Getting the all-electron density¶
The variational quantity of the PAW formalism is the pseudo-density
\(\tilde{n}\). This is also the density returned by the
get_pseudo_density()
method of the GPAW
calculator. Sometimes it is desirable to work with the true all-electron
density. The PAW formalism offers a recipe for reconstructing the all-electron
density from the pseudo-density, and in GPAW, this can be reached by
the method get_all_electron_density()
of the
GPAW
class:
- get_all_electron_density(spin=None, gridrefinement=2, pad=True)¶
Return reconstructed all-electron density array.
The get_all_electron_density()
method is used in
the same way as you would normally use the
get_pseudo_density()
method, i.e.:
from gpaw import GPAW
from ase.build import molecule
calc = GPAW(mode='fd')
mol = molecule('C6H6', calculator=calc)
mol.center(vacuum=5)
E = mol.get_potential_energy()
nt = calc.get_pseudo_density()
n_ae = calc.get_all_electron_density()
would give you the pseudo-density in nt
and the all-electron
density in n_ae
.
As the all-electron density has more structure than the
pseudo-density, it is necessary to refine the density grid used to
represent the pseudo-density. This can be done using the
gridrefinement
keyword of the get_all_electron_density
method for n_ae_fine
:
n_ae_fine = calc.get_all_electron_density(gridrefinement=2)
Current only the values 1, 2, and 4 are supported (2 is default).
The all-electron density will always integrate to the total number of electrons of the considered system (independent of the grid resolution), while the pseudo density will integrate to some more or less arbitrary number. This fact is illustrated in the following example.
See also
Example: NaCl¶
As an example of application, consider the three systems Na, Cl, and
NaCl. The pseudo- and all-electron densities of these three systems
can be calculated with the script NaCl.py
:
# web-page: all_electron.csv
import numpy as np
from ase.build import molecule
from gpaw import GPAW
from ase.parallel import paropen
unitcell = np.array([6.5, 6.6, 9.])
gridrefinement = 2
f = paropen('all_electron.csv', 'w')
for formula in ('Na', 'Cl', 'NaCl',):
if formula in ['Na', 'Cl']:
hund = True
else:
hund = False
calc = GPAW(mode='fd',
xc='PBE',
h=0.18,
hund=hund,
convergence={'eigenstates': 1e-8},
txt=formula + '.txt')
sys = molecule(formula, cell=unitcell, calculator=calc)
sys.center()
sys.get_potential_energy()
# Get densities
nt = calc.get_pseudo_density()
n = calc.get_all_electron_density(gridrefinement=gridrefinement)
# Get integrated values
dv = sys.get_volume() / calc.get_number_of_grid_points().prod()
It = nt.sum() * dv
I = n.sum() * dv / gridrefinement**3
print('%-4s,%4.2f,%5.2f' % (formula, It, I), file=f)
f.close()
The result for the integrated pseudo- and all-electron densities of the three systems is:
formula |
ñ |
n |
---|---|---|
Na |
1.88 |
11.00 |
Cl |
7.50 |
17.00 |
NaCl |
9.36 |
28.00 |
From which we see that the all-electron densities integrate to the total number of electrons in the system, as expected.