3. Analysis of the Data
The analysis of the large amount of data taken on CO2 required careful organization of the data and the development of a reliable method of extracting the highly detailed information on vibrationally resolved asymmetry parameters and branching ratios that the data contained. The raw data must first be corrected for the differing detection efficiencies of the two electron energy analyzers. This correction is a function of the kinetic energy of the electron and is accounted for by the determination of an energy dependent correction factor from angular distribution data measured by the two electron spectrometers for rare gases as discussed in the previous section. The correction function was frequently checked during the course of the experiment by measuring the photoelectron spectra of a rare gas as a function of wavelength over a wavelength range sufficient to generate photoelectrons with kinetic energies equivalent to those produced in the photoionization of CO2. The system exhibited excellent stability in this regard as, once the lens voltages and other operational parameters of the electron spectrometers were set, the relative efficiencies of the analyzers did not vary in any significant manner.
The 2Πg ground state of CO2+ is spin-orbit split into two levels that are 19 meV apart and the bending mode populated in the resonance regions is split a similar amount by Renner-Teller coupling. This structure was not resolved and hence the effect of these interactions was to broaden the spectral peaks observed in the CO2+ photoelectron spectra. This could be accounted for by examining the shape of the photoelectron peaks seen in the rare gas spectra and appropriately adding two such peaks that are offset by the 19 meV splitting causing the broadening. Using this procedure and some testing on fitting some actual CO2+ spectra resulted in obtaining 47 meV and 56 meV as the best effective resolutions of the two analyzers to be used in the fitting routine. The fitting routine allowed for the resolution to be a free parameter or a constrained parameter but the best overall fittings were obtained when the resolutions were fixed at these values and not allowed to vary during the fitting process. The fitting routine used as a model a series of peaks whose widths were fixed and whose relative positions were fixed by the spectroscopic positions of the vibrational energy levels. The fitting routine was a nonlinear least squares analysis  that was customized at NIST for the type of spectra observed.
The spectroscopic values of the vibrational parameters were taken from the literature [35,36] and fitted to our data using ω1 = 156.7 meV for the symmetric stretch, ω2 = 0.060 meV for the bending mode, and ω3 = 182.3 meV for the asymmetric stretch. These values gave good fits to the data over a broad range of electron kinetic energies and represent an averaging over the levels split by the Renner-Teller interaction. The transitions to the asymmetric stretch and bending mode with odd number of quanta are forbidden by selection rules from the ground state of the neutral molecule but have been observed in other photoelectron work reported in the literature . Though these transitions are not allowed by the Franck-Condon principle, their existence is well established experimentally and can be attributed to anisotropic interactions of the escaping electron with the molecular core. The presence of these transitions and the Renner-Teller splitting of the levels gives rise to an overlapping of the possible transitions, and this makes unique identification of the photoelectron peaks we observe impossible except in some prominent circumstances. Chambaud, Gabriel, Rosmus, and Rostas have identified many of the low lying levels of CO2+, providing a basis for estimating the positions and compositions of the levels we observe . Using the notation (v1,v2,v3) where v1 is the quantum number of the symmetric stretch, v2 the quantum number of the bending mode, and v3 that of the asymmetric stretch, we find the spectral features can be adequately assigned to the following vibrational levels:
|(v1,0,0) v1||= 0, 1...|
|(v1,1,0) v1||= 0, 1...|
|(v1,2,0) v1||= 0, ..., 4|
|(v1,0,1) v1||= 0, 1, ..., 5|
|(v1,1,2) v1||≥ 3|
The upper value of v1 is determined by the span of the energy range covered by the photoelectron spectra. In some cases it was not necessary to include the (v1,0,1) or (v1,1,2) levels, as the intensity in the high binding region was small and the structure was adequately represented by the simpler set of levels. Table 1 gives the vibrational levels used in the fitting model in the energy range of one eV of vibrational excitation. The levels are identified as discussed above and the energy position is given with respect to the (0,0,0) vibrational ground state of CO2+.
It is possible other types of excitations may contribute to the spectra observed. The appearance of finite values for intensity at an energy determined by the above energy levels does not constitute an unambiguous identification of a particular vibrational level for CO2+.
As an example, our data would not separate out the (2,0,0) components from those of components of the split (1,0,1) and the (1,2,0) levels. The (1,2,0) level is split into 4 levels by the Renner-Teller and spin-orbit interactions, the upper two of which are approximately coincident with the (2,0,0) levels. The electron energy resolution of the present experiment was not sufficient to warrant attempts at resolving the levels due to the splittings. Similarly, the intensity observed for "forbidden" transitions to levels of type (n,1,0) should not be regarded as definitive. In an earlier publication where we studied CO2 photoionization with an electron spectrometer resolution of about 19 meV, members of the (v1,0,2) progression were clearly visible  which indicates that the asymmetric stretch excitations can play a role in the photoabsorption of CO2. In this lower resolution study, these excitations could not be resolved from those excitations shown in Table 1. Our goal in the fitting was to identify peak positions which would best account for the electron intensity observed and which could be identified with predicted vibrational structure.
To accomplish the fitting, relative peak positions were fixed by the spectroscopic model chosen, the peak widths were fixed as determined in the previous discussion and the heights of the photoelectron peaks were allowed to vary in order to minimize the fitting errors with the least squares technique. After many trials it soon became evident which vibrational modes were the most important to achieve a reasonable fit. This set of vibrational modes was adopted as a basis for the analysis. At energies 2 eV above the threshold, it was not possible to resolve the individual vibrational peaks and they were often seen as a modulated background whose intensity was accounted for by high members of the above progressions. In the wavelength region of 652.08 Å to 680.98 Å it was sufficient to use only the first three vibrational progressions shown above; in the region 689.7 Å to 713.9 Å it was necessary to use the first four levels; and in the remainder the complete set was used. The only exception to this was the use of levels of the sort (v1,1,1) in the 689.77 Å to 694.64 Å region where this addition seemed to improve the fit somewhat but did not suggest itself elsewhere and hence will not be pursued in this report.
After fitting, each data set and the fit were inspected, and the model varied by adding or removing vibrational modes as required to achieve the best fit. Fig. 3 shows the quality of the fit obtained by the 0° analyzer at a photon energy of 17.180 eV. The lower portion of Fig. 3 shows the fitting results for the same analyzer at a photon energy of 17.218 eV. Both fits were accomplished using the same model for vibrational state composition even though the structure is substantially different in the two closely spaced regions of the absorption spectrum. The vertical lines in the figures are the calculated amplitude of the contributions from the progression listed. The solid line, labeled FIT, represents the fit of the calculated spectrum to the raw data (DATA in the figure). The data points are for the most part obscured by the fitted line, an indication of the quality of the fit.
It can be seen that the data are of sufficient quality to give reliable intensities for the vibrational components marked, allowing vibrational branching ratios down to the 5 % level to be measured. In the figures shown later, the uncertainties include the statistical element (type A uncertainty) derived from the fitting procedure, and further contributions due to uncertainty in the polarization of the incoming light and the calibration of the tungsten photodiodes (type B uncertainty) . In general, the statistical uncertainty gave the largest effect. The error bars (coverage factor, k = 1) shown on the data represent the combination in quadrature of the uncertainties of type A and type B propagated with the appropriate error propagation for the equations given earlier in this paper.
Having obtained the intensities for the vibrational levels for each of the two analyzers, we then used these data, together with the measured light polarization and the electron spectrometers' energy efficiency curves, to calculate the asymmetry parameters and branching ratios. The total electron count as a function of wavelength is shown in Fig. 4. This was obtained by summing over the values of the calculated Nv for all the levels calculated for the X ground state. The intensities were corrected for the varying efficiency of the tungsten photodiode used to measure the light intensity and can be seen to agree closely with the results shown for the photoionization efficiency in Fig. 2 .
The three dimensional plot shown in Fig. 5 shows the electron spectra where the most intense members have been truncated in order to reveal the detail in the higher vibrational modes. Examination of this figure reveals the amount of CO2+ vibrational excitation occurring as a result of autoionization which populate molecular energy levels well above 0.5 eV. This excess population of excited state CO2+ would have consequences on the reactivity of this molecule in atmospheres where CO2+ is formed by uv irradiation. When the number of electrons as a function of molecular energy are summed as shown in Fig. 5, the total electron count as shown in Fig. 4 is obtained.