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Radiative lifetimes of the singlet oxygen molecule
Volume 78, number 1
RADIATfVE
CHEMICAL
LIFETflCfES
OF THE SINGLET
PHYSICS
LETTERS
OXYGEN
15 February
1981
hfOLECULE
Jr-Kang ZiiU *. Junqmg LI ** and Yuh-Kang PA.hI Department Recewsd
of Chmnst~
20 August 1980,
R;idxxlre
hfetlmes
Boston College
Chestnut HIII Massaarhvsetts O-7167. US-4
m fkd
Nolembrr I980
form 14
of the a IA, zmd b ‘Si stxes of the oxygen moiecule hz~vebeen calcukzted b> nnpravmg the approucalcuhtlon gives the correct order oi magruruds for the radrame Iiierrme of both
mx1ons used b) Carr et al The pkent
states
Smce singlet-tnplet transttron probab~ties are very small quantrtres and not readdy obtasned expenmenrally, rt IS desirable to be able to obtain the values of the transrtron probabrlirres theoretrcally. All calculatrons on smdet-triplet rransrtion probabrhrtes in the hrerature appear to be partial sum-over approaches [i-3]. The transrtron probablry IS usualiy expressed as an mtinite sum of spectral expansion by applymg perturbation theory, and only a few terms are taken m the actual calculatron. These approaches requrre exphcrt wavefunctrons of other states m addition to zerothorder ~va~efunctIons_ Carr et al. have developed a nea method by usmg double perturbarron rheory [S] and the exchange theorem [6] for offdragonai matrix elements. In their method more accurate calculatrons are feasible by the perturbation-variation approach. The many-electron equattons can be reduced to one-electron equations whrch can be solved by a Hylleraas-type variational procedure [5] whrch requues omy the zerothorder wavefunctions. Their method has been apphed to the calculations of the radiative lifetimes of the singlet oxygen molecule [4] by using the srmple SCF wavefunct~ons of Kotani et aI. [7] and indudmg only one-electron terms in the spin-orbit mteraction harcliftontan Hs, _Using the formula
for the hfetrme [4], they obtained the correct order of rnagruttide for the hferrme of the b ‘Xi state. whrie the radrattve bfetrme of the a IA_, state IS underestrmat,:d by two orders of magnitude. In the present work we use the accurate SCF wavefunctions of Cade ar.d WahI [S] and mclude rwo-eiectron terms m the spm-orbit hamrltoman, H,, = zl)Zrlll-s, I
+ .%,-Sl I
.
where
G! and Bj = (eWlm+‘)
X
k+J
r-’
Ix.
[(& - :P,) X rjk]
represent the two-electron rerms. We follow rhe notation used m ref. 141, and define
.
1= eWZ,14mzc2 A = r,r,3 + l/G &
c = eW12mac2
.
,
=WX’)@~--&I,,
B,=Wi’)@+Q),.
,
TV = (3h/64d-3 Q[) (&J&) (W~llZ) *
Vtsrttng scholar, on leave from Annul Inswure of Optics and Fme Mecbamcs, Academia Sinica, China. ** Visiting schokw, on leave from Department of Chemtitry, University of Science and Technology of Chma. Hefel. Anhui, china
(Ia = (7P’lk(Z g-
x + II y )IAp”‘>-
’ 129
Table 1
15 February 1981
CHESIIC4L PHYSKS LETTERS
Volume 78. number 1
--ra tsf
%(s)t-orb’r+i33Sp
fOr3”Jg-XEX~
67 20 67 05 3 7-12
_1938 3916 1-I 3 88 x 103
been unproved considerably by tmprowxg the approtinntlons used by Csrr et al. [4] However. It has been iound that accurate zeroth-order wavefunctions are more ~po~ant than the two-electron terms m the spin-orbxt hanultoman for the perturbatton-vsrlatton calculattons of radlatne hfex~mes of molecules The tk\oelectron terms come from spin-ocher-orbit mteraction and the siectron’s own orbtral motion uxth respect to the other electrons W&e they eust m pnnc~ple. they are usually small. The vduzs ior 3“ V commg from the two-electron terms are typtcaIiy of the order of IO-5 - 10m6 ~0. These are compared wth typicat miues for
References (11 H F Hsmcka, &d\ancrd quantum chemntry (AddrsonW&e), Readmg 1965) ch IO [2] H F Ham&s, m The triplet state. ed A B Zablsn
(Csmbrtdge Unrv Press, London, 1967) [3] S P hfcCIynn, T Azurm and hl Kmoshna, hfolecular
t $‘(
v, f
“‘6 - ‘c-7 - “8 +
vg+ I’,0
+ CTll + V,)
of Cade and Wahl
[S] , since
we take the SCF values we use these wavefunctions
E xx = 13153,ev
fora’Ag
)
EbS = 3 -* 6559eV
for b IXg _
For mean
e\cn~tlon
The computed
130
energies,
hfertmes
are m table
I _The results
have
specrroscop) of the trlplet state (Prentxe-Hall, Englewood CM-t-s. 1969). I-S] C E Carr, Y X Pan and T.-Y. Chans, Chem Phys 18 (1976) 25 1 IS 1 J 0 ~Kschfe~d~r, IV. Byers Brown and S T Epsrem, 4dxm Quantum C&m. I (1964) 255 [6] A. Dalguno. Adv%n Phys 11 (1962) 281. [7] hl Kotam, Y hltzuno, K. Kayama and E Ishxguro, J Phss Sot Japan 12 (1957) 707 [g j P.E Cade and AC. WahI, At Data Nuci. Dara Tabfzs 13 (197-I) 340 [9] P hi Badger, XC. Wrxght and R F. Whxtlock, J. Chem Phys 43 (1965) 4345. IlO] D.Q Wark and D.M. Mercer, Appl Opt 4 (1965) 839 fl f ] H. WaRace and D M Hunter, J. Geophys Res 73 (l968)
4813
RADIATfVE
CHEMICAL
LIFETflCfES
OF THE SINGLET
PHYSICS
LETTERS
OXYGEN
15 February
1981
hfOLECULE
Jr-Kang ZiiU *. Junqmg LI ** and Yuh-Kang PA.hI Department Recewsd
of Chmnst~
20 August 1980,
R;idxxlre
hfetlmes
Boston College
Chestnut HIII Massaarhvsetts O-7167. US-4
m fkd
Nolembrr I980
form 14
of the a IA, zmd b ‘Si stxes of the oxygen moiecule hz~vebeen calcukzted b> nnpravmg the approucalcuhtlon gives the correct order oi magruruds for the radrame Iiierrme of both
mx1ons used b) Carr et al The pkent
states
Smce singlet-tnplet transttron probab~ties are very small quantrtres and not readdy obtasned expenmenrally, rt IS desirable to be able to obtain the values of the transrtron probabrlirres theoretrcally. All calculatrons on smdet-triplet rransrtion probabrhrtes in the hrerature appear to be partial sum-over approaches [i-3]. The transrtron probablry IS usualiy expressed as an mtinite sum of spectral expansion by applymg perturbation theory, and only a few terms are taken m the actual calculatron. These approaches requrre exphcrt wavefunctrons of other states m addition to zerothorder ~va~efunctIons_ Carr et al. have developed a nea method by usmg double perturbarron rheory [S] and the exchange theorem [6] for offdragonai matrix elements. In their method more accurate calculatrons are feasible by the perturbation-variation approach. The many-electron equattons can be reduced to one-electron equations whrch can be solved by a Hylleraas-type variational procedure [5] whrch requues omy the zerothorder wavefunctions. Their method has been apphed to the calculations of the radiative lifetimes of the singlet oxygen molecule [4] by using the srmple SCF wavefunct~ons of Kotani et aI. [7] and indudmg only one-electron terms in the spin-orbit mteraction harcliftontan Hs, _Using the formula
for the hfetrme [4], they obtained the correct order of rnagruttide for the hferrme of the b ‘Xi state. whrie the radrattve bfetrme of the a IA_, state IS underestrmat,:d by two orders of magnitude. In the present work we use the accurate SCF wavefunctions of Cade ar.d WahI [S] and mclude rwo-eiectron terms m the spm-orbit hamrltoman, H,, = zl)Zrlll-s, I
+ .%,-Sl I
.
where
G! and Bj = (eWlm+‘)
X
k+J
r-’
Ix.
[(& - :P,) X rjk]
represent the two-electron rerms. We follow rhe notation used m ref. 141, and define
.
1= eWZ,14mzc2 A = r,r,3 + l/G &
c = eW12mac2
.
,
=WX’)@~--&I,,
B,=Wi’)@+Q),.
,
TV = (3h/64d-3 Q[) (&J&) (W~llZ) *
Vtsrttng scholar, on leave from Annul Inswure of Optics and Fme Mecbamcs, Academia Sinica, China. ** Visiting schokw, on leave from Department of Chemtitry, University of Science and Technology of Chma. Hefel. Anhui, china
(Ia = (7P’lk(Z g-
x + II y )IAp”‘>-
’ 129
Table 1
15 February 1981
CHESIIC4L PHYSKS LETTERS
Volume 78. number 1
--ra tsf
%(s)t-orb’r+i33Sp
fOr3”Jg-XEX~
67 20 67 05 3 7-12
_1938 3916 1-I 3 88 x 103
been unproved considerably by tmprowxg the approtinntlons used by Csrr et al. [4] However. It has been iound that accurate zeroth-order wavefunctions are more ~po~ant than the two-electron terms m the spin-orbxt hanultoman for the perturbatton-vsrlatton calculattons of radlatne hfex~mes of molecules The tk\oelectron terms come from spin-ocher-orbit mteraction and the siectron’s own orbtral motion uxth respect to the other electrons W&e they eust m pnnc~ple. they are usually small. The vduzs ior 3“ V commg from the two-electron terms are typtcaIiy of the order of IO-5 - 10m6 ~0. These are compared wth typicat miues for
References (11 H F Hsmcka, &d\ancrd quantum chemntry (AddrsonW&e), Readmg 1965) ch IO [2] H F Ham&s, m The triplet state. ed A B Zablsn
(Csmbrtdge Unrv Press, London, 1967) [3] S P hfcCIynn, T Azurm and hl Kmoshna, hfolecular
t $‘(
v, f
“‘6 - ‘c-7 - “8 +
vg+ I’,0
+ CTll + V,)
of Cade and Wahl
[S] , since
we take the SCF values we use these wavefunctions
E xx = 13153,ev
fora’Ag
)
EbS = 3 -* 6559eV
for b IXg _
For mean
e\cn~tlon
The computed
130
energies,
hfertmes
are m table
I _The results
have
specrroscop) of the trlplet state (Prentxe-Hall, Englewood CM-t-s. 1969). I-S] C E Carr, Y X Pan and T.-Y. Chans, Chem Phys 18 (1976) 25 1 IS 1 J 0 ~Kschfe~d~r, IV. Byers Brown and S T Epsrem, 4dxm Quantum C&m. I (1964) 255 [6] A. Dalguno. Adv%n Phys 11 (1962) 281. [7] hl Kotam, Y hltzuno, K. Kayama and E Ishxguro, J Phss Sot Japan 12 (1957) 707 [g j P.E Cade and AC. WahI, At Data Nuci. Dara Tabfzs 13 (197-I) 340 [9] P hi Badger, XC. Wrxght and R F. Whxtlock, J. Chem Phys 43 (1965) 4345. IlO] D.Q Wark and D.M. Mercer, Appl Opt 4 (1965) 839 fl f ] H. WaRace and D M Hunter, J. Geophys Res 73 (l968)
4813