The rotational constant of \(^{1} \mathrm{H}^{35} \mathrm{Cl}\) is \(10.4400 \mathrm{cm}^{-1}\) in the ground vibrational state and \(10.1366 \mathrm{cm}^{-1}\) in the state \(v=1 .\) Plot the wavenumbers of the \(\mathrm{P}\) -, \(\mathrm{Q}\) -, and \(\mathrm{R}\) -branches against \(J\) as a representation of the structure of the \(1-0\) transition. Take \(\tilde{v}=2990.95 \mathrm{cm}^{-1}\) and neglect anharmonicity. (The Q-branch is not observed.)

We are provided the rotational constants \(B_0\) and \(B_1\) for a molecule in its ground and first vibrational states respectively. We are also given the vibrational wavenumber (\(\tilde{v} \)). Our task is to calculate and plot the wavenumbers (\(\bar{\nu}\)) for the P, Q and R branches of the \(1-0\) transition against the rotational quantum number \(J\). The P, Q, and R branches usually correspond to \(\Delta J = -1, 0, +1\) transitions respectively. For the P and R branches, the wavenumber \(\bar{\nu}\) is given by \(\bar{\nu} = \tilde{v} + (2B_1 - 4B_0)J + 2B_1\) for the R branch and \(\bar{\nu} = \tilde{v} - (2B_0 + 4B_1)J - 2B_0\) for the P branch. Here, \(\Delta J = J_{\text{final}} - J_{\text{initial}}\). For the Q branch, \(\Delta J = 0\), so the wavenumber is just the vibrational frequency \(\tilde{v}\). As given in the problem, the Q branch is not observed. Assuming the anharmonicity is negligible, we can ignore any effects due to this.

Now using the formulas for the wavenumbers of the P and R branches and the given values, we can evaluate these for different \(J\). For example, for \(J = 0\), the wavenumber for P branch will be \(\tilde{v} - 2B_0\) and for R branch will be \(\tilde{v} + 2B_1\). For \(J = 1\), it will be \(\tilde{v} - (2B_0 + 4B_1)\) for P branch and \(\tilde{v} + (2B_1 - 4B_0)\) for the R branch. Similarly, we can continue computing these for higher \(J\) to get the set of data points. Remember, \(J\) must be a positive integer.

Now, we can plot the wavenumbers calculated above against the corresponding \(J\). This will give us a representation of the structure of the \(1-0\) transition. There will be two lines, one for each of the P and R branches. As \(\Delta J\) increases (moving away from \(J=0\)), the wavenumbers will decrease for the P branch and increase for the R branch, forming a characteristic 'X' shape. The wavenumbers for the Q-branch are not observed.