A. An Excitonic Model for the Amide I Band
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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.
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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
ikj,i0 t3 t2 Ci
t3
t2
υkj,i0
d
i,j,k
ð
0j t3
j0 t1
00
0i
0
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i0 t2
i
26
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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
Copyright © 2001 by Taylor & Francis Group, LLC