Vibration-rotational and rotational intensities for CO isotopes

JOURNAL OF MOLECULAR Sf’ECTROSCOPY 99, 431-449 (1983) Vibration-Rotational and Rotational Intensities for CO Isotopes C. CHACKERIAN,JR. NASA Ames ...

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JOURNAL

OF MOLECULAR

Sf’ECTROSCOPY

99, 431-449 (1983)

Vibration-Rotational and Rotational Intensities for CO Isotopes C. CHACKERIAN,JR. NASA Ames

Research

Center,

Mofett Field, Ca&rnia

94035

AND

R. II. TIPPING’ Air

Force Geophysics Laboratory, Hanscom Air Force Base, Bedford* Massachusetts OI 731

Dipole moment matrix elements have been computed for the five most abundant isotopes of CO. The wave functions utilized were obtained from a direct solution of the ~hr~inger equation with an accurate RKR potential. The dipole moment function, in the form of a Pad& approximant, was chosen to reproduce the experimental measurements near equilibrium, to have the proper united and separated atom limits, and to have the correct long-range asymptotic functional dependence on internuclear separation. Because of the huge number of transitions involved, and to facilitate applications, the squares of the dipole moment matrix elements were fitted by a least-squares procedure to polynomials in u and J. Predictions for the 5-O and 6-O rotationless matrix elements and Herman-Wallis coefficients are given, and their dependence on the isotopic reduced mass is discussed. For the pure rotational band, u = u’ = 0, explicit Einstein A values and transition frequencies were calculated for the three most abundant isotopes for J up to 55. The corresponding dipole moment matrix elements were also fitted to simple polynomials in m and the dependence of the coefficients on the reduced mass given. The present results incorporate the most accurate and extensive intensity measurements and theoretical dipole moment function data for any heteronucl~ diatomic molecule. fn view of this, because of the importance of the CO laser, the accuracy of the spectral frequencies, and the ubiquity of the CO molecule, it is reasonable to expect that some of these lines, in particular, in the fundamental band of ‘*C?O, can serve as laboratory standards for intensity measurements. I. INTRODU~ION

Carbon monoxide line intensities are important for a number of applications involving quanti~tive spectroscopy. In the area of astrophysics, some of these are isotopic ratios in stellar atmospheres (1) and the interstellar medium (Z), stellar atmospheric structure (3), thermal dynamics of molecular clouds (4), etc. In the area of laboratory chemical physics, the CO gas laser is useful, for example, as a probe in the examination of nascent population dist~butions produced in chemical reactions (5), or in the study of relaxation processes behind shock waves (6). Outside of the laboratory, the fundamental vibration-rotation and overtone bands are important in atmospheric work for, among other things, monito~ng pollution, or for tracking troposphere-stratosphere exchange as, for example, in the Space Shuttle based gas filter experiment (7). Because of the wealth and accuracy of the intensity measure’ Permanent address: Dept. of Physics and Astronomy, University of Alabama, University, Ala. 35486. 431

0022-2852/83 $3.00 Copyright @ I983 by Academic f+esr. Inc. Ail ri$hts of reproduction in any form reserved

432

CHACKERIAN

AND TIPPING

ments for 12C60 and its more common isotopes, it is hoped that these spectral lines, in particular, the fundamental lines of 12C’60, will be useful as absolute intensity standards (8). In this paper we report calculations of line intensities for five isotopic species of CO for transitions with Av = v’ - v = 0 through 4, v = 0 through 27, and every tenth J for both R and P lines from J = 0 through 150. For the convenience of application, and in view of the large number of lines involved, we have fitted our results to polynomials in the vibrational and rotational quantum numbers v and m, where m is J + 1 for R lines, and -J for P lines. We have also computed Einstein A values and frequencies for all lines in the pure rotational spectrum (v’ = v = 0; and 0 G J G 55) for the “C60, ‘3C’60, and ‘2C’80 isotopes. We feel that the present results are more accurate than previous results (many of which have been summarized recently by Werner (9)) for a number of reasons. (1) Recent, more accurate measurements of the fundamental line intensities (10) have resulted in changes of the order of 6%; this leads primarily to an increase in the fundamental rotationless matrix element, with concomitant changes in the dipole moment function around the equilibrium internuclear separation. (2) The availability of extensive frequency data and Dunham coefficients (II, 12) has enabled more accurate calculations (RKR) of classical turning points for the vibrational potential of the ground electronic state. Using these potentials, we have computed the vibration-rotational wave functions numerically, thus minimizing any errors associated with the wave functions in the calculation of the dipole moment matrix elements. The principal uncertainties at present arise from inaccuracies in the electric dipole moment function (EDMF). (3) Our representation of the dipole moment function, a Pad& approximant, is chosen so as to reproduce the most accurate experimentally determined value and slope around the equilibrium internuclear separation, to have the correct separated atom limit and slope, and is constrained to agree with the ab initio results in the intermediate region near 3.4 A. Refinements in this function may, of course, still be needed and can be made when accurate intensity data on the line or band intensities of the higher overtones (5-0,6-O, etc.) become available. The improvement (I) leads to more accurate intensities for the low v and small AV transitions, while the other two refinements lead to more realistic intensities for high v and large Av transitions. II. NUMERICAL

WAVE

FUNCTIONS

The vibration-rotational wave functions utilized in the present calculations were obtained via a direct integration of the radial Schrijdinger equation using the procedure described in Ref. (13). The appropriate classical turning points of the vibrational potential were calculated for each isotope using the RKR method (14). Vibrational energy term values needed for the latter calculations were obtained through the reduced Dunham coefficients reported by Dale et al. (II). The range of integration (0.45-3.0 A) of the wave equation was chosen to extend well beyond the limit of classical oscillation of the highest level considered, and the dissociation energy of the rotationless potential (zero-point vibration included) was set at 90 544.1 cm-’ (IS). Since our knowledge of the high-order Dunham constants is limited, there is some uncertainty in the magnitude of the effective potential and therefore in the wave-

INTENSITIES FOR CO ISOTOPES

433

functions for states having the largest VJ products. We have not made specinc numerical tests, but the dipole moment matrix elements and the curves fitted to these could be somewhat in error for these transitions. Fortunately, for most applications, these high uJ product matrix elements are not needed with an accuracy comparable to the lower VJ product results. III. DIPOLE MOMENT FUNCTION Previous work on the EDMF of CO, both ab initio and ex~~mental has been summa~zed recently by Werner (9), and earlier by Tipping (15). Using this information plus new experimental intensities for the fundamen~ band lines (IQ), we have obtained a “semiempirical” representation of the dipole moment function in the form of a Pad6 approximant (17). The near equilibrium region of the EDMF, out to an internuclear separation of approximately I .45 A, was well dete~in~ from laboratory measurements of vibrational band intensities for the fundamen~~ and first three ove~ones. The form of our EDMF is chosen so as to have an Rp4 asymptotic dependence which is appropriate for the ground electronic state dissociation of CO. The magnitude of the EDMF in the intermediate range was constrained to match the ab initio results near 3.4 J%(9, 18). Except for the improved 1-O band intensity, the vibrationat band intensities used in the present calculation differ only slightly from those used previously to determine the EDMF (9, 16, 17). The experimentally determined rotationless dipole moment matrix elements used in determining the coefficients in the Padi approximant were obtained from the following sources. Based on the recent results of Varanasi and Sara@ (19) and Varghese and Hanson (IO), we have increased the rotationless matrix element for the fundamental band by 2.75% (intensives by approximately 7.5%) over previous results. Many measurements on the 2-O band have recently been discussed by Chandraiah and Herbert (20), and most of the recent results are in substantial agreement; consequently, we have adopted the value of 2.10 cm-’ agt-’ for the total intensity of this band. Data for the 3-O and 4-O vibrations bands were taken from Refs. (21) and (X?), and the ground state result from Ref. (23). These input data are summarized in Table I and are reproduced by our EDMF which is expressed by

-0.122706(1

- 28.5291x - 29.5291x2)

M(x) = i 4 1.10102~ + 1.0643~~ C 0.43789~” + 3.55499x6

(1)

where x = (& - R)f& is the reduced disp~a~me~t from the ~uilib~um intern~c~~r separa~on. In Fig, I we compare the Padi EDMF, Eq. (I), with the two most recent (and probably most accurate) ab initio functions. The curves have the same general shape near equilibrium; however, at x = 0, the three curves are displaced from each other. The “‘semiempirical” curve (with the permanent moment equal to -0.1227 D) is approximately 0.07 D below the Cooper and Langhoff value (I@, and approximately 0. I1 D above that of Werner (9). This shift would result in a different distribution of intensities within vibrational bands, i.e., different He~an-Wa~lis factors (N), Because the vibrational matrix elements are dependent primarily on the shape of the EDMF around the equilibrium separation, we note that the Werner EDMF yields a

434

CHAC~RIAN

AND TIPPING

TABLE I Rotationless Matrix Elements M!(O) in Debye Used in the Determination of the EDMF

M;(O)

”

0

-0.10982 a

1

0.1084

2

-6.6

4.29 x 10-4

3 4

a.

x 1O-3

-2.016

x 1O-s

The quantity determined experimentally

Vibration-rotational and rotational intensities for CO isotopes

Vibration-rotational and rotational intensities for CO isotopes

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