def A_func(k, p):
# Initialise A at zero
A = 0
# Sum over mu - upper limit is k+2 so the expression is evaluated for mu=k+1
for mu in range(1, k + 2):
A += bang(k) / ((p**mu) * bang(k - mu + 1))
# Now multiply by the exponential term
A *= np.exp(-p)
return A
def hybrid_permutation(y, z, max_comparision=100000):
n = y.size + z.size
r = y.size
exact_comparisions = bang(n) / bang(r) / bang(n-r)
if exact_comparisions > max_comparision:
logging.warn('Using approximate test with %d samples' % max_comparision)
return permutation(y, z, max_comparision)
else:
return exact_permutation(y, z)
def B_func(k, pt):
# Initialise B at zero
B = 0
# B(0) can be evaluated much more straightforwardly (see Mulliken or Stewart)
# so include a check here to avoid unnecessary calculations
if pt == 0:
if k % 2 == 1: # k odd
B = 0
elif k % 2 == 0: # k even
B = 2 / (k + 1)
else:
# Calculate each of the sums seperately for ease of understanding, then
# multiply them by their exponential factors, then subtract them from B
# Label the two sums B_1 and B_2 - calculate them both in the same loop
# since the sums run over the same indices
B_1 = 0
B_2 = 0
for mu in range(1, k + 2):
B_1 += bang(k) / ((pt**mu) * bang(k - mu + 1))
B_2 += (bang(k) * (-1)**(k - mu)) / ((pt**mu) * bang(k - mu + 1))
B -= (B_1 * np.exp(-pt) + B_2 * np.exp(pt))
return B
def calc_prefactor(atoms, positions, basis, a, b):
# Extract orbital properties from the basis set
zeta_a = float(basis[a, -2])
zeta_b = float(basis[b, -2])
n_a = float(basis[a, 3])
n_b = float(basis[b, 3])
l_a = float(basis[a, 4])
l_b = float(basis[b, 4])
# Use atoms dictionary to match orbitals to atoms,
# then get their coordinates from the list of positions
# to calculate the bond length
# label = string, index = integer
atom_label_a = basis[a, 1]
atom_label_b = basis[b, 1]
atom_index_a = atoms[atom_label_a]
atom_index_b = atoms[atom_label_b]
R_ab = bond_length(positions, atom_index_a, atom_index_b)
# bang = factorial
prefactor = 0.5 * (zeta_a ** (n_a + 1/2) * (zeta_b ** (n_b + 1/2))) * \
((((2*l_a + 1) * (2*l_b + 1)) / (bang(2*n_a) * bang(2*n_b))) ** 0.5) \
* R_ab ** (n_a + n_b + 1)
return prefactor
def calc_radial_overlap(atoms, positions, basis, a, b, m):
# Extract orbital properties from the basis set
zeta_a = float(basis[a, -2])
zeta_b = float(basis[b, -2])
na = int(basis[a, 3])
nb = int(basis[b, 3])
la = int(basis[a, 4])
lb = int(basis[b, 4])
# Use atoms dictionary to match orbitals to atoms,
# then get their coordinates from the list of positions
# to calculate the bond length
# label = string, index = integer
atom_label_a = basis[a, 1]
atom_label_b = basis[b, 1]
atom_index_a = atoms[atom_label_a]
atom_index_b = atoms[atom_label_b]
R_ab = bond_length(positions, atom_index_a, atom_index_b)
# Define p and t
p = (R_ab * (zeta_a + zeta_b)) / 2
t = (zeta_a - zeta_b) / (zeta_a + zeta_b)
# Initialise overlap at 0
radial_overlap = 0
# Write each sum as a for loop
for ja in range(int((la - m) / 2) + 1):
for jb in range(int((lb - m) / 2) + 1):
for ka in range(m + 1):
for kb in range(m + 1):
for Pa in range(na - la - 2 * ja + 1):
for Pb in range(nb - lb - 2 * jb + 1):
for qa in range(la - m - 2 * ja + 1):
for qb in range(lb - m - 2 * jb + 1):
# Write each term on a different line and
# multiply them all together
radial_overlap += (
get_C_lmj(la, m, ja) *
get_C_lmj(lb, m, jb) *
((bang(lb - m - 2 * jb)) /
(bang(lb - m - 2 * jb - qb) *
bang(qb))) *
((bang(la - m - 2 * ja)) /
(bang(la - m - 2 * ja - qa) *
bang(qa))) *
((bang(na - la + 2 * ja)) /
(bang(na - la + 2 * ja - Pa) *
bang(Pa))) *
((bang(nb - lb + 2 * jb)) /
(bang(nb - lb + 2 * jb - Pb) *
bang(Pb))) * ((bang(m))**2) /
(bang(m - ka) * bang(ka) *
bang(m - kb) * bang(kb)) *
(-1)**(ka + kb + m + Pb + qb) *
B_func(2 * ka + Pa + Pb + qa + qb,
p * t) *
A_func(
2 * kb + na - la + 2 * ja + nb - lb
+ 2 * jb - Pa - Pb + qa + qb, p))
radial_overlap *= calc_prefactor(atoms, positions, basis, a, b)
return radial_overlap
def ncr(n,r):
return bang(n)/(bang(r)*bang(n-r))
def stirling_num_of_2nd_kind(n,k): #'''stirling_num_of_2nd_kind(number_of_elements, number_of_sets)''' from math import factorial as bang from scipy.misc import comb return (1./bang(k))*np.sum(map(lambda j: (-1)**(k-j)*comb(k,j,exact=True)*j**n, range(k+1)))