CHAPTER 1
Theory and Energetics of Mass Spectrometry
BY T. BAER
1 Introduction
A review of the theory and energetics in mass spectrometry is a formidable task because the field is so broad. Consistent with the theory and energetics chapters of previous volumes, I have tried to limit the review to those aspects of the literature during the past two years which are relevant to the fundamental understanding of ion dynamics. Particular emphasis has been given to those developments which will help us in our ultimate quest, the ability to predict qualitatively or quantitatively the behaviour of energized ions.
2 Ion Thermochemistry
As dynamical experiments and theories are becoming more sophisticated and precise, the need for accurate thermochemical data on molecules, ions, and fragments continues to grow. During the past few years the experimental effort has been supported by numerous calculations, most of which are of the ab initio type. With the advent of readily available high-level programs, numerous groups are now performing calculations. In combination with good experimental information, these results are of great value in extending our chemical knowledge because accompanying the calculated energy is an assumed structure. Although some of the theoretical work will be treated under a separate subheading, a large portion of it will be mixed in with the review of experimental results.
Molecular Orbital Calculations. — Ab initio calculations have decreased in cost to such an extent that few calculations are now being done with semiempirical programs. This is also partly as a result of the fact that the semiempirical programs are usually parametrized to do one job well, but at the expense of their predictive ability for other properties.
The most commonly used ab initio program is the STO-3G (Slater-type orbitals with 3 gaussian functions). This uses a minimal, split-valence set of basis functions. The split valence means that two basis functions are used for each valence atomic orbital. More sophisticated basis sets are ones belonging to the K-LMG family, in which K is the number of gaussians used to describe the inner-shell s-type functions, L is the number of gaussians for the s- and p-type valence functions, and M is the number of gaussians for the outer sp-type functions. A commonly used basis set has been the 4-31G which is available through the Quantum Chemistry Program Exchange (QCPE) of the University of Indiana. This and the other K-LMG programs have been developed by Pople and his co-workers.
Some new K-LMG programs have been developed and are, or will soon be, available through the QCPE. Two of these are ones which use the 6-21G and 3-21G basis sets. Either of these is claimed to be as good as the 4-31G or the PFPB 4-21G basis set. The K-21G split-valence basis sets are definitely superior to the STO-3G minimal basis set. Equilibrium geometries are about as good as those of the 4-31G but superior with regard to the description of the bond angles involving heteroatoms. Vibrational frequencies are also equal to, or better than, those of the 4-31G. Similarly, electric dipole moments are better with either the 6- or the 3-21G than with the 4-31G. Happily, because the 3-21G has fewer primitive gaussian functions it is faster than the 4-31G set. It appears to be inferior to the 4-31G only in the calculation of reaction energies. Comparisons for over 20 molecules are given.
Halgren et al.4 have compared the speed and accuracy of a number of semiempirical and ab initio programs for calculations of various properties. The overall effectiveness versus speed curve is shown in Figure 1. This paper also introduces a new semiempirical program, the PRDDO (partial retention of diatomic differential overlap). It is 16 times faster than one of the simplest ab initio programs, the STO-3G, yet it agrees very well with this program in relative energies, atomic charges, and dipole moments. In a follow-up to this paper, Dewar and Ford have added their MNDO program to this comparison. They compared seven MO methods by listing the root mean square (r.m.s.) error of the energy, ionization energy (Koopmans' theory), and dipole moment with respect to experimental results. These are listed in Table 1 and should serve as a guide to experimentalists. Although impressive, the calculations must be used with care. Furthermore it is doubtful that single configuration calculations will reach experimental accuracies of say 1 kJ mol-1. For such precision elaborate configuration interaction (CI) calculations must be carried out. These are still the domain of the theoreticians.
The most studied molecular ion during the past two years has been CN+. No less than four separate investigations were reported dealing primarily with the identity of the ground state. As with the isoelectronic C2, the two states 1Σ+ and 3Π are very close in energy. Wu did an SCF calculation and concluded that the 3Πis lower in energy by 0.33 eV. The fact that this was a single configuration calculation makes it somewhat suspect. Yet Ha using CI also found that the 3Π is lower than the 1Σ+ state by 0.41 eV. Murrell et al. would not commit themselves, stating that the two states are extremely close. This caution is certainly justified because Hirst using the ATMOL SCF calculation with CI found that either state could be made the ground state depending on the number of configurations used. Yet the bond distances are quite different (1.20 Å and 1.28 Å). Other diatomics studied are the mixed alkali metals and alkali salts such as NaK+, NaRb+, NaCs+, KBr+, KCs+, RbCs+, Na2+, K2+, Rb2+, and Cs2+. A number of stable excited electronic states were found.
Often the most stable structure for an ion is not the same as the most stable neutral structure. These situations are sometimes difficult to establish experimentally, but they are quite amenable to calculations. In fact the calculation of energies of isomers is one of the most fruitful uses of MO calculations. Murrell and Derzi have concluded that, although HCN is 0.5 eV more stable than HNC, in the ionic form HNC+ is more stable than HCN+ by 0.9 eV, and that the activation for HCN+ going to HNC+ is about 0.45 eV. These results are based on an ab initio SCF calculation in which the basis functions consisted of contracted gaussians augmented by polarization functions. Another study of HCN+ addressed the problem of the [??] and [??] states and, in particular, the slow predissociation rate at 20.3 eV which results in sharp vibrational structure in the spectra from photoelectron spectroscopy (PES).
The isomers HNO+ and NOH+ have been investigated,' and in both cases HNO+ was found to be more stable. Bruna and Marian using the MRD-CI (multi-reference double-excitation) program developed by Buenker et al. found the energy difference only about 12 kJ mol". Both ions are bent; the angle in HNO+ is 131° while in NOH+ it is 124°. The other SCF study with Cl found the energy difference to be 54 kJ mol" and the angles 126° and 116°, respectively.
The MRD-CI program was also used to calculate the relative stability of the HCS+-CSH+ system. HCS+ was found more stable by 465 kJ mol-1. In a follow up to this study, CNDO/2 investigation of the same isomers found an energy difference of 520 kJ mol-1. Yet another study with the MRD-CI program was the calculation of the low-lying states of NH2+. A rather strange result is that two states, the linear 3Σg- and the bent 3B1 (150°), are candidates for the ground state. They differ in energy by only 330 cm-1 while the barrier is about 900 cm-1. The reaction N+ + H2 -> NH+ + H was also investigated with CI by calculating triplet states of NH2+ which might correlate with the dissociation products. In a study similarly relevant to dynamics, Hansoul et al. investigated the higher-lying states of HCN+ by CI. They found that the third state ([??]) at 19 eV cannot be assigned to the removal of a single Aar electron. Instead at least two configurations, one of which is a two-electron excitation, are involved. On the basis of the experimentally observed H+ onset the authors conclude that the [??] state is strongly coupled with the [??] state.
One of the most remarkable predictions comes from an INDO calculation on N2O+. The neutral N2O is linear as is also the 2Π state of the ion, which is thought to be the ground state. Yet Barber et al. have found that for a bent state (N — N — O angle of 61.1°) the A" is 306 kJ mol-1 more stable than the 2Π state. The barrier is 220 kJ mol-1 which would explain why no experimental evidence for such a low-lying state of N2O+ has been reported. In view of these calculations it would be extremely interesting to carry out ab initio CI calculations and to determine onsets of N2O+ fragments from the dissociation of molecular ions containing N2O. The 4A" state of N2O+ has been carefully studied by Hopper who found that it is stable but has an energy considerably above the linear 2Π state.
Intermediate-size molecules whose structures and energies have been studied include four isomers of C2H4O+. At the 4-31G level, vinylalcohol+ was found most stable. Acetaldehyde+, ethylene-oxide+, and open ethylene-oxide+ were found to be 46, 195, and 133 kJ mol-1 less stable, respectively. The barrier for inversion in N2H3+ has been calculated with an STO basis set to be 190 kJ mol- 1, which is much greater than the experimental value in solution of 55 kJ mol-1. This indicates that the transition state in solution is greatly stabilized by the solvent. LCAO MO SCF Cl calculations have been carried out on the much studied cyclopropenyl cation C3H3+. The C — C bond length was found to be 1.389 Å. Substituent effects on the related propargyl ion have been carried out at the STO-3G level to determine the changes in the π electron charge distributions upon substitution of such groups as H, Me, F, and NH2. The formaldehyde and hydroxymethylene tautomeric ions [structures (1) and (2)] have been calculated. Structure (1) was found to be more stable than structure (2) by 41 kJ mol-1, and the inversion barrier via a 1,2 hydrogen shift was estimated to be 248 kJ mol-1.
Other investigations carried out by MO calculations include such diverse studies as the origin of the molecular shapes of Me+ and NH3+, the potential energy surface for cyclopentadienyl cation, the interactions of Li+ ions with ethers, thioethers, and amide systems, and the role of hyperconjugation in isotope effects. Ab initio calculations on the structure and stability of SiHn+ (n = 0 — 4), the structure of C4H7+, and the inductive effects of repelling groups such as Me, CCH, H, etc. on ions have been performed.
Jefford et al. calculated the various isomers of C8H9+ by the MINDO/3 method, while a semiempirical X-alpha calculation on C2H4+ and H2O+ was carried out by Barrow et al. Semiempirical methods have been compared with ab initio calculations for C2H7+. The authors conclude that MINDO/3 is not useful for predicting the most stable structure of five-co-ordinate carbon atoms. The most stable structure predicted by the ab initio method is a protonated ethane in which the proton bridges the two carbon atoms. On the other hand, Lischka and Köhler concluded that, in the case of Et+ and C2H3+ in which the nonclassical bridged structures are more stable by 16 kJ mol-1 as calculated by ab initio techniques, the MINDO/3 program significantly overestimates the bridged structure. Another comparison of ab initio with INDO was done by Mayr et al. on 3,3-dimethylallyl cations. The ab initio results suggested a planar structure while INDO gave a 30° twisted ground state.
MO calculations in support of PES have continued. Single configuration calculations of ionization energies may be adequate at low energies. However, at high energies the effect of doubly excited states and mixing of states becomes important so that either many configurations must be included or the many-body Green's function approach must be used. A comparison of the two methods has been carried out on the ionization energy of C2H4. Both predict energies close to the experimental value but comparison for the higher-lying states has not been made. The Green's function method has been applied to the states between 20 and 70 eV in HCN+ and HCOOH+, CnHm+ (n, m = 2 — 6), and the isoelectronic CO2+ and N2O+ ions. Many of the high-lying states are weak and diffuse and therefore cannot be described in terms of a 'quasi hole' or a one-electron excitation. This is especially true of the unsaturated hydrocarbons. Configuration interaction has been used in semiempirical calculations by Lauer et al. to calculate the low-lying IE's of isobutene, cis-butene, ethylene, etc. Certainly the most esoteric study on IE's was that of Koller et al. who determined that a quark in an H2O or SO2 molecule would significantly alter the IE. The problem lies in the experimental verification of this effect in view of the fact that the probability of finding a quark is only one in 10. An important word of caution has been raised by Heilbronner concerning the fitting of a series of calculated IE's to the measured peaks in a spectrum. By the very fact of assuming the same ordering of states in the calculations and experiment, a significant statistical correlation is built into the comparison.
A rather useful application of MO calculations has been the determination of molecular polarizabilities and electric dipole moments. Both of these properties are difficult to measure experimentally, but are extremely important in calculating ion–molecule collision cross-sections and intensities of rotational and vibrational spectra.
