ayan chattopadhyay 3rd integrated msc chemistry...
TRANSCRIPT
Ayan Chattopadhyay Mainak Mustafi
3rd yr Undergraduates Integrated MSc Chemistry
IIT Kharagpur
Under the supervision of: Dr. Marcel Nooijen Associate Professor
Department of Chemistry University of Waterloo
1
It is widely used to solve the electronic Schrodinger equation to obtain the potential energy surface, for different molecules, thus to study their : • Structures • Energetics • Statistical Mechanical Properties
INTRODUCTION INTRODUCTION
2
The Born-Oppenheimer approximation is a very important method in theoretical chemistry. Its is based on slower movement of nuclei than electrons.
It is used to study the electronic excitation spectra of different molecules, which gives information about their structural characteristics, rates of any reactions, and other physical properties
Courtesy : google images
Limitations of Born Oppenheimer Approximation
• Doesn’t incorporate the vibrational interactions of different excited electronic states.
• Conical Interaction
Courtesy : Wikipedia image
• Avoided Crossing
3
• BO Approximation breaks down for molecules with Jahn-Teller distortion
Vibronic Model
• Non-adiabatic dynamic study, where electronic Hamiltonian is solved in the diabatic basis
• Presence of a coupling term which takes into account of the different vibronic interactions
H E h
HO q
q E hHO
Vibronic Hamiltonian
Coupling Parameters: Δ,μ
E 1
2 q
2 q
q E 1
2 q
2
The Vibronic Model :
The Potential Energy Matrix :
1 Normal Mode and 2 Electronic States
4
μ = 0.3, Δ = 0.1 eV
Courtesy : Prateek Goel
Nuclear coordinate
Ener
gy
Inte
nsi
ty
Inte
nsi
ty
eBE
eBE
Full Born-Oppenheimer Franck-Condon
5
Questions at hand
• Obtain the Vibronic models for large systems, i.e., for molecules with many electronic states and many normal modes
• Also want to do Statistical mechanics study on these systems.
• Developing an efficient technique to solve for the Statistical Mechanics for these systems .
6
• To compare the Born-Oppenheimer and the Vibronic Models
• These models can be used to simulate Spectra
Models to be solved
Model1x1: 1 normal mode and 1 electronic state: • V(q) = ½*ω(q-a)2 ; The displaced Harmonic Oscillator • V(q) = D(1-e-αq)2 ; Morse Potential
7
Model1x2: 1 normal mode and 2 electronic states:
2
1
2
2
1
2
1
2
E q q q
q E q q
V(q) =
8
Model2x2: 2 normal modes and 2 electronic states:
2 2
1 1 2 2 1 1 2
2 2
2 1 1 2 2 2 2
1 1
2 2
1 1
2 2
E q q q q
q E q q q
V(q1 ,q2) =
How to solve these questions???
9
DVR Approach
• DVR or Discrete Variable Representation is a widely used method to discretize the Schrodinger equation, with less complicated calculations
• This method of defining the potential is easy in the eigenvector basis of the position operator
• The basis function is represented by points, and the potential function is a diagonal matrix, where the diagonal elements, are simply the potential at those points
10
• The Kinetic energy has to be evaluated in H.O. basis and transformed to the DVR basis, and is strictly non-diagonal.
• This is a convenient way to evaluate the eigenstates for any potential
In this method, considering a function f(Ȃ) : f(Ȃ) can be transformed to the original basis as:
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' '' 2
' '' 2
1( ) (0 ) (0 ) (0 ) ......
2
( ) [ (0 ) (0 ) (0 ) .....] ( )
i i i
i i i i i i
A a a a
f A f f A f A
f A a f f a f a a f a a
,
| | ( ) | |
|
T h u s , | ( ) | ( )
i i j i
i j
i
n a a f A a a m
m a U
n f A m U f a U
The Method
Eigen values are generated in HO basis
The Potential Matrix for each eigenvalue is
calculated in DVR basis
Kinetic Energy Matrix is calculated in HO basis
Kinetic Energy Matrix is tranformed into DVR
basis
The Potential and Kinetic energy matrices are added and
diagonalized to obtain the total energy matrix
The eigenvalues and the eigenvectors are used to calculate
the statistical mechanical properties for the model
For Non-adiabatic case: in diabatic basis
For Adiabatic (B.O. Approximation) case: in adiabatic basis
Here the Potential energy Matrix is diagonalized before calculating the total Hamiltonian 12
Statistical Mechanics
• Partition function (Q) = Σexp(-εi*β ) ; where β = 1/(kT) and k is the Boltzmann constant • Population probability for each state ‘i’ Pi = exp(-εi*β )/Q • Helmholtz free energy (A) = -kT*lnQ • Internal energy (U) = Σpiεi
• Entropy (S) = -k*ΣPi*lnPi = (U-A)/T • Specific Heat Capacity (Cv) = 1/kT2*Σ(εi – U)2*Pi
• Expected value of Potential (Ṽ) = Σ<εi|V|εi>*Pi ; where |εi> are the energy
eigenvectors
13
Δ Internal Energy (U)
(in eV)
Expected value of Potential
<V>(in eV)
0.0 5.93x10-2 -2.95x10-2
0.2 6.57x10-2 -2.3x10-2
0.3 6.21x10-2 -2.66x10-2
0.5 5.54x10-2 -3.33x10-2
1.0 5.37x10-2 -3.49x10-2
Δ Internal Energy (U)
(in eV)
Expected value of
Potential <V>(in eV)
0.0 5.94x10-2 -2.93x10-2
0.2 6.58x10-2 -2.27x10-2
0.3 6.23x10-2 -2.64x10-2
0.5 5.56x10-2 -3.29x10-2
1.0 5.39x10-2 -3.48x10-2
μ = 0.1 , E0 = 0.1 eV, Temperature = 2000 K
Non-adiabatic Adiabatic
For 1 Normal mode and 2 Electronic States
14
Comparison of Adiabatic and Non-adiabatic results based on DVR
2
1
2
2
1
2
1
2
E q q q
q E q q
V(q) =
The Internal Energy distribution for different values of delta.
The Difference in Internal Energies for adiabatic and non-adiabatic
15
Inte
rnal
En
ergy
(in
eV
)
Dif
fere
nce
in In
tern
al E
ner
gy (
in e
V)
The Expected Potential Energy distribution for different values of delta.
The Difference in Expected values of Potential for adiabatic and non-adiabatic
16
Exp
ecte
d v
alu
e fo
r P
ote
nti
al (
in e
V)
Dif
fere
nce
in E
xpec
ted
val
ue
for
Po
ten
tial
(in
eV
)
Difference is not that large
Δ Internal Energy (U)
(in eV)
Expected value of Potential
<V>(in eV)
0.0 -0.125 -0.214
0.2 -9.81x10-2 -0.187
0.3 -8.89x10-2 -0.178
0.5 -7.32x10-2 -0.162
1.0 -3.95x10-2 -0.128
Δ Internal Energy (U)
(in eV)
Expected value of
Potential <V>(in eV)
0.0 -0.124 -0.213
0.2 -9.71x10-2 -0.186
0.3 -8.79x10-2 -0.177
0.5 -7.23x10-2 -0.161
1.0 -3.9x10-2 -0.128
μ = 0.2 , E0 = 0.1 eV, Temperature = 2000 K
Non-adiabatic Adiabatic
17
The Internal Energy distribution for different values of delta.
The Difference in Internal Energies for adiabatic and non-adiabatic
18
Inte
rnal
En
ergy
(in
eV
)
Dif
fere
nce
in In
tern
al E
ner
gy (
in e
V)
The Expected Potential Energy distribution for different values of delta.
The Difference in Expected values of Potential for adiabatic and non-adiabatic
19
Exp
ecte
d v
alu
e fo
r P
ote
nti
al (
in e
V)
Dif
fere
nce
in E
xpec
ted
val
ue
for
Po
ten
tial
(in
eV
)
Difference is greater with increasing μ, more prominent in Δ =0.2-0.5 region
Δ Internal Energy (U)
(in eV)
Expected value of Potential
<V>(in eV)
0.0 1.95x10-2 3.42x10-2
0.2 1.65x10-2 3.17x10-2
0.3 6.87x10-3 2.23x10-2
0.5 -9.48x10-3 6.39x10-3
1.0 -2.36x10-2 -7.16x10-3
μ = 0.2, E0 = 0.1 eV, Temperature = 2000 K
Non-adiabatic Adiabatic
Δ Internal Energy (U)
(in eV)
Expected value of Potential
<V>(in eV)
0.0 1.88x10-2 3.4x10-2
0.2 1.64x10-2 3.18x10-2
0.3 6.79x10-3 2.25x10-2
0.5 -9.5x10-3 6.57x10-3
1.0 -2.36x10-2 -7.07x10-3
For 2 Normal mode and 2 Electronic States
20 Difference is much more observed than the 1x2 model
2 2
1 1 2 2 1 1 2
2 2
2 1 1 2 2 2 2
1 1
2 2
1 1
2 2
E q q q q
q E q q q
V(q1 ,q2) =
Δ Internal Energy (U)
(in eV)
Expected value of Potential
<V>(in eV)
0.0 -7.19x10-3 -5.9x10-3
0.2 3.15x10-3 7.9x10-3
0.3 -1.14x10-3 5.47x10-3
0.5 -1.1x10-2 -1.31x10-3
1.0 -2.12x10-2 -7.46x10-3
μ = 0.3, E0 = 0.1 eV, Temperature = 2000 K
Non-adiabatic Adiabatic
Δ Internal Energy (U)
(in eV)
Expected value of Potential
<V>(in eV)
0.0 -8.74x10-3 -6.63x10-3
0.2 2.05x10-3 7.51x10-3
0.3 -2.09x10-3 5.17x10-3
0.5 -1.17x10-2 -1.44x10-3
1.0 -2.16x10-2 -7.44x10-3
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Difference is greater with increasing μ, more prominent in Δ =0.2-0.5 region
Even though the DVR method is very useful in solving the electronic Schrodinger equation, but yet it has some limitations:
• It is possible for only small systems
• Even for a little increase in the size of the system, the computational time increases exponentially
• For molecules with large number of electronic states and normal modes, like 10 electronic states and 30 normal modes, there will be order of 10*2030 states, which is impossible to compute
Limitations
22
How to overcome this problem, and find a more efficient method for such
computations???
Path Integral Monte Carlo Approach
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• Provides a numerically exact solution to the time-dependent Schrodinger equation (hence can be made arbitrarily accurate)
• For Boltzmann systems in imaginary time importance sampling methods are ideally suited, which corresponds to Boltzmann averaged equilibrium statistical mechanical properties.
• Requires computational effort that grows comparatively slowly with the dimensionality of the system
PATH INTEGRAL METHOD
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DVR
PI
Computationally impossible (Hilbert space dimension of
2030 )
No. of beads*normal modes*electronic
states*No. of samples
• To make statistical mechanical study of LARGE systems involving multiple electronic states based on Path Integral studies (which are not possible to study with exact models like DVR approach)
• Typically models with about 1. 30 normal modes 2. 10 electronic states
The Advantage
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eiHt /
y (x,t) = xy (t) = dx ' x e-iHt / x'
ò x ' y 0 = dx 'K(x,x',t)y 0(x')ò
Ref. : Nancy Makri, Computer Physics Communications 63 (1991) 389-414
THE PROPAGATOR IN PI REPRESENTATION
• The quantum time evolution function
• Important in deriving semi classical approximations to quantum dynamical phenomena.
• The time evolution of a system can be expressed simply as:
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e-iHt / º e-bH
Time evolution operator The quantum density
operator useful in semi classical calculations of
quantum dynamical phenomena and statistical
mechanical studies
TIME EVOLUTION AND THE QUANTUM DENSITY OPERATOR
PI formalism of the time evolution operator can be generalized to the quantum density operator (also known as the Boltzmann operator)
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e-iHt / = e-iHDt / e-iHDt / ...e-iHDt / = e-iHDt /
k=1
N
Õ
Contd.
ebH = ebPHebPH ...ebPH = ebPH
i=1
P
Õ
Time Slices
• Time slicing is done as follows, for representing the long term propagator in terms of the short term propagators:
• Thus with the equivalence between the time evolution operator and the quantum density operator we can express it as:
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e-bPH @ e-bPKe-bPV +O(bP2 )
Where bP = b /P
TIME SLICES AND THE TROTTER APPROXIMATION… (CONTD.)
Thus in the Trotter approximation the error gets reduced quadratically with increasing P.
29
RESULTS FOR THE ADIABATIC CALCULATIONS (B.O. APPROXIMATION)
30
Simple case : Born Oppenheimer
• The Path Integral calculations for the Born Oppenheimer case, i.e. with the adiabatic approximation is first calculated.
• It is much simpler as it involves only the diagonalization of the potential matrix and the computation of the kinetic and potential energy functions in the PI discretization.
31
THE ADIABATIC CALCULATIONS (B.O. APPROXIMATION)
32
Temp (K) V (in eV) U (in eV)
300 -0.138 -0.113
600 -0.126 -0.093
1000 -0.101 -0.054
1500 -0.073 -0.006
2000 -0.050 0.038
Temp (K) V (in eV) U (in eV)
300 -0.139 -0.113
600 -0.126 -0.093
1000 -0.101 -0.054
1500 -0.073 -0.006
2000 -0.050 0.037
The DVR and the PIMC results match with each other
The adiabatic potential The state probabilities for energy distribution
Temp (K) V (in eV) U (in eV)
300 -0.324 -0.297
600 -0.314 -0.281
1000 -0.289 -0.242
1500 -0.254 -0.186
2000 -0.224 -0.136
Temp (K) V (in eV) U (in eV)
300 -0.324 -0.298
600 -0.314 -0.281
1000 -0.289 -0.242
1500 -0.254 -0.191
2000 -0.228 0.140
The DVR and the PIMC results match with each other 33
The adiabatic potential The state probabilities for energy distribution
Temp (K) V (in eV) U (in eV)
300 -0.617 -0.590
600 -0.609 -0.575
1000 -0.588 -0.540
1500 -0.552 -0.484
2000 -0.514 -0.426
Temp (K) V (in eV) U (in eV)
300 -0.617 -0.590
600 -0.610 -0.575
1000 -0.588 -0.539
1500 -0.551 -0.481
2000 -0.512 0.422
The DVR and the PIMC results match with each other 34
The adiabatic potential The state probabilities for energy distribution
1. THE GENERAL N LEVEL HAMILTONIAN A general N-level Hamiltonian is described by:
H = h0(R,P)+ Vn,m
n,m=1
N
å (R) y n y m
Nuclear kinetic energy + State independent part of
the potential energy
T (P)+V0(R)
The non-adiabatic potential energy matrix
elements
THE NONADIABATIC PATH INTEGRAL
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1. The General N-level Hamiltonian
Ref: N. Ananth and T.F. Miller III , J. Chem. Phy., 133, 234103(2010)
y n y m ® an+am
y n ® 0102.....1n.....0N
Bosonic creation and annihilation operators
The singly excited oscillator(SEO) states which acts as the basis for our calculations.
Equivalent to a system of N oscillators with a single quantum of excitation.
H = h0(R,P)+ an
+Vn,m
n,m=1
N
å (R)am
• The N level system is represented by N uncoupled Harmonic oscillators
• The mapping relations :
• The mapping transforms the Hamiltonian into:
• Transforming the boson operators into the Cartesian representation we obtain the Hamiltonian in the Cartesian form:
H = h0(R,P)+
1
2(xnxm + pnpm -d nm )
n,m=1
N
å Vnm (R)
36
2. The Stock Thoss mapping
Ref: G. Stock and M. Thoss, Phy. Rev. Lett.,78, 578 (1997)
xn =1
2(an + an
+ )
pn =1
2(an
+ - an )
H = h0(R,P)+
1
2(xnxm + pnpm -d nm )
n,m=1
N
å Vnm (R)
• Transforming the boson operators into the Cartesian representation we obtain the Hamiltonian in the Cartesian form:
3. The Stock Thoss mapping contd…
37
Ref: G. Stock and M. Thoss, Phy. Rev. Lett.,78, 578 (1997)
,n m nm
a a
Z = Tr e-bHéë ùû
I = dR R,nn=1
N
å R,nò
Z = d{Ra }ò Rana
a=1
P
Õ{na }=1
N
å e-bPH Ra+1na+1
d{Ra }ò º a=1
P dRaòÕ( ) º{na }=1
N
åna =1
N
åa=1
P
Õæ
èç
ö
ø÷
• The canonical partition function is defined from the trace of the Boltzmann operator.
• The resolution of identity operator for this space looks like:
• Repeated insertion of the completeness relation yields the PI discretization of the partition function:
where and
38
4. The Path Integral Formulation of Hamiltonian
Ref: N. Ananth and T.F. Miller III , J. Chem. Phy., 133, 234103(2010)
Applying the Trotter approximation we can get the partition function in the following form:
Z = limP®¥
d{Ra }òMP
2b
æ
èçö
ø÷a=1
P
Õf /2
e-bPV0 (Ra )
´exp -MP
2bRa - Ra+1( )
TRa - Ra+1( )
é
ëê
ù
ûú
´ na e-bPn (Ra )
a=1
P
Õ na+1
{na =1}
N
å
4. The Path Integral Formulation of Hamiltonian contd…
39
Ref: N. Ananth and T.F. Miller III , J. Chem. Phy., 133, 234103(2010)
4. THE PATH INTEGRAL FORMULATION OF THE HAMILTONIAN (contd.)
d{xa }ò xa
a=1
P
Õ e-bPn (Ra )
P xa+1
nn=1
N
å n = dxiò xi xiéë
ùû
i=1
N
Õ P
• Using the projection operator the SEO basis can be transformed into the Cartesian coordinate basis whereby the last term transforms into :
• The projection operator being:
40
4. The Path Integral Formulation of Hamiltonian contd…
Ref: N. Ananth and T.F. Miller III , J. Chem. Phy., 133, 234103(2010)
4. THE PATH INTEGRAL FORMULATION OF THE HAMILTONIAN (contd.) x e-bPn (R)P x ' = x n Mnm (R)n,m=1
N
å m x '
Mnm(R) = n e-bPn (R) m
• Thus effectively the electronic matrix elements reduce to the form:
where
• This representation is helpful as in our mapping we have used the SEO basis that consists of N-1 ground h.o. wave functions and 1 first-excited state H.O. wave function.
41
4. The Path Integral Formulation of Hamiltonian contd…
Ref: N. Ananth and T.F. Miller III , J. Chem. Phy., 133, 234103(2010)
Z = limP®¥
2MP
bp N+1
æ
èçö
ø÷
fP/2
d{Ra }ò d{xa }ò AaFaGa
a=1
P
Õ
Aa = e-MP
2b(Ra -Ra+1 )T (Ra -Ra+1 )
e-bPV0 (Ra )
Fa = xa
TM (Ra )xa+1
T
Ga = e- xa
T xa
• Finally putting all the terms together we obtain the final PI representation of the partition function as:
Where :
42
5. The Final Path Integral
Ref: N. Ananth and T.F. Miller III , J. Chem. Phy., 133, 234103(2010)
Importance sampling
W ({xa },{Ra }) = Aa
a=1
P
Õ Ga Fa
PI-MC Calculations
• The simulation has been performed using standard path integral Monte Carlo techniques.
• In our simulation the weight function for important sampling is given by the function:
43
Importance Sampling
1. The nuclear probability distribution
Calculations and Results for Non-adiabatic Systems
44
The nuclear distribution is calculated as: P(R) =
d (R- RP )sgn(F)W
sgn(F)W
Nuclear probability distribution for Path Integral Nuclear probability distribution for DVR
The Nuclear Probability Distribution
Ambiguity in the nuclear probability distribution function
45
Nuclear probability distribution for Path Integral Nuclear probability distribution for DVR
( ) 1
D V R
D V R
i
i
q ( ) 1q d q
Is it true that ? ( ) ( )D V R
i iq q
2. Calculating the energies: potential and total
E = -1
Z
¶Z
¶b
E =
P
2b+ F -
¶A
¶b
æ
èçö
ø÷sgn(F)
W
sgn(F)W
F =
xa
T -¶M (Ra )
¶bxa
xa
TM (Ra )xaa=1
P
å
• The average total energy operator is given by this well known statistical mechanical formula:
• This gives us the form to calculate the total energy from the partition function as:
where
46
Calculating the energies: Potential & Total
Ref: N. Ananth and T.F. Miller III , J. Chem. Phy., 133, 234103(2010)
Results for Non-adiabatic PI Calculations
Temp(in K) Total Potential energy (in eV)
DVR values (in eV)
600 -0.71 -0.092
1000 -0.68 -0.049
1500 -0.64 0.006
2000 -0.61 0.059
Model: V (q) =
E + l1q +1
2wq2 mq
mq E + l2q +1
2wq2
é
ë
êêêê
ù
û
úúúú
E = 0.1eV
l1 = 0.2
l2 = -0.1
w = 0.1
m = 0.1
47
So something is very wrong in our calculations and needs more work.
Future Directions
• Finding a more numerically stable algorithm suited for the non-adiabatic systems
• Using V0 as the harmonic potential and using H.O. in the PI scheme, the PI formulation for which is exactly known
• Using different schemes other than the Stock Thoss
• Implementing the final scheme with efficient vibronic models to solve larger systems
48
• Introduction to FORTRAN • Introduction to vibronic models • Introduction to DVR techniques • Introduction to the Path Integral Methodology • Using the path integral methodology to solve simple systems • Extending our codes to non-adiabatic problems WE HAVE A PIMC CODE THAT GENERATES NUMBERS THAT ARE SUPPOSED TO BE PROPERTIES OF NON ADIABATIC SYSTEMS… BUT UNFORTUNATELY ARE NOT THE CORRECT ONES !!!
Summary of Work
49
We are thankful to the following people for their guidance and support:
Prof. Marcel Nooijen Dr. Toby Zeng Prateek Goel
The Nooijen Research Group &
The Whole Theoretical Chemistry Group of University of Waterloo
Acknowledgement
50
Thank You
51