Spectroscopic studies of C2

(1)

Spectroscopic studies of C ₂ modulation in the B850 ring of light harvesting 2 complex of

purple bacteria

by

Tenzin Kunsel S2435306 Supervisors:

Dr. T.L.C. Jansen Prof. dr. Jasper Knoester

Referent:

Dr. R.W.A. Havenith

A thesis submitted in partial fulfillment for the degree of Master of Science

in the

Faculty of Mathematics and Natural Sciences

21 July 2015

(2)

(3)

Abstract

Investigating the processes in the natural light harvesting systems not only promote understanding of photosynthesis, it can also serve as potential solution to our energy need through integrating with the solar technology. Since the chromophores used by the purple bacteria as the light harvesting complex is structurally much simpler than the corresponding system in plants, purple bacteria is a tractable choice for the studies. Previous spectroscopic studies on light harvesting 2 (LH2) complex have suggested presence of C2 modulation in it’s B850 ring. This has been investigated in this project discriminating among various models using single molecule spectroscopy, linear absorption spectroscopy and two-dimensional spectroscopy in hope to find one model that can explain all the experimental results.

(4)

I would like to thank Prof. dr. Jasper Knoester and Dr. Thomas la Cour Jansen for the wonderful opportunity to work in their group on the very interesting project and supervising me through out the year. It was a great scientific experience where I got exposure to various areas of physics and developed scientific temperament. I would like to extend my sincerest thanks to Dr. Remco Havenith for accepting to be my referent and for all his effort. I would also like to express my gratitude to The Zernike Institute for Advanced Materials for supporting me in many ways to receive education for the two years of Top master programme in nanoscience. Last but not the least, I would like to thank all members of theory of condensed matter group of Zernike insitute and my friends for making my stay here joyous.

2

(5)

1.3 Fluorescence-excitation spectrum of the long-wavelength region of an individual LH2 complex for mutually orthogonal polarized excitation. Two peaks corresponds to the superradiant states of the B850 band. This figure is reproduced from the paper [3] 3 2.1 AFM image of a patch of membrane from cell of purple bacteria. The bigger circular structures are the LH1 complexes containing the RC and the smaller circular structures are the LH2 complexes. This figure is taken from the paper [4] . . . 4

2.2 The structure of LH2 showing the α− and β−subunits of each α − β dimer in yellow and pink. This figure is taken from the paper [5] . . . 5

2.3 Chemical structure and VMD representation of BChl a, composed of a five ring planar segment which binds a magnesium atom and a long phytol tail. . . 6

2.4 Chemical structure and VMD representation of carotenoids which contain long carbon chains. Carotenoids in varying species of purple bacteria are different. a)Rhodopin glucoside found in Rps. acidophila. b)Lycopene found in Rs. molischianum. c)Speroidene found in Rb. sphaeroides. . . 6

2.5 Geometrical arrangement of the BChl a molecules of the LH2 complex of Rps. Aci- dophila obtained using x-ray crystallography. Blue ones depict the B800 BChl a molecules and red ones depict the B850 BChl a molecules. Phytol chains of BChl a molecules are omitted for clarity. This figure is taken from the paper [3] . . . 7

2.6 Fluorescence-excitation spectra for an ensemble of LH2 complexes (top trace) and several individual LH2 complexes at 1.2K. Vertical scale applies to the bottom spectrum. cps,counts per second. This figure is reproduced from the paper [3] . . . 8

2.7 System of interacting two level molecules representing the B850 ring . . . 9

2.8 Band of frenkel exciton states of the system of interacting molecules . . . 10

2.9 Unperturbed B850 ring model. . . 13

2.10 a)Elliptical deformation model with uniform inter-pigment distance present in first and second model. b)Third elliptical deformation model where inter-pigment distance increases at the minor axis and decreases at the major axis with increase in eccentricity. c)Fourth elliptical deformation model where inter-pigment distance decreases at the minor axis and increases at the major axis with increase in eccentricity. ’e’ refers to eccentricity in the graphs. . . 20

2.11 The Feynman diagram for Linear absorption. . . 21

2.12 Experimental setup to measure 2D-spectra. This figure is taken from [6] . . . 21

2.13 Feynman diagrams of 2D spectra . . . 22

2.14 2DVIS Spectra . . . 22 4

(7)

List of Figures 5

2.15 Dimer system and it’s electronic structure . . . 23 2.16 Change in lineshapes with increase in waiting time showing loss of frequency memory 23 2.17 Flow diagram for calculation of linear absorption spectra and two dimensional spectra 23 3.1 Energy of exciton states varying with eccentricity for a) first model b) second model

c) third model d) fourth model and varying with E_mod for e) fifth model . . . 25 3.2 a) Histogram of energy separation between k = ±1 exction states for 144 complexes.

The figure is reproduced from the paper [7]. Energy separation between k = ±1 exction states varying with eccentricity for b) first model c) second model d) third model e) fourth model and varying with E_mod for f) fifth model . . . 26 3.3 a) Histogram of intensity ratio of exciton state with k = +1 and 0 states to exciton

state with k = −1 state for 88 complexes. This figure is reproduced from the paper [7]. Intensity ratio of exciton state with k = +1 state to exciton state with k = −1 state varying with eccentricity for b)third model c)fourth model and varying with E_mod for d)fifth model . . . 28 3.4 a) Histogram of relative orientation of the transition dipole moments of exciton

state with k = +1 state to exciton state with k = −1 state for 144 complexes. This figure is reproduced from the paper [7]. Relative orientation of the transition dipole moments of exciton state with k = +1 state and exciton state with k = −1 state varying with eccentricity for b) fourth model and varying with Emodfor c) fifth model 29 3.5 a) Experimental 77 K absorption spectrum (dashed line) of Rps. Acidophila. This

figure is reproduced from the paper [8]. Calculated linear absorption spectrum with experimental result for b) fourth model and c) fifth model . . . 30 3.6 a)Two dimensional spectra at varying waiting times from experiment[2]. Calculated

two dimensional spectra for waiting times for 0 and 250 fs from b) model with varying radii of LH2 complexes[1] c) fourth model and d) fifth model . . . 32 3.7 Energy shift observed in the 2D spectra when waiting time is changed from 0 fs to

250 fs from a) fourth model b) fourth model zoomed in to obtain the energy shift magnitude c) fifth model and d) fifth model zoomed in to obtain the energy shift magnitude. . . 33 3.8 Population change with time of BChl a molecules averaged over all the sites from

fourth model and fifth model. . . 34

(8)

Introduction

Photosynthesis plays a significant role in sustaining life on earth [9,10]. In addition to this immense significance, the process of harvesting sunlight and transferring it to convert the resulting electronic excitation into more stable forms of energy is highly efficient [11,12]. The internal quantum yield of energy transport is almost 100%[13]. That makes it subject of many studies in order to find its usage as a potential solution to our energy need. For instance the biomimetic strategies borrowed from plants and photosynthetic organism have been utilized for solar energy conversion. Through this, processes that occur during photosynthesis such as dynamic self-repair, quantum effects and highly efficient energy transfer can be integrated into man-made photovoltaic devices[14].

1.1 Related works and Motivation

For possible integration into solar cell technologies, in depth understanding of the initial stage of the light harvesting process is essential. In this initial stage solar photons are absorbed by a complex system of membrane associated pigment proteins (Light harvesting (LH) antenna) and the absorbed energy is efficiently transferred to the reaction center (RC) where it is converted into chemical energy[15]. Light harvesting complexes which act as the antenna are required because chlorophyll which actually absorb the sunlight has cross section close to 1˚A² which amounts to absorbing about 1 photon per second in a realistic situation. Hence this LH complex containing chlorophyll pigments increase the cross-section and thereby the amount of photons absorbed.

Chromophores used by purple bacteria as the light harvesting pigment-protein complexes is structurally much simpler than the corresponding systems from the evolutionarily more advanced cyanobaceria and plants[4]. Despite the simplicity in it’s structure, it’s photosynthetic processes are similar to that of other photosynthetic organisms. For this reason investigating the natural light harvesting system of purple bacteria is a tractable choice.

1

(9)

Chapter 1: Introduction 2

There have been numerous studies on the individual light harvesting systems of purple bacteria resulting in a detailed understanding of the structure and dynamics of the individual proteins[3,16].

In one such prominent study[1], the author has simulated linear absorption spectra and all parallel polarization two dimensional spectra for the LH2 complex ensemble of varying radii. Although there is good match between the results from the simulation and experimental data for linear absorption spectra and two dimensional spectra with zero waiting time, the low energy band in the measured spectra for non-zero waiting times exhibits an energy shift of about 110cm⁻¹ as shown in figure 1.2 which is unobserved in the simulation as shown in figure 1.1. Authors have speculated that the observed energy shift in the experiment results from intraband relaxation within B850 band of the LH2 complex. Eventhough relaxation of excitations have been included in the simulation, the reason why it’s not observed from the simulation was speculated to be due to the degeneracy of superradiant states within B850 band which are involved in the intraband relaxation due to inherent C8 symmetry(there are 8 pigments in the B850 ring of their model).

This postulation is substantiated by single molecule fluorescence-excitation spectroscopy on LH2 from another study[3] which shows that degeneracy of the two superradiant states in B850 band is indeed lifted. As outlined in figure 1.3, energy splitting between the two superradiant states within the B850 band was found to be 110cm⁻¹ consonant to the observed energy shift in the two dimensional spectra from the experiment. This motivates spectroscopic studies of the B850 ring of LH2 complex considering causes for the energy splitting between the superradiant states within the B850 band.

Figure 1.1: Calculated two dimensional spectra from model with varying radii of LH2 complexes.This figure is reproduced from [1]

1.2 Outline

The remainder of this thesis is outlined as follows. In the chapter 2, theory elaborating on the system of study, models considered in the investigation and methods used for the spectroscopic study are developed. In the chapter 3, spectroscopic results from various models are discussed comparing with the experimental results to discriminate among the models. In the final chapter,

(10)

Figure 1.2: Phased two-dimensional spectra at varying waiting times from experiment. This figure is reproduced from the paper [2]

Figure 1.3: Fluorescence-excitation spectrum of the long-wavelength region of an individual LH2 complex for mutually orthogonal polarized excitation. Two peaks corresponds to the superradiant

states of the B850 band. This figure is reproduced from the paper [3]

conclusions are drawn based on the discussion of obtained results and possible future studies are also mentioned which could give deeper understanding of the system.

(11)

Chapter 2

Theory

2.1 Complexes of the natural light harvesting system of purple bacteria: LH2,LH1,RC

The two most important components of natural light harvesting system are the antenna system, which absorbs the solar photons and the reaction center (RC) where the absorbed energy is converted into chemical energy. In particular the light harvesting system of the purple bacteria consist of the light harvesting complex 1 (LH1), light harvesting complex 2 (LH2) and RC [5]. Using atomic force microscopy the arrangement of the pigment-protein complexes are clearly observed as shown in the figure2.1. LH1 which is the largest light harvesting complex surrounds the reaction center.

The LH1 complexes are in turn surrounded by multiple LH2 complexes which expand the light harvesting capacity both in terms of area available and in the spectrum of wavelength that can be used[17].

Figure 2.1: AFM image of a patch of membrane from cell of purple bacteria. The bigger circular structures are the LH1 complexes containing the RC and the smaller circular structures are the

LH2 complexes. This figure is taken from the paper [4]

4

(12)

2.1.1 Structure of LH2 complex

The project will be particularly focused on the LH2 complexes that form the primary antenna system. The LH2 complexes are structurally studied in great detail with X-ray crystallography. The study shows that LH2 is formed by a circular aggregate (of eight subunits in the case of the Rho- dospirillium (RS.) Molischianum[18] and nine subunits in the case of the Rhodopseudomonas(Rps.) Acidophila[19]). Each subunit is called a α − β heterodimer as it contain a pair of short peptides commonly referred to as the α− apoprotein and the β− apoprotein which noncovalently binds three bacteriocholorophyll (BChl) molecules and one carotenoid (Car)[20] as outlined in figure2.2.

The α− apoprotein form the inner ring with radius of 18˚A and the β− apoprotein form the outer ring with radius of 34˚A[17].

Figure 2.2: The structure of LH2 showing the α− and β−subunits of each α − β dimer in yellow and pink. This figure is taken from the paper [5]

The BChl molecules and Carotenoids are the pigments primarily responsible for the absorption of light and subsequent excitation transfer events[5]. BChl molecules are of many different types and the one found in purple bacteria is called bacteriochlorophyll a (BChl a). Structure of BChl a and carotenoids found in varying purple bacterial species are as shown in figure2.3and figure 2.4 respectively. BChl a comprises of a five ring planar structure which has magnesium at it’s center and a long hydrophobic phytol tail. Carotenoids are formed of long hydrophobic molecules with a 40-carbon polyene chains and a head region that is specific to each species of purple bacteria[17].

From the near-IR absorption spectrum of LH2, which shows two distinct peaks at 800 and 850 nm, it can be learned that BChl a molecules are arranged in two distinct structures. In the case of Rps. Acidophila, eighteen of the BChl a molecules form a closely interacting ring[3](outlined in figure 2.5). This tightly packed nature allows strong interaction between the pigments by which delocalized exciton states are formed that shifts the absorption peak of the ring to 850 nm[5]. For this reason, its called a B850 ring. The remaining nine BChl a molecules form a ring where the molecules have their tetrapyrrole rings nearly perpendicular to the eighteen BChl a molecules in B850 ring. These BChl a molecules give rise to the 800 nm peak and its therefore called the B800

(13)

Chapter 2: Theory 6

ring. Rhodopin glucoside which is the carotenoid in Rps. acidophila, absorb at around 500 nm[21].

Since the energy splitting between superradiant states mentioned in the introduction belong to B850 band, following chapters will be focused on the B850 ring of LH2 complex.

Figure 2.3: Chemical structure and VMD representation of BChl a, composed of a five ring planar segment which binds a magnesium atom and a long phytol tail.

2.1.2 Electronic excitation in B850 ring: Frenkel exciton formalism

Optical response of aggregates can be very different from the addition of the optical responses of the individual molecules that make the aggregate depending on the interplay of the disorder of site energies (∆) and inter-molecular interaction (V ) between them. For weak coupling between the molecules corresponding to small ^V_∆ ratio, excitations are expected to be mainly localized on the

(a) (b)

(c)

Figure 2.4: Chemical structure and VMD representation of carotenoids which contain long carbon chains. Carotenoids in varying species of purple bacteria are different. a)Rhodopin glucoside found in Rps. acidophila. b)Lycopene found in Rs. molischianum. c)Speroidene found in Rb.

sphaeroides.

(14)

Figure 2.5: Geometrical arrangement of the BChl a molecules of the LH2 complex of Rps.

Acidophila obtained using x-ray crystallography. Blue ones depict the B800 BChl a molecules and red ones depict the B850 BChl a molecules. Phytol chains of BChl a molecules are omitted for

clarity. This figure is taken from the paper [3]

individual molecules. In contrast, if the ratio ^V_∆ is large, corresponding to strong coupling between the molecules, excitonic interactions have to be considered and excitation energy is delocalized among the molecules.

B800 and B850 rings, being aggregates of BChl a molecules bear witness to this phenomena as evident from their spectroscopic signature shown in figure2.6. The ensemble spectrum features two broad structure-less bands at around 800 nm and 860 nm corresponding to the absorption of the B800 and B850 rings. However, when the single complexes were observed using the single-molecule spectroscopic technique, remarkable difference in the spectral features are detected corresponding to the absorption of the two ring aggregates. The difference actually arises from the difference in interplay of ∆ and V in the two aggregates as a consequence of difference in the number of constituent pigments. In these ring aggregates, inter-molecular interaction V is determined by the intermolecular distance and the relative orientation of the molecular dipole moments. The site- energy variations ∆ is attributed to the changes in the electro-static interaction of BChl a molecules with the surrounding protein as a consequence of variations in the structure of the environment[3].

This can be estimated from the inhomogeneous width of the absorption lines of the ring aggregates.

For the B800 ring, diagonal disorder ∆ ∼ 125cm⁻¹ was extracted from the width of the B800 ensemble line and V ∼ −24cm⁻¹ was calculated using point-dipole approximation and structure parameters from the x-ray structure[22]. This amounts to k^V_∆k ≈ 0.2 which corresponds to electronic excited states that are localized on individual pigments. This explains the narrow sharp features around 800 nm attributing to absorption of individual BChl a molecules in the B800 ring.

Conversely, in the B850 ring, V ∼ 300cm⁻¹ is more than twice the estimated ∆ ∼ 125cm⁻¹. This amounts to k^V_∆k ≈ 2.4 calling for consideration of excitonic interactions. Therefore to ratio- nalize the observed spectral features for B850 ring, formalism of Frenkel exciton provides a good description.

Frenkel Excitons

(15)

Chapter 2: Theory 8

Figure 2.6: Fluorescence-excitation spectra for an ensemble of LH2 complexes (top trace) and several individual LH2 complexes at 1.2K. Vertical scale applies to the bottom spectrum. cps,counts

per second. This figure is reproduced from the paper [3]

In this description when site basis is used, | ni represents a state where molecule ”n” is in elec- tronically excited state and all other molecules are in the ground state. Then the hamiltonian for the simplest situation of the system without any kind of disorder looks as shown below in equation (2.1):

H =

N

X

n=1

E0| nihn | +

N

X

n=1

X

m6=n

Vnm| nihm | (2.1)

V_nm = hn | V | mi denotes the interaction between the molecules in excited state located on molecule ”n” and ”m”. Using point-dipole approximation, Vnmwere obtained as shown in equation

(16)

(2.2).

V_nm= d~_nd~_m− 3(ˆr_nm· ~d_n)(ˆr_nm· ~d_m)

r_nm³ (2.2)

d~n refers to transition dipole moment of BChl a molecule ”n” and ~rnm refers to distance vector from BChl a molecule n to m. Here it is also assumed that all the molecules are two level systems like shown in the figure2.7which is a reasonable assumption if one of the many possible electronic transitions in the molecules has an oscillator strength that dominates and we restrict ourselves to using light of frequency close to this dominant transition. In this particular case, homogeneous two level molecules have transition energy E₀.

Figure 2.7: System of interacting two level molecules representing the B850 ring

The eigenstates of this hamiltonian which contains the intermolecular interaction are called Frenkel exciton states and they are given by equation (2.3).

| ki = 1

√N

N

X

n=1

e^i2πk^Nⁿ | ni where k = 0, 1, ....N − 1 (2.3)

In such an ideal system with all molecules being equivalent, exciton wavefunctions have equal amplitudes on all the pigment which means excitation is distributed equally over all the molecules and energy is coherently shared by all molecules as shown in equation (2.4).

| hm | ki |²=| hm | 1

√ N

N

X

n=1

e^i2πk^Nⁿ | nii |²= 1

N (2.4)

Energy of these exciton states depicted in figure2.8 can be obtained using equation (2.5):

ε k = hk | H | ki = 1 2

X

n=1

X

m6=n

Vnme^i2πk^n−m^N (2.5)

(17)

Chapter 2: Theory 10

With this, initial degeneracy of the localized excited states is lifted and generates manifolds of energy levels called exciton bands.

Figure 2.8: Band of frenkel exciton states of the system of interacting molecules

In order to determine the optical response of this system, total transition dipoles from the ground state to each of the exciton states should be known. It can be obtained using equation (2.6):

M (k) = hg | ~~ D | ki = 1

√ N

N

X

n=1

e^i2πk^Nⁿm~_n (2.6)

D refers to the transition dipole moment operator and ~~ m_nrefers to the transition dipole of the two level system representing n^th molecule. As noticeable from the equation (2.6),M (k) is determined~ by both the magnitude and mutual arrangement of the individual transition dipole moments.

Therefore, for an optical transition to be allowed, there should be constructive interference of the individual dipole moments.

In the B850 ring exciton states with k = 0, ±1 have non-zero transition dipole moments due to the circular symmetry in arrangement of it’s pigments. The out of plane components of the transition dipole moments of individual pigments give rise to oscillator strength in the k = 0 level. The in- plane components of the transition dipole moments of the individual pigments give rise to oscillator strength in the k = ±1 exciton states which is much stronger than that of the k = 0 state since transition dipole moments of the individual pigments are oriented nearly in the plane of the ring structure [22,23]. This explains why there are two main peaks with higher intensity than the rest in region of 860 nm in figure2.6. These two peaks are attributed to the k = ±1 exciton states. This speculation is corroborated by the results from fluorescence-excitation spectrum of an individual LH2 complex for mutually orthogonal polarized excitation as shown in figure1.3which shows that the two peaks correspond to states with mutually perpendicular transition dipole moments. This is predicted from the exciton model for the circular ring [23].

(18)

Although this all fits properly, the exciton model for the unperturbed ring predicts two non- degenerate (k = 0 and k = 9) and eight pairwise degenerate (k = ±1 to k = ±8) states. So it can’t explain the splitting in energy that was observed for the two sharp peaks which are attributed to the k = ±1 states in figure 1.3. Non-degeneracy of k = ±1 states observed from the experiment can be due to varied causes. Random diagonal disorder in the hamiltonian representing random disorder in the site energy of the pigments caused by stochastic variations in the protein environment is the first thing we can think of. Physically it is quite reasonable due to the inhomogeneity of the pigments forming the B850 ring. Among it’s many effects like mixing of the different exciton levels and redistribution of oscillator strength to nearby states, the one that is important here is the modification of the spacing between the exciton levels which lifts the pair-wise degeneracy[24].

Eventhough this could be a plausible explanation, the quantitative studies showed that even un- reasonably large value of the width of distribution of site energies ∆ ≈ 500cm⁻¹ can’t give energy separation between k = +1 and k = −1 states as obtained experimentally (δE ∼ 126cm⁻¹)[3].

Random off-diagonal disorder representing random variations in the dipolar coupling is found to have indistinguishable effects on the exciton manifold[24].

This insufficiency of random disorder suggests presence of regular modulations in the B850 ring.

There have been studies done on effect of such regular modulation on the electronic structure of LH2 which showed that splitting of k = ±1 states can be done with a C₂ type modulation[25–27].

Such a perturbation can couple exciton states that have a difference of their quantum number by two which means the dominant effect would be observed in coupling between the degenerate k = ±1 states. The positions of the k = ±1 states can be affected by their interactions with the respective k = ±3 states. But this effect is much smaller than the former one due to the energy difference between the k = ±1 and k = ±3 states. This phenomena also applies to all the other interactions between the different k-levels. This modulation also causes redistribution of the oscillator strength from k = ±1 states to k = ±3 states[24]. This can then explain the splitting we observe in the two peaks attributed to k = ±1 states in figure2.6.

The C2 modulation can be in the site energy which can be physically interpreted as deviation of the bacteriochlorine macrocycle from planarity and/or reorientation of side groups as it is theoretically known that the exact value of the energy difference between the electronic ground state and the first excited singlet state of a BChl a molecule depends crucially on the π−conjugation of the bacteriochlorine macrocycle. It can be also due to influence of the protein backbone on the site energies of the pigments which is evident from the site-directed mutagenesis[7]. The C₂modulation can be also in the interaction between the BChl a pigments which in the simplest form can be physically interpreted as structural deformation of the B850 ring from circle to an ellipse since the leading term of the deformation of a circle into ellipse has the same cos(2θ) dependence.

(19)

2.2 Models

In the end of section 2.1.2, two main models that should be considered to explain the observed fluoresence-excitation spectra for individual LH2 complex have been briefly introduced. There are subcategory models, especially for the model with C₂ modulation of interaction between the pigments which will be shown in detail in this section.

Foremost the unperturbed ring is modeled in simplest way as having C₁₈ symmetry although in reality the nanomeric structure of LH2 is reflected in the B850 ring by having a C9 symmetry consisting of nine-repeating pairs of α− and β− bound BChl a molecules[16]. These α− and β−

bound BChl a molecules have different orientation of the transition dipole moments as well as varying excitation energy due to their unique local environment of the binding sites. Although this difference has importance for the circular dichroism, it has been found to have little influence on the absorption spectrum[28]. This added with the reason for simplicity by reducing the number of free parameters makes it reasonable to neglect the difference between α− and β− bound BChl a molecules. Moreover, as the deformation which are considered later are much larger than the orientational difference between the molecules,it further justify the assumption of C₁₈ symmetry of the ring which has been used. Therefore with the two level molecule approximation for all the BChl a molecules, in this unperturbed B850 ring 18 BChl a molecules are identical with site energy

∼ 12050cm⁻¹obtained from a previous study [1]. Moreover, the transition dipole moment along the symmetry axis carrying less than 2% of the total absorption intensity is also neglected. So we have a C₁₈symmetric ring with transition dipole moment of all the pigments lying in the plane of the ring.

Due to the assumption that α− and β− bound BChl a molecules are identical, the orientation of the transition dipole moment of the pigments is taken as average orientation of those of the α− and β− bound BChl a molecules. Consequently, the transition dipole of the BChl a molecules are titled 24.5 degrees from the local tangent. For the magnitude of the transition dipole moments, average nearest-neighbour interaction determined from the experiment of V₀= −240cm⁻¹was used[24]. By using this information of nearest neighbour interaction added with above assumptions and point dipole approximation, magnitude for the transition dipole moments was obtained ∼ 4.5Debye.

Taking all these into account, the unperturbed ring considered in the project is as outlined in figure2.9.

This unperturbed ring model is subjected to C₂modulation in mainly two different ways. The first kind of model considered is the model of B850 ring with random diagonal disorder and C2 modulated off-diagonal disorder induced by elliptical deformation of the ring. When the unperturbed ring is deformed elliptically, the area inside is kept constant. In this general model, four different models are studied which are enumerated below and shown in figure2.10.

(20)

Figure 2.9: Unperturbed B850 ring model.

1. Elliptical deformation is such that the inter-pigment distance is uniform. In addition to that, the angle between the local tangent and transition dipole moments is kept same as in the case of the unperturbed ring even after deformation.

2. Elliptical deformation is such that the inter-pigment distance is uniform. Contrary to the first model, the orientation of the transition dipole moments in the unperturbed ring is used for all the elliptical deformation. Therefore, the angle between local tangent and transition dipole moments is not conserved anymore after elliptical deformation.

3. Elliptical deformation where the inter-pigment distance at the minor axis increases and inter- pigment distances at the major axis decreases as the eccentricity increases. In this model the angle between the local tangent and transition dipole moments is kept same as in the case of the unperturbed ring even after deformation.

4. Elliptical deformation where the inter-pigment distance at the minor axis decreases and inter- pigment distances at the major axis increases as the eccentricity increases. In this model the

(21)

angle between the local tangent and transition dipole moments is kept same as in the case of the unperturbed ring even after deformation.

In addition to that, another model with both random and C₂ modulation on diagonal disorder was also implemented which will be referred to in rest of the study as the fifth model.

2.3 Methods

In all the investigations in this project, the initial unperturbed B850 ring model is developed as mentioned in section 2.2. The unperturbed C₁₈symmetric ring is then subjected to C₂ modulation through the five models mentioned in the end of section 2.2. From each of them, hamiltonian was obtained using equation(2.1) where the inter-molecular interactions are taken only upto second nearest neighbours since they are most dominant. From this energy of exciton states are then obtained by diagonalizing the hamiltonian and energy separation between k = ±1 states is determined. Further more, intensity of k = ±1 states are obtained by taking square of transition dipole moment from equation(2.6) and their ratio is calculated. In addition to that angle between k = ±1 states are also calculated. These single molecular spectroscopic results could help discriminate the models with elliptical deformation. However in order to discriminate the model with random and correlated diagonal disorder from the model with random diagonal disorder and C₂ modulated off-diagonal disorder induced by elliptical deformation which has good fit with the experimental results, linear absorption spectra and two dimensional spectra had to be obtained.

There are many different approaches to obtain the linear absorption and two dimensional spectra.

However for this project the numerical integration of the Schr¨odinger equation scheme[29–34] has been used which is computationally much less demanding than more exact method like hierarchical equations of motion (HEOM). In this scheme, the fluctuation in the hamiltonian of the B850 ring representing the inhomogeneity of the environment of individual BChl a molecules and their dynamics is satisfied using a stochastic model.

H_diagonal = E₀+ ∆E (2.7)

∆E is obtained stochastically from a random number generator with mean of 0cm⁻¹ and standard deviation of 250cm⁻¹. This value of standard deviation that correspond to magnitude of the diagonal disorder is used according to a previous study[7] where there were good match with experimental spectroscopic studies. This is how random diagonal disorder is added to the unperturbed ring hamiltonian.

For first to fourth models, off-diagonal disorder varies for varying eccentricity due to the structural deformation of the ring and change in orientation of the transition dipole moment. These variations

(22)

are manifested through equation(2.2) in the off-diagonal elements of the hamiltonian. E0 is taken as 12050cm⁻¹ assuming that the elliptical deformation doesn’t change the site energy.

For the fifth model, C2 modulation on the diagonal is implemented by adding a term for the diagonal elements of the hamiltonian with cos(2θ) as shown below in equation(2.8):

E0= 12050cm⁻¹+ E_modCos(2θ) (2.8)

E_mod corresponds to C2 modulation amplitude and θ correspond to angle of position of each BChl a molecule in the ring from the horizontal axis depicted in figure 2.9. In this model since there is no off-diagonal disorder, the off-diagonal elements are unchanged from the case of unperturbed ring. This way, the hamiltonian of the system without interaction with electromagnetic field is obtained just from the structure. On subjecting the system to external electromagnetic field, the hamiltonian would be then given by equation(2.9):

H = H_system+

N

X

n=1

~

µ_n(t) · ~E(t) (2.9)

On obtaining the hamiltonian on interaction with external electromagnetic field, the time evolution operator which is used in the response function is obtained by solving the time-dependent Schr¨odinger equation(2.10):

∂Φ(t)

∂t = −ı

~ HΦ(t) (2.10)

Φ(t) is the wave function at time ’t’ and it can be written in terms of the site basis functions φ as shown below in equation (2.11)

Φn(t) = X

m=1

φmcmn(t) (2.11)

The initial condition is then c_mn(0) = δ_mn. On inserting equation(2.11) in the time-dependent Schr¨odinger equation, equation for the expansion coefficient in the matrix form is obtained as shown below in equation(2.12).

∂c(t)

∂t = −ı

~ Hc(t) (2.12)

For the numerical integration of the equation using small time increments, hamiltonian can be treated as constant during each time step so that the time evolution operator during this time step is given by the solution of the time-independent Schr¨odinger equation as following:

(23)

c((i + 1)∆t) = exp −ı

~ H(i∆t)∆t

c(i∆t) = U ((i + 1)∆t, i∆t)c(i∆t) (2.13)

’i’ labels the time step. Combining all the time evolution operators from the starting time to any given i^th time gives:

U (i∆t, 0) =





j=i

Y

j=1

U (j∆t, (j − 1)∆t)



 (2.14)

To obtain linear absorption spectrum, the system is first excited through interaction with a laser pulse at a time τ1 after which signal is generated at time τ2 = τ1 + t1. Feynmann diagram corresponding to the linear absorption is given in figure 2.11.

Then the absorption spectrum is calculated by taking the imaginary part of the Fourier transform of the two point correlation function of the transition dipoles over time t₁. The lifetime T₁ of the excited state can be accounted for by multiplying the response function with Γ_LA(t₁) = exp

−t1

2T1

[6].

I_A(ω) = Im





∞

Z

0

dt₁i

~

hh0 | µ⁰¹(τ₂)U¹¹(τ₂, τ₁)µ¹⁰(τ₁) | 0ii_Eexp(−iωt₁)Γ_LA(t₁)



 (2.15)

Eventhough, linear absorption spectrum can relay important information of system like it’s electronic structure, life time of the excited states involved and oscillator strength of the allowed transitions, it may be insufficient in certain cases. For instance, to probe non-linear optical responses and resolve ultrafast structural and excitation dynamics down to femtosecond timescale, multi-dimensional spectroscopy is necessary. Two dimensional spectroscopy is a method where three lasers pulses interact with the system to probe the third order responses. Schematic diagram of set up for the two dimensional spectroscopy is depicted in figure 2.12.

In this method, system is initially pumped and allowed to interact with three laser pulses at time τ1, τ2 and τ3 with time delays t1 and t2. Then the signal is measured at time τ4after time delay t3. Often, the laser frequencies are kept equal and the second time delay is kept fixed. Then by fourier transforming after varying the time delay t₁ and t₃ between the laser pulses which corresponds to scanning over all probe frequencies for each pump frequency, 2D-spectrum for the fixed waiting time t₂ is obtained.

Usually two phase-matching conditions are used to obtain the two dimensional spectra: ~kI =

−k~₁+ ~k₂+ ~k₃ and ~k_II = ~k₁+ ~−k₂+ ~k₃[34].

(24)

Applying rotating wave approximation[35], Feynman diagrams associated with the the two phase matching conditions which have contributions are as shown in figure 2.13. Feynman diagrams associated with phase-matching condition ~kI are called the rephasing or the photon echo diagrams.

The Feynman diagrams associated with phase-matching condition ~k_II are called the non-rephasing diagrams. Both rephasing and non-rephasing diagrams incorporate three diagrams each denoted the ground state bleach (GB), the stimulated emission (SE) and excited state absorption (EA).

The responses of these process associated to the six diagrams can be calculated as following:

S_GB^R (t₁, t₂, t₃) = − i

~

3

hh0 | µ⁰¹(τ₁)U¹¹(τ₁, τ₂)µ¹⁰(τ₂)µ⁰¹(τ₄)U¹¹(τ₄, τ₃)µ¹⁰(τ₃) | 0ii_E×Γ(t₃, t₂, t₁) (2.16)

S_SE^R (t1, t2, t3) = − i

~

3

hh0 | µ⁰¹(τ1)U¹¹(τ1, τ3)µ¹⁰(τ3)µ⁰¹(τ4)U¹¹(τ4, τ2)µ¹⁰(τ2) | 0iiE×Γ(t₃, t2, t1) (2.17)

S_EA^R (t₁, t₂, t₃) = i

~

3

hh0 | µ⁰¹(τ₁)U¹¹(τ₁, τ₄)µ¹²(τ₄)U²²(τ₄, τ₃)µ²¹(τ₃)U¹¹(τ₃, τ₂)µ¹⁰(τ₂) | 0ii_E×Γ(t₃, t₂, t₁) (2.18)

S_GB^{N R}(t1, t2, t3) = − i

~

3

hh0 | µ⁰¹(τ4)U¹¹(τ4, τ3)µ¹⁰(τ3)µ⁰¹(τ2)U¹¹(τ2, τ1)µ¹⁰(τ1) | 0iiE×Γ(t₃, t2, t1) (2.19)

S_SE^{N R}(t1, t2, t3) = − i

~

3

hh0 | µ⁰¹(τ2)U¹¹(τ2, τ3)µ¹⁰(τ3)µ⁰¹(τ4)U¹¹(τ4, τ1)µ¹⁰(τ1) | 0ii_E×Γ(t₃, t2, t1) (2.20)

S_EA^{N R}(t₁, t₂, t₃) = i

~

3

hh0 | µ⁰¹(τ₂)U¹¹(τ₂, τ₄)µ¹²(τ₄)U²²(τ₄, τ₃)µ²¹(τ₃)U¹¹(τ₃, τ₁)µ¹⁰(τ₁) | 0ii_E×Γ(t₃, t₂, t₁) (2.21)

The lifetime of the excited states is included in an ad hoc way through the relaxation factor Γ [34]

[6].

Γ(t3, t2, t1) = exp −(t3+ 2t2+ t1) 2T₁

(2.22)

(25)

The signals in the frequency domain for fixed waiting time t2 are then generated using the 2D fourier transformation with respect to the time delays t₁ and t₃ as shown below in equation (2.23) and equation (2.24).

S^(R)(ω₃, t₂, ω₁) =

∞

Z

0

∞

Z

0

(S_GB^R (t₁, t₂, t₃)+S_SE^R (t₁, t₂, t₃)+S_EA^R (t₁, t₂, t₃))×exp(−i(ω₁t₁−ω₃t₃))dt₁dt₃ (2.23)

S^{(N R)}(ω3, t2, ω1) =

∞

Z

0

∞

Z

0

(S_GB^{N R}(t1, t2, t3)+S_SE^{N R}(t1, t2, t3)+S_EA^{N R}(t1, t2, t3))×exp(i(ω1t1+ω3t3))dt1dt3

(2.24) Then 2D spectrum is the imaginary part of the sum of the responses from photon-echo and non- rephasing diagrams[36].

I2D(ω3, t2, ω1) = Im(S^(R)(ω3, t2, ω1) + S^{(N R)}(ω3, t2, ω1)) (2.25)

An illustration of a two dimensional electronic spectrum of a dimer system (outlined in figure2.15) with parallel transition dipole moment is given in figure 2.14. ωpump corresponds to ω1 and ω_probe corresponds to ω₃. Red peak on the diagonal results from the ground state bleaching(GB) and stimulated emission(SE) diagrams when the system is in the same coherence during the two time delays. There would have been red cross peak arising from the same diagram when the system is in the coherence between the single excited state with energy ω− | V | and the ground state during the first time delay and in coherence between the single excited state with energy ω+ | V | and the ground state during the third time delay. However in the given dimer system, the oscillator strength is mainly concentrated in the exciton state with energy ω− | V |. Therefore, the red crosspeaks are not observed.

The blue peak corresponds to excited state absorption(EA) diagram which has opposite sign for absorption than the red peaks. In the EA diagrams, the system is in coherence between the double excited state and single excited state during the final time delay. Since the double excited state has higher energy than twice the energy of the single excited state with energy ω− | V | in the given dimer system, the blue peaks are obtained at energy ω+ | V |. The difference in the energy is denoted the anharmonicity and if it was zero, EA diagrams will overlap the GB and SE diagrams which will lead to undetectable signals. Therefore blue peaks can give information on the anharmonicity in the system.

(26)

Even though these are already important information that can be obtained from the two dimensional spectra, the main advantage is control over waiting time t₂. This can be used to study all kinds of dynamical processes. One prominent example is the dynamics due to the local environment of the chromophore pigments. The nuclear motion of the atoms in the surroundings causes the transition frequency and transition dipoles of the molecules to fluctuate through variation of electric fields commonly known as stark effect. If a molecule is excited during the first time delay and it is probed during the third time delay, the signal of the molecule at a point in the spectrum connected with the frequency it had during the first delay is observed along the ω1 or ωpump axis and the frequency it had during the third time delay is observed along the ω₃ or ω_probeaxis. If there is no frequency change during the waiting time, an elongated line will be observed on the diagonal axis whereas if the dynamics was faster than the waiting times, peaks take a round shape showing loss of memory of their initial frequency as shown in figure2.16. This is known as spectral diffusion.

This study can therefore, actually help to track the time scale of the changes in environment of the molecule.

To summarize, the linear absorption spectrum and two dimensional spectrum can be calculated in this NISE scheme by going through the steps in flow diagram of figure2.17. Even though the NISE scheme has been successfully applied to the vibrational and electronic systems[29, 30], there are limitations to this scheme. For instance, this method doesn’t take into account the influence of the quantum state of the system to classical bath which violates the conservation of total energy. The infinite temperature approximation is valid only when the energy difference between the states in the quantum system are smaller or comparable to the thermal energy.

Even though from the two dimensional spectrum, by varying the waiting time information of all kind of dynamics such as rotational motion, chemical exchange, population transfer and spectral diffusion can be obtained[30], in this project two dimensional spectroscopy will be mainly used to discriminate the model with random and correlated diagonal disorder from the model with random diagonal disorder and correlated off-diagonal disorder.

(27)

(a)

(b)

(c)

Figure 2.10: a)Elliptical deformation model with uniform inter-pigment distance present in first and second model. b)Third elliptical deformation model where inter-pigment distance increases at the minor axis and decreases at the major axis with increase in eccentricity. c)Fourth elliptical deformation model where inter-pigment distance decreases at the minor axis and increases at the

major axis with increase in eccentricity. ’e’ refers to eccentricity in the graphs.

(28)

Figure 2.11: The Feynman diagram for Linear absorption.

Figure 2.12: Experimental setup to measure 2D-spectra. This figure is taken from [6]

(29)

Figure 2.13: Feynman diagrams of 2D spectra

Figure 2.14: 2DVIS Spectra

(30)

Figure 2.15: Dimer system and it’s electronic structure

Figure 2.16: Change in lineshapes with increase in waiting time showing loss of frequency memory

Figure 2.17: Flow diagram for calculation of linear absorption spectra and two dimensional spectra

(31)

Chapter 3

Results and Discussion

In this chapter, as motivated in the introduction for the need of models that can explain the energy shift in two dimensional spectrum, five models which can possibly explain the phenomena are discriminated according to results from experiments. The five models are as listed in section 2.2 .

For the unperturbed ring, site energy of 12050cm⁻¹is assigned to the 18 identical BChl a molecules and using the point dipole approximation, nearest-neighbour interaction of ∼ −240cm⁻¹ and second nearest neighbour interaction of ∼ −30cm⁻¹ were obtained. Then on introducing various disorders, energetic separation of the of the k = ±1 states, intensity ratio of k = ±1 states , mutual orientation of the transition-dipole moments of k = ±1 states, linear absorption spectra and two dimensional spectra of ensemble are investigated to discriminate between the models.

3.1 Energetic separation of the k = ±1 states

To study this the energy of exciton states for varying eccentricity and amplitude of modulation for correlated diagonal disorder E_mod are plotted as shown in figure3.1.

In all the models it can be clearly observed that the degeneracy of the exciton states are lifted where there is contribution from both diagonal disorder and the C₂type modulation. Consequently, energy separation between the k = ±1 states can be calculated from the models and results are presented in figure 3.2.

In these plots, it can be observed that the magnitude of energy separation between the k = ±1 exciton states vary significantly between different models. This actually helps in discriminating these models comparing to the experimental result for the energy separation shown in figure 3.2 a). In the experimental result, blue refers to the energetically higher absorption band attributed

24

(32)

to k = −1 exciton state and red refers to the energetically lower absorption band attributed to k = +1 exciton state as shown in figure 1.3. The experimental distribution shows that the energetic separation of the k = ±1 states is centered at around 126cm⁻¹ and has a full-width half maximum(FWHM) of 101cm⁻¹. Since the average separation is around 126cm⁻¹, the first and second model with the elliptical deformation where BChl a molecules are uniformly distributed around the ellipse can be ruled out. These two models give maximum of ∼ 20cm⁻¹ and ∼ 50cm⁻¹ respectively. Intuitively, elliptical deformation with the uniform distribution of BChl a molecules looks more reasonable since it’s likely that once there is perturbation, the pigments move around to keep equal distance between each other since they all have identical forces of attraction towards each other. However, with this energy separation value for k = ±1 exciton states, these two models can be ruled out. But for the third, fourth and fifth model, energy separation of 126cm⁻¹ can be obtained for reasonable C₂ modulations. The corresponding eccentricity was obtained ∼ 0.53 for the third model and ∼ 0.5 for the fourth model. For the fifth model, energy separation of 126cm⁻¹

(a) (b)

(c) (d)

(e)

Figure 3.1: Energy of exciton states varying with eccentricity for a) first model b) second model c) third model d) fourth model and varying with Emod for e) fifth model

(33)

Chapter 3: Results and Discussion 26

corresponds to Emod∼ 132cm⁻¹.

The plots also show that the exciton states with k = ±2, ±3, ±4 also have their degeneracy lifted although not as strong as that of k = ±1 states. If there was only C2 modulation, there will be mixing of states with difference of k of 2 which means, we should see the effect only for k = ±3 states. The effect in k = ±2 and k = ±4 is observed due to the random diagonal disorder which mixes all exciton states. However the reason why energy separation is strongest for k = ±1 is due to the C₂ modulation.

(a) (b)

(c) (d)

(e) (f)

Figure 3.2: a) Histogram of energy separation between k = ±1 exction states for 144 complexes.

The figure is reproduced from the paper [7]. Energy separation between k = ±1 exction states varying with eccentricity for b) first model c) second model d) third model e) fourth model and

varying with Emod for f) fifth model

(34)

3.2 Intensity ratio of k = ±1 states

To discriminate the third, fourth and fifth model, intensity of k = ±1 exciton states can be used.

The experimental result in figure 3.3 a) shows that the intensity ratio of the k = −1 state which has higher energy to k = +1 state which has lower energy is centered around 0.73 with FWHM of 0.54. This means that on average, the k = −1 has lower intensity than the k = +1 state.

Comparing the results from the models, we can see that the third model doesn’t have a match with the experimental result since k = +1 state has lower intensity than the k = −1 state as the unperturbed ring is subjected to elliptical deformation. However the results from the fourth and fifth model has the behaviour matching the experimental result. Considering that energy separation of 126cm⁻¹ was obtained for e ∼ 0.5 for the fourth model and E_mod ∼ 132cm⁻¹ for the fifth model, the intensity ratio for these values are obtained ^I_I⁻¹

+1 = 0.86 for the fourth model and ^I_I⁻¹

+1 = 0.96 for the fifth model. Eventhough, both the models have results which match the experimental result qualitatively, quantitatively the results are not a good match. It could be due to effects from the other exciton states which are not considered in the calculation of the intensity ratio. For instance, due to the C2 modulation and diagonal disorder, oscillator strength is redistributed to the other exciton states and in the experiment, there will be intensity contribution of higher exciton states to the k = −1 state and intensity contribution of k = 0 state to the k = +1 state. Considering these effects might give a better quantitative fit to the experimental value.

Another important feature that can be observed in the results for fourth and fifth model is that oscillator strength which is lost from k = ±1 states is gained by k = ±3 states which is primarily due to the C₂ modulations. Although diagonal disorder leads to distribution of oscillator strength to all the neighbouring exciton states, it’s effect is not as strong as that of the C2 modulation.

3.3 Mutual orientation of the transition-dipole moments of k = ±1 states

To check if both the fourth and fifth models are consistent with the experimental result for mutual orientation of the transition dipole moments of the k = ±1 states, the angle between transition dipole moment of the k = ±1 states is plotted as a function of the eccentricity and E_mod in figure 3.4. Experimentally, it was observed that the mutual orientation of the transition-dipole moments is centered around 91^◦ with FWHM of 19^◦. This is consistent with the results from the two models where orthogonality is mantained for all magnitude of C₂ modulation within 3^◦. So this quantity doesn’t help to discriminate between the two models. However an interesting feature that can be observed in the result from the fourth model is that at around eccentricity of 0.7, a peak is observed which can be due to strong mixing of exciton state with k = −1 state to exciton states with higher energies which doesn’t happen in the fifth model.

(35)

3.4 Linear absorption spectra

Linear absorption spectrum for an ensemble of LH2 complex where the distribution of eccentricity and E_mod were obtained considering that energy separation for k = ±1 states originate from C2

modulations was calculated to check it’s consistency with the experimental result. Results from both the models could be fitted well with the experimental result of Rps. acidophila at 77K as shown in the figure 3.5. Since the diagonal disorder of 250cm⁻¹ which was used to fit the results was common in both the models, it seems that the contribution for width of the spectrum primarily comes from the random diagonal disorder and distribution of eccentricity and E_mod do not have large influence here.

3.5 Two dimensional spectra

Up to now considering the mentioned experimental results, the fourth and fifth model are both good candidates. As primary motivation of the study was to explain the energy shift in the two dimensional spectrum, both the models were tested if they could reproduce this effect from the

(a) (b)

(c) (d)

Figure 3.3: a) Histogram of intensity ratio of exciton state with k = +1 and 0 states to exciton state with k = −1 state for 88 complexes. This figure is reproduced from the paper [7]. Intensity ratio of exciton state with k = +1 state to exciton state with k = −1 state varying with eccentricity

for b)third model c)fourth model and varying with Emod for d)fifth model

(36)

experiment. Experimentally a 110cm⁻¹ energy shift has been observed on a time scale of 400f s as shown in figure 3.6 a). However authors of paper[1] have reported not observing any such phenomena although intraband relaxation has been taken into account in the model with varying radii of the LH2 complexes shown in figure 3.6 b). It was speculated to be due to lack of energy separation between k = ±1 states since the C₁₈symmetry is retained when only radius of the B850 ring is varied. Results of two-dimensional spectroscopy for the fourth and fifth model for waiting times of 0f s and 250f s are given in figure3.6 c) and 3.6d) respectively.

Qualitatively changes in the two dimensional spectrum on the time scale of 250f s could be observed like small shift in energy and change in shape. To quantify the energy shift, a simple approach was taken where the excitation frequency for which there is minimum ground state bleaching was identified at waiting time of 0f s and then minimum ground state bleaching at that frequency was observed at waiting time of 250f s. The difference between the frequencies correponding to minimum ground bleaching for two waiting time was taken as the energy shift. Results from this approach are presented in figure 3.7where it could be observed that after waiting time of 250f s, energy shift is observed for fifth model where as it wasn’t observed for the fourth model. So com- paratively the energy shift is observed much bigger in the fifth model with the correlated diagonal

(a)

(b) (c)

Figure 3.4: a) Histogram of relative orientation of the transition dipole moments of exciton state with k = +1 state to exciton state with k = −1 state for 144 complexes. This figure is reproduced from the paper [7]. Relative orientation of the transition dipole moments of exciton state with k = +1 state and exciton state with k = −1 state varying with eccentricity for b) fourth model

and varying with E_mod for c) fifth model

(37)

disorder than the model with correlated off-diagonal disorder. To compare with experimental results, it would be wise to make two-dimensional spectra for waiting time of 400f s. However as it is evident from the population plots in figure 3.8, no big differences are expected for the waiting time of 250f s compared to waiting time of 400f s. At 250f s, the system seems to have already reached steady state. Theoretically, since both the models can generate the energy separation between k = ±1 states, the energy shift in two-dimensional spectra is expected from both the models although there might be difference in magnitude of shift. The reason why energy shift is not observed in the model with the C2 modulated diagonal disorder can have more fundamental reasons or the method used to quantify this shift can be too rudimentary to draw conclusion this way. This is still not clearly understood and need further analysis.

However there are features of the calculated two-dimensional spectrum which substantiates the frenkel exciton model used to explain the dynamics in B850 ring. For instance, the excited state absorption peak which was observed with the negative absorption peak corresponding to ground

(a)

(b) (c)

Figure 3.5: a) Experimental 77 K absorption spectrum (dashed line) of Rps. Acidophila. This figure is reproduced from the paper [8]. Calculated linear absorption spectrum with experimental

result for b) fourth model and c) fifth model

(38)

state bleaching is also observed in the 2D-spectra of J-aggregate which is an indication of strong delocalization of the states[37,38]. Such features are absent in the 2D-spectra of B800 ring where the molecules are weakly coupled resulting in localization of excitations on the molecules. Moreover, the more rounded shape observed at waiting time of 250f s indicate that there is frequency change during the waiting time. Usually the rounded peak shape results from molecules’ loss of memory of the initial frequency which in this case could be due to dynamics from fluctuating environment and/or transition between the exciton states with degeneracy lifted by C₂ modulations. In case this round shape is due to transition between the k = ±1 exciton states, then the fact that it is observed in both the models is consistent with our expectation. With that it can be then inferred that either the way the energy shift is quantified is not accurate or only strong energy separation can be observed as a shift. This means that if the effects from energy fluctuation due to the dynamics of environment and effect due to transition between k = ±1 states are disentangled, then deeper understanding of the phenomena can be achieved. Therefore this requires further studies.

(39)

(a)

(b)

(c)

(d)

Figure 3.6: a)Two dimensional spectra at varying waiting times from experiment[2]. Calculated two dimensional spectra for waiting times for 0 and 250 fs from b) model with varying radii of LH2

complexes[1] c) fourth model and d) fifth model

(40)

(a)

(b)

(c)

(d)

Figure 3.7: Energy shift observed in the 2D spectra when waiting time is changed from 0 fs to 250 fs from a) fourth model b) fourth model zoomed in to obtain the energy shift magnitude c)

fifth model and d) fifth model zoomed in to obtain the energy shift magnitude.

Spectroscopic studies of C2