A table of energy level patterns for rigid asymmetric rotors is given, by means of which this approximation to the rotational energies of all molecules up to J = 10 may be readily evaluated. The symmetry classification of each level is determined and expressed in terms of the K values of the limiting prolate‐ and oblate‐symmetric rotors. A simple method is developed for calculating the transformation which diagonalizes the energy matrix and is applied to the derivation of perturbation formulas.
REFERENCES
1.
2.
3.
H. A.
Kramers
and G. P.
Ittmann
, 58
, 217
(1929
);H. A.
Kramers
and G. P.
Ittmann
, 60
, 663
(1930
).4.
5.
H. B. G. Casimir, Rotation of a Rigid Body in Quantum Mechanics (J. B. Wolter’s, The Hague, 1931).
6.
7.
8.
9.
10.
We may, therefore, make use of the symmetry determinations of Mulliken in our work. The actual phases of the symmetric rotor wave functions do not affect the energy, but must be clearly defined when symmetry is concerned. The confusion resulting from the many arbitrary choices of signs, phases, and “Eulerian” angles in the different quantum mechanical formulations has been cleared up by Van Vleck, especially in footnotes 9, 20, 21, 25, and 29. The phase to be determined here has only to be consistent with that of previous work which we wish to use. From the rigid phase relation between and given by (5), we see that has the same real, imaginary, or complex character as This quantity is very simply expressed in the Eulerian angles used by Mulliken Applying this operator to the symmetric rotor wave functions used by Mulliken and first clearly defined by Van Vleck, we find and hence to be imaginary. The complex conjugate must be taken on the first factor.
11.
12.
See Mulliken, reference 9, Eq. (3), and Van Vleck, reference 10, footnote 25.
13.
See Mulliken, reference 9, Appendix I and Table III. The identifications are made by examining the behavior of the under the operations of the group , , .
14.
It may, however, be displayed as two submatrices whose indices involve, respectively, only even and only odd K’s.
15.
That the symmetry classification in is given uniquely in terms of the parity of follows from the assignments of a, b, c to x, y, z for these cases ( and respectively). From Tables VI and V, even requires the symmetry A or or odd, or or On the other hand, even requires A or or odd, or or Hence it must follow that and It is immaterial whether right‐ or left‐handed Types I and III are chosen for the prolate and oblate representations.
16.
We are using and as the absolute magnitude of the corresponding limiting K values, as is customary in the labeling of symmetric rotor energy levels.
17.
18.
corrected and extended through by
H. M.
Randall
, D. M.
Dennison
, Nathan
Ginsburg
, and Louis R.
Weber
, Phys. Rev.
52
, 160
(1937
). a The term “orthonormal” as used here means that the rows and columns are orthogonal and normalized. The term “orthogonal” as used here means that the columns are orthogonal but not normalized (hence the rows are not orthogonal).19.
Note that the f’s used here with running subscript indices for each submatrix are not the ’s of Table I, although they are closely related.
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© 1943 American Institute of Physics.
1943
American Institute of Physics
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