# DFT+DMFT tutorial: Ce with Hubbard-I approximation

In this tutorial we will perform DFT+DMFT Wien2k
calculations from scratch, including all steps described in the
previous sections. As example, we take the high-temperature
\(\gamma\)-phase of Ce employing the Hubbard-I approximation for
its localized *4f* shell.

## Wien2k setup

First we create the Wien2k `Ce-gamma.struct`

file as described in the
Wien2k manual
for the \(\gamma\)-Ce fcc structure with lattice parameter of 9.75 a.u.

```
Title
F LATTICE,NONEQUIV.ATOMS: 1225_Fm-3m
MODE OF CALC=RELA unit=bohr
9.750000 9.750000 9.750000 90.000000 90.000000 90.000000
ATOM 1: X=0.00000000 Y=0.00000000 Z=0.00000000
MULT= 1 ISPLIT= 2
Ce NPT= 781 R0=0.00001000 RMT= 2.5000 Z: 58.0
LOCAL ROT MATRIX: 1.0000000 0.0000000 0.0000000
0.0000000 1.0000000 0.0000000
0.0000000 0.0000000 1.0000000
0 NUMBER OF SYMMETRY OPERATIONS
```

We initalize non-magnetic Wien2k calculations using the **init** script as
described in the same manual. For this example we specify 3000 \(\mathbf{k}\)-points
in the full Brillouin zone and LDA exchange-correlation potential (*vxc=5*), other
parameters are defaults. The Ce *4f* electrons are treated as valence states.
Hence, the initialization script is executed as follows

```
init -b -vxc 5 -numk 3000
```

and then LDA calculations of non-magnetic \(\gamma\)-Ce are performed by launching
the Wien2k **run** script. These self-consistent LDA calculations will typically
take a couple of minutes.

## Wannier orbitals: dmftproj

Then we create the `Ce-gamma.indmftpr`

file specifying parameters for construction
of Wannier orbitals representing *4f* states:

```
1 ! Nsort
1 ! Mult(Nsort)
3 ! lmax
complex
1 1 1 2 ! l included for each sort
0 0 0 0 ! l included for each sort
0
-.40 0.40 ! Energy window relative to E_f
```

As we learned in the section Supported interfaces, the first three lines
give the number of inequivalent sites, their multiplicity (to be in
accordance with the *struct* file) and the maximum orbital quantum
number \(l_{max}\). The following four lines describe the treatment of
Ce *spdf* orbitals by the **dmftproj** program:

```
complex
1 1 1 2 ! l included for each sort
0 0 0 0 ! l included for each sort
0
```

where complex is the choice for the angular basis to be used (spherical complex harmonics),
in the next line we specify, for each orbital quantum number, whether it is treated as correlated (‘2’)
and, hence, the corresponding Wannier orbitals will be generated, or uncorrelated (‘1’). In the latter
case the **dmftproj** program will generate projectors to be used in calculations of
corresponding partial densities of states (see below). In the present case we choose the fourth
(i. e. *f*) orbitals as correlated. The next line specify the number of irreducible representations
into which a given correlated shell should be split (or ‘0’ if no splitting is desired, as in the
present case). The fourth line specifies whether the spin-orbit interaction should be switched
on (‘1’) or off (‘0’, as in the present case).

Finally, the last line of the file

```
-.40 0.40 ! Energy window relative to E_f
```

specifies the energy window for Wannier functions’ construction. For a
more complete description of **dmftproj** options see its manual.

To prepare input data for **dmftproj** we execute lapw2 with the -almd option

```
x lapw2 -almd
```

Then **dmftproj** is executed in its default mode (i.e. without
spin-polarization or spin-orbit included)

```
dmftproj
```

This program produces the following files:

`Ce-gamma.ctqmcout`

and`Ce-gamma.symqmc`

containing projector operators and symmetry operations for orthonormalized Wannier orbitals, respectively.

`Ce-gamma.parproj`

and`Ce-gamma.sympar`

containing projector operators and symmetry operations for uncorrelated states, respectively. These files are needed for projected density-of-states or spectral-function calculations.

`Ce-gamma.oubwin`

needed for the charge density recalculation in the case of a fully self-consistent DFT+DMFT run (see below).

Now we have all necessary input from Wien2k for running DMFT calculations.

## DMFT setup: Hubbard-I calculations in TRIQS

In order to run DFT+DMFT calculations within Hubbard-I we need the corresponding python script, Ce-gamma.py. It is generally similar to the script for the case of DMFT calculations with the CT-QMC solver (see Single-shot DFT+DMFT), however there are also some differences. First difference is that we import the Hubbard-I solver by:

```
from triqs.applications.impurity_solvers.hubbard_I.hubbard_solver import Solver
```

The Hubbard-I solver is very fast and we do not need to take into account the DFT block structure
or use any approximation for the *U*-matrix. We load and convert the **dmftproj** output
and initialize the `SumkDFT`

class as described in Supported interfaces and
Single-shot DFT+DMFT and then set up the Hubbard-I solver

```
S = Solver(beta = beta, l = l)
```

where the solver is initialized with the value of beta, and the orbital quantum number l (equal to 3 in our case).

The Hubbard-I initialization Solver has also optional parameters one may use:

n_msb: the number of Matsubara frequencies used. The default is n_msb=1025.

use_spin_orbit: if set ‘True’ the solver is run with spin-orbit coupling included. To perform actual DFT+DMFT calculations with spin-orbit one should also run Wien2k and

dmftprojin spin-polarized mode and with spin-orbit included. By default, use_spin_orbit=False.Nmoments: the number of moments used to describe high-frequency tails of the Hubbard-I Green function and self energy. By default Nmoments = 5

The Solver.solve(U_int, J_hund) statement has two necessary parameters, the Hubbard U parameter U_int and Hund’s rule coupling J_hund. Notice that the solver constructs the full 4-index U-matrix by default, and the U_int parameter is in fact the Slater F0 integral. Other optional parameters are:

T: matrix that transforms the interaction matrix from complex spherical harmonics to a symmetry adapted basis. By default, the complex spherical harmonics basis is used and T=None.

verbosity: tunes output from the solver. If verbosity=0 only basic information is printed, if verbosity=1 the ground state atomic occupancy and its energy are printed, if verbosity=2 additional information is printed for all occupancies that were diagonalized. By default, verbosity=0.

Iteration_Number: the iteration number of the DMFT loop. Used only for printing. By default Iteration_Number=1

Test_Convergence: convergence criterion. Once the self energy is converged below Test_Convergence the Hubbard-I solver is not called anymore. By default Test_Convergence=0.0001.

We need also to introduce some changes in the DMFT loop with respect that used for CT-QMC calculations in Single-shot DFT+DMFT. The hybridization function is neglected in the Hubbard-I approximation, and only non-interacting level positions (\(\hat{\epsilon}=-\mu+\langle H^{ff} \rangle - \Sigma_{DC}\)) are required. Hence, instead of computing S.G0 as in Single-shot DFT+DMFT we set the level positions:

```
# set atomic levels:
eal = SK.eff_atomic_levels()[0]
S.set_atomic_levels( eal = eal )
```

The part after the solution of the impurity problem remains essentially the same: we mix the self energy and local Green function and then save them in the hdf5 file. Then the double counting is recalculated and the correlation energy is computed with the Migdal formula and stored in hdf5.

Finally, we compute the modified charge density and save it as well as correlation correction to the total energy in
`Ce-gamma.qdmft`

file, which is then read by lapw2 in the case of self-consistent DFT+DMFT calculations.

You should try to run your script before setting up the fully charge self-consistent calculation (see this page).

## Fully charge self-consistent DFT+DMFT calculation

Instead of doing only one-shot runs we perform in this tutorial a fully self-consistent DFT+DMFT calculations. We launch such a calculations with

run -qdmft 1

where -qdmft flag turns on DFT+DMFT calculations with Wien2k, and one computing core. We use here the default convergence criterion in Wien2k (convergence to 0.1 mRy in energy).

After calculations are done we may check the value of correlation (‘Hubbard’) energy correction to the total energy:

```
>grep HUBBARD Ce-gamma.scf|tail -n 1
HUBBARD ENERGY(included in SUM OF EIGENVALUES): -0.012866
```

In the case of Ce, with the correlated shell occupancy close to 1 the Hubbard energy is close to 0, while the DC correction to energy is about J/4 in accordance with the fully-localized-limit formula, hence, giving the total correction \(\Delta E_{HUB}=E_{HUB}-E_{DC} \approx -J/4\), which is in our case is equal to -0.175 eV \(\approx\)-0.013 Ry.

The band (“kinetic”) energy with DMFT correction is

```
>grep DMFT Ce-gamma.scf |tail -n 1
KINETIC ENERGY with DMFT correction: -5.370632
```

One may also check the convergence in total energy:

```
>grep :ENE Ce-gamma.scf |tail -n 5
:ENE : ********** TOTAL ENERGY IN Ry = -17717.56318334
:ENE : ********** TOTAL ENERGY IN Ry = -17717.56342250
:ENE : ********** TOTAL ENERGY IN Ry = -17717.56271503
:ENE : ********** TOTAL ENERGY IN Ry = -17717.56285812
:ENE : ********** TOTAL ENERGY IN Ry = -17717.56287381
```

## Post-processing and data analysis

Within Hubbard-I one may also easily obtain the angle-resolved spectral function (band structure) and integrated spectral function (density of states or DOS). In difference with the CT-QMC approach one does not need to do an analytic continuations to get the real-frequency self energy, as it can be calculated directly in the Hubbard-I solver.

The corresponding script Ce-gamma_DOS.py contains several new parameters

```
ommin=-4.0 # bottom of the energy range for DOS calculations
ommax=6.0 # top of the energy range for DOS calculations
N_om=2001 # number of points on the real-energy axis mesh
broadening = 0.02 # broadening (the imaginary shift of the real-energy mesh)
```

Then one needs to load projectors needed for calculations of corresponding projected densities of states, as well as corresponding symmetries:

```
Converter.convert_parpoj_input()
```

To get access to analysing tools we initialize the
`SumkDFTTools`

class

```
SK = SumkDFTTools(hdf_file=dft_filename+'.h5', use_dft_blocks=False)
```

After the solver initialization, we load the previously calculated chemical potential and double-counting correction. Having set up atomic levels we then compute the atomic Green function and self energy on the real axis:

```
S.set_atomic_levels(eal=eal)
S.GF_realomega(ommin=ommin, ommax=ommax, N_om=N_om, U_int=U_int, J_hund=J_hund)
```

put it into SK class and then calculated the actual DOS:

```
SK.density_of_states(broadening, proj_type="wien2k")
```

We may first increase the number of **k**-points in BZ to 10000 by executing the Wien2k
program **kgen**

```
x kgen
```

and then by executing the Ce-gamma_DOS.py with **python**:

```
python Ce-gamma_DOS.py
```

In result we get the total DOS for spins up and down (identical in our paramagnetic case)
in `DOScorrup.dat`

and `DOScorrdown.dat`

files, respectively, as well as the projected DOS
written in the corresponding files as described in Tools for analysis. In our case, for example, the files
`DOScorrup.dat`

and `DOScorrup_proj3.dat`

contain the total DOS for spin *up* and the
corresponding projected DOS for Ce *4f* orbital, respectively. They are plotted below.

As one may clearly see, the Ce *4f* band is split by the local Coulomb interaction into the filled lower
Hubbard band and empty upper Hubbard band (the latter is additionally split into several peaks due to the
Hund’s rule coupling and multiplet effects).