Theory of the radiative lifetime of the 3B1 state of SO2

337

Theory of the Radiative Lifetime of the 3B1 State of SO2

Philip Phillips* and Ernest R. Davidson Department of Chemistry, University of Washington, Seattle, Washington 98195 Received 13 September 1982; accepted 27 December 1982

Interchange and many-body perturbation theory are applied to the calculation of the radiative lifetime of the :3Bl state of SOz. The radiative lifetime of :%B1 SO2 is predicted to be 7.6 msec, in excellent agreement with the experimental collision-free phosphorescence lifetime. As a separate calculation, interchange perturbation theory is used to determine the radiative lifetime of the 'A1 state of CH2.

INTRODUCTION

In recent papers, Phillips and Davidson192 have developed novel ab initio approaches for the calculation of the singlet-triplet radiative rate. In this paper we apply these techniques, with modifica- tions, to the important problem of the radiative lifetime of the 3B1 state of S02.

Past experimental results on the "1 SO2 lifetime showed a large nonradiative e f f e ~ t ~ - ~ leading to a short (1 msec) lifetime with a low quantum yield. These experimental results generated much confusion among theoreticians since a small mol- ecule like SO2 is not expected to give rapid nonradiative internal conversion. The most recent experimental results,6-8 however, found a longer (8 msec) lifetime and a quantum yield near unity. I t now appears that the earlier experimental results were dominated by collision-induced transitions. Bendazolli and Palmierig reported the only ab initio estimate of the radiative lifetime. Using a small basis set and a small configuration inter- action (CI) with the spin-orbit operator treated as part of the Hamiltonian, they obtained 24 msec for the lifetime.

An additional source of confusion for SO2 has been the nature of the low-lying triplet states. The Walsh diagram is usually interpreted to imply that the "B1 (n+ - T * ) state is clearly lowest. Past calculations?Jo however, have found the 3B2 (T - T * ) state to be lower. Further, calculationdo and experiments on neat SO2 crystalsll have shown a very low-lying :3A2 (n- - T * ) state. Since the

* Danforth-Compton fellow and Chevron Summer fellow. Present address: Department of Chemistry, University of California, Berkeley, California 94720.

radiative lifetime of the lowest triplet state de- pends critically on the order and energies of these states, we have examined these energies more carefully in this paper.

METHODOLOGY

In a recent paper, Phillips and Davidsonl proved a general theorem for the application of interchange perturbation theory (IPT) to transition matrix elements. Use of this theorem transforms the general equation for the first-order (in some perturbation A ) transition matrix element of an operator B ,

Bltk = c + c ( $ 8

I f G

I # E

into

Bk% = ($81Al$kB9 + ( $ F ) I A l W + A"k?I$kB') - <$F'I$P')l -t AE- '<$$IAI$~>[<B)G - ( B ) E ] - ~ - ' ( $ 8 I B I $ k ) [ ( A ) G - ( A ) E ] (2)

In (1) and (2), $8 and $$ are the unperturbed ground- and excited-state spin-free Born-Op- penheimer wavefunctions, respectively, and $lu) is the first-order correction to the wavefunction induced by a perturbation 8,

Journal of Computational Chemistry, Vol. 4, No. 3,337-344 (1983) 0 1983 by John Wiley & Sons, Inc. CCC 0192-8651/83/030337-08$01.80

Phillips and Davidson 338

The transition amplitude for radiative triplet- singlet transitions requires that +$ be replaced by 3fiq, the q th sublevel of the excited triplet state, +%by l$, the singlet ground state, B by p, the dipole moment operator, and A by hs, t,he spin-orbit operator. Upon substituting these definitions into (2), the IPT formula for the spin-orbit-induced dipole transition amplitude is obtained:

(3a) M(l+?+q) = M D M + M I Y O + MOJ + MlJ

with

MDM = AE-l(l+lhsl 3 + s ) A ( p ) (3b)

and

AW = (3+,1p1 3+,) - ( v i p i I+) (3f)

For formaldehyde,l MlJ could be calculated with sufficient accuracy from the four simple Feynman diagrams for MIJpo, the uncorrelated approxima-

tion to M1J. Finite-field coupled Hartree-Fock (CHF) wavefunctions were used to compute MoJ and Ml.0. Lastly, accurate CI wavefunctions were used for the A(p) and AE factors in MDM, while a simple one-configuration approximation was used for the direct mixing spin-orbit matrix element. In this paper, we evaluate eq. (3a) for SO2 and CH2 using techniques similar to those utilized for CHzO in order to obtain the radiative lifetime of 3B1 SO2 and 1Al CH2.

RADIATIVE LIFETIME RESULTS FOR CHz

Phillips and Davidson2 reported radiative lifetime calculations on the lAl state of CH2 using many-body perturbation theory (MBPT). However, a programming error caused errors in the reported transition amplitudes and lifetimes. The corrected transition amplitudes and lifetimes are shown in Table I. In contrast to the old value2 of 100 s, the MBPT adjustment of denominators for higher- order correlation (ADHOC~) (Zeff = 2.8) estimate of the radiative lifetime of 'A1 CH2 is now 24 s. However, the principal conclusions of ref. 2 still

Table I. MBPT transition amplitude and lifetime results for CH2.'

ADHOC Sublevel Polarity Unresmed Resmedb AOHOCb Z,ff=2.8d

i Mo V Z -0.00036 -0.0002 -0.0002 -0.000093 Z Y to. 00009 0.0025 0.016 0.0075

1 i?

2 *0.00063 0.00007 0.00007 0.000033 2 Y - .00125 -0.0013 -0.0092 -0.0043

i? Y Z 0.00027 -0.00013 -0.00012 -0.000056 Z Y -0.0012 0.0013 0.0072 0.0034

missive b tesC A0

Y 295.2 0.05 0.0002 0.000044 2 20.1 7.5 1 .o 0.22

A V 165.2 0.02 0.00006 0.000013 2 3010.6 1.92 0.19 0.041

1 0.0003 0.52 5.1 23.4

a Mo and MI are the zeroth-order and first-order (in the electron correlation) contributions to the transition amplitudes, respectively, defined in ref. 2.

h R 1 IVOS were used in the evaluation of Mo, MI, and M. c A: and A,, are the zeroth-order and first-order Eistein A

coefficients computed from Mo and M, respectively, for emission from the 9th sublevel.

d Computed effective charge on the carbon atom.

Radiative Lifetime of the "1 State of SO2 339

Table 11.

Sublevel Polarity H.F. Level TCSCF for 1 ~ 1

IPT results for C H Z . ~

i M 1 ,o Y 2 -0.00006 -0.00007 2 Y 0.00000 0.00002

iMo.l

Y z -0.00001 0.00000 2 Y -0.00000 -0.00002

i MDM

Y 2 0.00000 0.00000 2 Y 0.00654 0.0043

iM1 >1 ,o Y 2 0.00000 0.00000 2 Y 0.00001 0.00001

i h j

Y 2 -0.00007 -0.00006 z Y 0.0065 0.0044

Rate Constants

Y 0.000 0.000 2 0.161 0.071

'I 6.2 14.0

a All transition amplitudes and rates are in atomic units. T is in seconds. M1J9O was computed with IVO lA1 orbitals. A Zeff of 2.8 was used in the spin-orbit operator.

hold: The direct mixing of the 'A1 and 3B1 states contributes 95% of the transition amplitude, and the second important configuration in the lA1 wavefunction contributes roughly half of the first-order electron correlation correction to the transition amplitude.

For CH201, improved estimates of 7 were obtained by computing the ' A I - ~ B ~ transition amplitude from eqs. (3a)-(3f). Hence such calculations were performed for CH2 at the experimental 'Al geometry and with the DZP basis set used in ref. 2. One column of Table I1 reports the lifetime and transition amplitude calculations from the IPT equations (3a)-(3f) in which IVO 'A1 orbitals were used in the computation of M1Jy0, and Hartree- Fock wavefunctions were used to evaluate ( '+hb :$bq ) , MOJ, and Ml.0. Accurate CI wavefunctions were used to calculate A(p> and AE. At this level of theory, the radiative lifetime is 6.21 s (2,ff = 2.8).

However, as is well known, two configurations (Dl = la]22a'21b223a'2,D~ = la122a121b221b12) are necessary to represent the 'A1 state of CH2; hence the results presented above are not accurate. Unlike the MBPT formalism in which the reference functions are restricted to single Slater determi- nants, multiconfiguration self-consistent field (MCSCF) (or CI) wavefunctions can be used in the

IPT formalism to evaluate the transition amplitude. Use of MCSCF wavefunctions for the reference states necessitates a finite-field coupled MCSCF calculation for the generation of liC/(P).

Column 3 of Table I1 shows the results for M when a two-configuration self-consistent field (TCSCF) wavefunction is used to describe the lA 1 state. Our best estimate of the radiative lifetime of the 'A 1

state, 14.0 s, is 40% lower than the MBPT estimate of 24 s. In the evaluation of MIJ,O, the ADHOC treatment was invoked by replacing the ' A 1-3B1 energy denominator by the best estimate (0.0056 a.u.) reported in ref. 2. Since MIJ*O contributed less than 2% to any component of M, small errors caused by molecular orbital (MO) choice and higher-order correlation effects in this quantity are not important.

RADIATIVE LIFETIME OF SO2

Figure 1 shows the experimental 3B1 geometry of Merer12 at which all calculations were done (Rso = 1.494 A, 0 = 126.3). In this axis system, the components of the dipole moment transform as B l ( x ) , B2(U), Al(z). The spatial components of the spin-orbit operator transform as B ~ ( x ) , B l ( y ) , A ~ ( z ) . From eq. (1) it follows that the transition from the 3B1 excited state to the 'A1 ground state cannot be x -polarized. The y -polarized transition from the 3 q Z sublevel can proceed via 'Bz or 3A2 intermediate states. Similarly, the z -polarized transitions from 3& proceeds through 3B1 and lA1 intermediates.

The Dunning13 [4s,2p] contraction of Huzina- ga7sI4 (9s75p) primitive set was used as the basis set for oxygen. A [6s,4p] contraction15 of the Huzinaga'6 (11s77p) primitive set was placed on sulfur. Finally, a set of polarization d functions were placed on each atom (a$ = 0.85 and aS&= 0.6).

Table I11 contains the SCF and CI energies of the ground state and several excited states of S02. The reference configurations used in the CI calculations are shown in Table IV. With this basis and geom-

L

Figure 1. = 1.494 A, 8 = 126.3').

The :jBl experimental geometry of Merer' (Rso

340 Phillips and Davidson

Table 111. excited states of SO*."

SCF and CI energies of the ground and several Table V. Estimates of energy differences.a

AE(sCF) AE(IVO)~ AE(SOQ)'

State SCF SD-CI SDQ est.

ground -547.188 -547.543 -547.699 A1

B1

1

8a1+3bl -547.099 -541.438 -547.594 3

1 a2+3bl -541.112 -541.41 2 -547.571

5b2+3bl -547.055 -547.383 -547.547

1 la2-3b, -546.951 -541.298 -547.401

382

A2

82

3

a All energies are reported in atomic units at the experimental 3B1 geometry.

b 1000 space-orbital products representing single and double excitations from the reference configurations shown in Table IV. Each state was computed with its parent-configuration SCF orbitals.

c These energies are the full singles-doubles and estimated quadruples energy obtained by using the full CI extrapola- tion formulas in ref. 9.

etry the lA1 and 3B1, SCF energies are, respectively, -547.1876 a.u. (ref. 17) and -547.0992 a.u., and the SCF excitation energy is 2.4 eV. A better estimate of the lA1-3B1 excitation energy can be obtained using the difference of the estimated full CI [AE(SDQ)] energies which include corrections for quadruple excitations.ls Table V reports the AE(sDQ) for the 3B1-1A1 excitation energy to be 2.82 eV, 0.4 eV lower than the experimental T, of 3.2 eV (minimum to minimum on each potential energy surface).12 As Table V indicates, the 3B1-1A1 AE computed, with 'A1 IVO orbitals,lgas expectation values of H with single configuration wavefunctions is in remarkable agreement with the

Table IV. used in CI calculations.8

Orbital occupations in reference configurations

7al 8a1 9a1 2bl 3bl 5 b ~ l a 2

States

2 2 0 2 0 2 2 2 2 0 2 2 2 0

A1 1

381 2 1 0 2 1 2 2

2 2 0 2 1 2 1 382

2 2 0 2 1 1 2 1 2 1 2 1 1 2

3 A2

2 2 0 2 1 2 1 2 2 1 1 1 1 2 2 2 1 2 0 1 2

I 8 2

a All orbitals lower in energy than those listed are doubly occupied.

~ ( ' 8 , ) - E ( ~ A ~ ) 2.39 2.03 2.83

E(3B2) - E(361) -0.54 0.49 0.62

E(3A2) - E(3Bl) 1.20 1.85 1.31

E('B2) - E(3B1) 4.60 6.45 3.07

a Energies in electron volts at the experimental 3B1 geom-

Excited states were calculated with single configurations

The difference of the estimated full CI energies.

etry.

made from ' A 1 Ivos.

~ ( S D Q ) number. Consequently, only 'A1 IVO orbitals were used in the diagram summations.

I t should be noted that two distinct solutions exist to the restricted Hartree-Fock (RHF) equations for the 3B1 state of S02: the symmetric one discussed previously with an energy of -547.0992 a.u. and the symmetry broken solution with a RHF energy of -547.1115 a.u. The fundamental problem encountered here is analogous to the doublet instability problem in a radicals. For SO2, the analogue of the doublet instability problem stems from the. proclivity of RHF wavefunctions to lo- calize the hole on one oxygen. This phenomenon arises from an intrinsic inadequacy of RHF to describe the correlation in position of the B and a lone pairs of the oxygens. This evidence for symmetry breaking in the "1 wavefunction echoes early spectroscopic studies on 3B1 SO2 which found the S-0 bond lengths to be slightly unequal.20 Hence MCSCF geometry optimizations that correlated the S-0 bonds as well as the full a space were performed. Our results from such optimizations strongly suggest that 3B1 SO2 is symmetrical. Thus the symmetry breaking noted above appears to be simply an artifact of RHF.

Table I11 also indicates that the 3B2(aa*) and 3A2(n-a*) states are nearly degenerate with the 3B~(n+a*) state. A t the SCF level, the "2 state is lower than the "1 state. However, if correlated wavefunctions are used to describe these states, the order of the "2 and "1 states is reversed, and the AE is 0.62 eV. The 3A2-3B1 ~ ( S D Q ) is 1.31 eV. In fair agreement with the AE(sDg) values are the low-oder estimates of AE reported in column 3 of Table V. Because of its low energy, the :3A2 state is expected to mix strongly with the ground state, thus contributing significantly to M(l$, 3$*).

Radiative Lifetime of the 3B1 State of SO2 341

The best estimate of lE - E(3A2) (the ~ ( S D Q ) value) was used in the ADHOC treatment.2

The remaining state that was investigated at the CI level was the 'B2 state whose E(SDQ) energy is -547.4811 eV, 5.88 eV above the ground state. Although 'A1 IVO orbitals were the best MO choice, they produced a one-configuration 'B2-3B1 energy difference twice as large as the ~ ( S D Q ) value. Hence the ADHOC treatment2 was also invoked here by replacing 3E - E(lB1) with the corresponding ~ ( S D Q ) value. The 'B2 state is expected to mix strongly with the 3B1 state in

These energies should be compared with pre- vious theoretical work on the order of these low- lying states of S02. Hillier and SaunderslO found that both the single determinant, virtual orbital treatment and limited CI among a few singly excited configurations placed 3Bz lowest, followed by 3B1 and 3A2, much like our M ( S C F ) results. Their results, however, were reported only for T, differences and did not give the state order at the 3B1 geometry. Bendazolli and Palmieri9 likewise reported results from a low-order CI involving only a few singly excited configurations. At the ground-state geometry, they found :3B1 and 3R2 nearly degenerate, with 3A2 substantially higher.'.?

These past theoretical energies, and our own, stand in contrast to the experimental interpreta- tion. Snow, Hovde, and Colsonll quote unpub- lished results by Merer which show that 3A2 is only 300 cm-' above 3B1 (AT,). Like NOz21 and S02+,l0 the state with n+ singly occupied has a much larger bond angle than the state with n- singly occupied. In "neat" SO2 crystals, Snow, Hovde, and Colsonll found evidence for a triplet state 83 cm-l below

In analogy with the gas phase results, they argue that this state should be of 3A2 symmetry. No experimental spectrum has yet been assigned to the 3B2 state, and Walsh's rules are often cited to show that this state should be much higher. The evidence from molecules such as NO:!21 and HCO2 22 indicates, however, that single occupancy of the a2 a orbital is only about 0.5 eV above the n+ or n- minima. Further, the n+ and n- minima are part of a complicated Jahn-Teller double cone when asymmetric distortions are considered. Thus, while there is no evidence from S02, experimental and theoretical data on similar molecules support the placement of 3B2 about 0.5 eV above 3B1. The analogous vertical n-,n+ excitation of NO2 lies near 3.0 eV, so the 1.5 eV result found here is plausible. As a function of angle bending, the 3B1

M('J/,"* 1.

/c--so n-- so

Figure 2. The zeroth-order diagrams used in the evaluation of MOJ.

and 3A2 curves will probably cross, and the 3A2 state T, could easily be within a few hundred wavenumbers of 3B1.

As noted for CH20, calculation of MoJ can be effected by the computation of CHF wave functions. However, because the 3B1 RHF wavefunction is unstable and breaks symmetry, an electric field in the direction that removed the left-right symmetry of SO2 could not be added to the symmetry-restricted Fock operator. Hence, accurate 3 J / ( Y )

wavefunctions were unobtainable. As a result, MOJ could not be calculated in the manner described for CH20 and CH2. Thus a standard diagrammatic approach was implemented in the evaluation of MOJ. Figure 2 illustrates the two zeroth-order diagrams used in the calculation of MOJ.

For the sake of comparison, the radiative lifetime of 3B1 SO2 was first computed using the MBPT formalism presented in ref. 2. Table VI presents the values for the MBPT transition am-

Table VI. Transition amplitude and lifetime results from the MBPT formalism with IVO ' A 1 orbitals."

Sublevel P o l a r i t y Resumed AOHOC

-in0 Y z -0.0007 -0.0007

z Y 0.0048 0.0052

-id Y 2 -0.0005 -0.0005

2 Y -0.0067 -0.0086

-iY Y 2 -0.0012 -0.001 2

z Y -0.0018 -0.0033

Rates(sec") Y 2

r(msec) Y 2 avg.

33.4 33.6 77 .6 274.1

30.0 30.0 13.0 3.6 27.0 9.8

a Mo and M' are the zeroth-order and first-order contributions to the transition amplitudes defined in ref. 2. All moments are reported in atomic units. The E(SDQ) values in Table V were used in the ADHOC results. 20 = 5.45,Zs = 14.05.


plitude computed with lAl IVO orbitals. In all calculations effective charges were used, that is, 20 = 5.46 and 2s = 14.05.23 For the resummed and ADHOC formalisms, the first-order correction for electron correlation to the transition amplitude, Mt,(1$,3$z), is at least 30% larger than the uncorrelated result, M;(l$,3G2), and opposite in sign. Hence the zeroth-order average lifetime, 1.5 msec, is much shorter than the first-order estimate of 9.8 msec. Here, as for CH2 and CH20, we see that MBPT is slowly convergent.

Unlike CH2O and CH2 in which the direct mixing dominated the transition amplitude, the direct mixing term enters only M:(11),3$~) whose total contribution is an order of magnitude smaller than Mb(1$,3$z). As indicated earlier, the lowest 3A2 and lB2 intermediate states mix strongly into M(l$, 3$2), the former through spin-orbit coupling (SOC) with the ground state and the latter through SOC with the 3B1 state. The remaining important excitations which mix in strongly are the 2lB2, 3lB2, and Z3A2 high-lying states formed from the excitations 8~1(n+) +6b2,8~1’7bz, and 4b2(n-) - 3bl. These orbitals are shown in Figures 3(a)- 3 ( 4 .

Figures 3(b) and 3(c) indicate that 6b2 and 7b2 are primarily left-right correlating c* orbitals. It was verified that the SOC matrix elements of 8 ~ 1 with 6b1 and 7b2, and 4b2 with 3bl were sizeable. Also contributing to the first-order (in electron correlation) transition amplitude are configurations of 3B1 and lA1 symmetries, which correlate the single determinant representation of the ground and 3B1 states.

Rendazolli and Palmierig calculated the lA I - ~ B ~ transition moment at the lA1 geometry with small CI wavefunctions using the full spin-orbit plus spin-other-orbit operator. They report a transition moment for 3Bly of 1.3 X and for 3Blz of 1.9 X which are in fair agreement with the results in Table IV. They also noted that their results were extremely wavefunction dependent. In agreement with our results, they found several low-energy states of 3A2 and lB2 symmetry which contributed to the 3Blz transition moment. They also noted that the direct mixing of 3B1 and ‘A1 was the largest contribution to the 3BlY moment. Their transition amplitudes predict an average lifetime of 24 msec.9

Contained in Table VII are the lifetime and transition amplitude calculations, from the IPT equations. In these calculations, the ADHOC treatment was used in the diagrammatic evaluation of M$’( l$, 3+z) and M1J.O. It was verified that

Table VII. u1ts.a

IPT transition amplitude and lifetime res-

~

Sublevel P o l a r i t y Ef fect ive Chargesb

Y

Z

Y 2

Y 2

Y 2

Y z

Y

2

Y

2

- iM

z Y

Rates (sec)

T (msec)

-0.0002 0.0009

0.0020 -0.0000

0.0004 0.0000

0.0008

0.0029

-0.0001

0.0038

22.5 355.7

44.4 2.8

7.93

a ~ ( S D Q ) energy differences were used for denominators associated with the 3A2, “1, and ‘I32 states.

b 2 0 = 5.45,Zs = 14.05.

M$’(l$, 3$y) calculated from zeroth-order diagrams agreed to with 5% of the corresponding CHF result. Hence the diagrams shown in Figure 3 should offer an accurate evaluation of M$’(l$, 3$2 ). This method predicts a radiative lifetime of 3B1 SO2 to be 7.9 msec.

Table VII indicates that, as in the MBPT calculations, the contribution from the direct mixing term MDM is an order of magnitude smaller than My (l$, 3$2 ). Also, partly because of the large soc matrix elements introduced by the sulfur atom, M1J?O is the principal contributor to My (1$,3$2).

In fact, My (l$, 3$z) is opposite in sign and a factor of 3 larger than the corresponding MBPT values. This discrepancy is clearly due to the magnitude of M1(1$,3$z) relative to M0(1$,3$z). The IPT transition amplitude is, however, in close agreement with the zeroth-order MBPT transition amplitude. The slow convergence of MBPT indicates that it is doubtful whether including higher-order (in the electron correlation) terms in the IPT transition amplitude would produce results more reliable than those tabulated here.

Unlike CH2O and CH2 for which there are no reported gas-phase phosphorescene studies available, a wealth of experimental data is avail-

Radiative Lifetime of the State of SO2 343

,.... I

Figure 3. 30,50,70, and 90% of the density.

The 8al,4b2,662, and 762 orbitals of SOz. Contour levels represent values of the orbital corresponding to 10,

able for SO2. Early experiments on gas phase "1 SO2 appeared to indicate an unusually large per- centage of the "1 molecules decayed via nonradiative relaxation."* However, later, more re- fined experiments which were able to achieve a collision-free environment at extremely low pressures observed almost entirely radiative decay of :3B1 S02.6-8 The most recent of these experiments reports a collision-free phosphorescent lifetime of 8.1 f 2.5 msec. This value is in excellent agreement with both the IPT (7.9 msec) and MBPT (9.6 msec.) estimates of the radiative lifetime of :3Bl SO2, and hence reinforces the experiments which found radiative decay to be the principal route for depopulating the 3B1 state of SOz. The

descrepancy between our lifetime values and those of Bendazzoli and Palmieri (24 msec) appears to stem from their omission of a few important states of 1B2 and 3Ag symmetry in their evaluation of the ' A 1-:3B1 transition amplitude.9 Further, the results from an analysis of the intensity of the rotational lines, which showed that the :3BlZ transition moment is indeed larger than the 3Bl, ,24 are in accord with our results.

CONCLUSION

The radiative lifetime of the 'A1 state of CH2 and the state of SO2 have been calculated from the IPT formalism. Except for the use of a TCSCF


reference wavefunction in the description of the IA 1 state, IPT was applied to CH2 in precisely the same manner as it was for CH20. However, for SO2 a CHF evaluation of Mot1 was not feasible because of the unavailability of 3@'). Hence, Mop1 was evaluated with the diagrams shown in Figure 2. SO2 was further complicated because of the large number of excited states that mix in and the near degeneracy of 3B1 and 3B2. It was found that the 3A2 state is 1.3 eV above 3B1, and 3Bz 0.6 eV above 3B1, at the 3B1 geometry.

P. Phillips would lie to thank Dr. D. Feller for his sug- gestions regarding the symmetry breaking problem in SO*. The authors would also like to thank the National Science Foundation for partially funding this research.

References

1. P. Phillips and E. R. Davidson, J. Phys. Chem., 86, 3729 (1982).

2. P. Phillips and E. R. Davidson, J. Chem. Phys., 76,516 (1982). The ADHOC treatment is discussed therein.

3. L. Herman, J. Akriche, and H. Grenat, J. Quant. Spectrosc. Radiat. Transfer, 2,215 (1962).

4. T. N. Rao, S. S. Collier, and J. G. Calvert, J. Am. Chem. SOC., 91,1609 (1969).

5. K. Otsuka and J. G. Calvert, J. Am. Chem. SOC., 93, 2581 (1971).

6. F. Su, J. W. Bottenheim, D. L. Thorsell, J. G. Calvert, and E. K. Damon, Chem. Phys. Lett., 49,305 (1977).

7. J. P. Briggs, R. B. Caton, and M. J. Smith, Can. J. Chem., 53,2133 (1975).

8. R. B. Caton, A. B. F. Duncan, J. Am. Chem. SOC., 90,

9. G. L. Bendazolli and P. Palmieri, Znt. J. Qauntum

10. I. H. Hillier and V. R. Saunders, Mol. Phys., 22,193

11, J. B. Snow, D. C. Hovde, S. D. Colson, J. Chem. Phys.,

12. A. J. Merer, Discuss. Faraday SOC., 35,127 (1963). 13. T. H. Dunning, Jr., J. Chem. Phys., 53,2823 (1970). 14. S. Huzinaga, J. Chem. Phys., 42,1293 (1965). 15. H. F. Schaeffer 111, Methods of Electronic Structure

Theory, Plenum, New York, 1977. 16. S. Huzinaga, J. Chem. Phys., 53,451 (1970). 17. This energy is expected to be higher than the

Rothenberg-Schaeffer value of -547.2089 [J. Chem. Phys., 53,3014 (1970)] since he used a larger basis and performed his calculations at the ' A 1 equilibrium geometry.

18. E. R. Davidson, in The World of Quantum Chemistry, R. Daudel and R. Pullman, Eds., Reidel, Dordrecht, 1974.

19. The electron was removed from the 8al orbital and the virtual orbitals defined according to the scheme of W. J. Hunt and W. A. Goddard, Chem. Phys. Lett., 3,414 (1969).

1945 (1968).

Chem., 9,537 (1975).

(1971).

76,3956 (1982).

20. R. S. Mulliken, Can. J. Chem., 36,lO (1958). 21. C. Jackels and E. R. Davidson, J. Chem. Phys., 65,2941

(1976). 22. D. Feller, W. T. Borden, and E. R. Davidson, private

communcation. 23. S. Frage and G. Malli, Many Electron Systems:

Properties and Interactions, W.B. Saunders, Phila- delphia, 1968.

24. J. C. D. Brand, J. T. Jones, and C. DiLauro, J. Mol. Spectrosc., 40,616 (1971).

Documents

Theory of the radiative lifetime of the 3B1 state of SO2