def calc_E_kin_bound(orbs):
"""
Computes the kinetic energy contribution from the bound electrons
Inputs:
- orbs (object) : the orbitals object
Returns:
- E_kin_bound (float) : kinetic energy
"""
# compute the grad^2 component
grad2_orbs = mathtools.laplace(orbs.eigfuncs, orbs._xgrid)
# compute the (l+1/2)^2 component
l_arr = np.array([(l + 0.5)**2.0 for l in range(config.lmax)])
lhalf_orbs = np.einsum("j,ijkl->ijkl", l_arr, orbs.eigfuncs)
# add together and multiply by eigfuncs*exp(-3x)
prefac = np.exp(-3.0 * orbs._xgrid) * orbs.eigfuncs
kin_orbs = prefac * (grad2_orbs - lhalf_orbs)
# multiply and sum over occupation numbers
e_kin_dens = np.einsum("ijk,ijkl->l", orbs.occnums, kin_orbs)
# integrate over sphere
E_kin_bound = -0.5 * mathtools.int_sphere(e_kin_dens, orbs._xgrid)
return E_kin_bound
def E_xc(density, xgrid, xfunc, cfunc):
"""
Wrapper function which computes the exchange and correlation energies
"""
# initialize the _E_xc dict (leading underscore to distinguish from function name)
_E_xc = {}
# get the total density
dens_tot = np.sum(density, axis=0)
# compute the exchange energy
ex_libxc = calc_xc(density, xgrid, xfunc, "e_xc")
_E_xc["x"] = mathtools.int_sphere(ex_libxc * dens_tot, xgrid)
# compute the correlation energy
ec_libxc = calc_xc(density, xgrid, cfunc, "e_xc")
_E_xc["c"] = mathtools.int_sphere(ec_libxc * dens_tot, xgrid)
# sum to get the total xc potential
_E_xc["xc"] = _E_xc["x"] + _E_xc["c"]
return _E_xc
def calc_E_en(density, xgrid):
"""
Computes the electron-nuclear energy
E_en = \int dr v_en(r) n(r)
Inputs:
- density (np array) : density
"""
# sum the density over the spin axes to get the total density
dens_tot = np.sum(density, axis=0)
# compute the integral
v_en = Potential.calc_v_en(xgrid)
E_en = mathtools.int_sphere(dens_tot * v_en, xgrid)
return E_en
def calc_E_ha(density, xgrid):
"""
Computes the Hartree energy
E_ha = 1/2 \int dr \int dr' n(r)n(r')/|r-r'|
Uses the pre-computed hartree potential
E_ha = 1/2 /int dr n(r) v_ha(r)
Inputs:
- density (np array) : the density object
- pot (object) : the potential object
Returns:
- E_ha (float) : the hartree energy
"""
# sum density over spins to get total density
dens_tot = np.sum(density, axis=0)
# compute the integral
v_ha = Potential.calc_v_ha(density, xgrid)
E_ha = 0.5 * mathtools.int_sphere(dens_tot * v_ha, xgrid)
return E_ha
def check_conv(self, E_free, pot, dens, iscf):
"""
Computes the changes in energy, integrated density and integrated potential
and checks if they satisfy the proscribed convergence parameters
Parameters
----------
E_free : float
the total free energy
v_s : ndarray
the KS potential
dens : ndarray
the electronic density
iscf : int
the iteration number
Returns
-------
conv_vals : dict
Dictionary of convergence parameters as follows:
{'conv_energy' : float, 'conv_rho' : ndarray,
'conv_pot' : ndarray, 'complete' : bool}
"""
conv_vals = {}
# first update the energy, potential and density attributes
self._energy[0] = E_free
self._potential[0] = pot
self._density[0] = dens
# compute the change in energy
conv_vals["dE"] = abs(
(self._energy[0] - self._energy[1]) / self._energy[0])
# compute the change in potential
dv = np.abs(self._potential[0] - self._potential[1])
# compute the norm
norm_v = mathtools.int_sphere(np.abs(self._potential[0]), self._xgrid)
conv_vals["dpot"] = mathtools.int_sphere(dv, self._xgrid) / norm_v
# compute the change in density
dn = np.abs(self._density[0] - self._density[1])
# integrate over sphere to return a number
# note we add a small constant to avoid errors if there are no electrons in one spin channel
conv_vals["drho"] = mathtools.int_sphere(
dn, self._xgrid) / (config.nele + 1e-3)
# reset the energy, potential and density attributes
self._energy[1] = E_free
self._potential[1] = pot
self._density[1] = dens
# see if the convergence criteria are satisfied
conv_energy = False
conv_rho = False
conv_pot = False
conv_vals["complete"] = False
if iscf > 3:
if conv_vals["dE"] < config.conv_params["econv"]:
conv_energy = True
if np.all(conv_vals["drho"] < config.conv_params["nconv"]):
conv_rho = True
if np.all(conv_vals["dpot"] < config.conv_params["vconv"]):
conv_pot = True
if conv_energy and conv_rho and conv_pot:
conv_vals["complete"] = True
return conv_vals