The microwave spectrum of the OH X211 radical in the ground and vibrationally-excited (u 5 6) levels

J. A. COXON Deport~~~c,t i t of Che171i.str.v, Drrlho~isic~ U ~ ~ i c ~ e r s i t y , Hcrl if~s, N . S . , Co~~nr / ( l B3H 4J3

AND

K. V. L. N . SASTRY Deprirt~?rer~t of Ph)l.rics, U~ricer.sil.y of N n v Brrr~iswick, Fredericton, N.B. , Crrrl~do E3B 5A3

AND

J. A. AUSTJN~ A N D D. H. LEVY The Jti~lles Frrc~~cX. I~~st i tr i te , U~iiuersity of Chicrrgo, Chiccrgo. IL 60637, U . S . A .

Received October 25, 1978

The microwave absorption spectrum of the OHX'II radical has been observed in all vibrational levels up to 11 = 6. Experimental details are described of the tunable cavity and parallel plate Stark-modulated spectrometers err~ployed for transitions below and above 23 GHz, respectively. The observed line frequencies, together with those reported by other workers, have been fitted using a non-linear least-squares routine with numerical diagonalization of the Hamiltonian matrix. Pseudo high-order corrections forcent~.ifugal distortion of the A-doubling are required for the u = 0 data, which now extend to J = 1912 in the TI,,, component. The various adjustable parameters of the Hamiltonian are compared with those of other similar models. The A-doubling and magnetic hyperfine constantsfor the different vibrational levels are reported, and several line frequencies of potential astrophysical significance are predicted.

Le spectre d'absorption des micro-ondes par le radical OH X'n a ete observe dans tous les niveaux de vibration jusqu'8 u = 6. Des details experimentaux sont donnes, pour decrire la cavite accordable ainsi que les spectrometres i plaques paralleles et B modulation Stark utilises respec- tivement au-dessous et au-dessus de 23 GHz. Les frequences observies pour les raies, reunies B cellesque d'autres chercheurs ont deji ete rapportees, ont fait I'objet d'un ajustement i I'aide de traitement de moindres carrts non lineaire, avec diagonalisation numerique de la matrice hamil- tonienne. Les donnees u = 0, qui s'itendent maintenant jusqu'i J = 1912 dans la composante 'IIl,Z requiereni des pseudo-corrections d'ordre supirieur pour la distorsion centrifuge du doublage A. Les divers parametres ajustables de I'hamiltonienne font I'objet d'une comparaison avec ceux de modeles similaires. Le doublage A et les constantes hyperfines magnltiques pour les differents niveaux de vibration sont donnees, de mkme que des predictions sur les frkquences de plusieurs raies susceptibles de presenter un grand interkt pour I'astrophysique.

Can. J . Phys., 57,619 (1979) [Traduit par le journal]

Introduction Fourier transform data on OH vibration-rotation The OH radical is known to be an important bands in the near infrared (15). Until recently, the

species in numerous and sys- ~ ~ c I - o w ~ I v ~ work was confined to the lowest u = 0 terns; it is not surprising, therefore, that an under- vibrational level, and a summary of these investi- standing of its spectroscopy has been the concern gations can be found in the recent review by of many workers over several decades. OH was the Beaudet and Poynter (16). However, in pal-allel first free radical to be investigated in the microwave with the present investigation, Meerts et a/ . (17) region (1, 2); and laser magnetic reso- and Destombes et al. (18) have obtained the first nance can be expected to sub- frequency measurements for z) = 1,2 and for r ) = 1, stantia] contributions to a more complete descrip- respectively. Even for these levels, as well as for 0 tion of the ground electronic state, the most reliable = 0, the Present work provides the first data for data at present have been obtained fi-o~n analyses of higher values of J ; for zl 2 3, the present results are the A2C+-X2n band system i n the re- the only data from microwave spectroscopy and gion (3, 4), from several investigations of the' complement those from the epr spectra of vibra- microwave spectrum (l,2,5-14, Table I), and from tionally excited OH, first studied by Clough et 01.

(19) and Lee et 01. (20,21). [Present address: Kimberly-Cla1.k COI-p., ~ ~ ~ ~ ~ h , WI 54956, The relatively extensive spectroscopic data on

u.S.A. OH and OD have been of much value traditionally

0008-420417910506 19- 16$0 1 .OO/O @ 1979 National Research Council of CanadaIConseil national de recherches du Canada

620 CAN. J . PHYS. VOL. 57, 1979

in testing and developing theoretical models for the coupling of the va~ious internal angular momenta with the nuclear rotation. Added impetus to this endeavour has come recently from the present ease of direct non-linear least-squares fitting of measured frequencies with an exact numerical diagonalization of the Hamiltonian matrix. With increasingly more extensive and accurate data, this approach reveals the need for careful consideration of the many minor and often subtle effects which can make significant contributions to the matrix elements. In the present work, this aspect is con- sidered in some detail, and the present Hamiltonian is found to give results for the A-doubling and magnetic hyperfine constants of OH which are in good agreement with other similar treatments.

Experimental Two types of spectrometer were used to study

the microwave spectra of OH in excited vibrational states: a Stark modulated tunable cavity spectl-om- eter for transitions below 23 GHz and a 4-ft palxllel plate Stark modulation spectrometer in the region above 23 GHz. A brief description of the cavity spectrometer was reported earlier (22).

C~zi i ty Spectrometer The tunable cavity is described in cross-sectional

view in Fig. 1. Its basic shape is modelled after the design of Dymanus (23). A plunger [A] drives the piston [B] up and down inside the main cavity housing [C]. The piston is provided with high vol- tage square wave modulation via an electric feed- through [Dl, a length of Teflon insulated wire (E), and a brass screw (F). The piston is insulated from the cavity body by a Teflon line (G) and from the plunger by a Lucite plate (H). The high voltage screw is insulated from the plunger by a ceramic insulator (I).

The plunger rides up and down in a housing (J) and is fitted with two O-rings [K] for vacuum tight- ness. A pump out port [L] provides for any leakage past the first O-ring of the moving seal. The plunger is provided with 2in. of threads [MI of pitch 11 threads per 1 in. and is prevented from rotating by a flat [N] and a nylon key [O]. An inside threaded tuning cap [PI fits on the top of the plunger and its vertical movement is prevented by bearings [Q]. The bearing housing is attached to the cavity by four screws [S]. A motor drive (Erwin Halstrup M64D) is attached to the tuning cap and its variable gear ratios provide for changes in scan speeds. The bearings provide a smooth sulface for tuning cap rotation. To obtain a mechanical movement which is vibration-free it is important that all threads are

0 MOTOR DRIVE

P

FIG. 1. Cross-section of the cavity spectrometel.

clean and well-oiled and that the plunger O-rings are greased and in good condition.

The entrance of the reactants into the cavity is provided by ports [TI and [U]. Port [U] is larger to allow higher pumping speeds for the free radical reactants. The reaction products are pumped out through port [V]. The microwaves are coupled in through slot [W] in end plate [XI. Vacuum tightness of the cavity body is provided by O-rings [Y] and [ZI .

A block diagram of the spectrometer is shown in Fig. 2. The frequency of the backward wave oscil- lator (BWO) is locked to the cavity using a 70-KHz automatic frequency control (AFC). This control provides adequate locking for small frequency drifts of the order of 1 MHz. Once the BWO fre- quency drifts beyond the half-width of the cavity resonance the cavity Q curve no longer acts as a discriminator and there is no dc correction voltage. For frequency scans of several hundred megahertz there must be a long range adjustment of the BWO frequency to keep the AFC correction voltageclose to zero. This is accomplished by feeding the dc correction voltage of the AFC system to the servo

COXON ET AL. 62 1

T u n a b l e C a v i t y S p e c t r o m e t e r

Power Servo Frequency

FIG. 2. Block diagram of the tunable cavity spectrometer.

frequency controller shown in Fig. 3. The function of the controllel- is to convert the dc correction voltages to a resistance change in the 100 kR heli- pot. The helipot is in turn connected into the resis- tance chain of a Fluke 415A high voltage power supply which provides the BWO cathode-helix voltage. The AFC control and the servo frequency control operate simultaneously. The AFC effects 'short range' fast response corrections in the BWO frequency and the servo controller effects 'long range' slow response corrections.

Parallel Plate S~LI I -k Modul~itiotz Spect~.ometer The block diagram of this spectrometer is shown

in Fig. 4. The microwave frequency sources are klystrons, and the klystron frequency is phase locked to a stable X band BWO which is in turn phase locked to the harmonic of the sum frequency of a Hewlett-Packard 8640B oscillator and 18 MHz synthesizer. The klystron frequency is swept by varying the frequency of the synthesizer frequency in the analog mode using the ramp voltage from the signal averager. The klystron in this system can be swept up to 20MHz at a time. The frequency sta- bility of the klystron is better than 1 ppm in the swept mode. The absorption cell consists of two aluminum parallel plates of 5 cm width and 120cm length. The spacing between the plates is 3.6mm.

The parallel plates are placed in a glass tube which is enclosed in p metal casing. The p metal reduces the earth's magnetic field by 20 times. The insides of the parallel plates are coated with fluoroglide. Figure 5 shows the cross-sectional view of the ab- sorption cell. Microwave power is coupled to the parallel plates by microwave horns. Stark modula- tion at 40 kHz is used to detect the absorption lines.

P~.otluction of OH OH radicals are produced by reacting 0, with

atomic hydrogen. This method is found to be the most efficient to produce OH in vibrationally ex- cited states. Atomic hydrogen is produced by pass- ing H, through a microwave discharge in a quartz tube. Ozone is collected by selective adsorption at --78°C on 3-8 mesh silica gel using the technique of Cook et (11. (49). After the gel was loaded with -7g ozoneI100g gel, the gel was pumped until the pressure fell to about -30 Torr to purge it of excess 0, . Cook et 01. (49) measure a partial pressure of 30Torr ozone above silica gel when the gel is loaded at -6g ozone/lOOg gel. It was found that additional pumping increased the OH sign a I ~nten- '

sity. During operation, the can containing the silica gel + ozone was warmed to increase the ozone pressure.

In the cavity spectrometer the pressures for op- timum signals are Hz - 1 Torr and 0, - 3 Torr. In the parallel plate cell a total gas pressure of 50-60 rnTol-1. is found to give the highest signals. The full width at half maximum for the absorption lines in this cell is approximately 0.5MHz. In the cavity spectrometer, the line widths are approximately 1.5 MHz.

Theory The microwave absol-ption spectrum of OH is

due to transitions between the two Kronig + and - symmetry A- doublet levels within each J rotational state of either the TI, , , or 'II,,, spin-orbit compo- nents of the X211 ground electronic state. However, owing to the non-zero nuclear magnetic moment of the proton (I = 1/2), nuclear spin-orbit and dipolar interactions split each A-doublet level into two components, with F = J f 112. The microwave transitions at high resolution thus appear as four hype~fine components.

There has been considerable success recently in developing model 211 Hamiltonians with adjustable parameters to which experimental microwave data can be fitted to within the estimated accuracy. It is necessary that such Hamiltonians should describe accul-ately and simultaneously both the fine struc- ture (A-doubling) and magnetic hyperfine struc- ture. Unfortunately, there is still no uniformly

CAN. J . PHYS. VOL. 57, 1979

SERVO FREQUENCY CONTROLLER FIG. 3. Block diagram of the servo frequency controller

TO BACKWARD WAVE OSCILLATOR POWER SUPPLY

FROM 70KC A A

PHASE SENSITIVE TIME

- -

18 0000-181000 MHz Synthesizer V C O Input

VlSCOUS DAMPER

Power Supply

Tuned Balanced Hewlett-Packard 86LOB

Synchron~zer &I Amphf~er H Mlxer S~gna l Generator 1-512 MHz

FIG. 4. Block diagram of the parallel plate Stark modulation spectrometer.

adopted notation for the various adjustable param- One of the most significant steps in the develop- eters; moreover, the parameters for a particular ment of the theory of A-doubling and magnetic Hamiltonian are usually only effective constants hyperfine structure was made by Meerts and which can be related to those of other models by Dymanus (24) in a now almost classical paper. In forming linear combinations. For these reasons, order to account for interactions or admixtures of comparisons of results from different laboratories other doublet electronic states, these authors are not always straightforward. In this section, we applied degenerate perturbation theory, as de- shall only attempt a brief review of the more recent veloped by Freed (25). Perturbation terms were and successful approaches. Later, we suggest that considered up to third order, both for the fine and the recent effective Hamiltonian approach of hyperfine structure, and were associated with sets Brown et 01. (34) should be adopted in all future of adjustable parameters ai and xi, respectively. work. Several of these parameters were found to be

8-12 L GHz

Synchron~zer

Hewlett-Packard 86908 Sweep Dsc~l la lor

A Square Wave

Generator

Ref

COXON ET AL. 623

These authors treated the effects of distant pertur- bers (vibrational or electronic) by means of the Van Vleck transformation, applied to second order. Although an extension of this treatment to third order presents no difficulties, it was considered more straightforward to introduce corrections for the A-doubling separations beyond second order by defining effective parameters p, and qJ for each J-state,

v = 2 v = 3 [l] p J = p + p D J ( J + l ) + p H J Z ( J + I)'+ ...

v = I vo = 12527 rnc vo=l 1631 mc vo = 10748rnc

V= 4 V = 5 V.6 vo = 9874mc v o = 9 0 0 3 m c vo=8133mc

OH 'n3,- J = 7 ~ Z A DOUBLING TRANSITION IN ,'

EXCITED VIBRATIONAL STATES ( TIME CONSTANT = 3 s )

FIG. 5. Spectra of OH 2n,,2, J = 712 A-doubling transitions in excited vibrational levels using the cavity spectrometer (time constant = 3 s).

strongly correlated. In addition, although the model Hamiltonian of Meerts and Dymanus was an impressive advance, it was still not completely suc- cessful; i.e., it failed to reproduce accurately the experimental frequency data, not only for OH (14) but also for the heavier NO molecule where cen- trifugal distortion plays a smaller role. In a later paper, Meerts (26) demonstrated convincingly that the difficulty in fitting the NO data was due to neglect of hyperfine terms off-diagonal in J. For the lighter OH molecule, however, off-diagonal matrix elements are about an order of magnitude smaller and do not account for the imperfect results ob- tained with the third-order perturbation treatment. For this reason, Meerts (27) has recently consid- ered the extension of perturbation theory to fourth order for the A-doubling terms. However, such fourth-order corrections were not derived explicitly; rather, from the form of the rigorous third-order results, the A-doubling terms were ex- panded as a power series in ( J + 1/2), and the parameters a, of the earlier work were replaced by the set p , q (second order) D,, D, (third order), and 6,, 6, (fourth order).

The representation of A-doubling terms to high order by a polynomial expansion is reminiscent of a similar approximation (for centrifugal distortion) suggested in an earlier paper by Zare et 01. (28).

Matrix elements for the A-doubling given by such an effective Hamiltonian for a 'II state which is perturbed by a 2Z+ state (or states) were given by Coxon (4), who also demonstrated the success of the method in an analysis of optical data for the A2Z+-X211i system of OD. The more complex situ- ation of simultaneous interactions with lZ+ and ?Z- states has been considered by Saenger et 01. (29) and more recently by Amiot et 01. (30) in applica- tions to the 211 ground states of HCl+ and NO, re- spectively. Unfortunately, however, the separate effects of 2Z+ and ZZ- interactions are difficult to distinguish, and it is found that many of the fitted parameters of the simpler model, such as B, A , y, p , and q are to be interpreted in terms of new effective parameters.

An extension of the techniques outlined by Zare et a l . (28) to the treatment of hyperfine structure was recently described by Destombes eta l . (31) in a study parallel to that of Meerts (26, 27) on the microwave data for OH X211, zi = 0. Since the unique perturber approximation has been found to hold well for OH XZII in numerous investigations, including some ab init io calculations by Hinkley et a l . (32) and Coxon and Hammersley (33), Des- tombes et a l . (31) chose to treat the problem by direct diagonalization in the X211u, A211v subspace. The Van Vleck transformation was then used merely to introduce the effects of mixing with other vibrational levels. This leads to the definition of new centrifugal distortion constants, (DL+), (HL+), and a constant (ADL+) associated with cor- rection matrix elements off-diagonal in A. In addi- tion, there are second-order correction terms to the hype~line structure, which lead to the definition of four electronic interaction parameters, h , , h2, h,, and h,, and to nine vibrational parameters, K, , K2, .. . , K , , from cross-terms between the rotational and magnetic parts of the Hamiltonian. Experi- mentally, however, none of the hi and only three of the Ki (K2, K,, K,) could be fitted significantly. It is interesting to note, however, that the most

624 CAN. J . PHYS. VOL. 57, 1979

significantly determined second-order hyperfine constant ( K , ) was also necessary in the work of Meerts as the constant x 6 ; however, whereas K , is essentially a correction for centrifugal distortion to the hyperfine A-doubling parameter, t i , the con- stant x6 was introduced via excited 'C states.

Very recently Brown et (11. ( 3 4 ) have reported a new study of the epr spectrum of OH X211, u = 0 in the J = 912 and 1112 rotational states. In analysing their data these authors reconsidered the 211 prob- lem and developed a new effective Hamiltonian using a formal tensor operator approach. While retaining the use of traditional spectroscopic nomenclature, this approach operates only within the spin and rotational levels of a single 211 vibra- tional level. Any admixture with other electronic states is introduced implicitly via the operator techniques. Some important points which can be noted from this novel method are as follows:

(i) At the second-order level, the traditional definitions of Mulliken and Christy ( 3 5 ) of the A- doubling para meters,^ and q , are retained.

( i i ) Higher-order corrections t o p and q can arise not only from mixing with neighbouring vibrational levels (usually denoted as centrifugal distortion of A-doubling), but also from non-adiabatic mixing of electronic states, as fo rp and q themselves. How- ever, since the two contributions have indistin- guishable effects, the less rigorous approach of [ l ] and [ 2 ] is expected to hold, with the constants p,, q ~ , pH, q ~ , ... having an uncertain physical in- terpretation.

(ii i) A physically more meaningful set of 211 hyperfine constants is a , b , , c , and d, rather than Frosch and Foley's ( 3 6 ) set n , b , c , and d . The constant b , is associated directly with the Fermi contact term, and is defined as b , = b + c / 3 .

( io ) Higher-order corrections to hyperfine in- teractions introduce five additional parameters; thus C, , C , ' , D,, and D,' are associated with terms from mixing with other electronic states through Coriolis coupling and/or centrifugal distortion of n , b, and c ; and Dd is introduced as a centrifugal distortion constant of the A-doubling hyperfine in- teraction parameter d. Of the set of five parameters, however, D,' is correlated totally with C I ' , and D, is strongly correlated with ( b , - c I 3 ) . Accordingly, for real experimental data, it is found that only three of the high order parameters can be deter- mined, namely Dd, which is the equivalent of x6 of Meerts and Dymanus ( 1 4 , 24) and of K , of Des- tombes et nl. ( 3 1 ) , and C I and C,' whose physical significance is doubtful.

( 1 1 ) Brown et nl. ( 3 4 , 37) have derived linear re- lationships between the effective parameters of

their Hamiltonian and those of Meerts ( 2 7 ) and of Zare et (11. ( 2 8 ) . In particular, the Mulliken and Christy ( 3 5 ) parameters p and q are expected to have identical values in the Hamiltonians of Brown et (11. ( 3 4 , 37) and of Zare et (11. ( 2 8 ) , while all the A-doubling parameters p , q , D,, D,, F,, F, of Meerts ( 2 7 ) are expressed as linear combinations of the parameters defined by Brown et 01. ( 3 4 ) . It should be noted, however, that in the more recent work of Meerts et 01. ( 1 7 ) , a newly modified Hamiltonian has been employed which maintains the traditional definitions o fp and q , and which will require new relationships to the parameters of Brown et trl. ( 3 4 ) . Furthermore, the present work indicates that when centrifugal distortion is in- cluded, p and q in the Hamiltonian of Zare et (11.

( 2 8 ) do not have exactly the same meaning a s p and q in the Hamiltonian of Brown et 01. ( 3 4 ) .

Computer Fitting In the present work, we have used an effective

Hamiltonian of the type described by Zare et trl. ( 2 8 ) , in which A-doubling is treated by the second- order Van Vleck transformation on the assumption of non-adiabatic mixing with the A 2 C + and other lC+ states, but in which the parametersp and q are replaced by their rotationally-dependent counter- parts, p, and q,, as defined in [ l ] and [ 2 ] . Although this model is found to provide a satisfactory de- scription of the A-doubling, even for the new, rota- tionally more extensive data on OH, v = 0, we propose later that the equally satisfactory, but physically more meaningful parameter set of the recent treatment by Brown et nl. ( 3 4 ) be adopted in future work. Hyperfine structure was included by adding the first-order Frosch and Foley ( 3 6 ) matrix elements, but with the a , b , , c , d parameter set of Brown et (11. ( 3 4 ) . Finally corrections were included for interactions off-diagonal in J, as given by Meerts ( 2 6 ) , and for the higher-order hyperfine terms involving C, , C,' and Dd, as given by Meerts et 01. ( 1 7 ) . The effective Hamiltonian matrix then contained 22 adjustable parameters: B , D , H, L , A , A D , Y , P , P D , P H ~ P L , ~ , ~ D , ~ H , Y L , ~ ~ , ~ F , ~ , ~ , ~ I , ~ I ' , and Dd. The 'fifth-order' parameters pL and q, were required for the present data set on u = 0. The fine structure parameters ( B , D , H, L, A and the correlated pair A D and y ) were always constrained at fixed values, taken from the preliminary results of a parallel study at Dalhousie University. This latter work is concerned with merging ( 3 8 , 3 9 ) most of the available microwave, infrared, and A-X op- tical data after fitting with the same effective 211 Hamiltonian.

The 4 x 4 Hamiltonian matrix for each J and

COXON ET AL. 625

parity block factorizes into two 2 x 2 matrices corresponding to F = J $. 112. The adjustable parameters were varied iteratively in reaching a f i t , in the usual least-squares sense, of the experimen- tal data to frequencies calculated after numerical diagonalization from term value differences. The complete set of magnetic and second- and higher- order A-doubling parameters could be determined only foro = 0. For higher vibrational levels, several or many of the parameters were held at fixed val- ues, as described later. The data were weighted (wi ) in accord with the experimentalists' estimated pre- cisions (pi), where w i = 1lpi2, and the estimated dimensionless standard deviation (6,") of each fit was calculated from [3] ,

where vOi and vCi are the measured and fitted fre- quencies and tn is the number of fitted parameters. If the values ofpi are good estimates of the standard deviations of the distributions of measurement er- rors, 6 , should be close to the value of unity for sufficiently large degrees of freedom, f = t7 - m, and for random measurement errors. For small val- ues off in a weighted least-squares fi t of this type, 6,, can be considerably larger or smaller than unity, even for random errors (39). However, from the fits described here, i t is apparent that the precisions estimated by several experimental groups can be interpreted as standard deviations. Thus, in accord with the results of Coxon (39), the estimated errors for the fitted parameters reported in this work are (PiiTL')'12, where P" is the output dispersion matrix of the weighted fi t . These en-or estimates are to be compared with the 'standard deviations', 6,, x (ViiW)'l2 of other workers. Despite this effort to obtain meaningful error estimates, i t has to be em- phasized, however, that the errors estimated for some of the parameters are unrealistically small owing to correlations with the constrained fine structure constants, B, D, H, L, A , and A D (or y).

Results In this section, we discuss our experimental re-

sults, other available data, and the least-squares fitting procedure for individual vibrational levels in the range v I 6. Except for u = 6 , the observed and fitted frequencies for each level are summarized in Tables 1-6. Values in parentheses are estimated accuracies; predicted frequencies of unobserved lines are given in square brackets.

With the cavity spectrometer, it was possible to

observe transitions for all vibrational levels in the range 1 I a I 6 ; however, except for v = 1, all the transitions belong to the lower 'n3/, component. Even for v = 1, the transitions were approximately 20 times weaker than in 1) = 0. The data at higher frequencies from the parallel plate spectrometer are confined to 11 I 3, and include six new frequen- cies for the 211,12, 1) = 0 level with J = 1512, 1712, and 1912. Figures 6 and 7 show typical signals ob- served with the two spectrometers. As mentioned above, while this work was in progress, we came to know of two complementary investigations (17, 18) which are restricted to v = 1, 2 only. We have included the latter data in our fits, so that the molecul:u- constants given in Table 7 are the best available at the present time. Where our data over- lap those of Destombes et crl. ( I s ) , the frequencies are in good agreement and within the estimated errors.

Before discussing the fits for each level, we note that with the parallel plate spectrometer we ob- served two lines at low pressures which are 180" out of phase with the normal absorption lines. These emission lines occur in the region where the ' n 3 / , , u = 5 , J = 1312 transitions are expected. As dis- cussed later, however, these two lines could not be included satisfactorily in the fi t for v = 5 , so that a population inversion in the J = 1312 A-doublet of this level from the H + 0, reaction cannot be confirmed at the present time. Further work is in progress to study these lines.

v = O Satisfactory fits of the relatively large data set on

OH X 2 n , v = 0 have been reported previously by several workers (27, 31, 34). In the present work, we have obtained the first A-doubling frequencies for three high-rotational levels of the 211,1, compo- nent, J = 1512, 1712, and 1912. The new data set requires at least one of the high-order A-doubling parameters, p, and q,. A comparison between the observed and calculated frequencies when both these parameters are included is given in Table 1 . The least-squares fit in Table 1 covers all available data except for the unpublished ' n3 / , , J = 1712 measurements cited in ref. 16. However, the ' n 3 / , , J = 712, F+ = 4 t F- = 3 transition at 13 433.93 MHz of Destombes and Marliere (40) is included erroneously; this frequency was calcu- lated (40) from a combination sum. Removal of this line would make a negligible difference to the fit since i t is fitted well within the estimated accuracy. The standard deviation (6,) of our fi t in Table 1 is 1.041, comparable with the values of 6 , = 1.07 of Meerts (27) and 6 , = 1.15 of Brown et 01. (34), and

626 CAN. J. PHYS. VOL. 57, 1979

TABLE 1. Observed and fitted hyperfine A-doubling transitionsa for O H XZn, v = 0

z n s / z znt/z

J F ' c F VO v, - v, Reference F' c F VO v, - v, Reference

This work This work

This work This work

This work This work

OExcept for entries in square brackets (calculated frequencies), v, are observed frequencies with estimated uncertainties in parentheses; v, - v, are the residuals between the observed and fitted frequencies; + o r - in the F a n d F' entries denote the level parities.

is obtained for a larger data set and without the flexibility of an additional fitted parameter (7) of the latter work. We note, however, that for the 'II,,,, J = 912, F- = 4 t F+ = 4 and F- = 5 t F+ = 5 transitions, the experimental uncertainties were taken (incorrectly) as 0.002 and 0.003MHz, re- spectively, as given by Meerts and Dymanus (14), rather than the value of 0.0002 MHz indicated in a later paper (27). In addition, the parameter qL is not well determined. Thus in a fi t when q, was con- strained to zero, the standard deviation was only a little higher,6,, = 1.072; however, ifp, and q, were both omitted, 6,, was considerably larger (6, = 1.824), and the data are not fitted to within their

experimental uncertainties. The numerical values of our constants in Table 7 are compared with those of other workers in the later discussion. Finally, we note that Destombes has measured the 'II,,,, J = 1512 transition in a parallel study (48). The measured frequencies (48), 32 879.80(5) and 32 921.32(5) MHz are in very good agreement with our calculated values in Table 1.

1) = 1 In addition to our own new measurements, the

data for o = 1 have been extended considerably in the two recent publications of Meerts et a / . (17) and Destombes et 01. (18). The molecular beam electric

COXON ET AL.

TABLE 2. Observed and fitted hyperfine A-doubling transitions4 for OH X2n, v = I

2n3/2 2nl/2

J F ' c F VO v, - v, Reference F' + F VO v, - v, Reference

17 17 17 17

17 17 17 17

This work This work This work This work

b

b

0.6608 This work 0.7865 This work

OExcept for entries in square brackets (calculated frequencies), v. are observed frequencies with estimated uncertainties in parentheses; v. - v, are the residuals between the observed and fitted frequencies; + or - in the F and F' entries denote the level parities.

bMean values of this work and Destombes cr 01. (18).

TABLE 3. Observed and fitted hyperfine A-doubling transitions4 for OH X211, v = 2

2n3/2 2n1/2

J F ' c F V, v, - v, Reference F' e F VO v, - v, Reference

112 0- 1+ 4233.346(.005) 0.0012 17 1- 1+ 4319.938(.005) -0.0024 17 1- O + 4338.543(.006) 0.0018 17

312 1+ 2- 1 371.377(.003) -0.0017 17 1 + 2 - 7146.208(.006) -0.0007 17 I + 1- 1412.850(.003) -0.0029 17 I + 1- 7160.046(.007) -0.0020 17 2+ 2- 1 414.424(.003) 0.0006 17 2+ 2- 7212.791(.006) 0.0028 17 2+ 1- 1455.896(.003) -0.0017 17 2 + 1 - 7226.627(.005) -0.0005 17

512 2 - 3 + 5 161.489(.007) 0.0065 17 2- 3+ [7694.11] 2- 2+ 5 168.657(. 006) - 0.0013 17 2- 2+ [7713.50] 3- 3+ 5 172.192(.006) 0.0002 17 3- 3+ [7762.20] 3- 2+ 5 179.376(.008) 0.0085 17 3- 2+ [7781.60]

712 3+ 3- 11 628.9(0.3) 0.2017 This work 3+ 3- [5563.5] 4+ 4- 11 634.0(0.3) - 0.2833 This work 4+ 4- [5609.3]

912 4 - 4 + 20 791 . O(0.3) -0.4191 This work 5- 5+ 20 798.7(0.3) -0.2180 This work

1112 5+ 5- 32 505.22(. 05) 0.0048 This work 6+ 6- 32 514.36(.05) -0.0034 This work

1312 6- 6+ 46 584.86(.05) 0.0468 This work 7- 7+ 46 595.38(. 05) -0.0360 This work

'Except for entries in square brackets (calculated frequencies), v. are observed frequencies with estimated uncertainties in parentheses; v. - v. are the residuals bztween the observed and fitted frequencies; + o r - in the F and F' entries denote the level parities.

CAN. J . PHYS. VOL. 57. 1979

TABLE 4. Observed and fitted hyperfine A-doubling transitionsa for O H X Z i I , u = 3

'n3/2 znl/z J F ' + F VO v, - v, Reference F' + F VO

- -

112 0- I + [4003 .6] 1- I + [4088.9] 1- O+ [4109.7]

312 1 + 2 - [l 257.501 1+ 2- [6806.5] 1+ 1- [ l 292.581 1+ 1- [6821.6] 2+ 2- [l 293.991 2+ 2- [6871.8] 2+ 1; [l 329.071 2+ 1- [6886.9]

1 293.15(.2) -0.1330 b

512 2- 3+ [4 749.131 2- 3+ [7419.9] 2- 3+ [4 752.371 2- 2+ [7440.4] 3- 3+ [4 755.551 3 - 3+ [7486.8] 3- 2+ [4 758.791 3 - 2+ [7507.3]

712 3 + 3 - 10 745.q.3) 0.0128 This work 4+ 4- 10 750.5(.3) -0.0448 This work

912 4 - 4 + 19 296.4(.6) 0.1335 This work 5- 5+ 19 303.2(.6) 0.1068 This work

1112 5+ 5- [30 275.61 6+ 6- [30 284.01

1312 6- 6+ 43 511.93(.5) -0.0481 This work 7- 7+ 43 521.72(. 5) 0.0203 This work

OExcept for entries in square brackets (calculated frequencies), v. are observed frequencies with estimated uncer- tainties in parentheses; v. - v, are the residuals between the observed and fitted frequencies; + or - in the F a n d F' entries denote the level parities.

bMean value o f data o f Clough cl a/. (19) and Lee rl a/ . (20) from epr spectra.

TABLE 5. Observed and fitted hyperfine A-doubling transitionsa for O H X211, v = 4

'n3/2 2nl/2

J F ' c F VO v, - v, Reference F' + F VO

112 0- 1+ [3753.8] 1- 1+ [3838.7] 1- O+ [3861.5]

312 1+ 2- [l 147.431 1+ 2- [6424.2] 1+ 1- [l 175.761 I + 1- [6440.4] 2+ 2- [l 177.031 2+ 2- [6488.8] 2+ 1- [ l 205.371 2+ 1- [6505.0]

1 176.29(.20) -0.1081 b

512 2- 2+ [4 344.211 2- 3+ [7079.2] 2- 3+ [4 345.241 2- 2+ [7100.6] 3- 2+ [4 346.061 3- 3+ [7145.5] 3- 3+ [4 347.091 3- 2+ [7166.9]

712 3 + 3 - 9 871.6(0.3) 0.0443 This work 4+ 4- 9 876.2(0.3) 0.0216 Thiswork

912 4- 4+ 17 805.2(0.7) - 0.0793 This work 5- 5+ 17 811.5(0.9) -0.0540 This work

'Except for entries in square brackets (calculated frequencies), v. are observed frequencies with estimated uncer- taint~es In parentheses. v - v, are the residuals between the observed and fitted frequencies; + or - in the F and F' entries denote the levei iarities.

"can value o f data o f Clough el a/. (19) and Lee st a / . (20) from epr spectra.

resonance (MBER) experiment of Meerts et 01. (17) similar to our own new data. The fitted frequencies has provided very accurate frequencies for 2113,2, J in Table 2 are our own data for J = 712 and 912, = 312 and 512, and for ZII,n, J = 112 and 312, mean values of our data and those of ref. 18 for J = whereas the data of Destombes et nl. (18) for Zll,,z, 1112 and 1312, and of ref. 18 for J = 1512. For 11 = 1 J = 7/2,9/2,11/2, . -. 1312, and 1512 is less accurate and and higher levels, it was not possible to determine

COXON ET AL.

TABLE 6. Observed and fitted hyperfine A-doubling transitions" for O H X211, v = 5

2n3/2 2n1/2 J F ' + F VO v, - v, Reference F' + F VO

0- 1+ 1- 1+ 1- o+ 1+ 2- 1+ 1- 2+ 2- 2+ 1-

-0.6754 20

2- 3+ 2- 2+ 3- 3+ 3- 2+

-0.1607 This work - 0.0953 This work

1.1074 This work 0.4646 This work

"Except for entries in square braekets (calculated frequencies), v. are observed frequencies with estimated un- certainties in parentheses; v. - v, are the residuals between the observed and fitted frequencies; + or - in the F and F' entries denote the level parities.

I 3L738.15 MHz

V FIG.^. SpectrumoftheOH, 211, ,2 ,v = I , J = 11/2,F'= 5 + F

= 5 A-doubling transition observed using the parallel plate spectrometer. The phase sensitive detection time constant is 100 ms. and the signal is averaged 300 times.

p ~ , p L , or q ~ , and these constants have been con- strained at the values determined in the o = 0 fit.

0 = 2

The data and type of fit for o = 2 are similar to those for 1) = 1; precise MBER data are now avail- able from Meerts et 01. (17) ~ O I - ~ I I , ~ , , J = 312 and 512 and for 2111,2, J = 112 and 312, while our own data extend to J = 1312 for the 211312 component.

u = 3and4 For o 2 3, relatively few microwave data are

available, and we have therefore included the 211312, J = 312 A-doubling separations estimated from the epr data of Clough et (11. (19), Lee et al . (20), and Lee and Tam (21). For a = 3 and 4, the results of refs. 19 and 20 agree to within about 0.2MH2,

FIG. 7. Variation of the ratios (R"), /(Rn),=, and of the ratios X,/X,=, , where X , represents the magnetic hyperfineconstants, a,, 6,". c,, and d,, for the X I I state of OH.

despite the conservative estimate of 1.3 MHz for the uncertainty in the latter work. In the present work on u = 3 and 4, the A-doublingfrequencies for 211312, J = 312 are the mean values of the results of Clough et al. and of Lee et al., and the uncertainty

TABLE 7. Molecular constants" for OH X211 determined from least-squares fits of microwave and epr data

Value

Constant v = O v = l v = 2 v = 3 v = 4 v = 5

B (cm-l) [18.5504] [17.8386] [I 7.13631 [I 6.44091 [15.7491] [IS. 05661 103D (cm-I) [ I . 91 561 [ I . 87691 [ I . 84181 [1.8116] [I ,78761 [ I . 76941 107H (cm-') [ I . 4261 [I ,3761 [I . 3021 [ I . 2251 [1.111] [O. 9371

10llL (cm-') [- 1 .30] [-I ,381 [-1.321 [- 1 ,591 [-1.281 [-1.31

A (cm-') [-139.2771 [- 139.5491 [- 139.8131 [- 140.0691 [- 140.3021 [- 140.505] O lo4 AI, (cm-I) [-5.011 [-3.961 [-4.191 [-3.151 [-3.871 [-4.441 > z

o (cm-') [-0.2211 [- 0.2201 [-0.2181 [-0.2161 [-0.2131 [-0.2091 > P (MHz) 7056.4480(11) 6739.0955(40) 641 6.829(47) [6080.2] [5732.0] [5360.3] 2 PD (MHz) -0.66685(17) - 0.6770(10) - 0.6930(11) [-0.70331 [-0.71561 [- 0.72791 <

?= 1 0 ~ ~ ~ (MHZ) - 1 .862(78) [- 1.8621 [- 1.8621 [- 1.8621 [- 1.8621 [- 1.8621

2.68(88) <

107pL (MHZ) [2.68] [2.68] [2.68] [2.68] [2.68] 0 !-

q (MHz) - 1162.4160(1) - 1109.8926(10) - 1057.0045(15) - 1003.752(31) - 949.965(90) - 893.860(23) VI

4~ (MHz) 0.442331(12) 0.43511(12) 0.42750(20) 0.42210(71) 0.423 l(47) [0.4155] -4

105qH (MHZ) - 8.301(55) -8.97(17) - 8.48(37) [-8.391 [- 8.391 [- 8.391 ~3 -4 -

IOsqL (MHz) 1 .18(62) [I ,181 [I . 181 [ I . 181 [1.18] [ I . 181 ~3

a (MHz) 86.0123(19) 8 1 .9763(44) 78.0353(49) [74.20] [70.45] [66.81] b~ (MHz) - 73.1258(39) - 77.074(12) -81.914(13) [- 87.641 [- 94.261 [-101.771 c (MHz) 130.749(10) 125.438(27) 120.284(3 1) [115.29] [110.45] [105.77] d (MHz) 56.6320(37) 53.7743(85) 50.9953(90) 48.5(2.5) 46.6(4.0) 44.0(4.4) c1 (MHz) - 0.09948(54) - 0.0982(19) - 0.0962(20) [: 0.09491 [-0.09331 [-0.09171 CI' (MHz) 0.01 804(12) 0.0138(37) 0.0092(41) [O .0048] [0 .0007] [-0.00361 d~ (MHz) - 0.02277(18) - 0.0258(34) - 0.0262(36) [- 0.02891 [- 0.03091 [-O.O329]

Number of lines 57 29 23 7 5 5 8, 1.041 0.811 0.951 0.374 0.410 0.837

Tonstants in square brackets were constrained at these values. Values in parentheses are (? , ,w)1~2 , from the dispersion matrix o f the fit &.

COXON ET AL. 63 1

is taken as 0.2 MHz. Since there are no data for the 211,,2 component, only one of the A-doubling parameters p and q could be determined. One of these parameters ( p ) was therefore constrained to values determined in the concurrent work on OH X211 infrared and optical data. Similarly, since no weak AF = 0 tlxnsitions have been observed, none of the hyperfine parameters ( 1 , b,, c, C,, or C,' could be found. In addition, the data are not sufficiently precise to determine Dd, or sufficiently rotationally extensive to determine q ~ . Thus, the only parameters determined for 11 = 3 and 4 are q, q,, and d . Following Meerts et al. (17) , constrained values ofp,, a , b,, c , C,, C I 1 , MdDd were found by extrapolations of polynominal representations for the 1) = 0 , 1, and 2 constants. For example, (1 , can be represented as

Any error in such extrapolations are of little con- cern in the determination of fitted constants (q, q,, and d ) and in the quality of fit (standard deviations) for o = 3 and 4 . However, predicted frequencies for AF + 0 and for transitions are strongly depen- dent on the reliability of constrained extrapolated constants. Since it is difficult to estimate such ex- trapolation en-or, we are unable to report uncer- tainties for the calculated frequencies of unob- served t1;ansitions; however, we believe that un- certainties are unlikely to exceed the order of mag- nitudes of the last significant figures quoted for each frequency.

11 = 5 The available data for 11 = 5 are similar to those

for v = 4 , though of lower accuracy. An epr value is available for the 211312, J = 312 A-doubling separa- tion (21) . Accordingly, q, is constrained to an ex- trapolated value, and only two parameters, q and d, are fitted for the 1) = 5 level. Some comment is necessary here on the emission doublet observed in the present work at 3731 1.77(.05) and 37318.44(.05) MHz. Although it is tempting to as- sign these lines as OH X211, 11 = 5 , J = 1312, AF = 0 transitions, it was not possible to obtain a satisfac- tory fit when the two lines were included with any of the data of Table 6 . Thus, although the standard deviation of a fit including the emission doublet with q , q,, and d allowed to vary appears not un- reasonable (6, = 2.17), the estimated values ofG,, = 0.372(1) and 2, = 35.1(4) MHz are significantly different from those given by smooth extrapolation from 11 I 4 .

14832.5(1.5) MHz and a line at 8131.2(0.3) MHz which lie close to the expected frequencies for the 11

= 6 , J = 912 and 712 transitions, respectively. The second member of the J = 712 doublet would be overlapped by the intense 11 = 0 , J = 512 line at 8135.87 MHz. However, least-squares fits of these three lines with or without the epr 211,11, J = 312 frequency of 948.38(1.3) MHz of Lee and Tam (21) were quite unsatisfactory (6, = 8.8 and 12.1, respectively, with q and ti allowed to vary). It is apparent, therefore, that either the J = 712 line or the J = 912 doublet are misassigned. Equally satis- factory fits were obtained when either of these data was omitted and only the A-doubling parameter q was allowed to vary. If the J = 712 line is omitted,G, = -837.91(7) MHz,6, = 1.6, whereas ifthe J = 912 doublet is omitted, cj, = -836.24(4), $,, = 1 . 1 . In the latter case, there is only one degree of freedom, and the value of 6, is not significantly better than that for the first fit. Unfortunately, there is also no preference for either of the fitted values of 4,, and a clarification of this dilemma must await the acqui- sition of further data for o = 6 .

Discussion The present work represents a significant ad-

vance toward the determination of the vibrational dependences of the second-order A-doubling and the magnetic hyperfine constants of OH X2I I . The values of these constants, and their higher-order counterparts are collected in Table 7 and have been estimated by least-squares fits of all the available microwave and epr data in the range 0 I 11 I 5. Since each of the constants, X u , is an expectation value of the corresponding radially-dependent operator X ( R ) ,

it is possible in principle to perform inversions to determine the functional forms of X(r.). Thus, for example, in the unique perturber approxima- tion, p, and q, are related to the radially-depen- dent off-diagonal electronicA expectation values (X211 1 H,, I A 2C+) and (X211 1 L+ I A2C+). Similarly, the magnetic constants, a, b,, C , and d can establish the radial variation of the unpaired electron aver- ages, ( l/r3),, ($2(0))u, ( ( 3 cos2 x - l ) /r3), , and (sin2 x1r3),. Since all of these quantities and their radial variations can also be estimated by ub it~itio methods, the present results will be of considerable value in any stringent testing of theoretical tech- niques. The values for some of the constants in Table 7 differ sianificantlv from those re~orted re- .,

11 = 6 cently by other workers. In particular, our values In the present work, we have observed three for p and q are considerably larger in magnitude

microwave lines, a doublet a t ' 14 827.0(1.5) and than the values obtained by Brown et al. (34) for v

CAN. J . PHYS. VOL. 57. 1979

TABLE 8. Relationships between the A-doubling parameters of Brown et nl. (34) and those determined in the present work

Brown et 01. (34) This work"

HP PH - 4qH

H4 411

% ( p ~ , q , r ) and jb(prr ,qr , ) are negligible corrections.

TABLE 9. A-doubling parameters (MHz) for OH X21T, v = 0, 1, 2 for the Hamiltonian of Brown et 01. (34)

v = o v = l v = 2

Parameter This work Meerts et nl. (17) This work Meerts et 01. (17) This work Meerts et al. (17)

= 0 and by Meerts et ol. (17) for v = 0, 1, and 2. As indicated earlier, however, the work of Brown and co-workers (34,37) calls folcaution in making such comparisons, since the effective parameters of a particular model Hamiltonian can be defined somewhat differently from those of alternative but yet essentially equivalent Harniltonians. Thus, for example, Brown er (11. (34) were able to show how the parameters of the Hamiltonian of Meerts (27) are related to those of their own Hamiltonian. The present work uses an effective Hamiltonian of the type developed by Zare er (11. (28), for which Brown er (11. (34, 37) have indicated that p and q should have definitions equivalent to those of their own Hamiltonian. The present results for v = 0 show that such an equivalence is not obtained when cen- trifugal distal-tion is included, and we have examined further what relationships hold between the two separate parameter sets. It can be noted that the different values o fp and q obtained in the present work are not due entirely to the enlarged data set, nor to the use of somewhat different fine structure constants, nor to the representation of centrifugal distortion effects as polynominals in J ( J + 1) rather than (J + 1/2)2. In fact, a detailed examination of the contributions to the A-doubling separations from matrix elements diagonal and

off-diagonal in R shows that p and q , for example, do not have exactly identical definitions in the two representations. The relationships found by this approach are given in Table 8, and were confirmed by a fit in which the 11 = 0 data set of Brown er [I/. (34) was employed with our Hamiltonian after set- t i n g ~ ~ and qL to zero. The A-doublingconstants of this fit, after adjustment with the relationships of Table 8, were in close agreement with those of ref. 34. A similar comparison of the present results for v = 0, 1, and 2 can be made also with the most recent A-doubling constants reported by Meerts er 01. (17). This comparison is made in Table 9, in which both sets of constants have been based on the ef- fective Hamiltonian of Brown et (11. (34). Even with this common representation, however, significant (but much smaller) discrepancies are still found, especially for o = 1 and 2. An explanation is proba- bly afforded by the somewhat lower fixed value for the A / B ratio (A) in the work of Meel-ts er (11. (17), h = -7.4794, compared with h = -7.5045 (34) and h = -7.5080 (this work). In addition, i t can be noted that the values o fp , q , DPM and Dnb! were derived by Meerts et (11. (17) essentially from only four A-doublet separations, so that their 'least-squares' estimates of these parameters are effectively the values of an exact fit.

COXON ET AL.

TABLE 10. Expectation values" of R" for the X211 ground state of OH

u <RZ) (R) W2) ( R - 3 ) <R--')

"The units of R are angstroms.

The preceding discussion has indicated that the several Harniltonian models considered by various authors all give essentially equivalent results and quality of fit. It would be of much advantage, therefore, if a single model and notation were adopted henceforth in any future work. We pro- pose here that the model of Brown et 01. (34) is the most appl-opl-iate choice, as the higher-order matrix elements of this Hamiltonian are established in cor- rect operator form, rather than by the more arbi- trary polynomial expansion appl-oach. However, in agreement with B r o ~ n , ~ we prefer to retain the p , PD, PH, ... q , q ~ , qn, ... notation for the A-doubling parameters, rather than thep, D , , H,, ... q, D,, H,, ... notation of Brown et al. (34).

A comparison of the present determinations of the magnetic hyperfine constants with those of Brown et 01. (34) for 1) = 0 and of Meel-ts et 01 . (17) is also of interest. In making such comparisons, how- ever, it is found that d and C,' in both the present work and ref. 17 have been redefined by Brown et 01. (34) as ci (Brown) = ci - Dd and C,' (Brown) = C,' + id . It should be noted also that since the hyperfine constants are based effectively on the precise data for low J values, the present extended data set makes little change in their determinations and is restricted essentially to a small modification of the high-order A-doubling parameters. The val- ues for all seven constants, o , b,, c , d , C,, C , ' , and Dd derived by Brown et 01. (34) from the earlier work of Meerts (27) for v = 0 agree well with the most recent values of Meerts et crl. (17). However, except for the results for^,=(), our values in Table 7 are generally much closer to the constants deter- mined by the Hamiltonian of Brown et 01. (34) than by that of Meerts et 01. (17), particularly for the correlated pair, b, and c. Similarly, for 11 = 1 and 2, we obtained values f o ~ o , and (I, which are in good agreement with Meerts et 01. (17) whereas the dis- crepancies forb,., and c, are similar to those found for v = 0.

'J. M. Brown. Private communication. 1978.

As discussed by Meerts and co-workers (14, 17), there is considerable interest in comparing experi- mental magnetic hyperfine constants with values given by a h ifzitio methods. However, most a h initio calculations on OH have been pelformed at a single internuclear sepal-ation, and it appears that the LCAO MO CI calculations of Kayama (41) for two separations are the only I-esults which can be related to the observed vibrational dependences for a , bF, c, and d. Thus Meerts et 01. (17) have calcu- lated hyperfine constants for 1) = 0, 1, and 2 from Kayama's data. The effective magnetic constants Xu for each level were determined by linear inter- polation of the values for the operators X(R) at the two R-separations of Kayama (41), with R, for 1) = 0, 1, and 2 being defined by the corresponding B, values. We suggest here that this procedure is neither self-consistent, nor in accord with the ex- pected variations of X(R). First, if X(R) vary linearly with R , the appropriate effective R, values would be (R),, rather than (I/( 1/R2),)1'2. Second, as suggested by Churg and Levy (42), N(R) and c(R) (and probably d(R) also) can be expected to vary approximately as R", with f7 -- -3. In fact, if X(R) = XOR" is assumed for the functional form of Kayama's results, the values ofn are -2.74, - 1.73, and - 3.89 for a , c , and d , respectively, and + 1.89 for b,. In this simple model, Xu for individual vi- brational levels would be given by

Expectation values (R"), for n = 2, 1, - 1, -2, -3, and -4 based on RKR wave functions for OH X211 have been calculated in the present work and are listed in Table 10 for o I 6. Figure 8 shows a plot of the ratios, (R"),/(Rn), for u 5 2. It is interesting to note that these ratios show minima at about f7 = -3. Figure 8 also shows the ratios X,/X, forX = 0, b,, c , and d . Although the quantitative agreement is not

634 CAN. J . PHYS.

perfect, i t is gratifying to find that the data for rr, c , and d all corl-espond to 17 < -2, while the points for bF fill1 between the curves for t i = 1 and n = 2, in accord with the values of 17 deduced from Kayama's (41) results. Clearly, i t would be ex- tremely useful to perform further N D i t z i t i o calcula- tions over a wide range of R.

Acknowledgements J.A.C. gratefully acknowledges the award of a

Research Professorship at Dalhousie University by the Trustees of the Izaak Walton Killam Memol-ial Fund. K.V.L.N.S. wishes to thank Dr. K. P. Lee for his help during the initial stages of the experi- ment. K.V.L.N.S. and J.A.C. thank the National Research Council of Canada for support with Operating Grants. The work at the University of Chicago was supported by the United States Na- tional Science Foundation.

We are indebted to Dr. W. L . Meerts and Dl-. J . L. Destombes for communications of their results prior to publication and to Drs. J . M. Brown and J . L . Destombes for helpful correspondence.

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