An approximate theory of the interactions of hindered internal rotation with over‐all rotations of symmetric rotors is given. This treatment considers the interdependence of hindered internal rotation and vibrations and their effect upon the rotational energy levels. The resulting expression for the frequencies of ΔK = 0, ΔJ = 1 transitions is
where Bv, Fv, Gv, and Lv are constants independent of the rotational quantum numbers, m represents the basis that diagonalizes the Hamiltonian corresponding to pure internal rotation, πz is the internal angular momentum operator, and θ is the angle of internal rotation. Procedures for evaluating (m|1—cos3θ|m) and (mzn|m) in terms of a parameter α are given.

This theory has been applied to the J=0→1 transitions of methyl silane. The parameters Bv, Fv, Gv, and α were obtained empirically and were then used to calculate frequencies. The agreement between observed and calculated values was quite good. Furthermore, the anomalous ordering of the lines observed by Lide and Coles1 is explained by these calculations. Assuming a cosine potential, the barrier height V0 is proportional to the parameter α. The value of V0 was set at 558 cm‐1±17 cm‐1. The constant Bv, which is the rotational constant in the ground torsional state for the limiting case of V0=0, is 10985.79 Mc/sec and 9636.50 Mc/sec for CH3SiH3 and CH3SiD3, respectively.

1.
David R.
Lide
and
Donald K.
Coles
,
Phys. Rev.
80
,
911
(
1950
).
2.
J.
Sheridan
and
W.
Gordy
, (CH3SiF3),
J. Chem. Phys.
19
,
965
(
1951
);
Minden
,
Mays
, and
Dailey
, (CH3SiF3),
Phys. Rev.
78
,
347A
(
1950
);
Dailey
,
Shulman
, and
Minden
, (CH3CF3),
Phys. Rev.
75
,
1319
(
1949
);
H.
Minden
and
B.
Dailey
, (CH3CF3) and (CH3SiF3),
Phys. Rev.
82
,
338A
(
1951
);
Bak
,
Hansen
, and
Rastrup‐Andersen
, (CH3CCCF3),
J. Chem. Phys.
21
,
1612L
(
1953
).
3.
This is also true of the CH3NO2,CH3OH,CH3NH2 types of molecule but is not true if neither group is a symmetric rotor. See footnote 19.
4.
In this article vibrational modes mean all internal modes of motion other than torsion (hindered internal rotation).
5.
E. B.
Wilson
, Jr.
, and
J. B.
Howard
,
J. Chem. Phys.
4
,
260
(
1936
).
6.
E. C. Kemble, The Fundamental Principles of Quantum Mechanics (McGraw‐Hill Book Company, Inc., New York, 1937), p. 394.
7.
H. H.
Nielsen
,
Phys. Rev.
40
,
445
(
1932
).
8.
King
,
Hainder
, and
Cross
,
J. Chem. Phys.
11
,
27
(
1943
).
9.
If the molecule vibrates, the α axis is given by the Eckart conditions;
C.
Eckart
,
Phys. Rev.
47
,
552
(
1935
).
10.
Terms of order P4 can be lumped together with the usual centrifugal distortion terms. See
D.
Kivelson
and
E. B.
Wilson
, Jr.
,
J. Chem. Phys.
20
,
1575
(
1952
).
11.
J. S.
Koehler
and
D. M.
Dennison
,
Phys. Rev.
57
,
1006
(
1940
).
12.
C = h/8π2cIc, where Ic is the moment of inertia about the figure axis. C1 and C2 are defined in a similar way. C, C1, and C2 are proportional to μzz(0),ρzz(0),ηzz(0), respectively.
13.
K is the quantum number of the z component of total angular momentum.
14.
The technique for obtaining these terms involves holding σ constant, differentiating the Hamiltonian of Eq. (11) with respect to α. and combining the result with Eq. (18). See reference 10. Πz is given in units of ℏ.
15.
The values of the frequencies were given to the author by Dr. David R. Lide.
16.
D.
Kivelson
and
E. B.
Wilson
, Jr.
,
J. Chem. Phys.
21
,
1229
(
1953
).
17.
G. Herzberg, Molecular Spectra and Molecular Structure: II. Infrared and Raman Spectra of Polyatomic Molecules (D. Van Nostrand Company, Inc., New York, 1949), p. 456.
18.
K. B.
Pitzer
and
J. L.
Hollenberg
,
J. Am. Chem. Soc.
75
,
2123
(
1953
).
19.
Actually these constants may depend upon θ and, hence, upon m. If the vibrations distort the molecule so that neither functional group remains a symmetric rotor, the moments of inertia and, hence, the rotational constants depend directly upon θ.
20.
See section IV of reference 10.
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