RESULTADOS DE ADOPCIÓN DE PRÁCTICAS Y ACTITUDES EN LOS

We have seen that the simple model of an anharmonically coupled chain of amide

groups or molecules displays coherent propagation of an energy pulse when the

magnitudes of intra- and inter-molecule coupling coefficients are roughly equal. This has

been demonstrated using quantum dynamics and, under appropriate initial conditions,

using classical mechanics.

In the reduced units (Tz/oq) employed here, the energy pulse resulting from one or

two quanta initially localised at one end of a chain is transferred between adjoining

molecules or groups in approximately one time unit (or two time units in the classical

simulations). A crude estimate of this time unit can be obtained. The wag coordinate for

the NH bond is usually defined by

where CMe represents the methyl carbon in N-methylacetamide. If Pa is the conjugate

momentum, the classical kinetic energy contains, amongst others, the wag kinetic energy

w h e r e i s the reduced mass of NH (~ 1 amu) and r is the NH bond length (~ 1 Ä).

Expanding r about its equilibrium value, req, gives

Q. = (ZCNH - ZC M

c

NH) / V2

(3.44)

(3.45)

(3.46)

Comparing Eqs 3.46 and 3.2, we can identify q with the displacement of the NH bond,

S jn

(£

)

Qu

an

tu

m

A

ver

ag

e

Time

Figure 3.13.

(a) The quantum average

[see Eq. 3.42] versus

reduced time for each of the seven molecules, under the conditions

of Figure 3.4, with

=

ay

(b) The classical average, over 50

trajectories, of sin(%) [Eq. 3.41] versus time, for each of the seven

wagging motion, so we can identify Pa as being proportional to the momentum P of Eq.

3.2. If the proportion were one, the factor MXx in Eq. 3.6 is given by

MXx = \ /req

(3.47)

Takings = 1 amu and co - 3300 cm 1, Eq. 3.6 gives

h / a x~ O.lps

(3.48)

88

Eq. 3.48 gives a good estimate of the lower limit of the time scale. Normally, the

wagging motion is only one component of the lower frequency mode. Supposing that the

CO stretch and NH bend contribute to two normal modes such that

Pco — ^Pq ^ Pq'

^ = - ' H - e 2Pa + ePa.

r eq

then the factor MXx of Eq. 3.6 is given by

(3.49)

MX, = 2fico(l - e 2) / {r.,[e2n + 2 ^ co(l - £2)]}

(3.50)

A little arithmetic shows that ti/ax rises to 0.2 ps only when e is as large as 0.96, while

ti/ax diverges as e —» 1. While only a crude estimate, this calculation shows that the time

scale for energy transfer between modes in Fermi resonance via this mechanism is likely

to be sub picosecond.

This timescale is in clear contrast to that of the earlier Davydov model9 for

coherent vibrational energy transfer in an a-helix. This mechanism gives rise to coherent

soli ton motion on a time scale at least one order of magnitude longer than that estimated

above.

Of the two Fermi resonance mechanisms invoked in the model, the kinematic

coupling mechanism is now familiar. The kinematic mechanism has been observed,

though not explicitly understood, in lower energy simulations of amide dimers31. The

intermolecular coupling will be considered more fully in the following chapters where

estimates of its possible size, albeit very approximate, are considered.

vibrational spectra of secondary amides and polypeptides which has been associated with

Fermi resonance. In cis secondary amides32 this band has been identified with a Fermi

resonance between the NH stretch and the combination of CO stretching and NH bending

modes at approximately 1650 and 1450 cm '1 respectively, while in trans peptides32,33 it

has generally thought to be the result of a Fermi resonance between the NH stretch and

the overtone of the Amide-II mode ( 2x1550 cm '1). (Cis secondary amides do not exhibit

an Amide-II mode). A more recent study34 has broadened this resonance to include the

overtone of theAmide-I mode and the third overtone of the NH out-of-plane bend

780

cm'1). While there is evidence for Fermi resonance type interactions: the intensity of the

Amide-B band is too great to be just the overtone of the Amide-II, it is certainly not clear

that such a simple Fermi resonance interpretation is appropriate. We have seen that if an

extended Fermi resonance regime exists, the expected Fermi splittings may be diminished

and therefore a theoretical analysis in terms of a single resonance or a single molecule

may not be accurate. However, the Amide-B band does offer some experimental evidence

of Fermi resonance occurring between the NH stretching and NH bending motions in

peptide groups. It should noted that the overtone of the Amide-I mode corresponds to the

middle of the broad NH band so there is little chance of resolving it and determining any

Fermi shift.

The model described in this section is clearly too simple to be quantitatively

applicable to the vibrational dynamics of proteins or amides in condensed phases. The

assumptions that only two types of mode are relevant and that these modes are in exact

2:1 frequency ratio severely restrict the extent to which quantitative inferences about

dynamics can be sustained. These major simplifications are also removed in the

subsequent chapters.

However, the simple model is useful in a number of respects. It exposes the

unexpected relation between Fermi resonance processes and nonlinear acoustic

phenomena, leading to coherent propagation not usually associated with optical modes. It

also shows that competing Fermi resonance processes can give rise to unexpected values

for the energy shifts of the high frequency fundamental and low frequency overtone

states. The comparison of classical and quantum dynamics establishes the conditions

under which classical simulations can give a qualitatively reasonable description of the

quantum dynamics of these nonlinearly coupled chains: while under these conditions the

classical simulations display a time scale for energy transfer that is slow by a factor near

two. Finally, both classical and quantum simulations show that propagating pulses of

vibrational energy, related to soliton solutions of the dynamical equations, 'self form'

from a localised excitation and propagate in a stable fashion.

Chapter 4. Simulation Of Energy Transfer In

In document EVALUACION DE IMPACTO DEL PROMER (página 32-36)