Theoretical Characterization of Optical Processes in ...13151/FULLTEXT01.pdf · studying molecular...

Theoretical Characterization of

Optical Processes in Molecular

Complexes

Kai Liu

Department of Theoretical Chemistry

School of Biotechnology

Royal Institute of Technology

Stockholm, Sweden 2008

c© Kai Liu, 2008

ISBN 978-91-7178-857-3

Printed by Universitetsservice US AB, Stockholm, Sweden, 2008

3

Abstract

The main theme of this thesis is to study effects of different environments on geometric and

electronic structures, as well as optical responses, of molecules using time-(in)dependent

density functional theory. Theoretical calculations have been carried out for properties that

can be measured by conventional and advanced experimental techniques, including one-

photon absorption (OPA), two-photon absorption (TPA), surface-enhanced Raman scatter-

ing (SERS) and second order nonlinear optical (NLO) response. The obtained good agree-

ment between the theory and the experiment allows to further extract useful information

about inter- and intra-molecular interactions that are not accessible experimentally.

By comparing calculated one-photon absorption spectra of aluminum phthalocyanine chlo-

ride (AlPcCl) and AlPcCl -water complexes with the corresponding experiments, detailed

information about the interaction between water molecules and AlPcCl, and geometric

changes of AlPcCl molecule has been obtained. Effects of aggregation on two-photon ab-

sorption spectra of octupolar molecules have been examined. It is shown that the formation

of clusters through inter-molecular hydrogen bonding can drastically change profiles of TPA

spectra. It has also demonstrated that a well designed molecular aggregate/cluster, den-

drimer, can enhance the second order nonlinear optical response of the molecules. In collabo-

ration with experimentalists, a series of end-capped triply branched dendritic chromophores

have been characterized, which can lead to large enhancement of the second order NLO

property when the dipoles of the three branches in the dendrimers are highly parallelized.

Surface-enhanced Raman scattering has made the detection of single molecules on metal

surface become possible. Chemically bonded molecule-metal systems have been extensively

studied. We have shown in a joint experimental and theoretical work that stable Raman

spectra of a non-bonding molecule, perylene, physically adsorbed on Ag nano-particles can

also be observed at low temperature. It is found that the local enhanced field has a tendency

to drive molecule toward a gap of two closely lying nano-particles. The trapped molecule

can thus provide a stable Raman spectrum with high resolution when its thermal motion is

reduced at low temperature.

For the ever growing size of molecular complexes, there is always the need to develop

new computational methods. A conceptually simple but computationally efficient method,

named as central insertion scheme (CIS), is proposed that allows to calculate electronic

structure of quasi-periodic system containing more than 100,000 electrons at density func-

tional theory levels. It enables to monitor the evolution of electronic structure with respect

to the size of the system.

4

Preface

The work presented in this thesis has been carried out at the Department of Theoretical

Chemistry, School of Biotechnology, Royal Institute of Technology, Stockholm, Sweden.

List of papers included in the thesis

Paper I Origin of the Q-band splitting in the absorption spectra of aluminum phthalo-

cyanine chloride, K. Liu, Y.-H. Wang, J.N. Yao, and Y. Luo, Chem. Phys. Lett., 438

(2007) 36.

Paper II Aggregation effects on two-photon absorption spectra of octupolar molecules,

K. Liu, Y.-H. Wang, Y.Q. Tu, H. Agren, and Y. Luo, J. Chem. Phys., 127 (2007) 026101.

Paper III Two-photon absorption of hydrogen bonded octupolar molecular clusters, K.

Liu, Y.-H. Wang, Y.Q. Tu, H. Agren, and Y. Luo, J. Phys. Chem. A, 00 (2008) 000.

Paper IV Hyperpolarizabilities of end-capped triply branched dendrimers: a theoretical

study, Y.H. Wang, K. Liu, J. Heck, H.G. Kuball, H. Agren, and Y. Luo, J. Chem. Phys.,

submitted.

Paper V Controlling single molecular Raman behavior of a non-bonding molecule on Ag

nanoparticles, Z.X. Luo, Y. Luo, J. Li, K. Liu, H.B. Fu, Y. Ma, and J.N. Yao, submitted.

Paper VI Raman spectra of a single perylene molecule on silver clusters, K. Liu, Z.X.

Luo, J.N. Yao, and Y. Luo, in manuscript.

Paper VII An efficient first-principle approach for electronic structure calculations of

nanomaterials, B. Gao, J. Jiang, K. Liu, Z. Wu, W. Lu, and Y. Luo, J. Comput. Chem.,

29 (2008) 434.

5

List of papers not included in the thesis

Paper VIII An elongation method for first principle simulations of electronic structures

and transportation properties of finite nanostructures, J. Jiang, K. Liu, W. Lu, and Y.

Luo, J. Chem. Phys., 124 (2006) 214711.

Paper IX Hierarchical chiral supra-molecular structures by the self-assembly of achiral

azobenzene derivative with Water, X. Sheng, G. Zhang, J. Wang, Y. Wang, H. Fu, X. Gao,

K. Liu, Y.H. Wang, Y. Luo, W. Yang, A. Peng, and J.N. Yao, J. Am. Chem. Soc.,

Submitted.

Paper X Structure-property relationship in organometallic compounds regarding SHG, J.

Heck, M. H. Prosenc, T. Meyer-Friedrichsen, J. Holtmann, E. Walczuk, M. Dede, T. Farrell,

A. R. Manning, H.-G. Kuball, G. Archetti, Y.-H. Wang, K. Liu, and Y. Luo, Proc. SPIE,

6653 (2007) 66530R

Paper XI Three-branched dendritic NLOphores, more than three times a single-strand

chromophore? J. Holtmann, E. Walczuk, M. Dede, C. Wittenburg, J. Heck, G. Archetti, R.

Wortmann, H.G. Kuball, Y.-H. Wang, K. Liu, and Y. Luo, Chem. Euro. J., submitted.

6

Comments on my contribution to the papers included

• I was responsible for calculations and writing of manuscripts for papers I, II, III and

VI.

• I was responsible for part of the calculations for papers IV and VII.

• I was responsible for most of the calculations for paper V.

7

Acknowledgments

First of all, I would like to give my greatest thanks to my supervisor Prof. Yi Luo for giving

me the opportunity to study in Sweden and for teaching me a lot for my research work.

And I should thank Prof. Luo and his wife Dr. Xing together for their kindly help in my

daily life.

I am heartily grateful to Prof. Hans Agren for creating such a warm atmosphere in the

Department of Theoretical Chemistry. I have really enjoyed my time studying here.

I would like to give my special thanks to Prof. Margareta Blomberg, Prof. Boris Minaev,

Prof. Faris Gel’mukhanov, Docent Fahmi Himo and Docent Pawel Salek for their excellent

courses. And I am deeply thankful to Dr. Jun Jiang, Bin Gao and Dr. Barbara Brena for

helping me in my work. I would like to thank all my friends for your sincere friendship.

Many thanks to the Swedish National Infrastructure for Computing (SNIC) for computer

time at national supercomputer facilities.

I would like to express my warmest thanks to my parents, my brother, my wife Dr. Yanhua

Wang and all the other close relatives for their love and support.

Kai Liu

2008-01

8

Contents

1 Introduction 11

2 Density Functional Theory and Response Functions 15

2.1 Density Functional Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.1.1 Hohenberg-Kohn Theorems . . . . . . . . . . . . . . . . . . . . . . . 16

2.1.2 Kohn-Sham equations . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.1.3 Exchange and Correlation Functionals . . . . . . . . . . . . . . . . . 17

2.2 Response Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.2.1 Response Theory for Exact States . . . . . . . . . . . . . . . . . . . . 19

2.2.2 One-Photon Absorption . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.2.3 Two-Photon Absorption . . . . . . . . . . . . . . . . . . . . . . . . . 21

3 Linear and Nonlinear Optical Spectroscopy 23

3.1 One-photon Absorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.2 Two-photon Absorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.3 Selection Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.4 Hyperpolarizabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.4.1 Finite Field Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.4.2 Analytical Derivative Method . . . . . . . . . . . . . . . . . . . . . . 30

3.4.3 Response Theory Method . . . . . . . . . . . . . . . . . . . . . . . . 30

3.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.5.1 Q-band Splitting of Aluminum Phthalocyanine Chloride . . . . . . . 31

9

10 CONTENTS

3.5.2 Two-Photon Absorption of Octupolar Molecules . . . . . . . . . . . . 32

3.5.3 One- or Two-end-capped Dendritic Structures . . . . . . . . . . . . . 34

4 Raman Spectroscopy 37

4.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4.2 Basic Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4.3 Surface Enhanced Raman Spectroscopy . . . . . . . . . . . . . . . . . . . . . 40

4.3.1 Electromagnetic Enhancement . . . . . . . . . . . . . . . . . . . . . . 41

4.3.2 Chemical Enhancement . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

5 Central Insertion Scheme 47

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

5.2 Central Insertion Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

5.2.1 Basic Idea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

5.2.2 Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

5.2.3 Central Insertion Process . . . . . . . . . . . . . . . . . . . . . . . . . 50

5.2.4 Approximations and Error Control . . . . . . . . . . . . . . . . . . . 51

5.2.5 BioNano Lego . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

5.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

Chapter 1

Introduction

If we knew what we were doing, it wouldn’t be called research, would it?

Albert Einstein

Since there were human being, people kept trying to understand and to change everything

around them. It is no doubt that many common phenomena can be studied by watching,

listening, touching, smelling and tasting directly. However, there are still enormous details

and ultimate truths behind the phenomena can not be obtained by these normal means.

A wide variety of experimental tools and techniques have been developed and applied for

different purposes. Molecular optical spectroscopy is one of such tools that is designed for

studying molecular structures and properties.

Optics is an old science started several centuries ago. It is a branch of physics that ex-

plains the behavior and properties of light and the interaction between light and matter.

The research area of optics covers a wide spectrum of light, including infrared, visible and

ultraviolet. The beginning of optical spectroscopy is marked by the first spectrum that

was obtained by Newton in 1666 with the help of a triangular prism. The spectrum was a

rainbow of colors generated by dispersing sunlight. By analysis of the emitted light, one can

immediately get information about temperature, mass and luminosity of an object. Nowa-

days, many modern spectrometers have been developed that allow to effectively collect high

resolution spectra of different systems. In 1859, Kirchhoff showed that elements or com-

pounds have their own unique spectra, which means that one can determine the chemical

composition of an unknown system by studying its spectrum. Those information usually

come from the positions and the intensities of spectral lines. Optical spectroscopy has been

heavily used in physics, analytical chemistry, astronomy and remote sensing since it reveals

11

12 CHAPTER 1. INTRODUCTION

microscopic information of the matter without imposing physical damage. It can also pro-

vide information about chemical surrounding and intra/inter-molecular interactions of the

studied systems by comparing spectra gathered under different conditions.

The interpretation of various spectra requires good knowledge of electronic structures and

properties of the given systems. Physical and chemical intuitions might be useful for under-

standing the spectra of simple and small systems, but they become quickly inefficient for

large and complicated systems. Theoretical modeling has played a significant role in the de-

velopment and application of various spectroscopic techniques. With the fast development

of computer technology and computational methods, nowadays it is possible to calculate

systems of very large size with good accuracy in a reasonable time. In comparison with

experiments, theoretical study usually costs much less and can provide massive information

that are normally not accessible experimentally. For instance, geometric structures of single

polyatomic molecules can nowadays be accurately determined by highly correlated theoreti-

cal methods, which can largely replace experimental measurements. However, for molecular

complexes or clusters the most powerful approach would be a combination of theoretical

modelings and experimental measurements. A good agreement between theoretical and ex-

perimental results for certain molecular properties provides not only good interpretation

and understanding of observed phenomena, but also useful inputs that enable to carry out

further theoretical modelings to obtain a more complete picture of the whole system under

investigation.

In a technology-driven society, the need for new materials is becoming severer. Molecular

materials have become increasingly important owing to their exceptionally optical and elec-

tronic properties. One of the examples is the molecular nonlinear optical (NLO) materials

which can be used to change the color of the laser light and to storage information. Theo-

retical investigations have made significant contribution to the understanding of nonlinear

optical processes, and can give a mechanistic insight to the connections between molecular

structure and nonlinear optical properties. They have helped chemists to design and synthe-

size organic molecules with optimized performance. The technological applications require

to use a large number of NLO active molecules and the interaction between molecules could

thus affect the performance of the material. An extreme example is the so-called push-

pull molecule attached with strong electron donor and acceptor groups. A single push-pull

molecule often possesses very large second order nonlinear optical response. When a large

number of molecules get together (in the case of liquid or solution phase), the strong dipole

interaction between molecules can quench the second order NLO of the complexes com-

pletely. Therefore, a good understanding of single molecular behavior might not be always

relevant to molecular complexes. The inter-molecular interactions under different conditions

should thus be taken into account.

13

The main theme of this thesis is to study effects of different environments on geometric and

electronic structures, as well as optical responses, of molecules. Theoretical calculations

have been carried out for experimental observeables, such as one-photon absorption (OPA),

two-photon absorption (TPA), Raman spectra and second order NLO response, of various

systems to extract useful information about molecular interactions.

Experimentally it was observed that the well defined so-called Q-band of aluminum phthalo-

cyanine chloride (AlPcCl) molecule displays a splitting in water solution. By computing

OPA of different possible water-AlPcCl complexes with time-dependent density functional

theory (TD-DFT), one can reveal the origin of the splitting and the geometric change of

the molecule in water, something that can not be directly observed experimentally. By

constructing different water-AlPcCl complexes, it is found that the distortion of AlPcCl

molecule could be the major cause of the Q-band splitting.

To avoid the possible cancelation of dipoles, a molecule, named as octupolar molecule that

possesses zero permeant dipolar moment but large second order NLO response was proposed

by Zyss.1,2 We have used a model system to examine the possible aggregation effects on

TPA spectra of octupolar molecules. With the help of molecular dynamics simulations and

TD-DFT calculations, we have found that the thermal motion can induced intra- and inter-

molecular hydrogen bonds among molecules to form clusters of different sizes. This study

demonstrates that even for molecule without dipole moment, the aggregation effect can still

take place and result in significant changes in TPA spectra in comparison with the case of

a single molecule.

Molecular aggregates can sometime enhance certain nonlinear optical properties if they

are arranged in certain ways. One of these examples is the so-called dendrimer, in which

the molecules are organized through very connecting groups. A well defined architecture

could avoid the cancelation of dipoles and result in larger second order NLO properties.

In collaboration with experimentalists, a series of end-capped triply branched dendritic

chromophores have been designed and characterized. It is shown that large enhancement of

the second order NLO property can be expected, when the dipoles of the three branches in

the dendrimers are highly parallelized.

Raman scattering is a general process that can provide more useful information on molecu-

lar structures, orientations and interactions with neighboring species than one-photon and

two-photon absorptions/emissions. However, the ordinary Raman effect is often very weak

and can not be employed in the analysis of small number of molecules. Situation becomes

quite different when a molecule is adsorbed on a metal surface. In this case, Raman scat-

tering can be dramatically enhanced due to the presence of surface plasmon coupled with

chemical bonding between the molecule and the metal surface. A single molecule can thus

14 CHAPTER 1. INTRODUCTION

be effectively determined by the so-called surface enhanced Raman scattering (SERS). We

have used a system that consists of a non-dipolar molecule, perylene, physically adsorbed on

Ag clusters to examine the possibility of extending the applicability of SERS to non-bonding

systems.

For very large and complex systems, there are always needs to increase the capacity of

computational programs. We have recently proposed and implemented a conceptually simple

but computationally efficient method, named as central insertion scheme (CIS), to calculate

electronic structures of quasi-periodic systems containing more than 100,000 electrons at

density functional theory levels. With this method, it thus becomes possible to monitor the

evolution of electronic structure with respect to the size of the system and the formation of

band from discrete molecular orbitals.

Chapter 2

Density Functional Theory and

Response Functions

2.1 Density Functional Theory

Based on the fact that the masses of nuclei are much heavier than those of electrons, molec-

ular systems could be viewed as interacting electrons moving in the static potential of nuclei

according to Born-Oppenheimer approximation. With fixed nuclear geometry ~R, one can

solve the electronic Schrodinger equation,

He | Ψe(~r; ~R)〉 = Ee | Ψe(~r; ~R)〉, (2.1)

and get the corresponding electronic energy eigenvalue Ee and eigenfunctions (or orbitals)

| Ψe(~r; ~R)〉, ~r is coordinates of electrons. The potential energy surface (PES): Ee(~R) on

which the nuclei move can be obtained by changing the nuclear positions ~R and repeatedly

solving the electronic Schrodinger equations. By solving the nuclear Schrodinger equation,(−

M∑

A=1

1

2MA

∇2A + Ee(~R)

)| Ψnuc(~R)〉 = E | Ψnuc(~R)〉. (2.2)

the total energy E of the molecule can be obtained. And the total molecular wavefunction

can be written as,

| Ψtot(~r, ~R)〉 =| Ψe(~r; ~R)〉 | Ψnuc(~R)〉. (2.3)

It is relatively easy to define the Hamiltonian operator H. However, the wave function Ψ

for large system is difficult to determine because it depends on 3N spatial coordinates and

N spin coordinates, where N is the number of electrons.

15

16 CHAPTER 2. DENSITY FUNCTIONAL THEORY AND RESPONSE FUNCTIONS

Moreover, electrons repel each other and their motions are strongly correlated in reality.

In the mean field approximation, like Hartree-Fock,3 this correlation effect of electrons is

completely neglected. To improve the quality of the wavefunctions, many so-called post-

Hartree-Fock methods, such as Configuration Interaction (CI),4 Coupled Cluster (CC),5 and

Møller-Plesset perturbation theory6 (MP2, MP4, etc), have been developed to take into ac-

count the electron correlation at different levels. In general, highly correlated computational

methods require much more computational power and time.

Since the late eighties and early nineties, Density Functional Theory (DFT) has become

more popular in molecular physics and chemistry. Compare with wavefunction based ab

initio methods, the main advantages of DFT methods are the sufficiently high accuracy

and short computational time, in particular for large systems, since it reduces the spatial

dimension of the problem from 3N to three. This simplification is done by assuming the

energy as a function of the electron density ρ(~r) which is only related to 3 space variables.

2.1.1 Hohenberg-Kohn Theorems

Before the introduction of Hartree-Fock theory, L. H. Thomas and E. Fermi proposed the

first DFT model independently.7–9 But only from 1964, with the help of two important

theorems proved by Hohenberg and Kohn,10 the Thomas-Fermi theory started to give good

results. These two theorems are usually considered as the base of modern DFT.

The first theorem states that the external potential V (~r) of a system of N electrons is

uniquely determined by its density ρ(~r). And then the Hamiltonian

H = −1

2

∑

i

∇2i +

∑

i

V (~ri) +∑

i>j

1

~iij(2.4)

is determined by ρ(~r). ρ(~r) can be got with the equation,

ρ(~r) =N∑

i=1

|ψi(~r)|2 . (2.5)

The energy can be written as a functional of the density

E = E[ρ]. (2.6)

The second one proves that if ρ(~r) is the exact ground state density, the total electronic

energy of the system becomes a minimum.

E[ρ] ≤ E[ρt], (2.7)

where ρt is any trial density.

2.1. DENSITY FUNCTIONAL THEORY 17

2.1.2 Kohn-Sham equations

Since the kinetic energy functional T [ρ] is easily calculated from the wave function, we can

express the energy functional of an isolated molecular system as,

E[ρ] = T [ρ] + VNe[ρ] + Vee[ρ], (2.8)

where VNe[ρ] and Vee[ρ] account for the nuclear-electron attraction and the electron-electron

interaction functional, respectively.

It has been assumed by the Kohn-Sham method that a fictitious non-interacting system has

the same electron density distribution ρs(~r) as that of the real molecular system ρ(~r). The

energy functional of this non-interacting system can then be written as,

Es[ρ] = Veff [ρ] + Ts[ρ], (2.9)

where Veff is an effective potential functional and Ts[ρ] is the non-interacting kinetic energy

functional. From the relationship ρs(~r) = ρ(~r) and Es[ρ] = E[ρ], we can easily get

Veff [ρ] = VNe[ρ] + (T [ρ] − Ts[ρ]) + Vee[ρ]. (2.10)

In detail,

Veff (~r) = VNe(~r) +

∫ρ(~r′)

|~r − ~r′|d ~r′ + Vxc[ρ(~r)], (2.11)

where the second term is the electron-electron Coulomb repulsion functional, and Vxc[ρ(~r)] is

the so-called exchange correlation potential functional. By solving the Schrodinger equations

of this non-interacting system,

[−1

2∇2

i + Veff(~r)]ψi(~r) = ǫi ψi(~r), (2.12)

the orbitals ψi(~r) that reproduce the electron density of the real molecular system will be

obtained.

2.1.3 Exchange and Correlation Functionals

The key problem of DFT is to find efficient exchange correlation functional since there is

no straightforward way to systematically improve the exact form of Exc[ρ]. Exc[ρ] has often

been separated in two parts, a pure exchange part Ex[ρ] and a pure correlation part Ec[ρ].


Local Density Methods

Local Density Approximation (LDA) assumes that the density is local and can be treated

as a homogeneous electron gas. For a system that an infinite number of electrons N in an

infinite volume V, the density ρ can be remaining finite and written as,

ρ = N/V. (2.13)

Then we can get the exact exchange energy Ex[ρ]

Ex[ρ] =4

3(3π−1)

2

3

∫ρ(~r)

4

3d~r. (2.14)

The correlation energy functional of the electron gas has been developed by Vosko, Wilk

and Nusair (VWN).11 When considering α and β spins separately for an open shell system,

LDA is replaced by the Local Spin Density Approximation (LSDA).

Gradient Corrected Approximation

For a non-uniform electron gas, LDA approach has been extended to functionals depending

on the gradient of the density ▽ρ(~r) based on the Generalized Gradient Approximation

(GGA). The general form of the GGA can be written as

EGGAxc [ρα, ρβ] =

∫f(ρα, ρβ,▽ρα,▽ρβ)d~r. (2.15)

By considering boundary conditions and fitting to accurate numerical or experimental data,

several gradient corrected functionals have been proposed for the exchange energy such as

Perdew and Wang (PW86),12 Becke (B88),13 and correlation energy such as Perdew (P86).14

And Lee, Yang and Parr (LYP)15 developed new correlation functionals that are not based

on corrections of the LSDA.

Hybrid Functionals

Another way of constructing functionals is to combine the LSDA, exact exchange and the

gradient correction terms together. Among many approximations that have been developed

over the years, one of the most successful functionals is the hybrid B3LYP method which

combine the Becke-3 (B3) parameter exchange functional16 and LYP correlation functional.

The B3LYP exchange-correlation functional can be written as,

EB3LY Pxc = ELDA

xc + a(EHFx −ELDA

x ) + b(EGGAx −ELDA

x ) + c(EGGAc − ELDA

c ), (2.16)

2.2. RESPONSE THEORY 19

where a = 0.20, b = 0.72, and c = 0.81 are three empirical parameters; EGGAx and EGGA

c

are the generalized gradient approximation formulated with the B88 exchange functional

and the LYP correlation functional, and ELDAc is the VWN correlation functional. B3LYP

method can often give results in good agreement with experiments and thus be widely used

in many theoretical calculations.

2.2 Response Theory

Among many fundamentally different ways of describing the response of the electronic de-

grees of freedom in the presence of a perturbing field, response theory17 has enjoyed a great

success in providing accurate results for a wide variety of molecular properties. With re-

sponse theory, the exact information about the excited states is no longer needed. A set

of response functions and their residues are used to express molecular properties, such as

(hyper)polarizabilities, excitation energies, spin-orbit coupling matrix elements, phospho-

rescence lifetimes, nuclear shielding constants and (hyper)magnetizabilities, just to name a

few. Moreover, results of response theory can be compared with experimental measurements

directly, since calculations are performed for external fields of any frequencies. Response

theory deals with the time dependent and time independent properties on the same footing.

2.2.1 Response Theory for Exact States

In detail, we can write Hamiltonian H of a molecular system which is perturbed by a time

dependent perturbation field as the summation of the unperturbed Hamiltonian H0 and the

time dependent perturbation term V (t).

H = H0 + V (t), (2.17)

V (t) =

∫ ∞

−∞

V ω e(−iω+ǫ) t dω, (2.18)

where ǫ is a positive infinitesimal. When t = −∞, the system is in the unperturbed state

V (t) = 0. The unperturbed hamiltonian H0 can be used to solve

H0|0〉 = E0|0〉 (2.19)

and

H0|n〉 = E0|n〉 (2.20)


Let A be a Hermitian operator describing an observable molecular property. When the

perturbation V (t) caused by the external field is sufficiently small, one can write the time-

evolution of the expectation value of A as,

⟨A⟩

(t) =⟨A⟩

0

+

∫ ∞

−∞

⟨⟨A; V ω1

⟩⟩e−iω1t dω

+1

2!

∫ ∞

−∞

∫ ∞

−∞

⟨⟨A; V ω1, V ω2

⟩⟩e−i(ω1+ω2)t dω1 dω2

+1

3!

∫ ∞

−∞

∫ ∞

−∞

∫ ∞

−∞

⟨⟨A; V ω1 , V ω2 , V ω3

⟩⟩e−i(ω1+ω2+ω3)t dω1 dω2 dω3

+ · · · ,

(2.21)

⟨⟨A; V ω1, · · · , V ωn

⟩⟩is the response function, which describes the response of the molecular

properties A to the external perturbation field.

For example, if A represents the dipole moment operator µ of the molecular system and the

perturbing field is electric, the corresponding response functions are the molecular linear

polarizability and nonlinear hyperpolarizabilities α, β, γ, etc.

αij(−ωσ;ω1) = −〈〈µi; µj〉〉ω1, (2.22)

βijk(−ωσ;ω1, ω2) = −〈〈µi; µj, µk〉〉ω1,ω2, (2.23)

γijkl(−ωσ;ω1, ω2, ω3) = −〈〈µi; µj, µk, µl〉〉ω1,ω2,ω3. (2.24)

By choosing other operators, one can obtain other properties through the corresponding

response functions. And for time-independent perturbations, Eq. 2.21 naturally reduces

to the ordinary Taylor expansion of the static expectation value of A, where the response

function⟨⟨A; V ω1, · · · , V ωn

⟩⟩ω1=0,··· ,ωn=0

becomes the expansion coefficient.

When a system is subject to one or several fields, the effects of the fields can be also deter-

mined by response functions. Each molecular property A can be described within a frame-

work by suitable choice of fields. For instance, the linear response function⟨⟨A; V ω1

⟩⟩

ω1

contains all terms that are linear in V ω1 . And the quadratic and cubic response functions

functions⟨⟨A; V ω1 , V ω2

⟩⟩

ω1,ω2

and⟨⟨A; V ω1, V ω2, V ω3

⟩⟩

ω1,ω2,ω3

contain all contributions

to the expectation values that are linear in V ω1 , V ω2 and V ω1 , V ω2 , V ω3 , respectively.

2.2. RESPONSE THEORY 21

2.2.2 One-Photon Absorption

Electron transitions between different states of a molecule can also be obtained with the

response functions. For instance, the linear response function⟨⟨A; V ω

⟩⟩ω

can be written

as,

⟨⟨A; V ω

⟩⟩

ω=∑

n 6=0

⟨0∣∣∣A∣∣∣n⟩⟨

n∣∣∣V ω

∣∣∣ 0⟩

ω − (En − E0)−∑

n 6=0

⟨0∣∣∣V ω

∣∣∣n⟩⟨

n∣∣∣A∣∣∣ 0⟩

ω + (En − E0), (2.25)

where En is the energy corresponding to eigenfunction |n〉. For frequencies equal to plus or

minus the excitation energies of the unperturbed system, the linear response function has

poles shown as,

limω−ωk

⟨⟨A; V ω

⟩⟩ω

=⟨0∣∣∣A∣∣∣ k⟩⟨

k∣∣∣V ω

∣∣∣ 0⟩, (2.26)

limω+ωk

⟨⟨A; V ω

⟩⟩ω

= −⟨0∣∣∣V ω

∣∣∣ k⟩⟨

k∣∣∣A∣∣∣ 0⟩, (2.27)

where ωk = Ek −E0. Clearly one-photon absorption process can be described by these two

functions that contains information about excitation energies from the reference state |0〉

and the corresponding transition matrix elements. The transition dipole moments between

the ground state |0〉 to an excited state |f〉 can be written as

limω1→−ωf

(ω1 − ωf) 〈〈µi;µj〉〉ω1= 〈0 |µi| f〉〈f |µj| 0〉 . (2.28)

2.2.3 Two-Photon Absorption

The two-photon cross section σTPA is related to the second hyperpolarizability γ. If one

chooses ω as the half of the energy ωf which is the excitation energy from the ground state

to the final two-photon state |f〉

ω =1

2ωf , (2.29)


The second hyperpolarizability, γαβγδ(−ω;ω,−ω, ω), can be written as,

γαβγδ(−ω;ω,−ω, ω) =h−3∑

P1,3×

∑

nm

[〈0 |µα|n〉〈n |µγ| f〉〈f |µβ|m〉〈m |µδ| 0〉

(ωn − ω)(−iΓf/2)(ωm − ω)

+〈0 |µγ|n〉〈n |µα| f〉〈f |µβ|m〉〈m |µδ| 0〉

(ωn − ω)(−iΓf/2)(ωm − ω)]

=i2h−3

Γf

∑P−δ,2

∑

n

〈0 |µα|n〉〈n |µγ | f〉

ωn − ω

∑P1,3

∑

m

〈f |µβ|m〉〈m |µδ| 0〉

ωm − ω

=i2h−1

Γf

SαγS∗δβ .

(2.30)

And two-photon transition matrix elements Sαβ can be identified as

Sαβ = h−1∑

n

[〈0 |µα|n〉〈n |µβ| f〉

ωn − ω+

〈0 |µβ|n〉〈n |µα| f〉

ωn − ω

]. (2.31)

Other than getting the TPA cross section from γ, the TPA transition matrix elements can

also be deduced from the single residue of the quadratic response function, as shown below,

limω2→−ωf

(ω2 − ωf) 〈〈µi;µj, µk〉〉−ω1,ω2

= −∑

n

[〈0 |µi|n〉〈n |(µj − 〈0 |µj| 0〉)| f〉

ωn − ω2

+〈0 |µj|n〉〈n |(µi − 〈0 |µi| 0〉)| f〉

ωn − ω1

]〈f |µk| 0〉 .

(2.32)

And the transition dipole moments between excited states can be obtained from the double

residue of the same response function, which can be written as

limω1→ωf

(ω1 − ωf)

[lim

ω2→−ωm

(ω2 − ωi) 〈〈µi;µj, µk〉〉−ω1,ω2

]

= −〈0 |µi| f〉〈f |(µj − 〈0 |µi| 0〉)| i〉〈i |µk| 0〉 ,

(2.33)

where ωm = ω1 + ω2.

Chapter 3

Linear and Nonlinear Optical

Spectroscopy

This chapter gives a brief introduction about the linear and nonlinear optical processes

that have been studied in this thesis with special attention to one-photon and two-photon

absorptions.

Absorption spectroscopy is based on absorption of photons by pure substances or mixtures

in solid, liquid, or gas phases. An absorption spectrum of a particular substance is obtained

by relating the amount of absorbed photon with its wavelength. Every compound has

absorption lines at particular wavelengths. And if the structure of the compound is stable,

the normalized absorption spectrum does not change much at different concentrations and

thus can be taken as a kind of chemical “fingerprint”. For instance, C2H4 has its absorption

band of longest wavelength at bout 180 nm, while for COH2 it is at about 280 nm.

The absorbed wavelength of every compound is determined by the energy difference between

the initial and final states. Atomic states are defined by arranging electrons in different

atomic orbitals, while molecular states are defined by quantized molecular states mixed

with vibrational and rotational modes. By absorbing one or several photons whose total

energy is exactly equal to the energy difference between the initial and final states, an

electron will be excited to the final state of higher energy. After some time, the excited

electron will be relaxed to the initial state of the system, which is a process that can be

accompanied by light emitting. One of the combined absorption and emission processes

gives Raman spectroscopy which will be discussed in next chapter.

Absorption spectroscopy covers a wide range of wavelengths, going from infrared, optical to

ultraviolet. With the conventional light source, only one photon absorption is detectable.

With the help of powerful laser, multi-photon absorptions can be routinely achieved.

23

24 CHAPTER 3. LINEAR AND NONLINEAR OPTICAL SPECTROSCOPY

3.1 One-photon Absorption

When a molecule absorbs one photon whose energy matches the energy difference between

the ground state and an excited state, electrons can be promoted from the ground state to the

excited state. This process is called one photon absorption (OPA) process as schematically

shown in Fig. 3.1. By tuning the frequency of the incoming light, electrons can be excited to

Figure 3.1: Scheme of one photon absorption process.

different excited states at where a strong absorption of the incoming light can be observed.

For a given final state f , the one-photon absorption cross section is associated with the

oscillator strength

δopa =2ω0f

3

∑

α

〈0 |µα| f〉2 (3.1)

ω0f is the energy deference between the ground |0〉 and the final states |f〉, 〈0 |µα| f〉 is

the electronic transition dipole moment between two states and the summation goes over

x, y and z axes of the molecule. These are the peak values of the absorptions. In reality,

each excited state has its own lifetime which give rise to the lifetime broadening of each

absorption peak. In the optical region, the lifetime of the charge transfer state, which often

possesses very large transition dipole moment, of a gas phase molecule is often in the range

of nanosecond. When the molecule is put into the solution, the lifetime of the state can

be drastically reduced, to the order of femto-second, because of the collision between the

molecule and the solvent. Furthermore, the involvement of vibronic structures can further

complicate the absorption spectrum. The experimental OPA spectra can be very broad and

3.2. TWO-PHOTON ABSORPTION 25

ill-defined, which make it difficult to extrapolate useful information about the molecular

system under investigation. Theory can in many cases provide much needed assistance.

3.2 Two-photon Absorption

Quantum mechanically, the possibility of absorbing several photons simultaneously by an

atom or a molecule is always existed, although it could be too small to be observed under

the excitation of conventional light. Two-photon absorption (TPA) was first predicted by

Goppert-Mayer in 193118 but had to wait for 30 years to be observed experimentally with

the help of newly invented laser sources. The TPA processes are schematically drawn in Fig.

3.2 which can be utilized by absorption of two photons of the same or different energies.

Two-photon absorption has been primarily used as a spectroscopic tool that complements

Figure 3.2: Scheme of TPA process.

one-photon absorption spectroscopy. It enables to study states that are inaccessible for

ordinary one-photon excitations due to symmetry or parity selection rules. It furthermore

stretches the accessible range of conventional lasers by using a frequency half of the actual

energy gap. In recent years, TPA has been found many technological applications, such

as two-photon excited fluorescence microscopy,19,20 optical limiting,21–23 three-dimensional

optical data storage,24 two-photon induced biological caging studies,25 and nano- or mi-

crofabrication.26 All these applications take the advantage of two unique features of TPA,

namely the ability to create excited states with photons of half the nominal excitation en-


ergy, which can provide improved penetration in absorbing or scattering media, and a high

degree of spatial selectivity in three dimensions because of its quadratic dependence on the

intensity. Moreover, because the wavelength used for TPA is much longer than that of OPA,

the influence of scattering on beam intensity is greatly reduced. These are clear advantages

for applications in imaging of absorbing or scattering media, like biological tissues.

As we shown in previous chapter, TPA matrix element for the resonant absorption of two

photons with identical energy can be expressed as

Sαβ = h−1∑

i

[〈0|µα|i〉〈i|µβ|f〉

ωi − ω0f/2+

〈0|µβ|i〉〈i|µα|f〉

ωi − ω0f/2

], (3.2)

in a sum-over-states form, where ωi is the excitation energy for the intermediate state |i〉,

ω0f is the energy difference between the initial and final states, and thus ω0f/2 means the

fundamental frequency of the laser beam. TPA cross section can be given as,

δtpa =∑

αβ

[F × SααS

∗ββ +G× SαβS

∗αβ +H × SαβS

∗βα

], (3.3)

by orientational averaging over the TPA probability for molecules in gas phase and solu-

tion,27 where the coefficients F , H , and G are related to the polarization of the radiation

source and the summation goes over the molecular x, y, and z axes. For linearly polarized,

the values of F , H , and G are 2, 2 and 2, for the circular case, F , H , and G are −2, 3 and

3. In this thesis, only linearly polarized case has been considered so that the microscopic

TPA cross section becomes,

δtpa = 6(S2xx + S2

yy + S2zz) + 8(S2

xy + S2yz + S2

zx) + 4(SxxSyy + SyySzz + SzzSxx) (3.4)

To be comparable with the experiment directly, the TPA cross section needs to be re-defined

as,

σtpa =4π3a5

0α

15c

ω2

Γf

δtpa, (3.5)

where a0 is the Bohr radius, c is the speed of light, α is the fine structure constant, hω is

the photon energy, and Γf is the lifetime broadening of the final state. The unit of TPA

cross section is GM named after Goppert-Mayer ( 1 GM= 10−50 cm4s/photon).

3.3 Selection Rules

In an atomic or molecular system, there are huge numbers of excited states. However, not all

states can be detected by one particular absorption process within dipole approximation.

3.3. SELECTION RULES 27

Only the states that obey dipole selection rules are allowed. For instance, for a center-

symmetric molecule, if the initial state has a parity of grade (g), only the final state with

parity of ungrade (u) can be reached by OPA since the dipole operator has always an ungrade

parity (u). In general, if the product of symmetry groups of the initial state, the dipole

operator, and the final state contains the symmetry group of the initial state, the transition

between the initial and final states is allowed. TPA is a product of two OPA processes,

involving the transition element of 〈0 |r| i〉〈i |r| f〉, which should obey a rule different from

that of OPA. For instance, starting from a ground state with parity g, the final states with

parity of g become accessible by TPA.

Here we take benzene molecule, shown in Fig. 3.3, which has very high symmetry of D6h as

an example. The calculated OPA and TPA spectra using DFT at B3LYP level with 6-31G*

Figure 3.3: Optimized structure of benzene (C6H6) molecule.

basis set are shown in Fig. 3.4. In a D6h group, there are twelve symmetry elements, namely

A1g, A1u, A2g, A2u, B1g, B1u, B2g, B2u, E1g, E1u, E2g and E2u. The dipole operators, x, y, z,

belong to symmetries of E1u and A2u, respectively. The ground state of benzene molecule

is of symmetry A1g, therefore, states with symmetries E1u and A2u can be reached by OPA,

but are not accessible with TPA. As shown in the calculated spectrum, the degenerate state,

1E1u, at 7.4 eV, has large OPA cross section and zero TPA strength. At the same time,

degenerate states, 1E2g, at 9.1 eV, and 2E1g, 10.0 eV, show considerable TPA cross section

with zero OPA strength. It is noticed that for the high symmetry molecule, like benzene,

there are many states with right parity can be symmetry forbidden. For instance, the lowest


single state 1B2u is a both OPA and TPA forbidden state.

2 3 4 5 6Fundamental energy (eV)

0

1

2

3

4

5

6

7

OPA

(R

el. I

nt)

TPA

cro

ss-s

ectio

n (G

M)OPA TPA

Figure 3.4: Calculated OPA (solid lines) and TPA (dashed lines) spectra of optimized

benzene molecule. Lifetime broadening of 0.1 eV is used. Energy scale in TPA is normalized

to coincident with that of OPA.

3.4 Hyperpolarizabilities

Nonlinear optics is a way of manipulation light through the interaction between the light

and the matter. The frequency, the polarization or the phase of an incoming laser light

can be altered after interacting with materials possessing strong nonlinear optical proper-

ties. Nonlinear optical (NLO) materials have wide potential applications in various photonic

technologies such as optical switching, telecommunications and optical computing, just name

a few.28–32 Therefore, finding materials with unique NLO properties has attracted consid-

erable attention in last decades.

3.4. HYPERPOLARIZABILITIES 29

From a microscopic view, the induced dipole moment µ of the molecular system and the

external electric field strength F have the relationship

µ = µ0 + αF (3.6)

in linear optics, where µ0 is the permanent dipole moment of the molecular system, and α

is the polarizability. While in nonlinear optics, the influence of the high power terms of the

electric field strength should be considered on the induced dipole moment. And then the

induced dipole moment can be written as,

µ = µ0 + αF +1

2!βF2 +

1

3!γF3 + · · · . (3.7)

i.e. by a Taylor series expansion. β and γ are the molecular first hyperpolarizability and

second hyperpolarizability that control the second and third order NLO processes.

If we write the external electric in a Fourier expansion form of the sum of electric fields with

discrete frequencies,

F(t) =∑

p

F(ωp) e−i ωp t. (3.8)

µi for component i can be got

µi =µ0i +

∑

p

αij(−ωσ1;ωp)Fj(ωp) e

−i ωσ1t

+1

2!

∑

pq

βijk(−ωσ2;ωq, ωp)Fj(ωq)Fk(ωp) e

−i ωσ2t

+1

3!

∑

pqr

γijkl(−ωσ3;ωr, ωq, ωp)Fj(ωr)Fk(ωq)Fl(ωp) e

−i ωσ3t

(3.9)

by using the Einstein summation convention. ωσ1= ωp, ωσ2

= ωq + ωp and ωσ3= ωr + ωq +

ωp showing different frequencies of the output light. This expression represents different

frequency dependent NLO processes.

3.4.1 Finite Field Method

Finite Field (FF) method is used to calculate linear and nonlinear polarizabilities at the

static limit. It is easily to be understood and to be realized in programming.

The total molecular energy E of a molecular system can be expanded into Taylor series over

the electric field strength F,

E = E0 − µ0iFi −

1

2!αijFiFj −

1

3!βijkFiFjFk −

1

4!γijklFiFjFkFl − · · · , (3.10)


in uniform electrostatic fields, where E0 is the total energy of the molecular system without

the external electric field, µi is the ith component of the molecular dipole moment vector

and Fi is the component in the ith direction of the external uniform electrostatic field with

each i goes over all Cartesian coordinates x, y and z.

In FF method, a series of field dependent total energy of the molecular system can be

obtained by applying electrostatic fields with different strengths at certain directions. And

then the corresponding coefficients in Eq. 3.10 can be determined by polynomial fitting.

For example, the dipole moment of the excited charge transfer state can be obtained with

hωeg(Fz) = hωeg(0) − (µzee − µz

gg)Fz −1

2!(αzz

ee − αzzgg)F

2z − · · · . (3.11)

3.4.2 Analytical Derivative Method

Analytical Derivative (AD) method is a generalized approach that can avoid the numerical

uncertainty problem of the FF method. In this case, µ, α, β, and γ are expressed in

derivative forms of the total molecular energy E over the electric field strength F.

µi = −∂E

∂Fi

,

αij = −∂2E

∂Fi∂Fj

,

βijk = −∂3E

∂Fi∂Fj∂Fk

,

γijkl = −∂4E

∂Fi∂Fj∂Fk∂Fl

,

(3.12)

...

Still, AD method can only obtain static properties.

3.4.3 Response Theory Method

This method is a very powerful tool to study dynamic NLO properties and has been well

described in Chapter 2.

3.5. APPLICATIONS 31

3.5 Applications

3.5.1 Q-band Splitting of Aluminum Phthalocyanine Chloride

Molecular phthalocyanines have been the subject of theoretical and experimental stud-

ies over the past decades owing to a variety of applications in areas such as xerography,

chemical sensors, and photodynamic therapy.33–41 The properties of phthalocyanine can be

effectively tuned by inserting different metal atom in the center of the molecule. Most met-

allophthalocyanines are nontoxic, water, air and thermally stable, which made them very

attractive for biotechnology. For instance, the second generation photosensors are mainly

based on the phthalocyanines, because metallophthalocyanine molecules absorb intensely in

the long-wavelength side of the visible region, which is particularly useful for diagnostics

and therapy of malignant neoplasms.42

One of important absorption bands in metallophthalocyanines is the so-called Q-band, lo-

cated at around 600-700 nm.43 Many studies have been devoted to understand the properties

of this band. One of the most interesting experimental observations is the sensitivity of this

band to the changes of the surrounding. A separation as large as 10.14 nm for the Q-band

splitting has been observed in fluorescence excitation spectra of aluminum phthalocyanine

tetrasulfonate (APT) in hyper-quenched glassy water. The aluminum phthalocyanine (APc)

molecule has been found to show a similar behavior in hyperquenched glassy ethanol with

APT.44 A study of Q-band splitting should provide useful information about the interaction

between the molecule and the surrounding.

We have carried out time-dependent density functional theory (TD-DFT) calculations for

AlPcCl molecule and AlPcCl-water complexes. Furthermore, the polarizable continuum

model (PCM) is employed for complex of AlPcCl and three water molecules, to explore the

effect of the long range inter-molecular interactions. It has been found that the interaction

with water molecules can only result in relatively smaller splitting for the Q-band, less than

3 nm in most cases. The dimerization of molecules also fails to induce larger splitting.

The most possible mechanism to explain the large splitting of the Q-band in the AlPcCl

(eventually the APT) molecule is found to be associated with the bond changes between the

metal atom and neighboring nitrogen atoms. Symmetrically changing the N−Al bonds by

reducing the α angles as shown in Fig. 3.5, we have found that the value of Q-band splitting

can be as large as 24 nm when a small geometry distortion that costs extra energy as less

as 2.8 kcal/mol occurs. This molecular geometry distortion is expected when the AlPcCl

molecule is mixed with the glasses. This result also shows that the interaction strength

between the surrounding and the AlPcCl molecule might be expressed by the magnitude of

the Q-band splitting.


Figure 3.5: Structure of AlPcCl monomer. α is a tuneable angle that represents the geometry

distortion.

3.5.2 Two-Photon Absorption of Octupolar Molecules

Strong conjugated molecules that are asymmetrically substituted with electron donor and

acceptor groups, such as para-nitroaniline (PNA) derivatives, substituted stilbenes, and

push-pull polyenes have been found to exhibit extremely large second order nonlinear optical

properties. However, they also possess some major drawbacks, such as a high tendency

toward unfavorable aggregation, difficult in non-centrosymmetric crystallization, and small

off-diagonal tensor components associated with their anisotropic dipolar character.45 To

avoid these weaknesses, octopolar molecules with zero permanent dipole moment and large

first hyperpolarizability have been designed and synthesized. They have been considered to

be the most promising compounds for second-order NLO applications.

With the help of molecular dynamics at room temperature, it is, however, found that the

charge transfer octupolar molecules can also form aggregates through intermolecular hy-

drogen bonds activated by thermal motion of the molecules.46 We have studied TPA cross

section of different clusters taken from snapshots of molecular dynamics simulations for a

model molecule 1,3,5-triamino-2,4,6-trinitrobenzene (TATB), including dimers, trimers and

tetramers. Aggregation effect on TPA cross section has been clearly demonstrated. Taken

the results of TATB dimers as an example, five representative dimer structures and their

TPA spectra are shown in Fig. 3.6. It can be seen that for all dimers, maxima of their TPA


Figure 3.6: Calculated TPA spectra of five representative dimers, together with those of the

optimized monomer. A lifetime broadening of 0.1 eV is used.


spectra are red shifted by as much as 0.6 eV to 0.9 eV comparing with that of the monomer.

Enhancement of TPA cross section for certain structure, like dimer E, could also be found.

3.5.3 One- or Two-end-capped Dendritic Structures

The concentration of nonlinear optical chromophores should be sufficiently high for realistic

applications. In that case, as we mentioned above the push-pull molecules turn to sponta-

neously form aggregates that then drastically reduce the nonlinear optical response of the

total system. A controlled aggregation of push-pull molecules could avoid this probelm. We

have constructed three-fold symmetric dendritic architectures of dipolar chromophores that

effectively enhance the local density of nonlinear optical molecules and their responses.

We have studied a series of end-capped triply branched dendritic chromophores including

one- and two-end-capped models as displayed in Fig. 3.7. It is found that the second order

nonlinear optical properties of one-end-cappedtype dendrimitic arrangements are strongly

dependent on the mutual orientations of the three chromophores and large enhancement of

the first hyperpolarizability can be obtained when the dipole moments of the three branches

in the dendrimers are highly parallelized. The increase of the conjugation length can also

improve the performance of the dendrimers. However, the use of two-end-caps is less useful

to improve the performance of these systems. The calculated structure-to-property relations

are successful in explaining various experimental observations.


Figure 3.7: Optimized molecular structures of one- and two-end-capped dendritic structures.


Chapter 4

Raman Spectroscopy

4.1 History

Quantum mechanically, light has the dual character of waves and particles. When consid-

ering light as particles, inelastic scattering processes could be expected. Compton showed

inelastic scattering of x-rays from a graphite target in 1923.47 And Smekal predicted the

inelastic scattering of light theoretically in the same year.48 It was known that the inten-

sity of inelastic scattering scales to the fourth power of the energy, which means that the

observed scattering cross section at optical wavelength of 500 nm should be 10 orders of

magnitude larger than that at x-ray wavelength of 0.7 nm. Raman focused a large telescope

on the sun to obtain strong green light through a filter. After the beam of green light

passed through a solution of chloroform, a weak yellow light was observed, resulting from

the inelastic scattering. In 1928, C. V. Raman and K. S. Krishnan published a paper in

the journal Nature to report their experiments which proved Smekal’s prediction of inelastic

scattering of light.49 The inelastic scattering of visible light from molecular transitions was

named after Raman. In 1930, Raman won the Nobel Prize in Physics.

Nowadays, with the help of a small HeNe laser and a CCD detector, modern Raman spec-

troscopy can be easily carried out in a matter of few seconds. Raman spectroscopy has been

widely used to identify chemical bonds as “fingerprint” of molecular compounds. Raman

spectra complements with infrared spectra, because some excitations can only happen in

one of them. And for the both allowed excitations, the intensities are always different.

37

38 CHAPTER 4. RAMAN SPECTROSCOPY

4.2 Basic Theory

When photons are scattered by molecular systems, most photons are elastically scattered

and this process can be called Rayleigh scattering as shown in Fig. 4.1. During this process,

Figure 4.1: Processes of Rayleigh scattering, Stokes Raman scattering and anti-Stokes Ra-

man scattering.

the emitted photon has the same energy as the incoming photon. Raman spectroscopy is

based on the inelastic scattering of photons. When light shines upon molecules and interacts

with the electron cloud of the molecules, a molecule can be excited from the ground state to

a virtual state by absorbing one incident photon and relaxes to an excited state. A photon

will then be emitted with less energy,

Eout = Ein − ωj, (4.1)

where ωj is energy level of the excited state. This is called Stokes Raman scattering. If

the molecule is in an excited state at initial, and relaxes to the ground state, the emitted

photon will have energy

Eout = Ein + ωj. (4.2)

Obviously, the energy of the scattered photon is smaller than the incident photon for the

Stokes scattering and while it is larger for the anti-Stokes case. When the initial and final

electronic states belong to the same electronic state, the energy difference in each process

4.2. BASIC THEORY 39

is related to the energy of the vibrational excited state. Therefore the Stokes or anti-Stokes

lines are a direct measurement of the vibrational energy levels of the molecule.

Although the anti-Stokes and Stokes lines are equally displaced from the Rayleigh line,

anti-Stokes lines are much less intense than the Stokes lines because only molecules that

are vibrationally excited at the initial state can give rise to the anti-Stokes lines. Therefore,

more intense Stokes lines are normally measured in Raman spectroscopy.

When a molecule exhibits vibrational Raman effect, there should be deformation of the

electron cloud or change of molecular polarizability. Raman intensity is determined by the

amount of the change of polarizability. Raman scattering transition moment can be written

as,

R = 〈i |α| j〉 , (4.3)

where α is the polarizability of the molecule, |i〉 and |j〉 are the initial and final states. α

can be expressed as,

α = α0 + (~r − ~re)

(dα

d~r

)+ · · · , (4.4)

where α0 is the polarizability at the equilibrium bond length ~re, ~r is the distance between

atoms. Since α0 is a constant and 〈i|j〉 = 0, R can be simplified to,

R =

⟨i

∣∣∣∣(~r − ~re)dα

d~r

∣∣∣∣ j⟩. (4.5)

It shows that to have a nonzero Raman scattering transition moment, there must be a

change in polarizability.

However, not all vibrational modes can take part in the Raman scattering. Here we use

the asymmetric stretch and the symmetric stretch of CO2 molecule, shown in Fig. 4.2, as

examples. The polarizability changes during vibration because it depends on how tightly

the electrons bind to the nuclei. Obviously, in the symmetric stretch mode, the strength of

electron binding to the nuclei is different with the minimum and maximum internuclear dis-

tances. Therefore the symmetric vibrational mode is Raman active. And for the asymmetric

stretch, there is no overall changes in polarizability so that it is Raman inactive vibration. In

general, Raman scattering obeys the same selection rule as that for two-photon absorption.

The vibrational Raman-scattering cross section is proportional to

σFI,αβ = |〈F, 0 |ααβ| I, 0〉|2 , (4.6)


Figure 4.2: Symmetric stretch and asymmetric stretch modes of CO2.

within the Born-Oppenheimer approximation, where |I, 0〉 is the initial vibrational state

and |F, 0〉 is the final state, and ααβ is the polarizability tensor. ααβ can be written as50,51

ααβ =∑

k

∑

J

〈0 |µα| k〉 |J, k 〉〈 J, k| 〈k |µβ| 0〉

EJk − EI

0 − ω − iΓ

+〈0 |µα| k〉 |J, k 〉〈 J, k| 〈k |µβ| 0〉

EJk − EI

0 + ωS + iΓ,

(4.7)

where EJk is the energy for the |J, k〉 state, ω is the frequency of the incident light, and ωS

is that of the scattered light. The summation of k and J is over all the electronic surfaces

|k〉 and all the vibrational levels of each surface |J, k〉. The above equation is only valid

for nonresonance scattering, because the polarizability is approximated by the electronic

polarizability tensor and expanded in a Taylor series about the equilibrium geometry in the

standard theory of Placzek.

4.3 Surface Enhanced Raman Spectroscopy

In 1974, Martin Fleischman and coworkers found large Raman signal from pyridine adsorbed

on electrochemically roughened silver surface.52 However, they did not notice that this was

a major enhancement effect. Three years latter, two groups realized that the large signal

was from Raman enhancement effect independently.53,54

It has been found that the Raman scattering from a compound adsorbed on a metal sur-

face can be 103-106 times greater than that in solution. Silver, gold and copper55 are the

most often used surface materials for surface enhanced Raman spectroscopy (SERS). It has

4.3. SURFACE ENHANCED RAMAN SPECTROSCOPY 41

also known that small particles or atomically rough surfaces can also produce large SERS

signals. Therefore, SERS can be used to study molecular mono-layer adsorbed on metals

or electrodes. Over the years, SERS technique has been applied to many metal systems,

such as metal films on dielectric substrates, colloids, and arrays of metal particles bound

to metal or dielectric colloids through short linkages. However, it has often been limited to

electron-rich molecules, like pyridine, aromatic amines or phenols, and carboxylic acids.

Two possible mechanisms for surface-enhanced Raman scattering that are substantially

different from each other have been proposed. One of them is an electromagnetic effect

proposed by Jeanmaire and Van Duyne,53 and the other is a charge-transfer effect or chemical

effect proposed by Albrecht and Creighton.54

4.3.1 Electromagnetic Enhancement

The electromagnetic enhancement relies upon the fact that the electric field provided by

particular surfaces where the molecules adsorbed on can be resonantly enhanced. When

light interacts with the metal surfaces, localized surface plasmons can be excited. When

the frequency of the plasmons ωp is in resonance with the incoming light, the greatest field

enhancement can be obtained. Moreover, the direction of the plasmon oscillation should be

perpendicular to the surface in order to realize the enhancement. Scattering will not occur if

the plasmon oscillations are in-plane with the surface.56 For this reason, roughened surfaces

or nano-particles are typically employed to get large SERS factor. Under these conditions,

surface areas in which the localized collective oscillations can take place are easily identified.

The wavelength of the incoming light is much longer than the features of the surfaces

or particles, only the dipolar contribution of the light needs to be considered. The field

enhancement can be divided into two steps. First, it magnifies the intensity of incident

light with a factor of E2. And then, it magnifies the intensity of the Raman signal by

the same mechanism with the same factor. For the total process, an enhancement factor

of E4 is obtained.57 Noticed, the maximum factor E4 is only for the Raman frequencies

that are slightly shifted from the incident light. In this case, the Raman signals are also

near resonance with the plasmon frequency. When there is larger Raman frequency shift,

the incident light and the Raman signal can not be both on resonance with ωp and thus

show smaller enhancement.58 Because most of SERS studies are in visible and near-infrared

radiation (NIR) ranges, the plasmon resonance frequency should be in these ranges to get

maximal enhancement. For this reason, silver, gold and copper55 are chosen to be typical

metals for SERS with plasmon resonance frequencies match the requirement.


4.3.2 Chemical Enhancement

Figure 4.3: Optimized structure of pyrazine (left) and structure of pyrazine molecule bonded

to two Ag10 clusters (right).

Figure 4.4: Calculated Raman spectra of pyrazine (left) and pyrazine molecule bonded to

two Ag10 clusters (right).

Although the electromagnetic theory gives a general explanation for the enhancement of

SERS regardless of the molecular system involved, it has difficulty to explain experimental

observations for certain specific systems, for instance, molecules with a lone pair of electrons

bonded to the surface. It was found that the enhancement magnitude of such systems are

too large to be explained by the electromagnetic mechanism alone. A chemical enhancement

mechanism involves charge transfer between the adsorbate and the metal surface has then

been proposed. Because electronic transitions of many charge transfer complexes are in the

visible range, resonance enhancement can easily occur. Here we take pyrazine molecule, see

Fig. 4.3, bonded to two Ag10 clusters as an example to demonstrate the possible chemical

enhancement of SERS. From the calculated Raman spectra as shown in Fig. 4.4, a large

enhancement, a factor of 20000, due to the presence of chemical bonds between the molecule

and metal clusters can be easily observed. It is also shown that Raman active modes for

the molecule in metal junction are different from that of a single molecule.


The chemical enhancement occurs mostly for chemically bonded molecule-metal complexes.

For physically adsorbed molecule-metal systems, the chemical enhancement can be occasion-

ally observed for certain specific configurations. It should also mentioned that the chemi-

cal enhancement mechanism is strongly associated with the electromagnetic enhancement

mechanism.59

4.4 Applications

The possibility of using SERS to probe single molecular behavior has attracted considerable

attentions in recent years.60–68 However, it is not a trivial task to determine the existence of

single molecule from SERS measurements due to many uncertainties.69–75 Up to now single

molecular SERS studies have been limited to a few dye molecules that often form strong

chemical bonds with the metal substrate. The formation of the chemical bond between

the molecule and the substrate can lead to relatively stable spectra with strong signal

as a result of the so-called chemical enhancement. Even with these systems, there is no

consensus on how to determine the structure of the single molecule, which has limited

the applicability of single molecular (SM-) SERS in general. In a joint experimental and

theoretical work, we have conducted SM-SERS measurements for a non-bonding molecule,

perylene, physically adsorbed on uniformly assembled colloidal Ag nanoparticles on glass

cover-slips. We have demonstrated that it is possible to control the thermal motion of the

single molecule by lowing the temperature of the sample, and to obtain very stable spectra

with high resolutions. With the help of first principles calculations, the position of perylene

molecule inside the Ag nanoparticles has been determined unambiguously. Our work has

not only significantly widened the scope of practical applications of SM-SERS technique,

but also been conceptually important since it shows for the first time that the chemical

enhancement is not a necessary condition for the generation of SM-SERS spectrum, hence

resolving the long standing debate in the field about the role of the chemical enhancement.

We have tested many model systems with DFT calculations. It is found that only when the

molecule is placed in between two Ag clusters, the calculated Raman spectrum can repro-

duce the most stable experimental spectrum. The agreement between the theory and the

experiment is remarkable, as nicely demonstrated in Fig. 4.5B. These results are consistent

with the observation of Xu et al63 that the molecule likes to sit in between two nanoparticles

with small gap in SM-SERS, which might due to the fact that in this situation the electric

field gradient force for a molecule could become zero and the molecule can safely remain

there.

Based on the evolutions of laser power dependent and temperature dependent experimental


spectra, we could propose a model to describe the dynamic process of a single perylene

molecule on Ag nanoparticles under low temperature as given in Fig. 4.5. When a perylene

molecule is spread on the nanoparticules, the most stable position for the perylene is to lie

on the surface of a nanoparticle. It is energetically unfavorable for a molecule to drop in

between two nanopartciles with a gap of less than 1 nm. Calculations indicate that when the

distance between molecule and the nanoparticle is sufficiently small, around 0.6 nm, more

vibrational modes become Raman active because of the interaction between the molecule

and the metal. As shown in Fig. 4.5A, the calculated spectrum resembles reasonably well

the experimental spectrum taken at the very beginning of the measurement. Under the

guidance of the strong electric field gradient force, the molecule eventually moves into a

gap of two nanoparticles as described by Fig. 4.5C. It should be mentioned that Raman

intensity of the single molecule in the gap shown in Fig. 4.5 is quite similar to that of in the

gas phase, indicating clearly that the chemical enhancement is not the necessary condition

for the observation of single molecular Raman spectrum.


Figure 4.5: (A) Calculated Raman activity of perylene based on the model by assuming a

single free molecule perylene adsorbed near Ag10 clusters, (B) and trapped into two pyra-

midal Ag10 clusters, in comparison with low-temperature experimental SM-SERS spectra.

(C) Schematic draws for the behavior of a perylene molecule near a pair of electronegative

Ag nano-particles: (K and E0 refer to wave vector and electric vector of incident light) the

molecule is initially adsorbed on top of one of the Ag particle, and then moved into the gap

of two particles driven by electric field gradient force.


Chapter 5

Central Insertion Scheme

5.1 Introduction

Nanomaterials in many cases could be considered as molecular complexes. In these systems,

electrons are often confined in a finite region. For this reason, nanostructures have been

expected to possess unique physical and chemical properties different from the corresponding

bulk materials. Nanotechnology rapidly became a growing interdisciplinary research area

since it was first envisioned in the 1980’s. And novel quantum phenomena, such as single-

electron charging effects, quantization of conductance, just name a few, have been exploited

near room temperature.76–78 For instance, a single carbon nanotube can be used to make a

transistor that works at room temperature,79,80 and a single C60 molecule can be operated as

an amplifier.81 Generally, nanomaterials can lead to very good performance with decreased

size, weight, and cost.

It is important to understand the structures and properties of nano-sized systems for the

development of nanotechnology. However, when nanomaterials are lack of infinite period-

icity, it is difficult to get the electronic structures of these relatively large systems with ab

initio or first principle methods because of the O(N3) or worse scaling behavior.

Following the first proposal of Yang,82,83 many computational methods with linear scaling

behavior have been developed. These methods can be divided in two main categories. The

first category is to calculate the whole system at once, combining several efficient techniques:

the “order-N exchange”84 and “near-field-exchange”85 methods for exchange matrix,86 the

fast multipole methods for the Coulombic matrix,87–90 and density matrix search methods91

for replacing the diagonalization of the Fock matrix. The second category is to divide the

whole system into a series of fragments, and construct the properties of the whole system

47

48 CHAPTER 5. CENTRAL INSERTION SCHEME

with the results of quantum chemical calculations of all the fragments.

5.2 Central Insertion Scheme

5.2.1 Basic Idea

The central insertion scheme (CIS) is a method that belongs to the second category.92 It

is based on a simple fact that for a large enough finite periodic system, the interaction

between different units in the middle of the system should be converged. In another word,

the Hamiltonian of those units in the middle becomes identical. It is thus possible to

obtain the Hamiltonian of a larger system by continuously adding the identical units in the

middle of the initial system. Apparently the application of CIS requires to have an initial

Hamiltonian possessing identical central parts, which can only be achieved by computing a

fairly large initial system. Fortunately this condition can be fulfilled routinely by modern

quantum chemistry programs. It should be noted that this central insertion process can be

applied to three-dimensional systems.

5.2.2 Hamiltonian

In CIS method, the initial system can be considered with several parts, including two end

parts, namely L and M , and n central uniform units Ui with periodic structures as described

in Fig. 5.1 (a)I. The wavefunction | Ψη〉 for a certain state of the system obeys

H | Ψη〉 = εη | Ψη〉, (5.1)

where H is the Hamiltonian of the system. When describing a periodic system, the wave-

functions are more suitable to be expanded in terms of the site basis: | L〉 and | M〉 for two

end parts, and | Ki〉 (i runs over U1, U2, ... , Un) for the middle parts. Each of them can

be described as linear combination of atomic orbitals (LCAO) shown as,

| L〉 =∑

Lα

c(L)Lα | ψLα〉;

|M〉 =∑

Mα

c(M)Mα | ψMα〉;

| Ki〉 =∑

iα

c(i)iα | ψiα〉. (5.2)

By defining

Hi,j = 〈ψi | H | ψj〉 , Si,j = 〈ψi | ψj〉 (5.3)

5.2. CENTRAL INSERTION SCHEME 49

Figure 5.1: (a) I and II: Schematic draw of the geometry and Hamiltonian matrix of a n

unit periodic structure, respectively; (b) I and II: The geometry and Hamiltonian matrix

of two stretched periodic structures from (a), respectively; (c) I and II: The geometry and

Hamiltonian matrix of the elongated n+ 1 unit periodic structure, respectively.


where i and j run over units L, U1, U2, · · · , Un and M , Eq. 5.1 can be described as,

HL,L HL,1 ... HL,n HL,M

H1,L H1,1 ... H1,n H1,M

... ... ... ... ...

Hn,L Hn,1 ... Hn,n Hn,M

HM,L HM,1 ... HM,n HM,M

CηL

Cη1

...

CηN

CηM

= εη

SL,L SL,1 ... SL,n SL,M

S1,L S1,1 ... S1,n S1,M

... ... ... ... ...

Sn,L Sn,1 ... Sn,n Sn,M

SM,L SM,1 ... SM,n SM,M

CηL

Cη1

...

CηN

CηM

(5.4)

where Cηi = (cηi1 c

ηi2 ... c

ηiα)T is the coefficient submatrices of unit L, Ui (i = 1, 2, . . . , n) and

M in the molecular orbital εη, respectively.

5.2.3 Central Insertion Process

The Hamiltonian of the initial system can be represented by a (n+ 2)× (n+ 2) matrix (L,

Ui, M), as shown schematically in Fig. 5.1 (a)II. It should be noted that the using of one

unit to represent each end part is only to simplify the description. And the sizes of them

are not constrained. The structures of the matrices for those two mismatched systems are

illustrated in Fig. 5.1 (b)II and are denoted as (L′

, U′

i , M′

) and (L′′

, U′′

i , M′′

), respectively.

If we put these two mismatched systems together, a new (n+ 3)× (n+ 3) matrix (L′′′

, U′′′

i ,

M′′′

) with the structure of Fig. 5.1 (c)II will be obtained for the elongated system. And in

the constructing of the new matrix, the following relationship,

U′′′

i,j = αU′

i,j + (1 − α)U′′

i−1,j−1, (5.5)

is adopted, where α is a weighting factor depending on the system under investigation. In

most cases, α = 0.5 is considered.

As shown in Fig. 5.1(c)II, there are still many matrix elements can not be generated

by the combination. However, these elements normally represent the long-rang interaction

between two ends of the system, and thus can be set to zero without losing any accuracy. By

repeating this process, Hamiltonian of even larger systems can be constructed. With these

new Hamiltonian, molecular orbitals energies and wavefunctions of the elongated systems

with high accuracy can be calculated.

5.2. CENTRAL INSERTION SCHEME 51

5.2.4 Approximations and Error Control

CIS method relies on two key approximations.92 First, the central parts of the initial system

are assumed to have converged electronic structures. For instance, as shown in Fig. 5.1 (a)I,

the wavefunction, charge density and the Hamiltonian of unit Un2−1 is approximately the

same as that of unit Un2. Second, interaction between long range (e.g., > 20A) units can be

neglected without loss of accuracy. For example, the interaction between two ends (L′′ +U ′′1

and U ′′n+1 +M ′′ as shown in Fig. 5.1 (c)II).

If we define the possible error arises from the first approximation as δ′ and that of the second

approximation as δ′′, the error matrix ∆ can be written as,

∆ = H −H =

δL,L δL,1 ... δL,n δL,M

δ1,L δ1,1 ... δ1,n δ1,M

... ... ... ... ...

δn,L δn,1 ... δn,n δn,M

δM,L δM,1 ... δM,n δM,M

, (5.6)

where H is the true Hamiltonian of the new system, H is the Hamiltonian obtained from

the CIS method, δi,j = δ′i,j + δ′′i,j is the total error.

Let u and λ be the eigenvector and eigenvalue of H , and the Euclidean norm of vector u

satisfies ‖u‖2 = 1. And let r = Hu− λSu = Hu − Hu be the residual vector. Then there

exists an eigenvalue λ of H ,93

|λ− λ| ≤ ‖r‖2

= ‖Hu− Hu‖2

= ‖∆u‖2

≤ ‖∆‖2‖u‖2

= ‖∆‖2

= ρ(∆), (5.7)

where ρ(∆) is the spectral radius of ∆.

Using Gerschgorin’s theorem,93 an approximated estimation of ρ(∆) can be obtained,

ρ(∆) ≤ maxi

∑

j

|δi,j| = ‖∆‖∞, (5.8)

where ‖∆‖∞ is the infinite norm of matrix ∆.

From Eq. 5.8, we can see that, for a large enough system whose central parts have converged

interaction energies, the insertion of a new unit at the center of the system would hardly

affect the convergence and consequently reduce remarkable errors.


5.2.5 BioNano Lego

“BioNano Lego” is an efficient tool package to proceed CIS using effective parallelization

technique.94 It can be divided into four parts as shown bellow.

Step 1

1. Calculate Hamiltonian H0 and overlap matrix S0 of the initial system L+⋃n

i=1 Ui +M

with well-converged central parts. H0 and S0 are usually sparse Matrices.

2. Perform the screw operator i times and calculate the analytical form of rotation matrix

for coefficient matrices of molecular orbitals RiC .

3. Calculate the weighting factor α for each unit according to the distance from the center

of mass of the system.

4. Compute S−10 by using SuperLU package.95

5. Calculate S−10 H0 by using Sparse BLAS package.96 S0 and H0 are saved in compressed

sparse row (CSR) format, while S−10 H0 is in full matrix format.

6. Check whether the initial system is converged.

Step 2

1. Perform the screw operator N times successively on the initial system to get Hi (i =

1, · · · , N).

Step 3

1. Get the large CIS Hamiltonian HNCIS by combining Hi (i = 0, 1, 2, · · · , N) together.

2. Then calculate the overlap matrix of the large system with the same sparse pattern

of HNCIS.

Step 4

1. Solve the general eigenvalue problems HNCISC

NCIS = SNCN

CISΛNCIS.


5.3 Applications

Figure 5.2: (a) Spatial distributions of wavefunction for LUMO of CNT31 calculated by

GAUSSIAN03 (I) and the CIS method (II). (b) Molecular orbitals of (5, 5) SWCNTs with

N = 9 + 12 × i (i=0, 1, ... ,134) units, corresponding to 1.1 nm to 200 nm in length.

Carbon nanotubes (CNT)are considered to be the most promising material for future molec-

ular and nono-electronics. Single-walled carbon nanotubes (SWCNTs) with finite lengths,

ranging from 10 to 50 nm, have found to possess unique electron transport properties.80

Since the electrons in the CNTs can only propagate in the direction of the tube axis,79,97

CNTs are therefore interesting systems for studying quantum behavior of electrons in one

dimension (1D). However, the finite-length CNTs present a great challenge for the first

principle modeling because of both the involvement of vast number of electrons and the


breakdown of periodic boundary conditions.

Here we show the result of a (5, 5) metallic SWCNT with 31 layers (CNT31, 3.7nm in length)

with CIS method using 19 layers SWCNT as the initial system. The spatial distribution

of the LUMO of the CNT31 by mapping the Cartesian(xyz)-coordinate into a cylindrical

one has been illustrated in Fig. 5.2(a), which it is almost identical as the one calculated

by GAUSSIAN03 program directly. Very good agreement has also found for other orbitals.

With the CIS, it is possible to calculate very large systems. The electronic structures of

sub-200-nm-long single-walled (5,5) carbon nanotubes (SWCNT(5,5)) are shown in Fig.

5.2(b). As expected, the density of states (DOS) gets higher for longer SWCNTs. It is also

interesting to see that the energy gaps (Eg) oscillate periodically with the increase of the

length of the tube.

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