PWDFT.jl

PWDFT.jl is a package to solve electronic structure problems based on density functional theory (DFT) and Kohn-Sham equations. It is written in Julia programming language.

The Kohn-Sham orbitals are expanded using plane wave basis. This basis set is very popular within solid-state community and is also used in several electronic structure package such as Quantum ESPRESSO, ABINIT, VASP, etc.

Features

• Total energy calculation of molecules, surfaces, and crystalline system within periodic unit cell (however, no corrections are implemented for non-periodic systems yet).
• SCF with electron density mixing (for semiconducting and metallic systems)
• Direct minimization method using conjugate gradient (for semiconducting systems)
• GTH pseudopotentials (included in the repository)
• LDA-VWN and GGA-PBE functionals (via Libxc.jl)

Requirements

• Julia version >= 0.7, with the following packages installed:
  • FFTW
  • SpecialFunctions
  • Libxc (a wrapper to Libxc)
  • LightXML (for parsing UPF file)

These packages are registered so they can be installed by using Julia's package manager.

using Pkg
Pkg.add("FFTW")
Pkg.add("SpecialFunctions")
Pkg.add("Libxc")
Pkg.add("LightXML")

These packages should be automatically installed PWDFT.jl is installed as local package (see below).

Installation

Currently, this package is not yet registered. So, Pkg.add("PWDFT") will not work (yet).

We have several alternatives:

1. Using Julia's package manager to install directly from the repository URL:

Pkg.add(PackageSpec(url="https://github.com/f-fathurrahman/PWDFT.jl"))

2. Using Julia development directory. We will use $HOME/.julia/dev for this. To enable $HOME/.julia/dev directory, we need to modify the Julia's LOAD_PATH variable. Add the following line in your $HOME/.julia/config/startup.jl.

push!(LOAD_PATH, expanduser("~/.julia/dev"))

After this has been set, you can download the the package as zip file (using Github) or clone this repository to your computer.

If you download the zip file, extract the zip file under $HOME/.julia/dev. You need to rename the extracted directory to PWDFT (with no .jl extension).

Alternatively, create symlink under $HOME/.julia/dev to point to you cloned (or extracted) PWDFT.jl directory. The link name should may not contain the .jl part. For example:

ln -fs /path/to/PWDFT.jl $HOME/.julia/dev/PWDFT

3. Install PWDFT.jl as local package. Firstly, get into Pkg's REPL mode by tapping ], and activate a independent environment activate . .

Install the PWDFT.jl package in this environment:

(PWDFT) pkg> develop <path/to/PWDFT.jl>

To make sure that the package is installed correctly, you can load the package and verify that there are no error messages during precompilation step. You can do this by typing the following in the Julia console.

using PWDFT

Change directory to examples/Si_fcc and run the following in the terminal.

julia run.jl

The above command will calculate total energy of hydrogen atom by SCF method.

The script will calculate total energy per unit cell of silicon crystal using self-consistent field iteration and direct energy minimization.

Units

PWDFT.jl internally uses Hartree atomic units (energy in Hartree and length in bohr).

A simple work flow

• create an instance of Atoms:

atoms = Atoms(xyz_file="CH4.xyz", LatVecs=gen_lattice_sc(16.0))

• create an instance of Hamiltonian:

ecutwfc = 15.0 # in Hartree
pspfiles = ["../pseudopotentials/pade_gth/C-q4.gth",
            "../pseudopotentials/pade_gth/H-q1.gth"]
Ham = Hamiltonian( atoms, pspfiles, ecutwfc )

• solve the Kohn-Sham problem

KS_solve_SCF!( Ham, betamix=0.2 )  # using SCF (self-consistent field) method
# or
KS_solve_Emin_PCG!( Ham ) # direct minimization using preconditioned conjugate gradient

More examples on creating an instance of Atoms

GaAs crystal (primitive unit cell), using keyword xyz_string_frac:

# Atoms
atoms = Atoms( xyz_string_frac=
    """
    2

    Ga  0.0   0.0   0.0
    As  0.25  0.25  0.25
    """,
    in_bohr=true,
    LatVecs = gen_lattice_fcc(10.6839444516)
)

Hydrazine molecule in extended xyz file

atoms = Atoms(ext_xyz_file="N2H4.xyz")

with the following N2H4.xyz file (generated using ASE):

6
Lattice="11.896428 0.0 0.0 0.0 12.185504 0.0 0.0 0.0 11.151965" Properties=species:S:1:pos:R:3:Z:I:1 pbc="T T T"
N       5.94821400       6.81171100       5.22639100        7 
N       5.94821400       5.37379300       5.22639100        7 
H       6.15929600       7.18550400       6.15196500        1 
H       5.00000000       7.09777800       5.00000000        1 
H       5.73713200       5.00000000       6.15196500        1 
H       6.89642800       5.08772600       5.00000000        1 

Lattice vectors information is taken from the xyz file.

More examples on creating an instance of Hamiltonian

Using 3x3x3 Monkhorst-Pack kpoint grid (usually used for crystalline systems):

Ham = Hamiltonian( atoms, pspfiles, ecutwfc, meshk=[3,3,3] )

Include 4 extra states:

Ham = Hamiltonian( atoms, pspfiles, ecutwfc, meshk=[3,3,3], extra_states=4 )

Using spin-polarized (Nspin=2 ):

Ham = Hamiltonian( atoms, pspfiles, ecutwfc, meshk=[3,3,3],
    Nspin=2, extra_states=4 )

NOTES: Currently spin-polarized calculations are only supported by specifying calculations with smearing scheme (no fixed magnetization yet), so extra_states should also be specified.

Using PBE exchange-correlation functional:

Ham = Hamiltonian( atoms, pspfiles, ecutwfc, meshk=[3,3,3],
    Nspin=2, extra_states=4, xcfunc="PBE" )

Currently, only two XC functional is supported, namely xcfunc="VWN" (default) and xcfunc="PBE". Future developments should support all functionals included in LibXC.

More examples on solving the Kohn-Sham problem

Several solvers are available:

• KS_solve_SCF!: SCF algorithm with density mixing

• KS_solve_SCF_potmix!: SCF algorithm with XC and Hartree potential mixing

• KS_solve_Emin_PCG!: using direct total energy minimization by preconditioned conjugate gradient method (proposed by Prof. Arias, et al.). Only the version which works with systems with band gap is implemented.

Stopping criteria is based on difference in total energy.

The following example will use Emin_PCG. It will stop if the difference in total energy is less than etot_conv_thr and it occurs twice in a row.

KS_solve_Emin_PCG!( Ham, etot_conv_thr=1e-6, NiterMax=150 )

Using SCF with betamix (mixing parameter) 0.1:

KS_solve_SCF!( Ham, betamix=0.1 )

Smaller betamix usually will lead to slower convergence but more stable. Larger betamix will give faster convergence but might result in unstable SCF.

Several mixing methods are available in KS_solve_SCF!:

• simple or linear mixing

• linear_adaptive

• anderson

• broyden

• pulay

• ppulay: periodic Pulay mixing

• rpulay: restarted Pulay mixing

For metallic system, we use Fermi smearing scheme for occupation numbers of electrons. This is activated by setting use_smearing=true and specifying a small smearing parameter kT (in Hartree, default kT=0.001).

KS_solve_SCF!( Ham, mix_method="rpulay", use_smearing=true, kT=0.001 )

Band structure calculations

Please see this as an example of how this can be obtained.

Citation

• Fadjar Fathurrahman, Mohammad Kemal Agusta, Adhitya Gandaryus Saputro, Hermawan Kresno Dipojono PWDFT.jl : A Julia package for electronic structure calculation using density functional theory and plane wave basis. Comp. Phys. Comm. 256 107372 (2020).

Some references

Articles:

• M. Bockstedte, A. Kley, J. Neugebauer and M. Scheffler. Density-functional theory calculations for polyatomic systems:Electronic structure, static and elastic properties and ab initio molecular dynamics. Comp. Phys. Commun. 107, 187 (1997).

• Sohrab Ismail-Beigi and T.A. Arias. New algebraic formulation of density functional calculation. Comp. Phys. Comm. 128, 1-45 (2000)

• C. Yang, J. C. Meza, B. Lee, L.-W. Wang, KSSOLV - a MATLAB toolbox for solving the Kohn-Sham equations, ACM Trans. Math. Softw. 36, 1–35 (2009)

Books:

• Richard Milton Martin. Electronic Structure: Basic Theory and Practical Methods. Cambridge University Press, 2004.

• Jorge Kohanoff. Electronic Structure Calculations for Solids and Molecules: Theory and Computational Methods. Cambridge University Press, 2006.

• Dominik Marx and Jürg Hutter. Ab Initio Molecular Dynamics: Basic Theory and Advanced Methods. Cambridge University Press, 2009.