1. Integrated Computational Materials Engineering Education Calculation of Equation of State Using Density Functional Theory
Mark Asta1, Katsuyo Thornton2, and Larry Aagesen3
1Department of Materials Science and Engineering, University of California, Berkeley
2Department of Materials Science and Engineering, University of Michigan, Ann Arbor
3Idaho National Laboratory
This afternoon, we will spend some time going over the DFT Module, which is based on the density functional theory, or DFT.
This module can be introduced to undergrads in a materials physics course that teaches quantum mechanics and possibly band structure. Other opportunities are mechanical behavior course (to link elastic properties to atomic interactions) or in laboratories where mechanical properties are measured.
Here, we collected slides that are at the appropriate level for undergraduate courses, but presentation can be modified more quantitative or less, depending on the level of the course or students’ background.

2. Purposes of Density Functional Theory Module
Understand fundamentals of Density Functional Theory (DFT)
Apply DFT to calculate:
Equilibrium lattice constant
Bulk Modulus
Components of elastic constant tensor
Understand how to check for convergence of results

3. Equation of State A Probe of Interatomic Interactions
Schematic Energy vs. Volume Relation
Diamond Cubic Structure of Si
Energy per atom a
Draw picture of interatomic potential. Point out compression and tension.
Volume per atom (=a3/8 for Si)

4. Equation of State What Properties Can we Learn from It?
Pressure versus Volume Relation
Equilibrium Volume (or Lattice Constant)
Bulk Modulus
Given E(V) one can compute P(V) by taking derivative
Recall 1st Law of Thermo: dE = T dS - P dV and consider T = 0 K
Volume corresponding to zero pressure = Volume where slope of E(V) is zero
≈ Volume measured experimentally at P = 1 atm
We start with the first law of thermodynamics. As DFT calculations are at zero temperature, the energy is solely due to PV work.
So what’s the lattice constant? It’s the size of the unit cell that is in equilibrium with the external pressure, with is nominally 1 atm which can be approximated as zero since it’s small compared to typical stress that results from elastic deformation.
As you can see, that can be also interpreted from the picture as the minimum of the energy.
On the other hand, bulk modulus is a measure of how stiff the material is, and is related to how quickly the counter-pressure/force increases in response to elastic deformation.
B related to curvature of E(V) Function

5. Generalize to Non-Hydrostatic Deformation Example of Uniaxial DeformationLz
Lz(1+e) Ly Ly Lx Lx
So, how do we calculate the equation of state? We can calculate the total energy for a given arrangement....
We can calculate the energy based on QM governing equations theoretically, but in practice we need to make approximations. In particular, we will use the density functional theory and numerically implement this because of the complexity.
(What we need to do, How we can do it, Computation is needed)
Definition of Deformation In Terms of Strain:
(All other strains are zero)

6. Linear-Elasticity for Single Crystals
General form of Hook’s Law (Linear Elasticity):
Stress Tensor
Elastic Constant Tensor
Strain Tensor
Voigt Notation:
11 → 1, 22 → 2, 33 → 3, 23 → 4, 13 → 5, 12 → 6
Elastic Energy:
So, how do we calculate the equation of state? We can calculate the total energy for a given arrangement....
We can calculate the energy based on QM governing equations theoretically, but in practice we need to make approximations. In particular, we will use the density functional theory and numerically implement this because of the complexity.
(What we need to do, How we can do it, Computation is needed)
In example from previous slide:
(All other strains are zero)
Note: for cubic crystal C11=C22=C33, C12=C13=C23

7. Equation of State How to Calculate from Density Functional Theory
Formulation: for a given arrangement of nuclei defined by the lattice constant, crystal structure, and non-hydrostatic strains, compute the total energy corresponding to the optimal arrangement of the electron density
Theoretical Framework: Quantum mechanical calculation of energy of electrons and nuclei interacting through Coulomb potential
Practical Implementation: Density functional theory
So, how do we calculate the equation of state? We can calculate the total energy for a given arrangement....
We can calculate the energy based on QM governing equations theoretically, but in practice we need to make approximations. In particular, we will use the density functional theory and numerically implement this because of the complexity.
(What we need to do, How we can do it, Computation is needed)

8. Total Energy in Density Functional Theory
Electron Density
Electron Wavefunctions
DFT is an approximation of the quantum mechanics. The total energy is a sum of the kinetic energy and the potential energy.
It’s involves electron density in calculating ion-electron interactions and electrostatic part of electron-electron interactions via electron density. Exchange correlation takes any electron-electron interactions that are not accounted for in these terms.
Potential Electrons Feel from Nuclei
Exchange-Correlation Energy
Form depends on whether you use Local Density Approximation (LDA)
or Generalized Gradient Approximation (GGA)

9. Kohn-Sham Equations Schrödinger Equation for Electron Wavefunctions
Exchange-Correlation Potential
The governing equation for quantum mechanics is basically to say the kinetic energy plus the potential energy gives the total energy that is quantized. With the approximation described earlier, you arrive at this form of equation, which is referred to as the Kohn-Sham equation. Here the phi_i s are the effective single electron wavefunction that obeys this form of the schodinger equation. The key point here is that the operator only explicitly depends on the electron density, but since the electron density is the magnitude of wavefunction. Therefore, the solution must be sought in an integrative manner.
Electron Density
Note: fi depends on n(r) which depends on fi 
Solution of Kohn-Sham equations must be done iteratively

10. Self-Consistent Solution to DFT Equations
Set up atom positions
Make initial guess of “input” charge density (often overlapping atomic charge densities)
Solve Kohn-Sham equations with this input charge density
Compute “output” charge density from resulting wavefunctions
If energy from input and output densities differ by amount greater than a chosen threshold, mix output and input density and go to step 2
Quit when energy from input and output densities agree to within prescribed tolerance (e.g., 10-5 eV)
Input Positions of Atoms for a Given Unit Cell and Lattice Constant
guess charge density
compute effective potential
compute Kohn-Sham orbitals and density
different
compare output and input charge densities
So, this is the typical iterative steps involved in calculating the equation of state. First one fixes the lattice parameter, which allows to set the atomic positions. Guess the electron density, n(r), then evaluate the Kohn-Sham equation to obtain the wave function. That wavefunction can then be used to calculate the electron density. So, now you have the input n(r) and output n(r), so you can make a direct comparison. If the difference is small, then your guess is good enough, so you have a solution for the given atomic configuration.
What is the convergence criteria?
same
Energy for Given Lattice Constant
Note: In this module the positions of atoms are dictated by symmetry. If this is not the case another loop must be added to minimize energy with respect to atomic positions.

11. Implementation of DFT for a Single Crystal
Crystal Structure Defined by Unit Cell Vectors and Positions of Basis Atoms
Example: Diamond Cubic Structure of Si a
Unit Cell Vectors
a1 = a (-1/2, 1/2 , 0)
a2 = a (-1/2, 0, 1/2)
a3 = a (0, 1/2, 1/2)
Basis Atom Positions
0 0 0
¼ ¼ ¼
All atoms in the crystal can be obtained by adding integer multiples of unit cell vectors to basis atom positions

12. Electron Density in Crystal Lattice
Unit-Cell Vectors
a1 = a (-1/2, 1/2 , 0)
a2 = a (-1/2, 0, 1/2)
a3 = a (0, 1/2, 1/2)
Electron density is periodic with periodicity given by
Electron density is periodic.
Lattice vectors.
beta = band index
Translation Vectors:

13. Electronic Bandstructure Example for Si
Brillouin Zone
Bandstructure
Electronic wavefunctions in a crystal can be indexed by
point in reciprocal space (k) and a band index (b)

14. Why? Wavefunctions in a Crystal Obey Bloch’s Theorem
For a given band b
Where is periodic in real space:
Translation Vectors:
The envelope function represents delocalized distribution of electron density

15. Representation of Electron Density
Integral over k-points in first Brillouin zone
In practice the integral over the Brillouin zone is replaced with a sum over a finite number of k-points (Nkpt)
beta = band index
Band occupation (e.g., the Fermi function)
One parameter that needs to be checked for numerical convergence is number of k-points

16. Representation of Wavefunctions Fourier-Expansion as Series of Plane Waves
For a given band:
Recall that is periodic in real space:
can be written as a 3D Fourier Series:
Basically, this is the 3D version of Fourier series, which in case you need a refresher,
where the are primitive reciprocal lattice vectors

17. Recall Properties of Fourier Series
Black line = (exact) triangular wave
Colored lines = Fourier series truncated at different orders
General Form of Fourier Series:
It is to write a periodic function in terms of an
For a smooth function, convergence (as you keep more terms) is quite rapid.
For Triangular Wave:
Typically we expect the accuracy of a truncated Fourier series to improve as we increase the number of terms

18. Representation of Wavefunctions Plane-Wave Basis Set
For a given band
Use Fourier Expansion
In practice the Fourier series is truncated to include all G for which:
Another parameter that needs to be checked for convergence is the “plane-wave cutoff energy” Ecut

19. Examples of Convergence Checks
Effect of Ecut
Effect of Number of k Points
Note: the different values of kTel corresponds to different choices for occupation function (wj in slide 14)

20. DFT Module
Problem 1: Calculate equilibrium volume and bulk modulus of diamond cubic Si using Quantum Espresso on Nanohub (
Outcome 1: Understand effect of numerical parameters on calculated results by testing convergence with respect to number of k-points and plane-wave cutoff
Outcome 2: Understand the effect of the theoretical model for exchange-correlation potential on the accuracy of the calculations by comparing results from Local Density Approximation (LDA) and Generalized Gradient Approximation (GGA) with experimental measurements

21. DFT Module
Problem 2: Calculate the single-crystal elastic constants C11 and C12
Outcome 1: Understand how to impose homogeneous elastic deformations in a DFT calculation
Outcome 2: Understand the effect of the theoretical model for exchange-correlation potential on the accuracy of the calculations by comparing results from Local Density Approximation (LDA) and Generalized Gradient Approximation (GGA) with experimental measurements

22. (Fractional Coordinates)
DFT Module
For problem 1 you will make use of the unit cell for diamond-cubic Si shown below. You will vary only the lattice constant a.
Unit Cell Vectors
a1 = a (-1/2, 1/2 , 0)
a2 = a (-1/2, 0, 1/2)
a3 = a (0, 1/2, 1/2) a
Basis Atom Positions
(Fractional Coordinates)
0 0 0
¼ ¼ ¼

23. (Fractional Coordinates)
DFT Module
For problem 2 you will impose a homogeneous tensile strain (e) along the [001] axis (see slide 4)
Such a strain results in the x3 coordinate of all atoms changing to x3*(1+e)
This homogeneous deformation can be represented by changing the unit cell vectors as follows:
Unit Cell Vectors
a1 = a (-1/2, 1/2 , 0)
a2 = a (-1/2, 0, (1+e)/2)
a3 = a (0, 1/2, (1+e)/2) a
Basis Atom Positions
(Fractional Coordinates)
0 0 0
¼ ¼ ¼