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IV. VIBRATIONAL EXCITONS IN CYCLIC PENTAPEPTIDE

IV. VIBRATIONAL EXCITONS IN CYCLIC PENTAPEPTIDE

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IR Spectroscopy in Polypeptides



359



structures (42). The structure is not unique, and several conformations are

compatible with the NMR and x-ray measurements. In our study we used

the crystallographic structure to obtain the atomic coordinates of cyclo(AbuArg-Gly-Asp-Mamb) shown in Fig. 2. The backbone conformation traces

out a rectangular shape with a ˇ-turn centered at the Abu-Arg bond.



Figure 2



3D structure of the pentapeptide. (From Ref. 42.)



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360



Piryatinski et al.



To obtain the one-exciton Hamiltonian we used the central frequencies

for the peptide CO vibrations reported in Ref. 41 and assigned 1 D

1588 cm 1 to Abu-Arg, 2 D 1671 cm 1 to Arg-Gly, 3 D 1648 cm 1

to Gly-Asp, 4 D 1610 cm 1 to Asp-Mamb, and 5 D 1618 cm 1 to

Mamb-Abu. The dipole-dipole couplings among CO vibrations were

calculated from

Jmn D



m



Ð



n



ˆ Ð

3m

jRmn j3



ˆ Ð

m



m



n



m, n D 1, . . . , 5



(24)



˚ from the carbon atom

by assigning each dipole on a CO bond 0.868 A

°

and forming 25 angle with respect to the bond. The absolute value of

each dipole moment is 0.37 D (1,2,15,41). Using these parameters, the

one-exciton Hamiltonian (in cm 1 ) assumes the form





1588

7.2

5.7

1.7

7.0

0.7

0.6

7.6 

 7.2 1671





(25)

h D  5.7

0.7 1648

2.2

6.2 





1.7

0.6

2.2 1610

0.3

7.0

7.6

6.2

0.3 1618

Since j n

m j > Jnm , each one-exciton state is a weakly perturbed

localized CO vibration. The one-exciton eigenstates are

je1 i D

je2 i D

je3 i D

je4 i D

je5 i D



0.98j1i 0.07j2i

0.05j1i 0.03j2i

0.16j1i 0.15j2i

0.10j1i C 0.07j2i

0.10j1i 0.98j2i



0.07j3i C 0.08j4i C 0.18j5i

0.08j3i C 0.98j4i 0.17j5i

0.21j3i 0.17j4i 0.94j5i

0.97j3i 0.05j4i C 0.19j5i

0.03j3i 0.01j4i C 0.15j5i



26



where jni, n D 1, . . . , 5 represents first excited vibrational state of the nth

CO vibration. The corresponding one-exciton energies are ε1 D 1586 cm 1 ,

ε2 D 1610 cm 1 , ε3 D 1617 cm 1 , ε4 D 1650 cm 1 , and ε5 D 1673 cm 1 .

The two-exciton manifold consists of two types of doubly excited

vibrational states. The first are overtones (local), where a single bond is

doubly excited. The other are collective (nonlocal), where two bonds are

simultaneously excited (43,50). We denote the former OTE (overtone twoexcitation) and the latter CTE (collective two-excitation). A pentapeptide

has 5 OTE and 10 CTE. The two-exciton energies are determined by the

parameters gn in the Hamiltonian [Equation (17)], which in turn depend

on the peptide group energies n , the anharmonicity n , and dipole

moment ratio Än , n D 1, . . . , 5. We set them equal for all CO units



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IR Spectroscopy in Polypeptides



361



p

and  D 16 cm 1 adopted from the experiment (15,41) and Ä D 2.

The two-exciton energies are obtained from the poles of the exciton

scattering matrix (rather than a direct diagnolization of the 15 ð 15 twoexciton Hamiltonian, which is much more expensive). They are ε1 D

3157 cm 1 , ε2 D 3195 cm 1 , ε3 D 3200 cm 1 , ε4 D 3205 cm 1 , ε5 D

3220 cm 1 , ε6 D 3227 cm 1 , ε7 D 3235 cm 1 , ε8 D 3258 cm 1 , ε9 D

3259 cm 1 , ε10 D 3264 cm 1 , ε11 D 3283 cm 1 , ε12 D 3287 cm 1 , ε13 D

3288 cm 1 , ε14 D 3322 cm 1 , and ε15 D 3331 cm 1 . In general the twoexciton eigenstates are linear combinations of the OTE and the CTE.

However, since j 0m 2 n j > Ä2 Jnm , most two-exciton states can be

classified as weakly perturbed OTE or CTE type.

Having introduced the one- and two-exciton states, we next turn to

the line broadening. We denote the dephasing rate of the first vibrational

transition by  and the overtone by 2 . In all calculations  and 2 are

set identical for all peptide groups. The anharmonicity  D 16 cm 1 is

fixed and independent of disorder. We have employed six models:

A.

B.



C.



Small homogeneous dephasing rates  D 0.2 cm 1 and 2 D

0.4 cm 1 .

Large homogeneous dephasing rates  D 5 cm 1 and 2 D

10 cm 1 , which correspond typically to experimental values

(15,41).

Static diagonal disorder. The n’th peptide energy is

represented as

n



D.



E.



D



n



C



n



n D 1, . . . , 5



(27)



where n is average energy of the n’th peptide group set

to the central frequencies in the one-exciton Hamiltonian

[Equation (25)]. The random variables n , representing energy

disorder, are assumed to be uncorrelated random Gaussian

variables with variance d D 12 cm 1 equal for all peptide

groups:  D 0.2 cm 1 , 2 D 0.4 cm 1 .

Same as model 3 except that the homogeneous dephasing rates

of each peptide are adopted from experiment and set to  D

5 cm 1 , 2 D 10 cm 1 .

Static off-diagonal disorder. The exciton coupling is given by

Jmn D Jmn C



mn



n 6D m;



n, m D 1, . . . , 5



(28)



where Jmn is average coupling energy between the m’th and the

nth peptide groups, whose values are given in Equation (25). mn

are uncorrelated Gaussian random variables with equal variances



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362



Piryatinski et al.



D 12 cm 1 for all mn , (n 6D m; n, m D 1, . . . , 5). The homogeneous dephasing rates are  D 0.2 cm 1 , 2 D 0.4 cm 1 .

Same as model 5 except that the homogeneous dephasing rates

are taken from experiment and set to  D 5 cm 1 , 2 D

10 cm 1 .

od



F.



Different peptide groups in the pentapeptide have inhomogeneous

broadening, which varies in the range of ¾ 3–12 cm 1 (41). Some of

the experimentally observed lines are therefore dominated by homogeneous

and others by inhomogeneous broadening. We introduced these different

models in order to study the signature homogeneous and inhomogeneous

broadening on the 2D PE spectra. In models (A), (C), and (E) we used

small homogeneous broadening in order to resolve all resonances. Some of

these resonances are not resolved in models (B), (D), and (F) which use

larger, more realistic, homogeneous broadening.

The degree of one-exciton state localization can be described by the

inverse participation ratio (52–54):

1



5



Pε D



j



ε



nj



4



(29)



nD1



where ε n is the n’th component of one-exciton wavefunction with energy

in the interval [ε, ε C dε]. For our model P ε may vary between P D

1 (localized state) and P D 5 (delocalized state). The participation ratio

distribution as well as the density of states are shown in Fig. 3. For models

A and B the exciton states are well localized, since j n

m j < Jnm . In

models C and D diagonal disorder slightly increases the one-exciton state

delocalization, resulting in the participation ratio ¾1.3 in the maximum

of the density of states (dotted line in the plot), and in models E and F

the off-diagonal disorder corresponds to the state delocalization within ¾2

peptide groups near the density of states (dotted line) maxima.

The linear (1D) absorption spectra of all models are presented in

Fig. 4. Model A shows five well-resolved one-exciton lines. In model B

the lines ε2 and ε3 are poorly resolved due to the increased homogeneous

broadening. Diagonal disorder in models C and D further broadens the

spectra. Since off-diagonal disorder induces state delocalization, the oneexciton resonances shift for models E and F and become ε01 D 1578 cm 1 ,

ε02 D 1605 cm 1 , ε03 D 1618 cm 1 , ε04 D 1652 cm 1 , and ε05 D 1679 cm 1 .

Additional information related to the one- and two-exciton dynamics

can be obtained from the 2D spectra, as will be shown in the following

section.



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IR Spectroscopy in Polypeptides



363



Figure 3 Solid line: Inverse participation ratio of one-exciton states for models

A–F. Dotted line: Density of one-exciton states for models C–F.



Copyright © 2001 by Taylor & Francis Group, LLC



364



Figure 4



Piryatinski et al.



Infrared absorption (1D) spectra for models A–F.



V. 2D PHOTON ECHOES OF A CYCLIC PENTAPEPTIDE



In a 2D three-pulse spectroscopy, two of the three pulses are time-coincident

and differ only by their wave vector. The system thus interacts once with a

single pulse and twice with a pulse pair. In the 2D photon echo technique

we set t2 D 0. We consider the heterodyne signal [Equation (10)]. This



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IR Spectroscopy in Polypeptides



365



signal can be measured by mixing the third-order signal with the heterodyne

pulse arriving with delay time t3 in the direction determined by the phase

matching conditions ks D k3 C k2 k1 . We shall display the 2D PE signal

in the frequency domain by performing a double Fourier transform:

S



2,



1



1



D



1



dt3



dt1 exp i



0



2 t3



Ci



1 t1



S t3 , 0, t1 .



(30)



0



The 2D PE signal is computed using the response function given by

Equations (21)–(23). Since we consider the 2D response on the time

scale smaller than the dephasing times, only the coherent component of

the response function [Equation (21)] contributes to the signal. The 2D

Fourier transform PE signal determined by Equations (22) and (30) has the

following form (17):

S



2,



1



DS1



2,



1



CS2



2,



(31)



1



The first component,

2,



S1



1



D



i

2

ð



a



b



c



d

2



abcd



1

εd C i



1

1 C εc C i



cd,ab εc C εd

εc C εd

εa C εb C 2i



32



represents correlations between one-exciton states shown by the Feynman

diagram (Fig. 5(1)). The second component,

2,



S2



1



D



a



b



c



d



abcd



ð

ð



1

1

1 C εc C i 2



1

ω

ω



εc

1

εc C ε d

2



i C

i



0



ω



dωcd,ab ω



1

εa C εb C i 2



i



0



0



33

0



is induced by correlations between one- and two-exciton states and is

represented by the Feynman diagram in Fig. 5(2).Ł

Below we first analyze the absolute value of the 2D PE signal and then

consider its phase by looking at the real and the imaginary parts separately.

Ł



The components of the signal calculated according to these diagrams, using the

sum-over-state approach presented in Appendix A, coincide with Equations (32)

and (33) in the narrow line limit  − 0 , J



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