葉綠素內轉換之理論研究

國立臺灣大學理學院化學系 碩士論文

Department of Chemistry College of Science

National Taiwan University Master Thesis

葉綠素內轉換之理論研究

A Theoretical Study on the Internal Conversion Dynamics of Chlorophylls

施 欣妤 Petra Shih

指導教授：鄭原忠 博士

Advisor: Yuan-Chung Cheng, Ph.D.

中華民國 107 年 6 月

June, 2018

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doi:10.6342/NTU201803678

中

中中文文文摘摘摘要要要

分子激發態的非絕熱躍遷在許多光物理和光化學過程中起著重要

作用。特別是，葉綠素中Qx → Q_y的超快內部轉換對於光合作用中光

捕獲的高效率至關重要，但其機制尚未明確闡明。在這項工作中，

我們通過評估振動耦合和電子耦合來構建研究非絕熱Qx → Q_y動力

學的有效哈密頓算符，從理論上探索了葉綠素a和細菌葉綠素a的內轉 換過程。通過結合含時密度泛函理論和透熱化(diabatization)方法，最 大化兩個低激發態的組態均勻性，實現了無輻射弛豫的第一原理研

究。使用費米黃金律計算的弛豫速率表明，快於100飛秒Qx → Qy過

程可以用具有弱振動耦合的透熱模型充分描述。此外，我們還確

定了一些支配Qx → Q_y弛豫的關鍵振動模式，並依此解釋葉綠素

中Qx → Q_y的弛豫速率與溶劑種類與取代基效應有高度相關性的現象

可歸因於Qy/Q_x能階間隙的變化。預計所開發的方法能廣泛應用於分

子系統中能量弛豫的詳細動力學研究，並且還使我們能夠深入了解自 然界光合作用中最重要的發色團的設計。

doi:10.6342/NTU201803678

doi:10.6342/NTU201803678

Abstract

Non-adiabatic transitions in molecular excited states play significant roles in many photophysical and photochemical processes. In particular, the ultra- fast Q_x → Q_y internal conversion of in chlorophylls is crucial to the high effi- ciency of light harvesting in photosynthesis, yet the mechanism has not been clearly elucidated. In this work, we theoretically explored the internal conver- sion processes of chlorophyll a and bacteriochlorophyll a by evaluating the vibronic couplings and electronic couplings to construct effective Hamiltoni- ans for the non-adiabatic Qx → Q_ydynamics. The first principle study of the radiationless relaxation was achieved by combining time dependent density functional theory (TD-DFT) and diabatization method through enforcement of configuration uniformity for the two low lying excited states. The relax- ation rates calculated using Fermi’s Golden rule suggest that the sub-100 fs Qx → Qy process can be fully described by a diabatic model with weak vi- bronic couplings. In addition, we also identified a few key vibrational modes that dominate the Qx → Q_y relaxation, and show that the highly solvent and substituent dependent relaxation rates can be attribute to variations in Qy/Q_x energy gaps. The methodology developed in this work is expected to enable detailed dynamical study of energy relaxation in general molecular systems, and it also allows us to gain insights into the natural design of the most im- portant chromophores in photosynthesis.

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Contents

中 中

中文文文摘摘摘要要要 i

Abstract iii

List of Figures ix

List of Tables xiii

1 Introduction 1

1.1 Light Harvesting in Photosynthesis . . . 1

1.2 Pigments in Light-Harvesting Complex . . . 3

1.2.1 Chlorophylls and Bacteriochlorophylls . . . 3

1.2.2 Studies of Chlorophyll Excited States . . . 5

1.2.3 Internal Conversion in Chlorophylls . . . 7

1.3 Non-Adiabatic Transition . . . 9

1.3.1 Recent Developments . . . 10

1.3.2 Summary . . . 11

2 Theoretical Background 13 2.1 Molecular Hamiltonian . . . 14

2.1.1 Separation of the Electronic and Nuclear Motions . . . 14

2.1.2 Born-Oppenheimer Approximation . . . 15

2.1.3 Non-Adiabatic Coupling . . . 17

2.2 Theoretical Treatments of Non-Adiabatic Dynamics . . . 18

2.2.1 Multi-Configuration Time-Dependent Hartree . . . 19

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2.2.2 Fewest Switch Surface Hopping . . . 19

2.2.3 Ehrenfest Mean Field . . . 20

2.2.4 Path Integral for Non-Adiabatic Dynamics . . . 21

2.3 Diabatic Representation . . . 21

2.3.1 Definition of Diabatic Basis . . . 23

2.3.2 Nonexistence of Strictly Diabatic Basis for General Molecules . . 23

2.3.3 Quasi-diabatization Methods Overview . . . 24

2.4 Enforcement of Configuration Uniformity . . . 26

2.4.1 Configurational Expansions of Adiabatic and Diabatic States . . . 26

2.4.2 Deformation of Configurations and Molecular Orbitals . . . 28

2.4.3 Uniformity of Electronic Structures . . . 29

2.4.4 Diabatization Criterion . . . 30

3 Electronic Excitations of Chlorophylls 33 3.1 Methods . . . 33

3.1.1 Computational Details . . . 33

3.1.2 Transition Density and Transition Dipole Moment . . . 34

3.2 Excited States Properties . . . 36

3.2.1 Energy Levels . . . 36

3.2.2 Configurations . . . 39

3.2.3 Transition Density and Transition Dipole Moment . . . 43

3.3 Solvent and Coordination Dependences . . . 46

3.3.1 Structures . . . 46

3.3.2 Energy Levels . . . 48

3.4 Diabatization . . . 49

3.4.1 Molecular Orbitals Deformations . . . 49

3.4.2 Transformation Matrix . . . 52

3.4.3 Dominant CI Coefficients in Adiabatic and Diabatic Basis . . . . 53

3.5 Analysis of Potential Energy Surfaces . . . 55

3.6 Transition Density of the Diabatic States . . . 56

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3.7 Summary . . . 57

4 Vibronic Couplings of Chlorophylls 59 4.1 Theory . . . 59

4.1.1 Coupling of Electronic and Nuclear Motion . . . 59

4.2 Vibronic Coupling in Adiabatic Basis . . . 64

4.2.1 Four-Point Method . . . 64

4.2.2 Huang-Rhys Factor Method . . . 67

4.2.3 Spectra Simulations . . . 76

4.3 Vibronic Coupling in Diabatic Basis . . . 84

4.3.1 Evaluate Coupling Constant from Potential Energy Surfaces . . . 84

4.3.2 Analysis of Off-Diagonal Strong-Coupled Modes . . . 85

4.4 Summary . . . 86

5 Internal Conversion Dynamics 91 5.1 Theory . . . 92

5.1.1 Spin-Boson Hamiltonian . . . 92

5.1.2 Fermi’s Golden Rule for Vibronic Transitions . . . 93

5.1.3 Studying Internal Conversion using Fermi’s Golden Rule . . . 95

5.2 Results and Discussion . . . 98

5.2.1 Characteristic Frequencies . . . 98

5.2.2 Energy Gap Dependence . . . 102

5.2.3 Mode-Specific Contributions to Rate . . . 105

5.3 Reduced Vibrational Models . . . 107

5.3.1 Selected Vibrational modes . . . 108

5.3.2 Effective Underdamped Harmonic Oscillators . . . 110

5.4 Summary . . . 113

6 Concluding Remarks 115

Appendix A. Molecular Structures 119

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Appendix B. Basis Set Benchmark 127

Appendix C. Vibronic Coupling in Diabatic Basis 129

Bibliography 139

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List of Figures

1.1 Molecular structures of (a) Chl a (b) BChl a. The five rings are lettered

A, B, C, D and E conversionally. . . 4

1.2 Linear spectra of Chl a in diethyl ether[1]. (a) absorption spectrum (b) fluorescence spectrum . . . 6

1.3 Linear spectra of BChl a in diethyl ether[1]. (a) absorption spectrum (b) fluorescence spectrum . . . 7

2.1 Potential energy curves for the diatomic NaCl. . . 22

3.1 The reduced molecular structures in our model. The phytyl tails of mo- lecules shown in Fig. 1.1 are replaced by methyl group. (a) Chl a (b) BChl a . . . 34

3.2 The traditional and modern assignments of the Q_x origin from the MCD spectra of Chl a in ether (magenta) and pyridine (brown)[2]. The vibronic coupling assignments are the interpretation proposed by Reimers et al.[2]. 38 3.3 Frontier molecular orbitals of Chl a. . . 40

3.4 Frontier molecular orbitals of BChl a. . . 41

3.5 Transition densitys of single configuration for Chl a. . . 41

3.6 Transition densitys of single configuration for BChl a. . . 42

3.7 S₁and S₂ transition densities relative to ground state of Chl a. . . 43

3.8 S1and S2 transition densities relative to ground state of BChl a. . . 43

3.9 The molecular axes defined by Gouterman four orbital model. (a) Chl a (b) BChl a . . . 44

doi:10.6342/NTU201803678 3.10 Transition dipole moments of (a) Chl a and (b) BChl a. ~µ_y and ~µ_xdenote

the transition dipole moment of Q_y and Q_x excitation, respectively. . . 44 3.11 Geometries of three kinds of solvents considered in this section. The

calculations do not include PCM. (a) diethyl ether (b) tetrahydrofuran (c) pyridine. . . 47 3.12 Geometries of the solvent-coordinate Chl a (geometry optimizations without

PCM). (a) 5CO-Diethyl ether. (b) 6CO-Tetrahydrofuran. (c) 6CO-Pyridine. 47 3.13 Absolute value of correlation matrix between two points on q1346 of Chl

a. (a) Total MOs. (b) Occupied MOs. (c) Frontier orbitals. . . 51 3.14 Diabatization matrix of chlorophyll a along vibrational normal mode q₁₃₄₆. 52 3.15 Square of CI coefficients of dominant configurations in adiabasis basis

for (a) S₁ and (b) S₂ states in chlorophyll a system. Black: HOMO- 1→LUMO+1; Blue: HOMO→LUMO; Magenta: HOMO-1→LUMO;

Red: HOMO→LUMO+1. . . 53 3.16 Square of CI coefficients of diabasis basis for (a) Q_y and (b) Q_xstates in

chlorophyll a system. Black: HOMO-1→LUMO+1; Blue: HOMO→LUMO;

Magenta: HOMO-1→LUMO; Red: HOMO→LUMO+1. . . 54 3.17 Potential energy surfaces of low-lying excited states of chlorophyll a in

(a) adiabatic and (b) diabatic basis along the vibrational normal mode, q1346. 55 3.18 Diabatic energy gap and electronic coupling along the vibrational normal

mode, q₁₃₄₆. . . 56 3.19 Transition densities in adiabatic basis. The isovalues are set to be 0.001.

(a) q₁₃₄₆ = −1, S₀ → S₁; (b) q₁₃₄₆ = −1, S₀ → S₂; (c) q₁₃₄₆ = −0.1, S₀ → S₁; (d) q₁₃₄₆ = −0.1, S₀ → S₂ . . . 57 3.20 Transition densities in diabatic basis. The isovalues are set to be 0.001.

(a) q1346 = −1, Qy excitation; (b) q1346 = −1, Qx cxcitation; (c) q1346 =

−0.1, Q_y excitation; (d) q1346 = −0.1, Q_xexcitation . . . 57

4.1 Schematic representation of Displaced harmonic oscillator model. . . 61 4.2 Schematic representation of Four-Point Method. . . 64

doi:10.6342/NTU201803678 4.3 Mode correlation matrixes between ground state and low-lying excited

state of Chl a. (a) S₁ (b) S₂state . . . 68 4.4 Mode correlation matrixes between ground state and low-lying excited

state of BChl a. (a) S1 (b) S2state . . . 69 4.5 Distribution of reorganization energy of Chl a excited states with respect

to ground state vibrational modes. . . 70 4.6 Distribution of reorganization energy of BChl a excited states with respect

to ground state vibrational modes. . . 70 4.7 Simulated absorption spectra of Chl a in (a) 1-propanol (σ_Qy=σ_Qx=150

cm⁻¹, ∆E_Qy=1.83192 eV, ∆E_Qx=1.90160 eV, f_Qy/f_Qx ≈4.84 (tuned));

(b) diethyl ether (σ_Qy=σ_Qx=180 cm⁻¹, ∆E_Qy=1.83192 eV, ∆E_Qx=1.90160 eV, fQy/fQx ≈27.1 (tuned)). The gaussian broadening factors, excitation energies and Qy/Qx relative oscillator strength are determined by fitting experimental spectra[3]. . . 78 4.8 Simulated absorption spectra of BChl a in triethylamine (σ_Qy=σ_Qx=250

cm⁻¹, ∆E_Qy=1.60747 eV, f_Qy=0.3868 (untuned), ∆E_Qx=2.14877 eV, f_Qy/f_Qx ≈3.51 (untuned)). The gaussian broadening factors, excitation energies are determined by fitting experimental spectra[4]. . . 79 4.9 Simulated ∆FLN spectra of Chl a in (a) 1-propanol (b) Diethyl Ether.

The n=0 and n=1 states of 15 most strongly-coupled vibrational modes are considered. The Gaussian broadening factors are set to be 5 cm⁻¹ for all peaks. . . 80 4.10 Simulated ∆FLN spectra of BChl a in (a) 1-propanol (b) diethyl ether.

The n=0 and n=1 states of 15 most strongly-coupled vibrational modes are considered. The Gaussian broadening factors are set to be 5 cm⁻¹ for all peaks. . . 81 4.11 Distribution of reorganization energy of Chl a excited states with respect

to ground state vibrational modes. . . 83

doi:10.6342/NTU201803678 4.12 (a) Diagonal and (b) Off-diagonal vibronic coupling constants in diabatic

basis for Chl a. . . 88 4.13 (a) Diagonal and (b) Off-diagonal vibronic coupling constants in diabatic

basis for BChl a. . . 89 4.14 Visualization of off-diagonal vibronic stong-coupled mode for (a) Chl a

(ω=1755 cm⁻1) and (b) BChl a (ω=1749 cm⁻1). . . 90 4.15 Scheme of the projection of the stong-coupled mode on the conjugation

path for (a) Chl a (ω=1755 cm⁻1) and (b) BChl a (ω=1749 cm⁻1). . . 90 5.1 Mode-specific contribution to Qx → Q_y rate of Chl a. (a) Γ0=100 cm⁻¹,

∆=680 cm⁻¹; (b) Γ₀=100 cm⁻¹, ∆=1200 cm⁻¹ . . . 100 5.2 Mode-specific contribution to Q_x → Q_yrate of BChl a. (a) Γ₀=100 cm⁻¹,

∆=680 cm⁻¹; (b) Γ₀=100 cm⁻¹, ∆=1200 cm⁻¹ . . . 101 5.3 Total relaxation rate of Chl a as a function of the S₁/S₂ energy gap. (a)

Γ₀=50 cm⁻¹;(b) Γ₀=100 cm⁻¹; (c) Γ₀=200 cm⁻¹; (d) Γ₀=300 cm⁻¹. . . . 102 5.4 Total relaxation rate of BChl a as a function of the S₁/S₂ energy gap. (a)

Γ0=50 cm⁻¹; (b) Γ0=100 cm⁻¹; (c) Γ0=200 cm⁻¹; (d) Γ0=300 cm⁻¹. . . . 103 5.5 2D contour plot of mode-specific contributions to rate withΓ0=100 cm⁻¹

for (a) Chl a and (b) BChl a. . . 105 5.6 Rate distribution of BChl a calculated in ∆=3240 cm⁻¹. (a) Γ0=100 cm⁻¹,

IC time=668 fs; (b) Γ0=200 cm⁻¹, IC time=328 fs; (c) Γ0=300 cm⁻¹, IC time=216 fs . . . 107 5.7 Spectral density of off-diagonal vibronic coupling in adiabatic basis for

(a) Chl a and (b) BChl a. Γ_i=100 cm⁻¹ . . . 111 5.8 Fitted spectral density of off-diagonal vibronic coupling in adiabatic basis

for (a) Chl a and (b) BChl a, three lorentzian functions are considered. . . 112 5.9 Comparison of the relaxation rate constant dependence on the S₁/S₂ en-

ergy gap in the reduced models (blue and red) and the exact model (black).

Blue: 3 strongest-coupled modes considered; Red: 3 effective modes by fitting the spectral density considered (a) Chl a and (b) BChl a . . . 113

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List of Tables

2.1 The conceptual scheme for the dominant configurations of adiabatic wave- functions in different nuclear region. . . 30 2.2 The conceptual scheme for the dominant configurations of diabatic wave-

functions in different nuclear region. . . 30

3.1 S₁ and S₁ vertical electronic transition energyies of Chl a and BChl a calculated by TD-DFT, all calculations are done with 6-31G(d) basis set.

The experimental results are extracted from the absorption spectra meas- ured by Freiberg et al. in 1-propanol and triethylamine for Chl a[3] and BChl a[4], respectively. . . 38 3.2 The contributions of configurations correspond to the first and second

electronic excitations in chlorophyll a calculated by TD-DFT in different functionals. In this table, H-1, H, L, L+1 are the frontier molecular orbit- als which denotes HOMO-1, HOMO, LUMO, and LUMO+1, respectively. 39 3.3 The contributions of configurations correspond to the first and second

electronic excitations in bacteriochlorophyll a calculated by TD-DFT in different functionals. In this table, H-1, H, L, L+1 are the frontier mo- lecular orbitals which denotes HOMO-1, HOMO, LUMO, and LUMO+1, respectively. . . 40 3.4 The S1/S2 energy gap of chlorophyll a without coordination of solvent

molecules calculated by TD-DFT in different functionals. The solvation effect is considered implicitly by PCM. . . 48

doi:10.6342/NTU201803678 3.5 The S₁/S₂ energy gap of chlorophyll a with coordination of solvent mo-

lecules calculated by TD-DFT in different functionals. The solvation ef- fect is considered implicitly by PCM. . . 48

4.1 Total reorganization energy of excited states relative to ground state cal- culated using Four-Point Method for Chl a in three DFT and TD-DFT functionals. . . 65 4.2 Total reorganization energy of excited states relative to ground state cal-

culated using Four-Point Method for BChl a in three DFT and TD-DFT functionals. . . 65 4.3 Ten strongest S0/S1 coupled vibrational modes of Chl a between 200

to 2000 cm⁻¹. Their frequencies, Huang-Rhys factors, reorganization energies and the assignments of vibrational feature are listed. . . 71 4.4 Ten strongest S₀/S₂ coupled vibrational modes of Chl a between 200

to 2000 cm⁻¹. Their frequencies, Huang-Rhys factors, reorganization energies and the assignments of vibrational feature are listed. . . 71 4.5 Ten strongest S₀/S₁ coupled vibrational modes of BChl a between 200

to 2000 cm⁻¹. Their frequencies, Huang-Rhys factors, reorganization energies and the assignments of vibrational feature are listed. . . 72 4.6 Ten strongest S₀/S₂ coupled vibrational modes of BChl a between 200

to 2000 cm⁻¹. Their frequencies, Huang-Rhys factors, reorganization energies and the assignments of vibrational feature are listed. . . 72 4.7 Comparison of the theoretical and experimental total reorganization en-

ergies of Chl a between Φg and ΦS1 state. (∗ Estimated Stokes shift is extracted from Abs. and FL. spectra measured in 1-propanol[3]) . . . 73 4.8 Comparison of the theoretical and experimental total reorganization en-

ergies of BChl a between Φg and ΦS1 state. (∗ Estimated Stokes shift is extracted from Abs. and FL. spectra measured in triethanolamine[4]) . . . 73

doi:10.6342/NTU201803678 4.9 Comparison of the total reorganization energies of Chl a evaluated in dif-

ferent solvent conditions. For pyridine, the dielectric constant, ε, used in

PCM equals to 12.978. . . 81

5.1 Comparison of internal conversion time (IC time) in different solvents. The energy gaps are provided by Reimers et al.[2]. Γ₀is parametrized to be 150 cm⁻¹ by fitting absorption spectrum as shown in Fig. 4.7. . . 104

5.2 Comparison of internal conversion time (IC time) in different solvents estimated in Ref. [2]. The energy gaps are also provided by Reimers et al.[2]. . . 104

5.3 Ten strongest-coupled modes sorted by diabatic off-diagonal vibronic coup- ling strength for Chl a and BChl a. . . 108

5.4 Estimated relaxation time considered limited number of strongest-coupled modes for Chl a. . . 108

5.5 Estimated relaxation time considered limited number of most contributed modes for Chl a. . . 109

5.6 Parameters used for the fitting of spectral densities in Figure 5.8. . . 112

1 The optimized Chl a S₀ state structure. . . 120

2 The optimized Chl a S₁ state structure. . . 121

3 The optimized Chl a S₂ state structure. . . 122

4 The optimized BChl a S₀state structure. . . 123

5 The optimized BChl a S1state structure. . . 124

6 The optimized BChl a S2state structure. . . 125

7 The Chl a adiabatic energy (eV) calculated using CAM-B3LYP functional with different basis sets. . . 128

8 The diagonal vibronic coupling constants (unitless) of Chl a in diabatic basis corresponds to the vibrational frequencies (cm⁻¹). . . 130

9 The off-diagonal vibronic coupling constants (unitless) of Chl a in dia- batic basis corresponds to the vibrational frequencies (cm⁻¹). . . 132

doi:10.6342/NTU201803678 10 The diagonal vibronic coupling constants (unitless) of BChl a in diabatic

basis corresponds to the vibrational frequencies (cm⁻¹). . . 134 11 The off-diagonal vibronic coupling constants (unitless) of BChl a in dia-

batic basis corresponds to the vibrational frequencies (cm⁻¹). . . 136

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Chapter 1 Introduction

1.1 Light Harvesting in Photosynthesis

Being one of the most important and complicated processes in the world, photosyn- thesis has attracted tremendous scientific interest over hundreds of years[1]. Photosyn- thesis is the process by which organisms convert pure energy of light into chemical en- ergy and provide the foundation for essentially all life. These organisms capture radiant energy of the sun and, by utilizing carbon dioxide and water, convert it to chemical en- ergy stored in carbohydrates. It takes place in all chlorophyll-containing plants, algae, and cyanobacteria. It is extremely widespread and act as the source of life.

Light harvesting is the first step of photosynthesis carried out by all photosynthetic organisms[5, 6]. It is initiated by capturing sun light using light-absorbing pigments (chromophores). The energy then transfer among pigments in the form of exciton. Through these processes, photon absorption creates an exciton that eventually leads to charge sep- aration in the reaction center. Overall, after the absorption, photosynthesis consists of a sequential processes of excitation energy transfer by antenna systems, primary electron transfer in reaction centers, energy stabilization by secondary processes, synthesis and export of stable products.

The vast majorty of pigments function as antenna which collect light and then deliver energy to the reaction center. Using pigments with various substituents and in different environments, the extensive spatial distribution of photosynthetic organisms utilize light energy in different frequency regions subject to the environmental constraints. Light-

doi:10.6342/NTU201803678 harvesting antennae absorb solar energy efficiently contributed by their large variation in

structure and pigment compositions, which enables them to adapt to the light conditions in their natural habitat[7, 8, 9, 10]. These pigments are constrained by proteins with dif- ferent but specific orientations. In these pigment-protein complex, it has been discovered that the nature is able to pack high density chromophores while avoiding concentration quenching in them.

In photosynthetic systems, the excited state relaxation of chlorophylls is extremely fast to facilicate the efficiency of light harvesting. In the primary step of photosynthesis in plants, absorption of photon by chlorophylls and carotenoids in light-harvesting complexes leads to electronic excitation. The spectral region for efficient absorption is tuned by the orient- ations of carotenoids and chlorophylls with different functional groups [7, 11]. After the absorption, the molecule undergoes various intramolecular relaxation pathways and may also transfer the excitation energy to nearby molecule in the antenna complex. The energy attained from pigments transfers in the nature-design delicate network and centralizes to the photosynthetic reaction center in this manner. The efficiency of light harvesting is thus strongly depends on the relative rates of these radiative and non-radiative processes.

The delicate networks of these chromophores are responsible for the subsequent funnel- ing of the excitation energy to the reaction centers. That is, the different energy transfer pathways act as timescale funnel such that the energy are sufficiently prepared for the reaction center in various environmental conditions and at all the time. The remarkably high efficiency of light harvesting of antenna systems is crucial to the energy conversion of photosynthesis in nature[12]. To exploit the solar energy, it is imperative to further unravel the contrivance of nature photosynthesis.

Due to the importance of light-harvesting and its complicated mechanisms, the struc- tures of light-harvesting complexes and their dynamics have been studied. The structure- function relationship has long been one of the most thrilling topic for chemists. Crys- talization of protein structures, transient spectroscopy, two-dimensional electronic spec- troscopy (2DES), and coarse graining models were combined to investigate the intra- and inter-molecular excitation energy transfer dynamics in photosynthesis[13]. However,

doi:10.6342/NTU201803678 there are still abundant of mechanisms waited to be further understood.

1.2 Pigments in Light-Harvesting Complex

The lifeblood of a photosynthetic organism is its pigments. These photosynthetic pigments are excited by photon to excited states. Without them, solar energy cannot be absorbed and transfered among the light harvesting complexes. Therefore, these pigments are essential to convert the solar energy to chemical energy and to stored for life. There are a remarkable number of pigments found in different photosynthetic organisms which serve a variety of functional roles. The functions they perform in the photosynthetic pro- cess are determined by their intrinsic chemical structures and their interactions with the protein in light harvesting complexes (pigment-protein complexes). There are four major classes of photosynthetic pigments, chlorophylls, bacteriochlorophylls, carotenoids, and bilins. Here, we focus on two classes of pigments with similar structures but have distinct photochemical and photophysical performances.

1.2.1 Chlorophylls and Bacteriochlorophylls

Chlorophylls (Chls) are the most well known class of photosynthetic pigments among the four classes. The study of Chls has a very long history. The name was first used by Pelletier and Caventou in 1818 to describe the green pigments which are involved in photosynthesis in higher plants. Three Nobel prizes have been given for the studies on its structure. Richard Wilst¨atter was awarded the 1915 Nobel prize for his work that established the major features of the chlorophyll structure, including the existance of mag- nesium and the empirical formula. Hans Fisher was honors in 1930 in part because his determination of the complete structure. In 1965, Robert Woodward received prize in part for his work that culminated in the total synthesis of chlorophyll.

The Chls are named a-d in order of their discoveries. Among the chlorophylls family, only Chlorophyll a (Chl a) exists in all known eukaryotic photosynthetic organisms, cy- anobacteria, and some anoxygenic bacteria. The reason of the ubiquity of this impressive

doi:10.6342/NTU201803678 molecule is not yet understood.

The empirical formula for Chl a is C₅₅H₇₂N₄O₅Mg, while its chemical structural for-

(a) (b)

Figure 1.1: Molecular structures of (a) Chl a (b) BChl a. The five rings are lettered A, B, C, D and E conversionally.

mula is shown in Fig. 1.1(a). It is a squarish planar molecule, about 10 ˚Aon each side.

The Mg atom in the center of the planar portion is coordinated to four nitrogens atoms and is an ion assigned to be +2 +2 in charge. The substructures of this planar molecule are four derivatives of pyrrole which contain one nitrogen separately. Accordingly, Chls and their related compounds are often referred to as tetrapyrroles. They possess a fifth ring in the lower-left corner as shown in Fig. 1.1(a) and a long hydrocarbon tail attached to the upper-left ring. Chemically, the Chls are related to the porphyrins, which are also tetrapyrroles but with higher symmetry.

The five rings in Chls are lettered A through E. By convention, as shown in Fig. 3.9(a), the y molecular axis of Chls is defined as passing through the N atoms of rings A and C, while the x axis passing through the N atoms in rings B and D. The z zxis is perpendicu- lar to the plane of the macrocycle. An extensive delocalized π conjugated system extends over most of the molecule, with the exception of ring D, in which one pyrrole double bond is reduced to a single bond. The tail, often called the phytol tail or the isoprenoid tail, is formed by condensation of four isoprene units and is then esterified to ring D.

doi:10.6342/NTU201803678 In addition, the chemical structure of Bacteriochlorophyll a (BChl a) is shown in Fig.

1.1(b). It is the principal chlorophyll-type pigment exists in the majority of anoxygenic photosynthetic bacteria. The chemical differences between the structures of chlorophylls and bacteriochlorophylls are the acetyl group at ring A and the single bond in ring B for BChls which reduces the degree of conjugation in the macrocycle and also reduces the molecular symmetry compared with Chls. These structural changes exert major effects on the spectral properties, which we will discuss below.

1.2.2 Studies of Chlorophyll Excited States

The excited states of chlorophylls have been intensely discussed both theoretically and experimentally. For sake of simplicity, one may consider Mg metal center and the substituents of chlorophylls on the ring as the perturbation on the structure of porphyrin.

Considering its higherly symmetric geometry, the molecular orbitals of porphyrin can be described clearly. In the 1960s, Matrin Gouterman first proposed a four-orbital model to explain the absorption spectra of porphyrins[14]. According to this model, the absorption bands in porphyrins system are assigned to the transitions between four frontier molecu- lar orbitals, i.e. HOMO-1, HOMO, LUMO and LUMO+1. The relative energies of these transitions are affected by the identities of the metal center and the substituents on the ring. Traditionally, the violet-blue absorption bands of porphyrins are named as Soret bands, where the lower energy bands are known as Q bands.

Shown in Fig. 1.2(a) are the absorption spectrum of Chl a. This magnesium-containing reduced porphyrin absorbs most energy with narrow bands in the violet-blue (near 430 nm) and orange-red (near 662 nm) spectral ranges[15, 16, 3, 17]. The lack of a signi- ficant absorption in the green region gives the Chls their characteristic blue-green color.

At the lower frequency region, the Q bands of porphyrin were described in Gouterman four-orbital model as Qy (S1) and Qx (S2) by virtue of their orientation of polarization upon the macrocycle plane. When applying to chlorophyll a, there exists distortion on the originally perpendicular polarizations and these two electronic states are vibronically

doi:10.6342/NTU201803678

650 700 750 800 850 900 950 1000

Wavelength (nm)

Intensity

300 400 500 600 700

Wavelength (nm)

Absorbance

300 400 500 600 700

Wavelength (nm)

Absorbance

(a) (b)

Figure 1.2: Linear spectra of Chl a in diethyl ether[1]. (a) absorption spectrum (b) fluor- escence spectrum

mixed.[18, 2]

Furthemore, the vibrational overtone transitions can be observed in the spectrum, espe- cially on the Qy band. These represent a simultaneous vibrational and electronic trans- ition, with the final state being an excited vibrational state of the excited electronic state.

A progression of vibrational states can be observed, with the most intense band given by (0,0) transition, and the higher energy satellites termed as (0,1), (0,2) etc. The first number is the vibrational quantum number of the ground electronic state before the light absorption, and the second number denotes the vibrational state of the excited electronic state after the transition. As shown in Fig. 1.2(a), the Qxband is embedded in the vibronic bands of Qy. What makes it worse, there are many vibrations in a chlorophyll molecule, giving the assignments of absorption peaks and the interpretation of its vibronic charac- ters even more difficult.

The absorption spectrum of BChl a is shown in Fig. 1.3(a), which exhibits very similar characters as the Chl a spectrum. However, the Q_y band of BChl a locates at much longer wavelength, which means the Qy excitation energy of BChl a is much smaller than the one in Chl a. So without overlaid with the Qy vibronic sidebands, the Qx origin can be simply identified. The relative simple spectral character of BChl a enables a clearer as-

doi:10.6342/NTU201803678

700 800 900 1000 1100 1200

Wavelength (nm)

Intensity

300 400 500 600 700 800

Wavelength (nm)

Absorbance

300 400 500 600 700

Wavelength (nm)

Absorbance

(a) (b)

Figure 1.3: Linear spectra of BChl a in diethyl ether[1]. (a) absorption spectrum (b) fluorescence spectrum

signment of its absorption peaks. The most intense peak at about 773 nm is the Qy band, while the Q_x peak locates at about 577 nm.

The fluorescence spectra of both Chls exhibit peaks at slightly longer wavelengths than the absorption maximum (see Fig. 1.2(b) and 1.3(b)). The fluorescence emission is po- larized along the y molecular axis, as it is emitted from the Q_y transition[19]. During the emission, the initial state is the ground vibrational state of the excited elecronic state, and the final state is the excited vibrational state of the ground electronic state. This causes a shift of the emission to the longer-wavelength side of the main transition comparing to the first absorption band, in what is known as the Stokes shift.

1.2.3 Internal Conversion in Chlorophylls

To unravel the secrets of natural photosynthesis, it is important to find a good expla- tion of the mechanisms of these remarkble phenomena. At introduced previously, most works studying light harvesting focused on the intermolecular energy transfer, whereas the process of Chl Qx → Qy intramolecular relaxation is often overlooked. In this study, we focus on the first step of light harvesting, the internal conversion in the Chl photosyn- thetic pigment.

doi:10.6342/NTU201803678 Internal conversion is a non-radiative relaxation process for molecules or atoms from

a higher to a lower electronic states of the same spin multiplicity, converting the elec- tronic excitation energy into vibrational excitation energy[20]. Internal conversion plays a crucial role in many photophysical and photochemical processes. For instance, in biosystems, it enables the photoprotection molecules such as DNA[21] and melanin for skin[22], and it expands the energy range and efficiency of light harvesting in photosyn- thesis. For artificial molecular designs such as the photovoltaics and the polymers for photostabilization[23], internal conversion is also pivotal. The timescale for these pro- cesses, 10 fs to 10 ps[24], extends a broad range from systems to systems.

The ultrafast Q_x → Q_y interncal conversion in chlorophylls plays an significant role in the high efficiency of light harvesting in nature. The process facilitates the intermolecular excitation energy transfer between chlorophylls in antenna complex and debilitates the energy transfer from Qx to carotenoid S2[25, 26, 27]. It has been discussed in previous studies that the Q bands of chlorophylls play important role in light-harvesting and ex- citon transfer in photosynthesis. All exciton transfer and quantum coherence aspects of antenna complex are governed by the properties of the Q bands of chlorophyllides. Since inter-chlorophyll a Q_y state energy transfer are the major channel in excitation energy transfer dynamics among the antenna complex, the ultrafast Q_x to Q_y relaxation can fa- cilitate the dynamics competing to other relaxation pathways. Moreover, the portion of solar energy with wavelength shorter than 600 nm is significant, thus the utilization of energy except Qy band of Chl a cannot be negleted.

Due to the ultrafast timescale, the dynamics of this internal conversion is very difficult to probe. The experimental measurement of Q_x → Q_y rate constants were reported in a broad range previously[28, 29]. More recently, femtosecond time-resolved stimu- lated emission pumping fluorescence depletion (FS-TR-SEP-FD) enables rather accurate measurement of the relaxation rates[30, 31, 32, 33, 34]. About 100 fs relaxation times are reported depend on the solvents. Besides, in 2017, Collini et al. exploited 2D electronic spectroscopy (2DES) to provide insights into the mechanism of this internal conversion process involving vibronic levels of the Qy state and suggested the timescale to be about

doi:10.6342/NTU201803678 170 fs[35].

Theoretically, a detailed ab initio study of Q_x → Q_y internal conversion of chlorophylls was deficient and entails a significant challenge because of its complexity. Much ef- forts have been devoted to provide a simplified model that can be used to interprete the Qx → Qy relaxation.

In 2013, Reimers et al. combined a broad range spectral assignments to construct a theor- eticaly model for the estimation of the relaxation time of different Chls[2]. They suggest that the relaxation process is dominated by the interaction between the Q_xorigin and the vibrational line of Q_y excited by one quantum of one dominant vibronic-coupling mode.

The frequency and coupling strength of this effective mode is determined by fitting the spectra. One single empirical parameter which represents the Franck-Condon weighted coupled density of states is used in the simulation[36]. In their model, the vibrational fre- quency, coupling strength, and the relaxation tme of this effective mode are determined phenomenologically.

In 2015, Y. Zhao et al. applied non-adiabatic excited state molecular dynamics simu- lations and combined with ultrafast transient absorption spectroscopy measurements to modeled the internal converesion of Chl a and Chl b[37]. They proposed that the efficient dissipation of electronic energy into heat is achieved via selective participation of specific atomic groups and complex global migration of the wavefunction from the outer to inner ring.

However, despite being such a common chromophore and various experimental and com- putational studies have been continuously reported, to date a physical model based on first principle study to account for its foundamental dynamics is yet to be developed.

1.3 Non-Adiabatic Transition

The internal conversion between molecular excited states is one type of non-adiabatic transitions. The adiabatic states are function of nuclear coordinate and describe the mo- lecular states very well in general. However, in some region where non-adiabatic coup- lings between two or more adiabatic potential energy surfaces become large, the energy

doi:10.6342/NTU201803678 transfer between the electronic degrees of freedom and those of nuclei may occurs. That

is, the transition between the adiabatic electronic states can be achieved by gaining the energy from the nuclear motion. This is the so-called non-adiabatic transition.

Non-adiabatic quantum dynamics is an ubiquitous phenomenon common to a wide range of physical, chemical, and biological processes[38]. Especially, it is responsible for many fundamental excited state dynamics, such as photodissociation, photoisomerization[39], radiationless relaxation, collision processes with energy higher than the excitation threshold[40], charge and energy transfer.

1.3.1 Recent Developments

Non-adiabatic transition between PESs plays a pivotal role in numerous chemical pro- cesses of current interest. Fortunately, an ever increasing array of powerful experimental tools are provided recently. Especially, the recent progress of laser technology is remark- able and opened new possibilities of controlling various molecular processes. Ultrafast lasers make it possible to follow the course of chemical reactions in real time. Molecu- lar beam and laser technologies allow quantum state preparation of reactions and state- resolved detection of products, revealing detailed and quantitative information sufficient to challenge any theory. Novel multi-dimensional spectroscopies promise hitherto inac- cessible information about atomic motion and energy flow. Single molecule spectroscopy removes the obscuring effects of ensemble averaging to focus on an individual chemical event. These and other advances have propelled us to think in new ways about chemical reaction dynamics. This broader view, in turn, is now permeating many fields of chem- istry.

Within the context of photochemical reactions in polyatomic systems, ultrafast non-adiabatic dynamics often determines the underlying molecular relaxation after UV-visible photoex- citation. These processes produce broad and structureless photoabsorption spectra, mask- ing the spectroscopic features that could provide a structural and dynamical description of the system at the molecular level. Therefore, rigorous interpretations of spectroscopy often require detailed investigation using theoretical models and computational methods

doi:10.6342/NTU201803678 where quantum effects are explicitly considered.

Theoretically, the computational simulation of photochemical and photophysical pro- cesses governed by non-adiabatic dynamics in polyatomic molecules is one of the main driving forces in the field of molecular excitonics. For sufficient small systems, a com- plete quantum treatment involving multiple electronic states is feasible. For complex systems involving many degrees of freedom, an exact quantum mechanical treatment is impractical at present. The approximated methods must be employed. Two classes of the this kind of methods are the semi-classical approach and the mixed quantum-classical approach. Some brief introductions of the common methods will be given in Chapter 2.

1.3.2 Summary

In this study, we aim to investigate the mechanism of the remarkable Qx → Qy in- ternal conversion in Chls and uncover the secrets of the solvent environments and chem- ical structures dependences of relaxation rate for different Chls. The thesis is organized in the following manners.

In Chapter 2, the theoretical background in treating non-adiabatic transitions is provided.

First, the non-adiabatic coupling is derived from the standard molecular Hamiltonian and Born-Oppenheimer approximation. Some common approaches to simulate the non- adiabatic dynamics are breifly reviewed. Given the enormous computational efforts re- quired to directly evaluate the non-adiabatic couplings, we introduce the concept of dia- batic states and several property based diabatization methods. Among them, the diabatiz- ation method through enforcement of configuration uniformity proposed by Atchity and Ruedenberg is thoroughly demonstrated.

The electronic excitations of Chls are examined in Chapter 3. More specifically, the en- ergy levels, configurations of excited states in adiabatic basis are obtained using TD-DFT calculations and discussed. The different characters of the transition properties of the two low-lying excited state are emphasized. Furthemore, the diabatization method is applied on these molecules near the Qx equilibrium geometry. One vibrational normal mode is exemplified to demonstrate our procedure of diabatization method. The analysis of dia-

doi:10.6342/NTU201803678 batic potential energy surfaces are presented and the physical quantities in diabatic basis

show uniformity at the same situation where configuration uniformity is achieved.

The coupling between electron and nuclear degrees of freedom are evaluated in Chapter 4. The vibronic properties of the two molecules are investigated in adiabatic basis using reorganization energy. The absorption and ∆FLN spectra are simulated to judge the ap- plicability of calculation level used in our systems. By the analysis of potential energy curves in diabatic basis shown in Chapter 3, the diagonal and off-diagonal diabatic vi- bronic coupling between Q_y and Q_x are estimated for the first time. The strong-coupled modes which expected to be dominant in dynamics are discussed.

Finally, in Chapter 5, the rate of internal conversion is simulated using Fermi’s Golen rule with the effective Hamiltonian constructed throughout previous chapters. The vibronic coupling is considered as perturbation terms in the calculation. The results are carefullty analyzed considering various parameters region and compared with experimental life- time. The mechanism of the ultrafast internal conversion in Chls is uncovered and its dependences on environments and structural differences are discussed.

doi:10.6342/NTU201803678

Chapter 2

Theoretical Background

In quantum chemistry, the Born-Oppenheimer (BO) approximation is applied to define molecular electronic states. Since the mass of an electron is so light compared to that of nuclei, and the electron moves so quickly, the internuclear coordinates can be con- sidered as a good adiabatic parameter. If two electronic states are well separated ener- getically, non-adiabatic couplings (i.e., coupling introduced by nuclear motion) between these states may be neglected. This leads to the popular Born-Oppenheimer adiabatic approximation[41]. However, if the energy gap between the electronic states becomes small the non-adiabatic coupling can be large and the BO approximation breaks down. It is then desirable to find new electronic states, called diabatic states[42, 43, 44, 45, 46], in which the non-adiabatic couplings remain small and may thus be neglected or treated perturbatively.

In this chapter, start from the general molecular Hamiltonian, we introduce the BO and adiabatic approximation and go beyond them to consider the non-adiabatic dynamics. In addition, the theoretical approaches for studying non-adiabatic dynamics are briefly re- viewed. Given the difficulties of calculating the non-adiabatic coupling explicitly, we consider the concepts of diabatization and quasi-diabatization. Finally, we thoroughly introduce the strategy to enforce the configuration uniformity as a mean for determing diabatic states, which is the quasi-diabatization method used in this work.

doi:10.6342/NTU201803678

2.1 Molecular Hamiltonian

2.1.1 Separation of the Electronic and Nuclear Motions

For a molecule with N nuclei and n electrons, let R = (R1, ..., Rα, ..., RN) and r = (r1, ..., ri, ..., rn) denote the position vectors of the coordinates of the nuclei and the electrons, respectively. The molecular Hamiltonian is written as:

H^mol(r, R) = T^el(r) + T^nu(R) + V^nu−el(r, R) + V^el−el(r) + V^nu−nu(R). (2.1)

The various terms occurring, T^el(r), T^nu(R), V^nu−el(r, R), V^el−el(r), and V^nu−nu(R), are the kinetic energy operator for the electrons, and the kinetic energy operator for the nuclei, the electrostatic attraction between the electrons and the nuclei, the electrostatic repulsion between the electrons, and the electrostatic repulsion between the nuclei, re- spectively. The time-independent Schr¨odinger equation corresponding to the molecular Hamiltonian reads,

H^mol(r, R)Ψ^mol_l (r, R) = E_lΨ^mol_l (r, R). (2.2) It provides a prescription for computing the molecular eigenenergies, E_l, and eigenfunc- tions, Ψ^mol_l (r, R).

The Eq. (2.1-2.2) are extremely difficult to solve and several approximations and separ- ations of variables are required to make it easier to handle and to understand. Here, we introduce the most important one i.e. the separation of the electronic and nuclear motions.

Let us split the molecular Hamiltonian operator as follows:

H^mol(r, R) = T^nu(R) + H^el(r; R), (2.3)

where

H^el(r; R) = T^el(r) + V^nu−el(r, R) + V^el−el(r) + V^nu−nu(R) (2.4) denotes the electronic Hamiltonian for a given nuclear geometry R. For this Hamilto- nian, the nuclear positions are considered simply as parameters since the nuclear kinetic energy, T^nu(R), has been left out. Using an orthonormal basis set of nuclear functions, {ϕ^nu_λ (R)}, and an orthonormal basis set of electronic functions,Φ^el_m(r; R) , the eigen-

doi:10.6342/NTU201803678 function in Eq. (2.2) can be expanded and expressed as:

Ψ^mol_l (r, R) =X

m

X

λ

d^m_λ,lϕ^nu_λ (R)Φ^el_m(r; R), (2.5) with

Z

ϕ^nu∗_λ (R)ϕ^nu_µ (R)dR = δ_λµ (2.6) and

Z

Φ^el∗_m (r; R)Φ^el_n(r; R) = δ_λµdr = δ_mn, (2.7) for each nuclear geometry R. Note that the electronic basis function are parametrized by the nuclear coordinates. The Eq. (2.5) can be written as

Ψ^mol_l (r, R) = X

m

Ψ^l_m(R)Φ^el_m(r; R), (2.8) with

Ψ^l_m(R) =X

λ

d^m_λ,lϕ^nu_λ (R). (2.9)

Let us now define a specific electronic basis set frequently used in molecular physics and quantum chemistry, where the electronic basis functions are the eigenfunctions of the electronic Hamiltonian (Eq. (2.4). They are called adiabatic electronic states and form the so-called adiabatic basis and are denoted asn

Φ^el/adm (r; R)o

. By definition,

H^el(r; R)Φ^el/ad_m (r; R) = E_m^el(R)Φ^el/ad_m (r; R). (2.10) It is worth noting that the electronic energy, E_m^el(R), is a function of the nuclear coordin- ates, R, and as such, is commonly regarded as a potential energy surfaces (PES) for the nuclear motion.

The use of the adiabatic basis allows one to introduce the notion of PESs for each elec- tronic state and simplifies the coupled differential equations by suppressing the potential couplings. Nevertheless, the equations are still coupled through the adiabatic coupling terms, T_nm^ad(R) (for n 6= m).

2.1.2 Born-Oppenheimer Approximation

To this end, it is possible to expand the total molecular wavefunction in terms of a set of electronice wavefunctions as a function of R, Plug Eq. (2.8) into time-independent

doi:10.6342/NTU201803678 Schr¨odinger equation (Eq. (2.2), multiply by Φ^el∗_n (r; R) on the both sides, and then integ-

rate over r, we obtain:

−1 2

( _N X

I=1

1

M_I∇²_IΨ^l_n(R) + 2X

m

F_nm(R) ~∇_IΨ^l_m(R) +X

m

G_nm(R)Ψ^l_m(R) )

+X

m

E_nm^el (R)Ψ^l_m(R)

= Eel,nuΨ^l_n(R),

(2.11)

where

F_nm(R) = Z

Φ^el∗_n (r; R)

N

X

I=1

1 M_I

∇~_IΦ^el_m(r; R)dr, (2.12)

G_nm(R) = Z

Φ^el∗_n (r; R)

N

X

I=1

1 MI

∇²_IΦ^el_m(r; R)dr, (2.13) and

E^el_nm(R) = Z

Φ^el∗_n (r; R)H_el(r, R)Φ^el_m(r; R)dr

=E^el_nm(R)δ_nm.

(2.14)

We have (

−1 2

N

X

I=1

1

M_I∇²_I+ E^el_nn )

Ψ^l_n(R) +

∞

X

m

C_nmΨ^l_m(R) = E_nm^el (R)Ψ^l_n(R), (2.15)

where C_nn =

Z

Φ^el∗_n (r; R)

N

X

I=1

1

M_I∇²_IΦ^el_m(r; R)dr + Z

Φ^el∗_n (r; R)

N

X

I=1

1 M_I

∇~_IΦ^el_m(r; R)dr ~∇_I. (2.16) Born-Oppenheimer approximation assumes that all C_nm = 0, therefore:

(

−1 2

N

X

I=1

1

M_I∇²_I+ E^el_nn )

Ψ^l_n(R) = E_nm^el (R)Ψ^l_n(R). (2.17)

It is clear that under the BO approximation, the eigen energy of electronic Hamiltonian at a set of coordinate R, E_nm^el (R) yields the potential energy governing the nuclear motions.

Under the BO approximation, nuclear wavefunction is given by the expansion coefficient:

H_nu(R)Ψ^l_n(R) = E_nm^el (R)Ψ^l_n(R) (2.18)

where

Hnu(R) = (

−1 2

N

X

I=1

1

M_I∇²_I+ V(R) )

(2.19)

doi:10.6342/NTU201803678 Moreover, the potential energy that the nucleus feel, V(R), is the energy of the electron,

E^el_nn, at that geometry, R. The motion of electron and nucleus have been separated.

At this point, it is worth mentioning that the Born-Huang approximation, which is also a adiabatic approximations and is closely related to the BO approximation to the point that it is often confused with it, includes the diagonal coupling term, Cnn.

2.1.3 Non-Adiabatic Coupling

Going beyond the Born-Oppenheimer approximation, the molecular states can no longer be written as single products of nuclear and electronic contributions. In constract, we must write them as a so-called Born expansion,

Ψ^mol_l (r, R) =X

m

X

λ

d^m_λ,lϕ^nu_λ (R)Φ^el_m(r; R), (2.20)

which is equivalent to Eq. (2.8). The first-order non-adiabatic couplings, Fnm(R), also named as the derivative coupling, are defined in Eq. (2.12). And the second-order non- adiabatic couplings, G_nm(R), the scalar couplings, is defined in Eq. (2.13).

This shows that neglecting C_nm(R) is equivalent to the Born-Oppenheimer approxima- tion. Neglecting C_nm(R) for n 6= m while keeping C_nn(R) is equivalent to the adiabatic approximation mentioned in the previous subsection.

Applying the Hellmann-Feynman theorem, the derivative coupling can be further derived, for n 6= m,

F~_mn( ~R) = −1 2hm|

NA

X

I=1

1 MI

∇_I|ni = −1 2

NA

X

I=1

1 MI

hm| − ¹₂∇_IH_el|ni

Em(R) − En(R), (2.21) where Em(R)−En(R) is the energy gap between the two states. This shows that T_nm^ad(R) diverges when the electronic states become degenerate. Similarly, the scalar coupling is written as

G_mn( ~R) = hm| − 1 2

NA

X

I=1

1

M_I∇²_I|ni

= − 1 2

NA

X

I=1

1

M_I∇Ihm| ∇I|ni + 1 2

NA

X

I=1

1 M_I

∞

X

γ=1

hγ| ∇I|mi hγ| ∇I|ni .

(2.22)

doi:10.6342/NTU201803678 However, the curse of dimensionality of these third order derivative terms make it ex-

tremely difficult to calculated even though we have the exact analytical form of the non- adiabatic coupling.

2.2 Theoretical Treatments of Non-Adiabatic Dynamics

The non-adiabatic dynamics plays significant roles in many photochemical and pho- tophysical reactions, such as radiationless relaxation, charge and energy transfer. Thess processes pass through the region where non-adiabatic couplings are important. As a res- ult, going beyond Born-Oppenheimer approximation becomes an extraordinarily critical issue in the field of chemical physics.

Given the importance of non-adiabatic dynamics and its adversity in understanding and interpretation of experimental phenomena, a variety of theoretical methods used to simu- late these processes have been developed[47]. Here we will introduce some of the most widely used methods. The comparison between these methods and other methods can be found in Ref [48, 49]. For example, the full quantum methods such as split operator approach[50, 51], Gaussian wavepacket dynamics[52, 53].

The direct solution of time-dependent Schr¨odinger equation leads to wavepacket propaga- tion methods[54, 55]. In wavepacket dynamics, the system is represented in its initial state by a wavepacket, which is a nonstationary superposition of eigenstates. The time- evolution of this wavepacket followed the total Hamiltonian is then calculated numeric- ally. Not only can all the required information be extracted from the evolving wavepacket, but it provides a very pictorial description of the process of interest. Unfortunately, the standard wavepacket dynamics suffers from its need of tremendous computer resources, which scales exponentially with the number of degrees of freedom in the system. For systems with more than four to six degrees of freedom, the simulations are typically pre- cluded. This limitation on dimensionality forces us to search for approximate methods.

doi:10.6342/NTU201803678

2.2.1 Multi-Configuration Time-Dependent Hartree

Multi-configuration time-dependent Hartree (MCTDH) is one of the powerful wave- packet dynamics algorithms at present[53]. It was first introduced in 1990 by Meyer, Manthe, and Cederbaum[56, 57]. The idea of the MCTDH approach is to expand the MCTDH wavefunction in a set of basis, which is smaller than the basis set used in standard wavepacket method but time-dependent.The time-dependent weighted Hartree products with time-dependent single-particle functions is written as

Φ(Q₁, ..., Q_f, t) =

n1

X

j1=1

· · ·

nf

X

jf=1

A_j1...jf(t)

f

Y

k=1

ϕ^(k)_j

k (Q_k, t). (2.23) By solving the time-dependent Schr¨odinger equation using a variational principle, one obtain the equations of motion for the expansion coefficients and the single-particle func- tions. The basis thus follows the evolving wavepacket. The high-accuracy potential en- ergy surfaces and non-adiabatic couplings are then obtained. Importantly, the result con- verges on the exact result as the basis is increased in size. MCTDH is a full quantum and numerically exct approach which can be applied in system with up to 24 nuclear degrees of freedom. More recently, multi-layer multi-configuration time-dependent Hartree (ML- MCTDH) has been developed for large systems with hundreds of degrees of freedom.

However, those full quantum approaches require a significant amount of computational efforts and thus only applicable in small systems. Accordingly, many semi-classical meth- ods and mix quantum classical methods have been developed.

One class of the mix quantum-classical methods is the nonadiabatic molecular dynamics which involves the separation of electron and nuclear subsystems by propagating nuclear trajectories on an electronic potential energy surface defined by electronic states.

2.2.2 Fewest Switch Surface Hopping

Surface hopping is one of the mixed quantum-classcal technique that includes quantum mechanical effects into molecular dynamics (MD) simulations[58]. It is developed to deal with the dynamics where Born-Oppenheimer approximation breads down. By including

doi:10.6342/NTU201803678 excited adiabatic surfaces in the calculation and allowing sudden changes (trajectory hop)

between these surfaces, surface hopping incorporates the non-adiabatic effects to the sim- ulations. For a large number of trajectories, the following procedure is repeated to have the nuclear wavepacket information. Along the propagation of one trajectory, for a fixed nuclear geometry, one solves the time-independent Schr¨odinger equation for electrons and evaluates the energy gradient. The energy gradient is then used to update the nuclear geometry classically according to the Newton’s equation. For the new step, nonadiabatic coupling is calculated and a hopping between electronic states is performed if necessary, while the transition probability is determined according to the coupling.

One of the most widely used surface hopping method was introduced by J. C. Tully, called Fewest Switch Surface Hopping (FSSH)[58]. In this approach, the non-adiabatic effect which treated by the trajectory hop is included. The popularity of this method cannot be understated. However, there exist many problems of surface hopping. The most import- ant among them is the unphysical way of determining the distribution of kinetic energy by adjusting the component of velocity in the direction of the nonadiabatic coupling after the trajectory hop.

2.2.3 Ehrenfest Mean Field

Another simulation method is Ehrenfest mean field (EMF) method[59] proposed in 1927. This method is not originally derived to study the non-adiabatic dynamics but has been proved to be an acceptable method in simulating non-adiabatic dynamics.

In Ehrenfest mean field method, the nuclear degrees of freedom is propagated along an average potential, hence the name mean field:

d²

dt²R_nu(t) = −∇_nuhH_el(R_nu)i. (2.24) Unlike FSSH, there is no hopping between different potential energy surface. This is because nuclear degrees of freedom is propagated along mean potential and thus non- adiabatic transistion is already included. EMF is a relatively cheap simulation method compare to FSSH and is useful in simulating dynamics over less important states. How-