Examination of the structure, Λ doubling, and perturbations in the 11Πg state of Li2

JOURNAL OF MOLECULAR

SPECTROSCOPY

142, 340-375 (1990)

Examination of the Structure, A Doubling, and Perturbations in the 1 ‘I& State of Lip C. LINTON, ’ F. MARTIN, AND R. BACIS Laboratoire de SpectrometrieIonique et Molkulaire, Universitt!Claude Bernard, Lyon I, 43 boulevarddu II Novembre 1918, 69622 VilleurbanneCedex, France

AND J. VERGES LaboratoireAimk Cotton, Centre Nationalde la Recherche ScientijiqueII, Universite’de Paris-Sud (Paris Xl). 91405 Orsay Cedex, France Fluorescence in the C’II,- I ‘n, and 2 ‘2:: - 1‘IIg transitions of ‘Liz and 6Li2,excited by ultraviolet lines of argon and krypton ion lasers, has been examined at high resolution in the region 700012 000 cm-’ using a Fourier transform spectrometer. The analysis has provided molecular constants for the 1 ‘f&.state which are isotopically consistent and in good agreement with ab initio predictions. An RKR potential curve, extending to r,,,, = 24 A, has been constructed. The dissociation energy, calculated by several methods using long-range analysis, is found to be 1422.03 f 0.05 cm-’ and the long-range terms in the potential energy expansion are C, = 1.805( 10) X lo5 cm-’ A3 and C, = 7.77(60) X lo6 cm-’ A6. The precision in determining D,( 1‘LI,) has allowed us to lower the error limits on the ground state dissociation energy to 0.10 cm-’ (D,(X’Zf) = 85 16.78( 10) cm-‘). The interaction between the 1 ‘TIgand 2’Zf states is studied in detail and is shown to account for all the irregularities observed in the 1‘II, state A doubling and for the strong perturbations that are observed. The detailed study of the interaction and the intensities of transitions to the perturbed levels has led to the determination of electronic transition moment ratios for the 2 ‘Z :- 1 ‘II, and 2 ’ Z : -2 ‘Bg+transitions and shown that they are in accord with ab initio calculations. 0 1990 hdmic Press. hc. 1. INTRODUCTION

In a recent publication ( I ), we described the observation and analysis of the inner limb of the 2 ‘L:: “double minimum” state of Liz, excited by the ultraviolet lines of argon and krypton ion lasers. It was detected, using Fourier transform infrared spectroscopy, via fluorescence progressions in the 2 ‘Z i-2 ‘Zg transition. In addition, we also listed several C’II,-1 ‘I$ progressions (two in ‘Liz, six in 6Li2) that were observed in the same spectra and a 2 ‘Z :- 1 ‘I& progression in ‘Liz. The 1 ‘I& state has previously been observed only in ‘Liz (2) using an X’Z,+ + A ‘Z: --f 1 ‘I& double resonance scheme. Coupled with our recent observations of the 2 ‘Xl and 1’2: states, this completes the observation of all the states dissociating ’ On leave from Physics Department, University ofNew Brunswick, Fredericton, New Brunswick, Canada E3B 5A3. 0022-2852190 $3.00 Copyright

0

1990 by Academic Press. Inc.

All rights of reproduction

in any fwm reSewed.

340

341

1 ‘I& STATE OF Li2

to the two lowest dissociation limits, 2s + 2s, 2s + 2p. We have now completed a more thorough investigation of the 1 ‘I& state examining in detail, via C’II,-1 ‘IIn and 2 ‘Z :- I ‘I& transitions, its rotational structure, dissociation energy, and potential energy curve. The A doubling in the 1 ‘II, state and perturbations in 1 ‘I& and 2 ’ 2: states have been studied in detail and are discussed in terms of the interaction between the two states. This work, which is described in the following sections, has concentrated mainly on the 6Li2 isotopomer although the investigation of A doubling includes transitions in ‘Li2. The experiment is the same as that used to observe the 2 ’ Z : state ( 1) and will not be described here. 2. RESULTS

A. General Observations Of the observed 6Li2 C’II,- 1 ‘II, transitions that were listed in the previous paper (Ref. ( 1 ), Table II), one has been reassigned to 6Li7Li (see below). Six fluorescence progressions in the 1 ‘I$ state were observed, the longest covering the range of v( 1 ‘II,) from 0 to 34 at J’ = 5 and 1 to 29 at J’ = 31. There were also several other long progressions. The spectrum excited by the krypton laser lines, shown in Fig. 1. is dominated by long progressions of RP doublets which appear to continue almost to dissociation. There was very little detectable relaxation. Several very weak lines were detected around the very intense R( 28) and P( 30) lines in the progression from C’TI,,,

7700

7900

FIG. 1. Low-dispersion spectrum, between 7500 and 9100 cm-‘, of ‘Liz fluorescence excited by the ultraviolet lines of a krypton ion laser. Three vibrational progressions in the C’Il,-1 ‘I& transition are marked, from C’II., u = 0, J = 31 (arrows); u = 1, J = 29 (solid circles); u = 1, J = 5 (crosses). The numbers above the transitions are the vibrational quantum numbers of the 1 ‘TIgstate (marked every fifth level). The un = 5, .I = 30 perturbation can be seen near 8680 cm-’ (P(30). l-5) and 8510 cm-’ (R(30), O-5) and the on = 4, J = 4 perturbation is near 8525 cm -’ (R( 4). l-4). The other lines in the spectrum are from other Cl&l ‘II,transitionsas wellas2’Z~-2’Zd and 2’.Z:-1 rl&. The frequencies(shown below the spectrum) are in cm-’ and the scale has been changed by a factor of 2.5 at 8200 cm-‘.

342

LINTON

ET AL.

u = I, J = 29. These were assigned as relaxation lines from J’ = 27, 28, 30, and 31 as described below. The J’ = 28 and 30 lines were -20 times weaker than the J’ = 29 lines and the J’ = 27 and 3 1 lines were a factor of 2 weaker still. The low-dispersion spectrum in Fig. 1 shows some obvious signs of perturbations. The most apparent are at P( 30), l-5 and R( 4), l-4 which are much weaker than their R(28) and P( 6) counterparts and are accompanied by extra lines. Thus the v = 5, J = 30 and u = 4, J = 4 levels of the 1 ‘IIs state are obviously strongly perturbed. Many perturbations were observed and are discussed in Section 5. The intensity distribution in the progressions is evidence of further perturbations. In a normal ‘II-‘II transition, the Honl-London factors indicate that the P( J + 1) line should be more intense than the R( J - 1) line. This is most marked at low J, and at higher J, the two branches should have almost equal intensity. In Fig. I, in the J’ = 5 progression, P( 6) is more intense than R( 4) as expected but in both the J’ = 29 and 3 1 progressions, the R line is clearly more intense than the P line. Thus, it appears as if there is a Jdependent perturbation that significantly affects the intensities but not the positions of the lines. This is discussed in more detail in Section 6. The J and v assignment of the main lines in the various progressions was straightforward and is discussed in Ref. (1). The assignment of the relaxation lines in the J’ = 29 progression was based on upper state combination differences, A2F’( J) = R(J) - P(J) . These were calculated using known C ’ II, state constants (3)) and were first used to find the weak P(28), R(30) lines accompanying the intense R(28), P(30) fluorescence lines and then to find other pairs of weak lines. As the upper state was common to all members of the 0” progression, the upper state combination differences must be the same at all Y” and this acted as a definitive test of the assignment. For example, R(29) and P(29) were assigned for V” = 2-10 and all nine upper state combination differences agreed within a range of 0.008 cm-‘. The lines assigned from the C’II,, J’ = 29 excitation are listed in Table I along with upper and lower state combination differences. The lines in Table I have been listed as R( J - I), P( J + 1) rather than R(J), P(J). There are large differences in the lower state combination differences near the perturbations whereas the upper state combination differences remain constant. The analysis and the reduction of the data to molecular constants were more difficult than those for the 2 ‘2 Z-2 ’ 2: transition because of the A doubling in the II states and the necessity of determining which ef parity component of the C’II, state was excited by the laser. Of the five excited C state levels used in the analysis, four were e levels excited via P- or R-branch C’lI,-X ‘2; transitions and only one was an j” level, excited via a Q-branch transition, Q( 12). Thus, in the resulting C’II,- 1 ‘I& fluorescence, which consisted almost entirely of R- and P-branch lines, there were only data forthe l*II,Slevelsat J= 11 and 13. There are two primary reasons why it is important to obtain data for the f levels of the 1 ‘I& state. In order to determine the A doubling, it is necessary to be able to calculate the separation of e and f levels, which requires sufficient data for both sets of levels. As will be shown in Section 4 the A doubling is caused by the 2 ‘Zl state perturbing the e levels. Thus, the f levels represent the true unperturbed positions of the rotational levels and are therefore crucial to a proper analysis. To find more f levels required a search of the weak features in the spectra for (i) Q lines in the C’II,-

343

1 ‘I& STATE OF Liz TABLE I Observed Line Positions (cm-‘) and Combination Differences (cm-‘) in C’II.-1 ‘II, Fluorescence from C’II., u = 1, J = 29 J

”

R(J-1)

P(J+l)

A,P’(J)’

A,F”(.J)’

Cl

29

9109.6306

9073.2558

36.3748

1

29

9022.2500

8987.1429

35.1071

2

28 29 30

8921.0379 8938.6063 8953.1420

8887.9812 8904.7067 8918.3998

30 31

8840.3676 8858.4082 8864.4379” 8873.8927 8894.1979

8795.0165 8808.7272 8826.7143 8822.0890’ 8840.4406 8859.4515

27 28 29 30 31

8748.8755 8763.7036 8782.6554 8798.4295 8818.7455

27 28 29

8675.0802 8690.8426 8709.6055 8723.7720’ 8726.7803

3

4

5

27 28 29

30 31

6

7

8

9

10

11 ‘A,F’(J)

I

? 1

67.4836

8719.2706 8733.2625 8751.2652 8766.2631 8785.6512

g I

8646.2168 8661.6122 8677.3916 8685.03853 8695.9080

63.3917 65.1655

63.3848 65.1670 67.4803

2

63.3887 65.1681

I

33.0567 33.8596 34.7422

31.6404 31.6939 42.3489 33.4521 34.7464 29.6049 30.4411 31.3902 32.1664 33.0943 28.8634 29.2304 32.2139 38.7335 30.8723

27 28 29 30 31

8606.1811 8621.8214 8642.3179 8658.9688 8680.9517

8578.9289 8593.8060 8613.4704 8629.3964 8650.5823

63.3893 65.1628 67.4813

28.2522 28.0154 28.8475 29.5724 30.3694

27 28 29 30 31

8540.3039 8556.6645 8577.5910 8595.0347

63.3843 65.1670 67.4790

8617.6410

8514.2067 8529.8677 8550.1620 8566.7736 8588.5983

26.0972 26.7968 27.4290 28.2611 29.0427

27 28 29 30 31

8478.4388 8495.3957 8516.9463 8534.9984 8558.1251

8453.5604 8469.8300 8490.6526 8508.0601 8530.4972

63.3859 65.1684 67.4725

2L.8784 25.5657 26.2937 26.9383 27.6279

27 28 29 30 31

8438.0331 8460.3247 8478.8755 8502.7366

8396.9348 8413.7080 8435.2580 8453.2755 8476.4774

63.3899 65.1675 67.4786

28 29 30

8384.5814 8407.2997 8426.6693

8361.5097 8383.54182 8402.4341

65.1596

29

8358.5082

8336.1140

= R(J)-P(J);

‘Perturbed ‘Extra

2

65.1608

lines. lines.

A,F”(J)

= R(J-l)-P(Jt1).

24.3251 25.0667 25.6000 26.2592 21.0717 23.7579 24.2352 22.3942

344

LINTON ET AL.

1 ‘I& transition, (ii) 2 ‘2 :- 1 ‘I& transitions, and (iii) appropriate relaxation lines in the C’II,-1 ‘I& transition. (i). Q-branch lines originating in an e component of the upper state will end on f levels of 1 ‘I$.. In ‘II- ‘II transitions, the Q branch is only observable at very low J. A search of the spectra revealed very weak lines between the strongest R( 4), P( 6) doublets from C’II,, v = 1, J = 5. These have been assigned as the Q( 5) lines as their intensities agree well with those expected from the Hiinl-London factors (e.g., at J = 5, I,/I, - 14). (ii). In 2 ‘Z:-1 ‘I& fluorescence, the R and P branches will go to e levels of 1 ‘I& whereas the more intense Q branch will access the f levels. From the analysis of 2 ‘Z z-2 ‘Zi ( I), it is known which 2 ‘Z: levels were excited by the laser. The expected frequencies of 2 ‘Z i- 1 ‘III, transitions from these levels were calculated and a search of the spectrum revealed several of these transitions in both 6Li2 and ‘Liz. The observed 2 ‘25-l ‘II8 progressions are listed in Table II. However, not all of these were useful for A-doubling calculations as, in several transitions, only the Q line was observed. One progression in ‘Li2, at J’ = 30, contained only Q lines. (iii). Several of the weak relaxation lines from C’II,, v = 1, J = 29e involved f levels of 1 ‘III,. Because of the need to preserve s-a symmetry in collisions, AJ = + 1 collisions must involve a change in ef parity. Thus the rotational relaxation process will be J = 29(-, e, s) --f J = 28( -, f,s).The subsequent very weak R- and P-branch fluorescence thus provided some data on the 1 ‘I&. f levels for J = 27, 29, and 3 1. B. Analysis The 6Li2 analysis was thus based on 9 rotational levels (3 f,6 e) of the C’II, state at four different D’S,four rotational levels of 2 ’ 2: : at 3 different D’S,long vibrational TABLE II Observed 2 ‘Z :- 1 ‘IIS Fluorescence Progressions” 211’

l’ll

”

”

‘Li,

‘Li,

%

several

bQ b ranch

.I

1

14

2-9

2

25

O-6

3

7

o-3

3

31

o-5

0

19

6-12

1

17

3-9

1

38

4-8

2

30

3-bb

3

40

2-6

transitions, only.

g ”

Unable

only to

use

Q branch for

lines

A doubling.

were

observed.

1 ‘II, STATE OF Li2

345

progressions from C’II,-1 ‘I& at 16 J’s (6 f, 10 e), and short progressions from 2 ‘Z,$1 ‘I& at 8 additional J’s (4 f,4 e) in the 1 ‘IIK state. The molecular constants (Dunham coefficients) for the 1 ‘IIg state were determined by simultaneously fitting the lines in the C’II,-1 ‘I& and 2 ‘Z c-1 ‘I& transitions to a Dunham expansion. It was necessary to fix the C’II, state coefficients to the values quoted by Ennen et al. (3) as only at u’ = 1 were there data for more than one rotational level. A reasonable fit could only be obtained by changing the sign of sol, the A-doubling coefficient. quoted by Ennen et al. (3) (probably a typographical error). For the 2 ‘Z 2 state, only term energies could be determined as, except at zi = 3, only one rotational level was observed at each vibration and there were thus insufficient data to calculate molecular constants. The ‘II state expressions used in the fit are slightly different from those used by Ennen et al. (3). The rotational terms were still fitted to an expansion in [ .I( J + 1) - l] but the A doubling was expanded in terms of J(J + 1) rather than [ J( J + 1) - I]. This is because the rotational Hamiltonian involves the diagonal operator, J’ - 55 with eigenvalues J( J + 1) - 52*, whereas the A doubling is determined by the J-L+ operator which, for a ‘II state, gives eigenvalues depending on J(J + 1). The C state constants were adjusted to reflect this change. The adjustments are very small but allow us to preserve the Hamiltonian in its correct form. Fitting the data also proved more difficult than for the 2 ‘2:-2’2,+ transition. .4 global fit to all the lines in the C’II,-1 ‘I& and 2 ‘Z:-1 ‘IIg transitions was first attempted. However, it order to achieve a reasonable fit, it was necessary to omit many lines and, even when this was done, the standard deviation in the fit, -0.04 cm-‘, was larger than normal for Fourier transform spectra. At first, it was thought that this was entirely due to the small data set available but a careful investigation revealed that there is a fundamental reason pertaining to perturbations and A doubling in the 1 ‘I& state (see Section 4). It is apparent that there are numerous small perturbations and that the A doubling does not follow a regular pattern with vibration. The A doubling was fitted to [sol + al 1(21+ 4 )].I( J + 1). This has the effect of smoothing the effect of the perturbations and the irregularity of the doubling, and the constants are merely fitting parameters without any real physical significance. In fact, the global fit masks the most interesting and informative physical phenomena that can be revealed by the A doubling. This is discussed in detail in Section 4. The second attempt involved two separate fits, one to transitions involving the 1 ‘II, e levels and the other to the f levels. The e-level fit used a larger data set and involved the most intense and accurately measured lines. The standard deviation of 0.04 cm -’ with many lines omitted again rellected the irregularities caused by the A doubling. The S-level fit, even though the data set was smaller and the lines were weaker, was far more satisfactory, with a standard deviation of 0.007 cm-’ and all lines included. This reflects the fact that the A doubling occurs as a result of perturbations of the e levels. The f levels are unperturbed and the constants obtained from these levels represent the true molecular constants for the 1 ‘I& state. For completeness, both the e- and thef-level constants for the 1 ‘I& are listed in Table III. The e-level constants are good fitting parameters which do not have a great deal of physical significance but will allow calculation of new transitions involving the e levels that are not strongly perturbed.

346

LINTON ET AL. TABLE III Dunham Coefficients (cm-‘) for the I ‘I& State of 6Liz Ab Initio

Experimental

T

f

e Component

Ref(4jc

21998.25(20)

21998.800(20)

21998.828(54)

e

componenta

Ref(2)b

Ref(51c

21998

y,o

100.193(35)

100.206(15)

100.82(12)

99.20

99.79

y2,

-1.9480(70)

-1.9657(44)

-2.186(32)

-1.894

-1.75

0.3379

0.3424

lO'Y,, lO’Y, o

n.s.

7.08(54)

E.S(l.2)

lOV,,

-4.40(26)

106Y,,

6.44(60)

IO’Y,,

0.742(39)

1.12(12) 0.33969(32)

0.33983(10)

lOV,,

-1.1084(56)

-1.1318(44)

n.s.

106Y,,

-1.45(15)

loeY,I

4.712(72)

1O’“Y 61

-6.lCl.O)

105Y02

1O”Y

C‘I 5‘

From

isotopically

Not

are

5.5C3.3)

“.S.

“.C$. -16.6C6.9)

unperturbed

adjusted adjusted

-1.62(27)

S.l(l.6)

1.19(16)

‘Isotopically “.S.

“.S.

-1.03(14)

aThe f components V%XlWS.

1.22(26)

-5.4Cl.l)

3.01(42)

10’5Y,;

1.05(19)

n.s.

-3.02(44)

lO’“Y,,

29.OC5.6) -27.7(5.0)

4.5C2.0)

2.20(26)

n.s.

-1.149

-6.3C4.2)

-1.5139(92)

1.68(52)

10’2Y,,

b

n.s.

-1.526(X?)

lo’Y,g IO’OY

0.252(32)

“.S.

81

0.3405(15) -1.19481(44)

“.S..

lO”Y,,

n.s. n.s.

8.13(58)

9.8Cl.l)

1O”Y

1.122(78)

-0.61(15)

Yo,

lOV,,

1.037(68)

“.S.

-1.41(14)

10VB,

31(17)

n.s.

-1.19(13)

41.0(38)

from

and

these

‘Li,

data.

‘Li,

calculations.

constants

represent

the

true

significant.

The Dunham expression was fitted to very high order in vibration, but low order in rotation. This is because observations were made over 34 vibrational levels whereas only 24 rotational levels, clustered into two main groups around J = 6 and 30, were assigned. Several Yjk coefficients were found to be very poorly determined (the fit appeared to favor even orders in j) and were omitted. The rotational constants are compared in Table III with the isotopically adjusted ‘Liz constants of Miller et al. (2) and the ab initio values of Schmidt-Mink et ul. (4) and the ab initio values of Konowalow and Fish (5), both of which were isotopically adjusted from ‘Liz calculations. There is good agreement, particularly for the loworder constants, between the two sets of experimental data and any differences, especially the large differences in some of the higher-order constants, are partially caused

l’&

STATE

OF Li2

341

by fitting the two sets of data to a different number of highly correlated constants. The agreement with the ab initio predictions, especially those of Schmidt-Mink d al. (4), is very good. As a check on internal consistency, the Kratzer relation, Y,, = -4Y&/Y&, gives Y,, = -1.56 X 10e5 cm-’ compared with the fitted value of - 1.5 1 X lo-’ cm-‘. The small discrepancy is probably a reflection of the fact that the date set is small and the higher-order parameters, which are strongly correlated with the lower-order terms, are not very accurate. There was one long progression, excited in the 6Li2 source by the argon laser, which had been identified and listed in Ref. ( 1) as coming from J’ = 35, u = 0 of the C’II, state. However, all attempts to include the progression in the least-squares fit failed, producing calculated values that were several cm-’ removed from the observed frequencies. The possibility was examined that the transition may possibly come from the 6Li7Li mixed isotopomer as there was about 6-7s ‘Li in the ‘Li source. The isotope effect was used to calculate expected frequencies for 6Li7Li and a very good agreement was found for S = 37. Examination of the 7Li2 spectrum revealed the presence of the same lines (much weaker and of poorer quality as the signal-tonoise ratio was worse in ‘Liz). This confirms the assignment as 6Li7Li and the progression thus arises from C’II,, v = 0, J = 37. 3. POTENTIAL

CURVE

AND DISSOCIATION

ENERGY

A. Calculation of RKR Potential Curve Ideally, the RKR potential for the I ‘UR state should be calculated using well-determined molecular constants obtained from the unperturbed f levels. Unfortunately, because of the small data set and the fact that S levels were observed only up to 2: = 22, it was not possible to construct an accurate potential curve in this way. Because of the fluctuations in the A doubling, it was not felt that the constants from the t’ levels were of high enough quality to produce a reliable RKR curve. After careful examination, it was decided that the best RKR curve should be obtained from the single R(4), P( 6) progression from J = 5 in the 6Liz C’II,, state. This progression extends from u( 1 ‘IIg) = 0 to 34 which is energetically close to dissociation. There are very few perturbed levels and. when these are removed and the progression is fitted by itself, the standard deviation is 0.003 cm-’ . Even though it samples the e levels in 1 ‘II,, the A doubling at J = 4 is very small (-0.008 cm-‘) and the fluctuations are negligible. The energies will thus be very close to the “true” J = 4 energies. At such low J, centrifugal distortion effects will be small, although significant, but after taking these into account, errors caused by the uncertainties in distortion constants will be negligible. The following procedure was used to construct the RKR curve. Neglecting distortion effects, B, was calculated for each vibrational level from the R( 4) - P( 6) combination difference. This B, was used to extrapolate each R( 4) line to J” = 0 (this actually gives the hypothetical J’ = 5-J” = 0 transition frequency). By subtracting all these hypothetical frequencies from that at u” = 0, the vibrational energies (without rotation) of all the 1 ‘IIg levels relative to V” = 0 were obtained. The RKR curve was calculated from these G, and B, values in the usual way and was then used with the program of Hutson (6) to generate new constants, including centrifugal distortion, D,. These D, values were then used in conjunction with the combination

348

LINTON ET AL.

differences to generate new, more realistic B, values, and the extrapolation to zero rotation was repeated, including centrifugal distortion effects. A new RKR potential was computed and the process repeated. Convergence was achieved after two iterations. The RKR curve reproduced the experimental B, values to within 0.0008 cm-’ at the highest D’Sand were much better at lower U. This is very satisfactory considering that they were determined from a progression from a single rotational level and that the small fluctuations in energy at the high levels caused by the interaction with 2 ‘2; are unknown. The turning points (at J = 0) of the RKR potential are listed in Table IV and the potential curve, drawn at J = 30 to highlight resonances with 2 ‘Zi (see Section 5), is shown in Fig. 2. B. Dissociation Energy The dissociation methods.

energy of the 1 ‘I$. state was determined

using three different

(i) Using X’Z: state dissociation energy. The ground X’Z: state dissociates to Li( 2s) + Li( 2s) and the 1 ‘I& state to Li( 2s) + Li(2p,,z) (8). The ground state

TABLE 1V RKR Potential Curve for the 1 ‘II, State of 6Li2 ”

0 1

4

5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34

Gv(cm-') 49.1317

146.0459 230.4594 327.0141 411.7278 492.6062 569.6484 642.8506 712.2075 777.7147 839.3712 897.1830 951.1660 1001.3503 1047.7834 1090.5330 1129.6891 1165.3644 1197.6943 1226.8346 1252.9594 1276.2570 1296.9258 1315.1698 1331.1938 1345.1999 1357.3833 1367.9295 1377.0130 1384.7955 1391.4262 1397.0421 1401.7693 1405.7241 1409.0156

R,,,~,,(A) 3.755991

3.576159 3.467272 3.387288 3.323908 3.271866 3.228239 3.190915 3.158288 3.129293 3.103396 3.080400 3.060193 3.042546 3.027064 3.013282 3.000819 2.989500 2.979353 2.970496 2.962952 2.956534 2.950867 2.945584 2.940574 2.936109 2.932641 2.930320 2.928559 2.926460 2.924957 2.923691 2.922629 2.921744 2.921009

Inax

R

4.430012 4.764417 5.027961 5.267104 5.495143 5.718803 5.942164 6.167853 6.397853 6.634086 6.878604 7.133982 7.402358 7.686114 7.987467 a.308688 8.652282 9.021134 9.418536 9.848090 10.313563 10.818798 11.367796 11.964977 12.615533 13.325685 14.102675 14.954516 15.889855 16.918731 18.055049 19.313706 20.712160 22.269261 24.002474

1 ‘fIg STATE OF Liz

349

2500

1500

-

1000

-

500

-

FIG. 2. Potential energy curves of the I ‘II, and 2 ‘Xi states of Liz at J = 30. The 6Li2 vibrational levels (at J = 30) are marked and clearly show the resonances at un = 3, uz: = 14 and un = 5, ur = 15. The inset shows all the singlet states dissociating to the lowest two dissociation limits and the two states ( C'Il. and 2 ‘Z:) excited in the present experiments. The outer well of the 2 ‘2: state (dashed lines) has not yet been observed.

dissociation energy is well known ( 7), and may be used to determine that of 1 ‘ILK, as D,( 1 ‘IIJ = &(X’Z:g+) + Y(2&,2 - 2s) - r,( 1 ‘rI,). Using the known values of D,(X’Z,t) = 8516.78(54) cm-’ and ~(2~3~2 - 2s) = 14904.00(01) cm-’ and the value of 7’J 1 ‘I&) = 21998.80(03) cm-’ obtained from the global fits to both e and f levels, we find that D,( 1 ‘II,) = 142 1.98 ( 55 ) cm-‘. Using the same method for 7Li2, Miller et ul. (2) found D, = 1422.5 + 0.4 cm-‘. Their value of T, = 21998.25 cm-’ was calculated from T,(A ‘Z:) whereas our value was calculated from Te( C’II,). The relative accuracies thus depend on the relative accuracy of the T, values for A ‘Z: and C’II,. The above discrepancy gives a difference of 0.55 cm-’ in D, and accounts for the difference of 0.52 cm-’ in the quoted values.

350

LINTON ET AL.

(ii) Long-range power series expansion. At large internuclear separations, the potential energy, V(r), may, in this case, be expanded as an inverse power series in r. V(r)

= D, - F $

(n = 3, 6, 8, 10).

(3.1)

Using the LeRoy criterion ( 9), the long-range approximation is expected to be valid above I = 9 A. As the outer turning points of the highest observed levels of the 1 ‘I& state extend well into the long-range region (up to 24 A), the C’, term should be the dominant term in the expansion with, perhaps, a small contribution from the Cs term. When the last 10 levels (V = 25-34) were fitted to the above expression including the C’sand Cs terms, the dissociation energy was found to be 142 1.942 ( 50) cm-r. The standard deviation of the fit was 0.0060 cm-‘. The values of C, and C, are 1.783 l(40) X lo5 cm-’ A3 and 7.77(60) X lo6 cm-’ A6, respectively. When only the last four points (v = 3 l-34) were included in the fit, the C6 term was not significant and the potential was linear in r -3. The dissociation energy was 1422.035(50) cm-’ and C3 = 1.8007(40) X lo5 cm-’ A3. (iii) LeRoy method. LeRoy (9) has shown that, over a limited section in the longrange region, the potential energy can be written = D, - $,

V(r)

(3.2)

where n is a weighted average which accounts for all the contributing terms in the power series expansion. He also showed that, with n defined in this way,

-n -dG(v) = p,,2(: dv n ),,n

[D, - G(z))](“+~)‘~?

(3.3)

In the previous section we showed that, when only the last four levels are considered, the potential energy is linear in re3 and it should therefore be valid to use the above equation, with n = 3, with these four levels. The equation can then be rewritten (3.4)

where K3 = 34.5429. A fit of G(v) vs (dG(v)/dv)‘.2 for the last four points was linear and gave a dissociation energy of 1422.02 1 ( 33 ) cm -I. From the slope of the plot, C3 was calculated to be 1.808 ( 19) X lo5 cm-’ A3. The plot, shown in Fig. 3, illustrates that, on the scale of the graph, Eq. (3.4) remains valid for levels extending well down into the potential well ( - 300 cm -I below dissociation ) . LeRoy also showed that another convenient way of expressing the long-range behavior is given by 20(n)

[~“(c,)~]‘/(“-~)

(vD - v)(“+2V(n-2)

where vD represents the vibrational quantum number at the dissociation limit.

(3.5)

351

1 ‘I& STATE OF Liz

1100

1000

1200

1300

1400

1500

G(cm-‘) FIG. 3. LeRoy plot of (dG/d~)~” vs G used in determining the dissociation energy and long-range contributions to the potential energy of “Liz.

For n = 3, this gives dG( v) -zP dv

6&,( 3) cL3w3>*

(vLJ- v)5,

(3.6)

where x0(3) = 36410. Thus, (dG( v)/dv) 1’5is a linear function of v and, as shown in Fig. 4, the linearity appears to extend over a large number of vibrational levels. From the slope and intercept of the least-squares linear fit to the last four points, we found that C, = 1.805 ( 19) X 105cm-‘~3andvD=60.11(14). LeRoy derived two further expressions. The first involves the second derivative of the vibrational energy and the second involves the rotational constant, B,.

2.4

‘.

0.6 \

‘\ ‘\

‘\\

- 1.6 -\

20

30

40

50

‘\

‘E 2

\--

60\

0.0 10

20

FIG. 4. LeRoy plots of (dG/du)“’ a function of u for 6Liz.

30

40

50

60 V

(arrows), B:‘4 (solid circles), and -(dG/dv)/(dZG/dv2)

(inset) as

352

LINTON ET AL.

dG(v) dv

I

d2G( v)

-=

dv2

G’(v) _L( G”(v)

B J(3) (%I - N4, ” P3CZ

UD- v)

(3.7)

(3.8)

whereX’(3) = 60221. Plots of - [G’(u)/G”(u)] vs u and Bt25 vs u are shown in Fig. 4. While, on the scale of the graphs, the plots appear linear, they are not as linear as those given by Eqs. (3.4) and (3.6), the errors are larger, and they were not used in the final determination of uD and C,. C. Summary

The results of the various calculations of D,, C3, and Cs are summarized in Table V. The dissociation energies calculated from the ground state dissociation energy, the potential curve (for 4 and 10 levels), and the LeRoy method are all within 0.10 cm-’ of each other (0.007%) and I6 cm-’ higher than the ab initio value of Schmidt-Mink et al. (4). The C3 values from the potential curve (4 and 10 levels) and the two LeRoy plots all agree to within 1.4% and are in excellent agreement (< 1%) with the theoretical value (8). Cs was determined only once, from the 10 level fit to the potential curve, and is not very well determined. However, it is close to the theoretical value (8). The agreement between the four level fits to the potential curve, which depend on the values of r determined from the RKR potential, and the LeRoy method, which uses only the experimental vibrational energies, attests to both the quality of the RKR curve and the validity of the assumption that, near dissociation, the potential energy is linear in rp3, as expected when the dissociation products are Li ( 2s) and Li (2~). It is felt that the best values of D, and C, will be those determined from the fits to the last four levels. The “best” values are D, = 1422.03 f 0.05 cm-’ and C, = ( 1.805 + 0.010) X lo5 cm-’ A3. The fact that the dissociation energy obtained from the potential curve is within 0.05 cm-’ of that calculated from the ground state value attests to the accuracy of the ground state dissociation energy and allows us to lower the uncertainty (previously 0.54 cm-‘) on this determination to 0.10 cm-‘. However, we do not feel that the accuracy of our determination of D,( 1 ‘IQ,) at present justifies changing D,(X’Z,+) from its previously determined value of 8516.78 cm-’ to the value, 8516.83 cm-‘, which is calculated using the 1 ‘I& data. The value of vD = 60.1 I, calculated from the second LeRoy fit, shows that, even though the last observed level is only 13 cm-’ below the dissociation limit, there are still 26 unobserved vibrational levels at the top of the potential well. This is due to the very slow progression toward dissociation of the outer limb of the potential curve as a result of the C,/r3 long-range term. Calculation shows that, at r = 50 A, the binding energy is 1.44 cm-’ and, at 100 A, the potential curve is still bound by 0.18 cm-‘.

353

I ‘IIg STATE OF Liz TABLE V Comparison of Dissociation Energies and Long-Range Expansion Coefficients, C,, for the 1‘I& State of Liz Method

C,(cm

De(cm-‘)

I

’

h’)

C,(cm-’

A”)

1421.98(55)= 1422.41(60jb

II

1421.942(50) 1422.46(34)

1.7831(40)~10~ 1.781(14) x105

III

1422.035(50) 1422.79(62)

1.8007(40)x105 1.817(21) ~10’

1422.021(33) 1421.55(89)

1.808(19) 1.738(74)

x10’ x10’

1.805(19) 1.723(74)

~10’ x105

IV

V

VI

1406

VII

1421

7.77(60)x10’ 8.4(1.3)x106

1.792x10=

VIII

6.02

I.

From

D,(X’Tg).

II.

From

long

range

fit

to

RXR potential,

last

10

III.

From

long

range

fit

to

RUR potential,

last

4 levels.

levels.

[ 1

1.2

IV.

From

LeRoy

fit,

V.

From

LeRoy

fit,

G(v)

vs.

v

1

l/5

[

v

VI.

Ab

initio,

ref.

4.

VII.

Ab

initio,

ref.

5.

VIII.

Ab

initio,

ref.

8.

‘Upper

entry.

‘Li

entry,

‘Li,,

b

Lower

For

‘Li

2’

“D

2,

present

V.

results.

calculated

= 60.11(14);

vs.

For

from ‘Li,,

ref.

2.

vD = 65.3C2.9).

From the results of their experiments, Miller et al. (2) have calculated an RKR potential for the 1 ‘I& state of ‘Liz. Using these data, we have repeated all the longrange calculations outlined in the preceding two sections. The values of D,, C3, and c6 obtained from these calculations are compared with our 6Li2 results in Table V. Because of the lower resolution of their experiments and the fact that the last observed ‘Liz level, u = 32, was 26 cm-’ further from dissociation than the last 6Li2 level (u = 34), the uncertainties in the ‘Liz values are about an order of magnitude larger than those for 6Li2. In general, the agreement between the two sets of data is very good. The value of uD = 65.28 which we obtained for 7Li2 from Eq. (3.6) shows that there are five more vibrational levels of ‘Li2 than of 6Li2 within the potential well.

354

LINTON ET AL. 4. A DOUBLING

A. Introduction A-doubling parameters for the 1 ‘I& state were at first obtained from the global least-squares fit to all the line frequencies. While as fitting parameters they reproduce the observed line frequencies to within 0.04 cm -‘, it is important not to place a great deal of physical significance on their values for two reasons. The first reason is purely practical. The parameters were calculated from a very limited number of disconnected e and f levels arising from transitions from isolated levels in the C’II, state. A large number of the transitions to the J levels were very weak relaxation lines. The doubling parameters for the 1 ‘I& state greatly depend on the precision of the C state constants which were fixed in the present calculations. Allowing the C state constants to float did not significantly change the calculated line positions but had a significant effect on the doubling constants. Thus, without a much larger data set it is not possible to obtain meaningful A-doubling parameters. The second reason is more fundamental and raises the question of the physical validity of assuming that the A doubling will vary regularly with vibration and using a Dunham-type expansion [a~~ + all( II + 1) - - - ] to account for this variation. To answer this question it is necessary to examine the causes of the A doubling in more detail. The A doubling in the 1 ‘IIg state is the general manifestation of the interaction with the 2 ‘2: state. The perturbations discussed in the following section are special cases in which the states are close to resonance and second-order perturbation theory breaks down. A first intuitive glance at the Liz energy diagram (Fig. 2, inset) would indicate that, as the 2 ‘Zi state is 1890 cm-’ below 1 ‘I&, the e levels of the IT state will be pushed up by the interaction and lie above the f levels. However, because the potential curves and internuclear separations are so different for the two states, a single level in the 1 ‘IQ state will interact with many vibrational levels in the 2 ‘2: state and a “unique perturber” approach is invalid. Above 2 ‘I?:, o = 12, the energy levels of the two states of 6Liz are intermingled, as shown by the potential curves in Fig. 2, and it would seem feasible that, in a situation where the 2 ’ 2: level above a given 1 ‘I& level is closer than the one below and there is a favorable overlap of the eigenfunctions (i.e., significant Franck-Condon factor), the e levels would be below the f levels.

B. Theoretical Treatment The A doubling is given, to a first approximation, ATef= q,;J(J+

by 1).

(4.1)

Using second-order perturbation theory, we can write, in the case where the A doubling is caused by a single distant Z level, ox, (4.2) where AE is the separation of the ‘II and ’ 2 states and Hnz is the matrix element JCnnecting the two states.

355

1‘II, STATE OF Liz

The matrix element connecting single vibrational levels, On and vz, of a ‘II and ‘Z state is Hnz=

(‘II,,

Vn1(1/Z!p.R2)J-L+I’Z+,

(Vn11/2~R21V,)(‘n,IJ-L+I’Z:+).

-

Using the r-centroid approximation, Hnz

N

02)

2’lL&. c1 f

(4.3)

we may approximate the matrix element by

(vnl%)(‘KI

1)]“2,

L+I’z+)[J(J+

(4.4)

where R, is the r centroid. In the case where L is well determined, (‘IN ,+y,+>

(4.5)

= [L(L + 1)]“2.

Both the 1 iI& and the 2 ‘2: states dissociate to Li (2s) + Li( 2~)) and thus L = 1 and (‘II] ,+I’,+) = 2”2. Thus Hnz -2

s&s(VnlVx)[J(J+ c

1)1”2-

(4.6)

However, because there will be contributions from several vibrational levels of 2 ’ Zz, these must all be included in the calculations. The expression for q”, must include a summation over all interacting Z state levels and becomes 4 vn

_

4

c

%

(1/2~R,(VnVz)2)“Q”,“, &,, - ~5,

(4.7 1 ’

The dominant factors determining the A doubling will be the Franck-Condon factors Q”,“, 7 and the energy separation, E,, - E,,. One would expect the Z levels closest to the II level to have the greatest influence, but only if there is a significant overlap of the eigenfunctions, i.e., significant Qv,“, . Franck-Condon factors and r centroids were computed from the RKR potentials and the energy separation from the molecular constants for the two states. Because of the need to extrapolate the 2 ‘2: state beyond the highest observed levels (v = 13 for 6Li2, 15 for ‘Liz), there could be large uncertainties in both the Franck-Condon factors and the energy separations, particularly at the higher vibrational levels. Another possible source of error in calculating the matrix element is in assigning L = 1. Because of the influence of the ion-pair interaction, Li+( 2s) + Li-( 2s)) on the potential curves, especially 2 ‘2: (5), the assignment as pure 2s + 2p is likely to lead to errors. However, it is felt that calculation of qon using Eq. (4.7) will give reasonable order of magnitude estimates and will reflect the variation with vibration. Theoretical A-doubling constants, listed in Table VI, have been computed for all levels in 6Li2 and ‘Liz for which there were experimental data (see below). For low On = O-2, the Franck-Condon factors with the high uz are negligible so that the A doubling is caused by the lowest vibrational levels of 2 ‘Zz and the constants are all in the region of 2.6 X 10T4cm-’ (for 6Li2). At Vn = 3, there are significant contributions from a larger number of Z state levels but, for low J, the contribution near resonance

LINTON ET AL.

356

TABLE VI Observed’ and Calculatedb A Doubling in the 1‘II, State of Li2 10*q(cm~')C 7Li,

‘Li, J=

”

5

7

14

25

31

0

4.46 2.56

1

4.82 2.58

4.52 2.45

4.31 2.43

4.65 2.62

4.58 2.69

2

8.00 2.64

5.89 2.64

3

6.67 2.86

4.11 2.86

4

5

4.67 3.15

6

14.67 10.55

4.86 2.92

5.95 3.65

2.71 1.52

4.23 2.64

5.33 3.49

8.66 6.04

19

1.10 -0.05 4.67 1.05

0.52 0.49

9

38

40

3.64 2.52 3.20 2.44

5.14 3.11

3.20 1.71 4.22 3.06

-0.54 -1.32

7

0

17

1.11 0.44 0.95 0.13

1.37 0.30

1.75 0.82

1.16 0.13 2.66 1.59

10

-1.13 -1.65

11

2.34 1.30

12

0.29 _ _.

aUpperentry. bLover entry. =q

is

defined

as

ATef

= c, J(J+l).

is still very small. The A-doubling constant for J = 5, 7, 14 is -2.9 X 1O-4 cm-‘. However, at J = 25, there is a large contribution from the u2 = 14 level which is 17.2 cm-’ below on = 3, and the A doubling is considerably larger, qn - 3.7 X 10e4 cm -’ . At J = 30, the resonance is very close and there is a large perturbation (see Section 5). As un increases, contributions from Z levels above on become increasingly significant and the energy separation at the crossover becomes more critical. The A doubling is now the sum of all the positive contributions from lower lying Z levels and the negative contributions from the higher levels. The result is that qn is calculated to vary erratically with both un and J and one can no longer legitimately express the A doubling at a given Y as a function of J( J+ 1). A good example of the fluctuation is shown by the behavior at J = 25. At un = O-2, the doubling is essentially independent

357

1 ‘II, STATE OF Liz

of Jandq - 2.6 X lo-“ cm-‘. At on = 3, as mentioned above, the contribution from the close lying vz = 14 level below it increases qn to -3.7 X 10P4 cm -’ . At on = 4, the contributions from the two closest Z levels ( vL: = 14 below and 15 above) are equal and the doubling drops back to q - 2.6 X lop4 cm-‘. At v = 5, the vz = 15 level lies just below and gives a very large positive contribution and the doubling increases dramatically to qn - 6.0 X lop4 cm-‘. At v = 6, the largest contribution is from the higher lying vvz = 16 level and is negative. The negative contributions dominate and the doubling is therefore predicted to be negative (i.e., f above e) with 4 - - 1.3 X 10e4 cm-‘. For the higher levels, there is often near cancellation of the positive contributions from the lower lying and the negative contributions from the higher lying Z levels resulting in very small A doubling. For example, at vn = 7, J = 14, the cancellation is almost complete, resulting in a calculated q - 5.1 X 10 m6 cm-’ (an undetectable splitting of 0.00 1 cm-‘). This is the result of a contribution of +2.19 X 10m4 cm-’ from the lower lying levels and -2.24 X 10m4 cm-’ from the higher levels. In this situation, small errors in Q”,“, and AE near resonance will be critically important. All the calculated A-doubling constants that are listed in Table VI clearly show that, above vn = 2, the A doubling is expected to vary erratically with both vibration and rotation, and that it is physically unrealistic to attempt to fit it to a Dunham expansion either in vibration or in rotation. C. Experimental Observations Because the statistical fluctuations in the fit (u - 0.04 cm-‘) are, in general, of the same order as or larger than the A doubling, other methods must be used to experimentally determine the doubling. It is necessary to find transitions in which both components are accessed from a single upper level. We therefore require fluorescence giving R ,Q, and P branches in which the Q-branch transition will go to the opposite parity component. As mentioned earlier, we were able to observe several Q( 5) lines in C’II,-1 ‘I& fluorescence from v’ = 1 and several progressions in the 2 ‘Z i-1 ‘I& transition (four in ‘jLi2 and four in ‘Liz). The A doubling was determined using combination differences. Using standard expressions ( JO), the expected position of Q(J) can be determined from the known R (J - 1) and P( J + 1) frequencies as

Q(J) -

P(J

+ 1) = (J+ ‘) .[R(J(2

J

+

1)

where X(J) = 1 to a first approximation nificant, [I -(J+ x(J)

= [I -(2J+

1) - P(J+

1)1.X(J),

(4.8)

and, if centrifugal distortion terms are sig-

1)2*20/B] l)*. D/2B]

’

(4.9)

The separation of the measured and calculated Q line positions then gives both the magnitude and the sense of the A doubling. As the line positions are accurate to -0.003 cm-‘, the measured combination differences are probably accurate to -0.005 cm -’ and it is felt that the A doubling obtained in this way is realistic. It was found that, for Li2, the contribution of centrifugal distortion to the A doubling was significant

LINTON ET AL.

358

above J = 7. For example, in 6Li2, the centrifugal distortion term decreased the calculated combination difference by 0.013 cm-’ for J = 14 and 0.040 cm-’ for J = 25. Thus, this term was included in all the calculations. All the observed values of the A-doubling constant, qn, for 6Liz and 7Liz are tabulated in Table VI along with the theoretical values. It can be seen that, even though the observed values always seem to be a factor of - 1.5 to 2 times larger than the theoretical values, the calculated values follow the variations in the observed values extremely well. This is clearly illustrated in Fig. 5 where both the theoretical and the experimental 14,and7Li2, J= 19, values of qn are plotted as a function of u for 6Li 2, J=25and and the two distributions are almost parallel. In 6Li2 at J = 25, the observed doubling at u = 1,2, and 4 is found to be almost the same, increases slightly at o = 3, increases substantially at o = 5, and is very small and negative at u = 6 (i.e., the f levels are above the e levels). At J = 14, the doubling decreases significantly from 2, = 3 to 4, increases at u = 5, and is very small at u = 7 and 8. The calculated values follow exactly the same pattern, including the changes in sign. The erratic behavior of the doubling at higher u is clearly demonstrated in 7Li2 where the 2)’= 0, S = 19 progression in the 2 ‘Z :- 1 ‘I& transition was observed from un = 6 to 12. As in all the transitions, 10 a 6

0 ;- -2 E 27 u

1

06

J=14

0

0

2

4

6

a

6Li 2

10

12 v

FIG. 5. Variation of the A-doubling constant, q, with vibration at different rotational quantum numbers in the 1 ‘I& state of 6Li2 and ‘Liz. The experimental points are joined by dashed lines and the theoretical values by solid lines.

l’&

STATE OF Liz

359

the Q lines are unperturbed and follow a regular progression. The doubling was generally small and changed sign at v = 10 and 12, the latter being nearly zero, a pattern which was precisely followed by the calculated values. At v = 6, the P and R lines were far from their expected positions but were in accord with the calculations which showed a very close resonance with vx = 17 and a Franck-Condon factor -0.1 which led to a strong perturbation of the e levels. There were several similar situations, both in 6Li2 and in 7Li2, where the shift in the P and R lines relative to the Q line was larger than the normal A doubling and, in each case, the conditions ( AE and Qv,v,) were found to be favorable to significant perturbations. These are discussed in detail in the next section. The results in Table VI clearly show that, in nearly all cases, the observed A doubling is greater than that predicted by the theoretical calculations. It is apparent that, in most cases, especially at low vu, the ratio of observed to calculated doubling is almost constant and in the range 1.6-2.0. The largest deviations occur when the doubling is very small and the precision of AE and Q”,“, is more critical. It seems likely that the main cause of the discrepancy is the error in assigning L = 1 in the ( ‘II 1L+ ) ‘8) matrix element and that this matrix element is probably -( 1.6)“‘-(2.0) “* i.e., - 1.26-1.41 times greater than assumed in our calculations. However, this does not account for the observation that the observed magnitude of the splitting is less than the calculated magnitude in cases where q is negative. This suggests the possibility that the calculated 2 ‘Zi state energies are too low relative to 1 ‘I&. This is still speculative and a lot more data are required to obtain a clearer understanding of the quantitative differences between observation and theory. The above discussion has clearly shown that, even though the data set is small, the A doubling is well described by the interaction of the 1 ‘II, level with all vibrational levels of the 2 ’ 2; state and that a global fit to the data is invalid, smoothing out the important physical interactions and producing physically meaningless A-doubling parameters. There is clearly a need for more data to gain a clearer understanding of the interaction, especially at high vibration. The double resonance experiments of Miller et al. (2) on 7LiZ produced RQP progressions to v = 3 1 in ‘Liz but the resolution was not high enough to determine either the sense or the magnitude of the A doubling. 5. PERTURBATIONS

A. Observations As mentioned in Section 2, several perturbations were observed in the ‘IIg state. In Sections A to D we shall discuss in detail the analysis of the two obvious very strong perturbations, at v = 5, J = 30 and v = 4, J = 4, observed in the C’II,-1 ‘II, transition of 6Li2 and outline the procedure for detecting the others. By a fortunate coincidence, tlie perturbation at v = 5, J = 30 has been accessed by fluorescence following three different laser excitations, to (a) C’II,, v = 1, J = 29; (b) C’II,, v = 0, J = 31; and (c) 2 ‘Z:, v = 3, J = 3 1. Figure 6a shows the three spectra involving the perturbed levels and Fig. 6b shows a schematic energy diagram of the transitions. In Section E, there will be a discussion of perturbations observed via 2 ‘Z :- 1 ‘I& and 2 ‘2 i-2 ‘2 ,’ transitions, concentrating in particular on the above v = 5, J = 30 perturbation and othersatv=3,J=30andv=6,J=13and15.

360

LINTON ET AL.

C'll,-17-l,(1 - 5) 8500

*8480

Main

P St

II 1

4+!?~ t P,*

8820

c'n, _l'll,(O - 5)

%7 -I

h

8840

2'.E::-2'1;(3 - 15)

VMain 30

FIG.6a. High-resolution spectra showing the perturbation between 1 ‘II,, u = 5, J = 30 and 2’Z,C, u = 15, and J = 30. The top two spectra were excited by UV krypton laser lines and the bottom by a UV argon laser. The perturbation is illustrated for the P( 30) C’fI.-1 ‘II,, l-5 (top), R( 30) C’fI.-1 ‘II,, O-5 (middle), and R( 30) 2 ‘2 i-2 ’ Zt, 3- 15 (bottom ) transitions in which both main and extra lines are seen. The top spectrum also shows the perturbation in R(28) C’fI.-1 ‘II<, 1-5. The line (a) at 8838.7 cm-’ (bottom) is Q(31) 2 ‘XL-1 ‘IIS, 3-5. Other lines in the spectra come from other C’II,-1 ‘II, or 2 ‘XL-2 ‘Z: transitions. For example, the intense line at 8490 cm-’ (middle) is P( 30) C’II,-1 ‘II,, l-8 and the intense line at 8809 cm-’ (bottom) is R( 13) C’II,-1 ‘II,, 2-4. The frequency scale is in cm-’ and the intensity scale was increased by a factor of 5 for the bottom two spectra.

The fluorescence from (a.) is by far the most intense. Figure 6a clearly shows the perturbation in P(30) of the C’II,-1 ‘II, l-5 band. The R(28) line at 8709.6055 cm-’ is very strong with an intensity of Z = 284.4 (arbitrary units). The perturbed P( 30) transition consists of two less intense lines at 8677.3916 cm-’ (I = 148.5) and 8685.0385 cm-’ (I = 105.8), representing the main and extra lines. The sum of the intensities, 263.3, gives an R/P intensity ratio that is consistent with that of the other, unperturbed, vibrational levels. The separation of the main and extra lines, representing the separation of the perturbed 1 ‘I& and perturbing states, is 7.6469 cm-’ . Examination of the relaxation lines enabled us to observe the R( 30) main line at 8744.8748 cm-’ (I = 4.8) and a weaker extra line at 8752.5325 cm-’ (I = 4.3), a separation of 7.6577 cm-‘, very close to that in P( 30). The upper state combination differences, R( 30) - P( 30) are 67.4832 and 67.4940 cm-‘, respectively, for the main and extra lines, obviously in good agreement. They are also in agreement with the combination difference for all the other unperturbed vibrational levels (see Table I), thus confirming the assignment. The fluorescence from (b), also shown in Fig. 6a, is much weaker. The C’II,1 ‘I&, O-5 transition consists of a single P( 32) line accompanied by two weaker R( 30)

361

1 ‘fI, STATE OF Liz

V _- ___________._..J=28

vs.3

1 ‘&

‘u’ \%

_. AE~=lZ.XCM:__

---

\ . 8,d \ ’

M4.9XlP _ _.

FIG. 6b. Schematic energy level diagram (not to scale) showing the transitions in Fig. 6a. The actual levels of the I ‘II, and 2’Zf states are represented by solid lines and the unperturbed positions by dashed lines. The main R and P transitions are represented by solid vertical arrows while the extra lines are shown as dashed arrows. The energy separations of the interacting levels. AE. and the energy shifts, 6E, caused by the perturbations are all shown. The levels are all of e parity except for 1 ‘IIn, J = 3 I, which is an unperturbed flevelobservedviaQ(31)2’2:-I’&..

lines separated by 7.643 1 cm-‘, in excellent agreement with those mentioned in the preceding paragraph. The intensities of the main (lower frequency) and extra (higher frequency) lines are almost equal because the extra line coincides with the P( 28) line in the l-7 band which arises as a result of relaxation from J’ = 29 to J’ = 27. Before discussing (c) , it is necessary to examine the perturbation in more detail to obtain some qualitative information about the perturbing state. The frequency order of the main and extra lines indicates that the perturber is below 1 ‘I&,, II = 5, J = 30. The fact that the perturbation is very strong at .I = 30 and not at .I = 28 and 32 indicates that (i) it is a (J-dependent) heterogeneous perturbation and (ii) the B value of the perturber is considerably different from that of 1 IfIg. The only possible state of the correct symmetry which satisfies the above conditions is 2 ‘2:. The perturbations are therefore special, near resonant, cases of the interaction responsible for the A doubling. Calculation of all 2 ‘2: and 1 ‘I& levels of the same .I lying within 20 cm --’ of each other showed that 1 ‘I&, v = 5, J = 30 is very close and slightly above 2 ‘Zl, u = 15, J = 30. The perturbing state is therefore 2 ‘Zg, u = 15, J = 30. This near resonance is clearly seen in Fig. 2 where the potential curves and vibrational levels at J = 30 are shown for both states.

362

LINTON

ET AL.

The fluorescence from (c) allows us to observe emission to J = 30 in the 2 ‘2: state. The argon laser (at 333.6 nm) excites 2’2:, u = 3, J = 31 and a vibrational progression in the 2 ‘Zg state was observed in the resulting fluorescence. The R( 30) and P( 32) transitions to 2 ‘Xi, v = 15 are shown in Fig. 6a. P( 32) is a single line and, at first glance, so is R( 30). However, the difference between the observed and calculated frequency for R( 30) was too large and the line had to be left out of the least-squares fit as it was obviously perturbed. Closer examination shows a very weak line (I = 3.9) whose frequency is 7.6476 cm-’ lower than that of R( 30). This is exactly the same separation as that in the 1 ‘I& state perturbation and the fact that the extra line (to the 1 ‘I& state) is at lower frequency is consistent with the previous observation that the 2 ‘Zg state must be below 1 ‘II,. The same perturbation has thus been observed in transitions to both states and clearly confirms that the interacting statesare l’II,,v= 5, J= 30and2’2i,v= 15, J= 30.TheQ(31)lineof2’X:1 ‘IZIghas also been observed and the transitions from 2 ‘Z : will be discussed in detail in Section 5E. The second strong perturbation at v = 4, J = 4 can be seen in Fig. 1 where the R(4) line of the l-4 band is clearly split into two lines. The more intense “main” line has an intensity of I = 60.4, while the weaker “extra” line has an intensity of 3 1.1. The extra line is at lower frequency than the main line (separation = 0.4366 cm-‘) showing that the perturbing state must be above 1 ‘I&,, II = 4, J = 4. The P( 6) line appears to be single and unperturbed. Using the same arguments as those for the J = 30 perturbation, the perturber has been identified as 2 ‘2:) II = 15, J = 4. B. Analysis Because of lack of experimental data on the 2 ‘2: state of 6Li2 above v = 12, it is not possible to accurately determine the separation of the two interacting states directly from the constants. However, the perturbation information can be combined with the 1 ‘II, state constants obtained from the fit, or with the intensity information, to determine the energies of the perturbing 2 ‘2: levels. We analyzed the 1 ‘I&, u = 5, J = 30 perturbation using both the P( 30), l-5 and the R( 30), O-5 transitions. The R( 30), l-5 relaxation line is too weak to be reliable. For the P( 30) transition, the frequency calculated using the C’II, and 1 ‘I& state constants was 3.136 cm-’ greater than the observed frequency. Thus, the lower state level has been shifted upward by 6E = 3.136 cm-‘. The corresponding downward shift in the 2 ‘2: state will lead to an upward shift in frequency and the calculated frequency will be 3.136 cm-’ below the observed value. The difference between the two calculated frequencies then gives the deperturbed energy separation, AE = 1.375 cm-‘, between the two states. For R( 30), O-5, the same technique gives 6E = 3.128 cm-’ and AE = 1.387 cm-‘. Thus the average values are 6E = 3.132 cm-’ and AE = 1.381 cm-’ with the 2 ‘Zi level below 1 ‘I& as shown in the schematic in Fig. 6b. If we assume an uncertainty of 0.08 cm-’ (twice the standard deviation of the fit) in the calculated frequencies, this gives uncertainties of 0.08 cm-’ (2.5%) in 6E and 0.11 cm-’ (8.0%) in AE. Knowing both the energy shift, 6E, and the separation, AE, the matrix element, Hnx, connecting the two states was determined from

1 ‘IIs STATE OF Li2

Hnz = ( 6E2 + GEAE) ‘I2

363 (5.1)

from which a value of Hnz = 3.760 + 0.082 cm-’ was obtained. It is now possible to determine the degree of mixing between the two states and check the intensities of the main and extra lines. If we label the eigenfunctions of the mixed states 1II* ) and 1Z* ), then these can be written in terms ofthe pure eigenstates III)and IZ)as in*> = QIII)

+ c&z)

(5.2a)

+ c,lZ),

(5.2b)

IX*) = -c2III) where cl and c2 are the mixing coefficients and c:+c;= Using standard perturbation

1.

(5.3)

theory,

c: = O.S{l + [I + (2Hny/AE)2]-“2},

(5.4)

from which we determine that and

c: = 0.590 + 0.008

c?_ = 0.410 + 0 .008 .

Thus, the nominal 1 ‘I& level is, in fact, a mixture of 59% 1 ‘I& and 41% 2 ‘2:. If there is no transition probability for the C’II, (u = 1, J = 29)-2 ‘2; (u = 15, J = 30) transition, then the intensity ratio of the main to extra lines in P( 30) will be given by Imain _

Zextra

C: - 2 =

1.44 + 0.03.

cz

The Franck-Condon factor for C’II,-2’Z~, P(30), I-15, is calculated to be 10 orders of magnitude less than that for C’II,-1 ‘IIg, l-5, so this would appear to be a reasonable assumption. From the measured intensities, the ratio is 1.40 + 0.04 which is in excellent agreement with the calculated value. Thus, the calculated energy separation is consistent with the observed intensities. It is also possible to work from the intensities to calculate the matrix element and the energy separation. This will be very useful when there is not a good set of constants for either state so that energies cannot be determined directly. In the present case, the uncertainties in the experimental and calculated intensity ratios are similar so that the two methods should be equivalent. Starting with intensities gives c: = 0.584 f 0.018, AE = 1.25 + 0.13 cm-‘, 6E = 3.107 -t 0.065 cm-‘, and HnI; = 3.68 + 0.40 cm-‘. Using the molecular constants to examine the 1 ‘I&, u = 4, J = 4 - 2 ‘LX:, v = 15, J = 4 interaction, we find an energy shift, 6E = 0.156 f 0.080 cmP1, a separation, AE = 0.125 f 0.110 cm-’ (2’2: above 1 ‘II,), and a matrix element, Hnz = 0.209 f 0.073 cm-’ _ The mixing fractions, c? = 0.64 -t 0.12 and c: = 0.36 + 0.12, show that, at u = 4, J = 4, the nominal 1 ‘I& state is a mixture of 64% 1 ‘I& and 36% 2 ‘2:. The intensity ratio for the main to extra lines is calculated as 1.80 +- 1.44 compared with an observed value of 1.95 + 0.18.

364

LINTON

ET AL.

Starting with the intensity ratio of 1.95 & 0.18 gives c: = 0.660 -t 0.073, AE = 0.140 f 0.070 cm-‘, 6E = 0.149 + 0.053 cm-‘, and Hnz = 0.217 + 0.100 cm-‘. The two methods give results that are in very good agreement. However, it would appear that, because of the uncertainty in the fit and the closeness of the interacting states at J = 4, the intensities probably give a more reliable estimate of the energy separation than the constants. C. Theoretical Calculations As previously mentioned, the same interaction responsible for the A doubling is also responsible for the perturbations. The latter are specific cases in which the interaction with a single close lying 2 ‘Zg level is so much stronger than that with the other vibrational levels that these latter may be neglected and a near resonant, unique perturber approach may be used. The matrix element, H nz, between the perturbing levels is exactly the same as that used in the A-doubling calculations for individual 2 ‘2: state levels and is given by Eq. (4.6). However, the energy shifts and separations may no longer be calculated using second-order perturbation theory and Eq. (5.1) must be used. For J = 30, 2) = 5, a value of 3.358 cm-’ was obtained for the matrix element. Considering the uncertainties involved in the calculations, the agreement with the experimental value of 3.760 cm-’ is good. For J = 4, v = 4, the calculated matrix element was Hnz = 0.18 cm-‘, in good agreement with the experimental value of 0.209 cm-‘. D. Other Perturbations Because of the overlap in energy of the 1 ‘IIg and 2 ‘Xi states (Fig. 2), it seems likely that there would be many perturbations. The J = 30 perturbation described above is particularly strong because of the near resonance of the two states and the favorable Franck-Condon factor. It is expected that the R( 28) line of the C’II,-1 ‘I& l-5 band should also show signs of a perturbation as the matrix element and FranckCondon factor will be similar to those of the P( 30) line. The expected line position was calculated from the constants and, on comparison with the observed frequency, it was determined that the J = 28 level was pushed up by 6E = 0.95 14 cm-‘. Thus, at J = 28, the 1 ‘I& state is above 2’2:. As, at J = 30, H - 3.761 cm-‘, then at J = 28, Hnz - 3.514 cm-‘. From this, the energy se&ation was calculated, using Eq. (5-l), to be AE - 12.028 cm-’ and the main: extra line intensity ratio - 14. Thus, there should be an extra line of I - 20 at around 8723.536 cm-‘. A search of the spectrum showed a previously unidentified line of intensity = 16.8 at 8723.7720 cm-‘. This is almost certainly the extra line accompanying the R(28) line. From this, it was calculated that, at J = 28, the 1 ‘I& state is 12.264 cm-’ above 2 ‘2; , they are connected by a matrix element H - 3.545 cm-‘, and the state is 93.3% 1 ‘I& and 6.7% 2 ‘2:. By extrapolating the molecular constants, the energy separation has been calculated as - 12.781 cm-‘. The R( 28) main and extra lines are shown in Fig. 6a. The same technique was used to search for more perturbations. All cases in which 1 ‘I& and 2 ’ 2: levels of the same J were calculated to be within 20 cm -’ of each

365

1 ‘II8 STATE OF Liz

other were listed and, for those with favorable Franck-Condon factors, theoretical energy shifts and matrix elements were computed. These transitions were then examined more closely and, in many cases, the lines appeared perturbed although it was not always possible to assign the extra line which was normally expected to be very weak. Of particular interest was the P( 30) line in the C’II,-1 ‘II*, l-3 band. From the constants, the 2’Zi (u = 14) state was calculated to be 3.515 cm-’ above 1 ‘I& (II = 3) (the near resonance is shown on the potential energy diagram in Fig. 2 ) and the Franck-Condon factor was 0.0 176. There was therefore expected to be a significant perturbation. The extra line was found 4.625 cm -’ from the main line and the matrix element was in very good agreement with the theoretical value (see Table VII). The perturbing state is 2 ’ Z g’, 2)= 14, and a search of the 2 ‘Z z -2 ‘2: fluorescence revealed a very weak line 4.6 18 cm-’ from the R( 30) line of the 3-14 band, confirming the

TABLE VII Observed Perturbations in the i ‘& and 2 ‘Si States of Liz J

“II Strong

‘Li,

Hnx(theory)d

H,&e~p)~

Perturbations

with

Extra

Notes

Lines

30

5

15

1.38

3.76

3.36

28

5

15

12.26

3.55

3.14

a.b

30

3

14

3.60

1.45

1.50

a,b

32

3

14

-13.14e

1.47

1.63

b

28

3

14

5.24

1.49

1.30

0.21

4

4

15

0.13

0.18

a

13

6

16

0.51e

1.50

c

15

6

16

4.46e

1.72

c

Weaker

‘Li,

AEd

“L

Perturbations

with

no Observed

Extra

Lines

32

5

15

9.54e

3.54

6

4

15

1.64e

0.27

30

10

17

30

12

18

28

12

18

32

13

18

16

6

17

2.aoe

1.50

18

6

17

2.27e

1.68

20

6

17

7.97e

1.86

a.

Very

strong

b.

Observed

in

c.

Observed

only

and extra

perturbations

in

2’Z;

-

at

.l=13.

in

cm ‘.

d.

BE and Hnz are

e.

BE calculated from the perturbation.

2.04 1.76

9.61ae 7.83e

1.64

10.2e

with

transitions

line

28.0e -

intense

from

C’nu

l’ng

transition

constants.

Not

and

1.78

extra 2’~;

lines. states.

with

enough

only

main

information

line

to

at

.I=15

calculate

from

LINTON

366

ET AL.

assignment of the perturbation. The energy separation, AE = 3.642 cm-‘, is in good agreement with that calculated from the constants. Further investigation also revealed the Q( 3 1) and P( 32) lines of 2 ‘Z :- 1 ‘I&, 3-3 and this is discussed in Section 5E( iii). The R( 28) line accompanying the perturbed P( 30) line was also found to be perturbed and the extra line was detected within 0.1 cm-’ of the position predicted by the preliminary calculations. Many perturbations have thus been detected and are listed in Table VII. Considering that the experiment samples only six rotational levels in the 1 ‘I& state, it is obvious that, in 6Liz, there will be a very large number of perturbations affecting all vibrational levels of the 1 ‘I& state and the o > 12 levels of the 2 ’ Zg state and that many of these perturbations will be very strong. E. Intensities of Perturbed 2’2 :-2%:

and 2’2 L-1 I& Transitions

(i) We showed above, in the detailed discussion of the 1 ‘II,, v = 5, J = 30 - 2 ‘Zl, v = 15, J = 30 perturbation, that the intensity ratio of the main to extra lines in the P( 30) transition of the C’II,-1 ‘I&, l-5 band of 6Li2 is consistent with theoretical predictions assuming no transition probability to the perturbing state. This, however, will not be true of transitions from 2 ‘Z:. Transitions to neighboring 2 ‘2: and 1 ‘I& levels are clearly seen in the spectra (Fig. 6a), indicating a significant transition probability to each state. In this situation, the main/extra line ratio will not simply be the ratio of the mixing fractions but there will be a transfer of intensity from the main to the extra line in one branch and vice versa in the other branch of the 2 ‘2:-2 ‘Zz transition involving the perturbed level. This is clearly illustrated by the R ( 30) transition of the 2 ’ 2 c-2 ‘2: 0- 15 band where the main/extra line intensity ratio is 11.3 (k4.0). This is very different from the 1.40 ratio observed in the C’II,1 ‘IQ,, l-5, P( 30) transition and shows that there is a significant transfer of intensity from the extra line to the main line. For the purposes of this discussion, unless otherwise specified, in situations where transitions to both perturbing levels are observed, we shall define 2 ‘2:-2 ‘Xi as the main line and the transition to 1 iI& as the extra line. The intensity information in these perturbations is very useful as it enables us to determine the ratio of the transition moments for the two transitions. In this section, we shall briefly outline the theory and discuss in detail the ( vn = 3, vz = 14, J = 30), ( vn = 5, v2: = 15, J = 30) perturbations in 6Li2 and outline some other examples. (ii) Theoretical outline. If we let p zz and pLznrepresent the dipole moments of the 2 ‘I; i-2 ‘2: and 2 ’ 2: :- 1 ‘I& transitions, respectively, then it can be shown ( 1 I ) that the ratio of the main to extra line intensities will be given by Imain -=-= I extra

I22 I zn

c2/.l& + (1 - c*)& f 2c( 1 - C2)“*pzz&n (1 - c2)& + c2/.l;* T 2c( 1 - C*)l’*/JLzz~~n’

(5.5)

where c ( = cl ) is given by Eq. (5.4) and is determined from the matrix element and separation of the interacting states. If we define the dipole moment ratio, r = pzz/pzn, the above expression can be rewritten -Imi” = c*r* + ( 1 - c2) 1- 2c( 1 - c*)l’*r I extm ( 1 - c2)r2 + c* T 2c( 1 - c*)l’*r .

(5.6)

I ‘II, STATE OF Li2

367

The sign of the last term depends on the sign of the matrix element ( 11)) and the signs of the transition moments will also determine whether the last term is added or subtracted. The sign is normally determined experimentally. For a given J, the signs for the P and R branches will be opposite as will be the signs of the numerator and denominator. Thus, measurement of the intensity ratio may allow us to determine r. In some cases, when all the lines in the transition to both states are observed, r can be determined directly in which case the intensity ratio can be predicted. The value of r will consist of ratios of vibrational and rotational factors, K2”n”211?t~J’Jw2, and of the electronic transition moments, R,. Both the vibrational and the rotational parts are known from Franck-Condon factor and HGnlLondon factor calculations so that the intensity ratio will lead to determination of the electronic transition moment ratio. 3-14 and 2’2:-1 ‘I&, 3-3 (iii) Vn = 3, vz = 14, J = 30, 32. The 2’2:-2’2R+, transitions are the only ones involving a perturbation in which all five lines, R( 30 ). P( 32) to 2 ‘2: and R( 30), Q( 3 I), P( 32) to 1 ‘I&, are clearly observed. This is shown in Fig. 7. The intensity transfer, from II to Z in R( 30) and Z to JIIin P( 32). is clearly seen. The transition to the 2; state is intrinsically more intense than that to the Il state but the R-branch intensity ratio I&I_ pII is 30.6 whereas for the P branch, it is 1.53. Thus, in Eq. (5.6), the positive sign will be in the numerator for R( 30) and, for P( 32 ). the sign will be negative. As all five lines are observed, the transition probability ratio can be determined directly. The intensities are 2’z+-1 u

‘n g.*

2’Z:-2’Z;:

IR = 3.2,

z, = 43.1,

I, = 49

IR = 98.

I, = 75.

Thus, the total intensity of the R + P branches, which are mixed by the perturbation, is 225.2 and the Q branch, which involves the 1 ‘II, f levels and is unaffected by the perturbation, has I = 43.1. This can then be used to calculate the unperturbed R and P intensities in the X-II transition. For the transitions to On = 1 and 2, both of which

2’z: - 2’X: ( 3 - 14 )

P32

8950

2’z:

8970

-

l’ll, ( 3 - 3 )

8990

FIG. 7. High-resolution spectrum showing the interaction, at J = 30 and 32, between 1 ‘II,, u = 3 and Z’Z:,u= 14.Thespectrumshowsallthetransitionsfrom2’2~.~= 3, J= 31 to2’2i,u= 14and 1’11,. u = 3. At R( 30), transition probability is transferred from I ‘IIg to 2’Z,f whereas, at P( 32). the transfer is in the opposite direction. The frequency scale is in cm -‘.

LINTON

368

ET AL.

are “unperturbed,” the ratio (Z, + Z,)/Zo = 1.35 and Zp/ZR = 1.1. The latter ratio is consistent with the Hiinl-London factors, but the ratio relative to the Q (normally unity) is larger than expected. There is thus an unrelated interaction suppressing the Q branch relative to the P and R branches. This seems to be consistent for all the transitions unaffected by the 2 ‘2: - 1 ‘II, interaction and can be used to determine the unperturbed intensities. As Zo = 43.1, the above information gives ZcP+Rj= 58.2 leading to Zp = 30.5 and Z, = 27.7. As the total P + R intensity is 225.2, the total unperturbed intensity for 2 ‘Z:2’2: is 167 and Z, = 84.8, ZR = 82.3. For each branch, the ratio of the transition probabilities ( =r2) is the intensity ratio; thus 84.8 r2[ P( 32)] = 3. = 2.78:

r[P(32)]

= 1.67

82.3 = 27 = 2.97:

r[R(30)]

= 1.72.

r2[R(30)]

In Section 5D, it was mentioned that the calculated energy separation of the interacting states is 3.5 15 cm-’ and the experimental value from C’II,-1 ‘I&, 1-3, P( 30) is 3.642 cm- ’ . There were three other observations involving the vri = 3, J = 30 level [R(30), C’II,-1 ‘I&, 1-3; R(30), Cl&-l ‘I&, O-3; R(30), 2’2:-1 ‘II,, 3-31 and, by calculating the deviations of each of these lines from the calculated values, the average experimental energy shift was found to be 6E = 0.5 12 k 0.029 cm-‘. As the average value of the main-extra line separation was 4.622 + 0.004 cm-‘, the average observed energy separation is AE = 3.598 + 0.058 cm-‘. From the calculated separation, the matrix element is ZZnz = 1.501 cm-’ and the mixing fractions are c: = 0.880 and c: = 0.120. From the average observed aE, Hnz = 1.45 1 cm-‘, cf = 0.889, c$ = 0.111. Using Eq. (5.6), the calculated intensity ratio for R( 30) is I,,/ Zzn = 32.5 using the calculated AE and 27.7 using the average observed M. These are in excellent agreement with the observed ratio of 30.6 which has a large ( -30%) uncertainty because of the low intensity (3.2) of the extra line. For J = 32, the perturbation is much weaker and it is not possible to determine AE experimentally from the fit as the shift, 6E, is much smaller than that at J = 30. From the constants, aE = - 13.143 cm-’ and the separation of main and extra lines was measured at 13.469 cm-‘. The shift is thus calculated to be 6E = 0.163 cm-’ from which the matrix element Hnx = 1.473 cm-‘, c: = 0.988, c$ = 0.012. The mixing is therefore very small ( 1.2%) but still has a significant effect on the intensities. As the transfer is from 2 ‘Zl to 1 ‘IIg, Eq. (5.6) must be used with the negative sign in the numerator and, combined with the value of r = 1.67 for P( 32)) gives an intensity ratio Zzz/Zzn = 1.63 compared to the experimental ratio of 1.53. Considering the uncertainty in determining AE and H nz, the agreement is good. The observed R- and P-branch intensities are thus well explained by the interference effects caused by the 212+g - 1 ‘II, interaction. (iv) vn = 5, vx = 1.5, J = 30, 32. This interaction, which, at J = 30, has already been treated in detail in Section 5B, has to be treated slightly differently from the vn = 3, vz = 14 interaction as not all the lines in the 2 ‘Xi-2 ’ 2; and 1 ‘I& transitions

1 ‘IIg STATE

OF Li2

369

have been observed. The transition probability ratio can therefore not be determined directly. The transition to 2’2: is strong with IR = 43.9, Z, = 55.0. However, the transition to 1 ‘I& is weak. For the unperturbed Q line, 1, = 12.5. Using the same reasoning as that in the previous section, we would expect the unperturbed intensities of the R and P lines to be IR = 8.0,1, = 8.9. However, we find that Z, = 3.9 and there is no obvious P line. Thus, in this transition, the intensity transfer is from the 1 ‘IIq to 2’2: for both R(30) and P(32) and Eq. (5.6), with the positive sign in the numerator, must be used for both! This is not inconsistent with the earlier statement that the transfer is in the opposite directions for R and P branches. In the present case, we are dealing with two separate perturbations, at J = 30 and 32. At J = 30, 1 ‘II, is above 2’2: whereas, at J = 32, the order is reversed and AE changes sign. This is consistent with a change in sign of the interference term in the perturbation treatment ( I I ), and the transfer of intensity will be in the same direction for both lines. For the 011= 3, 02 = 14 perturbation the Z state is above the II at both J’s and the transfer is opposite in the P and R branches. It is interesting to note that, for P( 32), the Z state is above the II for both transitions, yet the intensity transfer is in the opposite direction. This suggests that there has been a phase change, probably in one of the transition moments, between the two transitions. The values of c: and c: have already been calculated (Section 5B) for J = 30. On substituting these in Eq. (5.6) along with the measured Zrr/Ixn ratio of 11.3, we can solve for r which gives a value of r = 2.33 for the R( 30) transition. For P( 32)) r will be slightly different because the ratio of H6nl-London factors of the P and R branches is slightly different in ‘Z- ‘Z and ‘Z- ‘II transitions. Assuming that the Franck-Condon factors and electron transition moment do not change between R( 30) and P(32), we calculate r = 2.26 for P( 32). The transition probability is r2 times greater for the ‘Z-‘Z transition than for ‘2;‘II. Thus the unperturbed intensities in the ‘Z-‘S transitions should be R( 30) = 2.33’ X 8.0 = 43.4; P( 32) = 2.262 X 8.9 = 45.5. The total intensity ofthe R( 30) and P( 32) lines in both transitions is thus calculated to be 105.8. The actual total intensity of the three observed lines is 102.8. Thus, the P( 32) extra line of 2’2:-1 ‘II, should have an intensity of about 3.0. Because of the uncertainties involved in the calculations, this simply means that the P( 32) extra line should be very weak or nonexistent. Using the value of AE calculated from the constants (- -10.23 cm-‘) and the matrix element (proportional to [ J( J + 1 )] “2) calculated from that at J = 30, we find c: = 0.893, cs = 0.107, and ZzSn/Zzn- 142. Thus, the extra line is not expected to be observed. The extra line was calculated to be - 13.00 cm-’ from the main line and at higher frequency, and nothing was observed in this position. To confirm the above, it would be useful to observe R and P transitions to the same J where the signs of the interference terms should definitely be opposite. This will involve relaxation lines which, unfortunately, are very weak. For the ul, = 3, uy = 14 interaction, the R( 32) main line was observed with Z = 3.0. There was no line observed in the calculated position of the extra line. The transfer of intensity is obviously from II + x:, opposite to that observed in P( 32) and thus in accord with expectations. In P( 30), we expect the transfer to be from main to extra lines with the extra line more intense. It was not possible to assign either line with any degree of confidence. In the transition to ox = 15, Q, = 5, for both P( 30) and R( 32). the transfer is expected to

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be from main to extra lines and Eq. ( 5.6) should be used with the negative sign in the numerator. For P( 30) the extra line was calculated to be four times more intense than the main line. No line was observed less than 0.02 cm -I from the calculated extra and main line positions. For R( 32), the main and extra lines are calculated to have similar intensities and there were weak lines observed within 0.02 cm-’ of the main (I = 1,4) and calculated extra (I = 1.6) line positions. If these are in fact the main and extra lines, it confirms the change in sign in going from J = 30 to J = 32. However, the lines are so weak that one cannot really place any confidence in the assignments and spectra of much higher signal-to-noise ratio are required. We can use the transition probability ratios to determine ratios of electronic transition moments. For example, we showed for the un = 5,~~ = 15, J = 30 perturbation that the intensity ratio gave I = 2.33. The vibrational contribution to this ratio is 0.90 and the rotational contribution is 1.44. Thus, the ratio of the electronic moments is 1.80. Schmidt-Mink et al. (4) calculated the ab initio dipole moment functions for both transitions. Although the numerical data were not published, examination of the published transition moment curves shows that the ab initio calculations predict a ratio of -2. Thus our observation of the main and extra lines in the 2 ’ z z-2 ‘2 g’ transition has enabled us to determine the electron transition moment ratio and show that it is in accord with ab initio predictions. A similar result was obtained for the un = 3, vz = 14 perturbation. (v) Other perturbations. The only other perturbation in which an extra line was clearly observed involved the fluorescence from 6Li2, 2 ‘Z :, v = 1, J = 14. Progressions to both 2 ‘Z g’ and 1 ‘I& were observed and the ZI= 6 level of 1 ‘II, was clearly perturbed. Calculation showed that the perturbing state is 2 ‘Zi, u = 16, but the progression to 2 ‘2: was observed only to u = 11. Using the techniques described in the previous section, it was possible to calculate very approximate expected main and extra line positions for R( 13) and P( 15) where, in this case, the main line is to the 1 ‘I& state. From this, it was found that only the extra line in R( 13) and the main line in P( 15) were observed. Thus the intensity transfer was from II + Z in the R branch and t: + II in the P branch, the same as that for the vn = 3, 2)x= 14, J = 30,32 interaction. Calculation showed that the absence of the main R ( 13 ) and extra P( 15 ) line is consistent with expectations. (vi) Summary. We have shown in this section that the observed perturbations can all be explained in terms of intensity borrowing mechanisms and the interference term which arises between transitions with significant transition probabilities to two interacting states. The intensity information allowed us to determine ratios of the electron transition moment and show that they are consistent with ab initio predictions (4). A complete list of all observed perturbations is given in Table VII. 6. INTENSITY DISTRIBUTION

A. Vibration From the RKR potential curve, Franck-Condon factors were computed for the C’II,-1 ‘I$ transitions in 6Li2 for which long progressions were observed. The measured intensities and Franck-Condon factors (both normalized to 100 for the strongest transition) for the progression from C’II,, t, = 1, J = 5 are listed in Table VIII and

371

1 ‘I& STATE OF Liq TABLE VIII Comparison of Observed Intensities and Calculated Franck-Condon Factors (Normalized to 100 at u” = 4) for the R(4) Lines in the C’II. (u = 1, J = 5)-l ‘rIp Transition v(l’I&)

Intensity

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29

10.6 35.4 63.3 84.6 100.0 92.5 74.3 55.4 32.3 17.6 6.2

6.4 8.3 9.8 11.5 11.4 11.9 11.3 10.4 9.9 8.2 7.8 6.3 5.4 5.4 3.3

Franck-Condon Factor 8.6 32.0 63.6 89.6 100.0 93.6 75.6 53.6 32.9 17.1 6.9 1.7 0.0 0.6 2.2 4.1 5.8 7.1 8.0 8.4 8.4 8.2 7.8 7.2 6.6 5.9 6.6 5.8 5.0 4.6

are shown schematically in Fig. 8. It can be seen that the experimental and calculated intensity distributions are in excellent agreement. The same is also true of the other observed progressions. B. Rotation It was mentioned earlier that the intensities in the C’II,-1 ‘I& transition are anomalous in that, at higher J, J’ = 29 and 3 1, the R branch is clearly more intense than the P branch while at J’ = 5, the P branch is more intense, in accord with the HGnlLondon factors. At J’ = 12, the P lines are slightly more intense than the R lines, again in accord with the HGnl-London factors. For S = 29 and 3 1, there were slight variations in the R/P ratio with vibration but these were normally within the range of the experimental uncertainty. For J’ = 29, the R/P ratio was - 1.15 and for J = 31, - 1.30. The J’ = 37 progression in 6Li7Li, which was discussed in Section 2B, had the expected R/P ratio - 1. All the above suggest that there is a weak, J-dependent, perturbation and that an intensity borrowing mechanism, similar to that discussed in Section 5E, is in effect. It is unlikely that the interaction between 2 ‘2: and 1 ‘II, is responsible, as most of the transition probability in the C’II,-2 ‘2: transition is in the region of Av - 0. Thus, the only interactions of importance to the R/P intensity ratios are those between

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ET AL.

100

m OBSERVED 90

I

CALCULATED

80

FIG. 8. Comparison of observed intensities and calculated Franck-Condon factors for the transitions from C’LI,, u = 1, J = 5 to o = O-29 in the I’ll, state. Both observed and calculated values have been normalized to a value of 100 at u = 4. Possible variations in the electron transition moment have not been taken into account in the calculations.

the 1 ‘I& level and the 2 ‘Zi level with the same u as the upper state. These are usually - 1500- 1800 cm -’ apart and there is less than 0.1% mixing, too small to account for the observed intensity distribution. It therefore seems likely that the perturbation is in the C’II, state. The fact that J' = 12 shows no anomalies could be because this is the one transition assigned to the f levels of the C’II, state. The other transitions from the f levels were all very weak relaxation lines from which it was difficult to obtain useful intensity information. The perturbation thus seems to affect only the e levels of the C state and must therefore be caused by a ’ 2; state. The A doubling in the C state is an order of magnitude larger than that in the 1 ‘I& state with the e levels above the f levels suggesting that it is caused by a lower lying ‘I: : state that is not too distant. It is certainly feasible that the same state is responsible for both the C’II, state A doubling and the intensity anomalies and it is possible that this could be the 2 ‘Z : state. We can examine the possible interaction with the 2 ‘Z: state more closely. The 2 states are separated by AE - 450 cm-’ and have B values of -0.5 cm-‘. For these states, it is unlikely that L is a well-defined quantum number but, if we assume that L- 1, then we can obtain an order of magnitude estimate for the A-doubling coefficient, q, as q _ 2B2(H1 L+lZ+) _ 2B2L(L + 1) _ 9 (6.1) AE AE AE which gives a value of q - 2.2 X 10m3cm-‘, compared to the experimental value of

1 ‘rip STATE

373

OF Lit

2.38 X 10m3cm-‘.

The agreement is much better than is warranted by the approximations used, but is nevertheless an indicator that the interaction is feasible and capable of producing the observed A doubling. From the experimental A-doubling constant, the matrix element connecting the two states can be calculated without making any assumptions about L. The A doubling is given by AT,, = qJ(J + 1) = F;

(6.2)

from which H = (qAE)“*[J(J+ l)]“* = l.O35[J(J+ l)]“*. At J= 29, the matrix element is -30.5 cm-’ which gives c* - 0.9955, 1 - cz - 0.0045. Thus the CIII, state at J = 29 is 99.55% C’II, and 0.45% 2 ‘2:. The mixing is thus very small but is sufficient to significantly affect the R/P-branch intensity ratio. The same method can now be used to estimate the transition moment ratio as was used for 2 ‘Z z-2 ‘Zg _ The interaction transfers intensity in the same way, and if we assume that the HGl-London factors for the R and P branches are approximately equal, we can write IR -= IP

c*r* + (1 - c*) + 2c( 1 - c2)“*r c*r* + (1 - c2) - 2c( 1 - c*)“‘r

’

(6.3)

where r is the ratio of the total dipole moments

Using the intensity ratio - 1.15, we find r - 1.2. At the moment, this is very speculative. It is quite possible that both upper and lower state perturbation may contribute to the intensity anomaly and also that interactions with other states may play a role. There is not sufficient information available to be able to perform a more rigorous analysis but the above treatment clearly demonstrates that only a small degree of mixing is required to cause noticeable intensity anomalies. 7. DISCUSSION

The rotational analysis of the C’II,-1 ‘II8 and 2 ‘Z :- 1 ‘I& fluorescence in 6Li2 has yielded 1 ‘I$ state molecular constants which are in very good agreement with ab initio predictions. Because of the interaction with the 2 ‘2; state, any global analysis of data involving the e levels of 1 ‘II, is of limited accuracy and dubious validity. The true molecular constants can only be obtained by fitting the S levels. The constants, obtained from the f levels and presented in Table III, are currently the best available for the 1 ‘IIg state. However, the present data set is very limited and, in order to improve the constants, it will be necessary to obtain complete bands, over a large range of J, for many vibrational transitions. An experiment similar to the double resonance experiment on ‘Li2 reported by Miller et al. (2) could yield this information if the resolution were improved. The RKR potential curves, even though obtained from data for a single J, appear to be of high quality. The dissociation energy and long-range expansion coefficient, CX, are the same whether obtained directly from the potential curve or from LeRoy

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fits to the upper vibrational levels. Our determination of the 1 ‘I& state dissociation energy has confirmed the previous value for the ground state dissociation energy and reduced its uncertainty by a factor of 5. The interaction between the 1 ‘I& and 2 ‘2: states has been examined in detail. The A doubling was observed to be irregular, both in magnitude and in sign, but the irregularities follow precisely the trends predicted by calculations of the interaction of a single 1 ‘I’& level with all levels of 2 ‘2; for both 6Li2 and ‘Liz. All situations in which the doubling appeared so large as to initially cast doubt on the correctness of the assignment were found to correspond to situations in which the nearest 2 ‘I;: level was very close to the 1 ‘I$ level and the interaction was much greater than that with the other 2 ‘Zi levels. The irregularities were large enough and numerous enough that it was not feasible to fit the doubling to a Dunham-type expansion in vibration or even, for vn > 3, to the usual expansion in J( J + 1) . The values of qn quoted in Table VI are not meant to imply proportionality to J(J + 1) but are a convenient means of comparing the doubling at different u and J. The numerous observed perturbations are all completely explained in terms of the 2’z+ - 1 ‘I& interaction for cases in which there is a close resonance between the twottates. Both the frequency shifts and the extra line to main line intensity ratios are consistent with the calculations. In situations where transitions from the 2 ‘I;: state to both interacting levels are observed, the intensity ratios are consistent with a perturbation treatment including intensity interference effects ( II ). In treating the stronger perturbations as an interaction between single, near resonant, 1 ‘I$ and 2 ’ 2: levels, we have omitted the contributions from all other vibrational levels of 2 ‘2;. However, in all these cases, the resonant interaction is so strong compared to all other interactions that the effect of the latter will be negligible. For weaker perturbations in which the resonance effect is dominant but not so strong, we have included all 2 ‘2: levels in the calculation. It is interesting to note that, in our earlier analysis of the 2 ‘Xi-A ‘2: transition (12, 13), the bands were assigned only up to 2, = 13 ( 6Li2) and 15 (‘Liz) in the 2 ‘2: state. Our present work has shown that the strong interactions with the 1 ‘III, state start at u = 14 for 6Li2 and 16 for ‘Lil. It is therefore probable that these perturbations, which were not understood at the time, may have prevented assignment of the higher vibrational levels. Reexamination of the earlier spectra, which are completely relaxed, may yield further valuable information on the interaction. The anomalous RIPintensity ratios in the C’II,-1 ‘I& progressions have tentatively been linked to the C’II, state A doubling and have been used to calculate transition moment ratios for the C’II,-1 ‘II, and 2 ‘Z L-1 ‘I’& transitions. It is felt that other interactions may also be contributing to the anomaly and that more data are needed before this phenomenon can be properly understood. In conclusion, we have provided the first spectroscopic information on the 1 ‘I& state of ‘jLiz and the first detailed study of the interaction between the 2 ‘2: and 1 ‘I$ states for any of the alkali dimers. There is an obvious need for more data to provide more precise molecular parameters, potential curves, and A-doubling parameters. We have highlighted some of the hazards of “global” fits to spectroscopic data in that, even though a global fit to the 1 ‘II, state reproduces the line frequencies reasonably well, it smooths over all the irregularities in A doubling that provide so much insight

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into the interaction with the 2 ‘2: state. We have shown that a careful investigation of the 1 ‘I& state provides information not only on spectroscopic constants but also on transition moments and on the properties of the 2 ‘I;: state. It is likely that the same will probably be true, to a lesser extent, for the other alkali dimers, Naz, KZ, etc., in which the 1 ‘II, state has been observed and treated globally. ACKNOWLEDGMENTS This work has been funded by grants from NATO and the Natural Sciences and Engineering Research Council of Canada. We thank Dr. A. J. Ross and S. Poulat for their assistance with the computations and Professor R. A. Bemheim for sending results of their work (2) prior to publication.

RECEIVED:

April

23, 1990 REFERENCES

I. C. LINTON, F. MARTIN, R. BACIS,AND J. VERGES, .I. Mol. Spectrosc. 137,235-241 ( 1989 ). 2. D. A. MILLER, L. P. GOLD, P. D. TRIPODI. AND R. A. BERNHEIM,J. Chem. Phys.. in press ( 1990). 3. G. ENNEN, CH. OTTINGER, K. K. VERMA, AND W. C. STWALLEY, J. Mol. Spectrosc. 89, 413-420 (1981). 4. I. SCHMIDT-MINK,W. MULLER, AND W. MEYER, Chem. Phvs. 90,263-285 ( 1985 ). 5. D. D. KONOWALOWAND J. L. FISH, Chem. Phys. 84,463-477 ( 1984). 6. J. M. HUTSON, .I. Phys. B. 14,851-857 (1981). 7. B. BARAKAT, R. BACIS, F. CARROT, S. CHURASSY, P. CROZET, AND F. MARTIN, Chem. Phys. 102, 215-227 (1986). 8. B. BUSSERYAND M. AUBERT-FRECON,Chem. Phys. Lett. 105,64-71 ( 1984). 9. (a) R. J. LEROY, in “Specialist Periodical Reports, Molecular Spectroscopy” (R. F. Barrow, D. A. Long, and D. J. Millen, Eds.), Vol. 1. pp. 113-176, The Chemical Society, London, 1973: (b) in “Semiclassical Methods in Molecular Scattering and Spectroscopy” (M. S. Child, Ed.), pp. 109-126. Reidel, Dordrecht, 1980: (c) J. Chem. Phys. 73,6003-60 12 ( 1980). 10. G. HERZBERG,“Molecular Spectra and Molecular Structure. I. Spectra of Diatomic Molecules.” Van Nostrand-Reinhold, Princeton, NJ, 1950. II. H. LEFEBVRE-BFUON AND R. W. FIELD,“Perturbations in the Spectra of Diatomic Molecules,” Academic Press, San Diego, 1986. 12. B. BARAKAT, R. BACIS, S. CHURASSY,R. W. FIELD, J. Ho, C. LINTON, S. MCDONALD, F. MARTIN. AND J. VERGES, J Mol. Spectrosc. 116, 271-285 ( 1986). 13. F. CARROT, R. BACIS,S. CHURASSY,J. Ho, C. LINTON. S. MCDONALD, F. MARTIN, AND J. VERGES, .I. Mol. Spectrosc. 119, 38-50 (1986).