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A. An Excitonic Model for the Amide I Band

A. An Excitonic Model for the Amide I Band

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308



Hamm and Hochstrasser



We have adopted the excitonic band model not only because it

describes conventional absorption spectroscopy (linear spectroscopy), but

because it enables an extremely convenient description of nonlinear

experiments, such as pump-probe, dynamical hole burning, or photon

echoes. In these third-order experiments one has to consider not only

transitions from the ground state to the one-excitonic states but also

transitions from the one-excitonic to the two-excitonic states (see Fig. 13).

These additional transitions reveal the required information to deduce, at

least in principle, the complete coupling scheme.



Figure 13 Energy level scheme for a system of two coupled oscillators. The

isolated peptide states (left side) are coupled by some weak interaction, which

mixes them to generate the excitonic states (right side). Anharmonicity, which is

crucial for understanding the 2D pump probe spectra, is introduced into this model

by lowering the energies of the double excited monomeric site states ji2 i and jj2 i

by  from their harmonic energies 2εii . This anharmonicity mixes into all coupled

states, giving rise to diagonal anharmonicity εkk and off-diagonal anharmonicity

(mixed-mode anharmonicity, εkl ) in the basis of the normal modes discussed in

the text.



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Proteins and Peptides



309



To describe such an excitonic system, the eigenstates of

n



H D H0 C V D



Hi C

iD1



Vij



(22)



i


are needed, where the Hi describes the n noninteracting (monomeric)

peptide groups and the Vij the interpeptide potential. Whether the coupling

term V consists only of dipole-dipole coupling [as proposed by Tasumi

et al. (123)] or includes higher-order multipole interactions or additional

through-bond effects is not vital at this point. The only assumption made

is that Vij is bilinear in the coordinates of site i and j. The diagonalization

is readily accomplished in the basis of site excitations involving 0,1,2

quanta: the ground state j0i, having no vibrational quanta excited, the states

fji1 ig, have one vibrational quantum on the ith site, and the two-particle

states fji1 j1 ig, i 6D j, having one quantum at two different sites i and j, and

fji2 ig, the biphonon (overtone) states, having two quanta at a single site i,

respectively. In this basis, the matrix elements of H are:

h0jHj0i Á 0

hi1 jHji1 i D εi

hi1 jHjj1 i D ˇij

hi1 j1 jHji1 j1 i D εi C εj

hi2 jHji2 i D 2εi 

hi1 j1 jHji1 k1 i D ˇjk

p

hi1 j1 jHji2 i D 2ˇij



23a

23b

23c

23d

23e

23f

23g



where the εi are the vibrational zero-order frequencies of the uncoupled

states,  the anharmonicity of these states (assumed to be the same for all

groups), and ˇij the excitonic resonance couplings. For the sake of clarity,

the vacuum-to-aggregate shift terms D, which are required in a fully

general form of Equation (23) (131), are omitted

here and are included into

p

an effective anharmonicity. (30) The factor 2 in Equation (23g) originates

from a harmonic approximation. Site i is de-excited D 1 ! D 0, while

site j undergoes a D 1 ! D 2 transition so that the

p matrix element of

Equation (23g) is proportional to hi1 jqi ji0 ihj1 jqj jj2 i D 2hi1 jqi ji0 ihj0 jqj jj1 i,

where qi and qj are the oscillation coordinates of the amide I mode on sites

i and j, respectively. The resonance term in Equation (23f), on the other

hand, includes only D 0 ! D 1 transitions so that no corresponding

factor appears.



Copyright © 2001 by Taylor & Francis Group, LLC



310



Hamm and Hochstrasser



If the individual amide modes were harmonic oscillators, the excitonically coupled states of the polypeptide would also be a set of n independent

harmonic normal modes (under the assumption of a bilinear coupling term),

which are completely decoupled from each other. In this case, the twoexciton states could be obtained directly without explicit diagonalization

of the two-exciton matrix, for they are merely product states of the oneexcitonic states. However, the third-order response of such a harmonic

system would be exactly zero, since all transitions depicted in Fig. 13 would

cancel. Therefore, it is essential to consider anharmonicity in order to understand the third-order response of the amide I band (as it is in any nonlinear

spectroscopy on vibrational systems). It is nevertheless instructive to think

about the problem in a picture of “almost harmonic” excitonically coupled

states and consider anharmonicity as a perturbation. The anharmonicity 

lowers the zero-order energies of the doubly excited states ji2 i from their

harmonic values 2εi [Equation (23e) and dotted lines in Fig. 13], while the

energies of states ji1 j1 i, having two excitations at two different sites, are

not affected by anharmonicity. The anharmonicity  of the monomeric

sites has been determined with the help of IR-pump–IR-probe experiments

on NMA to be  D 16 cm 1 (30). This anharmonicity is mixed into all

coupled excitonic states, and the resulting normal modes now are associated with a diagonal anharmonicity εkk D εkk 2εk and off-diagonal

(or mixed-mode) anharmonicity εkl D εkl εk εl (see Fig. 13). In other

words, the right-hand side of the level scheme in Fig. 13, and as a consequence all nonlinear IR experiments on the amide I band, can be readily

understood in terms of ordinary anharmonicity. The crucial point is that

the excitonic coupling model provides a convenient and extremely simple

way to relate these anharmonicities to the coupling Hamiltonian (ultimately

related to the three-dimensional structure of the protein). The anharmonicities would otherwise have to be determined from unrealistically computer

expensive quantum chemistry methods, requiring the calculation of the 2nd,

3rd, and 4th derivatives of the ground state potential surface [the largest

molecule for which this has been demonstrated, is benzene (132)].

The picture of “almost harmonic” excitonically coupled states is

particularly appropriate in the localization or weak coupling limit. This limit

will be valid in smaller peptides that do not have the rather strict symmetries

of helices or sheets. It is very likely that the vibrational frequencies of

each amide unit will be different even in the absence of any coupling. An

example of this limit is found in the pentapeptide discussed below. If the

frequency separations between the uncoupled modes are large compared

with the individual coupling terms, jˇij / εi εj j < 1, the coupled states



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Proteins and Peptides



311



are predominantly localized on single peptide units. In weak coupling, the

diagonal anharmonicities are given to first order by εkk D , and the

off-diagonal anharmonicities are calculated from perturbation theory (in

lowest order of  and ˇkl ; see Appendix):

εkl D



4



2

ˇkl

εk εl



(24)



2



In addition, one obtains for the transition dipole moments

the

p

relation E j k i!j k ij l i D E j0i!j l i , k 6D l, and E j k i!j k ij k i D 2 E j0i!j k i ,

respectively, and all other transitions j k i ! j l ij m i, k 6D l, m, are

forbidden.

In the strong coupling limit, for example, for a symmetrical ˛- helix,

where all peptide groups are essentially equivalent so that the zero-order

energies εi are all identical, it is more complicated to find simple pictures for

the oscillators. Not only is a numerical diagonalization of the two-exciton

matrix then needed to find the diagonal and off-diagonal anharmonicities,

but also the transition dipoles between E j0i!j l i and E j k i!j k ij l i change

considerably and even transitions such as j k i ! j l ij m i, k 6D l, m, which

are forbidden both in the harmonic and the localization limit, become

weakly allowed. In the case when anharmonicity is large compared with

excitonic couplings  > ˇij , the two-excitonic states completely lose their

identity of being products of one-excitonic states. This situation will most

likely prevail for the N-H vibrations of the peptides.

B. Response for N Coupled Oscillators



An alternative perspective on third-order responses of N coupled vibrators,

which will be particularly helpful to describe spectral diffusion processes in

such coupled systems (see Section IV. D), can be developed by assuming

that R1 and R2 are the same and writing the total response function as:

RD



h2



0j t3



j0 t1



00



0i



0



i0 t2



i,j,k

ik t3



jk t2



j0 t1



00



0i



0i



25



After factoring out the parts involving the anharmonicity, the response

becomes

RD2



h1



ij,k e



ikj,i0 t3 t2 Ci



t3

t2



υkj,i0



d



i,j,k



ð



0j t3



j0 t1



00



0i



0



Copyright © 2001 by Taylor & Francis Group, LLC



i0 t2



i



26



312



Hamm and Hochstrasser



where i and j each take N values corresponding to the one quantum state,

the t are given by Equation (3), and k runs over all of the 12 N N C 1

two quantum states. The anharmonicity has both diagonal and off diagonal

contributions, kj,i0 D ωnj ωi0 . The transition dipole factor ij,k is:

ij,k



D



ik



2



0i



kj



(27)



j0



In the absence of coupling and electrical anharmonicity, ij,k D 12

when k is one of the 12 N N 1 two-particle states, and ij,k D 1 when k

is one of the N biphonon states.

C. Two-Dimensional IR Spectroscopy on the Amide I Band



We will describe in the following a novel two-dimensional IR (2D-IR)

spectroscopic technique on some peptide samples. These experiments have

been performed in order to verify the excitonic coupling and to establish the

nature and strength of the coupling. Our intention is the use the foregoing

principles to develop a novel structure analysis method with the potential

of ultrahigh time resolution. In our first approach to 2D-IR spectra, the

transient response of the sample was measured with the help of a broadband, ultrashort probe pulse (120 fs) as a function of the peak frequency of

a narrower band pump pulse (ca. 10 cm 1 ). The pump pulse was generated

from the output of the femtosecond IR light source (see Sec. II) by means

of an adjustable IR-Fabry-Perot filter (30,42). The essence of the experiment is to selectively populate individual one-excitonic levels with the

pump pulse and probe the response of the sample by means of transitions

back to the ground state and to the two-excitonic states. In that way, diagonal anharmonicity εkk is sensed along the diagonal of the 2D spectra,

where the negative bleach and stimulated emission signals are observed

at the frequency εk of the pumped state j k i and excited state absorption

involves transitions to the double excited state j k ij k i. In the off-diagonal

region corresponding to the kl cross-peak, one finds a bleach of the probed

transition εl , arising from the depopulation of the common ground state and

an excited state absorption to the mixed state j k ij l i, which is red-shifted

from the bleach by the off-diagonal anharmonicity εkl .

In order to picture the underlying principles of nonlinear 2D-IR

spectra, a model calculation of an idealized system consisting of two

coupled vibrators is shown in Fig. 14b together with their linear absorption

spectrum (Fig. 14a). The frequencies of these transitions were chosen

as 1615 cm 1 and 1650 cm 1 , the anharmonicity  D 16 cm 1 , and the

coupling ˇ12 D 7 cm 1 . We used a homogeneous dephasing rate of T2 D



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