Comparison of the contour for the band illustrated in Fig. 1.9 with that of the origin band of BTD [Christoffersen, Hollas and Wright
1969] shows a close similarity in the gross features present in both the high and low wavenumber series. The transition in BTD is 1B2 which is
the same as that predicted for IBF by the molecular orbital considerations of Section 1.2. Moreover, the alternation in the
intensity of successive R branch features strongly suggests that these are groups of transitions each of which may be labelled by successive values of the (pseudo-) quantum number K " . The intensity alternation is
(even K M : odd K" = 7:9). These maxima c o n f o r m to the predicted energies
of R s u b b r a n c h transitions in a prolate symmetric top. F r o m Eq. (1.1), and the a p p r o p r i a t e selection rules,
AE = ABJ"(J" + 1) + 2 B ’ (J" + 1) + A(A - B ) K " 2 + 2 ( A ’ - B ’)K" + A' - B' , (1.5)
w h e r e A implies the difference between an excited and ground state quantity, and B = ^ (B + C). Therefore
2ABJ + 2B' + B .
Tu r n i n g points in the function AE(J), w h i c h c orrespond to reversals in the R subbranches (subbranch h e a d s ) , are a c h ieved whe n
AB = 2Bf
2 J + 1
and so AB must clearly be negative in the present case. The w avenumbers of the subbranch heads are given by Eq. (1.8), where
v = v 0 + 2 (A* - B ?)k" + A ( A - B ) K " (1.8)
Vn is the w a v e number of the K " = 0 band.
The wave n u m b e r s of the observed series were fitted by least squares to a quadratic w h i c h returned the coefficient of the squared ter m in Eq. (1.8); this coefficient is independent; of the n u m b ering of the series members. The v a lue obtained w a s 0.00195 cm - 1 . This figure should, if these bands are R subbranch heads, provide a reasonable trial v a l u e for the quantity A ( A - B ) . The ground state infertial constants wer e c alculated from the geometry illustrated in Fig. 1.5, so yielding a
trial v a l u e for A ? - B f. The numbering of the series w a s then adjusted to force the coefficient of the linear t e r m in Eq. (1.8) to the value of 2 (A 1 - B ’). This gave K = 1 8 for the first observed band. (In BTD
[Christoffersen 1969a] the first observed R head has K ~ 9 . ) a
1.6.7 The P Branch
The long series of line-like features to low wavenumber of the main pea k was expected to result from the commonly o c c urring high J, low K s t r u cture [Mcllugh 196obj. ide kinds of m o l e cules w h i c h show this type of structure have been reviewed by Ross  and Brown  and the
reason for its formation is now outlined. Gora  obtained a formula for the energy levels of an oblate top w h i c h have high J values and low K values. This is roughly the region to the right of the upward running d i a gonal in Fig. 1.6. Brown  obtained similar expressions for pr o l a t e tops. For planar molecules, M c Hugh [1968b] showed that G o r a ’s equation may be considerably simplified to
E = C(n + 1 ) 2 - S (1.9) w h e r e m = J - K c n = J + m = 2 J - K c S = %U2(A-!-B) m2 = 2 m 2 + 2m + 1 .
In terms of m and n the selection rules for the r^P and subbranches of a type B band, w h i c h are degenerate in the oblate top region, become:
Am = AJ - AK = 0
An = -1
A p p l y i n g these rules to Eq. (1.9) yields, for these transitions,
AE = A C n 2 -- (2n + 1)C" - AS , w h e r e
AS = ^ 2 (a a + a b ) •
The v a l u e of AS, for a fixed value of n and various ra, is small in
comp a r i s o n wit h the n C M terms. So the transitions wit h the same value of n = 2 J - are almost coincident in energy. For example, the transitions
75 74 73 1,75 2,73 3,71 ■*- 76 <- 75 74 0,76 ; 1,74 ; 2,72 ’ 0,75 1,73 73 2,71 76 •*- 75 74 1,76 2,74 3,72 e t c .
are all ver y nearly coincident; the transitions wit h initial and final J values because they have the usual oblate top d e g e neracy for low K , and those w i t h different sets of J values b e c ause they have the same n value.
The members of the high wavenumber series were thus fitted by least squares to a quadratic, and yielded -0.000523 cm 1 for the
coefficient of the squared term. This was equated with AC from Eq. (1.11). Not all the series members were included in the fit, for the region towards the end of the band becomes irregular in both the spacing and the intensities of the peaks, as will be discussed in a later section.