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vibronic_sparse.py
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vibronic_sparse.py
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# vibronic sparse diagonalization
# by Pierre-Nicholas Roy, 2022
# models from
# THE JOURNAL OF CHEMICAL PHYSICS 148, 194110 (2018)
import numpy as np
from scipy import sparse
from scipy.sparse.linalg import eigsh
from scipy.sparse.linalg import LinearOperator
from numpy.random import default_rng
import matplotlib.pyplot as plt
import matplotlib.tri as tri
import sys
# functions
def q_matrix(size):
# q coordinate in the harmonic oscillator eigenstate basis
qmat=np.zeros((size,size),float)
for i in range(size):
for ip in range(size):
if ip==(i+1):
qmat[i,ip]=np.sqrt(float(i)+1.)
if ip==(i-1):
qmat[i,ip]=np.sqrt(float(i))
qmat[i,ip]/=np.sqrt(2.)
return qmat
def B(size):
Bmat=np.zeros((size,size),float)
for i in range(size):
for ip in range(size):
if ip==(i+1):
Bmat[i,ip]=1.
if ip==(i-1):
Bmat[i,ip]=1.
Bmat[0,size-1]=1.
Bmat[size-1,0]=1.
return Bmat
# constants
eV_per_K=8.617333262e-5
kB=eV_per_K
#displaced model parameters (all in eV)
displaced = {
'energy': [0.0996, 0.1996],
'gamma': [0., 0.04, 0.08, 0.12, 0.16, 0.20],
'lambda': 0.072,
'w1': 0.02,
'w2': 0.04,
}
jahn_teller = {
'energy': [0.02999, 0.00333, 0.07666, 0.20999, 0.39667, 0.63135],
'lambda': [0.00, 0.04, 0.08, 0.12, 0.16, 0.20],
'w1': .03,
'w2': .03,
}
def main(model, system_index,distributions=False):
def Ea_v(v): # act with diagonal Ea term
N =len(v)
u=np.multiply(Elist_vec,v)
return u
def h01_v(v): # act with h01 term
N =len(v)
u=v.copy()
vtemp=np.zeros((n1,n2*na),float)
utemp=np.zeros((n1,n2*na),float)
for a in range(na):
for i2 in range(n2):
for i1 in range(n1):
vtemp[i1,a*n2+i2]=v[((a*n1+i1)*n2+i2)]
utemp=np.matmul(h01_dvr,vtemp)
for a in range(na):
for i1 in range(n1):
for i2 in range(n2):
u[((a*n1+i1)*n2+i2)]=utemp[i1,a*n2+i2]
return u
def h02_v(v): # act with h02 term
# optimize with blas
N =len(v)
u=v.copy()
vtemp=np.zeros((n2,n1*na),float)
utemp=np.zeros((n2,n1*na),float)
for a in range(na):
for i2 in range(n2):
for i1 in range(n1):
vtemp[i2,a*n1+i1]=v[((a*n1+i1)*n2+i2)]
# use blas through dot?
#utemp=np.dot(h0D2_dvr,vtemp)
utemp=np.matmul(h02_dvr,vtemp)
for a in range(na):
for i1 in range(n1):
for i2 in range(n2):
u[((a*n1+i1)*n2+i2)]=utemp[i2,a*n1+i1]
return u
def h0_fbr_v(v): # act with h01 term
N =len(v)
u=v.copy()
for a in range(na):
for i1 in range(n1):
for i2 in range(n2):
u[((a*n1+i1)*n2+i2)]=(h01_fbr[i1]+h02_fbr[i2])*v[((a*n1+i1)*n2+i2)]
return u
def q1_v(v): #act with displaced q1
N =len(v)
u=np.multiply(lamb_grid1_vec,v)
return u
def q1_fbr_v(v):
N =len(v)
u=v.copy()
vtemp=np.zeros((n1,n2),float)
utemp=np.zeros((n1,n2),float)
a=0 # plus contribution
for i2 in range(n2):
for i1 in range(n1):
vtemp[i1,i2]=v[((a*n1+i1)*n2+i2)]
utemp=np.matmul(qmat1,vtemp)
a=1 # minus contribution
for i2 in range(n2):
for i1 in range(n1):
vtemp[i1,i2]=-v[((a*n1+i1)*n2+i2)]
vtemp=np.matmul(qmat1,vtemp)
for i1 in range(n1):
for i2 in range(n2):
a=0
u[((a*n1+i1)*n2+i2)]=utemp[i1,i2]
a=1
u[((a*n1+i1)*n2+i2)]=vtemp[i1,i2]
return u
def q2_fbr_v(v):
#
# <a|q2|a'> . v[a']
#
# | 0 q2 | |v0| |q2.v1|
# | q2 0 | |v1| = |q2.v0|
#
N =len(v)
u=v.copy()
vtemp=np.zeros((n2,n1),float)
utemp=np.zeros((n2,n1),float)
a=0
for i2 in range(n2):
for i1 in range(n1):
vtemp[i2,i1]=v[((a*n1+i1)*n2+i2)]
utemp=np.matmul(qmat2,vtemp)
a=1
for i2 in range(n2):
for i1 in range(n1):
vtemp[i2,i1]=v[((a*n1+i1)*n2+i2)]
vtemp=np.matmul(qmat2,vtemp)
for i1 in range(n1):
for i2 in range(n2):
a=0
u[((a*n1+i1)*n2+i2)]=vtemp[i2,i1]
a=1
u[((a*n1+i1)*n2+i2)]=utemp[i2,i1]
return u
def q2_v(v): #act with displaced q1
#
# <a|q2|a'> . v[a']
#
# | 0 q2 | |v0| |q2.v1|
# | q2 0 | |v1| = |q2.v0|
#
N =len(v)
u=v.copy()
for i1 in range(n1):
for i2 in range(n2):
a=0
u[(((a+1)*n1+i1)*n2+i2)]=param_times_grid2[i2]*v[((a*n1+i1)*n2+i2)]
a=1
u[(((a-1)*n1+i1)*n2+i2)]=param_times_grid2[i2]*v[((a*n1+i1)*n2+i2)]
return u
# basis sizes (store in dictionary for easy passing to functions)
n1, n2, na =80, 80, 2
nmodes=2
basis = {'n1': n1, 'n2': n2, 'a': na}
# total size of product basis
N = n1*n2*na
basis['N'] = N
# modes only basis size
n12=n1*n2
# allocate memory for the hamiltonian terms
h01=np.zeros((n1,n1),float) # diagonal matrix
h02=np.zeros((n2,n2),float) # diagonal matrix
h01_fbr=np.zeros(n1,float)
h02_fbr=np.zeros(n2,float)
if model=='Displaced':
w1=displaced['w1']
w2=displaced['w2']
if model=='Jahn_Teller':
w1=jahn_teller['w1']
w2=jahn_teller['w2']
# initialize these matrices
for i1 in range(n1):
h01[i1,i1]=w1*(float(i1)+.5)
h01_fbr[i1]=h01[i1,i1]
for i2 in range(n2):
h02[i2,i2]=w2*(float(i2)+.5)
h02_fbr[i2]=h02[i2,i2]
# define dimentionless q matrices for each mode (basis sizes could be different)
qmat1=q_matrix(n1)
qmat2=q_matrix(n2)
# define the dvr grid
grid1, T1 = np.linalg.eigh(qmat1)
grid2, T2 = np.linalg.eigh(qmat2)
param_times_grid2=grid2.copy()
if model=='Displaced':
param_times_grid2=displaced['gamma'][system_index]*grid2
qmat2=displaced['gamma'][system_index]*qmat2
qmat1=displaced['lambda']*qmat1
if model=='Jahn_Teller':
param_times_grid2=jahn_teller['lambda'][system_index]*grid2
qmat2=jahn_teller['lambda'][system_index]*qmat2
qmat1=jahn_teller['lambda'][system_index]*qmat1
# convert h01 and h02 to the DVR
h01_dvr=np.dot(np.transpose(T1),np.dot(h01,T1))
h02_dvr=np.dot(np.transpose(T2),np.dot(h02,T2))
#prepare vectors for fast multiplies
# allocate memory for Ea_list
Elist_vec=np.zeros(N,float)
lamb_grid1_vec=np.zeros(N,float)
for a in range(na):
for i1 in range(n1):
for i2 in range(n2):
if model=='Jahn_Teller':
Elist_vec[((a*n1+i1)*n2+i2)]=jahn_teller['energy'][system_index]
if (a==0):
lamb_grid1_vec[((a*n1+i1)*n2+i2)]=jahn_teller['lambda'][system_index]*grid1[i1]
if (a==1):
lamb_grid1_vec[((a*n1+i1)*n2+i2)]=(-jahn_teller['lambda'][system_index])*grid1[i1]
if model=='Displaced':
Elist_vec[((a*n1+i1)*n2+i2)]=displaced['energy'][a]
if (a==0):
lamb_grid1_vec[((a*n1+i1)*n2+i2)]=displaced['lambda']*grid1[i1]
if (a==1):
lamb_grid1_vec[((a*n1+i1)*n2+i2)]=(-displaced['lambda'])*grid1[i1]
hEa = LinearOperator((N,N), matvec=Ea_v)
h01 = LinearOperator((N,N), matvec=h01_v)
h02 = LinearOperator((N,N), matvec=h02_v)
hq1 = LinearOperator((N,N), matvec=q1_v)
hq2 = LinearOperator((N,N), matvec=q2_v)
h0_fbr = LinearOperator((N,N), matvec=h0_fbr_v)
hq1_fbr = LinearOperator((N,N), matvec=q1_fbr_v)
hq2_fbr = LinearOperator((N,N), matvec=q2_fbr_v)
# DVR Hamiltonian
H_total=hEa+h01+h02+hq1+hq2
H_fbr_total=hEa+h0_fbr+hq1_fbr+hq2_fbr
kmax=100
niter=100
#evals, evecs = eigsh(A_total, k=kmax,which = 'SA', maxiter=niter)
evals, evecs = eigsh(H_total, k=kmax,which = 'SA')
# test norm
'''
for k in range(kmax):
norm=0
for i in range(N):
norm+=evecs[i,k]**2
print(norm)
'''
# test of fbr calculation
'''
evals_fbr,evecs_fbr = eigsh(H_fbr_total, k=kmax,which = 'SA')
for i in range(kmax):
print(i,evals[i]-evals_fbr[i])
'''
# it works
if model=='Displaced':
delta_E=evals[1]-evals[0]
if model=='Jahn_Teller':
delta_E=evals[2]-evals[0] # use next gap because Jahn_Teller is degenerate
Theta=delta_E/eV_per_K
# choose temperatures between 0.1 and 10 time the characteristic Theta=delta_E/eV_per_K
Tmin=.1*Theta
Tmax=3.*Theta
nT=1000 # number of temperature values
deltaT=(Tmax-Tmin)/float(nT)
T=np.zeros(nT,float)
Probs=np.zeros((nT,kmax),float)
Z=np.zeros(nT,float)
E=np.zeros(nT,float)
E2=np.zeros(nT,float)
Cv=np.zeros(nT,float)
A=np.zeros(nT,float)
S=np.zeros(nT,float)
for t in range(nT):
T[t]=Tmin+t*deltaT
# estimators, <E>, Cv, S, and Z
Z[t]=0.
for i in range(kmax):
Ei=(evals[i]-evals[0])/eV_per_K
Z[t]+=np.exp(-Ei/T[t])
E[t]+=np.exp(-Ei/T[t])*Ei
E2[t]+=np.exp(-Ei/T[t])*Ei*Ei
E[t]/=Z[t]
E2[t]/=Z[t]
Cv[t]=(E2[t]-E[t]**2)/(T[t]**2)
A[t]=-T[t]*np.log(Z[t])
S[t]=(E[t]-A[t])/T[t]
for i in range(kmax):
Ei=(evals[i]-evals[0])/eV_per_K
Probs[t,i]=np.exp(-Ei/T[t])/Z[t]
if model=='Displaced':
labels='D, gamma='+str(displaced['gamma'][system_index])
if model=='Jahn_Teller':
labels='JT, E='+str(jahn_teller['energy'][system_index])+' lambda='+str(jahn_teller['lambda'][system_index])
logfile=open(str(model)+str(system_index)+'.log','w')
logfile.write('Model: '+str(model)+'; System_index: '+str(system_index)+ '\n')
logfile.write('Basis: n1 n2 na = '+str(n1)+' '+str(n2)+' '+str(na)+'\n')
logfile.write(' Theta(n1,n2,na) = '+str(Theta)+' K'+'\n')
logfile.write('Prob(n,Theta)= '+str((np.exp(-(evals[0:kmax]-evals[0])/(kB*5.*Theta))))+'\n')
fig_EV, ax_EV = plt.subplots()
fig_E, ax_E = plt.subplots()
fig_Cv, ax_Cv = plt.subplots()
fig_S, ax_S = plt.subplots()
fig_A, ax_A = plt.subplots()
ax_EV.plot([i for i in range(kmax)],(evals-evals[0])/eV_per_K,'o',label=labels)
ax_E.plot(T,E,label=labels)
ax_Cv.plot(T,Cv,label=labels)
ax_Cv.plot([Theta,Theta],[0,np.max(Cv)],'--')
ax_A.plot(T,A,label=labels)
ax_S.plot(T,S,label=labels)
ax_EV.set(title='E(n) vs n '+str(model),xlabel='n',ylabel='E(n)/kB (K)')
ax_E.set(title='<E> vs T '+str(model),xlabel='T (K)',ylabel='<E>/kB (K)')
ax_Cv.set(title='Cv vs T '+str(model),xlabel='T (K)',ylabel='Cv/kB ')
ax_A.set(title='A vs T '+str(model),xlabel='T (K)',ylabel='A/kB (K)')
ax_S.set(title='S vs T '+str(model),xlabel='T (K)',ylabel='S/kB')
fig_EV.savefig('Evsn_'+str(model)+str(system_index)+'.png')
fig_E.savefig('EvsT_'+str(model)+str(system_index)+'.png')
fig_Cv.savefig('CvvsT_'+str(model)+str(system_index)+'.png')
fig_A.savefig('AvsT_'+str(model)+str(system_index)+'.png')
fig_S.savefig('SvsT_'+str(model)+str(system_index)+'.png')
if distributions==True:
# calculate distributions for each temperature
Tlist=[0.1, 1., 2., 5., 10.,300./Theta,400./Theta]
for tindex in Tlist:
rho1=np.zeros((n1),float)
rho2=np.zeros((n2),float)
rho12=np.zeros((n1,n2),float)
rho1a=np.zeros((n1,na),float)
rho2a=np.zeros((n1,na),float)
w12=np.zeros((n1,n2),float)
rhoa=np.zeros((na,na),float)
#output files
rho1_out=open('h1_T'+str(tindex)+str(model)+str(system_index)+'.dat','w')
rho2_out=open('h2_T'+str(tindex)+str(model)+str(system_index)+'.dat','w')
rho12_out=open('h12_T'+str(tindex)+str(model)+str(system_index)+'.dat','w')
rho1a_out=open('h1a_T'+str(tindex)+str(model)+str(system_index)+'.dat','w')
rho2a_out=open('h2a_T'+str(tindex)+str(model)+str(system_index)+'.dat','w')
w12_out=open('w12_T'+str(tindex)+str(model)+str(system_index)+'.dat','w')
rhoa_out=open('a_T'+str(tindex)+str(model)+str(system_index)+'.dat','w')
t=tindex*Theta
Z=0.
for i in range(kmax):
Ei=(evals[i]-evals[0])/kB
Z+=np.exp(-Ei/t)
for a in range(na):
for ap in range(na):
for i1 in range(n1):
for i2 in range(n2):
rhoa[a,ap]+=evecs[(a*n1+i1)*n2+i2,i]*evecs[(ap*n1+i1)*n2+i2,i]*np.exp(-Ei/t)
for i1 in range(n1):
for i2 in range(n2):
for a in range(na):
sum=(evecs[(a*n1+i1)*n2+i2,i]**2)*np.exp(-Ei/t)
rho1[i1]+=sum
rho2[i2]+=sum
rho12[i1,i2]+=sum
rho1a[i1,a]+=sum
rho2a[i2,a]+=sum
rho1=(1./Z)*rho1
rho2=(1./Z)*rho2
rho12=(1./Z)*rho12
rho1a=(1./Z)*rho1a
rho2a=(1./Z)*rho2a
rhoa=(1./Z)*rhoa
w12=-t*eV_per_K*np.log(rho12)
#grid1, T1
#convert to grid
for i1 in range(n1):
h1=rho1[i1]
# multiply by gauss hermite weight
h1*=np.exp(-grid1[i1]**2)/(np.sqrt(np.pi)*T1[0,i1]**2)
rho1_out.write(str(grid1[i1])+' '+str(h1)+' '+str(-t*eV_per_K*np.log(h1))+'\n')
for i2 in range(n2):
h2=rho2[i2]
# multiply by gauss hermite weight
h2*=np.exp(-grid2[i2]**2)/(np.sqrt(np.pi)*T2[0,i2]**2)
rho2_out.write(str(grid2[i2])+' '+str(h2)+' '+str(-t*eV_per_K*np.log(h2))+'\n')
for i1 in range(n1):
for i2 in range(n2):
# multiply by gauss hermite weight
rho12[i1,i2]*=((np.exp(-grid2[i2]**2)/(np.sqrt(np.pi)*T2[0,i2]**2))*(np.exp(-grid1[i1]**2)/(np.sqrt(np.pi)*T1[0,i1]**2)))
rho12_out.write(str(grid1[i1])+' '+str(grid2[i2])+' '+str(rho12[i1,i2])+'\n')
w12_out.write(str(grid1[i1])+' '+str(grid2[i2])+' '+str(w12[i1,i2])+'\n')
for a in range(na):
for ap in range(na):
rhoa_out.write(str(a)+' '+str(ap)+' '+str(rhoa[a,ap])+'\n')
for i1 in range(n1):
for a in range(na):
rho1a[i1,a]*=np.exp(-grid1[i1]**2)/(np.sqrt(np.pi)*T1[0,i1]**2)
rho1a_out.write(str(grid1[i1])+' '+str(rho1a[i1,0])+' '+str(rho1a[i1,1])+'\n')
for i2 in range(n2):
for a in range(na):
rho2a[i2,a]*=np.exp(-grid2[i2]**2)/(np.sqrt(np.pi)*T2[0,i2]**2)
rho2a_out.write(str(grid2[i2])+' '+str(rho2a[i2,0])+' '+str(rho2a[i2,1])+'\n')
rho1_out.close()
rho2_out.close()
rho12_out.close()
rho1a_out.close()
rho2a_out.close()
w12_out.close()
rhoa_out.close()
logfile.close()
if (__name__ == "__main__"):
# choose the model
model = ['Displaced', 'Jahn_Teller'][0]
system_index = 5 # 0..5 for Displaced and Jahn-Teller
distributions=True
# run
system_index=int(sys.argv[1])
main(model,system_index,distributions)