Overview

This document describes the most important objects used for a DFT calculation. More information can be found in the code.

PAW

This object is the central object for a GPAW calculation:

                 +----------+
                 |GPAWLogger|         +-----------+
                 +----------+     --->|Hamiltonian|
                        ^        /    +-----------+
                        |    ----       +------+
                        |   /      ---->|Setups|
+-----+              +------+     /     +------+
|Atoms|<-------------| GPAW |-----
+-----+              +------+     \
                    /   |    \     \            +-----------+
 +-------------+   /    |     ---   ----------->|Occupations|
 |WaveFunctions|<--     v        \              +-----------+
 +-------------+     +-------+    \   +-------+
                     |Density|     -->|SCFLoop|
                     +-------+        +-------+

The implementation is in gpaw/calculator.py. The GPAW class doesn’t do any part of the actual calculation - it only handles the logic of parsing the input parameters and setting up the necessary objects for doing the actual work (see figure above).

A GPAW instance has the following attributes: atoms, parameters, wfs, density, setups, hamiltonian, scf, log, timer, occupations, initialized, world and observers.

The GPAW inherits from:

• ase.calculators.calculator.Calculator

  This implements the ASE calculator interface.

Note

GPAW uses atomic units internally (\(\hbar=e=m=1\)) and ASE uses Ångström and eV (units).

Generating a GPAW instance from scratch

When a GPAW instance is created from scratch:

calc = GPAW(mode='fd', xc='LDA', nbands=7)

the GPAW object is almost empty. In order to start a calculation, one will have to do something like:

atoms = Atoms(...)
atoms.calc = calc
atoms.get_potential_energy()

ASE will then arrange to call the calculate() method with the correct arguments. This will trigger:

1. A call to the initialize() method, which will set up the objects needed for a calculation: Density, Hamiltonian, WaveFunctions, Setups and a few more (see figure above).

2. A call to the set_positions() method, which will initialize everything that depends on the atomic positions:

   1. Pass on the atomic positions to the wave functions, Hamiltonian and density objects (call their set_positions() methods).

   2. Make sure the wave functions are initialized.

   3. Reset the SCFLoop.

Generating a GPAW instance from a restart file

When a GPAW instance is created like this:

calc = GPAW('restart.gpw')

the initialize() method is called first, so that the parts read from the file can be placed inside the objects where they belong: the effective pseudo potential and the total energy are put in the Hamiltonian, the pseudo density is put in the density object and so on.

After a restart, everything should be as before the restart file was written. However, there are a few exceptions:

• The wave functions are only read when needed … XXX

• Atom centered functions (\(\tilde{p}_i^a\), \(\bar{v}^a\), \(\tilde{n}_c^a\) and \(\hat{g}_{\ell m}^a\)) are not initialized. … XXX

WaveFunctions

We currently have two representations for the wave functions: uniform 3-d grids and expansions in atom centered basis functions as implemented in the two classes FDWaveFunctions and LCAOWaveFunctions. Both inherit from the WaveFunctions class, so the wave functions object will always have a GridDescriptor, an Eigensolver, a Setups object and a list of KPoint objects.

  +--------------+     +-----------+
  |GridDescriptor|     |Eigensolver|
  +--------------+     +-----------+
              ^           ^
              |gd         |
               \          |
+------+        +-------------+ kpt_u   +------+
|Setups|<-------|WaveFunctions|-------->|KPoint|+
+------+        +-------------+         +------+|+
                       ^                 +------+|
                      /_\                 +------+
                       |
                       |
            --------------------------------
           |                                |
  +-----------------+            +-----------------+
  |LCAOWaveFunctions|            | FDWaveFunctions |
  +-----------------+            +-----------------+
        |        |              /    |           |
        v        |tci          |     |kin        |pt
+--------------+ |             v     |           v
|BasisFunctions| |        +-------+  |         +----------+
+--------------+ |        |Overlap|  |         |Projectors|
                 v        +-------+  |         +----------+
  +------------------+               v
  |TwoCenterIntegrals|     +---------------------+
  +------------------+     |KineticEnergyOperator|
                           +---------------------+

Attributes of the wave function object: gd, nspins, nbands, mynbands, dtype, world, kpt_comm, band_comm, gamma, bzk_kc, ibzk_kc, weight_k, symmetry, kpt_comm, rank_a, nibzkpts, kpt_u, setups, ibzk_qc, eigensolver, and timer.

Exchange-correlation functionals module

The gpaw.xc module contains all the code for XC functionals in GPAW:

+--------------+
| XCFunctional |
+--------------+
    ^         ^
   /_\       /_\
    |         |
  +-------+   |    +------------------------+
  |  LDA  |    ----|vdW-DF/HybridXC/SIC/GLLB|
  +-------+        +------------------------+
    ^
   /_\
    |
  +---+
  |GGA|
  +---+
    ^
   /_\
    |
  +----+
  |MGGA|
  +----+

An XCFunctional object is usually created using the gpaw.xc.XC() function:

gpaw.xc.XC(kernel, parameters=None, atoms=None, collinear=True, xp=<module 'numpy' from '/tmp/gpaw-docs/venv/lib/python3.10/site-packages/numpy/__init__.py'>)

Create XCFunctional object.

kernel: XCKernel object, dict or str

Kernel object or name of functional.

parameters: ndarray

Parameters for BEE functional.

Recognized names are: LDA, PW91, PBE, revPBE, RPBE, BLYP, HCTH407, TPSS, M06-L, revTPSS, vdW-DF, vdW-DF2, EXX, PBE0, B3LYP, BEE, GLLBSC. One can also use equivalent libxc names, for example GGA_X_PBE+GGA_C_PBE is equivalent to PBE, and LDA_X to the LDA exchange. In this way one has access to all the functionals defined in libxc. See xc_funcs.h for the complete list.

Example:

# Implementation of PBE from LibXC:
from gpaw.xc import XC
xc = XC('PBE')
# alternative call:
from gpaw.xc.libxc import LibXC
from gpaw.xc.gga import GGA
xc = GGA(LibXC('PBE'))
# or, explicitly:
xc = GGA(LibXC('GGA_X_PBE+GGA_C_PBE'))

In this example, calling the calculate method of the xc object passing in a GridDescriptor, an input density array and an output array for the potential, the GGA object will calculate the gradient of the density and pass that and the density on to the libxc kernel.

Refer to Exchange-Correlation functional for other examples.

GPAW also has a few non-libxc kernels that one can use like this:

from gpaw.xc.kernel import XCKernel
xc = XC(XCKernel('PBE'))

Naming convention for arrays

A few examples:

name	shape
spos_c	(3,)	Scaled position vector
nt_sG	(2, 24, 24, 24)	Pseudo-density array \(\tilde{n}_\sigma(\vec{r})\) (t means tilde): two spins, 24*24*24 grid points.
cell_cv	(3, 3)	Unit cell vectors.

Commonly used indices:

index	description
a	Atom number
c	Unit cell axis-index (0, 1, 2)
v	xyz-index (0, 1, 2)
k	k-point index
q	k-point index (local, i.e. it starts at 0 on each processor)
s	Spin index (\(\sigma\))
u	Combined spin and k-point index (local)
G	Three indices into the coarse 3D grid
g	Three indices into the fine 3D grid
M	LCAO orbital index (\(\mu\))
n	Principal quantum number or band number
l	Angular momentum quantum number (s, p, d, …)
m	Magnetic quantum number (0, 1, …, 2*l - 1)
L	l and m (L = l**2 + m)
j	Valence orbital number (n and l)
i	Valence orbital number (n, l and m)
q	j1 and j2 pair
p	i1 and i2 pair
r	CPU-rank

Array names and their definition

name in the code	definition
wfs.kpt_u[u].P_ani	\(\langle\tilde{p}_i^a|\tilde{\psi}_{\sigma\mathbf{k}n} \rangle\)
density.D_asp	\(D_{s i_1i_2}^a\)
hamiltonian.dH_sp	\(\Delta H_{s i_1i_2}^a\)
setup.Delta_pL	\(\Delta_{Li_1i_2}^a\)
setup.M_pp	\(\Delta C_{i_1i_2i_3i_4}^a\) eq. (C2) in [1] or eq. (47) in [2]
wfs.kpt_u[u].psit_nG	\(\tilde{\psi}_{\sigma\mathbf{k}n}(\mathbf{r})\)
setup.pt_j	\(\tilde{p}_j^a(r)\)
wfs.pt	\(\tilde{p}_i^a(\mathbf{r}-\mathbf{R}^a)\)

The Setup instances are stored in the Setups list, shared by the wfs, density, and Hamiltonian instances. E.g. paw.wfs.setups, paw.density.setups, or paw.hamiltonian.setups.

Parallelization over spins, k-points domains and states

When using parallelization over spins, k-points, bands and domains, four different MPI communicators are used:

• mpi.world

  Communicator containing all processors.

• domain_comm

  One domain_comm communicator contains the whole real space domain for a selection of the spin/k-point pairs and bands.

• kpt_comm

  One kpt_comm communicator contains all k-points and spin for a selection of bands over part of the real space domain.

• band_comm

  One band_comm communicator contains all bands for a selection of k-points and spins over part of the real space domain.

These communicators constitute MPI groups, of which the latter three are subsets of the world communicator. The number of members in the a communicator group is signified by comm.size. Within each group, every element (i.e. processor) is assigned a unique index comm.rank into the list of processor ids in the group. For instance, a domain_comm rank of zero signifies that the processor is first in the group, hence it functions as a domain master.

For an example on how to use an MPI communicator to perform simple data communication, please refer to parallelization.py.

To investigate the way GPAW distributes calculated quantities across the various MPI groups, simulating an MPI run can be done using gpaw-mpisim:

$ gpaw-mpisim -v --dry-run=4 --spins=2 --kpoints=4 --bands=3 --domain-decomposition=2,1,1

Simulating: world.size = 4
    parsize_c = (2, 1, 1)
    parsize_bands = 1
    nspins = 2
    nibzkpts = 4
    nbands = 3

world: rank=0, ranks=None
    kpt_comm    : rank=0, ranks=[0 2], mynks=4, kpt_u=[0^,1^,2^,3^]
    band_comm   : rank=0, ranks=[0], mynbands=3, mybands=[0, 1, 2]
    domain_comm : rank=0, ranks=[0 1]
world: rank=1, ranks=None
    kpt_comm    : rank=0, ranks=[1 3], mynks=4, kpt_u=[0^,1^,2^,3^]
    band_comm   : rank=0, ranks=[1], mynbands=3, mybands=[0, 1, 2]
    domain_comm : rank=1, ranks=[0 1]
world: rank=2, ranks=None
    kpt_comm    : rank=1, ranks=[0 2], mynks=4, kpt_u=[0v,1v,2v,3v]
    band_comm   : rank=0, ranks=[2], mynbands=3, mybands=[0, 1, 2]
    domain_comm : rank=0, ranks=[2 3]
world: rank=3, ranks=None
    kpt_comm    : rank=1, ranks=[1 3], mynks=4, kpt_u=[0v,1v,2v,3v]
    band_comm   : rank=0, ranks=[3], mynbands=3, mybands=[0, 1, 2]
    domain_comm : rank=1, ranks=[2 3]

For the case of a \(\Gamma\)-point calculation without band-parallelization, all parallel communication is done in the one domain_comm communicator, which in this case is equal to mpi.world.

[1]

J J. Mortensen and L. B. Hansen and K. W. Jacobsen, Phys. Rev. B 71 (2005) 035109.

[2]

C. Rostgaard, The Projector Augmented Wave Method.