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from __future__ import division
from functools import reduce
from itertools import combinations, chain
from math import factorial
from operator import mul
import numpy as np
from ase.units import kg, C, _hbar, kB
from ase.vibrations import Vibrations
class FranckCondonOverlap:
"""Evaluate squared overlaps depending on the Huang-Rhys parameter."""
def factorial(self, n):
try:
return self._fac[n]
except AttributeError:
self._fac = [1]
return self.factorial(n)
except IndexError:
for i in range(len(self._fac), n + 1):
self._fac.append(i * self._fac[i - 1])
return self._fac[n]
def directT0(self, n, S):
"""Direct squared Franck-Condon overlap corresponding to T=0."""
return np.exp(-S) * S**n / self.factorial(n)
def direct(self, n, m, S_in):
"""Direct squared Franck-Condon overlap."""
if n > m:
# use symmetry
return self.direct(m, n, S_in)
S = np.array([S_in])
mask = np.where(S == 0)
S[mask] = 1 # hide zeros
s = 0
for k in range(n + 1):
s += (-1)**(n - k) * S**float(-k) / (
self.factorial(k) *
self.factorial(n - k) * self.factorial(m - k))
res = np.exp(-S) * S**(n + m) * s**2 * (
self.factorial(n) * self.factorial(m))
# use othogonality
res[mask] = int(n == m)
return res[0]
def direct0mm1(self, m, S):
"""<0|m><m|1>"""
sum = S**m
if m:
sum -= m * S**(m - 1)
return np.exp(-S) * np.sqrt(S) * sum / self.factorial(m)
def direct0mm2(self, m, S):
"""<0|m><m|2>"""
sum = S**(m + 1)
if m >= 1:
sum -= 2 * m * S**m
if m >= 2:
sum += m * (m - 1) * S**(m - 1)
return np.exp(-S) / np.sqrt(2) * sum / self.factorial(m)
class FranckCondon:
def __init__(self, atoms, vibname, minfreq=-np.inf, maxfreq=np.inf):
"""Input is a atoms object and the corresponding vibrations.
With minfreq and maxfreq frequencies can
be excluded from the calculation"""
self.atoms = atoms
# V = a * v is the combined atom and xyz-index
self.mm05_V = np.repeat(1. / np.sqrt(atoms.get_masses()), 3)
self.minfreq = minfreq
self.maxfreq = maxfreq
self.shape = (len(self.atoms), 3)
vib = Vibrations(atoms, name=vibname)
self.energies = np.real(vib.get_energies(method='frederiksen')) # [eV]
self.frequencies = np.real(
vib.get_frequencies(method='frederiksen')) # [cm^-1]
self.modes = vib.modes
self.H = vib.H
def get_Huang_Rhys_factors(self, forces):
"""Evaluate Huang-Rhys factors and corresponding frequencies
from forces on atoms in the exited electronic state.
The double harmonic approximation is used. HR factors are
the first approximation of FC factors,
no combinations or higher quanta (>1) exitations are considered"""
assert(forces.shape == self.shape)
# Hesse matrix
H_VV = self.H
# sqrt of inverse mass matrix
mm05_V = self.mm05_V
# mass weighted Hesse matrix
Hm_VV = mm05_V[:, None] * H_VV * mm05_V
# mass weighted displacements
Fm_V = forces.flat * mm05_V
X_V = np.linalg.solve(Hm_VV, Fm_V)
# projection onto the modes
modes_VV = self.modes
d_V = np.dot(modes_VV, X_V)
# Huang-Rhys factors S
s = 1.e-20 / kg / C / _hbar**2 # SI units
S_V = s * d_V**2 * self.energies / 2
# reshape for minfreq
indices = np.where(self.frequencies <= self.minfreq)
np.append(indices, np.where(self.frequencies >= self.maxfreq))
S_V = np.delete(S_V, indices)
frequencies = np.delete(self.frequencies, indices)
return S_V, frequencies
def get_Franck_Condon_factors(self, order, temp, forces):
"""Return FC factors and corresponding frequencies up to given order.
order= number of quanta taken into account
T= temperature in K. Vibronic levels are occupied by a
Boltzman distribution.
forces= forces on atoms in the exited electronic state"""
S, f = self.get_Huang_Rhys_factors(forces)
n = order + 1
T = temp
freq = np.array(f)
# frequencies
freq_n = [[] * i for i in range(n - 1)]
freq_neg = [[] * i for i in range(n - 1)]
for i in range(1, n):
freq_n[i - 1] = freq * i
freq_neg[i - 1] = freq * (-i)
# combinations
freq_nn = [x for x in combinations(chain(*freq_n), 2)]
for i in range(len(freq_nn)):
freq_nn[i] = freq_nn[i][0] + freq_nn[i][1]
indices2 = []
for i, y in enumerate(freq):
ind = [j for j, x in enumerate(freq_nn) if x % y == 0]
indices2.append(ind)
indices2 = [x for x in chain(*indices2)]
freq_nn = np.delete(freq_nn, indices2)
frequencies = [[] * x for x in range(3)]
frequencies[0].append(freq_neg[0])
frequencies[0].append([0])
frequencies[0].append(freq_n[0])
frequencies[0] = [x for x in chain(*frequencies[0])]
for i in range(1, n - 1):
frequencies[1].append(freq_neg[i])
frequencies[1].append(freq_n[i])
frequencies[1] = [x for x in chain(*frequencies[1])]
frequencies[2] = freq_nn
# Franck-Condon factors
E = freq / 8065.5
f_n = [[] * i for i in range(n)]
for j in range(0, n):
f_n[j] = np.exp(-E * j / (kB * T))
# partition function
Z = np.empty(len(S))
Z = np.sum(f_n, 0)
# occupation probability
w_n = [[] * k for k in range(n)]
for l in range(n):
w_n[l] = f_n[l] / Z
# overlap wavefunctions
O_n = [[] * m for m in range(n)]
O_neg = [[] * m for m in range(n)]
for o in range(n):
O_n[o] = [[] * p for p in range(n)]
O_neg[o] = [[] * p for p in range(n - 1)]
for q in range(o, n + o):
a = np.minimum(o, q)
summe = []
for k in range(a + 1):
s = ((-1)**(q - k) * np.sqrt(S)**(o + q - 2 * k) *
factorial(o) * factorial(q) /
(factorial(k) * factorial(o - k) * factorial(q - k)))
summe.append(s)
summe = np.sum(summe, 0)
O_n[o][q - o] = (np.exp(-S / 2) /
(factorial(o) * factorial(q))**(0.5) *
summe)**2 * w_n[o]
for q in range(n - 1):
O_neg[o][q] = [0 * b for b in range(len(S))]
for q in range(o - 1, -1, -1):
a = np.minimum(o, q)
summe = []
for k in range(a + 1):
s = ((-1)**(q - k) * np.sqrt(S)**(o + q - 2 * k) *
factorial(o) * factorial(q) /
(factorial(k) * factorial(o - k) * factorial(q - k)))
summe.append(s)
summe = np.sum(summe, 0)
O_neg[o][q] = (np.exp(-S / 2) /
(factorial(o) * factorial(q))**(0.5) *
summe)**2 * w_n[o]
O_neg = np.delete(O_neg, 0, 0)
# Franck-Condon factors
FC_n = [[] * i for i in range(n)]
FC_n = np.sum(O_n, 0)
zero = reduce(mul, FC_n[0])
FC_neg = [[] * i for i in range(n - 2)]
FC_neg = np.sum(O_neg, 0)
FC_n = np.delete(FC_n, 0, 0)
# combination FC factors
FC_nn = [x for x in combinations(chain(*FC_n), 2)]
for i in range(len(FC_nn)):
FC_nn[i] = FC_nn[i][0] * FC_nn[i][1]
FC_nn = np.delete(FC_nn, indices2)
FC = [[] * x for x in range(3)]
FC[0].append(FC_neg[0])
FC[0].append([zero])
FC[0].append(FC_n[0])
FC[0] = [x for x in chain(*FC[0])]
for i in range(1, n - 1):
FC[1].append(FC_neg[i])
FC[1].append(FC_n[i])
FC[1] = [x for x in chain(*FC[1])]
FC[2] = FC_nn
"""Returned are two 3-dimensional lists. First inner list contains
frequencies and FC-factors of vibrations exited with |1| quanta and
the 0-0 transition.
Second list contains frequencies and FC-factors from higher
quanta exitations. Third list are combinations of two normal modes
(including combinations of higher quanta exitations). """
return FC, frequencies
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