In an asymmetrical top molecule, all the three moments of inertia about the three
principal inertial axes, a, b and c, are unequal to one another.
The axes are designated as a, b and c so that the moments of
inertia about them satisfy the condition:
Ia < Ib < Ic. Obviously,
an asymmetrical top molecule has no preferential axis, and therefore the treatment of
the molecule can be
done in six different representations (Chandra 1987). The rotational
energy levels in an asymmetrical top molecule may be represented
by
or
Jka, kc,
where
,
ka and kc are sub-quantum numbers and are related through the relations:
and
Thus,
can assume an integer value ranging from -J to +J. However, ka and kc can,
independently, assume positive integer values ranging from 0 to J.
In case of a planar asymmetrical top molecule, the electric dipole of the
molecule
lies in the ab-plane. When the electric dipole of the molecule is along the a-axis of inertia, the molecule is known
as a-type asymmetrical top molecule, and when it is along the b-axis of inertia,
the molecule is said to be b-type asymmetrical top molecule. In case
the electric dipole lies in between the
a and b axes, it can be resolved into two components along the two axes, and such molecule has
both a and b type transitions. In an
a-type asymmetrical top molecule, the rotational energy levels are grouped
into two distinct categories:
(ka, kc): | (odd, even) or (odd, odd) | one category |
(even, even) or (even, odd) | another category |
whereas in a b-type asymmetrical top molecule, the rotational energy levels are grouped into the following two categories:
(ka, kc): | (even, odd) or (odd, even) | one category |
(even, even) or (odd, odd) | another category |
This grouping of energy levels is independent of the aforesaid representations. Spectroscopists generally use Ir representation in which the a-axis of inertia is used as the preferential direction (Sharma & Chandra 1996; Chandra & Rashmi 1998). However, for calculations of the collisional rate coefficients, the direction of the electric dipole of the molecule is generally taken as the preferential direction (Green et al. 1987; Palma & Green 1987). Consequently, for collisional transitions in an a-type asymmetrical top molecule, one has to use Ir representation, whereas in the b-type asymmetrical top molecule, one has to use IIr representation.
Rotational wave-functions for an asymmetric top molecule can be expressed as linear combinations of wave-functions for a symmetric top molecule (Chandra & Rashmi 1998; Chandra et al. 1984):
Values of the expansion coefficients
depend on the representation (Chandra 1987).
The C3H2 is a b-type asymmetrical top molecule, and therefore
IIr representation is used. We have accounted for 47 and 48 rotational
energy levels for ortho- and para-C3H2,
and for the rotational constants for the molecule,
A=35092.508332 MHz,
B=32212.946832 MHz, and
C=16749.028632 MHz, the
wave-functions are given in Tables 1A and 1B, respectively. The SiC2 is an a-type
asymmetrical top molecule, and therefore
Ir representation is used. For ortho-SiC2, we have accounted for 40
rotational energy levels, and for the rotational constants for the molecule,
A=52473.6664 MHz,
B=13158.65426 MHz, and
C=10441.61928 MHz, the
wave-functions for one category are given in Table 2. The wave-functions for
the second category of SiC2 are not reported here as these levels do not
exist due to the spin statistics for the identical carbon nuclei.
The coefficients with negative value of K are related to those with positive value of
K through the relation
where the value of
for the levels is given in the table
for wave-functions. Thus, the coefficients with negative value of K can be obtained.
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