def lda(molecule, psi_0, E_0, grid, T, num_iter, eps, ind):
pr = None
E_correct = molecule.energy
eps_exchange = eps
a = grid[0]
b = grid[1]
N = grid[2]
h = (b - a) / (N - 1)
x = np.zeros(N)
for i in xrange(N):
x[i] = a + i * h
accuracy = np.zeros(num_iter)
timing = np.zeros(num_iter)
sf = h**(3. / 2) # scaling factor
Norb = molecule.orbitals
num_atoms = molecule.num_atoms
##### Output #####
output_E = np.zeros((Norb + 1, num_iter))
output_r = np.zeros((3, num_iter))
fname = molecule.name + '_' + 'lda' + '_' + str(N) + '_' + str(eps)
############################# Programm body ######################################
psi = psi_0
E = E_0
V = [0] * Norb
psi_new = [0] * Norb
################## Coulomb potential calculation ####################
prod = lambda A, B, C: tuck.cross.multifun(
[A, B], eps, lambda (a, b): a * b, y0=C)
E_nuc = 0
for i in xrange(num_atoms):
vec_i = molecule.atoms[i].rad
charge_i = molecule.atoms[i].charge
for j in xrange(i):
vec_j = molecule.atoms[j].rad
charge_j = molecule.atoms[j].charge
E_nuc = (E_nuc +
charge_i * charge_j / np.sqrt((vec_i[0] - vec_j[0])**2 +
(vec_i[1] - vec_j[1])**2 +
(vec_i[2] - vec_j[2])**2))
print "Coulomb potential..."
pot_coulomb = tuck.zeros((N, N, N))
for i in xrange(num_atoms):
vec = molecule.atoms[i].rad
charge = molecule.atoms[i].charge
pot_coulomb = pot_coulomb - charge * qt.pots.coulomb(x, vec, ind, eps)
pot_coulomb = tuck.round(pot_coulomb, eps)
################## Iteration process ####################
print "Iteration process..."
time_hartree = 0
time_exchange = 0
time_other = 0
time_whole = 0
time_iter = 0
check = 0
E_full = 0
E_full_0 = 0
E_full_1 = 1
for k in xrange(num_iter):
print "############################################ "
print "Iteration Number %s" % (k + 1)
print "############################################ "
time_hartree_iter = 0
time_exchange_iter = 0
start_iter = time.time()
start_hartree_iter = time.time()
density = tuck.zeros((N, N, N)) # density calculation
for i in xrange(Norb):
density = density + prod(tuck.conj(psi[i]), psi[i],
psi[i]) # !can be faster!
density = tuck.round(density, eps)
density = 2 * density
pot_hartree = tuck.cross.conv(T, density, eps, pr=None)
pot_hartree = tuck.round(tuck.real(pot_hartree), eps)
#density = tuck.real(density)
#density = tuck.round(density , eps)
end_hartree_iter = time.time()
time_hartree_iter += (end_hartree_iter - start_hartree_iter)
time_hartree += (end_hartree_iter - start_hartree_iter)
E_electr = 0
e_xc = tuck.cross.multifun([density, density],
3 * eps,
lambda (a, b):
((cen_fun(np.abs(np.real(a))) + xen_fun(
np.abs(np.real(a)))) * np.abs(b)),
y0=density,
pr=pr)
de_xc = tuck.cross.multifun(
[density, density],
3 * eps,
lambda (a, b):
((cpot_fun(np.abs(np.real(a))) + xpot_fun(np.abs(np.real(a)))) *
(b)),
y0=density,
pr=pr)
e_h = 0.5 * prod(pot_hartree, density, density)
E_h = sf**2 * tuck.dot(e_h, tuck.ones(e_xc.n))
E_xc = sf**2 * tuck.dot(e_xc, tuck.ones(e_xc.n))
dE_xc = sf**2 * tuck.dot(de_xc, tuck.ones(e_xc.n))
#print E_xc, dE_xc, E_h
for i in xrange(Norb):
V[i] = prod(tuck.round(pot_coulomb + pot_hartree, eps), psi[i],
psi[i])
psi[i] = tuck.real(psi[i])
psi[i] = tuck.round(psi[i], eps)
list = [None] * (Norb + 1)
list[1:] = psi
for i in xrange(Norb):
list[0] = psi[i]
Vxc = tuck.cross.multifun(
list,
10 * eps,
lambda (a): ((cpot_fun(density_fun(a[1:])) + xpot_fun(
density_fun(a[1:]))) * (a[0])),
y0=density,
pr=pr
) # Somehow it can not converge if precision eps. That is why 3*eps is used
V[i] = tuck.round(V[i] + Vxc, eps)
# Full energy computation
#V_energ = prod(pot_hartree, psi[i], psi[i])
E_electr += 2 * E[
i] # - sf**2*tuck.dot(psi[i], V_energ)/2 #- sf**2*tuck.dot(e_xc, tuck.ones(Exc.n))
psi_new[i] = -qt.poisson(2 * V[i], -2 * E[i], N, h, eps)
E_full_0 = E_full
E_full = E_electr - E_h + E_nuc + E_xc - dE_xc
E_full_1 = E_full
print 'Full Energy = %s' % (E_full)
print 'Correct Energy = %s' % (E_correct)
accuracy[k] = ((E_correct - E_full) / abs(E_correct))
print 'Relative Precision of the full Energy = %s' % (accuracy[k])
################# Fock matrix ###################
L = np.linalg.cholesky(qt.bilinear(psi_new, psi_new) *
sf**2) # orthogonalization
psi_Q = qt.UT_prod(psi_new, H(np.linalg.inv(L)), eps)
# Fock matrix
V_new = [0] * Norb
for i in xrange(Norb):
V_new[i] = prod(tuck.round(pot_coulomb + pot_hartree, eps),
psi_Q[i], psi_Q[i])
psi_Q[i] = tuck.real(psi_Q[i])
psi_Q[i] = tuck.round(psi_Q[i], eps)
list = [None] * (Norb + 1)
list[1:] = psi_Q
for i in xrange(Norb):
list[0] = psi_Q[i]
Vxc_new = tuck.cross.multifun(list,
eps * 10,
lambda (a):
((cpot_fun(density_fun(a[1:])) +
xpot_fun(density_fun(a[1:]))) *
(a[0])),
y0=density,
pr=pr)
V_new[i] = tuck.round(V_new[i] + Vxc_new, eps)
Energ1 = np.dot(np.diag(E), H(np.linalg.inv(L)))
Energ2 = qt.bilinear(psi_new, psi_new) * sf**2
Energ2 = np.dot(np.linalg.inv(L), Energ2)
Energ = np.dot(Energ2, Energ1)
F = qt.bilinear(psi_Q, V_new) * sf**2 - np.dot(
qt.bilinear(psi_Q, V) * sf**2, H(np.linalg.inv(L))) + Energ
print 'Fock Matrix:'
print F
#F = np.real((F + F.T)/2)
E_new = np.zeros(Norb, dtype=np.float64)
E_new, S = np.linalg.eigh(F)
output_E[:Norb, k] = E_new
output_E[Norb, k] = E_full
output_r[:, k] = density.r
np.save('experiments/' + fname, [output_E[:, :k], output_r[:, :k]])
print 'Orbital Energies:'
print np.array(E_new)
psi = qt.prod(psi_Q, S.T, density, eps)
for i in xrange(Norb):
psi[i] = qt.normalize(psi[i], h)
psi[i] = tuck.round(psi[i], eps)
E = E_new.copy()
for i in xrange(Norb):
if E[i] > 0:
E[i] = -0.0
end_iter = time.time()
print '1 Iteration Time: %s' % (end_iter - start_iter)
print 'Hartree Time on this Iteration: %s' % (time_hartree_iter)
#print 'Exchange Time on this Iteration: %s' % (time_exchange_iter)
time_whole += (end_iter - start_iter)
timing[k] = end_iter - start_iter
if abs((E_full_1 - E_full_0) / E_full_1) <= 10 * eps:
check += 1
if check == 3:
break
print 'The Whole Time: %s' % (time_whole)
return psi, E, accuracy, timing, E_full
def Hartree(molecule, psi_0, E_0, grid, T, num_iter, eps, ind):
alpha = 0.0
a = grid[0]
b = grid[1]
N = grid[2]
h = (b-a)/(N-1)
x = np.zeros(N)
for i in xrange(N):
x[i] = a + i*h
sf = h ** (3./2) # scaling factor
Norb = molecule.orbitals
num_atoms = molecule.num_atoms
############################# Programm body ######################################
psi = psi_0
E = E_0
V = [0]*Norb
psi_new = [0]*Norb
################## Coulomb potential calculation ####################
prod = lambda A,B: tuck.cross.multifun([A,B], eps, lambda (a,b): a*b)
print "Coulomb potential..."
pot_coulomb = tuck.zeros((N,N,N))
for i in xrange(num_atoms):
vec = molecule.atoms[i].rad
charge = molecule.atoms[i].charge
pot_coulomb = pot_coulomb - charge * qt.pots.coulomb(x, vec, ind, eps)
pot_coulomb = tuck.round(pot_coulomb, eps)
################## Iteration process ####################
print "Iteration process..."
for k in xrange(num_iter):
density = tuck.zeros((N,N,N)) # density calculation
for i in xrange(Norb):
density = density + prod(tuck.conj(psi[i]), psi[i]) # !can be faster!
density = tuck.round(density, eps)
pot_hartree = tuck.cross.conv(T, density, eps)
pot_hartree = tuck.round(tuck.real(pot_hartree),eps)
for i in xrange(Norb):
V[i] = prod(tuck.round(pot_coulomb + pot_hartree, eps), psi[i])
V[i] = tuck.round(V[i], eps)
psi_new[i] = -qt.poisson(2*V[i], -2*E[i], N, h, eps)
################# Fock matrix ###################
L = np.linalg.cholesky(qt.bilinear(psi_new, psi_new)*sf**2) # orthogonalization
psi_Q = qt.UT_prod(psi_new, H(np.linalg.inv(L)), eps)
# Fock matrix
V_new = [0]*Norb
for i in xrange(Norb):
V_new[i] = prod(tuck.round(pot_coulomb + pot_hartree,eps), psi_Q[i])
V_new[i] = tuck.round(V_new[i], eps)
Energ1 = np.dot(np.diag(E), H(np.linalg.inv(L)))
Energ2 = qt.bilinear(psi_new, psi_new)*sf**2
Energ2 = np.dot(np.linalg.inv(L), Energ2)
Energ = np.dot(Energ2, Energ1)
F = qt.bilinear(psi_Q, V_new)*sf**2 - np.dot(qt.bilinear(psi_Q, V)*sf**2, H(np.linalg.inv(L))) + Energ
print F
#F = np.real((F + F.T)/2)
E_new = np.zeros(Norb, dtype = np.complex128)
E_new, S = np.linalg.eigh(F)
#E_new = alpha*E + (1-alpha)*E_new
print np.array(E_new)
psi_new = qt.prod(psi_Q, S.T, eps)
for i in xrange(Norb):
psi[i] = alpha*psi[i] + (1-alpha)*qt.normalize(psi_new[i], h)
psi[i] = tuck.round(psi[i], eps)
E = E_new.copy()
for i in xrange(Norb):
if E[i]>0:
E[i] = -0.0
return E, psi
def hf1iter(molecule,
psi,
E,
grid,
T,
pot_coulomb,
eps,
poisson_solver,
num_iter=1):
E_correct = molecule.energy
eps_exchange = eps
a = grid[0]
b = grid[1]
N = grid[2]
h = (b - a) / (N - 1)
x = np.zeros(N)
for i in xrange(N):
x[i] = a + i * h
sf = h**(3. / 2) # scaling factor
Norb = molecule.orbitals
num_atoms = molecule.num_atoms
V = [0] * Norb
psi_new = [0] * Norb
prod = lambda A, B, C: tuck.cross.multifun(
[A, B], eps, lambda (a, b): a * b, y0=C)
for k in xrange(num_iter):
density = tuck.zeros((N, N, N)) # density calculation
for i in xrange(Norb):
density = density + prod(tuck.conj(psi[i]), psi[i],
psi[i]) # !can be faster!
density = tuck.round(density, eps)
pot_hartree = tuck.cross.conv(T, density, eps, pr=None)
pot_hartree = tuck.round(tuck.real(pot_hartree), eps)
E_electr = 0
for i in xrange(Norb):
V[i] = prod(tuck.round(pot_coulomb + 2 * pot_hartree, eps), psi[i],
psi[i])
exchange = qt.pots.exchange(psi, psi, i, eps_exchange, T, molecule)
V[i] = tuck.round(V[i] - exchange, eps)
# Full energy computation
#V_energ = tuck.round(prod(2*pot_hartree, psi[i],psi[i]) - exchange, eps)
#E_electr += 2*E[i] - sf**2*tuck.dot(psi[i], V_energ)
psi_new[i] = -qt.poisson(
2 * V[i], -2 * E[i], N, h, eps, solver=poisson_solver)
#if abs( (E_full_1-E_full_0)/E_full_1 ) <= 3 * eps:
#check += 1
#if check == 3:
#break
################# Fock matrix ###################
L = np.linalg.cholesky(qt.bilinear(psi_new, psi_new) *
sf**2) # orthogonalization
psi_Q = qt.UT_prod(psi_new, H(np.linalg.inv(L)), eps)
# Fock matrix
V_new = [0] * Norb
for i in xrange(Norb):
V_new[i] = prod(tuck.round(pot_coulomb + 2 * pot_hartree, eps),
psi_Q[i], psi_Q[i])
exchange_new = qt.pots.exchange(psi_Q, psi, i, eps_exchange, T,
molecule)
V_new[i] = tuck.round(V_new[i] - exchange_new, eps)
Energ1 = np.dot(np.diag(E), H(np.linalg.inv(L)))
Energ2 = qt.bilinear(psi_new, psi_new) * sf**2
Energ2 = np.dot(np.linalg.inv(L), Energ2)
Energ = np.dot(Energ2, Energ1)
F = qt.bilinear(psi_Q, V_new) * sf**2 - np.dot(
qt.bilinear(psi_Q, V) * sf**2, H(np.linalg.inv(L))) + Energ
#F = np.real((F + F.T)/2)
E_new = np.zeros(Norb, dtype=np.float64)
E_new, S = np.linalg.eigh(F)
psi = qt.prod(psi_Q, S.T, density, eps)
for i in xrange(Norb):
psi[i] = qt.normalize(psi[i], h)
psi[i] = tuck.round(psi[i], eps)
E = E_new.copy()
for i in xrange(Norb):
if E[i] > 0:
E[i] = -0.0
return psi, E
def lda(molecule,
psi,
E,
density,
grid,
T,
eps,
pot_coulomb,
max_iter=3,
pr=None):
E_correct = molecule.energy
a = grid[0]
b = grid[1]
N = grid[2]
h = (b - a) / (N - 1)
x = np.zeros(N)
for i in xrange(N):
x[i] = a + i * h
accuracy = np.zeros(max_iter)
timing = np.zeros(max_iter)
sf = h**(3. / 2) # scaling factor
Norb = molecule.orbitals
num_atoms = molecule.num_atoms
##### Output #####
#output_E = np.zeros((Norb+1, max_iter))
#output_r = np.zeros((3, max_iter))
#fname = molecule.name + '_' + 'lda' + '_' + str(N) + '_' + str(eps)
############################# Programm body ######################################
V = [0] * Norb
psi_new = [0] * Norb
prod = lambda A, B, C: tuck.cross.multifun(
[A, B], eps, lambda (a, b): a * b, y0=C)
################## Iteration process ####################
pot_hartree = tuck.cross.conv(T, density, eps, y0=density, pr=None)
pot_hartree = tuck.round(tuck.real(pot_hartree), eps)
for k in xrange(max_iter):
for i in xrange(Norb):
V[i] = prod(tuck.round(pot_coulomb + pot_hartree, eps), psi[i],
psi[i])
psi[i] = tuck.real(psi[i])
psi[i] = tuck.round(psi[i], eps)
for i in xrange(Norb):
Vxc = tuck.cross.multifun([density, psi[i]],
10 * eps,
lambda (a, b):
((cpot_fun(np.abs(np.real(a))) +
xpot_fun(np.abs(np.real(a)))) * (b)),
y0=psi[i],
pr=None)
V[i] = tuck.round(V[i] + Vxc, eps)
psi_new[i] = -qt.poisson(2 * V[i], -2 * E[i], N, h, eps)
################# Fock matrix ###################
L = np.linalg.cholesky(qt.bilinear(psi_new, psi_new) *
sf**2) # orthogonalization
psi_Q = qt.UT_prod(psi_new, H(np.linalg.inv(L)), eps)
# Fock matrix
V_new = [0] * Norb
for i in xrange(Norb):
V_new[i] = prod(tuck.round(pot_coulomb + pot_hartree, eps),
psi_Q[i], psi_Q[i])
psi_Q[i] = tuck.real(psi_Q[i])
psi_Q[i] = tuck.round(psi_Q[i], eps)
for i in xrange(Norb):
Vxc_new = tuck.cross.multifun([density, psi_Q[i]],
10 * eps,
lambda (a, b):
((cpot_fun(np.abs(np.real(a))) +
xpot_fun(np.abs(np.real(a)))) *
(b)),
y0=psi_Q[i],
pr=None)
V_new[i] = tuck.round(V_new[i] + Vxc_new, eps)
Energ1 = np.dot(np.diag(E), H(np.linalg.inv(L)))
Energ2 = qt.bilinear(psi_new, psi_new) * sf**2
Energ2 = np.dot(np.linalg.inv(L), Energ2)
Energ = np.dot(Energ2, Energ1)
F = qt.bilinear(psi_Q, V_new) * sf**2 - np.dot(
qt.bilinear(psi_Q, V) * sf**2, H(np.linalg.inv(L))) + Energ
print 'Fock Matrix:'
print F
#F = np.real((F + F.T)/2)
E_new = np.zeros(Norb, dtype=np.complex128)
E_new, S = np.linalg.eigh(F)
#output_E[:Norb,k] = E_new
#output_E[Norb,k] = E_full
#output_r[:,k] = density.r
#np.save('experiments/' + fname, [output_E[:,:k], output_r[:,:k]])
print 'Orbital Energies:'
print np.array(E_new)
psi = qt.prod(psi_Q, S.T, density, eps)
for i in xrange(Norb):
psi[i] = qt.normalize(psi[i], h)
psi[i] = tuck.round(psi[i], eps)
if np.linalg.norm(E - E_new) / np.linalg.norm(E_new) < 2 * eps:
break
E = E_new.copy()
for i in xrange(Norb):
if E[i] > 0:
E[i] = -0.0
density = tuck.zeros((N, N, N)) # density calculation
for i in xrange(Norb):
density = density + prod(tuck.conj(psi[i]), psi[i],
psi[i]) # !can be faster!
density = tuck.round(density, eps)
density = 2 * density
return psi, E, density, F
def lda_scf(solver, molecule, psi_0, E, grid, T, eps, pot_coulomb, pr=None, max_iter=1):
E_correct = molecule.energy
a = grid[0]
b = grid[1]
N = grid[2]
h = (b-a)/(N-1)
x = np.zeros(N)
for i in xrange(N):
x[i] = a + i*h
accuracy = np.zeros(max_iter)
timing = np.zeros(max_iter)
sf = h ** (3./2) # scaling factor
Norb = molecule.orbitals
num_atoms = molecule.num_atoms
##### Output #####
#output_E = np.zeros((Norb+1, max_iter))
#output_r = np.zeros((3, max_iter))
#fname = molecule.name + '_' + 'lda' + '_' + str(N) + '_' + str(eps)
############################# Programm body ######################################
psi = psi_0
V = [0]*Norb
psi_new = [0]*Norb
prod = lambda A,B,C: tuck.cross.multifun([A,B], eps, lambda (a,b): a*b, y0=C, pr=pr)
density = psi2dens(psi_0, eps)
################## Iteration process ####################
pot_hartree = tuck.cross.conv(T, density, eps, y0=density, pr=pr)
pot_hartree = tuck.round(tuck.real(pot_hartree), eps)
pot_V = tuck.round(pot_coulomb + pot_hartree, eps)
for k in xrange(max_iter):
#print "1 scf iter", E
for i in xrange(Norb):
V[i] = prod(pot_V, psi[i], psi[i])
psi[i] = tuck.real(psi[i])
psi[i] = tuck.round(psi[i], eps)
for i in xrange(Norb):
Vxc = tuck.cross.multifun(psi_0 + [psi[i]], 10*eps, lambda a: ((cpot_fun(density_fun(a[:-1])) + xpot_fun(density_fun(a[:-1])))*(a[-1])), y0 = psi[i], pr=pr)
V[i] = tuck.round(V[i] + Vxc, eps)
psi_new[i] = -qt.poisson(2*V[i], -2*E[i], N, h, eps)
################# Fock matrix ###################
L = np.linalg.cholesky(qt.bilinear(psi_new, psi_new)*sf**2) # orthogonalization
psi_Q = qt.UT_prod(psi_new, H(np.linalg.inv(L)), eps)
# Fock matrix
V_new = [0]*Norb
for i in xrange(Norb):
V_new[i] = prod(pot_V, psi_Q[i], psi_Q[i])
psi_Q[i] = tuck.real(psi_Q[i])
psi_Q[i] = tuck.round(psi_Q[i], eps)
for i in xrange(Norb):
Vxc_new = tuck.cross.multifun(psi_0 + [psi_Q[i]], 10*eps, lambda a: ((cpot_fun(density_fun(a[:-1])) + xpot_fun(density_fun(a[:-1])))*(a[-1])), y0 = psi_Q[i], pr=pr)
V_new[i] = tuck.round(V_new[i] + Vxc_new, eps)
Energ1 = np.dot(np.diag(E), H(np.linalg.inv(L)))
Energ2 = qt.bilinear(psi_new, psi_new)*sf**2
Energ2 = np.dot(np.linalg.inv(L), Energ2)
Energ = np.dot(Energ2, Energ1)
F = ((qt.bilinear(psi_Q, V_new)*sf**2 - np.dot(qt.bilinear(psi_Q, V)*sf**2, H(np.linalg.inv(L))) + Energ ))
#print 'Fock Matrix:'
#print F
#F = np.real((F + F.T)/2)
#E_new = np.zeros(Norb, dtype = np.float64)
E_new, S = np.linalg.eigh(F)
#S+= 1e-15
#output_E[:Norb,k] = E_new
#output_E[Norb,k] = E_full
#output_r[:,k] = density.r
#np.save('experiments/' + fname, [output_E[:,:k], output_r[:,:k]])
#print 'Orbital Energies:'
#print np.array(E_new)
#psi = copy.copy(psi_Q)
psi = qt.prod(psi_Q, S.T, density, eps)
for i in xrange(Norb):
psi[i] = qt.normalize(psi[i], h)
psi[i] = tuck.round(psi[i], eps)
#if np.linalg.norm(E - E_new)/np.linalg.norm(E_new) < 2*eps:
# break
E = E_new.copy()
for i in xrange(Norb):
if E[i]>0:
E[i] = -0.0
#density = tuck.zeros((N,N,N)) # density calculation
#density = tuck.cross.multifun(psi, eps/10, lambda a: density_fun(a), y0 = psi_Q[i], pr=pr)
return psi, E