Spectral branches

A detailed analysis of the quantum mechanics of simultaneous vibrational and rotational changes shows that the rotational quantum number J changes by ±1 during the vibrational transition of a diatomic molecule. If the molecule also possesses angular momentum about its axis, as in the case of the electronic orbital angular momentum of the paramagnetic molecule NO, then the selection rules also allow ∆J = 0. The appearance of the vibration–rotation spectrum of a diatomic molecule can be discussed in terms of the combined vibration–rotation terms, S:

S(v,J) = G(v) + F(J)

If we ignore anharmonicity and centrifugal distortion,

S(v,J) = (v + )v + BJ(J + 1)

In a more detailed treatment, B is allowed to depend on the vibrational state because, as v increases, the molecule swells slightly and the moment of inertia changes. We shall continue with the simple expression initially. When the vibrational transition v + 1 ←voccurs , J changes by ±1 and in some cases by 0 (when ∆J = 0 is allowed). The absorptions then fall into three groups called branches of the spectrum. The P branch consists of all transitions with ∆J =−1:

vP(J) = S(v +1, J − 1) − S(v,J) = v −2BJ

This branch consists of lines at # − 2B, # − 4 B,... with an intensity distribution reflecting both the populations of the rotational levels and the magnitude of the J −1 ← J transition moment (Fig. 13.35). The Q branch consists of all lines with ∆J = 0, and its wavenumbers are all

vQ(J) = S(v +1,J)−S(v,J) = v

for all values of J. This branch, when it is allowed (as in NO), appears at the vibrational transition wavenumber. In Fig. 13.35 there is a gap at the expected location of the Q branch because it is forbidden in HCl. The R branch consists of lines with ∆J =+1:

vR(J) = S(v +1, J +1)−S(v,J)=v+2B(J+1)

This branch consists of lines displaced from # to high wavenumber by 2B, 4B, .... The separation between the lines in the P and R branches of a vibrational transition gives the value of B. Therefore, the bond length can be deduced without needing to take a pure rotational microwave spectrum. However, the latter is more precise.

Fig. 13.35 The formation of P, Q, and R branches in a vibration–rotation spectrum. The intensities reflect the populations of the initial rotational levels.

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Units

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Working galvanic cells

Electrolysis

Experimental techniques

The rate laws

The electric potential at the interface

The structure of the interface

Catalytic activity at surfaces

Mechanisms of heterogeneous catalysis

Mobility on surfaces

The rate of desorption