Local modes in molecules

Journal Elmvier

A.

of Molecular Liquids, 41(1989) Science Publishers B-V, e

L. Ban&in1

cl Sam’.

‘Program in Appkd %qxartment

1~6-111

105

-Printiin7%eNe&.imrlande

and l_ C Eilbc&

Mathunatkm,

of Mathematic+

The University

of Arixona,

Tucron,

Unlvemity, Ricuuton.

Herlot-Watt

AZ

J367?21(USA)

EdMnugh

ga

14 4AS (UK)

Iaxliwd

in comparcd

SUMMARY The

time-for

which

mole&u

vibrational

energy

can

remain

far

two

nonlinenrityz i) m which arisa from a nocdincar mct0ring force (8uch an Sn the CH stretching modar of ene), and ii) m which comes about through coupling to low fiWpleucy phononr

typks of clan&d

INTRODUCTION At the preunt time it is not nuxsary fundamental

proparty of &s&al

conurved quantity (en-,

for

to amp-

no&near

-phj

ina

to the physics community at lead, thnt a

dynamical Ioudii

m&nw

4011.

[l. 2J but for a much wider dana of nonlinear dynamid vibrational

energy)

moleada

such

par&&Cal linear,

because

should

detaiJod

AE)

methana,

these molecules

am not localixed,

will

messured seconds,

remain

in wave

localized number0

mod&

in the etc

packet

IocaHlstion.

quantum

that

The

This

paradax

theory

it can

of a &u&ally

at some

initial

this

chapter)

of

a rtstricted

and

being

through

the

dynamical

energy

of tima.

to AE.

m

be undumtnod

nonlinear

value

proportional

whkh,

is resoIvcd

[4-61, and

(lying within

(loc&xatio~~

obaervation8

by the quantum

states

(am ie the caue throughout

effect

of curtah-aymmetrIcal

exprrimcntal

them-istry

IO inversely

same

spectra

Such

to be governed

can be constructed (T)

[S]_

mne

(4-q.

of physicnl

coma

This b true not only for 8ollLon m

to select mcverd of them

for a time or cni’

$hwrv

the stationary

it is pomible

such that a local~ued wave

packet

of energy

mahQ

Although

bcnzens,

am expected

the property

of the w

am foIlown_

cxperimantally

observed

ammonia.

ndt e&bit

c&ulations

qualitatively system

ae

Is also

In the ability to concent&

range,

Such a wave

Indeed. if AE IO if T

in mearmred

in

then

where c Is the vdocity of light in UII/MC The dcpendr

aim of thin chapter is to d&be

upon

parameters

of

the

dawkal

how tha AE

of an appropriately

nonlinear system and upon the quantum

corresponding wavu pck&

01&22/89/$03&0Q

i989

constructed

ELsavisr Science

Publishem

B.V.

wave

packet

.IWCIS of the

106 In this demcdption it b nec~y can &ve~iemly

.*

be C&C-

nonwty,

m-isa from

particuIar moteeuIar bond.

A lina~

molecule.

Tks

k

of dausical ~onllnearity which

molecular vIb&ionr

The first tm

which WC potantial)

An exnmpte k provided by the CH stretching vibrzation iu buuena Tha second h

which we call m

arks

cali

of m [8] for

in thu following

molecular vibration L coupled to a low frequency macbanlcal dk&xtion

of the

mechanical dIstortion reacts back upon the original molecular vibration, inOuencing ita

dynamica in I nonllaear manner. c~work&

of

the nonlinear force &mutant (or nonquadmtic

which the ratorlng force is sublinear. n?‘ilner.

to condder two distinct typa

in tha context

the infrar&

The efkct of extritic

wtrum

of &line

nontinearit~r hau been observed by Careri and

acetauilidu (ACN),

a synthetic polypeptida [9-12]_

Ike tha CO stretching vibration (at 1685 Cm’) t coupled to pbonon mrxics of the crystal (at about 70 -1 cm ) to produce a local moda at 1660 cm-‘Thb erkct. is of general -~porkulce lxcause it has been proponed as a machanikn for energy storage in

natural

proteln [13-E].

To explain the distinction in

soliton Jargon. we point out that the nonlinear ScbrZidlnger equation [16] is an example of intrinsic nonlinearity, and the Zakharov system [17] Is an example of extriruic nonlinearity. We discuss two mud&,

one of intrinmic and the other of extrinsic nonIinearity.

am chosen to be as aimplt and aa similar m ponalbla in order to facllihb

INTRINSIC

Thcne models

comparativa conchuionr

NONLINEARITY

As a simple cxnrnpla of a dasaical, nonlinear dynnmicd choose the diite

&trapping

where A = col (A,,- --,AM) symmettic, &,

system with intrinsic nonlinearity. we

(DST) equation[3]

is a column vector of M complex mode amptituda

M x M d-on

and D =

[dijJ is a

matrix that ti normaliaed through adjustment of tha parameter c

such that max ldi$ = 1. Solutions or (2) ConWrve the norm (or number)

N=

E IAil’

co

j=l

and the energy

H=RoN-c

diiA;AS+

IAjl’-

(4)

The DST equation in exactly integrable for M 5 2, but for M 2 3 solution trqjectories can be periodic, quaoipcrlodlc or chaotic depending upon equation parametun and initial conditioau [S, IS]. Under quantiktion

the complex mode amplitudes Aj and AI become boson annihilation and

creation operators fij and fii [19]. Th e number and tha energy becoma tha numbur operator

107

A=

5 It+8. ’ ’

j=l

and the energy operator

y, Planck’~ constant is equal to uuity, and

whmcgroundstaklavalshavt~sattozarqRo=P)Ored Ill wa&llumbcrs.

allcncrgks(Qorrandy)aramcasu Since fi and i% commute, condittons

we seek stationary state wavu functions I+) which satisfy two

&I$) = Nl+) and ii) fil#) = El*)-

i)

lpb) = c,IN)lO) ---IO) + --- -I- cp IO) --where p = (M+N-I)!/(&I)1

lO)ll)

Condition 1) is sat&f&

by choosing (for M > N)

--- 11)

N! Condition ii) requirea that the set {c;) satMy tha matrix equation

tiE=EE

(8)

where fi is a readtry computed. symmatdc.

real. p x p matrix and E’ = co1 (cl,---,cp)-

Unlss

p is

inconveniently large. tham is no need to amploy app roximate methodr of solution PO]Whcu

thH formdinm

is uecd to calculab

liquid benzene [8, 211 it is found that c = 4 cni’ thecma<<
Iti) = c,lWlO)

The M

---

statesof

IO) + c, lO)lN)

tha overtone spectrum of CH stretching modu and 7‘ = 117 an-‘-

loweut energy at tha ,a

---

10) f --- -I-

cM

in

Thus it in of Interest to con&k

axcited level hava the form

lo)to) --- IN)

+temmofo&rcorlus.

(9)

It is from thepa M stationary states that local mode wave packeti am cond.n~ctd. We

lint here some stahmunk

stationary staks

[21, 221.

about the maximum

Ccrtaln of thaa

which is defined as follows--

&tcmcnb

energy diicrcncc

(AE)

involve the concept of

A system bar transitive symmetry

if all the modes hava the ssma

uncoupled frequency, and the same set of interaction energies with the other madea choow

A, to reprucn t tha amplitude of any o&llator

the governing cqztion

(2) remains invariant,

hetwecn these

Thus

ouu

UUI

in the system aud label thu other modes su that

This Is equivalent to the requirement that the d-on

matrix D be Invariant under the pcrmuta&on of its rows and columns by the clemants of a tranaiti~ permutation group [23].

108

Ci)

At

thu first

excited

level

th arasraon]yM~b~nar~~~aadAEfrob~ar~.

(N=l).

(Tlim ii tvident

from

the rtructure

(6) giv&

Thii

AE

zero.)

normal

moda

of

a

is the suns

linear

Open0011

of fi_

on a ningk

m tho maximum

systah.

Thum

ttm

energy

buw

state

difkmnm

nonlinearity

does

4th

the last turn

bet weun thu.M not

cm~ta-ibuta

of

quantized to

energy

Io&zation*

@)

At

(iii)

For&DSTsystcmatieNth

(iv)

For a -tan

the necond

en-

excited

diff’ucnm

kvel

between

(N=2)

the lowest

there

are

M(M+l)

M of them atatec (AE)

excited led,

with tra+tivs

1

symmetry

AE

stath~nary

Tba

maximum

ir of order ~‘/7_

is no Iargar than orda

at tha Nfh excit&

SW

~‘/7_

.

level,

AE

(10) Conuida

(v)

the

condition*.

dti

I&SSJJ,

a

with

equal

nearest

neighbor

intuactions

and

.+adic

end

Thou

zlfoorl-j==ltimodM = 0 otherwba

Thb

t a q&al

of tran&ive

a

symmetry.

At the Nti

excited

level,

(11) At the first excited level (Nzl),

13) =

sll)lO)

---

IO) -I- ---

+ CM

p =

M and

lO)lO) --- 11)

(12)

w-hem

ci = M

-1/a

and k = energy

E,(k)

-P

Prv/M

WE)

(13)

with

Y

=

0. f

1, f

2. ---,

&M/2

(f(M-1)/2)

for hi even (odd).

The

corwponding

levels are [al]

=

n,

-

y -

2

At tha second

l

ux

k.

excited

(14)

level (N=2),

p =

@(M+l)

and

109

j03)

= cJ2?lO)

---

IO) -I- 4O)l2)

---

IO)+ --- + Qw)---12)

Mtermmofordacorlea.

+p-

Fro&

apandon

a perturbation

EXTRlNsIC

(15)

i6 small

l

[21]

NONLlNEAFtlTY pk of a dynamical

Aeaa3mpIeexam Hamtltonian

system

with uctrindc

nonliuenrity,

WC choose the FrBhkh

1241

(17)

f 18)

Pnm~~~eter~ and opuators meaning,

In

oscillaton

which

constant

x-

addition, are

Evidently

and (19)

in (18) $

and

I$

individually

am

haw

annihilation

conpkd

% commutcn

that

to the

with thu number

the me and

high

notation

creation frequency

In (6)

operatom moda

and (6)

for

(a,)

ha~a

phkmtr through

tb

of the

~IIIC

Efnddn Coupling

operator

(20)

Assuming calculate

Itim)

again

tha energy_

Thus

the

mokcular

the zero order wavu

we

turn

functiono

Lo a

perturbation

at tba fiat

excibd

expan8ion level (N=l)

in wall

the coherent

Motivated

c to

am

(ii)

= E =j lwa --- lo---lw~j) j=L J& Ph=

where tha cj are given by (IS)_

I+

chain,

by Davydov

ilS-X5]

we choaae the phonon

ZBckrs

14,

to be

statem

= axp(-qy’)m~o

ET

WI?

--- Im)---IW 3* PIa=

(22)

110 It should be emphasize6 that eisenfunctions of fi] displayed

..; (21) while those of 61 3

~j are displayed

are

in (22).

Writing

I&, in the form

(23)

and noting that 41”j)

= yl~j)

necessary condition for l@(k))

Y =

-t

and also that bj

has no nonzero eigenfunction,

it is evident that a

to be an eigenfunction of fi, is

$.

(24)

Then the order c estimate of the energy is

l+)(k)

30s k.

=

Upon comparing

(?5) with (14) WC note that AE

If we consider the limit w. + packet remains localized.

(25)

0 and x*/w,

has been reduced by the factor exp(-x2/w:).

-+ constant, then AE

-,

0 and an initially localized wave

Since Davydov assumes extrinsic coupling to acoustic phonons (13-151, this is

the limit in which his analysis becomes valid. Finally, we note that the phonon wave function in (22) is d special tasc.

More generally, (21)

takes the form

where

(27)

and L?-“(w)

is an Rssociatcd Lagucrre polynomial

[25].

The corresponding

cncrgy cigcnvalues of

ii,

are

E(‘)(N

,n) = N 0, +

For ACN,

nwo -

N2 go.

measurements

of the overtone spectrum

phonon modes are coupled to each CO stretching vibration

dctcrminc x*/w,, [12].

Following

or G

X:/Wi if several

Krum ilansl [2G] WC have

111

us&lib e,.

t

h&nation

to interpret

l~~+Ii’)

an+ con&da

a$+++

$h

the dudy

the tampcraiyrs

dependence of the intauiry

of tha local mods (at

that sever& phonou modes near 70 Cm’ am involved [2fl. by Alexander and Krumiusnsl. [28$

Th3s in in guneral

Wa expect aeuiIarcffectinnatnml

p&in.

ACKNOWLEDGEMENTS It is S pleasure to thank H. Bo&raucr partly r~ppmtcd by the Hational

and B. R Henry for bclpt’nl discussions. Thin work VM

Scianes Foundat3on.

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