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Oscillator strengths of the Schumann-Runge bands of isotopic oxygen molecules
J. Quam. Sptwrosc. Iiadiat. Transfer Vol. 43, No. 3, pp. 225-238, Printed in Great Britain. All rights reserved
1990
0022-4073/90
$3.00 + 0.00 Press plc
Copyright 0 1990Pergamon
OSCILLATOR STRENGTHS OF THE SCHUMANN-RUNGE BANDS OF ISOTOPIC OXYGEN MOLECULES R. S. FRIEDMAN? Harvard-Smithsonian
Center for Astrophysics and Department of Chemistry, Harvard University, Cambridge, MA 02138, U.S.A. (Received 29 June 1989)
Abstract-Oscillator strengths of the Schumann-Runge bands of the isotopic molecules 1602 (u’ = O-17, u” = O-2), ‘60’80 (u’ = O-16, v” = O-2), and ‘*02 (v’ = O-18, v” = O-2) have been calculated. The dependence of the band oscillator strength on the reduced mass and the vibrational level is discussed. The variation of the rotational oscillator strength with the rotational quantum number is explored. The calculated values agree with experimental data.
INTRODUCTION The Schumann-Runge (S-R) absorption bands arise from the transition B(v’)~C;--X(U”)~C~ in OZ. Detailed descriptions of these bands have been given by Krupenie,’ Creek and Nicholls,2 and Huber and Herzberg.’ The S-R band system occupies a central place in the photochemistry of the atmosphere. Because the B3C,y state undergoes predissociation, Schumann-Runge band absorption of ultraviolet (u.v.) radiation results in the production of two ground state (‘P) oxygen atoms. The predissociation of the B’C; state has been studied theoretically (Julienne and Krauss;4 Julienne;’ Sink and Bandrauk;6 Cheung et a17) and experimentally (Lewis et a18). Knowledge of the photoabsorption cross-sections and oscillator strengths for the S-R bands is required for the calculation of photodissociation rates of molecular oxygen. Band oscillator strengths of the three isotopic molecules 1602, 1802, and ‘60’80 have been reported by Lewis et al9 and by Yoshino et al.“,” The band oscillator strengths of Yoshino et al”,” are determined by direct numerical integration of measured absolute cross-sections and those of Lewis et al9 are extracted from absorption measurements using an equivalent width fitting procedure. We have recently tabulated” theoretical band oscillator strengths for 160, (14,0)-(0,O); l8O2 (14,0)-(0,O); and ‘60’80 (12,0)-(0,O). Agreement between the theoretical values and the experimental data of both Lewis et al9 and Yoshino et allo.” is good; the calculated and measured oscillator strengths usually agree to within 10%. In this paper, we extend these calculations to higher u’ levels and we compare line oscillator strengths to measured values. Oscillator strengths for hot bands (u’, u” > 0) are also of interest. Although the population in the t”’ = 1 level at 300 K is only 0.06% of the total 0, X state population, (v’, 1) band oscillator strengths are more than an order of magnitude larger than those of (v’, 0) and the u” = 1 transitions will affect the transmission of solar radiation through the atmosphere for example in spectral regions where the 21”= 0 transitions are weak. The thermal population in the u” = 2 level is very small (4 x 10P5%); however, other mechanisms for producing vibrationally excited ground state molecules in the atmosphere, such as the photodissociation of ozone,12 may lead to a nonequilibrium population. In combustion and discharge experiments, in which X3Z; u” > 0 levels are readily populated, hot bands are observed.13 For these reasons, we also have calculated band and line oscillator strengths for (v’, 1) and (rl’, 2). tPresent address: Department of Chemistry, University of Minnesota, Minneapolis, MN 55455, U.S.A. 225
R. S.FRIEDMAN
226
RESULTS
AND
DISCUSSION
The rotational line oscillator strength for a transition from the state (II”, N”, J”) to the state (t”, W, J’), N being the rotational angular momentum of the nuclei and J the total angular momentum, is given by fJY =
S,.,..
8rt’mv,,y,~~~ x Sr.J.ps,/. x
3he2
(25” + 1)’
.iL
where v~,,,,~,~is the photon frequency, m and e are the electronic mass and charge, respectively, S,..,.,...,.. is the band strength, S,.,.. is the Hbnl-London factor normalized to obey the sum rule cS,.,..=(25”+ J
I),
(2)
and g, is the statistical weight of the lower state, g, = (2 - 6,./j-,)(2S” + l),
(3)
where S” is the spin and ,4” is the projection quantum number for the initial electronic state. The band strength S,..,.,...,.. is related to the dipole moment matrix element ~(~,,~(R)(D(R)Ix,.,,,(R))( byi S,.,,,.,,. = (2 - &M,,V)(2S’ + 1) x ((x,.,~(R)(D(R)Ix,..,~(R)>(2,
(4)
where D(R) is the dipole moment, R is the internuclear distance, S’ and II’ are the spin and projection quantum numbers of the upper electronic state, respectively, and x,,sy(R) and x,,~(R) are the discrete vibrational wave functions, normalized to unity, of the initial and final states. For the transition B3C; - X’C; ,
s
JJ" fYY = 3.0375595 x 10-6 x vJ.,L.J'.,V... x I(x~.,/(R)JD(R)Ix~,,~(R))J~ x (2J” + 1)’
where v is in cm-’ and the dipole matrix element is in atomic units. Band oscillator strengths & can be determined by summing the rotational line oscillator strengths weighted by the Boltzmann population of the initial rotational level; that is .f;,,w = Cj& YJ"
x POP(J"),
(6)
l)exp( -E(J")/kT) l)exp( -E(J")/kT)'
(7)
where (2J” +
pop(J")= Cr(2J”
+
E(J")being the energy of the (N”, J")level. The equivalent vibrational band oscillator strength f (c', c",J',J")is determined from a single rotational line oscillator strength via f(C’, PI’,J',J")= (2J” + I)v,. ,...fJ.J../‘(~J.,Y.J..N.’ SYr),
(8)
where r,.,,.,,is the vibrational frequency. Because v,.,,.,,and vJ.N.J’.N.. are approximately equal, the equivalent vibrational band oscillator strength for a band in the Schumann-Runge system can be expressed as
f(k,VI', J',~'1) = 3.0375595
x lo-6 x vJ..w,I.N.. x I(xl,J(~)I~(~)(~l.,J(~))(?.(9)
Dependence of the dipole moment matrix element on J' and J" will be reflected in the variation of .f(r’, P”, J',J")with angular momentum. The initial state wave function x~,,~(R) is calculated via numerical integration of the radial Schrodinger equation for the ground X’C; state. The ground state potential is taken from Albritton” and is given in Table 1. At small R (R < 1.443a,,) the potential (in atomic units) is given by V(R)=
15.002872 Rg 3
(10)
Oscillator
strengths
of
the S-R
bands of isotopic
oxygen
molecules
227
and at large R (R 2 3.465a,) by V(R) =
- 17.57 x [1 + 38.906 exp( -0.68 R)] R6
(11)
This long range form has been chosen to attach smoothly onto the RKR data as well as to have the asymptotically correct - C,/R6 form.16 The final state wave function x,.,~(R) is calculated via numerical integration of the radial Schrodinger equation for the upper B3C; state. An RKR potential for the B state has been constructed from the vibrational-rotational term values of Cheung et al.]’ Values of the potential energy curve along the repulsive wall above the dissociation limit are taken from Wang et al.” The attractive RKR potential well is extended to smaller internuclear distances so that it smoothly attaches onto the values of Wang et al. I8 Values of the potential energy at large internuclear distances have been obtained from the term values recalculated by Creek and Nicholls2 for the high vibrational levels. The potential energy values are given in Table 2. At small internuclear distances (R < 1.795a,), the potential is given by V(R) = 5.97338 x IO3x exp( - 5,769546R),
(12)
and at large distances (R > 11.Oa,) by 1.1 V(R)=zTable
R (ao)
0.1442878 0.1488151 0.1533424 0.1578697 0.1623970 0.1669243 0.1714516 0.1759789 0.1805062 0.1850335 0.1853417 0.1856575 0.1859811 0.1863130 0.1866534 0.1870030 0.1873619 0.1877308 0.1881100 0.1885001 0.1889016 0.1893151 0.1897413 0.1901807 0.1906342 0.1911025 0.1915864 0.1920870 0.1926054 0.1931425 0.1936998 0.1942786 0.1948807 0.1955077 0.1961618 0.1968453 0.1975609 0.1983119 0.1991018 0.1999352 0.2008172 0.2017543 0.2027544 0.2038276 0.2049868 0.2062491 0.2076382 0.2091884 0.2109522 0.2130186
1. Potential
energies for the O2 A”Z;
V(R) t(a.u.) +I+ +I +l +l +l +l +I +l +l +I +l +l +l +l +l +l +l +l +1 +l +1 +I +l +l +I +l +l +I +l +l +l +l +l +1 +l +l +1 +l +I +l +l +l +l +1 +l +l +l +l +l +l
17.06858 ~6 .
+0.7986142 +0.6237548 to.4763910 MI.3524676 +0.2485212 to.1615952 +0.8916670 +0.2908350 &2049020 -0.6110570 -0.6358080 -0.6608410 -0.6861530 -0.7117430 -0.7376090 -0.7637500 -0.7901620 -0.8168460 -0.8438000 -0.8710210 -0.8985100 -0.9262640 -0.9542820 -0.9825640 -0.1011108 -0.1039914 -0.1068981 -0.1098308 -0.1127894 -0.1157739 -0.1187843 -0.1218206 -0.1248826 -0.1279704 -0.1310840 -0.1342234 -0.1373885 -0.1405795 -0.1437963 -0.1470389 -0.1503075 -0.1536021 -0.1569227 -0.1602694 -0.1636423 -0.1670414 -0.1704670 -0.1739190 -0.1773976 -0.1809030
R (ad
-1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1
TEnergy above O()P) + O()P). IRead, e.g., as 0.1442878 x IO’.
0.2155614 +I 0.2190311 +l 0.2215959 +l 0.2281982 +l 0.2354259 +l 0.2386177 +l 0.2433468 +l 0.2471559 +l 0.2504957 +l 0.2535402 +l 0.2563785 +l 0.2590634 +I 0.2616294 cl 0.2641002 +l 0.2664932 +l 0.2688217 +I 0.2710958 +l 0.2733237 +l 0.2755121 +l 0.2776664 +l 0.2797914 +I 0.2818910 +l 0.2839685 +l 0.2860271 +I 0.2880694 +l 0.2900978 +I 0.2921143 +l 0.2941209 +l 0.2961195 +l 0.2981116 +l 0.3000987 +l 0.3020824 +l 0.3040639 +l 0.3060445 +l 0.3080255 +I 0.31ooO8~ +l 0.3119933 +1 0.3139823 +l 0.3159761 +l 0.3179760 +l 0.3199828 +I 0.3219977 +l 0.3240218 +l 0.3260560 +l 0.3281014 +l 0.3301591 +l 0.3322301 +l 0.3343157 +1 0.3404213 +I 0.3465269 +l
(13) state. V(R) t(a.u.) -0.1844352 -0.1879944 -0.1897842 -&1915818 -0.1897842 -0.1879944 -0.1844352 -0.1809030 -0.1773976 -0.1739190 -0.1704670 -0.1670414 -0.1636423 -0.1602694 -0.1569227 -0.1536021 -0.1503075 -0.1470389 -0.1437963 -0.1405795 -0.1373885 -0.1342234 -0.1310840 -0.1279704 -0.1248826 -0.1218206 -0.1187843 -0.1157739 -0.1127894 -0.1098308 -0.1068981 -0.1039914 -0.1011108 -0.9825640 -1 -0.9542820 -1 -0.9262640 -1 -0.8985100 -1 -0.8710210 -1 -0.8438OLHl-1 -0.8168460 -1 -0.7901620 -1 -0.7637500 -1 -0.7376090 -1 -0.7117430 -1 -0.6861530 -1 -0.6608410 -1 -0.6358080 -1 -0.6110570 -1 -0.5414010 -1 -0.4755810 -1
R.
228
S.
FRIEDMAN
With the asymptotic difference between the X and B states set at the measured 3P-'Dsplitting of 1.9673 eV,19 the potentials have been constructed so that the dissociation limit of the B3Z; state with respect to the v” = 0 level of the ground state is 57127.5 cm-‘, the value advocated by Brix and Herzberg.*’ This dissociation limit is close to the value of 57136.0cm- reported by Lewis et al.*’ For the B3C;-X3C; dipole moment, we adopt the function calculated by Allison et al (see Table 1 of Ref. 22) but shifted by -O.O49a, since the excited state electronic eigenfunction used in the evaluation of D(R) corresponded to an electronic potential with an R,value 0.049~” larger than the experimental value. At small internuclear distances (R d 1.75la,), the dipole moment is taken to be 0.2661 a.u. For large R (R > 3.251~~) the dipole moment is given by
D(R)=5.52097x exp( -0.7529213).
(14)
The dipole moment that we adopt here is in good agreement with that determined by Lewis et al from experimental oscillator strengths.’ Within the Born-Oppenheimer approximation, the ground and excited state potentials as well as the dipole moment are isotopically invariant. Due to the very different equilibrium separations of the X and B states, 2.282 vs 3.032a,,, 3 the oscillator strengths are primarily determined by the values of the dipole moment and the overlap of the wave functions between 2.3 and 3.0~2,. Band oscillator strengths are determined for transitions emanating from the F2component of the X3C; state, but the dependence off,,... on the fine-structure component is weak.’ Thus, of the 14 possible rotational lines for 3Z-3C transitions, we shall consider only the P,,'Q12, "Q3*and R, branches. Furthermore, since we have omitted spin-dependent terms from the radial Schrbdinger equation, the P,and ‘Q,* branches will have identical photon frequencies and band strengths; the same applies for the Rz and “Q3* branches. The band oscillator strength is thus given by
f;.,?,c = 3.0375595 x lO-6 x ; ;$(+N;j
x {QN’.+,.w + Q,,c,.,v},
(154
where Qw+,.N’. =
VW+I.N.
x I(x,sN’.+I (R)(D(R>~x,wW>(* x LGZ+ &jJ
(15b)
Qw _,,w
vw _ ,.,v
x I(x~,N.‘-,(R)ID(R)~x~“N’.(R))I* x VP,+ %,J
(15c)
and =
Table 2. Potential energies for the 0, B3Z; state R (a,,) 0.1795240 +1+ 0.1833035 +l 0.1870829 +l 0.1908624 +l 0.1946418 +l 0.1984213 +l 0.2022008 +l 0.2059802 +l 0.2097597 +1 0.2135391 +1 0.2173186 +l 0.2210980 +l 0.2248775 +l 0.2286569 +l 0.2324364 +l 0.2362158 +l 0.2399953 +1 0.2437747 +l 0.2541380 +l 0.2551113 +l 0.2562356 +l 0.2575259 +1 0.2590356 +l 0.2607859 +l 0.2628232 +l 0.2651812 +l 0.2679747 +l 0.2713182 +l 0.2754653 +l
V(R)t(a.u.) t0.1896354 +0.1524815 +0.1247345 +0.1059804 +0.9279569 -1 t0.8348717 -1 +0.7704655 -1 +0.7350726 -1 +0.7202481 -1 +0.7089404 -1 +0.6964457 -1 +0.6615012 -1 +0.5902670 -1 +0.4943889 -1 +0.3912897 -1 +0.2909246 -1 +0.1980784 -1 +0.1148567 -1 -0.6118781 -2 -0.7692308 -2 -0.9460505 -2 -0.1141362 -1 -0.1353818 -1 -0.1581754 -1 -0.1824744 -1 -0.2080680 -1 -0.2350191 -1 -0.2631541 -1 -0.2923718 -1
tEnergy above O(jP) + O(‘D). SRead, e.g., as 0.1795240 x 10’.
R (a,,) 0.2809420 +l 0.2896516 +l 0.3031613 +l 0.3189504 cl 0.3323837 +I 0.3428314 +l 0.3522586 +l 0.3612630 +l 0.3701847 +l 0.3792048 +l 0.3886126 +l 0.3985462 +l 0.4092581 +I 0.4209680 +l 0.4338827 +l 0.4410442 +l 0.4487473 +I 0.4570745 +1 0.4661173 +1 0.4759752 +1 0.4867532 +I 0.5115898 +l 0.5410834 +l 0.5764620 +l 0.6190967 +l 0.6726680 +l 0.7449987 +l 0.8593388+l 0.11ooooo+2
V(R)to -0.3226786 -1 -0.354OQ64-1 -0.3700498 -1 -0.3540064 -1 -0.3226786 -1 -0.2923718 -1 -0.2631541 -1 -0.2350191 -1 -0.2080680 - 1 -0.1824744 -1 -0.1581754 -1 -0.1353818 -1 -0.1141362 -1 -0.9460505 -2 -0.7692200 -2 -0.6879600 -2 -0.6117ooO-2 -0.5405600 -2 -0.4746300 -2 -0.4139400 -2 -0.3584700 -2 -0.2628441 -2 -0.1858431 -2 -0.1253216 -2 -0.7879281 -3 -0.4429217 -3 -0.2032583 -3 -0.5781992 -4 -0.2804634 -5
Oscillator strengths of the S-R bands of isotopic oxygen molecules
229
The Ho&London factors for the AN = f 1 transitions were taken to be those for a coupling case intermediate between Hund’s cases (a) and (bJz3 Accurate molecular parameters for the B states of the three isotopic oxygen molecules are taken from Cheung et a1.7.‘7,24 As for the ground X state, spectroscopic constants for the u = 0 level of all three isotopic molecules are taken from Steinbach and Gordy.2s For the u = 1 level of the homonuclear molecules, the values reported by Endo and Mizushimaz6 are employed. For the u = 1,2 levels of the heteronuclear molecule and the u = 2 level of the homonuclear molecules, the values predicted by Mizushima” are utilized. We have also calculated band oscillator strengths using case (b) H&l-London factors, whereby23 SRI+ SR~,~= N” + 1
and
SP, + SPY,,= N”.
(16)
Although rotational line oscillator strengths (especially for low N) are sensitive to the coupling case employed, the band oscillator strengths calculated using case (b) and using intermediate coupling rotational line strengths differ by ~0.2%; this is true even at high u’ where it is expected23 that, because the spin-splitting constant I increases with U’ while the rotational constant B decreases,” the intermediate case coupling will be important. Therefore, for the u’ = 0 and 1 levels of the ‘60’s0 and ‘sO2 isotopic molecules, for which accurate constants are not available, the adoption of case (b) Honl-London factors in the calculation offD,csfintroduces an error of ~0.2%. The X3C; state vibrational level spectroscopic constants are also used in the determination of the Boltzmann rotational population distribution; that is, for the F2 component (see, for example, Cheung et al”), E(N~) = BN"(N" + 1)- D[N~(N"+ ip+;n -7 +$vy~"
+ i)1, -NAN"+
I)?,. (17)
All the oscillator strengths we report refer to T = 300 K. The transition frequencies that appear in the line oscillator strength expressions are computed from the X and B state eigenvalues. The calculations of band oscillator strengths for the heteronuclear and homonuclear molecules differ only in that, for the ‘60’80 isotopic molecule, due to the distinguishability of the nuclei, the summation of rotational line oscillator strengths includes initial rotational levels with both even and odd values of N. In the homonuclear molecules, only odd N initial rotational levels are populated. In two recent publications, ” theoretical band oscillator strengthsf,,, have been reported for 1602 (u’ = O-14) 1802 (u’ = O-14) and ‘60’80 (u’ = O-12) and have been compared with experimental values. In Table 3, we have extended the calculation to higher u’ levels and we also have computed hot band (u” = 1,2) oscillator strengths. Lewis et al9 and Berzins** reported values off,,’ for the 1602 and 1802 isotopic molecules and we list them in Table 3. The experimental numbers are systematically larger than the theoretical ones with an average difference of 9.7% for 1602and 7.3% for 1802. Hudson and Carter29 have determined oscillator strengths of the S-R bands of 1602with 2~’= 5-13 and u” = O-2 by recording photoelectric spectra at 300,600, and 900 K. Their u” = 1 and 2 band oscillator strengths are, on average, 15 and 24%, respectively, larger than the present theoretical values. An earlier theoretical calculation by Allison et a130for the S-R bands of 1602 with u’ = O-21 and u” = O-2 resulted in band oscillator strengths consistently larger than the values reported here, by, on average, 1415%. The present calculation should be more accurate for two reasons. First, our RKR potential for the B state has been constructed from more accurate spectroscopic constants. And secondly, the dipole moment of Allison et a130is derived from low resolution experimental absorption and emission studies using the R-centroid approximation. For all three isotopic molecules, the band oscillator strengths, for fixed u”, initially increase with u’, reach a maximum at u’ = 13, 14, or 15 and then decrease. This behaviour can be explained by examining the initial and final state radial wave functions. In Figs. 1 and 2 we present the radial wave functions for 160, for (u’ = 14, N’ = 2; u” = 0, N” = 1) and (u’ = 0, N’ = 2; u” = 0, N” = 1). As u’ increases, the upper state wave function penetrates into regions of smaller nuclear separation, increasing its overlap with the ground state wave function. In Fig. 3 we present the 1602radial wave functions (u’ = 17, N’ = 2; u” = 0, N” = 1). Although the u’ = 17 wave function does indeed penetrate into a region of slightly smaller internuclear distance than does the u’ = 14 wave function, the amplitude of the u’ = 17 wave function is smaller. There is also more destructive interference in the overlap between the ground state and u’ = 17 wave functions than in the (u’ = 14, u” = 0)
R. S. FRIEDMAN
230
overlap. Both of these factors, but primarily the first, cause a smaller Franck-Condon factor and smaller band oscillator strength. In agreement with the experimental band oscillator strengths of Lewis et al,9 the theoretical&, for the 1602isotopic molecule reaches a maximum value at v’ = 14. Where comparisons can be made, the maximum experimental and the maximum theoretical band oscillator strengths, for fixed v”, are usually found at the same value of v’. For fixed v’, the theoretical and experimental band oscillator strengths increase as u” increases. With increasing v”, the position of the outer classical turning point moves to larger internuclear distances, allowing the radial wave function to penetrate into regions of larger R,thereby increasing the overlap of the initial and final state wave functions. Table
3. Band oscillator
strengths
1;,,,
‘602
2.471 2.987 1.858 7.916 2.593 6.955 1.588 3.174 5.655 9.121 1.344 1.821 2.265 2.584 2.719 2.645 2.416 2.074
”
1 :
4 5 !: !
10 11 12 13 14 15 16 17
tTaken #Read,
g&
meay -lO$ -9 -8 -8 -7 -7 -6 -6 -6 -6 -5 -5 -5 -5 -5 -5 -5 -5
7:438 4.198 1.624 4.840 1.183 2.467 4.515 7.389 1.099 1.501 1.895 2.209 2.381 2.386 2.228 1.969 1.646
” =2
1
I
v”=O
v’
meaty 8952 8 8:800 :7 4.460 -6 1.551 -5 4.156 -5 9.146 -5 1.720 -4 2.843 -4 4.215 -4 5.700 4 7.105 -4 8.237 -4 8.877 -4 8.915 -4 8.403 -4 7.453 -4 6.315 -4 5.104 -4
Expt
-8 -7 -6 -6 -5 -5 -5 -5 -4 -4 4 -4 -4 -4 -4 -4 -4
1.76 5.03 1.24 2.68 5.03 8.57 1.20 1.71 2.05 2.61 2.58 2.93 2.52 2.09
-6 -6 -5 -5 -5 -5 -4 -4 -4 -4 -4 -4 -4 -4
from Reference 9. e.g., as 2.471 x IO-‘“. ‘60% v”=l
v-=0
v’
i!zT!F
0 1 2
v”=2
*
1.116 4.888 1.644 4.530 1.062 2.178 3.984 6.5% 9.986 1.393 1.790 2.114 2.310 2.331 2.202
43 2 7 ; 10 11 12 13 14 15 16
-8 -8 -7 -7 -6 -6 -6 -6 -6 -5 -5 -5 -5 -5 -5
21693 1.075 3.299 8.308 1.784 3.362 5.667 8.684 1.223 1.595 1.932 2.166 2.267 2.212 2.038
-7 -6 -6 -6 -5 -5 -5 -5 -4 -4 -4 -4 -4 -4 -4
zz? 2.967 1.067 2.955 6.719 1.304 2.225 3.405 4.750 6.115 7.322 8.190 8.541 8.385 7.742 6.813
-6 -5 -5 -5 -4 -4 -4 -4 -4 -4 -4 -4 -4 -4 -4
‘802 v’ U : : :
I ; 10 11 12 13 14 15 16 17 18
tTaken
v”=l
0
G&y
Theory 2.212 -9 2.586 -8 1.554 -7 6.400 -7 2.026 -6 5.258 -6 1.163 -5 2.254 -5 3.906 -5 6.147 -5 8.887 -5 1.189 -4 1.481 -4 1.711 -4 1.841 -4 1.850 -4 1.746 -4 1.566 -4 1.333 -4
3 559 9:695 1:; 6.388 -9 2.880 -8 9.970 -8 2.825 -1 6.811 -7 1.436 -6 2.700 -6 4.595 -6 7.156 -6 1.027 -5 1.365 -5 1.672 -5 1.8% -5 1.993 -5 1.954 -5 1.808 -5 1.578 -5
from Reference
28
1
Expt
I
5.51 -6 1.27 -5 2.46 -5 4.32 -5 6.82 -5 1.02 -4 1.36 -4 1.63 -4 1.1 -4 1.9 -4 1.9 -4
v”=2 Theory 3.113 -8 3.288 -7 1.787 -6 6.657 -6 1.908 -5 4.487 -5 9.003 -5 1.586 -4 2.503 -4 3.597 -4 4.166 -4 5.871 -4 6.165 -4 7.273 -4 7.343 -4 6.978 -4 6.282 -4 5.420 -4 4.469 -4
231
Oscillator strengths of the S-R bands of isotopic oxygen molecules
-1.5
2
3 internuclear
2.5
4 (a,)
3.5 distance
5
4.5
55
Fig. I. Radial wave functions for the X and B states of 1602.The dotted line represents the ground state (1”’= 0, N” = I) wave function and the solid line represents the upper state (~1’= 14, N’ = 2) wave function.
As observed,‘.” the band oscillator strength increases as the reduced mass of the isotopic molecule decreases. The heavier the isotopic molecule, the deeper in the potential well lies the vibrational level; therefore the radial wave function is constrained to a smaller range of internuclear distances. The smaller the reduced mass, the greater the amplitude of the ground and excited state
I,
2.5,,,,,/,,,,,,,,,,,,,:,,
.;I,/ , 0 1.5
( ~ j
,
,
i,
,JJ , /, ,yL,,, ,i
/’
2
2.5 Internuclear
3 distance
3.5
4
(a,)
Fig. 2. Radial wave functions for the X and B states of 1602.The dotted line represents the ground state (L.”= 0, N” = I) wave function and the solid line represents the upper state (u’ = 0, N’ = 2) wave function.
R. S. FRIEDMAN
232
; -1”““““““““” 2
3
4
internuclear
distance
?a01
6
7
Fig. 3. Radial wave functions for the X and E states of “Oz. The dotted line represents the ground state (P” = 0, N” = I) wave function and the solid line represents the upper state (D’ = 17, N’ = 2) wave function.
Table 4. Absorption
intensity
ratios
‘%-“0’“O.
(v’. v”=O) Fran&-Con&n Factor Ratios
v’ U
1
32 4 5 6 I
8 9 :: :z
14
1.712 1.665 1.619 1.577 1.535 1.495 1.451 1.419 1.383 1.346
tzz 1:222 1.177 1.135 1.097
1761 1:710 1.661 1.615 1.572 1.530 1.490 1.451 1.413 1.377 1.340 1.304 1.262 1.220 1.177 1.136 1.100
(v’, V=l) Franck-Condon Factor Ratios
1.51 1.47 1.42 1.38 1.34 1.30 1.26 1.23 1.19 1.14 1.10 1.05 1.01 0.97
171 1:66 1.61 1.56 1.51 1.47 1.42 1.39 1.35 1.31 1.27 1.24 1.20 1.16 1.12 1.09 1.05
Oscillator strengths of the S-R bands of isotopic oxygen molecules
233
radial wave functions in the 2.3-3.0a, region. The increase in the overlap of the Xand B state radial wave functions with decreasing reduced mass has also been discussed by Lewis et al.’ We have shown previously” that the theoretical ratios of the band oscillator strengths&’ of the isotopic molecules 1602and “02 agree closely with the ratios of the rotationless Franck-Condon factors, implying that the effect of the dipole moment function on the ratios is small. In Table 4, we make a similar comparison for 160, and ‘60’80; the ratios of the Franck-Condon factors and the ratios of the band oscillator strengths are also in close agreement. The very small disagreement Table 5. Line oscillator strengths
for (16,O) bands of 0,.
‘602 R
9
25 27 29
1.0232 1.1610 1.1975 1.1991 1.1820 1.1523 1.1128 1.0655 1.0115 9.5208 8.8809 8.2040 7.4984 6.7719 6.0315
-5’t -5 -5 -5 -5 -5 -5 -5 -5 -6 -6 -6 -6 -6 -6
-ztk1:3797 1.3336 1.2842 1.2310 1.1736 1.1120 1.0466
-5 -5 -5 -5 -5 -5 -5
_ _ _
9.7711 -6
9.0613 8.3215 7.5650 6.7979 6.0263 5.2561
-6 -6 -6 -6 -6 -6
tRead, e.g., as 1.0232 x 10m5.
1% P2
15 17 19 ;: 22: 29 71
8.2022 9.3243 9.6466 9.7007 9.6151 9.4369 9.1888 8.8840 8.5319 8.1397 7.7139 7.2606 6.7852 6.2933
R2
-6 a -6 a -6 -6 -6 -6 -6 -6 -6 -6 -6 -6
5.7901 -6
10738 1:1395 1.1117 1.0793 1.0451 1.0087 9.6964 9.2788 8.8351 8.3674 7.8789 7.3732 6.8544 6.3266 5.7943 5.2616
5 :5 -5 -5 -5 -5 -6 -6 -6 -6 -6 -6 -6 -6 -6 -6
R2
26 27 E
7.3330 9.2911 1.0126 1.0552 1.0783 1.0901 1.0946 1.0939 1.0893 1.0814 1.0708 1.0579 1.0429 1.0260 1.0075 9.8737
-6 -6 -5 -5 -5 -5 -5 -5 -5 -5 -5 -5 -5 -5 -5 -6
KEY 9:1916 8.9414 8.6815 8.4128 8.1362 7.8526 7.5628 7.2677 6.9681 6.6647 6.3582
f -6 -6 -6 -6 -6 -6 2 d d -6 -6
6.0494 -6
12163 1:2915 1.2895 1.2744 1.2561
5 15 -5 -5 -5 1.2368 -5 1.2169 -5 1.1964 -5 1.1752 -5 1.1532 -5 1.1304 -5 1.1068 -5 1.0823 -5 1.0570 -5 1.0309 -5 1.0039 -5 9.7623 -6 ;I;;: 8:8905 8.5882 8.2810 7.9697 7.6548 7.3370 7.0169 6.6951 6.3723 6.0490 5.7257 5.4031
:; -6 -6 -6 -6 -6 -6 -6 -6 -6 -6 -6 -6
234
R. S.
FRIEDMAN
for the ratios for the v” = 1 progression should not be attributed to the variation of the dipole moment (the band strength ratios agree very closely with the Franck-Condon factor ratios) but rather to the N-dependence of the rotational oscillator strengths. The dependence of the line oscillator strengths on the rotational quantum number is responsible for the temperature dependence of the band oscillator strengths. We have calculated” that the&, at 300 K are about 3% smaller than those at 79 K due to the decrease in rotational oscillator strengths within a given band with increasing N. This decrease is also responsible for the observed N-dependence of the equivalent vibrational band oscillator strengths.9,3’ To explore this further, we have calculated P2 and R, rotational line oscillator strengths for all the bands listed in Table 3. In Table 5, we present the results for the (16,O) band of the three isotopic oxygen molecules. Although the vibrational band is degraded to the red (that is, vVN..decreases with N), the primary reason that the line oscillator strengths decrease is the accompanying decrease with N of the band strength (and the dipole moment matrix element). [The increase in SMw/(2N” + 1) is responsible for the observed increase in fwN'. at low N”.] In Fig. 4, we plot the square of the dipole moment matrix element as a function of N for the (16,O) R, lines of the three isotopic molecules. As the reduced mass increases, the centrifugal energy is reduced and the decrease of the band strength with N is less. The decrease in the overlap of the X and B state radial wave functions with increasing angular momentum has been addressed by Lewis et al9 in their discussion of the variation of the measured equivalent vibrational band oscillator strength with N. To explore this further, we look in detail at the radial wave functions. We plot the (v’ = 16, N’ = 2; v” = 0, N” = 1) and (v’ = 16, N’ = 32; v” = 0, N” = 31) radial wave functions of 160, in Figs. 5 and 6. The first peak of the upper state N’ = 32 wave function is of slightly lower amplitude than that of the N’ = 2 wave function. This difference, albeit a small one, is enough to cause the band strength for high N” to be less than that for low N”. A similar argument involving the ground and excited state wave function overlap explains why the band strength for
Fig. 4. Square of the dipole moment matrix element (in atomic units) between the X (0” = 0, N”) and B (c’ = 16, N’ = N” + I) rovibronic states as a function of N”. The solid, dashed and dotted lines correspond to 1602, ‘601*0, and ‘*O,, respectively.
Oscillator strengths of the S-R bands of isotopic oxygen molecules
235
L
-l~‘!““‘!““““r”“““!““li 2
3
4
5
internuclear
distance
(a,,)
6
7
8
Fig. 5. Radial wave functions for the X and E states of 160,. The dotted line represents the ground state (t ” = 0, N” = 1) wave function and the solid line represents the upper state (c’ = 16. N’ = 2) wave function.
2.5 n
,
III
2
1.5
I,,,,
:.> : :: :: :j i i. :: i i : :
1
1
!
:
:
:
;
i
:
t
;
i
:
:
:
:
:
;
;
:
1 ;
: :
i
:
:
;
:
:
I
/
/ , , , / , , , , , , , I /
I I I I
I I I I
.5 I t
-1”’
! ’ ” 2
I / 3
’ ”
1” 4
”
internuclear
1 “‘I
1 “I’
1”
5
6
7
distance
(a,)
”
”
’ ”
8
Fig. 6. Radial wave functions for the X and B states of 1602.The dotted line represents the ground state (1.”= 0, N” = 31) wave function and the solid line represents the upper state (~1’= 16, N’ = 32) wave function.
9
236
R. S. FRIEDMAN
the R, branch is less than that for the Pz branch of the same N”‘. However, the R2 H&r-London factor is larger than the Pz H&l-London factor for all N” and as a result, R, line oscillator strengths are larger than P2 line oscillator strengths at small N. Even though we have omitted spin-dependent terms from the radial Schrodinger equation, we still can calculate rotational line oscillator strengths for all 14 allowed branches by assuming that oscillator strengths for branches pertaining to the same N values but different J values (for example, R,(N"), Rz(N"), and R,(N")) differ only because of the S,.,./(ZJ” + 1) factor. Oscillator strengths for some low N” rovibronic lines of the (13,0)-( 16,0) bands of 1602have been measured by Smith et a1.32For low N”, it is important that intermediate case coupling rotational line strengths be used. The largest discrepancy between the experimental and calculated transition frequencies is found to be 5.5 cm-‘, a difference that introduces little error into the calculatedf,, . In Table 6 we present the theoretical and experimental line oscillator strengths. The agreement is good and is usually within the lo-20% uncertainty that is typical of the experimental values. Table 6. Rotational oscillator strengths of lines in the (v’, 1%” = 0) bands of “0 2 fYJ
-SF 2.463 0.102 1.226 0.213 1.533 1.383 0.315 1.151 1.057 0.760 0.250 1.474 1.471 1.474 0 902 2:703 0.141 1.345 0.227 1.607 1.471 0.253 1 551 1:563 1.534 0 850 21742 0.179 1.363 0.223 1.537 1.447 0.168 1.149 1.106 0.708 0.261 1.501 1.536 1.458 0.059 0.747 2.598 0.206 1.291 1.366 1.339 1.023 0.610 1.353 1.419 1.282 0.088
tTaken from Reference 32
6) -+%?--2173 0.10 1.11 0.20 1.39 1.31 0.32 1.01 0.95 0.68 0.25 1.29 1.17 1.17 15 fL:91 0.14 1.30 0.24 1.62 1.46 0.27 33 ::49 1.47 0 90 3:38 0.24 1.46 0.23 1.64 1.73 0.78 1.19 1.18 0.93 0.34 1.45 1.45 1.38 0.11 0.85 2.80 0.32 1.43 1.46 1.47 06 ;:62 1.39 1.40 1.28 0.10
Oscillator strengths of the S-R bands of isotopic oxygen molecules
237
We also can compare theoretical and experimental values of equivalent vibrational band oscillator strengths. Lewis et a13’have recently measured rotational line oscillator strengths for P, and R, branches in the (17,0) band of i602 using high-order anti-Stokes Raman shifted radiation in Hz of 0.15 cm-’ FWHM bandwidth. Values of f (v’ = 17, v” = 0, N’, N”) have then been determined from the measured line oscillator strengths using Eq. (8). The measured equivalent band oscillator strengths decrease significantly with rotation, and oscillator strengths that are derived from R-branch measurements are slightly less than those derived from P-branch measurements. These observed results are consistent with the theoretical calculations. Our theoretical (17,O) equivalent vibrational band oscillator strengths are plotted in Fig. 7. The agreement between theory and experiment is very good; for example, Lewis et a13’ perform a least-squares fit to their measurements giving f(17,0,
N’, N”)=(2.27_+0.06)
while we find for the P-branch
x 10-5-(1.89f0.14)
x 10-*x N”(N”+
l),
(18a)
theoretical values,
f (17,0, N’, N”) = (2.36) x lo-‘-
(1.62) x 1O-8 x N”(N” + l),
(18b)
and for the R-branch,
CONCLUSIONS In this paper, we have reported theoretical calculations of Schumann-Runge band oscillator strengths for 160, (v’ = O-17, v” = O-2), 1802(v’ = O-18, v” = O-2), and ‘60’80 (v’ = O-16, v” = O-2); several rotational line and equivalent vibrational band oscillator strengths have also been given. The agreement between theoretical and measured values is good. We have also described in some I’
l
I
l
I
I
I
I
1’1
I
I
’
I
1
I
’
I
’
I
’
I
I
I’1
..
1.2x1o-s
.‘.,
-
..
‘.._ ‘.._
lO-s-
1 0
’ 50
11
11 100
11 150
1 200
I 250
”
1 300
” 350
11 400
11 450
11 500
11 550
11 600
._
A . .._ 11 650
N”(N”+ 1) Fig. 7. Theoretical equivalent vibrational band oscillator strengths of the (17.0) band of 160z, as a function of rotational quantum number. The open squares represent P-branch values and the open triangles represent R-branch values, The solid and dotted lines are the best linear least-squares fits to the calculated values.
/ 700
238
R. S. FRIEDMAN
detail the dependence of the oscillator strength on the initial and final vibrational level, the reduced mass of the isotopic molecule, and the rotational quantum number. Our calculation of S-R oscillator strengths has invoked a single channel representation of the upper B3Z; state although the upper state wave function also contains contributions from repulsive continuum states which are responsible for the observed predissociation. However, a single channel description is justified because the couplings between the B state and the continuum states are weak;4,5.7the amount of intensity alteration in the S-R bands should therefore be minimal. Acknowledgements-1 am pleased to acknowledge useful discussions with A. Dalgarno, K. Yoshino and D. Freeman. I am also very grateful to B. R. Lewis for communication of unpublished work on measured oscillator strengths. The work was supported by the National Science Foundation under Grant ATM 87-13204 to Harvard University.
REFERENCES 1. P. H. Krupenie, J. Phys. Chem. Re$ Data 1, 423 (1972). 2. D. M. Creek and R. W. Nicholls, Proc. R. Sot. Lond. A 341, 517 (1975). 3. K. P. Huber and G. Herzberg, Molecular Spectra and Molecular Structure. IV: Constants of Diatomic Molecules, Van Nostrand-Reinhold, New York, NY (1979). 4. P. S. Julienne and M. Krauss, J. Molec. Spectrosc. 56, 270 (1975). 5. P. S. Julienne, J. Molec. Spectrosc. 63, 60 (1976). 6. M. L. Sink and A. D. Bandrauk, J. Chem. Phys. 66, 5313 (1977). 7. A. S-C. Cheung. K. Yoshino, D. E. Freeman. R. S. Friedman, A. Dalgarno, and W. H. Parkinson, J. Molec. Spectrosc. 134, 362 (1989). 8. B. R. Lewis, L. Berzins, J. H. Carver, and S. T. Gibson, JQSRT 36, 187 (1986); B. R. Lewis, L. Berzins, and J. H. Carver, JQSRT 37, 229, 243 (1987). 9. B. R. Lewis, L. Berzins, and J. H. Carver, JQSRT 36, 209 (1986); B. R. Lewis, L. Berzins, and J. H. Carver, JQSRT 37, 219, 255 (1987). 10. K. Yoshino, D. E. Freeman, J. R. Esmond, and W. H. Parkinson, Planet. Space Sci. 31, 339 (1983); ibid. 35, 1067 (1987). 1 I. K. Yoshino, D. E. Freeman, J. R. Esmond, R. S. Friedman, and W. H. Parkinson, Planet. Space Sci. 36, 1201 (1988); ibid. 37, 419 (1989). 12. C. E. Fairchild, E. J. Stone, and G. M. Lawrence, J. Chem. Phys. 69, 3632 (1978); R. K. Sparks, L. R. Carlson, K. Shobatake, M. L. Kowalczyk, and Y. T. Lee, J. Chem. Phys. 72, 1401 (1980). 13. A. M. Wodtke, L. Huwel, H. Schhiter, H. Voges, G. Meijer, and P. Andresen, J. Chem. Phys. 89, 1929 (1988); A. M. Wodtke, L. Huwel, H. Schliiter, H. Voges, G. Meijer, and P. Andersen, 22nd Symp. (Znt.) on Combustion, The Combustion Institute, Pittsburgh, PA (1988). 14. A. Schadee, JQSRT 19, 451 (1978).
15. 16. 17. 18. 19.
20. 21. 22.
23. 24. 25.
26. 27. 28.
D. Albritton, private communication (1985). H. P. Kelly, Phys. Lett. 29A, 30 (1969). A. S.-C. Cheung, K. Yoshino, W. H. Parkinson, and D. E. Freeman, J. Molec. Spectrosc. 119, 1 (1986). J. Wang, D. G. McCoy, A. J. Blake, and L. Torop, JQSRT 38, 19 (1987). C. E. Moore, “Atomic Energy Levels. Vol. I,” National Bureau of Standards, Washington, DC (1949). P. Brix and G. Herzberg, Can. J. Phys. 32, 110 (1954). B. R. Lewis, L. Berzins. J. H. Carver, and S. T. Gibson, JQSRT 33, 627 (1985). A. C. Allison, S. L. Guberman, and A. Dalgarno, J. Geophys. Res. 91, 10193 (1986). J. B. Tatum and J. K. G. Watson, Can. J. Phys. 49, 2693 (1971). A. S.-C. Cheung, K. Yoshino, D. E. Freeman, and W. H. Parkinson, J. Molec. Spectrosc. 131,96 (1988). W. Steinbach and W. Gordy, Phys. Rev. A 11, 729 (1975). Y. Endo and M. Mizushima, Jap. J. Appl. Phys. 21, L379 (1982); ibid. 22, L534 (1983). M. Mizushima, Jap. J. Appl. Phys. 26, 645 (1987). L. Berzins, Ph.D. Thesis, Australian National University, Canberra (1986). R. D. Hudson and V. L. Carter, JOSA 58, 1621 (1968). A. C. Allison, A. Dalgarno, and N. W. Pasachoff, Planet. Space Sci. 19, 1463 (1971).
29. 30. 3 I. B. R. Lewis, S. T. Gibson,
32. P. L. Smith, H. E. Griesinger,
K. G. H. Baldwin, and J. H. Carver, JOSA B 6, 1200 (1989). J. H. Black, K. Yoshino, and D. E. Freeman, Astrophys. J. 277, 569 (1984).
1990
0022-4073/90
$3.00 + 0.00 Press plc
Copyright 0 1990Pergamon
OSCILLATOR STRENGTHS OF THE SCHUMANN-RUNGE BANDS OF ISOTOPIC OXYGEN MOLECULES R. S. FRIEDMAN? Harvard-Smithsonian
Center for Astrophysics and Department of Chemistry, Harvard University, Cambridge, MA 02138, U.S.A. (Received 29 June 1989)
Abstract-Oscillator strengths of the Schumann-Runge bands of the isotopic molecules 1602 (u’ = O-17, u” = O-2), ‘60’80 (u’ = O-16, v” = O-2), and ‘*02 (v’ = O-18, v” = O-2) have been calculated. The dependence of the band oscillator strength on the reduced mass and the vibrational level is discussed. The variation of the rotational oscillator strength with the rotational quantum number is explored. The calculated values agree with experimental data.
INTRODUCTION The Schumann-Runge (S-R) absorption bands arise from the transition B(v’)~C;--X(U”)~C~ in OZ. Detailed descriptions of these bands have been given by Krupenie,’ Creek and Nicholls,2 and Huber and Herzberg.’ The S-R band system occupies a central place in the photochemistry of the atmosphere. Because the B3C,y state undergoes predissociation, Schumann-Runge band absorption of ultraviolet (u.v.) radiation results in the production of two ground state (‘P) oxygen atoms. The predissociation of the B’C; state has been studied theoretically (Julienne and Krauss;4 Julienne;’ Sink and Bandrauk;6 Cheung et a17) and experimentally (Lewis et a18). Knowledge of the photoabsorption cross-sections and oscillator strengths for the S-R bands is required for the calculation of photodissociation rates of molecular oxygen. Band oscillator strengths of the three isotopic molecules 1602, 1802, and ‘60’80 have been reported by Lewis et al9 and by Yoshino et al.“,” The band oscillator strengths of Yoshino et al”,” are determined by direct numerical integration of measured absolute cross-sections and those of Lewis et al9 are extracted from absorption measurements using an equivalent width fitting procedure. We have recently tabulated” theoretical band oscillator strengths for 160, (14,0)-(0,O); l8O2 (14,0)-(0,O); and ‘60’80 (12,0)-(0,O). Agreement between the theoretical values and the experimental data of both Lewis et al9 and Yoshino et allo.” is good; the calculated and measured oscillator strengths usually agree to within 10%. In this paper, we extend these calculations to higher u’ levels and we compare line oscillator strengths to measured values. Oscillator strengths for hot bands (u’, u” > 0) are also of interest. Although the population in the t”’ = 1 level at 300 K is only 0.06% of the total 0, X state population, (v’, 1) band oscillator strengths are more than an order of magnitude larger than those of (v’, 0) and the u” = 1 transitions will affect the transmission of solar radiation through the atmosphere for example in spectral regions where the 21”= 0 transitions are weak. The thermal population in the u” = 2 level is very small (4 x 10P5%); however, other mechanisms for producing vibrationally excited ground state molecules in the atmosphere, such as the photodissociation of ozone,12 may lead to a nonequilibrium population. In combustion and discharge experiments, in which X3Z; u” > 0 levels are readily populated, hot bands are observed.13 For these reasons, we also have calculated band and line oscillator strengths for (v’, 1) and (rl’, 2). tPresent address: Department of Chemistry, University of Minnesota, Minneapolis, MN 55455, U.S.A. 225
R. S.FRIEDMAN
226
RESULTS
AND
DISCUSSION
The rotational line oscillator strength for a transition from the state (II”, N”, J”) to the state (t”, W, J’), N being the rotational angular momentum of the nuclei and J the total angular momentum, is given by fJY =
S,.,..
8rt’mv,,y,~~~ x Sr.J.ps,/. x
3he2
(25” + 1)’
.iL
where v~,,,,~,~is the photon frequency, m and e are the electronic mass and charge, respectively, S,..,.,...,.. is the band strength, S,.,.. is the Hbnl-London factor normalized to obey the sum rule cS,.,..=(25”+ J
I),
(2)
and g, is the statistical weight of the lower state, g, = (2 - 6,./j-,)(2S” + l),
(3)
where S” is the spin and ,4” is the projection quantum number for the initial electronic state. The band strength S,..,.,...,.. is related to the dipole moment matrix element ~(~,,~(R)(D(R)Ix,.,,,(R))( byi S,.,,,.,,. = (2 - &M,,V)(2S’ + 1) x ((x,.,~(R)(D(R)Ix,..,~(R)>(2,
(4)
where D(R) is the dipole moment, R is the internuclear distance, S’ and II’ are the spin and projection quantum numbers of the upper electronic state, respectively, and x,,sy(R) and x,,~(R) are the discrete vibrational wave functions, normalized to unity, of the initial and final states. For the transition B3C; - X’C; ,
s
JJ" fYY = 3.0375595 x 10-6 x vJ.,L.J'.,V... x I(x~.,/(R)JD(R)Ix~,,~(R))J~ x (2J” + 1)’
where v is in cm-’ and the dipole matrix element is in atomic units. Band oscillator strengths & can be determined by summing the rotational line oscillator strengths weighted by the Boltzmann population of the initial rotational level; that is .f;,,w = Cj& YJ"
x POP(J"),
(6)
l)exp( -E(J")/kT) l)exp( -E(J")/kT)'
(7)
where (2J” +
pop(J")= Cr(2J”
+
E(J")being the energy of the (N”, J")level. The equivalent vibrational band oscillator strength f (c', c",J',J")is determined from a single rotational line oscillator strength via f(C’, PI’,J',J")= (2J” + I)v,. ,...fJ.J../‘(~J.,Y.J..N.’ SYr),
(8)
where r,.,,.,,is the vibrational frequency. Because v,.,,.,,and vJ.N.J’.N.. are approximately equal, the equivalent vibrational band oscillator strength for a band in the Schumann-Runge system can be expressed as
f(k,VI', J',~'1) = 3.0375595
x lo-6 x vJ..w,I.N.. x I(xl,J(~)I~(~)(~l.,J(~))(?.(9)
Dependence of the dipole moment matrix element on J' and J" will be reflected in the variation of .f(r’, P”, J',J")with angular momentum. The initial state wave function x~,,~(R) is calculated via numerical integration of the radial Schrodinger equation for the ground X’C; state. The ground state potential is taken from Albritton” and is given in Table 1. At small R (R < 1.443a,,) the potential (in atomic units) is given by V(R)=
15.002872 Rg 3
(10)
Oscillator
strengths
of
the S-R
bands of isotopic
oxygen
molecules
227
and at large R (R 2 3.465a,) by V(R) =
- 17.57 x [1 + 38.906 exp( -0.68 R)] R6
(11)
This long range form has been chosen to attach smoothly onto the RKR data as well as to have the asymptotically correct - C,/R6 form.16 The final state wave function x,.,~(R) is calculated via numerical integration of the radial Schrodinger equation for the upper B3C; state. An RKR potential for the B state has been constructed from the vibrational-rotational term values of Cheung et al.]’ Values of the potential energy curve along the repulsive wall above the dissociation limit are taken from Wang et al.” The attractive RKR potential well is extended to smaller internuclear distances so that it smoothly attaches onto the values of Wang et al. I8 Values of the potential energy at large internuclear distances have been obtained from the term values recalculated by Creek and Nicholls2 for the high vibrational levels. The potential energy values are given in Table 2. At small internuclear distances (R < 1.795a,), the potential is given by V(R) = 5.97338 x IO3x exp( - 5,769546R),
(12)
and at large distances (R > 11.Oa,) by 1.1 V(R)=zTable
R (ao)
0.1442878 0.1488151 0.1533424 0.1578697 0.1623970 0.1669243 0.1714516 0.1759789 0.1805062 0.1850335 0.1853417 0.1856575 0.1859811 0.1863130 0.1866534 0.1870030 0.1873619 0.1877308 0.1881100 0.1885001 0.1889016 0.1893151 0.1897413 0.1901807 0.1906342 0.1911025 0.1915864 0.1920870 0.1926054 0.1931425 0.1936998 0.1942786 0.1948807 0.1955077 0.1961618 0.1968453 0.1975609 0.1983119 0.1991018 0.1999352 0.2008172 0.2017543 0.2027544 0.2038276 0.2049868 0.2062491 0.2076382 0.2091884 0.2109522 0.2130186
1. Potential
energies for the O2 A”Z;
V(R) t(a.u.) +I+ +I +l +l +l +l +I +l +l +I +l +l +l +l +l +l +l +l +1 +l +1 +I +l +l +I +l +l +I +l +l +l +l +l +1 +l +l +1 +l +I +l +l +l +l +1 +l +l +l +l +l +l
17.06858 ~6 .
+0.7986142 +0.6237548 to.4763910 MI.3524676 +0.2485212 to.1615952 +0.8916670 +0.2908350 &2049020 -0.6110570 -0.6358080 -0.6608410 -0.6861530 -0.7117430 -0.7376090 -0.7637500 -0.7901620 -0.8168460 -0.8438000 -0.8710210 -0.8985100 -0.9262640 -0.9542820 -0.9825640 -0.1011108 -0.1039914 -0.1068981 -0.1098308 -0.1127894 -0.1157739 -0.1187843 -0.1218206 -0.1248826 -0.1279704 -0.1310840 -0.1342234 -0.1373885 -0.1405795 -0.1437963 -0.1470389 -0.1503075 -0.1536021 -0.1569227 -0.1602694 -0.1636423 -0.1670414 -0.1704670 -0.1739190 -0.1773976 -0.1809030
R (ad
-1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1
TEnergy above O()P) + O()P). IRead, e.g., as 0.1442878 x IO’.
0.2155614 +I 0.2190311 +l 0.2215959 +l 0.2281982 +l 0.2354259 +l 0.2386177 +l 0.2433468 +l 0.2471559 +l 0.2504957 +l 0.2535402 +l 0.2563785 +l 0.2590634 +I 0.2616294 cl 0.2641002 +l 0.2664932 +l 0.2688217 +I 0.2710958 +l 0.2733237 +l 0.2755121 +l 0.2776664 +l 0.2797914 +I 0.2818910 +l 0.2839685 +l 0.2860271 +I 0.2880694 +l 0.2900978 +I 0.2921143 +l 0.2941209 +l 0.2961195 +l 0.2981116 +l 0.3000987 +l 0.3020824 +l 0.3040639 +l 0.3060445 +l 0.3080255 +I 0.31ooO8~ +l 0.3119933 +1 0.3139823 +l 0.3159761 +l 0.3179760 +l 0.3199828 +I 0.3219977 +l 0.3240218 +l 0.3260560 +l 0.3281014 +l 0.3301591 +l 0.3322301 +l 0.3343157 +1 0.3404213 +I 0.3465269 +l
(13) state. V(R) t(a.u.) -0.1844352 -0.1879944 -0.1897842 -&1915818 -0.1897842 -0.1879944 -0.1844352 -0.1809030 -0.1773976 -0.1739190 -0.1704670 -0.1670414 -0.1636423 -0.1602694 -0.1569227 -0.1536021 -0.1503075 -0.1470389 -0.1437963 -0.1405795 -0.1373885 -0.1342234 -0.1310840 -0.1279704 -0.1248826 -0.1218206 -0.1187843 -0.1157739 -0.1127894 -0.1098308 -0.1068981 -0.1039914 -0.1011108 -0.9825640 -1 -0.9542820 -1 -0.9262640 -1 -0.8985100 -1 -0.8710210 -1 -0.8438OLHl-1 -0.8168460 -1 -0.7901620 -1 -0.7637500 -1 -0.7376090 -1 -0.7117430 -1 -0.6861530 -1 -0.6608410 -1 -0.6358080 -1 -0.6110570 -1 -0.5414010 -1 -0.4755810 -1
R.
228
S.
FRIEDMAN
With the asymptotic difference between the X and B states set at the measured 3P-'Dsplitting of 1.9673 eV,19 the potentials have been constructed so that the dissociation limit of the B3Z; state with respect to the v” = 0 level of the ground state is 57127.5 cm-‘, the value advocated by Brix and Herzberg.*’ This dissociation limit is close to the value of 57136.0cm- reported by Lewis et al.*’ For the B3C;-X3C; dipole moment, we adopt the function calculated by Allison et al (see Table 1 of Ref. 22) but shifted by -O.O49a, since the excited state electronic eigenfunction used in the evaluation of D(R) corresponded to an electronic potential with an R,value 0.049~” larger than the experimental value. At small internuclear distances (R d 1.75la,), the dipole moment is taken to be 0.2661 a.u. For large R (R > 3.251~~) the dipole moment is given by
D(R)=5.52097x exp( -0.7529213).
(14)
The dipole moment that we adopt here is in good agreement with that determined by Lewis et al from experimental oscillator strengths.’ Within the Born-Oppenheimer approximation, the ground and excited state potentials as well as the dipole moment are isotopically invariant. Due to the very different equilibrium separations of the X and B states, 2.282 vs 3.032a,,, 3 the oscillator strengths are primarily determined by the values of the dipole moment and the overlap of the wave functions between 2.3 and 3.0~2,. Band oscillator strengths are determined for transitions emanating from the F2component of the X3C; state, but the dependence off,,... on the fine-structure component is weak.’ Thus, of the 14 possible rotational lines for 3Z-3C transitions, we shall consider only the P,,'Q12, "Q3*and R, branches. Furthermore, since we have omitted spin-dependent terms from the radial Schrbdinger equation, the P,and ‘Q,* branches will have identical photon frequencies and band strengths; the same applies for the Rz and “Q3* branches. The band oscillator strength is thus given by
f;.,?,c = 3.0375595 x lO-6 x ; ;$(+N;j
x {QN’.+,.w + Q,,c,.,v},
(154
where Qw+,.N’. =
VW+I.N.
x I(x,sN’.+I (R)(D(R>~x,wW>(* x LGZ+ &jJ
(15b)
Qw _,,w
vw _ ,.,v
x I(x~,N.‘-,(R)ID(R)~x~“N’.(R))I* x VP,+ %,J
(15c)
and =
Table 2. Potential energies for the 0, B3Z; state R (a,,) 0.1795240 +1+ 0.1833035 +l 0.1870829 +l 0.1908624 +l 0.1946418 +l 0.1984213 +l 0.2022008 +l 0.2059802 +l 0.2097597 +1 0.2135391 +1 0.2173186 +l 0.2210980 +l 0.2248775 +l 0.2286569 +l 0.2324364 +l 0.2362158 +l 0.2399953 +1 0.2437747 +l 0.2541380 +l 0.2551113 +l 0.2562356 +l 0.2575259 +1 0.2590356 +l 0.2607859 +l 0.2628232 +l 0.2651812 +l 0.2679747 +l 0.2713182 +l 0.2754653 +l
V(R)t(a.u.) t0.1896354 +0.1524815 +0.1247345 +0.1059804 +0.9279569 -1 t0.8348717 -1 +0.7704655 -1 +0.7350726 -1 +0.7202481 -1 +0.7089404 -1 +0.6964457 -1 +0.6615012 -1 +0.5902670 -1 +0.4943889 -1 +0.3912897 -1 +0.2909246 -1 +0.1980784 -1 +0.1148567 -1 -0.6118781 -2 -0.7692308 -2 -0.9460505 -2 -0.1141362 -1 -0.1353818 -1 -0.1581754 -1 -0.1824744 -1 -0.2080680 -1 -0.2350191 -1 -0.2631541 -1 -0.2923718 -1
tEnergy above O(jP) + O(‘D). SRead, e.g., as 0.1795240 x 10’.
R (a,,) 0.2809420 +l 0.2896516 +l 0.3031613 +l 0.3189504 cl 0.3323837 +I 0.3428314 +l 0.3522586 +l 0.3612630 +l 0.3701847 +l 0.3792048 +l 0.3886126 +l 0.3985462 +l 0.4092581 +I 0.4209680 +l 0.4338827 +l 0.4410442 +l 0.4487473 +I 0.4570745 +1 0.4661173 +1 0.4759752 +1 0.4867532 +I 0.5115898 +l 0.5410834 +l 0.5764620 +l 0.6190967 +l 0.6726680 +l 0.7449987 +l 0.8593388+l 0.11ooooo+2
V(R)to -0.3226786 -1 -0.354OQ64-1 -0.3700498 -1 -0.3540064 -1 -0.3226786 -1 -0.2923718 -1 -0.2631541 -1 -0.2350191 -1 -0.2080680 - 1 -0.1824744 -1 -0.1581754 -1 -0.1353818 -1 -0.1141362 -1 -0.9460505 -2 -0.7692200 -2 -0.6879600 -2 -0.6117ooO-2 -0.5405600 -2 -0.4746300 -2 -0.4139400 -2 -0.3584700 -2 -0.2628441 -2 -0.1858431 -2 -0.1253216 -2 -0.7879281 -3 -0.4429217 -3 -0.2032583 -3 -0.5781992 -4 -0.2804634 -5
Oscillator strengths of the S-R bands of isotopic oxygen molecules
229
The Ho&London factors for the AN = f 1 transitions were taken to be those for a coupling case intermediate between Hund’s cases (a) and (bJz3 Accurate molecular parameters for the B states of the three isotopic oxygen molecules are taken from Cheung et a1.7.‘7,24 As for the ground X state, spectroscopic constants for the u = 0 level of all three isotopic molecules are taken from Steinbach and Gordy.2s For the u = 1 level of the homonuclear molecules, the values reported by Endo and Mizushimaz6 are employed. For the u = 1,2 levels of the heteronuclear molecule and the u = 2 level of the homonuclear molecules, the values predicted by Mizushima” are utilized. We have also calculated band oscillator strengths using case (b) H&l-London factors, whereby23 SRI+ SR~,~= N” + 1
and
SP, + SPY,,= N”.
(16)
Although rotational line oscillator strengths (especially for low N) are sensitive to the coupling case employed, the band oscillator strengths calculated using case (b) and using intermediate coupling rotational line strengths differ by ~0.2%; this is true even at high u’ where it is expected23 that, because the spin-splitting constant I increases with U’ while the rotational constant B decreases,” the intermediate case coupling will be important. Therefore, for the u’ = 0 and 1 levels of the ‘60’s0 and ‘sO2 isotopic molecules, for which accurate constants are not available, the adoption of case (b) Honl-London factors in the calculation offD,csfintroduces an error of ~0.2%. The X3C; state vibrational level spectroscopic constants are also used in the determination of the Boltzmann rotational population distribution; that is, for the F2 component (see, for example, Cheung et al”), E(N~) = BN"(N" + 1)- D[N~(N"+ ip+;n -7 +$vy~"
+ i)1, -NAN"+
I)?,. (17)
All the oscillator strengths we report refer to T = 300 K. The transition frequencies that appear in the line oscillator strength expressions are computed from the X and B state eigenvalues. The calculations of band oscillator strengths for the heteronuclear and homonuclear molecules differ only in that, for the ‘60’80 isotopic molecule, due to the distinguishability of the nuclei, the summation of rotational line oscillator strengths includes initial rotational levels with both even and odd values of N. In the homonuclear molecules, only odd N initial rotational levels are populated. In two recent publications, ” theoretical band oscillator strengthsf,,, have been reported for 1602 (u’ = O-14) 1802 (u’ = O-14) and ‘60’80 (u’ = O-12) and have been compared with experimental values. In Table 3, we have extended the calculation to higher u’ levels and we also have computed hot band (u” = 1,2) oscillator strengths. Lewis et al9 and Berzins** reported values off,,’ for the 1602 and 1802 isotopic molecules and we list them in Table 3. The experimental numbers are systematically larger than the theoretical ones with an average difference of 9.7% for 1602and 7.3% for 1802. Hudson and Carter29 have determined oscillator strengths of the S-R bands of 1602with 2~’= 5-13 and u” = O-2 by recording photoelectric spectra at 300,600, and 900 K. Their u” = 1 and 2 band oscillator strengths are, on average, 15 and 24%, respectively, larger than the present theoretical values. An earlier theoretical calculation by Allison et a130for the S-R bands of 1602 with u’ = O-21 and u” = O-2 resulted in band oscillator strengths consistently larger than the values reported here, by, on average, 1415%. The present calculation should be more accurate for two reasons. First, our RKR potential for the B state has been constructed from more accurate spectroscopic constants. And secondly, the dipole moment of Allison et a130is derived from low resolution experimental absorption and emission studies using the R-centroid approximation. For all three isotopic molecules, the band oscillator strengths, for fixed u”, initially increase with u’, reach a maximum at u’ = 13, 14, or 15 and then decrease. This behaviour can be explained by examining the initial and final state radial wave functions. In Figs. 1 and 2 we present the radial wave functions for 160, for (u’ = 14, N’ = 2; u” = 0, N” = 1) and (u’ = 0, N’ = 2; u” = 0, N” = 1). As u’ increases, the upper state wave function penetrates into regions of smaller nuclear separation, increasing its overlap with the ground state wave function. In Fig. 3 we present the 1602radial wave functions (u’ = 17, N’ = 2; u” = 0, N” = 1). Although the u’ = 17 wave function does indeed penetrate into a region of slightly smaller internuclear distance than does the u’ = 14 wave function, the amplitude of the u’ = 17 wave function is smaller. There is also more destructive interference in the overlap between the ground state and u’ = 17 wave functions than in the (u’ = 14, u” = 0)
R. S. FRIEDMAN
230
overlap. Both of these factors, but primarily the first, cause a smaller Franck-Condon factor and smaller band oscillator strength. In agreement with the experimental band oscillator strengths of Lewis et al,9 the theoretical&, for the 1602isotopic molecule reaches a maximum value at v’ = 14. Where comparisons can be made, the maximum experimental and the maximum theoretical band oscillator strengths, for fixed v”, are usually found at the same value of v’. For fixed v’, the theoretical and experimental band oscillator strengths increase as u” increases. With increasing v”, the position of the outer classical turning point moves to larger internuclear distances, allowing the radial wave function to penetrate into regions of larger R,thereby increasing the overlap of the initial and final state wave functions. Table
3. Band oscillator
strengths
1;,,,
‘602
2.471 2.987 1.858 7.916 2.593 6.955 1.588 3.174 5.655 9.121 1.344 1.821 2.265 2.584 2.719 2.645 2.416 2.074
”
1 :
4 5 !: !
10 11 12 13 14 15 16 17
tTaken #Read,
g&
meay -lO$ -9 -8 -8 -7 -7 -6 -6 -6 -6 -5 -5 -5 -5 -5 -5 -5 -5
7:438 4.198 1.624 4.840 1.183 2.467 4.515 7.389 1.099 1.501 1.895 2.209 2.381 2.386 2.228 1.969 1.646
” =2
1
I
v”=O
v’
meaty 8952 8 8:800 :7 4.460 -6 1.551 -5 4.156 -5 9.146 -5 1.720 -4 2.843 -4 4.215 -4 5.700 4 7.105 -4 8.237 -4 8.877 -4 8.915 -4 8.403 -4 7.453 -4 6.315 -4 5.104 -4
Expt
-8 -7 -6 -6 -5 -5 -5 -5 -4 -4 4 -4 -4 -4 -4 -4 -4
1.76 5.03 1.24 2.68 5.03 8.57 1.20 1.71 2.05 2.61 2.58 2.93 2.52 2.09
-6 -6 -5 -5 -5 -5 -4 -4 -4 -4 -4 -4 -4 -4
from Reference 9. e.g., as 2.471 x IO-‘“. ‘60% v”=l
v-=0
v’
i!zT!F
0 1 2
v”=2
*
1.116 4.888 1.644 4.530 1.062 2.178 3.984 6.5% 9.986 1.393 1.790 2.114 2.310 2.331 2.202
43 2 7 ; 10 11 12 13 14 15 16
-8 -8 -7 -7 -6 -6 -6 -6 -6 -5 -5 -5 -5 -5 -5
21693 1.075 3.299 8.308 1.784 3.362 5.667 8.684 1.223 1.595 1.932 2.166 2.267 2.212 2.038
-7 -6 -6 -6 -5 -5 -5 -5 -4 -4 -4 -4 -4 -4 -4
zz? 2.967 1.067 2.955 6.719 1.304 2.225 3.405 4.750 6.115 7.322 8.190 8.541 8.385 7.742 6.813
-6 -5 -5 -5 -4 -4 -4 -4 -4 -4 -4 -4 -4 -4 -4
‘802 v’ U : : :
I ; 10 11 12 13 14 15 16 17 18
tTaken
v”=l
0
G&y
Theory 2.212 -9 2.586 -8 1.554 -7 6.400 -7 2.026 -6 5.258 -6 1.163 -5 2.254 -5 3.906 -5 6.147 -5 8.887 -5 1.189 -4 1.481 -4 1.711 -4 1.841 -4 1.850 -4 1.746 -4 1.566 -4 1.333 -4
3 559 9:695 1:; 6.388 -9 2.880 -8 9.970 -8 2.825 -1 6.811 -7 1.436 -6 2.700 -6 4.595 -6 7.156 -6 1.027 -5 1.365 -5 1.672 -5 1.8% -5 1.993 -5 1.954 -5 1.808 -5 1.578 -5
from Reference
28
1
Expt
I
5.51 -6 1.27 -5 2.46 -5 4.32 -5 6.82 -5 1.02 -4 1.36 -4 1.63 -4 1.1 -4 1.9 -4 1.9 -4
v”=2 Theory 3.113 -8 3.288 -7 1.787 -6 6.657 -6 1.908 -5 4.487 -5 9.003 -5 1.586 -4 2.503 -4 3.597 -4 4.166 -4 5.871 -4 6.165 -4 7.273 -4 7.343 -4 6.978 -4 6.282 -4 5.420 -4 4.469 -4
231
Oscillator strengths of the S-R bands of isotopic oxygen molecules
-1.5
2
3 internuclear
2.5
4 (a,)
3.5 distance
5
4.5
55
Fig. I. Radial wave functions for the X and B states of 1602.The dotted line represents the ground state (1”’= 0, N” = I) wave function and the solid line represents the upper state (~1’= 14, N’ = 2) wave function.
As observed,‘.” the band oscillator strength increases as the reduced mass of the isotopic molecule decreases. The heavier the isotopic molecule, the deeper in the potential well lies the vibrational level; therefore the radial wave function is constrained to a smaller range of internuclear distances. The smaller the reduced mass, the greater the amplitude of the ground and excited state
I,
2.5,,,,,/,,,,,,,,,,,,,:,,
.;I,/ , 0 1.5
( ~ j
,
,
i,
,JJ , /, ,yL,,, ,i
/’
2
2.5 Internuclear
3 distance
3.5
4
(a,)
Fig. 2. Radial wave functions for the X and B states of 1602.The dotted line represents the ground state (L.”= 0, N” = I) wave function and the solid line represents the upper state (u’ = 0, N’ = 2) wave function.
R. S. FRIEDMAN
232
; -1”““““““““” 2
3
4
internuclear
distance
?a01
6
7
Fig. 3. Radial wave functions for the X and E states of “Oz. The dotted line represents the ground state (P” = 0, N” = I) wave function and the solid line represents the upper state (D’ = 17, N’ = 2) wave function.
Table 4. Absorption
intensity
ratios
‘%-“0’“O.
(v’. v”=O) Fran&-Con&n Factor Ratios
v’ U
1
32 4 5 6 I
8 9 :: :z
14
1.712 1.665 1.619 1.577 1.535 1.495 1.451 1.419 1.383 1.346
tzz 1:222 1.177 1.135 1.097
1761 1:710 1.661 1.615 1.572 1.530 1.490 1.451 1.413 1.377 1.340 1.304 1.262 1.220 1.177 1.136 1.100
(v’, V=l) Franck-Condon Factor Ratios
1.51 1.47 1.42 1.38 1.34 1.30 1.26 1.23 1.19 1.14 1.10 1.05 1.01 0.97
171 1:66 1.61 1.56 1.51 1.47 1.42 1.39 1.35 1.31 1.27 1.24 1.20 1.16 1.12 1.09 1.05
Oscillator strengths of the S-R bands of isotopic oxygen molecules
233
radial wave functions in the 2.3-3.0a, region. The increase in the overlap of the Xand B state radial wave functions with decreasing reduced mass has also been discussed by Lewis et al.’ We have shown previously” that the theoretical ratios of the band oscillator strengths&’ of the isotopic molecules 1602and “02 agree closely with the ratios of the rotationless Franck-Condon factors, implying that the effect of the dipole moment function on the ratios is small. In Table 4, we make a similar comparison for 160, and ‘60’80; the ratios of the Franck-Condon factors and the ratios of the band oscillator strengths are also in close agreement. The very small disagreement Table 5. Line oscillator strengths
for (16,O) bands of 0,.
‘602 R
9
25 27 29
1.0232 1.1610 1.1975 1.1991 1.1820 1.1523 1.1128 1.0655 1.0115 9.5208 8.8809 8.2040 7.4984 6.7719 6.0315
-5’t -5 -5 -5 -5 -5 -5 -5 -5 -6 -6 -6 -6 -6 -6
-ztk1:3797 1.3336 1.2842 1.2310 1.1736 1.1120 1.0466
-5 -5 -5 -5 -5 -5 -5
_ _ _
9.7711 -6
9.0613 8.3215 7.5650 6.7979 6.0263 5.2561
-6 -6 -6 -6 -6 -6
tRead, e.g., as 1.0232 x 10m5.
1% P2
15 17 19 ;: 22: 29 71
8.2022 9.3243 9.6466 9.7007 9.6151 9.4369 9.1888 8.8840 8.5319 8.1397 7.7139 7.2606 6.7852 6.2933
R2
-6 a -6 a -6 -6 -6 -6 -6 -6 -6 -6 -6 -6
5.7901 -6
10738 1:1395 1.1117 1.0793 1.0451 1.0087 9.6964 9.2788 8.8351 8.3674 7.8789 7.3732 6.8544 6.3266 5.7943 5.2616
5 :5 -5 -5 -5 -5 -6 -6 -6 -6 -6 -6 -6 -6 -6 -6
R2
26 27 E
7.3330 9.2911 1.0126 1.0552 1.0783 1.0901 1.0946 1.0939 1.0893 1.0814 1.0708 1.0579 1.0429 1.0260 1.0075 9.8737
-6 -6 -5 -5 -5 -5 -5 -5 -5 -5 -5 -5 -5 -5 -5 -6
KEY 9:1916 8.9414 8.6815 8.4128 8.1362 7.8526 7.5628 7.2677 6.9681 6.6647 6.3582
f -6 -6 -6 -6 -6 -6 2 d d -6 -6
6.0494 -6
12163 1:2915 1.2895 1.2744 1.2561
5 15 -5 -5 -5 1.2368 -5 1.2169 -5 1.1964 -5 1.1752 -5 1.1532 -5 1.1304 -5 1.1068 -5 1.0823 -5 1.0570 -5 1.0309 -5 1.0039 -5 9.7623 -6 ;I;;: 8:8905 8.5882 8.2810 7.9697 7.6548 7.3370 7.0169 6.6951 6.3723 6.0490 5.7257 5.4031
:; -6 -6 -6 -6 -6 -6 -6 -6 -6 -6 -6 -6
234
R. S.
FRIEDMAN
for the ratios for the v” = 1 progression should not be attributed to the variation of the dipole moment (the band strength ratios agree very closely with the Franck-Condon factor ratios) but rather to the N-dependence of the rotational oscillator strengths. The dependence of the line oscillator strengths on the rotational quantum number is responsible for the temperature dependence of the band oscillator strengths. We have calculated” that the&, at 300 K are about 3% smaller than those at 79 K due to the decrease in rotational oscillator strengths within a given band with increasing N. This decrease is also responsible for the observed N-dependence of the equivalent vibrational band oscillator strengths.9,3’ To explore this further, we have calculated P2 and R, rotational line oscillator strengths for all the bands listed in Table 3. In Table 5, we present the results for the (16,O) band of the three isotopic oxygen molecules. Although the vibrational band is degraded to the red (that is, vVN..decreases with N), the primary reason that the line oscillator strengths decrease is the accompanying decrease with N of the band strength (and the dipole moment matrix element). [The increase in SMw/(2N” + 1) is responsible for the observed increase in fwN'. at low N”.] In Fig. 4, we plot the square of the dipole moment matrix element as a function of N for the (16,O) R, lines of the three isotopic molecules. As the reduced mass increases, the centrifugal energy is reduced and the decrease of the band strength with N is less. The decrease in the overlap of the X and B state radial wave functions with increasing angular momentum has been addressed by Lewis et al9 in their discussion of the variation of the measured equivalent vibrational band oscillator strength with N. To explore this further, we look in detail at the radial wave functions. We plot the (v’ = 16, N’ = 2; v” = 0, N” = 1) and (v’ = 16, N’ = 32; v” = 0, N” = 31) radial wave functions of 160, in Figs. 5 and 6. The first peak of the upper state N’ = 32 wave function is of slightly lower amplitude than that of the N’ = 2 wave function. This difference, albeit a small one, is enough to cause the band strength for high N” to be less than that for low N”. A similar argument involving the ground and excited state wave function overlap explains why the band strength for
Fig. 4. Square of the dipole moment matrix element (in atomic units) between the X (0” = 0, N”) and B (c’ = 16, N’ = N” + I) rovibronic states as a function of N”. The solid, dashed and dotted lines correspond to 1602, ‘601*0, and ‘*O,, respectively.
Oscillator strengths of the S-R bands of isotopic oxygen molecules
235
L
-l~‘!““‘!““““r”“““!““li 2
3
4
5
internuclear
distance
(a,,)
6
7
8
Fig. 5. Radial wave functions for the X and E states of 160,. The dotted line represents the ground state (t ” = 0, N” = 1) wave function and the solid line represents the upper state (c’ = 16. N’ = 2) wave function.
2.5 n
,
III
2
1.5
I,,,,
:.> : :: :: :j i i. :: i i : :
1
1
!
:
:
:
;
i
:
t
;
i
:
:
:
:
:
;
;
:
1 ;
: :
i
:
:
;
:
:
I
/
/ , , , / , , , , , , , I /
I I I I
I I I I
.5 I t
-1”’
! ’ ” 2
I / 3
’ ”
1” 4
”
internuclear
1 “‘I
1 “I’
1”
5
6
7
distance
(a,)
”
”
’ ”
8
Fig. 6. Radial wave functions for the X and B states of 1602.The dotted line represents the ground state (1.”= 0, N” = 31) wave function and the solid line represents the upper state (~1’= 16, N’ = 32) wave function.
9
236
R. S. FRIEDMAN
the R, branch is less than that for the Pz branch of the same N”‘. However, the R2 H&r-London factor is larger than the Pz H&l-London factor for all N” and as a result, R, line oscillator strengths are larger than P2 line oscillator strengths at small N. Even though we have omitted spin-dependent terms from the radial Schrodinger equation, we still can calculate rotational line oscillator strengths for all 14 allowed branches by assuming that oscillator strengths for branches pertaining to the same N values but different J values (for example, R,(N"), Rz(N"), and R,(N")) differ only because of the S,.,./(ZJ” + 1) factor. Oscillator strengths for some low N” rovibronic lines of the (13,0)-( 16,0) bands of 1602have been measured by Smith et a1.32For low N”, it is important that intermediate case coupling rotational line strengths be used. The largest discrepancy between the experimental and calculated transition frequencies is found to be 5.5 cm-‘, a difference that introduces little error into the calculatedf,, . In Table 6 we present the theoretical and experimental line oscillator strengths. The agreement is good and is usually within the lo-20% uncertainty that is typical of the experimental values. Table 6. Rotational oscillator strengths of lines in the (v’, 1%” = 0) bands of “0 2 fYJ
-SF 2.463 0.102 1.226 0.213 1.533 1.383 0.315 1.151 1.057 0.760 0.250 1.474 1.471 1.474 0 902 2:703 0.141 1.345 0.227 1.607 1.471 0.253 1 551 1:563 1.534 0 850 21742 0.179 1.363 0.223 1.537 1.447 0.168 1.149 1.106 0.708 0.261 1.501 1.536 1.458 0.059 0.747 2.598 0.206 1.291 1.366 1.339 1.023 0.610 1.353 1.419 1.282 0.088
tTaken from Reference 32
6) -+%?--2173 0.10 1.11 0.20 1.39 1.31 0.32 1.01 0.95 0.68 0.25 1.29 1.17 1.17 15 fL:91 0.14 1.30 0.24 1.62 1.46 0.27 33 ::49 1.47 0 90 3:38 0.24 1.46 0.23 1.64 1.73 0.78 1.19 1.18 0.93 0.34 1.45 1.45 1.38 0.11 0.85 2.80 0.32 1.43 1.46 1.47 06 ;:62 1.39 1.40 1.28 0.10
Oscillator strengths of the S-R bands of isotopic oxygen molecules
237
We also can compare theoretical and experimental values of equivalent vibrational band oscillator strengths. Lewis et a13’have recently measured rotational line oscillator strengths for P, and R, branches in the (17,0) band of i602 using high-order anti-Stokes Raman shifted radiation in Hz of 0.15 cm-’ FWHM bandwidth. Values of f (v’ = 17, v” = 0, N’, N”) have then been determined from the measured line oscillator strengths using Eq. (8). The measured equivalent band oscillator strengths decrease significantly with rotation, and oscillator strengths that are derived from R-branch measurements are slightly less than those derived from P-branch measurements. These observed results are consistent with the theoretical calculations. Our theoretical (17,O) equivalent vibrational band oscillator strengths are plotted in Fig. 7. The agreement between theory and experiment is very good; for example, Lewis et a13’ perform a least-squares fit to their measurements giving f(17,0,
N’, N”)=(2.27_+0.06)
while we find for the P-branch
x 10-5-(1.89f0.14)
x 10-*x N”(N”+
l),
(18a)
theoretical values,
f (17,0, N’, N”) = (2.36) x lo-‘-
(1.62) x 1O-8 x N”(N” + l),
(18b)
and for the R-branch,
CONCLUSIONS In this paper, we have reported theoretical calculations of Schumann-Runge band oscillator strengths for 160, (v’ = O-17, v” = O-2), 1802(v’ = O-18, v” = O-2), and ‘60’80 (v’ = O-16, v” = O-2); several rotational line and equivalent vibrational band oscillator strengths have also been given. The agreement between theoretical and measured values is good. We have also described in some I’
l
I
l
I
I
I
I
1’1
I
I
’
I
1
I
’
I
’
I
’
I
I
I’1
..
1.2x1o-s
.‘.,
-
..
‘.._ ‘.._
lO-s-
1 0
’ 50
11
11 100
11 150
1 200
I 250
”
1 300
” 350
11 400
11 450
11 500
11 550
11 600
._
A . .._ 11 650
N”(N”+ 1) Fig. 7. Theoretical equivalent vibrational band oscillator strengths of the (17.0) band of 160z, as a function of rotational quantum number. The open squares represent P-branch values and the open triangles represent R-branch values, The solid and dotted lines are the best linear least-squares fits to the calculated values.
/ 700
238
R. S. FRIEDMAN
detail the dependence of the oscillator strength on the initial and final vibrational level, the reduced mass of the isotopic molecule, and the rotational quantum number. Our calculation of S-R oscillator strengths has invoked a single channel representation of the upper B3Z; state although the upper state wave function also contains contributions from repulsive continuum states which are responsible for the observed predissociation. However, a single channel description is justified because the couplings between the B state and the continuum states are weak;4,5.7the amount of intensity alteration in the S-R bands should therefore be minimal. Acknowledgements-1 am pleased to acknowledge useful discussions with A. Dalgarno, K. Yoshino and D. Freeman. I am also very grateful to B. R. Lewis for communication of unpublished work on measured oscillator strengths. The work was supported by the National Science Foundation under Grant ATM 87-13204 to Harvard University.
REFERENCES 1. P. H. Krupenie, J. Phys. Chem. Re$ Data 1, 423 (1972). 2. D. M. Creek and R. W. Nicholls, Proc. R. Sot. Lond. A 341, 517 (1975). 3. K. P. Huber and G. Herzberg, Molecular Spectra and Molecular Structure. IV: Constants of Diatomic Molecules, Van Nostrand-Reinhold, New York, NY (1979). 4. P. S. Julienne and M. Krauss, J. Molec. Spectrosc. 56, 270 (1975). 5. P. S. Julienne, J. Molec. Spectrosc. 63, 60 (1976). 6. M. L. Sink and A. D. Bandrauk, J. Chem. Phys. 66, 5313 (1977). 7. A. S-C. Cheung. K. Yoshino, D. E. Freeman. R. S. Friedman, A. Dalgarno, and W. H. Parkinson, J. Molec. Spectrosc. 134, 362 (1989). 8. B. R. Lewis, L. Berzins, J. H. Carver, and S. T. Gibson, JQSRT 36, 187 (1986); B. R. Lewis, L. Berzins, and J. H. Carver, JQSRT 37, 229, 243 (1987). 9. B. R. Lewis, L. Berzins, and J. H. Carver, JQSRT 36, 209 (1986); B. R. Lewis, L. Berzins, and J. H. Carver, JQSRT 37, 219, 255 (1987). 10. K. Yoshino, D. E. Freeman, J. R. Esmond, and W. H. Parkinson, Planet. Space Sci. 31, 339 (1983); ibid. 35, 1067 (1987). 1 I. K. Yoshino, D. E. Freeman, J. R. Esmond, R. S. Friedman, and W. H. Parkinson, Planet. Space Sci. 36, 1201 (1988); ibid. 37, 419 (1989). 12. C. E. Fairchild, E. J. Stone, and G. M. Lawrence, J. Chem. Phys. 69, 3632 (1978); R. K. Sparks, L. R. Carlson, K. Shobatake, M. L. Kowalczyk, and Y. T. Lee, J. Chem. Phys. 72, 1401 (1980). 13. A. M. Wodtke, L. Huwel, H. Schhiter, H. Voges, G. Meijer, and P. Andresen, J. Chem. Phys. 89, 1929 (1988); A. M. Wodtke, L. Huwel, H. Schliiter, H. Voges, G. Meijer, and P. Andersen, 22nd Symp. (Znt.) on Combustion, The Combustion Institute, Pittsburgh, PA (1988). 14. A. Schadee, JQSRT 19, 451 (1978).
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