# 15.4: Time Correlation Function Expressions for Transition Rates

The first-order E1 "golden-rule" expression for the rates of photon-induced transitions can be recast into a form in which certain specific physical models are easily introduced and insights are easily gained. Moreover, by using so-called equilibrium averaged time correlation functions, it is possible to obtain rate expressions appropriate to a large number of molecules that exist in a distribution of initial states (e.g., for molecules that occupy many possible rotational and perhaps several vibrational levels at room temperature).

## State-to-State Rate of Energy Absorption or Emission

To begin, the expression obtained earlier

$R_{i,f} = \left( \dfrac{2\pi}{\hbar^2}\right) g(\omega_{f,i}) | \textbf{E}_0 \cdot{\langle} \phi_f | \mu | \Phi_i \rangle |^2,$

that is appropriate to transitions between a particular initial state $$\Phi_i$$ and a specific final state $$\Phi_f$$ , is rewritten as

$R_{i,f} = \left( \dfrac{2\pi}{\hbar^2}\right) \int g( \omega ) | \textbf{E}_0 \cdot{\langle} \phi_i | \mu | \phi_i \rangle |^2 \delta (\omega_{f,i} - \omega )\text{ d}\omega .$

Here, the $$\delta (\omega_{f,i} - \omega )$$ function is used to specifically enforce the "resonance condition" that resulted in the time-dependent perturbation treatment given in Chapter 14; it states that the photons' frequency $$\omega$$ must be resonant with the transition frequency $$\omega_{f,i}$$ . It should be noted that by allowing $$\omega$$ to run over positive and negative values, the photon absorption (with $$\omega_{f,i}$$ positive and hence w positive) and the stimulated emission case (with $$\omega_{f,i}$$ negative and hence $$\omega$$ negative) are both included in this expression (as long as g($$\omega$$) is defined as g(|$$\omega$$|) so that the negative-$$\omega$$ contributions are multiplied by the light source intensity at the corresponding positive $$\omega$$ value).

The following integral identity can be used to replace the $$\delta$$-function:

$\delta (\omega_{f,i} - \omega ) = \dfrac{1}{2\pi} \int\limits_{-\infty}^{\infty} e^{i(\omega_{f,i} - \omega )t} \text{ dt}$

by a form that is more amenable to further development. Then, the state-to-state rate of transition becomes:

$R_{i,f} = \left(\dfrac{1}{\hbar}\right) \int g(\omega ) | \textbf{E}_0 \cdot{\langle}\phi_f | \mu | \phi_i \rangle |^2 \int\limits_{-\infty}^{\infty}e^{i(\omega_{f,i} - \omega)t} \text{ dt d}\omega .$

## Averaging Over Equilibrium Boltzmann Population of Initial States

If this expression is then multiplied by the equilibrium probability $$\rho_i$$ that the molecule is found in the state $$\Phi_i$$ and summed over all such initial states and summed over all final states $$\Phi_f$$ that can be reached from $$\Phi_i$$ with photons of energy $$\hbar \omega$$, the equilibrium averaged rate of photon absorption by the molecular sample is obtained:

$R_{\text{eq.ave.}} = \left( \dfrac{1}{\hbar^2} \right) \sum\limits_{i,f} \rho_i$

$\int g(\omega ) | \textbf{E}_0 \cdot{\langle}\Phi_f | \mu | \Phi_i \rangle |^2 \int\limits_{-\infty}^{\infty} e^{i(\omega_{f,i} - \omega )t} \text{dt d}\omega .$

This expression is appropriate for an ensemble of molecules that can be in various initial states $$\Phi_i$$ with probabilities $$\rho_i$$. The corresponding result for transitions that originate in a particular state ($$\Phi_i$$) but end up in any of the "allowed" (by energy and selection rules) final states reads:

$R_{\text{state i.}} = \left( \dfrac{1}{\hbar^2} \right) \sum\limits_f \int g(\omega ) | \textbf{E}_0 \cdot{\langle}\Phi_f | \mu | \Phi_i \rangle |^2$

$\int\limits_{-\infty}^{\infty} e^{i(\omega_{f,i} - \omega )t}\text{ dtd}\omega .$

For a canonical ensemble, in which the number of molecules, the temperature, and the system volume are specified, $$\rho_i$$ takes the form:

$\phi_i = \dfrac{g_i}{Q}e^{-\dfrac{E_i^0}{kT}}$

where Q is the canonical partition function of the molecules and $$g_i$$ is the degeneracy of the state $$\Phi_i$$ whose energy is $$\text{E}_i^0.$$

In the above expression for $$\text{R}_{\text{eq.ave.}}$$, a double sum occurs. Writing out the elements that appear in this sum in detail, one finds:

$\sum\limits_{i,f} \phi_i \textbf{E}_0 \cdot{\langle} \Phi_i | \mu | \Phi_f \rangle \textbf{E}_0 \cdot{\langle}\Phi_f | \mu | \Phi_i \rangle e^{i(\omega_{f,i})t}.$

In situations in which one is interested in developing an expression for the intensity arising from transitions to all allowed final states, the sum over these final states can be carried out explicitly by first writing

$\langle \Phi_f | \mu | \Phi_i \rangle e^{i(\omega_{f,i})t} = \langle \Phi_f | e^{\dfrac{iHt}{\hbar}} \mu e^{-\dfrac{iHt}{\hbar}} | \Phi_i \rangle$

and then using the fact that the set of states {$$\Phi_k$$} are complete and hence obey

$\sum\limits_k | \Phi_k \rangle \langle \Phi_k | = 1.$

The result of using these identities as well as the Heisenberg definition of the time dependence of the dipole operator

$\mu (t) = e^{\dfrac{iHt}{\hbar}} \mu e^{-\dfrac{iHt}{\hbar}},$

is:

$\sum\limits_i \rho_i \langle \Phi | \textbf{E}_0 \cdot{\mu} \textbf{E}_0\cdot{\mu}(t) | \Phi_i \rangle .$

In this form, one says that the time dependence has been reduce to that of an equilibrium averaged (n.b., the $$\sum\limits_i \rho_i \langle \Phi_i | | \Phi_i \rangle$$) time correlation function involving the component of the dipole operator along the external electric field at t = 0 ($$\textbf{E}_0\cdot{\mu}$$) and this component at a different time $$t(\textbf{E}_0\cdot{\mu}(t))$$.

## Photon Emission and Absorption

If $$\omega_{f,i}$$ is positive (i.e., in the photon absorption case), the above expression will yield a non-zero contribution when multiplied by $$e^{-i \omega t}$$ and integrated over positive $$\omega$$ values. If $$\omega_{f,i}$$ is negative (as for stimulated photon emission), this expression will contribute, again when multiplied by $$e^{-i\omega t}$$, for negative $$\omega$$-values. In the latter situation, $$\rho_i$$ is the equilibrium probability of finding the molecule in the (excited) state from which emission will occur; this probability can be related to that of the lower state $$\rho_f$$ by

$\rho_{\text{excited}} = \rho_{\text{lower}} e^{-\dfrac{(\text{E}^0_{\text{excited}} - \text{E}^0_{\text{lower}})}{kT}}$

$= \rho_{\text{lower}}e^{-\dfrac{\hbar \omega}{kT}}.$

In this form, it is important to realize that the excited and lower states are treated as individual states, not as levels that might contain a degenerate set of states.

The absorption and emission cases can be combined into a single net expression for the rate of photon absorption by recognizing that the latter process leads to photon production, and thus must be entered with a negative sign. The resultant expression for the net rate of decrease of photons is:

$\text{R}_{\text{eq.ave.net}} = \left( \dfrac{1}{\hbar^2} \right) \sum\limits_i \rho_i \left( 1 - e^{-\dfrac{\hbar \omega}{kT}} \right)$

$\iint g(\omega ) \langle \Phi_i | (\textbf{E}_0 \cdot{\mu} ) \textbf{E}_0 \cdot{\mu}(t) | \Phi_i \rangle e^{-i\omega t} \,d\omega \,dt.$

## The Line Shape and Time Correlation Functions

Now, it is convention to introduce the so-called "line shape" function $$I(\omega)$$:

$I(\omega ) = \sum\limits_i \rho_i \int \langle \Phi_i | (\textbf{E}_0 \cdot{\mu}) \textbf{E}_0\cdot{\mu} (t) | \Phi_i \rangle e^{-i\omega t}\, dt$

in terms of which the net photon absorption rate is

$\text{R}_{\text{eq.ave.net}} = \left( \dfrac{1}{\hbar^2} \right) \left( 1 -e^{-\hbar \omega/kT}\right) \int g(\omega )I(\omega ) \text{ d}\omega .$

As stated above, the function

$\text{C}(t) = \sum\limits_i \rho_i \langle \Phi_i | \textbf{E}_0 \cdot{\mu} ) \textbf{E}_0\cdot{\mu}(t) | \Phi_i \rangle$

is called the equilibrium averaged time correlation function of the component of the electric dipole operator along the direction of the external electric field $$\textbf{E}_0$$. Its Fourier transform is $$I(\omega)$$, the spectral line shape function. The convolution of $$I(\omega)$$ with the light source's (\g(\omega\)) function, multiplied by $$\left(1 - e^{-\frac{h \omega}{kT}} \right)$$, the correction for stimulated photon emission, gives the net rate of photon absorption.

## Rotational, Translational, and Vibrational Contributions to the Correlation Function

To apply the time correlation function machinery to each particular kind of spectroscopic transition, one proceeds as follows:

### Pure Rotational transitions

For purely rotational transitions, the initial and final electronic and vibrational states are the same. Moreover, the electronic and vibrational states are not summed over in the analog of the above development because one is interested in obtaining an expression for a particular $$\chi_{\text{iv}} \psi_{\text{ie}} \rightarrow \chi_{\text{fv}} \psi_{\text{fe}}$$ electronic-vibrational transition's lineshape. As a result, the sum over final states contained in the expression (see earlier)

$\sum\limits_{\text{i,f}} \rho_i \textbf{E}_0 \cdot{\langle} \Phi_{\text{i}} | \mu | \Phi_{\text{f}} \rangle \textbf{E}_0 \cdot{\langle} \Phi_{\text{f}} | \mu (t) | \Phi_{\text{i}} \rangle e^{i(\omega_{\text{f,i}})t}$

applies only to summing over final rotational states. In more detail, this can be shown as follows:

$\sum\limits_{\text{i,f}} \rho_{\text{i}} \textbf{E}_0 \cdot{\langle}\Phi_{\text{i}} | \mu | \Phi_{\text{f}} \rangle \textbf{E}_0 \cdot{\rangle} \Phi_{\text{f}} | \mu (t) | \Phi_{\text{i}} \rangle$

$=\sum\limits_{\text{i,f}} \rho_{\text{i}} \textbf{E}_0 \cdot{\langle} \phi_{\text{ir}} \chi_{\text{iv}} \psi_{\text{ie}} | \mu | \phi_{\text{fr}} \chi_{\text{iv}} \psi_{\text{ie}} \rangle \textbf{E}_0 \cdot{\langle} \phi_{\text{fr}} \chi_{\text{iv}} \psi_{\text{ie}} | \mu (t) | \phi_{\text{ir}} \chi_{\text{iv}} \psi_{\text{ie}} \rangle$

$=\sum\limits_{\text{i,f}} \rho_{\text{ir}} \rho_{\text{iv}} \rho_{\text{ie}} \textbf{E}_0 \cdot{\langle} \phi_{\text{ir}} \chi_{\text{iv}} | \mu (\textbf{R}) | \phi_{\text{fr}} \chi_{\text{iv}} \rangle \cdot{\langle} \phi_{\text{fr}} \chi_{\text{iv}} | \mu (\textbf{R},t) | \phi_{\text{ir}} \chi_{\text{iv}} \rangle$

$=\sum\limits_{\text{i,f}} \rho_{\text{ir}} \rho_{\text{iv}} \rho_{\text{ie}} \textbf{E}_0 \cdot{\langle} \phi_{\text{ir}} | \mu_{\text{ave.iv}} | \phi_{\text{fr}} | \mu_{\text{ave.iv}}\text{(t)} | \phi_{\text{ir}} \rangle$

$=\sum\limits_{\text{i}} \rho_{\text{ir}} \rho_{\text{iv}} \rho_{\text{ie}} \textbf{E}_0 \cdot{\langle} \phi_{\text{ir}} | \mu_{\text{ave.iv}} \textbf{E}_0 \cdot{\mu}_{\text{ave.iv}}\text{(t)} | \phi_{\text{ir}} \rangle .$

In moving from the second to the third lines of this derivation, the following identity was used:

$\langle \phi_{\text{fr}} \chi_{\text{iv}} \psi_{\text{ie}} | \mu \text{(t)} | \phi_{\text{ir}} \chi_{\text{iv}} \psi_{\text{ie}} | e^{\left(\dfrac{\text{iHt}}{\hbar}\right)} \mu e^{\left(-\dfrac{\text{iHt}}{\hbar}\right)} | \phi_{\text{ir}} \chi_{\text{iv}} \psi_{\text{ie}} \rangle$

$=\langle \phi_{\text{fr}} \chi_{\text{iv}} \psi_{\text{ie}} | e^{\left(\dfrac{\text{itH}_{v,r}}{\hbar}\right)} \mu (\textbf{R}) e^{\left(-\dfrac{\text{itH}_{v,r}}{\hbar}\right)} | \phi_{\text{ir}} \chi_{\text{iv}} \psi_{\text{ie}} \rangle ,$

where H is the full (electronic plus vibrational plus rotational) Hamiltonian and $$\text{H}_{\text{v,r}}$$ is the vibrational and rotational Hamiltonian for motion on the electronic surface of the state $$\psi_{\text{ie}}$$ whose dipole moment is $$\mu (\textbf{R})$$. From the third line to the fourth, the (approximate) separation of rotational and vibrational motions in $$\text{H}_{\text{v,r}}$$

$\text{H}_{\text{v,r}} = \text{H}_{\text{v}} + \text{H}_{\text{r}}$

has been used along with the fact that $$\chi_{\text{iv}} \text{ is an eigenfunction of } \text{H}_{\text{v}}$$:

$H_v \chi_{iv} = E_{iv} \chi_{iv}$

to write

$\langle \chi_{iv} | \mu (\textbf{R},t) | \chi_{iv} \rangle = e^{\left(\dfrac{\text{itH}_r}{\hbar}\right)} \langle \chi_{iv} | e^{\left(\dfrac{\text{itH}_v}{\hbar}\right)} \mu (\textbf{R}) e^{\left(-\dfrac{\text{itH}_v}{\hbar}\right)} | \chi_{iv} \rangle e^{\left(-\dfrac{\text{itH}_r}{\hbar}\right)}$

$= e^{\left( \dfrac{\text{itH}_r}{\hbar} \right)} \langle \chi_{\text{iv}} | e^{\left(\dfrac{\text{itE}_{iv}}{\hbar}\right)} \mu (\textbf{R}) e^{\left(-\dfrac{\text{itE}_{iv}}{\hbar}\right)} | \chi_{iv} \rangle e^{\left(-\dfrac{\text{itH}_{r}}{\hbar}\right)}$

$= e^{\left(\dfrac{\text{itH}_{r}}{\hbar}\right)} \langle \chi_{iv} | \mu (\textbf{R}) | \chi_{iv} \rangle e^{\left(-\dfrac{\text{itH}_{r}}{\hbar}\right)}$

$= \mu_{\text{ave.iv}}\text{(t)}.$

In effect, $$\mu$$ is replaced by the vibrationally averaged electronic dipole moment $$\mu_{\text{ave,iv}}$$ for each initial vibrational state that can be involved, and the time correlation function thus becomes:

$\text{C(t)} = \sum\limits_{\text{i}} \rho_{\text{ir}} \rho_{\text{iv}} \rho_{\text{ie}} \langle \phi_{\text{ir}} | \textbf{E}_0 \cdot{\mu}_{\text{ave, iv}}\text{(t)} | \phi_{\text{ir}} \rangle ,$

where $$\mu_{\text{ave, iv}}\text{(t)}$$ is the averaged dipole moment for the vibrational state $$\chi_{\text{iv}}$$ at time t, given that it was $$\mu_{\text{ave, iv}}$$ at time t = 0. The time dependence of $$\mu_{\text{ave, iv}}\text{(t)}$$ is induced by the rotational Hamiltonian $$\text{H}_r$$ , as shown clearly in the steps detailed above:

$\mu_{\text{ave, iv}}\text{(t)} = e^{\left( \dfrac{\text{itH}_r}{\hbar} \right)} \langle \chi_{\text{iv}} | \mu (\textbf{R}) | \chi_{text{iv}} \rangle e^{\left(-\dfrac{\text{itH}_r}{\hbar}\right)}.$

In this particular case, the equilibrium average is taken over the initial rotational states whose probabilities are denoted $$\rho_{\text{ir}}$$ , any initial vibrational states that may be populated, with probabilities $$\rho_{\text{iv}}$$, and any populated electronic states, with probabilities $$\rho_{\text{ie}}$$.

### vibration-rotation transitions

For vibration-rotation transitions within a single electronic state, the initial and final electronic states are the same, but the initial and final vibrational and rotational states differ. As a result, the sum over final states contained in the expression $$\sum\limits_{i,f} \phi_i \textbf{E}_0 \cdot{\langle}\Phi_i | \mu | \Phi_f \rangle \textbf{E}_0 \cdot{\langle} | \mu | \Phi_i \rangle e^{i(\omega_{f,i})t}$$ applies only to summing over final vibrational and rotational states. Paralleling the development made in the pure rotation case given above, this can be shown as follows:

$\sum\limits_{i,f} \rho_i \textbf{E}_0 \cdot{\langle} \Phi_i | \mu | \Phi_f \rangle \textbf{E}_0 \cdot{\langle} | \mu \text{(t) }| \Phi_i \rangle$

$= \sum\limits_{i,f} \rho_i\textbf{E}_0 \cdot{\langle}\phi_{\text{ir}} \chi_{\text{iv}} \psi_{\text{ie}} | \mu | \phi_{\text{fr}} \chi_{\text{fv}} \psi_{\text{ie}} \rangle \textbf{E}_0 \cdot{\langle} \phi_{\text{fr}} \chi_{\text{fv}} \psi_{\text{ie}} | \mu \text{(t) }| \phi_{\text{ir}} \chi_{\text{iv}} \psi_{\text{ie}} \rangle$

$= \sum\limits_{i,f} \rho_{\text{ir}} \rho_{\text{iv}} \rho_{\text{ie}} \textbf{E}_0 \cdot{\langle}\phi_{\text{ir}} \chi_{\text{iv}} | \mu (\textbf{R}) | \phi_{\text{fr}} \chi_{\text{fv}} \rangle \textbf{E}_0 \cdot{\langle} \phi_{\text{fr}} \chi_{\text{fv}} | \mu (\textbf{R},t)| \phi_{\text{ir}} \chi_{\text{iv}} \rangle$

$= \sum\limits_{i,f} \rho_{\text{ir}} \rho_{\text{iv}} \rho_{\text{ie}} \textbf{E}_0 \cdot{\langle}\phi_{\text{ir}} \chi_{\text{iv}} | \mu (\textbf{R}_e) + \sum\limits_a (R_a - R_{a, eq}) \dfrac{\partial \mu}{\partial R_a} | \phi_{\text{fr}} \chi_{\text{fv}} \rangle \textbf{E}_0 \cdot{\langle} \phi_{\text{fr}} \chi_{\text{fv}} | e^{\left(\dfrac{\text{itH}_r}{\hbar}\right)}\left( \mu (\textbf{R}_e) + \sum\limits_a(R_a - R_{a, eq}) \dfrac{\partial \mu}{\partial R_a} \right) e^{\left(-\dfrac{\text{itH}_r}{\hbar}\right)} | \phi_{\text{ir}} \chi_{\text{iv}} \rangle e^{\left(i\omega_{fv, iv}t\right)}$

$\sum\limits_{i,f} \rho_{\text{ir}} \rho_{\text{iv}} \rho_{\text{ie}} \langle \phi_{\text{ir}} | \textbf{E}_0 \cdot{\mu_{\text{i,f}}}(\textbf{R}_e)\textbf{E}_0 \cot{\mu_{\text{i,f}}}(\textbf{R}_e,t) | \phi_{\text{ir}} \rangle | \langle \chi_{\text{iv}} | \chi_{\text{fv}} |^2 e^{i\omega_{\text{fv, iv}}t + i\Delta E_{\text{i,f}}\dfrac{t}{\hbar}},$

where

$\mu_{\text{i,f}}(\textbf{R}_e,t) = e^{\left( \dfrac{\text{itH}_r}{\hbar} \right)} \mu_{\text{i,f}}(\textbf{R}_e) e^{\left( -\dfrac{\text{itH}_r}{\hbar} \right)}$

is the electronic transition dipole matrix element, evaluated at the equilibrium geometry of the absorbing state, that derives its time dependence from the rotational Hamiltonian $$\text{H}_r$$ as in the time correlation functions treated earlier.

This development thus leads to the following definition of C(t) for the electronic, vibration, and rotation case:

$\text{C(t)} = \sum\limits_{\text{i,f}} \rho_{\text{ir}} \rho_{\text{iv}} \rho_{\text{ie}} \langle \phi_{\text{ir}} | \textbf{E}_0 \cdot{\mu_{\text{if}}}(\textbf{R}_e)\textbf{E}_0 \cdot{\mu_{\text{if}}}(\textbf{R}_e,t) | \phi_{\text{ir}} \rangle | \langle \chi_{\text{iv}} | \chi_{\text{fv}} \rangle |^2 e^{\left( i\omega_{\text{fv, iv}}t + \dfrac{i\Delta E_{\text{i,f}}t}{\hbar} \right)}$

but the net rate of photon absorption remains:

$\text{R}_{\text{eq.ave.net}} = \left(\dfrac{1}{\hbar^2}\right)\left( 1 - e^{\left( -\dfrac{\hbar \omega}{kT} \right)} \right) \int\limits g(\omega ) I (\omega )\text{ d}\omega .$

Here, I($$\omega$$) is the Fourier transform of the above C(t) and $$\Delta E_{\text{i,f}}$$ is the adiabatic electronic energy difference (i.e., the energy difference between the v = 0 level in the final electronic state and the v = 0 level in the initial electronic state) for the electronic transition of interest. The above C(t) clearly contains Franck-Condon factors as well as time dependence

$e^{\left( it\omega_{\text{fv, iv}} + i\Delta E_{\text{i,f}}\dfrac{t}{\hbar} \right)}$

that produces $$\delta$$-function spikes at each electronic-vibrational transition frequency and rotational time dependence contained in the time correlation function quantity

$\langle \phi_{\text{ir}} \textbf{E}_0 \cdot{\mu_{\text{i,f}}} (\textbf{R}_e)\textbf{E}_0\cdot{\mu_{\text{i,f}}}(\textbf{R}_e,t) |\phi_{\text{ir}} \rangle .$

To summarize, the line shape function I($$\omega$$) produces the net rate of photon absorption

$R_{\text{eq.ave.net}} = \left( \dfrac{1}{\hbar^2} \right)\left( 1 - e^{\left(-\dfrac{\hbar \omega}{kT}\right)}\right) \int\limits g(\omega ) I(\omega )\text{ d}\omega$

in all of the above cases, and I($$\omega$$) is the Fourier transform of a corresponding time-dependent C(t) function in all cases. However, the pure rotation, vibration-rotation, and electronic-vibration-rotation cases differ in the form of their respective C(t)'s. Specifically,

$\text{C(t)} = \sum\limits_i \rho_{\text{ir}} \rho_{\text{iv}} \rho_{\text{ie}} \langle \phi_{\text{ir}} | (\textbf{E}_0 \cdot{\mu}_{\text{ave, iv}} ) \textbf{E}_0 \cdot{\mu}_{\text{ave, iv}}(t) | \phi_{\text{ir}} \rangle$

in the pure rotational case,

$\text{C(t)} = \sum\limits_i \rho_{\text{ir}} \rho_{\text{iv}} \rho_{\text{ie}} \langle \phi_{\text{ir}} | (\textbf{E}_0 \cdot{\mu}_{\text{trans}} ) \textbf{E}_0 \cdot{\mu}_{\text{trans}}(t) | \phi_{\text{ir}} \rangle e^{it\omega_{\text{fv, iv}}}$

in the electronic-vibration-rotation case.

All of these time correlation functions contain time dependences that arise from rotational motion of a dipole-related vector (i.e., the vibrationally averaged dipole $$\mu_{\text{ave, iv}}(t)$$, the vibrational transition dipole $$\mu_{\text{trans}}(t)$$, or the electronic transition dipole $$\mu_{\text{i,f}}(\textbf{R}_{\text{e,t}})$$) and the latter two also contain oscillatory time dependences (i.e., $$e^{\left( it\omega_{fv,iv} \right)}$$ or $$e^{\left( it\omega_{fv,iv} + i\Delta E_{\text{i,f}}\frac{t}{\hbar} \right)}$$) that arise from vibrational or electronic-vibrational energy level differences. In the treatments of the following sections, consideration is given to the rotational contributions under circumstances that characterize, for example, dilute gaseous samples where the collision frequency is low and liquid-phase samples where rotational motion is better described in terms of diffusional motion.

## Line Broadening Mechanisms

If the rotational motion of the molecules is assumed to be entirely unhindered (e.g., by any environment or by collisions with other molecules), it is appropriate to express the time dependence of each of the dipole time correlation functions listed above in terms of a "free rotation" model. For example, when dealing with diatomic molecules, the electronicvibrational-rotational C(t) appropriate to a specific electronic-vibrational transition becomes:

$\text{C(t)} = \left( \dfrac{1}{\text{q}_r\text{q}_v\text{q}_e\text{q}_t}\right) \sum\limits_J (2J+1) e^{\left( -\dfrac{-h^2J(J+1)}{8\pi^2IkT} \right)}e^{\left( -\dfrac{hv_{\text{vib}}vi}{kT} \right)}g_{ie} \langle \phi_J | \text{E}_0 \cdot{\mu}_{\text{i,f}}(\textbf{R}_e)\textbf{E}_0\cdot{\mu}_{\text{ie}}(\textbf{R}_e,t) | \phi_J \rangle | \langle \chi_{\text{iv}} | \chi_{\text{fv}} \rangle |^2 e^{\left( ithv_{\text{vib}} + i\Delta E_{\text{i,f}}\dfrac{t}{\hbar} \right)}.$

Here,

$\text{q}_r = \left( \dfrac{8\pi^2 IkT}{h^2} \right)$

is the rotational partition function (I being the molecule's moment of inertia $$I = \mu R_e^2, \text{ and } \dfrac{h^2J(J+1)}{8\pi^2I}$$ the molecule's rotational energy for the state with quantum number J and degeneracy 2J+1)

$\text{q}_v = \dfrac{ e^{\left( -\dfrac{hv_{\text{vib}}}{2\text{kT}} \right)} }{1 - e^{\left( -\dfrac{hv_{\text{vib}}}{\text{kT}}\right) } }$

is the vibrational partition function ($$v_{\text{vib}}$$ being the vibrational frequency), $$\text{g}_{\text{ie}}$$ is the degeneracy of the initial electronic state,

$\text{q}_t = \sqrt[3]{\left( \dfrac{2\pi \text{mkT}}{h^2} \right)}V$

is the vibrational partition function ($$v_{\text{vib}}$$ being the vibrational frequency), $$\text{g}_{\text{ie}}$$ is the degeneracy of the initial electronic state,

$\text{q}_t = \sqrt[3]{\dfrac{2\pi mkT}{h^2}}\text{V}$

is the translational partition function for the molecules of mass m moving in volume V, and $$\Delta E_{\text{i,f}}$$ is the adiabatic electronic energy spacing.

The functions $$\langle \phi_J | \textbf{E}_0 \cdot{\mu}_{\text{i,f}}(\textbf{R}_e)\textbf{E}_0 \cdot{\mu}_{\text{i,f}}(\textbf{R}_e,t) | \phi_J \rangle$$ describe the time evolution of the dipole-related vector (the electronic transition dipole in this case) for the rotational state J. In a "free-rotation" model, this function is taken to be of the form:

$\langle \phi_J | \textbf{E}_0 \cdot{\mu}_{\text{i,f}}(\textbf{R}_e)\textbf{E}_0\cdot{\mu}_{\text{i,f}}(\textbf{R}_e,t) \phi_J \rangle = \langle \phi_J | \textbf{E}_0 \cdot{\mu}_{\text{i,f}}(\textbf{R}_e)\textbf{E}_0\cdot{\mu}_{\text{i,f}}(\textbf{R}_e,0) \phi_J \rangle \text{Cos}\left( \dfrac{\text{hJ(J + 1)}}{4\pi\text{I}} \right)$

where

$\omega_J = \left( \dfrac{\text{hJ(J + 1)}}{4\pi\text{I}} \right)$

is the rotational frequency (in cycles per second) for rotation of the molecule in the state labeled by J. This oscillatory time dependence, combined with the $$e^{\left( it\omega_{\text{fv,iv}} + i\Delta E_{\text{i,f}} \frac{t}{\hbar} \right)}$$ time dependence arising from the electronic and vibrational factors, produce, when this C(t) function is Fourier transformed to generate I($$\omega$$) a series of $$\delta$$-function "peaks" whenever

$\omega = \omega_{\text{fv,iv}} + \dfrac{\Delta E_{\text{i,f}}}{\hbar}$

The intensities of these peaks are governed by the

$\text{C(t)} = \left( \dfrac{1}{\text{q}_r\text{q}_v\text{q}_e\text{q}_t}\right) \sum\limits_J (2J+1) e^{\left( -\dfrac{-h^2J(J+1)}{8\pi^2IkT} \right)}e^{\left( -\dfrac{hv_{\text{vib}}vi}{kT} \right)}g_{ie}.$

Boltzmann population factors as well as by the $$| \langle \chi_{iv}| \chi_{fv} \rangle |^2$$ Franck-Condon factors and the $$\langle \phi_J | \textbf{E}_0 \cdot{\mu}_{\text{i,f}}(\textbf{R}_e)\textbf{E}_0\cdot{\mu}_{\text{i,f}}(\textbf{R}_e,0) | \phi_J \rangle$$ terms.

This same analysis can be applied to the pure rotation and vibration-rotation C(t) time dependences with analogous results. In the former, $$\delta$$-function peaks are predicted to occur at

$\omega \pm \omega_J$

and in the latter at

$\omega = \omega_{\text{fv,iv}} \pm \omega_J ;$

with the intensities governed by the time independent factors in the corresponding expressions for C(t).

In experimental measurements, such sharp $$\delta$$-function peaks are, of course, not observed. Even when very narrow band width laser light sources are used (i.e., for which g($$\omega$$) is an extremely narrowly peaked function), spectral lines are found to possess finite widths. Let us now discuss several sources of line broadening, some of which will relate to deviations from the "unhindered" rotational motion model introduced above.

In the above expressions for $$C(t)$$, the averaging over initial rotational, vibrational, and electronic states is explicitly shown. There is also an average over the translational motion implicit in all of these expressions. Its role has not (yet) been emphasized because the molecular energy levels, whose spacings yield the characteristic frequencies at which light can be absorbed or emitted, do not depend on translational motion. However, the frequency of the electromagnetic field experienced by moving molecules does depend on the velocities of the molecules, so this issue must now be addressed.

Elementary physics classes express the so-called Doppler shift of a wave's frequency induced by movement either of the light source or of the molecule (Einstein tells us these two points of view must give identical results) as follows:

$\omega_{\text{observed}} = \omega_{\text{nominal}} \left(\dfrac{1}{1+\dfrac{v_z}{c}}\right) \approx \left( 1 - \dfrac{v_z}{c} + ...\right) .$

Here, $$\omega_{\text{nominal}}$$ is the frequency of the unmoving light source seen by unmoving molecules, $$v_z$$ is the velocity of relative motion of the light source and molecules, c is the speed of light, and $$\omega_{\text{observed}}$$ is the Doppler shifted frequency (i.e., the frequency seen by the molecules). The second identity is obtained by expanding, in a power series, the $$\left(1 + \dfrac{v_z}{c}\right)^{-1}$$ factor, and is valid in truncated form when the molecules are moving with speeds significantly below the speed of light.

For all of the cases considered earlier, a C(t) function is subjected to Fourier transformation to obtain a spectral lineshape function I($$\omega$$), which then provides the essential ingredient for computing the net rate of photon absorption. In this Fourier transform process, the variable $$\omega$$ is assumed to be the frequency of the electromagnetic field experienced by the molecules . The above considerations of Doppler shifting then leads one to realize that the correct functional form to use in converting C(t) to I($$\omega$$) is:

$I(\omega ) = \int C(t) e^{-t\omega \left( 1 - \dfrac{v_z}{c} \right)} \text{dt} ,$

where $$\omega$$ is the nominal frequency of the light source.

As stated earlier, within C(t) there is also an equilibrium average over translational motion of the molecules. For a gas-phase sample undergoing random collisions and at thermal equilibrium, this average is characterized by the well known Maxwell-Boltzmann velocity distribution:

$\sqrt{\left( \dfrac{m}{2\pi kT} \right)^3} e^{\left( \dfrac{-m\left( v_x^2 + v_y^2 + v_z^2 \right)}{2kT}\right)}dv_x dv_y dv_z .$

Here m is the mass of the molecules and $$v_x, v_y, \text{ and }v_z$$ label the velocities along the lab fixed cartesian coordinates.

Defining the z-axis as the direction of propagation of the light's photons and carrying out the averaging of the Doppler factor over such a velocity distribution, one obtains:

$\int\limits_{-\infty}^{\infty} e^{-it\omega \left( 1- \dfrac{v_z}{c} \right)} \sqrt{\left( \dfrac{m}{2\pi kT} \right)^3} e^{\left( \dfrac{-m(v_x^2 + v_y^2 + v_z^2)}{2kT} \right)}dv_x dv_y dv_z$

$= e^{-\omega t} \int\limits_{-\infty}^{\infty}\sqrt{\dfrac{m}{2\pi kT}}e^{\left(\dfrac{i\omega tv_z}{c}\right)} e^{\left( \dfrac{-mv_z}{2kT} \right)} dv_z$

$=e^{-\omega t} e^{\left( \dfrac{-\omega^2t^2kT}{2mc^2} \right)}$

This result, when substituted into the expressions for C(t), yields expressions identical to those given for the three cases treated above but with one modification. The translational motion average need no longer be considered in each C(t); instead, the earlier expressions for C(t) must each be multiplied by a factor $$e^{\frac{-\omega^2 t^2kT}{2mc^2}}$$ that embodies the translationally averaged Doppler shift. The spectral line shape function I($$\omega$$) can then be obtained for each C(t) by simply Fourier transforming:

$I(\omega) = \int\limits_{-\infty}^{\infty} e^{-i\omega t}C(t)\text{dt}.$

When applied to the rotation, vibration-rotation, or electronic-vibration-rotation cases within the "unhindered" rotation model treated earlier, the Fourier transform involves integrals of the form:

$I(\omega) = \int\limits_{-\infty}^{\infty} e^{-\omega t}e^{\dfrac{-\omega^2 t^2kT}{2mc^2}}e^{it \left(\omega_{\text{fv,iv}} + \Delta \dfrac{E_{\text{i,f}}}{\hbar} \pm \omega_J \right)} \text{dt}.$

This integral would arise in the electronic-vibration-rotation case; the other two cases would involve integrals of the same form but with the $$\frac{\Delta E_{\text{i,f}}}{\hbar}$$ absent in the vibration-rotation situation and with $$\omega_{\text{fv, iv}} + \frac{\Delta E_{\text{i,f}}}{\hbar}$$ missing for pure rotation transitions. All such integrals can be carried out analytically and yield:

$I(\omega ) = \sqrt{\dfrac{2mc^2\pi}{\omega^2 kT}}e^{\left( \dfrac{-\left(\omega - \omega_{\text{fv, iv}} - \dfrac{\Delta E_{\text{i,f}}}{\hbar} \pm \omega_J\right)^2 mc^2}{2\omega^2 kT} \right)}.$

The result is a series of Gaussian "peaks" in $$\omega$$-space, centered at:

$\omega = \omega_{\text{fv,iv}} + \dfrac{\Delta E_{\text{i,f}}}{\hbar} \pm \omega_J$

with widths ($$\sigma)$$ determined by

$\sigma^2 = \dfrac{\omega^2kT}{mc^2},$

given the temperature $$T$$ and the mass of the molecules $$m$$. The hotter the sample, the faster the molecules are moving on average, and the broader is the distribution of Doppler shifted frequencies experienced by these molecules. The net result then of the Doppler effect is to produce a line shape function that is similar to the "unhindered" rotation model's series of $$\delta$$-functions but with each $$\delta$$-function peak broadened into a Gaussian shape

To include the effects of collisions on the rotational motion part of any of the above C(t) functions, one must introduce a model for how such collisions change the dipolerelated vectors that enter into C(t). The most elementary model used to address collisions applies to gaseous samples which are assumed to undergo unhindered rotational motion until struck by another molecule at which time a randomizing "kick" is applied to the dipole vector and after which the molecule returns to its unhindered rotational movement.

The effects of such collisionally induced kicks are treated within the so-called pressure broadening (sometimes called collisional broadening) model by modifying the free-rotation correlation function through the introduction of an exponential damping factor $$e^{\frac{-|t|}{\tau}}$$:

$\langle \phi_J | \textbf{E}_0\cdot{\mu}_{\text{i,f}}(\textbf{R}_e)\textbf{E}_0\cdot{\mu}_{\text{i,f}}(\textbf{R}_e,0) | \phi_J\rangle Cos \left( \dfrac{hJ(J+1)t}{4\pi I} \right)$

$\rightarrow \langle \phi_J | \textbf{E}_0 \cdot{\mu}_{\text{i,f}}(\textbf{R}_e)\textbf{E}_0\cdot{\mu}_{\text{i,f}}(\textbf{R}_e,0) | \phi_J\rangle Cos\left( \dfrac{hJ(J+1)t}{4\pi I} \right) e^{\dfrac{-|t|}{\tau}}.$

This damping function's time scale parameter $$\tau$$ is assumed to characterize the average time between collisions and thus should be inversely proportional to the collision frequency. Its magnitude is also related to the effectiveness with which collisions cause the dipole function to deviate from its unhindered rotational motion (i.e., related to the collision strength). In effect, the exponential damping causes the time correlation function $$\langle \phi_J | \textbf{E}_0\cdot{\mu}_{\text{i,f}}(\textbf{R}_e)\textbf{E}_0\cdot{\mu}_{\text{i,f}}(\textbf{R}_e,t)|\phi_J\rangle$$ to "lose its memory" and to decay to zero; this "memory" point of view is based on viewing $$\langle \phi_J | \textbf{E}_0 \cdot{\mu}_{\text{i,f}}(\textbf{R}_e)\textbf{E}_0\cdot{\mu}_{\text{i,f}}(\textbf{R}_e,t) | \phi_J\rangle$$ as the projection of $$\textbf{E}_0\cdot{\mu}_{\text{i,f}}(\textbf{R}_e,t)$$ along its t=0 value $$\textbf{E}_0\cdot{\mu}_{\text{i,f}}(\textbf{R}_e,0)$$ as a function of time t.

Introducing this additional $$e^{\frac{-|t|}{\tau}}$$ time dependence into C(t) produces, when C(t) is Fourier transformed to generate I($$\omega$$),

$I(\omega ) = \int\limits_{-\infty}^{\infty}e^{-i\omega t}e^{\dfrac{-|t|}{\tau}}e^{\dfrac{-\omega^2 t^2kT}{2mc^2}}e^{it\left( \omega_{\text{fv,iv}} + \dfrac{\Delta E_{\text{i,f}}}{\hbar} \pm \omega_J \right)}\text{dt} .$

In the limit of very small Doppler broadening, the $$\frac{\omega^2t^2kT}{2mc^2}$$ factor can be ignored (i.e.,$$e^{\frac{-\omega^2t^2kT}{2mc^2}}$$ set equal to unity), and

$I(\omega) = \int\limits_{-\infty}^{\infty}e^{-i\omega t}e^{-\left(\dfrac{|t|}{\tau}\right)}e^{\left( it\left( \omega_{\text{fv,iv}}+\dfrac{\Delta E_{\text{i,f}}}{\hbar} \pm \omega_J \right)\right)} \text{dt}$

results. This integral can be performed analytically and generates:

$I(\omega) = \dfrac{1}{4\pi}\left( \dfrac{\dfrac{1}{\tau}}{\left( \dfrac{1}{\tau} \right)^2 + \left( \omega - \omega_{\text{fv,iv}} -\dfrac{\Delta E_{\text{i,f}}}{\hbar} \pm \omega_J\right)^2} + \dfrac{\dfrac{1}{\tau}}{\left( \dfrac{1}{\tau} \right)^2 + \left( \omega + \omega_{\text{fv,iv}} + \dfrac{\Delta E_{\text{i,f}}}{\hbar} \pm \omega_J \right)^2} \right)$

a pair of Lorentzian peaks in $$\omega$$-space centered again at

$\omega = \pm \left[ \omega_{\text{fv,iv}} + \dfrac{\Delta E_{\text{i,f}}}{\hbar} \pm \omega_J \right].$

The full width at half height of these Lorentzian peaks is $$\frac{2}{\tau}$$. One says that the individual peaks have been pressure or collisionally broadened.

When the Doppler broadening can not be neglected relative to the collisional broadening, the above integral

$I(\omega) = \int\limits_{-\infty}^{\infty}e^{-\omega t}e^{-\dfrac{|t|}{\tau}}e^{-\dfrac{\omega^2t^2kT}{2mc^2}}e^{\left(it \left( \omega_{\text{fv,iv}} + \dfrac{\Delta E_{\text{i,f}}}{\hbar} \pm \omega_J \right) \right)} \text{dt}$

is more difficult to perform. Nevertheless, it can be carried out and again produces a series of peaks centered at

$\omega = \omega_{\text{fv,iv}} + \dfrac{\Delta E_{\text{i,f}}}{\hbar} \pm \omega_J$

but whose widths are determined both by Doppler and pressure broadening effects. The resultant line shapes are thus no longer purely Lorentzian nor Gaussian (which are compared in the figure below for both functions having the same full width at half height and the same integrated area), but have a shape that is called a Voight shape.

Figure 15.4.1: Insert caption here!

## Rotational Diffusion Broadening

Molecules in liquids and very dense gases undergo frequent collisions with the other molecules; that is, the mean time between collisions is short compared to the rotational period for their unhindered rotation. As a result, the time dependence of the dipole related correlation function can no longer be modeled in terms of free rotation that is interrupted by (infrequent) collisions and Dopler shifted. Instead, a model that describes the incessant buffeting of the molecule's dipole by surrounding molecules becomes appropriate. For liquid samples in which these frequent collisions cause the molecule's dipole to undergo angular motions that cover all angles (i.e., in contrast to a frozen glass or solid in which the molecule's dipole would undergo strongly perturbed pendular motion about some favored orientation), the so-called rotational diffusion model is often used. In this picture, the rotation-dependent part of C(t) is expressed as:

$\langle \phi_J | \textbf{E}_0\cdot{\mu}_{\text{i,f}}(\textbf{R}_e)\textbf{E}_0\cdot{\mu}_{\text{i,f}}(\textbf{R}_e,t)|\phi_J\rangle = \langle \phi_J | \textbf{E}_0\cdot{\mu}_{\text{i,f}}(\textbf{R}_e)\textbf{E}_0\cdot{\mu}_{\text{i,f}}(\textbf{R}_e,0) |\phi_J\rangle e^{-2D_{\text{rot}}|t|},$

where D$$_{\text{rot}}$$ is the rotational diffusion constant whose magnitude details the time decay in the averaged value of $$\textbf{E}_0\cdot{\mu}_{\text{i,f}}(\textbf{R}_e,t)$$ at time t with respect to its value at time t = 0; the larger D$$_{\text{rot}}$$, the faster is this decay.

As with pressure broadening, this exponential time dependence, when subjected to Fourier transformation, yields:

$I(\omega) = \int\limits^{\infty}_{-\infty}e^{-i\omega t}e^{-2D_{\text{rot}}|t|}e^{\dfrac{-\omega^2t^2kT}{2mc^2}}e^{\left(it \left( \omega_{\text{fv,iv}} + \dfrac{\Delta E_{\text{i,f}}}{\hbar} \pm \omega_J \right) \right)}\text{dt}.$

Again, in the limit of very small Doppler broadening, the $$\frac{\omega^2t^2kT}{2mc^2}$$ factor can be ignored (i.e., $$e^{-\frac{\omega^2t^2kT}{2mc^2}}$$) set equal to unity), and

$I(\omega) = \int\limits_{-\infty}^{\infty}e^{-i\omega t}e^{-2D_{\text{rot}}|t|}e^{\left(it \left( \omega_{\text{fv,iv}} + \dfrac{\Delta E_{\text{i,f}}}{\hbar} \pm \omega_J \right) \right)} \text{dt}$

results. This integral can be evaluated analytically and generates:

$I(\omega) = \dfrac{1}{4\pi}\left( \dfrac{2D_{\text{rot}}}{(2D_{\text{rot}})^2 + \left( \omega -\omega_{\text{fv,iv}} - \dfrac{\Delta E_{\text{i,f}}}{\hbar} \pm \omega_J \right)^2} + \dfrac{2D_{\text{rot}}}{(2D_{\text{rot}})^2 + \left( \omega + \omega_{\text{fv,iv}} + \dfrac{\Delta E_{\text{i,f}}}{\hbar} \pm \omega_J \right)^2} \right),$

a pair of Lorentzian peaks in $$\omega$$-space centered again at

$\omega = \pm \left[ \omega_{\text{fv,iv}} + \dfrac{\Delta E_{\text{i,f}}}{\hbar} \pm \omega_J \right].$

The full width at half height of these Lorentzian peaks is 4D$$_{\text{rot}}$$. In this case, one says that the individual peaks have been broadened via rotational diffusion. When the Doppler broadening can not be neglected relative to the collisional broadening, the above integral

$I(\omega) = \int\limits_{-\infty}^{\infty}e^{-i\omega t}e^{-2D_{\text{rot}}|t|}e^{-\dfrac{\omega^2t^2kT}{2mc^2}}e^{\left(it \left( \omega_{\text{fv,iv}} + \dfrac{\Delta E_{\text{i,f}}}{\hbar} \pm \omega_J \right) \right)} \text{dt.}$

is more difficult to perform. Nevertheless, it can be carried out and again produces a series of peaks centered at

$\omega = \pm \left[ \omega_{\text{fv,iv}} + \dfrac{\Delta E_{\text{i,f}}}{\hbar} \pm \omega_J \right]$

but whose widths are determined both by Doppler and rotational diffusion effects.

Whenever the absorbing species undergoes one or more processes that depletes its numbers, we say that it has a finite lifetime. For example, a species that undergoes unimolecular dissociation has a finite lifetime, as does an excited state of a molecule that decays by spontaneous emission of a photon. Any process that depletes the absorbing species contributes another source of time dependence for the dipole time correlation functions C(t) discussed above. This time dependence is usually modeled by appending, in a multiplicative manner, a factor $$e^{-\frac{|t|}{\tau}}$$. This, in turn modifies the line shape function I($$\omega$$) in a manner much like that discussed when treating the rotational diffusion case:

$I(\omega) = \int\limits_{-\infty}^{\infty}e^{-i\omega t}e^{-\dfrac{|t|}{\tau}}e^{-\dfrac{\omega^2t^2kT}{2mc^2}}e^{\left( it\left( \omega_{\text{fv,iv}} + \dfrac{\Delta E_{\text{i,f}}}{\hbar} \pm \omega_J \right) \right)} \text{dt}.$

Not surprisingly, when the Doppler contribution is small, one obtains:

$I(\omega) = \dfrac{1}{4\pi} \left( \dfrac{\dfrac{1}{\tau}}{\left(\dfrac{1}{\tau}\right)^2 \left( \omega - \omega_{\text{fv,iv}} - \dfrac{\Delta E_{\text{i,f}}}{\hbar} \pm \omega_J \right)^2 } + \dfrac{\dfrac{1}{\tau}}{\left( \dfrac{1}{\tau} \right)^2 + \left( \omega + \omega_{\text{fv,iv}} + \dfrac{\Delta E_{\text{i,f}}}{\hbar} \pm \omega_J \right)^2} \right).$

In these Lorentzian lines, the parameter $$\tau$$ describes the kinetic decay lifetime of the molecule. One says that the spectral lines have been lifetime or Heisenberg broadened by an amount proportional to $$\frac{1}{\tau}$$. The latter terminology arises because the finite lifetime of the molecular states can be viewed as producing, via the Heisenberg uncertainty relation $$\Delta E\Delta t > \hbar$$, states whose energy is "uncertain" to within an amount $$\Delta$$E.

## Site Inhomogeneous Broadening

Among the above line broadening mechanisms, the pressure, rotational diffusion, and lifetime broadenings are all of the homogeneous variety. This means that each molecule in the sample is affected in exactly the same manner by the broadening process. For example, one does not find some molecules with short lifetimes and others with long lifetimes, in the Heisenberg case; the entire ensemble of molecules is characterized by a single lifetime.

In contrast, Doppler broadening is inhomogeneous in nature because each molecule experiences a broadening that is characteristic of its particular nature (velocity $$v_z$$ in this case). That is, the fast molecules have their lines broadened more than do the slower molecules. Another important example of inhomogeneous broadening is provided by so called site broadening. Molecules imbedded in a liquid, solid, or glass do not, at the instant of photon absorption, all experience exactly the same interactions with their surroundings. The distribution of instantaneous "solvation" environments may be rather "narrow" (e.g., in a highly ordered solid matrix) or quite "broad" (e.g., in a liquid at high temperature). Different environments produce different energy level splittings

$\omega = \omega_{\text{fv,iv}} + \frac{\Delta E_{\text{i,f}}}{\hbar} \pm \omega_J$

because the initial and final states are "solvated" differently by the surroundings and thus different frequencies at which photon absorption can occur. The distribution of energy level splittings causes the sample to absorb at a range of frequencies as illustrated in the figure below where homogeneous and inhomogeneous line shapes are compared.

The spectral line shape function I($$\omega$$) is further broadened when site inhomogeneity is present and significant. These effects can be modeled by convolving the kind of I($$\omega$$) function that results from Doppler, lifetime, rotational diffusion, and pressure broadening with a Gaussian distribution P($$\Delta E$$) that describes the inhomogeneous distribution of energy level splittings:

$I(\omega) = \int\limits I^0(\omega ;\Delta E) P(\Delta E) d\Delta E.$

Here I$$^0(\omega ;\Delta E)$$ is a line shape function such as those described earlier each of which contains a set of frequencies (e.g., $$\omega = \omega_{\text{fv,iv}} + \frac{\Delta E_{\text{i,f}}}{\hbar} \pm \omega_J = \omega + \frac{\Delta E}{\hbar})$$ at which absorption or emission occurs.

A common experimental test for inhomogeneous broadening involves hole burning. In such experiments, an intense light source (often a laser) is tuned to a frequency $$\omega_{\text{burn}}$$ that lies within the spectral line being probed for inhomogeneous broadening. Then, a second tunable light source is used to scan through the profile of the spectral line, and, for example, an absorption spectrum is recorded. Given an absorption profile as shown below in the absence of the intense burning light source:

one expects to see a profile such as that shown below:

Figure 15.4.4: Insert caption here!

if inhomogeneous broadening is operative.

The interpretation of the change in the absorption profile caused by the bright light source proceeds as follows:

1. In the ensemble of molecules contained in the sample, some molecules will absorb at or near the frequency of the bright light source $$\omega_{\text{burn}}$$; other molecules (those whose environments do not produce energy level splittings that match $$\omega_{\text{burn}}$$) will not absorb at this frequency.
2. Those molecules that do absorb at $$\omega_{\text{burn}}$$ will have their transition saturated by the intense light source, thereby rendering this frequency region of the line profile transparent to further absorption.
3. When the "probe" light source is scanned over the line profile, it will induce absorptions for those molecules whose local environments did not allow them to be saturated by the $$\omega_{\text{burn}}$$ light. The absorption profile recorded by this probe light source's detector thus will match that of the original line profile, until
4. the probe light source's frequency matches $$\omega_{\text{burn}}$$, upon which no absorption of the probe source's photons will be recorded because molecules that absorb in this frequency regime have had their transition saturated.
5. Hence, a "hole" will appear in the spectrum recorded by the probe light source's detector in the region of $$\omega_{\text{burn}}$$.

Unfortunately, the technique of hole burning does not provide a fully reliable method for identifying inhomogeneously broadened lines. If a hole is observed in such a burning experiment, this provides ample evidence, but if one is not seen, the result is not definitive. In the latter case, the transition may not be strong enough (i.e., may not have a large enough "rate of photon absorption" ) for the intense light source to saturate the transition to the extent needed to form a hole.

This completes our introduction to the subject of molecular spectroscopy. More advanced treatments of many of the subjects treated here as well as many aspects of modern experimental spectroscopy can be found in the text by Zare on angular momentum as well as in Steinfeld's text Molecules and Radiation , 2$$^{nd}$$ Edition, by J. I. Steinfeld, MIT Press (1985).