5366
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J . Phys. Chem. 1985, 89, 5366-5370
Analysis of the Temperature Dependence of the 'A OCS, N20, and CS2
X'Z' Absorption Spectra of
Jeffrey A. Joens Department of Chemistry, Florida International University, Miami, Florida 33199 (Received: May 21, 1985)
The sum rules that apply for electronic/vibrational transitions in molecules are used to derive expressions for the mean frequency of an absorption spectrum for the case of a forbidden electronic transition that becomes allowed due to Herzberg-Teller coupling of the electronic motion with the bending vibration. The results are used to analyze data on the 225-nm system of OCS, the 182-nm system of NzO, and the 320-nm system of CSz, all of which are transitions of the type lA X I B + . For OCS, previous results are reexamined to determine the effect of vibrational excitation of the bending mode on the mean frequency of the absorption spectrum. For N 2 0 , the component absorption spectra, intensities, and mean frequencies are derived from experimental data for the (00'0) and (01*'0) vibrational states. Values for the curvature of the u p p r electronic state potential energy surface in the Franck-Condon region for OCS and NzO are used to construct bending mode potential energy curves which are compared with previous results for analogous transitions in isoelectronic molecules.
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Introduction Changes in the intensity, shape, and average frequency of a continuous vibronic spectrum that accompany vibrational excitation of the ground electronic state of a molecule can be used as a sensitive probe of the magnitude and distribution of vibrational energy. This has practical application in the determination of vibrational energy relaxation rates for highly excited for measurement of specific intermolecular V-V and V-T relaxation rate constants for molecules with low levels of vibrational and for determination of the average nascent vibrational excitation of molecules formed by photolysis or chemical reaction.* Knowledge of the relationship between vibrational excitation and absorption of light also has important applications to the atmospheric chemistry of the Earth9~lo and other planets." Since the properties of a continuous vibronic spectrum depend on the properties of the potential energy surfaces of the ground and excited electronic states, changes in continuous vibronic spectra with vibrational excitation can also be used to determine information on the shape of the excited-state potential energy surface in the Franck-Condon region of a m o l e ~ u l e . ' ~ - ~ ~ A number of theoretical approaches have been used in developing the relationship between properties of continuous absorption spectra and vibrational excitation.16-21 One promising approach has been to use the sum rules that govern electronic/vibrational (1) H. Hippler, J. Troe, and H . J. Wendelken, Chem. Phys. Lett., 84, 257 (1981). (2) H . Hippler, J. Troe, and H. J. Wendelken, J . Chem. Phys., 78, 6709 (1983). (3) H . Hippler, J . Troe, and H. J . Wendelken, J . Chem. Phys., 78, 6718 ( 1983). (4) I. C. McDade, and W. D. McGrath, Chem. Phys. Lett., 72,432 (1980). (5) S. M. Adler-Golden, and J. I. Steinfeld, Chem. Phys. Lett., 76, 479 (1980). (6) S. M. Adler-Golden, E. L. Schweitzer, and J. I . Steinfeld, J . Chem. Phys., 76, 2201 (1982). (7) J. A. Joens, J. B. Burkholder, and E. J. Bair, J . Chem. Phys.. 76, 5902 ( 1982). (8) J. A. Joens, and E. J. Bair, J . Phys. Chem., 88, 6009 (1984). (9) G. Selwyn, J. Podolske, and H. S. Johnston, Geophys. Res. Lett., 4, 427 (1977). (10) C. Hubrich, and F. Stuhl, J . Phorochem., 12, 93 (1980). (11) C. Y . R. Wu, and D. L. Judge, Ceophys. Res. Lett., 8, 769 (1981). (12) R. J. LeRoy, R. G . McDonald, and G. Burns, J . Chem. Phys., 65, 1485 (1976). (13) J. B. Burkholder, and E. J. Bair, J . Phys. Chem., 87, 1859 (1983). (14) J. A. Joens, and E. J. Bair, J . Phys. Chem., 87, 4614 (1983). (15) J. A. Joens, and E. J . Bair, J . Chem. Phys., 79, 5780 (1983). (16) P. Sulzer, and K. Wieland, Helu. Phys. Acta, 25, 653 (1952). (17) R. T. Pack, J . Chem. Phys., 65, 4765 (1976). (18) E. J. Heller, J . Chem. Phys., 68, 2066 (1978). (19) S.-Y. Lee, R. C. Brown, and E. J. Heller, J . Phys. Chem., 87, 2045 ( 1983). (20) E. Grunwald, J . Phys. Chem., 3409 (1981). (21) H. Hippler, J. Troe, and H . J. Wendelken, J . Chem. Phys., 78, 5351 (1983).
0022-3654/85/2089-5366$01.50/0
transitions in molecules to derive expressions for the first and higher moments of an absorption s p e c t r ~ m . ~While ~ ~ ~ this ~-~~ method does not provide detailed information on the shape of the absorption spectrum, it does give a simple relationship between the bulk properties of the spectrum (the spectral moments, defined as ( P )= (l/r)lband[t(V)v"-l] dv, where I = Sbandt(v)/v dv is the band intensity) and the potential energy surface of the excited electronic state. This has proven useful both in the analysis of absorption data6114*23 and in testing the limitations in the assumptions commonly used in more sophisticated theories.15 The purpose of the present paper is to derive an expression for the mean frequency of an absorption spectrum for the case of a forbidden transition in a linear triatomic molecule that becomes allowed due to Herzberg-Teller interaction of the electronic motion with the bending vibration. The results of the derivation are used to analyze temperature-dependent absorbance measurements of the 225-nm system of OCS,the 182-nm system of NzO, and the 320-nm system of CSz. Mean Frequency Sum Rule The expression for the potential energy of the ground electronic state of a linear triatomic molecule in the approximation that the molecular vibrations can be treated as separable harmonic oscillators is w" 2 z Q 1 +T '( Q 2 a '
w'/3
W"1
v"(Ql,Q2a*Q2b,Q3) =
+ QZb2) + (1)
where dI, w " ~ ,and w " ~are the frequencies of the normal mode and Q3 vibrations in the ground electronic state and QI, QZa, are the normal mode displacements. The eigenfunctions for this potential are *"n,
(Q1, Q2ar
.nla.nlb,n3
Q3) = i"'n,(Q,) Vnh(Q2a) 'Y'n2,(Q2b) 'Y'n3(Q3) (2) Q Z ~ ?
where nl, nza, n2b,and n3 are the vibrational quantum numbers, and $",,(Qj) = N,,H,,(Qi) exp(-Q;/2), with H,(Qi) the nth Hermite polynomial and N , a normalization factor for the eigenfunction. It is convenient to make the coordinate transformation Qz = (Qk2 + Qzb2)1/2and @ = arctan (QZb/Qza),as shown in Figure 1. The bending potential can then be written as V'12(Q2) = ~ ' ' ~ Q 2 / 2 , with eigenfunctionsZ5 Vn2m(Q2,@.) = Nn2mQJml e ~ p ( - Q ~ ~ /x2 ) 1Fl(lml - nz/2, Iml + 1; Q22) exp(im@) (3) (22) M. Lax, J . Chem. Phys., 20, 1752 (1952). (23) S. M. Adler-Golden, Chem. Phys., 64,421 (1982). (24) R. D. Coalson and M. Karplus, J . Chem. Phys., 81, 2891 (1984).
0 1985 American Chemical Society
IA
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X’Z+Absorption Spectra of OCS, N20, and CS2
The Journal of Physical Chemistry, Vol. 89, No. 25, 1985 5361 (n2,mIQ2(fi2 - Enz)Q21n2m)= 0 ” ~ / 2
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