def exchange(psi1, psi2, i, eps_exchange, T, molecule):
exchange = tuck.zeros(psi1[0].n)
for j in xrange(molecule.orbitals):
conv = tuck.cross.multifun([psi1[i], psi2[j]], eps_exchange, lambda (a,b): a*b, y0 = psi1[i])
conv = tuck.cross.conv(T, conv, eps_exchange, y0 = conv)
conv = tuck.round(tuck.real(conv),eps_exchange)
conv = tuck.cross.multifun([psi2[j], conv], eps_exchange, lambda (a,b): a*b,y0 = psi2[j])
exchange = exchange + conv
exchange = tuck.round(exchange, eps_exchange)
return exchange
def hf_full_energy(molecule, psi, E, grid, T, eps):
eps_exchange = eps
a = grid[0]
b = grid[1]
N = grid[2]
h = (b-a)/(N-1)
sf = h ** (3./2)
Norb = molecule.orbitals
num_atoms = molecule.num_atoms
prod = lambda A,B,C: tuck.cross.multifun([A,B], eps, lambda (a,b): a*b, y0=C)
density = tuck.zeros((N,N,N)) # density calculation
for i in xrange(Norb):
density = density + prod(psi[i], psi[i], psi[i]) # !can be faster!
density = tuck.round(density, eps)
pot_hartree = tuck.cross.conv(T, density, eps)
pot_hartree = tuck.real(pot_hartree)
pot_hartree = tuck.round(pot_hartree, eps)
V = [0] * Norb
for i in xrange(Norb):
V[i] = (prod(tuck.round(2*pot_hartree, eps), psi[i], psi[i]) -
qt.pots.exchange(psi, psi, i, eps_exchange, T, molecule))
V[i] = tuck.round(V[i], eps)
E_electr = 0
for i in xrange(Norb):
E_electr = E_electr + 2*E[i] - sf**2*tuck.dot(psi[i], V[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))
return E_nuc + E_electr
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 pyquante_hf(mol, x, eps, basis = "sto-3g"):
atoms = [None]*mol.num_atoms
for i in xrange(mol.num_atoms):
atoms[i] = (mol.atoms[i].charge,
(mol.atoms[i].rad[0],mol.atoms[i].rad[1],mol.atoms[i].rad[2]))
mol_pq = Molecule(mol.name, atoms, units='Bohr')
solver = SCF(mol_pq,method="HF",ConvCriteria=1e-7,MaxIter=40,basis=basis)
print 'SCF iteration - done'
solver.iterate()
norb=int(solver.solver.nclosed)
w=solver.basis_set.get()
cf=solver.solver.orbs[:,0:norb]
psi = [None]*norb
for i in xrange(norb):
psi[i] = gto2tuck(w,cf[:,i],x)
print psi[i].r
psi[i] = local(psi[i],1e-14)
psi[i] = tuck.round(psi[i],eps)
print psi[i].r
E = solver.solver.orbe[:norb]
return psi,E
def lda_full_energy(molecule, psi, E, grid, T, eps, eps_exchange):
a = grid[0]
b = grid[1]
N = grid[2]
h = (b-a)/(N-1)
sf = h ** (3./2)
Norb = molecule.orbitals
num_atoms = molecule.num_atoms
prod = lambda A,B: tuck.cross.multifun([A,B], eps, lambda (a,b): a*b)
density = tuck.zeros((N,N,N)) # density calculation
for i in xrange(Norb):
density = density + prod(psi[i], psi[i], psi[i]) # !can be faster!
density = tuck.round(density, eps)
pot_hartree = tuck.cross.conv(T, density, eps)
V = [0] * Norb
for i in xrange(Norb):
V[i] = (prod(tuck.round(2*pot_hartree, eps), psi[i], psi[i]) -
qt.pots.exchange(psi, psi, i, eps_exchange, T, molecule))
V[i] = tuck.round(V[i], eps)
E_electr = 0
for i in xrange(Norb):
E_electr = E_electr + 2*E[i] - sf**2*tuck.dot(psi[i], V[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))
return E_nuc + E_electr
def UT_prod(psi, R, eps): #psi by R - Upper Triangular matrix multiplication
Norb = len(psi)
psi_new = [tuck.zeros(psi[0].n)]*Norb
for i in xrange(Norb):
for j in xrange(i+1):
psi_new[i] = psi_new[i] + R[j,i]*psi[j]
psi_new[i] = tuck.round(psi_new[i], eps)
return psi_new
def LT_prod(psi, L, eps): #psi by L - Lower Triangular matrix multiplication
Norb = len(psi)
psi_new = [tuck.zeros(psi[0].n)]*Norb
for i in xrange(Norb):
for j in xrange(i+1):
psi_new[i] = psi_new[i] + L[i,j]*psi[j]
psi_new[i] = tuck.round(psi_new[i], eps)
return psi_new
def psi2dens(psi, eps):
density = tuck.zeros(psi[0].n)
for i in xrange(len(psi)):
density = density + tuck.cross.multifun([psi[i]], eps/10, lambda a: a[0]**2, y0 = psi[i])
density = tuck.round(density, eps/10)
density = 2*density
return density
def UT_prod(psi, R, eps): #psi by R - Upper Triangular matrix multiplication
Norb = len(psi)
psi_new = [tuck.zeros(psi[0].n)] * Norb
for i in xrange(Norb):
for j in xrange(i + 1):
psi_new[i] = psi_new[i] + R[j, i] * psi[j]
psi_new[i] = tuck.round(psi_new[i], eps)
return psi_new
def exchange(psi1, psi2, i, eps_exchange, T, molecule):
exchange = tuck.zeros(psi1[0].n)
for j in xrange(molecule.orbitals):
conv = tuck.cross.multifun([psi1[i], psi2[j]],
eps_exchange,
lambda (a, b): a * b,
y0=psi1[i])
conv = tuck.cross.conv(T, conv, eps_exchange, y0=conv)
conv = tuck.round(tuck.real(conv), eps_exchange)
conv = tuck.cross.multifun([psi2[j], conv],
eps_exchange,
lambda (a, b): a * b,
y0=psi2[j])
exchange = exchange + conv
exchange = tuck.round(exchange, eps_exchange)
return exchange
def LT_prod(psi, L, eps): #psi by L - Lower Triangular matrix multiplication
Norb = len(psi)
psi_new = [tuck.zeros(psi[0].n)] * Norb
for i in xrange(Norb):
for j in xrange(i + 1):
psi_new[i] = psi_new[i] + L[i, j] * psi[j]
psi_new[i] = tuck.round(psi_new[i], eps)
return psi_new
def psi2dens(psi, eps):
density = tuck.zeros(psi[0].n)
for i in xrange(len(psi)):
density = density + tuck.cross.multifun([psi[i], psi[i]], eps/10, lambda (a,b): a*b, y0 = psi[i])
density = tuck.round(density, eps)
density = 2*density
return density
def hf(molecule, psi, E, eps, grid, T, ind, max_iter, poisson_solver='Fourier'):
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
Norb = molecule.orbitals
num_atoms = molecule.num_atoms
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)
T0 = tuck.round(T, eps)
k=0
for i in xrange(max_iter):
psi, E_new = hf1iter(molecule, psi, E, grid, T0, pot_coulomb, eps, poisson_solver)
#print E
if max(abs((E-E_new)/E))<eps:
k += 1
if k == 4:
print 'Process converged with', i, 'iterations'
break
E = E_new
print E
if i == max_iter - 1:
print 'Process did not converge with eps precision'
E_full = hf_full_energy(molecule, psi, E, grid, tuck.round(T, eps*1e-3), eps*1e-3)
print E_full
return psi, E, E_full
def get_coulomb(self):
molecule = self.molecule
x = self.params.grid
N = len(self.params.grid)
Norb = self.molecule.orbitals
num_atoms = self.molecule.num_atoms
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, self.params.ind, self.params.eps)
pot_coulomb = tuck.round(pot_coulomb, self.params.eps)
self.pots.coulomb = pot_coulomb
def get_coulomb(self):
molecule = self.molecule
x = self.params.grid
N = len(self.params.grid)
Norb = self.molecule.orbitals
num_atoms = self.molecule.num_atoms
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, self.params.ind, self.params.eps)
pot_coulomb = tuck.round(pot_coulomb, self.params.eps)
self.pots.coulomb = pot_coulomb
def prod(psi, S, density, eps): #psi by S matrix multiplication
Norb = len(psi)
psi_new = [tuck.zeros(psi[0].n)] * Norb
for i in xrange(Norb):
for j in xrange(Norb):
psi_new[i] = psi_new[i] + S[i, j] * psi[j]
psi_new[i] = tuck.round(psi_new[i], eps)
#def psi_S_i(psi):
# res = 0.
# for j in xrange(Norb):
# res += (S[i, j]) * psi[j]
# return np.real(res)
#psi_new[i] = tuck.cross.multifun(psi, eps, lambda a: psi_S_i(a), y0 = density, pr = 1)
return psi_new
def prod(psi, S, density, eps): #psi by S matrix multiplication
Norb = len(psi)
psi_new = [tuck.zeros(psi[0].n)]*Norb
for i in xrange(Norb):
for j in xrange(Norb):
psi_new[i] = psi_new[i] + S[i,j]*psi[j]
psi_new[i] = tuck.round(psi_new[i], eps)
#def psi_S_i(psi):
# res = 0.
# for j in xrange(Norb):
# res += (S[i, j]) * psi[j]
# return np.real(res)
#psi_new[i] = tuck.cross.multifun(psi, eps, lambda a: psi_S_i(a), y0 = density, pr = 1)
return psi_new
def newton_galerkin(x, eps, ind):
# galerkin tensor for convolution as a hartree potential
if ind == 6:
a, b, r = -15., 10, 80
elif ind == 8:
a, b, r = -20., 15, 145
elif ind == 10:
a, b, r = -25., 20, 220
elif ind == 12:
a, b, r = -30., 25, 320
else:
raise Exception("wrong ind parameter")
N = x.shape
hr = (b-a)/(r - 1)
h = x[1]-x[0]
s = np.array(range(r), dtype = np.complex128)
s = a + hr * (s - 1)
w = np.zeros(r, dtype = np.complex128)
for alpha in xrange(r):
w[alpha] = 2*hr * np.exp(s[alpha]) / np.sqrt(pi)
w[0] = w[0]/2
w[r-1] = w[r-1]/2
U = np.zeros((N[0], r), dtype = np.complex128)
for alpha in xrange(r):
U[:, alpha] = ( func_int(x-h/2, x[0]-h/2, np.exp(2*s[alpha])) -
func_int(x+h/2, x[0]-h/2, np.exp(2*s[alpha])) +
func_int(x+h/2, x[0]+h/2, np.exp(2*s[alpha])) -
func_int(x-h/2, x[0]+h/2, np.exp(2*s[alpha])) )
newton = can2tuck(w, U, U, U)
newton = tuck.round(newton, eps)
return (1./h**3) * newton
def newton_galerkin(x, eps, ind):
# galerkin tensor for convolution as a hartree potential
if ind == 6:
a, b, r = -15., 10, 80
elif ind == 8:
a, b, r = -20., 15, 145
elif ind == 10:
a, b, r = -25., 20, 220
elif ind == 12:
a, b, r = -30., 25, 320
else:
raise Exception("wrong ind parameter")
N = x.shape
hr = (b - a) / (r - 1)
h = x[1] - x[0]
s = np.array(range(r), dtype=np.complex128)
s = a + hr * (s - 1)
w = np.zeros(r, dtype=np.complex128)
for alpha in xrange(r):
w[alpha] = 2 * hr * np.exp(s[alpha]) / np.sqrt(pi)
w[0] = w[0] / 2
w[r - 1] = w[r - 1] / 2
U = np.zeros((N[0], r), dtype=np.complex128)
for alpha in xrange(r):
U[:,
alpha] = (func_int(x - h / 2, x[0] - h / 2, np.exp(2 * s[alpha])) -
func_int(x + h / 2, x[0] - h / 2, np.exp(2 * s[alpha])) +
func_int(x + h / 2, x[0] + h / 2, np.exp(2 * s[alpha])) -
func_int(x - h / 2, x[0] + h / 2, np.exp(2 * s[alpha])))
newton = can2tuck(w, U, U, U)
newton = tuck.round(newton, eps)
return (1. / h**3) * newton
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 * self.rho) * (x[1] - x[0])**3 + dE_nuc
def coulomb(x, vec, ind, eps, beta=1.0): # 1/|r-vec|**beta in Tucker format
if ind == 6:
a, b, r = -15., 10, 80
elif ind == 8:
a, b, r = -20., 15, 145
elif ind == 10:
a, b, r = -25., 20, 220
elif ind == 12:
a, b, r = -30., 25, 320
else:
raise Exception("wrong ind parameter")
N = x.shape
h = (b-a)/(r - 1)
s = np.array(range(r))
s = a + h * (s - 1)
w = np.zeros(r, dtype = np.float64)
for alpha in xrange(r):
w[alpha] = 2*h * np.exp(beta*s[alpha]) / gamma(beta/2)
w[0] = w[0]/2
w[r-1] = w[r-1]/2
U1 = np.zeros((N[0], r), dtype=np.float64)
U2 = np.zeros((N[0], r), dtype=np.float64)
U3 = np.zeros((N[0], r), dtype=np.float64)
for alpha in xrange(r):
U1[:, alpha] = np.exp(-(x-vec[0])**2 * np.exp(2*s[alpha]))
U2[:, alpha] = np.exp(-(x-vec[1])**2 * np.exp(2*s[alpha]))
U3[:, alpha] = np.exp(-(x-vec[2])**2 * np.exp(2*s[alpha]))
newton = tuck.can2tuck(w, U1, U2, U3)
newton = tuck.round(newton, eps)
return newton
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*self.rho)*(x[1]-x[0])**3 + dE_nuc
def coulomb(x, vec, ind, eps, beta=1.0): # 1/|r-vec|**beta in Tucker format
if ind == 6:
a, b, r = -15., 10, 80
elif ind == 8:
a, b, r = -20., 15, 145
elif ind == 10:
a, b, r = -25., 20, 220
elif ind == 12:
a, b, r = -30., 25, 320
else:
raise Exception("wrong ind parameter")
N = x.shape
h = (b - a) / (r - 1)
s = np.array(range(r))
s = a + h * (s - 1)
w = np.zeros(r, dtype=np.float64)
for alpha in xrange(r):
w[alpha] = 2 * h * np.exp(beta * s[alpha]) / gamma(beta / 2)
w[0] = w[0] / 2
w[r - 1] = w[r - 1] / 2
U1 = np.zeros((N[0], r), dtype=np.float64)
U2 = np.zeros((N[0], r), dtype=np.float64)
U3 = np.zeros((N[0], r), dtype=np.float64)
for alpha in xrange(r):
U1[:, alpha] = np.exp(-(x - vec[0])**2 * np.exp(2 * s[alpha]))
U2[:, alpha] = np.exp(-(x - vec[1])**2 * np.exp(2 * s[alpha]))
U3[:, alpha] = np.exp(-(x - vec[2])**2 * np.exp(2 * s[alpha]))
newton = tuck.can2tuck(w, U1, U2, U3)
newton = tuck.round(newton, eps)
return newton
h = (b - a) / (N - 1)
x = np.zeros(N)
for i in xrange(N):
x[i] = a + i * h
def slater_fun((i, j, k)):
return np.exp(-(x[i]**2 + x[j]**2 + x[k]**2)**0.5)
def gaussian_fun((i, j, k)):
return np.exp(-(x[i]**2 + x[j]**2 + x[k]**2))
print 'Converting tensors in the Tucker format...'
print '(significantly faster version to be updated soon)'
a = tuck.cross.cross3d(slater_fun, N, eps)
b = tuck.cross.cross3d(gaussian_fun, N, eps)
print 'Converting is done'
print 'tensor a: %s' % (a)
print 'tensor b: %s' % (b)
print 'tensor 2a: %s' % (2 * a)
print 'tensor a+a: %s' % (a + a)
print 'tensor a+a after rounding: %s' % (tuck.round(a + a, eps))
print 'relative Frobenius norm of a-a: %s' % (tuck.norm(a - a) / tuck.norm(a))
# Warning! for big mode sizes (N >~ 256) full tensors may be out of memory.
# So, use tensor_full function carefully.
def mixing(solver, molecule, psi_0, E_0, eps, grid, T, ind, max_iter, m,
max_iter_inscf, pr):
count = 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
Norb = molecule.orbitals
num_atoms = molecule.num_atoms
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)
G = lambda psi, E, rho: solvers.dft_dens.lda(molecule,
psi,
E,
rho,
grid,
T,
eps,
pot_coulomb,
max_iter=max_iter_inscf,
pr=pr)[0:4]
Norb = molecule.orbitals
rho = psi2dens(psi_0, eps)
rho_m = []
Grho_m = []
D_m = []
rho_m.append(rho)
res = G(psi_0, E_0, rho)
psi = res[0]
E = res[1]
Grho_m.append(res[2])
D_m.append(tuck.round(Grho_m[0] - rho, eps))
rho = Grho_m[0]
err_and = np.zeros(max_iter)
for k in xrange(1, max_iter):
mk = min(k, m)
f = np.zeros(mk + 1)
f[mk] = 1.
A = np.ones((mk + 1, mk + 1))
A[mk, mk] = 0.
for p in xrange(mk):
for q in xrange(mk):
A[p, q] += 2 * qt.inner(D_m[p], D_m[q], h)
alpha = np.linalg.pinv(A).dot(f)
#alpha = np.array([0.5, 0.5, 1.])
if len(alpha) == 3:
if alpha[0] > 1. + 1e-2:
alpha[0] = .1
alpha[1] = .9
if alpha[0] < 0. - 1e-2:
alpha[0] = 0.1
alpha[1] = .9
rho_temp = tuck.zeros((N, N, N))
for p in xrange(mk):
rho_temp += tuck.round(alpha[p] * Grho_m[p], eps / 10)
rho = tuck.round(rho_temp, eps / 10)
#print alpha
rho_m.append(rho)
rho_m = rho_m[-min(k + 1, m):]
res = G(psi, E, rho)
psi = res[0]
E_new = res[1]
err_and[k] = max(np.abs(E - E_new) / np.abs(E_new))
#if err_and[k] < eps:
#break
solver.psi = psi
solver.orb_energies = E_new
print solver.psi[0].r, solver.psi[-1].r
print E_new
err = abs((E - E_new) / E)
E = copy.copy(E_new)
#solver.iterative.orb_energies.append(E_new)
#solver.iterative.convergence.append(err)
print 'Iteration', k, 'accuracy = %.2e' % max(err)
if max(err) < 4 * eps:
count += 1
if count == 4:
print 'Process converged with', i, 'iterations'
break
Grho_m.append(res[2])
Grho_m = Grho_m[-min(k + 1, m):]
D_m.append(tuck.round(Grho_m[-1] - rho_m[-1], eps))
D_m = D_m[-min(k + 1, m):]
Eel = solvers.dft_dens.lda(molecule,
psi,
E,
rho,
grid,
T,
eps * 1e-2,
pot_coulomb,
max_iter=max_iter_inscf,
pr=pr)[1]
Efull = solvers.dft_dens.lda_full_energy(
molecule, psi, E, rho, grid, T, eps * 1e-2, ind,
pr=None) + 2 * np.sum(Eel)
print Efull
solver.energy = Efull
return psi, E, rho, Efull, err_and
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 gaussian_fun((i,j,k)):
return np.exp(-(x[i]**2 + x[j]**2 + x[k]**2))
print 'Converting tensors in the Tucker format...'
print '(significantly faster version to be updated soon)'
a = tuck.cross.cross3d(slater_fun, N, eps)
b = tuck.cross.cross3d(gaussian_fun, N, eps)
print 'Converting is done'
print 'tensor a: %s' % (a)
print 'tensor b: %s' % (b)
print 'tensor 2a: %s' % (2*a)
print 'tensor a+a: %s' % (a + a)
print 'tensor a+a after rounding: %s' % (tuck.round(a+a, eps))
print 'relative Frobenius norm of a-a: %s' % (tuck.norm(a-a)/tuck.norm(a))
# Warning! for big mode sizes (N >~ 256) full tensors may be out of memory.
# So, use tensor_full function carefully.
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_full_energy(molecule, psi, E, density, grid, T, eps, 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
accuracy = np.zeros(max_iter)
timing = np.zeros(max_iter)
sf = h**(3. / 2) # scaling factor
prod = lambda A, B: tuck.cross.multifun([A, B], eps, lambda (a, b): a * b)
Norb = molecule.orbitals
num_atoms = molecule.num_atoms
density = tuck.zeros((N, N, N)) # density calculation
for i in xrange(Norb):
density = density + prod(psi[i], psi[i], psi[i]) # !can be faster!
density = tuck.round(density, eps)
pot_hartree = tuck.cross.conv(T, density, 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))
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)
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)
#accuracy[k] = ((E_correct - E_full)/abs(E_correct))
#print 'Relative Precision of the full Energy = %s' % (accuracy[k])
return E_full
def cross3d(func, M, eps_init, delta_add = 1e-5):
N = int((M+1)/2)
r1 = 2
r2 = 2
r3 = 2
GG = np.zeros((r1,r2,r3),dtype=np.complex128)
U1 = np.zeros((M,r1),dtype=np.complex128)
U2 = np.zeros((M,r2),dtype=np.complex128)
U3 = np.zeros((M,r3),dtype=np.complex128)
U1[:N,:] = np.random.random((N,r1))
U2[:N,:] = np.random.random((N,r2))
U3[:N,:] = np.random.random((N,r3))
U1, R = np.linalg.qr(U1)
U2, R = np.linalg.qr(U2)
U3, R = np.linalg.qr(U3)
eps_cross = 1
while True:
row_order_U1 = tuck.mv.maxvol(U1)
row_order_U2 = tuck.mv.maxvol(U2)
row_order_U3 = tuck.mv.maxvol(U3)
Ar = np.zeros((r1,r2,r3),dtype=np.complex128)
for i in xrange(r1):
for j in xrange(r2):
for k in xrange(r3):
Ar[i,j,k] = func((row_order_U1[i],row_order_U2[j],row_order_U3[k]))
U1_r = U1[row_order_U1,:]
U2_r = U2[row_order_U2,:]
U3_r = U3[row_order_U3,:]
G_UV = np.linalg.solve(U3_r,np.reshape(np.transpose(Ar,[2,0,1]),(r3,r1*r2),order='f'))
G_UV = np.reshape(G_UV,(r3,r1,r2),order='f')
G_UV = np.transpose(G_UV,[1,2,0])
G_U = np.linalg.solve(U2_r,np.reshape(np.transpose(G_UV,[1,2,0]),(r2,r1*r3),order='f'))
G_U = np.reshape(G_U,(r2,r3,r1),order='f')
G_U = np.transpose(G_U,[2,0,1])
G = np.linalg.solve(U1_r,np.reshape(G_U,(r1,r2*r3),order='f'))
G = np.reshape(G,(r1,r2,r3),order='f')
norm = np.linalg.norm(G)
eps_cross = (np.linalg.norm(GG-G))/norm
#print 'relative accuracy = %s' % (eps_cross), 'ranks = %s' % r1, r2, r3
G_Tucker = tuck.tensor(G, eps_init/10)
G = G_Tucker.core
U1 = np.dot(U1, G_Tucker.u[0])
U2 = np.dot(U2, G_Tucker.u[1])
U3 = np.dot(U3, G_Tucker.u[2])
(r1, r2, r3) = G_Tucker.r
if eps_cross < eps_init:
break
row_order_U1 = tuck.mv.maxvol(U1)
row_order_U2 = tuck.mv.maxvol(U2)
row_order_U3 = tuck.mv.maxvol(U3)
Ar = np.zeros((r1,r2,r3),dtype=np.complex128)
for i in xrange(r1):
for j in xrange(r2):
for k in xrange(r3):
Ar[i, j, k] = func((row_order_U1[i], row_order_U2[j], row_order_U3[k]))
A1 = np.reshape(Ar, [r1,-1], order='f')
A1_r = np.transpose(A1)
A1_r,R = np.linalg.qr(A1_r)
column_order_U1 = tuck.mv.maxvol(A1_r)
A2 = np.reshape(np.transpose(Ar, [1,0,2]), [r2,-1], order='f')
A2_r = np.transpose(A2)
A2_r,R = np.linalg.qr(A2_r)
column_order_U2 = tuck.mv.maxvol(A2_r)
A3 = np.reshape(np.transpose(Ar, [2,0,1]), [r3,-1], order='f')
A3_r = np.transpose(A3)
A3_r,R = np.linalg.qr(A3_r)
column_order_U3 = tuck.mv.maxvol(A3_r)
u1 = np.zeros((M, r1), dtype=np.complex128)
for i in xrange(r1):
for ii in xrange(M):
k1_order, j1_order = mod(column_order_U1[i], r2)
u1[ii,i] = func((ii, row_order_U2[j1_order], row_order_U3[k1_order]))
u2 = np.zeros((M, r2), dtype=np.complex128)
for j in xrange(r2):
for jj in xrange(M):
k1_order, i1_order = mod(column_order_U2[j], r1)
u2[jj,j] = func((row_order_U1[i1_order], jj, row_order_U3[k1_order]))
u3 = np.zeros((M, r3), dtype=np.complex128)
for k in xrange(r3):
for kk in xrange(M):
j1_order, i1_order = mod(column_order_U3[k], r1)
u3[kk,k] = func((row_order_U1[i1_order], row_order_U2[j1_order], kk))
u1, v, r11 = round_matrix(u1, delta_add)
u2, v, r22 = round_matrix(u2, delta_add)
u3, v, r33 = round_matrix(u3, delta_add)
u1 = u1[:,:r11]
u2 = u2[:,:r22]
u3 = u3[:,:r33]
U1_0 = np.zeros((M,r1+r11),dtype=np.complex128)
U2_0 = np.zeros((M,r2+r22),dtype=np.complex128)
U3_0 = np.zeros((M,r3+r33),dtype=np.complex128)
U1_0[:,:r1] = U1.copy()
U2_0[:,:r2] = U2.copy()
U3_0[:,:r3] = U3.copy()
U1_0[:,r1:r1+r11] = u1
U2_0[:,r2:r2+r22] = u2
U3_0[:,r3:r3+r33] = u3
U1 = U1_0.copy()
U2 = U2_0.copy()
U3 = U3_0.copy()
r1 = r1+r11
r2 = r2+r22
r3 = r3+r33
U1, R1 = np.linalg.qr(U1)
U2, R2 = np.linalg.qr(U2)
U3, R3 = np.linalg.qr(U3)
GG = np.zeros((r1,r2,r3),dtype=np.complex128)
GG[:(r1-r11),:(r2-r22),:(r3-r33)] = G.copy()
GG = np.dot(np.transpose(GG,[2,1,0]),np.transpose(R1))
GG = np.dot(np.transpose(GG,[0,2,1]),np.transpose(R2))
GG = np.transpose(GG,[1,2,0])
GG = np.dot(GG,np.transpose(R3))
#print 'ranks after rounding = %s' % r1, r2, r3
G_Tucker.n = (M, M, M)
G_Tucker.u[0] = U1
G_Tucker.u[1] = U2
G_Tucker.u[2] = U3
G_Tucker.r = [r1, r2, r3]
return tuck.round(G_Tucker, eps_init)
def get_rho(self):
self.rho = tuck.zeros(self.psi[0].n)
for i in xrange(len(self.psi)):
self.rho += tuck.cross.multifun([self.psi[i]], self.params.eps, lambda x: x[0]**2)
self.rho = tuck.round(self.rho, self.params.eps)
def cross3d(func, M, eps_init, delta_add=1e-5):
N = int((M + 1) / 2)
r1 = 2
r2 = 2
r3 = 2
GG = np.zeros((r1, r2, r3), dtype=np.complex128)
U1 = np.zeros((M, r1), dtype=np.complex128)
U2 = np.zeros((M, r2), dtype=np.complex128)
U3 = np.zeros((M, r3), dtype=np.complex128)
U1[:N, :] = np.random.random((N, r1))
U2[:N, :] = np.random.random((N, r2))
U3[:N, :] = np.random.random((N, r3))
U1, R = np.linalg.qr(U1)
U2, R = np.linalg.qr(U2)
U3, R = np.linalg.qr(U3)
eps_cross = 1
while True:
row_order_U1 = tuck.mv.maxvol(U1)
row_order_U2 = tuck.mv.maxvol(U2)
row_order_U3 = tuck.mv.maxvol(U3)
Ar = np.zeros((r1, r2, r3), dtype=np.complex128)
for i in xrange(r1):
for j in xrange(r2):
for k in xrange(r3):
Ar[i, j, k] = func(
(row_order_U1[i], row_order_U2[j], row_order_U3[k]))
U1_r = U1[row_order_U1, :]
U2_r = U2[row_order_U2, :]
U3_r = U3[row_order_U3, :]
G_UV = np.linalg.solve(
U3_r,
np.reshape(np.transpose(Ar, [2, 0, 1]), (r3, r1 * r2), order='f'))
G_UV = np.reshape(G_UV, (r3, r1, r2), order='f')
G_UV = np.transpose(G_UV, [1, 2, 0])
G_U = np.linalg.solve(
U2_r,
np.reshape(np.transpose(G_UV, [1, 2, 0]), (r2, r1 * r3),
order='f'))
G_U = np.reshape(G_U, (r2, r3, r1), order='f')
G_U = np.transpose(G_U, [2, 0, 1])
G = np.linalg.solve(U1_r, np.reshape(G_U, (r1, r2 * r3), order='f'))
G = np.reshape(G, (r1, r2, r3), order='f')
norm = np.linalg.norm(G)
eps_cross = (np.linalg.norm(GG - G)) / norm
#print 'relative accuracy = %s' % (eps_cross), 'ranks = %s' % r1, r2, r3
G_Tucker = tuck.tensor(G, eps_init / 10)
G = G_Tucker.core
U1 = np.dot(U1, G_Tucker.u[0])
U2 = np.dot(U2, G_Tucker.u[1])
U3 = np.dot(U3, G_Tucker.u[2])
(r1, r2, r3) = G_Tucker.r
if eps_cross < eps_init:
break
row_order_U1 = tuck.mv.maxvol(U1)
row_order_U2 = tuck.mv.maxvol(U2)
row_order_U3 = tuck.mv.maxvol(U3)
Ar = np.zeros((r1, r2, r3), dtype=np.complex128)
for i in xrange(r1):
for j in xrange(r2):
for k in xrange(r3):
Ar[i, j, k] = func(
(row_order_U1[i], row_order_U2[j], row_order_U3[k]))
A1 = np.reshape(Ar, [r1, -1], order='f')
A1_r = np.transpose(A1)
A1_r, R = np.linalg.qr(A1_r)
column_order_U1 = tuck.mv.maxvol(A1_r)
A2 = np.reshape(np.transpose(Ar, [1, 0, 2]), [r2, -1], order='f')
A2_r = np.transpose(A2)
A2_r, R = np.linalg.qr(A2_r)
column_order_U2 = tuck.mv.maxvol(A2_r)
A3 = np.reshape(np.transpose(Ar, [2, 0, 1]), [r3, -1], order='f')
A3_r = np.transpose(A3)
A3_r, R = np.linalg.qr(A3_r)
column_order_U3 = tuck.mv.maxvol(A3_r)
u1 = np.zeros((M, r1), dtype=np.complex128)
for i in xrange(r1):
for ii in xrange(M):
k1_order, j1_order = mod(column_order_U1[i], r2)
u1[ii, i] = func(
(ii, row_order_U2[j1_order], row_order_U3[k1_order]))
u2 = np.zeros((M, r2), dtype=np.complex128)
for j in xrange(r2):
for jj in xrange(M):
k1_order, i1_order = mod(column_order_U2[j], r1)
u2[jj, j] = func(
(row_order_U1[i1_order], jj, row_order_U3[k1_order]))
u3 = np.zeros((M, r3), dtype=np.complex128)
for k in xrange(r3):
for kk in xrange(M):
j1_order, i1_order = mod(column_order_U3[k], r1)
u3[kk, k] = func(
(row_order_U1[i1_order], row_order_U2[j1_order], kk))
u1, v, r11 = round_matrix(u1, delta_add)
u2, v, r22 = round_matrix(u2, delta_add)
u3, v, r33 = round_matrix(u3, delta_add)
u1 = u1[:, :r11]
u2 = u2[:, :r22]
u3 = u3[:, :r33]
U1_0 = np.zeros((M, r1 + r11), dtype=np.complex128)
U2_0 = np.zeros((M, r2 + r22), dtype=np.complex128)
U3_0 = np.zeros((M, r3 + r33), dtype=np.complex128)
U1_0[:, :r1] = U1.copy()
U2_0[:, :r2] = U2.copy()
U3_0[:, :r3] = U3.copy()
U1_0[:, r1:r1 + r11] = u1
U2_0[:, r2:r2 + r22] = u2
U3_0[:, r3:r3 + r33] = u3
U1 = U1_0.copy()
U2 = U2_0.copy()
U3 = U3_0.copy()
r1 = r1 + r11
r2 = r2 + r22
r3 = r3 + r33
U1, R1 = np.linalg.qr(U1)
U2, R2 = np.linalg.qr(U2)
U3, R3 = np.linalg.qr(U3)
GG = np.zeros((r1, r2, r3), dtype=np.complex128)
GG[:(r1 - r11), :(r2 - r22), :(r3 - r33)] = G.copy()
GG = np.dot(np.transpose(GG, [2, 1, 0]), np.transpose(R1))
GG = np.dot(np.transpose(GG, [0, 2, 1]), np.transpose(R2))
GG = np.transpose(GG, [1, 2, 0])
GG = np.dot(GG, np.transpose(R3))
#print 'ranks after rounding = %s' % r1, r2, r3
G_Tucker.n = (M, M, M)
G_Tucker.u[0] = U1
G_Tucker.u[1] = U2
G_Tucker.u[2] = U3
G_Tucker.r = [r1, r2, r3]
return tuck.round(G_Tucker, eps_init)
def lda_full_energy(molecule, psi, E, density, 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
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)
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_h + E_nuc + E_xc - dE_xc
print 'Full Energy = %s' % (E_full)
#accuracy[k] = ((E_correct - E_full)/abs(E_correct))
#print 'Relative Precision of the full Energy = %s' % (accuracy[k])
return E_full
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 mixing(solver, molecule, psi_0, E_0, eps, grid, T, ind, max_iter, m, max_iter_inscf, pr):
count = 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
Norb = molecule.orbitals
num_atoms = molecule.num_atoms
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)
G = lambda psi, E, rho: solvers.dft_dens.lda(molecule, psi, E, rho, grid, T, eps, pot_coulomb, max_iter = max_iter_inscf, pr=pr)[0:4]
Norb = molecule.orbitals
rho = psi2dens(psi_0, eps)
rho_m = []
Grho_m = []
D_m = []
rho_m.append(rho)
res = G(psi_0, E_0, rho)
psi = res[0]
E = res[1]
Grho_m.append(res[2])
D_m.append(tuck.round(Grho_m[0] - rho, eps))
rho = Grho_m[0]
err_and = np.zeros(max_iter)
for k in xrange(1, max_iter):
mk = min(k, m)
f = np.zeros(mk + 1)
f[mk] = 1.
A = np.ones((mk + 1, mk + 1))
A[mk, mk] = 0.
for p in xrange(mk):
for q in xrange(mk):
A[p, q] += 2 * qt.inner(D_m[p], D_m[q], h)
alpha = np.linalg.pinv(A).dot(f)
#alpha = np.array([0.5, 0.5, 1.])
if len(alpha) == 3:
if alpha[0]> 1. + 1e-2:
alpha[0] = .1
alpha[1] = .9
if alpha[0]< 0. - 1e-2:
alpha[0] = 0.1
alpha[1] = .9
rho_temp = tuck.zeros((N, N, N))
for p in xrange(mk):
rho_temp += tuck.round(alpha[p] * Grho_m[p], eps/10)
rho = tuck.round(rho_temp, eps/10)
#print alpha
rho_m.append(rho)
rho_m = rho_m[-min(k + 1, m):]
res = G(psi, E, rho)
psi = res[0]
E_new = res[1]
err_and[k] = max(np.abs(E - E_new)/np.abs(E_new))
#if err_and[k] < eps:
#break
solver.psi = psi
solver.orb_energies = E_new
print solver.psi[0].r, solver.psi[-1].r
print E_new
err = abs((E-E_new)/E)
E = copy.copy(E_new)
#solver.iterative.orb_energies.append(E_new)
#solver.iterative.convergence.append(err)
print 'Iteration', k, 'accuracy = %.2e' % max(err)
if max(err) < 4 * eps:
count += 1
if count == 4:
print 'Process converged with', i, 'iterations'
break
Grho_m.append(res[2])
Grho_m = Grho_m[-min(k + 1, m):]
D_m.append(tuck.round(Grho_m[-1] - rho_m[-1], eps))
D_m = D_m[-min(k + 1, m):]
Eel = solvers.dft_dens.lda(molecule, psi, E, rho, grid, T, eps*1e-2, pot_coulomb, max_iter = max_iter_inscf, pr=pr)[1]
Efull = solvers.dft_dens.lda_full_energy(molecule, psi, E, rho, grid, T, eps*1e-2, ind, pr = None) + 2*np.sum(Eel)
print Efull
solver.energy = Efull
return psi, E, rho, Efull, err_and
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
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_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 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
