Theory and Energetics

BY R. A. W. JOHNSTONE

This chapter covers theoretical developments in ion chemistry particularly with regard to application of theory to experimental work and the reflective effect of the latter on the development of theory. The last few years have seen renewed interest in the application of theory to the understanding of ion fragmentation and ion/neutral interactions. Frequently, developments in one area of mass spectrometry have spread over into others so that, although this chapter is divided into sections, extensive cross-references to other sections have had to be made to achieve overall coverage. For this reason, the division into sections has a strong flavour of convenience rather than strict logic but it is hoped the subject matter is more readily digestible and comprehensible treated this way. Photoelectron photoion coincidence spectroscopy is covered by the special review in Chapter 3 and is not dealt with as such here although results are referred to when necessary. The use of molecular orbital theory in mass spectrometry was comprehensively reviewed in Volume 4 of this series and, using criteria developed there, only the more significant applications and developments since then are reviewed in this Chapter.

1 Thermochemical Aspects

By well-known energy cycles, thermochemical data, such as heat of reaction and heat of formation, are interdependent in that one can be derived from others. Therefore, the division of this section into sub-sections describing for example, heats of formation, electron affinities, and proton affinities separately is artificial but has been done for convenience in dealing with the literature and for emphasizing particular points of interest.

A valuable compilation of thermochemical data for gaseous ions has appeared and a review on studies of metastable ions which lists advantages of their use as ions of low internal energy for determining thermochemical thresholds.

Free Energies of Reaction. — Total free energy changes in a reaction (ΔITL [Gθ) are dealt with here in discussion of the derivation of heat of reaction (ΔITL [Hθ) from equilibrium measurements on ion/molecule reactions. Fragmentation of isolated ions is dealt with in the section on RRKM theory. Free energy and heat of reaction are linked through the equation, ΔGθ = ΔHθ - TΔSθ, where T is temperature and ΔSθ the change in entropy. Also, - ΔGθ = RT ln K, in which K is the equilibrium constant for reaction. Usually, ΔSθ is very small and put equal to zero so that, - ΔGθ [congruent to] ΔHθ. An empirical correlation of exothermicity and activation energy is discussed later.

Equilibrium constants determined from ion/molecule reactions may be in error through competitive reaction, differential ion losses, or slow arrival at equilibrium. Suitable methods and precautions for obtaining equilibrium constants from measurements in ICR cells have been discussed. A further source of error is the extraction and trapping fields in many types of apparatus which give a non-Maxwell–Boltzmann distribution to the ions, i.e. the ions have an effective temperature greater than ambient. This point is discussed further for ICR and SIFf techniques in the section on ion/molecule reaction and for ion mobilities in the section of that name. After emphasizing this possible error, a satisfactory way of removing it from time-resolved experiments has been described. Equilibrium constants are measured for different values of the ratio, E/P, in which E is the strength of the electric extraction field and P, the pressure of gas in the apparatus. The value of K and therefore ΔGθ is obtained by extrapolation to E/P equal zero; this technique provided a heat of reaction for CO2H+(CH4, CO2)CH5+ in excellent agreement with other work using the flowing afterglow and ICR methods.

The common practice of putting ΔSθ = 0 for ion/molecule reactions has been examined and found satisfactory. Using the unimolecular reaction rate equation, K = ZP exp (E/RT), and making one or two assumptions, these authors showed that ΔSθ could be estimated through the expression, ΔSθ = R ln (Zf/Zp), where Zf, Zp are the collision rate constants for forward and back reactions; these collision rate constants can be calculated (see section on ion/molecule reactions). Estimates of ΔSθ for a number of reactions were shown to be small, of the right order of magnitude, and in the right direction.

Failure to ensure ions have been thermalized, i.e. have internal and kinetic energies corresponding to ambient temperatures, is a cause for concern when determining thermochemical quantities from equilibrium measurements. It has been shown that, at least for H+-transfer, the reaction, BH+ + B [??] [BHB+]* [??] B + BH+, is so efficient that the BH+ ions are rapidly relaxed.

The sign of ΔHθ can be inferred in ICR experiments from the variation of the double-resonance signal with variation in the irradiating field strength. Thus, by use of bracketing reactions, upper and lower limits can be set for ΔHθ; this point is illustrated in the later section on proton affinities.

All of these thermochemical quantities are determined in the dilute gas-phase in which there are no solvent effects (heat of solvation, dielectric, viscocity, and so on). However, solvent can greatly influence both the extent and nature of a reaction, with products changing and also rates by orders of magnitude from the gas-phase reaction. Further, ionic reactions in solution always have a gegenion which itself can modify the reaction. Bridging the gap between gas-phase and solution-phase has been attempted with considerable success. For example, the interaction of clusters of CH3CN molecules with Na+, K+, Rb+, or Cs+ has been examined, with ΔGθ, ΔHθ, and ΔSθ measured for such as reaction (1).

Na+(CH3CN)n-1 + CH3CN [??] Na+(CH3CN)n (1)

Calculations based on simple electrostatics indicate that the weak interaction of a single CH3CN molecule with a negative ion is due to the diffuse distribution of the positive end of its dipole over the carbon and hydrogen atoms. In contrast, the negative pole of the dipole, strongly localized on nitrogen, leads to strong interaction with a positive ion. Comparison of reaction (1) with a similar one for negative ions, shows that at n = 5, the overall interaction with negative ions becomes slightly more favourable. In a similar piece of work, the differences between the stabilities of complexes of K+ ions with nitrogen and oxygen bases were found to be very much smaller than the differences in the proton affinities of these bases. The exothermicity of hydration of C4H9+ ions (2) changes as the number of solvent molecules increases. For n = 1, the exothermicity was significantly lower than the values for n = 2, 3. The stability of the NO+N2 ion cluster has been examined from 178 to 273 K, giving ΔHθ, ΔSθ values and a bond-dissociation energy, D0(NO+ - N2) = 4.98 ± 0.12 kcal mol.- 1

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Electron Affinities. — Negative ion mass spectrometry has received considerably increased interest in the past few years, prompting the appearance of an extensive review and the issue of a third edition of a classic textbook on the subject. Because of this increased interest, it was thought opportune to discuss somewhat more fully than usual some of the more exciting advances in this area. As many of the techniques used in negative ion chemistry are similar to or identical to those used in positive ion chemistry, their fuller description in this section automatically implies less description in other later sections of this Chapter concerned with positive ions.

The simple reaction (3) requires an enthalpy of reaction equal to the difference in heats of formation of the negative ion (AB-) and the neutral (AB atom, radical, or molecule). This enthalpy, the electron affinity (EA), cannot be determined by direct electron attachment or exothermic electron transfer.

AB + e- -> AB-; EA(AB)=ΔHf(AB)-ΔHf(AB- )(3)

For the particularly stable radical, (CF3)2NO·, the corresponding negative ions were found to be about forty times longer lived than similar ions with similar numbers of degrees of freedom. This relative stability of a negative ion formed by direct electron attachment was ascribed to completion of pairing of electrons in the molecular orbitals of the radical; SCF molecular orbital calculations supported this hypothesis. Attachment began at 0 eV and exhibited a maximum at 1.2 ± 0.1 eV Some limits to electron affinities were set by observed dissociative attachment reactions of (CF3)2NO·. Reaction of a slow electron with a neutral (AB) frequency leads to dissociative resonance electron capture, whereby the electron is deposited in an anti-bonding molecular orbital, weakening bonds sufficiently and depositing sufficient energy to cause the ion to fragment as shown in reaction (4). Dissociative electron capture affords the basis of one method for determining electron affinities as in determination of these values for NaBO2 and KBO2 formed by electron ionization of complexes, M2BO2F(M = Na, K, Cs), present in the vapour phase of mixtures of MF and MBO2. The other main methods used in mass spectrometry include the (usually) very accurate photo- detachment of electrons from negative ions, emission of negative ions from surfaces bombarded with beams of neutral or charged species, charge-exchange between charged and uncharged species, and estimation of electron affinities from other measured thermochemical data or from ab initio molecular calculations.

AB + e- -> [AB-] -> A- + B (4)

Most ab initio molecular orbital calculations yield eigenvalues (ε) giving not only the energies of orbitals occupied by electrons but also the virtual energies of unoccupied or 'ghost' orbitals. By assuming no electron correlation or relaxation effects (Koopmans' theorem) on addition of an electron to the lowest unoccupied molecular orbital (LUMO), its eigenvalue may be equated to the electron affinity. Recently, such calculations have been carried out on simple molecules and led to an unexpected but simple relationship between electron affinity and permanent dipole moment: EA (calculated) = -ε(LUMO) = 0.562 + 0.129μ, (calculated), where the electron affinity is expressed in electron-volts and the dipole moment, μ in Debyes. Advanced molecular orbital calculations with correct choice of basis sets yield values of dipole moments close to those observed experimentally and this relationship can be used to predict electron affinities from dipole moments or vice versa, although it appears to be valid only in the interval, 5 < μ < 10D, and not so when μ ≤ (ca) 4.4 D. The correlation applies only to simple polar molecules like LiH and it would be useful to know whether or not a similar correlation holds for larger molecules with smaller dipole moments. These calculations have shown further that, barring exothermic dissociative electron attachment, all electronically non-degenerate polar molecules with μ > 1.625D will have positive electron affinities if the calculated Born–Oppenheimer (assumption of a 'frozen' molecule, i.e. separation of rotational and vibrational from electronic effects) electron affinities exceed about one twentieth to one tenth of the values of their rotational constants. Again, it would be useful to know how far this prediction can be extended to larger molecules.

Measurement of electron affinity by dissociative resonance electron capture has been greatly improved to yield not only accurate values for affinities but also knowledge of the electronic states of the fragments [reaction (4)] and of the partitioning of excess of energy in the transition-state complex (AB-) along the reaction co-ordinate into vibrational and translational energy in the fragments.

For any dissociative electron capture, the total thermochemical energy balance is given by equation (5) in which P is the appearance energy of the negative ion, D is the bond dissociation energy of the molecule, E*v,t is the energy contained in the transition-state complex in excess of the ionization threshold, and E*el is the electronic energy in excess of the ground-state. In experiments in which only the appearance potential is measured and no account taken of E*v,t, E*el, the derived electron affinity is only a limit. Such is the case for example in the measurements of EA(WF4) ≥ 0.95, EA(WF5) ≥ 0.4, and EA(WOF3) ≥ 0.3 eV, and of EA[Fe(CO)n; n = 0-5] and EA[Ni(CO)n; n = 0-3].

-EA(A) = P(A-)-D(A-B)-E*v,t - E*el (5)

In equation (5), generally E*el >> E*v,t and any excess of electronic excitation energy is readily apparent in the thermochemical balance and the problem of finding EA by this method lies in determining the excess, E*v,t This last energy is partitioned into vibrational energy, εv, and translational energy, εt. in the fragments from decomposition of the transition-state or activated complex. On the basis of the oscillator-set model of RRKM theory, it can be shown that E*v,t = N · [bar.ε]t where N is the number of oscillators in AB and [bar.ε]t is the mean translational energy in the fragments. Because not all of the oscillators appear to be effective in the reaction complex this equality has been adjusted empirically to E*v,t = αN · [bar.ε]t where 0 < α < 1. From conservation of momentum (in a centre-of-mass frame) and allowing for thermal motion in the original molecule, AB, the mean translational energy, [bar.ε]i deposited in the ion, A-, from the reaction complex is given by, [MATHEMATICAL EXPRESSION OMITTED], where mA, mB, and mAB are the masses of the species A, B, and AB. In earlier work, mainly with positive ions, α was averaged to 0.44 but, in the present work, α is determined by experiment directly. Equation (6) follows by replacing [bar.ε]t with E*v,t/αN and differentiating. The slope of the experimental line relating [bar.ε]i with electron energy in excess of the appearance potential for A- has a slope, mB/αNmAB allowing a value for α to be extracted. The mass spectrometrically detected mass peak for A- is broadened by thermal motion in AB and the extra translational energy, [bar.ε]i, imparted by the fragmenting complex, AB-; the broadening of the peak is measured by its full width at half maximum, W1/2. From known reference ions, a calibration curve of W1/2 and ion mass, mi is constructed. Knowing W1/2 for the ion, A-, under investigation, an apparent mass, mapp, is obtained from the calibration curve. It has been shown that, to within about 10% (well inside the larger experimental errors involved in appearance energy measurements), [bar.ε]i = 3/2kT(mapp/mA). This quantity is evaluated for electron energies above the appearance energy and is plotted against the latter to give α. Knowledge of [bar.ε]i near threshold gives [bar.ε]t and, because N, [bar.ε]i and α are known, E*v,t follows.

[MATHEMATICAL EXPRESSION OMITTED] (6)

[MATHEMATICAL EXPRESSION OMITTED] (7)