Lifetimes of collective resonance states (excimols) in polyatomic regular type molecules and clusters

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Lifetimes of collective resonance states (excimols) in polyatomic regular type molecules and clusters

H.-W. Fritsch, H. Jungclas, V. Komarov, L. Schmidt

To cite this version:

H.-W. Fritsch, H. Jungclas, V. Komarov, L. Schmidt. Lifetimes of collective resonance states (exci-

mols) in polyatomic regular type molecules and clusters. Journal de Physique II, EDP Sciences, 1994,

4 (4), pp.567-571. �10.1051/jp2:1994147�. �jpa-00247982�

Classification Physics Abstracts

30.00 34,10 36.40

Short Communication

Lifetimes of collective

_resonance

states (excimols) in polyatomic regular type ^{molecules and clusters}

H.-W. Fritsch

(~),

H.

Jungclas

(~>~>*), ^{V-V- Komarov} (~) ^{and L. Schmidt}

(~)

(~) Fachbereich Physikalische Chemie, Philipps-Universitit Marburg, D-35033 Marburg, Germany

(~) MZ Radiologie, Philipps-Universitit Marburg, D-35033 Marburg, Germany

(Received

lo December 1993, accepted 4

February1994)

Abstract A model to estimate lifetimes of excited collective _resonance states

(excimols)

for

large biomolecules _or clusters, which have _a regular _or close to regular structure, is presented.

These collective excited states arise due to _resonance excitation of _{one or a} few bonds in the constituents of the system. The model is based

on the quantum multi-particle theory and the exciton _or the excimol theory for the condensed _matter and molecules. The spectral analysis for non-self-adjoined _energy operators is used in the model. As _an example, lifetimes _are calculated for one-dimensional chain-like polyatomic molecules and for clusters formed by such molecules.

The resulting lifetimes of the entire excitation _are longer than the lifetimes of the initial trigger excitation.

Recently

it _was shown

experimentally

that the lifetime of

large

clusters' _or molecules' exci- tation is

longer-

than the lifetime of initial electron excitations of _{one or more} valence bonds 11, 2]. Dissociation and ionisation processes are

essentially delayed

and

large

molecules _or clus-

ters _can be considered _{as a} compartment ^{of excitation}

energies.

^{A model based} on methods of quantum

multi-particle theory

_[3] and exciton

theory

_{[4, 5]} for an estimation of this time

delay

is

presented.

We consider systems

(clusters

_or

large molecules) consisting

of similar constituents

(small

molecules _or valence

groups)

with

regular

structure and

equal

distances between the _con- stituents. From the

point

of view of the

spectral analysis

of energy operators for the systems under consideration the

large majority

of excited states in these systems are discrete states embedded in continuous spectra. The excitation energy of these _{states can} be described

by complex values,

and the relaxation _processes of bond vibrations _can be taken into account.

We consider _a

trigger

excitation of _{one or more} vibrational bonds

by

electron

impact

which leads to a quasi stationary state in the entire system. As _an

example

of

possible

constituents in

(*) ^Author to whom correspondence should be addressed.

568 JOURNAL DE PHYSIQUE II N°4

large

biomolecules _we take valence groups

(C=O, C=N,

etc.

).

For _our model the Harriltonian of the clusters

containing

chain like molecules is taken in the form:

'~HN

=

fl _hqn

₊

fl _Tn,pm

₊

£ ~Tn

₊

£

_Tqn

_(i)

qn qn#Pm n qn

where _{q, p E}

(1, 2,...,Q)

_are the numbers of valence _groups inside _a

molecule,

_{n, m} E

(1,2,...,N)

are the numbers of molecules inside _a

cluster, hqn

^{is the} Hamiltonian of _one valence group

(qn);

_Vqn,pm is

potential

between two valence groups

(qn)

and

(pm); ~Tn

and

Tqn

are kinetic energy operators ^of molecule

(n)

and valence group

(qn) correspondingly

in their centre of

mass.

If the time of

intergroup

excitation transfer in

large

biomolecules is

essentially

less than the time of reconstruction of their

equilibrium configurations,

then _{we can} consider the vibrational

excitations of valence groups

independently

of the

phonon

molecular vibrations. In the above

assumption

_we have

~Tn

= Tqn = 0 and the _new Hamiltonian

~fiN

without the last _{two terms} in

equation (I).

For the estimation of the system _energy ^{with the} Harniltonian

~fiN

we use a

perturbation theory assuming

_Vqn,pm to be weak. For the collective

ground

state energy we have _now:

E0

"

(~0 ~HN( ~0)

^"

^~0 ~ _hqn 01+

4~0

~

_Vqn,pm

_{~0 (2)}

qn qn#pm

where

~° _~

ii9'°,q"'

~~d hqn§~0,qn = £o§~o,qn,

qn

then

Eo

₌

Noeo

₊

Do, Do

=

40

~jvqn,pm ^{40 (3)}

In the

presented

model the structural _groups

forming

molecules _are

essentially

different from the molecular cells [4]. ^In our case the operators

hqn

^{of valence} groups are not

self-adjoined

and their excited states

might

be

quasi-stationary

_as well _{as resonance} states with definite lifetimes.

The _energy of the

ground

state of

hqn

is _eo and the _proper

eigenfunction

is qoo,qn. ^The first vibrational _state with the

frequency

_vi has the energy zi = El i~ti and the wave-function qoi,qn, which is not square

integrable.

Vibrational collective states of

large

molecules _are

quasi-stationary

states. It _means that for the

application

of the

perturbation theory

_we have to

develop

_a

special technique

which permits _us to _use

quasi-stationary

_wave functions

(normalization problem).

We _use the scale transformation _or the dilatation method [6], ^which

replace

the operator

~fiN

_{with quasi-}

stationary

states

by

the operator

~HN(8)

_with

_stationary

_{states at the}

same energy z.

~fiN(8)

=

U(8) ~fINU(8)~~

=

~j U(e)hinU(8)~~

₊

~j U(8)Vqn,pmU(8)~~ (4)

qn qn,pm

Here

U(8)

is _a dilatation operator ^which transfers the _wave function _qoi

qn

into _qoi

qn(8):

u(e)§7I,qn

_"

e'~§7I,qn (e~)

,

(s)

where 8 is _a dilatation parameter.

By

this transformation _we choose e

= I _{no no "} I arctan

(~tilei).

After the dilatation

procedure,

the state with the _{energy zi} of

hqn(8)

is _a

stationary

state with _a normalized

vector. Then _{we use} the

perturbation theory

to estimate the excimol's

complex

_energy from

the operator

~HN(8) _{equation (4),}

if _one of the valence groups is in _a

quasi-stationary

state with the

complex

_{energy zi}

If _we

neglect intergroup interactions,

then the cluster _wave function with _one excited valence group

(qn)

can be

represented

in the form

*1,qn(8)

=

v7i,qn(8) fl _{v7o,pm(8). (6)}

pm#qn

The total _energy in this _case is

El _"

(NQ I)eo

+ zi

(7)

The state is

NQ

times

degenerated,

because there is _no information which

particular

valence group is excited. One _can expect ^that the _resonance interaction between _groups takes _away the

degeneration.

A contribution to the excimol energy from the

intergroup

interaction _can be

estimated from the cluster _wave function with

arbitrary

located excited valence _groups

41(k, 8)

₌

fi _j ~ti,qn41,qn(8)e'~~°" (8)

where ei~~°" _{is the} translation operator

eigenvalue,

and ran defines the valence group location inside the

cluster,

with the _wave number

k=£~~u,~; _Ni=N, N2"Q,-'<u,§'; bi"b~~, b2"a~~

Here _a and b _are distances between groups in _a molecule and molecules in _a cluster _corre-

spondingly,

_{~ti,qn is} a

weight

coefficient for the function

41,qn(8)

with normalization

£~~

[~ti,qn(~

= l.

The translation symmetry ^{is violated} on the

crystal

_or molecule

boundary

_or in the end

points

of molecular chains of finite sizes. Nevertheless _{we can use} the translation symmetry operation

introducing cyclic boundary

conditions [5].

In _a real _case, if _we have

only

_a few valence groups in _one

molecule,

and _{a <} b, then the

excimol is localized inside this molecule and its energy Zi has _a discrete spectrum, ^I-e-

only Davydov's splitting

_[5]. The equation for the excimol's

complex

_energy

AfiN41(k, ₈₎ =Z141(k, 8). (9)

where

AfIN

=

~HN

_{Eo may} be used to determine the

weight

coefficient ~ti,qn and Zi

spectrum. We _now consider the

example

of

only

_one molecule with

regular

valence _groups.

The total

complex

_energy is in the form

zl

_{" (Zl E0)} +

Dl

+

Ll (k), (lo)

570 JOURNAL DE PHYSIQUE II N°4

with

Q

Di

=

L 1v7i,q(e)v7o,pie) lK,pie)1v7i,q(e)v7o,p(e))

Do

(II) q~f»

and

Li(k)

is _{a resonance}

integral

Q

Li(k)

₌

£

Mi,qpe'~~~~~~P~,

(12)

~ q>»

q # _» where

rp = ap, rq = aq, -x < ka < x,

(ka)max

_{= x,}

(13)

and

Mi,qp

=

1v7i,q(e)v7o,p(e) lvq,p(e)1v7o,q(e)v7i,p(e)1 (14)

In this

approach using complex dynamics variables,

the

integrals Di

and

Li (k)

_are

complex

and _we rewrite

equation (10)

in the form

Zi(k)

₌ Re

Zi

_{+ Im}

Zi

_{" El} i~ti _eo + Re

Di

+ iIm

Di

+ Re

Li(k)

+ iIm

Li(k) (Is)

As it is

well-known,

the value of the

imaginary

part ^of

quasi stationary

state's _energy

(Eq.

(Is)

permits _us to establish the lifetime _t of this state: t

=

h/ (2

_x Im

Zi).

The attractive interaction between valence _groups reduces Im

Zi

and thus increases the lifetime of the whole system ^and vice-versa.

The _resonance interaction energy between the valence _groups is

proportional

to the matrix

element of valence _group transition from the state with the _{energy El} to the

ground

state with the _{energy eo.} This matrix element _can be estimated

by using

^the

theory

of quantum systems in external fields [7].

Let _{us now} consider the results of such calculations for the

large

chain-like biomolecule

peptides,

with amide I valence _groups. We have to note that _a molecular interaction in the chain-like molecules is essential

only

between the

nearest-neighbour

valence _groups. The Coulomb interactions between these _{groups are} less than the _resonance interactions [8]. ^Thus

the value Im Zi is defined

by

Im Li

(Eq. (Is)).

The

dipole-dipole

term in the Coulomb

integral

expansion in

multipoles plays

_an

important

role in the

integral (14).

One _{can see} from above expressions

(lo-14)

that the

intergroup

interactions lead to the

splitting

of

Zi,

and from equation

(13) follows,

that

(Zi (k))~~~

arises

by (ka)max

= ~, which defines the sc-called _zone bottom in the excimol

splitting

_zone with the width L

=

[(Li)max (Li)mjn(.

The time of electron excitation transfer

along

the chain is _r Gt

h/(2xL).

If _we denote

by

6 _an

angle

between the chain axis and

dipole

momentum d of the valence group

direction,

then we have for the

imaginary

_part of the collective

quasi stationary

states of the whole chain

Im

AZI

m -'fi +

3/2~tie/~Wo

^O

(~t)e/~) (16)

with

Wo "

2e~d~a~~ (1-

₃ cos~

6)

From equation

(16)

_{one can see} that the lifetime of collective excited states of the whole system

provoked by trigger

excitation of _one valence _group is defined

by

the structural param-

eters of the system

(molecule

_or molecular

cluster)

and the vector of

dipole

transition inside

the valence _group. If _we have for the a-helix

protein

6 _m

90°,

_a m 5

I

_and

[do,i

m 0.3

D,

then

we obtain that the lifetime of the entire system excitation is _a hundred _times

longer

than the lifetime of amide I group

trigger

excitation.

For molecules _or molecular clusters with _a strong attractive _resonance interaction between the constituents _{we can} face _a situation when the

trigger

excitation of _{one or a} few constituents leads to the stationary excitation of the entire system ^with infinite lifetime. The lifetime of

excitations in molecules _or molecular clusters

depends

_on the

shape

of these systems and _on the number of inter-constituent bonds.

Certainly

in the real system we expect to have deviations from

regularity.

Nevertheless,

if chaotic

perturbations

and deviations from

regularity

_are not very

big,

we can represent ^the Hamiltonian in the form

H=HR+AHC,

where

HR

is _a

regular-case

Hamiltonian and

AHC

is _a chaotic

perturbation.

Then for the

corresponding

Green function _we obtain the

expansion

_[9]

Gc = GR

GRAHCGR

+

GRAHCGRAHCGR (17)

and the

partial

summation of this series is

possible.

The wave-functions for

practical

estimation _can be considered in their

asymptotic

form in the usual _manner of

multi-particle approaches

_[7].

Acknowledgements.

We thank Professor H.

Neumann, Philipps-Universitit Marburg,

for

helpful

scientific discus- sions. We _are also

grateful

to the Deutsche

Forschungsgemeinschaft, Bonn,

for financial _sup- port.

References

ill

Schmidt L., Fritsch H.W., K6hl P., Jungclas H., Ion formation from organic solids

(IFOS

V) L6vanger 18.-21.6 1989, Hedin, Sundqvist and Benninghoven Eds.

(J.

Wiley & _sons,

1990).

[2] Lykke K.R. and Wurz P., J. Phys. Chew. 96

(1992)

3191;

Gallogly E., Bao Y., Jackson W.M., XVIII ICPEAC, Aarhus, Denrnark 21.-27.7 1993, ^book of abstracts

(1993)

_p. 316.

[3] ^Komarov V-V-, Popova A.M. and Shablov VI., J. Math. Phys. 24

(1980)

854.

[4] ^Knox R-S-, Theory of Excitons

(Academic

Press, 1963).

[5] Davydov A.S., Theory of Molecular Excitons (Mc. Graw-Hill, N-Y-,

1962).

[6] ^{Reed M. and} Simon B., Analysis of operators

(Acadelnic

Press, 1978).

[7] ^Komarov V-V-, Popova A-M-, Shablov VI., Sov. J. Part. Nucl. 14

(1983)

138-154.

[8] ^{Human J.M. and} McLoughlin D.W., Physica 3D

(1981)

23.

[9] ^{Belousov M-V- and} Pogarev D.E., Dynamical Properties of Molecules and Condensed Matter,

A. Lazarev Ed. (NAUKA, Moscow, Russia, 1985) _p. lo?.

JOURNAL DE PHYS<DUEII -T 4, N'4 APRIL 1994

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