Statistical Rice-Ramsperger-Kassel-Marcus Quasiequilibrium ...core.ac.uk/download/pdf/82671559.pdfKassel-Marcus (RKKM) theory in a parallel fashion. In the original QET paper, the

ACCOUNT AND PERSPECTIVE

Statistical Rice-Ramsperger-Kassel-Marcus Quasiequilibrium Theory Calculations in Mass Spectrometry

Tomas Baer and Paul M. Mayer* Chemistry Department, University of North Carolina, Chapel Hill, North Carolina, USA

The statistical theory [ R i c e - R a m s p e r g e r - K a s s e l - M a r c u s quas iequi l ibr ium theory (RRKM/QET)] for calculating dissociation rate constants is explained and its implementation is outlined with sample computer programs. The energy deposition involved in various types of ionization processes is discussed and related to the appearance of the mass spectrum. The RRKM/QET calculations are used to explain the kinetic shift and its effect on the observed onset for fragmentations in the halobenzene ions. Direct dissociation versus rearrangement reactions are discussed in terms of the dissociation rates and the observation of metastable ions. Finally, it is shown how an average rate constant can be obtained from metastable peak intensities as a function of the ion extraction voltage in a conventional mass spectrometer. © 1997 Amer ican Society for Mass Spec trometry (J A m Soc Mass Spectrom 1997, 8, 103 -115 )

T he stat ist ical theory [ R i c e - R a m s p e r g e r -

K a s s e l - M a r c u s q u a s i e q u i l i b r i u m theo ry (RRKM/QET)] traces its origins to the 1920s

when the mechanism for thermally activated unimolecular reactions was first established by Hinshel- wood [1]:

A B + M ~ A B * + M activation-deactivation steps A B * ~ A + B unimolecular dissociation

This mechanism, which distinguishes the activation and dissociation steps, accounts for the peculiar transition from second order to first order as the inert gas pressure (typically argon) is increased. Although the activation and deactivation steps are still difficult to model, the rate of the unimolecular step can now be determined with excellent precision.

Rice and Ramsperger [2, 3] and Kassel [4] first proposed that the rate of the unimolecular reaction is dependent on the vibrational modes of AB, and not on how the molecule is activated. They treated the molecule as a collection of s identical harmonic oscillators. One of these, designated as the critical oscillator (v), was associated with the reaction coordinate. For instance, in the dissociation of benzene to the phenyl radical plus H, one of the C - - H stretch modes would be designated as the critical oscillator. They then assumed that energy flows freely among all the normal

Address reprint requests to Dr. Tomas Baer, Chemistry Department, University of North Carolina, Chapel Hill, NC 27599-3290. E-mail: [email protected] "Present address: Research School of Chemistry, The Australian Na- tional University, Canberra, Australia, ACT 0200.

© 1997 American Society for Mass Spectrometry 1044-0305/97/$17.00 PII $1044-0305(96)00212-7

modes of the molecule. Any configuration in which the critical oscillator contains more energy than the bond energy is assigned to the transition state. All other configurations are assigned to the molecule.

The computed probability for the critical oscillator with an energy E > E 0, is given by:

( n - m + s - 1) !n! probability = ( n - m ) ! ( n + s - 1)! (1)

where n is the total number of vibrational quanta and m is the number of quanta in the critical coordinate. This is a quantum mechanical expression because the vibrations are treated as discrete energy levels. If we multiply the probability by a frequency of passing through the transition state, we convert eq I into a rate constant. The accurate quantum expression in eq 1 can be simplified if the number of quanta (n) is much greater than the number of oscillators (s). This classical limit, usually referred to as the classical Rice-Rams- perger-Kassel (RRK) rate constant is then given by

k ( E ) = v = v (2)

where we have used the expression E = n h v to convert the number of quanta into an energy. Although this expression shows very nicely that the rate constant goes up as the energy E increases and goes down as the number of oscillators s increases, it is not useful

Received July 3, 1996 Revised October 8, 1996

Accepted October 8, 1996

104 BAER A N D MAYER J A m So(: Mass Spectrom 1997, 8, 103-115

for obtaining rate constants. Depending on the energy range, the errors can easily exceed 4 orders of magnitude, which is hardly useful even for qualitative information.

In the late 1930s Wigner and Hirschfelder [5, 6] developed what is now called classical transition state theory in which the reaction was viewed as a reaction flux in phase space. The reaction rate constant is pro- portional to the flux of phase space passing through the transition state, which is just a "plane" in the multidimensional phase space. An alternative derivation treats the transition state as a "particle in a box" [7]. Finally, in the early 1950s, Rosenstock et al. [8] developed the quasiequilibrium theory (QET) while Marcus and Rice [9] derived the Rice-Ramsperger- Kassel-Marcus (RKKM) theory in a parallel fashion. In the original QET paper, the application was to mass spectrometry and so for many years, mass spec- trometrists and other ion chemists referred exclusively to the QET, whereas publications dealing with neutral systems referred to the statistical theory as the RRKM theory. Today, we recognize that there is only one statistical theory, albeit with a number of variations to account for differences in the nature of the transition state and the amount of rotational energy incorporated into the system. Among these are the phase space theory, which is useful for describing reactions with no exit channel barriers [10-13], the variational transition state theory [14-16], and the closely related transition state switching model [17-19].

In this article, we treat only the simplest version of the statistical theory and show how to apply it to typical problems in mass spectrometry. More sophisti- cated approaches that take into account rotations, an- harmonicity, tunneling, and so forth are described by Baer and Hase [20]. In most cases, we are not justified in taking these effects into account unless the energy, structure, and vibrational frequencies of the reacting ion and transition states are well known.

The RRKM / QET Equation

Rather than deriving the RRKM/QET equation, we simply present it and concentrate on a physical understanding of its origin. Its derivation can be followed in Baer and Hase [20], Holbrook et al. [21], Gilbert and Smith [22], Forst [7], Robinson and Holbrook [23], or Steinfeld et al. [24]. It is very similar also to the derivation of the standard transition state theory in terms of partition functions found in many physical chemistry texts. The latter formulations deal with canonical systems at a given temperature and thus the expression is in terms of the canonical partition functions. In mass spectrometry, we generally do not deal with systems at a given temperature and thus it is more useful to use the expression as a function of the ion internal energy. However, the basic physical principles are identical.

The RRKM/QET equation, which yields the rate constant for a molecule at a given energy E and with an activation energy of E 0' is given by

o-N~( E - E 0) k(E) = (3)

hp(E)

where cr is the reaction degeneracy, N~(E - E o) is the transition state sum of states from 0 to E - Eo, h is Planck's constant, and p(E) is the parent ion density of states at an energy E. To utilize this equation it is essential to have a good understanding of the concept of sum and density of states.

The Sum and Density of States

If we ignore the ion's rotational energy, the sum and density of states refer only to the vibrational degrees of freedom. Suppose that we have a molecule with s vibrational degrees of freedom and an internal energy E (as measured from the molecule's zero point energy). The sum of states represents all of the possible ways to arrange energy among the s oscillators such that the total energy is equal to or less than E. Thus one configuration might be (0, 0, 0, 0, 0, 0 . . . . . 0), which is the ground state (all oscillators in their lowest energy state). Another arrangement might be (0, 0, 2, 0, 0 . . . . . 0) in which all oscillators except the third are in their ground state while this third oscillator is excited to the second vibrational level. We simply enumerate all of the possible configurations and add them up, giving equal weight to all configurations. Unlike in the original RRK theory, each oscillator has a different vibrational frequency and thus a different energy content. If E = 0, then N(0) = 1 because there is only one state that corresponds to zero energy. However, as the energy increases, the number of ways to arrange this energy increases rapidly. Because N(E) is just the total number of states, it is a unitless number.

The density of states at an energy E can be thought of as the derivative of the sum of states. It represents the number of vibrational configurations with an energy content between E and E + ~E. The sum of states is related to the density of states as the volume of a sphere is related to its surface area. The units for the density are number of states per energy interval, that is, E-1. As with the sum, the density is a rapidly increasing function of the energy. Consider again the sphere. As the radius (or energy) increases, the sum increases by E 3, while the surface area increases by E 2. A molecule with numerous vibrational oscillators has a much higher dimensionality than our three-dimen- sional sphere so that the sum and density of real molecules increase much more rapidly than our sphere. However, the picture of the sphere is nevertheless a good model to keep in mind to get some physical insight.

J Am Soc Mass Spectrom 1997, 8, 103-115 STATISTICAL RRKM / QET CALCULATIONS IN MS 105

Qualitative Interpretation of the R R K M / QET Equation

The s u m of s tates in the R R K M / Q E T equa t ion represents the n u m b e r of w a y s to pass t h rough the transi- t ion state that has a total ene rgy of E - E 0, whi l e the dens i ty of states at an ene rgy E represen ts the n u m b e r of w a y s to get lost in the molecu la r ion phase space. W h y is the n u m e r a t o r a s u m and d e n o m i n a t o r a den- s i ty of states? At the t rans i t ion state, a po r t i on of the ion ' s total ene rgy has been channe led into the crit ical coord ina te (the C - - H b o n d for the benzene ion). H o w m u c h ene rgy is in this crit ical C - - H b o n d ? Well , w e need to have at least E 0 in this bond . If we have just enough ene rgy to pass t o w a r d p roduc t s , w e w o u l d have a total of E - E 0 left to d i s t r ibu te to the o ther 29 v ib ra t iona l m o d e s in the benzene ion. This represen ts one w a y to pass t h rough the t rans i t ion state. W e can represen t its con t r ibu t ion to the total rate cons tant b y the first t e rm in the fo l lowing express ion:

k(E) p*(E - E o) p*(E - E o - ~l) +

hp(E) hp(E)

p * ( E - Eo - 82) + + . . . hp(E)

(4)

The second and subsequen t te rms involve pass ing th rough the t rans i t ion state w i th ene rgy 8 i above the r equ i red m i n i m u m ene rgy of E 0. Because the reac t ion coord ina te at the t rans i t ion state consists of t ransla- t ional ene rgy of the d e p a r t i n g f ragments , w e can also th ink of this ene rgy 8 i as the t rans la t iona l ene rgy of the f ragments . Thus, the total ene rgy of the t rans i t ion state E - E 0 is pa r t i t i oned b e t w e e n the t rans la t ional ene rgy 8 and the r ema in ing in terna l ene rgy E - E 0 - 8. This is s h o w n schemat ica l ly in F igure 1.

If w e n o w a d d up all of the t e rms in the n u m e r a t o r s of eq 4, w e obta in a s u m of states. It is equ iva len t to

l

R •

Figure 1. Reaction coordinate for a dissociation with a real barrier. E and E 0 are the total ion energy and activation energy. e is the translational energy in the reaction coordinate. The remaining energy, E - E 0 - 8 , is the energy available to be statistically distributed.

s u m m i n g all of the surface areas of our sphere f rom the m a x i m u m area in which the energy is E - E 0 to the m i n i m u m w h e r e E - E 0 - 8 = 0. This s u m is thus the s u m of states f rom 0 to E - E 0.

It is ev iden t that the m i n i m u m rate of d issoc ia t ion is at threshold . At this energy , the rate cons tan t is

1 kmi n = O'hp ( ~ (5)

It is in teres t ing that this equa t ion is i n d e p e n d e n t of the t rans i t ion state s t ruc ture and v ibra t iona l frequencies.

Evaluation of the Sum and Density of States

Let us a s sume that the ion is wel l desc r ibed b y a set of n o r m a l ha rmon ic oscil lators. This m a y seem to contra- dict the no t ion of free ene rgy f low since one of the basic pos tu la t e s of q u a n t u m mechanics s tates that normal v ibra t iona l m o d e s are o r thogona l to each o ther and thus d o not interact. W e n o w k n o w ( f rom h igh reso lu t ion spec t roscopy) that the n o r m a l m o d e pic ture , especia l ly at e leva ted energies , is no t correct [20]. Nev - er theless , it can still be h o p e d that the s u m and dens i ty of s ta tes are wel l desc r ibed b y the use of ha rmon ic oscil lators. Fur the rmore , s ince the same er ror is m a d e in bo th n u m e r a t o r and d e n o m i n a t o r there is some cancel la t ion of error. H o w e v e r , note that at ve ry low energies (i.e., for kmi~), the er rors def in i te ly do no t cancel so that kmt~(harmonic) > kmin(anharmonic). The extent of the er ror is no t easy to de t e rmine because it d e p e n d s u p o n the molecu le and the energy. In small molecules , w h e r e there are few n o r m a l modes , the er ror wi l l be grea ter than in larger molecules . The reason for this is that the b o n d energy , or crit ical ene rgy E 0, is nea r ly the same for all molecules or ions (i.e., b e tw e e n I and 2 eV). Thus the total ene rgy re- qu i r ed for reac t ion is also abou t the same. Cons ide r an ion w i th 3 eV ( --- 24,000 c m - 1) of in ternal energy. For a smal l ion such as acety lene wi th jus t seven n o r m a l modes , the average energy per n o r m a l m o d e is ~ 3500 cm -1. O n the o ther hand , for a large ion such as b u t y l b e n z e n e wi th its 66 n o r m a l modes , the average ene rgy per n o r m a l m o d e is jus t 350 cm -1. Since all osci l la tors are ha rmon ic at suff iciently low energy , it is clear that the bu ty lbe nz e ne dens i ty of s tates ca lcula ted wi th the ha rmon ic a p p r o x i m a t i o n is more l ikely to be correct than that of acetylene. Final ly , to take into account anha rmon ic i t y in any meaningf l f l w a y re- qui res m u c h m o r e in fo rmat ion abou t a molecu la r ion than is genera l ly avai lab le for mos t ions of interest .

There are severa l w a y s to calculate sums and dens i - ties of states. The easiest w a y b y far is the d i rec t count m e t h o d d e v e l o p e d b y Beyer and Swineha r t [25]. A s imple BASIC p r o g r a m , l is ted in the A p p e n d i x , can calculate these functions. A l t h o u g h the r equ i red pro- g r a m m i n g is ve ry s imple , it does have severa l d r a w - backs. First , it is the mos t t ime c o n s u m i n g app roach , wh ich becomes a p r o b l e m w h e n dea l ing w i th la rge

106 BAER A N D MAYER J A m Soc Mass S p e c t r o m 1997, 8, 103-115

ions. Second, in most calculations one is interested in the density or sum of states at a few energies that are well above the min imum energy for dissociation. However, in the exact count method one is obliged to calculate these functions from 0 up to the max imum energy of interest. Third, each frequency is treated separately so that frequencies cannot be bunched to save time in calculations. For these reasons, it is generally worthwhile to invest the additional programing time required for the following two approximate methods.

The Whitten-Rabinovitch [26, 27] approach is based on patching up the sum and density of states from classical mechanics by incorporating a scaled zero point energy into the classical equations. Above about 0.5 eV for a molecule such as cyclopropane, the results are very close to the exact count results. However, at lower energies it becomes inaccurate, especially for large molecules. The reason for this is that the number of vibrational quanta excited at, for example, 0.5 eV, for a large molecule is very small. Thus large ions with their many oscillators tend to be much closer to the quantum limit than a small molecule such as cyclopropane. Even a relatively small ion such as pentane has a total zero point energy (ZPE) of 4.2 eV. Whereas the Whitten-Rabinovitch method is best at energies E > 0.1 (ZPE), it becomes progressively less useful as the size of the molecule increases.

A third method is based on the inversion of the partition function (steepest descent) [7]. The density of states can be obtained by solving the inverse Laplace transform. Although the mathematics of this integration in the complex plane is challenging, it can be p rogrammed in about 10 lines of code. It is quite fast and accurate. Furthermore, it is easy to combine similar frequencies, which results in considerable computa- tion time savings. Both of these approximate methods along with the exact count method are described in more detail in the Appendix and the results are compared for pentane in Figure 2.

. - . 10 s '7,

lo` 6m~ w lO' =

-~ 10 t

• 10 0 "O

.~ 10 "t

lO-Z o.o

.~̀~*.~..~..~t..~..~.~'.~.-.̀'.~̀~̀~.~.~.~.''~.~'.~.~.~'''~.~̀ ........

1 I / /

J J

011 0~2 013 014 015 pentane internal energy (eV)

Figure 2. A comparison of three methods for calculating the density of states for pentane: steepest descent, Beyer-Swinehart (direct count), and Whitten-Rabinovitch (modified classical).

Entropy of Activation and the Transition State

The vibrational frequencies of the reacting ion and the transition state give the entropy of activation d~S ¢. The AS ¢ is a convenient way to describe the nature of the reaction. Transition states that are less ordered ("loose") than the reactant ion are characterized by positive AS¢ values. Simple bond cleavage reactions typically have loose transition states. Transition states that are more ordered than the reactant ion have negative values for AS ¢. These " t ight" transition states are usually associated with rearrangement processes, so, if the vibrational frequencies of the reactant ion and the transition state are known, the nature of the process can be surmised. Conversely, if the nature of the process is known, the transition state frequencies can be estimated so as to obtain AS t values of the appropriate sign. The entropy of activation is given by the expression

Q* U * - U . FIq/~ U * - U as*= Btn-ff +---- T - kB n-h- q +---- T -

(6)

where Q is the total partition function qlq2q3 . . . . and

1 (7)

qi = 1 - - e x p ( - h ~ i / k B T )

and U is the average internal energy. The average internal energies U and U* can be calculated from the usual formulas by using the vibrational partition functions. Typically, AS * values are reported at specific temperatures, either 600 or 1000 K.

The effects of both E 0 and AS ~ on the rate constant as a function of energy are complimentary. The E 0 is largely responsible for the magnitude of k(E) (Figure 3), whereas AS * is largely responsible for the slope (Figure 4). Tight transition states (AS * < 0) have k(E) curves that increase gradually with increasing energy,

10 to 10 9 .AS + (6001C) = +2 e.u.

l°S 1

1°' 2 ~ ~ V 106 lO~ 3 )

~'q 103104 4 / J 1 1 : :

J 3 2.2 I o2 4 2.4 101 5 5 2.6 10 0

2.6 f.8 3'.0 3'.2 3'.4 3'.e 3'.8 E (eV)

Figure 3. The effect of E 0 on the k(E) vs. E curve for the loss of "CH 3 from 2-butene ions.


lo ' 1 lO" ~ 2 3 4

~,--. 107

-~ 10 e ~ 1 +13e.u. 2.SeV

~ 2 +4 2.2

10' / : -'41 I : : I

104 2.6 2'.6 3~0 3'.2 3'.4 3'.6 318 E (e~

Figure 4. The effect of AS ~; on the k(E) vs. E curve for the loss of "CH 3 f rom 2-butene ions. The activation ene rgy ha s also been ad jus ted accordingly so that all cu rves pass t h r o u g h a c o m m o n point.

whereas loose transition states (AS s > 0) have k(E) curves that increase more rapidly. If neither E 0 nor the transition state structure are known, both of these parameters can be adjusted to obtain an appropriate fit to experimental rate constant versus energy data. The assignment of AS* affects the value of E 0, which needs to be used to obtain a satisfactory fit to the experimental data (Figure 4). Ideally, the transition state of a reaction can be calculated by ab initio calculations and vibrational frequencies therein obtained. This will fix AS*, leaving only E 0 to be adjusted to fit the experimental data, because at low levels of theory the calculated vibrational frequencies are more reliable than the relative energies. The transition state frequencies may themselves be estimated if necessary. The usual technique is to use the reacting ion vibrational frequencies less one to represent the reaction coordinate. The transition state can be made loose or tight by scaling the lowest five or six vibrational frequencies (which contribute the most to the calculated sum of states) by a common factor (less than 1.0 will produce AS* > 0, whereas greater than 1.0 will produce AS* < 0).

Energy Deposition and Ion Internal Energy in Mass Spectrometry

Photoionization

Photoionization with photons of selected energies can be a precise method for generating ions with known internal energies, but because it is a bound to contin- uum process the ions are produced in a distribution of internal energies. The ion energy is given by Eio n = h v - Etherm - IE - Eel, where IE is the ionization energy, Ether m is the molecule's average thermal energy prior to ionization, and E~t is the kinetic energy taken away by the departing electron. In the absence of autoionization, the deposition function for photoionization is just the photoelectron spectrum up to the photon energy.

Photoionization differs from electron impact ionization primarily because it is possible to generate a large number of ions with low energy photons. The threshold law for photoionization is a step function at the ionization threshold. Thus, the number of ions created in the ground state is independent of the photon energy so long as h v > IE. For this reason it is possible to ionize selectively one molecule in the presence of another simply by selecting photons with an energy in between the ionization energies of the two molecules. This is difficult to accomplish with electron impact because the cross section for electron ionization van- ishes as the electron energy approaches the ionization threshold.

Autoionization, in which the molecule first is excited to a neutral state above the ionization energy and then ionizes [ A + h v ~ A* ~ A ++ e-] , causes the photoionization deposition of energy to be different from that given by the photoelectron spectrum (PES). To what extent it differs depends upon the molecule. The only way to determine this is to measure the PES with a photon at the energy of interest. Few molecules aside from oxygen [28] have been investigated in this manner. Thus very little is known about the role of autoionization in large molecules.

Soft Ionization Techniques

The most common techniques that produce ions with relatively small internal energy distributions are chemical ionization and field ionization (FI).

Chemical ionization. Chemical ionization mass spectrometry was first introduced by Munson and Field [29] in 1966 and has been the subject of a series of handbooks authored by Harrison [30]. The purpose of doing chemical ionization, rather than electron ionization, is to have control over the internal energy of the created ions by controlling the enthalpy of the chemical ionization reaction. The two most widely used chemical ionization processes are charge transfer and proton transfer.

CHARGE TRANSFER. A charge transfer process is one in which an electron is transferred from a neutral molecule to an ion. At thermal ion source energies, the reaction will proceed if it is exothermic, that is, the IE of the molecule is less than the recombination energy (RE) of the reagent ion:

R+'+ M -o R + M ÷" A H - - IE(M) - R E ( R +" ) < 0

In principle, the resulting M +" ion internal energy Eio n can be obtained from the expression

Eio n = RE(R +') + Ethe~m(M) - IE(M) (8)

108 BAER AND MAYER J Am Soc Mass Spectrom 1997, 8, 103-115

where Ether m is the thermal internal energy of the molecule at the ion source temperature. However, there are two things that interfere with the application of eq 8. The first is that the recombination energy of the reagent ion RE(R +" ) is equal to the negative of the ionization energy of the reagent molecule - IE(R) only for atomic reagents. If the charge transfer reagent is diatomic or polyatomic, the recombination energy can be less than the IE because reagent molecules can be produced rotationaUy and excited vibrationally [30]. This will result in M +" ions with a lower internal energy than predicted by eq 8. A second factor that adds uncertainty to the M +" internal energy is collisional relaxation. The high ion source pressures required to efficiently produce M +" ions by charge transfer ultimately lead to numerous collisions between M +" and neutral R and M. If collisional relaxation occurs, the M +" internal energy again will be less than that predicted by eq 8. Lin and Dunbar [31] have shown from a study of dissociation rates that the molecular ions of iodotoluene produced by charge transfer with CS~-" must have internal energies about 0.4 eV lower than that predicted by eq 8, so care must be taken when applying eq 8 to determine molecular ion internal energies.

PROTON TRANSFER. Molecules can be ionized in the ion source of a mass spectrometer by the exothermic transfer of a proton from a reagent ion. The most common, and simplest, reagent used is CH~, which is formed by the reaction of CH~" with C H 4 :

CH~-" + CH 5 ~ CH~- + CH 3

M + CH~ ~ MH++ C H 4

The internal energy of MH + formed in this way will be determined by adding the initial internal thermal energy of M to the exothermicity of the proton transfer step. The greater the exothermicity, the higher the MH + internal energy. Again, care must be taken when applying this relationship due to the collisional relaxation that can occur at the high pressures present in the ion source required to efficiently produce MH +.

Field ionization. Field ionization [32] produces ionized molecules by exposing then to high electric fields, typically as high as 108 V / c m ( ~ 1 V/A). The mass spectra of field ionized molecules are typically domi- nated by molecular ions M +, indicating that most of the ions are formed below their fragmentation threshold. Unfortunately, there is no quantitative way to determine the internal energy distribution of the ions other than from an analysis of their fragmentation rates [32], which is the information we wish to determine and analyze, not which we have at hand. Never- theless, FI affords a method for elucidating mecha- nisms for reactions proceeding at very short (picosec- ond) time scales [33-35].

Electron Impact Ionization

The standard energy used in electron ionization (AB + e - ~ AB++ 2 e - ) is 70 eV. This has been chosen because the ion signal maximizes at about this electron energy. Because the ionization energies of typical polyatomic molecules are below 10 eV, the ions can be produced with internal energies ranging from 0 to 60 eV. That is, ions can be produced in a wide range of vibrational and electronic energy levels. According to the Wannier threshold law [36], the cross section for electron ionization (EI) varies linearly with the electron kinetic energy going to zero as the electron energy approaches the molecule's ionization energy. The probability P for producing an ion with internal energy Ein t

with an electron of energy KE(e-), is given by P --- c [ K E ( e - ) - (IE + E~,t)]. This means that low energy ions are preferentially produced by electron ionization. If we ignore autoionization, we can use the photoelectron spectrum (PES) of the molecule to determine the energy deposition in EI. An example is shown in Figure 5 for 1-propanol. Because the electron energy of 70 eV is so much greater than the ground state ion energy, the energy deposition is nearly identical to the PES.

Autoionization (A + e - ~ A* + e-, followed by A* A++ e- ) also competes with direct ionization in

electron ionization. Because ionization with 70-eV electrons can be thought of as ionization with "whi te" light up to h p - - 7 0 eV, autoionization is probably much more important in EI than in PI. However, even less is known about this process in electron ionization.

(M÷.) * ., /

M "÷i I~ ~ o / ~ 'ntem" "e-'r~+ d'~:"~:(nv E,

~ m l ~ _ W a n n l e r threshold law

0 1 2 3 4 5 6 ? 8 9 10 11

l -propanol ÷" internal energy (eV)

Figure 5. The internal energy distribution of 1-propanol ions formed by 70-eV electron impact ionization. The lower spectrum is the 1-propanol photoelectron spectrum that was multiplied by the Wannier threshold law (middle line) to obtain the final 1-propanol +" internal energy distribution (upper spectrum). The internal energy ranges of the three types of ions formed in the ion source are shown: M +" denotes intact molecular ions surviv- ing to the detector; (M+') * denotes metastable molecular ions; F + denotes fragment ions formed in the ion source.


Mass Spectrometric Methods for State Selection and Obtaining k(E) Data

To state select ions with photoionization, it is necessary to carry out a coincidence experiment. By collect- ing only ions in coincidence with initially zero energy electrons, the ion internal energy can be selected. This photoelectron-photoion coincidence (PEPICO) technique has been one of the major methods for measuring the dissociation rates of ions as well as breakdown diagrams as a function of ion internal energy [37, 38]. The other major method for generating ions in known internal energies is by photodissociation of ground state ions. Ground state ions can be produced either by laser photoionization, charge transfer ionization, or electron impact followed by thermal cooling in an ICR cell or some other ion trap.

The foregoing methods provide a means for measuring the dissociation rates of ions with selected energies. Numerous studies have been carried out by a variety of groups by using photoelectron-photoion coincidence (PEPICO) [37-41] photodissociation of ions in a time-of-flight mass spectrometer [42-44], an ion cyclotron resonance cavity [45-47], ion trap [48-50], or by photodissociation of energy-selected ions in a sector mass spectrometer [51, 52]. Such energy-selected decay rate constants are readily compared with calculated RRKM/QET rate constants by using eq 3.

The problem of extracting interesting dynamical or thermochemical information becomes much more difficult when the ions are formed and react in a poorly known distribution of internal energies, as in a normal electron impact ion source of a mass spectrometer. The distribution is certainly not given by a Boltzmann distribution. If it were, it would be possible to analyze the results by using the thermal transition state theory in terms of canonical partition functions.

portant parameter, which contains information about the rate constant, is the intensity of the metastable peak. As shown by Butler et al. [53], the average dissociation rate can be extracted from metastable ion intensities if the experiment is run at several different drawout voltages and if the metastable fragment peak intensity is compared with that of the parent ion (see Obtaining the Fragmentation Rate Constant by Vary- ing Vacc). Quantitative applications are given in the next section.

Applications

The Kinetic Shift in the Appearance Energies for Chlorine Loss from Chlorobenzene

Figure 6 shows the photoelectron and the C6H ~- fragment photoionization spectra of chlorobenzene [37]. Because the appearance energy or onset for the C6H ~- fragment appears abruptly at about 13 eV, it is tempt- ing to assign this onset to the dissociation limit for the chlorobenzene ion. However, it is now well under- stood that appearance energies (AE) are rather mean- ingless unless a careful rate analysis is carried out. The culprit is the kinetic shift [54, 55], which shifts the observed onset to an energy sufficiently high to permit the ion to dissociate before reaching the mass analyzer. How can we determine the magnitude of this shift?

The first step is to determine the sensitivity of the data collection system. Suppose that this is found to be 5%, which means that a signal is observable only if at least 5% of the ions dissociate before reaching the mass spectrometer. Second, we must determine the time required for the ion to move from the ionization region to the analyzer region of the mass spectrometer.

The Internal Energy Distribution of Metastable Ions

The observation of metastable ions in a sector instrument is a sign that an ion has a sufficiently large activation energy to cause the dissociation rates to be in the range from 104 to 106 s -1. Furthermore, metastable ions can be formed by a variety of ionization methods. How they are produced is not important. They will always have the same internal energy content, which is determined by the relationship between the ion's internal energy and its decay rate k(E). It is important to understand that just because the metastable ions dissociate in the field-free region between 10 and 20/~s after ion formation does not mean that the parent ion is dissociating with a rate constant of 5 x 104-10 s s -1. The dissociation rate could be much faster than this. If the rate were 5 x 10 s s -1, only 0.67% of the ions would dissociate in the drift region. However, this is easily in the range of the sensitivity of most mass spectrometers. Thus, an ira-

- I I" T r I ci @ /

r~

9 13 15 17 Ion Energy (eV)

Figure 6. The photoelectron spectrum of ch]orobenzene (solid line) and the threshold for the appearance of C6H ~ fragment ions. The difference between the apparent threshold and the actual E 0 (arrow) is due to kinetic shift. Figure reproduced with permission from Baer [37].


Suppose that this acceleration time is t a seconds. The m i n i m u m fraction of ions that mus t dissociate in this time ta in order to observe a signal is then given by

fraction = 0.05 < ~o t 'kexp(-kt) dt

o o

f0 k exp( - kt) dt = 1 - e x p ( - kta)

(9)

in which the numera tor is the n u m b e r of ions that dissociate in t a seconds, whereas the denomina tor represents the dissociation of all the ions. If the time prior to mass analysis is 5 ~s, then the rate constant mus t be greater than 104 s -1. Any ions with rate constants below this value wou ld not yield sufficient numbers of f ragment ions to be detectable. Thus, the measured onset is de termined by the ion residence time and the signal-to-noise ratio of the instrument.

It is possible to determine an approximate activation energy for the chlorobenzene ion by the use of the R R K M / Q E T calculation. The ionization energy is k n o w n to be 9.07 eV. We n o w s imply calculate the dissociation rates by using the activation energy as an adjustable parameter (Figure 7a). The transition state vibrational frequencies were est imated to be the frequencies of the chlorobenzene molecular ion less a mode due to the C- -C1 stretching frequency (assumed to be 702 cm -1) [56]. The lowest five modes were scaled by 0.7 to obtain a a S ~ of +2.2 e.u, that is, a loose transition state for the simple bond cleavage reaction. The E 0 = 3.03 eV, which yields an ion dissociation rate of 104 s - I at the appearance energy of 13 eV (apparent activation energy, A E - IE, 3.92 eV), should be close to the correct value for the activation energy. Such a plot of AE - IE versus E 0 is shown in Figure 7b. It is evident that the kinetic shift associated wi th the chlorine a tom appearance ene rgy of chlorobenzene is about 0.89 eV. A more detailed analysis involving the fitting of measured and RRKM calculated decay rates yields a kinetic shift of 0.74 eV [37]. From the latter analysis a more correct value E 0 = 3.19 eV can be obtained.

The kinetic shift can also be illustrated by vary ing the residence time of the ions in an ion trap prior to mass analysis as has been done by Lifshitz and co- workers [48, 57, 58]. A dramatic illustration of the shift in AE as the residence time in the mass spectrometer is varied is s h o w n in Figure 8 for the case of bromobenzene [58]. Most of the AE shift occurs be tween 6 ~s and about 0.5 ms. Beyond I ms, the AE remains constant. One might expect that w h e n the observed AE remains constant, AE - IE = E 0. However , this does not take into account infrared fluorescence of the excited ions which competes with dissociation at low energies [59-61]. This IR fluorescence rate of about 103 s-1 has been shown to be relatively constant for large ions. Thus, it is never possible to observe the true thermochemical onset for the dissociation of a large

t0 | :

10 s A

t0 4

10:

10:8

(a)

E o = 2 .0 2 .5 3 .0 3 .5 4 .0 eV

/

/ 3 4 6 8 7

E l n t e r n , i ( e V )

(b) 6

Q v

o s - r

oZ - - 4 A ÷tm -r

UJ <

C s H s C l ÷" --> CsHs + + Cl"

k l n e t i c s h i f t = 0 . 8 9 e V J

E o = 3 .03 eV

2 . , , , • . . . . , , • , • , • , • ,

2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0

E o (eV )

Figure 7. (a) A plot of k(E) vs. E for several values of E 0 for the dissociation of chlorobenzene ions (see text). The ion internal energy that results in a fragmentation rate constant of 1 0 4 s - 1 is defined as the AE for a particular choice of E 0. (b) A plot of the AE - IE vs. E 0 from the data in (a). The value of AE - IE is the apparent activation energy and the difference between AE - IE and E 0 is the kinetic shift.

ion. H u a n g and Dunbar [62] have termed this shift the intrinsic kinetic shift. H o w can one determine E 0 f rom such data? The only way is to model the k(E) versus E function with the RRKM theory and thus to extract E 0 f rom the calculation [19, 63, 64].

Simple Bond Cleavage Versus Rearrangement

A quest ion that often arises w h e n considering the origin of a metastable peak is whether the ion origi- nated f rom a simple bond cleavage reaction or a rear- r angement process. An energy d iagram illustrating these two possibilities is shown in Figure 9. The direct dissociation toward products PA has a small activation energy E 0 and m a y thus proceed with a relatively fast rate. In competi t ion wi th this s imple bond cleavage reaction is an isomerizat ion to a more stable ion struc- ~tre B. Once the ion finds itself in the lower energy


INTERNAL ENERGY (eV)

3.0 3.5 : I I i 1 I I | [

,oo ' , o : ' / , b Ib' 6 ' , 6" ,o CALCULATED RATE CONSTANTS (S -I ) ~

80 C6H~ (CGHsBr) O/~ 0

~."' 6 0 - / m s

4 0 -

o . . . . . . II 4 11.6 11.8 :2.O 12.2 12.4 1216

PHOTON ENERGY (eV) Figure 8. The photoionization efficiency plots for the C6H ~" product ion from bromobenzene at two ion trapping times. The shift toward lower energies at the longer trapping time is a result of the kinetic shift. Reproduced with permission from Malinovich et al. [58].

potential well, its activation energy to products (either PA or PB) is considerably larger and thus its rate greatly reduced. The rate constants for reaction from the A and B wells may differ by orders of magni tude [65]. Examples in which parent ions rapidly rearrange to more stable ion structures prior to dissociation are the butyne ions, which rearrange to the butadiene structure prior to loosing CH 3 [66], and 2,4- and 1,5- hexadiyne ions, which rearrange to the benzene ion isomer prior to dissociation [67-70].

Another example of dissociation and isomerization in competition is the loss of a C H 3 0 radical from ionized methylacetate which results in a strong fragment ion peak in the mass analyzed ion kinetic energy (MIKE) spectrum [71]. The ionization energy of meth-

O

PA

B

PB

Reaction Coordinate

Figure 9. The reaction potential for an ion that can decay by both direct dissociation and by prior isomerization to a lower energy well. Reaction from the latter will be slower than the direct dissociation.

ylacetate as well as the product energies for the simple bond cleavage to form CH30" are known [72], f rom which we deduce an E 0 = 0.84 eV. A statistical theory k(E) versus E curve can be calculated for this direct bond cleavage reaction to determine if the measured slow rate constant is consistent with RRKM theory. First, a set of frequencies must be estimated for the transition state. A good approximation uses the molecular ion frequencies (from an ab initio calculation), subtracting one mode to represent the reaction coordinate [for methylacetate, the C ( = O ) - - O stretch at 1248 cm -1 can be removed] and scaling the lowest five modes to obtain a positive AS* (loose transitions states are typical of simple bond cleavage reactions). The resulting k(E) versus E curve is shown in Figure 10. As can be seen from the figure, the rate constant for this reaction increases very quickly with internal energy and thus only a very small range of ion internal energies near threshold ( < 0.05 eV) will result in fragmentations occurring in the second field-free region of a sector mass spectrometer. This is an indication that the fragment ions observed in the MIKE spectrum probably result primarily from the rearrangement process to the lower energy B isomer. This has indeed been suggested to explain the loss of CH2OH in addi- tion to C H 3 0 [71, 73]. However, even without the observation of the CH2OH product, a similar conclusion can be reached just on the basis of the k(E) versus E curve. If the dissociation rate constant is measured, it is possible to determine the energy of the B well by use of the RRKM theory.

The loss of the same neutral (CH30) from ionized trimethylborate also proceeds via a metastable ion. However , the known activation energy of 1.30 eV is considerably larger than the E 0 in the case of the methyl acetate ion. As a result, the calculated dissociation rate constant for the direct loss of C H 3 0 is much smaller (see Figure 10), indicating that this simple

1011 1010 1 ~ . ~ - ~ - ~ - t h y l a c e t a t e Ion

3 ~ L~'+" +3 e'"" 101 0.84 eV

A 10 s

1.. -~ 10 3

101

10 o

10-1 10-2

10"= 0.0 0.1 0.2 0.3 0.4 O.S 0.6 0.7 0.8 0.9 1.0 1.1

energy above E o (eV)

Figure 10. The k(E) vs. E curves for the loss of CH30" via direct dissociation steps (from isomer A in Figure 9) for ionized methylacetate and trimethylborate. The internal energy regions in which the molecular ions fragment with rate constants of 104 to 106 s -1 are denoted by the dashed lines.


bond cleavage may well be responsible for a large proportion of the fragment ions present in the MIKE spectrum [71, 74]. PEPICO studies of this ion have shown that dissociation from both the A and B wells in Figure 9 lead to slow rate constants [74]. The use of RRKM calculations therefore can help identify some of the first steps in the fragmentation pathways of ions.

With some experience it is possible to deduce without the need for RRKM calculations when ions isomer- ize prior to dissociation. Recall that the minimum rate of dissociation is given by eq 5. Because the density of states increases with both E 0 and the number of normal modes, the minimum E 0 required to observe metastable ions depends on the size of the ion. For small ions such as C4H~', E 0 should be about 2 eV, whereas for large ions such as butylbenzene, an E 0 of 1 eV is sufficient to result in metastable ions. From these numbers it can be guessed that allyl cyanide ions, which have a strong metastable signal and an apparent E 0 (AE - IE) of 1 eV, must rearrange to a lower energy isomer prior to dissociation. The lower energy isomer turns out to be the pyrrole ion [75].

Obtaining the Fragmentation Rate Constant by Varying Vac c

It is possible to obtain rate constant data on a sector mass spectrometer by varying the ion source acceleration voltage Vac ~. Metastable precursor ions M fragment in a field-free region of a magnetic sector mass spectrometer (usually the second field-free region of a BE instrument) to produce a MIKE spectrum of the fragmentation products. Suppose that the total rate of production of metastable ions is [M] 0 ions per second. These ions dissociate by exponential decay with rate constants that depend upon their energy. Suppose we assign an average rate constant to the dissociating ions. The number of ions that remain undissociated after time t is given by

[M] = [M]0e -kt (10)

where k is the average total fragmentation rate constant of M. The number of metastable ions that dissociate in a given field-free region A[M] is then expressed a s

aiM] = [M]o~t~exp(-k t ) dt

[M]0 - k ( e x p ( - k t l ) - exp( -k t2 ) ) (11)

where t 1 and t 2 are the times it takes M to enter and exit the field-free region. The number of ions dissociating between t 1 and t 2, A[M], is given by the area of the fragment ion peaks in the MIKE spectrum. How- ever, we cannot extract k from this equation because we do not know [M] 0. The latter consists of an un- known combination of undissociated metastable ions and stable parent ions (those that have an internal energy below E0). This problem can be circumvented

by measuring the MIKE spectrum with different accelerating voltages, thereby changing t 1 and t 2. The new MIKE spectrum peak intensity will now be given by

AIM]' = [M]0 (exp(-kt~) - exp(-kt '2)) (12)

Dividing eqs 11 and 12 removes [M] 0 from the expression:

dl[M] e x p ( - k t 1) - e x p ( - kt 2) - - = (13) a[M] ' exp( -k t~) - exp( -k t~)

All of the quantifies in eq 13 are known except k. To avoid error due to the variation in the collection efficiency of fragment ions at different values of Vacc, A[M] is taken as the ratio of the fragment ion peak area to the precursor ion peak area in the MIKE spectra.

An example of this approach is the fragmentation of metastable butanoic acid ions [53]. MIKE spectra were collected on a VG (VG Analytical Ltd., Manchester, UK) ZAB double focusing mass spectrometer at four accelerating voltages: 8, 4, 2, and 1 keV. The resulting &[M] values t 1 and t 2 a r e shown in Table 1. Using eq 13, six values of k were determined from the six possible combinations of the four data sets. The values of k were 1.48, 1.01, 1.17, 1.2, 1.185, and 1.09 x 10 s s -1. The average of these values is the total average fragmentation rate constant of metastable butanoic acid ions, 1.2 x 105 s -1.

Conclusion

This article is an introduction to the application of statistical rate theory (RRKM/QET) to problems in mass spectrometry. Although a rigorous derivation of the RRKM/QET equation for the rate constant of a unimolecular reaction was outside the scope of this treatment, it is hoped that some physicochemical insight was provided for the various mathematical ex- pressions surrounding statistical theory. Three cases where the RRKM/QET equations can come in handy for the experimental mass spectrometrist were also discussed.

The RRKM/QET expression for k(E) versus E pre- sented in this paper (eq 3) is the simplest form of the theory. The equation becomes more complicated as other considerations are taken into account such as the role of rotational degrees of freedom in a reaction and

T a b l e 1. Relative fragment ion peak intensities and flight times for the MIKE spectra of metastable butanoic acid ions at several accelerating voltages

Vac c t 1 t 2 ~(fragment ion peak area)

(keV) (p.s) (/~s) precursor ion peak area

8 5.5 15.4 0.00234

4 7.4 21.3 0.00201

2 10.1 29.8 0.00175

1 13.8 41.6 0.00126

J Am Soc Mass Spectrom 1997, 8, 103-115 STATISTICAL R R K M / Q E T CALCULATIONS IN MS 113

the role of quantum mechanical tunneling. The reader is referred to refs 7 and 20-24 for detailed discussions of how to calculate k(E) for these more complex cases, as well as other aspects of statistical theory such as variational transition state theory.

A p p e n d i x : Ca lcu la t ing D e n s i t i e s and S u m s o f States

Direc t C o u n t

The most accurate method for determining the density and sum of harmonic vibrational states is the direct count method. The method described here is that proposed by Beyer and Swinehart [25]. If a system consists of s harmonic oscillators with frequencies of oo i = v i / c cm-1, where c is the speed of light, each will have a series of equally spaced states located at E i = n oa i (n = 0,1, 2 . . . . ). The zero of energy is the ion's zero point energy and the internal energy is divided into bins of size 1 cm -1. The s frequencies can then be expressed in wavenumbers , rounded off to the nearest wavenumber . The density of states p(E) is given by the following algorithm (in BASIC):

p(I) = [1, 0, 0, 0, 0 ... 0]

I FOR J = l T O s

2 FOR I = ~o(,1) TO M

3 p(1) = p(I) + p(I - to(J)) 4 NEXT I 5 NEXT J

initialize the p vector s = the number of oscillators M = max imum energy bin of interest

The sum of states N ( E ) can be obtained either by direct numerical integration of the density of states or computed directly by using the Beyer-Swinehart scheme:

N(I) = [1, 1, 1, 1 "-- 1] initialize the N vector

6 FOR J = 1 TO s. s = the number of oscillators

7 FOR I = ea(J) TO M M -- the maxi- m u m energy bin

of interest

8 N(I) = N(I) + N(I - ~oO)) 9 NEXT I

10 NEXT J

W h i t t e n - R a b i n o v i t c h M e t h o d

The Whit ten-Rabinovitch [26, 27] method is based on the classical sum of states calculation. The equation for

calculating the sum of states is

N(E) -- ( E + a ZPE) s

s!l-I( hv )

where E is the energy, ZPE is the ion's zero point energy, h is Planck's constant, v are the vibrational frequencies, s is the number of frequencies, and a is a constant which varies from 0 to 1 as the energy goes from 0 to infinity. The parameter a is given by

where

a = 1 - /3~(E' )

E E s ~ m

ZPE ( s - 1 ) ( v 2)

( v ) 2

co = [5.00E' + 2.73(E') °'5° + 3.51] -1

The equation for ~o is approximate and is good for the range of energies E' between 0.1 and 1.0. More accurate values are tabulated by Whitten and Rabinovitch [26].

The density of states can be calculated from the derivative of the sum of states equation:

(E + aZPE) s-1 [ dco(E') ] p(E)= t-s-

However, as pointed out in the text, there is a question as to how accurate this approach is for large molecules or ions.

Steepes t Descen t

The program that follows calculates the densities and sums of rovibrational states. It is a translation into BASIC of a p rogram written in APL by Forst [7].

Definition of variables (those terminating with # are double precision)

Nfreq = number of unique vibrational frequencies Nrot = number of free rotors DEN(E) = density of states and sums of states vectors E(I) = ion internal energy H = Planck's constant in cm -1, 3.3364E-11 cm -1 sec R = N r o t / 2 for density of states and (Nrot /2) + I for sums of states QR = reduced rotational partition function

The intermediate variables F0#, F I# , QV, SU#, T H # , T I # , and EP# are functions whose meanings are defined on pages 396-397 of Forst [7].

Input

Number of unique frequencies (Nfreq); Number of free rotors (Nrot); Initial energy (Einit);


Activat ion energy (Eo); Energy increment (Edel); N u m b e r of energies to be calculated (NUM);

Molecular ion frequencies (Freq) in cm-1 ; Molecular ion frequency degeneracies (Deg); Moments of inertia in amu-A 2 (MOM)

The program

DIM Freq(50), Deg(50), DEN(100), E(100), MOM(5), SUM(100)

T I # = 0.9995: E P # = 5E-08: QR = 1: H -- 3.33644E-11

FOR I = 1 TO Nrot: QR = QR,SQR(MOM(I) '0.18634): NEXT I

E(1) = Einit

FOR K = 1 TO N U M 'scan over the n u m b e r of energies

N = I

20 F0# = 0: S U # = 0: R = Nro t / 2 : QV = 1

FOR I = 1 TO Nfreq

F0# = F 0 # + Deg(I ) , Freq(I)* T I # ^Freq( I ) / (1 -T l# ^Freq(I))

SU# = S U # + Deg(I)* Freq(I)^2* T I # ^(Freq(I) - 1)/(1 - T I # ^Freq(I)) ^2

NEXT I

F0# = F0# + R / L O G ( 1 / T 1 # ) - E(K) ^

F I # = SU# + R / ( T I # , ( L O G ( 1 / T 1 # ) ) 2)

T2# = T I # - F 0 # / F I #

IF ABS(T2# - T I # ) < = EP# OR N > 10 THEN GOTO 30 'convergence test

T2# = ABS(T2#)

IF T2# < 1 THEN T I # = T2#

IF T2# > 1 THEN T I # = 0.9999 ' this insures that calculation w o n ' t diverge

N = N + l : G O T O 2 0 30 T H # = T I #

FOR I = 1 TO Nfreq: QV = QV*(1 - T H # ^Freq(I)) ^ ( -Deg(I ) ) : NEXT I ^ ^

DN = T H # E(K) , (LOG ( 1 / T H # ) ) R , S Q R ( 6 . 2 8 3 2 , F l # , T H # )

DEN(K) = Q v • Q R / D N

E(K + 1) = E(K) + Edel: NEXT K ' inc rement energy and repeat

The preceding calculation will produce a densi ty of states because R was set to N r o t / 2 . To obta in a s u m of states, SUM(K), change R to (Nro t /2 ) + 1 and redefine the energy scale as E = E - Eo. The RRKM rate constant is then g iven by

RATE(K) = S U M ( K ) / ( H * D E N ( K ) )

The three methods of eva lua t ing densi ty and s u m of states are compared to each other for pen tane molecules in Figure 2.

Acknowledgments The authors thank the Department of Energy for continuing financial support and P. M. M. would like to thank the Natural Sciences and Engineering Research Council of Canada for a Postdoctoral Fellowship during the tenure of which this work was completed.

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Statistical Rice-Ramsperger-Kassel-Marcus Quasiequilibrium ...core.ac.uk/download/pdf/82671559.pdfKassel-Marcus (RKKM) theory in a parallel fashion. In the original QET paper, the