The time-correlation function method

 

We define a time-correlation function as

where the angle brackets represent an ensemble average and A is the dynamic variable of interest. If we compare the value of A(t) with its value at zero time, A(0) the two values will be correlated at sufficiently short times, but at longer times the value of A(t) will have no correlation with its value at t=0. Information on relevant dynamical processes is contained in the time decay of C(t). The starting time is arbitrary so we can also discuss the ensemble average starting at any time, t.

Normalization may also be applied by dividing by 

The normalized function is a decay from a value of one to some lower value (not always zero). It represents the loss of correlation with the initial value. The short time value is proportional to áA2ñ. The asymptotic long time value is proportional to áAñ2 as shown in the figure below. 

The decay shown in the figure is the average of a large number of trajectories. We can see that C(t) decays from áA2ñ to áAñ2.

This means that initially C(0) = áA(0)A(0)ñ which is clear from our definition. Note that this value is shown as 1.0 on the y-axis. This corresponds to a normalized value since áA(0)A(0)ñ/áA(0)A(0)ñ = 1. As time goes on the value of A(t) changes and becomes less correlated with its value at time zero. At long enough times A(t) has no correlation and so its average value is áA(t)ñ. The average value of A at time zero is áA(0)ñ. These averages are both average values of A and should be the same. Thus, áA(0)ñáA(t)ñ = áA(0)ñáA(0)ñ = áA(0)ñ2. The value of zero shown in the Figure above is arbitrary. The normalized long time value is áA(0)ñ2/áA(0)A(0)ñ and is less than one, but is zero only if áA(0)ñ = 0.

 

Spectral lineshapes

Spectral lineshapes can be calculated by determining the time-correlation function for rotational, vibrational or other motion involved. These are examples of the spectral density, which is the Fourier transform of the correlation function. The most direct route to understanding the time-correlation function is to determine an expression for the intensity of absorption or emission of a photon. To determine the intensity a rather lengthy exposition is required. This involves a review of material already covered.

The intensity is defined to be:

The delta function in the Golden Rule and intensity expressions can be represented by the integral

Using the Einstein relation

The delta function in the intensity expression is

We can replace the delta function in the intensity expression with the above integral and apply

as an operator

The time dependent dipole operator is defined in Heisenberg representation where a time-dependent operator is related to a static operator by

The above transformation is valid if i and f are eigenfunctions of the same hamiltonian. This is the case for vibrational and rotational transitions, but not electronic transitions. Using the above formalism the intensity is 

The individual matrix elements ái|m|fñ are just numbers so the order of multiplication does not matter. Since the final states form a complete set we can use the closure relation

This results in

The sum over initial states is a weighted Boltzmann average (pi is the Boltzmann weighting factor) so this is nothing more than an ensemble average. Thus, we can write

Since this represents the ensemble averaged dipole moment, m here really represents the system dipole moment. For example, ám(0)m(¥)ñ = 0 since the isotropically averaged dipole moment points in all directions of space with equal probability.

 Since we have considered the intensity of absorbed radiation we can use the following definitions for m to relate explicitly to spectra.

  1. Microwave or far infrared: m = m0, the permanent dipole moment of the molecule.
  2. Infrared m = (¶m/Q)Q where Q is a normal coordinate of vibration
  3. Rayleigh scattering: m = mind µ eI×a×es where a is the ground state polarizability and ei and es are the direction of incident and scattered radiation, respectively.
  4. Raman scattering: m = mind µ ei×(¶a/Q)×es where (¶a/Q) is the polarizability derivative with respect to the normal coordinate Q.

    5. We can also apply the formalism to the following phenomena.

  5. Electronic absorption spectra: m is the electronic transition moment. It must be kept in mind that the hamiltonian in the ground state and excited state are not the same. The correlation function can be applied to electronic and nuclear motion together or following the Condon approximation it can be applied to the nuclear motion alone treating the electronic part as a constant.
  6. Solvent dynamics: the correlation function is C(t) = ádw(0)dw(t)ñ. The energy fluctuations due to electrostatic interactions of solvent molecules is described.
  7. Dielectric relaxation: The time-dependent response of a medium to an applied electric field can be described using a relaxation function, which is an auto-correlation function of the dielectric response.

Mathematical examples

We can consider three important examples that can be used to calculate spectral line shapes

  1. C(t) = C(0) then I = I0d(w - w0)
  2. C(t) = C(0)(1/Ö2pt)exp(-t2/2t2), I = I0exp(-w2t2/4)
  3. C(t) = C(0)exp(-t/t), I = t/(1 + w2t2)
  1. The delta function

    2.  

  2. A Gaussian function

    3. Once substituted into the Fourier transform the integral is no longer a Gaussian integral. However, the function can be converted into a Gaussian by completing the square. We complete the square by adding a term in the exponent that depends only on frequency and hence is a constant with respect to the integration over time.

    To find the unknown G we note that the cross term must be iwt so we must have

    so

    We rewrite the integral as

    Using the substitution

    we have

    which yields

  3. An exponential function

This is an exponential integral so

This can be decomposed into real and imaginary parts that represent in-phase and out-of-phase components, respectively.