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

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

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

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

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:

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

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

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

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:

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

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

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:

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

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

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

\[ \phi_i = \dfrac{g_i}{Q}e^{-\dfrac{E_i^0}{kT}} \nonumber \]

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:

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

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

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

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

\[ \sum\limits_k | \Phi_k \rangle \langle \Phi_k | = 1. \nonumber \]

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

\[ \mu (t) = e^{\dfrac{iHt}{\hbar}} \mu e^{-\dfrac{iHt}{\hbar}}, \nonumber \]

is:

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

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

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

\[ = \rho_{\text{lower}}e^{-\dfrac{\hbar \omega}{kT}}. \nonumber \]

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:

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

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

The Line Shape and Time Correlation Functions

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

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

in terms of which the net photon absorption rate is

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

As stated above, the function

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

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.