Ayan Chattopadhyay 3rd Integrated MSc Chemistry...

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:

11

' '' 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

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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)

27

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)

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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 :

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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