Page 291 - Academic Press Encyclopedia of Physical Science and Technology 3rd Analytical Chemistry

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Encyclopedia of Physical Science and Technology EN009N-447 July 19, 2001 23:3

808 Microwave Molecular Spectroscopy

TABLE III Spectroscopic Constants and Struc-
tures for Some Linear Molecules
Bond length ( ˚ A) B 0 (MHz) D 0 (kHz)

1.063 1.155
H C N 44,315.98 87.2
1.067 1.542
H C P 19,976.01 21.2
1.262 1.159
F C N 10,554.20 5.3
1.285 1.541
F C P 5,257.80 1.0
1.629 1.160
Cl C N 5,970.83 1.7
1.789 1.160
Br C N 4,120.22 8.8
1.129 1.189
N N O 12,561.64 5.4
1.160 1.560
O C S 6,081.49 1.3
FIGURE 6 The first few rotational energy levels for a linear 1.053 1.198 1.279
molecule. The allowed transitions and the resulting spectrum, with H C C F 9,706.19 —
1.055 1.204 1.637
approximate intensities, are also shown. H C C Cl 5,684.24 —
1.95 0.96
Na O H 12,567.05 28.7
A series of lines at 2B, 4B, 6B,... is thus expected for K 2.21 O 0.91 H 8,208.68 12.2
2.80
a rigid rotor. The energy levels, allowed transitions, and HCN···HF a 3,591.11 5.2
3.07
spectrum of a rigid linear rotor are illustrated in Fig. 6. OC···HF a 3,063.90 9.8
3.54
The molecule OCS, which is commonly used as a stan- Ar···HF a 3,065.71 70.9
3.65
dard for various purposes by microwave spectroscopists, Kr···HF a 2,392.41 31.9
has lines that occur at 12,162.97, 24,325.92, 36,488.80, [H 1.09 C 1.11 O] + 44,594.42 82.4
48,651.40 MHz,... for the most common isotope. For S 1.4840 O 21,523.56 33.9
a light molecule such as CO, B = 57,635.97 MHz, and
a For these complexes the bridge length r(X F) is
the rotational lines are spaced 115,271.94 MHz apart;
given with X = N, C, Ar, or Kr.
thus, high-frequency microwave techniques must be em-
ployed to measure even the 0 → 1 transition, which is 2
at 115,271.94 MHz. The effect of centrifugal distortion E J,K = BJ(J + 1) + (A − B)K , (29)
is to produce a small shift to lower frequency in each with the rotational constants defined as A = h /8π I a and
2
transition. Illustrative rotational constants are collected in B = h/8π I b . The energy levels are characterized by the
2
Table III. quantum numbers J, K, and M, with
J = 0, 1, 2,...,
B. Symmetric-Top Molecules
K = 0, ±1, ±2,..., ±J, (30)
The rotational Hamiltonian for a prolate symmetric top
has the form M = 0, ±1, ±2,..., ±J.
P 2 1 1
2 For an oblate top, the unique axis is denoted by c.By
= + − P , (27)
a replacement of A by C and a by c, the energy expression
2I b 2I a 2I b
and angular momentum matrix elements for an oblate top
2
2
2
2
where P = P +P +P is the total angular momentum.
a b c may be obtained. In particular, for the energy,
In addition to P Z , a symmetric top has a component of the 2
total angular momentum P a (P Z ) along the symmetry axis, E JK = BJ(J + 1) + (C − B)K , (31)
2
which is a constant of motion. The quantities , P , P Z , 2
with C = h/8π I c .
and P a commute with each other and hence have a com-
As apparent from Eqs. (29) and (31), the energy levels
mon set of eigenfunctions denoted by ψ JKM ≡|J, K, M).
increase with K for a prolate rotor (A > B) and decrease
The matrix elements in the symmetric-top basis,
with K for an oblate rotor (C < B). There are J + 1 dif-
2
2
(J, K, M | P J, K, M) = h J(J + 1), ferent rotational levels for each J value since the energy
does not depend on the sign of K. The rotational levels for
(J, K, M|P a |J, K, M) = hK, (28)
J 3 are illustrated in Fig. 7. Furthermore, in the absence
(J, K, M|P Z |J, K, M) = hM, of external fields each level is (2J + 1)-fold degenerate in
the space orientation quantum number M. For absorption
specify the values of the quantized angular momenta. It
of radiation, the important selection rules are
follows from Eq. (27) that the energy levels for a prolate
rotor are given by J → J + 1, K → K. (32)