Rotation-vibration energy levels of CO2 using effective normal coordinates : definition of the spectroscopic constants

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Rotation-vibration energy levels of CO2 using effective normal coordinates : definition of the spectroscopic

constants

G. Amat

To cite this version:

G. Amat. Rotation-vibration energy levels of CO2 using effective normal coordinates : def- inition of the spectroscopic constants. Journal de Physique, 1988, 49 (8), pp.1325-1328.

�10.1051/jphys:019880049080132500�. �jpa-00210814�

Rotation-vibration _energy levels of _CO2 using ^{effective normal}

coordinates : definition of the spectroscopic ^constants

G. Amat

Laboratoire de Physique Moléculaire et Atmosphérique, Université Pierre et Marie Curie et CNRS,

tour 13, ⁴ place Jussieu, ^{75252 Paris Cedex} 05, ^France

(Reçu ^{le 29 mars} 1988, accepti ^Ie 7juin 1988)

Résumé. 2014 Cet article traite du calcul des niveaux d’énergie ^de vibration-rotation de CO2 par la méthode des "coordonnées normales effectives". La formule donnant les éléments de matrice diagonaux

de l’hamiltonien transformé,

fonction des nombres quantiques ^{et des} constantes spectroscopiques, peut être écrite

forme plus simple que celle publiée précédemment,

qui facilite la solution du problème ^inverse.

Abstract.2014 This _paper deals with the calculation of rotation-vibration _energy levels of CO2 using

"effective normal coordinates". The formula giving ^the diagonal matrix elements of the transformed Hamiltonian in terms of quantum ^{numbers and} spectroscopic ^constants

be written in

form

simpler ^{than the}

previously published ^and

convenient for the solution of the inverse problem.

(1988)

Classification

Physics ^Abstracts

35.20P

Short communication

As Bord6 pointed ^out [1], ^the convergence of the

perturbation calculation used to compute ^the rotation-vibration _energy ^{levels of} C02 ^becomes

less and less satisfactory when the value of the quantum ^number v3 increases. This is due ^to the large ^{value of} k133, coefficient of q1 q3 ^{in the}

cubic term VI ^{of the} potential ^{function V.}

expanded ^{in a} power series with respect ^{to nor-} mal coordinates. In an attempt ^to improve ^the situation, ^{we have} proposed [2] ^{to replace the}

standard normal coordinates _qn by "effective normal coordinates" * qn according ^{to the fol-} lowing ^{scheme :}

where _Pn and _*pn are momenta conjugate to qn and _*qn respectively (ra = 1, 21, 22, 3) ^{and where}

a, b1, b2 ^and b3 ^are functions of _v3 defined by

the relations

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:019880049080132500

1326

Symbols W3, k33, 3" k333, 3’ (s ^{= s’ or} s 0 s’) ^desi-

gnate the coefficients appearing ⁱⁿ Vo, V1 and V2 respectively. The effect of ^a is to introduce in the first order Hamiltonian Hl ^an _operator pro-

portional ^to * ql whose matrix elements will _ap-

proximately ^{cancel out} the troublesome matrix

elemen ts (VI V3 k133 ^{* q1} * q3 V, ± 1, V3) ; ^{the ef-}

fect of 61) b2 ^and b3 ^{is to restore the} harmonicity

of Ho. ^{In this} formulation, the standard _power series expansion (1) ^{of the} potential ^{function V}

is replaced by power series expansions ^{with res-} pect ^to the effective normal coordinates _{* qn :}

the coefficients _w *k,,,,, ^and *k[j[j[jl[jl, appea-

ring ⁱⁿ * Yo, * Y1 ^and * V2 respectively, depend

upon the value of _{v3 ;} therefore we use the same

potential ^V throughout ^the calculation , ^{but we}

have a different expansion (right hand side of

Eq. (5)) for each value of _v3. The relations bet-

ween the new coefficients and the standard coef- ficients appearing ⁱⁿ equation (1) ^{can be found}

in reference [3]. ^{As an} example

As we have already said, *Vi contains, ^{in addi-}

tion to the cubic terms, ^{a linear} term . ki * ql .

Actually :

Grenier-Besson and Mahmoudi [3] ^{have cal-} culated, up to second order, the matrix ele- ments of the rotation-vibration transformed (1)

Hamiltonian H’ using effective normal coordi-

nates (ENC). ^The diagonal ^{matrix elements of} H’ , expressed ⁱⁿ cm-1, ^can be written as the

sum (* G + * F) ^{of a} vibrational and a rotational

spectral ^{term :}

The expressions ^{of the ENC} spectroscopic

constants appearing ⁱⁿ equations (7a, 7b) ^can

be found (2) in reference [3]:

The ENC spectroscopic constants * x,,, (S

^s’

or s # s’) ^and * g22 ^are given, ^{in terms of} *úJ"

*kss,s,2 * k"’r’r ^and Be by formulae ^identical

(3) ^{to those giving the} corresponding ^standard spectroscopic ^{constants in terms of} úJ" k"""

ks,s’s’ ^and Be. ^For example :

(1) ^{H’ is obtained} by performing

^contact

transformation H’

T H T-1

the rotation-

vibration Hamiltonian H. * C ^{is a function} of e3 :

The calculation described above requires only

that the quantity ^{written on} the left hand side of relation (3) is small. From now on, ^we shall

assume that this quantity ^is exactly ^{zero : sim-} plifications ^can then be made in formulae (7a, 7b).

1) ^{We can now write (compare relations (3),} (6a) and (6b))

As a consequence, the terms proportional ^to

* kl33

and * kl

will I

out . _{* x1} +

2014

and

will cancel out in * Xl +

*w1 *W1

* Xl3 (tJ3 + !), in * X2 + * z23 (v3 + )) , in *C +

* X3 v3 ⁺ 1 ^{+ *z33} (v3 ⁺ 1 2. Therefore, ^{in for-}

mula (7a), ^{we can} omit the coefficients _*Xs, re-

place ^{* Xl3 * z23 * X33} by * X13 * X23 * X33

resp ectively ^and replace ^{* C} by * C’ ⁺ * C" .

In the same way, it can be shown that the terms proportional ^{to k133}

1/2

in * B and in

w1

-

* a3 w3 ⁺ 2 ) ^{cancel out} in formula (7b). ^After

these simplifications, ^{all terms} involving * k133 ^*w1

have disappeared in the formulae giving ^the

energy (4) (*ú)1 ^is the zeroth order _energy dif- ference between the states Iv) and Iv - 1) ^which

are coupled by ^the operator * k133 * ql * q3 ).

2) By ^a redistribution of terms bet-

ween *Ú)1 and *x13 w3 + Z), ^{between *Ú)2}

and *X;3 (v3 +1 /2 ), ^{between *C’, *w3} (V3 + 1/2)

and *x33 tJ3 + 2 and between * B and

-

* cx3 (V3 ⁺ 1/2), ^equations (7a, 7b) ^{can be re-}

written as follows :

(2) ^There

^{few misprints}

^reference [3] :

in * a2 the term proportional to * k122 ^{within the}

brackets should be written with

+ sign ^instead

of the - sign; ^the expression (25) given ^{for the t-}

type doubling ^constant qo ^{should be} multiplied by c2

(3) ^{It is not}

^{for the} rotational spectroscopic

constants *B *ad *D.

with

(4) This situation can be compared ^{with the}

one

met in the

of Fermi resonance : there,

the states IVI V2) ^and Iv1 - 1, ^{v2 +} 2) ^are coupled by ^the operator k,22 q, (q22, ⁺ q222); ^{the energy}

difference between the two states is _{W1 -} 2W2- The modification of the contact transformation

performed

^the Hamiltonian,

proposed by

Nielsen [4] ^{in the case of Fermi} resonance, has the effect of removing ^{from the} diagonal part ^of the transformed Hamiltonian all terms involving

kl2’2

wi

-2W2

1328

where *C’ ^is given by equation (lOb)

and

In formulae (12a, 12b) the zeroth or-

der spectroscopic ^constants (Wj, ^{cv2, w3,} Be)

are identical to the standard spectroscopic

constants : they ^{do not} depend upon v3.

The 11 second order spectroscopic ^constants (*x *9 *Q *D) ^depend upon v,3; when _v3

- 1/2, the limits of these "constants" are identi- cal to the corresponding ^standard spectroscopic

constants. The function * C" ^{as defined} by equa- tion (loc) ^{does not} depend explicitly upon a;

when _v3 --+ - ’, *C" --* ^{2 C" ; this limit, C", is}

different from zero (5).

About the two formulations obtained in

paragraph 1) ^{and in} paragraph 2) above, ^{we can}

make the following ^{comments :}

-

when, in relation (3), ^the symbol = ^is replaced by ^a symbol = , it is clearly ^advanta-

geous ^to make the simplifications described in

paragraph 1);

-

the formulations obtained in paragraphs 1) ^and 2) ^are strictly equivalent. ^{In the direct} problem, ^when energy levels are computed ^from

the potential, ^{it makes no} difference to use one

formulation or the other. In the inverse vibra- tional problem however, ^{when one tries to deter-}

mine the potential ^{from the} energy levels, ^formu-

lation 2) ^{should prove to be more} convenient: the

dependence of effective spectroscopic ^constants

upon v3 has been concentrated in the second order constants (.*Xss’ ^{*g22 * a’} * D) ^{and in}

* C" ; ^{as a} consequence, the zeroth order _spec-

troscopic ^constants (w9 Be ) ^{do not} depend upon v3 and the limits of second order constants when

v3 --+ - -1/2 ^{are identical to} the standard values of these constants.

References

[1] ^{BORDE, J., Mol. Phys. 32} (1976) ^1027.

[2] ^{AMAT, G., Chem. Phys. 71} (1982) ^363.

[3] GRENIER-BESSON, ^{M.L. and} MAHMOUDI, A., ^J. Phys. ^{France 44} (1983) ^923.

[4] NIELSEN, H.H., Phys. ^{Rev. 68} (1945) ^181.

(5) ^{The constant C" is given by a} formula iden- tical to the

giving *C" (Eq. (10c)), ^{but all}

the ENC coefficients (with *) ^{must be} replaced by ^the corresponding ^standard coeflicients (wi-

thout *). ^{The constant C" is indeed present in}

the standard calculation (it ^{comes from the ma-}

trix elements of operators ^of degree ^{4 with}

pect ^to the vibrational operators in the second order transformed Hamiltonian). ^{C" is always}

omitted in the standard calculation because it

disappears ^{as the} positions ^{of lines}

^obtained

from the difference of two spectral ^terms. *C"

cannot be omitted in the ENC calculation be-

cause

it is

function of _v3.