# Miscellaneous objects and functions¶

gpaw.occupations.occupation_numbers(occ, eps_skn, weight_k, nelectrons)[source]

Calculate occupation numbers from eigenvalues in eV.

occ: dict

Example: {‘name’: ‘fermi-dirac’, ‘width’: 0.05} (width in eV).

eps_skn: ndarray, shape=(nspins, nibzkpts, nbands)

Eigenvalues.

weight_k: ndarray, shape=(nibzkpts,)

Weights of k-points in IBZ (must sum to 1).

nelectrons: int or float

Number of electrons.

Returns a tuple containing:

• f_skn (sums to nelectrons)

• fermi-level

• magnetic moment

• entropy as -S*T

class gpaw.lfc.NewLocalizedFunctionsCollection(gd, spline_aj, kd=None, cut=False, dtype=<class 'float'>, integral=None, forces=None)[source]

New LocalizedFunctionsCollection

Utilizes that localized functions can be stored on a spherical subset of the uniform grid, as opposed to LocalizedFunctionsCollection which is just a wrapper around the old localized_functions which use rectangular grids.

Add localized functions to extended arrays.

--  a     a
a (G) += >  c   Phi (G)
x       --  xi    i
a,i

Add derivative of localized functions on atom to extended arrays.

Parameters:

a: int

Atomic index of the derivative

v: int

Cartesian coordinate of the derivative (0, 1 or 2)

This function adds the following sum to the extended arrays:

--  a      a
a (G) += >  c   dPhi  (G)
x       --  xi     iv
i

where:

a        d     a
dPhi  (G) =  -- Phi (g)
iv       dv    i

is the derivative of the Phi^a and v is either x, y, or z.

derivative(a_xG, c_axiv, q=-1)[source]

Calculate x-, y-, and z-derivatives of localized function integrals.

/              a*
c_axiv =  | dG a (G) dPhi  (G)
/     x        iv

where:

a        d     a
dPhi  (G) =  -- Phi (g)
iv       dv    i

and v is either x, y, or z, and R^a_v is the center of Phi^a.

Notice that d Phi^a_i / dR^a_v == - d Phi^a_i / d v.

griditer()[source]

Iterate over grid points.

integrate(a_xG, c_axi, q=-1)[source]

Calculate integrals of arrays times localized functions.

/             a*
c_axi =  | dG a (G) Phi  (G)
/     x       i
second_derivative(a_xG, c_axivv, q=-1)[source]

Calculate second derivatives.

Works only for this type of input for now:

second_derivative(self, a_G, c_avv, q=-1)
2 a _ _a
/  _   _  d f (r-R )
c_avv =  | dr a(r) ----------
/             a  a
dR dR
i  j
class gpaw.spline.Spline(l, rcut, list) → Spline object[source]

The integer l gives the angular momentum quantum number and the list contains the spline values from r=0 to r=rcut.

The array f_g gives the radial part of the function on the grid. The radial function is multiplied by a real solid spherical harmonics (r^l * Y_lm).

get_angular_momentum_number()[source]

Return the angular momentum quantum number.

get_cutoff()[source]

get_value_and_derivative(r)[source]

Return the value and derivative.

class gpaw.poisson.FDPoissonSolver(nn=3, relax='J', eps=2e-10, maxiter=1000, remove_moment=None, use_charge_center=False, metallic_electrodes=False)[source]
create_laplace(gd, scale=1.0, n=1, dtype=<class 'float'>)[source]

Instantiate and return a Laplace operator

Allows subclasses to change the Laplace operator

iterate2(step, level=0)[source]

Smooths the solution in every multigrid level

class gpaw.xc.functional.XCFunctional(name, type)[source]
calculate(gd, n_sg, v_sg=None, e_g=None)[source]

Calculate energy and potential.

gd: GridDescriptor

Descriptor for 3-d grid.

n_sg: rank-4 ndarray

Spin densities.

v_sg: rank-4 ndarray

Array for potential. The XC potential is added to the values already there.

e_g: rank-3 ndarray

Energy density. Values must be written directly, not added.

The total XC energy is returned.

get_description()[source]

Get long description of functional as a string, or None.

summary(fd)[source]

Write summary of last calculation to file.

todict()[source]

Get dictionary representation of XC functional.

This representation works for libxc kernels; other classes should likely override this function and should probably not rely on this implementation.

tostring()[source]

Get string representation of XC functional.

This will give the name for libxc functionals but other data for hybrids.

class gpaw.xc.gga.GGA(kernel, stencil=2)[source]
get_description()[source]

Get long description of functional as a string, or None.

todict()[source]

Get dictionary representation of XC functional.

This representation works for libxc kernels; other classes should likely override this function and should probably not rely on this implementation.

gpaw.forces.calculate_forces(wfs, dens, ham, log=None)[source]

Return the atomic forces.

class gpaw.grid_descriptor.GridDescriptor(N_c, cell_cv=[1, 1, 1], pbc_c=True, comm=None, parsize_c=None)[source]

Descriptor-class for uniform 3D grid

A GridDescriptor object holds information on how functions, such as wave functions and electron densities, are discreticed in a certain domain in space. The main information here is how many grid points are used in each direction of the unit cell.

There are methods for tasks such as allocating arrays, performing symmetry operations and integrating functions over space. All methods work correctly also when the domain is parallelized via domain decomposition.

This is how a 2x2x2 3D array is laid out in memory:

3-----7
|\    |        | \   |         |  1-----5      z
2--|--6  |   y  |
\ |   \ |    \ |
\|    \|     \|
0-----4      +-----x

Example:

>>> a = np.zeros((2, 2, 2))
>>> a.ravel()[:] = range(8)
>>> a
array([[[0, 1],
[2, 3]],
[[4, 5],
[6, 7]]])

Construct grid-descriptor object.

parameters:

N_c: 3 ints

Number of grid points along axes.

cell_cv: 3 float’s or 3x3 floats

Unit cell.

pbc_c: one or three bools

Periodic boundary conditions flag(s).

comm: MPI-communicator

Communicator for domain-decomposition.

parsize_c: tuple of 3 ints, a single int or None

Number of domains.

Note that if pbc_c[c] is False, then the actual number of gridpoints along axis c is one less than N_c[c].

Attributes:

 dv Volume per grid point. h_cv Array of the grid spacing along the three axes. N_c Array of the number of grid points along the three axes. n_c Number of grid points on this CPU. beg_c Beginning of grid-point indices (inclusive). end_c End of grid-point indices (exclusive). comm MPI-communicator for domain decomposition.

The length unit is Bohr.

bytecount(dtype=<class 'float'>)[source]

Get the number of bytes used by a grid of specified dtype.

calculate_dipole_moment(rho_g, center=False, origin_c=None)[source]

Calculate dipole moment of density.

coarsen()[source]

Return coarsened $$GridDescriptor$$ object.

Reurned descriptor has 2x2x2 fewer grid points.

Collect distributed array to master-CPU or all CPU’s.

Return coordinates along one of the three axes.

Useful for plotting:

import matplotlib.pyplot as plt
plt.plot(gd.coords(0), data[:, 0, 0])
plt.show()
distribute(B_xg, out=None)[source]

Distribute full array B_xg to subdomains, result in b_xg.

B_xg is not used by the slaves (i.e. it should be None on all slaves) b_xg must be allocated on all nodes and will be overwritten.

Return new uninitialized 3D array for this domain.

The type can be set with the dtype keyword (default: float). Extra dimensions can be added with n=dim. A global array spanning all domains can be allocated with global_array=True.

find_center(a_R)[source]

Calculate center of positive function.

get_boxes(spos_c, rcut, cut=True)[source]

Find boxes enclosing sphere.

get_grid_point_coordinates(dtype=<class 'float'>, global_array=False)[source]

Construct cartesian coordinates of grid points in the domain.

get_grid_point_distance_vectors(r_v, mic=True, dtype=<class 'float'>)[source]

Return distances to a given vector in the domain.

mic: if true adopts the mininimum image convention procedure by W. Smith in ‘The Minimum image convention in Non-Cubic MD cells’ March 29, 1989

get_nearest_grid_point(spos_c, force_to_this_domain=False)[source]

Return index of nearest grid point.

The nearest grid point can be on a different CPU than the one the nucleus belongs to (i.e. return can be negative, or larger than gd.end_c), in which case something clever should be done. The point can be forced to the grid descriptors domain to be consistent with self.get_rank_from_position(spos_c).

integrate(a_xg, b_yg=None, global_integral=True, hermitian=False, _transposed_result=None)[source]

Integrate function(s) over domain.

a_xg: ndarray

Function(s) to be integrated.

b_yg: ndarray

If present, integrate a_xg.conj() * b_yg.

global_integral: bool

If the array(s) are distributed over several domains, then the total sum will be returned. To get the local contribution only, use global_integral=False.

hermitian: bool

Result is hermitian.

_transposed_result: ndarray

Long story. Don’t use this unless you are a method of the MatrixOperator class …

interpolate_grid_points(spos_nc, vt_g, target_n, use_mlsqr=True)[source]

Return interpolated values.

Calculate interpolated values from array vt_g based on the scaled coordinates on spos_c.

Uses moving least squares algorithm by default, or otherwise trilinear interpolation.

This doesn’t work in parallel, since it would require communication between neighbouring grid.

new_descriptor(N_c=None, cell_cv=None, pbc_c=None, comm=None, parsize_c=None)[source]

Create new descriptor based on this one.

The new descriptor will use the same class (possibly a subclass) and all arguments will be equal to those of this descriptor unless new arguments are provided.

plane_wave(k_c)[source]

Evaluate plane wave on grid.

Returns:

_ _
ik.r
e    ,

where the wave vector is given by k_c (in units of reciprocal lattice vectors).

refine()[source]

Return refined $$GridDescriptor$$ object.

Returned descriptor has 2x2x2 more grid points.

wannier_matrix(psit_nG, psit_nG1, G_c, nbands=None)[source]

Wannier localization integrals

The soft part of Z is given by (Eq. 27 ref1):

~       ~     -i G.r   ~
Z   = <psi | e      |psi >
nm       n             m

psit_nG and psit_nG1 are the set of wave functions for the two different spin/kpoints in question.

ref1: Thygesen et al, Phys. Rev. B 72, 125119 (2005)

Pad array with zeros as first element along non-periodic directions.

Array may either be local or in standard decomposition.

Return new zeroed 3D array for this domain.

The type can be set with the dtype keyword (default: float). Extra dimensions can be added with n=dim. A global array spanning all domains can be allocated with global_array=True.

class gpaw.scf.SCFLoop(eigenstates=0.1, energy=0.1, density=0.1, force=inf, maxiter=100, niter_fixdensity=None, nvalence=None)[source]

Self-consistent field loop.

collect_errors(dens, ham, wfs)[source]

Check convergence of eigenstates, energy and density.

log(log, niter, wfs, ham, dens, occ, errors)[source]

Output from each iteration.