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Franck-Condon InterfaceIn the absence of vibronic coupling, electronic spectra (excitation, fluorescence, ionization, detachment, etc.) can be quite effectively modelled with a harmonic Franck-Condon (FC) calculation. While CFOUR does not have a program to specifically do FC simulations (xsim can do them, but not efficiently, as it is a program primarily for vibronic coupling treatments), but a number of programs exist (Anna Krylov's ezSpectrum being one example) for this purpose. In essence what is really needed for such a calculation are: 1) the displacements between the two states; and 2) the so-called Duschinsky transformation matrix, which relates the normal coordinates of the two states. Translated to quantum chemistry, this means that you need to optimize the structures and calculate the harmonic force fields of the two states of interest. A utility program, xquadmodel (also used for xsim parameters), writes a file called fcinput.txt, the contents of which provide all the information needed to do an FC simulation. To run xquadmodel, do the following:
$spectrum FC $memory 500.00 $minenergy 0.00000000 $maxenergy 4000.00000000 $graining 1.00000000 $vibmodes 3 $transdip 1.0000000000 1.0000000000 1.0000000000 $initfreq 793.481 770.120 355.538 $finalfreq 1299.699 809.853 371.274 $duschins 0.9944320256 0.0000000000 0.0000000000 0.0000000000 0.9502953229 0.3113499628 0.0000000000 -0.3113499628 0.9502953229 $duschinsky (transpose) 0.9944320256 0.0000000000 0.0000000000 0.0000000000 0.9502953229 -0.3113499628 0.0000000000 0.3113499628 0.9502953229 $displacement in amu^{1/2} bohr (Q initial) 0.0000000000 -1.8385555605 -1.7033794267 $displacement in amu^{1/2} Angstrom (Q initial) 0.0000000000 -0.9729214629 -0.9013893511 $displacement in dimensionless coordinates (q initial) 0.0000000000 -4.6499033860 -2.9271315159 $displacement in amu^{1/2} bohr (Q final) 0.0000000000 -1.2168236289 -2.1911477076 $displacement in amu^{1/2} Angstrom (Q final) 0.0000000000 -0.6439151748 -1.1595051457 $delta displacement in dimensionless coordinates (q final) 0.0000000000 -3.1558667265 -3.8477473782 $end Once you generated your fcinput.txt file, execute the following command at the prompt: FCfast fcinput.txt > output & Results for the CCl2 -> CCl2+ system are given below: Relative intensity (FCF^2) ----------------------------------------- origin 1.00 3_0^1 9.19 3_0^2 42.02 3_0^3 127.59 ----------------------------------------- These results were obtained with FCfast, which was also checked against the xsim module of CFOUR, which uses a variational numerical method rather than being based on the much more efficient and accurate analytical approach pioneered by Sharp and Rosenstock in the Cretaceous age. Imaginary frequencies? There are cases in which the upper state will have imaginary frequencies, which is usually -- but not always -- attributable to vibronic coupling with a nearby state. In such a situation, a Franck-Condon simulation is inappropriate, since the harmonic potential is not bounded from below. Ideally, one has to work with a vibronic Hamiltonian, which is the purpose of the xsim module. However, an expedient can be achieved by removing the offending frequency (and all those of the same symmetry, to which it can directly couple), and then restrict the Franck-Condon simulation to avoid this symmetry. The capability to do this is provided by the aptly-named xfcfudge module, whose use is now described. You will discover that your upper state has one or more imaginary frequencies by a number of possible mechanisms. You might look at the output when you run the calculation (I would hope that you do), or see negative numbers in the fcinput.txt file under the $finalfreq card, or -- when you run the xquadmodel executable -- you might see the message: WARNINGWARNINGWARNING: Imaginary frequency or frequencies present at the very end of the xquadmodel output. If you then run FCfast, things will not work for the reasons mentioned above. To eliminate the effect of these modes and their symmetry-equivalent partners, invoke the executable xfcfudge as follows: xfcfudge < fcinput.txt which will produce a new file (fcinput-clean.txt) that can be used as input to FCfast. Visualizing the output FCfast produces four output files (two added by your narrator). These are spectrum.abs, spectrum.raw, fort.20 and fort.31. The first two files are x,y files of stick spectra (although produced by a binning procedure that I do not completely understand) and can just be plotted with gnuplot using the plot 'filename' w lines command type. The fort.20 and fort.31 files are formatted exactly like the same two files from the xsim program, and can be used by the xsimplot module to produce spectra convoluted with lineshape functions of a given width (file spectrum) and a nice stick spectrum of the usual sort (file stick). |