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)
