Electronic spectroscopy of yttrium monosulfide

Electronic spectroscopy of yttrium monosulfide: molecular beam studies and density functional calculations' ANDREWM . JAMES,R E N FOURNIER, ~ BENOITSIMARD, AND MARGOTD. CAMPBELL' Sreacie Ir~stit~ite for Molecular Scierzces, National Research Co~cr~cil of Carzrrda, 100 Sussex Drive, Orfawn, ON KIA OR6, Crrnada Received April 13, 1993

Can. J. Chem. Downloaded from www.nrcresearchpress.com by 213.209.107.178 on 06/03/13 For personal use only.

Thispc~per-is dedicated to Professor Gerald W. King on the occosiorz of his 65th birthday ANDREW M. JAMES, RENBFOURNIER, BENOIT SIMARD, and MARGOT D. CAMPBELL. Can J. Chem. 71, 1598 (1993) The yttrium monosulfide molecule (YS) has been investigated using the techniques of molecular beam fluorescence spectroscopy and density functional theory. Fluorescence spectra in the region of the electronic orgin of the B'C' + x'Z+ system (vw = 14 826.07 cnl-I) were recorded using a ring dye laser, the experimental resolution being 120 MHz. The B'C' + x'C+ (0,O) band, and a cold band of a hitherto unreported Jrl,,,2 +x'Cf system (v,,., = 14 926.02 cm-I) have been rotationally analysed. (The 1 /2 notation implies that the state has either or ''K ,/, symmetry .) Imroved molecular rotational constants were obtained for the v = 0 levels of the x'C' and B'C' states (ro(X)= 2.27191(17) A, r,(B) = 2.32202(19) A, yo(B! = -0.15150(14) c m ' (20 error bounds)). The magnetic hyperfine and spin rotation parameters determined for the X-Cf state were found to be in good agreement with previous work. An accurate bond length has been derived for the upper vibrational level of the 14 926.0 c m ' band (I-,,.= 2.49510(16) A). The v = 1 level of the BIZ+ state is found to be strongly perturbed by another vibrational level of the 'rIn,,,>st;te. The spin-forbidden In,,,, + x'Z+ transition gains intensity via spin-orbit mixing between the "rI state and the A-C' state. Radiative lifetimes of the observed bands were measured by digitizing the fluorescence decay traces obtained upon excitation with a pulsed dye laser. The results of the density functional treatment provide broad confirmation of the experimental measurements. A molecular orbital description of the bonding in the YS molecule is presented.

*

ANDREW M. JAMES, RENEFOURNIER, BENOIT SIMARD et MARGOT D. CAMPBELL. Can J. Chem. 71, 1598 (1993). On a CtudiC la molCcule de monosulfure d'yttriuin (YS) B l'aide de techniques de spectroscopic de fluorescence de rayons laser et par la theorie de la densite fonctionnelle. Pour enregistrer les spectres de fluorescence dans la rCgion de l'origine Clectronique du systkme B'C+ + x'C+ (urn = 14 826,07 cm-I) on a utilisC un laser 5 colorant cyclique pour lequel la resolution experimentale est de 120 MHz. On a analysC la rotation de la bande B'C+ + x'C' (0,O) et une bande froide d'un systkme 'IIlI2 + x 2 C f non rapport6 jusqu'a mainenant (v,,., = 14 926.02 cm-I). On a obten) des constantes de rot?tion ameliorees pour les niveaux v = 0 des Ctats x'C+ et B'C.' (r,(X) = 2,27191(17) A, ro(B) = 2,32202(19) A et y,(B) = -0,15150(14) cm-I (limites,d7erreurde 20)). On a trouvC que les paramktres magnCtiques hyperfins et de rotation de spin determinks pour 1'Ctat X-C ' sont en bon accord avec ceux obtenus au cours d'un travail antCrieur. On a dCterminCounelongueur de liaison prCcise pour le niveau vibrationnel supCrieur de la bande 2 14 926,O cm-' (ri,.= 2,495 lO(16) A). On a trouvC que le niveau v = I de 1'Ctat B'Cf est fortement perturb6 par un autre niveau vibrationnel de 1'Ctat "1,1,2. La transition JrI,y:! + x2Cf non permise par les spin augmente d'intensitC par le biais d'un mClange spin-orbite entre 1'Ctat "rI et I'Ctat B-Z'. On a mesurC les temps de vie de radiation des bandes observCes en procCdant a la digitalisation des courbes de dCgCnCrescence de la fluorescence obtenues lors de l'excitation h l'aire d'un laser B colorant pulse. Les rtsultats du traitement fonctionnel de la densit6 fournissent une large confirmation des mesures expCrimentales. On prksente une description de I'orbitale molCculaire de la liaison dans la ~nolCculede YS. [Traduit par la redaction]

Introduction Difficulties arise in the spectroscopic study of molecules containing a transition metal atom due to the high density of low-lying electronic states associated with partially occupied d orbitals (1, 2). In addition, the large nuclear spins of the transition elements can give rise to complex magnetic hyperfine structure patterns in the already congested spectra of these molecules. However, unravelling the spectroscopy of such systems can pay rich dividends when one considers how central to chemistry is an understanding of bonding involving transition metals. Our work focusses on the electronic spectroscopy of gas phase, transition-metal-containing diatomic molecules. Preparing the molecule of interest in a gaseous environment permits the extraction of detailed information on the properties of a single transition metal - ligand bond, free from interference by other ligands o r a host matrix. The versatile technique of chemical reaction in a laser-generated plasma, coupled with supersonic expansion 'Issued as NRCC No. 3525 1 . 'NRCC/WISE summer student 1992.

of the products, leads to the formation of intense, internally cold beams of refractory transition metal molecules, which may then be probed by laser spectroscopic techniques (3-6). We have used this approach to make detailed investigations of molecules containing a group three transition metal atom (Sc, Y , or La) bound to a simple ligand. The group three transition elements have the simplest open-d shell electronic structure [(n - 1)d' ns'], and should therefore give rise to less daunting spectra than their counterparts towards the middle of the transition periods. Experimental determination of the spectroscopic properties of these transition metal molecules furnishes the theoretician with data to check the accuracy of high level nb initio calculations. Yttrium monosulfide is a particularly apt candidate in this respect, since the metal-ligand bond is expected to contain both covalent and ionic contributions to a significant extent. T h e experimental database relating to the spectroscopy of yttrium monosulfide is rather limited. McIntyre et al. (7) have reported esr and infrared spectra of the YS molecule isolated in an argon marix at 4 K. Stringat, Fenot, and FLmCnias (8) have obtained electronic spectra of gas phase YS in the

1599


JAMES ET AL.

red and near-infrared spectral regions, using a hollow cathode discharge source. Several bands belonging to the A% + x'C' and B% + xX" systems were reported, la number of them perturbed. A rotational analysis of the B - 2 + x'C+ (0,O) band was effected, despite the limited resolution and high source temperature. In 1990, Azuma and Childs (9) performed an elegant laser - radio frequency double resonance study on the ground state of YS, pumping the B 2 C + x'C+ (0,O) band. The ultra-high resolution of this technique (10-15 kHz) permitted very precise determinations of the hyperfine structure and spin-rotation constants for the x'C' state, although no attempt was made to fit the optical frequencies in order to improve the upper state rotational constants. Recently, we employed molecular beam Stark spectroscopy to determine the permanent electric dipole moments of the x'C' and B'C+ states of YS (6). Values of 6.098(64) D (x'C+ state, v = 0 , 2 u error bound) and 4.572(92) D (B2C+ state, v = 0) were obtained. The marked reduction in the dipole moment ( k ) upon B'C + x'C+ excitation was unexpected, and warranted further investigation. The density functional calculations reported here demonstrate that the large difference between the dipole moments of the x'C' and B2C+ states of YS can be attributed to the differing extents of valence orbital hybridization in each state. We also present experimental and theoretical evidence for a 'II state in the vicinity of the B'C+ state.

Experimental A brief outline of the experimental procedure is given, as a detailed description can be found elsewhere (10). The metal atom plasma was formed by pulsed laser vaporisation of an yttrium target rod (Goodfellows, 99.9%). An excimer laser (Lurnonics model 430) operating on the 308 nrn line, or a Nd:YAG laser (Lumonics HY400) generating third hamonic radiation (355 nm), were employed as vaporization sources. There were no noticeable differences in the efficiency of production of YS when 355 nm radiation was substituted for 308 nm. Vaporisation took place in a Smalleytype cluster source fixture, in the presence of a flowstream of CS2seeded helium gas (0.5% concentration) from a molecular beam valve (Newport BV- 100). The post-reaction gas mixture, containing YS molecules, was expanded into a vacuum chamber, where laser-induced fluorescence was excited 4.3 cm downstream from the molecular beam orifice. Fluorescence was imaged through a monochromator (Spex, 0.75 m, 1200 lines/mm) on to a photomultiplier (cryo-cooled Hamamatsu R943). Low-resolution survey spectra and fluorescence decay profiles were recorded using a pulsed dye laser (Lumonics Hyperdye, 0.08 cm-' linewidth), operating on the dyes Kiton Red, DCM, and LD690. The decay traces were sampled by feeding the photomultiplier signal to a digitizer (Tektronix 7912AD). The high-resolution fluorescence spectra were obtained using a cw ring dye laser (Coherent CR699-29), operating on DCM dye. The resolution of the experimental arrangement was found to be 120 MHz. The high-resolution spectra were calibrated by the wavelength meter system incoporated in the ring laser, its operation being checked by a Burleigh WA-20 wavemeter. The absolute accuracy of the ring dye laser calibration system is 200 MHz.

Density functional calculations : computational details W e performed Kohn-Sham density functional calculations on the lowest doublet and quartet states of YS, and YO for comparison, using the Linear Combination of GaussianType Orbitals-Density Functional program deMon (1 1). We used the local exchange-correlation potential of Vosko et al. (12) throughout the self-consistent field procedure and in-

cluded gradient corrections to exchange (13) and correlation (14) in the final energy evaluation. The vibrational frequencies were calculated in the harmonic approximation from energy second derivatives obtained as finite differences of analytical first derivatives. The orbital basis sets we employed have the following contraction patterns, which show the number of primitive Gaussian functions in each contracted function, first for s symmetry, second for p , and then f o r d : oxygen : (631 1/31 1/1); sulfur: (731 11/611/1); yttrium: (633321/532111/531). The yttrium has two diffuse p functions to give extra flexibility in describing the polarization of the 5 s orbital. The auxiliary basis sets for fitting the charge density and exchange-correlation potential were fully uncontracted and consisted of eight s , f o u r p , and four cl functions on oxygen, nine s, four p, and four d functions on sulfur, and ten s , five p, and five d functions on yttrium. The formalism on which our method is based holds rigorously only for the lowest states of a given symmetry and spin multiplicity. The calculated doublet-quartet energy separations have physical meaning and are truly "first principles." It should be noted that methods based on the local spin density approximation seem to underestimate the energy of high-multiplicity states relative to low-multiplicity states, but the trend is apparently not systematic (15, 16). Results for states that are n o t the lowest of their symmetry and spin multiplicity should be viewed with some caution. They involve additional, and poorly understood, approximations. For a discussion of exact a n d approximate treatment of excited states in density functional theory, the reader should consult the articles listed in ref. 17.

Description of the spectra W e completed a low-resolution survey of the YS spectrum over the region 600-800 nm, observing numerous, vibronic bands of the A'II + x'C+ and B'C + X-2' transitions, as well as several new bands extraneous to these systems. In this work, we concentrate on the portion of the spectrum between 630 and 700 nm, in the vicinity of the electronic origin of the B'C+ + x'C' system, where two interesting new bands were discovered and investigated. Other new features and their interaction with the A'II +x'Z+ and B'C' + x'C' systems will be the subject of future publications. Figure l(n) shows a part of the low-resolution LIF spectrum in the region of the B'C' + x%+ (0,O) band. Approximately 100 cm-' to higher energy of this band, a new and considerably less intense band was detected, with rotationless origin at 14 926.02 cm-'. Dispersing the emission from this band confirmed that it originated from the v = 0 level in the ground state of YS. Figure I (b) shows a p o r t i ~ n of the pulsed dye laser LIF spectrum in the vicinity of the B-Z + x'C+ (1,O) band. In this region, two double-headed cold bands of comparable intensity were found, located at 15 264.2 cmpl and 15 300.1 c ~ n - (approximate ' location of rotationless origins). At the outset, it was not clear which of (1,0), since these bands should be designated B'C+ + it appeared that the (1,O) band was being strongly perturbed by an almost degenerate band from another system. (As we demonstrate below, the upper vibrational levels of the 14 926.0 cm-' and 15 300.1 cm-' bands are neighbouring vibrational levels of a 411,,,2state.) Lifetime measurements on the 15 264.2 cm-' and 15 300.1 cm-' bands (vide infra)

x"'


1600

CAN. 1. CHEM. VOL. 71, 1993

I

668

I

670

I

672

I

674

I

676

I

678

I

65 1

I

653

LASER WAVELENGTH (nm)

I

634

I

655

I

657

I

659

I

661


I

63 6

I

638

I

640

I

642

I

644


FIG. 1. Low-resolution LIF spectra recorded with the pulsed dye laser. The vertical scale is the LIF intensity. The ( a ' , u") labels refer to the B ' Z i t x'C' system. ( a ) Spectrum in the region of the electronic origin of the B 2 Z + t x'Z+ system. The feature at 670.0 nm is the (vl,O) band of the 4n,,,2t x ' Z t system discussed in the text. (b) Spectrum in the region of the (1,0) band of the B'Z' t x'C+ system. The feature at 653.2 nm is the (zl' + 1 , 0 ) band of the 'II,,,? t x'Z+ system. ( c ) Spectrum in the region of the (2,0) band of the B'Z+ + x'Z+ system. revealed that the 15 264.2 cm-' band derived more character from the B'C+ +-X'S+ system than did the 15 300.1 cm-' band. Thus for our purposes it will henceforth be referred to as the (1,O) band, recognizing the caveat that its upper vibrational level is strongly mixed7 with that of the 15 300.1 cm-' band. In Fig. l ( c ) , the B-C+ +- x2C+(2,0)band is shown, accompanied by neighbouring sequence bands, the identity of all features having been confirmed by recording dispersed emission spectra. Unlike the (1,O)band, the (2,O) band is clearly unperturbed and resembles the (0,O) band. Figures ~ ( L z )and 2(b) show portions of the c w - ring laser LIF spectra recorded for the B'C' +- X2C'(0,0) band and the 14 926.0 cm-' band, respectively. Unequivocal line assignment in these bands was greatly facilitated by identification of the hyperfine structure patterns associated with each rovibronic line. The hyperfine structure observed in these spectra is due entirely to the x'C' state. Our experimental resolution did not enable us to resolve the splittings due

to the small Fermi contact interaction in the B'C' state ( b = -78 MHz ( 9 ) ) .Figure 3 shows how the hyperfine struclture patterns arise in the B'C' +- x"' transition. In the X-C' state, at low rotational excitation, a bps coupling scheme I =,1 / 2 ) , is obtains in which the yttrium nuclear spin ('9 coupled to the spin of the unpaired electron ( S ) to give a resultant G , which takes values of 0 or 1. In this case, the three G = 1 hyperfine levels are almost degenerate, and the hyperfine structure is manifested in the spectra as a doublet with a 3 : 1 intensity ratio between the two components. The more intense component corresponds to transitions out of the G = 1 levels, the weaker one to transitions from the G = 0 level. The best example of bps coupling is furnished by the line R,(O)in the B'C+ +- x'C' (0,O)band (Fig. 2(n)).The splitting between the G = 0 and G = 1 components is equal to state. With the Fermi contact interaction for the x'C' increasing rotational excitation, bp, coupling sets in, the spinrotation interaction gradually decoupling S from I , and COU-


JAMES ET AL.

1

1

14826.0

1

1

1

1

t

1

1

1

1

8

1

1

1

1

'

1

1

1

1

i

l

l

l

l

l

r

1

1

l

l

1

1

1

14827.0

LASER WAVENUMBER (cm-')

l

l

14923.0

l

l

'

l

l

l

'

l

l

l

l

l

"

r

l

l

l

l

l

l

l

l

l

l

14924.0

LASER WAVENUMBER (cm-')

FIG. 2. Portions of the cw - ring laser LIF spectra recorded for (a) the (0,O) band of the B'Z' band of the 4rI,,,1 +- x'C+ system. Vertical scale is the LIF intensity.

I

pling it to N, the vector representing nuclear rotation. The three G = 1 levels split apart, giving, in the limiting case of fully developed bpJcoupling, two pairs of hyperfine levels. The primary splitting between the levels is due to the spinrotation interaction, the secondary splitting to the Fermi contact term. The R,(N) lines returning from the bandhead in the B2C+ +- x 2 C + (0,O) band (Fig. 2(a)) provide good examples of fairly well-developed bpJ coupling. Figure 3 shows the situation w h e r ~the coupling is intermediate between bp, and bpJin the X-2' state, the coupling in the B'Z+ state be~ngbpJeven in the absence of rotation on account of

+-

x 2 C + system, and ( b ) the

(vl,O)

the large spin-rotation parameter. With the selection rule A F = +- 1 in operation, only three hyperfine components are seen for lines of the P2 and R, branches of the B2C+ +-x 2 C + (0,O) band, while the P , and R2 branches exhibit four hyperfine components. The e l f symmetry of the upper state levels in this band could thus be established unequivocally from the number of hyperfine components associated with a line. Similar arguments may be applied to rationalize the hyperfine structure in the 14 926.0 cm-' band, which we demonstrate below to be a 411,,,2 + x 2 C + transition. Figure 4 shows a typical fluorescence decay trace obtained in


CAN. J . CHEM. VOL. 71, 1993

FIG. 3. Energy level diagram for the B'Z'

+ x?Cftransition,

showing allowed hyperfine transitions.

TABLE 1. Fluorescence lifetimes of some vibronic levels of the B'C' State

ZI

and 'rI?,,, states of Y S ~,,(cm-')

~(ns)

"Numbers quoted in parentheses are the uncertainties ( 2 0 ) in the units of the last quoted decimal place.

Data analysis

Time ( ns ) F I G . 4. Fluorescence decay trace obtained by exciting the R , bandhead of the B'Z' + x'C' (0,O) band.

this case by exciting the R , bandhead of the B'Z' + ~ ' 2 , ' (2,O) band. Decay profiles were recorded for all the bands whose LIF spectra are presented here, excepting the 14 926.0 cm-' band which was found to be too weak to permit recording of an adequate trace.

Fluorescence decay traces Lifetimes were extracted from raw digitized data by first performing a background subtraction and then subjecting a portion of the trace well clear of the excitation pulse to a simple linear least-squares analysis. Deconvolution of the excitation pulse was not carried out, since the observed lifetimes were substantially longer than the laser pulse duration (9 ns). All the observed decays appeared to follow a single exponential function, and were fitted as such. Results of the analysis are presented in Table 1. Measurements are unlikely to be affected by "field of view" problems, since the molecules remain within the fluorescence collection zone for about 1 ps. Each reported lifetime is the weighted average of several independent determinations. The uncertainties quoted in parentheses represent the spread in lifetimes over repeated measurements. The rather large error bounds found for some of the bands were due to electrical noise recorded


JAMES ET AL.

on the decay traces that affected the analysis. As measurements were made under collision-free, molecular beam conditions, and as the fluorescence quantum yield should be unity at these energies, the data in Table I may be regarded as intrinsic radiative lifetimes. The relatively short lifetimes exhibited by the 71 = 0 and z1 = 2 bands of the B'C' state are typical of a strongly allowed radiative transition. In a system where an electronically excited state decays to a single, isolated lower state, and in which the vibrational frequency of the upper state is less than that of the lower state, the radiative lifetime of the upper state should increase with increasing vibrational quantum number (18). Such behaviour is expected in YS, where w,(B'C+) (=449.7 cm-') < w,(x'C+) (= 492.7 cm-') (8). The contradictory experimental observation that the ~ ' 2 ' state u = 0 level is longer lived than the u = 2 level, is an indication that one of the vibrational levels is being perturbed by another state. Unlike the (0,O) band, which lies only 100 cm-' from the 14 926.0 cm-' band, the (2,O) band is quite isolated from any other experimentally observed cold bands. Thus a plausible explanation for the observed trend in lifetimes is that the u = 0 level is mixed with the presumably long-lived upper state level of the 14 926.0 cm-' band, leading to an increase in the lifetime of the u = 0 level relative to the u = 2 level. Unfortunately, since we were unable to measure the lifetime of the upper state level in the 14 926.0 cm-I band, we cannot comment further on this point. Inspection of the data in Table 1 suggests that there is a strong interaction between the upper state levels in the 15 264.2 and 15 300.1 cm-' bands, leading to an increase in the lifetime of the former, and a decrease in the lifetime of the latter. We present evidence below to demonstrate that the upper vibrational level of the 15 300.1 cm-I band belongs to a state. Since the upper level in the 15 264.2 cm-' band has the shorter lifetime, we assign it as the B'Z+ u = 1 level, and the other level as belonging to the manifold, even though the magnitudes of the lifetimes indicate that the extent of mixing is quite considerable. The interaction between the B2C+ state z1 = 1 level and the state may be neighbouring vibrational level of the 4n+,,2 understood by way of a simple two-state intensity borrowing mechanism (19). In the absence of an interaction between the B'C+ state and the 4rI+1,zstate, the oscillator +- x'C+ transition would be zero, and strength for the only a single bnd would be observed. If the measured radiative lifetimes of the upper state levels in the 15 264.2 and 15 300.1 cm-' bands are denoted by 7- and 7, respectively (the - and referring to the directions of the energy shifts experienced by the levels when the perturbation is switched on), and the lifetimes of the parent, unperturbed B'C+ and 'LIl2 state levels by 7' and 72, respectively, then by conservation of transition intensity we may write for the radiative decay of the upper state levels:

+

The radiative lifetime for the vibrational level of the Qn,,/? state in the 15 300.1 cm-I band, in the absence of the per, be calculated using the data in turbation ineraction ( T ~ )may Table 1. The unperturbed radiative lifetime for the B'C' state u = 1, 7,, is approximated by the lifetime of the 71 = 2 level,

1603

making a small correction for the differing u3 factors for transitions from u = 1 and u = 2 to the ground state. The lifetime of the unperturbed B'C+ u = 1 level is thus estimated to be 112.4(22) ns. Substituting this value and 7- = 188(11) ns, 7, = 233(24) ns in eq. [1] yields 7;' = (7.1 + 7.6) X 10-"s, implying that 7, could lie in the range 680 ns + infinity. The relative error on the result is very large due to the ill-conditioning of eq. [ J ] arising from the large errors associated with the determination of 7- and 7+. All we can say with certainty is that the lifetime of the perturbing state is very long, and possibly infinite. This result is quite consistent with the spin-forbidden nature of the 411,1,2+ X'S+ transition. We can not, however, rule out the possibility that the 411,1,2 state decays slowly to a nearby quartet state. Tlze B"+ +- x"+ (0,O) band Line positions for the B'C+ +- x 2 C + (0,O) band are reported in Table 2(a). Molecular constants were extracted from the rotational line data for the B'C+ +- x'C+ (0,O) band using a nonlinear least-squares procedure. There follows a description of how the pertinent eigenvalue expressions for the ~ ' 2 'and x'C+ state levels were obtained and incorporated into the fitting procedure. The effective Hamiltonian for a 'C state may be written as:

where B is the rotational constant, D the centrifugal distortion constant, y the spin-rotation constant, bF the Fermi contact parameter, and c the dipolar interaction parameter, the quantum numbers having their usual definition. Since with our experimental resolution there was no detectable hyperfine splitting in the B'C' state, the last two terms in eq. 121 were dropped for the evaluation of the upper state eigenvalues. The matrix elements for the first three terms are diagonal, and give rise to the well-known pair of expressions for the energies of e and f rotational levels in a 2C state (20):

(e levels)

(f levels) Equations 131 and. [4] were used directly to represent the BY state energy levels. Since the spin rotation constant in the upper state is large, it was represented by a centrifugal expansion thus:

In the x'C+ state, the spin-rotation parameter is smaller than the Fermi contact term, and therefore the hyperfine energy levels were determined by diagonalization of the appropriate 4 X 4 energy matrix, which is presented in Table 3. The matrix elements are those derived by Radford using a case b p , basis set (21). Our treatment was simplified by discarding all terms involving the spin-dipolar parameter, c, as our experimental resolution did not warrant their inclusion. The observed line positions of Table 2(a) were fitted using a

CAN. J . CHEM. VOL. 71, 1993

TABLE 2(a). Observed line wavenumbers and residuals (obs. - calc., in parentheses) for the B'C+ c x'Z* (0,O) band of "Y"S


P,(N)


JAMES ET AL


CAN. J. CHEM. VOL. 71, 1993

Residual mean square error = 109.6 MHz

modified Levenberg-Marquardt algorithm. The energy matrix for the ground state hyperfine levels was diagonalized and the resulting eigenvalues added to B1'N(N 1) - D"N'(N 1)' to generate the lower state term values. These were combined with term values calculated for the upper state using eqs. [3] and [4]. The values of the various spectroscopic parameters were varied until the best fit of calculated to experimental line positions was obtained. The results of the fitting procedure are presented in Table 4(a). Rotational perturbations in the upper state were found to set in beyond N = 36, so lines terminating on these levels were not included in the fit. The random error of measurement (109 MHz) is well below the experimental accuracy of 200 MHz. The obtained for the Fermi contact and spin-rotation ground state are in good agreement with those determined by Azuma and Childs (9). The sign and magnitude of the spin-rotation constant in the B'C+ state (yo = -0.15150(14) cm-I) are a manifestation of an interaction of the "unique perturber" type between this state and the A'TI state, where Stringat et al. (8) determined the A-doubling parameter p to be +O. 155 cm-I. The 14 926.0 cnz-' band The band located at 14 926.0 cm-' (see Fig. 2(b)) qualitiatively resembles the B'C' +- x%+ (0,O) band, in that it is red-degraded and possesses four discernible rotational branches with the same hyperfine structure patterns as the

+

+

(0,O) band. Similarly, there was no evidence for hyperfine structure in the upper state at the experimental resolution of 120 MHz. Line assignments were confirmed in two ways. Firstly, the combination differences A7Fv(J) were formed for the e and f manifolds. The B" values thus determined were found to be in good agreement with the value reported in Table 4(a). Secondly, the hyperfine splitting patterns of the lines in the 14 926.0 cm-l band were compared with those of the corresponding lines with the same N" values in the B2C+ +- X'C+ (0,O) band. The patterns were found to match, well within the limits of experimental error, confirming the assignments of the quantum number N " and the e l f symmetry labelling for lines of the 14 926,O cm-I band. A rotational energy level diagram for the upper state level of the 14 926.0 cm-' band, shown in Fig. 5 , was constructed as follows. Firstly, the rotational and hyperfine parameters determined for the x'C+ state v = 0 level (Table 4(a)) were used to calculate the term values of the rotational-hyperfine levels in the lower state. The upper state rotational term values were obtained by adding the appropriate ground state term values to the experimental line positions reported in Table 2(b). This procedure yielded either three of four determinations of the position of each upper state rotational level, which were found to coincide, well within the experimental uncertainty. These values were averaged and are plotted for the lowest rotational levels of the upper state in

1607

JAMES ET AL.

TABLE2(b). Observed line wavenumbers and residuals (obs. - calc., in parentheses) for the X'C' (ZI = 0) 14 926.0 cm-' band of R 9 ~ 3 2 ~


P I I+

(N)

+


CAN. J . CHEM. VOL. 71. 1993

JAMES ET AL.

TABLE2(b) (concluded)

+


@ ~ 1 2

Residual mean square error

=

Ql,

(N)

106.2 MHz.

TABLE3. Energy matrix for the rotational, spin-rotation, and magnetic hyperfine structure parts of the Hamiltonian of a '2' state, using case b p J basis functions

(f)

(e) J = N + 1/2

+ + +

+

+

J = N -

+ + + + +

+

A = (N 1)(N 2 ) - I(1 I ) - (N 1/2)(N 3 / 2 ) B = N ( N + 1) - I(I + 1) - ( N 1/2)(N + 3 / 2 ) C = ((2N - I 1 /2)(1 + 1 /2)(2N I 3 / 2 ) ( 1 + 1 /2))"' D = ((2N - I 1/2)(1 - 1/2)(2N I 1/2)(1 3/2)j1/' E = N ( N + I) - I(I + 1) - ( N - 1 / 2 ) ( N + 1 / 2 ) F = N(N 1) - I(I I) - ( N - 1 / 2 ) ( N 1/2) B IS the rotational constant and y the spln-rotatlon constant; b = b, parameter The quantum numbers have t h e ~ rusual defin~tion

+

+

+

1/2

+

+

-

Fig. 5 . The derived energy level pattern is quite characteristic of a state with non-zero orbital angular momentum in Hund's case (a). The splitting between e and f levels with the same J value, which is found to increase almost linearly with

r / 3 , where b,

IS

the F e n 1 contact parameter and c the d~polarInteractLon

J + 1/2, is due to A-doubling. The existence of a level with J = 1 /2 indicates that the R value for the state is either + 1/ 2 or - 1/2. There now arises the question of how the new fl = + 1/2 state should be assigned. The A'II state has been

1610

C A N . J . C H E M . VOL. 71, 1993

TABLE 4(n). Sqectroscopic constants for the B'C' X-Cf (0,O) band of 8 9 ~ 3 2 ~

X?C+ .(, Bo

=

0

0.138883(20)" 2.27191(17)

( A )

B?C+

=

c

0

0.132953(21) 2.32202(19)


Do 5.7(13) X lo-' 1.96(13) X lo-' X lo-' -0.15150(14) Yo.o,,".437(39) 3.16(44) x lo-" Y I .O,> 1.53(31) X 72.0 bo -0.02120(98) woo = 14 826.07260(65) "Numbers in parentheses are the uncertainties (2a) in the units of the last quoted decimal place. "Centrifugal expansion of spin-rotation constant (see text).

TABLE4(b). Spectroscopic constants for the 'IT,,/2 + x?C+ (ul,O)band of

Energy ( c ~ - ~ )

14928.0

89~32s

41~,1/2 U' B r(A) D n b T,,.,

0.115 1477(30)" 2.495 lO(16)" - 1.90(24) X lo-' -7.618(38) X lo-' -3.421(45) X 14 926.02420(65)

"Numbers in parentheses are the uncertainties (2a) in the units of the last quoted decimal place. "10a error bound (see text).

well characterized experimentally (8), so it can be ruled out immediately. Another possibility is that the new state is of doublet spin symmetry and is derived from the configura(The parentage of these molecular orbitals can tion u~.rr:u~. be found in the "Results of Calculations" section below.) Calculations on the isovalent molecule YO suggest that this "charge transfer" type state should have a bond length considerably longer than that in the ground state (22), as is found experimentally by taking combination differences for the upper state in the 14 926.0 cm-' band. However, such a dramatic bond-length change upon excitation would be expected to produce an extended vibronic progression in the upper state vibrational quantum number in the allowed 'n +X'S+transition. This expectation is at odds with the experimental observation of only one, or possibly two (including the new band at 15 300.1 cm-'), bands belonging to the new system in the energy region of interest. Furthermore, transitions to both the 0 = 1/2 and f l = 3/2 spin-orbit components of this doublet state would be allowed, and there is no evidence for any other bands that could be due to transitions from the ground state to the R = 3/2 component. Similar arguments may be applied to rule out assignment to other % states based on charge transfer type excitations. There remains another possibility: that the new state belongs to the quartet spin manifold. Calculations (vide infra) place the lowest lying quartet states in approximately the same energy region as the B'C+ state. On the basis of the above arguments we assign the symmetry of the new state . . to be 'II,,,,, arising from the configuration u:.rr:uS. The ro-

FIG. 5 . Rotational energy levels in the u' level of the 'lT,,/2 state, derived from the experimental data reported in Table 2(b).

tational branches in the 14 926.0 cm-' band are identified Q l l + 'R'?, and R , ,. The spin-forbidden as P I1 + " Q ~ ? , 411+1,2 +x'C+ transition acquires intensity through spin-orbit contamination of the R = + 1/2 spin-orbit components by the B'C+ state. The other spin-orbit components of the 'I3 state, 52 = 3/2 and fl = 512, cannot be observed in excitation from the X'S+state on account of the strict AR = 0 selection rule for the spin-orbit operator ( 19), which leads to non-zero matrix elements between the 'I3 (R = -t. 1/2) and B2C+ (R = 5 1/2) states only. The spin-orbit interaction mechanism explains why the 14 926.0 cm-I band can be observed, despite unfavourable Franck-Condon overlap with the ground state. This point is developed further in the Discussion section. One possible assignment for the 14 926.0 cm-I band is that excitation. In such a case, it arises from a 'n-,/' + x2C+ the new band at 15 300.1 cm-' band could be due to the corresponding 411+,/2+ x 2 C + vibronic transition, implying a spin-orbit splitting in the 'n state of approximately +374 cm-'. However, a calculation of the spin-orbit splitting for the 'II state using atomic spin-orbit parameters suggests that it should be of the order of +266.5 cm-', a considerable discrepancy. Furthermore, in a 'n state, the rotational constant for the R = 1/2 component should be greater than that of the R = - 1/2 component. A deperturbation analysis of the 15 264.2 and 15 300.1 cm-' bands is currently under way. From a preliminary centres of gravity analysis, using the data in Table 4 and the Pekeris relation to estimate the


JAMES ET AL.

rotational constant in the v = 1 level of the B state (0.13242 cm-I), we calculate the rotational constant of the perturbing level to be 0.1149 cm-I. This is appreciably smaller than the rotational constant determined for the upper level of the 14 926.0 cm-' band, contradicting the suggested assignment. A more plausible explanation is that the upper level in the 15 300 cm-' band belongs to the sam spin-orbit component of the 'II state as the upper level in the 14 926.0 cm-I band (see Discussion). Since we lack rotational line data on any of the other spinorbit components of the % state, a definitive assignment of + X'S+ or 'II-,/, + the 14 926.0 cm-' band to a "11+1/1 x'C+ transition cannot easily be made. Cheung et al. (23) have given a detailed treatment of the derivation of the matrix elements of the rotational Hamiltonian of a 'II state in case (a) coupling. In a 411+,/,state, the A doubling increases in an approximately linear fashion with J + 1 /2 on account of the p 2q term appearing in the diagonal matrix state, although there is no element. However, in a 411-,/2 diagonal term in J 1/2 (apart from a distortion constant), q term connecting the R = 1/2 the off-diagonal o p and R = - 1/2 components has a similar effect upon matrix diagonalization. Thus in fitting the 14 926.0 cm-' band, we decided at the outset to employ a simple expression for the rotational energies of the e and f levels of the 'IIn,,/, state:

+

+ + +

+

?

(J

+ 1/2)[a + bz]

where z = ( J + 1/2)', the upper and lower signs referring to the e and f levels respectively. B , D, a , and b are, respectively, effective rotational, centrifugal distortion, A-doubling, and A-doubling distortion parameters. Equation [6] is simply a reduced form of the diagonal matrix elements for a 'II+,/, state, as given by Cheung et al. The observed line positions in Table 2(b) were submitted to leasts uares analysis in the manner described for the B'C+ + 9 X-C+ (0,O) band. In the present case, the rotational and hyperfine parameters for the x'C' state v = 0 level were not varied, since these were already determined with high precision in the B'C' + x'C' (0,O) analysis. The results of fitting the 411,1,2+ x 2 C f band (423 hyperfine lines) are presented in Table 4(b). Since the rotational constant is contaminated by a number of effects, we quote a IOU error bound on the bond length, rather than the more typical 2u error. The negative centrifugal disortion constant and the anomalously large A-doubling distortion terms reflect the effects of S-uncoupling within the 'II state. The RMS error (283 MHz) is somewhat higher than the experimental resolution (120 MHz), reflecting the need for data on the other spin-orbit components. Rigorous analyses involving diagonalization of the complete Hamiltonian matrix were also attempted, using an estimated value of the spin-orbit parameter of $266.5 cm-l. Due to the lack of data on any other spin-orbit parameters of the 'II state, several of the parameters obtained from these fits were poorly determined and highly correlated. However, whether the upper state was assumed to be of 'II+,/, or '11-1/2symmetry, the rotational constant was always well determined and in reasonable agreement with the value obtained from the fit using Eq. [6]. The bond length in the 411,1,,state is some 9.8% longer than that in the x'C+ state. This is quite consistent with the results of the density functional treatment (vide infra) where

1611

the bond length in the lowest lying quartet state is calculated to be 10% longer than in the ground state (see Table 5(a)). The negative sign for the A-doubling terms linear in J + 1/2 indicate that the 4C- state, which is assumed to be responsible for the A-doubling, lies above the 'n,,,, state.

Results of calculations The calculated spectroscopic properties of YO and YS are shown in Table 5(a) along with those from the theoretical study of Langhoff and Bauschlicher (22) and experimental values. Gradient corrections are known to considerably improve the accuracy of bond energies relative to purely local functionals and bring the discrepancies down to typically a few kcal/mol (24). The generally very good agreement obtained here between experiment and theory is probably somewhat fortuitous. In YO, the lowest quartet state is calculated to lie 26 166 c m ' above the ground state, in excellent agreement with the value calculated by Langhoff and Bauschlicher. In YS, the lowest quartet state is calculated to be much more stable, lying only 17 439 cm-I above the ground state, on account of the lower electronegativity of sulfur relative to oxygen. The other calculated properties (w,, p , dp/dr) are not highly accurate, but sufficiently so to give the correct trends. Overall, the agreement of the present calculations with those of Langhoff and Bauschlicher and with experiment is quite good. It should be noted that the density functional method does not allow us to further break down the s , p , and d composition of a molecular orbital into constituent contributions from individual ns, np, and tzd orbitals. Thus in the ensuing discussion, we make the assumption that one particular ns, tzp, or nd orbital dominates the s , p , or d contribution to a particular molecular orbital. The bonding can be characterized from the Mulliken populations of Table 5(b). As expected, bonding in YO and YS is rather similar. The valence configuration can be de\ . the x'C+ state and as ufuf.rr:a:sl for noted as u ~ u j . r r ~ ufor the *II state. The a , orbital is a low-lying s orbital of sulfur (oxygen) with some participation from yttrium s , p a , and d u orbitals. The .rrl and a, orbitals are bonding combinations of sulfur (oxygen) valence p orbitals and yttrium 4 d orbitals, with substantially heavier weight on sulfur (oxygen). The participation of the yttrium 5s orbital in the u, molecular orbital is calculated to be less than 8% in each case (see Table 5(b)). In both YO and YS, the a, orbital closely resembles the yttrium 5s orbital, with only about 5% 5p character. The a, orbital is semi-occupied in the lowest doublet and quqrtet states of YS and YO, so that the states correlate to a 4d- 5s' yttrium atom. Note, however, that the binding energies given in Table 5(a) are calculated relative to the calculated (an: true) ground state of the yttrium atom, which has a 4 d ' 5sconfiguration. In the dissociation limit of YS (YO), there are four electrons in the 3p shell of sulfur (oxygen) and two in the 4d shell of yttrium. The "ideal covalent" populations on the sulfur for the .rrI and a, orbitals are therefore 4 / 6 = 0.67. The actual populations in YS are larger (0.73 and 0.69), and they are larger still on oxygen in YO (0.78 and 0.79), as expected on the basis of atomic electronegativities. The bond order in the YS ground state (2.54) is quite large and the Mulliken charges ( 5 0 . 2 0 ) fairly small. The bond in YS is thus largely covalent, with a small but nevertheless significant ionic contribution. As expected, the bond in YO is more ionic.

CAN. J . CHEM. VOL. 71, 1993

TABLE5(a). Calculated spectroscopic constants of the lowest doublet and quartet states of YO and YS


YO

Expt .

DF

LB'

DF"

LB"

Expt.

DF"

Expt .

LB~

DFa

Expt .

LB"

"This work. *Calculations of Langhoff and Bauschlicher (22). "Reference (26). "Reference (27). 'r, value. 'Reference (8). "Reference (6).

TABLE 5(b). Description of bonding in the YO and YS molecules

YO

Parameter Configuration Bond order Charge on Y

YS

x2C' u?&.rr4 1 2 1

2.43 +0.37

"4, I ~

x2C+

4rI, 44,

U~U$T~:U:~;

U f u;T$T:

u2u2.rr3 1 z 1

1.21 +0.27

2.54 +0.20

1.35 +0.27

3

1 I ~ 3

~

1

Mulliken populations

Formally, the lowest quartet state in these molecules is accessed from the ground state by "exciting" one electron from a bonding a,orbital, which is centred mostly on sulfur (oxygen), to a nonbonding, purely atomic, yttrium d orbital. The density functional calculation does not distinguish between the 4@ and 411 states that are formed by promotion into the 6, molecular orbital. A 411 state could also arise by excitation from the .rr, orbital into a higher lying a orbital,

but this transition would probably lie at significantly higher energy. If a "rigid-orbitaln description was strictly valid, upon excitation of the 'II state, the bond order would decrease by roughly 0.5, and the negative charge on the yttrium would increase by 1 - 0.25 = 0.75. Actually, the calculated bond order decreases by 1.2 and the charge remains essentially unchanged. This is because the change in configuration is accompanied by a substantial relaxation of the orbitals: the

1613


JAMES ET AL.

population on the sulfur (oxygen) atom goes from 0.73 to 0.91 (0.78 to 0.91) for the .rr, orbital and from 0.69 to 0.78 (0.79 to 0.84) for the u 2 orbital. The tendency towards achieving approximate charge neutrality makes the .rr, and a, orbitals more atomic-like and therefore weakens the bond. The reduction in bond order is manifested in the observed and calculated magnitudes of the bond length for the 4rI state, which indicate a bond substantially weaker than in the x'Cf state (see Table 5(a)). The first doublet excited state corresponds to excitation from the a, orbital, which is mostly yttrium 5 s with about 5% 5p character, to the u, orbital. The calculated composition of the u, orbital may be summarized in terms of Mulliken populations as (YS): 0.46 pz, 0.29 dz', 0.05 dx', 0.05 dy' on yttrium, 0.0 s , 0.13 pz on sulfur; (YO): 0.67 pz, 0.22 dz', 0.05 dx', 0.05 dy2 on yttrium, -0.08 s, 0.08 pz on oxygen. Clearly, the u, orbital is strongly hybridized and points away from the ligand. This polarization counteracts the principal component of the dipole moment. The net effect is that the dipole moments in the doublet excited states are much smaller than in the corresponding ground states, values of 3.19 D (YS) and 0.70 D (YO) being calculated. The precise values are not trustworthy, but they illustrate the general trend of excitation from the largely unhybridized u, orbital to the strongly hybridized u, orbital.

Discussion The small fraction of 5p character (5%) calculated for the a, orbital of YS is somewhat at odds with the interpretation of the hyperfine structure presented by Azuma and Childs (9). They used a simple treatment that assumed that the reduction-in the Fermi contact term from the atomic value was due entirely to ligand-induced 5s-5p hybridization effects. In this way, the orbital composition of a, was determined to be 53% 5 s , 4 7 % 5p. A possible explanation for the discrepancy between their analysis and our calculation is that s orbitals of higher principal quantum number are mixed into the a, orbital by the ligand field of sulfur, reducing the electron density at the nucleus. Such a process would reduce the magnitude of the contact interaction in the molecule relative to the atom, and is known to operate in &OH" and MnF (25). It is difficult to reconcile the analysis of Azuma and Childs with the experimentally observed trends in the dipole moments. Appreciable hybridization of both the u, and a, orbitals would not be expected to pr2duce the experimentally observed reduction in p upon B-C+ + x'C+ excitation. The small dipole moment calculated for the ,II state can be rationalized crudely in terms of the ,II + x'C+ excitation involving the back-transfer of an electron from a sulfur- (oxygen-)centred orbital (T,) to one localized on yttrium (6,). In language familiar to spectroscopists, this excitation constitutes a "charge transfer" transition. In diatomic molecules where the ionic contribution to bonding dominates the covalent contribution, the reduction in bond strength accompanying such a transition would be attributed primarily to a diminution in the electrostatic contribution to the bonding. In systems such as YS and YO, where calculation indicates that covalent bonding predominates, a molecular orbital description of the excitation is more appropriate. It seems likely that, in common with the upper level in the

'z.J. Jakubek and R.W.

Field. Private communication

14 926.0 cm-' band, the upper level in the 15 300.1 cm-' band also belongs to the 4rI,l,2 state. Analysis of the lifetimes data supports this assertion: the very long or possibly infinite lifetime determined for the upper level of the 15 300.1 cm-' band is consistent with a state that is not optically coupled to the ground state. For simplicity, we denote the upper levels of the two new bands as v' and v' + 1, respectively. Using the vibrational constants reported by Stringat et al. (8), the v = 1 level of the B'C+ + x'Cf system is calculated to lie 15 273.0 cm-I above the ground state v = 0 level, in the absence of perturbations. The downward shift experienced by the B2C+ v = 1 level is therefore 15 273.0 - 15 264.2 = 8 . 8 cm-I. Assuming that the v' + 1 level is shifted up by the same amount, its unperturbed position is calculated to be 15 300.1 - 8.8 = 15 29 1.3 cm- . Assuming that the u' level experiences only a small shift on account of interaction with the B'C+ v = 0 level, the difference in energy between the unperturbed vibrational levels of the 411,1,2state is 15 291.3 - 14 928.0 = 365.3 cm-I. The ratio of this vibrational interval to the ground state vibrational frequency is 365.3/492.7 = 0.741, in reasonable agreement with the ratio w,(411,,2)/w,(~'C+) = 0.770 computed from the density functional data in Table 5(a). The foregoing discussion begs the question as to why we observe only two vibronic bands of the ,II, +x'C+ system in the spectral region of interest. The explanation is related to the intensity distribution in the B2C++x'C+ system. There are basically two criteria to be met for a spin-forbidden + x'C+ vibronic transition to exhibit observable intensity. Firstly, the spin-orbit matrix element coupling the relevant levels of the 417,1,2 and B'C+ states must be significantly non-zero. Secondly, the two interacting vibrational levels must lie fairly close in energy. A calculation of the Franck-Condon factors (s$,,,) for the B2C+ + x'C+ system reveals that they drop off rapidly with increasing v': S& = 0.659; s:, = 0.280; s:, = 0.054; s:, = 0.006. Clearly, only the (0,O) and (1,O) bands are intense, as is borne out by experiment. Thus the only vibrational levels of the state which can be observed experimentally are those that are e;ergetically the closest to the v = 0 and v = 1 levels of the B-Z+ state. Explicit calculation using Slater determinants to and B'C+ states represent the wavefunctions of the reveals that, in a first-order approximation, the matrix element of the spin-orbit operator connecting these two states is zero. Clearly, a second-order mechanism must be invoked to explain the non-zero ,II * B2C+ interaction. Given that transitions to both the R = 1/2 and R = - 1/2 spin-orbit components gain intensity through the spin-orbit interaction, our failure to observe transitions to the other spin-orbit component must remain an open question.

'n,,,,

Conclusions The yttrium monosulfide molecule has been investigated using molecular beam fluorescence spectroscopy and density functional theory. Improved molecular constants for the v = 0 levels of the x'C+ and B'C+ states have been obtained. Two vibrational levels of a state with 4n,1,2symmetry have been observed in excitation from the ground state, the spin-forbidden 4rI,l,2 + X'S+ transition oaining intenS sity through spin-orbit mixing between the B-C+ state and vithe 'n,,,, state. The bond length for one of these ,nZ1l2 brational levels was determined to be 2.495 lO(16) A. The other 4rI,,,2 vibrational level is responsible for strongly

1614

CAN. J. CHEM.


perturbing the v = 1 level of the B'C+ state. Density functional calculations on low-lying states of YS (and YO) in the doublet and quartet spin manifolds have been carried out, confirming the experimental observations. 1. S.R. Langhoff and C.W. Bauschlicher, Jr. Annu. Rev. Phys. Chem. 39, 181 (1989). 2. M.D. Morse. Chem. Rev. 86, 1049 (1986). 3. B. Simard, S.A. Mitchell, L.M. Hendel, and P.A. Hackett. Faraday Discuss. Chem. Soc. 86, 163 (1988). 4. B. Simard, A.M. James, and P.A. Hackett. J. Chem. Phys. 96, 2565 (1992). 5. B. Simard and A.M. James. J. Chem. Phys. 97,4669 (1992). 6. A.M. James and B. Simard. J. Chem. Phys. 98, 4422, (1993). 7 . N.S. Mclntyre, K.C. Lin, and W. Weltner, Jr. J. Chem. Phys. 56, 5576 (1972). 8. R. Stringat, B. Fenot, and J.L. FCmCnias. Can. J. Phys. 57, 300 ( 1979). 9. Y. Azuma and W.J. Childs. J. Chem. Phys. 93, 8415 (1990). 10. B. Simard, S.A. Mitchell, M.R. Humphries, and P. A. Hackett. J. Mol. Spectrosc. 129, 186 (1988). 1 1. A. St.-Amant and D.R. Salahub. Chem. Phys. Lett. 169, 387 (1990); A. St.-Amant. These de doctorat, UniversitC de MontrCal, 1992; D.R. Salahub, R. Fournier, P. Mylnarski, I. Papai, A. St.-Amant, and J. Ushio. Proceedings of the Ohio Supercomputer Center workshop on the theory and applications of density functional theory to chemistry, May 1990. Edited by J. Labanowsky and J. Andzelm. Springer, New York. 1991.

12. S.H. Vosko, L. Wilk, and M. Nusair. Can. J. Phys. 58, 1200 (1 980). 13. A.D. Becke. Phys. Rev. A, 38, 3098 (1988). 14. J.P. Perdew. Phys. Rev. B, 33, 8822 (1986). 15. A. Selmani and D.R. Salahub. J. Chem. Phys. 89, 1529 (1988). 16. J. Andzelm and E. Wimmer. J. Chem. Phys. 96, 1280 (1992). 17. 0 . Gunnarson and B.I. Lundqvist. Phys. Rev. B, 13, 4274 (1976); L.N. Oliveira. Adv. Quantum. Chem. 21, 135 (1990); L. Fritsche. Physica B (Amsterdam), 172, 7 (199 1). 18. G.R. Fleming, O.L.J. Gijzemann, and S.H. Lin. Chem. Phys. Lett. 21, 527 (1973). 19. H. Lefebvre-Brion and R.W. Field. Peturbations in the spectra of diatomic molecules. Academic, Orlando. 1986. 20. G. Herzberg. Molecular spectra and molecular structure. Vol. 1. Spectra of diatomic molecules. van Nostrand, Princeton. 1950. 21. H.E. Radford. Phys. Rev. A, 136, 1571 (1964). 22. S.R. Langhoff and C.W. Bauschlicher, Jr. J. Chem. Phys. 89, 2160 (1988). 23. A.S.-C. Cheung, A.W. Taylor, and A.J. Merer. J. Mol. Spectrosc. 92, 39 1, (1982). 24. A.D. Becke. J. Chem. Phys. 97, 9173 (1992). 25. 0. Launila, B. Simard, and A.M. James. J. Mol. Spectrosc. 159, 161 (1993). 26. K.P. Huber and G. Herzberg. Molecular spectra and molecular structure 4. Constants of diatomic molecules. van Nostrand Reinhold, New York. 1979. 27. T.C. Steimle and J.E. Shirley. J. Chem. Phys. 92, 3292 (1990).