Atomistic Modelling and Simulation Hands On Report

SHANGHAI JIAO TONG UNIVERSITY, SHANGHAI CN Atomistic Modelling and Simulation Hands On Report 15-01-2018 Submitted to: Prof. Lingti Kong Submitted by: Muhammad Khizar Shafique 117050990004 410D, School of Materials Science and Engineering Table of Contents Task 1: Single point energy calculation for Cu ............................................................................................ 3 Method for energy calculation w/o any Convergence Tests .................................................................. 3 Results for energy calculation w/o Convergence Tests ...................................................................... 4 Method for energy calculation w.r.t. k point convergence .................................................................... 4 Results for energy calculation w.r.t. k point convergence ................................................................. 5 Method for energy calculation w.r.t. E-cut convergence ....................................................................... 6 Results for energy calculation w.r.t. E-cut convergence ..................................................................... 6 Method for energy optimization w.r.t. K-point & E-cut convergence ................................................... 7 Results for energy optimization w.r.t. K-point and E-cut convergence ............................................. 8 Analysis ..................................................................................................................................................... 8 Task 2: Lattice Constant, Bulk Modulus & Band Structure for Cu .............................................................. 9 Method for Finding Bulk Modulus and Lattice Constant for Cu ............................................................. 9 Results for Bulk Modulus and Lattice Constant for Cu ..................................................................... 10 Method for Finding Band Structure for Cu ............................................................................................ 11 Results for Band Structure for Cu ...................................................................................................... 12 Analysis ................................................................................................................................................... 12 Task 3: Lattice Constant, Bulk Modulus & Band Structure for Fe ............................................................. 13 Method for Finding Bulk Modulus and Lattice Constant for Fe ........................................................... 13 Results for Bulk Modulus and Lattice Constant for Fe ...................................................................... 14 Method for Finding Band Structure for Fe ............................................................................................ 16 Results for Band Structure for Fe ....................................................................................................... 17 Analysis ................................................................................................................................................... 17 Task 4: Energy Levels, Bond Length and Bond Energy for H ..................................................................... 18 Method of Finding Energy Levels for Hydrogen .................................................................................... 18 Results for Energy Levels of Hydrogen .............................................................................................. 19 Method of Finding Bond Length and Bond Energy for Hydrogen......................................................... 20 Results for Bond Length and Bond Energy of Hydrogen ................................................................... 21 Analysis ................................................................................................................................................... 21 Task 5: Band Structure for Graphene ........................................................................................................ 22 Method for constructing Band Structure for Graphene ....................................................................... 22 Results for Band Structure of Graphene............................................................................................ 23 Task 6: Lattice Constant, Bulk Modulus & Band Structure for Al ............................................................. 24 Method for Finding Bulk Modulus and Lattice Constant for Al ............................................................ 24 1|Page Results for Bulk Modulus and Lattice Constant for Aluminum ........................................................ 25 Method for Finding Band Structure ....................................................................................................... 26 Results for Band Structure of Al ........................................................................................................ 27 Analysis ................................................................................................................................................... 27 Task 7: Energy Level, Bond Length and Bond Energy for N ...................................................................... 28 Method of Finding Energy Levels for Nitrogen ..................................................................................... 28 Results for Energy Levels of Nitrogen ................................................................................................ 29 Method of Finding Bond Length and Bond Energy for Hydrogen......................................................... 30 Results for Bond Length and Bond Energy of Hydrogen ................................................................... 31 Analysis ................................................................................................................................................... 32 Summary ..................................................................................................................................................... 32 2|Page Task 1: Single point energy calculation for Cu In this task we have to do convergence with various methods to optimize energy values for Copper We will do this task by three methods: 1. Without convergence tests 2. With convergence test using K-point convergence a. Convergence is not variational and frequently oscillates. b. Even simple metals like Al need dense meshes for primitive cell. c. Finite-temperature smearing can accelerate convergence, but must extrapolate the result back to 0 K. d. In case of insulators some k-point error cancellation occurs but only between identical simulation cells. e. Consequently comparative phase stability energetics and surface energetics frequently demand high degree of k-point convergence. 3. With convergence test using Cut-off energy a. Cutoff determines highest representable spatial Fourier component of density. b. Energy differences converge faster than total energies. Electron density varies most rapidly near nuclei, where it influences bonding very weakly. c. Ecut (and Gmax) depend only on types of atoms, not numbers. d. Simulation cutoff is maximum of individual elements/pseudo-potentials. e. Required cutoff is system-independent but not property-independent. In the end we will see how the final optimization is done using convergence methods Method for energy calculation w/o any Convergence Tests The input code will be mostly the same for all the convergence test: &control calculation restart_mode pseudo_dir outdir prefix tstress tprnfor / &system ibrav celldm(1) nat ntyp ecutwfc ecutrho occupations smearing degauss = = = = = = = 'scf' 'from_scratch', '$PSEUDO_DIR/', '$TMP_DIR/' 'Cu' .true. .true. = = = = = = = = = 2, 6.73, 1, 1, 25.0, 300.0 'smearing', 'gaussian', 0.02 3|Page / &electrons diagonalization = 'david' conv_thr = 1.0e-6 mixing_beta = 0.7 / ATOMIC_SPECIES Cu 63.55 Cu.pbe-n-van_ak.UPF ATOMIC_POSITIONS Cu 0.0 0.0 0.0 K_POINTS (automatic) 8 8 8 0 0 0 Results for energy calculation w/o Convergence Tests Method for energy calculation w.r.t. k point convergence The code for k-point convergence is almost the same as the previous one with just one difference; the K-points are specifically defined at the end of the code as follows: K_POINTS (automatic) 8 8 8 0 0 0 K_POINTS (automatic) ${nk} ${nk} ${nk} 0 0 0 This sort of code will calculate an equally spaced mesh and put in the proper k-points automatically. Later we can also form a graph using this code for the required results by adding the command for an output data file. 4|Page Results for energy calculation w.r.t. k point convergence We can get two types of results in k-point convergence test. The first one is in the form of values: The second one is in the form of graph plotted using the info.dat file in GNUPLOT as follows: We can clearly see from the graph how the energy is converged with increasing iterations. 5|Page Method for energy calculation w.r.t. E-cut convergence In this method, we change the system values and replace the set a self variating value of ecutwfc as follows: &system ibrav celldm(1) nat ntyp ecutwfc ecutrho occupations smearing degauss = = = = = = = = = 2, 6.73, 1, 1, 25.0, 300.0 'smearing', 'gaussian', 0.02 &system ibrav celldm(1) nat ntyp ecutwfc ecutrho occupations smearing degauss = = = = = = = = = 2, 6.73, 1, 1, ${ec}, 300.0 'smearing', 'gaussian', 0.02 This will variate the appropriate value of ecut converge the energy to provide exact values Results for energy calculation w.r.t. E-cut convergence We get two types of results for this simulation. One is in the form of values of energy: 6|Page The other one is in the form of graph plotted by info.dat file using GNUPLOT We can observe the convergence with increasing iterations in the above graph. Method for energy optimization w.r.t. K-point & E-cut convergence In the K-point method, the system of selecting k-point coordinates set it to 15x15x15. In E-cut method, the system of selecting E-cut value set it to 50 Ry. If we apply both systems at the same time for optimization, we get the k-point coordinates to be 12x12x12 and the E-cut value to 30 Ry. It means that the most optimum value for Fermi Energy will be obtained on these set value. We’ll put these values in the source code and run the simulations to calculate results. 7|Page Results for energy optimization w.r.t. K-point and E-cut convergence We can get the value of Total Energy and Fermi Energy as a result of the last simulation. Analysis We have observed the following values in these simulations: Fermi Energy Total Energy w/o Convergence w.r.t K-point Convergence w.r.t. E-Cut Convergence 13.3999 -121.4857 13.4989 -121.4878 13.3880 -121.4883 w.r.t. K-point and E-cut Convergence 13.3868 -121.4879 We can observe that the optimized value using both K-point and E-cut Convergence test have given the best value for Fermi Energy. Usually it depends on the circumstances which methods is better for convergence, however optimizing by using both methods will always provide better values. 8|Page Task 2: Lattice Constant, Bulk Modulus & Band Structure for Cu Find the Equilibrium Lattice Constant and Bulk modulus for FCC Copper and compare with MD results. Also find the Band Structure for copper. Method for Finding Bulk Modulus and Lattice Constant for Cu For finding the lattice constant and bulk modulus, we used “Cu.pbe-n-van_ak.UPF” pseudo potential among various pseudo potentials provided by Espresso 5.0.2. Following is the code for the input file: &control calculation restart_mode pseudo_dir outdir prefix tstress tprnfor = = = = = = = 'scf' 'from_scratch', '/opt/espresso-5.0.2/pseudo/', '/tmp/ex02.46/' 'Cu' .false. .false. / &system ibrav celldm(1) nat ntyp ecutwfc ecutrho occupations smearing degauss = = = = = = = = = 2, 7.4, 1, 1, 30.0, 300.0 'smearing', 'gaussian', 0.01 / &electrons diagonalization = 'david' conv_thr = 1.0e-9 mixing_beta = 0.7 / ATOMIC_SPECIES Cu 63.55 Cu.pbe-n-van_ak.UPF ATOMIC_POSITIONS Cu 0.0 0.0 0.0 K_POINTS (automatic) 12 12 12 0 0 0 Later the results are fed to another environmental variable fit_pwscf which fits the output file to a graph and produce values accordingly 9|Page Results for Bulk Modulus and Lattice Constant for Cu We will obtain two type of results from this simulation. One will be the values of Bulk Modulus, Lattice Constant and various other parameters: Bulk Modulus Lattice Energy Equilibrium Volume Lattice Constant The second result is the graph between Equilibrium Volume and Lattice Energy 10 | P a g e Method for Finding Band Structure for Cu For finding the band structure, we used “Cu.pbe-n-van_ak.UPF” pseudo potential among various pseudo potentials provided by Espresso 5.0.2. Following is the code for the input file: &control calculation pseudo_dir outdir prefix verbosity / &system ibrav celldm(1) nat ntyp ecutwfc ecutrho nbnd occupations smearing degauss / = = = = = = = = = = = = 'bands' '/opt/espresso-5.0.2/pseudo/', '/tmp/ex03.51/', 'Cu' 'high' 2, 6.867, 1, 1, 30.0, 300.0, 7 = 'smearing', = 'gaussian', = 0.01 &electrons diagonalization = 'david' / ATOMIC_SPECIES Cu 63.55 Cu.pbe-n-van_ak.UPF ATOMIC_POSITIONS (crystal) Cu 0.0 0.0 0.0 K_POINTS crystal_b 6 0.0000000 0.0000000 0.5000000 0.0000000 0.5000000 0.2500000 0.5000000 0.5000000 0.0000000 0.0000000 0.3750000 0.3750000 0.0000000 0.5000000 0.7500000 0.5000000 0.0000000 0.7500000 25 10 15 20 30 10 11 | P a g e Results for Band Structure for Cu As a result of this simulation we obtain the band structure of FCC Copper: The band structure is obtained by using the same pseudo-potential as that of the Bulk Modulus and Lattice Energy values. The intermingling of band structure clearly shows the conductive nature of Aluminum. However these band structures can vary if different pseudo-potentials are used. Analysis Following is a comparison of Molecular Dynamics and Atomistic Calculation results for Copper Calculation Atomistic Molecular Dynamic Difference Bulk Modulus 136.19 137.23 0.7% Lattice Constant 3.63 3.61 0.5% There are a number of factors that can cause this difference e.g. the values largely depend on the pseudo-potentials used in the calculation. It is entirely possible that there other pseudopotentials that can provide values closer to those of MD. There is also possibility of error in the MD results. 12 | P a g e Task 3: Lattice Constant, Bulk Modulus & Band Structure for Fe Find the Equilibrium Lattice Constant and Bulk modulus for BCC Iron and compare with MD results. Also find the Band Structure for iron. Method for Finding Bulk Modulus and Lattice Constant for Fe For finding the lattice constant and bulk modulus, we used “Fe.pbe-sp-van_ak.UPF” pseudo potential among various pseudo potentials provided by Espresso 5.0.2. Following is the code for the input file: &control calculation restart_mode pseudo_dir outdir prefix tstress tprnfor / = = = = = = = 'scf' 'from_scratch', '/opt/espresso-5.0.2/pseudo/', '/tmp/ex04.37/' 'Fe' .false. .false. &system ibrav = 3, celldm(1) = 5.6, nat = 1, ntyp = 1, ecutwfc = 30.0, ecutrho = 300.0 nspin = 2, starting_magnetization = 2.5 occupations = 'tetrahedra', degauss = 0.01 / &electrons diagonalization = 'david' conv_thr = 1.0e-9 mixing_beta = 0.7 / ATOMIC_SPECIES Fe 55.85 Fe.pbe-sp-van_ak.UPF ATOMIC_POSITIONS Fe 0.0 0.0 0.0 K_POINTS (automatic) 11 11 11 0 0 0 Later the results are fed to another environmental variable fit_pwscf which fits the output file to a graph and produce values accordingly 13 | P a g e Results for Bulk Modulus and Lattice Constant for Fe We will obtain two type and two sets of results from this simulation. One would be for the magnetic properties of iron: Bulk Modulus Lattice Energy Equilibrium Volume Lattice Constant These values and the graph between energy and volume are for the magnetic state of Iron. But since iron also has a non-magnetic state, we will get entirely different set of values for the nonmagnetic state of iron: 14 | P a g e Bulk Modulus Lattice Energy Equilibrium Volume Lattice Constant Since we have two sets of values, we will only consider the magnetic state values, because magnetic state is more dominant and also more stable for the system. 15 | P a g e Method for Finding Band Structure for Fe For finding the band structure, we used “Fe.pbe-sp-van_ak.UPF” pseudo potential among various pseudo potentials provided by Espresso 5.0.2. Following is the code for the input file: &control calculation pseudo_dir outdir prefix verbosity / = = = = = 'bands' '/opt/espresso-5.0.2/pseudo/', '/tmp/ex05.65/', 'Fe' 'high' &system ibrav = 3, celldm(1) = 5.40389170599231, nat = 1, ntyp = 1, ecutwfc = 30.0, ecutrho = 300.0, nspin = 2, starting_magnetization = 2.0 nbnd = 12, occupations = 'smearing', smearing = 'gaussian', degauss = 0.01 / &electrons diagonalization = 'david' / ATOMIC_SPECIES Fe 55.85 Fe.pbe-sp-van_ak.UPF ATOMIC_POSITIONS (crystal) Fe 0.0 0.0 0.0 K_POINTS crystal_b 7 0.0000000 0.0000000 0.5000000 -0.5000000 0.0000000 0.0000000 0.0000000 0.0000000 0.2500000 0.2500000 0.5000000 -0.5000000 0.2500000 0.2500000 0.0000000 0.0000000 0.0000000 0.5000000 0.5000000 0.0000000 0.2500000 0.5000000 0.2500000 0.5000000 15 10 10 10 15 15 10 15 16 | P a g e Results for Band Structure for Fe As a result of this simulation we obtain the band structure of BCC Iron: Analysis First, we will compare the MD and Atomistic Computation Values for Iron (the magnetic ones only as explained before): Calculation Atomistic Molecular Dynamic Difference Bulk Modulus 146.63 177.53 17% Lattice Constant 2.86 2.86 0% As we have observed, there are two distinct bands of energy in the band structure for Iron. This is because one of them is for magnetic state and the other one is for non-magnetic state. If we observe closely, the magnetic band (blue color) is tracing the same way as non-magnetic band (red color) but just a bit higher energy showing the dominance of magnetic state in Iron. 17 | P a g e Task 4: Energy Levels, Bond Length and Bond Energy for H Find the Energy levels of Hydrogen Atom and find the Bond Length and Bond Energy for a Hydrogen Molecule Method of Finding Energy Levels for Hydrogen For finding the energy levels for hydrogen atom we used “H.pbe-rrkjus.UPF” pseudo-potential among various pseudo potentials provided by Espresso 5.0.2. Following is the code for the input file: &control verbosity calculation restart_mode pseudo_dir outdir prefix / = = = = = = 'high' 'scf' 'from_scratch', '/opt/espresso-5.0.2/pseudo/', '/tmp/ex06.86/' 'H' &system ibrav = 1, celldm(1) = 30, nat = 1, ntyp = 1, ecutwfc = 30.0, ecutrho = 300.0 occupations = 'smearing' smearing = 'gaussian' degauss = 0.01 nspin = 2 starting_magnetization(1) = 1 / &electrons diagonalization = 'cg' conv_thr = 1.0e-10 mixing_beta = 0.7 / ATOMIC_SPECIES H 1.00794 H.pbe-rrkjus.UPF ATOMIC_POSITIONS H 0.0 0.0 0.0 K_POINTS (gamma) 18 | P a g e Results for Energy Levels of Hydrogen Apart from visualizing the energy levels there is another interesting thing in the output file. The values of Spin Up and Spin Down are different: This difference indicates that there are some magnetic forces present which are disturbing the equilibrium of both spins. Moreover, magnetic state is more stable than non-magnetic state. 19 | P a g e Method of Finding Bond Length and Bond Energy for Hydrogen In this method we used “H.pbe-rrkjus.UPF” pseudo-potential among various pseudo potentials provided by Espresso 5.0.2. We will do relax calculation on a cubic lattice. Because of relaxation, the atoms will try to optimize the distance between the atoms. The following code is used as input: &control calculation restart_mode pseudo_dir outdir prefix = = = = = 'relax' 'from_scratch', '/opt/espresso-5.0.2/pseudo/', '/tmp/ex07.95/' 'H2' / &system ibrav celldm(1) nat ntyp ecutwfc ecutrho occupations smearing degauss = = = = = = = = = 1, 30, 2, 1, 30.0, 300.0 'smearing' 'gaussian' 0.02 / &electrons diagonalization = 'cg' conv_thr = 1.0e-10 mixing_beta = 0.7 / &ions / ATOMIC_SPECIES H 1.00794 H.pbe-rrkjus.UPF ATOMIC_POSITIONS (crystal) H 0.518 0.518 0.518 H 0.482 0.482 0.482 K_POINTS (gamma) We can observe that there are some differences as compared to the previous code 1. The calculation is changed from scf to relax 2. No verbosity is included 3. Atomic Positions are assigned in 3D space for both atoms 20 | P a g e Results for Bond Length and Bond Energy of Hydrogen For Bond Energy 𝐸𝑏 = 𝐸𝐻2 − 2𝐸𝐻1 = −2.3304 − 2(−0.9985) = −0.3334 𝑅𝑦 For bond length, we found the following coordinates: If we plot these points in the cartesian coordinates, we can see how the atoms are becoming closer by relaxation. For calculation of bond length we use the formula for finding distance between two points: 𝑑 = √(𝑥1 − 𝑥2 )2 + (𝑦1 − 𝑦2 )2 + (𝑧1 − 𝑧2 )2 𝑑 = 0.0426 𝐵𝑜ℎ𝑟 × 30 × 0.529 = 0.75Å 0.52 0.515 0.51 In this calculation “30” is the cell dimension used in the input code and “0.529” is the conversion factor for converting Bohr to Angstrom 0.505 0.5 0.495 0.49 0.485 0.48 0.48 0.485 0.49 0.495 0.5 0.505 0.51 0.515 0.52 Analysis Following is a table to compare the experimental and computational values of bond length and bond energy: Atomistic Computations Experimental Values Difference Bond Energy -0.3334 Ry -0.3329 Ry 0.15% Bond Length 0.75 Å 0.74 Å 1.3% The resultant computational values are quite similar to the experimental values showing that the computations were quite accurate. 21 | P a g e Task 5: Band Structure for Graphene Construct Band Structure for Graphene Method for constructing Band Structure for Graphene For finding the band structure, we used “Si.pbe-rrkj.UPF” pseudo potential among various pseudo potentials provided by Espresso 5.0.2. Following is the code for the input file: &control calculation pseudo_dir outdir prefix / = = = = &system ibrav celldm(1) celldm(3) nat ntyp ecutwfc ecutrho nbnd / = 4, = 9.68, = 5, = 2, = 1, = 25.0, = 300.0, = 7 'bands' '/opt/espresso-5.0.2/pseudo/', '/tmp/ex07.69/', 'graphene' &electrons diagonalization = 'david' / ATOMIC_SPECIES Si 12.00 Si.pbe-rrkj.UPF ATOMIC_POSITIONS (crystal) Si 0.0 0.0 0.0 Si 0.3333333333333333 0.6666666666666666 0.0 K_POINTS crystal_b 5 0.0000000 0.0000000 0.0000000 0.5000000 0.0000000 0.0000000 0.3333333 0.3333333 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 10 10 10 10 10 22 | P a g e Results for Band Structure of Graphene The above images show the band structure for Graphene. The intermingling of bands show the conduction properties of Graphene. Moreover, this structure is more ordered as compared to Aluminum and Copper showing the specific properties of Graphene. Another point of interest is the pseudo-potential used. This pseudo-potential is actually for Silicon but is used for Graphene due to certain similarities in structure and properties. 23 | P a g e Task 6: Lattice Constant, Bulk Modulus & Band Structure for Al Find the Lattice Constant, Bulk Modulus and Band Structure for FCC Aluminum and compare it with the molecular dynamics’ results Method for Finding Bulk Modulus and Lattice Constant for Al For finding the lattice constant and bulk modulus, we used “Al.pbe-n-van.UPF” pseudo potential among various pseudo potentials provided by Espresso 5.0.2. Following is the code for the input file: &control calculation restart_mode pseudo_dir outdir prefix tstress tprnfor / = = = = = = = 'scf' 'from_scratch', '/opt/espresso-5.0.2/pseudo/', '/tmp/ex02.12/' 'Al' .false. .false. &system ibrav celldm(1) nat ntyp ecutwfc ecutrho occupations smearing degauss / = = = = = = = = = 2, 7.4, 1, 1, 30.0, 300.0 'smearing', 'gaussian', 0.01 &electrons diagonalization = 'david' conv_thr = 1.0e-9 mixing_beta = 0.7 / ATOMIC_SPECIES Al 26.98 Al.pbe-n-van.UPF ATOMIC_POSITIONS Al 0.0 0.0 0.0 K_POINTS (automatic) 12 12 12 0 0 0 Later the results are fed to another environmental variable fit_pwscf which fits the output file to a graph and produce values accordingly 24 | P a g e Results for Bulk Modulus and Lattice Constant for Aluminum The values for various parameters are as follows: Bulk Modulus Lattice Energy Equilibrium Volume Lattice Constant The simulation also gives us a graph between Lattice Energy and Equilibrium Volume Here we can observe that the value of Equilibrium volume coincides with that of Lattice Energy. This was done by curve fitting using PWSCF method. 25 | P a g e Method for Finding Band Structure For finding the band structure, we used “Al.pbe-n-van.UPF” pseudo potential among various pseudo potentials provided by Espresso 5.0.2. Following is the code for the input file: &control calculation pseudo_dir outdir prefix verbosity / &system ibrav celldm(1) nat ntyp ecutwfc ecutrho nbnd occupations smearing degauss / = = = = = = = = = = = = 'bands' '/opt/espresso-5.0.2/pseudo/', '/tmp/ex03.a0/', 'Al' 'high' 2, 6.867, 1, 1, 30.0, 300.0, 7 = 'smearing', = 'gaussian', = 0.01 &electrons diagonalization = 'david' / ATOMIC_SPECIES Al 26.98 Al.pbe-n-van.UPF ATOMIC_POSITIONS (crystal) Al 0.0 0.0 0.0 K_POINTS crystal_b 6 0.0000000 0.0000000 0.5000000 0.0000000 0.5000000 0.2500000 0.5000000 0.5000000 0.0000000 0.0000000 0.3750000 0.3750000 0.0000000 0.5000000 0.7500000 0.5000000 0.0000000 0.7500000 25 10 15 20 30 10 26 | P a g e Results for Band Structure of Al The following band structure was obtained as a result of the simulation The band structure is obtained by using the same pseudo-potential as that of the Bulk Modulus and Lattice Energy values. Analysis Following is a comparison of Bulk Modulus and Lattice Constant found using MD Simulations: Calculation Bulk Modulus Lattice Constant Atomistic 100.01 4.00 Molecular Dynamic 77.705 4.05 Difference 23% 1.3% There are a number of factors that can cause this difference e.g. the values largely depend on the pseudo-potentials used in the calculation. It is entirely possible that there other pseudopotentials that can provide values closer to those of MD. There is also possibility of error in the MD results. The intermingling of band structure clearly shows the conductive nature of Aluminum. However these band structures can vary if different pseudo-potentials are used. 27 | P a g e Task 7: Energy Level, Bond Length and Bond Energy for N Find the Energy levels for hydrogen and also find the Bond length and Bond Energy for Nitrogen Molecule Method of Finding Energy Levels for Nitrogen For finding the energy levels for nitrogen atom we used “N.pbe-van_ak.UPF” pseudo-potential among various pseudo potentials provided by Espresso 5.0.2. Following is the code for the input file: &control verbosity calculation restart_mode pseudo_dir outdir prefix / = = = = = = 'high' 'scf' 'from_scratch', '/opt/espresso-5.0.2/pseudo/', '/tmp/ex06.86/' 'N' &system ibrav = 1, celldm(1) = 30, nat = 1, ntyp = 1, ecutwfc = 30.0, ecutrho = 300.0 occupations = 'smearing' smearing = 'gaussian' degauss = 0.01 nspin = 2 starting_magnetization(1) = 1 / &electrons diagonalization = 'cg' conv_thr = 1.0e-10 mixing_beta = 0.7 / ATOMIC_SPECIES N 14.007 N.pbe-van_ak.UPF ATOMIC_POSITIONS N 0.0 0.0 0.0 K_POINTS (gamma) 28 | P a g e Results for Energy Levels of Nitrogen Apart from visualizing the energy levels there is another interesting thing in the output file. The values of Spin Up and Spin Down are different: This difference indicates that there are some magnetic forces present which are disturbing the equilibrium of both spins. Moreover, magnetic state is more stable than non-magnetic state. 29 | P a g e Method of Finding Bond Length and Bond Energy for Hydrogen In this method we used “N.pbe-van_ak.UPF” pseudo-potential among various pseudo potentials provided by Espresso 5.0.2. We will do relax calculation on a cubic lattice. Because of relaxation, the atoms will try to optimize the distance between the atoms. The following code is used as input: &control calculation restart_mode pseudo_dir outdir prefix / = = = = = 'relax' 'from_scratch', '/opt/espresso-5.0.2/pseudo/', '/tmp/ex07.75/' 'N2' &system ibrav celldm(1) nat ntyp ecutwfc ecutrho occupations smearing degauss / = = = = = = = = = 1, 30, 2, 1, 30.0, 300.0 'smearing' 'gaussian' 0.02 &electrons diagonalization = 'cg' conv_thr = 1.0e-10 mixing_beta = 0.7 / &ions / ATOMIC_SPECIES N 14.007 N.pbe-van_ak.UPF ATOMIC_POSITIONS (crystal) N 0.518 0.518 0.518 N 0.482 0.482 0.482 K_POINTS (gamma) We can observe that there are some differences as compared to the previous code 1. The calculation is changed from scf to relax 2. No verbosity is included 3. Atomic Positions are assigned in 3D space for both atoms 30 | P a g e Results for Bond Length and Bond Energy of Hydrogen For Bond Energy 𝐸𝑏 = 𝐸𝑁2 − 2𝐸𝑁1 = −39.7861 − 2(−19.5346) = −0.7169 𝑅𝑦 For bond length, we found the following coordinates: If we plot these points in the cartesian coordinates, we can see how the atoms are becoming closer by relaxation. For calculation of bond length we use the formula for finding distance between two points: 𝑑 = √(𝑥1 − 𝑥2 )2 + (𝑦1 − 𝑦2 )2 + (𝑧1 − 𝑧2 )2 𝑑 = 0.0686 𝐵𝑜ℎ𝑟 × 30 × 0.529 = 1.09Å In this calculation “30” is the cell dimension used in the input code and “0.529” is the conversion factor for converting Bohr to Angstrom It can be observed here that with the passage of simulation, the atoms are coming closer in each iteration. For the calculation of bond length value, we have considered the closest atomic positions. 31 | P a g e Analysis Following is a table to compare the experimental and computational values of bond length and bond energy: Atomistic Computations Experimental Values Difference Bond Energy -0.7169 Ry -0.8692 Ry1 17.5 % Bond Length 1.09 Å 1.095 Å1 0.4 % The resultant computational value of bond length has negligible difference from the experimental value. However the Bond Energy Value is a bit different. There can be two reasons for this difference. The first one can be the pseudo-potential used in the calculations. The second one is that there might be a problem with the experimental value because it says in the reference that it is the “Estimated Average Bond Energy”. Summary In these Hands On sessions I have learnt how to find the values of Energy, Bulk Modulus, Equilibrium Volume and many other values by computational methods using a set pseudopotential. I have also learnt how to visualize the Energy Bands and Band Structure of different elements. These computations are very useful in the analysis of elements to be used for specific applications. I have also learnt how to use Linux operating systems and since it is preferred for computational purposes. There is just one suggestion for the DFT part: There should have been more Hands On sessions so that we could learn more about the Atomistic Computations. I would be delighted if the school or the relevant TA can inform me about any upcoming seminars or symposiums related to the Atomistic Computations. I have developed a keen interest in this field after studying this course, I never knew before that there is a field like this in Materials Engineering. Thanks to Professor Kong for the time he gave in teaching us efficiently and thanks to the teaching assistant, she guided me well during the hands on sessions. I hope to see more courses offered in this field since its quite revolutionary and highly effective in the development of materials especially the Materials Genome Initiative. 1 Estimated Bond Energies in Carbon, Nitrogen, Oxygen, and Hydrogen Compounds The Journal of Chemical Physics 19, 124 (1951); https://doi.org/10.1063/1.1747958 32 | P a g e