15.3: Vibrational and Rotational Spectroscopy

IR vibrational spectroscopy

As we originally observed in our classical model of absorption spectroscopy, absorption of mid-infrared light by vibrations arises from the light driving changes in charge distribution as a molecule vibrates. As we discussed in Section 10.2, we describe this effect by recognizing that the dipole operator depends on the displacement of nuclei. We can take the dipole moment to depend weakly on the displacement of vibrational coordinates \(q\) so that we expand the dipole operator as 

\[\boldsymbol{\mu} = \boldsymbol{\mu}_{0} + \boldsymbol{\mu}'q + \cdots \label{eq15.3.01}\]

where \(\boldsymbol{\mu}'=\left. \partial\boldsymbol{\mu}/ \partial q \right|_{q=q_0}\) is the dipole derivative, and \(\boldsymbol{\mu}_0\) is the permanent dipole moment. Then, eq. (15.2.07) can be expressed as a correlation function in the vibrational coordinate

\[C_{\text{vib}}(\tau ) = \left| \boldsymbol{\mu}' \right|^{2}\left\langle q(t)q(0) \right\rangle \label{eq15.3.02}\]

So, the time-dependent dynamics of the vibrational coordinate dictate the IR lineshape. Equation (\ref{eq15.3.02}) can be used to calculate absorption spectra for classical and quantum mechanical cases. 

As an example, consider the case of a quantum harmonic oscillator with Hamiltonian \(H=\hbar\omega_0(a^\dagger a+\frac{1}{2})\). Using the second expansion term in eq. (\ref{eq15.3.01}), the dipole correlation function can be written in terms of the harmonic oscillator correlation function

\[\begin{aligned}
 C_{\text{vib}}(\tau ) & = \sum\limits_{n} p_{n} \left\langle n \right|\mu (\tau)\mu (0)\left| n \right\rangle \\
 & = \mu '^{2}\sum\limits_{n} p_{n} \left\langle n \right|q(\tau)q(0)\left| n \right\rangle \\
\end{aligned} \label{eq15.3.03}\]

We discussed how to evaluate the harmonic oscillator correlation function earlier. Using this result, the vibrational dipole correlation function is written

\[C_{\text{vib}}(\tau ) = \frac{\mu '^{2}\hbar }{2 m_R \omega _{0}}\left[ (\bar n + 1)e^{ - \mathrm{i}\omega _{0}t} + \bar ne^{\mathrm{i}\omega _{0}t} \right] \label{eq15.3.04}\]

where \(\bar n = (e^{\beta \hbar \omega _{0}} - 1)^{ - 1}\) is the thermal occupation number. 

This quick result, while correct, glosses over the fact that these expressions include our evaluation of the transition dipole matrix elements

\[\begin{aligned}
 \mu _{mn} & = \left\langle m \right|\mu 'q\left| n \right\rangle \\
 & = \sqrt {\frac{\mu '^{2}\hbar }{2 m_R \omega _{0}}} \left( \delta _{n + 1,m}\sqrt {n + 1} \, +\, \delta _{n - 1,m}\sqrt {n} \right) \\
\end{aligned} \label{eq15.3.05}\]

This expression reminds us of the \(\Delta n=\pm1\) selection rules, indicates that \(\mu _{mn} = \mu _{nm}\), and defines the fundamental transition matrix element \(\mu _{10} = \sqrt {{ \mu '^{2}\hbar / 2m\omega _{0} }} \). It also shows us how transition dipole moments scale with quantum number relative to the fundamental transition

\[\left| \mu _{mn} \right|^{2} = \left| \mu _{10} \right|^{2}\left( (n + 1)\delta _{m,n + 1} + n \delta _{m,n - 1} \right) \label{eq15.3.06}\]

We can substitute this into the eigenstate form of the dipole correlation function to get eq. (\ref{eq15.3.04})

\[\begin{aligned}
 C_{\text{vib}}(\tau ) & \displaystyle = \sum\limits_{n,m} p_{n}\left| \mu _{mn} \right| ^{2}e^{ - \mathrm{i}\omega _{mn}t} \\
 & \displaystyle = \left| \mu _{10} \right|^{2}\left( (\bar n + 1)e^{ - \mathrm{i}\omega _{0}t} + \bar n e^{\mathrm{i}\omega _{0}t} \right) \\
\end{aligned} \label{eq15.3.07}\]

The absorption lineshape is then obtained from eq. (15.1.17) 

\[\sigma _{\text{vib}}(\omega ) = \frac{\mu '^{2}}{4 m_R \varepsilon _{0}c}\left( 1 - e^{ - \beta \hbar \omega } \right)\left[ \left( \bar n + 1 \right)\delta \left( \omega - \omega _{0} \right) + \bar n \delta \left( \omega + \omega _{0} \right) \right] \label{eq15.3.08}\]

The leading term describes absorption from state \(n\rightarrow n+1\), whereas the second term describes stimulated emission from \(n\rightarrow n-1.\) Most of the time, molecular vibrational frequencies are of high enough  energy that the low temperature limit applies at room temperature (\(\bar n \to 0\), \(e^{-\beta\hbar\omega}\approx0\)), and we have only a single absorption peak at the fundamental transition frequency

\[\sigma _{\text{vib}}(\omega) = \frac{\mu '^{2}}{4 m_R\varepsilon _{0}c}\delta \left( \omega - \omega _{0} \right) \label{eq15.3.09}\]

Rotational spectroscopy

Now let’s consider pure rotational spectroscopy, measured in the far infrared or microwave region of the spectrum. We will look specifically at the free rotation of a rigid rotor, as is characteristic of a gas-phase diatomic molecule. The Hamiltonian is expressed in terms of the rotational angular momentum operator \(\mathbf{J}\) and moment of inertia \(I\)

\[H_{\text{rot}} = \frac{\mathbf{J}^{2}}{2I} \label{eq15.3.10}\]

and the corresponding eigenvalue equations for the rotor’s angular momentum are

\[\begin{aligned}
 \mathbf{J}^{2}\left| J,M \right\rangle &= \hbar ^{2}J\left( J + 1 \right)\left| J,M \right\rangle \quad && J = 0,1,2... \\
 \mathbf{J}_{z}\left| J,M \right\rangle &= M\hbar \left| J,M \right\rangle &&M = - J, - J + 1,...,J
\end{aligned} \label{eq15.3.11}\]

\(J\) is the rotational quantum number, and \(M\) refers to its projection onto the \(z\) axis. Therefore, the energy eigenvalues for \(H_{\text{rot}}\) are 

\[E_{J} = BJ(J + 1) \label{eq15.3.12}\]

where the rotational constant, here in units of joules,\(\bibref{Cite1} %in-text-citation\) is

\[B = \frac{\hbar ^{2}}{2I} \label{eq15.3.13}\]

The rigid rotor approximation allows one to separate the rotational Hamiltonian from the vibrational and electronic degrees of freedom and evaluate a rotational lineshape in terms of a rotational correlation function without explicitly accounting for the other degrees of freedom. The dipole moment can be written in terms of a permanent dipole moment that is a product of its magnitude \(\mu _{0}\) and \(\hat{u}\), a unit vector that specifies its orientation: 

\[\boldsymbol{\mu} = \mu _{0}\hat u \label{eq15.3.14}\]

In some respects, the rigid rotor approximation can be viewed as an adiabatic approximation in which \(\mu _{0}\) reflects a value that is averaged over much faster electronic and vibrational degrees of freedom.

The rotational absorption lineshape will be related to the Fourier transform of the rotation correlation function for the permanent dipole vector in the laboratory frame, \(\hat{\varepsilon}\), which is set by the polarization of the light.

\[C_{\text{rot}}(t) = \mu _0^{2}\left\langle \left( \hat \varepsilon \cdot \hat u(t) \right)\left( \hat \varepsilon \cdot \hat u(0) \right) \right\rangle \label{eq15.3.15}\]

Setting the polarization of the field along the \(\hat{z}\) axis such that \(\left( \hat u \cdot \hat z \right) = \cos \theta \), we get

\[C_{\text{rot}}(t) = \mu _0^{2}\left\langle \cos \theta (t) \cos \theta (0) \right\rangle \label{eq15.3.16}\]

The leading \(\mu_0^2\) term means that a permanent dipole moment is required to get absorption. Remember that \(\theta(t)\) still refers to a time-dependent operator in the interaction picture, whose time-dependence is governed by the rotational Hamiltonian. That is, 

\[\cos \theta (t) = e^{\mathrm{i}H_{\text{rot}}t/\hbar }\cos \theta e^{ - \mathrm{i}H_{\text{rot}}t/\hbar } \label{eq15.3.17}\]

Thus, we can already anticipate that the frequency of rotational resonances will depend on the rate of rotation, which is determined by the angular momentum and the moment of inertia. Evaluating the correlation function will require that we average over both a Boltzmann distribution for the initial occupation of rotational states and the initial orientational distribution of dipoles, which is taken as isotropic, as we expect in a gas. 

Then the rotational correlation function in eq. (\ref{eq15.3.16}) can be written as

\[C_{\text{rot}}(t) = \mu _0^{2} \sum_{J,M} p_{J,M}\left\langle J,M \right| e^{\mathrm{i}H_{\text{rot}}t/\hbar }\cos \theta e^{ - \mathrm{i}H_{\text{rot}}t/\hbar } \cos \theta \left| J,M \right\rangle \label{eq15.3.18}\]

Here, \(|J,M\rangle\) refers to the initial state of the system and \(p_{J,M}\) is the probability of initially occupying a particular \(J, M\) level:

\[p_{J,M} = e^{ - \beta E_{J}}/Z_{\text{rot}} \label{eq15.3.19}\]\[Z_{\text{rot}} = \sum_{J,M} e^{ - \beta E_{J}} \label{eq15.3.20}\]

To evaluate this correlation function, we can insert the identity operator \(I=\sum_{J^\prime{,M}^\prime}{|J^\prime M^\prime\rangle \langle J^\prime M^\prime|}\) where marked by the arrow in eq. (\ref{eq15.3.18}) and apply \(e^{ - \mathrm{i}H_{\text{rot}}t/\hbar }\left| J,M \right\rangle = e^{ - \mathrm{i}E_{J}t/\hbar }\left| J,M \right\rangle \). We also need to evaluate the square of a matrix element of the form \(\left\langle J'M' \right| \cos \theta \left| J M \right\rangle \), which we described in Sec. 10.2:

\[\left| \left\langle J', M' \right| \cos  \theta \left| J,M \right\rangle \right|^{2} = s_{J,M}^{( + )} \delta _{J ' , J + 1}\delta _{M ' , M} + s_{J,M}^{( - )} \delta _{J ' , J - 1}\delta _{M', M} \label{eq15.3.21}\]

Here the constant coefficients are

\[ s_{J,M}^{( + )} = \frac{\left( J + 1 \right)^{2} - M^{2}}{4\left( J + 1 \right)^{2} - 1} \qquad s_{J,M}^{( - )} = \frac{J^{2} - M^{2}}{4J^{2} - 1} \label{eq15.3.22}\]

Equation (\ref{eq15.3.21}) defines the rigid-rotor selection rules linking initial states \(|J,M\rangle\) and final states \(|J^\prime \kern-.2em , \, M^\prime\rangle\), where \(J^\prime =J \pm 1\) and \(M^\prime=M\). Even though the energy levels of the rigid rotor, eq. (\ref{eq15.3.12}), are nonlinearly spaced, the restriction to changes in \(J\) by \(\pm 1\) means that the transition energies are spaced linearly in \(J\):

\[\begin{aligned}
 E_{J'} - E_{J} &= B\left[ J'\left( J' + 1 \right) - J\left( J + 1 \right) \right] \\
 & = \left\{ \begin{array}{ll}
 2B(J + 1) \quad & J' = J + 1 \\
 -2BJ & J' = J - 1
\end{array} \right. \\
\end{aligned} \label{eq15.3.23}\]

Spectroscopically, the \(\Delta J=\pm1\) changes in \(J\) correspond to the absorption or stimulated emission of radiation between adjacent energy levels (Figure 15.3.1). For gas phase molecules at room temperature, the thermally populated ensemble of rotational states extends to high \(J\), so we can expect both processes to occur and expect that the net absorption will be related to the population difference between the upper and lower state. These steps lead to the correlation

\[C_{\text{rot}}(t) = \mu _0^{2}\sum\limits_{J,M} p_{J,M}\left( s_{J,M}^{( + )}e^{ - 2\mathrm{i}\left( J + 1 \right)Bt/\hbar } + s_{J,M}^{( - )}e^{2\mathrm{i}JBt/\hbar } \right) \label{eq15.3.24}\]

which has two terms corresponding to the \(\Delta J=\pm1\) quantum number changes. 

Let’s continue by thinking about how to perform the thermal average. Since the rotational energy \(B\) isn’t dependent on \(M\), we can simplify things by performing the sum of \(s_{J,M}^{( \pm )}\) coefficients over \(M=-J\) to \( +J\). As described in Sec. 10.2, these sums result in a factor of \((J+1)/3\) for \(\Delta J=+1\) and \(J/3\) for \(\Delta J=-1\). With these steps, we obtain a correlation function in which we only need to sum over the initial \(J\):

\[C_{\text{rot}}(t) = \frac{1}{3}\mu _0^{2}\sum\limits_{J} p_{J}\left( (J + 1) e^{ - 2\mathrm{i}\left( J + 1 \right)Bt/\hbar } + J e^{2\mathrm{i}JBt/\hbar } \right) \label{eq15.3.25}\]

This is expressed in terms of the probability of being in a particular \(J\) state \(p_J = e^{-\beta E_J} / Z_{\text{rot}}\). The sum over \(M\) in the rotational partition function can also be eliminated by including the degeneracy of \(M\) states for a given \(J\): \(Z_{\text{rot}} = \sum\nolimits_{J} (2J + 1)e^{ - \beta E_{J}} \).

Using equation (15.1.16), the corresponding rotational correlation function in the frequency domain is 

\[C_{\text{rot}}(\omega) = \frac{\hbar \mu _0^{2}}{6\pi }\sum\limits_{J} p_{J}\left[ (J + 1) \delta \left( \hbar \omega - 2B\left( J + 1 \right) \right) + J \delta \left( \hbar \omega + 2BJ \right) \right] \label{eq15.3.26}\]

This spectral correlation function exhibits a series of delta-function peaks spaced by \(2B\), with the first term contributing peaks that arise from \(\Delta J=+1\) and appear at positive frequencies for \(\omega=2B(J+1)/\hbar\), while the peaks from the second term arise from \(\Delta J=-1\) and are observed at negative frequencies for \(\omega=-2BJ/\hbar\). These two terms reflect the processes of absorption and stimulated emission respectively, but in this form it’s not clear how the net absorption that we measure resulting from the sum of these processes comes about. This is where our expression for the cross-section, eq. (15.1.23), comes in. In calculating the imaginary part using \( C_{\mu \mu }(\omega ) - C_{\mu \mu }( - \omega )\), subtracting the conjugate spectrum \(C_{\mu \mu }( - \omega )\) removes the stimulated emission contribution from the total absorption terms to give the net absorption. So, the absorption cross section from eq. (15.1.23) is 

\[\sigma _{\text{rot}}(\omega) = \frac{\pi \mu _0^{2}}{6\varepsilon _{0}cZ_{\text{rot}}}\omega \sum\limits_{J} p_{J} \left[ (J + 1)\delta \left( \hbar \omega - 2B\left( J + 1 \right) \right) - J \delta \left( \hbar \omega + 2BJ \right) \right] \label{eq15.3.27}\]

The two terms reflect the fact that each thermally populated level with \(J \gt 0\) contributes both to absorptive and stimulated emission processes, and the observed intensity reflects the difference in populations. Since both terms enforce the \(\Delta M=0\) selection rule, we are assured that stimulated emission is properly accounted for and not solely dependent on the population of a \(J\) state.

Let’s look at the behavior of the rotational correlation function and absorption spectrum as a function of temperature. In Figure 15.3.2, we plot the correlation function, the initial population of \(J\) states, and the absorption cross section using the example of a diatomic molecule that has a rotational constant of \(\bar{B}=5 \, \text{cm}^{-1}\), which corresponds roughly to the value for \(\text{DCL}\). Since the occupation of initial states is proportional to \(e^{-\beta E_J}=e^{-\beta2BJ(J+1)}\), the temperature scale that defines low and high temperature is measured with respect to \(2B\), so we calculate the behavior of these functions at \(1\), \(20\), and \(300 \, \text{K}\), which correspond to \(k_BT/2B\) values of \(0.07\), \(1.4\) and \(21\). These are shown in Figure 15.3.2.

At low temperature, all of the population that is initially in \(J=0\) and \(C_{\text{rot}}(t)\) oscillates as a simple sinusoidal function with a rotational period of \(\tau_{\text{rot}}= h/2B=21  \, \text{ps}\). The corresponding absorption spectrum has one resonance with an absorption frequency of \(2B\), corresponding to absorption from \(J=0\) to \(J^\prime=1\). 

At the more elevated temperature of \(20 \, \text{K}\), on the same scale as the rotational constant, significant population is also initially observed in \(J=1\) and \(2\). The rotational correlation function now shows periodic peaks at \(\tau_{\text{rot}}\) but without the negative excursions of a simple sinusoid. One can rationalize this by recognizing that the correlation function is expressed in terms of a sum of harmonics with a base frequency \(2B\), which results in destructive interference unless \(t=\tau_{\text{rot}}\). These are referred to as rotational recurrences, since they arise from the periodic realignment of dipole moments within the ensemble of molecules. The corresponding absorption spectrum is a series of resonances separated by \(2B\) that reflect the weighting of these different harmonics. The spectral weights now account for the possibility of absorption and stimulated emission among the initially populated states. 

Finally, at high temperature, the ensemble has rotational states initially populated to high \(J\). It’s important to remember that the energy levels of the rigid rotor are unequally spaced, meaning that at high \(J\), the thermal population states falls off as a Gaussian in \(J\). The net effect on \(C_{\text{rot}}(t)\) is that the rotational recurrences are sharpened and that destructive interferences suppress any amplitude between them. The corresponding spectrum now has a large series of lines spaced by \(2B\) that extend to a frequency of \(30\bar{B}=150 \, \text{cm}^{-1}\). The envelope rises with frequency at low \(J\) due to the growing degeneracy of \(M\) states that scale as \(g_M (J)=(2J+1)\), but at high \(J\) the spectrum drops due to the falling thermal population in initial states. The high temperature behavior can be compared to the classical rotational correlation function that we described and plotted in Sec. 13.2. We find that the classical description reproduces the initial decay of the correlation function at times less that \(\tau_{\text{rot}}\), but not the recurrences. The periodic recurrences are the features that give rise to the fine structure in the rotational spectrum, and the classical correlation function recovers the overall envelope at high temperature.

Rotational-Vibrational Spectroscopy

Since we have done the work to describe the vibrational spectroscopy of a harmonic oscillator and the rotational spectroscopy of a rigid rotor, it is not much extra work to describe IR rotational-vibrational (or ro-vibrational) spectroscopy involving simultaneous transitions in vibrational and rotational quantum number. We can draw from the results above and put more attention into how separable Hamiltonians result in products of correlation functions. 

We will assume a separable rigid rotor/harmonic oscillator Hamiltonian:

\[H_{0} = H_{\text{rot}} + H_{\text{vib}} \label{eq15.3.28}\]

with the same form as those above. An eigenstate of the system will be described by the product state labeled by the vibrational and rotational quantum numbers \(\left| \psi \right\rangle = \left| n,J,M \right\rangle \), and the TISE is \(H_{0}\left| n,J,M \right\rangle = E_{n,J,M}\left| n,J,M \right\rangle \), with energy eigenvalues given by 

\[\begin{aligned}
 E_{n,J} & = E_{\text{rot}} + E_{\text{vib}} \\
 & = BJ\left( J + 1 \right) + \hbar \omega _{0}\left( n + \frac{1}{2} \right) \\
\end{aligned} \label{eq15.3.29}\]

The transition energies in the mid-IR we will be discussing, \(\Delta E = E_{f} - E_{i}\), involve changes in rotational and vibrational quantum number. Let’s use a prime to index the quantum numbers for the final state and no prime for the initial: \(\left| f \right\rangle = \left| n',J',M' \right\rangle \) and \(\left| i \right\rangle = \left| n,J,M \right\rangle \). Let’s consider the case of vibrational transitions involving a high frequency vibration with \(\hbar \omega _{0} \gg k_{B}T\), so that we only need to consider \(n=0\rightarrow1\) absorption from the ground vibrational state with \( \Delta n = + 1\). However, the temperature still exceeds the rotational energy: \(k_{B}T \gg B\). Incorporating the rotational selection rules, \(\Delta J = \pm 1\) and \(\Delta M = 0\), the transition energies are 

\[\begin{aligned}
 \Delta E & = E_{n' \kern-.2em , J'} - E_{n,J} \\
 & = \hbar \omega _{0} + B\left( J'\left( J' + 1 \right) - J\left( J + 1 \right) \right) \\
 & = \left\{ \begin{array}{ll}
 \hbar \omega _{0} + 2B(J + 1) \quad & J' = J + 1 \ (\text{R-branch}) \\
 \hbar \omega _{0} - 2BJ \quad & J' = J - 1 \ (\text{P-branch})
\end{array} \right. \\
\end{aligned} \label{eq15.3.30}\]

As illustrated in Figure 15.3.3, the transition involves a change in vibrational and rotational quantum number, with rotational frequency shifts added to the vibrational \(0 \rightarrow 1\) transition frequency \(\omega_0\). The transition frequency can involve a simultaneous increase in the vibrational and rotational state in the case of \(\Delta J=+1\), and it is also possible to have a drop in rotational state, \(\Delta J=-1\), accompanying the vibrational transition. Of course, a \(\Delta J=-1 \) transition can only occur from rotational excited states that are initially thermally populated, \(J \geq 1\).

To calculate the correlation function, we assume that the molecule’s dipole moment varies linearly with the displacement of the vibration, as in eq. (\ref{eq15.3.01}):

\[\boldsymbol{\mu} = \boldsymbol{\mu}_{0} + \boldsymbol{\mu}'q + \cdots \label{eq15.3.31}\]

Here again, \(\boldsymbol{\mu}^\prime\) is the dipole derivative, which we separate as a product of its magnitude and orientation:

\[\boldsymbol{\mu}' = \mu '\hat u \label{eq15.3.32}\]

Both of these factors carry a time dependence, but since the rotational and vibrational Hamiltonians commute, we can separate the time-dependence

\[\begin{aligned}
 \boldsymbol{\mu}'(t) &= e^{\mathrm{i}H_{0}t}\boldsymbol{\mu}'e^{ - \mathrm{i}H_{0}t} \\
 & = e^{\mathrm{i}H_{\text{vib}}t}\mu 'e^{ - \mathrm{i}H_{\text{vib}}t}e^{\mathrm{i}H_{\text{rot}}t}\hat ue^{ - \mathrm{i}H_{\text{rot}}t} \\
 & = \mu ' (t)\hat u(t) \\
\end{aligned} \label{eq15.3.33}\]

We can substitute this into the dipole correlation function, here written to emphasize the polarization of the incident light: 

\[\begin{aligned}
 C_{\mu \mu }(\omega) & = \left\langle \hat \varepsilon \cdot \mu (t) \hat \varepsilon \cdot \mu (0) \right\rangle \\
 & = \left\langle \mu '(t)(\hat \varepsilon \cdot \hat u(t)) \mu '(0)(\hat \varepsilon \cdot \hat u(0)) \right\rangle \\
 & = \left\langle \mu '(t)\mu '(0) \right\rangle \left\langle (\hat \varepsilon \cdot \hat u(t)) (\hat \varepsilon \cdot \hat u(0)) \right\rangle \\
 & = C_{\text{vib}}(t)C_{\text{rot}}(t) \\
\end{aligned} \label{eq15.3.34}\]

The separable nature of the Hamiltonian and the fact that the rotational Hamiltonian commutes with the vibrational position operator (and vice versa) allow us to separate the total rovibrational correlation function into a product of vibrational and rotational correlation functions.

Of course, these two correlation functions are essentially identical to those we evaluated in the preceding sections. Using eq. (\ref{eq15.3.07}) and eq. (\ref{eq15.3.25}), and setting \(\bar{n}\rightarrow0\), we find

\[\begin{aligned}
 C_{\mu \mu } \nobreak (t) & = \frac{1}{3}\left| \mu _{10} \right|^{2} \goodbreak \sum\limits_{n,J} p_{n}p_{J}\left( (\bar n + 1)e^{ - \mathrm{i}\omega _{0}t} + \bar n e^{\mathrm{i}\omega _{0}t} \right) \goodbreak \left( (J + 1)e^{ - 2\mathrm{i}\left( J + 1 \right)Bt} + Je^{2\mathrm{i}JBt} \right) \\
 & = \frac{\mu '^{2}\hbar }{6 m_R\omega _{0}} \goodbreak \sum\limits_{J} p_{J} \goodbreak \left( (J + 1)e^{ - \mathrm{i}(\omega _{0} - 2\left( J + 1 \right)B)t} + Je^{ - \mathrm{i}(\omega _{0} + 2JB)t} \right) \\
\end{aligned}\label{eq15.3.35}\]

We can see that the total dipole correlation function has a sum over two oscillating terms at the transition energies given in eq. (\ref{eq15.3.30}) which arise from \(n=0\rightarrow1\) vibrational transition and the \(\Delta J=+1\) and \(-1\) changes in rotational quantum number. When we calculate our absorption cross-section using eq. (15.1.23), we obtain

\[\begin{aligned}
 \sigma \nobreak (\omega ) & = \frac{\pi \omega }{\varepsilon _{0}\hbar c} \goodbreak \left[ C_{\mu \mu }(\omega ) \right.\left. - C_{\mu \mu }( - \omega ) \right] \\
 & = \frac{\mu '^{2}}{12 m_R\varepsilon _{0}\hbar c} \goodbreak \left( \frac{\omega }{\omega _{0}} \right) \goodbreak \sum\limits_{J} p_{J}\left( (J + 1) \goodbreak \delta \left( \omega - \omega _{0} - 2B\left( J + 1 \right) \right) \goodbreak + J \delta \left( \omega - \omega _{0} + 2JB \right) \right) \\
\end{aligned} \label{eq15.3.36}\]

We observe that we have a sum over delta function transitions for the \(\Delta J=\pm1\) terms. In this case, stimulated emission terms do not contribute since we ignored the negligibly small thermal population of the \(n=1\) vibrational level. However, it is still possible to have \(\Delta J=-1\) transitions with \(\Delta n=+1\) involving higher lying rotational states that are initially thermally occupied. 

An example of a rovibrational infrared correlation function and absorption spectrum from this model are shown in Figure 15.3.4, using the rotational parameters from Figure 15.3.2 and a vibrational frequency of \(\omega_0= 2000 \, \text{cm}^{-1}\). While the pure vibrational correlation function only oscillates at a single (high) frequency \(\omega_0\), the total correlation function is a product with the rotational correlation function. As a result, we observe a correlation function in which the high frequency vibrational oscillations are modulated by slower rotational recurrences. In the resulting absorption spectrum, we see two lobes—the P-branch (\(\Delta J=-1)\) and R-branch (\(\Delta J=+1\))—with periodic transitions spaced by \(2B\) on either side of the vibrational \(0 \rightarrow 1\) transition frequency, \(\omega_0\). The lobes rise from either side of \(\omega_0\) as a result of the growth in the degeneracy of \(M\) states with increasing \(J\); however, at higher detuning from \(\omega_0\), the lines switch to decaying as a result of decreases in thermal occupation of the initial rotational states which occur as \(J\) increases.

Figure \(\PageIndex{5}\): Temperature-dependence of IR rotational-vibrational spectroscopy, illustrating the change in \(J\) state populations, the dipole correlation function \(C_{\mu\mu}(t)\), and the absorption cross-section \(\sigma\).

Raman spectroscopy

Although we derived a description of the absorption spectrum in terms of a correlation function for the electric dipole operator, similar forms emerge for non-resonant light-matter interactions, which we can illustrate using Raman scattering. Technically, we need second-order perturbation theory to describe Raman interactions, because the transitions are induced by two electromagnetic fields whose frequency difference equals the energy splitting between states. But we can brush over this and summarize the time-correlation function formalism by replacing the dipole operator with an induced dipole moment generated by an incident field. The incident field \(\boldsymbol{\mathcal{E}}_i\) polarizes the molecule, inducing a dipole moment whose magnitude is determined by the polarizability \(\boldsymbol{\alpha}\)

\[\boldsymbol{\mu}_{\text{ind}}=\boldsymbol{\alpha} \boldsymbol{\mathcal{E}}_i(t) \label{eq15.3.37}\]

The scattered light field \(\boldsymbol{\mathcal{E}}_s\) results from an interaction with this induced dipole

\[\begin{aligned}
 V(t) &= - \boldsymbol{\mu}_{\text{ind}} \cdot \boldsymbol{\mathcal{E}}_{s}(t) \\
 & = \boldsymbol{\mathcal{E}}_{s}(t) \cdot \boldsymbol{\alpha} \cdot \boldsymbol{\mathcal{E}}_{i}(t) \\
 & = \mathcal{E}_{s}(t)\mathcal{E}_{i}(t)\left( {\hat \varepsilon }_{s} \cdot \boldsymbol{\alpha} \cdot {\hat \varepsilon }_{i} \right) \\
\end{aligned} \label{eq15.3.38}\]

Here we have written the polarization components of the incident (\(i\)) and scattered (\(s\)) light projecting onto the polarizability tensor \(\boldsymbol{\alpha}\). Equation (\ref{eq15.3.38}) leads to an expression for the Raman spectrum in terms of the time-correlation function for the polarizability :

\[\begin{aligned}
 \sigma (\omega) & \propto \int_{ - \infty }^{ + \infty } dt \, e^{\mathrm{i} \omega t} \left\langle \left( {\hat \varepsilon }_{s} \cdot \boldsymbol{\alpha}(t) \cdot {\hat \varepsilon }_{i} \right) \left( {\hat \varepsilon }_{s} \cdot \boldsymbol{\alpha}(0) \cdot {\hat \varepsilon }_{i} \right) \right\rangle \\
 & = \int_{ - \infty }^{ + \infty } dt \, e^{\mathrm{i}\omega t} \left\langle \boldsymbol{\alpha}(t) \boldsymbol{\alpha}(0) \right\rangle \\
\end{aligned} \label{eq15.3.39}\]

As with the dipole operator, the time-evolution of the polarizability operator arises from the Hamiltonian for the matter in the absence of the field \(\boldsymbol{\alpha}(t)=e^{\mathrm{i}H_0t/\hbar}\boldsymbol{\alpha}e^{-\mathrm{i}H_0t/\hbar}.\) To evaluate this, the polarizability tensor can also be expanded in the nuclear coordinates, as in eq. (\ref{eq15.3.01}),

\[\boldsymbol{\alpha} = \boldsymbol{\alpha}_{0} + \left. \frac{\partial \boldsymbol{\alpha}}{\partial q} \right|_{q = q_{0}}q + \cdots \label{eq15.3.40}\]

where the leading term would lead to Raleigh scattering and rotational Raman spectra, and the second term would give vibrational Raman scattering.

Also remember that the polarizability tensor is a second rank tensor that tells you how well a light field polarized along \(i\) can induce a dipole moment (light-field-induced charge displacement) in the \(s\) direction. For cylindrically symmetric systems which have a polarizability component \(\alpha _{||}\) along the principal axis of the molecule and a component \(\alpha _{ \bot }\) perpendicular to that axis, this usually takes the form

\[\boldsymbol{\alpha} = \left( \kern-.5em \begin{array}{c}
 \alpha _{||}&{}&{} \\
 {}&\alpha _{ \bot }&{} \\
 {}&{}&\alpha _{ \bot }
\end{array} \kern-.5em \right) = \alpha \mathbf{I} + \frac{1}{3}\beta \left( \kern-.5em \begin{array}{c}
 2&{}&{} \\
 {}& - 1&{} \\
 {}&{}& - 1
\end{array} \kern-.5em \right) \label{eq15.3.41}\]

where \(\alpha\) is the isotropic component of polarizability tensor, and \(\beta\) is the anisotropic component. 

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References

\[\tag {1} \label {Cite1}\]

The traditional form for the rotational energies is expressed as a Dunham expansion in powers of \(J(J+1)\) and expressed in units of wavenumbers, such that \(\bar{\nu}= E_J / hc=\bar{B}J\left(J+1\right)+\ldots\) where \(\bar{B}=h/8\pi^2Ic\). The bar over a variable indicates the \(\text{cm}^{-1}\) units convention.