CHAPTER 1
Theory and Energetics in Mass Spectrometry
BY B. N. McMASTER
1 Introduction
The previous Report in this series essayed a fairly critical discussion of the fundamental concepts underlying some of the newer theoretical and experimental approaches being used in mass spectrometry. The past two years have witnessed further application of these techniques, rather than any major new developments. This report therefore focuses more on the results obtained by these methods, and is intended to complement the previous Report which provides the relevant background material. The literature coverage is accordingly more selective, but gives a representative illustration of the current capabilities of reported techniques.
Ab initio calculations of the structures of ions and energy barriers for their rearrangements are reviewed. Accurate appearance potential measurements, and other experimental methods of determining the energetics of ion decompositions are also discussed. A brief review of some recent theoretical studies of unimolecular dissociation reactions then leads on to an examination of results from experimental studies which have provided important information about the rate constants and translational energy disposal in unimolecular ion decompositions.
2 Ab initio Calculations of Ion Structures
In view of the rapidly growing dissemination and use of ab initio quantum chemistry programs, such as GAUSSIAN 70, it may be wise to inject a blunt note of caution. At their lowest minimal basis set level (e.g. STO-3G), which is the most widely used level because it is cheapest, the results are not of predictive value with respect to their implicitly stated aims of determining reliable geometries and relative energies of ion structures. Such calculations represent a false economy, particularly when the cheaper MINDO/3 method gives more reliable predictions (see Chapter 2, Section 5). More expensive ab initio calculations using larger basis sets are necessary to achieve results which are of chemically useful predictive value. Because of its practical importance, this question of basis set quality is emphasized in the following discussion. But before proceeding, it is useful to clarify briefly some of the shorthand jargon commonly used to describe the basis sets.
For reasons of computational efficiency, the basis sets used for ab initio calculations on polyatomic species are almost invariably composed of contracted Gaussian-type orbitals (CGTO), which are simply fixed linear combinations of one or more GTOs. These may be derived from atomic calculations in which a primitive GTO basis set is first optimized and then broken into the required number of CGTOs. Alternatively the CGTOs may be derived by determining the best least squares fit of appropriate numbers of GTOs to optimized Slater-type orbitals (STO) from atomic calculations. The quality of the basis set may be roughly ranked in a convenient fashion according to how many CGTOs are used to represent each canonical atomic orbital. A single-zeta (SZ), or minimal, basis set uses only one CGTO per atomic orbital, whereas a double-zeta (DZ) basis uses two, and so on. One particular intermediate case is also commonly encountered, where the valence atomic orbitals are represented by two CGTOs and the core orbitals by only one CGTO. Examples of this type of basis set include the GAUSSIAN 70 4–31G and 6-31G sets, which may be regarded as roughly '1½-zeta' on this relative scale. All these basis sets may also be augmented by polarization functions consisting of a single GTO of the appropriate type for each atom (e.g. p for H, d for C), and this is often indicated by superscript asterisks (see below).
Most ab initio calculations on ionic species have so far been performed at the Hartree–Fock (HF) SCF level, which neglects electron correlation effects. Such neglect is likely to render unreliable the conclusions regarding the relative stabilities of different structures where small energy differences are involved. Furthermore, the wide disparity in the quality of the basis sets used can make it difficult to assess the reliability of the results obtained by different workers, particularly for those who are not closely familiar with these calculations. At the end of this section, an attempt has therefore been made to summarize the more important criteria of quality, and to suggest standards which should be met in reliable calculations of various properties (e.g. geometries, energy differences of stable structures, and potential energy surfaces).
Pople has recently reviewed ab initio calculations on small organic ions and second-period hydride cations performed at the HF-SCF level with various basis sets. The geometries of several AH+n species (A = C, N, O, or F) have been optimized under appropriate symmetry constraints with 6-31G* basis sets, which contain d-type polarization functions on the central atom. In the few cases where comparison is possible, the calculated bond lengths were within 0.1 — 0.2 Å of experimental values, and bond angles were within 1 — 2°. Other calculations on NH2[??], NH2[??], and H2O[??] using larger double-zeta plus polarization (DZ + P) basis sets were primarily concerned with the electronic reorganization upon ionization, although the structures of some excited states of H2O[??] have also been investigated.
The species H3+, LiH2+, BeH3+, BH4+, CH5+ may be regarded as complexes of H2 with the corresponding simple Lewis acids, and have recently been studied by Collins et al. using several basis sets. The geometries of these cations were first determined using an STO-3G basis, and then reoptimized for a 4-31G basis set. Single SCF calculations at these geometries were then done with 6-31G* and 6-31G** basis sets; the latter includes p-type polarization functions on hydrogen, as well as d-type on the second-period atoms. This inclusion of polarization functions gave a modest increase in the binding energies between H2 and the Lewis acids. Finally, unrestricted Moller–Plesset second-order perturbation theory (UMP2) was applied to these calculations to obtain an approximate estimate of the correlation energy. The UMP2/6-31G** results predicted the following decreasing order of binding energies to H2 (kcal mol-1) for H+ (104.5), CH3+ (43.2), BeH+ (21.3), Li+ (4.5), BH2+ (3.8), which may be compared with experimental estimates for H+ and CH3+ of 99 (± 1) and 37.9 kcal mol-1, respectively. These results provide encouragement in suggesting that correlation energies might be estimated to within ca. 5 kcal mol-1 by this comparatively simple method (cf. extensive Cl calculations) in appropriate cases.
A recent study of the stable structures of the C2H3+ cation by Weber et al. represents a 'state-of-the-art' ab initio calculation for polyatomic ions of this size. It gave predictions of chemical accuracy (i.e. within 1–2 kcal mol-1), and provides an opportunity for instructive comparisons with more approximate methods. A double-zeta plus polarization (DZ+P) basis set was used, since this was regarded as the minimum quality necessary to give a balanced description of the electronic structure at different geometries at the HF-SCF level. At selected geometries large-scale configuration interaction (CI) calculations were performed to obtain reliable estimates of electron correlation effects in the valence shell; correlation effects due to the carbon 1s core orbitals were ignored, since they were not expected to vary significantly with geometry.
The geometries of the classical and H-bridged structures were fully optimized at the HF-SCF level and gave an energy difference of 5.36 kcal mol-1, with the classical ion being more stable. Although the optimized geometries obtained with a 4-31G basis were very similar to these geometries, the calculated energy difference was much too large (19 kcal mol-1); however, much better relative stabilities of 7.4 and 5.7 kcal mol-1 were obtained from 6-31G* and 6-31G** SCF calculations, respectively. These results stress the importance of polarization functions in determining the relative stabilities of isomeric structures. But correlation effects were found to be of equal importance in this case, where the electronic structure changes extensively between the two geometries and the HF-SCF energy difference is small. The correlation energy changed by 5.35 kcal mol-1 favouring the bridged structure, so that both structures were calculated to have the same energy including correlation. Taking into account the small remaining errors due to basis set deficiencies and the lack of geometry optimization at the CI level, Weber et al. predicted that the relative energies of the classical and bridged structures were equal, to within 1–2 kcal mol-1, with the latter probably being more stable. They also determined the minimum energy path for rearrangement and predicted it to be planar with an energy barrier less than 1–3 kcal. mol-1 at a transition state about half-way along the rearrangement path (Figure 1). These large-scale Cl results also indicated that the more approximate UMP2 and IEPA methods had overestimated the change in correlation energy by ca. 5 kcal mol-1, resulting in the predictions that the bridged structure was significantly more stable than the linear structure.
The bridged structure of C2H3+ can be described as a π-protonated acetylene in which the C2H2 moiety scarcely distorts from its equilibrium geometry, and the positive charge is almost equally shared by the three hydrogens while the carbons are approximately neutral. In the linear structure, however, there is considerable charge transfer between the two carbon atoms. Because of this important electronic reorganization which occurs during the rearrangement, the electronic spectra of the two structures were predicted to be quite different. Separate HF-SCF calculations on the ground and excited states indicated that the two lowest vertical singlet-singlet excitation energies (π -> σ*, π -> π*) were much smaller in the linear (2.37, 6.95 eV) than in the bridged structure (6.53,11.36 eV). The low π -> σ* transition lies in the visible spectrum for the linear structure and may be detectable by trapped-ion photodissociation studies, provided the linear C2H3+ ion is not significantly less stable than the bridged structure and presuming that formation of this excited state leads to fragmentation.
Double-zeta quality SCF calculations indicated that the vertical IP of acetylene lay 0.12 eV above the adiabatic value, owing to a 0.05 Å longer C — C bond in the linear C2H2+ cation. The geometries of the closed-shell C3 cations C3H+, C3H3+, C3H5+, and C3H7+, have been optimized at the minimal STO-3G level, and the energy differences between the isomeric structures determined with 6-31G* basis sets at the SCF level. The cyclopropenyl structure (1) was found to be 34 kcal mol-1 more stable than the propargyl structure (2) for C3H3+10a, agreeing fairly closely with the difference of 31 kcal mol-1 obtained using a double-zeta basis set. The latter study also predicted an 83 kcal mol-1 energy barrier for the rearrangement (l) -> (2), in which the hydrogen atoms move out of the carbon-atom plane during the ring-opening owing to the formation of a deformed π-system. Lossing has experimentally estimated an energy difference of 25 kcal mol-1 between (1) and (2), which would only be consistent with the SCF predictions if correlation effects favoured (2) over (1).
The heats of formation of optimized geometries for C3H7+ have also been determined by the semi-empirical MINDO/3 method, and provide interesting comparison with the 6-31G* SCF results (Table 1). The two methods differ most significantly in their predicted stabilities of the edge-protonated cyclopropane cation (4) relative to the most stable 2-propyl cation (3). MINDO/3 predicts that these two structures represent the only local energy minima with all other structures collapsing to (4) without activation energy, whereas the 6-31G* calculations predict that both the corner-protonated cyclopropane (5) and 1-propyl (6) cations are slightly more stable than (4). In view of the demonstrated tendency of ab initio SCF calculations to underestimate the relative stability of H-bridged structures owing to the neglect of electron correlation, it seems probable that the MINDO/3 prediction is nearer the truth. Interestingly, the estimates of the relative energies of the other structures agree within 1–2 kcal mol-1 for both methods.
The geometries of the two valence-tautomers of C2H2F+ have been fully optimized at the HF-SCF level using double-zeta basis sets. The bridged fluorenium ion (7) corresponded to an energy saddle-point lying 31 kcal mol-1 above the 1-fluorovinyl cation (8), which had a normal C — F bond length of 1.323 Å (cf. 1.32 Å in fluoroethylene), whereas (7) possessed an anomalously long C — F bond (1.642 Å). The analogous structures of C2H4F+ ions were also optimized (6-31G) and the final energies calculated using the same DZ set as for C2H2F+. The classical 2-fluoroethyl cation (9) was 10.3 kcal mol-1 more stable than the bridged structure (10), but in this case (10) represented a local energy minimum with an energy barrier to ring-opening of 8.5 kcal mol-1. A Mulliken gross population analysis indicated that (10) was best described as involving delocalized three-centre bonding rather than a formally bivalent positively charged fluorine. These conclusions regarding the relative stabilities of (9) and (10), and the nature of the bonding in (10), were opposite to those reached in earlier HF-SCF calculations using STO-3G basis sets, although 4-31G calculations agreed with the DZ results and also indicated that (9) was probably unstable with respect to rearrangement via a hydrogen-bridged structure to form the 1-fluoroethyl cation.
For both the C2H2F+ and C2H4F+ cations it is almost certain that the inclusion of polarization functions would lower the energies of the bridged relative to the classical structures, perhaps by ca. 5 kcal mol.-1. Treatment of electron correlation effects would also be expected to result in a further lowering of the relative energy of the bridged fluorenium ions. Hence, although the prediction that (8) is more stable than (7) is fairly secure, with the proviso that the figure of 31 kcal mol-1 overestimates the energy difference, the situation is much less certain for the C2H4F+ structures. It is quite possible that in reality (10) has a very similar energy to (9), rather than being 10 kcal mol-1 less stable as predicted by these DZ calculations at the HF-SCF level.
The classical (11) and bridged (12) structures of C2H2SH+ were optimized using a minimal basis, and the total energies recomputed with a DZ+P basis. Both structures represent local minima of nearly the same energy, with a modest barrier to rearrangement of ca. 13 kcal mol-1. The thiol hydrogen in (12) subtends an angle of 80° with the molecular plane, and there is a large energy barrier (ca. 73 kcal mol-1) to inversion via the planar thiirenium ion which would involve the formation of an antiaromatic 4π-electron system over the sulphur and carbon atoms.
