Volume 104. number 1 CHEMICAL PHYSICS LETTERS 27 January 1984

TRENDS FOR INTENSITIES OF VIBRATIONAL OVERTONES IN METHANE

Kenneth FOX Department of Physics and Astronomy, University of Temessee, Knoxvilie. Te,messee 3 7996-1Z000. Uti

Received 12 August 1983; in fimal form 26 October 19S3

The concept of local vibrational modes is examined quantitatively for spherical-top molecules. New theoretical formula- tions of transition electric dipole moments are developed for diatomics and local-mode spherical tops. Transition moments inferred from high-resolution spectra of va, 2va and 311s in ‘*CHQ and 13CH4 constitute the data base from which compari- sons between theory and experiment, and predictions, are made.

1. Introduction

In recent years, various workers have sought to characterize certain higher vibrational states of poly- atomic molecules by the excitation of a single bond or bonds [l-3] _ This “local-mode” interpretation is con- trasted with the classical “normal-mode” description in which many or all molecular bonds play a role [4] _

The concept of normal vibrations emphasizes small (strictly speaking, infinitesimal) amplitudes of oscilla- tion_ However, in high-energy vibrations, it would ap- pear that intramolecular excursions should be large. And the thought is that some of these vibrations are localized on only one oscillator.

The concept of local modes in molecular vibrations has several attractive features. In high vibrational exci- tations of polyatomic molecules, many normal vibra- tions are required to properly describe the physical state of the system. The density of quantum energy levels in such a state increases rapidly. The local-mode view permits this complex situation to be described by the excitation of a single bond in the simplest case. This description has particular utility for molecular dis- sociation in which the breaking of a bond occurs some- where high on a vibrational ladder. Local modes may also facilitate the calculation of excitation energies and especially transition rates to high-lying vibrational states from the ground and intermediate states.

The purpose of the present work is to explore quan- titatively the concept of local modes of vibration as it

applies to spherical-top molecules in general and to methane in particular. Similar investigations have pro- duced preliminary results [ l-3,5] _ This paper takes, as its experimental data base, absolute intensities of vibration-rotation lines in infrared spectra of v3 [6], 2v3 [7] and 32~~ [S] of both 12CH4 and L3CH1_ The measured line intensities [6-S] have been convenient- ly characterized by transition moments for each band of the isotopic variants [9] _ Systematics of these quan- tities are studied in the local-mode context, and gener-

alizations are suggested for other spherical tops.

2. Transition moments

The electric dipole moment operator which charac-

terizes vibration-rotation transitions is a complicated

function of normal coordinates for spherical-top mole- cules [lo] . A local-mode, in contrast with a normal- mode, view takes this operator to be a function of a sin- gle coordinate. That is the generalization of the case for a diatomic molecule in which the dipole moment function n(x) is expressed as a power series in the dis- placement x from equilibrium internuclear separation_ The usual expression is

I.l(-r) = J.Q + I_11x •t l.lpx2 •t --_ ) (1)

where I-(~ is the permanent dipole moment and n 1, ~2, etc., are the appropriate dipole moment derivatives.

Modelling the diatomic or the local mode of the

0 009-2614/84/S 03.00 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

21

Volume 104. number 1 CHEMICAL PHYSICS LETTERS 27 January 1984

spherical top as a Morse oscillator [5,1 l] results in a

transition dipole moment of the form

G&L = tiLln)[u!(B,l~e) (Wexe/Oe)u-l I”* - t2) Here the leading coefficient is the first derivative of the dipole moment, and higher derivatives are ne-

glected *_ The quantum number u corresponds to the

excited vrbrational state_B, represents the equilibrium

rotational constant for either the diatomic or the

spherical top, while we and c+xe stand for the vibra-

tionai harmonic and anharmonic constants, respective-

ly [ 131. A refined approximation, taking into account

electrical as well as mechanical anharmonicity, yields

iS,L 11

where (~,& is given by eq. (2), p2 is the second de- rivative of-the dipole moment, and 6 is the Kronecker

delta. In an effort to test the local-mode hypothesis, the

analysis based upon eq. (2) proceeds relatively simply. The mechanical constants Be and w, are evaluated from the spectrum of the vibration-rotation funda-

mental (u = l), and o+xe is obtained from its first

overtone (u = 2). Then, from measured spectral inten-

sities in the fundamental, the values of (r_lol>r and con- sequently pl are obtained_ Successive applications of

eq. (2) result in predictions for intensities of overtones u > 2_ These may be compared with available experi- mental values.

An alternative approach, developed here, avoids the

need to evaluate the mechanical constants, at least in

leading approximation. This analysis also proceeds from eq. (2). The frost step is to express explicitly

(&)L)L = Ir$$&e)l/2 (4)

bO2)L = 011/2)[2!(Be/Gie)(Wexe/well 1’2 , (9

(pu3)L = 01~/3~~3~~B,/w~~~~~x./w321 U2 (6) The next step is to eliminate the mechanical constants

from eqs. (4)-(6). This is achieved readily by some

algebraic manipulation. Wbat results is the invariant exz

pression

* For this work, the permanent dipole monient of a spherical- top molecule is taken to bezero. But see, for example, ref.

1121.

22

~(/.QJ~)~(c(O~$ = 8(/-+-& -

The general results are

(7)

(clOu)J =(2~-~u!/u~)~‘*~~~z~~-~/~~o~~~-~. (8)

For u= 1 and 2, eq. (8) is an identity. In the u = 3 case,

eq. (8) reduces to eq. (7). For u > 4, the general ex- pression predicts intensities of higher overtones.

When electrical anharmonicity is included, the con- trolling result is eq. (3). It does not appear that an in- variant expression like eq. (7) can be found. Rather

there is the following relation among transition dipole

moments:

- 6-1’2(~e-xe/~e)(~uL)2 - (9)

It is noteworthy that while the rotational constant Be

does not appear in eq. (9) the vibrational constants we and wexe do.

The general result is

-+I(1 -w3 (1% where (pov)L is given by eq. (8). For u = 1, the out-

come of eq. (10) is simply QtuJ)2 = &L)J which is al- ready expressed in eq. (3)_ And for u = 2, an identity

is obtained from eq. (10). In the case ZJ= 3, the result

is eq. (9) For IJ> 4, the general expression eq. (10) predicts intensities of higher overtones to greater accu-

racy than does eq. (S), in principle.

3. Computations for methane

Transition dipole moments have been determined

recently [9] for v3, 2v3 and 3v3 of both isotopic spe-

cies 12CHq and 13CHq. The parameters inferred from

absolute intensities in high-resolution spectra [6-81 are summarized in table 1. The predicted values for <r_lo3)I from eq. (7) are 2.288 X 10m4 and 1.632

X 10m4 for L2CH4 and 13CH4, respectively. These re-

sults are only a factor of 2.1 different from the mea- sured parameters_ This suggests some support, but cer-

tainly not conclusive evidence, for the local-mode hy-

pothesis.

For higher overtones of the type uv3, the projection from eq. (8) is

Volume 104, number 1 CHEMICAL PHYSICS LETTERS 27 January 1984

Table 1 Transition electric dipole moments for methane % b)

“3 2v3 3v3

12CH4 5.334 x 10-a 2.734 x lO-3 1.069 x 1O-4 13CH4 5.312 x lO-2 2.304 x lO-3 7.705 x 10-s

al Taken from the analysis of ref. [9], based upon the data in refs. [6-8]_

b, Absolute values in units of lo-l8 esu cm.

(~~~11 = 5.334 X 10-2(7.249 X 10--2)u-1

x [(u - l)!/u] l/2 (11)

for 12CH4; while for 13CH4 the corresponding predic-

tion is

(pou)l = 5.312 X 10-2(6.134 X 10-2)u-1

x [(u - l)!/u] 112 . (12)

Numerical values for u = 4,5,6 and 7 are presented in

table 2. Inclusion of electrical anharmonicity in the locai-

mode analysis yields eqs. (3), (9) and (10). In principle,

the relation in eq. (9) may be applied to the values of QoV)2 in table 1 to obtain the quantity ~exe/we for

IzCH, and 13CH4. This result may be substituted into eq. (10) which then becomes a basis for predicting transition dipole moments for overtones u > 4 in the

local-mode approximation. But, in practice, the relative signs of the transition moments in eq. (9) are unknown, so that an inferred value of w,x,/w, would be prob- lematical.

Nevertheless, an estimate of the parameter

(~eLJ&) ‘~%&2/~1-‘o~ 2 1 may be made on the ba- sis of the transition moments in table 1 and calcula- tions of the anharmonic constants in refs. [3,5] . The

Table 2 Calculated transition dipole moments for methane a, b)

” 12CH4 13CH4

4 2.49 x 10-s 1.50x 10-s 5 3.23 x lo+ 1.65 x lo+ 6 4.77 x 10-V 2.06 x 10-7 7 7.85 x lO-8 2.87 x lo-*

a) Based upon the local-mode hypothesis as reflected in eqs (21, (4)-(8). and eweclaW (111 and (121.

bl Absolute values in units of 10wla esu cm.

numerical results suggest that the coefficient of u in eq.

(10) is of order unity. This in turn implies that (~ou)7/

(FOu)l - 1 =O(l)u. In other words, electrical anhar-- monicity introduces a mild, but non-negligible, addi- tional dependence of (Lou) on u.

The quantitative approach developed here neglects p3 and higher-order terms [ 141 in the dipole moment expansion of eq. (1). Consequently, eqs. (8) and (10)

have potentially serious limitations in their present

forms.

Acknowledgement

This research was supported by the Planetary Atmospheres Program of the National Aeronautics and

Space Administration, under grant NAGW-125. Con-

structive criticism from an anonymous referee is ap-

preciated-

References

111

121

131

141

151

161

171

PI

[91 1101

I. Schek, J. Jortner and M-L. Saae, Chem. Phys. Letters 64 (1979) 209, and references therein. I. Abram, A. de hlartino and R. Frey, J. Chem. Phys. 76 (1982) 5727. and references therein. L. Halonen and h1.S. Child, MoL Phys. 46 (1982) 239,

and references therein G. Heraberg, Molecular spectra and molecular structure, Vol. 2. Infrared and Raman spectra of polyatomic mole- cules (Van Nostrand, Princeton, 1945) ch 2. C. Chackerlan and R.H. TippIng, J. MoL Spectry. 93 (1982) 237. M. DanS-Nhu, AS. Pie and A.G. Robiette, J. hIoL Spectry- 77 (1979157_ K. Fox, G-W- Halsey, S.J. Daunt, W-E. Blass and D.E. Jennings, J. Chem. Phys. 72 (1980) 4657. J-W. Bra&, K. Fox, D.E. Jennings and J.S. hlargolis, Astrophyr J. 247 (1981) LlOl. K. Fox, J. Phyr Chem., to be published_ K. Fox in: The signiticance of nonlinearity in the natural

sciences, eds. B. Kursunogiu, A_ Perlrnutter and L.F. Scott (Plenum Press, New York, 1977) pp_ 265-292.

(111 H.S. Heaps and G. Herrberg. 2. Physik 133 (1952) 48; R.H. Tipping and J-F. Ogilvie, J. Mol. Struct. 35 (1976) 1.

[ 121 T. Oka, hx hiolecti spectroscopy: modem research, VoL 2, ed. K. Narahari Rao(Academic Press, New York, 1976) pp. X9-253.

[ 131 G. Henberg. Molecular spectra and molecular structure, VoL 1. Spectra of diatomic molecules (Van Nostrand, Princeton, 1950) ch. 3.

[ 141 B.L. Crawford and H.L. Dinsmore, J. Chem. Phys. 18

(1950) 983; C. Secroun, A_ Barbe and P. Jouve, J. McL Spectry. 45 (1973) 1.