def relax_continuum_orbital(atomlist,
phi0,
potential_ee,
E,
max_iter=100,
thresh=1.0e-6):
# DOES NOT WORK
"""
improves an initial guess phi0 for a continuum orbital with energy
E by iteratively solving the following linear equation
2 2
(H-E) dphi = -(H-E) phi
0
for the orbital correction dphi. The improved orbital is phi0+dphi.
The linear equation has the form
A.x = b
and can be solve using by modified Richardson iteration
(k+1) (k) (k)
x = x + w (b - A.x )
for some parameter w > 0.
Parameters
==========
atomlist : list of tuples (Zat,[x,y,z]) with atomic numbers
and positions
phi0 : callable, phi0(x,y,z) evaluates the initial orbital
potential_ee : callable, potential_ee(x,y,z) evaluates the effective
molecular Kohn-Sham potential WITHOUT the nuclear attraction,
Vee = Vcoul + Vxc
E : float, energy of continuum orbital
Optional
========
max_iter : int, maximum number of iterations
thresh : float, the iteration is stopped as soon as the error
falls below this threshold: |b-A.x| < thresh
Returns
=======
phi : callable, phi(x,y,z) evaluates the improved orbital
"""
w = 1.0
# right hand side of A.x = b
# b = (H-E)^2 phi0
residual1 = residual_ee_func(atomlist, phi0, potential_ee, E)
residual2 = residual_ee_func(atomlist, residual1, potential_ee, E)
b = residual2
def null(x, y, z):
return 0 * x
x = null
for i in range(0, max_iter):
# compute A.x^(k) = (H-E)^2 phi_k
print("residuals...")
residual1 = residual_ee_func(atomlist, x, potential_ee, E)
residual2 = residual_ee_func(atomlist, residual1, potential_ee, E)
Ax = residual2
# b - A.x^(k)
print("error...")
bmAx = add_two_functions(atomlist, b, Ax, 1.0, -1.0)
error = np.sqrt(overlap(atomlist, bmAx, bmAx))
print("iteration i= %d error |b-Ax|= %e" % (i, error))
if error < thresh:
print("CONVERGED")
break
# x^(k+1) = x^(k) + w * (b - A.x^(k))
x_next = add_two_functions(atomlist, x, bmAx, 1.0, w)
x = x_next
phi = add_two_functions(atomlist, phi0, x, 1.0, 1.0)
residual = residual_ee_func(atomlist, phi, potential_ee, E)
import matplotlib.pyplot as plt
plt.clf()
plt.title("Iteration i=%d" % (i + 1))
plt.xlabel("z / bohr")
plt.ylim((-0.6, +0.6))
r = np.linspace(-10.0, 10.0, 1000)
plt.plot(r, phi(0 * r, 0 * r, r), label=r"$\phi$")
plt.plot(r,
residual(0 * r, 0 * r, r),
ls="-.",
label="residual $(H-E)\phi$")
plt.legend()
plt.savefig("/tmp/relaxation_%3.3d.png" % (i + 1))
plt.show()
else:
msg = "Orbital relaxation did not converge in '%s' iterations!" % max_iter
print("WARNING: %s" % msg)
#raise RuntimeError(msg)
return phi
def improve_continuum_orbital(atomlist,
phi0,
potential_ee,
E,
max_iter=100,
thresh=1.0e-6):
"""
improves an initial guess phi0 for a continuum orbital with energy
E by repeatedly computing orbital corrections
phi_{i+1} = a_i * phi_{i} + b_i * dphi_{i}
The orbital correction delta_phi is obtained as the solution of
the inhomogeneous Schroedinger equation
(H-E) dphi = -(H-E) phi
i i
The Schroedinger equation is solved approximately by replacing the
effective potential with a spherical average around each atom. The
best linear combination of phi and the correction is chosen. The
coefficients a_i and b_i minimize the expectation value of the residual
2
R = (H-E)
2
a ,b = argmin <phi |(H-E) |phi >
i i i+1 i+1
Parameters
==========
atomlist : list of tuples (Zat,[x,y,z]) with atomic numbers
and positions
phi0 : callable, phi0(x,y,z) evaluates the initial orbital
potential_ee : callable, potential_ee(x,y,z) evaluates the effective
molecular Kohn-Sham potential WITHOUT the nuclear attraction,
Vee = Vcoul + Vxc
E : float, energy of continuum orbital
Optional
========
max_iter : int, maximum number of orbital correction
thresh : float, the iteration is stopped as soon as the weight
of the orbital correction falls below this threshold:
2
b_i <= thresh
Returns
=======
phi : callable, phi(x,y,z) evaluates the improved orbital
"""
# attraction potential between nuclei and electrons
nuclear_potential = nuclear_potential_func(atomlist)
# effective potential (electron-electron interaction + nuclei-electrons)
def potential(x, y, z):
return potential_ee(x, y, z) + nuclear_potential(x, y, z)
print("orbital corrections...")
phi = phi0
for i in range(0, max_iter):
residual = residual_ee_func(atomlist, phi, potential_ee, E)
def source(x, y, z):
return -residual(x, y, z)
# orbital correction
delta_phi = inhomogeneous_schroedinger(atomlist, potential, source, E)
a, b = variational_mixture_continuum(atomlist, phi, delta_phi,
potential_ee, E)
# The error is estimated as the norm^2 of the orbital correction:
# error = <delta_phi|delta_phi>
# For large radii, we cannot expect the solution to be correct since
# the density of points is far too low. Therefore we exclude all points
# outside the radius `rlarge` from the integration.
rlarge = 10.0
def error_density(x, y, z):
err = abs(delta_phi(x, y, z))**2
r = np.sqrt(x**2 + y**2 + z**2)
err[rlarge < r] = 0.0
return err
error = integral(atomlist, error_density)
print(" iteration i= %d |dphi|^2= %e b^2= %e (threshold= %e )" %
(i + 1, error, b**2, thresh))
# The solution is converged, when the weight of the
# correction term b**2 is small enough.
if b**2 < thresh:
print("CONVERGED")
break
# next approximation for phi
phi_next = add_two_functions(atomlist, phi, delta_phi, a, b)
### DEBUG
import matplotlib.pyplot as plt
plt.clf()
# plot cuts along z-axis
r = np.linspace(-40.0, 40.0, 10000)
x = 0.0 * r
y = 0.0 * r
z = r
plt.title("Iteration i=%d" % (i + 1))
plt.xlabel("z / bohr")
plt.ylim((-0.6, +0.6))
plt.plot(r, phi(x, y, z), label=r"$\phi$")
plt.plot(r, delta_phi(x, y, z), ls="--", label=r"$\Delta \phi$")
plt.plot(r, phi_next(x, y, z), label=r"$a \phi + b \Delta \phi$")
plt.plot(r, residual(x, y, z), ls="-.", label="residual $(H-E)\phi$")
plt.legend()
plt.savefig("/tmp/iteration_%3.3d.png" % (i + 1))
# plt.show()
###
# prepare for next iteration
phi = phi_next
else:
msg = "Orbital corrections did not converge in '%s' iterations!" % max_iter
print("WARNING: %s" % msg)
#raise RuntimeError(msg)
return phi
def test_lcao_continuum():
import matplotlib.pyplot as plt
# bond length in bohr
dist = 2.0
# positions of protons
posH1 = (0.0, 0.0, -dist / 2.0)
posH2 = (0.0, 0.0, +dist / 2.0)
atomlist = [(1, posH1), (1, posH2)]
# Set resolution of multicenter grid
settings.radial_grid_factor = 20
settings.lebedev_order = 23
# energy of continuum orbital
E = 1.0
# same functional as used in the calculation of pseudo orbitals
xc = XCFunctionals.libXCFunctional(Parameters.pseudo_orbital_x,
Parameters.pseudo_orbital_c)
dft = BasissetFreeDFT(atomlist, xc)
print("initial orbital guess from DFTB calculation")
orbitals = dft.getOrbitalGuess()
norb = len(orbitals)
# all orbitals are doubly occupied
nelec = 2 * norb
bound_orbitals = dft.getOrbitalGuess()
# effective potential
rho = density_func(bound_orbitals)
veff = effective_potential_func(atomlist, rho, xc, nelec=nelec)
ps = AtomicPotentialSet(atomlist)
r = np.linspace(-15.0, 15.0, 10000)
x = 0.0 * r
y = 0.0 * r
z = r
for lmax in [0, 1, 2, 3]:
bs = AtomicScatteringBasisSet(atomlist, E, lmax=lmax)
#test_AO_basis(atomlist, bs, ps, E)
R = residual2_matrix(atomlist, veff, ps, bs)
S = continuum_overlap(bs.bfs, E)
print("continuum overlap")
print(S)
print("residual^2 matrix")
print(R)
eigvals, eigvecs = sla.eigh(R, S)
print(eigvals)
print("eigenvector belonging to lowest eigenvalue")
print(eigvecs[:, 0])
# LCAO continuum orbitals
continuum_orbitals = orbital_transformation(atomlist, bs.bfs, eigvecs)
# improve continuum orbital by adding a correction term
#
# phi = phi0 + dphi
#
# The orbital correction dphi is the solution of the inhomogeneous
# Schroedinger equation
#
# (H-E)dphi = -(H-E)phi0
#
print("orbital correction...")
phi0 = continuum_orbitals[0]
phi = improve_continuum_orbital(atomlist, phi0, veff, E)
exit(-1)
residual_0 = residual_func(atomlist, phi0, veff, E)
def source(x, y, z):
return -residual_0(x, y, z)
delta_phi = inhomogeneous_schroedinger(atomlist, veff, source, E)
residual_d = residual_func(atomlist, delta_phi, veff, E)
a, b = variational_mixture_continuum(atomlist, phi0, delta_phi, veff,
E)
phi = add_two_functions(atomlist, phi0, delta_phi, a, b)
residual = residual_func(atomlist, phi, veff, E)
plt.plot(r, 1.0 / np.sqrt(2.0) * bs.bfs[0](x, y, z), label=r"AO")
plt.plot(r, phi0(x, y, z), label=r"$\phi_0$")
plt.plot(r, delta_phi(x, y, z), label=r"$\Delta \phi$")
plt.plot(r, phi(x, y, z), label=r"$\phi_0 + \Delta \phi$")
plt.legend()
plt.show()
"""
dphi = delta_phi(x,y,z)
imin = np.argmin(abs(r-1.0))
dphi[abs(r) < 1.0] = dphi[imin] - (dphi[abs(r) < 1.0] - dphi[imin])
plt.plot(r, dphi, label=r"$\Delta \phi$")
"""
plt.plot(r, residual_0(x, y, z), label=r"$(H-E) \phi_0$")
plt.plot(r, residual_d(x, y, z), label=r"$(H-E)\Delta \phi$")
plt.plot(r,
residual(x, y, z),
label=r"$(H-E)(a \phi_0 + b \Delta \phi)$")
plt.plot(r,
a * residual_0(x, y, z) + b * residual_d(x, y, z),
ls="-.",
label=r"$(H-E)(a \phi_0 + b \Delta \phi)$ (separate)")
plt.legend()
plt.show()
averaged_angular_distribution(atomlist, bound_orbitals,
continuum_orbitals, E)
# save continuum MOs to cubefiles
for i, phi in enumerate(continuum_orbitals):
def func(grid, dV):
x, y, z = grid
return phi(x, y, z)
Cube.function_to_cubefile(
atomlist,
func,
filename="/tmp/cmo_lmax_%2.2d_orb%4.4d.cube" % (lmax, i),
ppb=5.0)
#
for i, phi in enumerate(continuum_orbitals):
residual = residual_func(atomlist, phi, veff, E)
delta_e = energy_correction(atomlist,
residual,
phi,
method="Becke")
print(" orbital %d energy <%d|H-E|%d> = %e" % (i, i, i, delta_e))
l, = plt.plot(r,
phi(x, y, z),
label=r"$\phi_{%d}$ ($l_{max}$ = %d)" % (i, lmax))
plt.plot(r,
residual(x, y, z),
ls="-.",
label=r"$(H-E)\phi_{%d}$" % i,
color=l.get_color())
plt.legend()
plt.show()
def lithium_cation_continuum(l, m, k):
"""
compute continuum orbital in the electrostatic potential of the Li^+ core
Parameters
----------
l,m : angular quantum numbers of asymptotic solution
e.g. l=0,m=0 s-orbital
l=1,m=+1 px-orbital
k : length of wavevector in a.u., the energy of the
continuum orbital is E=1/2 k^2
"""
# Li^+ atom
atomlist = [(3, (0.0, 0.0, 0.0))]
charge = +1
# choose resolution of multicenter grids for bound orbitals
settings.radial_grid_factor = 20 # controls size of radial grid
settings.lebedev_order = 25 # controls size of angular grid
# 1s core orbitals for Li+^ atom
RDFT = BasissetFreeDFT(atomlist, None, charge=charge)
# bound_orbitals = RDFT.getOrbitalGuess()
Etot, bound_orbitals, orbital_energies = RDFT.solveKohnSham()
# choose resolution of multicenter grids for continuum orbitals
settings.radial_grid_factor = 120 # controls size of radial grid
settings.lebedev_order = 41 # controls size of angular grid
# show number of radial and angular points in multicenter grid
print_grid_summary(atomlist, settings.lebedev_order,
settings.radial_grid_factor)
print "electron density..."
# electron density of two electrons in the 1s core orbital
rho = density_func(bound_orbitals)
print "effective potential..."
# potential energy for Li nucleus and 2 core electrons
potential = effective_potential_func(atomlist, rho, None, nelec=2)
def v0(x, y, z):
r = np.sqrt(x * x + y * y + z * z)
return -1.0 / r
def v1(x, y, z):
return potential(x, y, z) - v0(x, y, z)
# The continuum orbital is specified by its energy and asymptotic
# angular momentum (E,l,m)
# energy of continuum orbital
E = 0.5 * k**2
# angular quantum numbers of asymptotic solution
assert abs(m) <= l
print " "
print "Asymptotic continuum wavefunction"
print "================================="
print " energy E= %e Hartree ( %e eV )" % (
E, E * AtomicData.hartree_to_eV)
print " wavevector k= %e a.u." % k
print " angular moment l= %d m= %+d" % (l, m)
print " "
# asymptotically correct solution for V0 = -1/r (hydrogen)
Cf = regular_coulomb_func(E, charge, l, m, 0.0)
phi0 = Cf
# right-hand side of inhomogeneous Schroedinger equation
def source(x, y, z):
return -v1(x, y, z) * phi0(x, y, z)
#
# solve (H0 + V1 - E) dphi = - V1 phi0
# for orbital correction dphi
atomic_numbers, atomic_coordinates = atomlist2arrays(atomlist)
print "Schroedinger equation..."
dphi = multicenter_inhomogeneous_schroedinger(
potential,
source,
E,
atomic_coordinates,
atomic_numbers,
radial_grid_factor=settings.radial_grid_factor,
lebedev_order=settings.lebedev_order)
# Combine asymptotically correct solution with correction
# phi = phi0 + dphi
phi = add_two_functions(atomlist, phi0, dphi, 1.0, 1.0)
# residual for phi0 R0 = (H-E)phi0
residual0 = residual_func(atomlist, phi0, potential, E)
# residual for final solution R = (H-E)phi
residual = residual_func(atomlist, phi, potential, E)
# spherical average of residual function
residual_avg = spherical_average_residual_func(atomlist[0], residual)
# The phase shift is determined by matching the radial wavefunction
# to a shifted and scaled Coulomb function at a number of radial
# sampling points drawn from the interval [rmin, rmax].
# On the one hand rmin < rmax should be chosen large enough,
# so that the continuum orbital approaches its asymptotic form,
# on the other hand rmax should be small enough that the accuracy
# of the solution due to the sparse r-grid is still high enough.
# A compromise has to be struck depending on the size of the radial grid.
# The matching points are spread over several periods,
# but not more than 30 bohr.
wavelength = 2.0 * np.pi / k
print "wavelength = %e" % wavelength
rmin = 70.0
rmax = rmin + max(10 * wavelength, 30.0)
Npts = 100
# determine phase shift and scaling factor by a least square
# fit the the regular Coulomb function
scale, delta = phaseshift_lstsq(atomlist, phi, E, charge, l, m, rmin, rmax,
Npts)
print "scale factor (relative to Coulomb wave) = %s" % scale
print "phase shift (relative to Coulomb wave) = %e " % delta
# normalize wavefunction, so that 1/scale phi(x,y,z) approaches
# asymptotically a phase-shifted Coulomb wave
phi_norm = multicenter_operation(
[phi],
lambda fs: fs[0] / scale,
atomic_coordinates,
atomic_numbers,
radial_grid_factor=settings.radial_grid_factor,
lebedev_order=settings.lebedev_order)
# The continuum orbital should be orthogonal to the bound
# orbitals belonging to the same Hamiltonian. I think this
# should come out correctly by default.
print " "
print " Overlaps between bound orbitals and continuum orbital"
print " ====================================================="
for ib, bound_orbital in enumerate(bound_orbitals):
olap_bc = overlap(atomlist, bound_orbital, phi_norm)
print " <bound %d| continuum> = %e" % (ib + 1, olap_bc)
print ""
# shifted regular coulomb function
Cf_shift = regular_coulomb_func(E, charge, l, m, delta)
# save radial wavefunctions and spherically averaged residual
# radial part of Coulomb wave without phase shift
phi0_rad = radial_wave_func(atomlist, phi0, l, m)
# radial part of shifted Coulomb wave
Cf_shift_rad = radial_wave_func(atomlist, Cf_shift, l, m)
# radial part of scattering solution
phi_norm_rad = radial_wave_func(atomlist, phi_norm, l, m)
print ""
print "# RADIAL_WAVEFUNCTIONS"
print "# Asymptotic wavefunction:"
print "# charge Z= %+d" % charge
print "# energy E= %e k= %e" % (E, k)
print "# angular momentum l= %d m= %+d" % (l, m)
print "# phase shift delta= %e rad" % delta
print "# "
print "# R/bohr Coulomb Coulomb radial wavefunction spherical avg. residual"
print "# shifted R_{l,m}(r) <|(H-E)phi|^2>"
import sys
# write table to console
r = np.linspace(1.0e-3, 100, 1000)
data = np.vstack((r, phi0_rad(r), Cf_shift_rad(r), phi_rad(r),
residual_avg(r))).transpose()
np.savetxt(sys.stdout, data, fmt=" %+e ")
print "# END"
print ""
def test_hmi_continuum():
"""
check that the continuum wavefunction of H2+ really are solutions
of Schroedinger's equation, i.e. have (H-E)\phi = 0 everywhere
starting from an LCAO guess for the continuum orbital, try to find
the exact solution by adding orbital corrections iteratively
"""
# First we compute the exact wavefunction of the hydrogen molecular ion.
from DFTB.Scattering import HMI
# The bond length and charges cannot be changed, since the
# separation constants were calculated only for the H2+ ion at R=2!
R = 2.0
Za = 1.0
Zb = 1.0
# energy of continuum orbital
E = 0.5
## sigma (m=0) orbital
m = 0
n = 0
trig = 'cos'
# separation constant
Lsep = HMI.SeparationConstants(R, Za, Zb)
Lsep.load_separation_constants()
Lfunc = Lsep.L_interpolated(m, n)
c2 = 0.5 * E * R**2
mL, nL, L = Lfunc(c2)
parity = (-1)**(mL + nL)
phi_exact = HMI.create_wavefunction(mL, L, R * (Za + Zb), 0.0, R, c2,
parity, trig)
# Old implementation of H2+ wavefunctions, the wavefunction
# looks indistinguishable from the exact wavefunction, but the
# non-zero residue shows that is contains large errors.
from DFTB.Scattering.hydrogen_molecular_ion import DimerWavefunctions
wfn = DimerWavefunctions(R, Za, Zb, plot=False)
delta, (Rfunc, Sfunc, Pfunc), wavefunction_exact = wfn.getContinuumOrbital(
m, n, trig, E)
def phi_exact_DW(x, y, z):
return wavefunction_exact((x, y, z), None)
# Set resolution of multicenter grid
settings.radial_grid_factor = 10
settings.lebedev_order = 41
# Next we compute the wavefunction using the basis set free method
atomlist = [(int(Za), (0.0, 0.0, -R / 2.0)),
(int(Zb), (0.0, 0.0, +R / 2.0))]
# no other electrons, only nuclear potential
def potential(x, y, z):
nuc = 0.0 * x
for Zi, posi in atomlist:
ri = np.sqrt((x - posi[0])**2 + (y - posi[1])**2 +
(z - posi[2])**2)
nuc += -Zi / ri
return nuc
# electron-electron interaction
def potential_ee(x, y, z):
return 0.0 * x
# Set resolution of multicenter grid
settings.radial_grid_factor = 10
settings.lebedev_order = 41
# residual of exact wavefunction (should be zero)
residual_exact = residual_func(atomlist, phi_exact, potential, E)
residual_ee_exact = residual_ee_func(atomlist, phi_exact, potential_ee, E)
residual_exact_DW = residual_func(atomlist, phi_exact_DW, potential, E)
# Laplacian
laplacian_exact = laplacian_func(atomlist, phi_exact)
import matplotlib.pyplot as plt
plt.clf()
r = np.linspace(-15.0, 15.0, 5000)
x = 0 * r
y = 0 * r
z = r
# plot exact wavefunction
plt.plot(r, phi_exact(x, y, z), label="$\phi$ exact")
#
phi_exact_xyz = phi_exact(x, y, z)
phi_exact_DW_xyz = phi_exact_DW(x, y, z)
scale = phi_exact_xyz.max() / phi_exact_DW_xyz.max()
plt.plot(r,
scale * phi_exact_DW_xyz,
label="$\phi$ exact (DimerWavefunction)")
# and residual
plt.plot(r, residual_exact(x, y, z), label=r"$(H-E)\phi$ (exact, old)")
plt.plot(r,
residual_exact_DW(x, y, z),
ls="-.",
label=r"$(H-E)\phi$ (exact, DimerWavefunction, old)")
plt.plot(r,
residual_ee_exact(x, y, z),
ls="--",
label=r"$(H-E)\phi$ (exact, new)")
# kinetic energy
plt.plot(r,
-0.5 * laplacian_exact(x, y, z),
ls="--",
label=r"$-\frac{1}{2}\nabla^2 \phi$")
# potential energy
plt.plot(r, (potential(x, y, z) - E) * phi_exact(x, y, z),
ls="--",
label=r"$(V-E)\phi$")
## The initial guess for the \sigma continuum orbital
## is a regular Coulomb function centered on the midpoint
## between the two protons.
#phi0 = regular_coulomb_func(E, +2, 0, 0, 0.0, center=(0.0, 0.0, 0.0))
## The initial guess for the \sigma continuum orbital is
## the sum of two hydrogen s continuum orbitals
bs = AtomicScatteringBasisSet(atomlist, E, lmax=0)
phi0 = add_two_functions(atomlist, bs.bfs[0], bs.bfs[1],
1.0 / np.sqrt(2.0), 1.0 / np.sqrt(2.0))
## start with exact wavefunction
#phi0 = phi_exact
plt.plot(r, phi0(x, y, z), ls="-.", label="guess $\phi_0$")
plt.legend()
plt.show()
#phi = improve_continuum_orbital(atomlist, phi0, potential, E, thresh=1.0e-6)
# place dummy atom at the center
atomlist_grid = atomlist + [(1, (0.0, 0.0, 0.0))]
phi = improve_continuum_orbital(atomlist_grid,
phi0,
potential,
E,
thresh=1.0e-6)
import matplotlib.pyplot as plt
plt.clf()
r = np.linspace(-15.0, 15.0, 5000)
x = 0 * r
y = 0 * r
z = r
phi_exact_xyz = phi_exact(x, y, z)
phi_xyz = phi(x, y, z)
# scale numerical phi such that the maxima agree
scale = phi_exact_xyz.max() / phi_xyz.max()
phi_xyz *= scale
print("scaling factor s = %s" % scale)
plt.plot(r, phi_exact_xyz, label="exact")
plt.plot(r, phi_xyz, label="numerical")
plt.legend()
plt.show()