Water vapor millimeter wave foreign continuum:
A Lanczos calculation in the coordinate representation
Q. Ma
a)
NASA/Goddard Institute for Space Studies and Department of Applied Physics, Columbia University,
New York, New York 10025
R. H. Tipping
Department of Physics and Astronomy, University of Alabama, Tuscaloosa, Alabama 35487
͑Received 19 July 2002; accepted 4 September 2002͒
The water vapor foreign-continuum absorption has been calculated theoretically from first principles
for the millimeter wave spectral region as a function of frequency f and temperature T. The
calculations are made using the Lanczos algorithm by writing the resolvent operator (
ϪL)
Ϫ1
as
continued fractions. In order to guarantee the quick convergence of the continued fractions, the line
space of H
2
O is divided into two subspaces: one consists of the positive resonance lines and the
other the negative ones. By ignoring the coupling between them, (
ϪL)
Ϫ1
is expressed as a sum
of two continued fractions. The parameters appearing in each of the fractions are functions of the
matrix elements of powers of the Liouville operator L between the starting vectors spanning the
corresponding subspaces. In the present work, we have taken into account all powers of L up to 5.
With the coordinate representation in which the orientations of the H
2
O–N
2
collision pair are chosen
as the basis functions in Hilbert space, the anisotropic interaction potential is diagonal, and
calculations of the matrix elements are transformed to multidimensional integrations. The latter are
evaluated with the Monte Carlo method. In order to reduce the lengthy calculations, we assume that
the anisotropic potential has rotational symmetry about the Z axis of H
2
O, and consists of the
long-range dipole–quadrupole part and a short-range repulsive site–site model. Once the parameters
of the continued fractions are known, one can calculate the poles and residues and then carry out the
ensemble average over the translational motion. Within the quasistatic approximation, one can treat
the latter classically and obtain contributions to the absorption coefficient at the poles. Finally, the
absorption coefficient at frequency f can be derived by an interpolation method. The results are fitted
to a simple function of f and T, and are compared with experimental data and with two different
versions of Liebe’s empirical model. In general, the theoretical results are in good agreement with
the experiment. Meanwhile, the magnitudes of the theoretical absorption are between those of the
1989 and 1993 versions, but the temperature dependence is closer to the latter one. © 2002
American Institute of Physics. ͓DOI: 10.1063/1.1516792͔
I. INTRODUCTION
A good knowledge of the water vapor millimeter wave
foreign-continuum absorption is important for atmospheric
applications, especially in dry air environments. At present,
our understanding of the problem is not satisfactory. Labora-
tory measurements of the foreign continuum made by differ-
ent groups differ by large amounts and various empirical
models proposed differ significantly from each other.
1
Mean-
while, there is a lack of theoretical work heretofore from
which one is able to predict the millimeter wave foreign
continuum quantitatively well. Collision-induced absorption
͑CIA͒ has been proposed to be partly responsible for this
continuum,
2
but there is no unambiguous laboratory evi-
dence or theoretical calculations to support this assertion. On
the other hand, although the recent far-wing line shape
theory works well in calculating continuum absorptions for
the infrared spectral region,
3
its applicability in the millime-
ter wave region is questionable. The main reason for this is
not the far-wing line shape theory itself, but the band average
approximation,
4
a usual procedure introduced to simplify
calculations. There are some alternative theoretical methods
available to calculate the millimeter wave continuum. One is
so-called ‘‘third-order linear absorption’’ which has been ap-
plied for calculating the self-continuum.
5
But this method is
not applicable for the foreign continuum, at least for that
caused by the H
2
O–N
2
collision pairs, because it is limited
to cases in which two interacting molecules undergo transi-
tions by sharing one photon energy cooperatively. The sec-
ond is the Lanczos algorithm, which was used successfully
to calculate the millimeter wave self-continuum.
6
With the
Lanczos algorithm, we showed that one can write the spec-
tral density as a continued fraction, and using the lowest-
order truncation, we can calculate the absorption. However,
an attempt to apply it for calculating the dominant contribu-
tion to the foreign continuum in the atmosphere caused by
the H
2
O–N
2
collision pairs has not been carried out. Because
the molecules are not identical, there is no contribution to the
a͒
Electronic mail: qma@giss.nasa.gov
JOURNAL OF CHEMICAL PHYSICS VOLUME 117, NUMBER 23 15 DECEMBER 2002
105810021-9606/2002/117(23)/10581/16/$19.00 © 2002 American Institute of Physics