J. Phys. Chem. 1995, 99, 1700-1710
1700
Multichannel Quantum Defect Theory and Double-Resonance Spectroscopy of Autoionizing Levels of Molecular Hydrogen Ch. Jungen" Laboratoire Aim& Cotton, Birtiment 505, 91405 Orsay Cedex, France
S . T. Pratt* Argonne National Laboratory, Argonne, Illinois 60439
S . C . Ross* Department of Physics, University of New Brunswick, P.O. Box 4400, Fredericton, N.B. E3B 5A3, Canada Received: August 18, 1994; In Final Form: October 24, 1994@
Multichannel quantum defect theory has been used to calculate vibrational and rotational branching ratios in the photoionization of H2 excited to the E,F IX; state. The calculations take into account the interaction with doubly excited channels in the E,F l2; lower state, the energy and R dependence of the transition to the upper state, and the rovibronic channel interactions in the final states. The input data consist of the ionic potential energy curves and the known R-dependent quantum defects for the gerade and ungerade channels of H2. All these parameters are used without adjustment. The theoretical results are compared with the results of earlier double-resonance experiments. This detailed comparison shows that the R dependence of the transition dipole moments has considerable influence on the resonance profiles and the branching ratios. The overall agreement with the experimental results is excellent, and most of the observed discrepancies can be accounted for by the competing decay processes of predissociation and fluorescence, which were not included in the calculations.
I. Introduction The purpose of this paper is to make a detailed comparison of experimental and theoretical partial cross sections for resonant photoionization of diatomic hydrogen. Molecular hydrogen has always served as an important benchmark for the comparison of experimental results with theory. In particular, multichannel quantum defect theory',2 (MQDT) has proven to be an extremely powerful technique for the analysis and understanding of highly excited and superexcited states of this m ~ l e c u l e . ~ -Indeed, '~ in the first application of MQDT to molecules, Fano3 combined Seaton's quantum defect methodI8 with the idea of frame transformations to explain the absorption spectrum of H2 in the region of the first ionization threshold. After Fano's initial discussion of rotational perturbations and autoionization,Herzberg and Jungen4 performed a detailed MQDT analysis of the H2 absorption spectrum in the region of the first ionization threshold, treating vibrational interactions in a perturbative fashion. Subsequently, Jungen and DilP9 adapted MQDT to treat simultaneous rotational and vibrational interactions, obtaining excellent agreement with the high-resolution photoionization experiments performed on H2 by Dehmer and Chupka.6 At about the same time, MQDT was applied to lower lying bound states of H2, yielding excellent agreement with spectroscopic data and accurate theoretical ab initio r e ~ u l t s . ~ , ~ In many respects, the photoionization mass spectrometry studies of H2 by Dehmer and Chupka6J9have provided the most comprehensive test of MQDT in molecular systems to date. In addition to high-resolution relative photoionization cross sections, Dehmer and Chupka6 obtained photoionization vs photodissociation branching ratios from a comparison of the photoabsorption and photoionization spectra; these branching @
Abstract published in Advance ACS Absfracts, January 15, 1995.
0022-365419.512099-1700$09.00/0
ratios were successfully accounted for in a recent extension of MQDT.I4 Dehmer and Chupka were also able to extract the details of the H2+ vibrational state distribution by creative use of ion-molecule reaction pr~babilities.'~ In a series of papers, Raoult and Jungen'o-'2 showed that MQDT provides excellent agreement with the relative cross sections and vibrational branching ratios. They also predicted rotational branching ratios and the wavelength dependence of photoelectron angular distributions. A number of experimental studies aimed at testing these predictions have been performed by using synchrotron light sources.2o In the most recent study of Dehmer et a1.,21 the wavelength resolution was sufficient to obtain vibrational branching ratios and angular distributions to provide a stringent test of the theory. The corresponding MQDT calculations of Stephens and Greene22were in semiquantitative agreement with experiment, although the discrepancies in the branching ratios and angular distributions were significantly greater than those in the total cross sections. In recent years, laser techniques employing double-resonance excitation schemes have allowed the photoionization dynamics of H2 to be studied with unprecedented detai1.23-41In a typical experiment, the first (pump) laser is used to excite ground-state molecules to a single rovibronic level by using a single- or multiphoton transition, and the second (probe) laser is used to excite single-photon transitions from the selected level to the energy region of interest. The overall process is then studied by the detection of photoelectrons, photoions, and/or photodissociation products. The double-resonance excitation scheme allows the selective enhancement and detection of the photoionization process of interest. By applying these techniques to H2, detailed studies of excited-state photoionization dynamics, photoelectron angular distributions, and vibrational and rotational branching ratios as a function of wavelength have all been
0 1995 American Chemical Society
Autoionizing Levels of Molecular Hydrogen
J. Phys. Chem., Vol. 99, No. 6, 1995 1701 between the product vibrational levels provide direct information on the mechanism for vibrational autoionization and serve to test the vibrational propensity r ~ l e . " ~ In - ~the ~ third experiment, corresponding to region 3 in Figure 1, transitions from the E2, N' = 1 level to the energy region between the H2+ X '2:. v+ = 2, Nf = 1 and 3 thresholds were studied.27 In this energy region the Rydberg series converging to the X 'X;, v+ = 2, N+ = 3 threshold can decay by rotational autoionization into the v+ = 2, N+ = 1 continuum and by vibrational X autoionization into the X '2;' v+ = 0 and 1 continua. The competition among these processes leads to complex interactions for which MQDT is ideally suited. Such detailed information on the photoionization dynamics of H2 was not available prior to these experiments. Taken together, these three experimental studies provide a stringent test of the ability of MQDT to account for the photoionization dynamics in this fundamental system.
)region 3
} region 2
'El,
) region 1
..... .....
'
I Figure 1. Schematic energy level diagram for the energy range of interest. Regions 1-3 correspond to the three regions discussed in the 124000
present study. performed. Experimental studies have also examined the effect of extemal electric fields on the decay p r o c e ~ s e s . ~ These ~-~~ new results are of sufficient detail and quality to provide a rigorous test of MQDT calculations similar to those of Raoult and JungenIo-l2 for the case of single-photon, ground-state photoionization studies. To date, the number of applications of MQDT to the results of laser studies of the photoionization dynamics of H2 has been sma11.32,40,42 In particular, Comaggia et al.42have used MQDT to calculate photoionization branching ratios for single-laser, two-photon resonant, three-photon ionization and four-photon resonant, five-photon ionization via the E,F, IZ; state, obtaining good agreement only in the latter case. More recently, Xu et al.32have calculated the relative cross sections for double-resonance excitation via the E,F, '2; state to the region just above the f i s t ionization threshold. They were able to reproduce the complex vibrational and rotational interactions in the same way that Jungen and Dillss9 accounted for this region in the single-photon ionization spectrum. In this paper MQDT is applied to the calculation of the three different photoionization processes in H2 that were studied experimentally in a series of papers by O'Halloran et and that are shown schematically in Figure 1. All three of these X processes involve pumping the two-photon E,F IXi,v' l2;, v" = 0 transition to populate individual rovibronic levels of the E,F IZ; state, and they are discussed in order of increasing energy and complexity. In what follows the vibrational levels of the E,F IZ; state are labeled as in ref 17, with EO, E l , and E2 corresponding to the E,F '2:' v' = 0, 3, and 6 levels, respectively. In the first experiment, corresponding to region 1 in Figure 1, transitions from the E l , N' = 1 level to the region below the X 2Zi, vf = 1, N+ = 1 ionization threshold were s t ~ d i e d . ~In~this , ~ ~case the resolution of the electron spectrometer was sufficient to determine the rotational branching ratios following vibrational autoionization of the states in this region into the X 2X;, v+ = 0, N+ continua. In the second experiment, corresponding to region 2 in Figure 1, transitions from the E2, N' = 1 level to the region just below the X 2X;, v+ = 2, N+ = 1 threshold were studied.26 In this energy region, Rydberg series converging to the X 2Zl, v+ 2 2 thresholds decay by vibrational autoionization into the X 2Zi, v+ = 0 and 1 continua. Here the branching ratios al.26327331333.34
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11. Calculations A. Possibility of Alignment in the Two-Photon Pump Transitions. The double-resonance experiments considered here are generally discussed in terms of two separate steps, corresponding to the pump and probe transitions. Because the pump laser is held at a fixed wavelength in any particular experiment, the only complication that could result from the two-photon transition is the production of alignment in the pumped E,F IZi,v', N' level. However, both experimental and theoretical results argue against any such alignment of the E,F '2;' v', N' level^.^^^^^ In particular, photoelectron angular distributions for one-color, two-photon resonant, three-photon ionization of HZ via the E,F IZ; state are consistent with an unaligned intermediate state. That is, the photoelectron angular distributions are well described by the form
qe) = 1 + p~,(cOse)
(1)
without the need for higher-order terms.47 (Here 8 is the angle between the detection axis and the laser polarization axis, and P2 denotes the second Legendre polynomial.) In addition, the polarization dependence of transitions from the El, N' = 1 state shows no dependence on the relative polarization of the pumpand-probe la~ers.4~ This is also consistent with the result that the M'N = 0 and M'N = f l levels of the E l , N' = 1 state are equally populated (i.e., that the intermediate state is unaligned). Finally, Comaggia et al.42argued that the two-photon transition amplitude for the pump transition is dominated by an M'Nindependent term and that the M'N-dependent terms are very small, again supporting an unaligned intermediate state. Thus, the problem is reduced to understanding the single-photon probe transition from the intermediate state to the autoionizing states in the energy region of interest. B. Calculation of the Intermediate- and Upper-State Energy Levels. The basic MQDT framework for the calculation of the probe transition has been discussed in detail p r e v i ~ u s l y , ' *and ~ ~we ~ ~will not reproduce it here. Instead, we focus on the approximations made in the present calculations and on the origin of the input parameters used. The present calculations are similar to those of Jungen et al.48349 in that both the upper and lower levels of the transition were calculated by using MQDT and in that the 7-quantum defects were used rather than the more common p-quantum defects.I8 However, as described below, the potential energy curves and quantum defects used as input parameters for the present calculation were obtained in a somewhat different manner. In addition, the possibility that both open and closed channels are present was included by following the approach described in ref 1.
1702 J. Phys. Chem., Vol. 99, No. 6,1995
I. EJ Intermediate Levels. The El and E2 levels discussed here correspond to the v' = 1 and 2 levels of the inner (E) well of the double-minimum p ~ t e n t i a l . ~ ~The , ~ ' E well is well described by the electron configuration 1sug2sug,and therefore the selection rules for transitions from the E,F lX; state into the ionization continuum are the same as those for transitions from the electronic ground state. Although both the El and E2 levels are below the barrier between the two wells, the E2 wave function displays some tunneling into the outer well, where the electronic character of the wave function is dramatically different, corresponding to luu2pau. This has been taken into account here in exactly the manner described by Ross and J ~ n g e n , ' ~ Jby ' including the doubly excited la,,Epu, channel and its coupling to the lugaugchannel explicitly in an electronic multichannel treatment. Further, Ross and Jungen showed that the lugedugchannel and its coupling to the luuepuuand lugEsag channels must be included as well. For N' = 1, BomOppenheimer channels with l-I symmetry also come into play, albeit very weakly in the region of the E,F levels considered here. All the required quantum defect functions have been taken from the work of Ross and J ~ n g e n . ' ~ ,In' ~ these papers the calculational details have also been described. 2. EP Final Levels. As discussed by Comaggia et a1.$2 over the range of intemuclear distances ( R ) sampled by the inner well of the E,F IX; state, the doubly excited, ungerade electronic states all lie well above the energy of interest for the experiments under discussion. Thus, only singly excited ungerade configurations need be considered. Given that the lsog2sugdescription of the E,F 'E: state is appropriate in the inner well, the most intense transitions will be to the npu and npn Rydberg series. We neglect f waves in this work because no resonances corresponding to f levels are apparent in the experimental spectrum. Thus, in the present calculations, the continuum channels are treated by considering the ejection of a p wave and by ignoring 1 mixing in the final state. This is an appropriate starting point for the present calculations, because the approximations are the same as those of the earlier work by Dill et a1.*-I2 on the photoionization of ground-state hydrogen. However, mixing between different partial waves and the competition between ionization and predissociation can be included within the framework of MQDT as necessary. In the present calculations, the vibrational channels used to describe the (X 2E;)~p, v+ levels consist of the H2+, X 2X:, 'v = 0-12 levels. We use the electron orbital angular momentum component A and the total component A without distinction here, because the core has A+ = 0 for all channels included in this work. Unlike in the earlier work on the p states, we have included a linear energy dependence of the q;&R) and q;&R) functions. Figure 2 displays q$R) and dqtddc(R) for I = 0 and 1. These values were detemned from the p' curves with n = 6 and 4 given by Raoult and Jungen" and by Jungen and Atabek,7 respectively, and from the px curves for n = 5 and 3 given by the same authors. The p defects given in the earlier papers were converted into q defects by using the relations given in ref 18. The X zXipotential curve of Wind52with the adiabatic corrections of K o l o ~was ~ ~used to calculate the vibrational wavefunctions of the H2+ ground state. As in the calculations of Jungen et al.,48.49the H2+, X v+, N+ energy levels obtaified from the nonadiabatic calculations of Wolniewicz and OrlikowskP4 were used as the limits of the Rydberg series. C. Calculation of the Probe Transition. The input parameters necessary for the calculation of the probe transition include the short-range (body-fixed) electronic transition dipole matrix elements d::f;",.(R) where E' and E are the electron
2Xi,
Jungen et al. 1.2
0.8
1 1'
-0.4
I
1
I
I
I
I
0
2
4
6
a
10
R (au)
Figure 2. R dependence of the 7 defects and their derivatives with respect to energy E for the EP final states. 40
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-40
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Figure 3. R dependence of the transition dipole matrix elements for the present calculations. Filled symbols correspond to the bound transition to the n=27 level, and open symbols correspond to the boundfree transition 0.04 Rydbergs into the continuum. The transition moments are normalized to the energies in a.u. of both the lower and upper state channels, E' and E, respectively.
energies in the intermediate and final state, respectively; these matrix elements have been calculated by using the phase-shifted Coulomb a p p r ~ x i m a t i o n . ~Note ~ - ~ ~that in this approximation the R dependence of the transition moment comes in directly through the corresponding dependence of the quantum defects in the combining channels. Cornaggia et al.42showed that the d:E(R) and d:?g(R) values calculated with this approximation are in good agreement with ab initio calculations of the EP 'q', 'l-IU E,F IEi electronic transition moments in the range R 5 3 au, where the E,F state has nearly pure s character. Figure 3 shows the transition moments calculated for the present study at two different values of E. The energy dependences of the transition moments, which were assumed to be linear, were determined from the values of the transition moments at these two energies and were included in the intensity calculations. The dipole transition moment from the doubly excited component of the intermediate channel to the upper channel is set to zero, because it involves simultaneous Rydberg-Rydberg and core transitions. D. Experimental Considerations. Several aspects of the experiment must be taken into account before a detailed
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J. Phys. Chem., Vol. 99, No. 6,1995 1703
Autoionizing Levels of Molecular Hydrogen
lo, it is also adjusted to give the best agreement between experiment and theory. Xu et al.32introduced M = Um4, where a,,,is the maximum cross section; in double-resonance experiments on H2 similar to those discussed here, they found M to typically take values of 10-100. At a given wavelength, depletion of the intermediate-state population is not expected to have an effect on the branching ratios for the different product channels, and thus the partial cross sections, including the effect of depletion, are proportional to
128400
128380 128360 ( 2 9 w2) ("1)
12
40
+
Figure 4. Theoretical X %,; v+ = 1 partial cross section from the E2, S = 1 state in region 2: (a) unconvoluted, unsaturated spectrum; (b) same as (a), but convoluted with a Gaussian line shape with a width of 0.75 cm-I; (c) same as (b), but including the effect of saturation with M D = 10.
comparison with the theoretical results is made. As an example, Figure 4a shows the theoretical results for the v+ = 1 partial cross section obtained from the E2, K = 1 level in the region between the X 2E:, v+ = 1 and 2 ionization thresholds. These results correspond to infinite wavelength resolution in the probe transition. The finite bandwidth of the experiment can be introduced- by convoluting the theoretical spectrum with a Gaussian Instrument function whose width corresponds to the reported experimental line width. Figure 4b shows the result of convoluting Figure 4a with a line width of 0.75 cm-I. The effect is to decrease the observed relative intensity of the extremely sharp features and to increase the relative intensity of the features with line widths that are better matched to the instrument function. The experimental spectra discussed below also display features that can be attributed to saturation effects. As discussed by Xu et al.,32 the signal in double-resonance experiments is proportional to the theoretical cross section and to the laser flux in the limit of low light intensity when the depletion of the population in the lower level of the probe transition is negligible. More generally, the total transition intensity is proportional to It, the population in the upper state, which is given by
I, = Z,(I - e-"@> where l o is the population in the intermediate state, a, is the total optical cross section (proportionalto the oscillator strength), and 4 is the integrated laser flux. Equation 2 shows that linear behavior is observed only when 04
