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...