Ab Initio Computation of Energy Deposition During Electron Ionization

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Ab Initio Computation of Energy Deposition During Electron Ionization of Molecules Karl K. Irikura J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.7b07993 • Publication Date (Web): 19 Sep 2017 Downloaded from http://pubs.acs.org on September 25, 2017

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The Journal of Physical Chemistry

Ab Initio Computation of Energy Deposition During Electron Ionization of Molecules Karl K. Irikura* Chemical Sciences Division, National Institute of Standards and Technology, Gaithersburg, MD 20899-8320 ABSTRACT: The dissociative ionization of molecules under electron impact forms the basis for analytical mass spectrometry of volatile compounds. It is also important in other situations, notably plasmas. Although qualitative theory for mass spectrometry was developed long ago, progress toward predictive theory has been slow. A major obstacle has been ignorance of the amount of energy deposited in the molecular ion, prior to its fragmentation. Here, we consider the Binary-Encounter Bethe (BEB) theory, which was originally developed for predicting total ionization cross sections. The energy deposition function constructed from BEB molecular-orbital cross sections compares well with the two comparable (e, 2e + ion) experimental measurements. When combined with experimental breakdown data from photoelectron-photoion-coincidence measurements, the BEB energy deposition function successfully reproduces library mass spectra for all but one of the six molecules studied here. This indicates that BEB molecular-orbital cross sections are physically meaningful and are useful for modeling the energy deposition during electron ionization of molecules.

ment ions, leading to systematic bias against light ions, often called “mass discrimination.” A smaller effect, although still important, is the temperature of the ion source.16-18 Thus, if the goal is to reproduce an ordinary mass spectrum, one preceding and one following step must be added. (0) The neutral target molecules are assumed to have a distribution of internal energies, corresponding somewhat with the temperature of the ion source. If one accepts a thermal distribution and a simple model for the molecular density of states, such as the rigid-rotor harmonicoscillator model, then this step is routine. (3) Instrumental measurement biases must be modeled, including mass discrimination and the time for dissociation of metastable ions.19-21 Computational modeling is of established importance in the design of mass spectrometers.22,23 The same modeling techniques can be used to estimate the biases of an existing instrument.12 At least one general correction has been suggested for mass discrimination,24 but it is very crude. Statistical theories of chemical kinetics are standard for modeling metastable lifetimes.25-27 The test of a predictive model is comparison with reliable experimental measurements. However, such a prediction for mass spectrometry requires modeling all four phenomena listed above. When theory and measurement disagree, as they certainly will to some extent, it will remain unknown which of the four models is deficient. To avoid this problem, each step should be tested independently. Here, we seek to test step (1), modeling of the distribution function for internal energy in the molecular ion. Direct comparison with experiments is problematic because the necessary (e, 2e) coincidence experiments are so challenging, especially

INTRODUCTION In contrast to analytical methods based upon electromagnetic spectroscopy, there is no ab initio theory of electronionization mass spectrometry (EIMS). Early efforts, based upon statistical theories of unimolecular dissociation,1 were frustrated by ignorance of transition-state properties and of the distribution of internal energies in the ion population. More recent attempts have sought to avoid the need for such detailed information.2-5 A good summary has been published recently by Bauer and Grimme.6 The conceptual model, which is generally accepted, is composed of two principal steps.6,7 Note that a mass spectrum requires a large number of molecules to be ionized, not only one. (1) The energetic electron strikes the target molecule and dislodges one or more of the bound electrons. This produces a molecular ion, in addition to the scattered electrons. The molecular ion will contain some amount of internal excitation energy. (2) The molecular ion may experience unimolecular reactions and fragmentation, as enabled by its internal energy. The ion may relax to its electronic ground state prior to, during, or after fragmentation.8,9 Such internal conversion is assumed to be faster than the experimental timescale. Fragment ions may retain enough energy to dissociate further. This two-step model accounts for the intrinsic properties of the target molecule. In analytical mass spectrometry, extrinsic factors are also important. The biggest effect is instrumental measurement bias. In particular, ions with high kinetic energy are not detected efficiently.10-15 These are usually light frag-

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at sufficiently high energies and integrated over all solid angles. However, some data exist and will be discussed below. Comparisons with EIMS measurements are only possible when step (2), the fate of the excited molecular ions, is known from experiment. Fortunately, the necessary data are available for some molecules from threshold photoelectron-photoion coincidence (TPEPICO) measurements. Those experiments provide the fragment-ion distribution for a molecular ion as a function of its internal energy. Combining those data with a theoretical energy deposition function yields a simulated mass spectrum that can be compared with a library spectrum. Note that this is the same strategy espoused long ago by Meisels et al.28 It is implicit in such a comparison that steps (0) and (3) are approximately equal in the TPEPICO and EIMS experiments. That is, it is assumed that the temperature of the target gas is reasonably similar in the two experiments, and that detection biases are similar as well. These may be strong assumptions, but they are made necessary by the lack of information about the experiments. It should be emphasized that the present study is far from the first to seek the energy deposition function (EDF) during the electron ionization of molecules. However, during the previous century the emphasis was upon rationalizing and understanding mass spectra, not predicting them. For example, Meisels et al. published a review of the best EDFs available in 1972.28 All those EDFs were measured experimentally; none could be used to predict the mass spectrum for a hypothetical molecule. EDFs were not available from first principles and even semi-empirical EDFs were very crude. For example, Yeo and Williams used the sum of two triangular distributions to approximate an experimental EDF.29 Only recently, with the goal of predicting mass spectra, has this subject been revisited. Grimme considered three empirical, parameterized EDFs, favoring a Poisson-like distribution, in his molecular-dynamics simulations of mass spectra.4

The energy deposited in the molecular ion is assumed equal to the binding energy (B) of the orbital that is ionized. That is, the zero of energy is taken to be the neutral molecule, not the cold ion. This choice is convenient because it corresponds to the energy axis in TPEPICO breakdown graphs (the photon energy), and because it avoids the need for a value of the adiabatic ionization energy. The probability of ionizing an orbital is proportional to its ionization cross section (σ). Thus, over a large number of ionization events the deposited energy may follow the probability distribution function given in eq. (2), where E is the internal excitation energy.

∑ δ ( E − B ) σ (T ) P( E; T ) = ∑ σ (T ) i

(2)

i

i

An important limitation of the BEB model is that it is fundamentally a one-electron theory. It only includes ionic states that correspond to simple, one-electron ionization from the neutral molecule. It is known experimentally that more than one target electron can be involved.37,38 For example, “ionization-excitation” corresponds to removing one electron and exciting a second electron. Unfortunately, there are not yet any inexpensive theories for predicting the corresponding cross sections. Hence, we are evaluating the simple BEB model here. A delta function is used in eq. (2) for each orbital, for simplicity. Other choices would be physically superior. For example, Franck-Condon factors may be used. However, as described below, limited testing shows no benefit from the additional complexity and effort. For comparison with library mass spectra, the incident energy of interest is usually T = 70 eV. Published TPEPICO data do not extend to such high energy. Fortunately, orbital cross sections generally decrease with increasing binding energy, so data truncated at lower energy may cause only small errors. TPEPICO data take the form of relative ion intensities (i.e., branching fractions) as a function of the energy deposited. (Because it is based upon threshold electrons, which have essentially no kinetic energy, the full photon energy is deposited in the target molecule.) Let fi be the branching fraction for the ith ion, as obtained from a TPEPICO experiment. Then the (non-negative) branching fractions may be expressed together as a vector (l1-normalized), f = { f i } , which is a function of

The energy deposition function being tested is that implied by the Binary-Encounter Bethe (BEB) theory, as developed by the late Yong-Ki Kim and coworkers for predicting total ionization cross sections.30-34 The BEB molecular ionization cross section is formulated as the sum of ionization cross sections for individual molecular orbitals. The ionization cross section, σ(T), for one orbital is given by eq. (1), where T is the incident electron energy, a0 is the Bohr radius, N is the number of electrons in the orbital, R is the Rydberg constant, B is the orbital binding energy (vertical ionization energy), U (not explicit) is the orbital kinetic energy, t = T B is the reduced incident

the internal energy. The simulated mass spectrum, s, is a similar vector and may be written formally as shown in eq. (3). T

s ( T ) = ∫ f ( E ) P ( E ; T ) dE 0

(3)

Because of the simple form chosen for the energy deposition function, eq. (2), the computation reduces to the weighted average shown in eq. (4). TPEPICO measurements are reported at discrete energies, so linear interpolation is used at the intermediate energies needed to evaluate f ( Bi ) . For energies

energy, and u = U B is the reduced orbital kinetic energy. The parameter n = 1 for all the molecules considered here, but may take other values when the target is a cation35 or contains heavy atoms.32,33,36 Values of B and U are obtained from ordinary ab initio electronic-structure calculations. Note that σ = 0 if T < B (i.e., if t < 1).

4π a02 N R 2  ln t  1 1 −  t + (u + 1) n B 2  2  t 2

i

i

METHODS

σ (T ) =

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above the reported TPEPICO range, the value of f at the highest reported energy was used. This amounts to assuming that the branching fractions stop changing at the highest reported energy.

1 ln t    +1− −  (1) t (t + 1)  

2


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∑ f ( B ) σ (T ) s (T ) = ∑ σ (T ) i

Simulated spectra are compared with reference spectra both visually and numerically. Two different measures of distance are used for comparing mass spectra. The Jensen-Shannon divergence metric,53 here given the symbol D, was designed for discrete probability distributions and is therefore mathematically suitable for l1-normalized, non-negative vectors. This distance between vectors p and q is defined in eq. (5), where the base of the logarithm is chosen to be 4 so that D ∈ [ 0,1] . The sum is over the combined support, that is,

i

i

(4)

i

i

The values of B (orbital binding energy) and U (orbital kinetic energy) were computed using standard electronic structure methods. Molecular geometries were obtained by energy minimization using the hybrid B3LYP density functional39-41 along with 6-31G(d) basis sets. Values of U are from HartreeFock (HF) calculations using 6-311G(d,p) basis sets. More attention was given to the values of B: HF/6-311G(d,p) values for core orbitals, third-order P3/6-311+G(d,p) values42 (or the closest successful approximation) for valence orbitals, and CCSD(T)/cc-pVTZ values43,44 for the highest-lying orbital (i.e., the molecular vertical ionization threshold). Core orbitals were frozen in the post-HF calculations. These computations were done using the Gaussian 09 quantum chemistry software.45,46 Note that BEB cross sections can be calculated from Hartree-Fock results alone, if deemed acceptable. The importance of vibrational structure, in the form of harmonic Franck-Condon factors, was tested by using the program ezSpectrum.47 Equilibrium geometries, harmonic vibrational frequencies, and vibrational eigenvectors were obtained from single-hole self-consistent-field calculations with 631G(d) basis sets, as performed using the GAMESS quantum chemistry software.48 The calculations were CASSCF(29,15) for THF (cyclo-C4H8O, commonly called tetrahydrofuran) and CASSCF(7,4) for methane (CH4). The same level of theory was used to determine the geometric relaxation energy (the difference between the vertical and adiabatic ionization energies) for each final state. The corresponding level of theory for the neutral molecule is Hartree-Fock (HF). Correlated values of the vertical ionization energies were obtained from equation-of-motion calculations with triple-zeta basis sets augmented with diffuse functions on heavy atoms, that is, frozencore EOM-IP-CCSD/aug’-cc-pVTZ, as performed using the CFOUR quantum chemistry software.49 To test the effects of multi-electron, “shake-up” ionization for orbitals with B > 18 eV, Nakatsuji’s SAC-CI general-R method50,51 (symmetry-adapted cluster configuration interaction including up to sextuple excitations) was used as implemented in Gaussian 09. These calculations were done at the neutral geometry (frozen-core CCSD/cc-pVTZ) with core orbitals uncorrelated. The cc-pVTZ basis sets44 were used for the SAC-CI calculations for methane, but only the cc-pVDZ basis sets were feasible for THF because of computer limitations. Molecular orbitals were mapped to the SAC-CI ionic states based upon state symmetry and energy proximity. For the inner-valence orbitals only, the earlier values of B were replaced by the dominant corresponding roots of the SAC-CI. The BEB cross sections were re-computed using these binding energies. Orbital cross sections were allocated among the matching SACI-CI states in proportion to the corresponding monopole intensities. 32 ion states (as counted using nondegenerate representations) were converged for methane and 24 states for THF. Reference EIMS data, assumed to correspond to T = 70 eV, were taken from the National Institute of Standards and Technology (NIST) Chemistry WebBook.52 Additional EIMS data were found elsewhere, as cited.

supp(p) ∪ supp(q). (The support of a distribution is the subset of its domain for which it takes nonzero values.)

 2 pi 2qi  D 2 ( p , q ) = ∑  pi log 4 + qi log 4  p + q p i  i i i + qi 

(5)

The L1 distance, also called “Manhattan” or “taxicab” distance, is defined in eq. (6) and here given the symbol L. The factor of ½ is applied so that L ∈ [ 0,1] .

L ( p, q ) =

1 ∑ pi − qi 2 i

(6)

Usually, the set of product ions reported from a TPEPICO experiment is smaller than the set of ions in the corresponding EIMS spectrum. When making numerical comparisons of mass spectra using eq. (5) or eq. (6), only common ions (i.e., the intersection of the two sets) are considered, and only after renormalizing. This is done because the goal is to assess the quality of the theoretical energy deposition function, not the completeness of the mass spectra from the TPEPICO experiments. The symbols D* and L* will be used here, instead of D and L, when the corresponding sums are over the intersection of the supports instead of the union. Note that D*(p, q) ≤ D(p, q) and L*(p, q) ≤ L(p, q).

RESULTS AND DISCUSSION BEB energy deposition function. The most direct comparisons with experimental measurements are for tetrahydrofuran (THF) and for methane (CH4). In their electron-scattering experiments, Ren et al. determined the energies of the three outgoing particles (2e + ion) in coincidence, as collected over essentially all solid angles, and obtained the energy deposition function with a resolution of 1.5 eV for THF54 and 2.0 eV for methane.55 Their results for THF (incident electron energy T = 26 eV) are shown as “experiment” in Fig. 1, as taken from their Fig. 3b (“Total BE”).54 The prediction from BEB theory, eq. (2), is displayed in Fig. 1 as vertical lines representing the delta functions. The dashed curve is the BEB prediction after smoothing with a Gaussian function, exp[-(E/ρ)2], with ρ = 1.5 eV to match the energy resolution of the experiment. The smoothed BEB prediction appears to agree with the experimental measurements. Based upon the BEB results, the mean energy deposited (Fig. 1) is 12.86 eV. As stated earlier, internal energies are expressed relative to the neutral molecule. Assuming complete relaxation to the electronic ground state of the ion, and taking the adiabatic ionization energy as 9.38 eV,56 the corresponding mean vibrational energy deposited is 3.5 eV.

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Figure 2. Energy deposition in electron ionization of methane. Vertical scale is truncated to show small features. The zero of energy is the neutral ground state.

Figure 1. Energy deposition in electron ionization of tetrahydrofuran. The zero of energy is the neutral ground state.

The quality of the BEB prediction can be assessed numerically. As computed over the values of E reported in the experiment (black points in Fig. 1), the distance, as measured using eq. (5), between the smoothed BEB prediction and the experiment is D = 0.066. To understand the meaning of this value, analogous distances were computed for random energy distributions. 105 random distributions were generated by replacing the BEB cross sections (but not the binding energies) in eq. (2) with random numbers taken from a uniform distribution, normalizing, and smoothing. The mean (± standard deviation) distance (D) between the random distributions and the experiment is 0.272 ± 0.064. The BEB distance from the experimental data is smaller (i.e., better) than 99.98% of the random distributions. The corresponding comparison for methane is shown in Fig. 2 (incident electron energy T = 66 eV).55 (The vertical scale of Fig. 2 is truncated to show detail better; the BEB relative cross section of the first orbital is nearly 0.9.) The smoothed BEB prediction includes a Gaussian function, exp[-(E/ρ)2], with ρ = 2.0 eV to match the energy resolution of the experiment. The distance between the smoothed BEB result and the experimental energy deposition function is D = 0.197. Repeating the statistical exercise described above, this value is smaller (i.e., better) than 95.4% of random distributions on the same support. Based upon the BEB results, the mean energy deposited (Fig. 2) is 15.28 eV. As stated earlier, internal energies are expressed relative to the neutral molecule. Assuming complete relaxation to the electronic ground state of the ion, and taking the adiabatic ionization energy as 12.61 eV,56 the corresponding mean vibrational energy deposited is 2.7 eV. The good match between the BEB and experimental energy deposition functions, for both THF and methane, indicates that BEB orbital cross sections are physically meaningful.

Franck-Condon factors. As mentioned in the Methods section using delta functions in eq. (2) ignores vibrational structure, which is ubiquitous in ionization spectroscopy of molecules.57 To test the effect of including such structure, harmonic Franck-Condon factors (FCFs) were computed for the first three valence-hole states of THF radical cation, starting from the vibrational ground state of neutral THF. (Higher electronic states were not computed because of technical difficulty converging the calculations.) Including FCFs spreads an orbital cross section among a multitude of vibrational transitions, sometimes moving the peak center significantly. This is shown in Fig. 3, which is the same as Fig. 1 except that FCFs are included for the first three ion states.

Figure 3. Energy deposition in THF with some FranckCondon factors included.

Agreement with the (e, 2e + ion) experiment for THF is visibly degraded by including the Franck-Condon factors, despite it being a more rigorous theoretical approach. In Fig. 3, the distance between the smoothed theory and the experiment has increased to D = 0.090. However, this value is still smaller than 99.8% of the random distributions described above (i.e., supported only at the energies {Bi}). It is unknown whether adding FCFs for the other ion states would improve the

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agreement with the experimental distribution, or degrade it further. For methane, FCFs were computed for the second, (2a1)-1 state of the ion. For the first state, (t2)-1, the orbital and vibrational degeneracies make the problem difficult. Fortunately, good calculations have been reported by Mondal and Varandas.58 Their results, as digitized from their Fig. 5, are used here. Fig. 4 shows the comparison with experiment. To the eye, the agreement may appear to be somewhat degraded by including the FCFs, as it is for THF. However, the numerical distance is essentially unchanged. With D = 0.195, it is closer to experiment than 95.8% of the random distributions described above. For both THF and methane, including vibrational FranckCondon factors has a noticeable effect upon the energy deposition function. However, it does not improve agreement with the experiments. This observation is qualified by noting that the comparison for THF included FCFs only for three ion states. It is surprising that including Franck-Condon factors, which is a more complete theoretical procedure, does not improve the comparison with experiments. The most likely, however speculative, explanation is that the effect of FCFs is small. The noticeable effect for THF would then be an artifact, blamed upon the lack of FCF data for most of the ion states. Multi-electron transitions. In Figs. 1 and 2, the BEB energy deposition function (EDF) matches the experimental distribution more closely below ~18 eV than it does above. These energy regions may be considered the outer-valence and innervalence orbitals, respectively. It is established that multielectron processes are most common for inner-valence orbitals.59 Thus, the poorer agreement for inner-valence orbitals might derive from the inability of BEB theory to address multi-electron (“shake-up”) transitions. Here, we estimate these effects by combining BEB orbital cross sections with monopole intensities from SAC-CI general-R calculations51 (see Methods section for details).

(compare Fig. 1). The main change is to shift the peaks near 20 eV slightly higher. The distance between theory and experiment is very similar, D = 0.069. However, this EDF is closer to the experiment than all 105 of the random distributions supported only at the binding energies.

Figure 5. Shake-up-corrected (above 18 eV) energy deposition in THF radical cation.

For methane, the shake-up-corrected energy deposition function is plotted in Fig. 6. The new peaks are barely visible near 31 eV (compare Fig. 2). However, the numerical agreement with experiment is improved to D = 0.173. This distance is smaller than all 105 of the corresponding random EDFs. For both THF and methane, including multi-electron effects for orbitals deeper than 18 eV, by means of SAC-CI general-R calculations, causes only small changes in the energy deposition function. In both cases, however, the agreement with experiment is better than possible for an EDF supported only at the orbital energies. Thus, shake-up corrections for the inner-valence orbitals appear to provide a small but consistent improvement.

Figure 4. Energy deposition in methane with Franck-Condon factors included. Figure 6. Shake-up-corrected (above 18 eV) energy deposition in methane radical cation.

For THF, the EDF above 18 eV was corrected for shake-up. The result is plotted in Fig. 5, along with the experimental EDF. The effect of including shake-up structure is minor

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Indirect comparisons with mass spectra. Indirect comparisons, with ordinary mass spectra, can be made for molecules for which available TPEPICO data extend to reasonably high energy. Such data were found for the six molecules listed in Table 1. As mentioned in the Introduction, this comparison relies upon the extrinsic factors being similar in the TPEPICO and EIMS experiments. Table 1. Energy range of TPEPICO data. Molecule

Range / eV

Ref.

tetrahydrofuran

9.9 to 34

60

methyl t-butyl ether

9.1 to 36

61

methyl trimethylsilyl ether

9.4 to 32

61

acetone

10.2 to 31

62

t-butyl nitrile

10.85 to 40

63

8.5 to 34

64

t-butyl amine

Figure 7. Simulated and experimental mass spectra of THF at 70 eV.

In the comparisons above, the experimental EIMS is from the NIST database and is presumably a “consensus” spectrum65 based upon data from multiple laboratories using different equipment. However, typical mass spectra are more variable. To quantify the variability, at least for THF, 13 mass spectra were obtained from the NIST data archive. They are of lower quality than the published spectrum, with which they can be compared. Their distances to the published spectrum are in the ranges D* ∈ [0.043, 0.203] and L* ∈ [0.040, 0.186], with mean values = 0.13 and = 0.11. The BEBbased simulated spectrum is closer to the library spectrum than are 8 (D*) or 7 (L*) of the 13 lower-quality experimental spectra. That is, the simulated spectrum is about as good as a typical measurement. Besides the canonical energy, T = 70 eV, EIMS spectra for THF have been measured by Dampc et al. from 20 to 140 eV.66 Spectra were simulated at all these energies for comparison. The results are summarized in Table 2. Agreement with experimental mass spectra is generally better at lower electron energies. This may reflect that the breakdown data for THF reach only to 34 eV, but the differences are small. Table 2. Comparison of experimental and simulated mass spectra of THF at various electron energies, T.

60

For THF, the TPEPICO measurements by Mayer et al. (for 19 ions) were combined with BEB orbital cross sections computed for T = 70 eV, by using eq. (4). The resulting, simulated mass spectrum is shown as the red circles in Fig. 7. Experimental EIMS data (for 31 ions) from the NIST database52 are also shown, for comparison. Although there are obvious differences, the agreement appears reasonable. It is unknown how much the simulated spectrum would change if TPEPICO data were available up to 70 eV. A numerical assessment of the agreement in Fig. 7 can be obtained by using eqs. (5) and (6), with consideration restricted to ions that were observed in both the TPEPICO and EIMS experiments. The resulting distances between the simulated and experimental mass spectra are D* = 0.118 and L* = 0.105. To understand the meaning of these values, analogous distances were computed for random values of orbital ionization cross sections. 105 random, normalized distributions were generated by replacing the BEB cross sections in eq. (4) with random numbers taken from a uniform distribution and normalized. The distances obtained using the BEB values are smaller (i.e., better) than 99.92% (D*) or 99.88% (L*) of the spectra derived from random distributions, again supporting the hypothesis that the BEB orbital cross sections are physically meaningful.

T/eV

Expt. ref.

# Ions

20

66

D*

L*

30

66

9

0.096

0.078

9

0.087

0.085

40

66

9

0.088

0.090

50

66

9

0.091

0.094

60

66

9

0.096

0.099

70

66

9

0.103

0.104

70

52

19

0.118

0.105

80

66

9

0.107

0.108

90

66

9

0.110

0.109

100

66

9

0.112

0.112

110

66

9

0.113

0.112

6


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120

66

9

0.114

0.115

130

66

9

0.111

0.114

140

66

9

0.113

0.116

For methyl tert-butyl ether (MTBE), TPEPICO breakdown data (for 21 ions) are available up to 36 eV.61 Combining them with orbital ionization cross sections from BEB calculations, computed for T = 70 eV, leads to the simulated mass spectrum shown by the red dots in Fig. 8. The distances between the simulated and experimental52 (37 ions) spectra are D* = 0.138 and L* = 0.108. 105 random energy deposition functions were generated as for THF, above. The BEBderived spectrum is closer to the experimental spectrum than are 98.9% (D*) and 98.7% (L*) of the spectra derived from random energy deposition functions. Figure 9. Simulated and experimental mass spectra of methyl trimethylsilyl ether at 70 eV.

For acetone, breakdown data (for 18 ions) are available up to 31 eV.62 The simulated mass spectrum derived from BEB orbital cross sections is shown in Fig. 10 as the red circles. An experimental52 mass spectrum (22 ions) is also displayed. The distances between the two spectra are D* = 0.231 and L* = 0.154. These distances are smaller than 83% (D*) and 94% (L*) of the spectra derived from random orbital cross sections.

Figure 8. Simulated and experimental mass spectra of MTBE at 70 eV.

For methyl trimethylsilyl ether (TMSOMe), breakdown data are available (for 12 ions) up to 32 eV.61 Combining them with BEB orbital cross sections at T = 70 eV results in the simulated mass spectrum shown by the red dots in Fig. 9. A corresponding experimental spectrum (with 39 ions) is also shown,52 as the vertical lines. The distances between the simulated and experimental spectra are D* = 0.151 and L* = 0.108. Repeating the randomized comparison as before, BEB-based simulation is closer than 79% (D*) and 63% (L*) of the randomly derived simulations.

Figure 10. Simulated and experimental mass spectra of acetone at 70 eV.

For tert-butylnitrile, there are breakdown data (for 21 ions) up to 40 eV.63 Combining them with BEB cross sections yields the simulated mass spectrum shown in Fig. 11 as red dots. An experimental52 spectrum (46 ions) is shown by the vertical lines. The distances between the two spectra are D* = 0.165 and L* = 0.132. These distances are smaller than 89% (D*) and 81% (L*) of the spectra derived from random orbital cross sections. For tert-butylamine, TPEPICO breakdown data (for 15 ions) are available up to 34 eV.64 Combining these data with BEB orbital cross sections yields the simulated mass spectrum shown in Fig. 12 as the red dots. The experimental mass spectrum52 (43 ions) is also displayed in Fig. 12, in the conventional manner. The distances between the simulated and experimental spectra are D* = 0.218 and L* = 0.166. These distanc-

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es are smaller than 9% (D*) and 42% (L*) of the spectra derived from random orbital cross sections. For this molecule, the BEB orbital cross sections appear worse than a random guess. To illustrate the sensitivity of the simulated mass spectrum upon the EDF, Fig. 12 also includes a “violin plot” of the distribution of peak intensities derived from 105 random EDFs that are supported at the orbital energies. This is the set of flame-like features in the figure. For each m/z value, the width of the “flame” at a given height represents the fraction of the randomly-derived spectra that yielded the corresponding branching fraction for that ion.

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D* [eq. (5) with restricted sum] or L* [eq (6) with restricted sum]. Table 3. Performance of BEB energy distributions when simulating mass spectra (T = 70 eV): percentiles compared with random distributions. Molecule

D* %ile

L* %ile

tetrahydrofuran

99.9

99.9

methyl t-butyl ether

99

99

methyl trimethylsilyl ether

79

63

acetone

83

94

t-butyl nitrile

89

81

t-butyl amine

9

42

Energy dependence of organic mass spectra. In addition to quantitative predictions for specific molecules, the BEBderived energy deposition function can explain the widely known, if poorly documented, observation that EI mass spectra of organic compounds are sensitive to the electron energy below about 20 eV, but show relatively little change as the electron energy increases beyond 25 or 30 eV.20,67-70 The rate of change of the EDF, eq. (2), with respect to the incident electron energy (T), can be assigned a numerical value by using a metric such as D or L to measure changes. If X is the metric used to measure the distance between two normalized EDFs, then the rate of change of the EDF can be denoted dX/dT and defined by eq. (7). This derivative, as approximated by differencing, is plotted for THF in Fig. 13. Plots for other molecules are similar. The energy deposition function changes rapidly at low electron energy, but changes little above 25 eV or so. Thus, although the ion yield (i.e., total ionization cross section) increases substantially up to about 70 eV, the distribution of internal energies, and therefore the mass spectrum, changes little.

Figure 11. Simulated and experimental mass spectra of tertbutylnitrile at 70 eV.

X [ P( E; T + ∆T ), P( E; T )] dX = lim ∆ T → 0 dT ∆T

(7)

Figure 12. Simulated and experimental mass spectra of tertbutylamine at 70 eV. Shaded patches represent simulations derived from random EDFs.

The results for these six molecules are summarized in Table 3. Each percentile in Table 3 represents the fraction of randomly-derived energy deposition functions that yields a simulated mass spectrum that is worse than the spectrum obtained by using BEB orbital cross sections as the energy deposition function. “Worse” is defined as a larger distance from the appropriate NIST library spectrum,52 as measured using either

Figure 13. Rate of change of BEB energy deposition function in tetrahydrofuran, with respect to incident electron energy.

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The Journal of Physical Chemistry discussions. I thank the anonymous reviewers for helpful suggestions. I am especially grateful to the late Dr. Yong-Ki Kim (NIST) for instruction in the BEB theory over many years. This work was supported fully by the National Institute of Standards and Technology.

SUMMARY AND CONCLUSIONS To test the energy deposition function (EDF) derived from BEB theory, a direct experimental comparison is possible only for tetrahydrofuran (THF) and methane (CH4) at incident electron energies of T = 26 eV and 66 eV, respectively. After smoothing the theoretical EDFs to match the resolution of the experiments,54,55 agreement is good (Figs. 1 and 2). To judge the utility of the BEB calculations, random EDFs were generated, also supported at the orbital binding energies. The BEB result is better than nearly all the random EDFs, which indicates that the BEB orbital cross sections are physically meaningful. This has not been demonstrated previously and is the principal conclusion from this study. Adding harmonic Franck-Condon factors (FCFs) for the first three ionic states of THF degrades the agreement somewhat (Fig. 3). FCF calculations for the remaining ion states were unsuccessful, so it is unknown how they would affect the comparison with the experiment. For methane, adding FCFs changes the EDF visibly (Fig. 4), but has little effect upon the numerical distance between theory and experiment. Thus, the effort to compute FCFs does not appear worthwhile. Including estimates for multi-electron (“shake-up”) effects in the inner valence slightly improves the results for both molecules (Figs. 5 and 6). However, the effects are small in both cases. By combining the BEB EDF with experimental measurements of the fragment-ion distribution as a function of internal energy, mass spectra can be simulated for comparison with ordinary, analytical mass spectra. Once again, the quality of the BEB result may be judged by comparing it against results from randomly-generated EDFs. Such comparisons are summarized in Table 3. The results are encouraging, but mixed. The BEB result looks excellent for THF and MTBE, but poor for t-butylamine. The other three molecules are intermediate. As described in the Introduction, there are difficulties making these comparisons. The main complication is that experimental conditions and instrumental response may be different in the TPEPICO and EIMS measurements. If these differences are large, then comparing simulated and measured mass spectra yields little information about the accuracy of the EDF. Another difficulty is that the TPEPICO data do not extend to the highest orbital energies. In the mass-spectral simulations described above, this problem is handled crudely. The branching fractions at the highest measured energy are taken to be the same at all higher energies. In other words, it is assumed that the branching fractions stop changing at the highest measured energy. The error introduced by this approximation is unknown, but it probably creates a bias in favor of heavier and against lighter fragment ions.

AUTHOR INFORMATION Email: [email protected]

REFERENCES (1) Rosenstock, H. M.; Wallenstein, M. B.; Wahrhaftig, A. L.; Eyring, H. Absolute Rate Theory for Isolated Systems and The Mass Spectra of Polyatomic Molecules. Proc. Natl. Acad. Sci. USA 1952, 38, 667. (2) Mayer, I.; Gömöry, A. Predicting primary mass spectrometric cleavages: a 'quasi-Koopmans' ab initio approach. Chem. Phys. Lett. 2001, 344, 553. (3) Popović, S.; Williams, S.; Vušković, L. Electron-impact dissociative ionization of ethylene. Phys. Rev. A 2006, 73, 022711. (4) Grimme, S. Towards First Principles Calculation of Electron Impact Mass Spectra of Molecules. Angew. Chem. Int. Ed. 2013, 52, 6306. (5) Danon, A.; Amirav, A.; Silberstein, J.; Salman, I.; Levine, R. D. Internal Energy Effects on Mass-Spectrometric Fragmentation. J. Phys. Chem. 1989, 93, 49. (6) Bauer, C. A.; Grimme, S. How to Compute Electron Ionization Mass Spectra from First Principles. J. Phys. Chem. A 2016, 120, 3755. (7) Rosenstock, H. M.; Krauss, M. Quasi-Equilibrium Theory of Mass Spectra. In Mass Spectrometry of Organic Ions; McLafferty, F. W., Ed.; Academic Press: New York, 1963, p 1. (8) Lünnemann, S.; Kuleff, A. I.; Cederbaum, L. S. Ultrafast electron dynamics following outer-valence ionization: The impact of low-lying relaxation satellite states. J. Chem. Phys. 2009, 130. (9) Wang, L. J.; Akimov, A.; Prezhdo, O. V. Recent Progress in Surface Hopping: 2011-2015. Journal of Physical Chemistry Letters 2016, 7, 2100. (10) Tian, C. C.; Vidal, C. R. Cross sections of the electron impact dissociative ionization of CO, CH4 and C2H2. J. Phys. B-At. Mol. Opt. Phys. 1998, 31, 895. (11) Singh, H.; Coburn, J. W.; Graves, D. B. Appearance potential mass spectrometry: Discrimination of dissociative ionization products. J. Vac. Sci. Technol. A-Vac. Surf. Films 2000, 18, 299. (12) Di Palma, T. M.; Apicella, B.; Armenante, M.; Velotta, R.; Wang, X.; Spinelli, N. Dissociative electron impact ionization of methyl tert-butyl ether: total ionization cross-section and kinetic energy distributions. Chem. Phys. Lett. 2004, 400, 191. (13) Feil, S.; Bacher, A.; Zangerl, M.; Schustereder, W.; Gluch, K.; Scheier, P. Discrimination effects for ions with initial kinetic energy produced by electron ionization of C2H2 in a Nier-type ion source. Int. J. Mass Spectrom. 2004, 233, 325. (14) Pandey, A.; Bapat, B. Effect of transmission losses on measured parameters in multi-ion coincidence momentum spectrometers. Int. J. Mass Spectrom. 2014, 361, 23. (15) Christophorou, L. G.; Olthoff, J. K. Fundamental Electron Interactions with Plasma Processing Gases; Kluwer Academic/Plenum: New York, 2004. (16) Berry, C. E. Temperature coefficients for the mass spectra of a number of hydrocarbons. J. Chem. Phys. 1949, 17, 1164. (17) Reese, R. M.; Dibeler, V. H.; Mohler, F. L. Temperature variation of mass spectra of hydrocarbons. J. Res. Natl. Bur. Stand. 1951, 46, 79. (18) Amirav, A.; Gordin, A.; Poliak, M.; Fialkov, A. B. Gas chromatography-mass spectrometry with supersonic molecular beams. J. Mass Spectrom. 2008, 43, 141.

Acknowledgments I am grateful to Prof. Paul M. Mayer (Univ. of Ottawa) for providing copious TPEPICO data in electronic form, and to Prof. Mariusz Zubek (Gdańsk Univ. of Technology) for providing energy-dependent mass spectra in tabular form. I thank Dr. Samer Gozem (Univ. of Southern California) for assistance in using the ezSpectrum software. I am grateful to Dr. Xueguang Ren (MPI für Kernphysik) for providing the total-binding-energy data for methane. I thank Dr. Yuri Mirokhin (NIST) for providing unpublished mass spectra, and Dr. Anzor Mikaia (NIST) for helpful

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Ab Initio Computation of Energy Deposition During Electron Ionization

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