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A. Absolute Value of the 2D Signal

A. Absolute Value of the 2D Signal

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IR Spectroscopy in Polypeptides



367



Figure 6 Absolute value jS 2 , 1 j of 2D infrared photon echo signal for

models A–F. (The different panels have different intensity codes. Light gray is

zero and maximum value is dark gray.)



correlations between one-exciton and two-exciton (OTE) states on the

same sites. The resonances representing correlations between different oneexciton states (off-diagonal peaks) and correlations between the one-exciton

and two-exciton states make a negligible contribution to the spectra of

models C and D. This is a consequence of the uncorrelated disorder



Copyright © 2001 by Taylor & Francis Group, LLC



368



Piryatinski et al.



Figure 7 Slices of the signal of Fig. 6 for models A and B for (1) 1 D ε1 ,

(2) 1 D ε2 , (3) 1 D ε1 , (4) 1 D ε4 , and (5) 1 D ε5 . Model A shows

detailed structure; the broader curve corresponds to model B. All resonances of

model A are identified in Table 1.



Copyright © 2001 by Taylor & Francis Group, LLC



IR Spectroscopy in Polypeptides

Table 1



Energy of the Resonances Identified in Fig. 7 for Model A



2



resonances

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.



369



1



D



ε1

ε2

ε3

ε5

ε6

ε7

ε8

ε10

ε12



ε1

ε3



ε1

ε1



1



D



ε2



ε1

ε1



ε2

ε4

ε5

ε6



ε1

ε1

ε1

ε1

ε1

ε1



ε11 ε2













ε4



ε2

ε2

ε2

ε2



1



D



ε2

ε3

ε5

ε6

ε9

ε10

ε11

ε13



ε1

ε3

ε4



ε3

ε3

ε3

ε3

ε3

ε3

ε3

ε3

ε3



1



D



ε4

ε5

ε6

ε7

ε9

ε10

ε11

ε14

ε15



ε3

ε4



ε4

ε4

ε4

ε4

ε4

ε4

ε4

ε4

ε4

ε4



1



D



ε6

ε7

ε8

ε10

ε11

ε13

ε14

ε15



ε5







ε5

ε5

ε5

ε5

ε5

ε5

ε5

ε5

ε5



distribution for different peptide groups, which does not allow the rephasing

of the PE signal during the t3 time delay (Fig. 5). We assumed that

the anharmonicity is independent of disorder. Otherwise, for uncorrelated

disorder, the OTE cross peaks would not be resolved.

Slices of the 2D signal of models C and D along 2 at 1 D εa ,

a D 1, . . . , 5, are shown in Fig. 8. The lines in this plot are homogeneously

broadened, since the photon echo technique removes the inhomogeneous

broadening. Each slice of model C contains two distinct peaks. The narrow

one (width ) represents self-correlation of the one-exciton states and the

broader one (width 2 C ) is due to the OTE state. Since the resonances

present in each panel represent self-correlation of single excitons and of

single exciton with the OTE excited on the same peptide group, the energy

splitting of their maxima should provide an anharmonicity close to the

anharmonicity of a single peptide unit . In panels (1), (2), and (5) the

anharmonicity is 15 cm 1 and in panel D it is 14 cm 1 . These are very

close to the anharmonicity of a single peptide unit  D 16 cm 1 , indicating that the corresponding OTE states are weakly perturbed by the weak

coupling to the other two-exciton states. The anharmonicity 16 cm 1 in

panel (3) indicates that the OTE with energy ε5 is decoupled from the other

doubly excited states. The structure of spectra of model C suggests that

fitting the spectrum of model D by two Lorentzian lines, one of each representing a diagonal peak, should provide the anharmonicity and dephasing

rate of the two-exciton states.



Copyright © 2001 by Taylor & Francis Group, LLC



370



Piryatinski et al.



Figure 8 Slices of the signal of Fig. 6 for models C and D for (1) 1 D ε1 ,

(2) 1 D ε2 , (3) 1 D ε1 , (4) 1 D ε4 , and (5) 1 D ε5 . Model C shows

detailed structure; the broader curve corresponds to model D.



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IR Spectroscopy in Polypeptides



371



The 2D PE signal (Fig. 6) of models E and F is dominated by inhomogeneous broadening along the

1 D 2 direction of the diagonal peaks

and the cross peaks associated with the OTE states. In addition, weak offdiagonal and the cross-peaks are clearly seen in the plot. Slices of the 2D

PE signal along 2 at 1 D ε0a , a D 1, . . . , 5, are shown in Fig. 9. As

in Fig. 8, these spectra are homogeneously broadened; the broader curves

represent model F and the underlying structure is clearly seen for model E.

Self-correlations of one-exciton states are represented by the narrow lines

and correlations between the one- and two-exciton states are given by the

broader lines. In contrast to models C and D, the energy differences between

the maxima of the one-exciton and strong two-exciton resonances are not

equal to , indicating that the relevant two-exciton states are not necessarily the localized OTE type. Panels (2) and (4) show two equally strong

two-exciton lines close to the one-exciton line. They are clearly seen in

the 2D PE plot of model E as two inhomogeneous resonances stretched in

different directions which are close to

1 D 2 . The strong two-exciton

lines seen in panels (1) and (4) of Fig. 9 correspond to the 2D resonances

stretched in the direction close (but again not exactly equal to)

1 D 2.

This indicates that the two-exciton states in models E and F depend on

the coupling Jnm and are delocalized. The weak signal observed in the 2D

plot represents the off-diagonal peaks and the cross peaks inhomogeneously

broadened in directions 1 ¾ 2 .

So far we have discussed the absolute value of the 2D PE signal.

The phase of the signal (which can be observed as well) carries additional

information (43). In particular, the phase (or equivalently the real and the

imaginary parts of the signal) can explain the weak intensity of the cross

peaks in models E and F. Below we examine the real and imaginary parts

of the 2D PE signal.

B. Real and Imaginary Parts of the 2D Signal



The real and imaginary parts of the 2D PE signal are displayed in the

left and the right columns, respectively, of Figs. 10 and 11. To clearly

show the 2D resonance structure in model A, we display the signal on

an expanded scale, showing 2 resonances only at 1 D ε4 . The diagε4 , ε12 due to the OTE

onal peak 1

ε4 , ε4 , and the cross peak 2

are marked in the plot. The real part of the signal is dispersive in the

1 D 2 direction across the resonances. It approaches zero and changes

sign (edge between the light gray and dark gray regions) along the line

connecting the resonances in the

1 ¾ 2 direction. The imaginary part

of the signal has positive (dark gray) and negative (light gray) maxima at the



Copyright © 2001 by Taylor & Francis Group, LLC



372



Piryatinski et al.



Figure 9 Slices of the signal of Fig. 6 for models E and F for (1) 1 D ε01 ,

(2) 1 D ε02 , (3) 1 D ε03 , (4) 1 D ε04 , and (5) 1 D ε05 . Model E shows

detailed structure; the broader curve corresponds to model F.



Copyright © 2001 by Taylor & Francis Group, LLC



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