A near-wing correction to the quasistatic far-wing line shape theory Q. Ma Department of AppIied Physics, Columbia University and Institute for Space Studies, Goddard Space Flight Center, New York New York 10025 R. H. Tipping Department of Physics & Astronomy, University of Alabama, Tuscaloosa, Alabama 35487 (Received 3 September 1993; accepted 10 November 1993) A new representation is introduced in which the rapidly varying time-dependent part of the time displacement operator can be factored out and the remaining part, which varies with time more slowly, can be expanded in the usual perturbational fashion. The lowest order approximation leads to the far-wing quasistatic line shape theory developed previously, whereas the next order approximation, related to the noncommutation of the Liouville operators describing the * unperturbed absorber and bath molecules and the interaction between them, leads to a near-wing correction. Explicit expressions are derived for both the corrections to the spectral density and the statistical band-average line shape function assuming an anisotropic dipole- dipole interaction. Detailed computations for the case of self-broadened Hz0 are carried out for the line-shapes and the corresponding absorption coefficients for several temperatures and for frequencies to 10 000 cm-‘. From these results, we conclude that the near-wing corrections generally increase the line shape function between 10 and 200 cm-‘, and that this increase is more important for lower temperatures than for higher ones. This in turn leads to increased absorption nearer the band centers, especially for lower temperatures, and thus to improved agreement between theory and experiment. I. INTRODUCTION In a series of recent paperslm3 we have developed a far-wing line shape theory based mainly on the quasistatic and binary collision approximations. We have applied this theory to calculate the absorption coefficient of pure water vapor as a function of frequency and temperature,tf2 and subsequently to water broadened by N2 and C02.3 More recently, we have’attempted to assess the accuracy of these results by comparisons between theory and experiment-for high temperature laboratory water vapor data4 and for at- mospheric transmission measurements.5 As a result of this work, we are able to draw some general conclusions re- garding the theoretical results: ( 1) Given that experimen- tal measurements of the absorption coefficients for frequen- cies close to the band centers and those in the window regions between the bands differ by many orders of mag- nitude and are difficult to measure accurately,69 the over- all agreement between theory and experiment is remark- ably good; (2) In general; this agreement is better in the window regions (especially in the important atmospheric window around 1000 cm-‘) than near the band centers where the theoretical results predict consistently less ab- sorption than that measured; (3) The agreement is better for higher temperatures than for lower ones. tion must be taken into account in the near-wing region in order to refine the theory. This correction arises because of the noncommutation of the Hamiltonian describing the un- perturbed absorber and bath molecules and that describing the interaction between them. We expect that this correc- tion will improve the line shape in the near-wing region and that this modification will be relatively more important for lower temperatures. The general formalism for the calculation of the spec- tral density and the absorption coefficient in terms of the time displacement operator is given in Sec. II A. By ex- pressing the time displacement operator in a new represen- tation in which the rapidly varying time-dependent part can be factored out, one is able as discussed in Sec. II B to obtain an approximate expansion for the remaining part which varies with time more slowly. This expansion is used in Sets. II C and II D to obtain more accurate approxima- tions for the spectral density and the absorption coefficient, respectively. The computational details and results for the continuum absorption of pure water vapor for several tem- peratures are given in Sec. III. There we also discuss the conclusions drawn from the present work and possible im- provements of the theory. In an effort to improve the agreement between theory and experiment discussed above, we improve one of the approximations made in our previous papers, i.e., the re- placement of the time displacement Liouville operator re- lated to the total Hamiltonian of the absorber and the bath molecules by an ordered product. of two Liouville opera- tors, one related to the’unperturbed Hamiltonian and the other to the interaction potential. This approximation has been justified in the far-wing region.’ However, its correc- A. The absorption coefficient and spectral density II. THE GENERAL FORMALISM For a low-density gas .sample we can divide it into absorber molecules and the remaining bath molecules and we can focus on one absorber molecule only. Then, with n, absorber molecules per unit volume, the absorption coeffi- cient per unit volume can be expressed by J. Chem. Phys. 100 (4), 15 February 1994 0021-9606/94/l 00(4)/2537/l O/$6.00 @ 1994 American Institute of Physics 2537 Downloaded 06 Dec 2003 to 131.215.252.208. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/jcpo/jcpcr.jsp2538 Q. Ma and R. H. Tipping: Quasistatic far-wing line shape theory ar(o) =g np tanh(tiw12kT) [F(w) +F(--o) 1. (1) In this expression, F(w), the spectral density, is given by F(o& ; J f?‘“f(pyqO> *j.p(f))df m =:ReTr J m P’(jp(O) */p(f))& 0 (2) where the angular brackets denote the ensemble average over all (one absorber molecule plus bath) variables. The dipole operator of an absorber molecule in the Heisenberg representation, P$~ (t) , is determined by the time displace-- ment operator U(t) &y(t) = uQ)fp(O) =e iHt/fip~EO (~)~-iHf/fi = eiLtpLm (0). (3) In the above expression, H is the total Hamiltonian which consists of the unperturbed Hamiltonian of the absorber molecule and the bath, Ho = H, 4 Hg, and the interaction V between them. In the last step of Eq. (3), the Liouville representation in which the total Liouville operator L cor- responds to the absorber molecule, the bath, and the inter-