A new Coriolis perturbation in the methane spectrum I. Vibrational-rotational Hamiltonian and wave functions
Authors: Jahn H. A. 
Published: December 2006

It is known that the infra-red absorption band of the low-frequency fundamental vibration (
v
4
) of the methane molecule has a rotational structure which is much more complex than one would expect for such a simple molecule (Nielsen and Nielsen 1935). It is our purpose to show that nevertheless this complex structure can be explained on the basis of a regular tetrahedral model for the molecule. We shall see that the rotational levels of the vibration
v
4
are perturbed by the rotational levels of the next nearest vibration
v
2
in such a way as to produce in the spectrum just the observed complex structure. The perturbation arises from a Coriolis (or vibrational gyroscopic) interaction between the rotational-vibrational levels of the two different modes of vibration. In this first part we derive these Coriolis coupling terms in the vibrational-rotational Hamiltonian and find also the correct rotational-vibrational wave functions with which to carry out the perturbation calculation. In Part II we evaluate the matrix elements of the perturbation and determine the theoretical rotational energy spectrum of
v
4
. In the third and final part we use this energy spectrum to calculate the optical spectrum and compare this with the observed spectrum. For this purpose we calculate the theoretical intensities of the rotational fine structure lines, taking into account the nuclear spin weights of the four equivalent hydrogen atoms. From this theoretical spectrum we calculate the theoretical envelope which would be observed with slit widths of approximately 0⋅5 cm
-1
. as used by Nielsen and Nielsen. This theoretical envelope is found to agree remarkably well with the experimental envelope, even without taking into account any vibrational or rotational change in the equilibrium configuration. 1. Pure deformation and orthogonal valency modes Before we can make any explicit calculation of the rotational-vibrational levels of the methane molecule we must first of all know the fundamental modes of vibration. In the following we make use of what we call the pure deformation and orthogonal valency modes (cf. Mecke 1930). These are not exact normal modes of vibration of the molecule, but we shall find that they are very good approximations to the true modes. They are based on the experimental fact that the energy required to stretch a C—H bond is considerably greater than the energy required to change the H—C—H angles. Thus we can obtain a good approximation to the low-frequency modes by introducing the condition that all the C—H distances remain invariant. This removes four degrees of freedom and enables us to find five different orthogonal deformation vibrations in which only the angles change. The remaining four high-frequency modes are then determined simply by the condition that they should be orthogonal to these deformation vibrations. These high frequency valency vibrations will involve essentially changes in the C—H distances; they will also, however, involve to a slight extent changes in the angles. In finding these modes it will help us considerably to make use of the fact that three normal modes of vibration of the methane molecule, for any force system whatever (consistent with the tetrahedral symmetry), are determined completely by symmetry. These are the totally symmetrical vibration of type A
1
(in the notation of Tisza 1933) and the twofold degenerate set of vibrations of type E. That these are completely determined is a consequence of the group-theoretical result that the methane molecule possesses one and only one of each of these irreducible types of vibration. We denote the vibration of type A
1
by
Q
1
and two suitably chosen normal modes of type E by
Q

2
a

and
Q

2
b

. In each of these modes of vibration the carbon atom remains at rest, in
Q
1
the hydrogen atoms move radially in phase either away or towards the carbon atom (so that this is a pure valency mode of vibration), whilst in
Q

2
a

and
Q

2
b

the hydrogen atoms move on a sphere about the carbon atom, so that the C—H distances remain unaltered. These latter give us therefore at once two of the deformation modes of vibration. In the three remaining deformation modes the carbon atom takes part in the vibration. These modes can be arrived at by regarding them as compounded out of an infinitesimal translation of the whole molecule, followed by a backward displacement of the hydrogen atoms tangential to the equilibrium sphere drawn with its centre at the displaced position of the carbon atom. This will clearly satisfy the condition that the C—H distances remain unchanged, since neither of the two compounded displacements change them. The directions of displacement of the hydrogen atoms tangential to the sphere are determined completely by the conditions of orthogonality to the translations, the rotations and the twofold degenerate vibrations. In the three remaining valency modes the carbon atom also moves and the hydrogen atoms move out or in radially towards the equilibrium position of the central atom (and not towards the carbon atom itself) so that the valency angles do change slightly.