DFT Calculations
Shaun Swanson
The Game Plan
DFT Basics
Job File Structure
Sample Structure: Solid Mg
Performing ‘scf’ Calculations on Solid Mg
Converging ‘scf’ Energies
Performing ‘relax’ calculations on Solid Mg
The Limits of DFT
Current work: GST structure and CNT properties
What is Density Functional Theory (DFT)?
• In Summary: “Density Functional Theory is a phenomenally
successful approach to finding solutions to the fundamental
equation for the quantum behavior of atoms and molecules,
the Schrodinger equation, in settings of practical value.”
• Problem: Solving the Schrodinger equation is a many-body
problem (3N bodies to be exact).
• Solution: Electrons are indistinguishable. So the quantity of
physical interest is instead the probability that N electrons in
any order have specific coordinates. So the electron density,
which is a function of only 3 coordinates, contains a great deal
of information that is physically observable from the full
solution to the Schrodinger equation.
The T in DFT
• The entire field of Density Functional Theory rests on two
fundamental mathematical theorems proven by Kohn and
Hohenberg, and the derivation of a set of equations by Kohn
and Sham.
• Theorem #1: There exists a 1-to-1 mapping between the
ground state wave function and the ground state electron
density.
• Theorem #2: The electron density that minimizes the energy
of the above functional is the true electron density
corresponding to the full solution to the Schrodinger equation.
Guess the functional form and vary the electron density iteratively
until the energy is minimized!
The Math
• A useful way to write down the functional described by
Theorem #1 is in terms of the single electron wave
functions (according to the Hartree Product approximation).
The Schrodinger equation then becomes the Kohn-Sham
equations:
Pseudocode
1. Define an initial, trial electron density.
2. Solve the Kohn-Sham equations using the trial electron
density to find the single particle wave functions.
3. Calculate the electron density defined by the KohnSham single particle wave functions from step 2.
4. Compare the calculated electron density with the
electron density used to solve the Kohn-Sham
equations. If they agree with some specified error, then
compute the total energy from this density, otherwise
update the trial electron density in some way.
5. Repeat.
This iterative method can lead to a solution of the Kohn-Sham
equations that is self-consistent!
Methods for Efficient Computation
• K-points (k)
– Discrete points specified in Brillouin Zone used to perform
numerical integration during calculation.
• Energy Cut-off Value (ecut)
– Energy value for maximum energy state included in a
summation over electron states.
• Pseudopotentials (pp)
– Offers specific exchange-correlation functional form which
represents frozen “core electrons.” They are based largely
on empirical data.
The Game Plan
DFT Basics
Job File Structure
Sample Structure: Solid Mg
Performing ‘scf’ Calculations on Solid Mg
Converging ‘scf’ Energies
Performing ‘relax’ calculations on Solid Mg
The Limits of DFT
Current work: GST structure and CNT properties
Job File Structure
Input variables that
control the amount of
I/O on the hard drive
and on the screen
Input variables that
specify the system
under study.
Input variables that
control the algorithms
used to reach a self-consistent
solution of the Kohn-Sham
equations.
Type and coordinates of each
atom in the unit cell
Coordinates and weights of the
K-points used for BZ integration
Name, mass, and pseudopotential
used for each atomic species present
in the system
The Game Plan
DFT Basics
Job File Structure
Sample Structure: Solid Mg
Performing ‘scf’ Calculations on Solid Mg
Converging ‘scf’ Energies
Performing ‘relax’ calculations on Solid Mg
The Limits of DFT
Current work: GST structure and CNT properties
Solid Mg (hcp)
a = 3.21 Angstrom
c = 5.21 Angstrom
Space group: P63/mmc
Inputting Structures into Quantum ESPRESSO
• There are multiple options in QE, for inputting structures.
• In the &SYSTEM namelist, there is a parameter called ‘ibrav’
– ibrav=4 refers to the index for an internally-specified hcp
Bravais lattice.
– Ibrav=0 allows one to self-define their unit cell.
Ibrav = 4
Ibrav = 0
•Internally specifies lattice vectors
•Requires the celldm() parameters
to be specified [alat = celldm(1)]
•Better recognizes symmetry operations
•Requires CELL_PARAMETERS card for
lattice vector input.
•alat = the length of the first lattice vector
Inputting Vectors into Quantum ESPRESSO
• For the required ATOMIC_POSITIONS card and the optional
CELL_PARAMETERS card, the input of basis and lattice
vectors follow a general format:
• Note: While the three lattice vectors in the CELL_PARAMETERS
card can only be specified in alat-normalized Bohr units, there are
four options for inputting basis vectors in ATOMIC_POSITIONS:
“alat” – alat-normalized Bohr units (default) “angstrom” – Angstrom units
“bohr” – Bohr units
“crystal” – fractional coordinates
Performing Self-consistent Energy Calculations on Solid Mg
LDA vs. GGA Pseudopotential Approximations
• “Local-density approximations (LDA) are a class of
approximations to the exchange-correlation (XC) energy
functional in DFT that depend solely upon the value of the
electronic density at each point in space (and not, for
example, derivatives of the density or the Kohn-Sham
orbitals).” This is more of a first-order approximation.
• “Generalized gradient approximations (GGA) are still local but
also take into account the gradient of the density at the same
coordinate.”
SCF Output
• As intended, the ‘mg.txt’ file only contains the following:
• Reading the full ‘mg.scf.out’ file, however, allows us to step
through the entire calculation, from initialization through each
iteration. This is useful to visualize sometimes if you’re unsure
of your results and would like to visually inspect the
convergence of your self-consistent energies.
Ensuring k and ecut Lead to a Converged Energy
• The most important skill in performing DFT calculations is the
ability to get converged energies. Since the appropriate
choice of k-points and ecut vary wildly among different
geometries (and even different required accuracies), it is
important to be able to form the following graphs every time
you perform ‘scf’ calculations on new geometries.
• Converged values of ecut and k should be reported any time
you publish DFT results, so that someone else may reproduce
your calculation and agree on the same numerical error.
Energy Cut-off Convergence Plot
Not only should the graph look converged, but the difference in energy between
the last two consecutive points should be smaller or equal to your required accuracy!
K-point Convergence Plot
Note: In an automatic
distribution of k-points,
the value of k specifies
how many discrete points
there are equally-spaced
along each lattice vector
to populate the Brillouin
Zone.
As we can see from the
convergence plots, the
presence of smearing does
little to ensure convergence
with fewer k-points.
K-point Convergence Plot (cont.)
• When unit cells do not have equal-length lattice vectors, it is
sometimes computationally rewarding to “geometricallyoptimize” your automatic k-point distribution.
• For example, if one had a unit cell that was four times taller in
one direction than its other two directions, one should specify
only a quarter as many k-points along the taller direction.
– This makes sense, because in reciprocal space, the taller
distance will only be a quarter as long as the other two
distances.
Performing Geometry Relaxation Calculations on Solid Mg
‘vc-relax’ differs from ‘relax’ only
in that the cell parameters can vary
during the relaxation.
VC-RELAX Output
• Reading the full ‘mg.rx.out’ file allows us to step through the
entire calculation, from initialization through each iteration.
• It becomes apparent that a relaxation calculation is simply a
series of scf calculations with forces specified at each step
and updated positions at each iteration that correspond to the
strength of the forces between the atoms in the system.
Comparing the Relaxed Structure to Literature
Comparing the Relaxed Structure to Literature (cont.)
• On top of comparing the ‘a’ and ‘c/a’ lattice parameters to
literature and to experiment, it is also useful to compare a
quantity called “cohesive energy”
– The difference between the average energy of the atoms of
a crystal and that of the free atoms.
• It is important to compare cohesive energies and not
specifically the self-consistent energies, because only
differences in energies are physically meaningful.
– The energy datum for an ‘scf’ calculation is specified by the
choice of pseudopotential.
The Game Plan
DFT Basics
Job File Structure
Sample Structure: Solid Mg
Performing ‘scf’ Calculations on Solid Mg
Converging ‘scf’ Energies
Performing ‘relax’ calculations on Solid Mg
The Limits of DFT
Current work: GST structure and CNT properties
What Density Functional Theory Can Do
• Applies to atoms, molecules, surfaces and periodic
structures...
• Self-consistent total energies, forces, stresses (ground state)
• Structural optimization
• Elastic modulus
• Electron band structure of many-body systems
• Phonon frequencies and eigenvectors at any wave vector
• Full phonon dispersions
• Electron-phonon interactions
• ab-initio MD
• etc..
What Density Functional Theory CanNOT Do
• DFT Calculations are not exact solutions of the full
Schrodinger equation (exact functional unknown).
• The Hohenberg-Kohn theorems apply only to ground state
energies. Excited states can be predicted, but these
predictions are not on the same footing as predictions for
ground-state energies.
• There is a well-known inaccuracy in DFT involving the
underestimation of calculated band gaps in semiconducting
and insulating materials.
• DFT gives inaccurate results involving weak van der Waals
attractions, which are a direct result of long range electron
correlation.
• It is computationally expensive to perform DFT calculations.
Simulations with hundreds of atoms are about the largest you
can get.
Current NTPL Use of DFT Calculations
• Determining the Structure of Ge2Sb2Te5 Phase-change Material
Stable
Structure
(hcp)
Meta-stable Structure
(rocksalt)
• Electron and Phonon Properties of Carbon Nanotubes
Any Questions?
Please visit the related pages on the NTPL Wiki for more information!