Category Archives: 1st Post 2018

Lattice Parameter Prediction of Hafnium

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For this problem we want to predict the lattice parameters of Hafnium and to compare them with the experimental values, given that Hafnium is observed to be a hcp metal, see Figure 1, with a ratio of c to a of 1.58.

Figure 1: Structure of Hf lattice with a =3.1 and c/a=1.58

Methods

To find the optimal values for the lattice parameters, energy calculations were be preformed at differing values of lattice parameter a and c. To turn this into a pseudo one parameter optimization the ratio of c to a was be held fixed while a was varied, then this was repeated for multiple c/a ratios.

The exchange-correlation functional used was PBE, the pseudopotential used was OTFG ultrasoft and the relativistic treatment was that that of Koelling-Harmon.

Convergence tests with respect to the number of k-points and the energy cutoff energy were preformed and the results of which can be found in the Appendix. Using those results the energy cutoff was set to 435.4eV, and the k-point grid was set to 10x10x6. Also, the SCF tolerance was set to 1.0e-6eV/atom with a convergence window of three steps, all other parameters where kept to the quality fine preset for CASTEP.

Material Studio was used to create the crystal of Hafnium as a hcp metal of group P63/MMC and change the lattice parameters, while CASTEP was used to perform the calculations.

Atomic electron configuration for Hf is:
1s2 2s2 2p6 3s2 3p6 3d10 4s2 4p6 4d10 4f14 5s2 5p6 5d2 6s2

And the pseudo atomic electron configuration for Hf used was:
4f14 5s2 5p6 5d2 6s2

Optimization at c/a=1.58

As a starting point, the c/a ratio of was set to 1.58 and the lattice parameter a was set to the range from 2.5 Å to 4.0 Å in increments of 0.1 Å to determine the range of a where the optimal value for a should lie. The results of those calculations can be found in Table 1 and Figure 2.

Table 1: Results of the calculations for c/a=1.58

Figure 2: Cohesive Energy per atom versus Lattice Parameter a, for c/a =1.58

From Table 1 and Figure 2 it can be seen that the minimum cohesive energy occurs at a = 3.2 Å, a volume per atom of 22.419 Å3.

Optimization of c/a

To optimize the ratio of c to a, energy calculations were preformed for a c to a ratio of 1.56 and 1.60 for lattice parameter a ranging from 3.0 Å to 3.4 Å, with the resulting cohesive energy versus volume plots are shown in Figure 3.

 

Figure 3: Cohesive energy per atom versus Volume per atom plots for the different c/a ratios tested.

From Figure 3 we can see that the cohesive energy was at it’s minimum with a c to a ratio of 1.58.

To obtain the optimum value for a, and by extension c, a quadratic function was fit to the plot of the cohesive energy versus lattice parameter a for the c to a ratio of 1.58. The minimum point of the function was used to find the optimal value of a.

Figure 4: Plot of Cohesive Energy per atom vs lattice parameter a with a c/a=1.58.

From the quadratic fit the optimal values of a and c were found to be 3.20 Å and 5.06 Å, respectively.

From experiment the lattice parameters a and c of Hafnium were found to be 3.1964 Å and 5.0511 Å, respectively.  This corresponds to a relative error for a and c of  0.2% and 0.2%, respectively.

Appendix

Validation of the Cutoff Energy

To validate that the cutoff energy was low enough to obtain the level of precision we want from the calculations we can look at the convergence of the free energy with respect to the cutoff energy. While this calculation was done at the start of this project, it was redone for an a of 3.2Å  and a c/a ratio of 1.58 as it will serve as a stronger validation for the results presented.

Figure 5: Free Energy vs. Cutoff Energy for a=3.2, c/a=1.58 and k-points: 10x10x6

From Figure 5, one can see that the free energy for the cutoff used, 435.4eV, was within 0.05eV of the converged value.

Validation of k-points

To validate that the number of k-points was high enough for the level of precision we want from the calculations, we can look at the convergence of the free energy with respect to the number of k-points.

Figure 6: Free Energy vs. Number of k-points, a=3.2Å, c/a=1.58 and Ecut=480eV

Figure 6 shows that the k-point values used, 10x10x6, was within 0.001eV of the converged value.

Cohesive Energy

The DFT program used, CASTEL does not report the cohesive energy, it reports the energy of the free atom and then the free energy of the crystal.  To correct for that we define the cohesive energy as the free energy of the crystal divided by the number of atoms in the crystal’s cell minus the energy of the free atom.

Determination of Pt crystal structure and corresponding lattice constant

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In this project, the energetically favorable crystal structure of Pt and corresponding lattice constant were determined using Density Functional Theory. The total energies were computed in Material Studio with CASTEP Calculation Package [1]. The functional of Perdew Burke, and Ernzerhoff was employed [2]. A plane wave basis set was used with a cut-off energy of 321.1 eV and OTFG ultrasoft pseudopotentials was solved using Koelling-Harmon treatment. The pseudo atomic calculation was performed for Pt 4f14 5s2 5p6 5d9 6s1.

Determine Energy Cut-off
The energy cut-off was determined by considering both calculation accuracy and computational cost. Pt fcc with lattice constant of 3.21 angstrom was randomly selected to perform a series of calculation, in order to determine energy cut-off. As shown in Fig.1, the fluctuation of cohesive energy becomes smaller as energy cut-off increases, while the computational cost has an increasing trend. In order to guarantee the accuracy and stability of our data and also keep computational cost acceptable, we choose energy cut-off 321.1 eV.


Fig.1 a.cohesive energy for 12 energy cut-off values b. computational time for 12 energy cut-off values

Determine K Points
Similar to energy cut-off determination, determining the number of K points was also based on calculation accuracy and calculational expense. From Fig.2, it is clear that computational cost increase with number of K points and cohesive energy becomes more stable with the energy difference of 0.05 eV between 20 and 28 kpoints


Fig.2 a.cohesive energy for 6 K points values b.Computational time for 6 K points values

Determine Pt Crystal Structure and Lattice Constant
Simple cubic structure
For simple cubic structure, 10 × 10 × 10 K points were sampled with 0.1 eV Gaussian smearing width. The energetically most favorable lattice constant is 2.6 Å with cohesive energy of -9.175 eV.

Fig.3 Cohesive energies of Pt in simple cubic structure as a function of lattice parameters

Face center cubic structure
For face center cubic structure, 12 × 12 × 12 K points were sampled with 0.1 eV Gaussian smearing width. The energetically most favorable lattice constant is 3.924 Å with cohesive energy of -9.667 eV.

Fig.4 Cohesive energies of Pt in fcc structure as a function of lattice parameters

Hexagonal Close Packed (hcp) Structure
When performing calculations of Pt hcp, two parameters had to be considered. We randomly selected three lattice constants of a and 12 c values for corresponding a. 12 × 12 × 8 K points were sampled with 0.1 eV Gaussian smearing width. The energetically most favorable lattice constant is 2.7 Å and height of 5.4 Å with cohesive energy of -9.512 eV.

Fig.5 Cohesive energies of Pt in hcp structure as a function of lattice parameters

Conclusion
From calculations performed above, we can conclude fcc structure with lattice constant of 3.924 Å is energetically most favorable for Pt, which is in a good agreement with experiment 3.912 Å [3].

Reference
[1] “First principle methods using CASTEP” Zeitschrift fuer Kristallographie 220(5-6) pp. 567-570 (2005)
[2] Perdew, J. P; Burke, K; Ernzerhof, M. Phys. Rev. Lett. 1996, 77, 3865-3868
[3]”Precision Measurement of the Lattice Constants of Twelve Common Metals” Davey, Wheeler, Physical Review. 25 753-761 (1925)

 

 

Determining the lattice constants for Hf

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   For Project 1,  the lattice constants for Hf are calculated. Since Hf is an hcp metal with c/a=1.58, we need to calculate the energy by changing the lattice constant c while keeping c/a fixed. And the optimized a, c can be obtained by minimizing the free energy.

1. The hcp structure of Hf

   Figure 1 shows the unit cell of Hf hcp structure. OA, OB, OC are three lattice vectors. The lengths and angles of the lattices vectors (OA, OB, OC) are a, b, c and α, β, γ, respectively. To keep the symmetry, the constraints for the unit cell are a=b, α=β=90 degrees, γ=120 degrees.

Figure 1 The unit cell of Hf

2. The parameters for the calculations

Some important parameters and inputs for the calculation are listed as following:

Atomic structure for Hf:1s2 2s2 2p6 3s2 3p6 3d10 4s2 4p6 4d10 4f14 5s2 5p6 5d2 6s2

The core configuration of the pseudopotential: 1s2 2s2 2p6 3s2 3p6 3d10 4s2 4p6 4d10

Functional: GGA Perdew Burke Ernzerhof (PBA) functional

Pseudopotential: OTFG ultrasoft

Smearing width: 0.1

Origin shift of K points: 0, 0, 0

The k points grid: investigated in the next section

The energy cutoff: investigated in the next section

3. The convergence of the calculations

   Before starting the actual calculation for the lattice constant, it is important to explore the convergence of the calculations with respect to the number of k points and the energy cutoff. So the most accurate and efficient results can be obtained. For the following calculation, the lattice constant is fixed at c= 5.0511Å, a= 3.1946Å, while the size of k point grid and the energy cutoff are varied.

   (1) The number of k-points in the irreducible part of the BZ

     In this section 3.1, the energy cutoff is fixed at 400 eV, and the size of k point grid changes and Number of k-points in the irreducible part of the BZ (the number of k points) varies from 1 to 162, shown in Table 1. According to Table 1, the relations between Energy/atom, calculation time and the number of k points can be plotted, which are shown in Figure 2. As shown in Figure 2, the energy/atom becomes stable with the increase of k points. However, the calculation time also increases rapidly when the number of k points increases. By combining the results of these two graphs, 9*9*6 k points with 36 k points is chosen to obtain an accurate result with less calculation time.

   A little discussion here, since my calculation involves unit cells with different volumes, the k points need to be chosen so the density of k points in reciprocal space is comparable for the different supercells. But as shown in section 4, since the volumes of unit cell change just a little during the calculation, fixing the number of k points will not make much difference. So in section 4, the number of k points is chosen as 36 for all calculations.

Table 1 The calculation of different sizes of k point grid

Figure 2 The plots between Energy/atom, calculation time and the number of k points. The solid lines are just used for guide the eye.

(2) The energy cutoff

In this section, the number of k point is fixed at 36, and the energy cutoff varies from 200 to 800, shown in Table 2. And Figure 3 shows the relation between energy/atom, calculation time and energy cutoff. Although the energy/atom keeps increasing with the increase of energy cut off, the differences between each energy/atom are becoming smaller and smaller. On the other hand, the calculation time increases rapidly with the increase of energy cut off. In order to save the time and keep the accuracy, the energy cutoff is chosen as 435.4 eV.

Table 2 The calculation of different energy cutoff

Figure 3 The plots between energy/atom, calculation time and energy cutoff. The solid lines are used to guide the eye.

4 The calculation of the lattice constants

Using the results from section 3, the k points and the energy cutoff are chosen as 9*9*6 and 435.4 eV. Then lattice constants c and a are changed to obtain the minimum energy, shown in Table 3. As shown in Figure 4, the unit cell and the lattice constants can be predicted. As shown in Figure 4a, the quadratic relation between energy/atom and volume/atom only valid for a small range of lattice constants around the equilibrium value. A small range of volume/atom is chosen and the relation is re-plotted in Figure 4b. According to the fitting equation in Figure 4b, the fitting volume/atom is 22.64 Å^3 and the lattice constant are c=5.0673 Å, a=3.2072 Å. And for the experiment data, the lattice constants are a=3.1964 Å, c=5.0511 Å (1). So the calculation is accurate with an error of 0.3%.

Table 3 The calculation of energy with different lattice constants.

Figure 4 The plots between energy/atom and volume/atom. the solid lines are used to guide the eye. The dashed line is the fitting line.

5. Some discussion and future works

Although the calculation result is accurate with an error of 0.3%, there are still some ways to further increase the accuracy. First, the energy cut off can be increased so the energy/atom can be more precise. Second, we can calculate more points around the value c=5.05 Å and maybe a more accurate result can be obtained. Third, instead of using the quadratic fitting, we can use a higher order polynomial fitting.

For some future works: first, it is worthwhile changing the ratio c/a to determine the optimum value of c/a. Second, it should be interesting to analyze the behavior of energy at small or large lattice constants.

6. Reference:

(1) https://www.webelements.com/hafnium/crystal_structure.html

Determination of Structure and Lattice Constant for Platinum

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Overview of the Problem

This post will be about determining both the structure and lattice constant for platinum. This will be done using density functional theory (DFT) to calculate the cohesive energies across a range of lattice parameters for platinum in the simple cubic (SC), face-centered cubic (FCC), and hexagonal close-packed (HCP) structures.

In order to make sure the results are feasible, convergence tests are first done for both the number of k-points and the cutoff energies.

All calculations are done in Materials Studio with the CASTEP DFT package [1]. The functional used was that of Perdew Burke Ernzerhof [2], and the pseudopotential was obtained using the Koelling-Harmon solver for the 4f14 5s2 5p6 5d9 6s1 outer shells. 

Cutoff Energy Convergence

The cutoff energy was tested by starting at a cutoff energy of 360 eV and then increasing by increments of 30 eV to look for convergence in the free energy. For this test the lattice parameters of the three structures were chosen as a = 3.92 Å for FCC, a = 2.62 Å for SC, and a = 3.02 Å, c = 4.83 Å for HCP. These lattice constants were chosen for this test by roughly estimating the location of the minimum of the three structures at an energy cutoff of 300 eV. The results are outlined in table 1 and figure 1.

 

Energy Cutoff (eV)FCC Free Energy (eV)SC Free Energy (eV)HCP Free Energy (eV)
300-13050.818-13050.421-13049.923
330-13050.927-13050.534-13050.046
360-13050.960-13050.568-13050.080
390-13050.970-13050.578-13050.116
420-13050.973-13050.580-13050.119
450-13050.973-13050.581-13050.120
480-13050.974-13050.581-13050.120
510-13050.974-13050.581-13050.120
Table 1: Data for the energy cutoff convergence test.

As can be seen from the figure and the table, a lower energy cutoff here will overestimate the free energy. It can also be see that the variation in the free energy is less than 0.01 eV past the 390 eV cutoff energy. For consistency, a cutoff energy of 420 eV has been chosen for all calculations involving the determination of the lattice parameters.

K-Point Convergence

The k-point convergence was tested using an HCP structure with a = 3.12 Å and c/a = 1.6, an SC structure with a = 2.62 Å, and an FCC structure with a = 3.92 Å. The results are outlined in tables 2, 3, and 4, as well as figures 2, 3, and 4.

K-point GridNumber of K-pointsFree Energy (eV)
7x7x432-13050.000
8x8x5156-13049.970
9x9x675-13049.970
10x10x6240-13049.965
11x11x7144-13049.955
12x12x8456-13049.975
13x13x9196-13049.970
Table 2: K-point convergence data for HCP.
K-point GridNumber of K-pointsFree Energy (eV)
6x6x628-13050.960
8x8x820-13050.94
10x1010110-13050.943
11x11x1156-13050.945
12x12x12182-13050.945
13x13x1384-13050.943
14x14x14280-13050.943
15x15x15120-13050.945
16x16x16408-13050.945
Table 3: K-point convergence data for FCC.
K-point GridNumber of K-pointsFree Energy (eV)
10x10x1035-13050.460
11x11x1156-13050.470
12x12x1256-13050.490
13x13x1384-13050.500
14x14x1484-13050.510
15x15x15120-13050.500
16x16x16120-13050.500
Table 4: K-point convergence data for SC.

Figure 2: K-point convergence plot for SC

Figure 3: K-point convergence plot for HCP

Figure 4: K-point convergence plot for FCC

 

For the SC lattice, having few k-points gives energies that start above the convergence level with a tendency to decrease. This is not the case for the HCP lattice, where the energies start low and then increase before decreasing again. Lastly, for the FCC lattice we see that the energy starts above the convergence level before sharply decreasing and then approaching convergence.

Based on the results from the test of the k-point convergence, grids of 15x15x15 were chosen FCC and SC, while a grid of 13x13x9 was chosen for HCP.

Lattice Parameter for SC

The simple cubic structure is shown in figure 5.

 

Figure 5: Simple cubic lattice structure

The calculations for the cohesive energy were done by first calculating the free energy per atom for the simple cubic structure, and then subtracting away the atomic energy of -13042.12 eV, which was obtained from the pseudopotential calculations. The results are shown in figure 6.

 

Figure 6: Cohesive energy vs lattice parameter for the simple cubic structure.

Pt in the simple cubic configuration is found to be stable with a lattice constant of a = 2.62 Å and a cohesive energy of -8.396 eV.

Lattice Parameter for FCC

The face-centered cubic structure is shown in figure 7.

Figure 7: The face-centered cubic lattice structure

 

The calculations of the cohesive energy were done in the same way as they were for the simple cubic lattice. The atomic energy will remain the same, so we simply calculate the free energy and take the difference. The results of these calculations are shown in figure 8.

Figure 8: Results from the calculation of cohesive energy for Pt in the FCC configuration for various lattice parameters

From the results of the calculation, it is found that Pt in the FCC configuration has a preferred lattice constant of about 3.96 Å, which gives a cohesive energy of -8.851 eV.

Lattice Parameter for HCP

The lattice of the hexagonal close-packed structure is shown in figure 9.

 

Figure 9: The hexagonal close-packed lattice structure

The HCP lattice has two lattice constants, so there is a much larger phase space to explore in order to locate the minimum cohesive energy. In order to sample this space, the ratio between the lattice constants, c/a, is held fixed at values of 1.57, 1.6, and 1.63. The parameters a and c are then varied in tandem to search for the preferred lattice constant. The results of these calculations are shown in figure 10.

Calculations for the cohesive energy of HCP Pt for c/a ratios of 1.57, 1.6, 1.63.

The results show that an HCP lattice of Pt is most stable with lattice constants of a = 2.98 Å and c = 4.77 Å for c/a = 1.6. These parameters give a cohesive energy of -7.996 eV.

Conclusions

The calculations done above give cohesive energies of -8.396 eV for SC,  -8.851 eV for FCC, and -7.996 for HCP. This implies that Pt is most stable in the FCC configuration with a lattice constant of 3.96 Å. Davey finds through experimental methods that Pt is most stable in the FCC configuration with a lattice constant of 3.91Å [3]. Using density functional theory, we have shown that we can correctly predicted that Pt naturally forms an FCC lattice. However, we have overestimated the lattice constant by about 0.05 Å.

 

References

[1] First principle methods using CASTEP Zeitschrift fuer Kristallographie 220(5-6) pp. 567-570 (2005).

[2] Perdew, J.P., K. Burke, and M. Ernzerhof, Generalized gradient approximation made simple.Physical review letters, 1996. 77(18): p. 3865.

[3] Davey, W. P. (1925). Precision Measurements of the Lattice Constants of Twelve Common Metals. Physical Review, 25(6), 753-761. doi:10.1103/physrev.25.753.

Determination of Lattice Constant of Platinum Cell

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Introduction

Platinum is a chemical element with symbol Pt and atmomic number 78. The electron configuration is [Xe] 4f145d96s1. Its crystal structure is face-centered cubic (fcc, see Fig. 1), and the lattice constant is 3.9239 Å[1].

The target for this computation experiment is to use density functional theory (DFT) to evaluate the energy of fcc lattice at different lattice constants. The estimation of the lattice constant is hence decided by minimizing the energy. The result is compared with the standard data based on experimental measurements.

Key Factors of Density Functional Theory (DFT)

Denstity functional theory (DFT) is a computational method based on calculating the energy of a condensed matter system as a functional of the electron density. Thereare a lot of references that give systematic introductions and discussions on the method, which we will not go into details in this report[2]. The key factors affecting the calculation in this report include (1) Energy cut-off and (2) k-point sampling number. Energy cut-off refers the maximum energy of the bands involved in our calculation. The number of the bands included for a typical calculation is about 20, and this number may increase when higher energy resolution is required. K-point sampling number refers the number of sampling points within the first Brillouin zone. Usually the higher the energy cut-off is and the larger the sampling number is, the better the accuracy is. Due to limitation on the computational resources, we have to decide lowest values of both the factors that could meet requirements.

We performed several trial calculations and compared their numerical variance to decide the appropriate values. (1) We fix the cut-off energy at 800 eV, which is high enough for a simple calculation, and explore the effect of k-point density on energy calculation and atomic energy to figure out the minimum sufficient value for k-point sampling number. (2) We use the number of k-point sampling mentioned above and explore the effect of cut-off energy and decide the best cut-off energy that balances the calculating speed and precision. Both tests are performed at conventional lattice constant a=4Å.

Energy Evaluation and Lattice

Applying the parameters decided above, several calculations of free energy at lattice constant ranging from 3.4 ~ 4.4Å were performed. Then a quadratic fitting was performed to minimizing energy. The corresponding lattice constant is our theoretical prediction of the true lattice, and it is compared with standard value. A more detailed discussion can be found in the experimental section in this report.

Experimental Facts

The basic parameters and key factors shared in all the calculations are listed below:

Lattice cell type: primitive cell of face centered cubic (fcc)

Functional Type: GGA PBE

Smearing width: 0.10

Pseudo Potential Orbitals: 4f14 5s2 5p6 5d9 6s1

Pseudo Type: OTFG ultrasoft

k-points origin shift: no

K-point Number Examination

Based on the discussion above, we calculated the free energy of the lattice at a=4Å. The results are shown in Fig. 2. Based on this result, it is sufficient to choose 16×16×16 as the appropriate sampling number for one wants acceptable precision without spending too much time.

Energy Cut-off Examination

The convergence of energy and pseudo atomic energy are displayed as follows. The difference between them is the binding energy per atom. Although it is not fully converged, the pseudo atomic energy variance is highly suppressed when the cut-off energy is higher than 800eV, which indicates that 800eV is a reasonable choice for our estimation of the binding energy (Fig.3 and Fig.4).

 

 Energy at Different Lattice Constants

To best fit the plots in the graph, a polynomial to the order of 3 was applied, and the minimum free energy was reached at lattice constant a=3.95Å. Compared to the standard result, this calculation has an error of 0.6%.

Reference

[1] https://en.wikipedia.org/wiki/Platinum

[2]Sholl, David, and Janice A. Steckel. Density functional theory: a practical introduction. John Wiley & Sons, 2011.

Comparing ScAl in CsCl and NaCl Structures and Determining the Optimal Lattice Parameter of the Preferred Structure

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The goal of this post is to identify if ScAl, which has AB stoichiometry, exists in the CsCl structure (figure 1) or NaCl Structure (figure 2 and figure 3). To determine this primitive cells of ScAl were produced for both types of structures. The plan is to plot the cohesive energy of the structure as a function of the volume per ScAl dimer. From these plots the optimal lattice parameters for each structure can be determined. Then by comparing the cohesive energies of the two structures with optimal lattice parameters we can determine which structure is preferred by ScAl. All energy calculations were carried out using plane-wave based DFT. The GGA based PBE functional was used to treat the exchange-correlation effects. The ion and core were treated using ultrasoft pseudopotentials generated on the fly (OTFG ultrasoft) with Koelling-Harmon relativistic treatments. Pseudo atomic calculations for Sc treated the 3s2 3p6 3d1 4s2 electrons as valence electrons and the electrons in lower levels were treated as part of the frozen core. Pseudo atomic calculations for Al treated the 3s2 3p1 electrons as the valence electrons.

Figure 1 : Primitive unit cell of ScAl in the CsCl structure. This a simple cubic structure with a two atom basis.

Figure 2 : (a) Conventional unit cell of ScAl in the NaCl structure. This a face centered cubic structure with a two atom basis. (b) Primitive unit cell of ScAl in the NaCl structure.

However before we begin calculating the data points to populate the plots we described in the previous paragraph, we need to select appropriate cutoff energy and \(\vec{k}\) point mesh for our calculations. The constraints on our choice of cutoff energy and \(\vec{k}\) point mesh are (1) computational cost, (2) convergence of results to a desired tolerance and (3) ensuring that we have approximately the same \(\vec{k}\) point density for each structure.

To determine the cutoff energy we use for our calculations we first determine the approximate lattice parameter of CsCl structure primitive unit cell. For this we use a cutoff energy of 460 eV and \(\vec{k}\) point mesh of \(8\times8\times8\) . The cutoff energy and \(\vec{k}\) point mesh chosen here are the default settings for the “ultra-fine” quality energy calculation using the CASTEP tool implemented in Material Studio. The cohesive energy vs volume per  ScAl dimer plot (figure 4) indicates that cohesive energy is minimized when the lattice parameter is \(\sim 3.5\;\mathring{A}\).

Figure 3 : Plot of Energy per ScAl dimer vs lattice parameter for ScAl in the CsCl structure, used to determine the approximate value for the optimal lattice parameter.

Next we investigated the convergence of the total energy of a primitive unit cell of ScAl in the CsCl structure with respect to the cutoff energy used for the calculation. The results were plotted as shown in figure 5.  From this plot note that we get a convergence of \(\sim\;1\;m\,eV\) for a cutoff energy of \(500\;m\,eV\).

Figure 4 : Calculated total energy of a unit cell of ScAl in the CsCl structure (lattice constant = 3.5 \(\mathring{A}\)) vs the cutoff energy used for the calculation.

Next to obtain the most suitable \(\vec{k}\) point mesh, we plot the energy per ScAl dimer for ae a fixed cutoff energy of \(500\;eV\) while varying the number of \(\vec{k}\) points used to sample the first brillouin zone. Since all the reciprocal lattice vectors ( and real space lattice vectors) have the same length, we can specify the \(\vec{k}\) mesh by specifying the number of \(k\)-points used along each reciprocal lattice vector. Figure 6 shows this plot; and we can see that a \(8\times8\times8\) \(\vec{k}\) point is sufficient for the energy per ScAl dimer to have a convergence of \(\sim\;10\;m\,eV\). The resulting spacing between sampled \(k\) points is \(0.0357\;\mathring{A}^{-1}\). To ensure our subsequent calculations have the same degree of convergence, we will impose that the separation between two adjacent \(k\) points that are sampled along a reciprocal lattice vector is at most \(0.0357\;\mathring{A}^{-1}\).

Figure 5 : Plot of energy per ScAl dimer vs volume per ScAl dimer.

Now we are ready to calculate the energy per ScAl dimer and the corresponding volume per dimer, for both structures and various lattice parameters. Figure 6 shows the plot of energy per ScAl dimer vs volume per dimer, for ScAl in the CsCl and NaCl structure. From the plots in figure 6 it is clear that ScAl prefers CsCl structure over the NaCl structure. From the best fit line we obtain the optimal lattice parameter in the CsCl structure to be \(3.38\;\mathring{A}\). If ScAl were to be found in the NaCl structure the optimal lattice parameter would be \(4.00\;\mathring{A}\).

Experimentally ScAl has been verified to exist in CsCl structure with a lattice parameter of \(3.450\;\mathring{A}\) [1]. Our results verify this and estimate the lattice parameter within \(\sim\;2%\) of the experimentally determined lattice constant.

 

[1] O. Schob and E. Parthe. Ab Compounds with Sc Y and Rare Earth Metals. I. Scandium and
Yttrium Compounds with Crb and Cscl Structure. Acta Crystallographica, 19:214-&, 1965.

 

Determining the Lattice Constants of Hafnium

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Experimental results have reported that hafnium is a hcp metal with a lattice parameter ratio of c/a=1.58. [1] We verify this result with ab initio calculations; our calculations suggest that the lattice parameters of hafnium are a=3.12Å, c=4.94Å, assuming that hcp is the stable crystal structure of hafnium.

Figure 1 – The unit cell of hafnium

Methods and Results

We used the CASTEP code with the CA-PZ functional (Ceperley and Alder 1980, Perdew and Zunger 1981) for density functional theory calculations in the local density approximation (LDA). Ultrasoft pseudopotentials with core radii of 2.096Å were generated on-the-fly, and calculations in reciprocal space used a cutoff energy of 500eV and a 13×13×10 k-point mesh.

Atomic calculations were performed for the following core orbitals: 1s2 2s2 2p6 3s2 3p6 3d10 4s2 4p6 4d10 4f14 5s2 5p6 5d2 6s2

Pseudopotentials were used for the following valence orbitals: 4f14 5s2 5p6 5d2 6s2

Using twice the empirical atomic radius (1.59Å) as a starting point, we began searching for a minimum in the free energy with an energy cutoff of 435.4eV and 9×9×6 k-point mesh until we found an approximate minimum at a=3.12Å, to three significant figures. c was initially fixed with respect to a by the ratio c/a=1.58 and was updated as a varied to maintain this ratio. Once we had found this minimum, we increased energy cutoff and k-point sampling fineness until the final free energy had converged with a spead of less than ~0.001eV. We checked convergence by increasing cutoff energy first, then by increasing k-point sampling, and then checked that the cutoff energy had not been consequentially altered by the change in k-point sampling. We then returned to the minimization process to check that the same value of a minimized free energy. Having confirmed this, we minimized free energy by varying c with fixed a. After finding a minimum at c=4.94Å, as shown in Figure 2, we checked whether a=3.12Å still minimized free energy, and we confirmed that it did up to the level of precision established earlier. This is shown in Figure 3.

Figure 2 – Final minimization of the free energy over c

 

Figure 3 – Final minimization of the free energy over a

Discussion

Previously published results [3] indicate that a=3.20Å and c/a=1.582; our results corroborate the latter ratio (our c/a=1.583) but disagree significantly with the actual magnitudes of the lattice parameters. Given that our colleagues matched published values much more closely by using different functionals and parameters, such as the generalized-gradient approximation, we suspect that the LDA functional is inadequate for precision calculations, but a closer examination of LDA parameters reveal a way for this functional to be comparatively useful.

[1] D. Sholl and J. Steckel, Density Functional Theory: A Practical Introduction. (Wiley 2009)
[2] S. J. Clark, M. D. Segall, C. J. Pickard, P. J. Hasnip, M. J. Probert, K. Refson, M. C. Payne, “First principles methods using CASTEP”, Zeitschrift fuer Kristallographie 220(5-6) pp. 567-570 (2005)
[3] K. Hermann, Crystallography and Surface Structure: An Introduction for Surface Scientists and Nanoscientists. (Wiley-VCH, 2011)

Determination of the crystal structure with optimal lattice constant for Pt

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  1. Description of the problem

For this first project, we aim to predict the most-favored crystal structure of Pt and calculate the optimal lattice parameters for these structures.

Usually, the metal crystals can have simple cubic (sc), face centered cubic (fcc), and hexagonal close packed (hcp) structures. In this work, we will firstly examine the optimal lattice parameter for sc Pt based on the energy of bulk Pt, followed by the tests of on the fcc structure. Both optimal lattice parameter a and the ratio a/c will be determined for hcp Pt. Convergence tests will be done with respect to the number of k-points and the cutoff energy for all studies.

The Vienna Ab initio Simulation Package (VASP) is used to perform the periodic DFT calculations,1-3 employing the projected augmented-wave (PAW) pseudopotentials,4,5 as well as generalized gradient approximation with the exchange-correlation functional by Perdew, Burke, and Ernzerhof (PBE).6

  1. Simple cubic

The sc structure of Pt is built using the software Material Studio as shown in Figure 1. In each unit cell, there is one Pt atom.

               Figure 1. Unit cell of sc Pt                       Table 1. Results of k-points convergence for sc

Before we can calculate the energy of this whole system, the convergence tests are required with respect to the number of k-points and the cutoff energy. The criteria for choosing the number of k-points and energy cutoff in the following calculation is set to have the energy difference within 0.01 eV.

Firstly, in order to test the convergence of the number of k-points, we initially set the cutoff energy to 400 eV. The results of using number of k-points ranging from 1 to 120 are summarized above in Table 1. Figure 2 shows the trend of bulk energy as well as the computational cost versus the number of k-points.

Figure 2. Bulk energy and computational cost versus the number of k-points for sc

We can see the bulk energy becomes more stable with increased number of k-points while the computational cost keeps increasing. Considering the balance between higher accuracy and cost, the k-points sampling of 14x14x14 is chosen for further calculations.

The convergence of the cutoff energy is also tested as summarized in Table 2 with fixed k-points sampling shown above. An interesting thing is that we can see the (pseudo) atomic energy keeps becoming lower while the cutoff energy is increased (later we will see the same behavior for all other structures). Figure 3 shows the relationship of bulk energy and computational costs depending on the cutoff energy.

Table 2. Results of the cutoff energy convergence for sc

Figure 3. Bulk energy and computational cost versus the cutoff energy for sc

Similar as the cases for k-points sampling, the energy becomes stable with increased cutoff energy while the cost keeps increasing. The energy cutoff of 400 eV is chosen for further calculations.

In order to determine the optimal lattice parameter, we vary the lattice parameter to find the one resulting in lowest bulk energy. We firstly do a rough search using step size of 0.1 Å from 1.50 to 3.00 Å. With such rough search, we are able to determine the interval where the optimal value lies in and based on that, we can do a more precious search with step size of 0.01 Å. All the results are shown in Figure 4.

(a)                                                                            (b)

Figure 4. Bulk energy versus the lattice parameter for sc. (a) for rough search and (b) for detailed search.

From the first plot in Figure 4, we can see that the optimal value is in the interval from 2.50 to 2.70 Å. The detailed search is done in this interval with step size of 0.01Å. Finally, the optimal lattice parameter is determined to be 2.62 Å, which gives the lowest bulk energy of -5.655 eV for the simple cubic structure.

  1. Face centered cubic

The search for the optimal lattice parameter of fcc Pt is very similar to the study of sc Pt. The unit cell of fcc Pt is built with Material Studio as shown in Figure 5, containing on Pt atom.

               Figure 5. Unit cell of fcc Pt                    Table 3. Results of k-points convergence for fcc

As we did before, the convergence tests are firstly made and the results on k-points are summarized in Table 2. Figure 6 shows the relationship of bulk energy as well computational time to the number of k-points. Similarly, the k-points sampling of 12x12x12 is finally used.

Figure 6. Bulk energy and computational cost versus the number of k-points for fcc

The same tests on cutoff energy is done as summarized in Table 4 and Figure 7. The cutoff energy we use for further calculations of fcc Pt remains 400 eV.

Table 4. Results of the cutoff energy convergence for fcc

Figure 7. Bulk energy and computational cost versus the cutoff energy for fcc

The search for the optimal lattice parameter is carried out in a similar way, but roughly ranging from 3.10 to 4.30 Å. The results are shown in Figure 8. We can see from the left side of Figure 8 that the optimal lattice parameter lies in the interval from 3.90 to 4.10 Å. Thus, the optimal lattice parameter is determined with the detailed search using step size of 0.01Å. The optimal lattice parameter is 3.97 Å, which gives the lowest bulk energy of -6.097 eV for the face centered cubic structure.

(a)                                                                            (b)

Figure 8. Bulk energy versus the lattice parameter for fcc. (a) for rough search and (b) for detailed search.

  1. Hexagonal close packed

The difference in determining the optimal lattice parameter for hcp Pt is that there will be different optimums for different c/a. So in this section, we will compare the opmital lattice parameter for several potential c/a ratios (in this section, cases of c/a=1.57, 1.60, 1.63, 1.67, 1.70, 1.73 will be studied) and find the most-favored one which gives us the lowest bulk energy among them. Similarly, the hcp unit cell is consturcted using Material Studio as shown in Figure 9 containing 2 Pt atoms.

Figure 9. Unit cell of hcp Pt

The case of c/a=1.60 is chosen to test the convergence. The results are summarized below in Table 5. Figure 10 and 11 show the trends of bulk energy as well as computational cost with respect to the number of k-points and cutoff energy, respectively.

Table 5. Results of the convergence tests for hcp

Figure 10. Bulk energy and computational cost versus the number of k-points for hcp

Figure 11. Bulk energy and computational cost versus the cutoff energy for hcp

Accordingly, the k-points sampling is chosen to be 10x10x6 and the cutoff energy is 400 eV.

The search for optimal lattice parameter is achieved using the same method, but with different c/a values roughly ranging from 2.00 to 3.30 Å. Figure 12 shows the results of rough search for different c/a values.

Figure 12. Bulk energy versus the lattice parameter for hcp (rough search)

We can see that, for the case of c/a=1.56, the optimum lies in the interval from 2.80 to 3.00 Å. For all other cases, the optimal parameter is in the interval from 2.70 to 2.90 Å. The results for corresponding detailed search is shown in Figure 13 below. The final results reveal that the optimal lattice parameter for hcp Pt is 2.76 Å with c/a=1.73, giving the lowest bulk energy of -6.046 eV.

Figure 13. Bulk energy versus the lattice parameter for hcp (detailed search)

  1. Conclusion

According to the results above, we know that the optimal lattice parameters for these three different structures are 2.62, 3.97, and 2.72 Å, respectively. Among them, the fcc structure with lattice parameter of 3.97 Å gives the lowest bulk energy of -6.097 eV (-5.655 and -6.046 eV for sc and hcp). The experimental lattice constant is 3.92 Å. Our result is about 1.01% larger than the experimental observation value, which is commonly seen while using PBE functional as PBE tends to overestimate the lattice constant.

Reference

  1. Kresse, G. and J. Hafner, Ab initio molecular dynamics for liquid metals. Physical Review B, 1993. 47(1): p. 558.
  2. Kresse, G. and J. Hafner, Ab initio molecular-dynamics simulation of the liquid-metal–amorphous-semiconductor transition in germanium. Physical Review B, 1994. 49(20): p. 14251.
  3. Kresse, G. and J. Furthmüller, Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set. Physical review B, 1996. 54(16): p. 11169.
  4. Kresse, G. and D. Joubert, From ultrasoft pseudopotentials to the projector augmented-wave method. Physical Review B, 1999. 59(3): p. 1758.
  5. Blöchl, P.E., Projector augmented-wave method. Physical review B, 1994. 50(24): p. 17953.
  6. Perdew, J.P., K. Burke, and M. Ernzerhof, Generalized gradient approximation made simple. Physical review letters, 1996. 77(18): p. 3865.

Determination of the Lattice Parameter of ScAl in the CsCl Structure

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This Project aims to predict the lattice constant of ScAl with CASTEP calculation.

A. Project Description 

In this study, we focus on predicting the lattice constant of  ScAl based on the CsCl structure, and figuring out a converged energy cutoff and k-points in the CASTEP energy calculation. Atomic electron configuration for Al is 3s2 3p1, for Sc is 3s2 3p6 3d1 4s2. The energy calculation in CASTEP can provide a reasonable crystal structure for ScAl. The energy calculation in this study is based on the exchange correlation Perdew-Burke-Ernzerhof (PBE) density functionale, which is from the class of Generalized gradient approximation (GGA) functional. The relationships between energy and lattice parameters, energy cutoff, and k-points are discussed below.

B. Crystal Model

ScAl has CsCl structure, where Scandium (Sc) locates at the corner and Aluminum (Al) in the center of the unit cell. This structure belongs to the cubic system, the lattice parameter a=b=c, α=β=γ=90 degrees.

The cell vectors are along x, y, direction of the unit cell, with orthogonal \(a_i\) (1,0,0), \(a_j\)(0,1,0), and \(a_k\)(0,0,1) respectively. The equivalent fractional coordinate of the Sc is (0,0,0) whereas the Al is (1/2, 1/2, 1/2). The real coordinates of Sc and Al in the unit cell depend on the lattice parameter a, with Sc at (0,0,0), (a,0,0), (0,a,0), (0,0,a), (a,a,0), (a,0,a), (0,a,a), and (a,a,a); Al at (a/2,a/2,a/2). Figure 1 shows an example of Sc and Al positions with the lattice parameter a=b=c=3.379 Å.

Figure 1. Simple cubic structure of ScAl, where Aluminum locates in the center and Scandium in the corner of the unit cell.

C. Determine the Lattice Parameters of ScAl

In order to predict a reasonable lattice parameter of ScAl, the energy of the unit cell is calculated with the variation on lattice parameters. Given the structure is from the cubic system, the lattice parameters a, b, and C are equal and will be referred to “a” as below. Before the calculation, the lattice parameter a is estimated based on the atomic radius of Al (1.43 angstrom) and Sc (2.30 angstrom). To get a well packed structure along the body diagonal in (111) face, the lattice parameter should be smaller than 2(r(Al)+r(Sc))/√3, which is 4.3 Å.

As a starting point the energy of the ScAl structure was calculated witha lattice parameter of  4.3Å using the CASTEP calculation ( Energy cutoff 500 eV, k-points 10*10*10). With the decreased lattice parameter from the starting point, the free energy of the unit cell reached a minimum to some point and then increased with the lattice parameter decreased further (Figure 2). This shape of curve is caused by the relative atom positions, either too far or too close, generateing higher energy (less stable structure) than the minimum energy (the most stable structure by calculation). The lattice parameter \(a_0\)= 3.379 Å corresponds the minimum cohesive energy (Figure 2).

Figure 2. Cohesive energy for simple cubic ScAl, using 10*10*10 kpoints and 500eV energy cutoff, as a function of lattice parameter


D. The Energy Cutoff

Multiple calculations for ScAl structure were completed with a variation of energy cutoff \(E_{cut}\) from 200eV to 800eV. Lattice parameter and kpoints remained the same at 3.379 Å and 8*8*8 k-point grid for Brillouin zone integration respectively. An increase in the energy cutoff increases the number of plane-waves and improves the accuracy of ion cores, but costs longer computation time. Repeated calculations with higher energy cutoff aim to converge to a decent final free energy (Figure 3&4). Convergence is reached with an energy cutoff of 500eV providing an accuracy of the absolute energy better than 0.001eV (Figure 4).

Figure 3. Both Al and Sc atomic energy converged at 500eV with a lattice parameter 3.379 Å and 8*8*8 k-point grid for Brillouin zone integration.
Figure 4. Energy per cell and cohesive energy of ScAl as a function of energy cutoff with a lattice parameter 3.379 Å and 8*8*8 k-point grid for Brillouin zone integration.

The cohesive energy of a solid material is the energy to separate the condensed material into isolated free atoms.

\begin{equation}E_{coh}=(E_{total}-E_{atom})/N\end{equation}
with \(E_{total}\)is the total energy of the unit cell, \(E_{atom}\) is the total energy of the atom, and N as the number of atoms in the unit cell. Table 1 and Figure 4 indicate that the cohesive energy decreased with higher energy cutoff, converged to an accuracy of  0.01eV at energy cutoff of 500eV.

Table 1 Results from computing the total energy, atomic energy, and cohesive energy of simple cubic ScAl with 8*8*8 kpoints in different energy cutoffs

 

E. k-points

As k-point varied but energy cutoff (500 eV) and lattice parameter (3.379 Å) unchanged in the following calculation, the total energy of the cell converged with more kpoints.  The cell volume and atomic energy remained the same, because the cell volume only depends on lattice parameter and atomic energy depends on the energy cut-off.

Figure 5. Energy per cell and cohesive energy of ScAl as a function of M*M*M k-points, with a lattice parameter 3.379 Å and energy cutoff 500eV.

The used k points are reduced by symmetry operations using Monkhorst-Pack approach with M*M*M k points in ScAl cubic structure. Table 2 and Figure 5 shows the number of k poins in irreducible Brillouin zone (IBZ) and correspondent total energy and binding energy. Both the odd (2n+1) and even (2n+2) values of M have the same number of k points in IBZ, but even values (2n+2) of M converges better than odd values in regard to the same computational time. In our cases, both the 7*7*7 and 8*8*8 k points required 8.08 seconds to finish the calculation, but 8*8*8 converging better because all k points are inside of the IBZ (Sholl and Steckel, 2011). In ScAl structure, 12*12*12 k points are enough to get accurate energy.

Table 2 Energy calculation with k points varies, energy cutoff 500eV, lattice parameter 3.379 Å.

 

Conclusion:

The total energy of simple cubic ScAl minimized at lattice parameter a=3.379 Å. Energy per cell converges when energy cutoff is 500eV, k-points are 12*12*12.

 

Reference

[1] Sholl, D. and Steckel, J.A., 2011. Density functional theory: a practical introduction. John Wiley & Sons.

[2] BIOVA, 2014. CASTEP guide , Material Studio. http://www.tcm.phy.cam.ac.uk/castep/documentation/WebHelp/content/pdfs/castep.htm

 

Searching for the Lattice Parameter of ScAl

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Lattice Structure description: 

ScAl has the same type of structure that CsCl, which is simple cubic cell. The cell vector could be (1, 0, 0), (0, 1, 0), (0, 0, 1). Fractional coordinates for Cs could be (0, 0, 0) and for Al could be (0.5, 0.5, 0.5). Diagram for this structure is shown in Fig 1.

Fig1 Golden spheres are Sc, the brown shere is Al.

Since ScAl has simple cubic structure, We just need to adjust lattice parameter a in order to predict its structure. Idea is calculate total free energy for different parameters and find out the energetically favorable one.

DFT calculation is adopted for this search.

Cutoff energy test: 

A test is done to find a proper cutoff energy. Fixing other setting: functional: GGA PBE, k points: 6*6*6, lattice parameter a=4.0Å, we change cutoff energy and compare their total free energy results. Results are shown in Table 1. Energy difference between cutoff energy ‘460 eV’ and ‘560 eV’ is less than 0.02eV. If time cost in considered and total free energy resolution is controlled at 0.02 eV, using ‘460 eV’ for cutoff energy for following calculations is an acceptable choice.

cutoff energy(eV) total free energy(eV)
60 -1140.936931
160 -1347.963162
260 -1379.612688
360 -1383.327654
460 -1383.662277
560 -1383.681389
660 -1383.681690

Table 1

K point test:

functional: GGA PBE, cutoff energy: 460eV are fixed and lattice parameter a is changed.

K points in default will change with lattice parameter. (CASTEP tool is used here, ‘default’ meaning default number for k points in CASTEP tool)

Results are shown in Table 2 .

lattice parameter a(Å) k points total free energy (eV)
2.0000 14*14*14 -1336.132886
2.6000 10*10*10 -1376.299647
2.7000 10*10*10 -1379.066514
2.8000 10*10*10 -1381.168776
2.9000 10*10*10 -1382.723256
3.0000 8*8*8 -1383.831689
3.1000 8*8*8 -1384.578369
3.2000 8*8*8 -1385.031976
3.3000 8*8*8 -1385.254431
3.3600 8*8*8 -1385.298682
3.3700 8*8*8 -1385.300556
3.3750 8*8*8 -1385.300611
3.3800 8*8*8 -1385.300591
3.3850 8*8*8 -1385.299964
3.3900 8*8*8 -1385.29928
3.4000 8*8*8 -1385.296417
3.4200 8*8*8 -1385.287388
4.0000 6*6*6 -1383.662277
5.0000 6*6*6 -1379.859133
6.0000 4*4*4 -1377.783054

Table 2

If density of k points is defined as number of k points in one direction over k space parameter in that direction, this according change of k point might have a purpose of keeping density of k point unchanged. Since the lengths of lattice vector in cell and lattice vector in k space have inverse proportion relation. So in this simple cubic system, expectation would be that number of k points in one direction times lattice parameter ‘a’ should lead to a constant. Obviously, this expectation is not obeyed in this test.

K points will effect the precision and time cost of a calculation, so finding a balance point of precision and efficiency means  we need to find a suitable k points. This ‘finding a balance’ situation occurs as well when we deal with cutoff energy.

So which k point choice is suitable for this calculation? We can discuss this based on calculation results.

Fig 2 and Fig 3 show the search for lattice parameter. Relatively, one is rough, the other is fine.

Fig 2

Fig 3

We can see the parameter range which is located at energy valley is (3.36, 3.40). At this range, the k point is set as ‘8*8*8’ and in this range the finest search step is 0.005Å.

In ‘cutoff energy test’, ‘460 eV’ is used for cutoff energy so that resolution for total free energy is set to ‘0.02 eV’. Please notice that the ‘0.02 eV’ resolution actually also includes the setting of k points as ‘6*6*6’. And in the range we care about most adopts ‘8*8*8’ k point setting which should give precise enough results for this search. Energy numbers in table 2 for range (3.36, 3.40) do have difference less than 0.02 eV, which actually is less than 0.002 eV. So we can say that if ‘460 eV’ is adopted for cutoff energy, ‘8*8*8’ k point setting is ‘safe enough’ for this calculation. Of course, accordingly, it will be dangerous to make a prediction for lattice parameter beyond the precision of ‘0.005Å’.

Convergence test, however, is still done for k points, at a=4.0Å , cutoff energy=460 eV. Results are shown in Table 3.

k points total free energy(eV)
4*4*4 -1383.589307
5*5*5 -1383.619788
6*6*6 -1383.662277
7*7*7 -1383.635492
8*8*8 -1383.633134
9*9*9 -1383.639153
10*10*10 -1383.636616
11*11*11 -1383.63629
12*12*12 -1383.637744

Table 3

From data in this table, total free energy’s difference between ‘8*8*8’ and ‘9*9*9’ is less than 0.02 eV, which supports the point that ‘8*8*8’ setting for k points is precise enough under resolution of 0.02 eV for total free energy.  Consistent with expectation, with increasing number of k points, we have smaller energy difference.

‘8*8*8′ for k points is adopted for lattice parameters outside (3.36, 3.40) in order to constrain variables when comparing different parameters’ energy. And for parameters in (2.00, 2.90), calculations have larger k points so it would be meaningless to re-calculate these points. Just using ‘8*8*8’ k points re-calculate points with a=4.00, 5.00, 6.00 Å. Results and comparison are shown in Table 4.

lattice parameter(Å) total free energy with 8*8*8 k points(eV) total free energy with default k points(eV)
4.000 -1383.633134  -1383.662277
5.000 -1379.864649 -1383.662277
6.000 -1377.763773  -1377.783054

Table 4

From data in the table, we can see that with ‘8*8*8’ k points, total free energy for these points goes higher, which does not affect our search for lowest energy point.

Conclusion:

If ‘460 eV’ cutoff energy and ‘0.02 eV’ precision for total free energy is adopted, ‘8*8*8’ k points setting could provide precise enough for the search of lattice parameter. At the same time the precision of this parameter search is limited at ‘0.005Å’.

Based on the calculation results and just considering minimizing total free energy, ScAl should have a lattice parameter around 3.375Å.

If more decimal place is wanted for this prediction, larger cutoff energy and k points should be adopted.

Reference:

First principles methods using CASTEP. Zeitschrift fuer Kristallographie 220(5-6) pp. 567-570 (2005) S. J. Clark, M. D. Segall, C. J. Pickard, P. J. Hasnip, M. J. Probert, K. Refson, M. C. Payne