Study of the anharmonic resonance between ν2 + ν±1 4 and ν2 + 3 ν± 1 6, and between ν3 + ν±14 and ν3 + 3 ν±16 infrared bands of CH3Cl35

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Study of the anharmonic resonance between ν2 + ν±1 4 and ν2 + 3 ν± 1 6, and between ν3 + ν±14 and ν3 + 3

ν±16 infrared bands of CH3Cl35

N. Bensari-Zizi, C. Alamichel, G. Guelachvili

To cite this version:

N. Bensari-Zizi, C. Alamichel, G. Guelachvili. Study of the anharmonic resonance betweenν2 +ν±1 4 andν2 + 3ν±1 6, and betweenν3 +ν±14 andν3 + 3ν±16 infrared bands of CH3Cl35. Journal de Physique, 1981, 42 (6), pp.827-834. �10.1051/jphys:01981004206082700�. �jpa-00209068�

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Study ^{of the}anharmonic resonance between _03BD2 + 03BD±14 ^and^{03BD2 +}³03BD± 16,

and between _03BD3 + 03BD±14 ^and03BD3 + 3 03BD±16 infrared bands of CH3Cl35

N. Bensari-Zizi (*), C. Alamichel

Laboratoire de Photophysique Moléculaire du C.N.R.S. (**), ^Bâtiment213, Université de Paris-Sud, 91405 Orsay Cedex, ^France

and G. Guelachvili

Laboratoire d’Infrarouge ^associé^auC.N.R.S., Université de Paris XI, ⁹¹⁴⁰⁵Orsay, ^France

(Reçu ^{le 18}^décembre1980, accepté ^le³jévrier 1981)

Résumé. ²⁰¹⁴On a identifié environ 500 raies de _03BD2+ 03BD4 êt1 300 de _03BD3+ 03BD4 ^surûnspectre ôbtenuên^{très haute} résolution (0,007 ⁵cm-1). ^La^résonanceanharmonique, déjà ôbservéeêntre03BD±14 êt³03BD±16, âété trouvée également

entre 03BD2 + 03BD±14 et 03BD2 + 3 03BD±16, âinsiqu’entre 03BD3 ⁺03BD±14 êt03BD3 + 3 03BD±16: ^sonînfluence^sur^les^centres^de^bande,êt

donc sur la mesure des constantes d’anharmonicité _x24et x34, n’est pas négligeable. Dans les deux cas on a donné

une explication partielle ^duspectre, ^ne^tenantpas compte ^des^autresrésonances dépendant ^{de J.}

Abstract. ²⁰¹⁴About 500 lines of 03BD2 + 03BD4 and 1 300 of 03BD3 + 03BD4 have been assigned ^{on a}spectrum obtained in _very

high resolution (0.007 5 cm-1). ^Theanharmonic resonance, already observed between 03BD±14 ^{and 3}03BD±16, ^{has also}

been found between _{03BD2 +}03BD±14 and 03BD2 ⁺³03BD±16, and between _03BD3+ 03BD±14 and 03BD3 ⁺3 03BD±16: its influence of the band centres, and thus on the measurement of the x24 and x34 anharmonicity constants, ^is^notnegligible. ^In^both^cases

a partial explanation ^{of the}spectrum ^{has been}given, ^nottaking ^{the other}(depending ^onJ) perturbations ^into

account.

Classification

Physics ^Abstracts ^.

33.20E

1. Introduction. ^-The anharmonic resonance

between vi 1 ^{and 3}v6 ’ has been studied by ^different authors [1-5] ^{who found}^acoupling ^constant^about

3.5 cm-1. The same resonance of course happens

between 2 v+4 2 and v+41 ⁺³v6+1 [2, 6, 7] : ^however

the coupling ^constant^{should be}/2 ^bigger,^which

does not seem to be experimentally ^confirmed[7].

The same resonance should also exist, ^{with the}^same coupling ^constantaround 3.5 cm-1, ^betweenV2 ⁺vi 1

and _V2+ 3 v61, ^or^betweenV3 + v41 ^andV3 ⁺3 v± 6

it is the subject ^{of the}present work, ^{which is}only

concerned with the isotopic species CH3C135.

2. Experimental conditions. ^-The spectrum ^has been obtained in ^asingle recording from 3 500 _up to 6 000 cm-1 with a Fourier transform interfero- meter [8-11] ; only ^twoparts ^{of this}spectrum ^have been used for the present work : the 3 690-3 870 range

for v3 ⁺V4 and the 4 310-4 450 _rangefor _v2+ v4.

(*) Detachee de l’Universit6 Mohammed V, Faculte des Sciences, Charii Ibn Batouta, ^B.P.1014, ^Rabat(Agdal), ^Maroc.

(**) Laboratoire associ6 a l’Universit6 Paris-Sud.

The source is an iodine lamp and the detector used in our case is a PbS cell. The pressure, the optical path ^{and the}resolution are respectively ^0.8torr, 28 m and 0.007 5 cm-1.

3. The v2 + v4 band. - 3 .1 ASSIGNMENTS. ^-The v2 ⁺vi ¹ ^{band is}presumably ^submitted^tomany other interactions than the anharmonic ^resonance with _v2+ 3 v: 1. Âsâ^matterôffact, the v2 ândvs modes being coupled by âstrong Coriolis ^resonance

[12], ^thereforethe v2 + v4 ^{band is}linked to both components ^ofthe v4 + v5 ^band.^For^thesame reason

the _V2+ 3 v+61 ^{band is} coupled ^toboth components of _v.+ 3 v6+1. ^Besides,^thestudy of v2 ⁺V6 and _v5+ V6 [13] has shown that it is _necessaryto take into account a Fermi ^resonancebetween

with a coupling ^constant^{W’ - 0.6}^{cm - 1}^: ^so,^we

must expect ^a^similarcoupling ^betweenvs 1 ⁺³v6’

and 2 v3 + 3 v61. ^Thecorresponding scheme has been drawn on figure 1, ^{where the}l(2,2) ^resonances

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01981004206082700

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Fig. 1. - Scheme of interaction. The (v2 =1, V4 =11, K) ^{level is} coupled ^to^the(vs =1-1, v4 =11, K -1) ^and(v5 = v4 =11, K+1)

levels by ^a^Coriolisresonance; at low K, ^itis beneath these two

levels; ^when^Kincreases, ^itremains under the second one, but overtakes the first one at K - 9. The (v2 =1, v4 =1-’, ^{K )}^level

is coupled by ^thesame resonance to the (v5 =11, v4 =1-1, K + 1)

and (v5 = v4 =1-1, K -1 ) levels; ^at^lowK, it is also beneath these two levels; ^when^Kincreases, ^itremains under the first one, but overtakes the second one at K N 10. We remind that K is the absolute value of k.

of the v4 and v6 ^modeshave been omitted to make it more simple. ^We^did^not^intend^tostudy ^{all this}

very complicated ^bandsystem, but wanted only ^to explain ^{the Fermi}^resonancebetween v2 ⁺v+41 ^and

v2 ⁺3 v+61; ^so^we^haveonly considered the central part ^{of the}spectrum ^ofv2 + V4’

As a matter of fact, ârough calculation shows that the two centres of v2 ⁺v¡ 1 ândV2 ⁺3 v: 1 âre^close

and that a level crossing ^isexpected ^for^a^{low value}

of K, i.e. in this part ^{of the}spectrum where v2 ⁺V4 is well separated from V4 ⁺v5, whose centre is 100 cm -1 above (see ^commentôffig. 1). Îtîsexactly

what we observe on the spectrum ^{where the}Q ^branches stop being regularly spaced between 4 358 and 4 379 cm-1, ^whichobviously ^{is the}sign ^of^a^Fermi

resonance with level crossing. ^Thepositions ^{of the}

different Q ^branches^aregiven ^{in table}^I,^{with their} assignments ^wehave obtained according ^to^the

method explained just ^below.

As the B", D"j ^andD"JK ^constants^ofthe fundamental level are well known [13-15], ^wehave been able to confirm our assignments with the combination rela-

tionships connecting the three RK(J - 1), QK(J)

and PK( J ⁺1) ^lines^{of the}^same^K^{and AK.}^For

instance the relationship between the wavenumbers of the RK(J - 1) ^{and the}PK( J ⁺1) ^lines^can^be

written :

Table I. ^-Positions of ^theQ branch heads of ^V2⁺V4’

With this method it is _easyto identify the value of J without ambiguity; ^{but it is}impossible ^todistinguish

the values 0, ¹and 2 for K, because the quantity (4 J + 2) D"JK K2 îs^toosmall for these low values of K. Generally these values of K are identified by continuity : ^forexample, ⁱⁿ^caseôfâperpendicular band, ^wefollow ^theQ branches, starting ^fromâ higher ^{value of}^{K. It}^{is what}^wehave done for RQ2,

’Q ¹ândRQo (1), which have been assigned just âfter RQ4 ândRQ3. ^{On the AK}^{= -}¹^side,the combination relationships helped ûs^toassign ^the^K⁼⁵

and 4 sub bands and also two K ⁼3 sub bands;

but there were four Q branches between these two

’Q3 branches and the RQo branch : therefore these four Q ^branches^werenecessarily ^twoPQ2 ^and^two PQ1 ¹ branches. The figure ^2,corresponding ^to^the

Fig. ^2.^-Crossing ^{of the}v2 + v+41 ¹^and^{V2 +}³v+61 ¹ ^bands.

(1) ^{For K}⁼0, only the relationship RK(J - 1) ^-PK(J ⁺1) can be used ; because of the strong ^Coriolis^resonance^betweenv2 + V4 and _{v4 + vs,}and because of the 1 doubling ^{of the}v4 mode, ^the QK(J) ^lines^cannotbe connected to RK(J - 1) ^andPK(J ⁺1).

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already mentioned table I, give ^thepositions ^{of the}

heads (2) of the different Q branches and their assign-

ments. Several Q ^branches^are^shown^onfigures 3, 4, 5 and 6.

Fig. ^3.^-^TheRQ3 ^branchof vz ^{+ V4-}

Fig. ^4.^-^TheRQo ^{branch of}^{V2 + v4.}

Fig. ^5.^-^ThePQ1 ^andPQ2 branches of the interacting ^bands.

(2) ^We^{call «}Q ^branchhead » the first line PQK(K ) ^orRQK(K ⁺1), head and the sub band origin, considering ^that(B’ - B") J ( J ⁺1)

of low J, ^{for which}^weconsider the Coriolis interaction negligible; ^isnegligible for these low values of J.

as we shall explain ^it^later,^we^do^notdistinguish ^theQ ^branch

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830

Fig. ^6.^-^ThePQ, ^andPQ3 branches of the interacting ^bands.

3.2 INTERPRETATION. - a) ^When^twoênergy^levels E1 ândE2 âreonly coupled by â^Fermiresonance, the

perturbed ^energy^isgiven by :

where W is the coupling ^constant.

If _viand _V2are the unperturbed transitions between the same lower level and respectively E1 ^andE2,

then the perturbed transition is given by :

Except ^of^course^forvery high values of J, ^{all the} lines of the same K sub band give ^{more or}^{less the}

same information about the Fermi resonance; so, we can only consider the hypothetical ^J⁼^{0 levels}

in equation (1) ^or^{the sub}^bandorigins ⁱⁿequation (2).

And in the present ^case^it^wasabsolutely necessary to follow this method in order to get rid of the strong Coriolis interaction between v2 ⁺^V4^and^V4⁺^{v5 :}

we do not pretend that the sub band origins ^are^not

shifted at all by the Coriolis resonance, but, ^at^least if we consider K sub bands far enough ^{from the}^crossing of V2 ^{+ V4}^andv4 + v5, the influence of the Coriolis interaction is certainly very small at low

Fig. ^7.^-^TheRQo ^{branch of}v3 + v4.

Fig. ^8.^-Tbe RQ3 ^{branch of}^v3⁺^v4.

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values of J. Then, using ^therelationship (2) ^{for the}Q

branch heads as if they ^were^{sub band}origins, ^we

can take :

with in both cases :

In these conditions the equation (2) is that of an hyper-

bola whose asymptotes âre^thestraight ^linesgiven by equations (3). Ît^{is what}^weôbserveôn^thefigure 2,

where the b1 and b2 asymptote slopes ^arequite ^close

to those expected ^for V2 ⁺v4+1 ^andV2 ⁺3 v+61.

Besides; ^thefigure ²gives ^{W -}^2.65cm-1, ^while

the value 3.5 is expected.

b) ^Thepreceding interpretation may be easily improved ^because^our« Q branch heads » are not

exactly ^sub^bandorigins, and because in reality ^the equations (3) ^ofthe sub band origins ^are^notlinear;

thus, ^to^be^moreaccurate, ^wehave dealt with the Fermi resonance between the hypothetical ^J⁼⁰

upper energy levels, using this time the equation (1).

For this _purposewe have fitted the upper energy levels of the different lines of the same K" sub band with the following empirical ^{formula :}

It is quite possible ^to^{do this}fitting, ^since^we^know

the values of the constants A ", B", Dj’, D"JK ^andD"

of the fundamental level, which is the lower level of the involved transitions. In equation (5) ^we^have^used

up ^tosix parameters, ^butonly the first three of them have a physical meaning : ^forunperturbed ^levels,

CK, ^isequal ^{to -}D’J, BK, ^{is the}quantity B’ - D’JK K’2,

and ÈK, ^isequal ^to

For each sub band a least squares calculation has given

the optimal values of the parameters involved in the

equation (5) ; ^the^results^aregathered ^{in table}^II,

.while the observed wavenumbers are given ⁱⁿ

table III (3).

(3) The observed wavenumbers, with their assignments, ^of

tables III and V have been deposited ^assupplementary publication

No. SUP 90054 with the British Library ^at^BostonSpa, Wetherby,

West Yorkshire, ^LS237BQ, England. Copies âreâvailableâs

microfiches or as Xerox copies directly from the British Library

on a prepaid flat unit coupon basis.

Table II. ^-BK ^andEK parameters of V2 ⁺V4 band.

and for _V2+ 3 _v6the expected value of B’ is

So, if the anharmonic resonance was the only pertur- bation, ^theparameters Bx ^{of table}^IIshould be bet-

ween these two values. In fact we can see that the values of hK ^are^{in the}range 0.435 0-0.438 3; ^these particularly ^lowvalues, ^whichcorrespond ^to^the spreading ^{of the}Q ^branchestowards the low frequen-

cies on the spectrum (Figs. ^3,^{4, 5,}6), ^are^causedby ^the

Coriolis resonance with _V4+ vs. It is indeed well known that, ^when^two^levels^arecoupled by ^a^Coriolis

resonance, the B constant of the upper level ^seems to be increased of AB, ^{while the}^Bof the lower level

seems to be decreased of the same quantity. ^{This fact} explains why ^theBK parameters ^{of table}^II^{are so}low,

since the _upperlevels of v2 ⁺V4 ^arebelow the upper levels of _V4_{+ v5 to}which they ^arecoupled (see ^com-

ment of figure 1).

We have considered that the parameters EK,,

i.e. precisely ^thehypothetical ^J⁼⁰^upper^energy levels, ^were^not^shiftedby ^the^Coriolisresonance.

With this assumption, ^theparameters Ex- ^are^the eigenvalues of the secular equation :

where EE is the part ^{of the}unperturbed energy which does not depend ^on^{J :}

We have processed ^{the 13}EK, levels of table II in a

least _squarescalculation, according ^toequation (1’);

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for both bands we held DK ^andD& ^fixed^at^{the value}

8.342 ^x10-5 [13]. ^Thefollowing results have been obtained with a standard deviation of 0.016 cm-1 :

A " is given ⁱⁿ[13] and B" in [13, ^14,15] ; ^we^have

taken a6 ânda6 ât^reference[16], at and a4 ât^refe-

rences [1, 3, 4] ; ^wehave obtained a2 ^andaf by ^com- bining the results of [16] ^and[13].

The values obtained for A’ ^must^becompared ^to

those obtained for the fundamental bands : 0.319 for A’4 [4] and 1.310 0 for A’6 [16]. ^For(A’-B’) ^the agreement ^betweenexpected and observed values is

quite good for both bands. On the other hand, ^as^it

has been already said, ^thevalue obtained for W is lower than expected; this fact is not modified if

we take into account the weak z Coriolis resonance

which accompanies the anharmonic resonance, hold-

ing fixed its coupling constant at the value of refe-

rence [1].

4. The V3 + v4 ^band.^-^4.1ASSIGNMENTS. ^-We have assigned six sub bands (from K" = 1 up ^to K " ⁼6) ^on^{the AK}^{= -}¹^{side and}^ten^{sub bands} (from K" = 0 up ^toK" = 9) ^on^{the AK}⁼⁺¹

side. All these sub bands look unperturbed ; ^theQ

branches are regularly spaced ^andspread ^towards

the low frequencies, ^as^{it is}expected ^{since B’}⁼^0.439⁷

while B" ⁼0.443 4. We have assigned about 1 300 lines,

all belonging ^to^theCH3C135 species, using ^{the combi-}

nation relationships (except ^of^course^{for the}RQo

series because of the 1 doubling).

4.2 INTERPRETATION. ^-We know that _{the V3}+ v4 ’ ¹

band is coupled ^to^V3⁺³v+61 by ^an^anharmonic

resonance; ^anapproximate calculation shows that the band centres are respectively around 3 772 and 3 759 cm-1; ^on^{the AK}⁼⁺¹side, ^theupper level of _V3+ v4 is expected ^to^bealways ^{above the}^corresponding upper level of _V3+ 3 _v6,the distance bet-

ween them increasing ^{from 15}^cm-1up ^to20 when K’ increases from 1 _upto 10. But on the AK ^{= -}1

side, â^levelcrossing îsexpected ^for^K’⁼⁷ôr^8,

i.e. for ’Q8 ôr’Qg ^we^haveunfortunately ^notôbserv- ed, because they âre^{weak and}overlapped by vi ⁺V3

(Fig. 9).

As we did for _V2₊_v4,we have processed ^{the lines}

of each sub band in a least _squarescalculation accord-

ing ^toequation (5); ^{when this}equation is used with three parameters, the standard deviation _mayreach 0.019 cm-1; but when this equation is used with six parameters the standard deviation is always ^below

0.005 cm-1. The table IV gives the values of the

BK ^andEK parameters ^{when the}equation (5) ^{is used}

with three parameters. ^It^canbe noticed that the Bx

parameters ^arealways smaller than the expected ^value B’ v,+v4 ⁼^0.439^7;^thedifference

increases with K on the AK ⁼+ 1 side, and reaches 0.040 cm-1 for K ⁼10; this fact involves a shift

AB.J(J ⁺1) around 1.6 cm-’ at J ⁼20 on the spectrum. ^It^reveals^astrong Coriolis interaction : the perturbing level would always be above its homo-

logous ^{level of}V3 ⁺v4 ; the strong variation of BK,

for K N 9 or 10 on the AK ⁼+ 1 side, ^would^mean

that a level crossing ^occurs^{in this}range. The simplest

scheme would imply ^asingle perturbing band centred above v3 + v4 ; ^theperturbing (vx ⁼?, ^{K -}1) ^level

would be overtaken by ^the(V3 = 1, v4 ⁼11, K) perturbed ^levelât^K⁼¹⁰ôr^11.Âs^we^have^not

Fig. ^9.^-^ThePQ6 ^{branch of}v3 + V4 overlapped by V1 + v, [18]. CH,C]31 ^lines^aremarked with an asterisk.

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Table IV. ^-BK ^andEK parameters of V3 ⁺V4’

seen any perturbing ^band^on^thespectrum, ^we^have sought ^thisperturbing ^bandonly by calculation.

Among the different possibilities ^that^wehave consi-

dered, only ^the(v2 ⁼1, v5 ⁼11, V6 ⁼1-’, K - 1)

and (V3 ⁼2, v5 ⁼1-1, V6 ⁼11, K - 1) ^levels^can

cross the (v3 = 1, v4 ⁼1’, K) ^level^for^{K -}^10.

The first one must be excluded, because the (v3 ⁼^1,

v4 ⁼11, 3) level would be strongly perturbed, ^which

is untrue on the spectrum. ^In^theseconditions the

only perturbing ^bandshould be 2 _V3+ vs ⁺v6, which is coupled ^to2 V3 + v2 ⁺V6 by a ^Coriolis

resonance. We have then built the corresponding

model but we did not succeed to obtain a good ^fit.

More details about these trials will be given ^{in the}

thesis of N. Bensari-Zizi, that will be submitted at

Paris-Sud University. Âs^we^could^not^find^this strong resonance, depending ôn^J, ^whichmainly perturbs ^{the AK}⁼⁺¹^sideôfV3 ⁺v4, ^wehave tried to determine nevertheless the parameters ôf^the band by â^leastsquares calculation, êitherôver^the

EK ^levels,^as^we^didfor v2 ⁺v4, ^{or over}the low J lines of the AK ^{= -}1 side.

a) Least squares calculation over the Ex ^levels.^-

Two calculations have been done according ^to^the

method already applied to v2 ⁺v4 (Eq. ( 1’)), ^one^over

the 16 EK ^levels,and the other over the six levels of the AK ^{= -}1 side. The results are the following :

(a) Fixed value.

OCA 3 ândCXB 3 âre^takenât^reference[171, at and a6 ât^reference[16].

In both calculations the coupling ^constant^{has been}

held fixed at the value :

As we have not observed the crossing ^between^V3⁺v4 and _V3+ 3 _v6,our information about the perturbing ^{band is}poor, and it has been impossible ^to^leave

the coupling ^constant^freein the least _squarescal- culation.

b) ^Leastsquares calculation over all the lines of

the AK ^{= -}1 side with J 20. ^-Only ^{the anhar-}

monic interaction is taken into account. This calcula-

tion, ^over²³³lines, ^hasgiven ^thefollowing ^results

with a standard deviation of 0.012 cm -1.

As in the preceding calculation we have fixed W ⁼3.535 + 0.02 K AK. For the three levels we

have held fixed the centrifugal distortion constants at the values :

If the same 233 lines are processed ^{in the}least squares calculation, holding ^{this time}^W^fixed^at⁰(i.e. neglect-