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253 lines
12 KiB
ReStructuredText
.. _DFTDMFTtutorial:
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DFT+DMFT tutorial: Ce with Hubbard-I approximation
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==================================================
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In this tutorial we will perform DFT+DMFT Wien2k
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calculations from scratch, including all steps described in the
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previous sections. As example, we take the high-temperature
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:math:`\gamma`-phase of Ce employing the Hubbard-I approximation for
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its localized *4f* shell.
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Wien2k setup
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------------
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First we create the Wien2k :file:`Ce-gamma.struct` file as described in the
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`Wien2k manual <http://www.wien2k.at/reg_user/textbooks/usersguide.pdf>`_
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for the :math:`\gamma`-Ce fcc structure with lattice parameter of 9.75 a.u.
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.. literalinclude:: images_scripts/Ce-gamma.struct
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We initalize non-magnetic Wien2k calculations using the :program:`init` script as
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described in the same manual. For this example we specify 3000 :math:`\mathbf{k}`-points
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in the full Brillouin zone and LDA exchange-correlation potential (*vxc=5*), other
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parameters are defaults. The Ce *4f* electrons are treated as valence states.
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Hence, the initialization script is executed as follows ::
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init -b -vxc 5 -numk 3000
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and then LDA calculations of non-magnetic :math:`\gamma`-Ce are performed by launching
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the Wien2k :program:`run` script. These self-consistent LDA calculations will typically
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take a couple of minutes.
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Wannier orbitals: dmftproj
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--------------------------
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Then we create the :file:`Ce-gamma.indmftpr` file specifying parameters for construction
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of Wannier orbitals representing *4f* states:
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.. literalinclude:: images_scripts/Ce-gamma.indmftpr
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As we learned in the section :ref:`conversion`, the first three lines
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give the number of inequivalent sites, their multiplicity (to be in
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accordance with the *struct* file) and the maximum orbital quantum
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number :math:`l_{max}`. The following four lines describe the treatment of
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Ce *spdf* orbitals by the :program:`dmftproj` program::
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complex
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1 1 1 2 ! l included for each sort
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0 0 0 0 ! l included for each sort
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0
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where `complex` is the choice for the angular basis to be used (spherical complex harmonics),
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in the next line we specify, for each orbital quantum number, whether it is treated as correlated ('2')
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and, hence, the corresponding Wannier orbitals will be generated, or uncorrelated ('1'). In the latter
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case the :program:`dmftproj` program will generate projectors to be used in calculations of
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corresponding partial densities of states (see below). In the present case we choose the fourth
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(i. e. *f*) orbitals as correlated. The next line specify the number of irreducible representations
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into which a given correlated shell should be split (or '0' if no splitting is desired, as in the
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present case). The fourth line specifies whether the spin-orbit interaction should be switched
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on ('1') or off ('0', as in the present case).
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Finally, the last line of the file ::
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-.40 0.40 ! Energy window relative to E_f
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specifies the energy window for Wannier functions' construction. For a
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more complete description of :program:`dmftproj` options see its manual.
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To prepare input data for :program:`dmftproj` we execute lapw2 with the `-almd` option ::
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x lapw2 -almd
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Then :program:`dmftproj` is executed in its default mode (i.e. without
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spin-polarization or spin-orbit included) ::
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dmftproj
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This program produces the following files:
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* :file:`Ce-gamma.ctqmcout` and :file:`Ce-gamma.symqmc` containing projector operators and symmetry
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operations for orthonormalized Wannier orbitals, respectively.
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* :file:`Ce-gamma.parproj` and :file:`Ce-gamma.sympar` containing projector operators and symmetry
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operations for uncorrelated states, respectively. These files are needed for projected
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density-of-states or spectral-function calculations.
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* :file:`Ce-gamma.oubwin` needed for the charge density recalculation in the case of a fully
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self-consistent DFT+DMFT run (see below).
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Now we have all necessary input from Wien2k for running DMFT calculations.
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DMFT setup: Hubbard-I calculations in TRIQS
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--------------------------------------------
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In order to run DFT+DMFT calculations within Hubbard-I we need the corresponding python script,
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:ref:`Ce-gamma_script`. It is generally similar to the script for the case of DMFT calculations
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with the CT-QMC solver (see :ref:`singleshot`), however there are also some differences. First
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difference is that we import the Hubbard-I solver by::
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from triqs.applications.impurity_solvers.hubbard_I.hubbard_solver import Solver
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The Hubbard-I solver is very fast and we do not need to take into account the DFT block structure
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or use any approximation for the *U*-matrix. We load and convert the :program:`dmftproj` output
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and initialize the :class:`SumkDFT <dft.sumk_dft.SumkDFT>` class as described in :ref:`conversion` and
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:ref:`singleshot` and then set up the Hubbard-I solver ::
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S = Solver(beta = beta, l = l)
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where the solver is initialized with the value of `beta`, and the orbital quantum
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number `l` (equal to 3 in our case).
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The Hubbard-I initialization `Solver` has also optional parameters one may use:
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* `n_msb`: the number of Matsubara frequencies used. The default is `n_msb=1025`.
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* `use_spin_orbit`: if set 'True' the solver is run with spin-orbit coupling included.
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To perform actual DFT+DMFT calculations with spin-orbit one should also run Wien2k and
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:program:`dmftproj` in spin-polarized mode and with spin-orbit included. By default,
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`use_spin_orbit=False`.
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* `Nmoments`: the number of moments used to describe high-frequency tails of the Hubbard-I
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Green function and self energy. By default `Nmoments = 5`
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The `Solver.solve(U_int, J_hund)` statement has two necessary parameters, the Hubbard U
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parameter `U_int` and Hund's rule coupling `J_hund`. Notice that the solver constructs the
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full 4-index `U`-matrix by default, and the `U_int` parameter is in fact the Slater `F0` integral.
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Other optional parameters are:
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* `T`: matrix that transforms the interaction matrix from complex spherical harmonics to a symmetry
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adapted basis. By default, the complex spherical harmonics basis is used and `T=None`.
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* `verbosity`: tunes output from the solver. If `verbosity=0` only basic information is printed,
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if `verbosity=1` the ground state atomic occupancy and its energy are printed, if `verbosity=2`
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additional information is printed for all occupancies that were diagonalized. By default, `verbosity=0`.
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* `Iteration_Number`: the iteration number of the DMFT loop. Used only for printing. By default `Iteration_Number=1`
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* `Test_Convergence`: convergence criterion. Once the self energy is converged below `Test_Convergence`
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the Hubbard-I solver is not called anymore. By default `Test_Convergence=0.0001`.
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We need also to introduce some changes in the DMFT loop with respect that used for CT-QMC calculations
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in :ref:`singleshot`. The hybridization function is neglected in the Hubbard-I approximation, and only
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non-interacting level positions (:math:`\hat{\epsilon}=-\mu+\langle H^{ff} \rangle - \Sigma_{DC}`) are
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required. Hence, instead of computing `S.G0` as in :ref:`singleshot` we set the level positions::
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# set atomic levels:
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eal = SK.eff_atomic_levels()[0]
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S.set_atomic_levels( eal = eal )
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The part after the solution of the impurity problem remains essentially the same: we mix the self energy and local
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Green function and then save them in the hdf5 file.
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Then the double counting is recalculated and the correlation energy is computed with the Migdal formula and stored in hdf5.
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Finally, we compute the modified charge density and save it as well as correlation correction to the total energy in
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:file:`Ce-gamma.qdmft` file, which is then read by lapw2 in the case of self-consistent DFT+DMFT calculations.
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You should try to run your script before setting up the fully charge self-consistent calculation
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(see :ref:`this<runpy>` page).
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Fully charge self-consistent DFT+DMFT calculation
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-------------------------------------------------
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Instead of doing only one-shot runs we perform in this tutorial a fully
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self-consistent DFT+DMFT calculations. We launch such a calculations with
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`run -qdmft 1`
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where `-qdmft` flag turns on DFT+DMFT calculations with Wien2k,
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and one computing core. We use here the default convergence criterion
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in Wien2k (convergence to 0.1 mRy in energy).
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After calculations are done we may check the value of correlation ('Hubbard') energy correction to the total energy::
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>grep HUBBARD Ce-gamma.scf|tail -n 1
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HUBBARD ENERGY(included in SUM OF EIGENVALUES): -0.012866
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In the case of Ce, with the correlated shell occupancy close to 1 the Hubbard energy is close to 0, while the
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DC correction to energy is about J/4 in accordance with the fully-localized-limit formula, hence, giving the
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total correction :math:`\Delta E_{HUB}=E_{HUB}-E_{DC} \approx -J/4`, which is in our case is equal
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to -0.175 eV :math:`\approx`-0.013 Ry.
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The band ("kinetic") energy with DMFT correction is ::
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>grep DMFT Ce-gamma.scf |tail -n 1
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KINETIC ENERGY with DMFT correction: -5.370632
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One may also check the convergence in total energy::
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>grep :ENE Ce-gamma.scf |tail -n 5
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:ENE : ********** TOTAL ENERGY IN Ry = -17717.56318334
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:ENE : ********** TOTAL ENERGY IN Ry = -17717.56342250
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:ENE : ********** TOTAL ENERGY IN Ry = -17717.56271503
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:ENE : ********** TOTAL ENERGY IN Ry = -17717.56285812
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:ENE : ********** TOTAL ENERGY IN Ry = -17717.56287381
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Post-processing and data analysis
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---------------------------------
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Within Hubbard-I one may also easily obtain the angle-resolved spectral function
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(band structure) and integrated spectral function (density of states or DOS).
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In difference with the CT-QMC approach one does not need to do an
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analytic continuations to get the real-frequency self energy, as it can be
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calculated directly in the Hubbard-I solver.
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The corresponding script :ref:`Ce-gamma_DOS_script` contains several new parameters ::
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ommin=-4.0 # bottom of the energy range for DOS calculations
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ommax=6.0 # top of the energy range for DOS calculations
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N_om=2001 # number of points on the real-energy axis mesh
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broadening = 0.02 # broadening (the imaginary shift of the real-energy mesh)
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Then one needs to load projectors needed for calculations of
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corresponding projected densities of states, as well as corresponding
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symmetries::
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Converter.convert_parpoj_input()
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To get access to analysing tools we initialize the
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:class:`SumkDFTTools <dft.sumk_dft_tools.SumkDFTTools>` class ::
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SK = SumkDFTTools(hdf_file=dft_filename+'.h5', use_dft_blocks=False)
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After the solver initialization, we load the previously calculated
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chemical potential and double-counting correction. Having set up
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atomic levels we then compute the atomic Green function and
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self energy on the real axis::
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S.set_atomic_levels(eal=eal)
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S.GF_realomega(ommin=ommin, ommax=ommax, N_om=N_om, U_int=U_int, J_hund=J_hund)
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put it into SK class and then calculated the actual DOS::
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SK.density_of_states(broadening, proj_type="wien2k")
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We may first increase the number of **k**-points in BZ to 10000 by executing the Wien2k
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program :program:`kgen` ::
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x kgen
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and then by executing the :ref:`Ce-gamma_DOS_script` with :program:`python`::
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python Ce-gamma_DOS.py
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In result we get the total DOS for spins `up` and `down` (identical in our paramagnetic case)
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in :file:`DOScorrup.dat` and :file:`DOScorrdown.dat` files, respectively, as well as the projected DOS
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written in the corresponding files as described in :ref:`analysis`. In our case, for example, the files
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:file:`DOScorrup.dat` and :file:`DOScorrup_proj3.dat` contain the total DOS for spin *up* and the
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corresponding projected DOS for Ce *4f* orbital, respectively. They are plotted below.
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.. image:: images_scripts/Ce_DOS.png
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:width: 700
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:align: center
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As one may clearly see, the Ce *4f* band is split by the local Coulomb interaction into the filled lower
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Hubbard band and empty upper Hubbard band (the latter is additionally split into several peaks due to the
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Hund's rule coupling and multiplet effects).
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