A correlation function tells us how a quantity varies with time when compared to a starting value. There are many types of correlation functions such as velocity, dipole, torsional coordinate, diffusion and so on. Correlation functions are useful in spectroscopy to describe a time-dependent fluctuation that affects the power spectrum of a molecule. The relationship between the time decay tne power spectrum is a Fourier transform. For example, dipolar rotation in solution can give us a way to model microwave absorption in solution. Unlike the gas phase where molecule rotate freely, rotation in solution is hindered by collisions. Often the rotation is so restricted that we call it libration. Regardless of the name, we can construct a dipole-dipole correlation function that tells us how the dipole at time is oriented relative to the dipole at time zero. The rate of reorientation and the dynamics both govern the position and shape of the absorption line for a modified rotational spectrum in this case. Similar correlation functions can be constructed for dipolar oscillation in order to model infrared spectra. We are most interested in a specific correlation function will be used to understand absorption and Raman spectra in the following lectures.
The wave packet formalism is shown in the title figure of this web page. This is a correlation function that describes nuclear oscillation in the electornic excited state of a molecule. The frequency of this motion gives rise to the Franck-Condon line shape that we have considered previously using a sum-over-states formalism. The bredth of the line is determined by the exponential decay time, which governs the Lorentzian line width. There are several reasons for using the wave packet formalism. First, the time-dependent picture can help us to understand excited state dynamics. Second, it provides a convenient computational method that can accomodate any number of vibrational modes. Third, it is easily adapted to a calculation that includes low frequency modes. These "thermalized" modes are quite cumbersome to treat in a sum-over-states formalism.
The absorption cross section is calculated using the integrated time correlation function multipled by the square of the transition dipole moment. The line shape has previously been discussed based on the sum-over-states picture for calculation of the Franck-Condon factors. The time correlation function provides an alternative method.
The accepting mode correlation function is the form of the wave packet picture that is used to calculate the absorption spectrum. The name accepting mode refers to a Franck-Condon active mode. A promoting mode is a vibronically active mode. The accepting mode correlation function assumes a vertical transition. Then the wave packet in the excited state moves in a oscillatory fashion in response to the force due to the shifted potential energy surface in the excited state.
The most convenient method for derivation of the accepting mode correlation function involves the use of second quantization. This approach is based on raising and lowering operatores. In the case of the harmonic oscillator these operators provide an alternative way to treat the solutions of the Schrodinger equation. What is more important is that these operators can easily be adapted to explain motion on a different potential energy surface. Specifically, we can use the raising and lowering operators to describe the displacement of the nuclei in the excited state. A time-dependent form of these operators can be used to derive a correlation function.
Timetherm: a program to calculate the time correlator
The methods described in this lecture can be employed in the program timetherm to calculate the absorption spectra for a molecule once we have determined the electron-vbibration coupling, S for each mode of frequency omega. The frequencies of the input modes are given in cm-1.
