Page 1

Submitted to J. Chem. Phys.

Energy relaxation of the amide-I mode in hydrogen-bounded peptide units : a route

to conformational change

Vincent Pouthier∗

Institut UTINAM, Universit´ e de Franche-Comt´ e,

UMR CNRS 6213, 25030 Besan¸ con cedex, France.

(Dated: November 7, 2007)

A one-site Davydov model involving a C=O group engaged in a hydrogen bond is introduced

to study the amide-I relaxation due to Fermi resonances with a bath of intramolecular normal

modes. In the amide-I ground state, the hydrogen bond behaves as a harmonic oscillator whose

eigenstates are phonon number states. By contrast, in the amide-I first excited state, the hydrogen

bond experiences a linear distortion so that the system eigenstates are superimpositions of number

states. By assuming the hydrogen bond in thermal equilibrium at biological temperature, it is

shown that the amide-I excitation favors the population of these excited states and the occurrence

of coherences. Due to the interaction with the bath, the vibron decays according to an exponential

or a biexponential law depending on whether the Fermi resonance is wide or narrow. Therefore, each

excited state relaxes over a set of number states according to specific pathways. The consequence is

twofold. First, the relaxation leads to a redistribution of the number state population which differs

form the initial Boltzmann distribution. Then, it allows for coherence transfers so that, although

the vibron has disappeared, the hydrogen keeps the memory of its initial distortion and it develops

free oscillations.

PACS numbers:

I.INTRODUCTION

In living systems, the energy released by the hydroly-

sis of adenosine triphophate (ATP) is a universal energy

source allowing many biological processes such as enzyme

catalysis, cell motion, active ion transport or muscle con-

traction. This biological activity results from the trans-

duction of the released energy into a mechanical work

which usually originates from a conformational change of

the proteins that perform the required task. Although in

some cases the final conformation has been determined,

the detailed mechanisms for both energy transduction

and conformational change remain mysterious.

To elucidate this puzzle, a recent attention has been

paid by Cruzeiro and co-workers to apply the so-called

vibrational excited states hypothesis [1–4]. It is based

on the fact that the first step in protein function is the

storage of the released energy in the vibrational excited

states of some groups of the protein. This idea, which can

be traced back to the seminal works of McClare [5] in the

1970s, has been improved by Davydov and co-workers to

explain bioenergy transport in α-helices (see for instance

Ref. [6–8]).

At the present time, both theoretical and experimental

evidences suggest that energy flow in α-helices may result

from a polaron mechanism [11–23]. The released energy

is responsible for the excitation of the amide-I mode of a

peptide group which mainly involves the stretching vibra-

tion of the C=O group. This resonant coupling is prob-

ably mediated by the intermediate vibrational excitation

∗Electronic address: vincent.pouthier@univ-fcomte.fr

of water. Due to the dipole-dipole coupling, the amide-I

vibration delocalizes along the helix leading to the occur-

rence of vibrational excitons called vibrons. The polaron

formation originates from the strong coupling between

the vibrons and the phonons describing the vibrations of

the hydrogen bonds which stabilize the helix backbone.

Since the vibron bandwidth is lower than the phonon

cutoff frequency, the quantum behavior of the phonons

plays a crucial role and the nonadiabatic strong coupling

regime is reached. The creation of a vibron is thus ac-

companied by a virtual cloud of phonons describing a

localized lattice distortion which follows instantaneously

the vibron. The vibron dressed by the lattice distortion

forms the small polaron.

Although the coherent nature of small polarons is cer-

tainly lost at biological temperature [24–27], the Davy-

dov model is suitable for explaining how the energy re-

leased at an active site can propagate to other regions of

the protein. Nevertheless, it is unable to describe the en-

ergy relaxation of the amide-I mode since it conserves the

number of amide-I quanta. As pointed out by Cruzeiro

and co-workers [2–4], this weak point of the model is

fundamental because the energy released by the decay of

the amide-I mode can be used by the protein to realize a

conformational change.

The vibrational energy relaxation in proteins is very

fast and the amide-I lifetime is typically about 1.5 ps.

Indeed, time resolved spectroscopy of myoglobin revealed

that this lifetime varies from 1.8 ps at 10 K to 1.3 ps at

310 K [28, 29]. Note that Austin and co-workers have

reported a lifetime of about 2.7 ps. Moreover, they ob-

served both a blue shifted peak and a red shifted self-

trapped state whose lifetime at low temperature is 15 ps

and 30 ps, respectively [30, 31]. Pump-probe and hole

burning spectroscopies applied to three small globular

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peptides showed that the amide-I relaxation of all inves-

tigated peptides occurs over a similar time scale of about

1.2 ps [32, 33]. In addition, the 2D spectroscopy of ala-

nine based 21-residue revealed a lifetime lower than 5 ps

at room temperature [34]. In the same way, the amide-

I mode decays after about 2 ps at 93 K in Acetanilide

(ACN), a molecular crystal whose structural properties

are similar to those of α-helices [35]. Similarly, time re-

solved spectroscopies were applied to N-methylacetamide

(NMA), a model compound for a single peptide unit (see

for instance Ref.[36]). It has been shown that the amide-

I mode decays according to a biexponential law which

exhibits a very fast energy transfer over a time scale of

about 0.2 ps.

These experimental features cannot be explained in

terms of a direct energy transfer between the amide-I

mode and the solvent or the surrounding. They clearly

suggest that the amide-I lifetime results from intramolec-

ular energy redistribution due to the anharmonic cou-

pling with intramolecular normal modes whose displace-

ments are strongly localized on the C=O group. This sce-

nario has been corroborated by recent theoretical anal-

ysis. Indeed, the low temperature molecular dynamics

simulation of myoglobin revealed that a fast energy trans-

fer between the amide-I mode and a lower frequency

mode is mediated by Fermi resonances [37]. Similarly,

the numerical simulation of the relaxation in NMA shows

that the energy redistribution involves the Fermi reso-

nance between the amide-I mode and either the overtone

of a high frequency mode or a combination band between

two high frequency modes [38].

In that context, the aim of the present paper is to ad-

dress a comprehensive theory to describe how the amide-I

relaxation, due to the coupling with a set of intramolec-

ular normal modes, can modify the quantum state of a

conformational degree of freedom of a protein. To pro-

ceed, we consider a one-site Davydov model in which a

single C=O group is engaged in a hydrogen bond. There-

fore, the hydrogen bond plays the role of the conforma-

tional degree of freedom since hydrogen bonding is a fun-

damental element in protein structure and function. For

instance, it stabilizes the helical structures of α-helices

and it participates in the formation of β-turns allowing

the reversal of polypeptide chains. Therefore, breaking a

single hydrogen bond may impair the stability of a pro-

tein. Although this model is rather unrealistic to describe

the dynamics of a real protein, it gives a simple approach

to clearly understand how energy relaxation provides a

route to a conformational change. A more realistic model

will be presented in forthcoming works.

Note that the present work has been inspirited by the

recent developments of Cruzeiro and co-workers [2–4].

Nevertheless, three points are fundamentally different.

First, we develop a formalism at biological temperature.

Then, instead of the semi-classical approach in which the

hydrogen bond is treated classically, we use a full quan-

tum description. Finally, in the spirit of the experimental

results, the amide-I decay results from a coupling with

a thermal bath. In a marked contrast, Cruzeiro and co-

workers have introduced an extra term in the Davydov

model to break its conservative nature. This term does

not characterize a relaxation mechanism and it corre-

sponds to a time independent linear perturbation of the

amide-I mode.

The paper is organized as follows. In section II, the

one-site Davydov model is described and the correspond-

ing Hamiltonians are defined. In Section III, the amide-I

relaxation is described in terms of the reduced density

matrix whose time evolution is characterized by a gen-

eralized master equation established in Sec.

equation is solved numerically in Section V where a de-

tailed analysis of the vibrational dynamics is performed.

Finally, these results are discussed and interpreted in Sec-

tion VI.

IV. This

II.MODEL AND HAMILTONIANS

In a general way, the dynamics of a protein exhibits

a tremendous complexity due to the rather large num-

ber of degrees of freedom. For instance, in α-helices, the

3D conformation is stabilized by the hydrogen bond be-

tween the carboxyl oxygen (CO) of an amino acid and

the amide hydrogen (NH) of a second amino acid that

is situated four residues ahead in the linear sequence

of the polypeptide chain. The helix is thus formed by

three spines of hydrogen-bonded peptide units connected

through covalent bonds. To overcome this difficulty, we

consider a simplified system involving only a single C=O

group engaged in a hydrogen bond. This system con-

tains a amide-I mode which behaves as a high frequency

oscillator described by the boson operators b†and b and

whose internal frequency and anharmonicity are labeled

ω0and A, respectively. This mode interacts with the low

frequency vibration of the hydrogen bond which is mod-

eled by a harmonic oscillator with frequency Ω and with

boson operators a†and a. The coupling, whose strength

is specified by the parameter ∆, accounts for the mod-

ulation of the amide-I frequency which depends linearly

on the coordinate of the hydrogen bond. In that context,

the Hamiltonian of the one-site Davydov model is defined

as (using the convention ? = 1)

HA= ω0b†b − Ab†b†bb + Ωa†a + ∆(a†+ a)b†b

A natural basis set to describe the dynamics of HAis

formed by the number states { |v,p? } in which v refers

to the number of amide-I quanta, also called vibrons, and

p defines the number of quanta connected to the vibra-

tion of the hydrogen bond. These latter quanta will be

called optical phonons so that the name vibron-phonon

system (VPS) will be used to describe the one-site Davy-

dov model. Therefore, for non vanishing ∆ values, HA

can be solved exactly by performing a Lang-Firsov trans-

formation [42].The corresponding eigenstates are ex-

pressed in terms of the number states as

(1)

|Ψvp? = θv|v,p?(2)

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3

where θ = exp(−(∆/Ω)(a†−a)) is the so-called dressing

operator. Its matrix elements are defined as [8]

?p|θ|¯ p? =

?

p!

¯ p!u¯ p−pe−u2

2L¯ p−p

p

[u2](3)

where u = ∆/Ω and Ln

Laguerre polynomials. The associated eigenenergies are

defined in terms of the small polaron binding energy

EB= ∆2/Ω as

m[x] stands for the generalized

ωvp= v(ω0− EB) − v(v − 1)(A + EB) + pΩ (4)

According to Eq.(2), the VPS eigenstates are phonon

number states when the amide-I mode is in its ground

state (v = 0). By contrast, when v > 0, the potential

energy of the hydrogen bond experiences a linear pertur-

bation. The corresponding eigenstates are linear super-

impositions of phonon number states and they describe

a distortion of the hydrogen bond.

In addition to the standard Davydov model, we in-

troduce a set of high frequency intramolecular normal

modes. Each mode is described by a harmonic oscillator

whose frequency is ωαand whose displacement is defined

in terms of the boson operators b†

modes form a thermal bath whose Hamiltonian is written

as

αand bα. These normal

HB=

?

α

ωαb†

αbα

(5)

The thermal bath contains the normal modes of the

biopolymer whose displacements are strongly localized on

the C=O group. These modes are thus responsible for the

amide-I relaxation which results from energy exchanges

mediated by the anharmonicity of the potential energy of

the system. Because the density of normal modes in the

frequency range of half the amide-I frequency is rather

large, even for small biomolecules [38, 40, 41], the anhar-

monicity facilitates the occurrence of Fermi resonances.

To mimic the effect of such resonances, we follow Gerber

et co-workers [39] and assume that the potential between

the amide-I and the thermal bath is represented by a sum

of separable pair coupling terms as

∆H =

?

α

χα(b†2

αb + b2

αb†)(6)

where χα denotes the strength of the coupling between

the αth bath mode and the amide-I vibration.

Finally, the full Hamiltonian H = HA+ HB+ ∆H

yields a rather simple model for the VPS dynamics. It

allows for an exact quantum treatment of both the amide-

I mode and the hydrogen bond vibration and it includes

the coupling with a bath of intramolecular normal modes.

In the next sections, this model Hamiltonian will be used

to characterize the energy relaxation of the amide-I mode

and the evolution of the hydrogen bond quantum state.

III.VIBRATIONAL DYNAMICS AND

REDUCED DENSITY MATRIX

In a general way, the vibrational dynamics of a molec-

ular system coupled to a complex bath is described by

the so-called reduced density matrix. In our previous

works, a special attention has been paid to characterize

the vibron dynamics, only [25–27]. The reduced density

matrix was determined by performing a trace over the hy-

drogen bond degrees of freedom so that the phonons were

assumed to belong to the thermal bath. In the present

paper, a fully different approach is used since we intend

to understand the way the quantum state of the hydro-

gen bond is modified when the amide-I mode relaxes.

Therefore, the partial trace is done over the intramolec-

ular normal modes, only, and all the information that is

desired on the hydrogen bond can be extracted form the

reduced density matrix.

In that context, by performing a trace over the thermal

bath, the representation of the reduced density matrix in

the VPS eigenstate basis is written as

σv¯ v

p¯ p(t) = ?Ψvp|TrB[e−iHtρeiHt]|Ψ¯ v¯ p?

where ρ denotes the initial density matrix of the whole

system ” VPS + thermal bath ”.

Without any perturbation, the system is in thermal

equilibrium at the biological temperature T. Since ω0≈

1660 cm−1, the vibrons cannot be thermally excited.

However, this is no longer the case for the other degrees

of freedom whose true eigenstates are not well defined.

As a result, a statistical average is required so that the

optical phonons and the thermal bath are described by

using standard Boltzmann distributions ρphand ρB, re-

spectively.

To study the amide-I relaxation, we consider that the

energy released by the hydrolysis of ATP acts as an ex-

ternal source which brings the system in a state out of

equilibrium. This source is resonantly coupled with the

amide-I mode, only, and it does not affect significantly

the remaining degrees of freedom. Moreover, it is sup-

posed to act during a very short time scale so that the

population of the amide-I mode increases adiabatically

without any change in the quantum states of both the op-

tical phonons and the thermal bath. Consequently, the

released energy allows for the occurrence of v0 vibrons

on the amide-I mode. Note that since an ATP molecule

releases 0.49 eV under normal physiological conditions,

the vibron number is typically equal to v0= 1 or v0= 2.

Therefore, the whole system is prepared in an initial state

specified by the full density matrix

(7)

ρ = ρA⊗ ρB

(8)

where ρA= |v0??v0|⊗ρphdefines the VPS density matrix.

Because ρA does not commute with HA, it does not

refer to a statistical mixture of the VPS eigenstates. It

characterizes the presence of v0amide-I quanta, the hy-

drogen bond being described by a statistical mixture of

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4

number states. In the VPS eigenstate basis, the con-

sequence is twofold. First, the initial population of the

eigenstates with v0 vibrons differs from the Boltzmann

distribution. Then, ρAexhibits non vanishing coherences

which result from the fact that |Ψv0p? is a superimposi-

tion of the number states |v0¯ p?. Consequently, ρA has

diagonal and non diagonal matrix elements defined as

(see Eq.3)

?Ψv0p|ρA|Ψv0¯ p? =

?

p!

¯ p!(1 − w)¯ p−p+1wp(−u)¯ p−p

× e−u2(1−w)L¯ p−p

where u = v0∆/Ω and w = exp(−Ω/kBT).

From the knowledge of the reduced density matrix, we

can compute in principle the average value of any ob-

servable required to characterize the VPS dynamics. In

that context, to get information about the decay of the

amide-I mode, we first define < v(t) >= Trσ(t)b†b as the

average vibron number at time t. It is written as

p

[−u(1 − w)2/w](9)

< v(t) >=

?

vp

vσvv

pp(t) (10)

Similarly, the average phonon number at time t is de-

fined as < p(t) >= Trσ(t)a†a. To calculate < p(t) >,

let us remind that, depending of the vibron number, the

VPS eigenstates characterize a distortion of the hydro-

gen bond. The boson operators, a†and a, experience a

translation so that the phonon number matrix elements

are expressed as

?Ψvp|a†a|Ψ¯ v¯ p? = δv¯ v?p|

?

a†−v∆

Ω

??

a −v∆

Ω

?

|¯ p? (11)

By performing a trace according to Eq.(7), < p(t) > is

thus defined as

< p(t) >=

?

(12)

?

vp

p +

?v∆

Ω

?2?

σvv

pp(t) −2v∆√p + 1

Ω

ℜσvv

pp+1(t)

Finally, to describe the modification of the distortion

of the hydrogen bond, let us introduce the reduced hy-

drogen bond coordinate as < u(t) >= Trσ(t)(a†+ a).

After evaluating the phonon operators in the eigenstate

basis, it is expressed as

< u(t) >=

?

vp

−2v∆

Ω

σvv

pp(t) + 2?p + 1ℜσvv

pp+1(t) (13)

Eqs.(10), (12) and (13) show that the reduced density

matrix is the central object of the present study.

knowledge allows us to characterize the dynamics of both

the amide-I mode and the hydrogen bond. Nevertheless,

these equations reveal that only particular elements are

required. Indeed, the desired observables only depend on

Its

the population of the eigenstates σvv

coherences σvv

pp+1(t). In the following of the text, these

elements will be defined as

pp(t) and on specific

Pvp(t) = σvv

Qvp(t) = σvv

pp(t)

pp+1(t)(14)

In that context, the next section is devoted to the deriva-

tion of a Generalized Master Equation (GME) to study

the time evolution of Pvp(t) and Qvp(t).

IV. MASTER EQUATION

In this section, a standard projector method is used

to derive a GME for the reduced density matrix (see for

instance Refs. [43–47]). To proceed, two simplifying as-

sumptions are invoked. First, the coupling between the

amide-I mode and the thermal bath is supposed to be

sufficiently weak to perform a second order perturbation

theory with respect to ∆H. Second, we assume that the

bath exhibits correlations over a very short time scale so

that the Markovian limit is used.

Within these assumptions, the derived GME mixes in

a complicated manner both diagonal and non diagonal

matrix elements. To overcome this difficulty, we apply

the secular approximation which allows us to separate

the dynamics of matrix elements whose time evolution is

rather different [45]. Indeed, the dynamics of σv¯ v

typically governed by the energy difference |ωvp− ω¯ v¯ p|.

Since the VPS does not exhibit degenerate eigenstates,

the populations are clearly uncoupled with the coher-

ences whose smallest frequency is equal to Ω. Moreover,

the VPS supports two kinds of coherences. First, the

coherences involving states referring to the same vibron

number (v = ¯ v) evolve typically according to a frequency

of about a few times the phonon frequency (|p − ¯ p|Ω).

By contrast, coherences between states characterized by

a different vibron number (v ?= ¯ v) involve high frequen-

cies about a few times the amide-I frequency. As a re-

sult, these two kinds of coherences evolve almost inde-

pendently.

In that context, after some algebraic manipulations,

the GME yields two independent equations to character-

ize the time evolution of both the populations Pvp(t) and

the coherences Qvp(t). The dynamics of the populations

is governed by a standard Pauli Master equation written

as

dPvp(t)

dt

¯ v¯ p

p¯ p(t) is

=

?

W¯ v¯ p→vpP¯ v¯ p(t) − Wvp→¯ v¯ pPvp(t)(15)

where Wvp→¯ v¯ p denotes the rate for the transition from

|Ψvp? to |Ψ¯ v¯ p? due to the coupling with the thermal bath.

It is defined as

Wvp→¯ v¯ p = vδ¯ vv−1|?¯ p|θ|p?|2g1

?ωvp− ω¯ v¯ p

2

?ω¯ v¯ p− ωvp

?

(16)

+ (v + 1)δ¯ vv+1|?¯ p|θ†|p?|2g0

2

?

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5

where gi(ω) is the coupling distribution expressed in

terms of the Bose number pB(ω) = (exp(ω/kBT) − 1)−1

as

gi(ω) =

?

α

2πχ2

αδ(ω − ωα)(pB(ω) + i)2

(17)

Similarly, the time evolution of the coherences Qvp(t)

is controlled by the equation

dQvp(t)

dt

= (iΩ − Γvp)Qvp(t) +

?

¯ v¯ p

Tv¯ v

p¯ pQ¯ v¯ p(t)(18)

where Γvp= (γvp+γvp+1)/2 is expressed in terms of the

full relaxation rate γvpdefined as

γvp=

?

¯ v¯ p

Wvp→¯ v¯ p

(19)

In Eq.(18), Tv¯ v

diated by the thermal bath. It is defined as

p¯ pdescribes the transfer of coherences me-

Tv¯ v

p¯ p = vδ¯ vv−1?¯ p + 1|θ|p + 1??p|θ†|¯ p?g0

?ωvp− ω¯ v¯ p

?ω¯ v¯ p− ωvp

2

?

(20)

+ (v + 1)δ¯ vv+1?¯ p + 1|θ†|p + 1??p|θ|¯ p?g1

2

?

Eqs.(15) and (18) show that the bath strongly modifies

the dynamics of both the amide-I mode and the hydrogen

bond. Indeed, it is responsible for incoherent transitions

between eigenstates involving a different vibron number.

These transitions favor a redistribution of the popula-

tions of the eigenstates which is described by the Pauli

Master equation Eq.(15). Consequently, a vibron can be

absorbed or emitted by the bath according to the transi-

tion rates Eq.(16). However, during such a mechanism,

the phonon number fluctuates since the potential energy

experiences by the hydrogen bond depends on the vibron

number. For instance, if the VPS is initially in the state

|Ψ1p?, the hydrogen bond is described by the superim-

position of number states θ|p?. The absorption of the

vibron projects the amide-I in its ground state in which

the hydrogen bond reaches a number state. From quan-

tum mechanics, this final number state is not well defined

since the hydrogen bond can reach all the number states

|¯ p? which belong to the initial superimposition θ|p?. Such

a process can only be characterized by the corresponding

quantum probability |?¯ p|θp?|2which is a measure of the

overlap between the initial state and the final state.

Similarly, the influence of the thermal bath on the co-

herences is twofold. First, since the bath induces tran-

sitions, each eigenstate has a finite lifetime. Therefore,

any coherence will be destroyed in the long time limit due

to the decay of the eigenstates involved in a superimpo-

sition. The bath is thus responsible for the well-known

dephasing mechanism which is described by the dephas-

ing constant Γvp. Then, the thermal bath allows for a

transfer of coherences. To understand this feature more

clearly, let us consider the situation in which the initial

state is a superimposition of |Ψ1p? and |Ψ1p+1?. Under

the coupling with the bath, these two eigenstates can, for

instance, decay in the ground state of the amide-I mode.

During this process, the hydrogen bond has a non vanish-

ing probability to reach a state formed by the superimpo-

sition of two number states |¯ p? and |¯ p + 1?. As shown in

Eq.(20), the corresponding probability involves the prod-

uct between the overlaps ?¯ p|θ|p? and ?¯ p+1|θ|p+1?. Con-

sequently, this mechanism clearly shows that the initial

coherence Q1p(t) can be partially transfered to the co-

herence Q0¯ p(t).

As shown in Eqs.(16) and (20), both incoherent transi-

tions and coherence transfers result from the competition

between two mechanisms. The first mechanism originates

from the overlap between the final state and the initial

state of the hydrogen bond during the vibron relaxation.

It involves the matrix elements of the dressing opera-

tor and it participates in the selection of the eigenstates

which are coupled due to the interaction with the thermal

bath. The second mechanism corresponds to the ability

of the bath to exchange energy with the amide-I mode via

the Fermi resonances. It is characterized by the coupling

distributions gi(ω), i = 0,1 (Eq.(17)).

measures the ability of the bath at thermal equilibrium

to supply the energy 2ω to the amide-I mode, whereas

g1(ω) accounts for its ability to absorb the energy 2ω.

At this step, the knowledge of the transition rates and

of the coherence transfers allows us to solve numerically

the GME Eqs.(15) and (18). The corresponding solutions

can thus be used to evaluate the different observables de-

fined in Eqs.(10), (12) and (13). This numerical proce-

dure, illustrated in the following section, will provide a

complete understanding of the VPS dynamics.

Indeed, g0(ω)

V. NUMERICAL RESULTS

In this section, the previous formalism is applied to

study the vibrational dynamics of the VPS for v0= 1. To

proceed, typical values for the parameters of the Davydov

model are introduced [16]. The frequency and the anhar-

monicity of the amide-I mode are fixed to ω0 = 1660

cm−1and A = 8.0 cm−1, respectively.

phonon frequency is equal to Ω = 50 cm−1and the

temperature is fixed to T = 310 K. Finally, the vibron-

phonon coupling ∆ will be considered as a free parameter

varying form 0 to 30 cm−1.

To characterize the interaction with the thermal bath,

the coupling distribution gi(ω) is modeled by a gaussian

law centered around ωc= ω0/2 and written as

The optical

gi(ω) = (pB(ω) + i)

G

√παexp

?

−

?ω − ωc

α

?2?

(21)

where G = 400 cm−2and where α defines the width of the

Fermi resonance. The coupling distribution is illustrated

in Fig. 1 for the transition between the ground state and

Page 6

6

g1(?) (cm-1)

0

2

4

6

8

10

12

?=20?cm-1

?=40?cm-1

?=75?cm-1

(a)

p-p

-10-8 -6-4-202468 10

2x103g0(?) (cm-1)

0

2

4

6

8

10

12

(b)

FIG. 1: Coupling distributions (a) g1(ω) and (b) g0(ω) vs

p − ¯ p for ω = (ω1p− ω0¯ p)/2.

the first excited state of the amide-I mode. Such a tran-

sition involves the states |Ψ1p? and |Ψ0¯ p? whose energy

difference is equal to 2ω = ω0−EB+(p−¯ p)Ω. Therefore,

Fig. 1 clearly shows that g0(ω) is about three orders of

magnitude smaller than g1(ω). The bath is thus unable

to supply energy to the amide-I mode because it occu-

pies its ground state at biological temperature. There-

fore, it plays the role of an empty reservoir only able to

absorb the energy released by the decay of the amide-

I mode. Moreover, this ability is entirely characterized

by the coupling distribution bandwidth α and it does

not significantly depend on the vibron-phonon coupling

strength ∆. As displayed in Fig. 1a, for small α val-

ues, the bath is very selective and it allows almost only

diagonal transitions p = ¯ p. By contrast, for large α val-

ues, the bath looses its selectivity. The Fermi resonance

is sufficiently large so that the thermal bath can absorb

the energy released during transitions in the course of

which the phonon number exhibits important variations.

For instance, for α = 75 cm−1, all the transitions with

|p − ¯ p| = 0,±1,±2,±3 and ±4 can be absorbed by the

thermal bath almost in a similar way.

In addition to the thermal bath selectivity, the inten-

sity of a transition depends on the overlap between the

VPS eigenstates involved in the relaxation pathway. This

overlap is described by the dressing operator θ (Eq.(3))

whose particular matrix elements |?¯ p|θ|p?|2are displayed

in Fig. 2. This figure describes how the distorted state

θ|p? is expanded over the different number states |¯ p?.

Although such an expansion is not symmetrically dis-

tributed around ¯ p = p, the weights of |p+i? and |p−i? are

p

051015202530

0.0

0.2

0.4

0.6

0.8

1.0

0.0

0.2

0.4

0.6

0.8

1.0

(a)

(b)

FIG. 2: Relevant matrix elements |?¯ p|θ|p?|2vs p for (a) ∆ =

5 cm−1and (b) ∆ = 25 cm−1. Full circles correspond to

diagonal terms ¯ p − p = 0. Open circles and full squares refer

to an average of the non diagonal terms ¯ p − p = ±1 and

¯ p − p = ±2, respectively.

usually similar, at least for both small p values and small

i values. For that reason, Fig. 2 accounts for a mean ef-

fect in which the contributions of |p+i? and |p−i? to θ|p?

have been averaged. Therefore, full circles correspond to

the diagonal matrix elements. By contrast, open circles

and full squares describe the average contribution of the

states |p ± 1? and |p ± 2?, respectively.

For small ∆ values, Fig. 2a shows that the expansion

of θ|p? mainly involves the number states |p? and |p±1?.

More precisely, for the p values typically smaller than 15,

|p? represents more than 70 % of θ|p? so that θ is almost

diagonal. By contrast, for larger p values, the weight

of |p ± 1? increases and both |p? and |p ± 1? participate

similarly to the expansion of θ|p?. For larger ∆ values,

the expansion of θ|p? is more complex since the weight of

each number state exhibits oscillations with respect to p

(Fig. 2b). Therefore, although θ remains almost diagonal

for p = 0 and p = 1, the weight of |p? becomes very small

when p ranges between 4 and 7. Then, it increases again

to reach a maximum for p = 15. Consequently, when

p ranges between 2 and 5, the expansion of θ|p? mainly

involves |p ± 1? whereas, when p is around 10, the non

diagonal matrix elements involving ¯ p = p ± 2 become

dominant.

At this step, the knowledge of both the coupling distri-

butions and the dressing operator, allows us to evaluate

the transition rates (Eq.(16)). The GME can thus be

solved numerically to determine to average vibron num-

Page 7

7

<v(t)>

0.0

1.0

0.2

0.4

0.6

0.8

1.0

?=5?cm-1

?=15?cm-1

?=25?cm-1

t?(ps)

0.02.5 5.07.510.0

<v(t)>

0.0

0.2

0.4

0.6

0.8

(a)

(b)

FIG. 3: < v(t) > vs time for (a) α = 75 cm−1and (b) α = 20

cm−1.

?i

0.0

0.2

0.4

0.6

0.8

1.0

? (cm-1)

05 10 15202530

?i(ps)

0

1

2

3

4

5

6

(a)

(b)

FIG. 4: Biexponential fit of < v(t) > vs ∆ (see the text). (a)

Parameters αi and (b) parameters τi for i = 1 and 2. Full

circles describe α1 (resp. τ1) for α = 75 cm−1, open circles

characterize α2 (resp. τ2) for α = 75 cm−1, full squares refer

to α1 (resp. τ1) for α = 20 cm−1and open squares describe

α2 (resp. τ2) for α = 20 cm−1. In Fig. 4b, the dashed lines

represent the theoretical value of the lifetime defined in the

discussion.

ber < v(t) > whose time evolution is displayed in Fig. 3.

Whatever the value of the parameters, < v(t) > decays

from its initial value over a few picoseconds to finally

vanishes. To characterize this decay, we first introduce a

phenomenological lifetime t1defined as < v(t1) >= v0/e.

Therefore, for a given α value, Figs. 3a and 3b reveal that

the lifetime increases with the vibron-phonon coupling ∆.

Similarly, it increases with α for a fixed ∆ value. For in-

stance, for α = 20 cm−1, t1is equal to 0.48 ps and 1.58

ps for ∆ = 5 cm−1and 25 cm−1, respectively. In the

same way, for α = 75 cm−1, t1increases form 1.70 ps for

∆ = 5 cm−1to 2.12 ps for ∆ = 25 cm−1. In other words,

the more α is small, the more the lifetime is sensitive

to the vibron-phonon coupling strength. Nevertheless, a

detailed analysis of Fig. 3 shows that a single parame-

ter t1is not sufficient to correctly describe the decay of

the average vibron number. Indeed, although < v(t) >

seems to decrease exponentially for small ∆ values, this

is no longer the case for a rather strong vibron-phonon

interaction.

To clarify this feature, the time evolution of < v(t) >

has been fitted according to a biexponential law defined

as

< v(t) >= α1e−t/τ1+ α2e−t/τ2

(22)

The behavior of the parameters αiand τi vs ∆ is illus-

trated in Fig. 4. For α = 75 cm−1, Fig. 4a reveals

that < v(t) > decreases almost exponentially whatever

the ∆ value. More precisely, an exponential decay occurs

for ∆ < 15 cm−1since α2almost vanishes whereas α1is

about unity. Nevertheless, for larger ∆ values, α2slightly

increases although it remains rather small. For instance,

for ∆ = 25 cm−1, α1 = 0.894 whereas α2= 0.106. As

displayed in Fig. 4b, the dominant exponential is char-

acterized by a lifetime τ1smaller than τ2indicating that

the weak correction to the exponential law accounts for

the long time behavior of the average vibron number. For

∆ = 25 cm−1, τ1= 2.00 ps whereas τ2= 3.67 ps.

A fully different behavior occurs for α = 20 cm−1. In-

deed, Fig. 4a reveals that the exponential decay takes

place for very small ∆ values, only. By contrast, when

∆ > 5 cm−1, the weight α1of the dominant exponential

decreases linearly with ∆ whereas α2 increases. There-

fore, for large ∆ values, α1and α2are very similar which

indicates that < v(t) > experiences a real biexponential

decay. Nevertheless, as illustrated in Fig. 4b, the two

components of this biexponential law are characterized

by a different lifetime. For instance, for ∆ = 25 cm−1,

τ1= 0.86 ps whereas τ2= 3.07 ps.

The behavior of the average phonon number < p(t) >

is illustrated in Fig. 5 for α = 75 cm−1(Fig. 5a) and

α = 20 cm−1(Fig. 5b). Whatever the value of the pa-

rameters, < p(t) > shows two distinct regimes. In the

short time limit, it first rapidly increases from its initial

value equal to the Bose number pB(Ω) = 3.827. Then,

< p(t) > shows damped oscillations so that it finally con-

verges to a constant value denoted p∞. The amplitude

of the short time oscillations increases nonlinearly with

Page 8

8

<p(t)>

3.8

4.0

4.2

4.4

4.6

4.8

t?(ps)

05101520

<p(t)>

3.8

4.0

4.2

4.4

4.6

4.8

(a)

(b)

?=5?cm-1

?=15?cm-1

?=25?cm-1

?=5?cm-1

?=15?cm-1

?=25?cm-1

FIG. 5: < p(t) > vs time for (a) α = 75 cm−1and (b) α = 20

cm−1.

∆. For instance, for α = 75 cm−1, the first maximum of

< p(t) > is equal to 3.863, 4.154 and 4.739 for ∆ = 5, 15

and 25 cm−1, respectively. By contrast, the frequency of

the damped oscillations is independent on both α and ∆,

and it reduces to the optical phonon frequency Ω. Note

that when compared with the time evolution of < v(t) >,

Fig. 5 clearly shows that the damping regime occurs dur-

ing the decay of the amide-I mode.

For α = 75 cm−1(Fig. 5a), the stationary value p∞is

greater than pBand it increases with ∆. In other words,

it is as if the temperature of the hydrogen bond was in-

creased due to the relaxation of the amide-I mode. There-

fore, by identifying p∞to a Bose number with an effective

temperature Te, Fig. 5a reveals that Te= 311.3, 320.8

and 332.6 K for ∆ = 5, 15 and 25 cm−1, respectively.

By contrast, for α = 20 cm−1(Fig. 5b), p∞ is rather

close to pB. In addition, although it increases with ∆ for

small ∆ values, p∞becomes lower than the Bose number

for ∆ = 25 cm−1. This surprising result indicates that

the hydrogen bond is cool down for a sufficiently strong

vibron-phonon interaction. Indeed, Te= 310.8 and 312.7

K for ∆ = 5 and 15 cm−1whereas it reaches Te= 307.1

K for ∆ = 25 cm−1.

The behavior of the effective temperature is illustrated

in Fig. 6 which displays the evolution of ∆T = Te− T

versus ∆. Whatever the α value, ∆T increases with ∆

for small ∆ values. It typically scales as ∆T ≈ z∆2ac-

cording to a parameter z which slightly increases with α.

Nevertheless, this is no longer the case for strong ∆ values

since two regimes take place. Indeed, for rather large α

values typically greater than 50 cm−1, ∆T still increases

? (cm-1)

0510 152025 30

?T (K)

-5

0

5

10

15

20

25

30

35

?=20?cm-1

?=30?cm-1

?=40?cm-1

?=50?cm-1

?=60?cm-1

?=70?cm-1

?=80?cm-1

FIG. 6: Variation of ∆T = Te− T vs ∆.

with ∆. For instance, for α = 80 cm−1and ∆ = 30 cm−1,

∆T reaches 28.92 K indicating a rather large warming up

of the hydrogen bond. By contrast, for smaller α values,

the initial increase remains until ∆T reaches a maximum

value whose amplitude and position depend on α. For in-

stance, for α = 20 cm−1, the maximum of ∆T is equal to

3.11 K and it takes place for ∆ ≈ 13 cm−1. By contrast,

for α = 50 cm−1, the maximum of ∆T occurs for ∆ ≈ 28

cm−1and it corresponds to a temperature difference of

about 14.12 K. Therefore, after the maximum has been

reached, ∆T slightly decreases with ∆. Nevertheless, for

α = 20 cm−1, this decay is stopped since ∆T reaches a

minimum after which it increases again. In that case, for

∆ > 20 cm−1, ∆T < 0 which indicates a cool down of

the hydrogen bond.

Finally, the time evolution of the hydrogen bond coor-

dinate < u(t) > is shown in Fig. 7. In that case, we have

verified that the behavior of < u(t) > is rather insensi-

tive to the α values. Therefore, only a single α value has

been considered, i.e. α = 75 cm−1. Fig. 7 reveals that

< u(t) > exhibits two regimes. In the short time limit,

it rapidly decreases from its initial value equal to zero.

Therefore, the coordinate shows oscillations whose ampli-

tude increases almost linearly with ∆. They take place

around a negative average value which decreases with ∆.

For instance, the first minimum of the coordinate is equal

to −0.36, −1.10 and −1.85 for ∆ = 5, 15 and 25 cm−1,

respectively. Nevertheless, as time pass, a second regime

occurs. Indeed, the negative average value of the oscilla-

tions increases to finally vanishes. Nevertheless, the os-

cillations remain and they show a sine dependence whose

frequency is equal to the optical phonon frequency. The

amplitude of these oscillations increases with ∆ and we

have verified that they typically behaves as 2∆/Ω.

VI.DISCUSSION

To discuss and interpret the previous observations,

we can take advantage of the numerical results to ob-

tain an almost exact solution of the GME. Indeed, as

shown in Fig. 1, the relaxation of the amide-I mode re-

Page 9

9

<u(t)>

-2

-1

0

1

2

<u(t)>

-2

-1

0

1

2

t?(ps)

05101520

<u(t)>

-2

-1

0

1

2

(a)

(b)

(c)

FIG. 7: < u(t) > vs time for (a) ∆ = 5 cm−1, (b) ∆ = 15

cm−1and (c)∆ = 25 cm−1and for α = 75 cm−1.

sults from Fermi resonances involving intramolecular nor-

mal modes whose frequency is localized around half the

amide-I frequency. In that context, the relevant modes

of the thermal bath cannot be thermally excited at bio-

logical temperature. The bath is thus unable to supply

energy to the amide-I mode so that it is just responsible

for vibron annihilation. Consequently, both the relax-

ation rates Eq.(16) and the coherence transfers Eq.(20)

are very asymmetric. Indeed, only the rates Wvp→v−1¯ p

describing a vibron annihilation take significant values

whereas the rates connected to a vibron creation almost

vanish. Similarly, the coherence transfer Tvv−1

neglected so that only contributions of the form Tvv+1

remain.

In that context, the GME reduces to a system of equa-

tions describing both population exchanges and coher-

ence transfers between the two subspaces which contain

the one-vibron and the zero-vibron eigenstates, respec-

tively. Therefore, it is straightforward to show that the

Pauli master equation Eq.(15) can be rewritten as

p¯ p

can be

p¯ p

∂P0p(t)

∂t

=

?

¯ p

W1¯ p→0pP1¯ p(t)

∂P1¯ p(t)

∂t

= −γ1¯ pP1¯ p(t)(23)

Eq.(23) leads to a rather simple interpretation of the

amide-I decay. Indeed, it shows that the excitation of a

vibron from the amide-I ground state yields a non vanish-

ing population P1p(0) of the VPS eigenstates. Due to the

coupling with the thermal bath, a populated eigenstate

|Ψ1p? relaxes over a set of number states |0¯ p? according

to a pathway specified by the corresponding relaxation

rate. Therefore, the population of each excited state ir-

reversibly decays according to an exponential law written

as

P1p(t) = e−γ1ptP1p(0)(24)

These decaying populations act as a source for the zero-

vibron eigenstate populations which increase as

P0p(t) =

?

¯ p

W1¯ p→0pγ−1

1¯ p(1 − e−γ1¯ pt)P1¯ p(0)(25)

In the long time limit, these populations converge to sta-

tionary values expressed as

P0p(∞) =

?

¯ p

W1¯ p→0pγ−1

1¯ pP1¯ p(0)(26)

Therefore, after the vibron has been absorbed by

the thermal bath, the hydrogen bond reaches a quasi-

equilibrium in which the population of the correspond-

ing phonon number states is defined by P0p(∞). Since

the vibron relaxation is accompanied by a fluctuation

of the phonon quantum numbers, this stationary popula-

tion differs from the Boltzmann distribution at biological

temperature. In other words, it is as if the redistribution

of the number state population originated form a modi-

fication of the hydrogen bond temperature, as observed

in the previous section.

Nevertheless, let us mention that this statistical de-

scription of the hydrogen bond is certainly transient. In-

deed, in real systems, the hydrogen bond is also coupled

with a thermal bath involving the low frequency modes of

the protein backbone as well as the low energetic motions

of the biological surrounding. These coupling, which have

been disregarded in the present work, will be responsible

for the return to the thermal equilibrium at biological

temperature. Nevertheless, the quasi-equilibrium char-

acterized by a modified effective temperature can be ob-

served over the time scale needed to reach the true equi-

librium. A similar effect has been reported by P. Hamm

and co-workers who have shown experimentally that the

energy released by the decay of the amide-I mode in ACN

reappears on a much slower time scale in the form of an

increase lattice temperature [35].

At this step, only a detailed analysis of the relaxation

rates will provide a clear understanding on the way the

population exchanges modify the behavior of the hy-

drogen bond. To proceed, let us discuss the nature of

the relaxation pathways. As shown in the previous sec-

tion, these pathways strongly depend on two parameters,

namely, the vibron-phonon coupling strength ∆ and the

coupling distribution bandwidth α. More precisely, they

result from the interplay between three fundamental ef-

fects. The first effect results from the initial population

Page 10

10

of the VPS excited states which specifies the nature of the

eigenstates involved in the relaxation. Form Eq.(9), we

have verified that the initial distribution P1p(0) slightly

depends on ∆. It corresponds to a discrete exponential

law similar to the Boltzmann distribution which basically

extends form p = 0 to p = 15. The second effect orig-

inates form the coupling distribution g1(ω) which mea-

sures the ability of the thermal bath to absorb the energy

released by the vibron decay. It strongly depends on α

(see Fig. 1). The last effect corresponds to the overlap

between an initial excited state and a final number state.

This effect, characterized by the matrix elements of the

dressing operator, strongly depends on ∆.

In that context, for small ∆ values, the results dis-

played in Figs. 1 and 2 show that the relaxation pathway

does not significantly depends on α and it mainly origi-

nates from the overlap effect. In that case, the dressing

operator is well described by its diagonal part, at least

for the p values selected by the initial population P1p(0).

Consequently, the relaxation mainly results from diag-

onal transitions |Ψ1p? → |0p? for which γ1p ≈ W1p→0p.

Eq.(26) shows that the stationary population is very close

to the initial population, i.e. P0p(∞) ≈ P1p(0), so that

the hydrogen bond keeps the memory of its initial lin-

ear distortion.Although the vibron has been annihi-

lated, the number state population is different from the

Boltzmann distribution. Therefore, it is straightforward

to show that the average phonon number is given by

p∞ = pB(Ω) + (∆/Ω)2, in a good agreement with the

numerical observations displayed in Fig. 5. This result

characterizes a small increase of the effective hydrogen

bond temperature. For instance, for ∆ = 5 cm−1, the

corresponding temperature is equal to 310.7 K in a close

agreement with the numerical value of the effective tem-

perature equal to 310.8 K and 311.3 K for α = 20 and

α = 75 cm−1, respectively.

For larger ∆ values, the relaxation pathway strongly

depends on the competition between α and ∆. Indeed,

for large α values, the relaxation pathway is still dom-

inated by the overlap effect. Nevertheless, as shown in

Fig. 2b, the nature of the transitions strongly depends on

the p values. For p = 0 and 1, the main pathway involves

diagonal transitions. When p ranges between 2 and 5,

the decay of the excited states results form non diagonal

transitions |Ψ1p+1? → |0p?. Then, for 5 < p < 11, non

diagonal transitions |Ψ1p+2? → |0p? are clearly the dom-

inant relaxation pathway. Finally, for the larger p values

selected by the initial population P1p(0), diagonal transi-

tions recur. As a consequence, we have verified that this

p dependence of the various pathways is responsible for a

strong modification of the stationary population P0p(∞).

Basically, when compared with P1p(0), the stationary dis-

tribution extends to larger p values. Therefore, it yields

an increase of the average phonon number which corre-

sponds to an increase of the effective hydrogen bond tem-

perature. Note that for very large α values, the thermal

bath looses entirely its selectivity so that γ1p≈ g1(ω0/2)

whereas W1¯ p→0p≈ |?p|θ|¯ p?|2g1(ω0/2). From Eq.(26), the

stationary population is thus approximately given by

P0p(∞) ≈

?

¯ p

|?p|θ|¯ p?|2P1¯ p(0) (27)

This equation allows us to evaluate the corresponding av-

erage phonon number which reduces to p∞ = pB(Ω) +

2(∆/Ω)2. For ∆ = 25 cm−1, the corresponding temper-

ature is equal to 346.1 K. This value overestimates our

numerical observation (Te= 332.6 K for α = 75 cm−1)

since, even large, a finite α value yields a finite selectivity

of the thermal bath.

Finally, for small α values, the nature of the relaxation

pathway is now dominated by the selectivity of the ther-

mal bath. Nevertheless, the influence of the overlap effect

cannot be ignored for the strong ∆ values. Indeed, when

the coupling distribution bandwidth is rather small, the

bath is mainly able to absorb the energy released during

a diagonal transition |Ψ1p? → |0p?. Such processes ef-

fectively occur for small p values although non diagonal

transitions |Ψ1p+1? → |0p? contribute slightly. However,

for larger p values, the overlap effect can drastically pre-

vent diagonal transitions. For instance, as shown in Fig.

2 for ∆ = 25 cm−1, the diagonal part of the dressing op-

erator almost vanishes when p ranges between 4 and 7. In

that case, the relaxation pathway involves the most prob-

able non diagonal transitions which are favored similarly

by both the bath selectivity and the overlap effect. For

the parametersused in our simulation, the resulting path-

way involves exclusively the transitions |Ψ1p+1? → |0p?

for 3 ≤ p ≤ 7. However, for larger p values, diagonal

transitions recur. Therefore, for small p values, P0p(∞)

is slightly greater than P1p(0). However, the singularity

of the relaxation pathway for 3 ≤ p ≤ 7 leads to a hole in

the stationary distribution since P0p(∞) ≈ P1p+1(0). In

that case P0p(∞) becomes smaller than P1p(0) so that the

stationary population clearly favors the low p values to

the detriment of the large p values. Consequently, in the

quasi-equilibrium, the average phonon number can be-

come smaller than pB(Ω) which characterizes a cool down

of the hydrogen bond. This feature has been observed in

the previous section for α = 20 cm−1and ∆ = 25 cm−1.

The knowledge of the relaxation pathways provides a

clear interpretation of the time evolution of the average

vibron number < v(t) > displayed in Fig. 3. By inserting

Eq.(24) into Eq.(10), < v(t) >=?

at time t as

pP1p(t) defines the

probability to observe the VPS in one of its excited states

< v(t) >=

?

p

e−γ1ptP1p(0)(28)

Eq.(28) reveals that < v(t) > is an average over the dif-

ferent p values of each exponential decay characterizing

a given relaxation pathway. This average is thus realized

according to the initial population P1p(0).

For small ∆ values, the full relaxation rate γ1pis a very

slowly decreasing function of p whatever the coupling dis-

tribution bandwidth α (see Fig. 8). As mentioned previ-

Page 11

11

p

051015 202530

?1p(cm-1)

0

2

4

6

8

10

12

?1p(cm-1)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

?=5?cm-1

?=15?cm-1

?=25?cm-1

(a)

(b)

FIG. 8: Variation of the full relaxation rate γ1p of the first

amide-I excited state vs p. (a) α = 75 cm−1and (b) α = 20

cm−1.

ously, this behavior originates from the relaxation path-

way which mainly refers to diagonal transitions whatever

p. Consequently, the different values taken by γ1pare well

represented by the average value γ defined as

γ =

?

p

γ1pP1p(0) (29)

In that context, Eq.(28) can be solved approximately by

using a cumulant expansion so that the average vibron

number reduces to < v(t) >= exp(−γt). As observed

numerically, it exhibits an exponential decay whose life-

time is given by T1= 1/γ. As illustrated by the dashed

line in Fig 4b, the ∆ dependence of T1is in a very good

agreement with the numerical results for small ∆ val-

ues. Indeed, T1is identical to τ1which characterizes the

lifetime connected to the dominant exponential of our

biexponential fit.

For large ∆ values, Fig. 8 clearly shows that two situ-

ations occur depending on α. For rather large α values,

we have shown that the relaxation pathway is dominated

by the overlap effect so that γ1pstill remains a slowly de-

creasing function of p. To understand more clearly this

feature, let us assume that g1(ω) behaves as a uniform

function over a finite size frequency range. The full re-

laxation rate is thus approximately written as

γ1p≈ g1(ω0/2)

∗

?

¯ p

|?¯ p|θ|p?|2

(30)

where the symbol ∗ denotes a sum over the allowed tran-

sitions, only. When α → ∞, all the transitions are al-

lowed and the sum occurring in Eq.(30) is equal to unity.

Therefore, γ1pbecomes p independent and it reduces to

its average value γ. Consequently the average vibron

number is still described by an exponential decay. How-

ever, for a large but finite α value, the sum in Eq.(30)

appears smaller than unity and it is a decreasing function

of p (see Fig. 8a). Therefore, γ is less representative of

the different γ1p values so that the cumulant expansion

fails in representing the behavior of < v(t) >. As illus-

trated by the dashed line in Fig. 4b, T1= 1/γ slightly

differs from τ1for large ∆ values and for α = 75 cm−1.

Finally, as displayed in Fig. 8b, a fully different behav-

ior occurs for small α values. In that case, the relaxation

pathway is dominated by the thermal bath selectivity

which mainly allows diagonal transitions. Consequently,

the p dependence of the full relaxation rate γ1p repro-

duces the behavior of the diagonal matrix elements of

the dressing operator. As illustrated in Fig.

square), it rapidly decays for small p values to finally ex-

hibits a kind of damped oscillation. In that context, the

average value γ is not at all representative of the behav-

ior of γ1p and the cumulant expansion cannot be used.

The average vibron number does not show an exponen-

tial decay but, as observed numerically, it behaves as a

biexponential function. Indeed, as illustrated in Fig. 8b

for ∆ = 25 cm−1, γ1p takes significant values for small

p values as well as for p = 14. It is typically about 6.2

cm−1for 0 ≤ p ≤ 2 whereas it reaches 1.7 cm−1for

p = 14. These two contributions dominate the behav-

ior of < v(t) > whose decay mainly involves two life-

time parameters about 0.85 ps and 3.12 ps, respectively.

These values are in a close agreement with our numeri-

cal fit whose lifetime parameters were τ1= 0.86 ps and

τ2= 3.07 ps.

8b (full

Note that such a biexponential decay has been ob-

served experimentally in different systems such as NMA

[36] and small globular peptides [32]. Nevertheless, its

origin has not been clearly established and it has been

suggested that it may result from the coupling between

the amide-I first excited state and one or several dark

states. In our approach, a fully different point of view is

used since the excitation of the amide-I mode induces the

population of a set of excited VPS eigenstates. Therefore,

when the Fermi resonance is wide, the amide-I popula-

tion decreases exponentially since all the excited states

relax over a similar time scale. By contrast, for a rather

narrow Fermi resonance, the relaxation of the excited

states involves different time scales so that the amide-I

population decreases according to a biexponential law.

Let us now focus our attention on the time evolution of

the coherences. By using the assumptions invoked to sim-

plify the Pauli master equation, it is straightforward to

show that the evolution of the coherences is governed by

a system of two equations whose solutions are expressed

as

Page 12

12

Q0p(t) =

?

¯ p

T01

p¯ pΓ−1

1¯ p(1 − e−Γ1¯ pt)eiΩtQ1¯ p(0)

Q1¯ p(t) = e−Γ1¯ pteiΩtQ1¯ p(0)(31)

Eq.(31) can be interpreted as follows. Initially, the hy-

drogen bond at thermal equilibrium is described by a

statistical mixture of number states. Therefore, the cre-

ation of a vibron induces a linear distortion of the hy-

drogen bond whose eigenstates become superimpositions

of these number states. Consequently, the VPS exhibits

non vanishing coherences Q1p(0). Nevertheless, as dis-

cussed previously, each excited state |Ψ1p? relaxes over

a set of number states |0¯ p? due to the coupling with the

thermal bath. The consequence is twofold. First, because

each excited state has a finite lifetime, the coherences in

the one-vibron subspace irreversibly tend to zero. Then,

starting form the superimposition of two excited states,

the hydrogen bond has a non vanishing probability to

reach a final state involving the superimposition of two

number states. In other words, the coherences in the

excited subspace play the role of a source for the coher-

ences in the zero-vibron subspace. These mechanisms

take place during the annihilation of the vibron so that,

in the long time limit, no coherence remains in the ex-

cited subspace. By contrast, in the amide-I ground state,

the hydrogen bond supports coherences which behave as

a sine function whose frequency is equal to the optical

phonon frequency.Consequently, although the vibron

has been absorbed by the thermal bath, the hydrogen

bond keeps the memory of its initial distortion. It reaches

a quasi-equilibrium which does not refer to a statistical

mixture of number states since it supports coherences.

In that context, the knowledge of both the popula-

tions and the coherences allows us to evaluate the time

dependence of the average phonon number. By inserting

Eqs.(24), (25) and (31) in Eq.(12), we obtain

< p(t) > =

?

?

?

p¯ p

pW1¯ p→0pγ−1

1¯ pP1¯ p(0)(1 − e−γ1¯ pt)

+

p

(p + (∆

Ω)2)e−γ1ptP1p(0) (32)

−

p

2∆

Ω

?p + 1e−Γ1ptQ1p(0)cos(Ωt)

The first and the second terms in the right hand side of

Eq.(32) define the contributions to < p(t) > of the pop-

ulations of the zero-vibron and of the one-vibron eigen-

states, respectively. By contrast, the last term character-

izes the influence of the coherences. Therefore, Eq.(32)

clearly shows that < p(t) > exhibits damped oscillations

in a very good agreement with the numerical results (see

Fig.5). As mentioned previously, it finally converges to

the average phonon number p∞connected to the station-

ary distribution P0p(∞). The coherences only influence

the short time limit of the phonon number. Indeed, when

t << 1/Ω, < p(t) >≈ pB(Ω) because the second and the

last term in the right hand side of Eq.(32), which have

the same order of magnitude, have a different sign. How-

ever, this is no longer the case when t = π/Ω for which

< p(t) > is maximum. By neglecting the influence of the

decay rates, this maximum reduces to pB(Ω)+4(∆/Ω)2.

It is equal to 3.867, 4.187 and 4.827 for ∆ = 5, 15 and

25 cm−1, respectively, in a very close agreement with

the numerical values equal to 3.863, 4.154 and 4.739 for

α = 75 cm−1.

By contrast, the coherences strongly influence both the

short time and the long time behavior of the hydrogen

bond coordinate. Indeed, by inserting Eqs.(24), (25) and

(31) in Eq.(13), it is straightforward to show that the

average coordinate < u(t) > is written as

< u(t) > = −2∆

Ω

< v(t) >(33)

+

?

?

p

2?p + 1e−Γ1ptQ1p(0)cos(Ωt)

2?p + 1T01

+

p¯ p

p¯ p

Γ1¯ p(1 − e−Γ1¯ pt)Q1¯ p(0)cos(Ωt)

According to the standard dressing mechanism, it is well

known that the creation of one vibron is responsible for

a contraction of the hydrogen bond whose amplitude is

equal to 2∆/Ω. The first term in the right hand side of

Eq.(33) generalizes this effect but it takes into account of

the decay of the vibron. The amplitude of the contrac-

tion is thus proportional to < v(t) > and it decreases to

finally vanishes in the long time limit, i.e. when the vi-

bron has been absorbed by the thermal bath. The second

term yields damped oscillations around the average con-

traction which originates from the existence of coherences

in the one-vibron subspace. Due to dephasing, this con-

tribution vanishes in the long time limit. Therefore, only

the last contribution in the right hand side of Eq.(33)

remains. It characterizes oscillations which result from

the transfer of the coherences of the one-vibron subspace

to the zero-vibron subspace. As illustrated in Fig. 9,

the amplitude of these oscillations in the long time limit,

denoted um=?

∆ and it scales typically as 2∆/Ω, as observed numeri-

cally.This surprising result can be understood easily

given that both < v(t) > and < u(t) > are not indepen-

dent one from each other. Indeed, from the initial VPS

Hamiltonian Eq.(1), it is straightforwardto show that the

Heisenberg equation which governs the time evolution of

< u(t) > is written as

p¯ p2√p + 1T01

p¯ p/Γ1¯ pQ1¯ p(0), does not sig-

nificantly depends on α. It increases almost linearly with

d2< u(t) >

dt2

+ Ω2< u(t) >= −2∆Ω < v(t) >

This equation reveals that the vibron population plays

the role of an external force F(t) which drives the mo-

tion of the hydrogen bond coordinate.

to the vibron relaxation, this force decays and it typ-

ically behaves as F(t) = −2∆Ωexp(−γt).

(34)

However, due

Note that

Page 13

13

? (cm-1)

05 1015202530

um

0.0

0.2

0.4

0.6

0.8

1.0

1.2

?=75?cm-1

?=20?cm-1

um=2?/?

FIG. 9: Long time amplitude of the hydrogen bond coordinate

vs ∆ for α = 75 cm−1(full line) and α = 20 cm−1(dahsed

line). Symbols refer to the law um = 2∆/Ω.

the use of a biexponential decay does not modify the

present discussion. Consequently, for Ω >> γ, the long

time limit of the hydrogen bond coordinate scales as

< u(t) >≈ 2∆/Ωcos(Ωt), in a close agreement with our

numerical observations. Therefore, although the vibron

has been absorbed by the thermal bath, it has induced

an initial distortion of the hydrogen bond which, in turn,

develops a free oscillation according to its natural fre-

quency Ω.

VII.CONCLUSION

In the present paper, the vibrational dynamics of a

one-site Davydov model consisting in a C=O group en-

gaged in a hydrogen bond has been studied in great de-

tails.According to this model, the energy relaxation

of the amide-I mode originates from Fermi resonances

with a thermal bath involving a set of high frequency in-

tramolecular normal modes. A special attention has thus

been paid to understand how this relaxation modifies the

quantum dynamics of the hydrogen bond.

According to the one-site Davydov model, the coupling

between the amide-I vibron and the hydrogen bond dis-

criminates between two kinds of eigenstates. In the zero-

vibron subspace, the hydrogen bond is described by a

harmonic oscillator whose eigenstates are the well known

phonon number states. By contrast, in the one-vibron

subspace, the hydrogen bond experiences a linear dis-

tortion so that its eigenstates correspond to superim-

positions of number states. By assuming the hydrogen

bond in thermal equilibrium at the biological tempera-

ture, the initial creation of one vibron is thus responsible

for the occurrence of both populations and coherences

in the one-vibron subspace. Due to the interaction with

the thermal bath, the vibron decays according to an ex-

ponential or a biexponential law depending on whether

the Fermi resonance is wide or narrow. Therefore, each

populated eigenstate relaxes over a set of number states

according to specific pathways whose properties depend

on both the Fermi resonance bandwidth and the strength

of the vibron-phonon coupling.

The consequence of the relaxation is twofold. First, it

leads to a redistribution of the number state population

which can strongly differ form the initial Boltzmann dis-

tribution. In other words, it is as if the hydrogen bond

experienced an effective temperature which can be either

lower or greater than the biological temperature. Then,

the relaxation allows for coherence transfers. Therefore,

although the vibron has been absorbed by the thermal

bath, the hydrogen keeps the memory of its initial dis-

tortion and it develops free oscillations according to its

natural frequency. These features clearly show that the

relaxation of the amide-I mode strongly modifies the dy-

namics of the hydrogen bond.

softening of the hydrogen bond and it may appear as

a precursor to the hydrogen bond breaking. Indeed, the

hydrogen-bond dissociation energy in proteins is typically

about a few kcal/mol [48, 49]. For instance, the enthalpy

change accompanying the α-helix to random coil transi-

tion in water has been determined to be 0.9 kcal/mol per

residue for a 50-residue sequence that contains primar-

ily alanine [50]. Such a rather small value, which cor-

responds basically to the excitation of about six optical

phonons per residue, can be reached following the amide-

I relaxation. Consequently, given that a typical value for

protein folding energy is about 10 kcal/mol [51], soften-

ing or breaking one or more hydrogen bonds may impair

the stability of a protein and may allow a conformational

change.

It is responsible for a

[1] L. Cruzeiro, J. Phys.: Condens. Matter 17, 7833 (2005).

[2] P.A.S. Silva and L. Cruzeiro, Phys. Rev. E74, 021920

(2006).

[3] P.A.S. Silva and L. Cruzeiro-Hansson,

A315/6, 447 (2003).

[4] L. Cruzeiro-Hansson and P.A.S. Silva, J. Biol. Phys. 27,

S6 (2001).

[5] C.W.F. McClare, Ann. N.Y. Acad. Sci. 227, 74 (1974).

[6] A. S. Davydov and N. I. Kisluka, Phys. Status Solidi 59,

465 (1973); Zh. Eksp. Teor. Fiz 71, 1090 (1976) [Sov.

Phys. Lett.

Phys. JETP 44, 571 (1976)].

[7] A.C. Scott, Phys. Rep. 217, 1 (1992).

[8] A.C. Scott, Nonlinear Science, (Oxford University Press

2003).

[9] J.C. Eilbeck, P.S. Lomdahl, and A.C. Scott, Phys. Rev.

B, 4703 (1984) .

[10] J.C. Eilbeck, P.S. Lomdahl, and A.C. Scott, Physica

D16, 318 (1985).

[11] D.W. Brown and Z. Ivic, Phys. Rev. B40, 9876 (1989).

[12] D.W. Brown, K. Lindenberg, and X. Wang, in Davydov’s

Page 14

14

Soliton Revisited,(Plenum, New York, 1990), edited by P.

L. Christiansen and A. C. Scott.

[13] Z. Ivic, D. Kapor, M. Skrinjar, and Z. Popovic, Phys.

Rev. B48, 3721 (1993).

[14] Z. Ivic, D. Kostic, Z. Przulj, and D. Kapor, J. Phys.

Condens. Matter 9, 413 (1997).

[15] J. Tekic, Z. Ivic, S. Zekovic, and Z. Przulj, Phys. Rev.

E60, 821 (1999).

[16] V. Pouthier, Phys. Rev. E68, 021909 (2003).

[17] V. Pouthier and C. Falvo, Phys. Rev. E69, 041906

(2004).

[18] C. Falvo and V. Pouthier, J. Chem. Phys. 123, 184709

(2005).

[19] C. Falvo and V. Pouthier, J. Chem. Phys. 123, 184710

(2005).

[20] D.V. Tsivlin and V. May, J. Chem. Phys. 125, 224902

(2006).

[21] D.V. Tsivlin, H. Meyer, and V. May, J. Chem. Phys. 124,

134907 (2006).

[22] J. Edler, R. Pfister, V. Pouthier, C. Falvo, and P. Hamm,

Phys. Rev. Lett. 93, 106405 (2004).

[23] J. Edler, V. Pouthier, C. Falvo, R. Pfister, and P. Hamm

in Ultrafast Phenomena XIV, edited by T. Kobayashi, T.

Okada, T. Kobayashi, K. Nelson, S. De Silvesti, Springer

Series in Chemical Physics, Vol. 79 (Springer, Berlin,

2005).

[24] L. Cruzeiro-Hansson and S. Takeno, Phys. Rev. E56, 894

(1997).

[25] V. Pouthier, Physica D221, 13 (2006).

[26] V. Pouthier, Phys. Rev. E75, 061910 (2007).

[27] V. Pouthier, Physica D (in press 2007).

[28] K.A. Peterson, J.R. Engholm, C.W. Rella, and H.H.

Schwettman in Accelerator-Based Infrared Source and

Application, edited by G.P. Williams and P. Dumas,

Proc. SPIE Vol. 3153 147 (1997).

[29] K.A. Peterson, C.W. Rella, J.R. Engholm, and H.H.

Schwettman, J. Phys. Chem. B203, 557 (1999).

[30] R.H. Austin, A.Xie, L. Van der Meer, M. Shinn and

G. Neil, Nuclear Inst. and Methods in Physics Research

A507, 561 (2003).

[31] A.Xie, L. Van der Meer, W. Hoff, and R.H. Austin, Phys.

Rev. Lett. 84, 5435 (2000).

[32] P. Hamm, M. Lim, and R.M. Hochstrasser, J. Phys.

Chem. B 102 6123 (1998).

[33] P.Hamm,

Hochstrasser, J. Chem. Phys. 112 1907 (2000).

[34] S. Woutersen and P. Hamm, J. Chem. Phys. 115, 7737

(2001).

[35] J. Edler and P. Hamm, J. Chem. Phys. 117 2415 (2002).

[36] M.F. DeCamp, L. DeFlores, J.M. MacCracken, A. Tok-

makoff, K. Kwac and M. Cho, J. Phys. Chem. B109,

11016 (2006).

[37] K. Moritsugu, O. Miyashita, and K. Kidera, Phys. Rev.

Lett. 85, 3970 (2000).

[38] H. Fujisaki, Y. Zhang, and J.E. Straub, J. Chem. Phys.

124, 144910 (2006).

[39] G. M. Chaban and R.B. Gerber, J. Chem. Phys. 115,

1340 (2001).

[40] N. G. Mirkin and S. Krimm, J. Mol. Struc. 377, 219

(1996).

[41] P. Derreumaux and G. Veroten, J. Chem. Phys. 102,

8586 (1995).

[42] I. G. Lang and Yu. A. Firsov, Sov. Phys. JETP 16, 1293

(1962) 1293.

[43] R. Zwanzig, Lect. Theoret. Phys. 3, 106 (1961) ; Physica

30, 1109 (1964) ; J. Chem. Phys. 33 1338 (1960).

[44] H. Mori Prog. Theoret. Phys. 33, 423 (1965) ; Prog. The-

oret. Phys. 34 399 (1965).

[45] V. May and O. Kuhn, Charge and Energy Transfer

Dynamics in Molecular Systems (WILEY-VCH Verlag,

Berlin, 2000).

[46] F. Shibata, Y. Takahashi, and N. Hashitsume, J. Stat.

Phys. 17, 171 (1977).

[47] C. Uchiyama and F. Shibata, Phys. Rev. E601, 2636

(1999).

[48] B. Nie, J. Stutzman and A. Xie, Biophys. J. 88, 2833

(2005).

[49] S.Y. Sheu, D.Y. Yang, H. L. Selzle, and E. W. Schlag,

Proc. Natl. Acad. Sci. 100, 12683 (2003).

[50] J.M. Scholtz, S. Marqusee, R.L. Baldwin, E.J. York, J.M.

Stewart, M. Santoro, and D.W. Bolen Proc. Natl. Acad.

Sci. 88 2854 (1991).

[51] B. Honig and and A. S. Yang, Adv. Protein Chem. 46,

27 (1995).

M.Lim,W.F.DeGrado,andR.M.