Ionization Processes and Ion Dynamics

BY I. POWIS

1 Introduction

Compared with previous volumes in the series this first chapter will be found to exclude coverage of simple determinations of thermochemical quantities. Nevertheless the field to be surveyed remains broad with many new developments, and the necessity for selective reporting remains. In choosing material, I have been guided by the desire to present work that advances our understanding of the fundamental processes experienced by ionized molecules.

2 Ionization Processes

Photoionization.- Methods for the calculation of molecular photoionization cross-sections are being actively developed and evaluated. The partial-channel cross-section for ionization to a given final state is given by equation (1) where [??] is the dipole-moment operator, ω the photon frequency, and Ψi and Ψf the initial-state and final-state (ion plus electron) wavefunctions. The function Ψi may be evaluated with the Born-Oppenheimer approximation by means of standard computational methodology; usually ab initio single-configuration functions are used, although semi-empirical and correlated initial-state functions have also been employed. The principal difficulties lie in constructing the final-state wavefunction, Ψf, which can be treated as the product of a bound ionic core and a continuum orbital for the ejected electron. A 'frozen-core' approximation may be invoked and the initial-state orbitals used to generate the ionic hole-state. In view of Koopman's theorem this is acceptable, at least for the outer-valence-shell ionizations. It is the treatment of the continuum function that serves to characterize most usefully the various calculational schemes. Early approaches made use of plane waves (with or without orthogonalization to the remaining orbitals), coulomb waves, and single-centre pseudo-potential methods.

The continuum orbital may in principle be obtained as a solution to the Schrödinger equation (2) where ε is the electron kinetic energy and V2-1 is the static-exchange potential of the N-1 orbitals in the ion core. In general, the non-central nature of the static-exchange potential makes the solution of equation (2) difficult, but single-centre expansion techniques applicable to diatomic and linear polyatomic species such as HCl, HF, N2, O2 and CO2 have been described. The main problem is to ensure convergence in the calculations, but the use of the Schwinger variational method in particular seems to be well behaved and to provide solutions of Hartree-Fock accuracy. Single-centre expansions with a simpler static potential approximation have also been employed.

A more tractable and generally applicable method involves replacing the non-local static-exchange potential with localized model potentials; typically the Xα local-exchange scheme is used in which a parameterized, statistically averaged exchange term, based upon the electron density, is used in the Hamiltonian. Several local-exchange schemes have been compared against static-exchange calculations for atomic photionization and found to be fairly accurate, result that is also said to be relevant to molecular systems. In the multiple-scattering method (MSM) for molecules, the non-central potential is partitioned into spherical regions and localized model potentials, the so-called muffin tin potential, employed for each region. Photoionization cross-sections are then readily calculated. Although less accurate than single-centre expansion techniques, MSM calculations may be applied routinely to polyatomics such as BF3, H2O, and H2S. The general reliability of such calculations appears to be fairly good, although in a comparison with experimental results for CS2 and COS it has been noted that the results for the polar COS molecule and the inner-valence orbitals of CS2 are less good than for the outer-valence orbitals.

In solving equation (1), either a length or velocity form of the dipole operator may be used; when ψi and ψf are exact eigenfunctions of the electronic Hamiltonian these are entirely equivalent, but when only approximate wavefunctions are available different results are obtained. It has been proposed that the discrepancy between the two forms of the calculation may be used as an indication of the quality of the approximations being used. Thiel has compared the use of both forms with MSM calculations for a number of diatomics and concludes that the dipole-length form is to be preferred. This result has since been adopted for MSM calculations of HCN photoionization.

A third major category of calculations is available that avoids the use of model potentials and yet circumvents the problems associated with obtaining a continuum wave in a non-central non-local molecular-ion potential. In this approach the system Hamiltonian is diagonalized over a very large basis set of both compact and diffuse functions. The bound-state, so-called improved virtual orbitals (IVO) that result are used to generate a pseudo-spectrum of discrete transitions. The photoionization cross-section may then be obtained by smoothing the pseudo-spectrum oscillator strengths using Stieltjes-Tchebycheff moment theory (STMT). The virtual orbitals in this STMT approach are usually computed at the Hartree-Fock level by the use of a static-exchange Hamiltonian. As such, the calculations are potentially more accurate than the MSM method but have the drawback that, since the continuum wavefunction is never generated, information on angular distributions and so on can not be obtained. Like MSM, this method can be applied to non-linear polyatomics such as H2O.

Explicit comparisons of the Schwinger single-centre expansion method with both MSM and STMT calculations have been made by Lucchese et al. Their results suggest that STMT calculations may depend critically on there being an appropriate distribution of bound virtual orbitals in order to avoid oversmoothing of the oscillator-strength distribution with a consequent loss or distortion of detailed structure in the photoionization cross-section. Implicit comparisons of these methods are also contained in the many results discussed below. To summarize the present position, it seems that for the most accurate calculations single-centre expansion techniques are preferred, where such methods are applicable. Otherwise, STMT calculations can provide detailed quantitative cross-sections. MSM calculations generate semi-quantitative cross-sections, are widely applicable, and, unlike STMT, can be used to investigate angular distributions and so on.

Much interest focuses on the ability of calculations to predict shape resonances in photoionization. A shape resonance arises when the outgoing electron is excited to a quasi-bound electronic state supported by a centrifugal barrier generated by the electron's angular momentum. It is a purely one-electron phenomenon and should be reproducible by the above calculations. Methods for the graphical representation of the continuum electron waves have been developed. Figure 1 shows the f (l = 3) partial-wave component of the σu channel in the [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] ionization at the resonant energy. The f-like nature is evident from the nodal structure. Significantly, there is a dramatic increase in electronic amplitude near the nuclei, which gives the enhanced ionization probability. This also means that the resonance is very sensitive to the internuclear separation, R, and can result in non-Franck-Condon vibrational distributions. Anisotropic effects arise because the resonances occur in specific channels, i.e. for a continuum orbital of given symmetry. Consequently, when a given symmetry channel is preferentially enhanced by resonance, one expects to see this reflected in the photoelectron angular distribution. Recently, Poliakoff et al. demonstrated that the degree of polarization of the photoionized B2Σ+uN2+ fluorescence also reflects contributions from different ionization channels and so will be sensitive to shape resonances. Finally, shape resonances may be of value in achieving specific molecular-orientation effects in photoionization processes, e.g. of absorbed molecules.

Experimental photoelectron asymmetry parameters, β, have been measured for a range of small molecules of interest in the theory of photoionization, including H2O and H2S. Possibly the most dramatic example of a shape resonance is found in the ionization of C2Σ+gCO2+, where the experimentally determined β parameter dips sharply at around 42 eV. STMT calculations predict such a resonance in the σu channel, but at a photon energy of around 35 eV, whereas Schwinger variational calculations of the photoionization cross-section do place the resonance at ~42 eV. MSM-Xα calculations are also found to be broadly in agreement with experiment and to predict resonances for the first four ionic states. However, the C-state β dip is predicted to be too narrow and too deep. This may be a vibrational effect as the discrepancy with R-averaged calculations is less marked. Vibrationally resolved measurements of the A2Πu ionization show that g decreases with increasing vibrational quantum number, again in accordance with calculations.

Vibrationally resolved photoelectron angular distributions have been reported for X2Σ+uN2+. In those regions that are free of autoionization β(υ = 1) is found to be 0.4 - 0.8 units greater than β(υ = 0), a phenomenon that can be interpreted in terms of shape resonance. Although Grimm has suggested that, for the υ = 2 level, autoionization may contribute to the variations of β, the shape resonances have been widely studied. Raseev et al. and Lucchese have performed single-centre, frozen-core, static-exchange calculations for σ-1g ionization via σu and πu channels that indicate a σu-channel shape resonance. Using the Schwinger variational method, Lucchese has further improved and extended these calculations and claims considerably better results than are achieved with MSM or STMT methods. One feature of the Schwinger calculation is that correlation was considered in the initial-state wavefunction. The STMT method has been extended to allow for coupling of the continuum states by replacing the separated-channel, static-exchange calculation of virtual orbitals with the time-dependent Hartree-Fock (RPAE) approximation. The theoretical treatment of inner-shell ionizations has been tackled for N2 (and CO) by the use of c.i. calculations of ionic hole-states in the static-exchange approximation. Very similar behaviour, displaying a o-channel resonance, is found for the ionization of X2Σ+CO+.

Gustafsson has reported cross-sections for O2 ionization in the 20 - 45 eV region. The b,B4,2Σ-g states in particular show evidence of resonances, the former at around 21 eV. However, a higher-resolution study claims that the resonance structure lies at ~19 eV, and vibrational branching ratios also indicate some sort of resonance at this energy. The 19 eV structure is only 0.4 eV wide, unusually narrow for a shape resonance but rather broad for an autoionization resonance. Various calculations predict a shape resonance in the σu channel, but at around 21 eV photon energies, and the exact nature of the σ-1g ionization remains uncertain. The ionization of both X3Σ-g and a 1Δg neutral O2 to the 2Πg ionic state has been treated in different calculations using an orthogonalized coulomb-wave continuum function.

Like O2, NO is an open-shell molecule whose ionization is perhaps less well understood than that of N2 and CO. Both MSM and STMT calculations have been performed for these ionizations with only moderate agreement with experimental data; in particular, the exact position of the σ-channel resonance for the ionization to c3Π is uncertain.

Of prime importance in the development of our understanding of ionization processes is the system represented in equation (3).

Measurements of the overall photoelectron angular distribution over a wide range of photon energies produced β values of around 1.7, somewhat lower than predicted by many previous calculations. However, it is known that asymmetry parameters for those ionizations in which the rotational angular momentum, N', changes by 2 units are very different from those transitions in which ΔN = 0 and are much more sensitive to the details of calculation. Pollard et al. have measured IβΔN = 2 values for υ' = 0 from rotationally resolved photoelectron spectra, and they compared their results with the theoretical values of Itikawa (see the Table). Unlike most previous calculations, these allow for the inclusion of l = 1 and l = 3 waves for the outgoing electron, and this clearly produces better agreement at 584 Å. The converse may seem to be true at 736 Å, although the discrepancies might be attributable to autoionization effects at this wavelength. Jungen and co-workers have performed calculations that allow for autoionization with the results included in the Table. So far these calculations only treat a pure p-electron wave (l = 1). Clearly more work is needed to understand results of this type.

The work of Jungen is part of a general development of multi-channel quantum defect theory (MQDT) to treat the phenomenon of vibrational and/or rotational autoionization in the hydrogen molecule. The initial application was to autoionization of rotationally cold para-hydrogen (J'' = 0) just above the υ' = 0, N' = 0 ionization threshold, and excellent agreement with experimental data was claimed. Subsequently, vibrational autoionization in the 790 - 760 Å region has been investigated.

Other developments of MQDT have been reported, but our general level of understanding of autoionization phenomena in molecules other than hydrogen remains on a much less quantitative basis. Eland has discussed the position regarding electronic autoionization in a review-type article. A number of papers have reported and assigned Rydberg series in the autoionization structure of SO2, NO, and acetonitrile, and examples of autoionization in the production of various electronic states of O2+ have appeared.

The question of how autoionization affects vibrational-level populations in ions has been tackled by determining branching ratios at photon wavelengths corresponding to autoionizing resonances in N2 and O2. The Fano configuration interaction theory of the Rydberg-continuum coupling can be extended to provide detailed branching ratios in the region of a resonance, with information on the coupling being derived from the autoionization peak shapes, although at the peak of a resonance a simplified equation is applicable. The full theory has yet to be evaluated fully. In studies of N2O and CS2 Eland notes that the influence of autoionization on the vibrational branching may be observed well outside the resonance peak.