Kinetic energy release distributions in mass ... - Wiley Online Library

JOURNAL OF MASS SPECTROMETRY J. Mass Spectrom. 2001; 36: 459–478

SPECIAL FEATURE: TUTORIAL

Kinetic energy release distributions in mass Spectrometry J. Laskin1 and C. Lifshitz2∗ 1 Pacific Northwest National Laboratory, William R. Wiley Environmental Molecular Science Laboratory, P.O. Box 999 (K8-96), Richland, Washington 99352, USA 2 Department of Physical Chemistry and The Farkas Center for Light Induced Processes, The Hebrew University of Jerusalem, Jerusalem 91904, Israel

Received 25 January 2001; Accepted 28 February 2001

Kinetic energy releases (KERs) in unimolecular fragmentations of singly and multiply charged ions provide information concerning ion structures, reaction energetics and dynamics. This topic is reviewed covering both early and more recent developments. The subtopics discussed are as follows: (1) introduction and historical background; (2) ion dissociation and kinetic energy release: kinematics; potential energy surfaces; (3) the kinetic energy release distribution (KERD); (4) metastable peak observations: measurements on magnetic sector and time-of-flight instruments; energy selected results by photoelectron photoion coincidence (PEPICO); (5) extracting KERDs from metastable peak shapes; (6) ion structure determination and reaction mechanisms: singly and multiply charged ions; biomolecules and fullerenes; (7) theoretical approaches: phase space theory (PST), orbiting transition state (OTS)/PST, finite heat bath theory (FHBT) and the maximum entropy method; (8) exit channel interactions; (9) general trends: time and energy dependences; (10) thermochemistry: organometallic reactions, proton-bound clusters, fullerenes; and (11) the efficiency of phase space sampling. Copyright  2001 John Wiley & Sons, Ltd.

KEYWORDS: kinetic energy release (KER); laboratory frame; center of mass frame; kinetic energy release distribution (KERD); metastable ions; mass-analyzed Ion Kinetic Energy (MIKE); electrospray ionization (ESI); matrix-assisted laser desorption/ionization (MALDI); RRKM/QET; phase space theory (PST); finite heat bath theory (FHBT); the Gspann parameter; surprisal; organometallic ions; clusters; fullerenes; biomolecules; Coulomb explosion; PEPICO; average KER; Maxwell–Boltzmann-like distribution; avoided crossing; conical intersection; angular momentum; reaction cross-section; detailed balance; density of states; centrifugal barrier; prior distribution; momentum gap law; energy randomization

INTRODUCTION AND HISTORICAL BACKGROUND Energy disposal is of prime interest in the field of molecular reaction dynamics.1,2 When unimolecular dissociation reactions of polyatomic ions take place, some of the excess internal energy of the ion is released as kinetic energy of the two fragments. The kinetic energy release (KER) carries very valuable information concerning the structures of the species involved and concerning the energetics and dynamics of the reaction. Similar information can in principle be obtained for unimolecular reactions of neutral systems. There are several advantages in studying ionic systems: the kinetic energy of charged particles can easily be determined experimentally. In a fragmentation of a singly charged ion the total kinetic energy release is imparted to the ionic as well as Ł Correspondence to: C. Lifshitz, Department of Physical Chemistry, The Hebrew University of Jerusalem, Jerusalem 91904, Israel. E-mail: [email protected]

DOI: 10.1002/jms.164

to the neutral fragment. The total is, however, calculable from the kinetic energy of the ionic fragment through simple momentum and energy conservation. Measured kinetic energies are so-called ‘laboratory’ energies, i.e. kinetic energies of the particles relative to the laboratory system at rest. The physically interesting quantity is the kinetic energy relative to the center-of-mass frame of the reaction system. In experimental ion beam instruments, for example in magnetic sector double-focusing tandem mass spectrometric (MS/MS) systems, the ions possess very high kinetic energies in the laboratory system. Under these conditions there is a very high ‘amplification factor’ of the KER in the center-of-mass frame3 when measured as the change, imparted by the dissociation, to the ion energy in the laboratory frame. This has made KER measurements for ion beams particularly attractive since very low KERs can be measured with a high degree of accuracy. Finally, in coincidence measurements such as PEPICO (photoelectron photoion coincidence),2 it is possible to define accurately the internal energy of the reactant ion

Copyright  2001 John Wiley & Sons, Ltd.

J. Laskin and C. Lifshitz

and to vary the excess energy above the dissociation threshold. The KER is then determined for energy selected ions and its interpretation is particularly instructive. Early measurements of KERs4,5 were mostly involved with diatomic molecular ions. The first major development in the determination of KERs for the dissociations of polyatomic ion–molecules came with the discovery by Hipple and co-workers6 of the so-called ‘metastable ions’, i.e. ions dissociating in flight in field-free regions of the mass spectrometer. Observation of ion dissociation in a mass spectrometer is essential for most mass spectrometric applications. If dissociation is statistical, the extent of fragmentation depends dramatically on the internal energy content of the ions. The statistical nature of the dissociation implies that the internal excitation is completely randomized among all the internal degrees of freedom of the parent ion prior to fragmentation. The unimolecular rate constant is then given by the RRKM/QET (Rice–Ramsperger–Kassel–Marcus/quasiequilibrium theory) expression:2 kE D

N‡ E E0 hE

1

where E is the density of vibrational states of the reactant, N‡ E E0 is the sum of states at the transition state, E0 is the critical energy, h is Plank’s constant and is the reaction path degeneracy. The dependence of the unimolecular rate constant on the internal energy is shown schematically in Fig. 1 (note the semi-logarithmic scale). Ions are typically produced in the ion source of the mass spectrometer with a wide distribution of internal energies. Ions are isolated and following their formation and exit from the ion source undergo no, or hardly any, collisional energy exchange. As a result, ions from a high-energy part of the energy distribution decompose in the ion source, whereas relatively ‘cold’ ions can reach the detector without decomposition. The intermediate part of the ion ensemble gives rise to the metastable ions. These ions leave the ion source

log (k)

460

E0

E

Figure 1. Schematic drawing of the dependence of the microcanonical rate constant, kE, (logarithmic scale) on the internal energy, E. E0 is the critical energy of activation (threshold energy).


and decompose on their way to the detector. Metastable peaks commonly observed in mass spectra are due to ion dissociation in a field-free region of a mass spectrometer. Single-focusing magnetic sector instruments were employed in the early experiments.6 It was realized that in these experiments metastable peaks occur in the normal mass spectrum at non-integer masses mŁ D m2 2 /m1 providing unequivocal evidence that a precursor ion mC 1 has dissociated to generate a fragment ion mC within the field-free region 2 preceding the magnetic analyzer. Metastable peaks were observed to be broader than normal peaks because of the translational energy release. The development of double-focusing mass spectrometers and particularly reverse-geometry instruments in which the magnetic sector precedes the electric sector has revolutionized the study of metastable ions.7 Early measurements of KERs were used to characterize isomeric ion structures and the scope of metastable peak observations was reviewed.8 The KERs are never single valued even when the reactant ions are energy selected. Rather, kinetic energy release distributions (KERDs) are obtained experimentally. These KERDs can be Maxwell–Boltzmann like, in which case a ‘temperature’ can be assigned to them. Various theoretical approaches have been developed. Some of these try to reproduce the full KERDs, others calculate the average KER and/or the temperature of the distribution.2 Measurements of KERs have existed since the early days of mass spectrometry. Renewed interest has developed recently in connection with measurements on fullerenes and biomolecules. Measurements in these systems have centered not only on singly charged but also on multiply charged and multiply protonated species leading to so-called Coulombic repulsion.9,10 KERs measured for Coulomb explosion lead to information on ion structures (intercharge distances) at the moment of explosion. This has been useful for biomolecules. The measurements of KERDs have led to information on binding energies for example in organometallic species11 and fullerenes.12 The extraction of binding energies from the KERDs requires elaborate theoretical modeling of the data. This present tutorial is planned along the following lines. The topic is introduced by giving some background on the kinematics of ion dissociation and kinetic energy release followed by a short introduction into possible potential energy profiles and their effect on resultant KERs. This is followed by a short introduction into the concept of the KERD. The next section is devoted to metastable peak observations and explains KER determinations using both magnetic sector and time-of-flight instruments. A section describing how the KERD is extracted from metastable peak shapes follows. Ion structure determination from KER measurements is exemplified for singly charged ions by the C2 H5 OC ! HCOC C CH4 reaction system studied some years ago. Some more recent examples covering peptides and fullerenes are given for Coulomb repulsion reactions of multiply charged ions. The section on structures incorporates a short interlude on KERs and reaction mechanisms. Two examples are discussed: the H2 elimination reaction CH2 NH2 C !

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Kinetic energy release distributions in MS

CHNHC C H2 and the iodotoluene reactions C7 H7 IC ! C7 H7 C C I that proceed via two parallel channels. Theoretical treatments of KERDs are discussed briefly with an appropriate introduction to additional literature. These include phase space theory (PST) and orbiting transition state phase space theory (OTS/PST), finite heat bath theory (FHBT), including decomposition in a spherically symmetric potential, a model-free approach and, finally, the maximum entropy method. The theoretical sections are followed by experimental results that include comparisons with theoretical modeling. These are arranged in the following order: exit channel interactions, time and energy dependences, thermochemistry (with examples from organometallic systems, clusters and fullerenes) and some of the recent studies concerning the degree of phase space sampling.

Kinematics Consider the decomposition of a parent ion m1 C of mass m1 , kinetic energy E1 and velocity v1 D 2E1 /m1 1/2 in a field-free region (FFR) of a mass spectrometer: C

m1 ! m2 C m3

2

The excess internal energy of the parent ion is distributed among all the internal degrees of freedom. It follows that part of this energy is released in the relative translation of departing fragments. This energy is called the kinetic energy release (KER). Let u2 and u3 be the vector velocities of the ionic and neutral fragments, respectively, in the centerof-mass frame. Then the corresponding velocities in the laboratory frame are (v1 C u2 ) for the daughter ion and (v1 C u3 ) for the neutral fragment. If ε is the kinetic energy release, the conservation of momentum and conservation of energy laws imply εC

mv21 m2 v1 C u2 2 m3 v1 C u3 2 D C 2 2 2 0 D m2 u2 C m3 u3

3 4

Combining Eqns (3) and (4) gives the following expression for the kinetic energy release: εD

m1 m2 2 u2 2m3

5

The maximum velocity of the daughter ion in the center-ofmass frame is then given by u2 D

2m3 ε m1 m2

6

and the velocity in the laboratory frame by v2 D v1 š u2 D v1 š


2m3 ε m1 m2

E2 D ³

7

m2 v22 m2 2 m2 D v1 š u2 2 D v C u22 š 2v1 u2 2 2 2 1

m2 2 v š 2v1 u2 2 1

8

In most experiments, the kinetic energy of precursor ions in the laboratory frame is 1–10 keV, whereas typical kinetic energy releases are in the range from 2 meV to 2 eV. Thus the contribution of the u22 term to the kinetic energy of the daughter ion is much less than 0.2% and can be safely neglected. It is the cross term 2v1 u2 that causes the large amplification factor mentioned in the Introduction. Combining Eqns (6) and (8) gives m2 E2 D 2

ION DISSOCIATION AND KINETIC ENERGY RELEASE

C

Finally, the kinetic energy, E2 , of the ionic fragment can be written as

v21 š 2v1

2m3 ε m1 m2

m2 E1 D m1

1š

4m3 ε m2 E1

9

The spread of kinetic energies of ions fragmenting in a fieldfree region of the mass spectrometer, E, caused by the kinetic energy release (KER) is E D

4m2 m3 εE1 m21

10

The spread of kinetic energies increases (Eqn (10)) with increase in the KER ε and in the kinetic energy of the parent ion E1 , whereas Eqn (7) indicates that the spread in velocities is independent of experimental parameters. Moreover, as can be seen from Eqn (8), the absolute kinetic energy spread, E, is larger than the absolute velocity spread, u2 , by a factor of m2 v1 . It follows that measurement of the kinetic energy spread is advantageous over measurement of the velocity spread.

Potential energy surfaces The amount of energy released to relative translation of products depends on the details of the potential energy surface. Two types of potential energy surfaces are shown schematically in Fig. 2. A type I surface corresponds to reactions proceeding via a very loose transition state, for example, simple bond cleavages. This type of reaction (1) requires no (or very little) rearrangement in the transition state and (2) is characterized by a negligibly small reverse activation energy. The excess internal energy, E‡ , is partitioned among the internal degrees of freedom of the transition state configuration. The kinetic energy release for type I reactions is small and reflects the amount of internal excitation of the parent ion. A potential energy surface of type II is characteristic of dissociation reactions requiring a substantial rearrangement in the transition state (a tight transition state), e.g. reactions involving isomerization. A substantial barrier for the backreaction, Er0 , characterizes the type II reactions. If there is no coupling between the reaction coordinate and other degrees of freedom, the reverse activation energy is fully converted to relative kinetic energy of the departing products. In this case the KER is large and is mainly due to Er0 . As

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characterized by a similar KERD shifted from zero by the amount corresponding to the reverse activation energy. It is important to note, however, that the position and shape of the KERD are determined not only by the shape of the potential energy surface but also by dynamic effects that occur as the products separate. These effects are frequently called exit channel interactions and may be due to electronic excitation of one of the products or some energy flow between the reaction coordinate and other modes caused by a nonnegligible coupling between them. Some examples of these effects will be given later. Figure 2. Schematic drawings of two potential energy profiles. Type I: the potential energy along the reaction coordinate is continuously rising towards the dissociation limit. There is no reverse activation energy. This profile is characteristic for simple bond cleavages. Type II: same as I but the reverse bimolecular reaction demonstrates an activation barrier. This profile characterizes dissociation reactions involving rearrangements.

Table 1. Characteristic properties of potential energy surfaces Potential energy surface

Transition state Type of reaction Reverse activation barrier Kinetic energy release Metastable peak

Type I

Type II

Loose Simple bond cleavage No

Tight Isomerization Yes

Small

Large

Narrow

Wide

mentioned earlier, large kinetic energy releases cause large spreads of kinetic energy and velocity of the fragment ion. This results in observation of a wider metastable peak in a mass spectrometer. The above qualitative discussion is summarized in Table 1.

METASTABLE PEAK OBSERVATIONS Sector instruments As we mentioned earlier, the kinetic energy release results in a substantial broadening of the kinetic energy distribution of metastable ions. The cross-term, 2v1 u2 , in Eqn (8) causes the increased effect of the KER on the kinetic energy spread as compared with the velocity spread. The kinetic energy spread of the fragment ions can be measured experimentally using an electrostatic energy analyzer. These measurements are typically carried out on sector instruments shown schematically in Figure 3. Ions of interest are selected by a magnetic sector and transferred to the FFR. The kinetic energies of precursor ions and resultant fragment ions are determined by scanning the electrostatic analyzer (ESA). The resulting spectrum is called the mass-analyzed ion kinetic energy (MIKE) spectrum. Fragment ions formed in the FFR will be centered at an ESA voltage corresponding to the kinetic energy E2 D m2 /m1 E1 and have the kinetic energy spread given by Eqn (10). In a more general case, an ion of mass m1 and charge z1 that fragments in the FFR to give a fragment ion of mass m2 and charge z2 will be positioned at the voltage m2 z1 V1 11 V2 D m1 z2 where V1 is the ESA voltage corresponding to the maximum of the parent ion peak. The KER is deduced from half the

THE KINETIC ENERGY RELEASE DISTRIBUTION Consider an ion with excess internal energy E‡ that fragments on a type I potential energy surface. The probability of partitioning a given energy ε into the kinetic energy of the fragments, Pε, equals the probability of redistributing the energy (E‡ ε) among all the degrees of freedom, excluding the reaction coordinate. This in turn is determined by the density of states of the fragments, fr E‡ ε). Because the density of vibrational states rises dramatically with increasing ion internal energy, Pε will decrease rapidly with increasing kinetic energy release. For rotating molecules the conservation of angular momentum restricts the formation of fragments with relative translational energy close to zero. As a result, the KERD exhibits a maximum at small KERs and a close to exponential decrease at higher KERs. If no additional constraints apply, type II reactions are


Figure 3. Schematic representation of a reverse geometry (BE) double-focusing mass spectrometer suitable for KER measurements on metastable ions dissociating in the second field-free region (FFR).

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width of the metastable peak expressed in ESA volts, V2 , its position V2 and the acceleration voltageVacc using εD

m2 z1 eVacc 4m3

V2 V2

2 12

Most commonly the value of the KER, corresponding to the metastable peak width at 50% height, T0.5 , is reported in the literature. (In this paper we employ the notation ‘ε’ for kinetic energy release instead of ‘T’ in order to be consistent with the customary notation in theoretical treatments which follow). It has been shown that for a Gaussian-like peak, the average KER, hεi, can be determined from the peak width at 22% height. In this case ε0.5 and hεi are related through8 hεi D 2.16ε0.5 . It is important to note that the experimental peak width must be corrected for instrumental broadening. If both the parent ion peak and the metastable peak are Gaussian-like the following correction is applied:8 wcorr D d

w2d w2p

13

where wd and wp correspond to the metastable and the parent ion peak width and wcorr is the metastable peak d width corrected for instrumental broadening. A more general procedure that can be used for any type of experimental peak shapes involves deconvolution of the metastable peak with the parent ion peak. This is commonly done using fast Fourier transformation (FFT).

Time-of-flight According to Eqn (7), the kinetic energy release results in an additional velocity spread of the fragment ions. The velocity spread can be monitored using the time-of-flight (TOF) distribution of ions. Although this technique for determining KERs is, probably, less accurate, it has been successfully applied to obtain both qualitative and quantitative information on reaction energetics and dynamics. It can be easily shown that the relative broadening of the TOF peak is directly related to the broadening in the velocity distribution of ions, i.e. v t D v t

εD

1 m2 E1 Wt 2 < L2 > 2m1 m3

15

where E1 , m1 , m2 and m3 are as defined earlier, Wt is the width of the TOF distribution obtained by deconvolution and L is the distance from the position where the daughter ion is formed to the detector. Expressions for hL2 i can be found in the literature15 and will not be reproduced here. Kinetic energy release distributions of internal energy selected ions have been studied by Baer and co-workers2,16,17 using a linear TOF in combination with the threshold photoelectron photoion coincidence technique. Ions are formed by single-photon ionization. Zero-energy electrons are transferred to an electron multiplier and serve both to determine the internal energy content of the precursor ion and to trigger the TOF measurements. Fragmentation of energy selected ions in the field free region results in symmetrically broadened TOF distributions. The TOF distribution of the parent ion is measured at low photon energy below the appearance energy of fragment ions. The method used to extract KERDs from the experimental data will be described in the following section.

EXTRACTING KERDS FROM METASTABLE PEAK SHAPES It has been shown that in the absence of instrumental discrimination the metastable peak shape corresponding to a single kinetic energy release is well approximated by a rectangle. When a range of translational energies is released, the overall metastable peak shape is made up of a collection of rectangles as shown in Fig. 4. In this case the KERD, Pε, can be obtained by differentiating the peak shape, IV: 1 dIV Pε D 16 2 dV and changing the variable ε D V2 V2 , where I is the signal intensity, V is the ESA voltage, V2 is the position of

14

However, since parent and fragment ions have the same velocities in the laboratory frame, it is difficult to separate the precursor and metastable peaks in TOF mass spectra.13 Using a TOF/reflectron14 system can solve the problem. In this case the reflectron is operated in a so-called hard reflection mode, meaning that the potential on the front plate of the reflectron is set to a voltage that is higher than the kinetic energy of the daughter ion but lower than the kinetic energy of the parent ion. Parent ions penetrate into the reflectron, are accelerated and escape to the walls, while fragment ions formed in a field free region are reflected back to the detector. In a different experiment the parent ion peak shape is measured by reflecting both parent and fragment ions to the detector. The metastable peak is then subtracted from the parent ion peak to obtain a real TOF distribution of precursor ions. Deconvolution of the metastable peak with the parent ion peak yields the broadening of the TOF


distribution due to the KER. The KER is calculated using the equation

Figure 4. Approximating the metastable peak shape, that is the result of a distribution of KERs, by a collection of rectangles each due to a single-valued KER. V2 D electrostatic analyzer (ESA) voltage at the metastable peak maximum; Vi D ESA voltage at an arbitrary point i along the metastable peak.

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the metastable peak maximum given by Eqn (11) and D

m2 z1 eVacc 4m3 V22

17

Unfortunately, this simple and straightforward approach cannot be applied to metastable peaks distorted by instrumental discrimination that is always present in practice. The discrimination occurs upon ion passage through various slits and ion optics elements (or is caused by the final detector width in the case of a TOF instrument). It is affected by the quality of the parent ion beam, field imperfections, the kinetic energy release and by the position along the FFR, where the fragmentation takes place. Ions having a substantial velocity perpendicular to the instrument axis do not reach the detector, and the metastable peak corresponding to a single KER is dished rather than rectangular. Extraction of the KERD from this type of metastable peak requires knowledge of peak shapes for several discrete energy releases, εj j D 1, . . . , n, regarded as a set of basis functions. The peak height, Ii , at an ESA voltage Vi is then given by: Ii D

n

BVi , εj Pεj εj

18

jD1

where BVi , εj ) is the basis function and εj is the interval between adjacent energy releases. Pεj can be determined using multiple linear regression or a recursive procedure suggested by Beynon and co-workers18 for a given set of basis functions. However, accurate evaluation of basis functions for sector instruments is rather tedious. A wide variety of methods have been developed by different groups to address this problem. Rumpf and Derrick19 introduced the most elaborate analytical approach (TRAMP), which takes into account discrimination at various ion optics elements and slits and assumes that dissociation occurs randomly along the FFR. Alternatively, individual ion trajectories can be evaluated numerically using programs for ion-optical calculations.20,21 A detailed comparison between different methods used for extracting KERDs from metastable peaks was performed. It was demonstrated19 that significant errors could be introduced into KERD determinations using simplified methods.18,22 An incorrect order for average KERs of a series of isotopomeric methaneiminium cations was obtained using the simplified methods. It is interesting that although the energy releases at 50% obtained using Eqn (12) were about 15–20% higher than the accurate KERs, they provided a reliable order for the series of isotopomeric reactions. An analytical expression for metastable peak shapes has been derived by Yeh and Kim23 and compared with ion-optical trajectory calculations. Kim and co-workers extensively used this expression to analyze MIKE peaks for both metastable and collisioninduced dissociations in an FFR and in a floated collision cell.

ION STRUCTURE DETERMINATION AND REACTION MECHANISMS Singly charged ions In gas-phase ion chemistry the word ‘structure’ signifies connectivity, i.e. which atoms are joined together and the probable formal locations of charge and radical sites. The study of


ion structures has concentrated in recent years on methods using collisional activation techniques—collision-induced dissociation (CID), charge reversal, charge-stripping, etc., combined with ab initio calculations. However, in the early days,8 unimolecular metastable ion (MIKE) spectra and KERs were methods of choice. The rationale behind using metastable peak shapes and KERs to characterize ion structures is that isomeric ions sample different portions of a complex potential energy surface. En route to dissociation, such ions will traverse different activation barriers and as a result the KERs will differ even if the elementary reaction is nominally the same. On the other hand, equal KERs for reactions of ions having the same elementary formulae but resulting from different precursors would, by the same token, signify equal structures.24 A nice example for isomeric ions undergoing the same nominal reaction with different KERs was found in the C2 H5 OC isomers.25 The 1-hydroxyethyl ion, CH3 CH OC H, undergoes methane elimination forming HCC O, giving a dish-topped metastable peak with an energy release at 50% height of ε0.5 D 0.86 eV. On the other hand, the methoxymethyl cation, CH3 OC CH2 , undergoes formally the same methane elimination reaction, but giving a Gaussian-like peak shape with an energy release at 50% height of only ε0.5 D 0.024 eV. The two isomeric ions can be formed from different precursors. The CH3 CH OC H ion is formed from ethanol and various substituted alcohols whereas CH3 OC CH2 is formed from ethylene glycol dimethyl ether. Unknown isomeric C2 H5 OC ions formed from other precursors can in principle be characterized as being either CH3 CH OC H or CH3 OC CH2 according to their KER upon methane elimination. A high isomerization barrier separates these two isomeric ions. This enables one to distinguish them through their different KERs. The final reaction barrier leading to the dissociation products normally determines the KER in a reaction. If the isomeric forms are freely interconvertible at energies below the threshold for the probe reaction, then they will not be distinguishable through their KERs and other methods such as CID have to be employed. For example, O-protonated oxirane (ethylene oxide) is indistinguishable from CH3 CH OC H through its KER upon elimination of methane. More recently, KER measurements were used to distinguish between diastereomeric structures of mono- and disaccharides in the gas phase.26 Unimolecular fragmentation of deprotonated tricoordinate complexes of four monosaccharides with nickel(II) resulted in different KER values depending on the orientation of a C-2 hydroxyl group. Larger KERs were obtained for complexes possessing an equatorial C-2 hydroxyl group.26a These results combined with labeling studies and theoretical calculations have been used to obtain structural information and study dissociation pathways of nickel–monosaccharide complexes.26b Another study demonstrated that disaccharides can be differentiated based on their glucosidic bond configuration (˛or ˇ-linkage) using KER measurements.26c Complexes containing ˛-isomers produced higher KER values than the corresponding ˇ-isomers. KER measurements have been combined in mechanistic studies with other methods such as ab initio calculations,

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classical trajectory calculations, CID and photodissociation experiments. A recent interesting study27 involves H2 loss from CH2 NH2 C . Like many other H2 eliminations that have been reviewed recently,28 the reaction is characterized by a type II potential energy surface with a reverse activation barrier. This gives rise to a translational energy release which corresponds to a large non-statistical fraction of the reverse critical energy and to a dish-topped metastable peak (see Fig. 5). Isotopic labeling experiments have demonstrated that the reaction is a specific 1,2-H2 elimination. Combining the KER measurements with ab initio and trajectory calculations27,29 has led to the conclusion that the reaction takes place in two steps. The first step is a hydrogen rearrangement giving the high-energy isomer CH3 NHC . The second step gives the products CHNHC and H2 via a part of the potential surface that provides a strong repulsive force between the two fragments. This gives rise to the large translational energy release. Reaction path bifurcation was pointed out to be important29 and the fact the transition state precedes bifurcation and not vice versa has made possible the understanding why hydrogen scrambling is avoided. Other examples of mechanistic studies involve reactions whose KERs are bimodal30 such as the iodine atom loss from iodotoluenes, C7 H7 IC ! C7 H7 C C I. The relative contributions of the low- and high-KER components to the metastable peaks was observed to be time dependent.31 (Time-dependent KERs will be discussed in greater detail in a later section.) o-Iodotoluene, m-iodotoluene and p-iodotoluene demonstrated different time and energy dependences31,32 of the branching ratios of the two channels. The iodotoluene system is thus a classical example32 of two distinct, competing pathways to products having the same stoichiometry, with the extent of competition depending in an interesting way on the internal energy of the ion and on which of the three parent-ion isomers is used. In addition to KER measurements the system has been studied by PEPICO,33 by time-resolved VUV photoionization,31 by timeresolved photodissociation32 and by RRKM calculations.32 All the data were fitted into a comprehensive two-channel model of the dissociation kinetics,32 assuming competitive dissociation to form tolyl ions by direct C—I bond cleavage, and either benzyl or tropylium ions by a rearrangement process. The three isomers differ mainly in the critical activation energies for the direct bond cleavages.

Multiply charged ions Fragmentation of multiply charged ions commonly results Cz in the formation of two charged fragments (m1 1 ! Cz3 Cz2 m2 C m3 ). It is the Coulomb repulsion between the charged species that gives rise to very large KERs in this type of reaction. The intercharge distance at the time of Coulomb explosion, R, is related to the KER via a simple equation:3 RD

14.40

19

where R is in a˚ ngstroms and hεi is in eV. Equation (19) is valid if (1) the reverse activation barrier is only determined by the Coulomb repulsion between the charges and (2) coupling between the reaction coordinate and other degrees


Figure 5. (a) MIKE peak for the reaction CH2 NH2 C ! CHNHC C H2 : relative abundance as a function of half of the ESA voltage; reactant ions were transmitted at an ESA half-voltage of 498.48 V. The full line is the experimental peak and the dashed line is a best-fit model. (b) KERD deduced from the metastable peak shape of (a): relative probability as a function of the KER denoted T (in eV). The full vertical line indicates the most probable value of the KER, 0.95 eV; the dashed line marks the average KER, 0.93 eV. Reproduced from International Journal of Mass Spectrometry, Vol. 167–168, Øiestad AML, Uggerud E, pp 117–126, Copyright (1997) with permission from Elsevier Science.

of freedom is negligible, assuring that the entire reverse activation barrier is released as relative translation of the reaction products. KERs associated with fragmentation of small multiply charged ions have been reviewed recently.34,35 In this section we will concentrate on the fragmentation of large ions, e.g. peptides and fullerenes.

Biomolecules The ability to ionize large biomolecules using electrospray ionization (ESI) and matrix-assisted laser desorption/ionization (MALDI) brought about a renaissance in mass spectrometry as a fast and powerful tool for obtaining sequence information on peptides and proteins. Fragmentation patterns of multiply charged peptides and proteins

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in the gas phase provide the most useful structural information used for sequence assignments. Understanding of the fragmentation mechanisms of multiply protonated peptides can facilitate the interpretation of the mass spectra of unknown compounds. KER measurements in combination with molecular mechanics simulations of ion structure have been utilized to determine the structures and protonation sites of multiply protonated peptides. In addition, fragmentation mechanisms have been proposed based on the structural information and the type of product ions formed. The reaction coordinate for charge separation of multiply protonated biomolecules involves the ground-state electronic surface of a closed-shell ion.36 Contrary to openshell multiply charged ions such as fullerenes (see the next section), diabatic curve crossing is not involved. The kinetic energy release determined experimentally equals the work required to bring the two charges from infinity to the distance between the charges in the transition state for the reaction. Adams et al.36 studied the fragmentation of doubly protonated angiotensin II (AspArgValTyrIleHisProPhe). The C peak shapes of the complementary fragments bC 6 and y2 were examined. A most probable KER of 0.902 eV was deduced from metastable peak shapes for both products. This ˚ value was translated into an intercharge distance of 16.0 A, using Eqn (19), and compared with the values obtained from molecular mechanics simulations. The most likely protonation sites for angiotensin II are the Arg side-chain, the N-terminal amine group and the His side-chain. However, in order to obtain the yC 2 fragment ion at least one proton must be located on one of the two C-terminal amino acids Pro or Phe. Three possible charge sites that could give rise to an intercharge distance close to that determined experimentally were the N-terminal amine, aspartic acid, and the arginine side-chain. Protonation on the basic histidine residue is very unlikely, because it gives rise to a much smaller intercharge ˚ A further study by the same group focused distance (8.89 A). on the decomposition of the yC 7 fragment of angiotensin II, which does not contain the aspartic acid and the N-terminal group originally present in the precursor ion.37 Using the same strategy it has been unequivocally demonstrated that the second proton is located on the Arg side-chain. A charge-remote fragmentation mechanism involving C rearrangement was proposed for the formation of bC 6 /y2 pair from angiotensin II, based on the consideration that a chargeinduced cleavage involving proton transfer to the amide nitrogen of the amide bond being cleaved would result in a ˚ However, the possibility small intercharge distance of 5.78 A. that the C-terminal proton is located on the carbonyl oxygen of the Pro–Phe amide bond and is involved in the decomposition process could not be ruled out. MIKE studies of the fragmentation of the doubly protonated octapeptide ProProGlyPheSerProPheArg38 confirmed the charge-remote mechanism proposed.36 It has been concluded that although the proton is not directly involved in the fragmentation of ProProGlyPheSerProPheArg, its presence on the amide (or amine) group adjacent to the amide bond being cleaved is required.


The higher order structure of biomolecules is directly related to their function. Conformational studies of peptides and proteins in the gas phase are aimed at determining intrinsic properties of these systems unaffected by the presence and the nature of the solvent. This topic has been thoroughly reviewed recently.39 For the purpose of this tutorial we will discuss only the use of KER measurements for the conformational analysis of biomolecules. The first application (to the best of our knowledge) of KER to evaluate the geometry of a biomolecule was carried out in 1992 and involved alkyl-substituted porphyrins.40 Conformations of doubly protonated peptides such as bradykinin were studied by charge separation reactions.41 The experimental results provided evidence that medium-sized doubly charged peptides have a folded and not an extended geometry in the gas phase. Kaltashov and Fenselau42 studied the stability of an ˛-helix in the gas phase. Mellitin, a 26residue polypeptide that assumes a helical structure in a number of different solution environments, was chosen as a good candidate for this type of study. Fragmentation of triply protonated mellitin in the gas phase results in the formation of a series of y2C ions. Very similar KERs n of 1.27 š 0.12 and 1.22 š 0.12 were obtained for y2C 18 and y2C 19 fragments, respectively. Protonation on Arg-22, Arg-24 and Lys-7 residues was assumed for molecular mechanics (MM) calculations. The most probable electrostatic repulsion between the proton on Lys-7 and the two protons on Arg-22 and Arg-24 obtained from MM calculations is 1.26 š 0.06 eV for the helical structure of mellitin ion. Based on the good agreement between the theoretically predicted and experimental KERs, the authors concluded that triply protonated mellitin preserves the helical structure in the gas phase. It follows that the helix structure of mellitin is maintained by intramolecular hydrogen bonds rather than stabilized by electrostatic interactions with the solvent. The same group studied the stability of single ˇ-strands and multistrand ˇ-pleated sheets in the absence of solvent using MIKE spectrometry.43 The results indicated that single ˇstrands form more compact (collapsed) structures in the gas phase, while multistrand ˇ-pleated sheets retain ‘native-like’ secondary structures.

Fullerenes Multiply charged positive fullerene ions undergo superasymmetric fission reactions:9 C602m zC ! C602m2 z1C C C2 C

20

with m ranging from 0 to 7 and z from 3 to 7. These reactions were demonstrated to take place in three stages. They are initiated by neutral C2 evaporation followed by an electron (charge) transfer process from the receding neutral fragment to the remaining highly charged fullerene ion cage leading finally to Coulomb repulsion between the two charged reaction products. The intercharge distance at the point of charge transfer was calculated from the KER, using Eqn (19), ˚ 9 to be about 7 A. The mechanism is similar to the avoided diabatic curvecrossing model of Gill and Radom.44 This model pictures the potential curve along a reaction coordinate for fragmentation

J. Mass Spectrom. 2001; 36: 459–478


Energy

Diabatic Crossing Point

A2+ + B

∆

T A+ +

B+

A−B Bond Length rmin

r TS

Figure 6. Schematic representation of the diabatic (dashed) and adiabatic (solid) potential energy curves for the ground state of a diatomic AB2C dication. T denotes the KER; TS, transition state; , difference between the adiabatic ionization energies of AC and B. Reproduced from Chemical Physics Letters, Vol. 147, Gill PMW, Radom LJ, pp 213, Copyright (1988) with permission from Elsevier Science.

of AB2C into AC and BC as arising from an avoided crossing between the repulsive AC C BC and attractive A2C C B diabatic states (see Fig. 6). When the fragment monocations are infinitely separated, the attractive diabatic state is higher in energy than the repulsive one by a quantity  D IEa AC IEa (B), where IEa is the adiabatic ionization energy. The kinetic energy release is less than  because of diabatic coupling and polarization effects. Extraordinarily late transition structures are encountered from avoided crossings at very long bond lengths. Dications lose a proton via a two-stage process (as discussed above for CC 2 eliminations from multiply charged fullerenes). Initially the departing unit is a hydrogen atom and only later, at some point further along the decomposition pathway, does a spontaneous electron transfer take place to form the eventual products.

THEORETICAL APPROACHES Product energy distributions are extremely sensitive to details of the potential energy surface of the departing fragments. Statistical distribution of energies in reaction products is most commonly observed for reactions proceeding via a loose transition state (type I potential energy surface), whereas energy partitioning in reactions characterized by a substantial reverse activation barrier is usually hard to describe using statistical theory. A comprehensive review on the theoretical modeling of product energy distributions (PED) was given elsewhere.2 We will not attempt to provide the readers with an exhaustive theoretical background but rather concentrate on practical aspects of statistical treatments of KERDs and outline the most commonly used approaches.


Phase space theory (PST) and orbiting transition state phase space theory (OTS/PST) Klots45 has proposed a reformulated QET with explicit and rigorous conservation of angular momentum, which is equivalent to phase space theory (PST) (a 6N-dimensional space giving the positions and momenta of each particle in three independent directions is known as the phase space of the system of N particles) and incorporates the long-range ion–induced dipole Langevin interactions. PST was originally developed for bimolecular reactions. Klots introduced PST to unimolecular reactions in ionic systems. It is applicable to reactions with a negligibly small reverse activation barrier. The unimolecular reaction is monitored from the perspective of the reaction products. The basic assumption is that the reaction proceeds through formation of a strong coupling complex. Decomposition of the complex is purely statistical, meaning that the mode of decomposition of this complex is independent of the mode of its formation. In other words it is assumed that the phase space is sampled uniformly subject to conservation laws. PST explicitly accounts for the conservation of energy, linear momentum and total angular momentum. The unimolecular rate constant is calculated using detailed balance and the cross section for the association reaction of the products, b :2,45 EE0 fr E E0 ε, Jdε kE, J0 D b 0 21 2 hE, J0 p where D h¯ / 2ε is the de Broglie wavelength associated with the relative motion of the fragments, h¯ D h/2, is the reduced mass of the fragments, E, J0 is the density of states of the parent ion having internal energy E and

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467

468


angular momentum J0 and fr E E0 ε, J is the convoluted density of states of fragments having angular momentum J. It follows that the PST formalism does not require knowledge of transition state properties. Evaluation of the unimolecular rate constant relies on the properties of the reaction products and the precursor ion that can be easily determined either experimentally or theoretically. However, an accurate interaction potential between the fragments is required to obtain a reliable cross-section for the bimolecular association reaction, b . PST assumes that the unimolecular rate for reactions proceeding via a loose transition state can be characterized by properties of the infinitely separated fragments. However, as the colliding particles approach one another, their relative kinetic energy is converted into rotation (the so-called centrifugal energy). The energy uptake by the rotational motion introduces an additional barrier (the centrifugal barrier) on the reaction path. The presence of the centrifugal barrier results in lower unimolecular rates than predicted by PST. In addition, formation of the two fragments with very low relative kinetic energy is disfavored. The orbiting transition state phase space theory (OTS/PST) developed by Chesnavich and Bowers46 extends the phase space formalism to a wider class of chemical reactions, for which the centrifugal barrier on the reaction path is not negligible. In this section we will consider the OTS/PST formalism and point out the differences between PST and OTS/PST. Calculated rate constants are usually overestimated by PST, but KERDs are calculated fairly well. The KERD can be calculated using the equation Pε dε D

FE, J0 , ε dε FE, J0

22

where FE, J0 , ε) is the flux through the orbiting transition state at energy E, angular momentum J0 and translational energy ε and FE, J0 is the integrated flux at energy E and angular momentum J0 (the flux is defined as the sum of quantum states). If metastable ions are produced with a distribution of internal energies, the above KERD should be averaged over the fragmentation probability function, PE, J0 , that defines the probability that a precursor ion with energy E and angular momentum J0 will dissociate in the FFR. However, the correction introduced by averaging the KERD over PE, J0 is less than 3%47 and can be safely neglected in most cases. Finally, the KERD in Eqn (22) is averaged over the initial rotational distribution of the precursor ion. The total flux (or the total system phase space) is separated into two parts, one corresponding to system vibrational motion and the other corresponding to system rotational–translational motion:

FE, J0 D

E ‡

v E Etr Etr , J0 dEtr

23

Etr

where Etr D Er C ε is the translational–rotational energy, ‡ Er is the rotational energy, Etr is the minimum value of Etr needed to surmount the centrifugal barrier and Etr , J0 is the rotational–orbital sum of states. The differential flux,


FE, J0 , ε is given by

FE, J0 , ε D

E ‡

v E Er εε, Er , J0 dEr

24

Er

where ε, Er , J0 is the density of rotational–orbital states ‡ at the rotational energy Er , Er is the minimum rotational energy required to generate product angular momentum J which can be coupled with orbital angular momentum L to produce J0 : J0 D L C J

25

‡

‡

In the PST formalism, ε‡ D0 and Etr D Er . Conservation of energy and angular momentum imposes the following boundaries on the rotational–orbital phase space: jL Jj J0 jL C Jj L2max

D 4εC4

1/2

/¯h

26 2

27

The kinetic energy in Eqn (27) is given by ε D Etr Er D Etr BJ2 , where B is the effective product rotational constant. Lmax can be obtained from the centrifugal barrier for any interaction potential between the departing fragments using the relation h¯ Lmax D µubmax , where is the reduced mass of the fragments, u is their relative velocity and bmax is the maximum impact parameter. Equation (27) is obtained assuming that the interaction potential between the departing fragments is of the form VR D C4 /R4 relevant for an ion–induced dipole interaction (the so-called Langevin model) and the fragments are spherical tops. (Equations (7.60) and (7.61) in Ref. 2 give results for more general cases. The derivation of Lmax for other tractable potentials was discussed.48 ) The accessible rotational–orbital phase space is shown in Fig. 7. The need to conserve total energy in the products limits the upper value of J (drawn as the vertical line). For a given J the upper value of L is given by either L D J C J0 [Eqn (26)] or by the Langevin model [Eqn (27)], depending on which of the two is lower. In order to evaluate Etr , J0 and ε, Er , J0 , the sum and density of rotational states are integrated over the J –L plane. Chesnavich and Bowers46 evaluated the sum and density of rotational states for different types of products. Finally, the available phase space is obtained using Eqns (23) and (24). An example of a system that can be accurately described using PST was given by Mintz and Baer.16 They studied the fragmentation of internal energy-selected CH3 IC and CD3 IC ions and showed that for these systems the OTS/PST curve for Lmax J (Fig. 7) crosses horizontally above the accessible rotational phase space. Owing to the high polarizability of the departing iodine atom, the Langevin model [Eqn (27)] does not limit this reaction. In this case a much simpler formalism for the derivation of the KERD first suggested by Klots49 can be employed.16 Experimental KERDs obtained at excess internal energies below 0.65 eV were in excellent agreement with PST predictions (Fig. 8). At higher energies the theoretically predicted product energy distributions were substantially wider than the experimental KERDs. It was suggested that at higher energies opening of other reaction channels could perturb the experimental distributions.

J. Mass Spectrom. 2001; 36: 459–478


decomposition in a spherically symmetric potential (SSP):53 Pε ¾ expε/kB T‡ [1 expBL2max /kB T‡ ]

28

where B is the rotational constant: BD

Figure 7. Angular momentum conservation in dissociation reactions. The orbital angular momentum L is plotted versus the product angular momentum J for a given initial reactant angular momentum J0 . In the case of OTS/PST the allowed range of L values lies within the shaded area limited by the curved line marked Lmax J and by three straight lines that follow Eqn (26), namely: (i) L D J C J0 ; (ii) L D J J0 and (iii) L D J0 J. The vertical PST line corresponds to the maximum product Jmax , which is limited by energy conservation. The curved line shows the additional constraint imposed by the interaction potential and the resulting centrifugal barrier [Eqn (27)]. Lmax J rises with decreasing J because the translational energy ε increases at the expense of product rotational energy. Adapted from Ref. 2.

Bowers and co-workers have extensively used OTS/PST calculations in modeling KERDs for products of organometallic ion–molecule bond activation11,50 and in studies of the dynamics of photodissociation processes51 of small molecular ions and ion clusters. OTS/PST calculations of KERDs were also employed as prior distributions for surprisal analyses52 (see the following sections).

Finite heat bath theory (FHBT) Unimolecular decomposition in a spherically symmetric potential

RELATIVE PROBABILITY

Klots derived a simple expression for the kinetic energy release distribution resulting from the unimolecular

h¯ 2 2I1 C I2

I1 and I2 are the moments of inertia of the products, kB is Boltzmann’s constant, T‡ is the transition state temperature defined by the average kinetic energy on passing through the transition state and, as before Lmax is the maximum value of the orbiting angular momentum. The above expression is valid for small values of J0 ; these are expected for unimolecular reactions of ions formed by ionization of the corresponding neutrals but not for reactions of complexes formed by ion–molecule reactions. The term in square brackets contains the effect of the interaction potential on the distribution. Lmax is given for the ion–induced dipole potential by Eqn (27). Equation (28) has been used to study fragmentation energetics of fullerenes and endohedral fullerenes. The experimental KERD is fitted with the function given in Eqn (28), where T‡ is the only free parameter. The binding energy is then determined from T‡ using the finite heat bath theory (FHBT) developed by Klots.54 The isokinetic bath temperature, Tb , is calculated using55,56 Tb D T‡

expG/C 1 G/C

where G is the Gspann parameter and C is the heat capacity in units of kB minus one. Tb is the temperature at which the canonical rate constant in an infinite heat bath, kTb ), equals the microcanonical rate constant, kE. Finally, the binding energy can be calculated using Trouton’s relation: Evap D GkB Tb

30

A Gspann parameter of 23.5 š 1.5 has been determined independently for clusters of different composition and size.57 The Gspann parameter can be defined using the Arrhenius equation as G D lnA lnkmp , where A is the preexponential factor and kmp is the most probable experimental rate constant. The value of 23.5 for the Gspann parameter

(a)

(b)

hν-AP = .04 eV

29

hν-AP = .23 eV

(c) hν-AP = 1.69 eV

0 0

0.2

0.4

0

0.2 0.4 0 KINETIC ENERGY RELEASE (eV)

0.2

0.4

Figure 8. KERDs for the reaction CH3 IC ! CH3 C C I obtained by PEPICO at the indicated excess energies. The solid lines are OTS/PST calculations and the dashed lines are the best-fit exponential curves (PST). Reproduced from Ref. 16 with permission of American Institute of Physics, Copyright  1976.


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469


is consistent57 with a most probable rate constant of 105 s1 and A D 1.6 ð 1015 s1 . However, it is clear that if either A or kmp differs significantly from the above values, the Gspann parameter is different from the ‘best value’ of 23.5. It has been recognized that in case of fullerene fragmentations G is much higher than the above value. G D 31 was obtained from the analysis of metastable fractions (MFs) of fullerenes.58 A even higher value of 33 was obtained recently59 from RRKM modeling of MFs and the breakdown curves. The uncertainty in the Gspann parameter leads to an uncertainty in absolute values of binding energies determined using the SSP approach. However, the relative values are expected to be correct. The effect of the incorporation of an atom into the fullerene cage on the stability of the cage was studied using the SSP approach.60 The results to be discussed below were found to be in agreement with MP2 calculations by Buhl ¨ et al.61

The model free approach Another approach developed by Klots, commonly called the model free approach (MFA), makes use of the following analytical form for the KERD:55,56 Pε ¾ εl expε/kB T‡ 0 < l < 1

31

where l is a parameter that ranges from 0 to 1 and T‡ is as defined earlier. In this case the experimental KERD is fitted with the function (31) using non-linear regression with l and T‡ being free parameters. This is probably the simplest approach for analysis of KERDs that does not require any prior knowledge of the form of the interaction potential between the fragments. The binding energy is determined in the same way as in the SSP approach using Eqns (29) and (30). The MFA was used to extract fragmentation energetics of fullerenes.56,60,62 It was found that for fullerene fragmentations l is very close to 0.5. In this case the KERD is well approximated by a three-dimensional Maxwell–Boltzmann distribution. This is the behavior observed for C2 elimination from C60 C (see Fig. 9). It is interesting that l D 0 gives a simple two-dimensional distribution corresponding to a negligibly small centrifugal barrier as predicted by PST. The iodine atom loss of CH3 IC belongs to this category16 (see Fig. 8). The high polarizability of the iodine atom causes a high ion–induced dipole interaction between the receding fragments that practically eliminates the centrifugal barrier. On the other hand, l D 1 yields a four-dimensional distribution that corresponds to reaction with a large centrifugal barrier. This is the case, for example, for H-loss reactions because of the low polarizability of the hydrogen atom. The comparison of the SSP and the model free approaches60 revealed that the former yields systematically lower binding energies than the latter. However, the difference between the two approaches decreases when a more realistic potential is assumed in SSP. It is important to note that both approaches show the same trend in binding energies for different cluster ions.


1.6

1.2 Intensity

470

C60+ -> C58+ + C2 0.8

0.4

0.0 0.0 0.5 1.0 1.5 2.0 CENTER OF MASS KINETIC ENERGY RELEASE, eV Figure 9. KERDs for the reaction C60 C ! C58 C C C2 . Solid line, the experimental KERD; dashed line, the KERD obtained using the model free approach; dot-dashed line, the KERD obtained from the SSP model. Reproduced from International Journal of Mass Spectrometry, Vol. 185–187, Laskin J et al., pp 61–73, Copyright (1999) with permission from Elsevier Science.

The maximum entropy method In the case of a purely statistical dissociation (a complete sampling of the phase space), the translational distribution of departing fragments is given by the so-called prior distribution, P0 ε, E. The prior distribution describes an ideal case, in which all quantum states of the system are equally probable. An interesting insight into the dynamics of the unimolecular dissociation can be obtained from the comparison of the prior distribution with the experimentally measured KERD.63 The deviation of the KERD from the prior distribution, described by a surprisal I [Eqn (32)], results from the existence of dynamic constraints that restrict the sampling of the phase space. Pε, E Iε, E D ln 0 32 P ε, E Averaging of the surprisal over the experimental distribution, Pε, E, yields the opposite of the (dimensionless) entropy deficiency, DS. DS is defined as the difference between the entropy of the prior distribution, S0 , and the entropy of the experimental distribution, S: DS D S0 S

33

This value is positive because the prior distribution is the maximum entropy distribution. The basic assumption of the maximum entropy method is that the experimentally obtained product energy distribution is of maximum entropy subject to dynamic constraints. It has been shown that the distribution Pε, E of maximum entropy is related to the prior distribution as1 n 0 Pε, E D P ε, E exp 0 i Ai 34 iD1

where i are Lagrange parameters, Ai are constraints and exp0 is a normalization factor. The surprisal is then

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given by I D 0 C

n

i Ai

35

iD1

and the entropy deficiency by DS D 0

n

i hAi i

36

iD1

Equation (35) shows that the surprisal is a linear function of the constraints, Ai . This allows identification of dynamic constraints by plotting I as a function of a power of ε. The most commonly observed dynamic constraint for reactions with negligible reverse activation barrier is the square root of the kinetic energy, corresponding to the momentum of the departing fragments.63 In this case the entropy deficiency is given by a simple expression: DS D 0 1 hε1/2 i

Eε

p Eε

sˇ0 ε

v xE ε xs1 dx

38

where v x is the density of vibrational states of the reaction products; s D r 1/2 and r is the number of rotational degrees of freedom of the products; ˇ0 is a parameter that depends on the interaction potential of the fragments and their rotational symmetry.45b The results demonstrated a


P0 ε, E D E 0

Ntr εNE ε

39

Ntr εNE εdε

where NE ε) denotes the rovibrational density of states of the fragments and Ntr ε is the translational density of states. Because the translational density of states is proportional to p ε, the prior distribution can be rewritten as p P0 ε, E D CE εNE ε

37

The existence of this constraint has been rationalized based on the ‘momentum gap law’.1 The law implies that excess internal energy is preferentially released into vibrational and rotational excitation of the fragments rather than into their relative translation. This can be explained based on the Frank–Condon principle, according to which large changes of nuclear momenta in vibrationally predissociating molecules are unlikely. As a result, dissociation with large kinetic energy release is strongly disfavored. Lorquet and co-workers63 have found that 1 obtained for reactions constrained by the momentum gap law is always positive. A negative sign of 1 indicates that the KER is larger than the statistically predicted value and the reaction is not controlled by the momentum gap law. For example, fragmentation of chlorobenzene gives rise to a composite KERD, 90% of which is characterized by a positive 1 and 10% by a negative 1 . It has been demonstrated that the major contribution to the KERD results from formation of Cl in its ground electronic state and is controlled by the momentum gap law. The large KER in the minor pathway leading to formation of a chlorine atom in its excited electronic state (2 P1/2 ) was rationalized by assuming the existence of a small, ¾0.12 eV, reverse activation barrier for this pathway. The barrier leads to an additional contribution to the observed translational release, so that the latter is larger than the statistical expectation. The prior distribution for the surprisal analysis can be chosen based on phase space theory. A simple analytical form for the phase space distribution was derived by Klots49 and used by Lifshitz64 in the analysis of KERDs for the metastable fragmentation of the enol ion of acetone: P0 ε, E /

substantial deviation from statistical behavior for the loss of methyl radical from the molecular ion, whereas the KERD corresponding to the loss of methane showed very minor deviation from the prior distribution. It should be mentioned that calculation of the prior distribution given by Eqn (38) requires knowledge of the interaction potential between the fragments. Lorquet and co-workers63 have used another approach. The most unbiased prior distribution is given by

40

where CE is a normalization factor. The above distribution does not take into account conservation of angular momentum and can be strictly applied only to reactions, for which the effect of angular momentum is small as is the case for the reactions studied by Lorquet et al.63 In the work of Lorquet et al., the prior distribution, Eq. (40) (including conservation of the total energy only) has been used to study dynamical constraints in various reactions proceeding via a loose transition state. These included fragmentation of halobenzene cations, pyridine and vinyl bromide ions. It has been observed that in all cases the deviation of the experimental KERD from the prior distribution could be described by a single constraint corresponding to the momentum of departing fragments (Fig. 10). Hoxha et al.63e performed an interesting comparison between the above formalism and Klots’ model free

15

2

∼ P

1

∼ I

10 ∼ P

0 ∼I ∼ P0

5

0

0

0.1

0.2

0.3

−1

0.4

−2

ε1/2 /eV1/2 Figure 10. The experimental kinetic energy release Q prior distribution, PQ 0 , and surprisal, QI, plotted as distribution, P: a function of the square root of the KER, ε1/2 , for the reaction C5 H5 NC ! C4 H4 C C HCN of the pyridine ion. The symbol ‘¾’ signifies that P, P0 and I are averaged over an internal energy distribution. Reproduced from International Journal of Mass Spectrometry, Vol. 185–187, Urbain P et al., pp 155, Copyright (1999) with permission from Elsevier Science.

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471

472


approach. They showed that l D 0.5 in Eqn (31) corresponds to the purely statistical prior distribution, for which 1 D 0. As has been mentioned earlier, l D 0.5 has been obtained for most of the cluster fragmentations studied so far. This implies that in the case of cluster evaporations the phase space is sampled statistically. The role played by angular momentum conservation in the maximum entropy method must still be examined in more detail.63f

EXIT CHANNEL INTERACTIONS An interesting property of the KERD is that it can deviate substantially from the distribution predicted by statistical theories even in the case the unimolecular fragmentation is statistical. Dynamic effects (frequently called exit channel interactions) that occur after passage through the transition state govern the extent of the deviation. The exit channel effects can result from coupling of the reaction coordinate with transitional modes that causes energy flow from the reaction coordinate and reduces the experimental KER. For example, in the absence of such coupling the KERD for type II reactions would be shifted from zero by the reverse activation energy and the shape of the distribution would be well described by OTS/PST formalism. However, most of the KERDs for type II reactions are substantially wider than theoretically predicted distributions and are shifted to lower energies. This indicates that the reverse activation energy, Erev , is only partially released into relative translation of the products. Another part of Erev is released as rotational and vibrational excitation of the reaction products. Yeh et al.65 used a ‘modified’ version of PST in order to quantify the KERD for the loss of molecular hydrogen from the benzene radical cation, C6 H6 Cž ! C6 HCž 4 C H2 . They assumed that the excess internal energy is partitioned among all the transitional modes, which are converted to rotational and translational degrees of freedom of the products. A fair agreement between the modified PST and the experiment indicated that all the transitional modes were involved in the energy partitioning after the system had passed through the transition state. However, the calculated distribution was wider than the experimental KERD. It was suggested that two major factors could lead to a narrower experimental KERD: (1) the relative motion of reaction fragments becomes faster at larger separations preventing efficient coupling of the reaction coordinate with the rest of the transitional modes and lowering the probability for small KERs; (2) preferential energy partitioning to the H2 stretching mode would suppress the high-energy tail of the distribution. The experimental distribution can be narrower than the OTS/PST distribution if an additional bottleneck for dissociation occurs on the potential energy surface. For example, formation of an intermediate complex and its further rearrangement prior to fragmentation can impose an additional constraint on the available phase space. Bowers and co-workers have demonstrated that in case there is an additional barrier with a tight transition state (TTS) along the reaction pathway, the experimental KERD is substantially narrower than the distribution obtained using


Figure 11. A schematic reaction coordinate diagram for insertion of CoC into a C—H bond of C3 H8 . The fluxes through the orbiting and tight transition states are depicted as F orb and F ‡ , respectively. Reproduced from Ref. 50c with permission from American Chemical Society, Copyright  1990.

Figure 12. Experimental and theoretical KERDs for metastable loss of CH4 from nascent Co(C3 H8 )C collision complexes. The ‘unrestricted’ PST curve assumes the entrance and exit channel contain only orbiting transition states and that there are no tight transition states in between that affect the dynamics. The CoC –ethene bond energy has been separately determined and is not a variable in the PST calculation. The ‘restricted’ PST calculation includes a tight transition state for insertion into a C—H bond located at E ‡ D 0.11 eV (see Fig. 11) below the asymptotic energy of the reactants. Collisions that surmount the centrifugal barrier are reflected at the tight transition state above a particular orbital angular momentum. This restricts the range of angular momentum available to the products and reduces the KER. Reproduced from Ref. 50c with permission from American Chemical Society, Copyright  1990.

unrestricted PST.50c,d As an example, consider the schematic potential energy surface for CoC insertion into a C—H bond of C3 H8 (Fig. 11). Although the formation of products is highly exothermic, the presence of the TTS barrier provides a possibility for the intermediate complex, CoC C3 H8 , to decompose back to reactants. Restricted PST calculations for CH4 loss from CoC C3 H8 , which included a TTS, located 0.11 eV below the energy of the reactants provided excellent agreement between experiment and theory (Fig. 12).50c,d The

J. Mass Spectrom. 2001; 36: 459–478


barrier reduces the contribution of high angular momentum states to the final products, thus reducing the high-energy portion of the product kinetic energy distribution. At the time it was the first documented example of a transition state remote from the exit channel strongly affecting product energy distributions.

GENERAL TRENDS: TIME AND ENERGY DEPENDENCES Metastable ions studied in FFRs correspond normally to lifetimes in the microsecond range. The development of modern techniques such as ion traps has allowed varying the time window for metastable dissociations and with it the measured KERs. PEPICO experiments have enabled the measurement of KERDs at different internal energies and to determine the variation of the average KER with excess energy above the dissociation threshold. There is a one to one correspondence between the unimolecular rate constant k (i.e. the inverse lifetime) and the non-fixed excess internal energy above threshold (see Fig. 1), E‡ D E E0 . Experimental results can be compared with expectations from theory. For reactions with no reverse activation energy that behave statistically, the KER is expected to rise with increasing excess energy and with decreasing ion lifetime. Classically the average KER is expected to increase linearly with excess energy above threshold, hεi D E‡ /N, where N is the number of vibrational degrees of freedom. An empirical linear relation, hεi D E‡ /0.44N, due to Franklin and coworkers,66 has been used in the past. This relation has no sound theoretical basis and Klots67 derived a better equation: E‡ D [r C 1/2]hεii C i hi [exphi /hεi 1]1

41

where hεi D kB T‡ , r, as before, is the number of rotational degrees of freedom in the reaction products and i are the frequencies of the N vibrational degrees of freedom. The experimental evaluation of the average kinetic energy release hεi as a function of E‡ at low values of E‡ was of particular interest since Eqn (41), contrary to Franklin’s equation,66 predicts that in this energy range the fraction of energy acquired (hεi/E‡ ) should increase rapidly as E‡ decreases. An experiment was performed68 in which hεi was measured for metastable ions of various reactions that have different excess energies E‡ in the usual microsecond time window of sector instruments. Excellent agreement was achieved between the experimental and calculated dependence of hεi/E‡ on E‡ using Eqn (41). One of the reactions studied68 was the formation of the cyclopentadienyl ion from aniline: C6 H5 NH2 C ! c C5 H6 C C HNC

42

This reaction had been studied before and it was known that a large excess energy is required in order to observe it on the microsecond time-scale. Baer and Carney69 pointed to an internal inconsistency in their conclusions that were based on coincidence (PEPICO) measurements. A loose transition state model gave an excellent fit to the rate energy kE dependence of the aniline reaction. This implied a type I


surface with no reverse activation energy. However, the transition state energy was in considerable excess above the dissociation limit corresponding to the most stable products—the cyclopentadienyl cation and neutral HCN. The metastable ion peak shape was observed70 to be pseudoGaussian and the KER was observed to decrease with increasing ion lifetime. This indicated that its origin was statistical and not due to a reverse activation energy. The inconsistency pointed out earlier69 , was resolved70 when it was realized that the cyclopentadienyl ion was indeed formed but together with HNC rather than HCN. That HNC is indeed being formed as the neutral fragment was later confirmed71 by collision induced dissociative ionization (CIDI) experiments. Reaction (42) has thus been demonstrated to be a good example of reactions demonstrating large kinetic shifts.72 The excess energy E‡ in the usual metastable time window is about68 1.5 eV. This is caused by a fairly high activation energy of about 3.3 eV and a slow rise of kE above threshold. Under these circumstances the variation of excess energy with ion lifetime leads to a clear variation in the resultant KER with ion lifetime. As ions of longer lifetimes are being observed experimentally, the excess energy is reduced and the resultant KER is lowered. The C2 elimination from C60 C and other large fullerene ions is a well-known example of reactions proceeding with a large kinetic shift: C60 C ! C58 C C C2

43

The earliest KER measurements were analyzed by PST73 and the average internal energy in the metastable parent ion was calculated to be 39 š 2 eV. Time-resolved KERs were measured74 for reaction (43) on an ion trap/reflectron mass spectrometer. The KERs were observed to decrease with increasing ion lifetime as expected for a KER of statistical origin in a reaction with no reverse activation energy. However, following a sharp decay in the KER at short times the KER was observed to level off at longer times (Fig. 13). These experimental results and KERs due to electroninduced decay of C60 zC ions have been modeled recently.75 The modeling included competition, with the dissociative decay reaction (43) of internally excited C60 C ions, from radiative decay in the visible. Radiative decay of C60 C ions becomes important in the microsecond time window and is the dominant decay process at longer times. This prevents the observation of reaction (43) at near threshold energies and does not enable to reduce the average KER below a certain limiting value even for very long-lived ions. Powis and co-workers76,77 were among the first to demonstrate, using PEPICO experiments, the statistical behavior of unimolecular reactions through energy dependences of the average KERs. The average KER was observed to increase smoothly as the excess energy increased and to represent a small fraction of this excess. Furthermore, there was good overall agreement between experimental and theoretically calculated dependences, e.g. for fragment ions from acetaldehyde and ethylene oxide. Similar dependences were observed by Baer and co-workers for the alkyl ions from alkyl iodides16,17b,c,72 and for ions from vinyl and ethyl bromide.17d

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473


THERMOCHEMISTRY

0.7

Organometallic reactions

0.6 0.5 KER, eV

0.4 0.3 0.2 0.1 0.0

0

20

40 tdelay , µs

60

80

Figure 13. Average kinetic energy release (KER) versus reaction time in microseconds for reaction 43. Open circles are experimental data from the ion trap/reflectron experiment; filled circles are from electron-induced decay experiments. The continuous line is the model calculation that includes dissociative and radiative decays. Reprinted by permission of IM Publications from kinetic energy releases electron-induced decay of C60 zC by Matt S et al., European Journal of Mass Spectrometry, Vol. 5, pp 477–484, Copyright IM Publications 1999.

An interesting and valuable property of the statistical KERDs is their scalability.2,78,79 Scaling of the KERDs brings out similarities among distributions obtained at different energies,17c helps to compact large amounts of data and facilitates comparison with theoretical models.77 When appropriately renormalized KERDs are plotted as a function of a reduced energy parameter they have quantitatively the same form over a range of values of the energy. The appropriate scaling parameter is either the total excess energy or the average KER. The reduced or scaled KERDs for all ions at all energies are plotted in Fig. 14 for the methyl iodide reaction.2

CH3I+

Comparison between theory and experiment allowed probing of the ion structures and energetics, and the mechanisms and dynamics of organometallic reactions. Ion–molecule reactions lead to insertion of metallic ions into hydrocarbon bonds and formation of chemically activated collision complexes. These dissociate unimolecularly. As was discussed earlier for ordinary unimolecular reactions, the resultant KERDs depend upon whether the potential energy profiles belong to type I or type II. KERDs that peak at low energy and fall off smoothly at higher energies are suggestive of statistical processes with no reverse activation energy in the exit channels. Bowers and co-workers modeled such KERDs by OTS/PST to deduce thermochemical information.50,80 A recent example11 involves reactions of ground-state TiC (4 F) and VC (5 D) with propane. KERDs were measured for H2 loss from TiC (C3 H8 ) and VC (C3 H8 ). In the calculation of the model KERDs, the only parameter allowed to freely vary was rxn H0 0 of the reaction MC C C3 H8 ! MC C3 H6 C H2

44

where MC is either TiC or VC . Since the heats of formation of all the species in reaction (44) except for the organometallic ion are known, the determination of rxn H0 0 enables one to deduce the organometallic bond energies. We have discussed earlier a documented example,50c,d CoC C3 H8 , of a tight transition state (TTS) remote from the exit channel strongly affecting product energy distributions. That example involved elimination of methane. In the case of the TiC and VC reactions (44) discussed here, the average KER calculated including a TTS between the electrostatic well (associated with the MC C3 H8 complex) and the exit channel was identical with the value calculated without inclusion of this TTS. Inclusion of the barrier does not significantly restrict the angular momentum for H2 loss (contrary to the methane loss discussed earlier)

CH3+ + I

Probability (Arb. Units)

474

0

1.0

2.0

3.0

Translational Energy (E/ E)

Figure 14. The KERD for CH3 IC ! CH3 C C I plotted versus the scaled kinetic energy ε/hεi. Each symbol represents data taken at different excess energy that ranges from 0.04 to 3 eV. Reproduced with permission from Ref. 72.


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because this process is already restricted to low angular momenta, owing to the low mass and polarizability of H2 . Dehydrogenation reactions are thus good candidates for deducing thermochemical information. Modeling the KERDs for dehydrogenation from propane11 and ethene80 led to the following bond energies: D00 TiC —C3 H6 D 34.5 š 3, D00 VC —C3 H6 D 30.7 š 2, D00 TiC —C2 H2 D 51 š 3 and D00 VC —C2 H2 D 41 š 2 kcal mol1 (1 kcal D 4.184 k.J).

(NH3)n-1 H+ + NH3

2

4

10

8 ε, meV

Proton-bound clusters

(NH3)n H+

6

81

Proton-bound clusters were reviewed by us several years ago. All the protonated clusters studied undergo unimolecular decompositions by ‘boiling-off’ one solvent molecule on a time scale of microseconds. For example, protonated ammonia clusters undergo the reaction C

NH3 n H ! NH3 n1 H C NH3

45

These reactions possess no reverse activation energies and their KERDs are statistical. The measured KERs have been used to deduce the binding energies of the consecutive desolvation steps. Gas-phase ion equilibria studies using high-pressure mass spectrometry have led to a wealth of thermochemical information on the solvation of the hydrogen ion. However large clusters cannot be studied using high-pressure mass spectrometry because of interference from condensation. The measured KERs and their modeling by statistical theories have therefore been fairly useful2 in deducing binding energies. Of special interest have been the dependences of the average KERs on cluster size. The average KER hεi for reaction (45) rises with cluster size between n D 2 and 5 reaches a maximum at n D 5 and then drops and levels off at higher n15a,82 [see Fig. 15(a) and (b)]. The cluster size n D 5 is a well-known ‘magic number’ for ammonia clusters, demonstrating special abundance in mass spectra. The special stability is due to closing of the first solvation shell at n D 5. The quantitative theoretical interpretation for magic numbers in kinetic energy releases has been carried out83 using both RRKM/QET and FHBT. The phenomenon can also be understood intuitively, remembering the rate/energy dependence [Eqn (1)]: the excess energy, E‡ , necessary to observe dissociation in the FFR increases with increasing cluster size and with increasing binding energy. The binding energy decreases with increasing n so the two effects counteract one another. The rise in cluster size is the dominant factor for n D 2–5 leading to a rise in the average KER, hεi. However, there is a sharp drop in the binding energy between n D 5 and 6 that causes the excess energy to level off between n D 5 and 6 and causes a decline in hεi. Average KERs were determined for ammonia clusters up to n D 17 and the binding energies were deduced15a,84 using two theoretical models due to Engelking85 and Klots,83 respectively (see also Ref. 2). The results are summarized in Fig. 16. The binding energies for the clusters with low n derived from the KERs are in good agreement with literature values based on equilibrium high-pressure mass spectrometry. Castleman and co-workers86 studied several other proton-bound cluster series using this method.


2

0

3

5

6

7

n 12

10

< Er >, meV

C

4

8

6

4

2

3

5

7

9 11 13 Number of Ammonia, n

15

17

19

Figure 15. (a) Average kinetic energy releases εD hεi) in meV for reaction (45) as a function of cluster size n for n D 2–7. Reproduced with permission from Ref. 82 Copyright  1989 American Chemical Society. (b) same as (a) for n D 4–17. Reproduced from Ref. 15a with permission of American Institute of Physics, Copyright  1990.

Fullerenes The current status of the C2 binding energy in C60 has been reviewed by us recently.12 The large number of degrees of freedom in C60 C and the competition from radiative decay lead to a very large kinetic shift and prevent the determination of the binding energy from a simple appearance energy measurement. A major method employed to determine the binding energy in C60 C has been the measurement of the KERD and the average KER for the reaction discussed above—evaporation of C2 , reaction (43).12,21,56,60,62,81,87 A typical experimental KERD and two theoretical KERDs are shown in Fig. 9. The latter were calculated by the approaches discussed earlier in the theoretical section—the model free and the SSP model, respectively. There is good agreement between the

J. Mass Spectrom. 2001; 36: 459–478

475

476


From high-pressure mass spectrometry Deduced from KERS using klots' theory83 Deduced values using Elgelking's model85(see text)

Figure 16. A plot of calculated binding energies of NH3 n HC , n D 4–17, as a function of cluster size. Reproduced from Ref. 84 with permission of American Institute of Physics, Copyright  1990.

experimental KERD and the models. The two models nearly overlap so that it is hard to distinguish between them.60 These theoretical fits then allow one to derive T‡ and using Eqns (29) and (30) one obtains the C2 binding energy. Using G D 33 leads to a C60 C binding energy,12 Evap , in excess of 9.5 eV. Although there are several other methods that lead to the same or to a similarly high binding energy,12,59 this value is still controversial. It has been shown that incorporation of noble gas atoms in C60 leads to a stabilization of the fullerene cage. Binding energies of the noble gas containing fullerenes increase with the size of the endohedral atom. The stabilization is due to the van der Waals interaction between 60 carbon atoms composing the cage and the endohedral atom. Endohedral metal atoms have a strong stabilizing effect on a C82 cage. It is well accepted that endohedral metal atoms donate electrons to a fullerene cage leading to the formation of a strong ionic bond between the fullerene and the metal atom. However, examination of KERDs associated with dissociation of metallofullerenes revealed that the formation of an ionic bond is not the only effect that needs to be considered. For example, although both Sc and Tb atoms are believed to donate two electrons to the C84 cage, it was found that two Sc atoms slightly destabilize the C84 cage, whereas incorporation of two Tb atoms leads to a substantial stabilization. The difference between the influences of these atoms on the carbon cage was attributed to the presence of a partly filled f-shell in the electron configuration of the Tb atom.

THE EFFICIENCY OF PHASE SPACE SAMPLING Powis has pointed out long ago78 that the surprisal analysis of KERDs provides the most rigorous test of the statistical


model for the dissociation of ions. Much of the early work64,88 and some of the recent work52 dealt with bimodal distributions. However, the real test of the theory63 is in the analysis of reactions proceeding via type I potential energy surfaces leading to seemingly statistical KERDs. It has been claimed63e that the KERDs provide a more severe test of the statistical theories of mass spectra than the measurement of unimolecular reaction rate constants. Lorquet discussed recently89,90 the efficiency of phase space sampling based on such analyses. One of the first reactions to be studied was63a,b C6 H5 BrC ! C6 HC 5 C Br

46

Reaction (46) is a simple bond cleavage91 characterized by a totally loose transition state. The KERD for this reaction had been determined experimentally before and found to be91 in very good agreement with PST calculations. Nevertheless, a careful surprisal analysis63b demonstrated incomplete randomization of the internal energy. A value of 0.26 š 0.02 was obtained for the entropy deficiency distribution at an internal energy of 0.85 eV above the reaction threshold. From this value it was concluded that about 77% of the transition state phase space is effectively sampled. Application of the surprisal analysis method to reactions of other halobenzene ions,63c pyridine,63d vinyl bromide,63e ethyl iodide47 and iodopropane63f led to the conclusion that about 75–80% of the available phase space is effectively sampled prior to dissociation. This is thought90 to be sufficient to make a statistical treatment meaningful. An interesting question raised63e was, ‘how does the efficiency of phase space sampling vary with internal energy?’ It was found that the fraction of phase

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space sampled by the pair of dissociating fragments, eDS , which is necessarily 100% at zero internal energy, first decreases, passes through a shallow minimum and then increases again, reaching almost 100% at high internal energies. This is a seemingly strange phenomenon90 because as E increases, the lifetime of the ion rapidly decreases, leaving less time to explore an increasing volume of phase space. This was interpreted63e,90 by proposing that for high energies energy randomization is enhanced by internal conversions due to conical intersections. These conical intersections between electronic potential energy surfaces are recognized as a source of chaos in the dynamics. A cascade of extremely rapid radiationless transitions brought about by a complicated network of surface crossings extending over a wide range of molecular geometries prepares the system in a random manner not necessitating exclusive dependence on anharmonic coupling among vibrational modes for intramolecular vibrational energy randomization. These conical intersections are also thought63e to be the reason why statistical theories are found to work extremely well in the case of ions.

CONCLUSION This review has demonstrated that KERDs continue to give extremely interesting information on structures, energetics and dynamics in ionic systems. Further interesting results are expected for many systems, but particularly for biomolecules. The question concerning the statistical sampling of phase space will have to be investigated further since the role played by angular momentum conservation in the maximum entropy method has to be examined in more detail. The present review emphasizes statistical dissociation on type I potential energy surfaces. Future theoretical efforts are called for including further dynamical studies and trajectory calculations on type II potential energy surfaces.

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