Rotation-vibration separability in the classical motion of triatomics

Journal of Molecular Structure, 142 (1986) 529-532 Elsevier Science Publishers B.V., Amsterdam

ROTATION-VIBRATION

SEPARABILITY

J . SANTAMAR I A’ , G. ALVAREZ 1

Oepto

de Q.

‘Depto 3 4

Inst.

Fisica,

Espectroscopia,

de

Estructura

Chemistry

R.

GALINOO’,

de

la

28040

CSIC,

G.S.

EZRA4

Madrid 28040

28006

University,

OF TRIATOMICS

and

Complutense,

Materia,

Cornell

MOTION

ESCRIBAN03

Complutense.

Universidad

Department,

in The Netherlands

IN THE CLASSICAL

Universidad

de

529

-Printed

Madrid

Ithaca,

Madrid (Spain)

N.Y.

14853(U.S.A.)

ABSTRACT The intramolecular vibrational-rotational energy transfer is followed in time along dassical trajectories of non-rigid(Ar3 clusters) and semi-rigid(H20) triatomics, where the exact amounts of rotational and vibrational energies are calculated for three body-fixed coordinate systems(Eckart, PAS, IMAS). It turns out that the Eckart system is allways a reasonable choice to partition such energies, although significative differences appear in comparisson with the while the PAS differs considerIMAS for highly rotationally excited molecules, ably from the other two. The mechanism for rotation-vibration energy transfer proceeds via centrifugal interactions in the cases studied.

INTRODUCTION Understanding (ref.1)

is

dynamics In

and

fact

of

small decay

1 is

and

lar

relaxation

regular

and

of Fermi

Previously

of

condition

highly

and

rotation

description

energy

can of

be separated

isolated

molecule

0022-2860/86/$03.50

and

more

low

polyatomics

about

lying

towards

molecules

Dynamical

and

spectroscopy

the

recently

excited

motion(ref.4),

in

studies

vibration

vibrorotational fluorescence

relation

~015

with

include

Corio-

intramolecu-

with

respect

to

the

problem

more

precisely

to

the

specific

transfer(ref.5)(Coriolis

those

studies

which

are

commonly

the

defined

momentum are

and

in

towards

couplings,

centrifugal

of

inter-

resonances).

most

that

studies

directed

rotationally

triatomics

kept

inertia by

the

equal

to

0 1986

we need

a function used

the

condition,

tem”(lMAS)

theoretical

(ref.2)

irregular

to

The

Eckart

vibration complete

interaction(ref.3).

ordinates(ref.6) system.

any

been

polyatomics

of

versus

actions,

which

to

have

centrifugal

mechanisms

to

spectroscopy.

interaction

rization

the

degree

essential

experimental

-rotation states

the

clearly

a clear of

systems(ref.7)

“principal

axis

tensor

becomes

condition

the

zero.

the

In

this

definition

of

particular are:

The

Eckart

system”(PAS) diagonal; three study

given and

the

components

of

we are

Elsevier Science Publishers B.V.

the

rotational

molecule-fixed

primarily

system by the “internal

co-

frame defined

by

holonomic axis

vibrational interested

sys-

angular in

530 the

technical

along tem

the

problem

classical

minimizes

situations the

ones(H20) the

of

with

vibrational

high

of

of

triatomics.

is

Waals

one

energies,

a good

Eckart

energy choice

doubt

or

many

follow

when

with

whether

sysin

to

nevertheless

clusters)

can

energies

the

Coriolis

energies,

der

vibrational

Although the

consequently

rotational

molecules(van

rotational

vs.

mOmenturn and

and

and

rotational

model

angular

molecules

non-rigid

at

best

partition

vibrational

semirigid

partition

dealing

the

trajectories

the of

of

we are

semi-rigid

Eckart

system

is

choice.

METHOD Classical

trajectories

equations

of

nard-Jones (two

computer

works

of The

where the

to

the

normal Euler

frame

x,n actual

angles

of

In

the

and

yea

I

Model

potential

the

the

angle

are

and

In

molecule

in

molecular

coordinates body-fixed

of

each

type

Euler

allways

plane

determined the

imposes

field

the

equilibrium a

of

that

For

in

the

condition

are

the

spe-

configuration; orientation.

system,

systems

in

triatomics each

the

space-fixed to

with

space-fixed

define

$I angles

expression

package(ref.8),

coordinate

energy.

Len-

start

random

the

Hamilton’s

pair-wise

order

it

is

rotating

spherical

system polar

an-

system.

partjcular

conditions

gives:

me(xOexu-yOuye))] components

intermediate and

force

Calculations

according

equilibrium an

of

angles B and

The

MERCURY universal

different

molecular

plane

condition

of

were:

I).

with

Cartesian

Eckart,

the

valence

mode at

the

is

the Table

to

the

integrating

models

normal

me(xOuye-yOexe))/(g

PAS the

TABLE

H2O(see

a modification

because

coordinates the

Ar3

amounts

system.

arctan[(;

for

each

determine

to

by numerically surface

for

added

absolute

IMAS),

third

each

+Eck=

in

specific

PAS,

in

then

introduce

easy

(Eckart,

of

is

program,

to

determine

relatively

bend)

energy

internally

we need

gles

one

energy

The

to

of

generated energy

function

and

amounts

rotational

were Potential

additive

stretches

cific

and

motion.

of

body-fixed

making

a atom system

and

x,

defined

ye

are

by B and

JI= 0.

now a relation

between

J, angles

parameters

Hz0

and

Ar3

Equilibrium bond distance/A 0.957 3.405 Equilibrium bond angle/degrees 104.52’ 60.0” Dissociation energy/Kcal/mol. 125.60 0.48 Frequencies/cm-T 1648.o;385o.o;39o6.o 32.24;22.79:22.79 Moments of inertia/amuA2 1.770;0.615;1.155 291.77;291.77;583.43 ----------------------------~~~~~~~~~~~~~~~-~~~~~~~_~~~~___~~________________

of

both

$

531

Eckart

and

qPAs=

PAS: arctan

JIEck + 0.5

where

x,

and

ye in

Finally, function nents

of (or

are the

(2~m~xaYa/~ma(x~-y~)]

the

coordinates

IMAS,

we don’t

coordinates,

between

but

in have

at

least

$ derivatives)

the

Exkart

any

analytical

we can

of

two

The

angle

and is

not

cal

any and

t

The

At)

It

systems

are

locity

-

is

the

of

the

component

instantaneous

Fig.l.Rotational

we obtain

that

Coriolis

J, due J, is

the

angles above

the

fact

to

in

each

in

this

system path

determined

when

is

are

Eckart

expression,

trajectory

the

system,

themselves

the

from

a

PAS.

& by

to

contributions remember

same vector

only

in

energy

ia

the

angular

differents

but

there

non-holo-

via

a numeri-

the

I/J angle,

the

molecular

from

one

system

along

the

conditions. the

projection

plane to

the

we know w com-

velocities

projections,

physical

on

w calculated inert

are

that

with

by different

systems varies

with

of

as

wx compo-

systems:

w components of

$tMAS

between

+ L$(t)At

differ

three

determine

determine

defined

systems

to

for

relation

as

important

vectors,

three

to

a

body-fixed

expression

IMAS case

At) and

not

IMAS coincides the

way

such

= $(t

ponents.

the

the

to

the

definition

only

rotational

different

of

analytical the

integration

$(t

the

In

PAS systems.

nomic

needed

derivatives, by derivation

expression

define

$IMAS’ +PAS + (~m,(yavxcc-rcrvycc)/~mcr(x~+v~)l where coordinates and velocities correspond

determined

frame.

is other.

trajectory

of

The

Due

to

the

fact

the

angular

and

it

angular

through

three

physically

of

constant,

the

but

the

is

vew2

velocity inversion

tensor.

for

At-3

in

Eckart(sharp

line)

and

IMAS(smooth

line).

of of

532 In to

summary,

the

are

going

to

artificial the

show

energies

a

semirigid

run

made

almost and

lie

the on

same z-axis

the

equivalent

take

such

time

both

rotational

in

the

because

axes

energy

plane.

while

vibration,

in-plane same

perpendicular

molecular

the

Coriolis

As we

PAS gives

in order

to

an orientation

order

and

to

conserve

The

water

an

cancel

that

some-

vibrational total

energy.

AND DISCUSSION trajectories

where

Eckart

calculations

at

Kcal/mol)

have

two

IMAS are

at

several

model,

We have

other

rotation

the

negative

RESULTS

We have

and

I,y,

systems

the

between

considerably

giving

NUMERICAL

body-fixed while

Eckart

product,

enhance

(-125

three

plane,

partition

inertia

times

the

molecular

and

where

for

system 120

the

H20

is

At-3.

expected

Kcal/mole,

initial

and

to

close

rotational

molecule

be a good

to

the

is

reference

frame

dissociation

energy

is

very

a

energy

high

(-60

Kca I/

/mol). We observe similar, can

gy

for

although

reach

in

In

the

is

only

expanded the

that

energies

transition The energy

of

0.48

precise goes

maximum.

Ar3,

to in

of

high

observe

through

a van

der

vibrations

the

discrepancies

of

in

energy when

a clear

of

and very

between

Eckart

than

distances energy

fig.

energy.

via

ener-

we have

IMAS. due

In to

a more

fact, the

sharp

1)

follows:

between

transfer

energy

dissociation and

Eckart

as

very

rotational

and

(See

IMAS are

Coriolis

slow

in

be described

the

where

very

system.

and

high

molecule

last

elongation

Eckart

At-3 are

profile

can

mechanism

of

Waals

this

flow

of

appear,

of

somooth

energy

a minimum

indicates

behavior

a consequence

The

Coriolis

mechanism

This

as

IMAS show a more

the

the

discrepancies

values

we have

Kcal/mol.

scale

molecule

significative

Eckart

case

water

atoms

Rotational reach

a

centrifugal

interactions.

Acknowledgement. This

work

collaboration

has in

been

supported

research

in

between

part two

by a NATO grant of

us

(J.S.

and

for

international

G.S.E.)

REFERENCES 1

E.B.Wilson,J.C.Decius

and

P.C.Cross,

“Molecular

Vibrat

ions”(Mc

Graw-Hi

11 ,N.Y.

1955) . 2 3 4 5 6 7 8

in “Molecular Spectroscopy: Modern Research”, eds. K-N. Rao and I.M. Mills, C.W. Mathews (Academic Press, N.T. 1972). G.M. Nathanson and G.M. McClelland, J.Chem.Phys. 81, 629 (1984); Chem Phys Lett. 114, 441 (1985). See recent articles by P. Brumer, M.J. Davis, W.P. Reinhardt, etc. J.H. Frederick, G.M. McClelland and P. Brumer, J. Chem. Phys (in press 1985) R. Meyer and H.H. Gunthard, J.Chem.Phys. 9, 1519 (1968). “Symmetry properties of molecules” (Springer-Verlag, Berlin, 1982) G.S. Ezra, W.L. Hase, QCPE, 3_, 453 (1983).