Lineshape Functions

We consider the mathematical description of the line shape for an absorptive transition. The transition may be electronic, rotational, or vibrational (i.e. visible, microwave or infrared radiation). In an ideal gas phase experiment where there is no Doppler broadening or lifetime broadening the energy at which transition occurs is unique. This implies that the line is infinitely narrow. How can we describe an infinitely narrow line? It turns out that this is done by defining the delta function d(x – x₀). The properties of a delta function are described in the math section of the website. The function defined is a line at x₀ that is infinitely narrow. The area under a delta function is unity.

In this expression e is the energy which we use anticipating expressions to be used in spectroscopy. But notice that if we define f(e) as a function of energy then the delta function returns a unique energy f(e₀).

The use of a delta function corresponds to a transition with no broadening. Thus, if we substitute the transition probability for f(e) we find that the integral with a delta function returns the magnitude of the transition probability at a unique energy.

Note that we observe a frequency (or wavenumber in an absorption spectrum). Therefore, we can also write the delta function in terms of the frequency w.

The Lorentzian function is a model of homogeneous broadening

Of course, real transitions always involve some line broadening. If nothing else there must be lifetime broadening since absorption of radiation always produces an excited state with a finite lifetime. Recall that the uncertainty principle states that there is a relationship between the lifetime and the energy width of the state. Usually, we describe the kinetic decay of an excited state using an exponential function. This emerges directly from first order kinetics. Looking ahead this means that if we define the rate of disappearance of the excited state E as

Then we can solve this equation to find the E(t) = E₀e^-t/T2 where T₂ is the excited state life time and E₀ is the initial concentration of the excited state.

The uncertainty principle states that the conjugate energy width can be obtained from the Fourier transform of the life time function.

This Fourier transform is easily solved since the integral is nothing more than an exponential integral. The solution for this integral is

which is complex. We can calculate real and imaginary parts by multiplying both numerator and denominator by 1/T₂ + iw. This gives

The real part of this integral is a Lorentzian line shape function. This is the line shape that will be observed for transitions that have only homogeneous broadening. Homogeneous broadening refers to broadening mechanisms in individual molecules. Examples include, NMR spectra, Lamb dip spectra in the gas phase, etc. Since homogeneous broadening is an intrinsic process to each molecule in the sample we model the process using a single exponential as above. Inhomogeneous broadening arises from matrix effects that cause molecules to have different relaxation rates.

We have used T₂ as the relaxation time. This brings up the important analogy between optical spectroscopy and NMR spectroscopy. The relaxation time T₂ has two components, T₁ the lifetime of the state or longitudinal relaxation time and T₂*, which is pure dephasing (optical) or the transverse relaxation time. The relationship between these times is:

The physical picture here is that we have excitation into an excited state followed by relaxation. The lifetime contributes to the relaxation time by decreasing the population of the excited state. Pure dephasing contributes to the T₂ relaxation time by destroying coherence between the ground and excited states.

The normalized Lorentzian function is (i.e. the real part of the above function L(w)).

It is of interest here to relate the relaxation time to the linewidth. The Lorentzian we have been discussing so far is centered about zero frequency. Therefore, the maximum is:

(obtained by substituting zero for w). The height at half-maximum is therefore:

The width at half-maximum is determined by solving

for w. The result is simply

A picture of these two solutions is shown in Figure below.

The width G is known as the full-width at half-maximum (fwhm). We see that G = 2/T₂ in this case. This corresponds to a homogeneous (Lorentzian) linewidth. Since T₂= 2T₁ we also have G = 1/T₁. Thus, the linewidth can be related to the excited state lifetime. Note that this is a manifestation of the Uncertainty principle as well since the frequency width G is proportional to an energy width De.

If T₁ alone is responsible for the relaxation (i.e. of the pure dephasing rate is zero) then the lineshape expression becomes).

The above considerations can be generalized by considering the fact that the Bohr frequency of a two level system is w₀ (elsewhere we have called this frequency w_if or the frequency difference between the initial and final states). Physically, we imagine that an oscillating electromagnetic field with frequency w is in near resonance with w₀ and results in a forced oscillation. Thus, the for this process will be the Fourier transform of

where we have considered explicitly the only the excited population change represented by the lifetime T₁.

The Fourier transform leads to

for the real part of the Lorentzian. The lineshape function is now a Lorentzian centered about the frequency w₀. This can also be expressed as:

Note that this is a normalized function so that the integral of L(w) from -¥ to ¥ is equal to one. Notice that the analogy with NMR is evident in the fact that our excited state decay function is a sinusoid times an exponential. The real and imaginary parts have the appearance

Note the similarity in appearance between this function and a free-induction decay in NMR spectroscopy.

The Gaussian function is a model for inhomogeneous broadening

A Gaussian function is also a useful lineshape function. Any source of inhomogneous broadening such as the Doppler shift or site differences of molecules in crystals or solution can be modeled as a Gaussian lineshape.

In this form of the Gaussian G is called the standard deviation. Since the inhomogeneous broadening is often significantly larger than the lifetime broadening or pure dephasing contributions to the linewidth, the Gaussian lineshape often dominates in solution.

The Gaussian distribution in frequencies can be thought of as corresponding to a Gaussian distribution in relaxation times of the excited state. The Gaussian distribution of relaxation times is usually assumed to arise from an inhomogeneous distribution of the molecules in the sample. The molecules experience different environments due to solvation or matrix effects and therefore they all have slightly different properties. In the gas phase, Doppler broadening serves as an excellent model for inhomogeneous broadening. Since the Fourier transform of a Gaussian is also a Gaussian, the distribution of the relaxation times gives rise to a distribution of energies (and therefore a distribution of frequencies in the observed spectrum).

The Gaussian spectral function can be thought of as the Fourier transform of a Gaussian temporal function. The Gaussian time decay in condensed phases has been observed experimentally and is due to the interactions between molecules. The Gaussian time decay

leads to an energy distribution

The full-width at half-maximum of a Gaussian is not as easily interprested as that of Lorentzian. However, it is often referred to as the inhomogeneous linewidth. The fwhm of a gaussian is determined in the same way as above for the Lorentzian. We begin with the normalized Gaussian:

The value at its maximum is:

So the value at half-maximum is

Thus, we can solve for w. For simplicity, we assume that w₀ = 0. The result will be the same for any value of w₀.

It is worth comparing the Gaussian and Lorentzian lineshapes. Shown below are normalized functions that demonstrate that a Gaussian has a much faster decay from the peak, but that a Lorentzian is a narrower with much wider wings.

Convolution

If the lineshape is intermediate between a Gaussian and a Lorentzian form, the spectrum can be fit to a convolution. A convolution of two functions G(w) and L(w) is

The function I(w) is called a Voigt profile. It can be used to resolve the lineshape into homogeneous (Lorentzian) and inhomogenous (Gaussian) components.

Spectral moments

Suppose we are confronted with an experimental spectrum and we wish to describe it. It may not have a particularly Gaussian or Lorentzian shape. Our models here will work for some systems, but there is often complexity that we need to account for in some general manner. One way to discuss the properties of a lineshape is in terms of its moments. The nth moment is defined as

The I (hat) function represents a lineshape normalized to unit area.

The first moment represents the average value of the lineshape function (this corresponds to w_max or l_max in wavelength space).

The second moment is called the variance and it represents the width of the function.

The third moment is called the skew and it represents the lowest asymmetry term in the band shape.

The fourth moment is the kurtosis. As we go higher the moments are harder to express in simple verbal terms. One means of using this type of information is to fit data to a higher order Gaussian which is also known as a cumulant expansion. The higher order Gaussian has the form:

In a Gaussian the variance is 2/a. The skew is proportional to b and the kurtosis is proportional to g.

The Kubo lineshape models both homogeneous and inhomogeneous broadening

The distinction between homogeneous and inhomogeneous is really a question of time scale. If any effect modulates the energy of a molecule on a time scale shorter than the relaxation time (T₂), then we think of an effect as homogeneous. If an effect that modulates the energy of a molecule occurs on a time scale long compared to the relaxation time we call the effect inhomogeneous. Kubo realized this and derived a single function that would account for both homogeneous and inhomogeneous broadening.

In Kubo's model the energy of a molecule is modulated by its surroundings. We can imagine a molecule in a polar solvent. As the solvent dipoles fluctuate the instantaneous transition energy (frequency) fluctuates as well. Kubo assumed that this fluctuation is a random function of time:

The average frequency is w₀. The Kubo model assumes that the correlation function for dw decays exponentially in time:

The variable

is the amplitude of the solvent-induced fluctuations in the frequency. T₂ is the relaxation time. In McHale the time is referred to as the pure dephasing time. Kubo theory is usually applied in situations where T₂ << T₁. In other words, solvent-induced fluctuations in the frequency are much faster than the intrinsic excited state decay. Under these conditions T₂ = T₂*.

Using these definitions the Kubo lineshape function is:

The limiting behavior of this function is determined by the competition between the overall decay exp{-(DT₂)²t/T₂*}, and the decay of the exponential within the exponential to exp{-t/T₂*}. If DT₂ >> 1, the amplitude of the frequency fluctuations is large compared to the rate of their decay. For T₂ sufficiently large relative to D, we can expand the exponential inside the exponential.

After muplication by T₂ the linear term (t/T₂) is cancelled leaving:

The Kubo formalism reduces to a Gaussian in this case. Since the Fourier transform of a Gaussian is also a Gaussian this form of the Kubo correlation function yields a Gaussian lineshape.

For T₂* sufficiently small the exponential exp{-t/T₂*} changes negligibly relative to exp{-(DT₂)²t/T₂*}. Another way to say this is that if DT₂ << 1 then we can neglect the term exp{-t/T₂*} and thus the exponential decay becomes:

The Kubo function is plotted for various values of DT₂ in the Figure below.

For an exponential decay, the lineshape function is a Lorentzian. Thus, the Kubo model treats the homogeneous (Lorentzian) and inhomogeneous (Gaussian) line broadening mechanisms as limiting cases of a single function. The trade-off that determines which of these dominates is the magnitude of the energy fluctuations D relative to the rate of relaxation 1/T₂. When the energy fluctuations (D) are large relative to the relaxation rate (1/T₂) there is a static inhomogeneity that is seen as inhomogeneous line broadening. When energy fluctuations are small relative to the relaxation rate each molecule in the ensemble appears to sample all of the fluctuations and there is no molecule-to-molecule variation. In this case, there is only homogeneous broadening.