Predissociation lifetimes of the b3Πu state of Li2 and the accidental predissociation of its A1Σu+ state

Volume

121, number

CHEMICAL

1.2

PREDISSOCIATION AND

THE

LIFETIMES

ACCIDENTAL

Bona SCHMIDT-MINK

and Wilfried Kaiseduurern.

Rcceiwd

UniuersirGr

1 November

LITTERS

OF THE b3111, STATE

PREDISSOCIATION

Fachhereich

Chemie,

PHYSICS

1985

OF Li,

OF ITS A’Z;

STATE

MEYER D - 6750

h~aiserduurern.

IVexr Germur~~

28 June 1985; in final rot-m 29 July 1985

&Cation4 coupling malrix elemenls between the b311, and n3X,’ states of Liz have hcen calculawd and b311, predissocieriun iifetimes are obtained using our previously reported accurak potemial curves. The Metimes are only about one twelfth or values derived previously from perturbed liretimes 01 the A’P; smte. The accidental predissockion of this srale is reinvestigated using theoretical spin-orbit coupling elements and good agreement is round with observed perturbed liktimes The spin-orbit coupling elements are only one third and one half of two earlier values determined rrom liretime mcasuremenw and by ab initia calculation. respectively. The discrepancies are due in part to incorrect defini&ns ol the coupling cons~3nts.

1. Introduction The b 3fIu state of Li2 has been the subject of several theoretical investigations and increasingly accurate potential curves have been reported [l-6]. It has been observed only quite recently by Engelke and Hage [7] by means of laser-induced fluorescence. The predissociation lifetimes, which are due to rotational coupling to the repulsive a 3Cf state, have been calculated by Uzer and Dalgamo [87. They also predicted accidental predissociation for various levels of the A ‘Z+ state resulting from its spin-orbit coupling to the b Y “u state. This has been investigated in more detail by Cooper et al. [9] using theoretical potentials and coupling matrix elements_ The calculated perturbed lifetimes of the A ‘Zz state were necessarily rather uncertain, however, because of their critical dependence on the precise matching of rovibrational levels of the two states. Evidence for this accidental predissociation has very recently been reported by Baumgartner et al. [lo] who performed accurate lifetime measurements for the A Zi state. About 40 rovibrational levels with significantly shortened lifetimes have been analysed in a multiparameter fitting procedure [ 1 l] based on the RovaEz-Budo model [12] _ This has produced -ather. precise data for the energy levels of the b 3fIu state for u >, 8. The resulting 0 009-2614/85/S Morth-Holland

03.30 0 Elsevier Science Publishers B.V. Phvsics Publishine Division)

spectroscopic constants derived from a Dunham analysis are in good agreement with our recent theoretical predictions [6] _ The information gained for the a 3Ci state is much less precise but is again in reasonable agreement with our potential curve. The effective electronic rotational coupling element determined by this procedure is in good agreement with the theoretical value OF Cooper et al. [9], both being slightly below the “pure precession” value [ 131, as is plausible. However, the empirical electronic spin-orbit coupling element is twice the theoretical value of ref. [9] _Even the latter is considerably larger than the coupling one would expect from simple arguments relating it to the observed spin-orbit splitting in the ‘P state of the Li atom. It appeared worthwhile, then, to recalculate the electronic coupling elements from our accurate wavefunctions and to reinvestigate the accidental predissociation of the A ‘Zi state. Since we found that the coupling elements were ill defined in refs. [9,11], a brief compilation of the pertinent formulas is given in section 2. The b 3fI, predissociation lifetimes are presented in section 3 and the accidental predissociation is discussed in section 4.

49

Volume 121. number 1,2

CHEMICAL PHYSICS LEJTERS

2. Coupling elements and model for accidental predksociation

IJ, 3/i (F;$2_N))

With a spin,orbit coupling constant of only 0.224 cm-l [9] for the Li atom, Liz certainly qualifies for Hund’s case (b) coupling_ The calculation of electronic energies and coupling matrix elements is always performed in a case (a) basis, however. These two basis sets are related by the transformation [ 131 lJSN,h)=

c

z

lJS,A~~n>C(Js2,S-Z;NA),

(1)

where J, a, S, Z, and A are the usual total and projection quantum numbers for total, spin and orbital angular momenta, respectively, and the C are the Clebsch-Gordan coefficients for coupling J and S to N= J-S [13]. The A ’ Ez = 1JO, 000) state is coupled to the b 311u state only by the spin-orbit interaction which obeys the selection rule An = 0. The b 311u state in turn is coupled to the a 3Zt state by the rotational interaction Hrot = -(1/2&)@+~-

+k-_E+)

NL

,

(2)

with non-vanishing matrix elements (JSN,

AIN’L’l

= [N(N+

JSN, A f 1)

l)-A(Ai

l)]ln

X (JSN,AILrIJSN,A

f 1) _

(3)

The case (b) matrix element for LT is easily related to pure electronic case (a) elements since the latter are independent of ZZso that the unitarity of the Clebsch-Gordan coefficients leads simply to (JSN, AIL’1 JSN, A 2 1) = (S, ASILT IS, (A + 1)s)

1 November 1985

.

A) 7 IJlN,

= 2+lJlN,

AfO.

-A>),

(4b)

Case (b) Z states IJlN, 0) have parity e for the F2 (N = J) component and parity f for the F,,, (N = J f 1) components. From AN = 0 for the interaction (2) follows that only the 311u components Flf >3 and F; are coupled to 32z and can predissociate. All nonvanishing case (b) rotational coupling elements result to (3l-le,flfirotI3x+e,f u u = [2N(N+

1

1)]“2(1,

lllL+ll,

01)/2@‘..

(5)

Since the A ‘Zi state is of e parity, only e components of 311u and 3Zi are involved in the accidental predissociation. The e-state coupling matrix elements obtained from (1) to (5) are collected in table 1. The eiectronic parts of the coupling elements, t and y, are defused to match the corresponding effective parameters of ref. [1 11. However, in this reference Hba/y is only half of what would result from table 1, a mistake which is partly responsible for the 311u predissociation lifetimes derived there being too long and the spin-orbit coupling parameter being too large_ As dis cussed later, the rate of accidental predissociation is essentially =[IHAbH,,al/(,!?A -&,)]‘. The numerator is found to be independent of the choice for the intermediate state b as b 3lI ,,,, or b 311u (F,). The former choice was made in refs. [8,9] although Eb is not well defined in this case. Note that their b 3IIu,-, predissociation lifetimes therefore have to be scaled by a factor l/2 when being compared with ours or those of ref. [8], which refer to case (b). The asymptotic forms of the wavefunctions are *(l~~)=f(ls,P~]

-

IsaPE] + Is&l

*(3”;,,)

I+ b,p; I - ba$

-

&,p~I),

(64

Both couplings preserve e/f parity. Case (a) states with proper parity have the form

IJ, 3A;$

= [2(1 + 6,,&&]

X (I Jl, AC!ZL) 7 IJl,

-A - 2 - SL)) ,

I - Is&

1) , (6b)

-1’2

*pq (44

where the Condon-Shortley phase convention for orbitals, 0~1 h) = (-1)” l--h), is implied [ 13]_ Trensformation (1) leads for case (b) to 50

= +(I?,$ = IS& I ,

(6~)

where, e.g., p,” denotes the 2pm atomic orbital at the Li atom a with spin /I_ If the spin-orbit interaction is approximated as a sum of one-electron terms for the valence electrons i,

Volume 121, number 1,2

CHEMICAL

PHYSICS

LEITERS

1 November

1985

Table 1 Coupkg

matrix elements of the states A ‘x:

(a) 32 = zz=l

0

n=2

a)

E=2(0,O 01Hs~ll.l al CoUPbrIg (r/2fiR2)eb

and a 3~he

with b %I:

for 05e

(al and case I.%) coupling, respectively

(AWsolb)

tilH,,tla,

E

-[J(J+ l)]~Y/zpR* -2--y/2pR2 [/(J+ 1) - 2]-Y/&R2

0 0

@) F1

0

F2 F3

0

-1)

[2J(J+

electronic spin-orbit coupling, for case (a) states IS, AZ); 21 of ref. [ 121.

corresponds t0

it is easily verified that the spin-orbit coupling constant &= (A ’ Ef IH,,I b 311Eo>, approaches asymptotically the value 2-1P-Zeff(2plr-3 12p), i.e. the spinorbit coupling constant of the Li atom divided by 2ln _From the short-range nature of the spin-orbit operator we expect the coupling constant to vary only weakly with ‘he internuclear separation, as is normally observed for diatomic molecules in the absence of avoided crossings. Table 2 presents t(R) as calculated from accurate wavefunctions but using only the approximate spin-orbit operator of eq. (7). Zefr has been adjusted to match the empirical splitting in the separated atom limit. This yields Zefr = 0569. A value smaller than 1 can be understood from the strong sh>e!ding by the core-valence exchange terms as is wkll known for the lighter alkali atoms [ 14,15]_ Our results for g(R) are in qualitative agreement with the calculations of Cooper et al. [9] as far as the shape of the curve is concerned but there is a deviation by a factor of ==2. The electronic part of the rotational coupling element, y =
Let usnow the accidental lifetimes have KovaEz-Budo tinuum-bound to be constant the size of the

a)

l)]‘n

Y = (1,-l

lIL_II.0

1) electronic rotation-

briefly discuss a theoretical model for predissociation. Usually the perturbed been interpreted in terms of tbe model [12] _ In this theory the concoupling matrix elements are assumed over an energy range comparable to coupling. This is normally, and certain-

Table 2 Calculated spin-orbit

and rotational coupling matrix elements t (cm-l)

Y

4.0 5.0 6.0 7.0 8.0 10.0

0.1423 0.1203 0.1090 0.1053 0.1078 0.1237

1.1797 12280 12802 1.3451 1.4347 1.6998

12.0 15.0

0.1414 0.1542

2.0123 2.4905

20.0 30.0

0.1578 0.1584

3.2981 49770

R (00)

6.80 484

=) a)

0.1055 11102.

Cooper et al. [9] 4.0 0.364 5.0 0266 6.0 0.186

1241 1268 1287

Preussand e6.8 a) ~49 a)

120

a) Crossing

Baumgartner [ll] 0.332 f 0.03 f 0.12

points R,. 51

ly in our case where only small interactions occur, sufficiently well satisfied. But this model has further been criticized on the grounds that it would not account for the interference effects between resonances with overlapping width, as obtained, e.g., in the configuration interaction theory of Fano [16]. In view of a predissociation width of up to 4 cm-l for those b 311U levels which are relevant for the miKing with the A ‘Zi state (LJ> 8) - see section 3 - such overlapping can be encountered in a few cases of close degeneracy. However, an analysis of Fano’s theory, similar to that given by Mies and Krauss for several special cases [ 171, reveals that the results for accidental predissociation are identical to the treatment by Kova& and Budo provided that proper account is made for the nature of the resonant state formed by the initial absorption_ In short, we have to consider the interaction between two states which are bound with respect to fin but each of which is coupled by an additional interaction to a continuum, i.e. the radiation field for A ‘Ci and the dissociation continuum of the repulsive a 3Xt for b 311U_As is well known [18], the continua can be eliminated from explicit consideration in a time-independent treatment by introducing optical potentials. They reduce to imaginary constants $I’ if the coupling to the continuum does not change appreciably over an energy interval comparable to r. Of course, RP1 is the lifetime of the state interacting with a continuum. One has then just to solve the eigenvalue equation E,

- $ir,

- A Veip

ye-ia

%

-

where E, r, V, c and s may be assumed to be real and s*+c*=l.WithAE=E,-EE,.AI’=I’,-I’,andA= AE’ + 4V2 + Ar2/4, the two solutions are given by

[=I 2s’~’

it=+aE(S - i/2(r,c2

+ (A2 - 4VzAl+)lr-]

-s?)+ SC(~V~+

r, s2),

,

(9)

sVAr2)-'~

52

*k,(c2e-i*ir/fi

+ *nsc(e-ihiflfi

+ &-iAzr/fi) - e--ixzr/fi)

.

(11)

The time dependence

of the fluorescence intensity is proportional to the absolute square of the coefficient of qz, i.e. with ri = -2 lm($)

I(r) (I c4e-rrr/n

+ s4e-

rzr/fi

+ 2sZc2e-(rl+rzlf/2fi

cos[Re(hl

- h+/fi]

.(12)

In most cases one has AE2 + iAr2 * 4V2 and therefore cz. 5 s2 so that an effective radiative width of

r=rsc2+r,G, =

(13)

rc +(I?" - r&Vz/(AE’

+

4AP

+4v')

(14)

is observed experimentally as the perturbed lifetime of the radiatively decaying state. One notes that this is valid even for AE --t 0 as long as iAr2 * 4V2, in contrast to the criticism of the KovaEz-Budo model in ref_ [IS] _ Only for 4V2 S AE + $Ar2 does one lind a decay with exp [-$(I’, + r,)t/fi] but modulated by 1 + cos(2Vt). In our case V is smaller than 0.15 cm-l, so that the latter situation can hardly occur.

3. Predk.sociation

lifetime

of the b %Iu state

Our theoretical potential curves have been documented and discussed in detail elsewhere [6] _From the singlet states that could be compared with experiment we may infer an accuracy of better than 1 cm-l for w,, better than 0.002 A for R, and 20-60 cm-l for De. The latter is still not sufficient for a quantitative treatment of the accidental predissociation which requires a precision in the relative energy positions in the order of 1 cm-l. We have therefore modeled the ab initio potential of the b state according to -

Vcxp(m)

a= 1.00271, The determination

(10)

where c-, = s, and s2 = -cl _A state which is formed at f = 0 as a pure C state then develops as (c c cl, s E Sl)

$-f(r) =

F(R)

= 4v2/[~

1 November 1985

CHEMICAL PHYSICS LEITERS

Volume 121. number 1.2

= a[ Vth(bR b = 0.99835

+ c) -

,

Vth(-)]

c = 0.0152

, . (15)

of the parameters a, b and c is discussed in section 4. The modifications are obviously very small. The resulting spectroscopic constants are compared in table 3 with our purely theoretical ones and the two sets of empirical constants: The constants derived from our modified potential are in vir-

Volume 121. number 1.2 Table 3 Spectroscopic Ref.

r111 1191 [71 [61 I5 I

CHEMICAL PHySICS

constants for the b 3~u state of ‘Liz

1 November

LEITERS

a)

Method

T,

4

=e

we

We%

DC

CI seal

11239.7 11240.1 11248(U)

0.7161 b, 0.7173 0.7 179(30) 0.7160) 0.7137

0.00606 b) 0.00597 0.00594(30) 0.006(2) 0.00568

345.42 b, 344.63 345.97(30) 345.6Q) 345.88 354.1

1904 b) 1.816 2.025(10) l-890(4) 1.980 252

12181.1(6) 12180.7(5) 12172(U) 12145@00) 12148 12443

exp. exp. exp. CPP + CI MP+c1

11256

1985

R,

=B~) =) =)

2591 2588 2587(5) 259 2595 2555

a) Energies in cm- ‘, distances in A; the number in paentheses indicate the unclertaintyin the last digit; hlP = model potential, CPP = core polarization potential b, Dunham analysis with 25 vibrational levels for J= 1. ‘) D, values are based on the D, for the ground state of ref. 1201. d)From the dissociation energy of 93525(6) cm-’ for the A ‘Xz state [20,21] follows: Te(3ll,) - T,(‘Iz~ = 2828.6 an-l_

tually perfect

agreement

with those

of Engelke

and

and Rai et al. [19]. The third empirical set, derived by Preuss and Baumgartner [ 111 from-fitting the accidentally predissociated lifetimes of the A state, is not strictly comparable because it refers to a quite different range of rovibrational levels. Agreement is nevertheless rather good. From the electronic rotational coupling elements given in table 2 and the factors of eq. (S), the predissociation width P,, = 27rH& is obtained as Hage

[7]

r N” = 2X2N(N

+ 1)

(16)

The continuum nuclear wavefunction has to be “energy normalized”, i.e. it has the asymptotic form xNk -+ (2p/fi27fk)lr-

sin(kR -

$-hr + q)

(17)

,

where k2 = Zp/fi2 (EN”(b) - Eel(a, R = -))_ The matrix elements (xNv(b)lr(R)/R21XNk(a)) are shown in fig. 1 for selected values of N. For those levels which are involved in the accidental predissociation of the A ‘Ci, the lifetimes are included in table 4. Our lifetimes differ considerably from both previous determinations. The theoretical lifetimes of Uzer and Dalgarno [S] were based on rather poor potentials so that the extrema of their P values are shifted to higher u by l-2 quanta. (As mentioned, their lifetimes refer to the case (a) b 3fIuo component and should be divided by two when being compared with our Fi lifetimes.) The selected lifetimes derived by Preuss and Baumgartner [ 111 follow closely the trend of ours but they are generally larger by a factx of z-12. A factor of 4 can be traced to the erroneous definition of the rotational coupling coefficient, the reason for the remaining factor of z3 is not clear to us.

4. Perturbed lifetimes of the A ‘Z z state I

0

1

5

10

15

20

25

Fig. 1. Calculated mati elements&f= W~17(R)/R21~~) VCKUS u for severalNnlues. The predissociation liIctime.s of the b %,, state can be obtaiued by ~b (ps) = 78.7 13 K2 N(N+ 1). The numerical values are available upon request. -N=l,_-N=23 ,___ N=32,-.---_N=39, ---N=45. . ..N=50.

Using formulas (9) and (13) from

the perturbed life-

times

are calculated

the electronic

ment

given in table 2 and the predissociation

coupling

ele-

lifetimes

discussed above. All the pertinent quantities are collected in table 4 for the rovibrational levels investigated in ref. [ 1 l] .‘Since x(R) and y(R)/R2 are rather 53

6Li2

I 2 3 4 4 2 2 3 3 3 4 5 6 7 B 9 6 6 7 8 9 10 11 12 13 14 13 14

a 9 10 ,I1 11 8 8 9 9 9 10 I1 12 13 14 15 II 11 12 13 14 15 16 17 18 19 i7 18

34 -I,4 29 0.8 23 2.0 14 -32 1s 2.7 50 -12,o 51 5.9 46 -17,o 47 -0,2 40 16.8 43 -2.4 39 -0.3 35 6.2 30 5,s 24 3.1 16 -0.7 53, -7,B 54 10.9 50 0.3 46 -62 43 6.7 38 -7.1 34 -0.9 29 -15 23 -3.4 16 0.3 45 -3.6 41 -65

0.0196 3.60 18.4 10,3 0.0272 16.05 18.4 8.1 0.0309 14580,OO 18,4 18.4 0.02,97 62800 18.4 1709 0.0297 58.35 18.4 17.7 0.0319 1.58 18.9 17.4 0.0319 1.6.1 HI,9 14.3 0.0287 1.73 18.9 18.3 0.0288 1.64 18.9 4.1 0.0289 156 18.9 18.3 0.019B 3.19 18.9 1389 0.0075 16.19 ItI,9 12.1 0.0050 638.00 18.9 18.9 0.0150 16,18 18,8 18.7 0.0207 10.08 La,8 17.4 18829 18.8 988 0.0213 O.b207 157 19.3 17.8 0.0206 1.46 19.3 18.5 0.0219 3.19 19.3 4.1 0.0174 22813 192 19.1 0.0092 209.00 192 19.2 0,0009 958 19.1 19.1 0,0097 568 19.1 14.6 6.29 1981 145 0.0159 12.62 19,O 18.3 0.0181 OS0163 53.66 1900 9.5 0.0079 8.03 19,4 19,l 0.0003 4.37 19,3 19.3

12.6 6Li’Li 2.0 18.4 17.7 17.8 16.9 14.0 18.0 3.0 18.0 15.6 ‘LIZ 169 1.9,ll 18,8 172 9.0 17.8 17,9 6.0 19.0 19.0 19.0 14,3 14.8 19.0 3.0 18.9 18,9

17 14 15 16 17 ‘18 19 20 20 8 9 10 9 IO 10 11 12 13 14 12 13 14 15 16 17 18 19 IS

11 9 10 11 12 13 14 15 16 0 1 2 2 3 3 4 5 6 7 6 7 B 9 10 11 12 13 10

31 24 16 46 41 42 37 32 26 18 49 45 41 37 32 26 19 7 52

11 49 45 41 37 32 27 20 40 4.1 -22 0.4 4.4 -6.0 7.3 1.8 2.7 1.5 -2.1 -5.1 -6.8 -4.1 2.3 1.9 -0.6 0.7 ,O.l -2.7

-1.6 1.3 -2.7 -2.8 1.0 -2.3 15 -2.7 -0.1

18.8 19.3 19.2 18.0 9.9 16.3 17.1 19.0 583 18.1 18.1 11.3 14,6 18.0 18.1 18.7 18.0 18.2 18.6 19.1 19.1 18.2 16.1 17.5 1y.9 l&l,2 18.2 19.4

18,9 19.3 19.3 19.2 19.2 19.1 19.1 19.1 19.3 18.2 18,2 18.2 18.8 18.7 18,8 18.7 18.7 IS,? lE.7 1902 19,l 19.1 19.1 19.0 19.0 19,o l&9 19.4

87.39 0.0050 324.40 0.0050 17.86 0,013o 5.92 0.0174 4.71 OS0177 6.40 0.0141 14,08 0.0075 88,85 0.0150 4.39 0.0070 9.69 0.0147 549.40 080224 95.29 080291 2.72 0,0306 11.98 0.0306 9.58 0.0264 98430.00 0,0176 20.85 0.0063 10.98 0.0053 17.93 0.0126 16.27 0.0198 190,60 0.0215 11.00 0.0180 5.97 6.39 0.0105 0.0012 12.68 56.43 0.0077 0.0144 6218.00 0.0093 84.54

o.oose

18.2 l&2 14.2 1287 18.0 18.0’ HI,6 18.6 18.6 18.6 18.9 l&9 . 17.9 is.3 16.6 1’9.0 17.8 1.9.0 19.0

18,9 19,o 19,O 17.5 9.3 15,s 14.7 19.0 786

Table 4 Calcuhted lihtimcs OCtllc A ‘Cz slate lcvcls pcrturbcd by accidentalprcdissociation, matrixclcrncntsandcncrgics.Tb(ps): cahht~d prcdissociatlon lifcthncs for lhc b3flu state,TX(IX): calcukd radiativeliktimcsol theA ‘c: stntc,TL (ns) nndT! (IX): calculatednndmcasurcd[ 111lifctimcs,rcspcctivcly,of the A I Ci state.CncrgicsandHAb OKsivcnin wovcnumbcrs

.Volume 121, number 1.2

‘1 November 1985.

CHEMICAL PHYSICS LETTERS

flat functions, the strong variations of the coupling elements HAb ‘and 311U predissociation lifetimes refleet mainly the variations of the corresponding Franck-Condon factors. Effective electronic coupling elements defied as (xAI~(R)&,)/(xAIXb) and (xblr(R)/R2 Ix,)l(xblxa) vary only slightly around the values E(Rtb) and y(REa)/(Rta)‘, respectively, except for cases with very small Franck-Condon factors_ More specifically, we find the following inequahties to be valid: l(~~lElx~)/~(R~~) (xAlxb)<(o.6 l(XblY/R2

- (xAlxb)l ;

lX,)(Rb,“)‘/Y(R~“)

(xblxa)65

au -

~0-002

, Wa)

- (xb Ix,> l G 0.06 au , (18b)

(Note that (xb Ix=,1-Z has the dimension of an enerW-1 It is therefore justified

to compare effective coupling elements derived from experiment with coupling elements calculated at the crossing point. The unperturbed lifetimes T: of table 4 are those calculated earlier [6] but scaled down by 1.5% in order to improve the overall agreement with the observed unperturbed Lifetimes [IO] _As already mentioned, the very sensitive dependence of the perturbed lifetimes on the small energy differences requires an adjustment of the purely theoretical 3fIU potential_ The three scaling parameters of eq. (15) are chosen to produce an optimal matching of calculated energies with the two rather accurately known F, energy levels 11 l] _ The latter have been determined from the observation of increased lifetimes for two particular A ‘Zi levels. This increase is due to significant mixing with b 3fIU (F,) levels which can not predissociate. From the size of the spin-orbit coupling it is clear that the A and b state levels should not be separated by more than 0.2 cm-rFurther input for the scaling parameters has been obtained from some perturbed lifetimes by solving cqs. (9) and (13) for A. With no other adjustable parameters, the observed perturbed lifetimes are nicely reproduced_ The mean deviation is in the order of 1 ns. Significantly larger deviations result for the 3fIu levels u,J=11,39andv,J=19,160f6Li2andv,J=25, 24 of ‘I_$, for which AE < 0.3 cm-l. Therefore the lifetime responds very sensitively to small errors (e.1 55

Volume 121, number 1.2

CHEMICAL PHYSICS LETTERS

cm-l) in the 311u energies. Apart from this, there remains a clear discrepancy in the measured lifetime only for the u, J = 9,29 level of 6Li2. Its energy would have to be changed by 0.5 cm-1 in order to match the reported lifetime but this appears to be ticompatible with the energy of the near-by u, J= 9,32 F, level. The tit of the lifetimes derived by Preuss and Baumgartner is slightly better but involved many more adjustable parameters. In particular, they used Dunham coefficients for the 311U state as unconstrained parameters and it is not guaranteed that they are consistent with a potential. As noted by these authors, their Dunham coefficients yield an energy for the precisely known u = 9, J = 50 F1 level of 6Li2 which is off by 1.8 cm-l _Our b state energies differ from theirs also by up to 3 cm-l for large J values. In this range r, reaches values of l-4 cm-l so that +I?: becomes comparable to LIE’ and has a significant effect on the lifetimes. This effect is missed in ref. [I 11 since r, is too small by one order of magnitude. In their case iAr2 is always negligible and their fit is effectively based on the approximation

X [~2/2~R~
(19)

This shows thaf an independent determination of,,& and -y, from the reduced lifetimes was in fact hardly possible. We note that the treatment of the accidental predissociation by Cooper et al. [9] was also based on the approximate eq. (19). During the preparation of this manuscript we were informed by Xie and Field [22] that they have determined the effective spin-orbit coupling constant ,$ for the vA = 2,N=33 andUA=9,N=20Fl kVdS from sub-Doppler spectra and the steady-state he-ticlineshape model. Their result of g = 0.114 * 0.006 cm-l is in very good agreement with our calculated value of 0.106 cm-l. The latter may well be off by about 10% due to the use of the approximate oneelectron spin-orbit Hamiltonian.

56

,l November 1985

References

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