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
"""
# The initial guess for the \sigma continuum orbital is
# a hydrogen continuum orbital in the center
bs = AtomicScatteringBasisSet([(1, (0.0, 0.0, 0.0))], E, lmax=0)
phi0 = bs.bfs[0]
plt.plot(r, phi0(x, y, z), ls="-.", label="guess $\phi_0$")
plt.legend()
plt.show()
#phi = improve_continuum_orbital(atomlist, phi0, potential_ee, E, thresh=1.0e-6)
phi = relax_continuum_orbital(atomlist,
phi0,
potential_ee,
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()
def test_hmi_lcao_continuum():
"""
"""
# First we compute the exact wavefunction of the hydrogen molecular ion.
from DFTB.Scattering import HMI
R = 2.0
Za = 1.0
Zb = 1.0
E = 0.5
## sigma (m=0) orbital
m = 0
n = 0
L = m + n
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)
"""
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(x,y,z):
return wavefunction_exact((x,y,z), None)
"""
# 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
# Set resolution of multicenter grid
settings.radial_grid_factor = 40
settings.lebedev_order = 21
# residual of exact wavefunction (should be zero)
residual_exact = residual_func(atomlist, phi_exact, 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")
# and residual
plt.plot(r, residual_exact(x, y, z), label=r"$(H-E)\phi$ (exact)")
# 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"$(V-E)\phi$")
# LCAO continuum orbitals
ps = AtomicPotentialSet(atomlist)
lmax = 4
bs = AtomicScatteringBasisSet(atomlist, E, lmax=lmax)
R = residual2_matrix(atomlist, potential, 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])
continuum_orbitals = orbital_transformation(atomlist, bs.bfs, eigvecs)
plt.cla()
plt.plot(r, phi_exact(x, y, z), ls="--", label="$\phi$ (exact)")
for i in range(0, len(continuum_orbitals)):
plt.plot(r, continuum_orbitals[i](x, y, z))
plt.legend()
plt.show()