Theory and Energetics of Mass Spectra

BY B. N. McMaster

1 Introduction

The quasi-equilibrium theory remains the cornerstone of discussions of mass spectral fragmentations in the literature, and has been well reviewed by Wahraftig. However, very significant advances have been made in the theoretical treatments of chemical dynamics in closely related fields. More sophisticated experimental methods are now being applied to studies of ion decompositions and promise to fill some of the wide gaps in our knowledge. Most emphasis has been therefore placed on these aspects and a review style has been adopted, since this is felt to be more useful in the midst of such developments. Where clarification of concepts or methods seems necessary, criticism has been made in a spirit of stimulating further work and discussion.

The topics reviewed follow a logical development. Theoretical calculations of ion structures and energies are briefly discussed, followed by ionization processes and their important relationship to energy deposition functions. Current developments in theories of unimolecular rate processes are reviewed and their relevance to ion decompositions is stressed. Results from photoelectron-photoion coincidence, charge-transfer mass spectra, field ionization kinetics, and meta-stable ion kinetic energy measurements are discussed with particular emphasis on fundamental aspects. Finally, methods of determining appearance potentials are critically evaluated, and their uses in thermochemical calculations are reported.

Although the literature coverage is as comprehensive as possible, some degree of selection has been necessary. Numerous reviews on specific topics are cited in the appropriate sections, but more general coverage of related work in mass spectrometry can be found in Vol. 2 of this series and in a very comprehensive literature survey covering 1972 — 73.

2 Calculations of Ion Structures and Energies

Ab initio Self-consistent Field Calculations including Electron Correlation. — An important distinction must be drawn between ab initio SCF calculations using extended basis sets and those using minimal basis sets. Only the former are expected to give results of Hartree–Fock accuracy in the single-particle approximation (which ignores electron correlation). These calculations are very expensive for polyatomic molecules, and minimal basis set calculations are therefore more generally performed. With judicious choice of basis functions for particular systems, results within a few kcal mol-1 of the Hartree-Fock limit can be obtained using minimal basis sets. But even with results of this accuracy erroneous conclusions may be drawn because electron correlation is ignored in the Hartree–Fock treatment. Theoretical calculations of electronic energies are generally used to determine the energy difference between species. If the correlation energy of each species is the same, then comparisons at the Hartree–Fock level will be quite accurate. However if it differs, even slightly, the relative energies may be drastically changed because the correlation energy is of a similar order of magnitude to the total chemical binding energy.

A number of techniques for estimating the correlation energy, or at least that part which differs between related species, have recently been developed to correct the Hartree–Fock energies. Some of these methods and their application to diatomic molecules have been reviewed by Wahl, including the molecular orbital configuration interaction (MO-CI), multiconfiguration self-consistent field (MCSCF), and independent electron-pair approximation (IEPA) treatments. The calculations of electron correlation effects reported below have used extended basis sets, unless stated otherwise.

Kutzelnigg et al. have calculated correlation energy corrections for vinyl and ethyl ions with classical and non-classical (H-bridged) structures (1), (2) and (3), (4) respectively using the IEPA method. They used the geometries corresponding to local energy minima obtained by slightly-extended basis set calculations, and found that the non-classical structures were more stable by ca. 8 kcal mol-1. Although the quantitative magnitude of the stability is uncertain because of the remaining small errors, its qualitative validity is assured and contrasts with calculations ignoring electron correlation which predicted the classical structure to be more stable in the case of the C2H3+ ions, and about equally stable in the case of the C2H5+ ion. These important differences in predictions have been attributed to the overestimation of electron repulsion in the Hartree–Fock approximation.

Similar calculations for the CH5+ ion indicated that the CS (5) and C2v (6) structures have almost identical energies, while the C4v (7) and D3h (8) structures were less stable by about 6 and 20 kcal mol-1 respectively. Minimal basis set calculations which ignored the correlation energy predicted all these structures to have very similar energies, although later calculations with extended basis sets predicted significant differences. A dissociation energy of 40 kcal mol-1 was obtained for the reaction CH5+ -> CH3+ H2 when the correlation energy was treated. The importance of correlation energy has also been noted in the derivation of accurate potential energy curves for proton transfer in H5O2+, and electrocyclic transformations of cyclopropyl and allyl cations, radicals, and anions.

The π-electron states of benzene have been extensively studied by Hay and Shavitt using an MO–CI method with a frozen σ-core. Excitation energies were obtained for many singlet, triplet, quintet, and Rydberg states of the molecule and several doublet states of the molecular ion. Good agreement with experiment was observed over all and the molecular ion states were well discussed. Very accurate calculations of correlation effects in the ground and ionized states of methane have been reported by Meyer using a pseudo-natural orbital (PNO–CI) treatment, and an interesting discussion of the energy surface of CH4+ has been given. Other CI calculations have been reported for neutral and ionic states of the silyl radical using a minimal basis set.

For some particular cases, which include the dissociation of a doublet molecular ion to a doublet and a singlet state product, the correlation energy does not change significantly with change in geometry, and calculations of Hartree–Fock accuracy then provide quite reliable dissociation energies and potential energy surfaces. By use of various approximations and symmetries in the integral evaluations, the computing time for such calculations has been shortened dramatically. It is now economical enough to study quite large polyatomic systems. Accurate calculations of dissociation energies for various ion decomposition pathways by these methods would be very useful in evaluating proposed fragmentation mechanisms, and would also provide important data for dynamical calculations using potential energy surfaces (see Section 4).

Electron correlation effects on energies may also be calculated by a conceptually different approach based on many-body theories of electronic structure. Rather than perform complete calculations of the different states to determine their energy differences, these methods evaluate excitation energies, ionization potentials, or electron affinities directly from the Hartree–Fock wavefunctions of the initial state. Two approaches are currently being developed, one using Green's functions and perturbation theory and the other using the equations-of-motion method.

The Green's function method has been used to calculate the 'Koopman's defect' corrections to vertical ionization potentials (obtained by Koopman's theorem), resulting from electronic reorganization and electron correlation differences in the ion compared with the molecule. The agreement between calculated values and photoelectron spectroscopy measurements for F2 and N2, HOF, H2O, and H2CO is impressive. For the last two examples the method was extended to calculate vibrational structure as well, and again gave excellent agreement with experimental vibrational energies and oscillator strengths. In the initial applications of the equations-of-motion method the excitation energies of formaldehyde and benzene were calculated. It has also been used to calculate vertical ionization potentials of N2 in good agreement with experiment.

The great advantage of these direct methods is that only the wavefunctions of the initial state are required to calculate the energies and oscillator strengths of any other states, thus leading to huge computational savings. For the best results the wavefunctions should be of Hartree–Fock quality, but reliable estimates of the ordering of excited states (at least) could be obtained with less exact wavefunctions. These new methods will be extremely useful for calculating the energies and oscillator strengths of excited states of ions.

Methods for the calculation of Rydberg levels using model potentials and the multiple-scattering-Xα technique have been applied to several polyatomic molecules. Rydberg states lying above the first ionization potential are likely to autoionize, and it is important to have some knowledge of their positions and oscillator strengths. It is becoming increasingly apparent that autoionization contributes substantially to the total ionization cross-section of polyatomic molecules, especially near the lower ionization potentials, and can have a marked influence on decomposition processes (see Sections 3 and 5).

Ab initio Self-consistent Field Calculations neglecting Electron Correlation. — Many of these calculations on polyatomic ions have used the minimal basis set method (STO–3G) developed by Pople and co-workers, who have made a full review of this field; more recent results using extended basis sets have been collected in a useful report. Studies have concentrated on the important problem of establishing the relative stabilities of structural isomers of several carbonium ions; e.g. C3H5+, C3H7+, C5H5+, and cyclopropylcarbinyl cations. Open (classical) structures have been found in most cases to be more stable than the corresponding bridged (non-classical) structures. However, relative stabilities favouring open structures which amount to only several kcal mol-1 must be considered uncertain, since electron correlation effects have been shown to favour bridged over open structures by this amount (see above).

It is also important to note that the accuracy of these calculations, with respect to the Hartree–Fock limit, depends critically on the basis set employed. Extended basis sets, including polarization functions, are absolutely necessary for best accuracy. Calculations on C2H2+ near the Hartree–Fock limit indicate that minimal basis set functions do not provide an adequate description of the severe electron reorganization between neutral and ionic states. The transferability of σ-orbitals to the ionic states is particularly poor. Other calculations (STO–3G) have been performed for haloethyl cations and acylium ions.

Semi-empirical Calculations. — Because of their great computational economy, semi-empirical molecular orbital treatments (e.g. MINDO, INDO, CNDO, EHT) are widely used but their accuracy is difficult, if not impossible, to determine a priori These methods supposedly take some account of electron correlation effects, but their success depends entirely on the parametrization used. They do not generally reproduce observed ionization potentials very well, because of the large changes which occur in electron correlation and reorganization upon ionization, and the reliability of conclusions based on the simpler approximations is very suspect.

The heats of formation of the C1 — C4 alkyl carbonium ion structures have been calculated by Dewar using his MINDO/2 method, and good agreement with experimental values was obtained for the higher homologues. The most stable forms of the C3H7+, C4H9+, and C5H11+ ions are predicted to have a bridged protonated cyclopropane structure. The C5H5+ ions have been studied using the more recent MINDO/3 version, and also by the CNDO and EHT treatments, providing interesting comparisons with the ab initio calculations. Semi-empirical treatments predict structures (9) and (10) to be local energy minima, but differ in the ordering of relative stabilities; MINDO/3 ordering agrees with the ab initio results. However, these calculations were only performed for a singlet state, whereas the ab initio calculations suggest that a triplet state cyclopentadienyl structure (D5h) is more stable than any singlet state structures. CNDO studies of several oxocarbonium ions, and INDO studies of fluorovinyl cations and 2-substituted allyl cations have also been reported.

Several workers have attempted to explain various features of mass spectral fragmentations from semi-empirical calculations of charge densities or bond densities at the (assumed) equilibrium geometry of the molecular ion, either in its ground state or 'pseudo-excited' states. These approaches take no explicit account of the dynamics of ion decompositions, and assume they are controlled by the properties of the ion at its equilibrium geometry. A clear presentation of the pitfalls awaiting such approaches has been given by Krier et al., with specific reference to the decomposition of the ethylamine molecular ion. A brief reiteration of their main points is worthwhile. Because the electron distribution changes markedly as the ion dissociates, both the charge densities and bond densities will also change, often in an unpredictable matter. Therefore the relation of the initial (equilibrium) charge or bond densities to the eventual fragmentation is by no means straightforward. On the other hand the potential energy surface of the molecular ion does control the decomposition process. However, the accuracy of current semi-empirical methods is quite inadequate to determine even the dissociation energy (let alone the surface) with any reliability. This is illustrated by some EHT and Iterative EHT calculations, where even the ordering of the dissociation energies changes with the method.

3 Ionization Processes and Energy Deposition Functions

It is vital that the various ionization processes and the behaviour of their cross-sections with energy be properly understood. Electron spectroscopy has contributed most towards our understanding of ionization phenomena, and Brion has comprehensively reviewed recent developments. Major points illustrated here involve photoionization (much more detailed information is available on this process), and particular reference is made to a brief, but excellent, discussion of photoionization processes by Chupka.