Exact Solution of Multi State Problems in Quantum and Statistical Mechanics

Exact solution of Multi state problems in Quantum and Statistical Mechanics

A THESIS

submitted by

DIWAKER

for the award of the degree

DOCTOR OF PHILOSOPHY

SCHOOL OF BASIC SCIENCES INDIAN INSTITUTE OF TECHNOLOGY MANDI. MANDI-175005 (H. P.) INDIA AUGUST, 2015 c 2015 - Diwaker

I dedicate this thesis to my parents, my brothers, my sisters and to my teachers.

Indian Institute of Technology BArtFy þODOEgkF s\-TAn Mandi, Mandi (H. P.)-175005, m\ÚF, m\ÚF,(Eh. þ.)-175005, India, Govt. of India BArt, BArt srkAr

THESIS CERTIFICATE

This is to certify that the thesis titled Exact Solution of Curve Crossing Problems in Quantum Mechanics, submitted by Diwaker, to the Indian Institute of Technology, Mandi, for the award of the degree of Doctor of Philosophy, is a bonaﬁde record of the research work done by him under my supervision. The contents of this thesis, in full or in parts, have not been submitted to any other Institute or University for the award of any degree or diploma.

Dr. Aniruddha Chakraborty Research Guide Associate Professor School of Basic Sciences IIT-Mandi, 175005 Phone-01905-237930 Email-achakraborty @iitmandi.ac.in http://www.aniruddhachakraborty.org/

Place: Mandi (H. P.) Date: 1st august 2015

Phone: 01905-237949/237944/267001, Fax: 01905-267009, www.iitmandi.ac.in Indian Institute of Technology BArtFy þODOEgkF s\-TAn Mandi, Mandi (H. P.)-175005, m\ÚF, m\ÚF,(Eh. þ.)-175005, India, Govt. of India BArt, BArt srkAr

DECLARATION BY RESEARCH SCHOLAR

I hereby declare that the entire work assimilated in this thesis entitled Exact Solution of Curve Crossing Problems in Quantum Mechanics is the result of investigations carried out by me in the School of Basic Sciences, Indian Institute of Technology Mandi, under the supervision of Dr. Aniruddha Chakraborty,and that it has not been submitted elsewhere for any degree or diploma. In keeping with the general practice, due acknowledgments have been made wherever the work described is based on ﬁnding of other investigators.

Diwaker Enrollment no. D10011 Research Scholar School of Basic Sciences IIT-Mandi, 175005 [email protected] https://sites.google.com/site/diwakeriitmandi/home

Place: Mandi (H. P.) Date: 1st august 2015

Phone: 01905-237949/237944/267001, Fax: 01905-267009, www.iitmandi.ac.in ACKNOWLEDGEMENTS

First, I would like to express my most sincere thanks and gratitude to my Ph.D. su- pervisor Dr. Aniruddha Chakraborty, Assistant Professor, School of Basic sciences, Indian Institute of Technology Mandi, Mandi (H. P.), for the seemingly endless hours he spent with me going over this work, for his patience when things did not always work out, for his support, encouragement, constant guidance and constructive advice which he has extended to me throughout all the stages of my research work. I also greatly appreciate the conﬁdence he has shown in my abilities and capabilities. Special thanks goes to him for encouraging me to think problems in multiple directions. Next, I am very grateful to Prof. Timothy A Gonsalves, Director, IIT Mandi for setting up a good research environmentin this Himalayan region. I also want to convey thanks to my Doc- toral committee members Dr. Pradumnya Pathak, Dr. Hari Verma, Dr. Prasanth Jose and Dr. Neeraj for their help, support, valuable suggestions and discussions. Iam also grateful to the MHRD India for providing me fellowship to carry out this research work. I am obliged to my parents Shri. Dina Nath and Smt. Brahmi Devi and my younger brother and sister, Akhilesh Kumar and Nivedita and above of all my lovely niece Hrutvika for their love, affection, moral and endless support that has enabled me to achieve this goal. I gratefully concede all my friends specially Manisha thakur and all other persons whose names do not appear here but whose love, cooperation and participation in various forms have motivated me and helped me to complete this work.

Diwaker Place: Mandi (H. P.)

i ABSTRACT

KEYWORDS: Curve crossing, Non adiabatic transitions, two state, three state, Multi state, scattering, Green’s function, Boundary conditions, Dirac Delta Potential, constant potential, linear potential, exponential potential, Morse potential, harmonic potential, Transfer matrix, Diabatic orbitals, Diabatization, Non adiabatic coupling matrix elements, MRCI, CASPT2. Non adiabatic transitions due to crossing of the potential energy curves is one of the most probable mechanism responsible for electronic transitions. This is purely an interdisciplinary topic which covers a wide range of fields like physics, chemistry and biology. Various spectroscopic, collisions processes and reactions are governed by such kind of transitions. Some of the examples involving such kind of transitions may include radationless transitions in condense matter physics, laser assisted collisions reactions, Zener transitions in flux driven metallic rings, super conducting Josephson junctions, reactions in nuclear physics and electron proton transfer processes in biological systems. Neutrino conversion in the sun, dissociation of molecules on the metal surfaces are some of other few examples which explains the importance of non adiabatic transitions. The first report on non adiabatic transitions was published in around 1932, when Landau, Zener, Stuckelberg and Rosen published pioneer work in the area of non adiabatic transitions which opens a pathway for solutions to the problems including such kind of transitions. The approach used by them is purely analytical which can be mapped to problems like electron detachments, ionization in slow atomic and ionic collisions and electronic transitions in crystals where one state of system is interacting with a group of states of different nature and many more. From 1932 onwards we have numerous citations in literature based on non adiabatic transitions which involves analytical as well as computational approach. The work presented here in this thesis pays attention to the use of analytical methods for problems involving non adiabatic ii transitions where one state of a system is interacting with a group of states of different nature through Dirac Delta interactions and we provide a simple analytical formula for calculation of transition probability between different interacting states. The present thesis is divided into 7 chapters. Chapter 1 includes the introduction of non-adiabatic transitions/Curve crossing methods in quantum mechanics in which different analytical models are described in brief. Chapter 2 includes the details of the different kind of analytical and computational methods used in study of non adiabatic transitions. Chapter 3 is devoted to the exact solution of curve crossing problems using boundary condition and Green’s function method. Chapter 4 includes the exact solution of curve crossing problems using Transfer matrix method while in Chapter 5 we have used time dependent approach for the study of non adiabatic transitions/curve crossing problems. Chapter 6 includes the computational approach to study the curve crossing problems in real molecules. Chapter 7 concludes this thesis by providing the summary of all six chapters and future prospects in this area.

iii TABLE OF CONTENTS

ACKNOWLEDGEMENTS i

ABSTRACT ii

LIST OF TABLES ix

LIST OF FIGURES xiii

ABBREVIATIONS xiv

NOTATION xv

1 INTRODUCTION 2 1.1 ClampednucleiHamiltonian ...... 2 1.2 Adiabatic and Born-Oppenheimer approximation ...... 3 1.3 Adiabatic and Diabatic approaches ...... 5 1.3.1 Adiabaticapproach ...... 5 1.3.2 Diabaticapproach ...... 6 1.4 CrossingofPotentialenergycurves ...... 8 1.4.1 NaClmolecule-: anexample ...... 9 1.5 Probability of a non-adiabatic transition ...... 14 1.6 Importantdefinitions ...... 17 1.6.1 Avoidedcrossing ...... 17 1.6.2 Diabaticstaterepresentation...... 18 1.6.3 Adiabaticstaterepresentation ...... 18 1.6.4 AdiabaticityorAdiabaticstates ...... 19 1.7 Examples of non adiabatic transitions in different fields of science 19 1.7.1 Non adiabatic transitions inPhysics...... 19 iv 1.7.2 Non adiabatic transitions in Chemistry ...... 20 1.7.3 Non adiabatic transitions in Biology and Economics . . . 21 1.8 Analytical Models of Non Adiabatic Transitions ...... 21 1.8.1 Landau-Zener-Stuckelberg Theory ...... 21 1.8.2 Rosen-Zener-DemkovTheory ...... 26 1.8.3 Nikitin’sExponentialmodel ...... 27 1.8.4 Non adiabatic Transition due to Coriolis coupling and Dy- namicalstaterepresentation ...... 29 1.8.5 Curve crossing type of Nonadiabatic transition: . . . . . 30 1.8.6 NonadiabticTunnelingtype ...... 30 1.8.7 Diabatically avoidedcrossing model ...... 31 1.8.8 ExponentialPotentialModel...... 32 1.8.9 LinearPotentialModel ...... 33 1.8.10 Demkov-kunikemodel ...... 34 1.8.11 Demkov-Osherovmodel ...... 35 1.8.12 Zhu-Nakamuratheory ...... 37 1.8.13 Ouranalyticalmodel:...... 37 1.9 Conclusions ...... 39

2 DIFFERENT APPROACHES USED TO STUDY NON ADI- ABATIC TRANSITIONS/CURVE CROSSING PROBLEMS 41 2.1 Boundaryconditionmethod ...... 41 2.1.1 Boundary condition method applied to curve crossing problems ...... 43 2.2 Green’sfunctionMethod...... 45 2.2.1 Green’s function method applied to the curve crossing problems ...... 46 2.3 Transfermatrixmethod ...... 49 2.3.1 Transfer matrix method applied to the curve crossing problems ...... 52 2.4 Time dependent Approach to the Non adiabatic transitions/curve crossingproblems ...... 54

v 2.4.1 Methodology1 ...... 54 2.4.2 Methodology2 ...... 57 2.5 Computational Approach to the Curve Crossing Problems . .. 58 2.5.1 CASPT2...... 58 2.6 Conclusions ...... 60

3 EXACT SOLUTION OF CURVE CROSSING PROBLEMS US- ING BOUNDARY CONDITION METHOD AND GREEN’S FUNCTION APPROACH. 62 3.1 Exact solution of curve crossing problems using boundary condition method ...... 62 3.1.1 Twochannelscatteringproblem ...... 62 3.1.2 Threechannelscatteringproblem ...... 64 3.1.3 N-channelscatteringProblems...... 66 3.2 Exact solution of two state curve crossing problems involving different potentials using boundary condition method ...... 66 3.2.1 Constantpotentialcase...... 66 3.2.2 Linearpotentialcase ...... 68 3.2.3 Exponentialpotentialcase ...... 74 3.3 Exact solution of three state curve crossing problems involving different potentials using boundary condition method ...... 76 3.3.1 Exact analytical solution for constant potential case: .. 78 3.3.2 Exact analytical solution for linear potential case: .... 81 3.3.3 Exact analytical solution for exponential potential case . 84 3.4 Exact solution of curve crossing problems using Green’s function approach...... 88 3.4.1 Green’s function approach for the two channel scattering problems...... 88 3.4.2 Green’s function approach for the three channel scattering problems...... 91 3.4.3 Green’s function approach for the N-channel scattering Prob- lems ...... 94 3.5 Green’s function approach for the curve crossing problem with Gaus- siantypecoupling...... 94 vi 3.5.1 Green’s function approach to calculate the eﬀect of curve crossing on electronic absorption spectra and Resonance Ra- manexcitationProﬁle ...... 98 3.6 Conclusions ...... 105

4 EXACT SOLUTION OF CURVE CROSSING PROBLEMS US- ING TRANSFER MATRIX METHOD 107 4.1 Transfermatrixmethod ...... 108 4.1.1 Transfer matrix approach for exact solution to the curve crossing of two constant diabatic potentials ...... 109 4.1.2 Transfer matrix approach for exact solution to the curve crossing of two linear diabatic potentials ...... 116 4.1.3 Transfer matrix approach to the curve crossing problem of two exponentialdiabatic potentials ...... 127 4.2 Conclusions ...... 142

5 EXACT SOLUTION TO THE CURVE CROSSING PROB- LEMS USING TIME DEPENDENT APPROACH 144 5.1 Exact solution of time-dependent Schrodinger equation for two state probleminLaplacedomain...... 145 5.1.1 Derivation of the propagator for constant b ...... 148 5.1.2 Consistency between derived operator and Green’s function usingFeynmanapproach ...... 149

st 5.1.3 Solution for constant k0: bound state representation for 1 potential...... 151 5.1.4 Partly unbound solutions for 1st potential ...... 152 5.2 Exact solution of Schrodinger equation for two state problem with timedependentcoupling...... 152 5.2.1 Case 1 : Linear time dependence of the strength of the Dirac Deltapotential ...... 154 5.2.2 Case 2: Solution for strength of the Dirac Delta potential having inversely proportional dependence on time . . . . 157 5.2.3 Case 3: Solution for strength of the Dirac Delta potential having exponential dependence on time ...... 160 5.3 Non-adiabatic transition probability with a moving δ potential coupling...... 163 vii 5.3.1 Non adiabatic transition probability for two state coupled by a moving δ potentialcoupling ...... 166 5.4 Conclusions ...... 171

6 AVOIDED CROSSING IN LITHIUM CHLORIDE USING CASPT2 AND MRCI LEVEL OF THEORY. 173 6.1 Computational details used in the calculations ...... 175 6.1.1 Diabaticorbitals ...... 176 6.1.2 NACME(Non-adiabatic coupling matrix elements) . . . . 177 6.1.3 Avoidedcrossing ...... 180 6.1.4 MRCI diabatic and adiabatic potential energy surfaces . 184 6.1.5 Furtherstudies ...... 187 6.2 Conclusions ...... 187

7 Summary 189 LIST OF TABLES

6.1 Diabatic orbitals along with occupancy and eigenvalue for lithium chloridecalculated usingDIABprocedure ...... 178 6.2 Non-adiabatic couplings for LiCl calculated using DIAB procedure. 180 6.3 RS2/CASPT2 energy extrapolation using serial basis set..... 181 6.4 Diabatic and Adiabatic energies for LiCl, obtained from CI-vectors and CI-vectorsplusorbitalcorrection ...... 186 6.5 Comparative study of avoided crossing for low lying electronic states oflithiumchloride ...... 186

ix LIST OF FIGURES

1.1 Schematic diagram of adiabatic approach...... 6 1.2 Schematic diagram of diabatic approach...... 7 1.3 Energy diagram showing the region of avoided crossing for Na-Cl + and Na and Cl− conﬁgurationcurves ...... 10 1.4 Energy diagram showing the avoided crossing of Na-Cl and Na+ and Cl− conﬁgurationcurves...... 12 1.5 Schematic plots of Diabatic and Adiabatic curves ...... 15 1.6 Schematic diagram representing Landau-Zener type of curve crossing...... 25 1.7 Rosen ZenerDemkovmodelof curvecossing ...... 27 1.8 Nonadiabatic tunneling type curve crossing...... 31 1.9 Crossing of many horizontal states with a slanted state (Demkov- Osherovmodel)...... 36 1.10 DiracDeltacouplingmodel...... 38

2.1 Schematic diagram showing the potential and the barrier .... 42 2.2 Tunneling through a rectangular barrier ...... 49 2.3 Arbitrary coupling modeled by one Dirac Delta potential .... 52 2.4 Arbitrary coupling modeled by three Dirac Delta potential ... 53

3.1 Two constant potential coupled by Dirac Delta coupling . ... 67 3.2 The plot of transition probability from one constant potential to another constant potential as a function of energy of incident particle (K0 =1)...... 69 3.3 Schematic diagram of the two state problem, where one linear potential is coupled to another linear potential in the diabatic representation...... 70 3.4 The plot of transition probability from one linear potential to another linear potential, as a function of energy of incident particle (K0=1.0)...... 72 x 3.5 Schematic diagram of the two state problem, where one exponential potential is coupled to another exponential potential in the diabatic representation...... 73 3.6 Transition probability plots from one exponential potential to another as a function of energy of incident particle (K0=1.0). . . . 73 3.7 Schematic diagram for three constant potential case coupled by Dirac Delta coupling in the diabatic representation...... 81

3.8 (Plot of Transition Probability, Plot Parameters areV1 = 0, V2 = 2, V3 = 3,m = 1, ~ = 1, K0 =1) ...... 82 3.9 Schematic diagram for three linear potential case coupled by Dirac Delta coupling in the diabatic representation...... 82

3.10 (Plot of Transition Probability,Plot Parameters are p1 = 0, p2 = 1, p = 0.5,m = 1, ~ = 1, K =1) ...... 85 − 3 − 0 3.11 Schematic diagram of the three channel problem, where one exponential potential is coupled to two other exponential potentials in thediabaticrepresentation ...... 86

3.12 (Plot of Transition Probability, Plot Parameters are a1 = 1,a2 = 1,b = 0.5,m = 1, ~ = 1, K = 0.1)...... 88 − 0 3.13 Schematic potential energy curves that illustrate both the diabatic plotted as dashed line and adiabatic potentials plotted as solid line in the case of the Gaussian type of coupling...... 95 3.14 Schematic view of diabatic potential energy curves that interpret the model. The forbidden state is denoted as F , while allowed state is denoted by A...... 99 3.15 Calculated electronic absorption spectra with coupling effect (solid line) and without coupling effect(dashed line). Here the values for 1 1 the simulations are E0 = EA = EF = 400cm− , Γ = 450cm− , εA = 1 1 1 2 10700 cm− , εF = 11500 cm− , K0 = 1200 cm− , α = 0.2594 A˚− and x = 0.1991A...... ˚ 100 c − 3.16 Simulated resonance Raman excitation profile for excitation from the ground vibrational state to the first excited vibrational state, with coupling effect (solid line) and without coupling effect (dashed line). Here the values for the simulations are E0 = EA = EF = 1 1 1 1 400 cm− , Γ = 450 cm− , εA = 10700 cm− , εF = 11500 cm− , 1 2 K = 1200 cm− , α = 0.2594 A˚− and x = 0.1991A...... ˚ 101 0 c −

xi 3.17 Calculated electronic absorption spectra with coupling effect (solid line) and without coupling effect (dashed line). Here the values for 1 1 the simulations are E0 = EA = EF = 400 cm− , Γ = 20 cm− , εA = 1 1 1 2 10700 cm− , εF = 11500 cm− , K0 = 1200 cm− , α = 0.2594 A˚− and x = 0.1991A...... ˚ 104 c − 3.18 Calculated resonance Raman excitation profile for excitation from the ground vibrational state to the first excited vibrational state, with coupling effect (solid line) and without coupling effect (dashed line). Here the values for the simulations are E0 = EA = EF = 1 1 1 1 400 cm− , Γ = 20 cm− , εA = 10700 cm− , εF = 11500 cm− , 1 2 K = 1200 cm− ,α = 0.2594 A˚− and x = 0.1991A...... ˚ 104 0 c −

4.1 Smooth Gaussian Function expressed as a collection of Dirac Delta potentials. Vertical lines represent the Dirac Delta potentials while curveisGaussianfunction ...... 109 4.2 Two diabatic potentials coupled by arbitrary coupling which is expressed as a collection of Dirac Delta potentials ...... 111 4.3 Plots of Non adiabatic Transition Probability for arbitrary coupling expressed by diﬀerent number of Dirac Delta potentials . . . . . 113 4.4 Converged Plots for Non adiabatic transition probability .... 114 4.5 Quantitative agreement between the analytical and exact numerical results for 21 number of Dirac Delta Potentials ...... 114 4.6 Schematic diagram for the crossing of two Linear potentials with Gaussian coupling (α =1)...... 116 4.7 Schematic diagram for Gaussian coupling expressed as a collection of Dirac Delta potentials α =1...... 117 4.8 Plot of transition Probability from 1st linear diabat to second linear as a function of energy of incident particle with arbitrary coupling expressed by one Dirac Delta function having numerical values as (m = 1, ~ = 1, k[n] = 1.76,a[n] = 0, p = 1, p = 1)...... 124 1 2 − 4.9 Plot of transition Probability from 1st linear diabat to second linear diabat as a function of energy of incident particle with arbitrary coupling expressed by diﬀerent number of Dirac Delta function . 125 4.10 Landau-Zener Transition probability for n=13,15,17 and 21 number ofDiracDeltapotentials ...... 126 4.11 Schematic diagram for the crossing of two exponential potentials with Gaussian coupling (α =1)...... 128 4.12 Schematic diagram for Gaussian coupling expressed as a collection of Dirac Delta Potentials α =1...... 128 xii 4.13 Converged Plots for Non adiabatic Transition probability . . . . 141

6.1 Shapes of electron density plots of diabatic orbitals under different irreduciblerepresentations...... 178 6.2 NACME for lithium chloride computed using finite differences. . 179 6.3 Avoided crossing between different levels of LiCl using single-state singlereferenceCASPT2 ...... 182 6.4 Avoided crossing between different levels of LiCl using multi-state multireferenceCASPT2 ...... 183 6.5 MRCI adiabatic and diabatic potential energy curves without or- bitalcorrection...... 185 6.6 MRCI adiabatic and diabatic potential energy curves with orbital correction...... 185

xiii ABBREVIATIONS

WKB Wentzel Kramers Brillouin LZ Landau Zener

xiv NOTATION

r Radius, m α Angle of thesis in degrees β Flight path in degrees

xv Chapter1 This chapter explains the basic concepts of Curve crossing(Potential energy curves)/Non adiabatic transitions by considering basic example of Sodium chloride molecule. A brief introduction explaining the concept of diabatic and adiabatic potential energy surfaces has been given. Apart from this different kind of exact analytically solvable models involving non adiabatic transitions has been discussed in brief.

1 CHAPTER 1

INTRODUCTION

1.1 Clamped nuclei Hamiltonian

We start with the deﬁnition of Hamiltonian by considering the case of a diatomic molecule with its nuclei labeled as p and q with m electrons. In space ﬁxed coordinate

system the position of nuclei is represented as Rp =(Xp, Yp, Zp) and Rq =(Xq, Yq, Zq)

0 0 0 while electrons has the coordinates xi, yi, zi. The Hamiltonian fo this system under consideration can be written as

~2 ~2 n ~2 ˆ 0 H = ∆p ∆q ∆i + V (1.1) −2Mp − 2Mq − 2m i=1 X In Equation 1.1, the ﬁrst two terms are kinetic energy operator for nuclei, third term is kinetic energy for all electrons while V is the Coloumbic potential energy operator given by the following equation which describes the interaction of all the particles involved in the system such as nucleus-nucleus, nuclei-electron, electron-electron.

Z Z e2 e2 e2 e2 V = p q + Z + Z + (1.2) R p r q r r pi qi i

~2 Hˆ = − ∆XY Z + Hˆ + Hˆ 0 (1.3) 2M 0 In above equation the ﬁrst term is kinetic energy operator while Hˆ0 is the electronic Hamiltonian (clamped nuclei Hamiltonian) given by

~2 Hˆ = ∆ + V (1.4) 0 − 2m i i X with ~2 Hˆ 0 = ∆ + Hˆ 00 (1.5) −2µ R and 2 ~2 ~2 1 1 00 Hˆ = i + R i (1.6) −8µ 5 2 Mp − Mq 5 5  i ! i X X   µ denotes the reduced mass of the system under consideration. The equation (1.4) is called as electronic or clamped nuclei Hamiltonian as it dos not contain kinetic energy operator for the nuclei. Equation (1.5) represents the kinetic energy operator for nuclei and equation (1.6) represents the coupled motion of nuclei as well as that of electrons. After seperation of the center-of-mass motion equation (1.3) can be rewritten as

0 Hˆ = Hˆ0 + Hˆ (1.7)

which is exactly equivalent to Schroedinger equation.

1.2 Adiabatic and Born-Oppenheimer approximation

The total wave function which describes both the electrons as well as nuclei can be proposed as following N

ψ(r, R)= ψi(r; R)fi(R) (1.8) i X

where ψi(r; R) are eigen functions of Hˆ0.

ˆ 0 H0(R)ψi(r; R)= Ei ψi(r; R) (1.9)

3 which depends parametrically upon R i. e. for each value of inter nuclear distance R,

we have different formula for ψi. Using the Hamiltonian as mentioned in equation (1.7) and wave function of equation (1.8), we start with the time independent Schroedinger equation of the following form

N N

Hˆ0 + Hˆ ψi(r; R)fi(R)= E ψi(r; R)fi(R) (1.10) i i X X

multiplying both sides by ψn∗(r; R) and integrating over the electronic coordinates r, we get N N ψ (Hˆ + Hˆ 0 (ψ f ) = E ψ ψ f (1.11) h n| 0 i i ie h n| iie i i i X X using the ortho normalization condition as well as considering that ψi is an eigen func-

tion of Hˆ0, we reach at the following equation

N E0f + ψ Hˆ 0 (ψ f ) = Ef (1.12) n n h n| i i ie n i X further simpliﬁcation of above equation will lead us to [2]

N ~2 ~2 E0f + (1 δ ) − ψ ψ δ ∆ f + H 0 f = Ef (1.13) n n − in µ h n| 5R iie − in 2µ R i ni i n i X

rewriting the above equation by grouping all the terms containing fn onthe L. H.S.we get ~2 N − ∆ + E0(R)+ H (R) E f = ρ f (1.14) 2µ R n nn0 − n − ni i i(=n) X6 for n=1, 2, .....N, where

~2 ρ = ψ ψ +H 0 (1.15) ni − µ h| n| 5R iie 5R ni

4 are non-adiabatic coupling operators. In the adiabatic approximation equation (1.14) reduces to

~2 N − ∆ + E0(R)+ H (R) E f = ρ f (1.16) 2µ R n nn0 − n − ni i i(=n) X6

which means that for i = n, ρ may be set equal to zero, and the curves E0(R) for 6 in n different values of n are well separated on energy scale with small coupling between

0 0 them. In the adiabatic approximation Hnn= ψn∗H ψndτe represents a small correction

0 0 to En, however if this correction is neglectedR i. e. Hnn ∼= 0 it will results into Born- Oppenheimer approximation. In the Born-Oppenheimer approximation the Potential energy for the motion of nuclei is independent of the nuclei while in the adiabatic approximation the potential energy depends upon the nuclei.

1.3 Adiabatic and Diabatic approaches

1.3.1 Adiabatic approach

The eigenfunctions ψ(r; R) along with eigenvalues Ei(R) of the Hamiltonian Hˆ (r; R) which depends upon electronic coordinates r and parametrically depends upon the nuclear conﬁguration of nuclei R (for practical applications Hˆ (r; R) Hˆ (r; R) the elec- ≡ 0 tronic clamped nuclei given by equation (1.9) are called as adiabatic. In the adiabatic approximation the total wavefunction is given by

ψ(r, R)= ψ(r; R)f(R) (1.17)

where f(R) is the rovibrational wave function which describes the rotations and vibra- tions of the molecule. The schematic diagram of adiabatic approach has been shown in Figure 1.1. The adiabatic approach shows the mixing of the diabatic states mainly at the crossing point. Each of the adiabatic state is an eigenfunction of the clamped nuclei Hamiltonian. If the nuclear motion is slow the electrons are able to adjust to it instanta-

5 Figure 1.1: Schematic diagram of adiabatic approach.

neously and the system follows the lower adiabatic curve which means that bond pattern changes qualitatively during this motion as shown in Figure 1.1 (Blue ball changes to green ball and system undergoes a transition from covalent to ionic going through in- termediate state shown as half green and half blue ball). The total wave function is a product of the adiabatic electronic state and a rovibrational wave function.

1.3.2 Diabatic approach

A basis set of expansion functions deﬁned as

M ψ(r; R) c ψ¯ (r; R) (1.18) ≈ i i i X where the basis set ψ¯i(r; R),i = 1, 2, 3.....M. termed as diabatic basis set which depends parametrically upon R. These diabatic wave functions are non-orthogonal and

6 diabatic energy is given by

E¯ (R)= ψ¯ Hˆ (R)ψ¯ (1.19) i h i| ii

. ψ¯1 may describe the ionic conﬁguration for a particlar diatomic molecule, while ψ¯2 may correspond to the covalent conﬁguration. The adiabatic wave function ψ(r; R)

of the diatomic molecule may be taken as superpoistion of ψ¯1 and ψ¯2. The schematic diagram for the diabatic appoach has been shown in Figure 1.2.

Figure 1.2: Schematic diagram of diabatic approach.

As shown in Figure 1.2, in the lower state the system is represented by a green ball (say ionic state), in the second the system is represented by a blue ball (say, covalent state). These balls oscillate in the corresponding wells preserving the chemical structure, so a state which preserve the chemical structure are generally termed as diabatic state ( is always ionic or covalent). The energies of these states are calculated by mean values of clamped nuclei Hamiltonian. It may happen that the two diabatic states cross to each other. In such a case if nuclear motion is fast then the electrons are unable to adjust to it and nuclear motion takes place on the diabatic curves which means that bond pattern do not change during this motion.

7 1.4 Crossing of Potential energy curves

We consider a diatomic molecule and let R0 be such an inter nuclear distance where the

two electronic adiabatic states ψ1(r; R0) and ψ2(r; R0) corresponds to non-degenerate but close on energy scale the eigenvalues of the clamped nuclei Hamiltonian

Hˆ0(R0)ψi(r; R0)= Ei(R0)ψi(r; R0),i = 1, 2. (1.20)

As Hˆ is Hermitian and E = E , so as per orthogonality relation Ψ ψ = 0. Further 0 1 6 2 h 1| 2i we are looking to solve the following equation

Hˆ0(R)ψ(r; R)= E(R)ψ(r; R) (1.21)

for R in the vicinity of R0 and check is it possible for energy eigen values to cross? The

eigenfunctions of Hˆ0 will be considered as a linear combination of ψ1 and ψ2 as given below

ψ(r; R)= c1(R)ψ1(r; R0)+ c2(R)ψ2(r; R0) (1.22)

for this distance R, Hamiltonian can be considered as

Hˆ0(R)= Hˆ0(R0)+ V (R) (1.23)

such that V(R) is small, as R is close to R0 and V (R0)=0. Using the Ritz Method [3], we arrive at two solutions with corresponding energies as

¯ ¯ ¯ ¯ 2 E1 + E2 E1 E2 2 E (R)= − + V12 (1.24) ± 2 ± s 2 | |

where V = ψ Vˆ (R)ψ and E¯ (R) = ψ (r; R ) Hˆ (R)ψ (r; R ) = E (R ) + ij h i| ji i h i 0 | 0 i 0 i i 0 Vii(R) The crossing of two energy curves at a particular nuclear coordinate means E+ = E , which means that expression under the root in equation (1.24) has to equal to zero. − Since the expression under the root is sum of two squares so for crossing to occur it

8 needs to satisfy the two conditions simultaneously i. e.

E¯ E¯ = 0 (1.25) 1 − 2 V = 0 (1.26) | 12|

we have two conditions and a single changeable parameter R, if we adjust the parameter to fulfill the first condition the other one is violated and vice versa. Let us consider that for some reason because of symmetry the coupling constant is automatically zero. i. e. V = 0 for all R, then we have a single condition to fulfill with only one parameter | 12| which can result into crossing. The coupling constant is equal to zero means

H = ψ Hˆ (R)ψ = 0 (1.27) | 12| h 1| 0 2i as Hˆ0(R) = Hˆ0(R0)+ Vˆ and [H0(R0)]12 = 0 due to orthogonality of both eigenfunctions of Hˆ0(R0). The Hamiltonian represents a fully symmetric object, however the wave functions ψ1 and ψ2 are not necessarily fully symmetric because they may belong to other irreducible representations of the symmetry group, so in order to make the coupling constant equals to zero, it is necessary that ψ1 and ψ2 transform according to different irreducible representations which means that they should have different symmetries. The adiabatic curves cannot cross if the corresponding wave function have same symmetry. Such cases shows an avoided crossing which means that the two curves repel each other and avoid the crossing.

1.4.1 NaCl molecule-: an example

We will consider the example of a simple molecule sodium chloride to understand the phenomenon of curve crossing. The sodium atom has 11 electrons while chlorine atom has 17 electrons. The solution of the Schroedinger equation for these 28 electrons is difﬁcult and we will concentrate on a single electron which goes from Sodium to the

+ Chlorine atom making it Na and Cl−. These ions attract each other and forms the familiar ionic bond. The question is which one of them is having the lower energy i. 9 + e. The two non interacting atoms Na and Cl or the two interacting ions Na and Cl−? From the above ﬁgure we conclude the following points as mentioned below

Figure 1.3: Energy diagram showing the region of avoided crossing for Na-Cl and Na+ and Cl− conﬁguration curves

(a) At inﬁnite separation, Na-Cl is more stable then its ionic conﬁguration by 1.5 eV which is actually

I A , (1.28) Na − Cl I and A being the ionization potential of sodium and electron afﬁnity of chlorine atom. (b) At the equilibrium bond distance ionic conﬁguration is more stable.

(c) In the region of crossing point, the mixing of two conﬁgurations i.e. (ionic and covalent) leads to a low and high energy conﬁgurations.

(c) State curves shown by dotted lines after mixing of the two conﬁgurations do not cross and this is basically termed as avoided crossing. 10 (d) The electron transfer takes place in the region of avoided crossing and actual point of electron transfer is approximated by the relation

e2 I A (1.29) Na − Cl ≈ r which means that as soon as the energy expenditure required to transfer electron I Na − ACl is provided by the electrostatic stabilization between a cation as well as an anion, e2 ( r ), an electron jump will takes place.

(e) Better the donor acceptor pair or smaller the covalent ionic conﬁguration gap, earlier is the electron jump.

(f) If the electrons are able to adapt instantaneously to the position of the nuclei (slow nuclear motion), the system will follow the adiabatic curve and the electron jump will occur. If the nuclear motion is very fast , the system will follow the diabatic curves and no electron transfer will takes place.

The true curves of the NaCl molecule occurs due to possibility of resonance between the ionic and covalent states. For better approximation let us write the wave functions for the covalent as well as ionic states of the NaCl molecule.

ψ = φ (1)φ (2)(α β α β ) (1.30) ionic Cl Cl 1 2 − 2 1 ψ =[φ (1)φ (2) + φ (2)φ (1)] (α β α β ) (1.31) covalent Na Cl Na Cl 1 2 − 2 1

In the above equation the functions φNa and φCl are 3s and 3p orbitals of Na and Cl molecule, α and β are electron spin functions while 1 and 2 are two electrons involved in the bonding. The true wave functions of the NaCl molecule is a hybrid of ionic and covalent wave functions written as

ψ = aψionic + bψcovalent (1.32)

11 + Figure 1.4: Energy diagram showing the avoided crossing of Na-Cl and Na and Cl− conﬁguration curves

12 The ﬁgure (1.4) shows the Potential energy curves for two true molecular states derived from wave functions of the covalent and ionic states will fail to cross with each other. From these states we will conclude that for more stable state 1, the NaCl molecule will be essentially covalent(b»a) at large internuclear distances and ionic(a»b) at smaller internuclear distances. For the second state 2 which is less stable state the reverse is true. At the crossing point one state has a=b giving the wave function

ψ = a(ψionic + ψcovalent) (1.33) with energy E = E0+e and one with state a = -b giving the wavefunction

ψ = a(ψ ψ ) (1.34) ionic − covalent with energyE = E0-e, E0 being the energy of two pure structures at the point of crossing and e is the resonance energy between the two forms at internuclear distance. The curve 1 gives the Potential energy curve if a sodium atom and a chlorine atom are slowly brought together. which means that at larger distances they will remain neutral and at smaller distances they will become charged ions. Curve 2 gives the energy obtained when a sodium ion and a chlorine ion are brought together slowly, here the ions retain there charges at large separations while become neutral at smaller distances. It requires a certain deﬁnite time for the electrons to tautomerize from the covalent to the ionic structure when the system is in the cross over region. If the two atoms are brought together so rapidly that they spend less time than a certain ﬁnite time in the cross over region they will remian neutral and will be no longer on curve 1 but will have gone a

+ transition to curve 2. In a similar fashion if Na and Cl− are brought together rapidly closer to each other they will undergo a transition fromcurve 2 to curve1 as they pass the cross over region. Such type of transition is called as non adiabatic transition. These transition are fairly common and occur in various interdisciplinary phenomena. Non adiabatic transition must be looked upon whenever the potential energy surfaces of different electronic states of a molecule intersect.

13 1.5 Probability of a non-adiabatic transition

The probability of a non-adiabatic transition can be easily estimated in terms of conﬁg- uration of the potential energy curves at the crossing point and velocity of the system past that point. The typical view of diabatic and adiabatic surfaces is shown in Figure 1.5. In the ﬁgure (1.5) the curves 1 and 2 are called as fast curves because they give the potential energy of the system when atoms move very rapidly. No resonance is permitted between the fast curves. On the other hand A and B represents the potential energy curves of the system when atoms move slowly and resonance is permitted between them. The minimum vertical separation of the two slow curves is equal to the resonance energy e and is at the crossing point. Let us suppose that a molecule is moving along curve A and its inter atomic distance is changing with velocity. Now we will look for the probability of the molecule to undergo a transition to the curve B after passing through the cross over region. The time required for the system to tautomerize

h from state 1 to state 2 is given by τ = 4e [4]. Let ∆ t be the effective time spent in the cross over region. If τ is much greater than this time , then the probability that a ∆t transition from state A to state B will not occur is approximately τ . The probability of a non adiabatic transition is therefore given by

∆t 4e∆t P = 1 = 1 (1.35) ∼ − τ − h

Next question is how to estimate this time ∆t?. It depend upon the shape of the cross over region which is determined by the energies of the fast curves (1) and (2), we denote them by E1 and E2 and the resonance energy e. Now according to the variational principle [5], we write the wave function as

ψ = aφ1 + bφ2 (1.36)

14 (a) Magniﬁed view of the diabatic and adiabatic curves

(b) Schematic construction for the calculation of transition probability 15 Figure 1.5: Schematic plots of Diabatic and Adiabatic curves where Φ1 and Φ2 are orthogonal wave functions of the two pure structures and a and b are variables. The energy of the slow curves can be written as

1 1 4e2 2 EA = (E1 + E2)+ (E1 E2) 1+ (1.37) 2 2 − (E E )2 1 − 2 1 1 4e2 2 EB = (E1 + E2) (E1 E2) 1+ (1.38) 2 − 2 − (E E )2 1 − 2

In the above equations E1 is the energy of structure Φ1 E1 = Φ1HΦ1dx , E2 is the

energy of the structure Φ2 E2 = Φ2HΦ2dx and e is the exchangeR integral given by

e = Φ1HΦ2dx = Φ2H Φ1dxR.From these equations we conclude that if E1 E2 − ≥ E1 E2 4e thenR E −E 0R.89. If we look at the ﬁgure (1.5) b then the points N and Q A− B ≥ corresponds to the equality sign in condition condition given as

E E 4e. (1.39) 1 − 2 ≥

To the left of point N and to the right of point Q, the slow and the fast curves merge with each other. The line NR is parallel to the fast curve 1 while NS is horizontal and PS is vertical. The time interval which the system seeks to move the distance NS is ∆t,

NS then ∆t = v where v is the rate of change of inter atomic distance. So the probability of a non adiabatic transition now can be rewritten as

NS4e P = 1 (1.40) ∼ − hv Now the distance NS is determined geometrically by the resonance energy and the inclinations or slopes of the two fast curves. Let p1 and p2 be the slopes of curve 1 and curve 2 denoted as dE dE p = 1 and p = 2 (1.41) 1 dr 2 dr

16 with r as inter atomic distance. hence we can further write as

NS = P S/p1 (1.42)

PS = SR + RQ + Qp (1.43)

SR = p2 NS (1.44) × RQ = QP = 4e (1.45) which ﬁnally gives us 8e NS = (1.46) p p | 1 − 2| therefore the non adiabatic transition probability becomes

32e2 P 1 (1.47) ≈ − hv p p | 1 − 2| which is valid for if P is close to unity. The exact calculations by Landau and Zener [6, 7]gives 4π2e2 P = exp − (1.48) hv p p | 1 − 2| which is close to the earlier result for values of P close to unity. Non adiabatic transitions are favored by small resonance energies, high velocities and large differences in the slopes of the intersecting problems. Before we start with the famous models of nona diabatic transitions let us review a few deﬁnitions related to adiabatic and non-adiabatic potential energy curves.

1.6 Important deﬁnitions

1.6.1 Avoided crossing

we know that electron mass is very less in comparison to he nuclear mass so at a ﬁxed nuclear coordinate electronic motion is solved and the so called electronic potential energy curves are obtained as a function of internuclear distance R. This R plays the role of adiabatic parameter and its motion induces a transition between different electronic 17 states when the electronic potential energy curves comes closer to each other. If we

consider two states V1(R) and V2(R) as a function of R which crosses somewhere at

Rxc and are coupled by V(R), then corresponding adiabatic states are obtained by the equation

1 1 E (R)= V (R)+ V (R) (V (R) V (R))2 + 4V (R)2 2 (1.49) 1,2 2 1 2 ± 1 − 2 h i Unless accidentally due to some reasons the coupling V(R)=0 at the crossing point

automatically, then E1(R) and E2(R) come close to each other but never cross on the real axis which is known as avoided crossing.

1.6.2 Diabatic state representation

The states V1(R) and V2(R) are termed as diabatic states while V(R)s termed as diabatic coupling. This representation is called as diabatic state representation.

1.6.3 Adiabatic state representation

The states E1(R) and E2(R) are termed as adiabatic states. This representation is called as adiabatic state representation. The adiabatic states are coupled by non adiabatic

rad coupling denoted as T12 which is given by

∂ ψa ∂Hel ψa T rad = ψa ψa = h 1 | ∂R | 1i (1.50) 12 h 1|∂R| 1i E (R) E (R) 1 − 2

a where ψj , j=1, 2,... are electronic wave functions of the adiabatic states. These states are coupled by the nuclear kinetic energy operator i. e. dependence of adiabatic electronic eigen functions on nuclear coordinate R. The adiabatic state representation is inconvenient in comparison to the diabatic state representation due to the ﬁrst derivative term and also nonadiabatic coupling term i a very sharp function of the coordinate.

18 1.6.4 Adiabaticity or Adiabatic states

The concept of adiabaticity implies that there are two set of variables which are used to describe the system of interest. Out of these two variables the system is well defined by eigen states defined at fixed value of one set of variables which are changing slowly as compared to another set of variables. These slowly varying variables are called as adiabatic parameters and the eigenstates are called as adiabatic states. Adiabaticity breaks down when the adiabatic parameters changes quickly. Transition between the adiabatic states are called as nonadiabatic transitions. The energy required to cause a non adiabatic transition is gained from the nuclear motion so non adiabatic transitions occurs effectively at high energies of nuclear motion. The basic equations which defines the adiabatic states are the time independent Schroedinger equations with respect to electronic coordinates at fixed inter nuclear coordinates, while the basic equations which defines the non adiabatic transitions along the nuclear motions are the coupled time independent Schroedinger equations with respect to inter nuclear coordinates. The coupling is termed as non adiabatic coupling which couples adiabatic electronic states which contain the derivative of electronic wave function with respect to the nuclear coordinate.

1.7 Examples of non adiabatic transitions in different ﬁelds of science

1.7.1 Non adiabatic transitions in Physics

In physics we have numerous examples where non adiabatic transitions plays an important role. In nuclear collisions the nuclear molecular orbtails and transitions between them can be analyzed in terms of non adiabatic transitions [8, 9]. Many dynamic processes on the surface of solids are initiated by the non adiabatic transitions and some of the examples are Molecular desorption from a solid surface [10]. Radiationless transitions in condense matter physics like quenching of F-color center and self-trapping of 19 exciton are another few examples where non adiabatic transitions plays an important role in solid state physics [11]. Transitions among Zeeman and Stark states in an external magnetic or electric field [12], transitions induced by laser [13, 14, 15], tunneling junction and josephon junction in an external electric or magnetic field [16, 17, 18] are some of the examples where we will find time dependent non adiabatic transitions. The non diabatic transitions are called as time dependent non adiabatic transitions which takes place in the presence of a certain applied time dependent external field where transitions are induced by by the change of the field with respect to time. Non adiabatic transitions are also reported in chaotic behavior and soliton like structures [19, 20, 21]. The connections between the two state Landau-Zener system with the quantum devices such as qubits,quantum dots and interferometry has been also of quite interest and importance in the latest days [22, 23]. The most exotic examples of all time is the neutrino conversion in the sun [24, 25] where the non adiabatic transition between different types of neutrinos is determined to judge the existence of finite masses of neutrinos.

1.7.2 Non adiabatic transitions in Chemistry

Various chemical reactions as well as many spectroscopic processes in molecules are induced by non adiabatic transitions due to the crossing of potential energy curves [26, 27, 28, 29, 30]. Many photochemical reactions as well as many organic reactions occurs in many steps which involve the non adiabatic transitions [31, 32]. One of the good example of non adiabatic transitions due to potential energy curve crossing is femto- second dynamics of molecules [33]. With the application of strong laser ﬁeld we can create molecules with dressed states (molecules dressed with photons)and shift up and down the molecular energy levels or potential energy curves by corresponding photon energies so with the help of lasers we can induce the non adiabatic transitions among dressed states [13, 14, 15]. Photo dissociation of bromoacethylchloride at the photon energy can be well understood in terms of non adiabatic transitions [34].

20 1.7.3 Non adiabatic transitions in Biology and Economics

The transfer of electron as well as proton plays an important role in many biological systems which are governed by non adiabatic transitions [35, 36, 37]. As an example photo synthetic reactions which transforms light energy to chemical energy by transfer of electrons or protons are caused through potential energy curve crossings [36]. Many other processes like eyesight, ﬂuorescent protein, and photo synthesis are caused or induced by non adiabatic transitions [38, 39, 40] Nonadiabatic transitions in biological system are natural since many chemical reactions occur in various biological systems and such reactions proceed effectively through non adiabatic transitions. Not only in sciences such kind of transitions also plays a crucial role in different ﬁelds like economics where an interaction between the spot market and future market or an interaction between the monetary supply and money market are quite interesting examples [41].

1.8 Analytical Models of Non Adiabatic Transitions

The interesting area of non adiabatic transitions had its origin in the year 1932 when Landau [42], Zener [43] and Stuckelberg [44] published pioneering papers independently. In the same year Rosen and Zener [45] published another pioneering work in the area of non adiabatic transitions which is known as Rosen-Zener Model. Since these pioneering works there are numerous citations in the literature on the area of non adiabatic transitions which simply reﬂects the importance of curve crossing. In the coming sections we will discuss the famous analytical models of non adiabatic transitions.

1.8.1 Landau-Zener-Stuckelberg Theory

Landau, Zener and Stuckelberg discussed potential energy curve crossing problems and formulated which is now known as Landau-Zener formula and Landau-Zener- Stuckel- berg(LZS) theory. 21 Landau analysis

Landau uses the contour integral method [46] to discuss the potential energy curve crossing problem. The transition probability is given by

P Υ (R)C(R)Υ (R)dR 2 (1.51) '| 1 2 | Z where C(R) represents the coupling between the two states and Υ1 and Υ2 are the nuclear wave functions. Further moving into the complex R plane and using the primitive WKB functions for nuclear wave functions Υ (n = 1, 2) exp[m R k (R)dR], he n ∼ n obtained the following expression R

R ∗ P PLS exp 2Im [k1(R) k2(R)]dR (1.52) ' ≡ − Re(R ) − Z ∗ and 2µ k (R)= [E E (R)] (1.53) n ~2 − n In the above equations µ is the mass of the system, Im and Re indicates the real and imaginary parts respectively. R indicates the complex crossing points of the adiabatic ∗

potentials i.e. E1(R )= E2(R ). The famous Landau-Zener formula given as ∗ ∗

2πA2 P = exp (1.54) LZ −~v ∆F | | can be derived from above by considering the assumptions that adiabatic potentials

En(R) are derived from the linear diabatic potentials with a constant coupling between them and the relative nuclear motion is descried by a straight line trajectory with constant velocity v. In the above formula known as Landau method given by equation (1.54), A is the diabatic coupling, ∆F = F F is the difference of the slopes of 1 − 2 the diabatic potentials. The Landau method is obtained in the diabatic representation however if we employ the adiabatic state representation the coupling C(R) given by

22 equation (1.51) is given by

∂ C(R) T rad , where (1.55) ∝ 12 ∂R ∂ ψa ∂Hel ψa T rad = ψa ψa = h 1| ∂R | 1 i (1.56) 12 h 1 |∂R| 1i E (R) E (R) 1 − 2

a where ψj , j=1, 2,... are electronic wave functions of the adiabatic states.

Zener analysis

Zener [7] uses the time dependent Schroedinger equation in the diabatic representation.

t i 0 d C1(t) 0 V (t)exp ~ (V1 V2)dt C1(t) i~ = − t dt  C (t)   V (t)exp i (V V )dt0 h R0 i   C (t)  2 − ~ 1 − 2 2 (1.57)    h R i    where the total wave function is expanded as

t t i (d) i (d) ψ = C (t)exp V (t0 )dt0 φ + C (t)exp V (t0 )dt0 φ (1.58) 1 −~ 1 1 2 −~ 2 2 Z Z

the coefﬁcients C1(t) and C2(t) depends upon the difference V1(t) - V2(t), which is assumed to be a linear function of time.

V (t) V (t)= ∆F vt (1.59) 1 − 2 | | further if the diabatic coupling is constant then equation (1.58) can be exactly solved in terms of Weber functions and the outcome transition probability is equivalent to equation (1.54). Linearity in time is very much different from linearity in coordinate R and the effects of turning points is completely neglected in former consideration. As time is unidirectionalso linear potential in time means energy changes forever to inﬁnity as time goes and there are no turning points so, in order to express the effect of turning points in the time independent linear potential model quadratic dependence in time is required. In Landau treatment the time dependent linear approximation corresponds to 23 the common classical strait-line trajectory with constant velocity. i. e. the coordinate R is assumed to increase linearly as a function of time, R = vt and is the reason that equation (1.54) presents the exact results for time dependent linear model but in case of time independent linear model it is valid only at collision energies higher than the crossing point. The coupled equations in the adiabatic representation can be written as

˙ d d1 E1(t) θ(t) d1 i~ = (1.60) dt       d2 θ˙(t) E2(t) d2      

In the above equation En(t)(n = 1, 2) are the adiabatic energies deﬁned by the equation (1.49) and θ (t) deﬁnes the diabatic adiabatic transformation matrix given by →

Cos θ(t) Sin θ(t) (1.61)  Sin θ(t) Cos θ(t)  −   with 1 2V (t) θ(t)= arctan (1.62) 2 V (t) V (t) 2 − 1 The non adiabatic coupling is given by

V˙ (t)(V (t) V (t)) V (t)(V˙ (t) V˙ (t)) θ(˙t)= 1 − 2 − 1 − 2 (1.63) (∆E(t))2

Stueckelberg Analysis

Stueckelberg [44, 45, 46] performed a detailed analysis by considering the approximate complex WKB to the fourth order differential equations obtained from original second order coupled Schroedinger equations and taking into account the so called stokes phenomena in the complex R plane associated with the asymptotic solutions. He derived the total inelastic transition probability along with the Landau Zener transition probability given by the following expressions

P 4p (1 p )sin2ρ (1.64) 12 ' LS − LS

24 where PLS is given by equation (1.52) and ρ is

Rx Rx ρ = k (R)dR k (R)dR (1.65) 1 − 2 ZT1 ZT2

In the above equation Tn are the turning points on the adiabatic potential En(R)(n = 1, 2).

Figure 1.6: Schematic diagram representing Landau-Zener type of curve crossing.

Defects in the Landau Zener Transition probability

Some of the drawbacks of Landau-Zener formula may be summarized as follows (a) The LZ formula does not work at energies near and lower than the crossing point. (b) No good formula exists for the transmission when the two diabatic potential crosses with opposite sign of slopes (c) The available formula contain inconvenient complex contour integrals and are not very useful for experimentalists. (c) The LZ formula requires the knowledge of diabatic potentials which uniquely can not be obtained from adiabatic potentials. 25 (e) The accurate phases to deﬁne scattering matrices are not available for all cases.

1.8.2 Rosen-Zener-Demkov Theory

The schematic diagram showing the potential energy curve of the Rosen-Zener-Demkov type is shown in Figure 1.6 [47, 48]. In this model the two diabatic potentials have very weak R-dependence and the diabatic coupling has a strong exponential R dependence. This is an important transition in case when the two potentials are in near resonance asymptotically. The name of Rosen Zener came from their work on the theory of Stern- Gerlach experiment [45] where the basic mathematics corresponds to that of a non adiabatic transition i. e. corresponding potential differenceand the diabatic couplingare constant and are sechyperbolic functions of time where they solved the time dependent Schroedinger equation exactly. In the year 1963 Demkov [48, 49] using the near charge transfer process in the time dependent formalism obtained the same formula as that of Rosen and Zener by assuming the constant energy difference and exponential coupling. The overall inelastic transition probability derived by him is given as

π∆ 2V P sech2 sin2 0 (1.66) 12 ' 2~βv παv

In the above equation ∆ is the constant difference between two potentials, βandV0 are the exponent and pre exponent factors of the diabatic coupling. Although in this model there is no clearly visible avoided crossing but non adiabatic transition occurs locally at Rx = Re(R ), where the adiabatic potentials starts diverging. R is the complex ∗ ∗ crossing point closest to the real axis although there are many complex points but it is wise able to take into account the one close to the real axis. Due to non crossing of the potential curves there is no change in the character of the electronic state. The non adiabatic transition probability is given by

1 π∆ − P 0 = 1+ exp (1.67) RZ ~βv

26 Figure 1.7: Rosen Zener Demkov model of curve cossing

and the total inelastic transition probability is expressed as

P = 4P (1 P )sin2σ = sech2δsin2σ (1.68) 12 ∼ RZ − RZ

LS with σ=σ0 +τ and π √2 1 σLS =( 4 )[2√2+ log − ] (1.69) 0 2~βv √2 + 1

Stoke phase φs do not appear in σ.

1.8.3 Nikitin’s Exponential model

In this model both diabatic potentials as well as diabatic coupling are exponentials and strongly depend on R or time. Nikitin [50] considers the following potentials and cou-

27 pling in the diabatic representation

1 x V (x)= V (x) δ[1 cos2θ e− ] (1.70) 1 0 − 2 − 0 1 x V (x)= V (x)+ δ[1 cos2θ e− ] (1.71) 2 0 2 − 0 1 x v(x)= δsin2θ e− (1.72) 2 0 with (1.73)

x α(R R ) (1.74) ≡ − p

The two diabatic potentials cross at x = ln(cos2θ ) when 0 θ < π and do not X 0 ≤ 0 4 cross when π θ < π . The complex crossing points of the adiabatic potentials are 4 ≤ 0 2 given by x = i( 2θ 2nπ), (n = 0, 1, 2,....) (1.75) c ± 0 ± ± ± Nikitin simpliﬁed the problem by considering the linear trajectory approximation R

= R+vpt,vp being the constant velocity and t is time. Deﬁning the new variables τ = αV t = x and χ = δ , the time-dependentequations as employed by Zener ( Equation p ~αvp 1.57) can be written as

τ dC1 τ τ i~ = χ (χ χ )e− exp i χ (χ 2χ )e− dτ C (1.76) dτ p − p − − p 2 Z0 q τ dC2 τ τ i~ = χ (χ χ )e− exp i χ (χ 2χ )e− dτ C (1.77) dτ p − p − − − p 1 Z0 q and these coupled equations are solved in terms of hypergeometric functions [50]. The transition probability using this model is given by

πχp e− Sinhπ(χ χ ) P = − p (1.78) N sinhπχ one of the nice featue of this model is that both Landau Zener and Rosen Zener transition probabilities can be derived from this model as a special limiting case [51].

28 1.8.4 Non adiabatic Transition due to Coriolis coupling and Dy- namical state representation

This is another class of non adiabatic transitions which are induced by coriolis or rota- tional coupling [52].These transitions shows a different type of character from the radially induced nonadiabatic transitions. These transitions are not localized at the crossing

2 point because the Coriolis coupling is proportional to R− and is dominant at the turning point instead of the crossing point. The theories developed for the curve crossing and the non crossing cases can not be applied directly to this case. However Nakamura introduced a new representation in which Coriolis coupling is diagonal, in that dynamic state representation all theories are applicable. we start with the Hamiltonian H for the Diatomic Molecule given in the body ﬁxed coordinate as

~2 1 ∂ ∂ H = R2 + H + H + H 0 + H (1.79) −2µ R2 ∂R ∂R rot cor el where

~2 ~2 ~2 1 2 2 1 0 1 2 Hrot = J 2κ , Hcor = (L+µ+ + L µ ) , H = L −2µ R2 − −2µ R2 − − −2µ R2 (1.80) J is total angular momentum, R is inter nuclear distance, µ is reduced mass, L ,µ are ∓ ∓ ladder operators.The Schroedinger equation and the Hamiltonian can be transformed as

Hψ = Eψ (1.81)

with (1.82) ~2 2 1 ∂ H = RHR− = + H + H + H 0 + H ,ψ = Rψ (1.83) −2µ ∂R2 rot cor el

The ordinary Born Oppenheimer adiabatic states are deﬁned by the eigen value problem of Hel :. H ψ(a) (r : R κ)= E (R : κ) ψ(a) (r : R κ) (1.84) el n | n n |

Nonadiabatic transitions among theses states are induced by either the Hcor. Transi- tions between the states of different electronic symmetries are induced by the coriolis 29 coupling and have quite different properties from the radially induced transitions. The coriolis coupling is usually not very strong unless the nuclear kinetic energy is very high, but it plays an important role as it couples the states which are not coupled by radial coupling. In spectroscopic problems such type of coupling is called as hetero- geneous perturbation in contrast with the homogeneous perturbation for the radial coupling case. In the radially coupling case the coupling has a pole of order unity at the complex crossing point while in case of coriolis coupling has a pole of order two at R = 0. This suggests that semi-classical theories developed for the radial coupling can not be applied directly to the Coriolis coupling problem however by introducing a new representation termed as dynamical state representation these problems can be treated.

1.8.5 Curve crossing type of Nonadiabatic transition:

In 1932 Landau [6]introduced the concept of analytical continuation of potential surfaces and dynamical variables that characterized the motion of atoms across these surfaces into the complex plane of coordinate and momenta. By this method he calculated the transition probability for near-adiabatic conditions. In the near adiabatic limit, when the non-adiabatic transition probability is exponentially small, he suggested a general method for estimating the exponential factor in the expression for the transition probability by constructing an analytical continuation of classical action integrals into the complex coordinate plane. Until the 1950s, the theory of non adiabatic transitions between electronic states was dominated by Landau-Zener-Stueckelberg model [42, 43, 44].

1.8.6 Nonadiabtic Tunneling type

In the nonadiabatic tunneling(NT) case, the two diabatic potentials cross with different sign of slopes as shown below in the ﬁgure This Linear NT-Case is a unique model. In this type the physical meaning of the matrix elements is quite different. The off-

diagonal elements represents transmission.The overall Transmission probability P12 and the nonadiabatic transition probability P for one passage of crossing point is given 30 Figure 1.8: Nonadiabatic tunneling type curve crossing.

by 2sin2(argU1) P = (1.85) 12 2sin2(argU1) + p2/2[(1 p)] − where U1 is a complex quantity which is given by Inﬁnite series and p has the physical meaning of nonadiabatic transition for one passage of crossing point only at certain energies higher than the bottom of the upper adiabatic potential. Complete reﬂection

P12=0 occurs when argU1= lπ (1.86) is satisﬁed and it suggests an interesting possibility of molecular switching and control of molecular processes.

1.8.7 Diabatically avoided crossing model

Another Interesting non curve crossing case is of diabatically avoided crossing model in which the two diabatic potentials do not intersect and have a constant coupling as 31 shown in the ﬁgure below. Here we have

α x U + Ve | | C V = (1.87)  α x  C U Ve | | −   where U,V,C and α are constants.A constant rotation by Π/4 reduces the above model to Rosen-Zener Model and the adiabtic potentials are given by

ua = U (U + Veα x )2 + C2 (1.88) 1,2 ± | | q Using some parameters and new variables this model can be exactly solvable in terms of Meijer’s G functions.

1.8.8 Exponential Potential Model

It present an Interesting generalization of the Ladau-Zener and Rosen Zener-Models.The diabetic model potentials are given by

αx V (x)= U V e− 11 1 − 1 αx V (x)= U V e− (1.89) 22 2 − 2 αx V (x)= U V e− 12 21 − 1 where

2 V1V2 = V (1.90) with the use of the variable

2m αx Z = (V + V )e− (1.91) −~2α2 1 2 we can transform the original coupled Schrodinger equation into the following fourth order single differential equation

4 d 2 d [ (z b ) z (z a + 1)]ψ(z) = 0 (1.92) dz − n − dz − m n=1 m=1 X X32 and this equation can be solved exactly in terms of Meijer’s G functions.

1.8.9 Linear Potential Model

In the linear potential model the diabetic potentials are linear functions of time and the diabetic coupling is constant which was solved by Zener.This Model works well at higher energies where the turning points are well seperated from the crossing points i.e transistion point,but when the energies become lower and the turning point become come close to the crossing point then this model fails as two crossing points can be treated independently and it corresponds to a two closely lying crossing point problem in the time dependent formulism.In this model the two coupled time dependent Schrodeinger’s equations for a quadratic potential model in which the diabatic potentials are quadaratic functions of time and diabatic coupling is constant.

d c1(t) 1(t) Vo c1(t) i~ = (1.93) dt       c2(t) Vo 2(t) c2(t)       where Vo is a constant diabatic coupling and 1(t),2(t) are quadaratcally time dependent potentials which are given by

2 1(t)= α1t + β1 (1.94) (t)= α (t γ)2 + β 2 2 − 1

Introducing the variables

2V (t) α γ τ = 0 (t + 2 ) ~ α α2 1 − δ(τ)= ατ 2 β (1.95) − α α ~2 3 α = 1 − 2 8Vo and 1 α α2γ2 β = (β β + 1 2 ) (1.96) 2V 2 − 1 α α 0 1 − 2 33 equation 29 can be written as

1 τ d c1(τ) 0 exp[ι ∆(τ)dτ] c1(τ) i~ = 2 τo dt    1 τ    c2(τ) exp[ ι ∆(τ)dτ]R 0 c2(τ) 2 − τo (1.97)    R    now it is clearly seen that this problem can be described completely by two parameters α and β. The parameter α can be assumed to be positive and β correspond to crossing case.The negative case is called as diabatically avoided crossing.Introducing the evolution operator F (z,z0)

c1(z) c1(z0) = F (z,z ) (1.98)   0   c2(z) c2(z0)     the exact solution of the transition matrix can be obtained by converting equation 33 into second order differential equation and it indicates that present time dependent quadratic potential model can be solved in the same way as landau Zener model and transition matrix correspond to reduced scattering matrix in the time independent problem.

1.8.10 Demkov-kunike model

In Demkov model the tanhyp potebtial are coupled by a constant or sechyp interaction.Out of two types of Demkov models ﬁrst one is represented by Hamiltonian matrix,

α(t) V (t) H(t)= (1.99)  V (t) α(t)  −   with t α(t)= a + btanh( ) T (1.100) t V (t)= csech( ) T where a,b,T and c are arbitrary constants. Taking into account that diabatic coupling V(t) is time dependent we obtain differential equation and calculate the transition probability.In the second case of Demkov model the Hamiltonian matrix is replaced by equation 1.100 with V(t)= c and get an hyper geometric equation.In this model the adiabatic 34 potentials are different from the diabatic ones at inﬁnite time and then nonadiabatic transition probability is calculated.

1.8.11 Demkov-Osherov model

This model basically discuss the one dimensional multichannel problems which contain many curve crossings [53]. In this model as shown below one slanted level with slope α crosses with many horizontal levels with the level dependent constant coupling. The subspace formed of the horizontal levels is assumed to be diagonalized. The corresponding Schroedinger equation in this case is

1 d2 + V φ = 0 (1.101) −2µ dx2 with

φ0   φ1    φ2  φ =   (1.102)    .       .       φN     and (1.103)

dx h1 h2 ......   h1 λ1 0 ...... V = (1.104)    h2 0 λ2......       ......      using the complex contour integral method Demkov and Osherov solved this problem exactly and obtain the following results for overall transition probability.

2 2πhn Pn = e− α (1.105)

35 Figure 1.9: Crossing of many horizontal states with a slanted state (Demkov-Osherov model).

36 1.8.12 Zhu-Nakamura theory

On of the biggest draw back of Landau-Zener theory is that it do not works in the cases where the incident energy is small in comparison to the off-diagonal coupling V. After around 60 years of the original Landau’s Zener work Zhu and Nakamura [54] achieved the complete solution of linear curve crossing model and its generalizations. The linear curve crossing model is converted into time independent form where the coupled Schroedinger equations are

~2 d2 F x φ Vφ = Eφ (1.106) −2m dx2 − 1 1 − 2 1 ~2 d2 F x φ Vφ = Eφ (1.107) −2m dx2 − 2 2 − 1 2 which are then solved in momentum representation. The solution is given in the form

dk 2 i k3 φ (x)= eikx A (k)exp (k ) (1.108) j 2π f j f − 3 Z | j| j

2mFj 2mE In the above equation j = 1 or 2, fj = ~2 and = ~2 is the converted force and energy respectively. Zhu-Nakamura theory is based upon the analysis of the Stokes phenomenon. This theory basically composed of many formulas which are dependent upon different physical situations under study. This theory also covers different cases such as low collision energy dynamics where incident energy is much lower than the barrier of the lower adiabatic potential. One great advantage of the Zhu-Nakamura theory is that its transition amplitude can be represented in terms of the parameters related to the adiabatic potentials and incident energy alone.

1.8.13 Our analytical model:

In our analytical model we consider two diabatic curves, crossing each other and there is a coupling between the two curves as shown below, which causes transitions from one curve to another. This transition would occur in the vicinity of the crossing point.

37 Figure 1.10: Dirac Delta coupling model.

38 In particular, it will occur in a narrow range of x, given by

V (x) V (x) V (x ) (1.109) 1 − 2 ' 12 c where x denotes the nuclear coordinate and xc is the crossing point. V1 and V2 are determined by the shape of the diabatic curves and V12 represent the coupling between them. Therefore it is interesting to analyze a model, where coupling is localized in space near xc. Thus we put

V (x)= K δ(x x ) (1.110) 12 0 − c here K0 is a constant. This model has the advantage that it can be exactly solved [56, 57, 58, 59, 60, 61, 62, 63].

1.9 Conclusions

Startingfrom the basic concepts of Clamped nuclei Hamiltonian and Born-Oppenheimer approximation this chapter brieﬂy explains the important terms like adiabatic and non adiabatic potential energy surfaces. Further the concept of curve crossing, avoided crossing are explained by taking an example of simplest diatomic molecule Sodium Chloride. A section is included in the chapter for possible applications of non adiabatic transitions. Different analytical models of non adiabatic transitions are discussed in brief

39 Chapter2 This chapter explains the different methods used in this thesis to study the non adiabatic transitions/Curve crossing problems like Boundary Condition methods, Green’s function approach, Transfer matrix approach, Time dependent approach and computational approach.

40 CHAPTER 2

DIFFERENT APPROACHES USED TO STUDY NON ADIABATIC TRANSITIONS/CURVE CROSSING PROBLEMS

In this chapter we will discuss the various types of approaches for the study of nonadiabatic transitions/Curve crossing problems. These approaches include both analytical as well as computational approaches. The boundary conduction method, Green’s function method, Transfer matrix method , Time dependent methods for the study of non-adiabatic transitions comes under the analytical approach while MRCI and CASPT2 comes under the computational approaches for the study of curve crossing problems. we will discuss these methods in detail as the chapter proceeds.

2.1 Boundary condition method

We will consider the problem of the type as shown below and will derive the two boundary conditions known as continuity of the wave function and discontinuity of the derivative of the wave function across the barrier. we will consider a small displacement η across the barrier and will consider the ﬁrst boundary condition which is the continuity of the wave function across the barrier and may be stated as

φ (x η)= φ (x + η) (2.1) 1 − 2 where φ1 and φ2is the solution of the time independent Schroedinger wave equation in region 1 and region 2 respectively. Similarly second boundary condition can be written as

~2 ∂2 φ x)+ V (x)φ(x)= Eφ(x) (2.2) − 2m ∂x2 ( Figure 2.1: Schematic diagram showing the potential and the barrier

42 integrating the above equation over the small displacement η the above equation can be rewritten as

0+η ~2 ∂2 0+η 0+η (2.3) 2 φ(x)dx + V (x)φ(x)dx = Eφ(x)dx 0 η −2m ∂x 0 η 0 η Z − Z − Z − or ~2 0+η dφ(x) 0+η 0+η + V (x)φ(x)[x]0 η = Eφ(x)[x]0 η (2.4) −2m dx 0 η − − − or ~2 dφ(x) dφ(x) +η η + V (0)φ(0)2η = Eφ(x)2η (2.5) −2m dx | − dx |− since η is a very small variation hence the second term and the term on the right hand side of above equation is equal to 0, hence the second boundary condition ﬁnally can be written as

dφ(x) dφ(x) η = +η (2.6) dx |− dx | or dφ (x) dφ (x) 1 = 2 (2.7) dx dx

2.1.1 Boundary condition method applied to curve crossing problems

We start with a general case where we have a crossing of the two potential energy curves and there is a Dirac Delta coupling between the two curves which will help in the transitions from one curve to another curve. We will derive the boundary conditions for this problem. We start with a particle moving on any of the two diabatic curves and the problem is to calculate the probability of the particle to be still on that diabatic curve after a time t. We write the probability amplitude for the particle as

ψ1(x) Ψ(x)= , (2.8)   ψ2(x)  

43 where ψ1(x,t) and ψ2(x,t) are the probability amplitude for the two states. The Hamil- tonian is given by

H11(x) V12(x) H = , (2.9)   V21(x) H22(x)   where H11(x), H22(x) and V12(x) are deﬁned by

2 H (x)= 1 ∂ + V (x), (2.10) 11 − 2m ∂x2 1 H (x)= 1 ∂2 + V (x) and 22 − 2m ∂x2 2 V (x)= V (x)= K δ(x x ). 12 21 0 − c

The above V1(x) and V2(x) are determined by the shape of that diabatic curve. V (x) is a coupling function which we assume to be a Dirac delta function. The time-independent .. Schrodinger equation is writen in the matrix form

H11(x) V12(x) ψ1(x) ψ1(x) = E . (2.11)       V21(x) H22(x) ψ2(x) ψ2(x)       This is equivalent to

H (x)ψ (x)+ K δ(x x )ψ (x)= Eψ (x) and (2.12) 11 1 0 − c 2 1 K δ(x x )ψ (x)+ H (x)ψ (x)= Eψ (x). 0 − c 1 22 2 2

Integrating the above two equations from x η to x + η (where η 0 and x is the c − c → c crossing point) we get the following two boundary conditions

2 xc+η ~ dψ1(x) + K0ψ2(xc) = 0 and (2.13) − 2m dx xc η − ~2 dψ (x) xc+η 2 + K ψ (x ) = 0. 2m dx 0 1 c − xc η −

44 where K0 is the strength of the Dirac Delta function, also we have two more boundary conditions

ψ (x η)= ψ (x + η) and (2.14) 1 c − 1 c ψ (x η)= ψ (x + η). 2 c − 2 c

Using the above four boundary conditions we derive the transition probability from one diabatic potential to the other

2.2 Green’s function Method.

Green’s functions are named after British Mathematician George Green who developed the concept in 1830. They played a variety of roles in different sciences like in mathematics they are studied largely to acquire fundamental solutions of linear partial differential equations, as a correlation functions in many body theory, and also as a propagators in quantum ﬁeld theory. The basic function in quantum mechanics is denoted by Φ termed as a probability amplitude or wave function associated with every quantum mechanical state. The wave function φ(x,t)is the total amplitude for a particle to arrive at a particular point (x,t)in space and time from the past. φ(x,t) 2 is the prob- | | ability density of ﬁnding the particle at a particular time and position. This wavefuntion is calculated by solving the time dependent Schroedinger equation given as

∂φ(x,t) i~ = Hφ(x,t) (2.15) ∂t

. The Green;s function of the above equation in the time evolution of the wave function can be written as

φ(x00 ,t00 )= dx0 J(x00,x0 ; t00 ,t0 )φ(x0 ,t0) (2.16) Z where J(x00 ,x0 ; t00 ,t0) is the Green’s function which determines the probability amplitude at a ﬁnal point x00 at time t00 in terms of probability amplitude φ(x0 ,t0 ) at an initial

point x0 at time t0 . This Green’s function plays the role of a kernal and is equivalent to 45 the quantum mechanical time evolution operator given as

i J(x00,x0 ; t00 ,t0 )= x00 exp( T H) x0 with T = t00 t0 > 0 (2.17) h | −~ | i −

In other words we can say that the probability P (x00 ,x0 ; t00,t0 ) to go from a point x0 at time t0 to the point x00 at time t00 is the absolute square given by

P (x00 ,x0 ; t00 ,t0 )= j(x00 ,x0 ; t00 ,t0 ) 2 (2.18) | | of an amplitude j(x00 ,x0 ; t00,t0 ) to go from (x0 ,t0 )to (x00 ,t00 ) which means sum of con- tributions from each path φ[x(t)] i. e.

j(x00 ,x0; t00 ,t0 )= φ[x(t)] (2.19) overallpathsfromX x0 to x00

Deﬁning the Fourier transform of the above mentioned kernel as given below

i ∞ T 00 0 [i(E+i) ~ ] 00 0 00 0 G(x ,x ; E)= ~ dT e j(x ,x ; t ,t ) (2.20) Z0 and by comparing equation (2.17) and (2.20) the Green’s function is deﬁned as [64]

1 G(x00 ,x0 ; E)= x00 x0 (2.21) h |(H E i)| i − −

2.2.1 Green’s function method applied to the curve crossing problems

.. We start with the time-independent Schrodinger equation for a two state system

H11(x) V12(x) ψ1(x) ψ1(x) = E . (2.22)       V21(x) H22(x) ψ2(x) ψ2(x)      

46 This equation can be written in the following form

1 ψ (x)= V (x)− [E H (x)] ψ (x) and (2.23) 2 12 − 11 1 1 ψ (x)=[E H (x)]− V (x)ψ (x). 2 − 22 21 1

Eliminating ψ2(x) from the above two equations we get

1 [E H (x)] ψ (x) V (x)[E H (x)]− V (x)ψ (x) = 0. (2.24) − 11 1 − 12 − 22 21 1

The above equations simplify considerably if V12(x) and V21(x) are Dirac Delta function at x , which in operator notation may be written as V = K S = K x x . The c 0 0 | ci h c| above equation now become

2 1 [E H (x)] K x x [E H (x)]− x x ψ (x) = 0. (2.25) − 11 − 0 | cih c| − 22 | cih c| 1 This may be written as

H (x)+ K2δ(x x )G0(x ,x ; E) ψ (x)= Eψ (x). (2.26) 11 0 − c 2 c c 1 1 The above equation can be written in the following form

[E H ] ψ (x)= K2δ(x x )G0(x ,x ; E)ψ (x), (2.27) − 11 1 0 − c 2 c c 1

where the right hand side is considered as an inhomogeneousterm. The general solution of this equation can be written as

2 0 0 ψ1(x)= ψ0(x)+ K0 G1(x,xc; E)G2(xc,xc; E)ψ1(xc), (2.28)

where ψ0(x) is a solution of the homogeneous equation

[E H ] ψ (x) = 0 and (2.29) − 11 0 0 [E H ] G (x,x0; E)= δ(x x0). − 11 1 −

47 In the above equation

∞ ψ1(x)= dx0G(x,x0; E)ψ0(x0) and (2.30) Z −∞ ∞ 0 ψ0(x)= dx0G1(x,x0; E)ψ0(x0). Z −∞

So Eq. (3.98) can be written as

∞ ∞ 0 dx0G(x,x0; E)ψ0(x0)= dx0G1(x,x0; E)ψ0(x0) −∞ −∞ (2.31) R2 0 0 R ∞ +K0 G1(x,xc; E)G2(xc,xc; E) dx0G(xc,x0; E)ψ0(x0). −∞R The solution in terms of Green’s function is as follows

0 2 0 0 G(x,x0; E)= G1(x,x0; E)+ K0 G1(x,xc; E)G2(xc,xc; E)G(xc,x0; E). (2.32)

In the above equation we put x = xc

0 2 0 0 G(xc,x0; E)= G1(xc,x0; E)+ K0 G1(xc,xc; E)G2(xc,xc; E)G(xc,x0; E). (2.33)

After simpliﬁcation, we get

0 G1(xc,x0; E) G(x ,x0; E)= , (2.34) c 1 K2G0(x ,x ; E)G0(x ,x ; E) − 0 1 c c 2 c c so that

2 0 0 0 0 K0 G1(x,xc; E)G2(xc,xc; E)G1(xc,x0; E) G(x,x0; E)= G (x,x0; E)+ . (2.35) 1 1 K2G0(x ,x ; E)G0(x ,x ; E) − 0 1 c c 2 c c In the above expression, we have the Green’s function for two state scattering problem

using delta function coupling model. Using the expression of G(x,x0; E) one can calculate wave function and from the wave function one can easily calculate the transition probability from one diabatic potential to the other.

48 2.3 Transfer matrix method

In early days transfer matrix method is used in optics and acoustics to analyze the propagation of electro magnetic and acoustic waves. Further it was applied to many branches of sciences including quantum mechanics to investigate quantum mechanical tunneling, statistical mechanics as well. This method is used when a total subsystem is broken down into many subsystems which interact with adjacent subsystems. In this section we will use the transfer matrix method to investigate the problem of quantum mechanical tunneling to get an overview of transfer matrix method and in further subsection this is applied to the problem of non adiabatic transitions. we will consider the tunneling thorough a rectangular barrier as shown below we divide our rectangular bar-

Figure 2.2: Tunneling through a rectangular barrier

rier problem into three regions and solve the time independent Schroedinger equation into each of these three regions. The solution of the time-independent Schroedinger equations into these regions may be considered as

ik1x ik1x φ1(x)= Pe + Qe− , (region 1) (2.36)

ik2x ik2x φ2(x)= Re + Se− , (region 2) (2.37)

ik x ik x φ (x)= Te 1 + Ue− 1 , (region 3) (2.38) 3 49 2mE 2m(E V0 where k1 = ~2 and k1 = ~2− . Let us consider this solution for region (2) in the form of theq dot product i.e.q

R ik x ik x ik x ik x φ2(x)= e 2 e− 2 = e 2 e− 2 ψ2 (2.39)  S   

where ψi are the coefficients vectors representing the wave function in each region. In order to find the relation among the coefficients of solutions of the wave function in different regions one require the wave function as well as its derivative to be continuous at the boundaries. Applying this at x=0 we get the following expressions as

P + Q = R + S (2.40)

ik P ik Q = ik R ik S (2.41) 1 − 1 2 − 2

in matrix notations above equations can be rewritten as

1 1 P 1 1 R = (2.42)  ik ik )   Q   ik ik )   S  1 − 1 2 − 2         P R A = B (2.43)  Q   S      P 1 R = A− B (2.44)  Q   S      P 1+ k2 1 k2 R 1 k1 k1 ψ1 = = − = d12ψ2 (2.45)  Q  2  1 k2 1+ k2   S  − k1 k1      

where matrix d12 is known as discontinuity matrix and connects the coefﬁcients of region (1) and region (2). Similarly at point x=a, we consider a new coordinate system such that transitions from region (2) to region (3) takes place at x0 = 0. The relation

between the two coordinates systems is given by x0 = x a. A similar relation as − shown in equation (2.45) can be obtained for the matrix notation which connects the

50 coefﬁcients of region (2) to region (3) given as

0 0 ψ2 = d21ψ3 (2.46)

The new wave function in region 2 must satisfy the equation given by φ (x)= φ0 (x a) 2 2 − i. e.

ik a R0 R0 e− 2 ik (x a) ik (x a) ik x ik x φ2(x)= e 2 − e− 2 − = e 2 e− 2  S0   S0 eik2a     (2.47) comparing equation (2.39) and (2.47) we get

ik a ik a R R0 e− 2 e− 2 0 R0 0 ψ2 = = = = P2ψ2 (2.48)  S   S0 eik2a   0 eik2a   S0         

where p2 is the propagation matrix in region (2). Similarly one can show that

eik1a 0 T 0 ψ3 = = P 1ψ3 (2.49)  ik1a    − 0 e− U    

where P 1ψ3 shifts the wave function back to original coordinate system. Now further − combining equations (2.45), (2.46), (2.48)and(2.49) and setting U = 0 we ﬁnally get

P t11T ψ1 = = d12P2d21P 1ψ3 = tψ3 = (2.50)   −   Q t22T     where t is the transfer matrix which relating the coefﬁcients of region (1) and region (3). In next subsection we will apply the transfer matrix method to the problem of non adiabatic/curve crossing problems.

51 2.3.1 Transfer matrix method applied to the curve crossing problems

In the problem of non adiabatic transition/curve crossing problems we consider that there is a crossing of two diabatic potentials and the coupling which we considered as arbitrary coupling is responsible for the transitions from one curve to another. We express this arbitrary coupling as a collection of Dirac Delta potentials and use the transfer matrix method to correlate the coefﬁcients in between these Dirac Delta potentials. The schematic formulation of our approach is shown as below where we modeled the arbitrary coupling by one Dirac Delta potential, then by two Dirac Delta potentials and so on to obtain a total transfer matrix equation. We will ﬁrst consider a case where we have two diabatic potentials which are coupled by an arbitrary coupling and this arbitrary coupling is modeled by one Dirac Delta potential as shown below. We will now

Figure 2.3: Arbitrary coupling modeled by one Dirac Delta potential

form a transfer matrix equation which will connect the coefﬁcients of the wave function from left side of Dirac Delta potentials to the right hand side of Dirac Delta potentials i.e. from region (1) to region (2). For the lower and the upper diabatic potential the 52 coefﬁcients in region (1) and region (2) are connected to each other by a matrix termed as T1 and the matrix equation can be written as

An+1 M11 M12 M13 M14 An       Bn+1 M21 M22 M23 M24 Bn = (2.51)        Cn+1   M31 M32 M33 M34   Cn               Dn+1   M41 M42 M43 M44   Dn              further we consider that if our arbitrary coupling is modeled by three Dirac Delta potentials as shown below then the matrix which related the coefﬁcients of region 4 with

Figure 2.4: Arbitrary coupling modeled by three Dirac Delta potential

region 3 is given as

An+2 M11 M12 M13 M14 An+1       Bn+2 M21 M22 M23 M24 Bn+1 = (2.52)        Cn+2   M31 M32 M33 M34   Cn+1               Dn+2   M41 M42 M43 M44   Dn+1             

53 using the value of equation (2.52) the above equation can be modeiﬁed as

An+2 M11 M12 M13 M14 M11 M12 M13 M14 An         Bn+2 M21 M22 M23 M24 M21 M22 M23 M24 Bn =          Cn+2   M31 M32 M33 M34   M31 M32 M33 M34   Cn                   Dn+2   M41 M42 M43 M44   M41 M42 M43 M44   Dn                 (2.53) which correlates the coefﬁcient of region (4) with region(1). Hence, if we modeled our arbitrary coupling by n number of Dirac Delta potentials then our matrix equation will look like as

MM = Tn 1Tn 2Tn 3.....T0 (2.54) − − − where MM is the total matrix obtained by multiplying all these transfer matrices.

2.4 TimedependentApproachtothe Nonadiabatictran- sitions/curve crossing problems

In the time dependent approach for non adiabatic transitions/curve crossing problems the problem which involves the partial differential equations in two variables is reduced to a single integral equation in Laplace domain and by knowing the wave fucntion at the point of coupling we can derive the wave function everywhere and if we know the wave function than non adiabatic transition probability from one diabatic potential to another can be easily calculated.

2.4.1 Methodology 1

We start with the case where two time dependent constant potential are coupled to each other by a time independent coupling. The Schroedinger equation in this case can be written as

∂ φ1(x,t) H11 V12 φ1(x,t) i = . (2.55) ∂t       φ2(x,t) V21 H22 φ2(x,t)    54    The above equation is equivalent to the following equation

∂φ (x,t) i 1 = H φ (x,t)+ V φ (x,t) (2.56) ∂t 11 1 12 2 ∂φ (x,t) i 2 = H φ (x,t)+ V φ (x,t) (2.57) ∂t 22 2 21 1

if V12 and V21 is the coupling between the potentials represented by V12 = V21 =

2k0δ(x), then the above equations reduces to

∂φ (x,t) i 1 = H φ (x,t) + 2k δ(x) φ (x,t) (2.58) ∂t 11 1 0 2 ∂φ (x,t) i 2 = H φ (x,t) + 2k δ(x) φ (x,t) (2.59) ∂t 22 2 0 1

taking the Laplace transform of equation (2.58), it can be written as

H φ (x,s) + 2k δ(x)φ (x,s)= is φ (x,s) iφ (x, 0) (2.60) 11 1 0 2 1 − 1

In a similar way equation (2.59) can also be written as

H φ (x,s) + 2k δ(x)φ (x,s)= is φ (x,s) iφ (x, 0) (2.61) 22 2 0 1 2 − 2

we put wave packet at time t = 0 on the ﬁrst potential, hence for second state in our

problem φ2(x, 0) = 0, hence above equation reduced to

H22 φ2(x,s) + 2k0δ(x)φ1(x,s)= is φ2(x,s) (2.62)

Equation (2.62) can be rewritten as

(is H ) φ (x,s) = 2k δ(x)φ (x,s) (2.63) − 22 2 0 1

or 1 φ (x,s)=(is H )− 2k δ(x)φ (x,s) (2.64) 2 − 22 0 1 55 using the value of φ2(x,s) from above equation, equation (2.60) can be rewritten as

H φ (x,s) + 2k2δ(x)G0(0, 0,s)φ (x,s)= is φ (x,s) iφ (x, 0) (2.65) 11 1 0 2 1 1 − 1

0 1 where G (0, 0,s) = δ(x) (is H )− δ(x) is the Green’s function for the second 2 h | − 22 | i state. We will now solve equation(11) by using a method similar to one discussed in Reference [65], hence the above equation at the point of coupling can be further reduced to the following two equation given by

~2 ∂2 = is φ (x,s) iφ (x, 0) (2.66) − 2m ∂x2 1 − 1

∂φ1(x,s) ∂φ1(x,s) 2 0 + = 2k G (0, 0,s)φ (0,s) ∂x |x=0 − ∂x |x=0− 0 2 1

In order to solve equation (12), we should consider the homogeneous solution in order to satisfy the discontinuity condition at the point of coupling. Its solution is given by

1 0 0 0 φ1(x,s)= η(s) exp(i√is x )+ dx exp(i√is x x )φ1(x , 0) (2.67) | | 2√is | − | Z where η(s) is an arbitrary function of s, which needs to be determined. Using the wave function given by equation(13) in equation (12), at the point of coupling we get

√ 2 0 2i is η(s) = 2k0G2(0, 0,s)φ1(0,s) (2.68) or (2.69) k2G0(0, 0,s)φ (0,s) η(s)= 0 2 1 (2.70) i√is with k = b (2.71) 0 − 0 b2G0(0, 0,s)φ (0,s) η(s)= 0 2 1 (2.72) i√is (2.73)

hence, the wave function can be written as

b2G0(0, 0,s)φ (0,s) 1 0 2 1 0 0 0 φ1(x,s)= exp(i√is x )+ dx exp(i√is x x )φ1(x , 0) i√is | | 2√is | − | Z (2.74) 56 at point of coupling i.e. x = 0, the above equation reduces to

b2G0(0, 0,s)φ (0,s) 1 0 2 1 0 0 0 φ1(0,s)= + dx exp(i√is x )φ1(x , 0) (2.75) i√is 2√is | | Z equation(21) is a integral equation in Laplace domain which will determine the wave function at the point of coupling, so if the wave function can be determined at the origin then it can be determined everywhere.

2.4.2 Methodology 2

In this section we consider the exact solution of the time dependent Schroedinger equation involving two potentials coupled by a time dependent Dirac Delta potential.We start with the case where two time dependent constant potential are coupled to each other by a time dependent coupling. The Schroedinger equation in this case can be written as

∂ φ1(x,t) H11 V12 φ1(x,t) i = . (2.76) ∂t       φ2(x,t) V21 H22 φ2(x,t)       The above equation is equivalent to the following

∂φ (x,t) i 1 = H φ (x,t)+ V φ (x,t) (2.77) ∂t 11 1 12 2 ∂φ (x,t) i 2 = H φ (x,t)+ V φ (x,t) ∂t 22 2 21 1

if V12 and V21 is the coupling between the potentials represented by V12 = V21 = 2k(t)δ(x), then the above equations reduces to

∂φ (x,t) i 1 = H φ (x,t) + 2k(t)δ(x) φ (x,t) (2.78) ∂t 11 1 2 ∂φ (x,t) i 2 = H φ (x,t) + 2k(t)δ(x) φ (x,t) ∂t 22 2 1

57 taking the Laplace transform of equation (2), it can be written as

H φ (x,s) + 2L[k(t)δ(x)φ (x,t)] = is φ (x,s) iφ (x, 0) (2.79) 11 1 2 1 − 1

In a similar way equation (3) can also be written as

H φ (x,s) + 2L[k(t)δ(x)φ (x,t)] = is φ (x,s) iφ (x, 0) (2.80) 22 2 1 2 − 2

The above equations can be reduced to a single integral equation and then by varying the time dependent coupling we can get the solution of these equations which can be solved by a method as described in reference [65] and can be used to calculate the wave function at the point of coupling. Once we know the wave function we can calculate the nonadiabatic transition probability further. The different cases for the variation of the coupling are discussed in the coming chapters.

2.5 ComputationalApproach to the Curve CrossingProb- lems

The work present in the coming chapters which included the computational approach to the curve crossing problems include two types of approaches namely Multi state CASPT2 and MRCI level of theory. We present an overview of CASPT2

2.5.1 CASPT2

Using Perturbative methods Hamiltonian can be written as

H = H0 + V (2.81)

where H0 is a zeroth order part and V is perturbation. Also we have

R + S = 1 (2.82) 58 i.e. Hilbert space is partitioned into reference space R and secondary space S. In the original single state CASPT2 method[66] the reference space is one dimensional and is spanned by β such that | i R = β β (2.83) | ih | which is an eigen function of H , H β = Eβ β . The wave operator ζβ operating on 0 0| i 0 | i the zeroth order state β produces the exact state of interest φ is a solution of the Bloch | i equation[67].

Eβ H ζβ β = QV ζβ β Qζβ β β V ζβ β (2.84) 0 − 0 | i | i− | ih | | i h i where β Φ = 1 is the normalization condition. The ﬁrst order equation for the wave h | i operator is given as Eβ H ζβ β = QH β (2.85) 0 − 0 1 | i | i h i with energy through second order is given as

Eβ = Eβ + β Hζβ β (2.86) 2 1 h | | i

The ﬁrst order wave operator ζβ is completely determined by H and β . Single state 1 0 | i CASPT2 chooses this β to be a CASSCF state and zeroth order Hamiltonian for this | i state is given by

Hβ = β β Fˆ β β β + k k Fˆβ k k + Qβ FˆβQβ + Qβ FˆβQβ (2.87) 0 | ih | | ih | | ih | | i × h | sd sd tq.... tq.... Xk

In the above equation Fˆβ is a generalized one body Hartree-Fock operator and sum β over k included all the states within the CAS(complete active space). Qsd is a subspace spanned by states pqrs; β which are produced by applying double excitation operators | i to β . The basis functions pqrs; β are internally contracted states that is given by | i | i pqrs; β = E E β where E are spin-averaged exciton operators i. e. E = | i pq rs| i pq pq

59 σ aqσ† apσ. Further using the following identities P QH β = Qβ H β | i sd | i β β β QsdH0 Qtq = 0 β β β β ˆβ β QsdH0 Qsd = QsdF Qtq (2.88)

equation (2.85) can be written as

Eβ Fˆβ ζβ β = Qβ H β (2.89) 0 − 1 | i sd | i h i β β β where ζ1 = Qsdζ1 CASPT2 method is a general method which uses a generalized Fock operator Fˆβ which is non diagonal and depends upon one particle density matrix of β . Equation (2.89) can be solved for many states. This algorithm is implemented in | i Molpro quantum chemistry package[68] which is used by authors to study the curve crossing in Lithium Chloride molecule.

2.6 Conclusions

Chapter 2 covers all the analytical as well as computational methods used in this thesis for the study of non adiabatic transitions. These methods include different approaches like Boundary condition approach,Green’s function approach, Transfer matrix approach, Time dependent approach and computational approaches for the study of non adiabatic transitions. All the methods are explained one by one with details. Further chapters will make use of all these approaches for the study of non adiabatic transitions.

60 Chapter3 In this chapter we have provided the exact solution of curve crossing problems using Boundary condition method as well as the Green’s function method. These studies include the case of constant, linear and exponential potentials which are solved using the Boundary condition method while the Green’s function method is used to study the curve crossing problem with Gaussian type of coupling.

61 CHAPTER 3

EXACT SOLUTION OF CURVE CROSSING PROBLEMS USING BOUNDARY CONDITION METHOD AND GREEN’S FUNCTION APPROACH.

Nonadiabatic transition due to crossing of the potential energy curves is one of the most fundamental mechanism which is responsible for inducing electronic transitions in collisions[69]. It is a multidisciplinary concept which appears in different ﬁelds of sciences including physics, chemistry and biology [70, 71, 72, 73, 74, 75, 76]. The theory of non-adiabatic transitions started nearly about 1932, when the pioneering works for curve-crossing and non-crossing were published by Landau [42], Zener [43] and Stueckelberg [44] and by Rosen and Zener [45] respectively. Osherov and Voronin solved the case where they took two constant diabatic potentials with an exponential coupling [81]. C. Zhu solved the case where two diabatic potentials are exponential with exponential coupling[77]. In this chapter we will discuss different type of curve crossing problems which are solved by boundary condition as well as Green’s function method using a Dirac Delta coupling model. The Dirac Delta coupling model has the advantage that it can be exactly solved [78, 79, 80, 81] if the uncoupled diabatic potential has an exact solution.

3.1 Exact solution of curve crossingproblemsusing boundary condition method

3.1.1 Two channel scattering problem

We start with a particle moving on any of the two diabatic curves and the problem is to calculate the probability of the particle to be still on that diabatic curve after a time t. We write the probability amplitude for the particle as

ψ1(x) Ψ(x)= , (3.1)   ψ2(x)  

where ψ1(x,t) and ψ2(x,t) are the probability amplitude for the two states. The Hamil- tonian is given by

H11(x) V12(x) H = , (3.2)   V21(x) H22(x)   where H1(x), H2(x) and V12(x) are deﬁned by

H (x)= 1 ∂2 + V (x), (3.3) 11 − 2m ∂x2 1 2 H (x)= 1 ∂ + V (x) and 22 − 2m ∂x2 2 V (x)= V (x)= K δ(x x ). 12 21 0 − c

H11(x) V12(x) ψ1(x) ψ1(x) = E . (3.4)       V21(x) H22(x) ψ2(x) ψ2(x)       This is equivalent to

H (x)ψ (x)+ K δ(x x )ψ (x)= Eψ (x) and (3.5) 11 1 0 − c 2 1 K δ(x x )ψ (x)+ H (x)ψ (x)= Eψ (x). 0 − c 1 22 2 2

Integrating the above two equations from x η to x + η (where η 0)we get the c − c → following two boundary conditions

2 xc+η ~ dψ1(x) + K0ψ2(xc) = 0 and (3.6) − 2m dx xc η − 2 xc+η ~ dψ2(x) + K0ψ1(xc) = 0. −2m dx xc η 63− Also we have two more boundary conditions

ψ (x η)= ψ (x + η) and (3.7) 1 c − 1 c ψ (x η)= ψ (x + η). 2 c − 2 c

Using the above four boundary conditions we can derive the transition probability from one diabatic potential to the other

3.1.2 Three channel scattering problem

We start with a particle moving on any of the three diabatic curves and the problem is to calculate the probability that the particle will still be in that diabatic curve after a time t. We write the probability amplitude for the particle as

ψ1(x)   Ψ(x)= ψ2(x) . (3.8)    ψ3(x)     

Where ψ1(x,t), ψ2(x,t) and ψ3(x) are the probability amplitude for the three states. The Hamiltonian matrix of this system is given by

H11(x) V12(x) V13(x)   H = V21(x) H22(x) 0 , (3.9)    V31(x) 0 H33(x)     

64 where H1(x), H2(x), H3(x), V12(x), V21(x), V31(x) and V13(x) are deﬁned by

1 ∂2 H (x)= + V (x), (3.10) 11 −2m ∂x2 1 1 ∂2 H (x)= + V (x), 22 −2m ∂x2 2 1 ∂2 H (x)= + V (x), 33 −2m ∂x2 3 V (x)= V (x)= K δ(x x ) and 12 21 0 − c V (x)= V (x)= K δ(x x ). 13 31 0 − c

In the above equation V1(x), V2(x) and V3(x) are determined by the shape of that di- .. abatic curve. The time-independent Schrodinger equation for this problem is given by

H (x) K δ(x x ) K δ(x x ) ψ (x) ψ (x) 11 0 − c 0 − c 1 1  K δ(x x ) H (x) 0   ψ (x)  = E  ψ (x)  . 0 − c 22 2 2        K0δ(x xc) 0 H33(x)   ψ3(x)   ψ3(x)   −           (3.11) This matrix representation is equivalent to the following three equations

H (x)ψ (x)+ K δ(x x )ψ (x)+ K δ(x x )ψ (x)= Eψ (x), (3.12) 11 1 0 − c 2 0 − c 3 1 K δ(x x )ψ (x)+ H (x)ψ (x)= Eψ (x) and 0 − c 1 22 2 2 K δ(x x )ψ (x)+ H (x)ψ (x)= Eψ (x). 0 − c 1 33 3 3

Integrating the above three equations from x η to x + η (where η 0) we get the c − c → following three boundary conditions

~2 dψ (x) xc+η 1 + K ψ (x )+ K ψ (x ) = 0, (3.13) 2m dx 0 2 c 0 3 c − xc η − 2 xc+η ~ dψ2(x) + K0ψ1(xc) = 0 and −2m dx xc η − 2 xc+η ~ dψ3(x) + K0ψ1(xc) = 0. −2m dx xc η − 65 Also we have three more boundary conditions

ψ (x η)= ψ (x + η) (3.14) 1 c − 1 c ψ (x η)= ψ (x + η) and 2 c − 2 c ψ (x η)= ψ (x + η). 3 c − 3 c

Using the above six boundary conditions we derive analytical expressions for transition probability from one diabatic potential to the other.

3.1.3 N-channel scattering Problems

3.2 Exact solution of two state curve crossing problems involving different potentials using boundary condition method

3.2.1 Constant potential case

The schematic diagram showing the case of two constant diabatic potentials coupled

by Dirac Delta coupling has been shown in Figure 3.1. In region 1 (x

1 ∂2ψ (x) 1 + V 1(x)ψ (x)= Eψ (x). (3.15) −2m ∂x2 1 1

The above equation has the following solution

ik1x ik1x ψ1(x)= Ae + Be− , (3.16)

66 Figure 3.1: Two constant potential coupled by Dirac Delta coupling

.. where k = 2m (E V 1). In region 2 (x>x ), the time-independent Schrodinger 1 ~2 − c equation forq the ﬁrst potential is given by

1 ∂2ψ (x) 1 = Eψ (x) (x) (3.17) −2m ∂x2 1 1

Physically acceptable solution of the above equation is given by

ik1x ψ1(x)= Ce . (3.18)

.. In region 1 (x

ik2 x ψ2(x)= De− , (3.20)

67 .. where k = 2m (E V ). In region 2 (x>x ), the time-independent Schrodinger 2 ~2 − 2 c equation for theq second potential is given by

1 ∂2ψ (x) 2 + V ψ (x)= Eψ (x). (3.21) −2m ∂x2 2 2 2

Physically acceptable solution is given by

ik2x ψ2(x)= Fe . (3.22)

Here we put xc = 0. Now using four boundary conditions we calculate

F 2 imK k ~2 2 = 0 1 . (3.23) A (k k ~4 + m2K2) 1 2 0

and D 2 imK k ~2 2 = 0 1 . (3.24) A (k k ~4 + m2K2) 1 2 0

So, the transition probability is given by

k imK k ~2 2 T = 2 2 0 1 . (3.25) k (k k ~4 + m2K2) 1 1 2 0

In our numerical calculation we use atomic units so that ~ = 1. In the atomic units, we set V1(x) = 0, V2(x) = 5, K0 = 1.0 and m = 1.0. The result of our calculation is shown in Figure 3.2.

3.2.2 Linear potential case

The schematic diagram showing the case of two Linear diabatic potentials coupled by .. Dirac Delta coupling has been shown in Figure 3.3. The time-independent Schrodinger equation for the case where a linear potential coupled to another linear potential through a Dirac delta interaction is given below (see Figure 3.3).

1 ∂2 2 + p1x K0δ(x xc) ψ1(x) ψ1(x) − 2m ∂x − = E . (3.26)  1 ∂2      K0δ(x xc) 2 p2x ψ2(x) ψ2(x) − − 2m ∂x − 68       Figure 3.2: The plot of transition probability from one constant potential to another constant potential as a function of energy of incident particle (K0 = 1)

69 Figure 3.3: Schematic diagram of the two state problem, where one linear potential is coupled to another linear potential in the diabatic representation.

The Eq. (4.27) can be split into two equations

1 ∂2 + p x ψ (x)+ K δ(x)ψ (x)= Eψ (x) and (3.27) −2m ∂x2 1 1 0 2 1 1 ∂2 p x ψ (x)+ K δ(x)ψ (x)= Eψ (x). −2m ∂x2 − 2 2 0 1 2 .. In our calculation, we use p1 = p2 = 1. The Time-independent Schrodinger equation for the ﬁrst diabatic potential is given below,

1 ∂2 + x ψ (x)= Eψ (x). (3.28) −2m ∂x2 1 1

In region 1 ( x

1 1 ψ (x)=(A + B)A 2 3 ( E + x) + i(A B)B 2 3 ( E + x) . (3.29) 1 i − − i − h i h i

Here Ai[z] and Bi[z] represent the Airy functions. In the above expression A denotes the probability amplitude for motion along the positive direction and B denotes the prob- 70 ability amplitude for motion along the negative direction. The physically acceptable

solution in region 2 ( x>xc), is given by

1 ψ (x)= CA 2 3 ( E + x) . (3.30) 1 i − h i In this region the net ﬂux is zero.

.. The time-independent Schro dinger equation for the second diabatic potential is given below 1 ∂2 x ψ (x)= Eψ (x). (3.31) −2m ∂x2 − 2 2

In region1 ( x

1 ψ (x)= DA 2 3 ( E x) . (3.32) 2 i − − h i

In this region the net ﬂux is zero. In region 2 ( x >xc) the physically acceptable solution is

1 1 ψ (x)= F A 2 3 ( E x) iB 2 3 ( E x) . (3.33) 2 i − − − i − − h i h i

Using the four boundary conditions mentioned above (here we put xc = 0), we have derived an analytical expression for the transition probability from one diabatic potential to the other diabatic potential and the ﬁnal expression is given below

N 2 T = , (3.34) D

where

2 3 3 3 3 3 3 N = 4i√2A √2E A √2E B 0 √2E A 0 √2E B √2E − i − i − i − − i − i − h i h i h i h i h (3.35)i

71 and

4 3 D = 4A √3 2E 8iA √3 2E B √3 2E (3.36) i − − i − i − 2 2 2 2/3 h 3 i h3 i h 3 i +2 A 0 √2E B √2E + A √2E i − i − i − h2 i h 2 i h i 2/3 3 3 5/3 3 2 B 0 √2E 4B √2E 2 A √2E i − − i − − i − h i 2 h i 2 h i 2/3 3 3 5/3 3 2 B 0 √2E 4B √2E 2 A √2E i − − i − − i − h i h i h i

In our numerical calculation we set p1 = 1, p2 = 1, K0 = 1 and m = 1 in atomic units. The result of our calculation is shown in Figure 3.4.

Figure 3.4: The plot of transition probability from one linear potential to another linear potential, as a function of energy of incident particle (K0=1.0).

72 Figure 3.5: Schematic diagram of the two state problem, where one exponential potential is coupled to another exponential potential in the diabatic representation.

Figure 3.6: Transition probability plots from one exponential potential to another as a function of energy of incident particle (K0=1.0).

73 3.2.3 Exponential potential case

The schematic diagram showing the case of two exponential diabatic potentials coupled by Dirac Delta coupling has been shown in Figure 3.5. We start with the time- .. independent Schrodinger equation for a two state system

1 ∂2 ax 2 + V0e K0δ(x xc) ψ1(x) ψ1(x) − 2m ∂x − = E . (3.37)  1 ∂2 ax      K δ(x x ) 2 + V e− ψ (x) ψ (x) 0 − c − 2m ∂x 0 2 2       Eq. (4.45) can be split into the following two equations

1 ∂2 + V eax ψ (x)+ K δ(x x )ψ (x)= Eψ (x) and (3.38) −2m ∂x2 0 1 0 − c 2 1 2 1 ∂ ax + V e− ψ (x)+ K δ(x x )ψ (x)= Eψ (x). −2m ∂x2 0 2 0 − c 1 2 .. In our calculation we took V0 = 1.0 and a = 1. Using the time-independentSchrodinger equation the ﬁrst diabatic potential is given below

1 ∂2 + V eax ψ (x)= Eψ (x). (3.39) −2m ∂x2 0 1 1

In the region x

√2E π √ x √ √2E π √ x √ ψ1(x)= Ae− I(2 i √2E )(2 2e ) Γ(1+2 i 2E )+Be I( 2 i √2E )(2 2e ) Γ(1 2 i 2E ). − − (3.40)

Here In(z) represent the modiﬁed Bessel function of the ﬁrst kind. In the above expression A denotes the probability amplitude for motion along the positive direction and B denotes the probability amplitude for motion along the negative direction. In the region,

where x>xc, the physically acceptable solution is

√ x ψ1(x)= CK( 2 i √2E )(2 2e ). (3.41) −

Here Kn(z) represent the modiﬁed Bessel function of the second kind. In this region the net ﬂux is zero.

74 .. For the second diabatic potential, the time-independent Schrodinger equation is given below 2 1 ∂ ax + V e− ψ (x)= Eψ (x). (3.42) −2m ∂x2 0 2 2

In the region, where x>xc, the physically acceptable solution is

i√2E √ x √ ψ2(x)= F ( 1)− I( 2 i √2E )(2 2e− ) Γ(1 2 i 2E ). (3.43) − − −

In the region, where x

√ x ψ2(x)= DK( 2 i √2E )(2 2e ) . (3.44) −

In this region the net ﬂux is zero. Using four boundary conditions and putting xc = 0 , as mentioned before, we derive an expression for transition probability from one exponential potential to the other exponential potential, and is given by

N 2 T = 1 E , (3.45) − D E

where

1 1 √ √ NE = I 2i√2E 1 2 2 I2i√2E+1 2 2 (3.46) −2√2 − − − 2√2 1 h i 1 h i √ √ I 2i√2E 2 2 + I 2i√2E 1 2 2 2√2 − 2√2 − − h i h i √ √ √ 2 +I 2i√2E 2 2 I2i√2E 2 2 K 2i√2E 2 2 − − h i h i h i and

√ √ DE = I1 2i√2E 2 2 + I 2i√2E 1 2 2 (3.47) − − − 1 h 1i h i √ √ √ I 2i√2E 2 2 K1 2i√2E 2 2 K 2i√2E 1 2 2 − −4 − − 4 − − h i 1 h i 1 h i √ √ 2 √ K 2i√2E 2 2 I 2i√2E 1 2 2 I 2i√2E 1 2 2 − − 8 − − − 4 − − h i h i 1 h i √ √ 2 I1 2i√2E 2 2 I1 2i√2E 2 2 − − 8 − h i h i 75 In our numerical calculation we use atomic unit. The value of K0 = 1 and m = 1. The result of our calculation is shown in Figure 3.6.

3.3 Exact solution of three state curve crossing problems involving different potentials using boundary condition method

The two state scattering or three state scattering models can be mapped to problems like like electron detachments, ionization in slow atomic and ionic collisions, electronic transitions in crystals where one state of a system is interacting with a group of states of differentnature. In this section we will consider the problem whereone state of a system is interacting with another states of differentnature through Dirac delta interaction’s and we provide a simple analytical formula for transition probability from one diabatic state to another state. Our method is applicable to general scattering problems and common applications to our model can be like it is completely valid for the interpretation of associative and dissociative ionization in helium where the transition probabilities can be evaluated quantum mechanically [82].Our model can also be useful in studying the collisional excitation of atoms modeled by the crossing of many valence states [83]. Our model can be useful in studying the pre dissociation in diatomic molecules which involves the crossing of many potential energy curves [84]. Our model can be useful in studying the dynamics of avoided curve crossing between the dissociative state and Rydberg series of a neutral molecule[85]. In further subsections we will discuss one by one different cases involving three state scattering. We consider a particle moving on any of the three diabatic curves and we calculate the probability of the particle on the same diabatic curve after a time t. The probability amplitude for the particle can be written as

ψ1(x,t)   Ψ(x)= ψ2(x,t) . (3.48)    ψ3(x,t)      76 where ψ1(x,t), ψ2(x,t) and ψ3(x,t) are the probability amplitude for the three states. The matrix equation for this system is given by

H11(x) V12(x) V13(x)   H = V21(x) H22(x) 0 , (3.49)    V31(x) 0 H33(x)      where H11(x), H22(x), H33(x), V12(x), V21(x), V31(x) and V13(x) are deﬁned by

~2 ∂2 H (x)= + V (x), (3.50) 11 −2m ∂x2 1 ~2 ∂2 H (x)= + V (x), 22 −2m ∂x2 2 ~2 ∂2 H (x)= + V (x), 33 −2m ∂x2 3 V (x)= V (x)= K δ(x x ) and 12 21 0 − c V (x)= V (x)= K δ(x x ). 13 31 0 − c

In the above equation V1(x), V2(x) and V3(x) are determined by the shape of that dia- .. batic curve. The time-independent Schrodinger equation for this problem will look like as given below

H (x) K δ(x x ) K δ(x x ) ψ (x) ψ (x) 11 0 − c 0 − c 1 1  K δ(x x ) H (x) 0   ψ (x)  = E  ψ (x)  . 0 − c 22 2 2        K0δ(x xc) 0 H33(x)   ψ3(x)   ψ3(x)   −           (3.51) converting this matrix equation into following three equations will give us

H (x)ψ (x)+ K δ(x x )ψ (x)+ K δ(x x )ψ (x)= Eψ (x), (3.52) 11 1 0 − c 2 0 − c 3 1 K δ(x x )ψ (x)+ H (x)ψ (x)= Eψ (x) and 0 − c 1 22 2 2 K δ(x x )ψ (x)+ H (x)ψ (x)= Eψ (x). 0 − c 1 33 3 3

77 Now, we obtain three boundary conditions by integrating the above three equations form x η to x + η (where η 0) and get the following c − c → ~2 dψ (x) xc+η 1 + K ψ (x )+ K ψ (x ) = 0, (3.53) 2m dx 0 2 c 0 3 c − xc η − 2 xc+η ~ dψ2(x) + K0ψ1(xc) = 0 and −2m dx xc η − ~2 dψ (x) xc+η 3 + K ψ (x ) = 0. 2m dx 0 1 c − xc η − In addition to above three boundary conditions, we have three more boundary conditions stated as

ψ (x η)= ψ (x + η) (3.54) 1 c − 1 c ψ (x η)= ψ (x + η) and 2 c − 2 c ψ (x η)= ψ (x + η). 3 c − 3 c

Now, in total we have six boundary conditions. Using these six boundary conditions we have obtained the transition probability from one diabatic potential to the other in the case of coupling between (a) three constant potentials, (b) three linear potentials and (c) three exponential potentials.

3.3.1 Exact analytical solution for constant potential case:

.. The time-independent Schrodinger equation for the ﬁrst potential in the region 1, i.e.

~2 ∂2ψ (x) 1 + V 1(x)ψ (x)= Eψ (x). (3.55) −2m ∂x2 1 1

The solution of the above equation is given by

ik1x ik1x ψ1(x)= Ae + Be− , (3.56)

78 .. where k = 2m(E V ). The time-independent Schrodinger equation for the ﬁrst 1 ~2 − 1 potential in regionq 2, i.e. (x>xc), is given by

~2 ∂2ψ (x) 1 + V 1(x)ψ (x)= Eψ (x). (3.57) −2m ∂x2 1 1

The solution of above equation in region 2, is given by

ik1x ψ1(x)= Ce . (3.58)

.. Similarly, for second potential the time-independent Schrodinger equation in region 1, i.e. (x

~2 ∂2ψ (x) 2 + V (x)ψ (x)= Eψ (x). (3.59) −2m ∂x2 2 2 2

The physically acceptable solution of the above equation is given by

ik2x ψ2(x)= D1e− , (3.60)

.. where k = 2m(E V ). Inregion2, i.e. (x>x ), the time-independent Schrodinger 2 ~2 − 2 c equation forq the second potential is given by

~2 ∂2ψ (x) 2 + V (x)ψ (x)= Eψ (x). (3.61) −2m ∂x2 2 2 2

The physically acceptable solution of the above equation is given by

ik2x ψ2(x)= F1e . (3.62)

.. The time-independent Schrodinger equation for the third potential in region 1, i.e. (x< xc), is given by ~2 ∂2ψ (x) 3 + V (x)ψ (x)= Eψ (x). (3.63) −2m ∂x2 3 3 3

79 The physically acceptable solution of the above equation is given by

ik3x ψ3(x)= D2e− , (3.64)

.. where k = 2m(E V ). Inregion2, i.e. (x>x ), the time-independent Schrodinger 3 ~2 − 3 c equation forq the third potential is given by

~2 ∂2ψ (x) 3 + V (x)ψ (x)= Eψ (x). (3.65) −2m ∂x2 3 3 3

The physically acceptable solution of the above equation is given by

ik3x ψ3(x)= F2e . (3.66)

Now we put xc = 0 and using six boundary conditions mentioned above we have derived analytical expression for transition probabilities from ﬁrst potential to the second and third potential. The transition probability from the ﬁrst potential to the second potential T1 2 is given by →

2 2k2 k1 1 T1 2 = (3.67) ~2 → k1 k2 k1 K0m K0m + ~2 + ~2 K0m k2 k3

The transition probability from the ﬁrst potential to the third potential T1 3 is given by →

2 2k3 k1 1 T1 3 = (3.68) ~2 → k1 k3 k1 K0m K0m + ~2 + ~2 K0m k2 k3

The total transition probability from the ﬁrst potential to both the second potential and the third potential is given by

T = T1 2 + T1 3 (3.69) → →

In our numerical calculation we use atomic unit. V1(x) = 0.0, V2(x) = 2.0 and

V3(x) = 3.0 . So ~ = 1 unit. The value of K0 we took is 1.0 unit. In our calculation m 80 Figure 3.7: Schematic diagram for three constant potential case coupled by Dirac Delta coupling in the diabatic representation.

is taken to be unity. The Schematic diagram for three constant potential case is shown

in Figure 3.7, while the result of our calculation are shown in Figure 3.8. T1 2 and →

T1 3 represents the plot of transition probability form 1st diabatic potential to 2nd and → 3rd diabatic potential respectively.

3.3.2 Exact analytical solution for linear potential case:

.. The time-independent Schrodinger equation for the case where a linear potential is coupled to another two linear potentials through a Dirac delta potential (Schematic diagram is shown in Figure 3.9).

2 2 ~ ∂ + p x K δ(x x ) K δ(x x ) ψ (x) ψ (x) − 2m ∂x2 1 0 − c 0 − c 1 1 2 2  K δ(x x ) ~ ∂ + p x 0   ψ (x)  = E  ψ (x)  . 0 − c − 2m ∂x2 2 2 2  ~2 ∂2       K0δ(x xc) 0 2 + p3x   ψ3(x)   ψ3(x)   − − 2m ∂x           (3.70)

81 (a) T1→2 (b) T1→3

Figure 3.8: (Plot of Transition Probability, Plot Parameters areV1 = 0, V2 = 2, V3 = 3,m = 1, ~ = 1, K0 = 1)

Figure 3.9: Schematic diagram for three linear potential case coupled by Dirac Delta coupling in the diabatic representation.

82 The Eq. (3.70) can be split into three equations

~2 ∂2 + p x ψ (x)+ K δ(x)ψ (x)+ K δ(x)ψ (x)= Eψ (x), (3.71) −2m ∂x2 1 1 0 2 0 3 1 ~2 ∂2 + p x ψ (x)+ K δ(x)ψ (x)= Eψ (x) and −2m ∂x2 2 2 0 1 2 ~2 ∂2 + p x ψ (x)+ K δ(x)ψ (x)= Eψ (x). −2m ∂x2 3 3 0 1 3 .. The time-independent Schrodinger equation for the ﬁrst diabatic potential is given below ~2 ∂2 + p x ψ (x)= Eψ (x). (3.72) −2m ∂x2 1 1 1

In region 1 ( x

1 1 ψ (x)=(A + B)A 2 3 ( E + x) + i(A B)B 2 3 ( E + x) . (3.73) 1 i − − i − h i h i

Here, Ai[z]andBi[z] denotes the Airy functions. In the above expression A denotes the probability amplitude for motion along the positive direction and B denotes the probability amplitude for motion along the negative direction. So the acceptable solution in region 2 ( x>xc), is 1 ψ (x)= CA 2 3 ( E + x) . (3.74) 1 i − h i In this region the net ﬂux is zero.

.. The time-independent Schrodinger equation for the second diabatic potential is given below ~2 ∂2 + p x ψ (x)= Eψ (x). (3.75) −2m ∂x2 2 2 2

In region 1 ( x

1 ψ (x)= D A 2 3 ( E x) . (3.76) 2 1 i − − h i

In this region the net ﬂux is zero. In region 2 ( x >xc) the physically acceptable

83 solution is given below

1 1 ψ (x)= F A 2 3 ( E x) iB 2 3 ( E x) . (3.77) 2 1 i − − − i − − h i h i .. The time-independent Schrodinger equation for the third diabatic potential is given below ~2 ∂2 + p x ψ (x)= Eψ (x). (3.78) −2m ∂x2 3 3 3

In region 1 ( x

ψ (x)= D A [( 0.5 0.866025 i) ( 2 E x)] . (3.79) 3 2 i − − − −

In this region the net ﬂux is zero. In region 2 ( x >xc) the physically acceptable solution is

ψ (x)= F (A [( 0.5 0.866025 i) ( 2 E x)] iB [( 0.5 0.866025 i) ( 2 E x)]) . 3 i − − − − − i − − − − (3.80)

Using the six boundary conditions mentioned above (here we put xc = 0), we calculate the transition probability from one diabatic potential to all the other diabatic potentials. The result of our calculation is shown in Figure 3.10. In our numerical calculation we use atomic unit. So ~ = 1 unit. The value of K0 we took is 1.0 unit. In our calculation m is taken to be unity.T1 2 and T1 3 represents the plot of transition probability form → → 1st diabatic potential to 2nd and 3rd diabatic potential respectively.

3.3.3 Exact analytical solution for exponential potential case

The schematic diagram which represents the one exponential potential coupled to other two exponential potentials in the diabatic representation is shown in Figure 3.11. Here,

84 (a) T1→2 (b) T1→3

Figure 3.10: (Plot of Transition Probability,Plot Parameters are p1 = 0, p2 = 1, p3 = 0.5,m = 1, ~ = 1, K = 1) − − 0

.. we start with the time-independent Schrodinger equation for a three state system

~2 2 ∂ + V ea1x K δ(x x ) K δ(x x ) ψ (x) ψ (x) − 2m ∂x2 0 0 − c 0 − c 1 1 ~2 2  K δ(x x ) ∂ + V ea2x 0   ψ (x)  = E  ψ (x)  . 0 − c − 2m ∂x2 0 2 2  ~2 ∂2 bx       K0δ(x xc) 0 2 + V0e   ψ3(x)   ψ3(x)   − − 2m ∂x          (3.81)  Eq.(3.81) can be split into

~2 ∂2 + V ea1x ψ (x)+ K δ(x x )ψ (x)+ K δ(x x )ψ (x)= Eψ (x(3.82)) −2m ∂x2 0 1 0 − c 2 0 − c 3 1 ~2 ∂2 + V ea2x ψ (x)+ K δ(x x )ψ (x)= Eψ (x) and −2m ∂x2 0 2 0 − c 1 2 ~2 ∂2 + V ebx ψ (x)+ K δ(x x )ψ (x)= Eψ (x). −2m ∂x2 0 3 0 − c 1 3 .. For the ﬁrst diabatic potential, the time-independent Schrodinger equation is given by

~2 ∂2 + V ea1x ψ (x)= Eψ (x). (3.83) −2m ∂x2 0 1 1

85 Figure 3.11: Schematic diagram of the three channel problem, where one exponential potential is coupled to two other exponential potentials in the diabatic representation

86 In the region 1, x

√2E π √ x √ √2E π √ x √ ψ1(x)= Ae− I(2 i √2E )(2 2e ) Γ(1+2 i 2E )+Be I( 2 i √2E )(2 2e ) Γ(1 2 i 2E ). − − (3.84)

where I is the Bessel’s function. In the region 2, where x>xc the physically acceptable solution is given by √ x ψ1(x)= CK( 2 i √2E )(2 2e ). (3.85) − where K is Bessel’s function. For the second diabatic potential, the time-independent .. Schrodinger equation is given by

~2 ∂2 + V ea2x ψ (x)= Eψ (x) (3.86) −2m ∂x2 0 2 2

In the region 1, where x

i√2E √ x √ ψ2(x)= F1 ( 1)− I( 2 i √2E )(2 2e− ) Γ(1 2 i 2E ) (3.87) − − −

In the region 2, where x>xc the physically acceptable solution is given by

√ x ψ2(x)= D1K( 2 i √2E )(2 2e ) (3.88) −

.. For the third diabatic potential, the time-independent Schrodinger equation is given by

~2 ∂2 + ebx ψ (x)= Eψ (x) (3.89) −2m ∂x2 3 3

In the region 1, where x

( 2.82843i)√E √ 0.5x √ ψ3(x)= F2( 1.) − I( 5.65685i√E)(5.65685 2.71828− )Γ 1 (5.65685i) E − − − h (3.90) i

In the region 2, where x>xc the physically acceptable solution is given by

√ x/2 ψ3(x)= D2K( 2i√2E)(2 2e− ). (3.91) −

87 (a) T1→2 (b) T1→3

Figure 3.12: (Plot of Transition Probability, Plot Parameters are a = 1,a = 1,b = 1 2 − 0.5,m = 1, ~ = 1, K0 = 0.1)

using six boundary conditions as mentioned above and putting xc = 0, we have calculated the transition probability from one exponential potential to all other exponential potentials. The result of our calculation is shown in Figure 3.12. In our numerical calculation we use atomic unit. So ~ = 1. The values of K0 we took is 0.1. In our calculation the value of m is taken to be unity.T1 2 and T1 3 represents the plot of transition prob- → → ability form 1st diabatic potential to 2nd and 3rd diabatic potential respectively.

3.4 Exact solution of curve crossingproblems using Green’s function approach

3.4.1 Green’sfunctionapproachforthe two channelscattering problems

.. We start with the time-independent Schrodinger equation for a two state system

H11(x) V12(x) ψ1(x) ψ1(x) = E . (3.92)       V21(x) H22(x) ψ2(x) ψ2(x)    88    This equation can be written in the following form

1 ψ (x)= V (x)− [E H (x)] ψ (x) and (3.93) 2 12 − 11 1 1 ψ (x)=[E H (x)]− V (x)ψ (x). 2 − 22 21 1

Eliminating ψ2(x) from the above two equations we get

1 [E H (x)] ψ (x) V (x)[E H (x)]− V (x)ψ (x) = 0. (3.94) − 11 1 − 12 − 22 21 1

2 1 [E H (x)] K x x [E H (x)]− x x ψ (x) = 0. (3.95) − 11 − 0 | cih c| − 22 | cih c| 1 This may be written as

H (x)+ K2δ(x x )G0(x ,x ; E) ψ (x)= Eψ (x). (3.96) 11 0 − c 2 c c 1 1 The above equation can be written in the following form

[E H ] ψ (x)= K2δ(x x )G0(x ,x ; E)ψ (x), (3.97) − 11 1 0 − c 2 c c 1

where the right hand side is considered as an inhomogeneousterm. The general solution of this equation can be written as

2 0 0 ψ1(x)= ψ0(x)+ K0 G1(x,xc; E)G2(xc,xc; E)ψ1(xc), (3.98)

where ψ0(x) is a solution of the homogeneous equation

[E H ] ψ (x) = 0 and (3.99) − 11 0 0 [E H ] G (x,x0; E)= δ(x x0). − 11 1 −

89 In the above equation

∞ ψ1(x)= dx0G(x,x0; E)ψ0(x0) and (3.100) Z −∞ ∞ 0 ψ0(x)= dx0G1(x,x0; E)ψ0(x0). Z −∞

So Eq. (3.98) can be written as

∞ ∞ 0 dx0G(x,x0; E)ψ0(x0)= dx0G1(x,x0; E)ψ0(x0) −∞ −∞ (3.101) R2 0 0 R ∞ +K0 G1(x,xc; E)G2(xc,xc; E) dx0G(xc,x0; E)ψ0(x0). −∞R The solution in terms of Green’s function is as follows

0 2 0 0 G(x,x0; E)= G1(x,x0; E)+ K0 G1(x,xc; E)G2(xc,xc; E)G(xc,x0; E). (3.102)

In the above equation we put x = xc

0 2 0 0 G(xc,x0; E)= G1(xc,x0; E)+ K0 G1(xc,xc; E)G2(xc,xc; E)G(xc,x0; E). (3.103)

After simpliﬁcation, we get

0 G1(xc,x0; E) G(x ,x0; E)= , (3.104) c 1 K2G0(x ,x ; E)G0(x ,x ; E) − 0 1 c c 2 c c so that

2 0 0 0 0 K0 G1(x,xc; E)G2(xc,xc; E)G1(xc,x0; E) G(x,x0; E)= G (x,x0; E)+ . (3.105) 1 1 K2G0(x ,x ; E)G0(x ,x ; E) − 0 1 c c 2 c c In the above expression, we have the Green’s function for two state scattering problem

90 3.4.2 Green’s function approach for the three channel scattering problems

.. We start with the time-independent Schrodinger equation for a three state system, given by

H11(x) V12(x) V13(x) ψ1(x) ψ1(x)       V21(x) H22(x) 0 ψ2(x) = E ψ2(x) , (3.106)        V31(x) 0 H33(x)   ψ3(x)   ψ3(x)              where

1 ∂2 H (x)= + V (x), (3.107) 11 −2m ∂x2 1 1 ∂2 H (x)= + V (x) and 22 −2m ∂x2 2 1 ∂2 H (x)= + V (x). 33 −2m ∂x2 3

This above matrix equation can be written in the following form

[H (x) E] ψ (x)+ V (x)ψ (x)+ V (x)ψ (x) = 0, (3.108) 11 − 1 12 2 13 3 [H (x) E] ψ (x)+ V (x)ψ (x) = 0 and 22 − 2 21 1 [H (x) E] ψ (x)+ V (x)ψ (x) = 0. 33 − 3 31 1

The above equation after rearranging is given below

[E H (x)] ψ (x) V (x)ψ (x) V (x)ψ (x) = 0, (3.109) − 11 1 − 12 2 − 13 3 1 ψ (x)=[E H (x)]− V (x)ψ (x) and (3.110) 2 − 22 21 1 1 ψ (x)=[E H (x)]− V (x)ψ (x). (3.111) 3 − 33 31 1

91 After eliminating both ψ2(x) and ψ3(x) from Eq. (97) we get

1 1 [E H (x)] V (x)[E H (x)]− V (x) V (x)[E H (x)]− V (x) ψ (x) = 0. − 11 − 12 − 22 21 − 13 − 33 31 1 (3.112)

The above equations are true for any general V12, V21, V13 and V31. The above equation simplify considerably if V12, V13, V31 and V21 are Dirac Delta functions, which we write in operator notation as V = V = K S = K x x and V = V = K S = 12 21 2 2| 2ih 2| 13 31 3 K x x . The above equation now becomes 3| 3ih 3|

[E H (x)] K2δ(x x )G0(x ,x ; E) K2δ(x x )G0(x ,x ; E) ψ (x) = 0. − 11 − 2 − 2 2 2 2 − 3 − 3 2 3 3 1 (3.113) This may be written as

[E H (x)] K2δ(x x )G0(x ,x ; E) ψ (x) = 0, (3.114) − 12 − 3 − 3 3 3 3 1 where H (x)= H (x)+ K2δ(x x )G0(x ,x ; E). (3.115) 12 11 2 − 2 2 2 2

For H12(x), one can ﬁnd the corresponding Green’s function G12(x,x0; E) using the method as we have used in two state case.

[E H ] ψ (x)= K2δ(x x )G0(x ,x ; E)ψ (x), (3.116) − 11 1 2 − 2 2 2 2 1 where the right hand side is considered as an inhomogeneousterm. The general solution of this equation can be written as

∞ 0 2 0 ψ (x)= ψ (x)+ dxG (x,x0; E)K δ(x0 x )G (x ,x ; E)ψ (x0), (3.117) 1 0 1 2 − 2 2 2 2 1 Z −∞ where ψ0(x) is a solution of the homogeneous equation

(E H )ψ (x) = 0, (3.118) − 11 0

92 where

0 (E H )G (x,x0; E)= δ(x x0). (3.119) − 11 1 − So

2 0 0 ψ1(x)= ψ0(x)+ K2 G1(x,x2; E)G2(x2,x2; E)ψ1(x2). (3.120)

In the above expression

∞ ψ1(x)= dx0G12(x,x0; E)ψ0(x0) and (3.121) Z −∞ ∞ 0 ψ0(x)= dx0G1(x,x0; E)ψ0(x0). Z −∞

So Eq. (108) can be written as ∞ dx0G12(x,x0; E)ψ0(x0)= −∞R

∞ ∞ 0 2 0 0 dx0G1(x,x0; E)ψ0(x0)+ K2 G1(x,x2; E)G2(x2,x2; E) dx0G12(x2,x0; E)ψ0(x0). Z Z −∞ −∞ (3.122) The solution in terms of Green’s function, extracted from last equation

0 2 0 0 G12(x,x0; E)= G1(x,x0; E)+ K2 G1(x,x2; E)G2(x2,x2; E)G12(x2,x0; E). (3.123)

In the above equation we put x = x2 to get

0 2 0 0 G12(x2,x0; E)= G1(x2,x0; E)+ K2 G2(x2,x2; E)G1(x2,x2; E)G12(x2,x0; E). (3.124) So after simpliﬁcation we get

0 G1(x2,x0; E) G (x ,x0; E)= , (3.125) 12 2 1 K2G0(x ,x ; E)G0(x ,x ; E) − 2 1 2 2 2 2 2

93 so that

2 0 0 0 0 K2 G1(x,x2; E)G2(x2,x2; E)G1(x2,x0; E) G (x,x0; E)= G (x,x0; E)+ (3.126) 12 1 1 K2G0(x ,x ; E)G0(x ,x ; E) − 2 1 2 2 2 2 2 Now we will incorporate the effect of third state which is coupled to ﬁrst state only, i.e. we will solve Eq. (102) in terms of Greens function.

2 0 K2 G12(x,x3; E)G2(x3,x3; E)G12(x3,x0; E) G (x,x0; E)= G (x,x0; E)+ (3.127) 13 12 1 K2G (x ,x ; E)G0(x ,x ; E) − 2 12 3 3 2 3 3 In the above expression, we have Green’s function for the three-state scattering problem using a delta function coupling model. Using the above expression one can calculate the wave function and from the wave function one can easily calculate the transition probability from one diabatic potential to the other.

3.4.3 Green’s function approach for the N-channel scattering Prob- lems

3.5 Green’s function approach for the curve crossing problem with Gaussian type coupling

In this section we will discuss the case of parabolic potentials coupled by Gaussian coupling which is further extended in calculating the effect of curve crossing on electronic absorption spectra as well as resonance Raman excitation proﬁle. Our model assumes the crossing of two parabolic diabatic potentials and there is a arbitrary coupling between them which is responsible for transitions from one curve to another. This arbitrary coupling is expressed in terms of Dirac Delta potentials having the advantage that it can be exactly solved. Figure 3.13 we have shown schematic potential energy curves that illustrate both the diabatic and adiabatic potentials in the case of a Gaussian type of coupling. An arbitrary coupling V (x) can be written as

94 Figure 3.13: Schematic potential energy curves that illustrate both the diabatic plotted as dashed line and adiabatic potentials plotted as solid line in the case of the Gaussian type of coupling.

∞ V (x)= dx0V (x0)δ(x x0) (3.128) − Z−∞ and the above integral can be discritized as

N V (x)= k δ(x x ), (3.129) j − j j=1 X

here kj are constants, given by

kj = wjV (xj). (3.130)

The weight factor wj varies depending on the scheme of discritization used [86].

We start with a particle moving on any of the two diabatic curves. The problem is to calculate the probability that the particle will still be on any one of the diabatic curves

95 after a time t. We write the probability amplitude as

ψ1(x,t) Ψ(x,t)= , (3.131)   ψ2(x,t)   where ψ1(x,t) and ψ2(x,t) are the probability amplitude for the two states. Ψ(x,t) .. obey the time dependent Schro dinger equation (we take ~ = 1 here and in subsequent calculations) ∂Ψ(x,t) i = HΨ(x,t), (3.132) ∂t where H is deﬁned by

H1(x) V (x) H = , (3.133)   V (x) H2(x)   where Hi(x) is 1 ∂2 H (x)= + V (x). (3.134) i −2m ∂x2 i We ﬁnd it convenient to deﬁne the half Fourier Transform Ψ(E) of Ψ(t) by

∞ Ψ(E)= Ψ(t)eiEtdt. (3.135) Z0 Half Fourier transformation of Eq. (3.132) leads to

1 − ψ (E) E H1 V ψ (0) 1 = i − − 1 . (3.136)  ψ (E)   V E H   ψ (0)  2 − − 2 2       This may be written as Ψ(E)= iG(E)Ψ(0). (3.137)

G(E) is deﬁned by (E H) G(E) = I. In the position representation, the above − equation may be written as

∞ Ψ(x, E)= i G(x,x0; E)Ψ(x0, E)dx0, (3.138) Z−∞

96 where G(x,x0; E) is

1 G(x,x ; E)= x (E H)− x . (3.139) 0 h | − | 0i

Writing

G (x,x0; E) G (x,x0; E) G(x,x ; E)= 11 12 (3.140) 0   G21(x,x0; E) G22(x,x0; E)   and using the partitioning technique [87] we can write

1 1 1 G (x,x ; E)= x [E H V (E H )− V ]− x . (3.141) 11 0 h | − 1 − − 2 | 0i

The above equation is true for any general V . This expression simplify considerably if V is expressed as a sum of delta functions [86, 89]. In that case V may be written as V = N K S = N K x x . Then j=1 j j j=1 j | jih j| P P N 2 0 1 G (x,x ; E)= x [E H K G (x ,x ; E)S ]− x , (3.142) 11 0 h | − 1 − j 2 j j j | 0i j=1 X where

0 1 G (x,x ; E)= x (E H )− x , (3.143) 2 0 h | − 2 | 0i and corresponds to propagation of the particle starting at x0 on the second diabatic curve, in the absence of coupling to the ﬁrst diabatic curve. Now we use the operator identity [89]

N 2 0 1 1 (E H K G (x ,x ; E)S )− =(E H )− + (3.144) − 1 − j 2 j j j − 1 j=1 X N N 1 2 0 2 0 1 (E H )− K G (x ,x ; E)S [E H K G (x ,x ; E)S ]− . − 1 j 2 j j j − 1 − j 2 j j j j=1 j=1 X X

Inserting the resolution of identity I = ∞ dy y y in the second term of the −∞ | i h | R 97 above equation, we arrive at

N 0 2 0 0 G11(x,x0; E)=G1(x,x0; E)+ Kj G1(x,xj; E)G2(xj,xj; E)G11(xj,x0; E). j=1 X (3.145)

Considering the above equation at the discrete points xi, we obtain a set of linear equations, which can be written as

AP = Q, (3.146)

where the elements of the matrices A =[aij], P =[pi] and Q =[qi] are given by

a = k2G0(x ,x ; E)G0(x ,x ; E)+ δ (3.147) ij − i 1 i j 2 j j ij

pi = G11(xi,x0; E)

0 qi = G1(xi,x0; E)

One can solve the matrix equation i.e. Eq. (3.146) easily and obtain G11(xi,x0; E)

for all xi. Eq. (3.145) then yield G11(x,x0; E). Similar one can derive expressions

for G12(x,x0; E), G22(x,x0; E) and G21(x,x0; E). Using these expressions for the Green’s function in Eq. (3.137) we can calculate Ψ(E) explicitly. The expressions that

we have obtained for Ψ(E) are quite general and are valid for any V1(x) and V2(x).

3.5.1 Green’sfunctionapproachtocalculatethe effectofcurve crossing on electronic absorption spectra and Resonance Raman excitation Proﬁle

In this section we apply the method as discussed above to the problem involving harmonic potentials. We consider a system of three potential energy curves, ground electronic state and two ‘crossing’ excited electronic states (electronic transition to one of them is assumed to be dipole forbidden and while it is allowed to the other) [90, 91]. 98 Figure 3.14: Schematic view of diabatic potential energy curves that interpret the model. The forbidden state is denoted as F , while allowed state is denoted by A.

We calculate the effect of‘crossing’ on electronic absorption spectra and on resonance Raman excitation proﬁle. The propagating wave functions on the excited state potential energy curves are given by solution of the time dependent Schrödinger equation

vib vib ∂ ψ (x,t) Hvib,e1(x) V12(x) ψ (x,t) i 1 = 1 . (3.148) ∂t  vib     vib  ψ2 (x,t) V21(x) Hvib,e2(x) ψ2 (x,t)      

In the above equation Hvib,e1(x) and Hvib,e2(x) describes the vibrational motion of the system in the ﬁrst electronic excited state (allowed) and second electronic excited state (forbidden) respectively

1 ∂2 1 H (x)= + mE2 (x a)2 (3.149) vib,e1 −2m ∂x2 2 A − and 1 ∂2 1 H (x)= + mE2 (x b)2. (3.150) vib,e2 −2m ∂x2 2 F − 99 Figure 3.15: Calculated electronic absorption spectra with coupling effect (solid line) and without coupling effect(dashed line). Here the values for the sim- 1 1 ulations are E0 = EA = EF = 400 cm− , Γ = 450 cm− , εA = 1 1 1 2 10700 cm− , εF = 11500 cm− , K0 = 1200 cm− , α = 0.2594 Å− and x = 0.1991Å c −

In the above m is the oscillator’s mass, EA and EF are the vibrational frequencies on the allowed and forbidden states and x is the vibrational coordinate. Shifts of the

vibrational coordinate minimum upon excitation are given by a and b, and V12 (V21) represent coupling between the two harmonic potentials which is taken to be

2 a(x xc) V21(x)= V12(x)= K0e− − , (3.151)

where K0 represent the strength of the coupling. The intensity of electronic absorption spectra is given by [91, 92]

vib IA(E) Re[ ∞ dx ∞ dx0Ψi ∗(x) ∝ −∞ −∞ vib iG(Rx,x0; ER+ iΓ)Ψi (x0)], (3.152)

100 Figure 3.16: Simulated resonance Raman excitation proﬁle for excitation from the ground vibrational state to the ﬁrst excited vibrational state, with coupling effect (solid line) and without coupling effect (dashed line). Here the val- 1 1 ues for the simulations are E0 = EA = EF = 400 cm− , Γ = 450 cm− , 1 1 1 2 εA = 10700 cm− , εF = 11500 cm− , K0 = 1200 cm− , α = 0.2594 Å− and x = 0.1991Å c −

101 where

1 G(x,x ; E + iΓ) = x [(E /2+ E E )+ iΓ H ]− x . (3.153) 0 h | 0 − eg − vib,e | 0i and

Hvib,e1(x) K0 xc xc Hvib,e = | ih | (3.154)  K x x H (x)  0| cih c| vib,e2   Here, Γ is a phenomenological damping constant which account for the life time effects.

vib Ψi (x, 0) is given by

vib χi(x) Ψi (x, 0) = , (3.155)  0    where χi(x) is the ground vibrational state of the ground electronic state, E0 is the

vibrational frequency on the ground electronic state, εA is the energy difference between the excited (allowed) and ground electronic state, and for the forbidden electronic state

it’s value is εF . Similarly resonance Raman scattering intensity can be expressed in terms of Green’s function and is given by [91, 92].

vib IR(E) ∞ dx ∞ dx0Ψf ∗(x, 0) ∝ | −∞ −∞ iGR(x,x ; ER + iΓ)Ψvib(x , 0) 2. (3.156) 0 i 0 |

vib In the above Ψf (x, 0) is given by

vib χf (x) Ψf (x, 0) = , (3.157)  0    0 where χf (x) is the ﬁnal vibrational state of the ground electronic state. As Gi (x,x0; E)

for the harmonic potential is known [64], we can calculate G(x,x0; E). We use Eq. (3.156) to calculate the effect of curve crossing on resonance Raman excitation proﬁle. The results from the above calculations can be interpreted in the following ways. As in [91], the ground state curve is taken to be a harmonic potential energy curve with its minimum at zero. The curve is constructed to be representative of the potential energy along a metal-ligand stretching coordinate. We take the mass as 35.4 amu and 102 1 the vibrational wavenumber as 400 cm− [91] for the ground state. The ﬁrst diabatic excited state potential energy curve is displaced by 0.2 and is taken to have a vibrational

1 wavenumber of 400 cm− . Transition to this state is allowed. The minimum of the

1 potential energy curve is taken to be above 10700 cm− of that of the ground state curve. The second diabatic excited state potential energy curve is taken to be an un- displaced excited state. On that potential energy curve, the vibration is taken to have

1 1 same wavenumber of 400 cm− . Its minimum is 11500 cm− above that of the ground state curve. Transition to this state is assumed to be dipole forbidden. The two diabatic

1 curves cross at an energy of 11765 cm− with x = 0.1991 . Value of K we use c − 0 1 in our calculation is K0 = 1200 cm− and the value of a we use in our calculation is 1 1 a = 0.2594 − . The lifetime of both the excited states are taken to be 450 cm− . The calculated electronic absorption spectra is shown in Figure 3.15. The profile shown by the dashed line is in the absence of any coupling to the second potential energy curve. The full line has the effect of coupling in it. The calculated resonance Raman excitation profile is shown in Figure 3.16. The profile shown by the full line is calculated for the coupled potential energy curves. The profile shown by the dashed line is calculated for the uncoupled potential energy curves. It is seen that curve crossing effect can alter the absorption and Raman excitation profile significantly. However it is the Raman excitation profile that is more effected. Further if we express the Gaussian coupling term by a collection of 100 Dirac Delta functions, it is enough for getting convergence up to eight decimal places in the intensity calculation and the answer was checked for convergence by comparing it with results obtained from considering the case of 200 Dirac Delta functions and also considering the case of 400 Dirac Delta functions.

1 We have also considered the case with lifetime 20 cm− in order to show the spectral proﬁle for vibrational energy levels around the crossing energy. The results of these calculations are shown in Figures 3.17 and 3.18.

103 Figure 3.17: Calculated electronic absorption spectra with coupling effect (solid line) and without coupling effect (dashed line). Here the values for the simula- 1 1 1 tions are E0 = EA = EF = 400 cm− , Γ = 20 cm− , εA = 10700 cm− , 1 1 2 εF = 11500 cm− , K0 = 1200 cm− , α = 0.2594 Å− and xc = 0.1991Å. −

Figure 3.18: Calculated resonance Raman excitation proﬁle for excitation from the ground vibrational state to the ﬁrst excited vibrational state, with coupling effect (solid line) and without coupling effect (dashed line). Here the val- 1 1 ues for the simulations are E0 = EA = EF = 400 cm− , Γ = 20 cm− , 1 1 1 2 εA = 10700 cm− , εF = 11500 cm− , K0 = 1200 cm− ,α = 0.2594 Å− and x = 0.1991Å. c − 104 3.6 Conclusions

Chapter 3 includes the exact solution of different type of curve crossing problems using both Boundary condition as well Greens function approach. Two state , Three state and multi state cases by using Boundary condition approach are discussed and analytical expressions for non adiabatic transition probability are derived. Green’s function approach for two channel, three channel and n channel scattering problems is used to ﬁnd the exact analytical expressions. Further the effect of curve crossing on electronic absorption spectra and Resonance Raman excitation proﬁle are discussed using Green;s function approach.

105 Chapter4 In this chapter we will discuss the transfer matrix method to study the exact solution of curve crossing problems. The already existing different types of transfer matrix methods works only for one state however the one discussed by us in the current chapter will consider the effect of two states coupled by arbitrary coupling. Using this kind of method we have derive exact analytical solutions to different type of curve crossing problems involving the two states.

106 CHAPTER 4

EXACT SOLUTION OF CURVE CROSSING PROBLEMS USING TRANSFER MATRIX METHOD

Various physical problems in interdisciplinary sciences do not allow simple analytical solutions, hence in such cases the Schroedinger equation is solved numerically [93]. The solution of the time-independent Schroedinger equation using numerical methods is of keen interest in early days of quantum mechanics [94] as there exists a limited number of potentials which are analytically solvable [95]. Different kind of numerical methods used to solve Schrodinger equation may includes methods like factorization [96, 97, 98, 99], the path integral [100], eigen value momentummethod [101] and power series expansion method [102] along with many more available in the literature. Out of all the existing methods both numerical and analytical the use of Dirac Delta potential to model different physical problems has its own established importance. Use of Dirac Delta potentials to model potential barriers has a long history when Kronig and Pen- ney [103] modeled the equidistant rectangular barriers by such functions to understand the quantum mechanics of electrons in crystal lattices. Dirac Delta potentials were also used to study the absorption spectra of organic dyes [104]. Existence of surface states is also predicted by the use of such models as reported by Tamm [105]. Saxon and hunter [106] expressed the atomic ﬁelds by Dirac Delta potentials and derived the wave functions and energy levels for mono atomic and diatomic Kronig-Peney models. Frost also reported in his ﬁndings the models of hydrogen like atoms [107], hydrogen-molecule ion [108] and more complex systems [109] by making use of single and multiple Dirac Delta potentials. These citations in the literature for the use of Dirac Delta potentials proved it to be useful as a analytical solvable model in variety of applications involving novel concepts of physics. We have also explored the area of non adiabatic transitions using analytical methods (Dirac Delta coupling model) and reported analytical solution in those cases where two or more arbitrary potentials are coupled by Dirac Delta interactions [55, 56, 57, 58, 59, 60, 61, 62]. This Dirac Delta function coupling model has a plus point that we can get exact analytical solution using such model. In the present chapter we will also discuss a similar kind of approach in which an arbitrary potential is expressed as a sum of Dirac Delta potentials. The present chapter deals with the case in which there is arbitrary coupling between the two different kind of diabatic states and this arbitrary coupling is expressed as a collection of Dirac delta potentials and hence the transition probability form one diabatic state to another state is calculated by using transfer matrix method.

4.1 Transfer matrix method

The main idea behind the transfer matrix method is to express any arbitrary potential as a sum of Dirac Delta potentials i.e.

n αx2 V (x)= e− = K δ(x x ) (4.1) n − n n=1 X In the above equation a potential represented by Gaussian function is expressed by a sum of Dirac Delta functions. xn represents the position of the Dirac Delta function and

Kn represents the strength of the delta function. The schematic diagram representing the problem is given in Figure 4.1. This method is applicable to any kind of coupling where the area under the coupling is represented as a sum of areas of the Dirac Delta potentials used to express the coupling along with the discretization scheme used. The pre-exponential factor for the Gaussian function is 1. The value of kn and xn can be calculated by the use of following equations as given below

a + c a = +(n 1) step (4.2) n 2 − ∗ step an+ 2 kn = V (x)dx. (4.3) step an Z − 2

108 Figure 4.1: Smooth Gaussian Function expressed as a collection of Dirac Delta potentials. Vertical lines represent the Dirac Delta potentials while curve is Gaussian function

a b where step = − and c = a+step with a = minimum of the barrier and b= maximum of Ndiv the barrier

4.1.1 Transfermatrixapproachforexactsolutionto the curve crossing of two constant diabatic potentials

We consider a particle moving on any of the two diabatic curves as shown in Figure 2. The probability amplitude for the particle can be written as

ψ1(x) Ψ(x)= , (4.4)   ψ2(x)  

where ψ1(x) and ψ2(x) are the probability amplitude for the two states. The Hamilto- nian is given by

H11(x) V12(x) H = , (4.5)   V21(x) H22(x)  

109 where H11(x), H22(x) and V12(x) are deﬁned by

2 2 H (x)= ~ ∂ + V (x), (4.6) 11 − 2m ∂x2 1 2 2 H (x)= ~ ∂ + V (x), 22 − 2m ∂x2 2

V12(x)= V21(x)= V (x).

where V1(x) and V2(x) are determined by the shape of the diabatic curve. V (x) is a coupling function which is a Gaussian function as discussed in section 2, and is approx- .. imated as a collection of Dirac Delta potentials. The time-independent Schrodinger equation in the matrix form for the general position of the Dirac delta potential at

x = xn can be written as.

H11(x) V12(x) ψ1(x) ψ1(x) = E . (4.7)       V21(x) H22(x) ψ2(x) ψ2(x)       This is equivalent to

H (x)ψ (x)+ K δ(x x )ψ (x)= Eψ (x) and (4.8) 11 1 n − n 2 1 K δ(x x )ψ (x)+ H (x)ψ (x)= Eψ (x). n − n 1 22 2 2

Integrating the above two equations from x η to x + η (where η 0)we get the n − n → following two boundary conditions

2 xn+η ~ dψ1(x) + Knψ2(xn) = 0 and (4.9) − 2m dx xn η − 2 xn+η ~ dψ2(x) + Knψ1(xn) = 0. −2m dx xn η − Also we have two more boundary conditions

ψ (x η)= ψ (x + η) and (4.10) 1 n − 1 n ψ (x η)= ψ (x + η). 2 n − 2 n

110 Figure 4.2: Two diabatic potentials coupled by arbitrary coupling which is expressed as a collection of Dirac Delta potentials

Using the above four boundary conditions we derive the transition probability from one diabatic potential to the other coupled by arbitrary coupling and this arbitrary coupling is expressed as a sum of Dirac Delta function potentials. The constant potential case where the two diabatic constant potentials coupled to each other through arbitrary coupling is shown in Figure 4.2.

.. In region 1 (x

ik1x ik1x ψ1(x)= Ane + Bne− , (4.12)

.. where k = 2m(E V ). In region 2 (x>x ), the time-independent Schrodinger 1 ~2 − 1 n equation for theq ﬁrst potential is given by

~2 ∂2ψ (x) 1 + V (x)ψ (x)= Eψ (4.13) −2m ∂x2 1 1

111 Physically acceptable solution of the above equation is given by

ik1x ik1x ψ1(x)= A(n+1)e + B(n+1)e− . (4.14)

.. In region 1 (x

ik2x ik2x ψ2(x)= Dne + Fne− , (4.16)

.. where k = 2m(E V ). In region 2 (x>x ), the time-independent Schrodinger 2 ~2 − 2 n equation for theq second potential is given by

~2 ∂2ψ (x) 2 + V ψ (x)= Eψ (x). (4.17) −2m ∂x2 2 2 2

Physically acceptable solution is given by

ik2x ik2x ψ2(x)= D(n+1)e + F(n+1)e− . (4.18)

In considering the smooth Gaussian potential expressed by a collection of Dirac Delta potentials the wave function between two consecutive Dirac Delta potentials is a free particle wave function and is given as

ik1x ik1x ψn(x)= Ane + Bne (4.19) where ψ (x) is the wave functionbetween the nthδ and (n+1)thδ. ~k = 2m(E V ) n 1 − 1 is the wave momentum. At each position of δ functions, the solution of thep Schrodinger equation has to satisfy the boundary conditions given by equations (4.9) and (4.10). These boundary conditions will give us a general (4 4) matrix for location of the δ × function at xn as given below which is termed as transfer matrix Tn. This transfer matrix correlate the coefﬁcient An+1, Bn+1, Dn+1,Fn+1 with the coefﬁcients An, Bn, Dn,Fn 112 (a) Numerical calculations for Non-adiabatic (b) Exact Numerical calculations for Non- transition probability using odd number adiabatic transition probability of Dirac Delta potentials

Figure 4.3: Plots of Non adiabatic Transition Probability for arbitrary coupling expressed by different number of Dirac Delta potentials by the following equation given below

An+1 An     Bn+1 Bn =(T ) (4.20)   n    Dn+1   Dn           Fn+1   Fn          where Tn is known as transfer matrix and is given by

i√2 5a[n] i√2a[n] i√2 5a[n] i√2a[n] 1 0 ie − − k[n] ie− − − k[n] − √2 − √2 2 i√2 5a[n]+i√2a[n] i√2a[n] i√2 5a[n] 3 0 1 ie − k[n] ie − − k[n] 6 √2 √2 7 Tn = 6 √ √ 7 6 iei 2a[n] i√2 5a[n] k[n] ie i√2 5a[n] i 2a[n] k[n] 7 6 − − − − − 1 0 7 6 − √2 5 − √2 5 7 6 −√ −√ 7 6 iei√2 5a[n]+i 2a[n] k[n] iei√2 5a[n] i 2a[n] k[n] 7 6 − − − 0 1 7 6 √2 5 √2 5 7 4 − − (4.21) 5 In terms of wavenumber as well as position of the Dirac Delta function the above transfer matrix can be written as

ia[n]k2() ia[n]k1() ia[n]k1() ia[n]k2() 1 0 ie − k[n] ie− − k[n] − k1() − k1() ia[n]k1()+ia[n]k2() ia[n]k1() ia[n]k2() 0 ie k[n] ie − k[n] 1 0 1 k1() k1() Tn = B ia[n]k1() ia[n]k2() ia[n]k1() ia[n]k2() C B ie − k[n] ie− − k[n] C B 1 0 C B − k2() − k2() C B ia[n]k1()+ia[n]k2() ia[n]k2() ia[n]k1() C B ie k[n] ie − k[n] 0 1 C B k2() k2() C @ (4.22) A In the above equation is the energy, a[n] is the position of the Dirac Delta potential and k[n] represents the strength of Dirac delta function positioned at a[n]. The total 113 (a) Converged Numerical Results for Non adia- (b) Converged Numerical Results for Non adiabatic transition Probability batic transition Probability using exact Numerical calculations

Figure 4.4: Converged Plots for Non adiabatic transition probability

Figure 4.5: Quantitative agreement between the analytical and exact numerical results for 21 number of Dirac Delta Potentials

114 transfer matrix can be considered as the multiplication of all transfer matrices denoted as,

Tn = Tn 1Tn 2Tn 3.....T0. (4.23) − − − The transition probability from one diabatic potential to another diabatic potential coupled by arbitrary potential (Gaussian potential expressed by a sum of Dirac Delta potentials is expressed as

k[n]2 k[n]2 T e 4Im[a[n]k1[]] 2 2 (Non Adiabatic) =1 − 1 (4.24) − − | k[n]2 | −| − k[n]2 | k1[]k2[] 1 + k1[]k2[] 1 + k1[]k2[] k1[]k2[] or in terms of total matrix MM designated as M, the above formula can be written as

and

2 M13M22 M12 (M23M41 M21M43) M M M23 T 2 = M 13 21 + − − (4.26) | |  11 − M M M M M  23 23 42 22 43 −  

In the above equation TNon adiabatic represents the transition probability from one dia- −

batic potential to another diabatic potential and Tn is the total transfer matrix obtained by using equation 23. We calculated the transition probability for this arbitrary coupling as a collection of N Dirac delta functions where N = n can be any number but in the current work n = 1, 3, 5, 7, 9, 13, 15 and 21 before the convergence reaches for 21 number of Dirac Delta functions. Hence our arbitrary coupling is best expressed by a collection of 21 Dirac delta functions. The Plots for transition probability for different number of Dirac Delta functions is represented in Figure 4.3, 4.4 and 4.5 respectively. Figure 4.5 provides a quantitative agreement between the analytical and exact numerical results for the 21 number of Dirac Delta potentials. From Figure 4.5 we can say

115 Figure 4.6: Schematic diagram for the crossing of two Linear potentials with Gaussian coupling (α = 1). that transition probability for 21 number of Dirac Delta potentials almost approaches equals to each other for both analytical as well as exact numerical results with very less deviation hence our arbitrary coupling is best expressed as a collection of 21 number of Dirac Delta potentials.

4.1.2 Transfermatrixapproachforexactsolutionto the curve crossing of two linear diabatic potentials

The schematic diagram representing the problem is shown in Figure 4.6 The time- .. independent Schrodinger equation for the case where a linear potential coupled to another linear potential through an arbitrary coupling is given below and is also shown in Figure 4.7.

1 ∂2 2 + p1x V (x) ψ1(x) ψ1(x) − 2m ∂x = E . (4.27)  V (x) 1 ∂2 p x   ψ (x)   ψ (x)  − 2m ∂x2 − 2 2 2       116 Figure 4.7: Schematic diagram for Gaussian coupling expressed as a collection of Dirac Delta potentials α = 1.

The Eq. (4.27) can be split into two equations

1 ∂2 + p x ψ (x)+ V (x)ψ (x)= Eψ (x) and (4.28) −2m ∂x2 1 1 2 1 1 ∂2 p x ψ (x)+ V (x)ψ (x)= Eψ (x). −2m ∂x2 − 2 2 1 2

In our calculation, we use p = p = 1. Integrating the above two equations from x η 1 2 n − to x + η (where η 0)we get the following two boundary conditions n →

2 xn+η ~ dψ1(x) + V (x)ψ2(xn) = 0 and (4.29) − 2m dx xn η − 2 xn+η ~ dψ2(x) + V (x)ψ1(xn) = 0. −2m dx xn η − Also we have two more boundary conditions

ψ (x η)= ψ (x + η) and (4.30) 1 n − 1 n ψ (x η)= ψ (x + η). 2 n − 2 n

In the schematic view as shown in Figure 4.7, we have expressed our Gaussian coupling (with α = 1) by a collection of Dirac Delta potentials. To derive different expressions for transfer matrices we express this coupling by three Dirac Delta potentials at the be-

117 ginning and then add more number of Dirac Delta potentials. The Dirac Delta Potential

positioned at xn = x2 as shown in ﬁgure 4.7 will correlate the ﬂux from region 2 to region 3. Region 2 is assigned to the left side of Dirac Delta Potential positioned at

xn = x2 while region 3 is assigned to the right side of of Dirac Delta potential posi-

tioned at xn = x2. Using the analytical equations as mentioned above we will calculate

a transfer matrix designated as MM23(4 4) which will connect the ﬂux from region 2 × to region 3 or in other words we can say that this transfer matrix will correlate the co-

efﬁcients An+1 and Bn+1, Dn+1 and Fn+1 with An+2 and Bn+2, Dn+2 and Fn+2 which are discussed in the next section in detail. Similar approach has been used to form the

transfer matrices MM12(4 4)and MM34(4 4) which will correlate the ﬂuxes in between × × respective regions. This will model our arbitrary coupling for three Dirac Delta potentials. Further we add more and more number of Dirac Delta Potentials till convergence value is reached. In the beginning of this section we will start with the formation of

matrix MM23(4 4) which will conclude with the formation of other matrices. For the × ﬁrst linear diabat in region 2, the physically acceptable solution is given below

(A + B ) Ai √3 2(x E) + i (A B ) Bi √3 2(x E) (4.31) n n − n − n −

Here Ai and Bi represent the Airy functions. In the above expression An denotes the probability amplitude for motion along the positive direction and Bn denotes the probability amplitude for motion along the negative direction. The physically acceptable solution in region 3, is given by

(A + B ) Ai √3 2(x E) + i (A B ) Bi √3 2(x E) (4.32) n+1 n+1 − n+1 − n+1 −

For second linear diabat in region2, the physically acceptable solution is

(D + F ) Ai √3 2(x E) + i (D F ) Bi √3 2(x E) (4.33) n n − n − n − In region 3, the physically acceptable solution is

3 3 (Dn+1 + Fn+1) Ai √2(x E) + i (Dn+1 Fn+1) Bi √2(x E) (4.34) − 118 − − Using the four boundary conditions mentioned above in equation (4.29) and (4.30) with x a[n],we arrive at the following equations as mentioned below n →

(A + B ) Ai √3 2(a[n] ep) + i (A B ) Bi √3 2(a[n] ep) (4.35) n n − n − n − =(A + B ) Ai √3 2(a[n] E) +i (A B ) Bi √3 2(a[n] E) n+1 n+1 − n+1 − n+1 −

3 3 3 3 √2(A + B ) Ai0 √2(a[n] E) + √2(A + B ) Ai0 √2(a[n] E)(4.36) − n n − n+1 n+1 − 3 3 3 3 i√2(A B ) Bi0 √2(a[n] E) + i√2(A B ) Bi0 √2(a[n] E) − n − n − n+1 − n+1 − = 2k Ai √3 2(a[n] E) (D + F ) + 2k iBi √3 2(a[n] E) (D F) n − n n n − n − n and

Ai √3 2(a[n] E) (D + F )+ iBi √3 2(a[n] E) (D F )(4.37) − n n − n − n = Ai √3 2(a[n] E) (D + F )+ iBi √3 2(a[n] E) (D F ) − n+1 n+1 − n+1 − n+1

3 3 3 3 √2Ai0 √2(a[n] E) (D + F )+ √2Ai0 √2(a[n] E) (D + F (4.38)) − − n n − n+1 n+1 3 3 3 3 i√2Bi0 √2(a[n] E) (D F )+ i√2Bi0 √2(a[n] E) (D F ) − − n − n − n+1 − n+1 = 2k(A + B ) Ai√3 2(a[n] E) + 2k i (A B ) Bi √3 2(a[n] E) n n n − n n − n − From the above four equations we form a (4 4) matrix as given below which connect × the ﬂux in region 2 to region 3.

M11 M12 M13 M14   M21 M22 M23 M24 MM23(4 4) = (4.39) ×    M31 M32 M33 M34       M41 M42 M43 M44      119 Similarly, (4 4) matrix designated as M and M , relates the ﬂux in region 1 and × 12 34 2 and region 3 and 4 respectively.

M11 M12 M13 M14   M21 M22 M23 M24 MM12(4 4) = (4.40) ×    M31 M32 M33 M34       M41 M42 M43 M44      and

M11 M12 M13 M14   M21 M22 M23 M24 MM34(4 4) = (4.41) ×    M31 M32 M33 M34       M41 M42 M43 M44     

120 The matrix elements of the MM23 matrix may be read as

3 3 3 3 3 3 √2Ai0 √2(a[n] E) Bi √2(a[n] E) √2Ai √2(a[n] E) Bi0 √2(a[n] E) M = − − − − − 11 3 “ 3 ” “ 3 ” “3 ” “3 ” √2 Ai √2(a[n] E) Bi √2(a[n] E) Ai √2(a[n] E) Bi √2(a[n] E) 0 − − − − 0 − “ “ ” “ ” “ ” “ ””

M12 = 0

2 2 ik[n]Ai √3 2(a[n] E) + iBi √3 2(a[n] E) k(n) M = − − 13 “ ” “ ” √3 2 Ai √3 2(a[n] E) Bi √3 2(a[n] E) Ai √3 2(a[n] E) Bi √3 2(a[n] E) 0 − − − − 0 − “ “ ” “ ” “ ” “ ””

2 2 ik[n]Ai √3 2(a[n] E) + 2Bi √3 2(a[n] E) k[n]Ai √3 2(a[n] E) iBi √3 2(a[n] E) k(n) M = − − − − − 14 “ ” “ ” “ ” “ ” √3 2 Ai √3 2(a[n] E) Bi √3 2(a[n] E) Ai √3 2(a[n] E) Bi √3 2(a[n] E) 0 − − − − 0 − “ “ ” “ ” “ ” “ ””

M21 = 0

3 3 3 3 2Ai0 √2(a[n] E) Bi √2(a[n] E) 2Ai √2(a[n] E) Bi0 √2(a[n] E) M = − − − − − 22 “ 3 ” “ 3 ” “ 3 ” “ 3 ” 2 Ai √2(a[n] E) Bi √2(a[n] E) Ai √2(a[n] E) Bi √2(a[n] E) 0 − − − − 0 − “ “ ” “ ” “ ” “ ””

2 2 i22/3 k[n]Ai √3 2(a[n] E) + 2 22/3 Bi √3 2(a[n] E) k[n]Ai √3 2(a[n] E) + i22/3 Bi √3 2(a[n] E) k(n) M = − − − − − 23 “ 3 ” 3“ ” 3 “ ” 3 “ ” 2 Ai √2(a[n] E) Bi √2(a[n] E) Ai √2(a[n] E) Bi √2(a[n] E) 0 − − − − 0 − “ “ ” “ ” “ ” “ ””

2 2 i22/3 k[n]Ai √3 2(a[n] E) i22/3 Bi √3 2(a[n] E) k(n) M = − − − − 24 “ ” “ ” 2 Ai √3 2(a[n] E) Bi √3 2(a[n] E) Ai √3 2(a[n] E) Bi √3 2(a[n] E) 0 − − − − 0 − “ “ ” “ ” “ ” “ ””

2 2 ik[n]Ai √3 2(a[n] E) + iBi √3 2(a[n] E) k(n) M = − − 31 “ ” “ ” √3 2 Ai √3 2(a[n] E) Bi √3 2(a[n] E) Ai √3 2(a[n] E) Bi √3 2(a[n] E) 0 − − − − 0 − “ “ ” “ ” “ ” “ ””

2 2 ik[n]Ai √3 2(a[n] E) + 2Bi √3 2(a[n] E) k[n]Ai √3 2(a[n] E) iBi √3 2(a[n] E) k(n) M = − − − − − 32 “ ” “ ” “ ” “ ” √3 2 Ai √3 2(a[n] E) Bi √3 2(a[n] E) Ai √3 2(a[n] E) Bi √3 2(a[n] E) 0 − − − − 0 − “ “ ” “ ” “ ” “ ””

3 3 3 3 3 3 √2Ai0 √2(a[n] E) Bi √2(a[n] E) √2Ai √2(a[n] E) Bi0 √2(a[n] E) M = − − − − − 33 “ ” “ ” “ ” “ ” √3 2 Ai √3 2(a[n] E) Bi √3 2(a[n] E) Ai √3 2(a[n] E) Bi √3 2(a[n] E) 0 − − − − 0 − “ “ ” “ ” “ ” “ ””

M34 = 0

2 2 i22/3 k[n]Ai √3 2(a[n] E) + 2 22/3 Bi √3 2(a[n] E) k[n]Ai √3 2(a[n] E) + i22/3 Bi √3 2(a[n] E) k(n) M = − − − − − 41 “ 3 ” 3“ ” 3 “ ” 3 “ ” 2 Ai √2(a[n] E) Bi √2(a[n] E) Ai √2(a[n] E) Bi √2(a[n] E) 0 − − − − 0 − “ “ ” “ ” “ ” “ ””

2 2 i22/3 k[n]Ai √3 2(a[n] E) i22/3 Bi √3 2(a[n] E) k(n) M = − − − − 42 3 “ 3 ” 3 “ ” 3 2 Ai √2(a[n] E) Bi √2(a[n] E) Ai √2(a[n] E) Bi √2(a[n] E) 0 − − − − 0 − “ “ ” “ ” “ ” “ ””

M43 = 0

3 3 3 3 2Ai0 √2(a[n] E) Bi √2(a[n] E) 2Ai √2(a[n] E) Bi0 √2(a[n] E) M = − − − − − 44 “ ” “ ” “ ” “ ” 2 Ai √3 2(a[n] E) Bi √3 2(a[n] E) Ai √3 2(a[n] E) Bi √3 2(a[n] E) 0 − − − − 0 − “ “ ” “ 121” “ ” “ ”” The matrix elements of matrix MM12 can be read as

3 3 3 3 3 3 √2Ai0 √2(a[n] E) Bi √2(a[n] E) √2Ai √2(a[n] E) Bi0 √2(a[n] E) M = − − − − − 11 “ ” “ ” “ ” “ ” √3 2 Ai √3 2(a[n] E) Bi √3 2(a[n] E) Ai √3 2(a[n] E) Bi √3 2(a[n] E) 0 − − − − 0 − “ “ ” “ ” “ ” “ ””

M12 = 0

iAi √3 2(a[n] E) k[n] + Bi √3 2(a[n] E) k(n) M = − − 13 “ ” “ ” √3 2 Ai √3 2(a[n] E) Bi √3 2(a[n] E) Ai √3 2(a[n] E) Bi √3 2(a[n] E) 0 − − − − 0 − “ “ ” “ ” “ ” “ ””

2 k[n]Bi √3 2(a[n] E) + iAi √3 2(a[n] E) k[n]Bi √3 2(a[n] E) M = − − − 14 3 3 “ 3” “ 3 ” “ 3 ” √2 Ai √2(a[n] E) Bi √2(a[n] E) Ai √2(a[n] E) Bi √2(a[n] E) 0 − − − − 0 − “ “ ” “ ” “ ” “ ””

M21 = 0

3 3 3 3 2Ai0 √2(a[n] E) Bi √2(a[n] E) 2Ai √2(a[n] E) Bi0 √2(a[n] E) M = − − − − − 22 “ ” “ ” “ ” “ ” 2 Ai √3 2(a[n] E) Bi √3 2(a[n] E) Ai √3 2(a[n] E) Bi √3 2(a[n] E) 0 − − − − 0 − “ “ ” “ ” “ ” “ ””

22/3 Bi √3 2(a[n] E) k[n] i22/3 Ai √3 2(a[n] E) k(n) M = − − − 23 3 “ 3 ” 3 “ ” 3 2 Ai √2(a[n] E) Bi √2(a[n] E) Ai √2(a[n] E) Bi √2(a[n] E) 0 − − − − 0 − “ “ ” “ ” “ ” “ ””

2 22/3Bi √3 2(a[n] E) k[n] i22/3 Ai √3 2(a[n] E) Bi √3 2(a[n] E) k(n) M − − − − 24 = “ ” “ ” “ ” 2 Ai √3 2(a[n] E) Bi √3 2(a[n] E) Ai √3 2(a[n] E) Bi √3 2(a[n] E) 0 − − − − 0 − “ “ ” “ ” “ ” “ ””

2 2 2k[n]Ai √3 2(a[n] E) 2Bi √3 2(a[n] E) k(n) M = − − − − 31 “ ” “ ” 2i √3 2Ai √3 2(a[n] E) Bi √3 2(a[n] E) 2i √3 2Ai √3 2(a[n] E) Bi √3 2(a[n] E) 0 − − − − 0 − “ ” “ ” “ ” “ ”

2 2 2k[n]Ai √3 2(a[n] E) + 4iBi √3 2(a[n] E) k[n]Ai √3 2(a[n] E) + 2Bi √3 2(a[n] E) k(n) M = − − − − − 32 3 “ 3 ” “3 ” 3 “ 3 ” “ 3 ” 2i √2Ai √2(a[n] E) Bi √2(a[n] E) 2i √2Ai √2(a[n] E) Bi √2(a[n] E) 0 − − − − 0 − “ ” “ ” “ ” “ ”

3 3 3 3 √2Ai0 √2(a[n] E) i √2Bi0 √2(a[n] E) M = − − − 33 “ ” “ ” 2i √3 2Ai √3 2(a[n] E) Bi √3 2(a[n] E) 2i √3 2Ai √3 2(a[n] E) Bi √3 2(a[n] E) 0 − − − − 0 − “ ” “ ” “ ” “ ”

3 3 3 3 3 3 √2Ai0 √2(a[n] E) Bi √2(a[n] E) √2Ai √2(a[n] E) Bi0 √2(a[n] E) M = − − − − − 34 “ ” “ ” “ ” “ ” 2i √3 2Ai √3 2(a[n] E) Bi √3 2(a[n] E) 2i √3 2Ai √3 2(a[n] E) Bi √3 2(a[n] E) 0 − − − − 0 − “ ” “ ” “ ” “ ”

2 2 2ik[n]Ai √3 2(a[n] E) + 4Bi √3 2(a[n] E) k[n]Ai √3 2(a[n] E) + 2iBi √3 2(a[n] E) k(n) M = − − − − − 41 3“ 3 ” “ 3 ” “ 3 ” 3“ ” 2 √2 Ai √2(a[n] E) Bi √2(a[n] E) Ai √2(a[n] E) Bi √2(a[n] E) 0 − − − − 0 − “ “ ” “ ” “ ” “ ””

2 2 2ik[n]Ai √3 2(a[n] E) 2iBi √3 2(a[n] E) k(n) M = − − − − 42 “ ” “ ” 2 √3 2 Ai √3 2(a[n] E) Bi √3 2(a[n] E) Ai √3 2(a[n] E) Bi √3 2(a[n] E) 0 − − − − 0 − “ “ ” “ ” “ ” “ ””

3 3 3 3 i √2Ai0 √2(a[n] E) √2Bi0 √2(a[n] E) M = − − − 43 “ ” “ ” 2 √3 2 Ai √3 2(a[n] E) Bi √3 2(a[n] E) Ai √3 2(a[n] E) Bi √3 2(a[n] E) 0 − − − − 0 − “ “ ” “ ” “ ” “ ””

3 3 3 3 3 3 i √2Ai0 √2(a[n] E) Bi √2(a[n] E) i √2Ai √2(a[n] E) Bi0 √2(a[n] E) M − − − − − 44 = “ ” “ ” “ ” “ ” 2 √3 2 Ai √3 2(a[n] E) Bi √3 2(a[n] E) Ai √3 2(a[n] E) Bi √3 2(a[n] E) 0 − − − − 0 − “ “ ” “ ” “ ” “ ””

122 and the matrix elements of matrix MM34 can be read as

3 3 3 3 3 3 3 3 √2Ai0 √2(a[n] E) + i √2Bi0 √2(a[n] E) √2Ai0 √2(a[n] E) i √2Bi0 √2(a[n] E) M = − − ,M = − − − 11 “ ” “ ” 12 “ ” “ ” √3 2Ai √3 2(a[n] E) √3 2Ai √3 2(a[n] E) 0 − 0 − “ ” “ ” 2Ai √3 2(a[n] E) k[n] + 2iBi √3 2(a[n] E) k(n) 2Ai √3 2(a[n] E) k[n] 2iBi √3 2(a[n] E) k(n) M = − − ,M = − − − 13 “ ” “ ” 14 “ ” “ ” √3 2Ai √3 2(a[n] E) √3 2Ai √3 2(a[n] E) 0 − 0 − “ ” “ ” iAi √3 2(a[n] E) Bi √3 2(a[n] E) 3 0 M = iBi √2(a[n] E) − − 21 “ 3 ” “ ” − − Ai √2(a[n] E) “ ” 0 − “ ” iAi √3 2(a[n] E) Bi √3 2(a[n] E) 0 3 M = − − iBi √2(a[n] E) 22 “ 3 ” “ ” Ai √2(a[n] E) − − 0 − “ ” “ 2 ” 22/3 k[n]Ai √3 2(a[n] E) i22/3 Bi √3 2(a[n] E) k[n]Ai √3 2(a[n] E) M − − − 23 = “ ” “ ” “ ” − Ai √3 2(a[n] E) − Ai √3 2(a[n] E) 0 − 0 − “ ” “ ” 2 i22/3 Ai √3 2(a[n] E) Bi √3 2(a[n] E) k(n) 22/3 Ai √3 2(a[n] E) k(n) M = − − − 24 “ ” “ ” “ ” Ai √3 2(a[n] E) − Ai √3 2(a[n] E) 0 − 0 − “ ” “ ” 2Ai √3 2(a[n] E) k[n] + 2iBi √3 2(a[n] E) k(n) 2Ai √3 2(a[n] E) k[n] 2iBi √3 2(a[n] E) k(n) M = − − ,M = − − − 31 “3 2E 2”a(n) 3 “ 2E 2a(n”) 32 “3 2E 2”a(n) 3 “ 2E 2a(n”) i √2Bi0 − − √2Ai0 − − i √2Bi0 − − √2Ai0 − − 22/3 − 22/3 22/3 − 22/3 “ ” “ ” “ ” “ ” 3 3 3 3 3 3 3 3 √2Ai0 √2(a[n] E) + i √2Bi0 √2(a[n] E) √2Ai0 √2(a[n] E) i √2Bi0 √2(a[n] E) M = − − ,M = − − − 33 3 “ 2E 2a(n)” 3 “ 2E 2a(n) ” 34 3 “ 2E 2a(n)” 3 “ 2E 2a(n) ” i √2Bi0 − − √2Ai0 − − i √2Bi0 − − √2Ai0 − − 22/3 − 22/3 22/3 − 22/3 “ ” “ ” “ ” “ ” 2E 2a(n) 3 3 2E 2a(n) 2Ai − − Ai √2(a[n] E) k(n) 2iAi √2(a[n] E) Bi − − k(n) 2/3 2/3 M = 2 − + − 2 41 − 3 “ 2E 2a(n”) “ 3 2E ”2a(n) 3 “ 2E 2a(n) ” “3 2E ”2a(n) i √2Bi0 − − √2Ai0 − − i √2Bi0 − − √2Ai0 − − 22/3 − 22/3 22/3 − 22/3 “ ” “ ” “ ” “ ” 2E 2a(n) 3 2E 2a(n) 3 2iAi − − Bi √2(a[n] E) k(n) 2Bi − − Bi √2(a[n] E) k(n) 22/3 − 22/3 − − 3 “ 2E 2a(n”) “ 3 2E ”2a(n) − 3 “ 2E 2a(n”) “ 3 2E ”2a(n) i √2Bi0 − − √2Ai0 − − i √2Bi0 − − √2Ai0 − − 22/3 − 22/3 22/3 − 22/3 “ ” “ ” “ ” “ ” Ai 2E 2a(n) Ai √3 a n E k n iAi √3 a n E Bi 2E 2a(n) k n 2 − −2/3 2( [ ] ) ( ) 2 2( [ ] ) − −2/3 ( ) M = 2 − + − 2 42 − 3 “ 2E 2a(n”) “ 3 2E ”2a(n) 3 “ 2E 2a(n) ” “3 2E ”2a(n) i √2Bi0 − − √2Ai0 − − i √2Bi0 − − √2Ai0 − − 22/3 − 22/3 22/3 − 22/3 “ ” “ ” “ ” “ ” 2E 2a(n) 3 2E 2a(n) 3 iAi − − Bi √ a n E k n Bi − − Bi √ a n E k n 2 2/3 2( [ ] ) ( ) 2 2/3 2( [ ] ) ( ) + 2 − + 2 − 3 “ 2E 2a(n”) “ 3 2E ”2a(n) 3 “ 2E 2a(n”) “ 3 2E ”2a(n) i √2Bi0 − − √2Ai0 − − i √2Bi0 − − √2Ai0 − − 22/3 − 22/3 22/3 − 22/3 “ ” “ ” “ ” “ ” 3 2E 2a(n) 3 i √2Ai − − Bi √2(a[n] E) 2/3 0 M = Ai √3 2(a[n] E) + iBi √3 2(a[n] E) 2 − 43 − − − 3 “ 2E 2a(n) ” 3 “ 2E 2a(n”) i √2Bi0 − − √2Ai0 − − “ ” “ ” 22/3 − 22/3 “ ” “ ” 3 2E 2a(n) 3 3 2E 2a(n) 3 √2Bi − − Bi0 √2(a[n] E) √2Ai − − Ai0 √2(a[n] E) 22/3 − 22/3 − − 3 “ 2E 2a(n) ” 3 “ 2E 2a(”n) − 3 “ 2E 2a(n) ” 3 “ 2E 2a(”n) i √2Bi0 − − √2Ai0 − − i √2Bi0 − − √2Ai0 − − 22/3 − 22/3 22/3 − 22/3 “ ” “ ” “ ” “ ” 3 3 2E 2a(n) i √2Ai √2(a[n] E) Bi − − 0 2/3 + − 2 3 “2E 2a(n) ”3 “ 2E 2a(n”) i √2Bi0 − − √2Ai0 − − 22/3 − 22/3 “ ” “ ” 3 2E 2a(n) 3 i √2Ai − − Bi0 √2(a[n] E) 3 3 2/3 M = Ai √2(a[n] E) iBi √2(a[n] E) + 2 − 44 − − − 3 “ 2E 2a(n) ” 3 “ 2E 2a(n”) i √2Bi0 − − √2Ai0 − − “ ” “ ” 22/3 − 22/3 “ ” “ ” 3 2E 2a(n) 3 3 2E 2a(n) 3 √ Bi − − Bi √ a n E √ Ai − − Ai √ a n E 2 2/3 0 2( [ ] ) 2 2/3 0 2( [ ] ) + 2 − 2 − 3 “ 2E 2a(n) ” 3 “ 2E 2a(”n) − 3 “ 2E 2a(n) ” 3 “ 2E 2a(”n) i √2Bi0 − − √2Ai0 − − i √2Bi0 − − √2Ai0 − − 22/3 − 22/3 22/3 − 22/3 “ ” “ ” “ ” “ ” 3 3 2E 2a(n) i √ Ai √ a n E Bi − − 2 0 2( [ ] ) 2/3 + − 2 3 “2E 2a(n) ”3 “ 2E 2a(n”) i √2Bi0 − − √2Ai0 − − 22/3 − 22/3 “ ” “ ”

Now the total ﬂux across our arbitrary coupling expressed by a collection of Dirac delta Potentials is obtained by multiplying these matrices M12, M23, M34 and obtaining

123 Figure 4.8: Plot of transition Probability from 1st linear diabat to second linear as a function of energy of incident particle with arbitrary coupling expressed by one Dirac Delta function having numerical values as (m = 1, ~ = 1, k[n]= 1.76,a[n] = 0, p = 1, p = 1) 1 2 −

124 Figure 4.9: Plot of transition Probability from 1st linear diabat to second linear diabat as a functionof energy of incident particle with arbitrarycoupling expressed by different number of Dirac Delta function a total transfer matrix equation i.e.

Ttot = MM12MM23MM34 (4.42) with the condition that R 2 + T 2 = 1 (4.43) | | | | where R and T are the reflection and transmission coefficients.The transfer matrix method is used to obtain the non-adiabatic transition probability from first diabatic potential to another diabatic potential by using the following equation given as

2 MM23MM41 MM21MM43 T(Non Adiabatic) = 1 Abs − (4.44) − − MM MM MM MM 23 42 − 22 43 Further we examine our approach by doing numerical calculations for the Landau-Zener transition probability by expressing our arbitrary coupling with n number of Dirac delta potentialswheren= 1, 3, 5, 7, 9, 13, 15, 17, 21. Theresultofour numerical calculations 125 Figure 4.10: Landau-Zener Transition probability for n=13,15,17 and 21 number of Dirac Delta potentials

126 are shown in Figure 4.10.

4.1.3 Transfer matrix approach to the curve crossing problem of two exponential diabatic potentials

The Schematic diagram showing the formulation of the problem is shown in Figure .. 4.11. We start with the time-independent Schrodinger equation for a two state system

1 ∂2 ax 2 + V0e V (x) ψ1(x) ψ1(x) − 2m ∂x = E . (4.45)  1 ∂2 ax      V (x) 2 + V e− ψ (x) ψ (x) − 2m ∂x 0 2 2       Eq. (4.45) can be split into the following two equations

1 ∂2 + V eax ψ (x)+ V (x)ψ (x)= Eψ (x) and (4.46) −2m ∂x2 0 1 2 1 2 1 ∂ ax + V e− ψ (x)+ V (x)ψ (x)= Eψ (x). −2m ∂x2 0 2 1 2

In our calculations we took V0 = 1.0 and a = 1. Integrating the above two equations from x η to x + η (where η 0)we get the following two boundary conditions n − n →

2 xn+η ~ dψ1(x) + V (x)ψ2(xn) = 0 and (4.47) − 2m dx xn η − 2 xn+η ~ dψ2(x) + V (x)ψ1(xn) = 0. −2m dx xn η − Also we have two more boundary conditions

ψ (x η)= ψ (x + η) and (4.48) 1 n − 1 n ψ (x η)= ψ (x + η). 2 n − 2 n

Simpliﬁed view of our model illustrating the problem is shown in ﬁgure 4.12.

In this schematic view we have expressed our Gaussian coupling ( with α = 1)bya collection of Dirac Delta potentials. To derive differentexpressions for transfer matrices we express this coupling by three Dirac Delta potentials at the beginning and then add 127 Figure 4.11: Schematic diagram for the crossing of two exponential potentials with Gaussian coupling (α = 1).

Figure 4.12: Schematic diagram for Gaussian coupling expressed as a collection of Dirac Delta Potentials α = 1. 128 more number of Dirac Delta potentials. The Dirac Delta Potential positioned at xn = x2 as shown in ﬁgure 4.12 will correlate the ﬂux from region 2 to region 3. Region 2 is

assigned to the left side of Dirac Delta Potential positioned at xn = x2 while region 3 is

assigned to the right side of of Dirac Delta potential positioned at xn = x2. Using the analytical equations as mentioned above we will calculate a transfer matrix designated as MAT23(4 4) which will connect the ﬂux from region 2 to region 3 or in other words × we can say that this transfer matrix will correlate the coefﬁcients An+1 and Bn+1, Dn+1

and Fn+1 with An+2 and Bn+2, Dn+2 and Fn+2 which are discussed in the next section

in detail. Similar approach has been used to form the transfer matrices MAT12(4 4)and ×

MAT34(4 4) which will correlate the ﬂuxes in between respective regions. This will × model our arbitrary coupling for three Dirac Delta potentials. Further we add more and more number of Dirac Delta potentials till convergence value is reached. We start with the formation of transfer matrix designated as MAT23(4 4) which will follow up × with the formation of other matrices. For the ﬁrst exponential diabat in region 2, the physically acceptable solution is given by

√ x/2 √ x/2 An+1I2i√2√E 2 2e + Bn+1I 2i√2√E 2 2e (4.49) − Here I represents the modiﬁed Bessel functions of ﬁrst kind. In the above expression

An+1 denotes the probability amplitude for motion along the positive direction and

Bn+1 denotes the probability amplitude for motion along the negative direction. The physically acceptable solution in region 3, is given by

√ x/2 √ x/2 An+2I2i√2√E 2 2e + Bn+2I 2i√2√E 2 2e (4.50) −

For second exponential diabat in region2, the physically acceptable solution is

√ x/2 √ x/2 Dn+1I2i√2√E 2 2e + Fn+1I 2i√2√E 2 2e (4.51) − In region 3, the physically acceptable solution is

√ x/2 √ x/2 Dn+2I2i√2√E 2 2e + Fn+2I 2i√2√E 2 2e (4.52) 129 − Using the four boundary conditions mentioned above in equation (4.47) and (4.48) with x a , a being the position of the Dirac Delta potential we arrive at the following n → n n equations as mentioned below

a[n] a[n] a[n] A I √ e 2 B I √ e 2 A I √ e 2 n+1 2i√2√E 2 2 + n+1 2i√2√E 2 2 = n+2 2i√2√E 2 2 (4.53) − a[n] B I √ e 2 + n+2 2i√2√E 2 2 −

1 a[n] a[n] a[n] e 2 A I √ e 2 I √ e 2 n+1 2i√2√E 1 2 2 + 2i√2√E+1 2 2 (4.54) 2√2 − 1 a[n] a[n] a[n] e 2 A I √ e 2 I √ e 2 n+2 2i√2√E 1 2 2 + 2i√2√E+1 2 2 −2√2 − 1 a[n] a[n] a[n] e 2 B B I √ e 2 I √ e 2 + ( n+1 n+2) 2i√2√E 1 2 2 + 1 2i√2√E 2 2 2√2 − − − − a[n] a[n] k n D I √ e 2 F I √ e 2 + ( ) n+2 2i√2√E 2 2 + n+2 2i√2√E 2 2 =0 − and

a[n] a[n] a[n] D I √ e 2 F I √ e 2 D I √ e 2 n+1 2i√2√E 2 2 + n+1 2i√2√E 2 2 = n+2 2i√2√E 2 2 (4.55) − a[n] F I √ e 2 + n+2 2i√2√E 2 2 −

a[n] a[n] k n A I √ e 2 B I √ e 2 [ ] n+2 2i√2√E 2 2 + n+2 2i√2√E 2 2 (4.56) − 1 a[n] a[n] a[n] e 2 D I √ e 2 I √ e 2 + n+1 2i√2√E 1 2 2 + 2i√2√E+1 2 2 2√2 − 1 a[n] a[n] a[n] e 2 D I √ e 2 I √ e 2 n+2 2i√2√E 1 2 2 + 2i√2√E+1 2 2 −2√2 − 1 a[n] a[n] a[n] e 2 F F I √ e 2 I √ e 2 + ( n+1 n+2) 2i√2√E 1 2 2 + 1 2i√2√E 2 2 =0 2√2 − − − − From the above four equations we form a (4 4) matrix as given below which connect ×

130 the ﬂux in region 2 to region 3.

M11 M12 M13 M14   M21 M22 M23 M24 MAT23(4 4) = ×   M31 M32 M33 M34     M41 M42 M43 M44    

Similarly, (4 4) matrix designated as MAT12(4 4) and MAT34(4 4), relates the ﬂux × × × in region 1 and 2 and region 3 and 4 respectively.

M11 M12 M13 M14   M21 M22 M23 M24 MAT12(4 4) = (4.57) ×  M M M M   31 32 33 34       M41 M42 M43 M44      and

131 M11 M12 M13 M14   M21 M22 M23 M24 MAT34(4 4) = (4.58) ×  M M M M   31 32 33 34       M41 M42 M43 M44     

132 M11a The matrix elements of above MAT can be read as M11 = where 23 M11b

a[n] a[n] a[n] a[n] a[n] a[n] e 2 I √ e 2 I √ e 2 2 e 2 I √ e 2 I √ e 2 2 2i√2√E 1 2 2 2i√2√E 2 2 2i√2√E+1 2 2 2i√2√E 2 2 − ! − ! ! − ! M11a = − 2√2 − 2√2

a[n] a[n] a[n] a[n] e 2 I √ e 2 I √ e 2 I √ e 2 2i√2√E 1 2 2 2i√2√E 2 2 2i√2√E 2 2 − − ! ! − ! + 2√2

a[n] a[n] a[n] a[n] 2 √ 2 √ 2 √ 2 e I1 2i√2√E 2 2e I2i√2√E 2 2e I 2i√2√E 2 2e − ! ! − ! + , 2√2

a[n] a[n] a[n] a[n] a[n] e 2 I √ e 2 I √ e 2 I √ e 2 I √ e 2 2i√2√E 1 2 2 + 1 2i√2√E 2 2 2i√2√E 2 2 2i√2√E 2 2 − − ! − !! − ! ! M11b = 2√2

a[n] a[n] a[n] a[n] e 2 I √ e 2 I √ e 2 I √ e 2 2 2i√2√E 1 2 2 + 2i√2√E+1 2 2 2i√2√E 2 2 − ! !! − ! − 2√2

M12 = 0,

M13a M13 = , M13b

where

a[n] a[n] M k n I √ e 2 2 I √ e 2 , 13a = ( ) 2i√2√E 2 2 2i√2√E 2 2 − − ! ! !

a[n] a[n] a[n] a[n] a[n] e 2 I √ e 2 I √ e 2 I √ e 2 I √ e 2 2i√2√E 1 2 2 + 1 2i√2√E 2 2 2i√2√E 2 2 2i√2√E 2 2 − − ! − !! − ! ! M13b = , 2√2

a[n] a[n] a[n] a[n] e 2 I √ e 2 I √ e 2 I √ e 2 2 2i√2√E 1 2 2 + 2i√2√E+1 2 2 2i√2√E 2 2 − ! !! − ! , − 2√2

M14a M14 = , M14b

where

a[n] M k n I √ e 2 3 , 14a = ( ) 2i√2√E 2 2 − − ! !

a[n] a[n] a[n] a[n] a[n] e 2 I √ e 2 I √ e 2 I √ e 2 I √ e 2 2i√2√E 1 2 2 + 1 2i√2√E 2 2 2i√2√E 2 2 2i√2√E 2 2 − − ! − !! − ! ! M14b = 2√2

a[n] a[n] a[n] a[n] e 2 I √ e 2 I √ e 2 I √ e 2 2 2i√2√E 1 2 2 + 2i√2√E+1 2 2 2i√2√E 2 2 − ! !! − ! , − 2√2

M21 = 0,

M22a M22 = , M22b

a[n] a[n] a[n] a[n] a[n] a[n] a[n] a[n] M ae− 2 √ e 2 I √ e 2 I √ e 2 ae− 2 √ √ e 2 I √ e 2 I √ e 2 22a = 2 2i√2√E 1 2 2 2i√2√E 2 2 + 2 2 2i√2√E+1 2 2 2i√2√E 2 2 − ! − 133 ! ! − !

a[n] a[n] a[n] a[n] a[n] a[n] a[n] a[n] ae 2 √ √ e 2 I √ e 2 I √ e 2 ae 2 √ √ e 2 I √ e 2 I √ e 2 , − 2 2 2i√2√E 1 2 2 2i√2√E 2 2 − 2 2 1 2i√2√E 2 2 2i√2√E 2 2 − − − ! ! − − ! ! a[n] a[n] a[n] a[n] M √ I √ e 2 I √ e 2 √ I √ e 2 I √ e 2 22b = 2 2i√2√E 1 2 2 2i√2√E 2 2 + 2 2i√2√E+1 2 2 2i√2√E 2 2 − ! − ! ! − !

a[n] a[n] a[n] a[n] √ I √ e 2 I √ e 2 √ I √ e 2 I √ e 2 , 2 2i√2√E 1 2 2 2i√2√E 2 2 2 1 2i√2√E 2 2 2i√2√E 2 2 − − − ! ! − − ! !

a[n] a[n] a[n] a[n] M I √ e 2 I √ e 2 I √ e 2 I √ e 2 23 = 2i√2√E 1 2 2 2i√2√E 2 2 + 2i√2√E+1 2 2 2i√2√E 2 2 − ! − ! ! − !

a[n] a[n] a[n] a[n] I √ e 2 I √ e 2 I √ e 2 I √ e 2 , 2i√2√E 1 2 2 2i√2√E 2 2 1 2i√2√E 2 2 2i√2√E 2 2 − − − ! ! − − ! !

M24a M24 = , M24b

a[n] a[n] a[n] M √ e 2 k n I √ e 2 I √ e 2 , 24a = 2 2 − ( ) 2i√2√E 2 2 2i√2√E 2 2 − − ! !

and

a[n] a[n] a[n] a[n] M I √ e 2 I √ e 2 I √ e 2 I √ e 2 24b = 2i√2√E 1 2 2 2i√2√E 2 2 + 2i√2√E+1 2 2 2i√2√E 2 2 − ! − ! ! − !

a[n] a[n] a[n] a[n] I √ e 2 I √ e 2 I √ e 2 I √ e 2 , 2i√2√E 1 2 2 2i√2√E 2 2 1 2i√2√E 2 2 2i√2√E 2 2 − − − ! ! − − ! !

M31a M31 = M32b

where

a[n] a[n] a[n] M √ e− 2 k n I √ e 2 I √ e 2 , 31a = 2 2 ( ) 2i√2√E 2 2 2i√2√E 2 2 − ! !

a[n] a[n] a[n] a[n] M I √ e 2 I √ e 2 I √ e 2 I √ e 2 31b = 2i√2√E 1 2 2 2i√2√E 2 2 + 2i√2√E+1 2 2 2i√2√E 2 2 − ! − ! ! − !

a[n] a[n] a[n] a[n] I √ e 2 I √ e 2 I √ e 2 I √ e 2 , 2i√2√E 1 2 2 2i√2√E 2 2 1 2i√2√E 2 2 2i√2√E 2 2 − − − ! ! − − ! !

M32a M32 = , M32b

where

a[n] a[n] 2 M √ e− 2 k n I √ e 2 , 32a = 2 2 ( ) 2i√2√E 2 2 − !

a[n] a[n] a[n] a[n] M I √ e 2 I √ e 2 I √ e 2 I √ e 2 32b = 2i√2√E 1 2 2 2i√2√E 2 2 + 2i√2√E+1 2 2 2i√2√E 2 2 − ! − ! ! − !

a[n] a[n] a[n] a[n] I √ e 2 I √ e 2 I √ e 2 I √ e 2 , 2i√2√E 1 2 2 2i√2√E 2 2 1 2i√2√E 2 2 2i√2√E 2 2 − − − ! ! − − ! !

M33a M33 = , M33b

where

134 a[n] a[n] a[n] a[n] a[n] a[n] a[n] a[n] M e− 2 √ e 2 I √ e 2 I √ e 2 e− 2 √ e 2 I √ e 2 I √ e 2 33a = 2 2i√2√E 1 2 2 2i√2√E 2 2 + 2 2i√2√E+1 2 2 2i√2√E 2 2 − ! − ! ! − !

a[n] a[n] a[n] a[n] a[n] a[n] a[n] a[n] e− 2 √ e 2 I √ e 2 I √ e 2 e− 2 √ e 2 I √ e 2 I √ e 2 , 2 2i√2√E 1 2 2 2i√2√E 2 2 2 1 2i√2√E 2 2 2i√2√E 2 2 − − − ! ! − − ! !

a[n] a[n] a[n] a[n] M √ I √ e 2 I √ e 2 √ I √ e 2 I √ e 2 33b = 2 2i√2√E 1 2 2 2i√2√E 2 2 + 2 2i√2√E+1 2 2 2i√2√E 2 2 − ! − ! ! − !

a[n] a[n] a[n] a[n] √ I √ e 2 I √ e 2 √ I √ e 2 I √ e 2 , 2 2i√2√E 1 2 2 2i√2√E 2 2 2 1 2i√2√E 2 2 2i√2√E 2 2 − − − ! ! − − ! !

M34 = 0,

M41a M41 = , M41b

where

a[n] √ 2 2 M41a = k(n) I √ √E 2 2e , − 2i 2 ! !

and

a[n] a[n] a[n] a[n] e 2 I √ e 2 I √ e 2 I √ e 2 2i√2√E 1 2 2 + 2i√2√E+1 2 2 2i√2√E 2 2 − ! !! − ! M41b = 2√2

a[n] a[n] a[n] a[n] e 2 I √ e 2 I √ e 2 I √ e 2 2i√2√E 1 2 2 + 1 2i√2√E 2 2 2i√2√E 2 2 − − ! − !! ! , − 2√2

M42a M42 = , M42b

where

a[n] a[n] M k n I √ e 2 I √ e 2 , 42a = ( ) 2i√2√E 2 2 2i√2√E 2 2 − − !! !

a[n] a[n] a[n] a[n] e 2 I √ e 2 I √ e 2 I √ e 2 2i√2√E 1 2 2 + 2i√2√E+1 2 2 2i√2√E 2 2 − ! !! − ! M42b = 2√2

a[n] a[n] a[n] a[n] 2 √ 2 √ 2 √ 2 e I 2i√2√E 1 2 2e + I1 2i√2√E 2 2e I2i√2√E 2 2e − − ! − !! ! , − 2√2

M43 = 0,

M44 = 1,

135 The matrix elements ofMAT12(4 4) can be read as ×

M11a M11 = , M11b

where

a[n] a[n] a[n] a[n] a[n] a[n] a[n] a[n] M e− 2 √ e 2 I √ e 2 I √ e 2 e− 2 √ e 2 I √ e 2 I √ e 2 11a = 2 2i√2√E 1 2 2 2i√2√E 2 2 + 2 2i√2√E+1 2 2 2i√2√E 2 2 − ! − ! ! − !

a[n] a[n] a[n] a[n] a[n] a[n] a[n] a[n] e 2 √ e 2 I √ e 2 I √ e 2 e 2 √ e 2 I √ e 2 I √ e 2 , − 2 2i√2√E 1 2 2 2i√2√E 2 2 − 2 1 2i√2√E 2 2 2i√2√E 2 2 − − − ! ! − − ! !

and

a[n] a[n] a[n] a[n] √ √ 2 √ 2 √ √ 2 √ 2 M11b = 2I2i√2√E 1 2 2e I 2i√2√E 2 2e + 2I2i√2√E+1 2 2e I 2i√2√E 2 2e − ! − ! ! − !

a[n] a[n] a[n] a[n] √ I √ e 2 I √ e 2 √ I √ e 2 I √ e 2 , 2 2i√2√E 1 2 2 2i√2√E 2 2 2 1 2i√2√E 2 2 2i√2√E 2 2 − − − ! ! − − ! !

M12 = 0,

M13a M13 = , M13b

where

a[n] a[n] a[n] M √ e− 2 k n I √ e 2 K √ e− 2 , 13a = 2 2 ( ) 2i√2√E 2 2 2i√2√E 2 2 − ! − !

and

a[n] a[n] a[n] a[n] M I √ e 2 I √ e 2 I √ e 2 I √ e 2 13b = 2i√2√E 1 2 2 2i√2√E 2 2 + 2i√2√E+1 2 2 2i√2√E 2 2 − ! − ! ! − !

a[n] a[n] a[n] a[n] I √ e 2 I √ e 2 I √ e 2 I √ e 2 , 2i√2√E 1 2 2 2i√2√E 2 2 1 2i√2√E 2 2 2i√2√E 2 2 − − − ! ! − − ! !

M14a M14 = , M14b

where

a[n] a[n] M √ e 2 k n I √ e 2 , 14a = 2 2 − ( ) 2i√2√E 2 2 − !

and

a[n] a[n] a[n] a[n] M I √ e 2 I √ e 2 I √ e 2 I √ e 2 14b = 2i√2√E 1 2 2 2i√2√E 2 2 + 2i√2√E+1 2 2 2i√2√E 2 2 − ! − ! ! − !

a[n] a[n] a[n] a[n] I √ e 2 I √ e 2 I √ e 2 I √ e 2 , 2i√2√E 1 2 2 2i√2√E 2 2 1 2i√2√E 2 2 2i√2√E 2 2 − − − ! ! − − ! !

M21 = 0,

M22 = 1, 136 M23a M23 = , where M23b a[n] a[n] M k n I √ e 2 K √ e− 2 , 23a = ( ) 2i√2√E 2 2 2i√2√E 2 2 − !! − !

a[n] a[n] a[n] a[n] e 2 I √ e 2 I √ e 2 I √ e 2 2i√2√E 1 2 2 + 2i√2√E+1 2 2 2i√2√E 2 2 − ! !! − ! M23b = 2√2

a[n] a[n] a[n] a[n] e 2 I √ e 2 I √ e 2 I √ e 2 2i√2√E 1 2 2 + 1 2i√2√E 2 2 2i√2√E 2 2 − − ! − !! ! , − 2√2

M24a M24 = , where M24b

a[n] M k n I √ e 2 , 24a = ( ) 2i√2√E 2 2 − !!

and

a[n] a[n] a[n] a[n] 2 √ 2 √ 2 √ 2 e I2i√2√E 1 2 2e + I2i√2√E+1 2 2e I 2i√2√E 2 2e − ! !! − ! M24b = 2√2

a[n] a[n] a[n] a[n] e 2 I √ e 2 I √ e 2 I √ e 2 2i√2√E 1 2 2 + 1 2i√2√E 2 2 2i√2√E 2 2 − − ! − !! ! , − 2√2

M31a M31 = , where M31b

a[n] a[n] M k n I √ e 2 I √ e 2 , 31a = ( ) 2i√2√E 2 2 2i√2√E 2 2 − ! !

a[n] a[n] a[n] a[n] e 2 I √ e 2 I √ e 2 I √ e 2 2i√2√E 1 2 2 + 2i√2√E+1 2 2 2i√2√E 2 2 − ! !! − ! M31b = , 2√2

a[n] a[n] a[n] a[n] e 2 I √ e 2 I √ e 2 I √ e 2 2i√2√E 1 2 2 + 1 2i√2√E 2 2 2i√2√E 2 2 − − ! − !! ! , − 2√2

M32a M32 = , where M32b

a[n] M k n I √ e 2 2, 32a = ( ) 2i√2√E 2 2 − !

a[n] a[n] a[n] a[n] 2 √ 2 √ 2 √ 2 e I2i√2√E 1 2 2e + I2i√2√E+1 2 2e I 2i√2√E 2 2e − ! !! − ! M32b = 2√2

a[n] a[n] a[n] a[n] e 2 I √ e 2 I √ e 2 I √ e 2 2i√2√E 1 2 2 + 1 2i√2√E 2 2 2i√2√E 2 2 − − ! − !! ! , − 2√2

M33a M33 = , where M33b

a[n] a[n] a[n] a[n] a[n] a[n] e− 2 I √ e 2 K √ e− 2 e− 2 I √ e 2 K √ e− 2 2i√2√E 2 2 2i√2√E 1 2 2 2i√2√E 2 2 1 2i√2√E 2 2 − ! − − ! − ! − ! M33a = + 2√2 2√2

a[n] a[n] a[n] a[n] a[n] a[n] e 2 I √ e 2 K √ e− 2 e 2 I √ e 2 K √ e− 2 2i√2√E 1 2 2 2i√2√E 2 2 1 2i√2√E 2 2 2i√2√E 2 2 − − ! − ! − ! − ! , − 2√2 137− 2√2 a[n] a[n] a[n] a[n] e 2 I √ e 2 I √ e 2 I √ e 2 2i√2√E 1 2 2 + 2i√2√E+1 2 2 2i√2√E 2 2 − ! !! − ! M33b = 2√2

a[n] a[n] a[n] a[n] e 2 I √ e 2 I √ e 2 I √ e 2 2i√2√E 1 2 2 + 1 2i√2√E 2 2 2i√2√E 2 2 − − ! − !! ! , − 2√2

M34a M34 = , where M34b

a[n] a[n] a[n] a[n] e 2 I √ e 2 e 2 I √ e 2 2i√2√E 1 2 2 1 2i√2√E 2 2 − − ! − ! M34a = , − 2√2 − 2√2

a[n] a[n] a[n] a[n] e 2 I √ e 2 I √ e 2 I √ e 2 2i√2√E 1 2 2 + 2i√2√E+1 2 2 2i√2√E 2 2 − ! !! − ! M34b = 2√2

a[n] a[n] a[n] a[n] e 2 I √ e 2 I √ e 2 I √ e 2 2i√2√E 1 2 2 + 1 2i√2√E 2 2 2i√2√E 2 2 − − ! − !! ! , − 2√2

M41a M41 = , where M41b

a[n] M k n I √ e 2 2 , 41a = ( ) 2i√2√E 2 2 − ! !

a[n] a[n] a[n] a[n] e 2 I √ e 2 I √ e 2 I √ e 2 2i√2√E 1 2 2 + 2i√2√E+1 2 2 2i√2√E 2 2 − ! !! − ! M41b = 2√2

a[n] a[n] a[n] a[n] e 2 I √ e 2 I √ e 2 I √ e 2 2i√2√E 1 2 2 + 1 2i√2√E 2 2 2i√2√E 2 2 − − ! − !! ! , − 2√2

M42a M42 = , where M42b

a[n] a[n] M k n I √ e 2 I √ e 2 , 42a = ( ) 2i√2√E 2 2 2i√2√E 2 2 − − !! !

a[n] a[n] a[n] a[n] e 2 I √ e 2 I √ e 2 I √ e 2 2i√2√E 1 2 2 + 2i√2√E+1 2 2 2i√2√E 2 2 − ! !! − ! M42b = 2√2

a[n] a[n] a[n] a[n] e 2 I √ e 2 I √ e 2 I √ e 2 2i√2√E 1 2 2 + 1 2i√2√E 2 2 2i√2√E 2 2 − − ! − !! ! , − 2√2

1 a[n] a[n] a[n] a[n] M iπe− 2 csch √ , π√E I √ e 2 K √ e− 2 K √ e− 2 43 = 2 2 2i√2√E 2 2 1 2i√2√E 2 2 + 2i√2√E+1 2 2 √2 ! − ! !! “ ”

a[n] a[n] a[n] ea[n] I √ e 2 I √ e 2 K √ e 2 , 2i√2√E 1 2 2 + 2i√2√E+1 2 2 2i√2√E 2 2 − − − ! !! !

a[n] a[n] a[n] iπe 2 csch √ π√E I √ e 2 I √ e 2 2 2 2i√2√E 1 2 2 + 2i√2√E+1 2 2 − ! !! M44 = “ ” , − √2

138 Further, the matrix elements of matrix MAT34(4 4) can be read as ×

a[n] a[n] I √ e 2 I √ e 2 2i√2√E 1 2 2 + 2i√2√E+1 2 2 − ! ! M11 = − a[n] a[n] K √ e 2 K √ e 2 , 2i√2√E 1 2 2 + 1 2i√2√E 2 2 − − ! − !

a[n] a[n] I √ e 2 I √ e 2 2i√2√E 1 2 2 + 1 2i√2√E 2 2 − − ! − ! M12 = − a[n] a[n] K √ e 2 K √ e 2 , 2i√2√E 1 2 2 + 1 2i√2√E 2 2 − − ! − !

a[n] a[n] √ e 2 k n I √ e 2 2 2 − ( ) 2i√2√E 2 2 ! M13 = − a[n] a[n] K √ e 2 K √ e 2 , 2i√2√E 1 2 2 + 1 2i√2√E 2 2 − − ! − !

a[n] a[n] √ e− 2 k n I √ e 2 2 2 ( ) 2i√2√E 2 2 − ! M14 = − a[n] a[n] K √ e 2 K √ e 2 , 2i√2√E 1 2 2 + 1 2i√2√E 2 2 − − ! − !

a[n] a[n] a[n] a[n] I √ e 2 K √ e 2 I √ e 2 K √ e 2 2i√2√E 1 2 2 2i√2√E 2 2 2i√2√E+1 2 2 2i√2√E 2 2 a[n] − ! − ! ! − ! M I √ e 2 21 = 2i√2√E 2 2 + + ! a[n] a[n] a[n] a[n] K √ e 2 K √ e 2 K √ e 2 K √ e 2 2i√2√E 1 2 2 + 1 2i√2√E 2 2 2i√2√E 1 2 2 + 1 2i√2√E 2 2 − − ! − ! − − ! − !

a[n] a[n] a[n] a[n] I √ e 2 K √ e 2 I √ e 2 K √ e 2 2i√2√E 1 2 2 2i√2√E 2 2 1 2i√2√E 2 2 2i√2√E 2 2 a[n] − − ! − ! − ! − ! M I √ e 2 22 = 2i√2√E 2 2 + + − ! a[n] a[n] a[n] a[n] K √ e 2 K √ e 2 K √ e 2 K √ e 2 2i√2√E 1 2 2 + 1 2i√2√E 2 2 2i√2√E 1 2 2 + 1 2i√2√E 2 2 − − ! − ! − − ! − !

a[n] a[n] a[n] √ e− 2 k n I √ e 2 K √ e 2 2 2 ( ) 2i√2√E 2 2 2i√2√E 2 2 ! − ! M23 = a[n] a[n] K √ e 2 K √ e 2 2i√2√E 1 2 2 + 1 2i√2√E 2 2 − − ! − !

a[n] a[n] a[n] √ e 2 k n I √ e 2 K √ e 2 2 2 − ( ) 2i√2√E 2 2 2i√2√E 2 2 − ! − ! M24 = a[n] a[n] √ 2 √ 2 K 2i√2√E 1 2 2e + K1 2i√2√E 2 2e − − ! − !

a[n] a[n] √ 2 √ 2 2 2e k(n)I2i√2√E 2 2e ! M31 = − a[n] a[n] I √ e− 2 I √ e− 2 2i√2√E 1 2 2 + 1 2i√2√E 2 2 − − ! − !

a[n] a[n] √ e 2 k n I √ e 2 2 2 ( ) 2i√2√E 2 2 − ! M32 = − a[n] a[n] I √ e− 2 I √ e− 2 2i√2√E 1 2 2 + 1 2i√2√E 2 2 − − ! − !

a[n] a[n] a[n] a[n] a[n] e 2 e 2 I √139e 2 e 2 I √ e 2 2i√2√E 1 2 2 + 2i√2√E+1 2 2 − ! !! M33 = − a[n] a[n] I √ e− 2 I √ e− 2 2i√2√E 1 2 2 + 1 2i√2√E 2 2 − − ! − ! a[n] a[n] a[n] a[n] a[n] e 2 e 2 I √ e 2 e 2 I √ e 2 2i√2√E 1 2 2 + 1 2i√2√E 2 2 − − ! − !! M34 = − a[n] a[n] I √ e− 2 I √ e− 2 2i√2√E 1 2 2 + 1 2i√2√E 2 2 − − ! − !

a[n] a[n] a[n] √ e 2 k n I √ e− 2 I √ e 2 2 2 ( ) 2i√2√E 2 2 2i√2√E 2 2 − ! ! M41 = a[n] a[n] I √ e− 2 I √ e− 2 2i√2√E 1 2 2 + 1 2i√2√E 2 2 − − ! − !

a[n] a[n] a[n] √ e 2 k n I √ e− 2 I √ e 2 2 2 ( ) 2i√2√E 2 2 2i√2√E 2 2 − ! − ! M42 = a[n] a[n] I √ e− 2 I √ e− 2 2i√2√E 1 2 2 + 1 2i√2√E 2 2 − − ! − !

a[n] a[n] a[n] a[n] a[n] a[n] e I √ e 2 I √ e− 2 e I √ e 2 I √ e− 2 2i√2√E 1 2 2 2i√2√E 2 2 2i√2√E+1 2 2 2i√2√E 2 2 − ! − ! ! − ! a[n] √ 2 M43 = + + I √ √E 2 2e a[n] a[n] a[n] a[n] 2i 2 ! I √ e 2 I √ e 2 I √ e 2 I √ e 2 2i√2√E 1 2 2 − + 1 2i√2√E 2 2 − 2i√2√E 1 2 2 − + 1 2i√2√E 2 2 − − − ! − ! − − ! − !

a[n] a[n] a[n] a[n] ea[n] I √ e 2 I √ e 2 ea[n] I √ e 2 I √ e 2 1 2i√2√E 2 2 2i√2√E 2 2 − 2i√2√E 1 2 2 2i√2√E 2 2 − − ! − ! − − ! − ! a[n] M I √ e 2 44 = + + 2i√2√E 2 2 a[n] a[n] a[n] a[n] − ! I √ e 2 I √ e 2 I √ e 2 I √ e 2 2i√2√E 1 2 2 − + 1 2i√2√E 2 2 − 2i√2√E 1 2 2 − + 1 2i√2√E 2 2 − − − ! − ! − − ! − !

The total ﬂux across this Gaussian coupling is obtained by multiplying all these matrices and obtaining a total transfer matrix equation given as

MM = MAT MAT MAT (4.59) 12 × 23 × 34

with the condition that R 2 + T 2 = 1. (4.60) | | | | where R and T are the reﬂection and transmission coefﬁcients. The transfer matrix method is used to obtained the non adiabatic transition probability from 1st diabatic potential to another diabatic potential by using the following equation given as

2 MM23MM41 MM21MM43 T(Non Adiabatic) = 1 Abs − (4.61) − − MM MM MM MM 23 42 − 22 43 where MM is the total matrix equation obtained by using equation (4.59). This analytical approach is examined by doing numerical calculations on our Gaussian coupling 140 (a) Converged numerical results for non adiabatic (b) Quantitative agreement between the present transition probability for different number of results(L.H.S)and exact numerical results for 21 Dirac Delta Potentials number of Dirac Delta potentials

Figure 4.13: Converged Plots for Non adiabatic Transition probability

with α=1 by expressing it using differentnumber of Dirac Delta potentials. Figure 4.13a represents the plot of non adiabatic transition probability for n=7, 9, 13, 15, 17 and 21 number of Dirac Delta potentials for convergence check and from these plots we ﬁnd that transition probability starts converging after we expresses our arbitrary coupling by n=7 number of Dirac Delta potentials. The converged numerical results with respect to number of Dirac Delta potentials are compared with exact numerical calculations and has been presented in next subsection.

Exact numerical approach

For exact numerical calculation purposes we deﬁne the limits of the barrier by two constants a and b which are 2 and 2 in our case. N is the number of segments in − div which the potential barrier has been divided. We deﬁne step which is given by

b a step = − (4.62) Ndiv

and is equivalent to the width of the individual segment used to approximate the po-

tential barrier. The position of Dirac Delta potential denoted by an is calculated by the 141 following equation as given below

1 a = (a + c) +(n 1) step (4.63) n 2| | − ∗ wherec = a+step and strength of the DiracDelta Potentialdenoted by kn at each position is given by the following equation

step an+ 2 kn = V (x)dx. (4.64) step an Z − 2

Using the above equations to calculate the position and strength of Dirac Delta poten-

αx2 tials we approximate the potential barrier V (x) = e− with α = 1 by a collection of Dirac delta potentials and calculate the transition probability from one diabatic potential to another diabatic potential.The result of our converged numerical calculations for different number of Dirac Delta potentials along with exact numerical calculations has been in ﬁgure 4.13a and 4.13b. The results obtained using exact numerical calculations are in quantitative agreement with the present results. Figure 4.13b shows a quantitative agreement between the present results and exact numerical calculations for Gaussian coupling approximated by 21 number of Dirac Delta Potentials

4.2 Conclusions

The current chapter explains the method of expressing the arbitrary coupling into number of Dirac delta potentials. This arbitrary coupling is used to calculate the non adiabatic transition probability between two diabatic potentials using Transfer Matrix method. A methodology of forming the transfer matrix between different Dirac Delta potentials is explained and non adiabatic transition probability is calculated for different kind of diabatic potentials. Exact analytical solutions for constant, linear and exponential cases are discussed along with numerical calculations for a quantitative agreement between the obtained results.

142 Chapter5 In the present chapter we consider the solution of the two state problem with time dependence. We solved the time dependent Schrodinger equations with a time dependent coupling which couples the two diabatic states and ﬁnd the exact analytical solution of the problem.Solutions for the different time variation of the strength of the Dirac Delta potentials has been derived.

143 CHAPTER 5

EXACT SOLUTION TO THE CURVE CROSSING PROBLEMS USING TIME DEPENDENT APPROACH

In time dependent Schrodinger equation which generally obey time dependent processes there are two types of time dependencies. One is where the time dependence is assumed for the adiabatic parameter while other is a quantum mechanical problem in a certain time dependent external field. As an example to explain the applicability of the former type is atomic collision processes where nuclear motion have time dependence and laser technology is of latter type. Literature survey reveals various analytical models where the time dependent Schrodinger equations are solved exactly. Some of the examples may include time dependent harmonic oscillator [110, 111, 112], infinite potential well with a moving boundary [113, 114, 115]and various other examples [116]. Two types of time-dependent problems which are commonly investigated includes the simple scattering problem where the potential has a periodic dependence on time [117]and amounts to a steady state solution while the other type includes the delta functional potential where strength varies with time which is solved using Floquet formalism [118]. Other time dependent problems includes the scattering problems involving delta potentials whose solutions are simple plane waves[116]. There also exists time dependent problems which are solved using diffusive solution to the Schroedinger wave equation. Different kind of approaches are adopted by different authors to solve the time dependent problems which includes a path integral of Feynman type and other is Laplace Transforms. The main disadvantage of the former method is slow convergence which is rectified by introducing a rapidly converging scheme[119]. The majority of the work involving the time dependent potentials is solved by using delta or a rectangular barrier. Intrinsically time dependent processes such as quantum mechanical processes in a certain time dependent external field becomes very important these days because of remarkable progress of Laser technology which means that Laser intensity and frequency can be now designed as a function of time. Our earlier work in this area was dedicated towards time independent coupling [55, 56, 57, 58, 59, 60, 61]. In thecurrent chapter we different subsections which are devoted to the study of two state problem involvingtime. Out of these section s one is devoted to the study of non adiabatic transitions involving time dependent Schroedinger equations for two state scattering problem coupled by Dirac Delta potential where the coupling is time independent and a solution is found in the Laplace domain. Other section belongs to the study of non adiabatic transitions involving time dependent potentials coupled by time dependent Dirac Delta potential where problem involving the partial differential equations in two variables is reduced to a single integral equation in Laplace domain and by knowing the wave function at the origin we can derive the wave function everywhere. Further solutions for the different time variation of the strength of the Dirac Delta potentials has been derived. Another section involves the case where we have studied the non adiabatic transition probability between two states which are coupled to each other by a moving δ function potential. Here we have used a similar methodology [120] by which we transfer our time dependent Schrodinger equation to independent one coupled by a moving δ potential by time dependent scaling factor and hence derived an analytical expression to calculate the non adiabatic transition probability thereby describing an important step to deal with the non adiabatic transition probability when we have moving δ potential coupling.

5.1 Exact solution of time-dependentSchrodingerequa- tion for two state problem in Laplace domain.

Methodology

We start with the case where two constant potential as represented by time dependent Schroedinger equations are coupled to each other by a time independent coupling. The

145 1D Schroedinger equation in this case can be written as

∂ φ1(x,t) H11 V12 φ1(x,t) i = . (5.1) ∂t       φ2(x,t) V21 H22 φ2(x,t)       The above equation is equivalent to the following equation

∂φ (x,t) i 1 = H φ (x,t)+ V φ (x,t) (5.2) ∂t 11 1 12 2 ∂φ (x,t) i 2 = H φ (x,t)+ V φ (x,t) (5.3) ∂t 22 2 21 1

if V12 and V21 is the coupling between the potentials represented by V12 = V21 =

2k0δ(x), then the above equations reduces to

∂φ (x,t) i 1 = H φ (x,t) + 2k δ(x) φ (x,t) (5.4) ∂t 11 1 0 2 ∂φ (x,t) i 2 = H φ (x,t) + 2k δ(x) φ (x,t) (5.5) ∂t 22 2 0 1

taking the Laplace transform of equation (5.2), it can be written as

H φ (x,s) + 2k δ(x)φ (x,s)= is φ (x,s) iφ (x, 0) (5.6) 11 1 0 2 1 − 1

In a similar way equation (5.3) can also be written as

H φ (x,s) + 2k δ(x)φ (x,s)= is φ (x,s) iφ (x, 0) (5.7) 22 2 0 1 2 − 2

we put wave packet at time t = 0 on the ﬁrst potential, hence for second state in our

problem φ2(x, 0) = 0, hence above equation reduced to

H22 φ2(x,s) + 2k0δ(x)φ1(x,s)= is φ2(x,s) (5.8)

146 Equation (5.8) can be rewritten as

(is H ) φ (x,s) = 2k δ(x)φ (x,s) (5.9) − 22 2 0 1

or 1 φ (x,s)=(is H )− 2k δ(x)φ (x,s) (5.10) 2 − 22 0 1

using the value of φ2(x,s) from above equation, equation (5.6) can be rewritten as

H φ (x,s) + 2k2δ(x)G0(0, 0,s)φ (x,s)= is φ (x,s) iφ (x, 0) (5.11) 11 1 0 2 1 1 − 1

0 1 where G (0, 0,s) = δ(x) (is H )− δ(x) is the Green’s function for the second 2 h | − 22 | i state. We will now solve equation(5.11) by using a method similar to one discussed in Reference [122], hence the above equation at the point of coupling can be further reduced to the following two equation given by

~2 ∂2 = is φ (x,s) iφ (x, 0) (5.12) − 2m ∂x2 1 − 1

∂φ1(x,s) ∂φ1(x,s) 2 0 + = 2k G (0, 0,s)φ (0,s) ∂x |x=0 − ∂x |x=0− 0 2 1

In order to solve equation (5.12), we should consider the homogeneous solution in order to satisfy the discontinuity condition at the point of coupling. Its solution is given by

1 0 0 0 φ1(x,s)= η(s) exp(i√is x )+ dx exp(i√is x x )φ1(x , 0) (5.13) | | 2√is | − | Z

147 where η(s) is an arbitrary function of s, which needs to be determined. Using the wave function given by equation(5.13) in equation (5.12), at the point of coupling we get

2 0 2i√is η(s) = 2k0G2(0, 0,s)φ1(0,s) (5.14) or (5.15) k2G0(0, 0,s)φ (0,s) η(s)= 0 2 1 (5.16) i√is with k = b where b is a positive real number (5.17) 0 − 0 0 b2G0(0, 0,s)φ (0,s) η(s)= 0 2 1 (5.18) i√is (5.19)

hence, the wave function can be written as

b2G0(0, 0,s)φ (0,s) 1 0 2 1 0 0 0 φ1(x,s)= exp(i√is x )+ dx exp(i√is x x )φ1(x , 0) i√is | | 2√is | − | Z (5.20) at point of coupling i.e. x = 0, the above equation reduces to

2 0 b0G2(0, 0,s)φ1(0,s) 1 0 0 0 φ1(0,s)= + dx exp(i√is x )φ1(x , 0) (5.21) i√is 2√is | | Z equation(5.21) is a integral equation in Laplace domain which will determine the wave function at the point of coupling, so if the wave function can be determined at the origin then it can be determined everywhere.

5.1.1 Derivation of the propagator for constant b

The solution in operator form can also be represented as

∞ 0 0 0 φ1(x,s)= dx G(x,x ,s)φ1(x , 0) (5.22) Z−∞

148 where G is an propagator. To ﬁnd the propagator we will make use of equation (5.12) and (5.21) to ﬁnd

2 0 2 0 b0 G (0, 0,s)φ1(0,s) b G (0, 0,s) 1 ∞ η(s)= 2 = − 0 2 dx0 exp(i√is x0 φ (x0 , 0) 2 0 1 i√is 2√s √s + √ib0G2(0, 0,s) | | Z−∞ (5.23) from equations (5.13) and (5.23) we get

b2G0(0, 0,s)exp(i√is( x + x0 ) exp(i√is x x0 ) G(x,x0 ,s)= − 0 2 | | | | + | − | (5.24) 2 0 2√s(√s + √ib0G2(0, 0,s) 2√is

further we will see that this propagator is equivalent to one derived by us in our previous research using Feynman approach [56] and is presented in next section

5.1.2 Consistency between derived operator and Green’s function using Feynman approach

We start with equation (5.11) given as

H φ (x,s) + 2k2δ(x)G0(0, 0,s)φ (x,s)= is φ (x,s) iφ (x, 0) (5.25) 11 1 0 2 1 1 − 1

further this equation can be written as

[ is + H k δ(x)]φ (x,s)= iφ (x, 0) (5.26) − 11 − 1 1 − 1

2 0 where k1 = 2k0G2(0, 0,s) further this can be written as

[is H k δ(x)]φ (x,s)= iφ (x, 0) (5.27) − 11 − 1 1 1

In terms of this the solution of above equation can be expressed as

∞ 0 0 φ1(x,s)= i G(x,x ,x)φ1(x, 0)dx (5.28) Z−∞

149 1 where G(x,x0 ,x)= x [is H k δ(x)]− x0 now further using the identity [123] h | − 11 − 1 | i Green’s function can be written as

1 1 1 G(x,x0 ,s)= x [is H ]− x0 x [is H ]− k δ(x)[is H k δ(x)]− x0 h | − 11 | i − h | − 11 1 − 11 − 1 | i (5.29)

now, inserting the resolution of the identity I = ∞ dy y y , between two inverses in −∞ | ih | the second term in the above equation we reach atR

∞ G(x,x0 ,s)= G (x,x0,s) k dyG (x,y,s)δ(y)G(y,x0,s) (5.30) 0 − 1 0 Z−∞

1 where G(x,x0 ,s)= x [is H ]− x0 further if δ(y)= δ(y x ), the above equation h | − 11 | i − s reduces to

G(x,x0 ,s)= G (x,x0,s) k G (x,x ,s)G(x ,x0 ,s) (5.31) 0 − 1 0 s s solving the above equation at x = xs, will gave us

0 0 1 G(xs,x ,s)= G0(xs,x ,s)[1 + k1G0(xs,xs,s)]− (5.32) putting equation (5.32) back in (5.31) will give us

1 G(x,x0 ,s)= G (x,x0 ,s) k G (x,x ,s)G (x ,x0,s)[1 + k G (x ,x ,s)]− (5.33) 0 − 1 0 s 0 s 1 0 s s in our calculations xs = 0, hence the above equation can be written as

1 G(x,x0 ,s)= G (x,x0,s) k G (x, 0,s)G (0,x0 ,s)[1 + k G (0, 0,s)]− (5.34) 0 − 1 0 0 1 0

For a free particle Green’s function this equation exactly reduces to as we have derived in last section given by equation no (5.24) which can be easily checked.

150 st 5.1.3 Solution for constant k0: bound state representation for 1 potential

If we choose k0 =-b0, where b0 is a positive real number and take

φ (x0 , 0) = √c exp ( b x0 ) (5.35) 1 0 − 0| | then choosing the above equation as initial condition in equation (5.21), we ﬁnd the following form of wave function in Laplace domain

2 0 c0G2(0, 0,s)φ1(0,s) 1 0 0 0 φ1(0,s)= + dx exp(i√is x )√c0 exp ( c0 x ) i√is 2√is | | − | | Z (5.36) The second term in above equation can be written and solved as

1 0 0 0 dx exp(i√is x )√c0 exp ( c0 x ) (5.37) 2√is | | − | | Z c √ 0 0 0 dx exp(i√is c0) x ) (5.38) 2√is − | | Z Solving this integral equation and putting it back into equation (5.23), we get the complete equation as

2 0 c0G2(0, 0,s)φ1(0,s) 2√c0 φ1(0,s)= (5.39) i√is − 2√is(i√is c ) − 0 further simpliﬁcation of the above equation will gave us

i φ (0,s)= (5.40) 1 0 2(i√is c0)c0√c0G (0, 0,s) i√is − 2 − Hence, the amplitude at the point of coupling is constant and the solution propagates as a bound state.

151 5.1.4 Partly unbound solutions for 1st potential

Let us look at the point that rather than starting from a bound state we start with an initial condition such that part of the wave function will be bound and a part will radiate

0 away. Further if we consider that our constant k0 = -c0 for t< 0, and k0 = -c for t

0 > 0 also,where both c0 and c are positive real numbers, lets us consider that our initial

condition is the bound state for k0 = -c0. We look for the wave function which is bound at t = and partly radiated away. In such kind of situation, the solution of equation ∞ (5.20) as given in reference number (122) can be written as

2 0 0 c G2(0, 0,s)φ1(0,s) 2√c0 φ1(0,s)= (5.41) i√is − 2√is(i√is c ) − 0 The solution is i√c0 φ (0,s)= (5.42) 1 2 0 (i√is c )c0 G (0, 0,s) i√is − 0 2 −

now to ﬁnd φ1(x,s), we will make use of equation (5.29) and (5.12) in the equation given below

1 0 0 0 φ1(x,s)= η(s)exp(i√is x )+ dx exp(i√is x x )φ1(x , 0) (5.43) | | 2√is | − | Z and hence the probability of ﬁnding the particle in bound state as well as total probability of radiating away can be calculated by the use of following expressions

∞ P = φ (x,s) 2 (5.44) | 1 | Z0 ∞ T = 1 P = 1 φ (x,s) 2 (5.45) − − | 1 | Z0

5.2 Exact solution of Schrodingerequation for two state problem with time dependent coupling.

We start with the case where two time dependent constant potential are coupled to each other by a time dependent coupling. The Schroedinger equation in this case can be 152 written as

∂ φ1(x,t) H11 V12 φ1(x,t) i = . (5.46) ∂t       φ2(x,t) V21 H22 φ2(x,t)       The above equation is equivalent to the following

∂φ (x,t) i 1 = H φ (x,t)+ V φ (x,t) (5.47) ∂t 11 1 12 2 ∂φ (x,t) i 2 = H φ (x,t)+ V φ (x,t) ∂t 22 2 21 1

if V12 and V21 is the coupling between the potentials represented by V12 = V21 = 2k(t)δ(x), then the above equations reduces to

∂φ (x,t) i 1 = H φ (x,t) + 2k(t)δ(x) φ (x,t) (5.48) ∂t 11 1 2 ∂φ (x,t) i 2 = H φ (x,t) + 2k(t)δ(x) φ (x,t) ∂t 22 2 1

taking the Laplace transform of equation (5.47), it can be written as

H φ (x,s) + 2L[k(t)δ(x)φ (x,t)] = is φ (x,s) iφ (x, 0) (5.49) 11 1 2 1 − 1

In a similar way equation (5.48) can also be written as

H φ (x,s) + 2L[k(t)δ(x)φ (x,t)] = is φ (x,s) iφ (x, 0) (5.50) 22 2 1 2 − 2

In the next coming sections we will discuss the different cases in which the strength of the Dirac delta Potential coupling the two states is varied in time and a solution for the single equation is obtained.

153 5.2.1 Case 1 : Linear time dependence of the strength of the Dirac Delta potential

We consider the equations (5.51) and (5.52) as given below.

H φ (x,s) + 2L[k(t)δ(x)φ (x,t)] = is φ (x,s) iφ (x, 0) (5.51) 11 1 2 1 − 1 H φ (x,s) + 2L[k(t)δ(x)φ (x,t)] = is φ (x,s) iφ (x, 0) (5.52) 22 2 1 2 − 2

we put wave packet at time t = 0 on the ﬁrst potential, hence for second state in our

problem φ2(x, 0) = 0, hence above equation reduced to

H φ (x,s) + 2L[k(t)δ(x)φ (x,t)] = is φ (x,s) iφ (x, 0) (5.53) 11 1 2 1 − 1

H22 φ2(x,s) + 2L[k(t)δ(x)φ1(x,t)] = is φ2(x,s) (5.54)

now we will consider the second term on the left hand side in both the equations which is nothing but

δ(x)2L[k(t)φ1(x,t)]

for k(t)= αt the above equations can be rewritten as

∂ H φ (x,s) 2δ(x)α φ (x,s)= is φ (x,s) iφ (x, 0) (5.55) 11 1 − ∂s 2 1 − 1 ∂ H φ (x,s) 2δ(x)α φ (x,s)= is φ (x,s) (5.56) 22 2 − ∂s 1 2

Equation (5.56) can be rewritten as

∂ (is H )) φ (x,s)= 2δ(x)α φ (x,s) (5.57) − 22 2 − ∂s 1 or 1 ∂ φ (x,s)= (is H )− 2δ(x)α φ (x,s) (5.58) 2 − − 22 ∂s 1

154 using the value of φ2(x,s) from above equation, equation (5.55) can be rewritten as

2 ∂ 0 H φ (x,s) + 4α δ(x) G (0, 0,s)φ0 (x,s) = is φ (x,s) iφ (x, 0) (5.59) 11 1 ∂s 2 1 1 − 1 0 1 where G (0, 0,s) = δ(x) (is H )− δ(x) is the Green’s function for the second 2 h | − 22 | i state. We will now solve equation(5.59) by using a method similar to one discussed in Reference [122], solving equation(5.59) is equivalent to solving the following two equations

~2 ∂2 = is φ (x,s) iφ (x, 0) (5.60) − 2m ∂x2 1 − 1

∂φ1(x,s) ∂φ1(x,s) 2 ∂ 0 + = 4α G (0, 0,s)φ0 (0,s) ∂x |x=0 − ∂x |x=0− ∂s 2 1

In order to solve equation (5.60), we should consider the homogeneous solution in order to satisfy the discontinuity condition at the point of coupling. Its solution is given by

1 0 0 0 φ1(x,s)= η(s) exp(i√is x )+ dx exp(i√is x x )φ1(x , 0) (5.61) | | 2√is | − | Z where η(s) is an arbitrary function of s, which needs to be determined. Using the wave function given by equation(5.60) in equation (5.60), at the point of coupling we get

2 ∂ 0 2i√is η(s) = 4α G (0, 0,s)φ0 (0,s) (5.62) ∂s 2 1 or

2 ∂ 0 0 2α ∂s G2(0, 0,s)φ1(0,s) η(s)= i√is (5.63)

hence, the wave function can be written as

2 ∂ 0 0 2α ∂s G2(0, 0,s)φ1(0,s) 1 0 0 0 φ1(x,s)= exp(i√is x )+ dx exp(i√is x x )φ1(x , 0) i√is | | 2√is | − | Z (5.64)

155 at point of coupling i.e. x = 0, the above equation reduces to

2 ∂ 0 0 2α ∂s G2(0, 0,s)φ1(0,s) 1 0 0 0 φ1(0,s)= + dx exp(i√is x )φ1(x , 0) (5.65) i√is 2√is | | Z

3 using the substitution y = s 2 the above equation can be modiﬁed and written as

2 3 3α ∂ 0 2 2 1 2 2 3 0 3 0 √ 3 0 0 φ1(0,y )= G2(0, 0,y )φ1(0,y ) + 2 dx exp(i iy x )φ1(x , 0) i√i ∂y 2√iy 3 | | h i Z (5.66) integrating above equation on both sides we get

2 ∞ 3 3α 0 2 ∞ 1 2 2 0 3 0 0 √ 3 0 0 φ1(0,y )dy = G2(0, 0,y )u00(y) + dy 2 dx exp(i iy x )φ1(x , 0) y i√i y 2√iy 3 | | Z h i Z Z (5.67) or

2 3α 0 2 ∞ 1 2 3 0 0 √ 3 0 0 u(y)= G2(0, 0,y )u00(y) + dy 2 dx exp(i iy x )φ1(x , 0) i√i y 2√iy 3 | | h i Z Z (5.68)

3 ∞ 2 0 where u(y) = y φ1(0,y )dy and so on. Dividing both sides of equation by the second term of aboveR equation we will get

2 0 1 0 0 0 2 i√i ∞ dy 2 dx exp(i√iy 3 x )φ1(x , 0) ∂ i√i y 2√iy 3 | | 2 u(y)+ 2 u(y) 2 2 = 0 ∂y 2 0 3 2 0 3 3 − 3α G2(0, 0,y ) − R 3α GR2(0, 0,y )2√iy (5.69) equation(5.69) is an integral equation which can be converted back to Laplace domain

3 by using the substitution given by y = s 2 and can be solved numerically [124]. This will determine the wave function at the point of coupling, so if the wave function can be determined at the origin then it can be determined everywhere and from the wave function we can calculate the nonadiabatic transition probability when strength of the Dirac Delta potential has linear variation in time. Further sub-sections will proceed with inverse and exponential variation in time.

156 5.2.2 Case 2: Solution for strengthof the Dirac Delta potential having inversely proportional dependence on time

We again start with equations (5.53) and (5.54) i.e.

H φ (x,s) + 2L[k(t)δ(x)φ (x,t)] = is φ (x,s) iφ (x, 0) (5.70) 11 1 2 1 − 1

H22 φ2(x,s) + 2L[k(t)δ(x)φ1(x,t)] = is φ2(x,s) (5.71)

considering

δ(x)2L[k(t)φ1(x,t)] (5.72) α with k(t)= (5.73) t α δ(x)2L[ φ (x,t)] (5.74) t 1

the above equations can be rewritten as

∞ H φ (x,s) + 2δ(x)α dsφ (x,s)= is φ (x,s) iφ (x, 0) (5.75) 11 1 2 1 − 1 Zs ∞ 0 0 H22 φ2(x,s) + 2δ(x)α ds φ1(x,s )= is φ2(x,s) (5.76) Zs0

From equation (5.76) we will ﬁnd the value of φ2(x,s) given as

1 ∞ φ (x,s)=(is H )− 2δ(x)α ds0 φ (x,s0) (5.77) 2 − 22 1 Zs0 putting this value back into equation (5.75) we get

∞ 1 ∞ H φ (x,s) + 2δ(x)α ds(is H )− 2δ(x)α ds0 φ (x,s0 )= is φ (x,s) iφ (x, 0) 11 1 − 22 1 1 − 1 Zs Zs0 or

∞ ∞ H φ (x,s) + 4α2δ(x) dsG0(0, 0,s) ds0 φ (x,s0 )= is φ (x,s) iφ (x, 0) 11 1 2 1 1 − 1 Zs Zs0 (5.78) 157 0 1 where G (0, 0,s) = δ(x) (is H )− δ(x) is the Green’s function for the second 2 h | − 22 | i state. We will now solve equation(5.78) by using a method similar to one discussed in Reference [122], hence the above equation at the point of coupling can be further reduced to the following two equation given by

~2 ∂2 = is φ (x,s) iφ (x, 0) (5.79) − 2m ∂x2 1 − 1 ∞ ∞ ∂φ1(x,s) ∂φ1(x,s) 2 0 0 0 + = 4α dsG (0, 0,s) ds φ (0,s ) ∂x |x=0 − ∂x |x=0− 2 1 Zs Zs0

In order to solve equation (5.79), we should consider the homogeneous solution in order to satisfy the discontinuity condition at the point of coupling. Its solution is given by

1 0 0 0 φ1(x,s)= η(s) exp(i√is x )+ dx exp(i√is x x )φ1(x , 0) (5.80) | | 2√is | − | Z where η(s) is an arbitrary function of s, which needs to be determined. Using the wave function given by equation(5.80) in equation (5.79), at the point of coupling we get

∞ ∞ 2 0 0 0 2i√is η(s) = 4α dsG2(0, 0,s) ds φ1(0,s ) (5.81) Zs Zs0 or 2 0 0 0 2α δ(x) ∞ dsG (0, 0,s) ∞ ds φ1(0,s ) η(s)= s 2 s0 R i√is R (5.82)

hence, the wave function can be written as

2 0 4α ∞ dsG (0, 0,s) ∞ ds0 φ (0,s0 ) ∞ s 2 s0 1 1 0 0 0 φ1(x,s)= exp(i√is x )+ dx exp(i√is x x )φ1(x , 0) R i√is R | | 2√is | − | Z−∞ (5.83) at point of coupling i.e. x = 0, the above equation reduces to

2 0 4α ∞ dsG (0, 0,s) ∞ ds0 φ (0,s0 ) ∞ s 2 s0 1 1 0 0 0 φ1(0,s)= + dx exp(i√is x )φ1(x , 0) R i√is R 2√is | | Z−∞ (5.84)

158 In the above equation let us consider that by solving the integral we will get a particular

∞ 0 value for the given Green’s function i.e. we put s dsG2(0, 0,s) = β then above equation will be reduced to R

2 ∞ ∞ 4α β 0 0 1 0 0 0 φ1(0,s)= ds φ1(0,s )+ dx exp(i√is x )φ1(x , 0) (5.85) i√is s0 2√is | | Z Z−∞ or 2 4α β ∞ 1 ∞ 0 00 00 0 0 0 φ1(0,s )= ds φ1(0,s )+ dx exp(i√is0 x )φ1(x , 0) (5.86) i√is s00 2√is0 | | Z Z−∞ integrating the above equation on both sides we get

2 ∞ ∞ 4α β ∞ ∞ 1 ∞ 0 0 0 00 00 0 0 0 0 ds φ1(0,s )= ds ds φ1(0,s )+ ds dx exp(i√is0 x )φ1(x , 0) s s i√is s00 s 2√is0 | | Z Z Z Z Z−∞ (5.87)

∞ 0 0 by considering the expression u(s) = s ds φ1(0,s ), the above expression considerably reduces to the following equationR given as

2 ∞ ∞ ∞ 0 4α β 0 0 1 0 0 0 u(s)= ds u(s )+ ds dx exp(i√is0 x )φ1(x , 0) (5.88) s i√is s 2√is0 | | Z Z Z−∞ or 2 ∞ ∞ 0 4α β 0 1 0 0 0 u(s)= ds u(s )+ dx exp(i√is0 x )φ1(x , 0) (5.89) s i√is 2√is0 | | Z Z−∞

2 8α β√s0 − multiplying the above equation on both sides by e i√i and after some mathematics above equation can be further written as

2 √ 8α β s0 ∞ ∂ 0 − 1 0 0 0 u(s ).e i√i = dx exp(i√is0 x )φ1(x , 0) (5.90) ∂s0 −2√is0 | | Z−∞ again integrating the above equation and using some mathematics we solved above equation and ﬁnally obtained

2 8α β√s0 − i√i ∞ 0 e 0 2 0 0 φ1(0,s )= dx exp √is0 (8α β + i x ) φ1(x , 0) (5.91) − 2√is0 | | Z−∞ h i equation(5.91) is an integral equation in Laplace domain which will determine the wave function at the point of coupling, so if the wave function can be determined at the origin 159 then it can be determined everywhere and from the wave function we can calculate the nonadiabatic transition probability when strength of the Dirac Delta potential has inversely proportional dependence in time.

5.2.3 Case 3: Solution for strength of the Dirac Delta potential having exponential dependence on time

In this case we will consider k(t) = -b for t< 0 and k(t) = βexp( αt) for t> 0, where 0 − b0 s a real number such that b0 > 0, β is real and α > 0. We will consider the bound state for k(t) = -b as the initial condition where φ (x0 , 0) = √b exp( b x0 ). We 0 1 0 − 0| | again start with equations (5.53) and (5.54)i.e.

H φ (x,s) + 2L[k(t)δ(x)φ (x,t)] = is φ (x,s) iφ (x, 0) (5.92) 11 1 2 1 − 1

H22 φ2(x,s) + 2L[k(t)δ(x)φ1(x,t)] = is φ2(x,s) (5.93)

considering

δ(x)2L[k(t)φ1(x,t)] (5.94)

with k(t)= βexp( αt) −

above equations can be rewritten as

H φ (x,s) + 2δ(x)βφ (x,s + α)= is φ (x,s) iφ (x, 0) (5.95) 11 1 2 1 − 1

H22 φ2(x,s) + 2δ(x)βφ1(x,s + α)= is φ2(x,s) (5.96)

From equation (5.96) we will ﬁnd the value of φ2(x,s) given as

1 φ (x,s + α)=(is + iα H )− 2δ(x)βφ (x,s + 2α) (5.97) 2 − 22 1

160 putting this value back into equation (5.95) we get

1 H φ (x,s) + 2δ(x)β(is + iα H )− 2δ(x)βφ (x,s + 2α)= is φ (x,s) iφ (x, 0) 11 1 − 22 1 1 − 1 or

H φ (x,s) + 4β2δ(x)G0(0, 0,s + α)φ (x,s + 2α)= is φ (x,s) iφ (x, 0) 11 1 2 1 1 − 1 (5.98)

0 1 where G (0, 0,s + α) = δ(x) (is + iα H )− δ(x) is the Green’s function for 2 h | − 22 | i the second state. We will now solve equation(5.98) by using a method similar to one discussed in Reference [122], hence the above equation at the point of coupling can be further reduced to the following two equation given by

~2 ∂2 = is φ (x,s) iφ (x, 0) (5.99) − 2m ∂x2 1 − 1

∂φ1(x,s) ∂φ1(x,s) 2 0 + = 4β G (0, 0,s + α)φ (0,s + 2α) ∂x |x=0 − ∂x |x=0− 2 1

In order to solve equation (5.99), we should consider the homogeneous solution in order to satisfy the discontinuity condition at the point of coupling. Its solution is given by

1 0 0 0 φ1(x,s)= η(s) exp(i√is x )+ dx exp(i√is x x )φ1(x , 0) (5.100) | | 2√is | − | Z where η(s) is an arbitrary function of s, which needs to be determined. Using the wave function given by equation(5.100) in equation (5.99), at the point of coupling we get

√ 2 0 2i is η(s) = 4β G2(0, 0,s + α)φ1(0,s + 2α) (5.101) or 4β2δ(x)G0(0, 0,s + α)φ (0,s + 2α) η(s)= 2 1 i√is (5.102)

161 hence, the wave function can be written as

2 0 4β G (0, 0,s + α)φ (0,s + 2α) 1 ∞ 2 1 0 0 0 φ1(x,s)= exp(i√is x )+ dx exp(i√is x x )φ1(x , 0) i√is | | 2√is | − | Z−∞ (5.103) at point of coupling i.e. x = 0, the above equation reduces to

2 0 ∞ 4β G2(0, 0,s + α)φ1(0,s + 2α) 1 0 0 0 φ1(0,s)= + dx exp(i√is x )φ1(x , 0) i√is 2√is | | Z−∞ (5.104) now we will consider the second term in the above equation which can be written as

1 ∞ 0 0 0 dx exp(i√is x )φ1(x , 0) (5.105) 2√is | | Z−∞ with φ (x0 , 0) = b exp( b x0 ) 1 0 − 0| | p the above equation will become

1 ∞ 0 0 0 dx exp(i√is x ) b0exp( b0 x ) (5.106) 2√is | | − | | Z−∞ p solving equation (5.104)can be rewritten as

2 0 2√b0 4β G2(0, 0,s + α)φ1(0,s + 2α) φ1(0,s)= − + (5.107) 2√is(i√is b ) i√is − 0 the above equation can be iterated to obtain a series solution i.e.

∞ p φ1(0,s)= β qp(s) (5.108) n=o X

162 using (5.108) in (5.107) we get

2√b0 qp(s)= − , p = 0 2√is(i√is b ) − 0 and q 1 q − 0 (√i) √b0 4G2(0, 0,s + α) qp(s)= , p> 0 i(s + 2nα)(i i(s + 2nα) b0) j=0 i i(s + 2jα) − Y p p p (5.109)

5.3 Non-adiabatic transition probability with a moving δ potential coupling

Methodology-:Transformation of time-dependent Schrodinger equation into time-independent Schrodinger equation using a time dependent scaling factor

We start with a similar methodology applied to our two state model as discussed by Lee [124] and also by Berry and Klein [125] where the problem of meta stability of a particle of mass µ trapped in a potential U(x,t) with a scaling form namely U(x,t) = x 1 U( R(t)) R2(t) with R(t) as a time dependent scaling factor is used to transform the time- dependent Schrodinger equation into independent one. We consider time dependent Schrodinger equation for a two state model as

∂ φ1(x,t) H11 U12 φ1(x,t) i~ = . (5.110) ∂t       φ2(x,t) U21 H22 φ2(x,t)      

163 where

~2 H = + V (x) (5.111) 11 −2µ 1 ~2 H = + V (x) (5.112) 22 −2µ 2 U x U = U = (5.113) 12 21 R2(t) R(t)

where the scaling factor R(t) is assumed to be a linear function of time R(t)= R0 + vt, x with v = constant. Further in the rescaled frame with rescaled coordinate x = R(t), equation (1) can be written as

∂ ~2 ∂2 U x i~ φ (x,t)= + V (x) φ (x,t)+ φ (x,t) ∂t 1 −2µ ∂x2 1 1 R2(t) R(t) 2 ∂ ~2 ∂2 U x i~ φ (x,t)= + V (x) φ (x,t)+ φ (x,t) ∂t 2 −2µ ∂x2 2 2 R2(t) R(t) 1 (5.114)

In the two state model our lower potential V1(x) = 0 hence above equations considerably reduces to

∂ ~2 ∂2 U x i~ φ (x,t)= φ (x,t)+ φ (x,t) ∂t 1 −2µ ∂x2 1 R2(t) R(t) 2 ∂ ~2 ∂2 U x i~ φ (x,t)= + V (x) φ (x,t)+ φ (x,t) ∂t 2 −2µ ∂x2 2 2 R2(t) R(t) 1 (5.115)

In the rescaled frame as mentioned above equation (5.115) can be further written as

∂ ~2 ∂2 v ∂ i~ φ (x,t)= φ (x,t)+ i~ x + U(x) φ (x, t) ∂t 1 −2µR2 ∂x2 1 R ∂x 2 ∂ ~2 ∂2 v ∂ i~ φ (x,t)= + V (x,t) φ (x,t)+ i~ x + U(x) φ (x, t) ∂t 2 −2µR2 ∂x2 2 2 R ∂x 1 (5.116)

164 Further the equation given below is obtained by making the substitution given in equation (5.118) and (5.119)

∂ ~2 ∂2 i~ φ (x, τ)= φ (x, τ)+ U (x)φ (x, τ) ∂t 1 −2m ∂x2 1 12 2 ∂ ~2 ∂2 i~ φ (x, τ)= + V (x) φ (x, τ)+ U (x)φ (x, τ) ∂t 2 −2m ∂x2 2 2 12 1 (5.117)

with

1 iµ 2 ( ~ )Rvx φ1(x,t)= e 2 ψ1(x, t) R(t)

1 iµ 2 ( ~ )Rvx φ2(x,t)= p e 2 ψ2(x, t) R(t) (5.118) p and introducing a new variable

t ds t τ = = (5.119) R2(s) R R(t) Z0 0 equation (5.117) resembles the Schrodinger equation with a stationary state potential which can be solved by the separation of variables technique. i.e.

i ( ~ )Eτ ψ(x, τ)= ψ(x)e− (5.120)

where Ψ(x) satisfy the eigen value equation

~2 d2 + U(x) ψ (x)= E ψ (x) (5.121) −2m dx2 k k k Above equation can be exactly solvable in the rescaled frame where the exact wave function in the rescaled frame is given by

1 iµ v 2 i 1 x ( ~ )( )x ( ~ )( )Et φk(x,t)= e 2 R e− RoR ψk (5.122) R(t) R p 165 where the set of solutions (5.122) is complete and orthonormal.

φ (x,t) φ (x,t) = φ (x) φ (x) = δ (5.123) h k | l i h k | l i kl furthermore if an initial state φ(x, 0) is expressible in the basis ψ as { k}

φ(x, 0) = c φ (x, 0),c = ψ (x, 0) ψ(x, 0) (5.124) k k k h k | i Xk then at a later time t the state is given as

ψ(x,t)= ckφk(x,t) (5.125) Xk

5.3.1 Non adiabatic transition probability for two state coupled by a moving δ potential coupling

Using the methodology discussed in the preceding section we will consider the case of crossing of two diabatic potentials coupled by moving δ potential. The time dependent Schrodinger equation in this case can be written as

∂ φ1(x,t) H11 U12 φ1(x,t) i~ = . (5.126) ∂t       φ2(x,t) U21 H22 φ2(x,t)       where

~2 H = + V (x) (5.127) 11 −2µ 1 ~2 H = + V (x) (5.128) 22 −2µ 2 U U x U = U = o δ(x a(t)) = o δ a (5.129) 12 21 R(t) − R2(t) R(t) − where a(t) = a R(t)> 0 gives the position of the δ potential. In our case we have V1 =0 and potential V is incorporated into energy E giving a new energy E0 i.e. E0 = E V 2 − 2 which is further equals to E for simplicity. In the rescaled frame as discussed in the 166 last section our coupling is transformed as U = U = U δ(x a) hence, our time 12 21 o − dependent equations will be converted into time independent one and in the matrix notation can be written as

H11 U oδ(x a) φ1(x) φ1(x) − = E (5.130)  U δ(x a) H   φ (x)   φ (x)  o − 22 2 2       The above equation is equivalent to the following

H φ (x)+ U δ(x a)φ (x)= Eφ (x) (5.131) 11 1 o − 2 1 H φ (x)+ U δ(x a)φ (x)= Eφ (x) (5.132) 22 2 o − 1 2 equation (5.132) can be written as

1 φ (x)=(E H )− U δ(x a)φ (x) (5.133) 2 − 22 o − 1 using equation (5.133) into equation (5.132) we get[122]

1 H φ (x)+ U δ(x a)(E H )− U δ(x a)φ (x)= Eφ (x) (5.134) 11 1 o − − 22 o − 1 1 which will gave us ﬁnally

2 H φ (x)+ U Go(0, 0, E)δ(x a)φ (x)= Eφ (x) (5.135) 11 1 o 2 − 1 1 to simplify further the above equation can be written as

H φ (x)+ V δ(x a)φ (x)= Eφ (x) (5.136) 11 1 o − 1 1

o 1 equation(5.136) is achieved by making the following substitutions G2(0, 0, E)= x (E H a and h | − 22 | i 2 o V o = U oG2(0, 0, E). Above equation is a time independent Schrodinger equation in

167 φ1(x) whose general solutions can be written as

φ1(x)= sin(kx), x< a  φ1(x)= A cos(kx + θ), a< x

 2µE where k = ~ , A is real constant and θ is a phase angle. The wavfunction and its derivative mustq satisfy the following boundary conditions at x = a i.e.

+ φ1(x = a )= φ1(x = a−)

and

dφ1(x) dφ1(x) 2µ + = V φ (x = a) (5.137) dx |(x=a ) − dx |(x=a−) ~2 o 1 from the above relations the coefﬁcient A as a function of k can be determined as

2 2µV A2(k)= sin2(ka)+ cos(ka)+ o sin(ka) (5.138) ~2k this value of A2(k) can be interpreted as the ratio of probability of ﬁnding the particle for a particular value of k. Taking the superposition of scattering states with k near kn ~2 2 2 kn using scattering state method and expanding A about En = 2µ we can write as

1 2 2 2 − 2 µa 2µV o 2 2µV o A (E) 1+ (E En + δ) + 1+ ≈ ~2k ~2k − ~2k n " n # " n # D2(∆ + δ)2 + H2(5.139) ≡

168 where

∆= E E − n and 2 µa 2 2µV D2 = 1+ o ~2k ~2k n " n # 2 1 2µV − H2 = 1+ o ~2k " n # 2 1 2V 2µV − δ = o 1+ o (5.140) a ~2k " n # now the scattering states having E near En can be written as

2 1 φ∆(x)= sin(kx), x< a J √D2(∆+δ)2+H2  q φ∆(x)= cos(kx + θ), a< x

 this system is quantized between [0, J], where J> a. From these scattering states an initial state is constructed in such a way that

ψ(x, τ =0)= ∆ c∆φ∆(x)= ψn(x), x< a  P ψ(x, τ =0)= ∆ c∆φ∆(x) = 0, x> a

 P coefﬁcient c∆ can be calculated from the orthogonality of states φ∆(x),

J c∆ = dxφ∆(x)ψ(x, 0) Zo 2 1 a = dxsin(kx)ψn(x) (5.141) 2 2 2 J D (∆+ δ) + H 0 r Z p If we choose 2 nπx ψ (x) sin , n = 1, 2, 3.. (5.142) n ≈ a a r

169 we get

1 1 a nπx c∆ dxsin(kx)sin 2 2 2 ≈ √Ja D (∆ + δ) + H 0 a Z a 1 p (5.143) 2 2 2 ≈ rJ D (∆ + δ) + H p hence initial state is then given by

a 1 ψ(x, τ = 0) φ∆(x) (5.144) ≈ J 2 2 2 r ∆ D (∆ + δ) + H X p from equation (5.122),(5.124) and (5.125), the solution at a later time τ is given by

a 1 i ( ~ )(En+∆)τ ψ(x, τ) φ∆(x)e (5.145) ≈ J 2 2 2 r ∆ D (∆+ δ) + H X p 0 π since the system is quantized in the interval [0, J], we have kJ = n 2 . Replacing the sum by an integral J 2µ d∆ ~ (5.146) → π sEn X∆ Z equation (5.145) can be written as

∞ J 2µ a 1 i ( ~ )(En+∆)τ ψ(x, τ) d∆ φ∆(x)e (5.147) ~ 2 2 2 ≈ π sEn J D (∆+ δ) + H Z∞ r p using the value of φ∆(x)from the preceding expressions we can approximate the wavefunction as

( i )∆τ 2 µa i ∞ e− ~ ( ~ )Enτ ψ(x, τ) sin(knx)e− d∆ ~ 2 2 2 ≈ π sEn × D (∆ + δ) + H Z∞

2 µa sin(k xp) i 1 H n ( ~ )Enτ ( ~ ) τ = e− e− | D | (5.148) ~ E HD s n | |

Now the probability of ﬁnding the particle is given by

P (t) exp [ α (t)] (5.150) ≈ − n and non adiabatic transition probability is given by

P(non adiabatic) = 1 P (t) = 1 exp [ αn(t)] (5.151) − − − − where H t α(t) = 2 (5.152) D R R(t) o

using equation (5.119) and (5.140) above equation can be further written as

1 ~2k 2µV − t α(t) = 2 n 1+ o (5.153) µa ~2k RoR(t) n

5.4 Conclusions

The current chapter involves the exact solution of the two state problems which have either time dependent Schroedinger equations under consideration or the coupling between two states is time dependent. It is covered in a range of different subsections in which the time dependent problems are treated explicitly. In the ﬁrst subsection we presented an alternative method for exact solution of time dependent Schrodinger equations for the two state scattering problem in Laplace domain. In the second subsection we consider explicit case of time dependent coupling and an exact solution in Laplace domain is presented along with few cases in which the strength of the Dirac Delta coupling has varied time dependence. In the last subsection we have provided an exact solution of the two state problem by calculating the non adiabatic transition probability with a moving δ potential coupling.

171 Chapter6 In the last few chapters we have consider differentmodel of curve crossing and provided an exact solution in different cases. In the present chapter we will discuss the case of a real molecule through computational packages to know weather there is any existence of such type of curve crossing in real molecules or not. We consider a simple example of Lithium chloride molecule and using CASPT2 and MRCI level of theory we have investigated the curve crossing in these molecules using MOLPRO -: A complete system of ab initio programs for molecular structure calculations. The generated adiabatic and diabatic potential energy surfaces can be used to understood the spectroscopy as well as non adiabatic transitions in Lithium chloride molecule.

172 CHAPTER 6

AVOIDED CROSSING IN LITHIUM CHLORIDE USING CASPT2 AND MRCI LEVEL OF THEORY.

Diatomic metal halides like LiCl have been of keen interest due to their important applications in semiconductor industry [127, 128, 129]. Such type of metal halides including the system under study served as a framework for understanding non adiabatic transitions as well as wave packet dynamics due to a special feature of the PEC that two adiabatic PEC exchange the ionic and covalent character at the avoided crossing point which can lead to a study of dissociation dynamics of various molecules. Other factors of investigating the system under study is the abundance of lithium atom in old stars which results from the spallation reactions during the decade of big-bang theory [130, 131]. The aqueous solution of lithium chloride have been found in various applications in different fields of biology, medicine and electro chemistry[132]. Apart from above, in quantum physics/chemistry such kind of metal halides plays an important role in understanding the curve crossing problems using Landau Zener method [42, 43]. Metal halides are also studied theoretically as well as experimentally for the quantum control of chemical reactions [133, 134, 135, 136]. Due to wide us- age of LiCl molecule in different fields, it was investigated not only experimentally [137, 138, 139, 140, 141, 142, 143] but also theoretically [144, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154, 155, 156, 157, 158, 159]. LiCl for the first time was studied experimentally by Klemperer and Rice in 1957 and he reported the infrared spectra of LiCl in his studies. The first theoretical research on lithium chloride was carried out by Matcha in 1967 who studied the electronic structure and molecular properties using molecular orbital approximation method. In the past few years with the development of quantum chemistry and different advance methods it is possible to ex- plore the detailed electronic structure of small molecule with high accuracy using ab initio calculations. There have been a lot of reports and citations in the literature on the different low lying electronic states of lithium molecule and its spectroscopic constants, but a few reports on avoided crossing /curve crossing which in one or the other way are necessary to study the wavepacket motion and non adiabatic transitions. Using the concept of avoided crossing, real time wave packet motion and decay processes in diatomic molecule have been studied by Zewails and his coworkers. Apart from this non adiabatic transitions also plays an important role in electron transfer mechanisms [53, 54, 55, 56, 57, 58, 59, 60, 61, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78]. Curve crossing is an interdisciplinary concept and it finds numerous applications in various fields of sciences like physics, chemistry and even in biology. In late 1932, the work on curve crossing and non-crossing were published by Landau, Zener , Stuckelberg and by Rosen-Zener respectively. In our earlier publications we have reported and found the exact analytical solutions where two or more arbitrary potentials are coupled by Dirac Delta interactions [55, 56, 57, 58, 59, 60, 61, 62, 63]. We have also found an exact analytical solution for the case where two or more potentials are coupled by arbitrary coupling i.e. Gaussian coupling. In these cases we have used the Green’s function technique to find the exact analytical solutions of curve crossing problems. While re- viewing the literature we found that there are very few papers which deals with the curve crossing/avoided crossing of real molecules so we choose a real system to investigate upon and to discuss about the avoided crossing and crossing of the potential energy curves. Another point of investigating the system under study is to construct the diabatic and adiabatic potential energy curves for a real molecule and then applying our Green’s function technique to calculate the transition probability form one diabatic state to another diabatic state. Also Curve crossing /avoided crossing helps in understanding the dissociation dynamics of a diatomic molecule as potential energy curve crossing plays an important role in the dissociation dynamics as well as pre dissociation dynamics[160]. In the present chapter we have used MRCI and CASPT2 level of theory to investigate the low lying electronic states of LiCl molecule to shows the region of avoided crossing . These are basically the 1 +ground state and B1 + first excited state potential energy curves in the region ofP avoided crossing. AdiabaticP and diabatic potential energy curves are also calculated for the low lying electronic states of Lithium Chloride molecule over an inter-nuclear separation ranging from 1Å to 10 174 Å. From these potential energy curves we found that there is crossing of the diabatic potential energy curves while repulsion of the adiabatic curves. Comparisons have been also made with previously calculated results using different/same level of theory.

6.1 Computational details used in the calculations

All the ab-initio calculations in the present paper have been done using the MOLPRO program package, version 2012.1, designed and maintained by Werner and Knowles. LiCl molecule belongs to the point group C V. All calculations have been performed ∞ within the point group C2v, which is a subgroup of C V due to the limitation of the ∞ MOLPRO program. In the said point group the electronic state symmetries are classi-

fied according to the four irreducible representations namely A1, B1, B2 and A2 respectively. Firstly, a reference calculation have been performed in C2v symmetry and then the diabatic orbitals for displaced geometries are obtained at Cs symmetry. For calculation of diabatic orbitals we have considered VTZ basis set for lithium atom while AVTZ set for chlorine atom. Reference calculations are considered at an inter-nuclear distance of 1Å, and then the diabatic orbitals are calculated at an inter-nuclear distance varying from2Å to 10Å as per the propertyof the procedure. The active space for including correlation effects consists of both active as well as inactive spaces. Electron density plots for diabatic orbitals have been visualized and plotted using GMOLDEN program and are given in Figure 6.1. The non-adiabatic coupling matrix elements have been computed by finite differences for Multi configuration self-consistent field method or CI wave functions using DDR procedure. These couplings basically involve the derivative of the electronic wave functions with respect to the single inter nuclear coordinate[161]. For calculating the NACME in case of system under study we have used VTZ and AVTZ basis set for lithium and chloride respectively over a range of inter nuclear distances varying from 2 to 10 Å. The typical strategy for calculating the NACME is as: The wave functions have been computed at the reference geometry i.e. 1Å. The matrix elements are computed for the MCSCF wave functions and further for CI wave functions as well. For a first order algorithm (two point method), the wave functions 175 at positively displaced geometries are calculated while for a second order algorithm (three point method) wave functions are calculated at the negatively displaced geometries. To compute the avoided crossing between different states of system under study the single state single reference (SS-SR-CASPT2) and multi state multi reference (MS- MR-CASPT2) level of theory have been used to find out the avoided crossing between different states using same basis set over a range of inter nuclear distances. Using quasi- diabatization method MRCI diabatic and adiabatic potential energy curves for Lithium Chloride also have been computed using ab initio methods. Quasi diabatic states have such property that the overlap between the states at the current geometry with those at reference geometry is maximized by performing a unitary transformation among the given states. Due to symmetry the diabatic and the adiabatic states are identical at the reference geometry. The Diabatization generally requires two steps. In the first step the active orbitals in the CASSCF calculation are rotated to maximize the overlap with the orbitals at the reference geometry and in the second by finding the transformation among the CI vectors. By using the unitary transformation

d ad ψm = ψn Unm (6.1) n X one can obtain the diabatic states from the adiabatic states. U is the transformation matrix and is chosen such that the non-adiabatic coupling matrix elements

∆ ψd ψd (6.2) h m|∆q n|i | are minimized over for all internal coordinates.

6.1.1 Diabatic orbitals

The diabatic orbitals needed for the construction of the diabatic states have been computed using DIAB procedure as implemented in molpro. For constructing diabatic states, it is compulsory to find the mixing of the diabatic states in the adiabatic wave functions. This mixing can be achieved by the integrationof the NACME (non-adiabatic 176 coupling matrix elements) or by inspection of the CI coefficients of the MCSCF( Multi configuration self-consistent wave functions) or CI (configuration Interaction) wave functions. For complete active space self-consistent wave functions it can be achieved by maximizing the overlap of the active orbitals with those of reference geometry, where wave functions are assumed to be diabatic. DIAB procedure as implemented in molrpo is also used to maximize the orbital overlap. Different diabatic orbitals/electron density plots for diabatic orbitals computed under different irreducible representations are plotted using GMOLDEN interface in Euclid mode of plotting. The grid point density for plotting the orbitals consists of (61x61) grid points along with maximum and minimum values as indicated in the plotting. The plotted electron density plots for diabatic orbitals are shown in Figure 6.1. The diabatic orbitals are numbered as 1.1, 1.2 etc. The labels 1.1 used for diabatic orbitals can be read as 1.1 means 1st orbital in 1st irreducible representation, while 1.2 means 1st orbital in second irreducible representation and so on. The axis labeling for such plots is done as to indicate the molecule lying along z-axis joined by inter nuclear distance and +/- x-axis represents the maximum and minimum value of electron density. From the electron density plots of diabatic orbitals under irreducible representation we can conclude that diabatic orbital no 1.1, 1.2, 4.1, 8.1, 9.1 have maximum electrons density along +ve axis, due to chlorine atom while 5.1 have due to lithium atom, however if we have a look at diabatic orbitals 6.1 as well as 2.1 it is in opposite direction due to chlorine atom. The value of electron density is equal in some of the diabatic orbitals. The occupancy and eigen values for different diabatic orbitals have been summarized in Table 6.1.

6.1.2 NACME(Non-adiabatic coupling matrix elements)

Non-adiabatic coupling matrix elements can be computed using the DDR procedure[162] described in MOLPRO by finite differences for MCSCF or CI wave functions. For com- puting the coupling matrix elements by finite differences, we have to compute and store the wave function at the displaced geometries. In the present work we have only used first order algorithm (two point method) to compute the wave functions at the positively displaced geometries starting from 2Å to 10Å with reference calculations at 1Å. Using 177 Figure 6.1: Shapes of electron density plots of diabatic orbitals under different irreducible representations.

Table 6.1: Diabatic orbitals along with occupancy and eigenvalue for lithium chloride calculated using DIAB procedure Orbital no. Orbital Occupancy Eigen Value 1 ¯1.1 ¯2.0 ¯-102.21 2¯ 2.1 ¯2.0 ¯-9.04 3 ¯3.1a ¯2.0 ¯-8.23 4¯ 4.1 ¯2.0 ¯-6.26 5¯ 5.1 ¯2.0 ¯-2.29 6¯ 6.1 ¯1.8 ¯-3.05 7¯ 7.1 ¯1.5 ¯-0.51 8¯ 8.1 ¯1.9 ¯-4.13 9¯ 9.1 ¯1.6 ¯-1.36 10¯ 1.2b ¯2.0 ¯-8.24 11¯ 2.2 ¯1.0 ¯-0.42

a3.1 implies 3rd orbital in 1st irreducible representation b st 1.2 implies 1 orbital in 2nd irreducible representation.

178 Figure 6.2: NACME for lithium chloride computed using ﬁnite differences. the trans directive of the CI program the transition density matrix between the states at the reference geometry and then at the displaced geometries also have been computed using ab initio methods. For the calculation of NACME, the diabatic CASSCF orbitals are generated in two CASSCF calculations at the displaced geometries by maximizing the overlap with the orbitalsat the reference geometry and to a very good approximation NACME are obtained from the CI-vectors. Calculated NACME for lithium chloride using different basis set for lithium and chlorine atom along with their values are shown in Figure 6.2 and Table 6.2. The full curves as represented in Figure 6.2 are the computed NACME. The full curves represents the computed NACME for forward difference ( 2-point formula) , backward difference (3-point formula) average of both of them and alone NACME computed using 3 point formula. By looking at the curves computed for NACME we found a slight variation in curves around 2 to 4 Å, which can be attributed to correlation effects. 179 Table 6.2: Non-adiabatic couplings for LiCl calculated using DIAB procedure. Ra NACME1Pb NACME1Mc NACMEAVd NACME2e 2¯ 0.688458¯ 0.74212¯ 0.715289¯ 0.715301 3¯ -0.06554¯ -0.06443¯ -0.06499¯ -0.06499 4¯ -0.06507¯ -0.06482¯ -0.06495¯ -0.06495 5¯ -0.05882¯ -0.05878¯ -0.0588¯ -0.0588 6¯ -0.06162¯ -0.06149¯ -0.06156¯ -0.06156 7¯ -0.06418¯ -0.06403¯ -0.06411¯ -0.06411 8¯ -0.07811¯ -0.0776¯ -0.07786¯ -0.07786 9¯ -0.15304¯ -0.15154¯ -0.15229¯ -0.15229 10¯ -0.36364¯ -0.36103¯ -0.36233¯ -0.36233

aInternuclear distance bNACME calculation using 2 point formula ( forward differences) cNACME calculation using 2 point formula ( backward differences) dNACME average of ( forward and backward differences) eNACME using 3 point formula

6.1.3 Avoided crossing

Single state single reference (SS-SR-CASPT2) [163] and multi state multi reference (MS-MR-CASPT2) [164] level of theory as implemented in molpro ab initio program package have been used to study the region of avoided crossing in case of system under study. We have used the VTZ and AVTZ basis set for lithium and chlorine atom over inter nuclear distance ranging from 2Å to 10Å to compute the different energy levels. The regions for avoided crossing between different states of the lithium chloride have been computed and shown in figure 6.3 and figure 6.4 respectively. These are basically the 1 +ground state and B1 + first excited state potential energy curves in the region of avoidedP crossing. If we comparebothP theories we found that avoided crossing in case of single state single reference occurs in the region from 5 Å to 6 Å, while in case of multi state multi reference the avoided crossing has been shifted to 7Å to 8Å. Afterusingthe same basis set over same number of inter nuclear distances we found that the region of avoided crossing has been shifted to nearly about 2 Å in case of multi reference multi state theory as compared to single state single reference which may be attributed to im- provement in the electron affinity of the chlorine atom as well as electron correlation effects. These methods SS-SR and MS-MR CASPT2 are preferred to calculate avoided crossing for the system under study over as in Malrieu et al [165] CASPT2 curves where 180 Table 6.3: RS2/CASPT2 energy extrapolation using serial basis set.

BASIS ¯EHF ECORR ETOT AVTZ ¯-466.969 ¯-0.123 ¯-467.092 AVQZ ¯-466.972 ¯-0.138 -467.111 AV5Z ¯-466.973 ¯-0.144 ¯-467.118 CBS ¯-466.974 ¯-0.151 ¯-467.125

Lithium Flouride shows a double crossing however in our system there is only single avoided crossing. One more important point which we can consider while comparing the results obtained by using (SS-SR-CASPT2) and (MS-MR-CASPT2) level of theory is that in the single state single reference type of avoided crossing we have crossing of the unmixed ground state and excited state CASPT2 energies and repulsion of the mixed ground state and excited state MS-CASPT2 energies but using multi state multi reference theory we obtained overlapping and repulsion of the mixed and unmixed energies without any crossing. For understanding the underlying physics for SS-SR-CASPT2 and MS-MR-CASPT2 level of theory readers are requested to read reference no 165 for more details since in the present chapter we are dealing with the applications of these methods to diatomic molecule. To check the reliability of our computed results the convergence tests have been done using different method with serial basis set and extrapolated to the complete basis set limit. Basis set extrapolation has been carried out for correlation consistent basis set using MOLPRO package of ab initio calculations. Basis set extrapolation or CASPT2 energy is extrapolated by using the following serial basis sets avtz:avqz:av5z along with the following extrapolation function given as

E = E + A exp[ C N] (6.3) n CBS × − ×

N is the cardinal number of the basis set. The results for the CBS limit extrapolation has been summarized in Table 6.3.

181 Figure 6.3: Avoided crossing between different levels of LiCl using single-state single reference CASPT2

182 Figure 6.4: Avoided crossing between different levels of LiCl using multi-state multi reference CASPT2

183 6.1.4 MRCI diabatic and adiabatic potential energy surfaces

In the present section we have used the DDR procedure available in MOLPRO package of ab initio programs to generate the MRCI diabatic and adiabatic potential energy curves and to study the avoided crossing between them. It is not easy to compute the nonadiabatic coupling matrix elements as they are varying functions of geometry and become infinite at conical intersections. To avoid such things it is desirable to use a diabatization scheme which is either based upon the diagonalization of some property matrix [166] or an analysis of CI vectors [167]. The method we used in the present calculations is of latter type. To compute the MRCI potentials [168] we have used the VTZ basis set for lithium atom while AVTZ basis set for chlorine atom. Firstly the wave function is calculated at the reference geometry i.e 1Å and then at the displaced geometries ranging from 2Å to 10Å respectively. At reference geometry evaluated states are unmixed due to their different symmetry while at the displaced geometries they have same symmetry and the evaluated states are strongly mixed. The figure 6.5 and figure 6.6 represents the MRCI diabatic and adiabatic potential energy curves where there is crossing of the diabatic potentials and repulsion of adiabatic potentials as it can be seen. The figure 6.5 represents the MRCI potential energy curves without the orbital correction while the other one is with the orbital correction however the results are almost identical due to the use of DIAB procedure to minimize the change of the active orbitals. By comparing the CASPT2/MRCI curves it can be noticed that dynamical electron correlation had significant effect on the relative energies as well on the crossing points which means that the crossing has been shifted to shorter distances in the latter type. Diabatic energies for LiCl, obtained from CI-vectors and orbital correction are summarized in table 6.4. We have also compared the avoided crossing points and corresponding energies for the low lying electronic states of lithium chloride [169] with the reported values by other authors and the comparative studies are included in table 6.5. It is found that our results for calculations of avoided crossing (at r = 7.5 Å) for the low lying electronic states of system under study are in close agreement with values reported by Cao et al ( r = 8.9 Å) reference no 168 reported at same/different level of theory. 184 Figure 6.5: MRCI adiabatic and diabatic potential energy curves without orbital correction.

Figure 6.6: MRCI adiabatic and diabatic potential energy curves with orbital correction. 185 Table 6.4: Diabatic and Adiabatic energies for LiCl, obtained from CI-vectors and CI- vectors plus orbital correction R E1 E2 H11CI H22CI H21CI MIXCI 1 ¯-461.594 ¯-461.415 ¯-461.594 ¯-461.415 ¯0 0 2 ¯-466.492 ¯-466.368 ¯-466.399 ¯-466.461 ¯-0.05358 ¯60.02 3 ¯-467.148 ¯-466.92¯ -467.131¯ -466.937 -0.06087 16.06 4 ¯-467.209 ¯-467.019 ¯-467.051 ¯-467.176 ¯-0.07142 ¯65.59 5 ¯-467.179 ¯-467.028 ¯-467.059 ¯-467.148 ¯-0.06075 ¯63.13 6 ¯-467.157 ¯-467.045 ¯-467.08¯ -467.122 -0.05187 56.02 7 ¯-467.168 ¯-467.092 ¯-467.114 ¯-467.146 ¯-0.03475 ¯57.19 8 ¯-467.142 ¯-467.094 ¯-467.109 ¯-467.127 ¯-0.02249 ¯55.67 9 ¯-467.118 ¯-467.095 ¯-467.101 ¯-467.113 ¯-0.00988 ¯60.74 10 ¯-467.104 ¯-467.096 ¯-467.096 ¯-467.103 ¯-0.00201 ¯75.26 R E1 E2 H11CI H22CI H21CI MIXTOT 1 ¯-461.594 ¯-461.415 ¯-461.594 ¯-461.415 ¯0 0 2 ¯-466.492 ¯-466.368 ¯-466.451 ¯-466.41¯ -0.05851 ¯35.48 3 ¯-467.148 ¯-466.92 ¯-467.114 ¯-466.954 ¯-0.0822¯ 22.95 4 ¯-467.209 ¯-467.019 ¯-467.051 ¯-467.176 ¯-0.07143 ¯65.59 5 ¯-467.179 ¯-467.028 ¯-467.048 ¯-467.159 ¯-0.05112 ¯68.63 6 ¯-467.157 ¯-467.045 ¯-467.08 -467.122 ¯-0.05187 ¯56.02 7 ¯-467.168 ¯-467.092 ¯-467.114 ¯-467.146 ¯-0.03475 ¯57.19 8 ¯-467.142 ¯-467.094 ¯-467.109 ¯-467.127 ¯-0.0223¯ 56.28 9 ¯-467.118 ¯-467.095 ¯-467.097 ¯-467.116 ¯-0.00625 ¯73.67 10 ¯-467.104 ¯-467.096 ¯-467.096 ¯-467.103 ¯-0.00198 ¯75.47

Table 6.5: Comparative study of avoided crossing for low lying electronic states of lithium chloride Aa Bb Cc Dd Ee Ff P.F.Weck[Ref 171] 16.2 MRSDCI SLATER TYPE FUNCTIONS -467.02 2.5-50.0 K.Yuzuru[Ref169] 6.0 MRSDCI AVnZ/ACVnZ -467.56 2.5-50.0 Y.F.Cao [Ref 168] 8.9 MRCI/CASSCF Aug-cc-PVQZ -467.14 1.2-20.0 CWg 5.6 SS-SR-CASPT2 Li=VTZ/Cl=AVTZ -467.10 2-10 CWh 7.5 MS-MR-CASPT2 Li=VTZ/Cl=AVTZ -467.10 2-10 CWi 3.5 MRCI/CASSCF Li=VTZ/Cl=AVTZ -467.10 2-10

aReference no. bAvoided crossing Point at r = in Angstrom cLevel of theory used dBasis Set used eCorresponding energy in Hartree fInter-nuclear distance in Angstrom/bohr gCurrent Work hCurrent Work iCurrent Work 186 6.1.5 Further studies

In literature there have been numerous citations where the dissociation dynamics of various molecules ranging from diatomic to polyatomic molecules have been studied and reported. Schinke et [170] in his work reported the PEC for the lowest electronic states of H2S using MRDCI method and showed that lower adiabatic potential energy surface is dissociative and non-adiabatic coupling effects must be included in order to describe the absorption and dissociation process in a realistic way. Further nuclear dynamics on the coupled diabatic states is treated with time dependent approach which gives a clear view of absorption and dissociation mechanisms. In view of above discussion the absorption and dissociation processes in lithium chloride will be subject of further studies however non adiabatic effects plays a very important role in dissociation dynamics so computation of diabatic orbitals, non-adiabatic coupling matrix elements, avoided crossing [171] and MRCI diabatic and adiabatic potential energy curves is a part of our studies which is reported in the present work.

6.2 Conclusions

In this work we have calculated the diabatic orbitals, non-adiabatic coupling matrix elements and avoided crossing in Lithium chloride using VTZ basis set for lithium and AVTZ basis set for chlorine atom over a number of inter nuclear distances and showed the regions of avoided crossing along with the calculation of MRCI diabatic and adiabatic potential energy curves. We have also compared our results with the reported values using same or different level of theory and report the region of avoided crossing using low level of theory. Using theses surfaces, and diabatization procedure, dissociation dynamics of the lithium chloride molecule can be studied and predicted and to an extent the quantitative agreement between the theoretical and experimental results can be explored.

187 Chapter7 In this chapter the entire work reported in this thesis is assimilated in brief.

188 CHAPTER 7

Summary

Chapter 7 summarizes the entire work that is reported in this thesis and is divided among different chapters. Chapter 1 is the introductory overview of the the work carried out in this thesis. Starting from the basic concepts of Clamped nuclei Hamilto- nian and Born-Oppenheimer approximation this chapter briefly explains the important terms like adiabatic and non adiabatic potential energy surfaces. The concept of curve crossing, avoided crossing are explained by taking an example of simplest diatomic molecule Sodium Chloride. A section is included in the chapter for possible applications of non adiabatic transitions. This chapter also covers different analytical models of non adiabatic transitions which are discussed in brief to gave an insight of the problem. Chapter 2 covers all the analytical as well as computational methods used in this thesis for the study of non adiabatic transitions. These methods include different approaches like Boundary condition approach, Green’s function approach, Transfer matrix approach, Time dependent approach and computational approaches for the study of non adiabatic transitions. Boundary condition approach included the derivation of the different boundary condition for the study of nonadiabatic transitions. Green’s function approach will include the derivation of analytical relations in form of Dirac Delta potentials along with its application to two state curve crossing problem. Trans- fer matrix approach is based upon the fact that a total subsystem is broken down in to different subsystems and how these systems interact with each other. A simplest analytical derivation for the transfer matrices is included in this chapter which is further expanded to the curve crossing problems which in our case means that this transfer matrix will correlate the coefficients of the solution of the wave function in different regions around the Dirac Delta potentials. Time dependent approach for the curve crossing problems will make use of finding the solution of the partial differential equations. Here the problem involving the partial differential equations in two variables is reduced to single integral equation and an expression for wave function is derived in Laplace domain. Time dependent approach is further supported by two methodologies where the one will be having time independent coupling between the two states while the other one have time dependent coupling. Computational approach for the curve crossing problems includes the MRCI and CASPT2 approaches. Chapter 3 includes the applications of Boundary condition and Green’s function approach discussed in chapter 2. In this chapter exact analytical derivation for the two state , three state and n state scattering are provided which are further used to study the problem of two state , three state and n state state scattering by taking example of constant, linear and exponential potentials where the non adiabatic transition probability is plotted using obtained analytical formulas. Green’s function approach for two channel, three channel and n channel scattering problems is used to find the exact analytical expressions. Further the effect of curve crossing on electronic absorption spectra and Resonance Raman excitation profile are discussed using Green’s function approach. Chapter 4 includes the methodology of expressing any arbitrary coupling as a collection of Dirac Delta potentials. In this chapter we have considered different cases of two state problems coupled by arbitrary coupling which is further expressed as a collection of Dirac Delta potentials and non adiabatic transition probability is calculated between the two potentials. Different cases of constant , linear and exponential potentials are considered and transfer matrices are formed for all of them which are further used in the calculation of non adiabatic transition probability. The formed transfer matrices are easy to handle with any of the computational softwares for a large number of Dirac Delta Potentials. Exact numerical treatment for different cases is also consider in the present chapter to check the validity of our obtained analytical results. Chapter 5 includes the solution of the two state problem with time dependence. In chapter 1 to 4 we have considered the two state problem without time dependence, it could be time independent coupling or time independent Schrodinger equations, however chapter 5 consider the time dependence explicitly. Time dependence is supported by two methods, one in which a solution for the problem involving the partial differential equations in two variables is reduced to single integral equation and an expression for wave function is derived in Laplace domain without time dependent coupling while the other method considers the time 190 dependent coupling. Further different cases in which the strength of the Dirac Delta coupling has varied dependence on time are discussed in details. Also a section for the calculation of non-adiabatic transition probability with a δ potential coupling has been included in the end of the chapter which is based upon the methodology of transformation of time dependent Schrodinger equation into time independent one by using a time dependent scaling factor. Chapter 6 will discuss the case of a real molecule through computational packages to know weather there is any existence of such type of curve crossing in real molecules or not. In the last chapters we have consider different model of curve crossing and provided an exact solution in different cases. In the present chapter we consider a simple example of Lithium chloride molecule and using CASPT2 and MRCI level of theory we have investigated the curve crossing in these molecules using MOLPRO -: A complete system of ab initio programs for molecular structure calculations. The generated adiabatic and diabatic potential energy surfaces can be used to understood the spectroscopy as well as non adiabatic transitions in Lithium chloride molecule.

191 AUTHOR LIST OF PUBLICATIONS

Published Work

1. Diwaker and A. Chakraborty, “Multi channel scattering problems:An analytically solvable model", Mol. Phys., 110, 2012, 2257-2267.

2. Diwaker and A. Chakraborty, “Curve crossing problem with Gaussian type coupling:analytically solvable model", Mol. Phys., 110, 2012, 2197-2203.

3. Diwaker and A. Chakraborty, “Transfer matrix approach to the curve crossing problems of two exponential diabatic potentials", Mol. Phys., 2015 (Accepted, In Press)

4. Diwaker and A. Chakraborty, “Two state problem with arbitrary coupling", Chinese Physics Letters., 2015 (Accepted,In Press)

5. Diwaker and A. K. Gupta, “Quantum Chemical and Spectroscopic Investiga- tions of(Ethyl 4 hydroxy-3-((E)-(pyren-1-ylimino)methyl)benzoate)byDFT Method", International J. Quantum Chem., 2014, 2014, Article ID 841593, 15 pages.

6. Diwaker “Spectroscopic (FT-IR, 1H, 13C NMR, UV), DOS and orbital overlap population analysis of copper complex of (E)-4-(2-(4-nitrophenyl) diazenyl)-N, N bis ((pyridin-2-yl) methyl) benzamine by density functional theory", Spec- trochimicia Acta Part A: Molecular and Biomolecular spectroscopy, 136, 2015, 1932-1940.

7. Diwaker “Quantum mechanical and spectroscopic (FT-IR, 1H, 13C NMR, UV) investigations of 2-(5-(4-Chlorophenyl)-3-(pyridin-2-yl)-4, 5-dihydropyrazol-1- yl)benzo[d]thiazole by DFT method", Spectrochimicia Acta Part A: Molecular and Biomolecular spectroscopy, 128, 2014, 819-829.

8. Diwaker, C. S. Chidan Kumar, Ashish Kumar, Siddegowda Chand raju, Ching Khen Quah, Hoong-Kun Fun “Synthesis, spectroscopic characterization, electronic and optical studies of (2Z)-5,6-dimethyl-2-[(4-nitrophenyl)methylidene]- 2,3-dihydro-1-benzofuran-3-one", J. Computational Science., 2015, (Acepted, 192 In Press)

9. Diwaker, C. S. Chidan Kumar, Ashish Kumar, Siddegowda Chand raju, Ching Khen Quah, Hoong-Kun Fun “Synthesis, spectroscopic characterization and computational studies of 2-(4-bromophenyl)-2-oxoethyl 3-methylbenzoate by density functional theory", J. Mol. Struc., 1092, 2015, 192-201.

10. Diwaker, C. S. Chidan Kumar, Ashwani Kaumar, “Spectroscopic (FT-IR, 1H, 13C NMR, UV) characterization and DFT studies of novel 8-((4-(methylthio)-2,5- diphenylfuran-3-yl)methoxy)quinoline", Spectrochimicia Acta Part A: Molec- ular and Biomolecular spectroscopy, 2015 (Accepted, In Press)

11. Diwaker, “Avoided crossing in lithium chloride using CASPT2 and MRCI level of theory.", Reviews in theoretical science, 2015 (Accepted, In Press)

12. Diwaker,“β-Al4.5FeSi: Structural, electronicand optical studies of a ﬂawed structure by DFT method.", AIP Conf. Proc. 2015 (Accepted, In Press)

Communicated Work

1. Diwaker and A. Chakraborty, “Multistate Curve Crossing Problems-: Exact Analytical Solution approach", Under Review.

2. Diwaker and A. Chakraborty, “A Simple Approach to Landau Zener Transition Probability", Under Review.

3. Diwaker and A. Chakraborty, “Exact solution of Schrodinger equation for two state problem with time dependent coupling", Under Review.

4. Diwaker and A. Chakraborty, “Exact solution of time-dependent Schrodinger equation for two state problem in Laplace domain", Under Review.

5. Diwaker and A. Chakraborty, “Non-adiabatic transition probability with a moving delta potential coupling ", Under Review.

6. Diwaker and A. Chakraborty, “Effect of curve crossing induced dissociation on absorption and resonance Raman spectra: An analytically solvable model", Under Review. 193 7. Diwaker and A. Chakraborty, “Diffusion in a Flat potential with a sink", Un- der Review.

8. Diwaker and Ashwani Kumar, “A DFT study of structural and electronic properties of Zn1 xSbxTe with x = (0.25, 0.50, 0.75)", Under Review. −

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