STUDIA UNIVERSITATIS BABEŞ-BOLYAI, PHYSICA, SPECIAL ISSUE, 2001
SOLITON SIGNATURE IN THE INFRARED SPECTRA OF
NONLINEAR QUASI-ONEDIMENSIONAL MOLECULAR
CRYSTALS
D. GRECU, ANCA VIŞNESCU
Institute of Physics and Nuclear Engineering "Horia Hulubei"
Bucharest, Romania, e-mail: dgrecu@theor1.theory.nipne.ro
ABSTRACT. Davydov’s soliton remains an attractive and possible mechanism for the
energy transport in proteins. Discussed initially in the context of alpha-helix structure, it was
applied later in several situations. An interesting system is the acetanilide (ACN), which has
many features in common with other hydrogen bonded polypeptide structures. An
unconventional amide I band was disovered in infrared and Raman spectra at low
temperatures, which was interpreted as a manifestation of the existence of soliton in this
structure. The theory of the infrared spectra is reviewed, with emphasize on the resonant
interaction between short waves (amide I excitations) and long waves (phonons), interaction
responsible for the soliton generation.
1. INTRODUCTION
The mechanism for energy transport in proteins has been an active area of
research for over 30 years, but still is not yet elucidated. Many years ago, an attractive
model was proposed by Davydov. In this model the amide I oscillations act as baskets
for the storage of biological energy released in the adenosine triphosphate hydrolysis.
The nonlinear interaction of these intramolecular excitations with the acoustic field,
describing the oscillations of the molecule along the α-helix chain, provides the
mechanism for generation of self-trapped excitations along the chain. In this way the
energy can travel over considerable molecular distances without being thermally dissipated.
Many authors have discussed and tried to improve this model, and in spite of many
controversial arguments it still remains in actuality [1], [2] (and references therein).
2. DAVYDOV’S MODEL
The model represents a very drastic approximation of the real situation, and in
spite of its simplicity it can not be solved exactly. Usually the quantum model is
discussed in a kind of coherent state approximation and a system of coupled nonlinear
equations for the classical field variables is found, namely

i a n  Ean   J p ( a n  p  a n  p )   n an
(1)
p 1
Mn  w(  n 1  2  n   n 1 )   ( a n 1  2 a n  a n 1 ) .
2
2
2
D. GRECU, ANCA VIŞINESCU
Here a n is the classical amplitude of the amide I oscillation in the n-th molecule,
E=E 0 -D, where E 0 is the oscillation energy and D a static shift due to lattice effects,
J p is the dipole-dipole interaction energy between C=O dipoles (usually only nearest
neighbours interaction is considered), ρ n is the relative displacement of two adjacent
molecules, M and w are the mass and the elastic spring, and χ is the coupling constant
for the mode-lattice interaction.
In studying this system we have to make distinction between two regimes of
space-time variations. The first corresponds to some fast motions described by space
and time variables of order of the distance between molecules and the inverse of a
characteristic frequency. These motions are described mainly by the linearized
equations and correspond to a decoupled system of intramolecular excitations (Asystem) of frequencies
 ( k )  E  2 J p cos klp
(2)

p 1
and a system of acoustical phonons (P-system) with the dispersion relation
(k )  2 0 sin
kl
2
,
0 
w
.
M
(3)
When no kind of resonance condition exists (this will be discussed latter) the
inclusion of the nonlinear coupling terms in the equations of motion will affect a little
these rapid motions, an effect which can be discussed perturbatively.
But there is also an another regime, characterized by some kind of slow space and
time variables, where the small cumulative effects of nonlinearity manifest and lead to a
modulation of the amplitude of the propagating excitation (Benjamin-Feir or modulation
instability) and the creation of a spatially localized excitation – the soliton. In the present
paper we are interested in this regime, and to discuss it we need an adequate mathematical
formalism. This is given by the asymptotic method of multiple scales (or reductive
perturbative) method [3].
If ε is a small parameter the slow variables are defined as
   ( nl  v g t ), t2   2 t, …
(4)
where v g is a group velocity to be specified latter. Then the separation of fast and slow
variables in the A-system is achieved assuming the following asymptotic expansion for a n
a n  e i ( knl t )   j A j ( , t2 ,...) .
(5)
j 1
As concerns the P-system one assumes that it depends only on the slow
variable (4) (the acoustical branch is assumed to be "slaved" to the optical branch). As
4
the right hand side of the second equation (1) is of ε order, an asymptotic expansion
2
for ρ starting with the ε term has to be used
   j Pj ( , t 2 ,...)
(6)

j 2
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SOLITON SIGNATURE IN THE INFRARED SPECTRA OF NONLINEAR QUASI-ONEDIMENSIONAL…
All these expansions are now introduced in (1) which has to be satisfied in any
1
order of ε. For the A-system in order ε the equation is satisfied if ω(k) is given by
2
the dispersion relation (3) and in order ε if the group velocity is given by
v g =dω/dk=ω 1
(ω n =1/n! d n ω(k)/dk n ).
In order ε
3
the first amplitude A 1 has to satisfy the equation
i
A1
 2 A1 
 2
 P2 A1  0
t2
 2

(7)
coupling the A and P systems.
Turning now our attention to the P-system the discussion starts with the
4
ε order. With the notation c p =Ω 0 l for the phase velocity of the acoustical phonons
one obtains
(v g
2
2
 2 P2
 2  A1
 cp )

l
 2
M
 2
2
2
(8)
If v g ≠c p (no resonance condition) from (1) one gets
P2  

w
1
1
2
A1
2
vg
cp
(9)
2
which introduced in (7) gives
i
A1
 2 A1
2
 2
 q A1 A1  0
2
t2

q
2
w
1
(10)
1
.
2
vg
cg
2
This is the complete integrable nonlinear Schrödinger equation (NLS). When
ω 2 q>0 (focusing case) the NLS equation admits soliton solutions [4], [5].
4
In the next order, ε , for the A-system one obtains
i
A2
 2 A2
2

2
 2
+ q ( A1 A2  2 A1 A2 ) 
2
t 2


i
A1
 3 A1
2 A1
2 A1
 i 3
 i ( A1
 A1
)
3
t3



(11)
  2q 2 v g /( c p 2  v g 2 )
165
D. GRECU, ANCA VIŞINESCU
This is an inhomogeneous linear equation for A 1 . In the left hand side (lhs)
one recognizes the linearized NLS equation (around a solution A 1 of the NLS
equation), and the inhomogeneity in the right side (rhs) depends only on A 1 . Possible
secular behaviours appear any time if in the rhs of (11) are identified solutions of the
linearized NLS equation (the symmetries of the NLS equation). They can be eliminated
if the t 3 dependence of A 1 is determined from the next equation in the NLS hierarchy
(the complex mKdV equation), namely
-
A1
 3 A1
2 A1
+ω 3 (
+6c A1
) =0
3
t3


c=
(12)
2
1
2w 2 1  v g 2 / c g 2
The remaining inhomogeneous linear equation for A 2 will be now free of
secularities and if A 1 is the one soliton solution it can be exactly integrated by a
suitable transformation [6].
The previous discussion is valid if v g ≠c p . Now let us discuss the case when
v g =c p and show that this corresponds to a long wave – short wave resonance
situation. Let us consider the resonance of a triad
k 1 =k 2 +k 3
ω 1 =ω 2 +ω 3
(13)
and suppose that k 1 and k 2 are very close, k 1 =k+εk, k 2 =k- εk. Then the wave number
triad is automatically satisfied if k 3 =2 εk , while the frequency triad is satisfied if
ω 3 = ω(k+εk)- ω(k-εk) ≈ 2εk dω/dk = k 3 dω/dk
(14)
But ω 3 /k 3 =c p is the phase velocity of the long wave, while dω/dk=v g is the
group velocity of the short one. Then this kind of resonance takes place any time when
c p = v g , a condition which is easily achieved if the short waves belong to an optical
branch, and the long waves to an acoustical one. When this situation happens the
4
equations (8) is no longer valid as the ε order is automatically satisfied. The lhs is
5
different of zero starting with the ε order, and this can be achieved if the asymptotic
expansion (5) is modified, namely
a n =e
i ( knl t )
  j Aj ( , t2 ,...)
(15)
j 1
the definition of the slow variables remaining unchanged. Then it is easily seen that in
5
the order ε the P-system has to satisfy the equation
166
SOLITON SIGNATURE IN THE INFRARED SPECTRA OF NONLINEAR QUASI-ONEDIMENSIONAL…
P
 2  A1
i 2 
l
t 2
2 Mv g

2
(16)
The equations (7) and (16) constitute an integrable system know as ZakharovBenney equations. [5], [7], [8]. Its soliton solution is only slightly modified from the
soliton solution of the NLS equation. Higher order approximations can be worked as
before; more details will be presented elsewhere.
3. CRYSTALINE ACETANILIDE CASE
Different types of collective excitations in quasi 1-D molecular crystals
consisting of weakly interacting molecules (groups of atoms) can exist. The most
studied are the excitons. They can be generated by light and characterize many
processes of absorbtion and luminiscence. In the beginning part of this paper a discussion
of another kind of collective excitation – the soliton – was given. It is supposed that
they can be excited in chemical reactions and by other local actions, playing an
important role in biological processes. But they cannot be excited directly by light.
But there is a very notable exception, namely the crystalline acetanelide
(CH 3 CONHC 6 H 5 ) (ACN), an organic quasi-one-dimensional crystal, whose structure
is very similar with the structure of typical polypeptide chain in protein. In the early
1
seventies Careri et al. found an unconventional amide I band at 1650 cm (shifted by
about 15 wave numbers from the primary band) at low temperature and assigned it to a
soliton type excitation in this system [1], [9], [10]. To explain the experimental results
a Davydov type model was used, in which the amide I oscillations are interacting
nonlinearly with a low frequency optical mode (describing the hydrogen bonded proton
motion). Treating the low frequency optical mode as an Einstein oscillator the
Hamiltonian is given by
H=
(17)
h n - JV




h 0 =[E 0 +λ 0 (b n +b n )]B n B n +ћω 0 b n b n
V=



( B n B n 1 + B n B n 1 )
(18)
(19)

Here B n (B n ) is the creation (annihilation) operator for an amide I oscillation

b n (b n ) the creation (annihilation) operator of the Einstein oscillator of frequency ω 0 , λ 0
the coupling constant of the mode-mode interaction and J the dipole-dipole interaction
energy between two amide I oscillations. In the ACN case J is small compared with
λ 0 , so the problem can be solved by using a perturbation theory in J. The unperturbed
problem is described by the energy operator of the "displaced harmonic oscillator" for
which exact eigenvalues and eigenfunctions are known [11]. A Bloch function can be
now written for the perturbed problem and a first order estimate of the energy gives
167
D. GRECU, ANCA VIŞINESCU
0
(
)
2
E ( k )  E0  0  2 J cos k e  .
 0
2
0
(20)
In an infrared experiment only k=0 states are observed. The peak of 1650
cm corresponds to a k=0 soliton type excitation (the name "soliton" is used in a
wider sense of a spatially localized solution) having the binding energy
1
0
EB 
(
)
0 2
 2 J [1- e  ]

2
0
(21)
One of the striking property of this peak is its strong temperature dependence,
2
the absorbtion strength varying as exp(-βT ). This fact is due to the interaction with
the acoustic phonon bath [1], [11]-[13]. In conclusion the appearance and the
1
temperature dependence of the experimentally observed amide I band at 1650 cm in
ACN can be explained along the lines proposed by Davydov.
Acknowledgments Helpful discussions with Dr. A.S. Cârstea are kindly
acknowledged. This work was supported by the Ministry of Education ads Research
under the grant nr. 5194/6008.
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