Journal of Molecular Structure
Elsevier Publishing Company. Printed in the Netherlands
470
Computerprogramsfor the asymmetricrotor
Computer programs which calculate various spectral characteristics of the
rigid asymmetric rotor have been described beforel - J. The most comprehensive
of these is that of Beaudet2. Alternative methods exist for writing such programs,
and those reported here differ from the above mentioned in many respects and
represent also certain extensions. Listing of the three programs (in Fortran) are
available from the authors upon request.
Calculation of the energy levels
The energy levels of the rigid, asymmetric rotor are given by4
W/h
= !(A+C)J(J+l)+!(A-C)E;(K)
(I)
Ef (K) are the eigenvalues of four symmetrical tridiagonal matrices E +, E-,
0+ and O-. In diagonalizing these matrices various methods have been usedl-6.
For microwave spectroscopic work, procedures yielding E;(K) of very high accuracy should be chosen. Givens' diagonalization technique7 using bisections to
locate the eigenvalues was used in this work. This method fully satisfies the requirement for accuracy. Error analysis7,Bshows that his procedure is exceptionally
stable, and that the accuracy of the eigenvalues is independent of the dimension
of the matrix.
The energymatrices E + , E-, O+ and O- are well suited for this technique
since they are tri diagonal and Givens' method exploits the Sturm sequence property of the principal minors of such matrices. The procedure is easy to program,
and the eigenvalues are found comparatively quickly.
Computation of line strength and angular momenta
To calculate line strengths and angular momenta, the eigenvectors were
needed. They were found using eqns. (47)-(54) of ref. 4.
Schwendeman9 has given the direction cosines in the Wang basis. Their
simple form makes it easy to apply the eigenvectors to them. Having thus
transforrned the direction cosines to the asymmetri c rotor basis, the line strengths
were calculated from eqns. (6) and (7) of ref. 9.
<Pz2) was calculated from the matrix element which in the symmetric rotor
basis is simply K2, and the corresponding eigenveetor. <p/) was obtained fram l o
<p/) = [W-yJ(J+ 1)-(a-y)<Pz2) J/CP-y)
and <p/> from the relation J(J + l) = <p/> +<p/> +<p/>.
J. Mol. Structure, 4 (1969) 470-472
(2)
SHORT
COMMUNICA
471
TIONS
Features of the programs
In the program MB05 the energy leveIs are calculated from egn. (l). They
are then labelled according to the JK-l K+1 convention and subtracted according
to the selection rules and I A. I ~ 6 (ref. Il). A test is perforrned to sort out the
freguencies lying in a specified spectral region. These transitions are listed in the
full J K-l K+ 1 ~ J~-'1K+'1 notation, Their corresponding line strengths are calculated and listed. If the dipole moment components and the absolute temperature
have been given, the relative intensity, R, is computed from egn. (3)2.
R = gS Jtl'
t'
e - w/kTV2/l g 2 . 10 - 7
(3)
MB04 calculates the second order Stark coefficients given by
Av/E2 =
L
(Ag + BgM2)/lg 2
(4)
g=a, b, c
The definition of the varioU3 terms of egn. (4) is found in ref. 12.
The computer generates all "interacting" energy 1evelsand their corresponding line strengths for specified transitions and dipole moment components, and
carries out the summations of egn. (4). Degeneracy or near degeneracy is not
treated, but a test is perforrned and those WJt' which are dose to WJt are not
induded
in egn. (4). However, all energy differences
WJt- WJ't' and their corre-
sponding line strengths are printed using the J K-l
K+ 1 notation.
if the secular eguation for a degeneracy problem
is to be solved.
In MB09
computed from
Eq
the guadrupole
interaction
energy,
They can b~ of use
Eq, of a single nudeus
is
13
= [2f(/, J, F)/J(J+ 1)]LXgg<p/)
g
(5)
The program accepts as input data the rotational and guadrupole coupling
constants, the spin, and the transition labels J, K_l, K+ l' The unperturbed centre
freguencies are calculated from egn. (l). From egns. (l) and (5) the guadrupole
fine structure components are obtained. They are listed according to the usual
F-notation. Their relative intensities are computed from egn. (6-6) of ref. 13 and
written out.
The three programs are general and flexible. They may easily be expanded
or rearranged.
Practical experiences
Calculations have been p<:rformed on the CDC 3300 computer in Oslo
using extended length arithmetic (14-15 significant digits). E;(K) is found to 10-10
or better. Comparisons with existing tables4,6,1l,12-15 have in all cases been in
complete agreement for E;(K), <P;), Ag,Bg, the relative intensities of the guadrupole fine structure components, and Casimir's function.
J. Mol. Sfrucfure, 4 (1969) 470-472
472
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COMMUNICATIONS
Mecke's sum rule13
(6)
'LE;(K) = tJ(J+ 1)(2J+ I)K
was excellently obeyed for J ~ 70.
Test ca1culations using literature data showed that the frequencies obtained
by MB05, the Stark coefficients from MB04, and the quadrupole perturbed
spectra computed by MB09, were in general agreement with the reported ones.
Acknowledgement
One of us, H.M., wishes to thank the Norwegian Council for Science and
the Humanities for a scholarship.
Department of Chemistry,
University of Oslo,
Blindern, Oslo 3 (Norway)
K.-M.
MARSTOKK
HARALD MØLLENDAL
I F. KNEUBuHL, T. GAUMANN AND Hs. H. GUNTHARD, J. Mol. Spectry., 3 (1959) 349.
2 R. A. BEAUDET, Ph.D. Thesis, Harvard
University,
1962, University
Microfilms,
Inc., Ann
Arbor, Michigan, U.S.A.
3 G. F. POLLNOW AND A. J. HOPFINGER, J. Chem. Educ., 45 (1968) 528.
4 G. W. KING, R. M. HAlNER AND P. C. CROSS, J. Chem. Phys., II (1942) 27.
5 J. M. BENNETT, F. G. Ross AND E. G. WELLS, J. Mol. Spectry., 4 (1960) 342.
6 M. SIDRAN, F. NOLAN AND J. W. BLAKER, Grumman Research Reports RE-155 (1962), RE-155
(1963), RE-I72 (1964), RE-189 (1964) and RE-I96 (1964).
7 W. GIVENS,Numerical Computation of the Characteristic Values of a Real Symmetric Matrix,
Oak Ridge National Laboratory,
ORN-1574
(1954); W. GIVENS, Simultaneous
Linear Equations and the Determination
of Eigenvalues,
Natl. Bur. Std. Applied Mathematics
Series, 29
(1953) 117.
8 J. H. WILKINSON, The Algebraic Eigenvalue Problem, C]arendon
Press, Oxford, 1965.
9 R. H. SCHWENDEMAN, J. Mol. Spectry., 7 (1961) 280.
O D. KIVELSON AND E. B. WILSON, JR., J. Chem. Phys., 20 (1952) 1575.
l P. C. CROSS, R. M. HAINER AND G. W. KING, J. Chem. Phys., 12 (1944) 2]0.
2 S. GOLDEN AND E. B. WILSON, JR., J. Chem. Phys., 16 (1948) 669.
3 C. H. TOWNES AND A. L. SCHAWLOW, Microwave Spectroscopy,
McGraw-Hill,
New York,
1955.
4 R. H. SCHWENDEMANAND V. W. LAURIE, Tables of Line Strengths,
Pergamon,
5 R. H. SCHWENDEMAN,A Table of <P;> and <P;> for Asymmetric Rotator Molecules,
of Chemistry,
Michigan State University,
East Lansing, Michigan.
~eceived July 8th, 1969
Mol. Structure, 4 (1969) 470-472
Oxford,
1958.
Department