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The photodissociation of Li2
Chemical Physics Sl(1980) 271-277 0 North-Holland Publishing Company
THE PHOTODISSOCIATION
OF Liz
T. UZER * and A. DALGARNO Center for Astrophysics,
Harvard-Smirhsonian
Cambridge. Massachusetts
02138. USA
Received 28 hfarch 1980
The absorption of photons by Liz from the X ‘xl state to the A ‘ZZ~and B %I,, states is considered and the mechanisms that lead to dissociation are studied quantitatively. Calculations are reported on the direct predissociation of the b 3t?,, state. The s&~ificance of accidental predissociation of the A ‘2: state is discussed and a quvltal theory of the pro-
cess is presented_
1_ Iniroduction
strength for a transition from the vibrational level u”of the Xr Zg state to the vibrational level u’ of the excited state is given by
The alkali metal dirners are useful candidates for the study of isotope separation and for the construction of visible lasers. Several calculations have been carried out of the potential energy curves of Li2 [i-5] and of the dipole moments connecting the ground X *Zi state with the excited A *Xi [6] and B ‘II, [73 states. Measurements have been reported of the lifetimes of specific vibration-rotation levels of the A’C: [S] and B’II, [9,10] states and spectroscopic analyses have been performed [l l--14]_ A tabulation of the wavelengths of lines of the B ‘II,X r Cg system that are accessib!e to an argon ion laser has been presented by Lightman and Mather [Is] _ We explore here the absorption of photons by Liz in the A *ZE-X’L’: and B ‘fI,-X’ZZ systems and we investigate quantitativeIy the mechanisms that lead to dissociated molecules. Accidental predissociation is significant and we present a quanta1 theory of the process.
2. Absorption
oscillator
dipole moment,
distance
- E’,,,) I(~~ID(R)Ixu”)~~,
where g is unity for a C-Z transition and two for a II-Z transition, E,. - Eg is the transition energy in atomic units and w and w are respectively the final and initial vibrational wavefunctions. In writing the formula for fu,*,c* we have ignored the dependence on the rotational quantum numbers_ We have verified that for low values of J”, the absorption oscillator strength is not sensitive to the initial rotational level when we sum over all the possible branches_ The transition dipole moments D(R) have been calculated [6,7] using the model potential method [S] _For the A ‘X:-X ‘xi transition, the calculated transition dipole moment leads to radiative lifetimes [67 in harmony with measurements [S] _For the B ‘l&-X ‘Ei transition, the calculated transition dipole moment leads to radiative lifetimes [7] about 5% larger than the measured values [9,10] _ We have slightly modified the calculated values ofD(R) and adopted the values listed in table 1. Baumgartner [9] and Renn [lOI have derived lifetimes of vibration-rotation levels (u’, J’) of the B ‘ff, state from studies of the fluorescent response, of Liz to an argon ion laser, which are in close agreement with our theoretical values, presented in table 2. Similar agreement is a!so found between theoretical radiative lifetimes and the measured lifetimes of levels
strengths
If R is the internuclear transition
= $g(E,, fu”,“’
and D(R) is the oscillator
the absorption
* Also Department of Chemistry, Harvard University. Cambridge, &achusett.s, USA. New at Department ciThec-
retical Chemistry, University of Oxford, 1 South Parks Road, Oxford OX1 3TG, UK.
271
272
T. iIzer,A.Dalgamo~The
TabIe 1
Modifiedvaluesof the dipole momentD(R) for the I3 *II,X 1X; transition of Liz, in au R(Qo)
3.0 4.0 5.0 6-O 7.0 8.0 10.5
12.0 16.0
D(R)
2.19 2.63 21.5 2.86 297 3.0s 3.20 3.23 3.27
photodissoctitionofLi+ Table 3 Emission band intensities for B ‘II&J’) -X tions of ‘Liz Band (u’, u”)
Measured
O,O 0,1 1,O 1,2 290
1.0 1.165 1.12 0.485 0.345 0.177
1 Z@“)
transi-
Calculated
[I61
2,1
0.240
2,2
1.0 l-156 1.10 0.485 0.353 0.172
0.242
with vibrational quantum numbers II’ from 0 to 12
absorption into the continuum is negligible_ We may
[lOIThe agreement between the calculated radiative Iifetimes and the measured decay lifetimes excludes any possibIe significant contribution to the decay of the B ‘II,, state by predissociation. Another test of the adopted dipole moment for the B ’ II,--X ‘Zg transition is provided by the data of Wu et al_ [16] who studied the fluorescence spectrum of %iz generated by white-light excitation of a ‘L& nozzle beam. The vibrational state populations have been determined by \Vu et al. [ i6] _The theoretical relative intensities of the emission bands (u’, I/‘) corresponding to the experimental populations are compared with observations in table 3_ The calculated absorption oscillator strengths fomv* of the A ‘C:-X ’ Eg transition are listed in table 4. Ah the absorption oscillator strength is taken up in transitions into discrete vibrational levels and
demonstrate this conclusion more explicitly by introducing the Franck-Condon factors
Table 2 Calculated lifetimes of vibration-rotation B 1 II, state of Liz in lo-’ s
6Li2
7
-
b2
6Li7Li
4”“,“’
=
ltkyl
xgw .
We have calculated the sum of qo,“* over ail the discrete vibrational levels u’ and find that it equals
unity to within IO-’ _ For an initial vibrational level u” = 20, we find that the sum of q 20,v*is 0.8648 so that direct photodissociation of highIy excited vibrational levels is not negligible. Tire absorption oscillator strengths f20,vs are presented in table 4. Table 5 lists the absorption oscillator strengths fo,,* for the B ‘II,-X’E: transition. The sum of the Franck-Condon factors for the discrete vibrational IeveIs is again unity to within 10e5 and direct photodissociation is negligible_ For absorption from higher vibrational levels, photodissociation may occur, the sum of 4”‘: upover u’ for u” = 10 being 0.5858. The oscillator strengthsfro,Un are listed in table 5_
Ievels u’, .I’ of the
V’,J’
c (10-9 s)
4. II 9.31 1.4
8.07 8.68 7.85
.9,4 4.24 2,31
8.43 8.09 8.013
%9
8.13
3. Spontaneous radiative dissociation Although for low-lying vibrational levels direct photodissociation is very slow, dissociation can also occur by absorption into discrete vibrational levels of the A ‘ZE and B ‘II, states followed by spontaneous radiative decay into the viirational continuum of the ground electronic state [17] _In practice emission occurs overwhelmingly into the discrete levels of the X ‘Zg’ state. Our calculations of the sum of the
T_ Uzer, A. Daigamo / The photodissociationbf Liz
273
Table4 Absorptionoscillatorstrengthsfrom the u” = 0 and d’ = 20 vIbrationallevelsof the X ‘Zi state to levelsU’of the A 'xz state of ‘Liz 0’
fo,;
II’
fzo,;
u'
f-20,“’
u’
f20,u
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
0.0254 0.0643 0.0891 0.0898 0.0737 0.0524 0.0335 0.0198 0_0109 0.0058 0.0029 0.0014 0.0007 0.0003 0.0002 0.0001
9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
0.0000 0.0023 0.0429 0.1389 0.0644 0.0122 0.0214 0.0198 0.0017 0.0188 0.0086 0.0004 0.0115 0.0127 0.0023 0.0010
2.5 26 21 28 29 30 31 32 33 34 3.5 36 37 38 39 40
0.0071 O-0094 0.0060 0.0014 0.0001 0.0021 0.0051 0.0066 0.0061 0.0044 O-0023 0.0008 0.0000 0.0002 0.0010 0.0019
41 42 43 44 4.5 46 47 48 49 50 51 52 53 54 55 56
0.0028 0.0034 0.0036 O-0034 0.0029 0_0030 0.0029 0.0023 0.0019 0.0017 0.0014 0.0012 0.0011 0.0009 0.0007 0.0006
Franck-Condon factors for discrete transitions originating in the u’ = 0 and u’ = 15 vibrational levels of the A ‘Zi state yield unity to within lo-‘. For v’ = 25, the sum is 0934 so some spontaneous radiative dissociation of the highest vibrational levels occurs. However, the probability of populating them by absorption is small. Similar results hold true for the B ‘II,, state. Table5 Absorptiontiscillatorstrengthsfrom the v” = 0 and U”= 10 vibrationallevelsof the X 1C; state to levelsv’ of the B ‘II, state of ‘Liz v-
d
f0,“’
0
0.3055
0
1 2 3 4 5 6 7 8 9
0.3192 0.1914 0.0871 0.0338 0.0119 0.0039 0.0012 0.0004 0.0001
1 2 3 4
fl0.U’
2 7 8 9 10 11 12 13 14
0.0000 0.0000 0.0001
0.0029 0.0365 0.1575 0.1385 0.0074 0.0858 0.0298 0.0131 0.05 86 0.0324 0.0015 0.0073
4_ Predissociation The potential energy curves of the A ‘xc, a ‘Zz and b 31&, states of Li,, shown schematicaliy in fig- 1, suggest another mechanism by which E2 can be dissociated by visible wavelength photons. Calculations [3,5] demonstrate that the A ‘2: state crosses the b 311Ustate. The b ‘IIn state cannot decay by an allowed radiative transition but it crosses the repulsive a “2: state which leads to dissociation into the ground state atoms. The phenomenon is known as accidental predissociation. The rCc rotational levels have e-parity and the 32g levels have f-parity so that only the 3Z: state participates iu the. predissociation of the A ‘Zi state. The A’ZZ, b 311Uand a3Zg states are coupled by the sum H’ of the spin-orbit interactions V,, and the interactions V, between the nuclear rotational momentum and the electron orbital and spin momenta- For V,, we may write v, = @z12/.LR2)[LTs-
+ Lx+]
- (fi2 /2/.&‘) [J*L- + J-L*] - @*~~/LR*)[J*S-
+ J-S+] ,
where p is the reduced mass of Li2, L’ and S* are the raisiig or lowering operators for the electronic orbital angular momentum S, respectively, in the body frame
T. Uzer. A. Dalgatno /The pitorodissociation of Liz
274
dissociation of the b ‘llg state through its interaction with the a 32: state. 4.1. Tke predkociation
of the b ‘II: state
The matrix element coupling the discrete vibration-rotation levels (u, J) of the b ‘llz state to the continuum levels of the a ‘Zz state are given by [ 181 i~sn~lH’132;)12 =J(J + l)P, 0 = l(fi2/2~2~b
31111L’l[a3C))Iz _
To estimate 0, we adopt the hypothesis of pure precession (CL ref. [19]) and write the vibrational coupling matrix element, which connects the initiaI discrete nuclear wavefunction x&R) to the fiial continuum nuclear wavefunction xx_=~(R),in the form Pt* = l(xuJltr2/2”‘lrR’Ixx_‘J)1*_ I
1
I
I
B
LO
1 6
5
I 12
Rta.u.1 Fig. 1. Potentti energy curve* far low-lying states of 7Li2_ adapted from ref. [ 3 ] _
and J’ is the raising or lowering operator for rotationai angular momentum J_ The A ‘Zz state mixes only with the 3FId component of the b 3flU state and we consider first the preTable 6 Predissociation -
If XuJ is normalised to unity and &lJ to the asymptotic form &*J
-
(Z,U/ZT%~E’)“~ sin(k’R - &Tlr + Q),
where (k*)l = &rE’/h*, E’ is the energy of relative motion of the dissociated atoms and VJ is the elastic phase shift, the predissociation lifetime, T,,J, is given by
lifetimes of vibration-rotation levels of the b 3ni state of 7Li2 in units of 10-l ’ s -___
J
u=Q
LJ=1
V=2
1;=3
v=4
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
274 91 46 28 19 13 10 79 63 5.2 4.4 3.8 3.3 2.9 2.6
154 50 2.5 15 IO 7.3 5.5 4.3 3.4 2.8 2.4 2-O 1.7 1.5 1.3
162 54 27 16 11 7.7 5.8 4.5 3.6 2.9 2-4 2-O 1.7 15 1.3
311 104 52 31 20 14 11 8.3 6.5 5.3
1735 571 281 165 108 7.5 54 40 31 24 19 16 13 IO 85
43
3.6 3.0 2.6 2.2
275
T. User. A. DcImrno / The photodissociation of Li2
ducing three projection operators PI, P2 and Ps such that PI projects onto the final state, here the a 3X: state, P2 onto the intermediate state, here the b 311u. state and P3 onto the initial state, here the A ‘2: state. We az.ume that no other states are significant so that PI + P2 + P3 = 1_ No direct coupling exists between the initial and fmal states so that P&PI 4 P,HP3 = 0, where His the hamiltonian of the system. Then if * is the wavefunction satisfying (H 2 E)9(r, R) = 0, it follows that P,(.-
E)PlQ
= -PIhiP**,
Pz (H - E) P2 \k = -PgYPl * - P&P3’F P,(H-
EJP3S = -P3HP~*
,
_
Let @i(r, R) be the solutions of the correspouding homogeneous equations P,(H - E)P,$ RLo.u.)
Fig_ 2. The potential energy curves of the b3n, and a “Xi states of’ 7Li2, adapted.from [3], and the vibrational levels of the b 3~U state.
= 0.
Formal solutions for Pi\k may be obtained using the Green’s functions Gi= {P&E-H+~c)P~}-‘. Then
We have calculated T~J for several vibrationrotation levels of the b “II: state. The discrete and continuum vibrational eigenfimctions were determined by numerical integration of the nuclear radial wave equations using for the interaction potentials of the b 311u and a 32z states the model potential calculations of Watson et al. [S] _ The resulting lifetimes are reproduced in table 6. Fig. 2 illustrates the intersecting potential energy curves and depicts the locations of the discrete vibrational levels (u, 0) of the b 3fI,state. The predissoiiation is close to case c’ [20] _ At u = I, the lifetimes pass through a minimum and the decay widths through a maximum and there is only a weak FranckCondon effect with rotational quantum number, most of the J-dependence arising from the external factor in the coupling matrix elements.
5-= (2nt24 Iz-[*)-‘,
4.2. Accidental predissociationof the A ‘Ez state
channel probleti to the pair of coupled equations
It is convenient to develop the theory of the phenomenon of accidental predissociation by intro-
The predissociation liietime is given by
where Tis the transition matrix, obtained from the asymptotic form of PI*, and is given by T= (CD1P1HP2W),
where the brackets denote an integration over the electron coordinates and the parentheses denote an integration over the nuclear coordinates_ By eliminating PI+, we may reduce the threeP2 (H f v&t - E)P*\k = -P2HPa *, P3(H + Vopt - E)P3’F = -PflP2
Q,
T. Uzer, A. DaIgamo/The photodissociationof Li2
776 where V,,, is an optical potential
.or.equivalently
V Opt =P~HP,G,PIHP,
and P3 V,,,Ps formally as
= 0. The solution P2q may be written gbnt =gopt
P,lk = G,,,P,HP,9,
can be defined equivalently as
where
g& = {E - ez (R) - TR - F(R) - J(R) f iel-’ .
Gopt = {P,(E-
H-
V,,,
tie)P2)-‘_
We now invoke the Born-Oppenheimer approtimation and write the eigenfunctions @i(r, R) in the separable form @i(rTR) = Qi(rlR)
The associated complete set of eigenfunctions c”(R) are solutions of the equation {T, +F(R)
vr, = (91
where g,a, is a nuclear Green’s function = (E - ez(R) - TR - F(R) f ie)-’ ,
in which Tn is the nucIear kinetic energy operator and F(R) = (Q2I Coopt I Qd 1211. Then if a(R) = (@21P,HP,IW, P2q and P3q may be written as Pz\I’=
IPIHP2
142).
vz,
= (92
IP*HP,
k&A
T becomes
which may be evaluated in the form
~~ilVt211"~~s"lv23133~
s
The Greeds function Gopt simplifies to Gopt = 192)gopt~92ir
+J(R) + e2(R) - I!?‘) S”(R) = O-
If we define
T=
g,,,
il - J@)gopt 1-r
n
E-E”
In practice, F(R) and J(R) are small in comparison with E(R) and may be ignored in the determination of c”(R) and E” _ The operator Vzs is the operator H’, responsible for the predissociation of the b 3 II: state. The operator VI2 is the spin-orbit interaction operator_ In Na,, the coupling.of the A ‘Ez and b 3ffu states shifts some of the rotational levels of the A ‘Zz state by about 1 cm-’ [22] and a marked decrease in the lifetime of the A ‘2: state has been observed for the V’ = 22, J’ = 14 vibration-rotation level [23] _For
IQ2~&**4~)
and Psyr = @s + G3P3H&,k,p+(R)By operating on P3U with Q2P2Hfr we obtain the equation
Table 7 . AccidentaI predissociarion lifetimes(7A*) of the A ‘Zz state of7Li2foru’=9
,A* (lo’s)
J’
a(R) = 11 -J(R)g,,,)-‘(Q+P2Hp3~s),
1
8.1
where J(R) is another optical potential
3 5
7.3 6.0
J(R) = ‘The transition matrix reduces to the expression T= (rr(QrPr~&~)go~t~)
10 13 15 12 20
1.7 0.09 0.24 1.7 7-S
277
T_Urer.A_DaIgamo/T~ephotodissociat~onofLi~ Ii2 there are no observations and the magnitude of the coupling matrix element (b 311$lvS’,,lA ‘Zz) is uncertain_ We adopt a value of A cm-’ and suggest tentatively that A may be of the order of unity. We assume that A varies slowly with internuclear distance in the region of overlap of the initial and intermediate nuclear wavefiurctions in which case the predissociation lifetimes are directly proportional to A-* _ The resulting predissociation lifetimes for several rotational levels J’ of the v’ = 9 vibrational state are presented in table 7. If A is of the order of unity, most of the lifetimes are long compared to the rathative lifetimes which are about 17 ns [6] and for most levels predlssociation Is a rate event. However, because of the resonant nature of the phenomenon those levels which lie close in energy to levels of the intermediate b 311u state may undergo rapid predissociation. For u’ = 9, the shortest lifetime for ‘Ii2 occurs for the J’ = 13 level where it is comparable to the radiative lifetimeThere are large uncertainties in our quantitative predictions which arise from inaccuracies in the spinorbit interaction matrix elements and in the b 311u potential energy curve and energy levels. The particular u’J’ levels which preferentially predissociate are not accurately predicted by our calculation. It is clear nevertheless that accidental predissociation will lead to the radiative destruction of specific rotationvibration levels of the A ‘Z: state and that the dissociation will be sensitive to the isotopic composition of the molecule.
Acknowledgement This work was supported in part by the Department of Energy under contract EY-76-S-02-2887. The absorption oscillator strengths.&, in table 4 were calculated by Dr. D.K. Watson. We are grateful to her for permission to quote them. We are indebted to Dr. R-W_ Field for a critical reading of an earlier version of the manuscript.
References [l] G. Das, J. Chem. Phys. 46 (1967) 1568.
121IV_Kutzelni~~, V. Staemmler and N. C&rs,
Chm.
Phys
Letters 13 (1972) 496. [3] ML. Olson and D.D. Konowalow, Chem. Phys. Letters 30 (1976) 281; Chem. Phys. 21(1977) 393; 22 (1977) [4] 2;. Kahn, TX Dunning, N.W_ Winter and W-A. Goddard, J_ Chem. Phys. 66 (1977) 1135. [5] D.K. Watson, CJ. Cerjan, S. Cuberman and A. Dalgamo, Chem. Phys. Letters 50 (1977) 181. [6] D.K. Watson, Chem. Phys. Letters 51(1977) 513. [7] T. Uzer, D.K. Watson and A. Dalgatno, Chem. Phys. Letters 55 (1978) 6. ]8] P.H. Wine and L.k Melton, Chem. Phys. Letters 45 (1977) 509. [9] G. Baumgartner, private communication (1978). [lo] A. Renn, private communication (1978). [ 1 l] R. VeIasco, CL. Ottinger and R.N. Zare. J. Chem. Phys. 51(1969) 5522. [ 121 G_ Ennen and CL. Ottinger, Chem. Phys. 3 (1974) 404. [ 131 MM. Hessel and CR. Vidd, J. Chem. Phys. 70 (1979) 4439. [14] P. Busch and M.M. Hessel, J. Chem. Phyr 67 (1977) 586. 1151 AJ. Lightmanand B.P. Mather, J. Chem. Phys. 69 (1978) 2262. [16] R-C-Y. Wu, J-B. Crooks, S-C. Yang, K.R. Way and WC. Stwalley, Rev. Sci. Inst. 49 (1978) 380. [17] A. Dalgarno, G. Heraberg and T.L. Stephens, Ap. J_ Letters 162 (1970) 49; T.L Stephens and A. Dalgamo, J. Quant. Spectry. Rad. Trans. 12 (1972) 569. [18] I. Kovacs, Rotational structure in the spectra of diiton% molecules (Elsevier, .Amsterdam, New York, 1969); RW_ Field, S-G. Tilford, R.A. Howard and J.D. Simmons, J. Mol. Spectry. 44 (1972) 347. [19] A.L. Ford, J. Mol. Spectry. 53 (1974) 369. 1201 R.S. Mull&en, J. Chem. Phys 33 (1960) 247; MS. Child, in Specialist periodical reports: molecular spectroscopy, Vol. 2 (The Chemical Society. London, 1974) n_ 446. [21] T-F_ O’NaUey, Phyr. Rev. 150 (1966) 14_ 1221 P. Kusch and MM. Hessel, J. Chem. Phys. 63 (1975) 4087. [23] M-E+ Kaminsky, R-T_ Hawkins, F-V. Kowalski and 4-L. Schawlow, Phys. Rev. Letters 36 (1976) 671.
THE PHOTODISSOCIATION
OF Liz
T. UZER * and A. DALGARNO Center for Astrophysics,
Harvard-Smirhsonian
Cambridge. Massachusetts
02138. USA
Received 28 hfarch 1980
The absorption of photons by Liz from the X ‘xl state to the A ‘ZZ~and B %I,, states is considered and the mechanisms that lead to dissociation are studied quantitatively. Calculations are reported on the direct predissociation of the b 3t?,, state. The s&~ificance of accidental predissociation of the A ‘2: state is discussed and a quvltal theory of the pro-
cess is presented_
1_ Iniroduction
strength for a transition from the vibrational level u”of the Xr Zg state to the vibrational level u’ of the excited state is given by
The alkali metal dirners are useful candidates for the study of isotope separation and for the construction of visible lasers. Several calculations have been carried out of the potential energy curves of Li2 [i-5] and of the dipole moments connecting the ground X *Zi state with the excited A *Xi [6] and B ‘II, [73 states. Measurements have been reported of the lifetimes of specific vibration-rotation levels of the A’C: [S] and B’II, [9,10] states and spectroscopic analyses have been performed [l l--14]_ A tabulation of the wavelengths of lines of the B ‘II,X r Cg system that are accessib!e to an argon ion laser has been presented by Lightman and Mather [Is] _ We explore here the absorption of photons by Liz in the A *ZE-X’L’: and B ‘fI,-X’ZZ systems and we investigate quantitativeIy the mechanisms that lead to dissociated molecules. Accidental predissociation is significant and we present a quanta1 theory of the process.
2. Absorption
oscillator
dipole moment,
distance
- E’,,,) I(~~ID(R)Ixu”)~~,
where g is unity for a C-Z transition and two for a II-Z transition, E,. - Eg is the transition energy in atomic units and w and w are respectively the final and initial vibrational wavefunctions. In writing the formula for fu,*,c* we have ignored the dependence on the rotational quantum numbers_ We have verified that for low values of J”, the absorption oscillator strength is not sensitive to the initial rotational level when we sum over all the possible branches_ The transition dipole moments D(R) have been calculated [6,7] using the model potential method [S] _For the A ‘X:-X ‘xi transition, the calculated transition dipole moment leads to radiative lifetimes [67 in harmony with measurements [S] _For the B ‘l&-X ‘Ei transition, the calculated transition dipole moment leads to radiative lifetimes [7] about 5% larger than the measured values [9,10] _ We have slightly modified the calculated values ofD(R) and adopted the values listed in table 1. Baumgartner [9] and Renn [lOI have derived lifetimes of vibration-rotation levels (u’, J’) of the B ‘ff, state from studies of the fluorescent response, of Liz to an argon ion laser, which are in close agreement with our theoretical values, presented in table 2. Similar agreement is a!so found between theoretical radiative lifetimes and the measured lifetimes of levels
strengths
If R is the internuclear transition
= $g(E,, fu”,“’
and D(R) is the oscillator
the absorption
* Also Department of Chemistry, Harvard University. Cambridge, &achusett.s, USA. New at Department ciThec-
retical Chemistry, University of Oxford, 1 South Parks Road, Oxford OX1 3TG, UK.
271
272
T. iIzer,A.Dalgamo~The
TabIe 1
Modifiedvaluesof the dipole momentD(R) for the I3 *II,X 1X; transition of Liz, in au R(Qo)
3.0 4.0 5.0 6-O 7.0 8.0 10.5
12.0 16.0
D(R)
2.19 2.63 21.5 2.86 297 3.0s 3.20 3.23 3.27
photodissoctitionofLi+ Table 3 Emission band intensities for B ‘II&J’) -X tions of ‘Liz Band (u’, u”)
Measured
O,O 0,1 1,O 1,2 290
1.0 1.165 1.12 0.485 0.345 0.177
1 Z@“)
transi-
Calculated
[I61
2,1
0.240
2,2
1.0 l-156 1.10 0.485 0.353 0.172
0.242
with vibrational quantum numbers II’ from 0 to 12
absorption into the continuum is negligible_ We may
[lOIThe agreement between the calculated radiative Iifetimes and the measured decay lifetimes excludes any possibIe significant contribution to the decay of the B ‘II,, state by predissociation. Another test of the adopted dipole moment for the B ’ II,--X ‘Zg transition is provided by the data of Wu et al_ [16] who studied the fluorescence spectrum of %iz generated by white-light excitation of a ‘L& nozzle beam. The vibrational state populations have been determined by \Vu et al. [ i6] _The theoretical relative intensities of the emission bands (u’, I/‘) corresponding to the experimental populations are compared with observations in table 3_ The calculated absorption oscillator strengths fomv* of the A ‘C:-X ’ Eg transition are listed in table 4. Ah the absorption oscillator strength is taken up in transitions into discrete vibrational levels and
demonstrate this conclusion more explicitly by introducing the Franck-Condon factors
Table 2 Calculated lifetimes of vibration-rotation B 1 II, state of Liz in lo-’ s
6Li2
7
-
b2
6Li7Li
4”“,“’
=
ltkyl
xgw .
We have calculated the sum of qo,“* over ail the discrete vibrational levels u’ and find that it equals
unity to within IO-’ _ For an initial vibrational level u” = 20, we find that the sum of q 20,v*is 0.8648 so that direct photodissociation of highIy excited vibrational levels is not negligible. Tire absorption oscillator strengths f20,vs are presented in table 4. Table 5 lists the absorption oscillator strengths fo,,* for the B ‘II,-X’E: transition. The sum of the Franck-Condon factors for the discrete vibrational IeveIs is again unity to within 10e5 and direct photodissociation is negligible_ For absorption from higher vibrational levels, photodissociation may occur, the sum of 4”‘: upover u’ for u” = 10 being 0.5858. The oscillator strengthsfro,Un are listed in table 5_
Ievels u’, .I’ of the
V’,J’
c (10-9 s)
4. II 9.31 1.4
8.07 8.68 7.85
.9,4 4.24 2,31
8.43 8.09 8.013
%9
8.13
3. Spontaneous radiative dissociation Although for low-lying vibrational levels direct photodissociation is very slow, dissociation can also occur by absorption into discrete vibrational levels of the A ‘ZE and B ‘II, states followed by spontaneous radiative decay into the viirational continuum of the ground electronic state [17] _In practice emission occurs overwhelmingly into the discrete levels of the X ‘Zg’ state. Our calculations of the sum of the
T_ Uzer, A. Daigamo / The photodissociationbf Liz
273
Table4 Absorptionoscillatorstrengthsfrom the u” = 0 and d’ = 20 vIbrationallevelsof the X ‘Zi state to levelsU’of the A 'xz state of ‘Liz 0’
fo,;
II’
fzo,;
u'
f-20,“’
u’
f20,u
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
0.0254 0.0643 0.0891 0.0898 0.0737 0.0524 0.0335 0.0198 0_0109 0.0058 0.0029 0.0014 0.0007 0.0003 0.0002 0.0001
9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
0.0000 0.0023 0.0429 0.1389 0.0644 0.0122 0.0214 0.0198 0.0017 0.0188 0.0086 0.0004 0.0115 0.0127 0.0023 0.0010
2.5 26 21 28 29 30 31 32 33 34 3.5 36 37 38 39 40
0.0071 O-0094 0.0060 0.0014 0.0001 0.0021 0.0051 0.0066 0.0061 0.0044 O-0023 0.0008 0.0000 0.0002 0.0010 0.0019
41 42 43 44 4.5 46 47 48 49 50 51 52 53 54 55 56
0.0028 0.0034 0.0036 O-0034 0.0029 0_0030 0.0029 0.0023 0.0019 0.0017 0.0014 0.0012 0.0011 0.0009 0.0007 0.0006
Franck-Condon factors for discrete transitions originating in the u’ = 0 and u’ = 15 vibrational levels of the A ‘Zi state yield unity to within lo-‘. For v’ = 25, the sum is 0934 so some spontaneous radiative dissociation of the highest vibrational levels occurs. However, the probability of populating them by absorption is small. Similar results hold true for the B ‘II,, state. Table5 Absorptiontiscillatorstrengthsfrom the v” = 0 and U”= 10 vibrationallevelsof the X 1C; state to levelsv’ of the B ‘II, state of ‘Liz v-
d
f0,“’
0
0.3055
0
1 2 3 4 5 6 7 8 9
0.3192 0.1914 0.0871 0.0338 0.0119 0.0039 0.0012 0.0004 0.0001
1 2 3 4
fl0.U’
2 7 8 9 10 11 12 13 14
0.0000 0.0000 0.0001
0.0029 0.0365 0.1575 0.1385 0.0074 0.0858 0.0298 0.0131 0.05 86 0.0324 0.0015 0.0073
4_ Predissociation The potential energy curves of the A ‘xc, a ‘Zz and b 31&, states of Li,, shown schematicaliy in fig- 1, suggest another mechanism by which E2 can be dissociated by visible wavelength photons. Calculations [3,5] demonstrate that the A ‘2: state crosses the b 311Ustate. The b ‘IIn state cannot decay by an allowed radiative transition but it crosses the repulsive a “2: state which leads to dissociation into the ground state atoms. The phenomenon is known as accidental predissociation. The rCc rotational levels have e-parity and the 32g levels have f-parity so that only the 3Z: state participates iu the. predissociation of the A ‘Zi state. The A’ZZ, b 311Uand a3Zg states are coupled by the sum H’ of the spin-orbit interactions V,, and the interactions V, between the nuclear rotational momentum and the electron orbital and spin momenta- For V,, we may write v, = @z12/.LR2)[LTs-
+ Lx+]
- (fi2 /2/.&‘) [J*L- + J-L*] - @*~~/LR*)[J*S-
+ J-S+] ,
where p is the reduced mass of Li2, L’ and S* are the raisiig or lowering operators for the electronic orbital angular momentum S, respectively, in the body frame
T. Uzer. A. Dalgatno /The pitorodissociation of Liz
274
dissociation of the b ‘llg state through its interaction with the a 32: state. 4.1. Tke predkociation
of the b ‘II: state
The matrix element coupling the discrete vibration-rotation levels (u, J) of the b ‘llz state to the continuum levels of the a ‘Zz state are given by [ 181 i~sn~lH’132;)12 =J(J + l)P, 0 = l(fi2/2~2~b
31111L’l[a3C))Iz _
To estimate 0, we adopt the hypothesis of pure precession (CL ref. [19]) and write the vibrational coupling matrix element, which connects the initiaI discrete nuclear wavefunction x&R) to the fiial continuum nuclear wavefunction xx_=~(R),in the form Pt* = l(xuJltr2/2”‘lrR’Ixx_‘J)1*_ I
1
I
I
B
LO
1 6
5
I 12
Rta.u.1 Fig. 1. Potentti energy curve* far low-lying states of 7Li2_ adapted from ref. [ 3 ] _
and J’ is the raising or lowering operator for rotationai angular momentum J_ The A ‘Zz state mixes only with the 3FId component of the b 3flU state and we consider first the preTable 6 Predissociation -
If XuJ is normalised to unity and &lJ to the asymptotic form &*J
-
(Z,U/ZT%~E’)“~ sin(k’R - &Tlr + Q),
where (k*)l = &rE’/h*, E’ is the energy of relative motion of the dissociated atoms and VJ is the elastic phase shift, the predissociation lifetime, T,,J, is given by
lifetimes of vibration-rotation levels of the b 3ni state of 7Li2 in units of 10-l ’ s -___
J
u=Q
LJ=1
V=2
1;=3
v=4
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
274 91 46 28 19 13 10 79 63 5.2 4.4 3.8 3.3 2.9 2.6
154 50 2.5 15 IO 7.3 5.5 4.3 3.4 2.8 2.4 2-O 1.7 1.5 1.3
162 54 27 16 11 7.7 5.8 4.5 3.6 2.9 2-4 2-O 1.7 15 1.3
311 104 52 31 20 14 11 8.3 6.5 5.3
1735 571 281 165 108 7.5 54 40 31 24 19 16 13 IO 85
43
3.6 3.0 2.6 2.2
275
T. User. A. DcImrno / The photodissociation of Li2
ducing three projection operators PI, P2 and Ps such that PI projects onto the final state, here the a 3X: state, P2 onto the intermediate state, here the b 311u. state and P3 onto the initial state, here the A ‘2: state. We az.ume that no other states are significant so that PI + P2 + P3 = 1_ No direct coupling exists between the initial and fmal states so that P&PI 4 P,HP3 = 0, where His the hamiltonian of the system. Then if * is the wavefunction satisfying (H 2 E)9(r, R) = 0, it follows that P,(.-
E)PlQ
= -PIhiP**,
Pz (H - E) P2 \k = -PgYPl * - P&P3’F P,(H-
EJP3S = -P3HP~*
,
_
Let @i(r, R) be the solutions of the correspouding homogeneous equations P,(H - E)P,$ RLo.u.)
Fig_ 2. The potential energy curves of the b3n, and a “Xi states of’ 7Li2, adapted.from [3], and the vibrational levels of the b 3~U state.
= 0.
Formal solutions for Pi\k may be obtained using the Green’s functions Gi= {P&E-H+~c)P~}-‘. Then
We have calculated T~J for several vibrationrotation levels of the b “II: state. The discrete and continuum vibrational eigenfimctions were determined by numerical integration of the nuclear radial wave equations using for the interaction potentials of the b 311u and a 32z states the model potential calculations of Watson et al. [S] _ The resulting lifetimes are reproduced in table 6. Fig. 2 illustrates the intersecting potential energy curves and depicts the locations of the discrete vibrational levels (u, 0) of the b 3fI,state. The predissoiiation is close to case c’ [20] _ At u = I, the lifetimes pass through a minimum and the decay widths through a maximum and there is only a weak FranckCondon effect with rotational quantum number, most of the J-dependence arising from the external factor in the coupling matrix elements.
5-= (2nt24 Iz-[*)-‘,
4.2. Accidental predissociationof the A ‘Ez state
channel probleti to the pair of coupled equations
It is convenient to develop the theory of the phenomenon of accidental predissociation by intro-
The predissociation liietime is given by
where Tis the transition matrix, obtained from the asymptotic form of PI*, and is given by T= (CD1P1HP2W),
where the brackets denote an integration over the electron coordinates and the parentheses denote an integration over the nuclear coordinates_ By eliminating PI+, we may reduce the threeP2 (H f v&t - E)P*\k = -P2HPa *, P3(H + Vopt - E)P3’F = -PflP2
Q,
T. Uzer, A. DaIgamo/The photodissociationof Li2
776 where V,,, is an optical potential
.or.equivalently
V Opt =P~HP,G,PIHP,
and P3 V,,,Ps formally as
= 0. The solution P2q may be written gbnt =gopt
P,lk = G,,,P,HP,9,
can be defined equivalently as
where
g& = {E - ez (R) - TR - F(R) - J(R) f iel-’ .
Gopt = {P,(E-
H-
V,,,
tie)P2)-‘_
We now invoke the Born-Oppenheimer approtimation and write the eigenfunctions @i(r, R) in the separable form @i(rTR) = Qi(rlR)
The associated complete set of eigenfunctions c”(R) are solutions of the equation {T, +F(R)
vr, = (91
where g,a, is a nuclear Green’s function = (E - ez(R) - TR - F(R) f ie)-’ ,
in which Tn is the nucIear kinetic energy operator and F(R) = (Q2I Coopt I Qd 1211. Then if a(R) = (@21P,HP,IW, P2q and P3q may be written as Pz\I’=
IPIHP2
142).
vz,
= (92
IP*HP,
k&A
T becomes
which may be evaluated in the form
~~ilVt211"~~s"lv23133~
s
The Greeds function Gopt simplifies to Gopt = 192)gopt~92ir
+J(R) + e2(R) - I!?‘) S”(R) = O-
If we define
T=
g,,,
il - J@)gopt 1-r
n
E-E”
In practice, F(R) and J(R) are small in comparison with E(R) and may be ignored in the determination of c”(R) and E” _ The operator Vzs is the operator H’, responsible for the predissociation of the b 3 II: state. The operator VI2 is the spin-orbit interaction operator_ In Na,, the coupling.of the A ‘Ez and b 3ffu states shifts some of the rotational levels of the A ‘Zz state by about 1 cm-’ [22] and a marked decrease in the lifetime of the A ‘2: state has been observed for the V’ = 22, J’ = 14 vibration-rotation level [23] _For
IQ2~&**4~)
and Psyr = @s + G3P3H&,k,p+(R)By operating on P3U with Q2P2Hfr we obtain the equation
Table 7 . AccidentaI predissociarion lifetimes(7A*) of the A ‘Zz state of7Li2foru’=9
,A* (lo’s)
J’
a(R) = 11 -J(R)g,,,)-‘(Q+P2Hp3~s),
1
8.1
where J(R) is another optical potential
3 5
7.3 6.0
J(R) =
10 13 15 12 20
1.7 0.09 0.24 1.7 7-S
277
T_Urer.A_DaIgamo/T~ephotodissociat~onofLi~ Ii2 there are no observations and the magnitude of the coupling matrix element (b 311$lvS’,,lA ‘Zz) is uncertain_ We adopt a value of A cm-’ and suggest tentatively that A may be of the order of unity. We assume that A varies slowly with internuclear distance in the region of overlap of the initial and intermediate nuclear wavefiurctions in which case the predissociation lifetimes are directly proportional to A-* _ The resulting predissociation lifetimes for several rotational levels J’ of the v’ = 9 vibrational state are presented in table 7. If A is of the order of unity, most of the lifetimes are long compared to the rathative lifetimes which are about 17 ns [6] and for most levels predlssociation Is a rate event. However, because of the resonant nature of the phenomenon those levels which lie close in energy to levels of the intermediate b 311u state may undergo rapid predissociation. For u’ = 9, the shortest lifetime for ‘Ii2 occurs for the J’ = 13 level where it is comparable to the radiative lifetimeThere are large uncertainties in our quantitative predictions which arise from inaccuracies in the spinorbit interaction matrix elements and in the b 311u potential energy curve and energy levels. The particular u’J’ levels which preferentially predissociate are not accurately predicted by our calculation. It is clear nevertheless that accidental predissociation will lead to the radiative destruction of specific rotationvibration levels of the A ‘Z: state and that the dissociation will be sensitive to the isotopic composition of the molecule.
Acknowledgement This work was supported in part by the Department of Energy under contract EY-76-S-02-2887. The absorption oscillator strengths.&, in table 4 were calculated by Dr. D.K. Watson. We are grateful to her for permission to quote them. We are indebted to Dr. R-W_ Field for a critical reading of an earlier version of the manuscript.
References [l] G. Das, J. Chem. Phys. 46 (1967) 1568.
121IV_Kutzelni~~, V. Staemmler and N. C&rs,
Chm.
Phys
Letters 13 (1972) 496. [3] ML. Olson and D.D. Konowalow, Chem. Phys. Letters 30 (1976) 281; Chem. Phys. 21(1977) 393; 22 (1977) [4] 2;. Kahn, TX Dunning, N.W_ Winter and W-A. Goddard, J_ Chem. Phys. 66 (1977) 1135. [5] D.K. Watson, CJ. Cerjan, S. Cuberman and A. Dalgamo, Chem. Phys. Letters 50 (1977) 181. [6] D.K. Watson, Chem. Phys. Letters 51(1977) 513. [7] T. Uzer, D.K. Watson and A. Dalgatno, Chem. Phys. Letters 55 (1978) 6. ]8] P.H. Wine and L.k Melton, Chem. Phys. Letters 45 (1977) 509. [9] G. Baumgartner, private communication (1978). [lo] A. Renn, private communication (1978). [ 1 l] R. VeIasco, CL. Ottinger and R.N. Zare. J. Chem. Phys. 51(1969) 5522. [ 121 G_ Ennen and CL. Ottinger, Chem. Phys. 3 (1974) 404. [ 131 MM. Hessel and CR. Vidd, J. Chem. Phys. 70 (1979) 4439. [14] P. Busch and M.M. Hessel, J. Chem. Phyr 67 (1977) 586. 1151 AJ. Lightmanand B.P. Mather, J. Chem. Phys. 69 (1978) 2262. [16] R-C-Y. Wu, J-B. Crooks, S-C. Yang, K.R. Way and WC. Stwalley, Rev. Sci. Inst. 49 (1978) 380. [17] A. Dalgarno, G. Heraberg and T.L. Stephens, Ap. J_ Letters 162 (1970) 49; T.L Stephens and A. Dalgamo, J. Quant. Spectry. Rad. Trans. 12 (1972) 569. [18] I. Kovacs, Rotational structure in the spectra of diiton% molecules (Elsevier, .Amsterdam, New York, 1969); RW_ Field, S-G. Tilford, R.A. Howard and J.D. Simmons, J. Mol. Spectry. 44 (1972) 347. [19] A.L. Ford, J. Mol. Spectry. 53 (1974) 369. 1201 R.S. Mull&en, J. Chem. Phys 33 (1960) 247; MS. Child, in Specialist periodical reports: molecular spectroscopy, Vol. 2 (The Chemical Society. London, 1974) n_ 446. [21] T-F_ O’NaUey, Phyr. Rev. 150 (1966) 14_ 1221 P. Kusch and MM. Hessel, J. Chem. Phys. 63 (1975) 4087. [23] M-E+ Kaminsky, R-T_ Hawkins, F-V. Kowalski and 4-L. Schawlow, Phys. Rev. Letters 36 (1976) 671.