Rotational Structure of Electronic Transitions

In general, rotational transitions accompany the excitation of the vibrational structure levels that accompany electronic excitation. We should expect $P$, $Q$, and $R$ branches for each vibrational transition (but for $\Sigma \rightarrow \Sigma$ transitions, see section 6.1) and therefore, electronic transitions have very reach structure. Let us consider the rotational structure for a given vibrational transition $v'\leftarrow v''$, where $v''$ belongs to the molecular ground state and $v'$ belongs to the molecular excited state. Denoting

$$\tilde{\nu_0} = E_{el}' - E_{el}'' + \omega_e'\left(v'+\frac{1}{2}\right) - \omega_e''\left(v''+\frac{1}{2}\right),$$ (69)

where $\nu_0$ is measured in the wave numbers, $cm^{-1}$, we can write the expressions for the $P$, $Q$, and $R$ branches as

Branch	$\Delta J$	Peak Position	for J
P	-1	$\tilde{\nu_P} = \tilde{\nu_0} + ({B_e}'-{B_e}'')J^2-({B_e}'+{B_e}'')J$	1,2,...
Q	0	$\tilde\nu_Q = \tilde{\nu_0} + ({B_e}'-{B_e}'')J(J+1)$	0,1,2,...
R	1	$\tilde\nu_R=\tilde{\nu_0} + (B_e'-B_e'')(J+1)^2 + (B_e'+B_e'')(J+1)$	0,1,2,...

Normally, the rotation constant ${B_e}$ is smaller for the excited state that for the ground state, ${B_e}'-{B_e}''<0$. This is because the electron excitation usually increases the bond length. Note, that ${B_e}=1/(8\pi^2 c I_e)$ and $I_e=\mu R_0^2$. In this case one can see from the formula above that the lines of the $R$ branch converge with increasing $J$, that is the line frequency increases at small $J$ values till $({B_e}''-{B_e}')(J+1) = ({B_e}'+{B_e}'')$ and then decreases at larger $J$ values. That is, the branch $R$ has a band head at $({B_e}''-{B_e}')(J+1) = ({B_e}'+{B_e}'')$ and is said to be red shadowed.

In more rare case when the electron excitation decreases the bond length ${B_e}'-{B_e}''>0$. Then, the lines of the $P$ branch begin to converge and go through a head at $({B_e}'-{B_e}'')J=({B_e}'+{B_e}'')$. This means that the line frequency decreases at low $J$ values before the band head and increases at higher $J$ values. In this case the branch ia said to be blue shadowed.

Auf diesem Webangebot gilt die Datenschutzerklärung der TU Braunschweig mit Ausnahme der Abschnitte VI, VII und VIII.