6.2 The DMFT-FLEX Solver
7.1.1 Determining K max and l max
The value ofKmaxdetermines the number of augmented plane waves used to represent a Kohn-Sham eigenfunction andlmax determines the number of radial functions used to augment the plane waves inside the muffin-tin regions. Both Kmax and lmax are
5.20 5.25 5.30 5.35 5.40 5.45 5.50 5.55
Lattice Constant [a
B] 0.000
0.001 0.002 0.003 0.004 0.005
Energy - E0 [htr]
Kmax = 3.90 Kmax = 4.05 Kmax = 4.20 Kmax = 4.35 Kmax = 4.50 Kmax = 4.65 a0 = 5.355
Figure 7.1: Data of convergence tests for Kmax and parabola fits from bulk crystal calculations of iron. To better display the data, the values are all shifted by a constant E0 = 1272.811 htr. The value for a0 is the converged equilibrium lattice constant.
obtained by calculating the total energy in FLEUR for a number of different values of the lattice constant below and above the equilibrium lattice constant a0 for a fixed choice of the exchange-correlation potential and a fixed number of k points. If the value for lmax is also kept fixed a different parabola for each choice of Kmax is obtained. Likewise, if Kmax is kept fixed, parabolas for different values of lmax can be plotted. I first examine the results for the convergence tests forKmax. Figure 7.1 shows as an example the calculations for iron for a fixed value oflmax= 8 and a total number of 322 k points.
The converged value forKmaxis obtained by converging the value of the equilibrium lattice constanta0. The equilibrium lattice constant depends on the chemical bonds, which are composed mainly by the s and p electrons in transition metals. Electrons in s and p states are fairly delocalized over the crystal lattice. Thus, the accuracy of the representation of the s and p wave functions in the LAPW basis depends primarily on the total number of augmented plane waves used in the expansion of the Kohn-Sham eigenfunctions. If the representation in the LAPW basis of thesand p wave functions that contribute to the chemical bonds inside the material becomes numerically exact, the value of a0 will not change further, if more plane waves are included in the expansion. Thus, both values for a0 and Kmax are converged if the
6.45 6.50 6.55 6.60 6.65 6.70 6.75 6.80 6.85 6.90
Lattice Constant [a
B] 0.000
0.001 0.002 0.003 0.004 0.005 0.006 0.007
Energy - E0 [htr]
Kmax = 3.60 Kmax = 3.80 Kmax = 4.00 Kmax = 4.20 Kmax = 4.40 Kmax = 4.60 a0 = 6.655
Figure 7.2: Data of convergence tests for Kmax and parabola fits from bulk crystal calculations of nickel. To better display the data, the values are all shifted by a constantE0 = 1520.8399 htr. The value fora0 is the converged equilibrium lattice constant.
7.1 Convergence Tests for the Parameters of the LAPW Basis 83 position of the minima of the parabolas does not change when Kmax is increased further. It can be seen in figure 7.1 that convergence of a0 is reached for a value of Kmax larger or equal to 4.2. This has been confirmed by calculating the minima of the parabolas obtained by fitting a quadratic function to the results of each FLEUR calculation. Results obtained from calculations for nickel with lmax = 8 and 280 k points are shown in figure 7.2. Again convergence is reached ifKmaxis larger or equal to 4.2.
It might seem odd that the minimum of the total energy still decreases, if Kmax
is further increased although convergence of a0 is reached. This is due to the fact that the total energy is calculated as a functional of the electron density of all elec-trons which is proportional to the sum of the squares of all occupied Kohn-Sham eigenfunctions (see equation (2.22)). The minimum of the total-energy functional can be obtained from the functional derivative of the total energy with respect to the electron density. Adding more variational degrees of freedom to the electron density by increasingKmaxwill therefore always yield a smaller value for the minimum of the total energy. Thus, the decrease of the total energy is independent of the convergence of a0 and Kmax.
5.20 5.25 5.30 5.35 5.40 5.45 5.50
Lattice Constant [aB] 0.0000
0.0005 0.0010 0.0015 0.0020 0.0025 0.0030
Energy - E0 [htr]
l_max: 6 l_max: 8 l_max: 10
a0 = 5.355
Figure 7.3: Data of convergence tests for lmax from bulk crystal calculations of iron. To better display the data, the values are all shifted by a constant E0 = 1272.811 htr. The value for a0 is the converged lattice constant.
In figure 7.3 calculations of the total energy for iron with Kmax = 4.2 and 322 k points are presented for different values of lmax. Since the electron density is proportional to the square of all occupied Kohn-Sham eigenfunctions, convergence of lmax is reached if the minimum of the total energy does not change further upon increasing the value of lmax for a given value of Kmax. The minimum of all three data sets in 7.3 does not change for the different values of lmax, hence, convergence is already reached with a choice of lmax = 6. This was to be expected, since the electronic states in 3d transition metals are occupied for l-quantum numbers up to 2. For free atoms with a spherically symmetric potential a value of 2 for lmax should hence be sufficient to account for all occupied electronic states contributing in the calculation of the total energy. Due to the anisotropic potential in a crystal environment a slightly higher value for lmax is necessary to describe distortions of the wave functions but theses deviations should only have a small impact on the description of the materials studied here. However, an additional difficulty arises from the fact, that the wave functions when represented by a finite sum over lmax
terms inside the muffin tins are not continuous at the sphere’s boundaries any more as they should be per definitionem. Although the total energy does not change any more for a choice oflmaxbetween 6 and 10 this additional problem has to be taken into account by choosinglmax large enough such that the continuation into the interstitial space is as smooth as possible while keeping the numerical effort at a tolerable level.
Therefore, lmax= 8 was used in the calculations for all materials.
0 200 400 600 800
# of k-points in irred. part of BZ 0.00
0.01 0.01 0.01 0.02 0.03 0.03 0.04
Energy - E0 [htr]
0 200 400 600 800
# of k-points in irred. part of BZ 0.000
0.001 0.002 0.003 0.004 0.005 0.006 0.007
Energy - E0 [htr]
Figure 7.4: Results of convergence tests for the number of kpoints from bulk crystal calculations of iron (left) and nickel (right). For Fe the parameters Kmax = 4.2 and lmax = 8 were used and for Ni Kmax= 4.0 and lmax= 8.
7.1 Convergence Tests for the Parameters of the LAPW Basis 85
5.00 5.05 5.10 5.15 5.20 5.25 5.30 5.35 5.40 5.45 5.50 5.55 Lattice Constant [a0]
0.010 0.013 0.015 0.018 0.020 0.022 0.025
∆E [htr]
PZ, LDA PBE, GGA PW91, GGA
aexp = 5.42
Figure 7.5: Results of convergence tests for the best choice of a parametrization for the exchange-correlation potential vxc from bulk crystal calculations of iron. The value for aexp denotes the lattice constant as obtained from experiments. Parameters used in the cal-culations: Kmax= 4.2, lmax= 8, number ofk points is 322.