In Figs. 6-20, the spectra for β and branching ratio are shown, for all the vibrational levels for which there was significant intensity. These have been selected from the complete data set because they show structure identifiable with the resonance positions; the complete data set is available from a NIST database as described below.
To proceed with further analysis, we consider the Born-Oppenheimer approximation and the Franck-Condon (FC) principle. These two approximations have been used successfully for the analysis of vibrationally-resolved photoelectron spectra. Within these well-known approximations, the branching ratio in the photoelectron spectrum, for ionization directly to the continuum, is given by the FC factor between the ground level and the vibrational levels of the ionic state. The β-values are expected to be independent of the vibrational quantum numbers of the ionic state. If photoelectron emission occurs through the resonance, on the other hand, one should take account of both the resonant and non-resonant components with the help of Fano's resonance theory . Then the branching ratio is governed not only by the FC factor between the ground level and the vibrational levels of the ionic state but also by the FC factor between the vibrational level of the excited state and the vibrational levels of the ionic state. If the resonant enhancement is so strong that the non-resonant component is negligible (i.e., |q| >> 1 in Fano's resonance formula ), the branching ratio is dominated solely by the FC factor between the vibrational level of the excited state and the vibrational levels of the ionic state. In this situation, the β-values are again independent of the vibrational quantum numbers of the ionic state.
In our previous publication , we made assumptions mentioned above and the approximation of neglecting the contribution from the non-resonant ionization on the basis that the line shapes were in general Lorentzian. In terms of the Fano  theory, this means that the |q|-value is very large, indicating that the continuum component is very small. Then the FC factors for the transition between the vibrational level of the resonant state and the vibrational levels of the final ionic state were calculated. The approximation was made that the potential parameters for the resonant state are the same as those for the ionic state to which the resonance converges. Clearly this approximation should improve the higher up the Rydberg series the resonance is, and some qualitative agreement with this method of calculation was found. For the (0,0,0) branching ratio, the An1 and An2 (n = 4-7) TO series are calculated to have a slowly increasing branching ratio as one proceeds up the Rydberg series approaching the theoretical value. However, there was poor agreement for the branching ratios as one proceeds along the members of the main vibrational progression (nv,0,0),as noted in , and it was clear that considerable intensity moves from this progression into other vibrational levels.
To take account of the intensity transfer from the main progression to the other vibrational modes as a first approximation, we have added the intensities of the nearest vibrational member above and below the (v,0,0; v = 1-7) vibrational members for the same resonances as in the earlier paper. For example, referring to Table 1, we added the values for (0,2,0) and (0,0,1) to the branching ratio for (1,0,0) and similarly for the other vibrational levels up to (5,0,0). Our labeling of the nearest levels changes for (6,0,0) and (7,0,0), but, as pointed out earlier, this labeling is not definitive given the resolution of our experiment. It was derived from the fitting process, but it would be impossible to confirm that intensity found in the (3,1,2) level could not in fact be attributed to the (5,2,0) level which was not included in our model. In the case of the (6,0,0) and (7,0,0) levels we added in the intensities from the vibrational levels just above and below as with the other vibrational levels.
The results are shown in Fig. 21 as a bar chart, where each group of intensities corresponds to the resonance shown. The heights of the bars correspond to the branching ratios for (0,0,0) to (7,0,0), going from left to right in each group, with the additional contributions added as described above. In Fig. 22, the theoretical values from the previous work are shown. In principle these should be the same for any given Rydberg series, and are labeled accordingly. Some similarities are immediately obvious:
- For the first set of resonances, A n0, n = 4-7, although theory puts (1,0,0) more intense than (0,0,0), the intensity dies away as one proceeds up the vibrational progression much as theory suggests.
- For the next set, A n1, n = 4-7, a minimum at (2,0,0) and a maximum at (5,0,0) is more or less reproduced. For A 71 the (0,0,0) intensity is about right, and here the theory looks quite good.
- For the next set, A n2, n = 4-7, a low value of (1,0,0) is not reproduced, but the minimum at (3,0,0) is there, together with higher values of (5,0,0), (6,0,0), and (7,0,0) compared with the other resonances. A 52 also looks particularly good; (0,0,0) is a bit low, but the overall structure is clearly there.
- In the last set, A n3, n = 4-7, two minima are predicted at (2,0,0) and (5,0,0). This fits well with A 53 and A 73, with some suggestion of it at A 43. Again theory puts (1,0,0) as most intense, which is not the case.
- The general trend for intensity to move to higher vibrational members as one moves to higher vibrational excitations in the resonances is also reproduced. This is evident in Fig. 5; in the resonance regions vibrational structure extends, with considerable intensity, well into the regions of high vibrational energy, in contrast to the off-resonance spectra where almost all the intensity is concentrated in the first few members of the main vibrational progression.
It seems, therefore, that our FC model, which neglects direct ionization and takes account of the intensity transfer to the nearest different vibrational levels, can be applied in resonance regions for some cases. We do not at the present time have any general rules for quantifying the intensity transfer to the nearest vibrational levels for all the spectral regions. Some disagreements may be attributed to the neglect of direct ionization.
An analysis on the basis of that carried out by Smith  for O2, in which the Fano parameters are derived by fitting the experimental line shape to the Fano formula, includes the direct ionization component. Given our assumption above based on Lorentzian line shapes, and the fact that there are many overlapping lines, it was not thought worthwhile to follow this procedure here, although for the Hopfield series in the region between 680 Å and 720 Å where the line shapes are not symmetric it may be more appropriate. Even here, however, our experiment has insufficient resolution to separate overlapping structure clearly and values of the Fano parameters obtained by any fitting procedure are likely to be in some doubt.
Turning our attention to the measured β-values, we note that the average value of β, where we define "average" as an underlying value of β outside resonant structure over the whole wavelength range (i.e., the regions corresponding to direct ionization), is between 0 and -0.5 in general for all the spectra shown. Even for the angular distribution measurements not shown here, because there was considerable scatter in the data, this still appeared to be the case. This would be consistent with, though not evidence for, validity of the Born-Oppenheimer approximation and the FC principle for direct ionization. The fact that the β-value is negative indicates the presence of "parity unfavored" transitions ; an analysis on the basis of the angular momentum theory proved partially successful for the oxygen molecule , where only one vibrational mode is present. Given the larger number of decay channels available for CO2, an analysis on this basis is unlikely to be conclusive.
In the resonance regions, however, we notice breakdown of the FC model applied as described earlier, since that model dictates that the β-values should not change for vibrational levels within the same progression. Note for example in Fig. 6 the markedly different response of β, in the case of the two vibrational levels shown, to the B 30(s) resonance at ≈ 751 Å. The breakdown of the simple picture may be just due to neglect of the non-resonant component in the region of the resonances. The B 30(s) peak is however the strongest among all the peaks in the total cross section spectrum, and it is hard to believe that the non-resonant component can contribute to this dramatic effect. The different response of the different vibrational levels within the same progression suggests breakdown of the Born-Oppenheimer approximation and/or the FC principle.
The results presented in this work show that there is intensity in the odd bending modes, for example (v1,1,v3), contrary to the selection rules. The way in which the branching ratios and β-values for the bending modes respond to the resonances varies dramatically: the strong response of the (0,1,0) β-value to the B30(s) resonance in Fig. 9, and in Fig. 10 the response of the (2,1,0) branching ratio to the B 50(s) resonance, the (3,1,0) branching ratio to the A 53(TO) resonance, the (4,1,0) branching ratio to the A 51(TO) resonance and the (5,1,0) branching ratio to the A 52(TO) resonance. However, our experiment cannot resolve these levels from the corresponding (0,0,2), (1,0,2), (2,0,2), and (3,0,2) levels, so it is not clear the selection rules are in fact being broken.
In general the response seems unpredictable; it does not appear to follow the generally accepted vibrational transition relative intensity expectations. For example, in Fig. 17 the branching ratios for the overtone modes (2,1,1) and (3,1,1) are enhanced at the B 80(s) resonance. The statistical errors in the corresponding values for β were too large for these modes to show any structure; note that the branching ratio is down at the 4 % level or less. On the other hand, in Fig. 18 where the branching ratios are down at the 2 % level, it was possible to discern some structure in the β-parameter measurements; the ability to do this depended very much on the quality of the fit, discussed above. However no clear pattern emerges, and it is not at all obvious what the connection is between one particular branching ratio or β-value being enhanced, and the resonant structure. Given the uncertainties of the identifications of the weaker vibrational levels, the conclusion must be that weak modes such as (n,1,0) are not necessarily enhanced by the resonances, a conclusion which is supported by the results of a higher resolution experiment which will be the subject of a forthcoming publication.
The detailed analysis of these data remains an outstanding task. The complicated nature of the interactions, and the fact that many of the resonances overlap or are incompletely resolved, makes theoretical modeling of these data very difficult. Nevertheless, one point emerges very clearly: it is necessary to make measurements of both the dynamical variables, the branching ratios and β parameters, in order to reveal weak structure not evident in the photoionization spectrum. The differing response of these two parameters is particularly evident in Fig. 6 for the (0,0,0) and (1,0,0) members. Furthermore, for the weaker modes, it is necessary for the experiment to have sufficient sensitivity to measure branching ratios at the 1 % level, with superior resolution to that for the data shown here, in order to be definitive.