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helper_functions.py
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# -*- coding: utf-8 -*-
"""
Created on Sat Aug 4 14:42:56 2018
@author: Roland Scheidel
"""
import numpy as np
import scipy
import matplotlib.pyplot as plt
import os
def franck_condon_analytic(mol_1, v1, mol_2, v2):
fc = 0
d = mol_2.Re - mol_1.Re
a1 = mol_1.ome * mol_1.M
a2 = mol_2.ome * mol_2.M
A = 2 * np.sqrt(a1 * a2) / (a1 + a2)
S = a1 * a2 * d ** 2 / (a1 + a2)
b1 = a2 * np.sqrt(a1) * d / (a1 + a2)
b2 = -a1 * np.sqrt(a2) * d / (a1 + a2)
for i in range(0, v1 + 1):
for j in range(0, v2 + 1):
if 0 == (i + j) % 2:
# print("i,j",i,j)
K = (i + j) // 2
I = scipy.special.factorial2(2 * K - 1, True) / ((a1 + a2) ** K)
B = scipy.special.binom(v1, i) * scipy.special.binom(v2, j)
H1 = scipy.special.eval_hermite(v1 - i, b1)
H2 = scipy.special.eval_hermite(v2 - j, b2)
C = ((2 * np.sqrt(a1)) ** i) * ((2 * np.sqrt(a2)) ** j)
fc = fc + B * H1 * H2 * C * I
norm = A * np.exp(-S) / (2 ** (v1 + v2) * scipy.special.factorial(v1, True) * scipy.special.factorial(v2, True))
fc = fc ** 2 * norm
return fc
def simpson(xi, fi):
'''
xi: spacial grid
fi: function values for on
'''
dx = xi[1] - xi[0]
N = fi.size # number of Points
boundaries = fi[0] + fi[-1]
sum_odd = np.sum(fi[1:N - 2:2], 0)
sum_even = np.sum(fi[2:N - 2:2], 0)
integral = dx * (1.0 / 3.0) * (boundaries + 2 * sum_even + 4 * sum_odd)
return integral
def solve_schroedinger_full_matrix(xi, Vi, M=1, normalize=True):
'''
:param xi: spatial grid vector
:param Vi: potential grid vector
:param M: reduced_mass
'''
dx = xi[1] - xi[0]
N = xi.size
diagonal = Vi + 1.0 / (M * dx * dx)
off_diag = np.ones(N - 1) * (-1.0 / (2 * M * dx * dx))
H = np.diag(diagonal) + np.diag(off_diag, -1) + np.diag(off_diag, 1)
# build the tridiagonal matrix as a sum of three matrices
ev, U = scipy.linalg.eigh(H)
if normalize:
C = (1.0 / simpson(xi, U ** 2)) ** 0.5
U = U * C
return ev, U
def solve_schroedinger_tridiagonal(xi, Vi, M=1, normalize=True):
'''
:param xi: spatial grid vector
:param Vi: potential grid vector
:param M: reduced_mass
:param normalize: normalize the eigenvectors?
'''
dx = xi[1] - xi[0]
N = xi.size
diagonal = Vi + 1.0 / (M * dx * dx)
off_diag = np.ones(N - 1) * (-1.0 / (2 * M * dx * dx))
ev, U = scipy.linalg.eigh_tridiagonal(diagonal, off_diag)
if normalize:
C = (1.0 / simpson(xi, U ** 2)) ** 0.5 # calculate the normalization constant for all eigenvactors
U = U * C
return ev, U
def sdg_nonequidistant(xi, Vi, M=1):
N = xi.size
H = np.zeros([N, N])
dx1 = xi[1] - xi[0]
dx2 = dx1
# xi[2]-xi[1]
p = dx2 / dx1
q = dx1 * dx2 * (1 + p)
B = 2 * (1 + p) / (2 * M * q) + Vi[0]
C = -1 / (2 * M * q)
H[0, 0] = B
H[0, 1] = C
for i in range(1, N - 1):
dx1 = xi[i] - xi[i - 1]
dx2 = xi[i + 1] - xi[i]
p = dx2 / dx1
q = dx1 * dx2 * (1 + p)
A = -2 * p / (2 * M * q)
B = 2 * (1 + p) / (2 * M * q) + Vi[i]
C = -2 / (2 * M * q)
H[i, i - 1] = A
H[i, i] = B
H[i, i + 1] = C
dx1 = xi[N - 1] - xi[N - 2]
dx2 = dx1
# xi[2]-xi[1]
p = dx2 / dx1
q = dx1 * dx2 * (1 + p)
A = -2 * p / (2 * M * q)
B = 2 * (1 + p) / (2 * M * q) + Vi[N - 1]
H[N - 1, N - 1] = B
H[N - 1, N - 2] = A
ev, U = np.linalg.eigh(H)
return ev, U
def find_limit(Vi, maxval):
# helper function to find the boundaries of a gridvector around the minimum
# so that all Values are within a certain range
start = np.argmin(Vi)
i = start
while (maxval - Vi[i]) / maxval > 0.02 and Vi[i] < maxval and i < Vi.size - 1 and i > 0:
i = i + 1
j = start
while (maxval - Vi[j]) / maxval > 0.02 and Vi[i] < maxval and i < Vi.size and i > 0:
j = j - 1
return j, i
def plot_trans(xi, xi_1, Vi_1, xi_2, Vi_2, psi1, psi2, ev1, ev2, FC, state_1, state_2):
'''
:param xi:
:param xi_1:
:param Vi_1:
:param xi_2:
:param Vi_2:
:param psi1:
:param psi2:
:param ev1:
:param ev2:
:param FC:
:return:
'''
minV1 = min(Vi_1)
minV1_index = np.where(Vi_1 == minV1)
n = FC.size
max_relevant_value = min(ev1[n] * 1.75, Vi_1.max() * 0.9)
s1, e1 = find_limit(Vi_1, max_relevant_value) # only draw relevant parts of the potentials
max_relevant_value = min(ev2[n] * 1.75, Vi_2.max() * 0.95)
s2, e2 = find_limit(Vi_2, max_relevant_value)
s = np.searchsorted(xi, xi_1[s1])
e = np.searchsorted(xi, xi_2[e2], 'right')
scale = 0.4 * (ev1[1] - ev1[0]) / max(psi1)
ax1 = plt.subplot(212)
# plot first state
plt.plot(xi_1[s1:e1], Vi_1[s1:e1], 'k-')
plt.fill(xi[s:e], ev1[0] + scale * psi1[s:e], c='r', alpha=1.0)
plt.axvline(x=xi[minV1_index])
plt.xlabel('Kernabstand ($a_0$)')
plt.ylabel("Energy ($E_h$)", horizontalalignment='left')
ax1.annotate(state_1.full_name, (xi_1[e1], 0.9 * Vi_1[e1]))
# plot second state
ax2 = plt.subplot(211, sharex=ax1)
plt.setp(ax2.get_xticklabels(), visible=False)
ax2.annotate(state_2.full_name, (xi_2[e2] - 0.15, state_2.min_electronic_energy + 0.84 * Vi_2[e2]))
Vi_2 = Vi_2 + state_2.min_electronic_energy
ev2 = ev2 + state_2.min_electronic_energy
plt.plot(xi_2[s2:e2], Vi_2[s2:e2], 'k-')
plt.subplots_adjust(hspace=0.01)
plt.axvline(x=xi[minV1_index])
for i in range(0, n):
alpha = 0.1 + 0.9 * FC[i] / max(FC)
print('alpha', alpha)
plt.fill(xi[s:e], ev2[i] + scale * psi2[s:e, i], c='r', alpha=min(0.1 + 0.9 * FC[i] / max(FC), 1))
print(FC[i])
filename = os.path.join('Output',
'wavefunctions_' + state_1.name + '_' + state_1.electronic_state + '_' + state_2.electronic_state + '.svg')
plt.savefig(filename, bbox_inches='tight', format='svg')
return
def compare_state_potentials(state_list):
for state in state_list:
xi = state.RKR_linspace()
Vi = state.RKR_potential(xi) + state.min_electronic_energy
plt.plot(xi, Vi)
raw_xi, raw_vi = state.RKR_raw()
plt.plot(raw_xi, raw_vi + state.min_electronic_energy, 'k.', markersize=2.5)
plt.annotate(state.full_name, [xi[-1] - 0.5, Vi[-1] - 0.0075])
plt.axhline(y=state.Ed + state.min_electronic_energy, c='black', linestyle=':', linewidth='1')
plt.xlabel('Kernabstand ($a_0$)')
plt.ylabel("Energy ($E_h$)", horizontalalignment='left')
filename = os.path.join('Output', 'compare_potentials_of_states.svg')
plt.savefig(filename, bbox_inches='tight', format='svg')
return
def compare_model_functions(xi, molecule):
ax = plt.figure()
plt.plot(xi, molecule.harmonic(xi), 'r-')
plt.plot(xi, molecule.morse(xi), 'b-')
plt.plot(xi, molecule.RKR_potential(xi), 'g-')
ax.legend(['Harmonisch', 'Morse', 'RKR'], loc='upper left', bbox_to_anchor=(0.125, 0.87))
plt.xlabel('Kernabstand ($a_0$)')
plt.ylabel("Energy ($E_h$)", horizontalalignment='left')
filename = os.path.join('Output', 'compare_model_functions_' + molecule.electronic_state + '_state.svg')
plt.savefig(filename, format='svg')
return
def double_linspace(range_a, range_b, n):
'''
helper function to create a linpspace for two overlapping intervals
:param range_a: min and max of the first grid vector
:param range_b: min and max of the second grid vector
:param n: total number of grid points
:return: grid vector and indizes for range_a, range_b,
'''
x_min = min(range_a[0], range_b[0])
x_max = max(range_a[-1], range_b[-1])
xi = np.linspace(x_min, x_max, n)
a_min = np.searchsorted(xi, range_a[0], side='right')
b_min = np.searchsorted(xi, range_b[0], side='right')
a_max = np.searchsorted(xi, range_a[-1])
b_max = np.searchsorted(xi, range_b[-1])
a_indices = [a_min, a_max]
b_indices = [b_min, b_max]
return xi, a_indices, b_indices