15.4: Line Broadening Mechanisms

The examples in the previous section illustrate how dynamical processes in thermal ensembles give rise to vibrational and rotational resonances; however, we still ended with spectra composed of delta function resonances. This is a natural consequence of a model with no damping of the correlation function. All spectra, particularly those for condensed matter but also those for the gas phase, are subject to irreversible processes that dissipate the energy absorbed from the light, even if that is purely radiative, as we saw in Sec. 11.3 and 11.4. Thus, the dynamics of irreversible dissipation processes influence the linewidth of a resonance. To illustrate, imagine that the dipole correlation function describing absorption between ground (\(g\)) and excited (\(e\)) states has an oscillatory, damped form

\[C_{\mu \mu }(t) \propto \left| \mu _{eg} \right|^{2}\exp \left[ - \mathrm{i}\omega _{eg}t - \Gamma t \right] \label{eq15.4.01}\]

This is the same as the eigenstate correlation function expression, eq. (15.1.10), but we have added a phenomenological damping term \(e^{-\Gamma t}\), where \(\Gamma\) is the damping rate. Then the corresponding absorption lineshape would be Lorentzian:

\[\sigma (\omega) \propto \left| \mu _{eg} \right|^{2}\frac{\Gamma }{\left( \omega - \omega _{eg} \right)^{2} + \Gamma ^{2}} \label{eq15.4.02}\]

Thus the absorption line is broadened to a half-width at half-maximum (HWHM) of \(\Gamma\). One can anticipate that measuring linewidths gives you an estimate of the relaxation rate. Indeed, the analysis of dissipative dynamics is closely related to the analysis of absorption lineshapes.

The difficulty with interpreting dynamics from absorption lineshapes is two-fold. One is practical. When multiple resonances are present, as with the rotational spectra in the previous section or with a complex polyatomic, then fast damping may result in spectral overlap. Second, and more complicated, is that numerous molecular processes can influence the lineshape, and these are generally inseparable in an absorption spectrum. These line-broadening processes can be separated into homogenous and inhomogeneous broadening. Homogenous broadening describes the dynamic processes that are intrinsic to individual constituents of the molecular system. The ensemble averaging effects that arise from the behavioral variation between different molecules are known as inhomogeneous broadening, and these mask dynamic information.

A careful assessment of these different factors is one of our primary objectives for the following chapters, but let’s here catalog some of these factors.

Homogeneous broadening

Several dynamical mechanisms can contribute to the decay of a correlation function and thereby cause line-broadening. These intrinsically molecular processes are often referred to as homogeneous broadening, and, when discussed phenomenologically, they are commonly assigned a collective time scale denoted by \(T_{2} = \Gamma ^{ - 1}\), a notation borrowed from NMR.

Population Relaxation. Population relaxation refers to decay of the correlation function caused by the limited lifetime of molecular states resulting from energy dissipating to the environment, often denoted \(T_{1}\). The rate of population relaxation \(1/T_1\) may have contributions both from the rate of radiative decay, i.e. spontaneous emission, and non-radiative processes, such as dissipating energy to other degrees of freedom or as heat.

\[\frac{1}{T_{1}} = \frac{1}{\tau _{\text{rad}}} + \frac{1}{\tau _{\text{NR}}} \label{eq15.4.03}\]

The observed population relaxation depends on the sum of the relaxation rates for the upper (\(m\)) and lower states (\(n\)) according to \(1/T_{1} = w_{mn} + w_{nm}\). The total rate is often written \(1/2T_{1} \), though, because when the energy splitting \(\omega_{mn}\) is large compared to \(k_{B}T\), only the downward rate \(w_{nm}\) contributes significantly to the population relaxation. We will take up the case of vibrational population relaxation in Chapter 18.

Dephasing. Dephasing is the randomization of phase within an ensemble caused by molecular interactions. Dephasing is a dynamic effect in which memory of a molecule’s phase of oscillation is lost due to intermolecular interactions randomizing the phase, for instance through collisions in a dense gas or fluctuations induced by a solvent, as illustrated in Figure 15.4.1. The process of pure dephasing does not change the populations of the states involved. Time-averaging this behavior results in the decay of a correlation function, which is commonly characterized by the time constant \(T_2^{*}\) in the phenomenological description. We will develop the formal framework for describing this in Chapter 17.

Figure \(\PageIndex{1}\): Illustration of dephasing induced by molecular interactions. (Left) A scenario in which the frequency of oscillation of a dipole moment \(\omega_{eg}\) can remain constant, but the phase is shifted due to random perturbations. (Right) Behavior of a dipole moment \(\mu\) whose frequency continuously varies around an average value, and as a result its phase \(\phi=\partial\omega_{eg}/\partial t\) also varies continuously.

Orientational relaxation. Orientational relaxation \(\left( \tau _{\text{or}} \right)\) also leads to decay of the dipole correlation function and thereby contributes to line-broadening. Since the correlation function depends on the projection of the dipole onto a fixed axis in the laboratory frame, the dephasing effect of the initial randomization of dipole orientations is averaged over the ensemble. We saw above how free rotational motion leads to complex rotational recurrences in the dipole correlation function, which result in a broad envelope of resonances around the lineshape of a vibrational-rotational spectrum. In solution, free rotation is interrupted and decay of the rotational correlation function is commonly treated as an orientational diffusion problem. For absorption lineshapes, the time-scale for the exponential decay of the correlation function is \(\tau _{\text{or}} = 1/2D_{\text{or}}\), where \(D_{\text{or}}\) is the orientational diffusion constant.

Multiple processes. If these homogeneous processes are independent, the rates for relaxation of the correlation function in eq. (\ref{eq15.4.03}) all contribute additively to the damping and line width caused by homogenous broadening:

\[\Gamma = \frac{1}{T_{2}} = \frac{1}{T_{1}} + \frac{1}{T_2^{*}} + \frac{1}{\tau _{\text{or}}} \label{eq15.4.04}\]

Inhomogeneous broadening

Absorption lineshapes can also be broadened by a static distribution of frequencies; this is inhomogeneous broadening. If static environmental variations have a greater influence on the variation of absorption frequencies within the ensemble than the inherent linewidth due to homogeneous broadening, then the observed lineshape reports on the distribution of those environments (Figure 15.4.2). As a result, inhomogeneous broadening is a static ensemble averaging effect that makes it impossible to obtain dynamical content from an absorption spectrum. The origin of the inhomogeneous broadening can be molecular (for instance, lattice defects in crystals) or macroscopic (i.e., an inhomogeneous magnetic field in NMR).

Figure \(\PageIndex{2}\): Illustration of how ensemble averaging over a static distribution of transition frequencies for different molecules leads to a decay of the dipole correlation function and corresponding broadening of the absorption spectrum.

To illustrate, let’s consider an ensemble of molecules with transition frequencies \(\omega _{eg}\) in a Gaussian distribution centered on \(\omega _{eg}^{0}\) with a standard deviation \(\Delta \):

\[P( \omega _{eg} ) = \frac{1}{\sqrt {2\pi \Delta^{2}} } \, \exp \left[ - \frac{\left( \omega _{eg} - \omega _{eg}^{0} \right)^{2}}{2 \Delta^{2}} \right] \label{eq15.4.05}\]

To calculate the correlation function, we need to integrate the dynamics of individual molecules over the distribution of transition frequencies in the ensemble. If the molecular relaxation rate is roughly constant for all members of the ensemble, then

\[C_{\mu \mu }(t) = \left| \mu _{eg} \right|^{2}\int P ( \omega _{eg} )\text{exp}\left( - \mathrm{i}\omega _{eg}t - \Gamma t \right)d\omega _{eg} \label{eq15.4.06}\]

Setting the integration limits from \( - \infty \) to \(\infty \), we find that the correlation function takes the form

\[C_{\mu \mu }(t) = \left| \mu _{eg} \right|^{2}\exp \left( - \mathrm{i}\omega _{eg}^{0}t - \Gamma t - \Delta^{2}t^{2}/2 \right) \label{eq15.4.07}\]

This correlation function oscillates at the central transition frequency of the ensemble \(\omega _{eg}^{0}\) and has both exponential and Gaussian damping factors. Note, when the distribution of frequencies is very small, \(\Delta \to 0,\) and we obtain the correlation function we wrote before for the homogeneous case, eq. (\ref{eq15.4.01}). On the other hand, when \(\Delta \gg \Gamma \), we can neglect the \(\Gamma t\) term in the exponential, and \(C_{\mu \mu }(t)\) decays purely as a Gaussian in time. The Fourier transform then gives an absorption lineshape that is Gaussian, peaked at \(\omega _{eg}^{0}\) with a standard deviation linewidth of \( \Delta\), for the expected behavior in the inhomogeneous limit:

\[\sigma (\omega) \propto \left| \mu _{eg} \right|^{2}\exp \left( - \frac{\left( \omega - \omega _{eg}^{0} \right)^{2}}{2 \Delta^{2}} \right) \label{eq15.4.08}\]

Total linewidth

The total linewidth of the absorption spectrum reflects contributions from all of these sources and more! Regardless of how diverse the different contributions to the dynamics of the dipole are, we will find that all line-broadening effects are ultimately expressed in terms of a time-dependent lineshape function \(g(t)\). The lineshape can then be written as the Fourier transform over the oscillating transition frequency, damped and modulated by a complex \(g(t)\):

\[\sigma (\omega) \propto \int_{ - \infty }^{ + \infty } dt \, e^{\mathrm{i}\omega t} e^{ - \mathrm{i}\omega _{eg}t - g(t)} \label{eq15.4.09}\]

For instance, in the case described by eq. (\ref{eq15.4.07}), we would assign \(g(t)=\Gamma t+\Delta^2t^2/2\).

At the end of this discussion, one might wonder how we learn anything dynamical from absorption spectroscopy, if numerous inseparable processes contribute to broadening the spectrum. It is true, there are only limited pieces of information that one can uniquely determine from the linewidth and lineshape. This is one of the biggest motivators for using nonlinear and two-dimensional spectroscopies. These techniques allow one to isolate different dynamic contributions to a lineshape through the use of time- and/or frequency-resolved measurements involving two or more incident fields.