def lda_full_energy(molecule, psi, E, pot_coulomb, grid, T, eps, ind, pr = None):
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
prod = lambda A,B,C: tuck.cross.multifun([A,B], eps, lambda (a,b): a*b, y0=C, pr=pr)
sf = h ** (3./2) # scaling factor
Norb = molecule.orbitals
num_atoms = molecule.num_atoms
density = psi2dens(psi, eps)
pot_hartree = tuck.cross.conv(T, density, eps)
pot_hartree = tuck.real(pot_hartree)
pot_hartree = tuck.round(pot_hartree, eps)
E_electr = 0
e_xc = tuck.cross.multifun([density, density], 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], 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, dtype=np.float64))
E_xc = sf**2*tuck.dot(e_xc, tuck.ones(e_xc.n, dtype=np.float64))
dE_xc = sf**2*tuck.dot(de_xc, tuck.ones(e_xc.n, dtype=np.float64))
#print E_xc, dE_xc, E_h
for i in xrange(Norb):
E_electr += 2*E[i]
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))
E_full = E_electr - E_h + E_nuc + E_xc - dE_xc
print 'Full Energy = %s' % (E_full)
print E_electr, - E_h + E_nuc + E_xc - dE_xc
#accuracy[k] = ((E_correct - E_full)/abs(E_correct))
#print 'Relative Precision of the full Energy = %s' % (accuracy[k])
return E_full
def gaussian(x, gamma): # Tucker representation of gaussian exp(-gamma * r**2)
n = x.shape[0]
a = tuck.ones((n, n, n))
a.u[0][:, 0] = np.exp(-gamma * x**2)
a.u[1][:, 0] = a.u[0][:, 0]
a.u[2][:, 0] = a.u[0][:, 0]
return a
def gaussian(x, gamma): # Tucker representation of gaussian exp(-gamma * r**2)
n = x.shape[0]
a = tuck.ones((n, n, n))
a.u[0][:,0] = np.exp(-gamma * x**2)
a.u[1][:,0] = a.u[0][:,0]
a.u[2][:,0] = a.u[0][:,0]
return a
def gradient_diatomic(self, smoothing=0.01):
# pot squared
#tensor_x = charge*tuck.round(vec[0]*tuck.ones((N, N, N)) - tensor_x, self.params.eps)
#qt.pots.coulomb(x, vec, self.params.ind, self.params.eps, beta=3.0)
#d_coulomb -= tuck.cross.multifun([tensor_x, pot_squared], self.params.eps, lambda x: x[0]*x[1])
molecule = self.molecule
x = self.params.grid
h = x[1] - x[0]
N = len(self.params.grid)
Norb = self.molecule.orbitals
num_atoms = self.molecule.num_atoms
i = 0
vec = molecule.atoms[i].rad
charge = molecule.atoms[i].charge
def fun_coulomb_squared((i, j, k)):
r = np.sqrt((vec[0] - x[i])**2 + (vec[1] - x[j])**2 + (vec[2] - x[k])**2)
return charge*(x[i] - vec[0]) / r * (-smoothed_coulomb_derivative(r/smoothing))/smoothing**2
d_coulomb = tuck.cross.cross3d(fun_coulomb_squared, N, self.params.eps)
d_coulomb = tuck.round(d_coulomb, self.params.eps)
self.get_rho()
dE_nuc = 0.0
j = 1
i = 0
vec_i = molecule.atoms[i].rad
charge_i = molecule.atoms[i].charge
vec_j = molecule.atoms[j].rad
charge_j = molecule.atoms[j].charge
dE_nuc = (dE_nuc + (vec_i[0] - vec_j[0])*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)**3)
return tuck.dot(d_coulomb, 2*tuck.ones(d_coulomb.n))*(x[1]-x[0])**3 #+ dE_nuc
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
# In[4]:
2*a - b
# In[16]:
x = sol.params.grid
# In[17]:
tuck.dot(sol.rho, tuck.ones(sol.rho.n))*(x[1]-x[0])**3
# In[4]:
from scipy.special import erf
def gradient_diatomic(self, smoothing=0.01):
# pot squared
#tensor_x = charge*tuck.round(vec[0]*tuck.ones((N, N, N)) - tensor_x, self.params.eps)
#qt.pots.coulomb(x, vec, self.params.ind, self.params.eps, beta=3.0)
#d_coulomb -= tuck.cross.multifun([tensor_x, pot_squared], self.params.eps, lambda x: x[0]*x[1])
molecule = self.molecule
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