def solve_radial_eigenproblem(n, l, r, u, relat=0, params=None):
"""
Solves the radial Schroedinger (Dirac) eigenproblem.
Input::
n, l ..... quantum numbers
r ........ radial mesh (NumPy array)
u ........ Potential on the radial mesh (for example -Z/r)
relat .... 0 solves Schroedinger equation
2 solves Dirac equation, spin up
3 solves Dirac equation, spin down
params ... optional dictionary with solver specific parameters (not all
parameters apply for each solver):
solver ... type of solver (dftatom, elk)
Z ... atomic number Z in the potential -Z/r
E_init, E_delta ... energy is sought in the interval
(Emin, Emax), where
E_min = E_init - E_delta
E_max = E_init + E_delta
eps ... accuracy for |Emax - Emin| < eps, default 1e-10
c ... speed of light in atomic units (default c = 137.035999037,
from http://arxiv.org/abs/1012.3627)
Returns (E, R), where E is the energy, and R(r) is the radial wave
function (normalized as \int R(r)**2 * r**2 \d r = 1)
"""
if params is None:
params = {}
solver = params.get("solver", "dftatom")
c = params.get("c", 137.035999037)
if solver == "dftatom":
if "Z" in params:
Z = params["Z"]
else:
# Disable the fragile estimation of Z below for now:
raise NotImplementedError("Z not specified. You can enable automatic determination of Z in qsnake/atom.py.")
# Estimate Z by assuming a coulombic potential u = -Z/r near the
# origin:
Z = -u[0] * r[0]
E_init = params.get("E_init", -3000)
E_delta = params.get("E_delta", 2000)
eps = params.get("eps", 1e-9)
from dftatom.rdirac import (solve_radial_eigenproblem,
ConvergeError as dftatom_ConvergeError)
try:
E, R = solve_radial_eigenproblem(c, n, l, E_init, E_delta, eps,
u, r, Z, relat)
except dftatom_ConvergeError, e:
raise ConvergeError(str(e))
return E, R
def solve_hydrogen_like_atom(Z, mesh_params, solver_params, verbose=False):
from sympy.physics.hydrogen import E_nl_dirac, E_nl
from sympy import TableForm
r_min = mesh_params["r_min"]
r_max = mesh_params["r_max"]
a = mesh_params["a"]
N = mesh_params["N"]
r = mesh_exp(r_min, r_max, a, N)
if verbose:
print "Mesh parameters:"
print TableForm([[r_min], [r_max], [a], [N]],
headings=(("r_min", "r_max", "a", "N"), ("Mesh parameters",)))
#print r
c = solver_params.get("c", 137.035999037)
solver_params["c"] = c
solver_params["Z"] = Z
dirac = solver_params.get("dirac", False)
# Potential:
vr = -Z/r
tot_error = -1
data = []
# (n, l):
# either k == l, or k == l + 1:
for n in range(1, 7):
for l in range(0, n):
if dirac:
relat_list = [2]
if l > 0:
relat_list.append(3)
else:
relat_list = [0]
for relat in relat_list:
try:
E, R = solve_radial_eigenproblem(n, l, r, vr, relat,
solver_params)
except ConvergeError:
if verbose:
print "Radial solver didn't converge"
raise
return 1e6
if dirac:
spin_up = (relat == 2)
E_exact = E_nl_dirac(n, l, spin_up=spin_up, Z=Z, c=c)
else:
E_exact = E_nl(n, Z=Z)
E_exact = float(E_exact)
delta = abs(E-E_exact)
if delta > tot_error:
tot_error = delta
if dirac:
k = int(spin_up)
else:
k = 0
data.append([n, l, k,
"%.6f" % E, "%.6f" % E_exact, "%.2e" % delta])
t = TableForm(data, alignment="right",
headings=(None, ("n", "l", "k", "E_calc", "E_exact", "delta")))
if verbose:
print t
return tot_error
