to a $4\times4\times4$ k mesh. - Adding floating orbitals: Note that the basis set for this calculation isn’t quite converged. For Si this is not much of an issue but it can matter a bit for other materials (making errors of order 0.1 eV). The atom-centered LMTO basis set is sufficient for LDA calculations, but it is not quite adequate for
*GW*(work is in progress for a next-generation basis which should address this limitation). To make the basis complete you should add floating orbitals (you cannot add APWs in the QSGW context because the self-energy interpolator does not work with delocalized orbitals). Floating orbitals are like the empty spheres often required by the ASA, but they have no augmentation radius. You can automatically locate them using lmchk (the same way the empty sphere locator works for the ASA).

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