Ayan Chattopadhyay Mainak Mustafi 3 rd yr Undergraduates Integrated MSc Chemistry IIT Kharagpur

Ayan Chattopadhyay Mainak Mustafi 3 rd yr Undergraduates Integrated MSc Chemistry IIT Kharagpur Under the supervision of: Dr. Marcel Nooijen Associate Professor Department of Chemistry University of Waterloo 1

INTRODUCTION The Born-Oppenheimer approximation is a very important method in theoretical chemistry. Its is based on slower movement of nuclei than electrons. 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 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 2

Limitations of Born Oppenheimer Approximation Doesn t incorporate the vibrational interactions of different excited electronic states. Conical Interaction Courtesy : Wikipedia image BO Approximation breaks down for molecules with Jahn-Teller distortion Avoided Crossing 3

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 The Vibronic Model : H E ĥ H O q q E ĥ H O Coupling Parameters: Δ,μ Vibronic Hamiltonian 1 Normal Mode and 2 Electronic States The Potential Energy Matrix : E 1 2 q 2 q q E 1 2 q 2 4

Energy Intensity Intensity μ = 0.3, Δ = 0.1 ev Nuclear coordinate Full Born-Oppenheimer Franck-Condon ebe ebe 5 Courtesy : Prateek Goel

Questions at hand Obtain the Vibronic models for large systems, i.e., for molecules with many electronic states and many normal modes These models can be used to simulate Spectra Also want to do Statistical mechanics study on these systems. To compare the Born-Oppenheimer and the Vibronic Models Developing an efficient technique to solve for the Statistical Mechanics for these systems. 6

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 Model1x2: 1 normal mode and 2 electronic states: V(q) = 1 2 E q q q 1 2 1 2 q E q q 2 2 7

Model2x2: 2 normal modes and 2 electronic states: V(q 1,q 2 ) = 1 2 1 2 E q q q q 2 2 1 1 2 2 1 1 2 q 1 2 E q 1 q 2 q 2 2 2 1 1 2 2 2 2 8

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 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 10

In this method, considering a function f(ȃ) : A a a a i i i ' 1 '' 2 f ( A ) f (0 ) f (0 ) A f (0 ) A... 2 ' '' 2 f ( A ) a [ f (0 ) f (0 ) a f (0 ) a...] a f ( a ) a f(ȃ) can be transformed to the original basis as: i i i i i i i, j n a a f ( A ) a a m i i j i m a U i T h u s, n f ( A ) m U f ( a ) U 11

The Method For Non-adiabatic case: in diabatic basis 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 The eigenvalues and the eigenvectors are used to calculate the statistical mechanical properties for the model The Potential and Kinetic energy matrices are added and diagonalized to obtain the total energy matrix Kinetic Energy Matrix is tranformed into DVR 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(-ε i *β ) ; where β = 1/(kT) and k is the Boltzmann constant Population probability for each state i P i = exp(-ε i *β )/Q Helmholtz free energy (A) = -kt*lnq Internal energy (U) = Σp i ε i Entropy (S) = -k*σp i *lnp i = (U-A)/T Specific Heat Capacity (Cv) = 1/kT 2 *Σ(ε i U) 2 *P i Expected value of Potential (Ṽ) = Σ<ε i V ε i >*P i ; where ε i > are the energy eigenvectors 13

Comparison of Adiabatic and Non-adiabatic results based on DVR For 1 Normal mode and 2 Electronic States μ = 0.1, E 0 = 0.1 ev, Temperature = 2000 K Non-adiabatic V(q) = 1 2 E q q q 1 2 1 2 q E q q 2 2 Adiabatic 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 14

Difference in Internal Energy (in ev) Internal Energy (in ev) The Internal Energy distribution for different values of delta. The Difference in Internal Energies for adiabatic and non-adiabatic 15

Difference in Expected value for Potential (in ev) Expected value for Potential (in ev) The Expected Potential Energy distribution for different values of delta. The Difference in Expected values of Potential for adiabatic and non-adiabatic Difference is not that large 16

μ = 0.2, E 0 = 0.1 ev, Temperature = 2000 K Δ Non-adiabatic 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 Adiabatic 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 17

Difference in Internal Energy (in ev) Internal Energy (in ev) The Internal Energy distribution for different values of delta. The Difference in Internal Energies for adiabatic and non-adiabatic 18

Difference in Expected value for Potential (in ev) Expected value for Potential (in ev) The Expected Potential Energy distribution for different values of delta. The Difference in Expected values of Potential for adiabatic and non-adiabatic Difference is greater with increasing μ, more prominent in Δ =0.2-0.5 region 19

For 2 Normal mode and 2 Electronic States V(q 1,q 2 ) = 1 2 1 2 E q q q q 2 2 1 1 2 2 1 1 2 1 2 1 2 q E q q q 2 2 2 1 1 2 2 2 2 μ = 0.2, E 0 = 0.1 ev, Temperature = 2000 K Non-adiabatic Adiabatic Δ Internal Energy (U) (in ev) Expected value of Potential <V>(in ev) 0.0 1.95x10-2 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 Difference is much more observed than the 1x2 model 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 20

μ = 0.3, E 0 = 0.1 ev, Temperature = 2000 K Δ Non-adiabatic 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 Δ Adiabatic 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 Difference is greater with increasing μ, more prominent in Δ =0.2-0.5 region 21

Limitations 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*20 30 states, which is impossible to compute 22

How to overcome this problem, and find a more efficient method for such computations??? Path Integral Monte Carlo Approach 23

PATH INTEGRAL METHOD Provides a numerically exact solution to the time-dependent Schrodinger equation (hence can be made arbitrarily accurate) Requires computational effort that grows comparatively slowly with the dimensionality of the system For Boltzmann systems in imaginary time importance sampling methods are ideally suited, which corresponds to Boltzmann averaged equilibrium statistical mechanical properties. 24

The Advantage 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 DVR PI Computationally impossible (Hilbert space dimension of 20 30 ) No. of beads*normal modes*electronic states*no. of samples 25

THE PROPAGATOR IN PI REPRESENTATION The quantum time evolution function e iht/ Important in deriving semi classical approximations to quantum dynamical phenomena. The time evolution of a system can be expressed simply as: y (x,t) = x y (t) = ò dx ' x e -iht/ x ' x ' y 0 = ò dx ' K(x, x ',t)y 0 (x ' ) Ref. : Nancy Makri, Computer Physics Communications 63 (1991) 389-414 26

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) 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 27

Time Slices Time slicing is done as follows, for representing the long term propagator in terms of the short term propagators: N Õ k=1 e -iht/ = e -ihdt/ e -ihdt/...e -ihdt/ = e -ihdt/ Thus with the equivalence between the time evolution operator and the quantum density operator we can express it as: e bh = e b PH e b PH...e b PH = P Õe b PH i=1 Contd. 28

TIME SLICES AND THE TROTTER APPROXIMATION (CONTD.) Thus in the Trotter approximation the error gets reduced quadratically with increasing P. e -b P H @ e -b P K e -b P V +O(b P 2 ) Where b P = b / 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) The adiabatic potential 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 The state probabilities for energy distribution 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 32

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 NONADIABATIC PATH INTEGRAL 1. The General N-level Hamiltonian A general N-level Hamiltonian 1. THE GENERAL is described N LEVEL by: HAMILTONIAN Ĥ = h 0 (R,P) + N å V n,m (R) y n y m n,m=1 Nuclear kinetic energy + State independent part of the potential energy T(P)+V 0 (R) The non-adiabatic potential energy matrix elements Ref: N. Ananth and T.F. Miller III, J. Chem. Phy., 133, 234103(2010) 35

2. The Stock Thoss mapping The N level system is represented by N uncoupled Harmonic oscillators The mapping relations : y n y m a n + a m Bosonic creation and annihilation operators y n 0 1 0 2...1 n...0 N The mapping transforms the Hamiltonian into: Ĥ = h 0 (R,P) + N å n,m=1 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. a + n V n,m (R)a m Transforming the boson operators into the Cartesian representation we obtain the Hamiltonian in the Cartesian form: Ĥ = h 0 (R,P) + 1 2 N å n,m=1 (x n x m + p n p m -d nm ) V nm (R) Ref: G. Stock and M. Thoss, Phy. Rev. Lett.,78, 578 (1997) 36

3. The Stock Thoss mapping contd Transforming the boson operators into the Cartesian representation we obtain the Hamiltonian in the Cartesian form: x n = 1 2 (a + a + n n ) p n = 1 2 (a + n - a n ) a, a n m n m Ĥ = h 0 (R,P) + 1 2 N å n,m=1 (x n x m + p n p m -d nm ) V nm (R) Ref: G. Stock and M. Thoss, Phy. Rev. Lett.,78, 578 (1997) 37

4. The Path Integral Formulation of Hamiltonian The canonical partition function is defined from the trace of the Boltzmann operator. Z = Tr éë e -bh ù û The resolution of identity operator for this space looks like: ò N å I = dr R,n n=1 R,n Repeated insertion of the completeness relation yields the PI discretization of the partition function: ò Z = d{r a } N P å R a n a e -bph R a +1 n a +1 Õ {n a }=1 a =1 where ( Õ ) N and å º {n a }=1 P ò d{r a } º a=1ò dr a æ ç è Õ N P a=1å n a =1 ö ø Ref: N. Ananth and T.F. Miller III, J. Chem. Phy., 133, 234103(2010) 38

4. The Path Integral Formulation of Hamiltonian contd Applying the Trotter approximation we can get the partition function in the following form: Z = lim P ò d{r a } P Õ a =1 æ MP ö è ç 2b ø é exp ê- MP 2b R - R a a +1 ë N å P f /2 e -b P V 0 (R a ) ( ) T ( R a - R a +1 ) n a e -b Pn (R a ) Õ n a +1 {n a =1} a =1 ù ú û Ref: N. Ananth and T.F. Miller III, J. Chem. Phy., 133, 234103(2010) 39

4. The Path Integral Formulation of Hamiltonian contd Using the projection operator the SEO basis can be transformed into the Cartesian coordinate basis whereby the last term transforms into : 4. THE PATH INTEGRAL FORMULATION OF THE HAMILTONIAN (contd.) ò P d{x a } Õ x a e -b Pn (R ) a P x a+1 a=1 The projection operator being: N å n n = Õé ëò dx i x i x i n=1 N i=1 ù û P Ref: N. Ananth and T.F. Miller III, J. Chem. Phy., 133, 234103(2010) 40

4. The Path Integral Formulation of Hamiltonian contd Thus effectively the electronic matrix elements reduce to the form: N 4. THE PATH INTEGRAL x e -b Pn (R) P FORMULATION x ' = x n M OF THE HAMILTONIAN (contd.) nm (R) m x ' å n,m=1 where M nm (R) = n e -b Pn (R) m 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. Ref: N. Ananth and T.F. Miller III, J. Chem. Phy., 133, 234103(2010) 41

5. The Final Path Integral Finally putting all the terms together we obtain the final PI representation of the partition function as: æ 2MP ö Z = lim P è ç ø bp N+1 fp/2 ò d{r a } ò d{x a } P Õ a =1 A a F a G a Where : A a = e - MP 2b (R a -R a +1 ) T (R a -R a +1 ) e -b P V 0 (R a ) F a = x T T a M (R a )x a +1 G a = e - x a T x a Ref: N. Ananth and T.F. Miller III, J. Chem. Phy., 133, 234103(2010) 42

PI-MC Calculations Importance Sampling The simulation has been performed Importance using standard sampling path integral Monte Carlo techniques. In our simulation the weight function for important sampling is given by the function: W ({x a },{R a }) = P Õ A a G a F a a =1 43

Calculations and Results for Non-adiabatic Systems The Nuclear Probability Distribution 1. The nuclear probability distribution The nuclear distribution is calculated as: P(R) = d (R - R )sgn(f) P W sgn(f) W Nuclear probability distribution for DVR Nuclear probability distribution for Path Integral 44

Ambiguity in the nuclear probability distribution function D V R D V R ( q ) 1 ( q ) d q 1 i i Nuclear probability distribution for DVR Nuclear probability distribution for Path Integral D V R Is it true that ( q ) ( q )? i i 45

Calculating the energies: Potential & Total The average total energy operator is given by this well known statistical mechanical formula: 2. Calculating the energies: potential and total E = - 1 Z Z b This gives us the form to calculate the total energy from the partition function as: E = æ P 2b + F - A ö è ç b ø sgn(f) W sgn(f) W where F = P å a =1 T - M (R x a ) a b x T a M (R a )x a x a Ref: N. Ananth and T.F. Miller III, J. Chem. Phy., 133, 234103(2010) 46

Results for Non-adiabatic PI Calculations é ê Model: V(q) = ê ê ê ë Temp(in K) E + l 1 q + 1 2 wq2 mq Total Potential energy (in ev) mq E + l 2 q + 1 2 wq2 DVR values (in ev) 600-0.71-0.092 1000-0.68-0.049 1500-0.64 0.006 2000-0.61 0.059 ù ú ú ú ú û E = 0.1eV l 1 = 0.2 l 2 = -0.1 w = 0.1 m = 0.1 So something is very wrong in our calculations and needs more work. 47

Future Directions Finding a more numerically stable algorithm suited for the non-adiabatic systems Using V 0 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

Summary of Work 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!!! 49

Acknowledgement 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 50

Thank You 51