MassKinetics: a theoretical model of mass spectra incorporating physical processes, reaction kinetics and mathematical descriptions

Abstract

A theoretical framework and an accompanying computer program (MassKinetics, www.chemres.hu/ms/ masskinetics) is developed for describing reaction kinetics under statistical, but non-equilibrium, conditions, i.e. those applying to mass spectrometry. In this model all the important physical processes influencing product distributions are considered: reactions, including the effects of acceleration, collisions and photon exchange. These processes occur simultaneously and are taken into account by the master equation approach. The system is described by (independent) product, kinetic energy and internal energy distributions, and the time development of these distributions is studied using transition probability functions. The product distribution at the end of the experiment corresponds to the mass spectrum. Individual elements in this scheme are mostly well known: internal energy-dependent reaction rates are calculated by transition state theory (RRK or RRKM formalisms). In the course of collisions, energy transfer and other processes may occur (the latter usually resulting in the ‘loss’ of ion signal). Collisions are characterized by their probability and by energy transfer in a single collision. To describe single collisions, three collision models are used: long-lived collision complexes, partially inelastic collisions and partially inelastic collisions with cooling. The latter type has been developed here, and is capable of accounting for cooling effects occurring in collision cascades. Descriptions of photon absorption and emission are well known in principle, and these are also taken into account, in addition to changes in kinetic energy due to external (electric) fields. These changes in the system occur simultaneously, and are described by master equations (a set of differential equations). The usual form of the master equation (taking into account reactions and collisional excitation) was extended to consider also radiative energy transfer, kinetic energy changes, energy partitioning and ion loss collisions. Initial results show that close to experimental accuracy can be obtained with MassKinetics, using few or no adjustable parameters. The model/program can be used to model almost all types of mass spectrometric experiments (e.g. MIKE, CID, SORI and resonant excitation). Note that it was designed for mass spectrometric applications, but can also be used to study reaction kinetics in other non-equilibrium systems. Copyright © 2001 John Wiley & Sons, Ltd.

INTRODUCTION

The desire to understand the physical processes governing the formation of mass spectra is nearly as old as mass spectrometry itself. The resulting efforts range from a qualitative understanding of fragmentation routes of organic compounds1–3 to the use of quantum chemistry for studying the structure and energetics of isolated ions4 and to the description of reaction rates leading to products. Soon after the development of transition state (or absolute reaction rate) theory,5 it was applied to mass spectrometry. The Rice–Ramsperger–Kassel (RRK)6, 7 and Rice–Ramsperger–Kassel–Marcus (RRKM) theories8, 9 have the same roots as the quasi-equilibrium theory (QET) developed by Rosenstock et al.10 QET and RRKM have successfully been used in many cases to evaluate energy-dependent reaction rates and to explain the results of various mass spectrometric experiments. These theories and their applications to mass spectrometry have been well reviewed11–16 and will not be described in detail here. Internal energy effects in mass spectrometry have also been reviewed recently.17

In addition to energy-dependent reaction rates, many other physical parameters influence the outcome of mass spectrometric experiments. Foremost among these is the internal energy distribution of the population of ions of interest and changes in this distribution. The latter may be intentional (i.e. collision-induced decomposition (CID)), or unintentional, e.g. infrared (IR) cooling of excited ions.18, 19 Of course, ‘intentional’ or ‘unintentional’ depends on the purpose of the experiments. Various excitation/de-excitation processes may (and do) occur in parallel. In some experiments (which are relatively simple to model) excitation is fast compared with fragmentation, and the two processes can be separated in time. In other cases, such as sustained off-resonance irradiation (SORI20) or blackbody infrared radiation-induced decomposition (BIRD21–23) experiments in Fourier transform ion cyclotron resonance (FT-ICR) instruments, excitation, de-excitation and fragmentation occur in parallel (have similar rate constants), and cannot be separated in time.

In recent years, a number of theoretical and numerical approaches have been developed to model mass spectrometric processes (in addition to ‘straightforward’ reaction rate calculations). A comprehensive discussion of these is outside the scope of this paper and only those which relate most closely to the algorithm described here will be mentioned. The motion of ions in electromagnetic fields can be well modeled by the SIMION program.24 This program takes into account ion–molecule collisions, but internal energy changes and fragmentation processes are only very crudely considered. More powerful but specialized programs have been developed to model ion motion in ion-trap instruments.25 The ion-trap program ITSIM has been extended to model fragmentation and collisional de-excitation in detail (W. Plass and R. G. Cooks, to be published). Collisional excitation and de-excitation processes were modeled by Goeringer and McLuckey,26, 27 in work also mainly oriented towards ion-trap applications. Most of these models use trajectory calculations and a random, Monte Carlo-type selection of initial conditions.

Collisional energy transfer is one of the many processes that determine the outcome of a collision event. This process has been studied for a long time in mass spectrometry. Many groups have made significant contributions to this field;28–39 a full review is not attempted here. Nearly without exception, it is assumed that the timeframe of collisional interaction is much shorter than that for chemical reaction. Collisional cooling (de-excitation) is important to consider in many applications. Under thermodynamic conditions, the concept of detailed balance provides the connection between the form of excitation and de-excitation.40 We have used the same physical principle to connect excitation and de-excitation processes under mass spectrometric (i.e. non-thermodynamic) conditions. In other words, knowing the form of collisional excitation and taking into account the principle of detailed balance, it is possible to determine the form of collisional de-excitation. One of the difficulties in understanding these energy transfer processes is that the range of internal and kinetic energies occurring in most mass spectrometric experiments is significantly different from, and much less well understood than, those occurring either under thermal conditions or in chemical activation. In spite of these differences, results and assumptions normally used in chemical kinetics are important to consider.41, 42

In addition to collisional energy exchange, IR radiation is also important for determining changes in the internal energy distribution. Depending on the experimental conditions, IR cooling or IR excitation may be predominant.18, 19, 43–47 Two typical examples of IR excitation are IR multiphoton dissociation (IRMPD)47–49 and BIRD.21–23 IR emission and absorption processes are well understood in principle (Einstein and Planck radiation laws50), but in practice energy exchange rates can be predicted only with large uncertainty.18 Collisional and radiative energy transfers occur in parallel with chemical changes, i.e. with fragmentation. This complicates discussion enormously, but in many cases (especially in the case of slow heating) it is essential that they be taken into account. This is done through the use of so-called master equations, which are a set of differential equations.51

Descriptions of mass spectrometric experiments based on fundamental physical processes and complex mathematics are very challenging. Our aim was to describe these experiments using predominantly molecular and instrumental data and as few adjustable parameters as possible. For this purpose, we have developed a computer program (MassKinetics), which is based on the physical model and the mathematical procedures described here. Most molecular parameters required as input data are relatively easy to calculate with reasonable accuracy or are relatively easy to determine experimentally. The instrumental parameters used are also straightforward. These relate to the time-scale of the experiments and to the acceleration of ions in electric fields. In a typical case, modeling mass spectrometric experiments requires no more than one or two adjustable parameters.

The aim of the paper is twofold. First, we present a comprehensive description of non-equilibrium reaction kinetics applicable to mass spectrometry. This consists of different aspects (models, methods, assumptions, etc.) that relate to various effects, such as reaction rate calculations, use of internal energy distributions, etc. We summarize them briefly, but our main effort is to synthesize our present understanding to create a general descriptive scheme for mass spectrometric reactions. We believe this to be the first case in which the simultaneous occurrence of all these processes has been considered in one model. We present the most relevant mathematical expressions used in mass spectrometry to describe reaction kinetics, collisions, IR radiation, energy distributions, etc. We believe that this could help in future research as a starting point, and may also be useful for teaching fundamental mass spectrometry at an advanced level.

Second, we want to show that, in spite of the complexity of the relevant processes (and of its mathematical description), the framework presented here could also be useful in practice. On the basis of the mathematical description presented, a computer program was developed (MassKinetics) which can be used to model reactions (predicting ion ratios) under various mass spectrometric conditions. In MassKinetics, our main concerns are changes in the product distribution and in the internal energy distribution for each ion considered. Note that in the program, equations and distributions characterizing the whole ion population are used, so it is unnecessary to use a random selection of initial parameters. The program is designed for ‘general’ use, i.e. not to simulate a particular experiment type, but to be easily applicable to most mass spectrometric experiments. Initial results indicate that ion ratios can be predicted very accurately using only a few adjustable parameters.

PHYSICAL PROCESSES CONSIDERED

An accurate mathematical description of the diverse physical processes occurring in a mass spectrometric experiment is complex, and involves various indices, variables and parameters, which are difficult to follow. To simplify discussion, the relevant physical processes and abbreviations used in the paper are summarized in Tables 1–5. Physical processes influencing product, internal energy and kinetic energy distributions are shown in Table 1. Mass spectrometric experiments can be built up using a series of events (such as ‘ionization’, etc.), listed in Table 2, which will be discussed in detail later. In Table 3 the most important features of various collision models are shown. Definition of symbols and abbreviations used in the equations and in the text are summarized in Tables 4 and 5. These will be repeated in the text only to facilitate discussion and to avoid misunderstanding. In general, parameters related to a given compound are distinguished by index i, those related to a compound pair (e.g. data characterizing a reaction) by indices i, j, and m indicates the mth oscillator of a compound. To simplify discussion to some extent, the variables of functions, distributions and probability functions are omitted from some equations, but these are listed in Table 4. Note that although the physical processes listed in Table 1 are discussed separately, they occur simultaneously. This simultaneity is taken into account using the master equation approach.

Table 1. Connection between physical processes considered in MassKinetics and distributions used to characterize the system

Physical process	Parameters used to characterize the system
	Product distribution	Internal energy distribution (IED)	Kinetic energy distribution
Acceleration	—	—	Calculated by well-known equations describing the movement of point-charges in electronic fields
Reaction	Determined by differential equations; the reaction rate is calculated by the transition state theory considering unimolecular reactions by RRK or RRKM formalisms	Fragmentation: depletes/changes the internal energy of precursor	Kinetic energy partitioning, based on well known kinematic equations
		Internal energy partitioning: determines the IED of the product
Collisions	Scattering, neutralization and other reactions, usually resulting in ion ‘loss.’ These occur within the lifetime of the collision complex, governed by non-statistical processes	Changes are described by the internal energy transfer functions, P^IET,SC. Reactions occur after energy randomization, described by statistical processes	Well-known kinematic equations, using the scattering angle (distribution) as a parameter; the latter is straightforward for spherical molecules
Photon emission/absorption	—	Radiative energy transfer, P^RET, based on Einstein and Planck radiation laws	—

Table 2. Experimental sequences (‘events’) considered in MassKinetics

Experimental event	Processes occurring	Description, comments
Formation of ions (ionization)	Ionization	Production of a given ion with a given internal energy distribution and (usually thermal or zero) kinetic energy. Assumed to be an ‘instantaneous’ process. Depending on the experiment, either ion formation or ions leaving the source may be taken as the beginning of the experimental sequence. If necessary, an initial product distribution (with accompanying internal and kinetic energy distributions) can also be specified
Field-free flight (with fragmentation but without collisions)	Reactions, and radiative energy exchange	Occurs for a certain time, usually the time necessary to traverse a distance
Single ion selection	Reactions and radiative energy exchange followed by deleting the concentration of all but one ion	Field-free flight without collisions for a certain period, followed by an ‘instantaneous’ change in product distribution—setting all but one ion concentration to zero. Depending on the experiment, with or without the loss of fragments formed during the field-free flight period. In special cases not only one but several ion concentrations may be left intact
Ion selection by ‘scanning’, followed by detection	Reactions and radiative energy exchange	Field-free flight without collisions for a certain period, with or without the loss of fragments formed during this time. The resulting product distribution forms the mass spectrum
Acceleration without collisions	Reactions, radiative energy exchange and change in kinetic energy	Same as field-free flight without collisions, but with a change in kinetic energy due to external acceleration (electric fields)
Acceleration with collisions	Reactions, radiative and collisional energy exchange and change in kinetic energy	The most general case; all energy transfer processes occur simultaneously. Note that the kinetic energy is changed not only by the external electric field, but also due to collisions
Field-free flight with collisions	Reactions, change in kinetic energy, radiative and collisional energy exchange	The general form of collisional processes, without external acceleration. Occurs for a certain time, usually the time necessary to traverse a given distance. Note that in collisions the kinetic energy (and therefore the velocity) also changes. Duration can be specified either by residence time or by the distance to be traversed
keV collisions in a collision cell (typical for sector/TOF instruments)	Reactions, radiative and collisional energy exchange. Scattering is important	A particular case of ‘field-free flight with collisions.’ There are usually only few collisions, and the collision energy can be considered as a constant value. Ion loss due to scattering and neutralization is often taken into account
eV collisions in a collision cell (typical for quadrupole instruments)	Reactions, change in kinetic energy, radiative and collisional energy exchange	A particular case of ‘field-free flight with collisions’. The collision number can be fairly high, reducing the initial kinetic energy (and velocity) of the ions significantly
Collision cascade (typical for resonant excitation in FT-ICR instruments)	Reactions, change in kinetic energy, radiative and collisional energy exchange	A particular case of ‘field-free flight with collisions,’ which is usually preceded by ion acceleration. The initial kinetic energy of the ion is reduced close to zero in a series of collisions (transferred to internal energy or to the collision gas)
Collisions occurring in parallel with periodical changes in ion velocity due to external radiofrequency fields (typical for SORI in FT-ICR or ‘tickling’ in ion-trap instruments)	Reactions, radiative and collisional energy exchange, and change in kinetic energy	A particular case of ion acceleration with collisions. The ion kinetic energy will depend on the external radiofrequency field, and will change rapidly and periodically during the experiment. Kinetic energy loss of the ions due to collisions can be neglected. The time dependence of kinetic energy can be converted into a time-independent kinetic energy distribution. At the end of the excitation event the ‘residual’ kinetic energy of the ion will be lost in a collision cascade (see above). Usually only one ion is accelerated by the field, other ions (e.g. fragments) behave as in a collision cascade

Table 3. Description of collision models used

Collision model	Internal energy change	Kinetic energy change	Comments
Long-lived collision complex	Statistical: Eqn (22) Equipartitioning: Eqn (25)	Eqn (27)	Assumes equilibration between internal and translational energies in every collision. Appropriate to describe some low-energy collisions
Completely inelastic collision	Ecom	Eqn (27)	A limiting case, all of the com collision energy is converted into internal energy, and the two collision partners stick together
Completely elastic	0	Eqns (28) and (29)	Another limiting case, only scattering occurs, no change in the internal energy
Partially inelastic	ηEcom	Eqns (31) and (32)	The most typically used model in mass spectrometry. A fraction of the com collision energy is converted into internal energy
Partially inelastic with cooling effect	Eqn (44)	Eqns (31) and (32)	Similar to the partially inelastic model, but de-excitation is also taken into account. Can describe adequately all collision types. All previous models (even the ‘long-lived collision complex’) may be regarded as a limiting case of this model

Table 4. Definition of symbols used in equations

Description of the parameter	Sign or abbreviated form	Unit	Usage in equations, full form with arguments	Comment, description
Compound	A	–	Ai	ith compound
			Ax	A complementary product in a reaction, not considered in further reactions
Einstein A coefficient	AEin	s^−1	AEin(ν,n,I0)	Einstein A coefficient
Pre-exponential factor	APE	s^−1		Pre-exponential factor for reaction Ai → Aj
Einstein B coefficient	BEin	m^3 Hz J^−1 s^−1	BEin(ν,n,I0)	Einstein B coefficient
Relative concentration	c	arb.	ci	Concentration of ith compound (the sum of initial concentration of all ions is usually normalized to unity; typically that of the molecular or precursor ion is unity and zero for the products)
			ci(t)	Time-dependent relative concentration of ith compound
Speed of light	c	m s^−1	c	2.997 × 10^8
Relative full width at half-height	δ	–	δ	Relative full width at half-height; relative to Ecom in the case of a Gaussian internal energy transfer distribution
Dirac function	δ	–	δ(x)	Mathematical Dirac delta function
Degree of freedom	D	–	Di	Degree of freedom (ith compound)
			DG	Degree of freedom (collision gas)
			DT,G	Translational degree of freedom of the collision gas
			Drot	Rotational degree of freedom
Electrostatic field strength	ε	V m^−1	ε	Electrostatic field strength
Internal energy transfer in a single collision	ΔEint	J		Internal energy transfer, inelastic collision
				Internal energy transfer, elastic collision
				Internal energy transfer, partially inelastic collision
				Internal energy transfer, partially inelastic collision with cooling
				Average internal energy transfer, partially inelastic collision with cooling
Critical energy	E0	J		Critical energy for reaction Ai → Aj
Collision energy (in a thermal system)	Ecoll,T	J	Ecoll,T	Collision energy (in a thermal system) at temperature T
			〈Ecoll,T〉	Mean collision energy (in a thermal system) at temperature T
Kinetic energy, center of mass	Ecom	J	Ecom,i	Center of mass kinetic energy of the ith compound
Center of the Gaussian distribution	Ecent	J	Ecent	Center of the Gaussian distribution
Internal energy	Eint	J	Eint,i	Internal energy of the ith compound (vibrational and internal rotational energy above the zero point)
			Eint,G	Internal energy of the collision gas
			〈Eint,i〉	Average internal energy of the ith compound
				Molecules with this amount of internal energy will be in equilibrium in a system characterized by a translational energy resulting in Ecom collision energy
Kinetic energy release	EKER	J		Kinetic energy release for reaction Ai → Aj
Non-statistical part of the kinetic energy release	EKER, NS	J		Non-statistical part of the kinetic energy release for reaction Ai → Aj
Statistical part of the kinetic energy release	EKER, stat	J		Statistical part of the kinetic energy release for reaction Ai → Aj
Kinetic energy, laboratory frame	Ekin	J	Ekin,i	Laboratory frame kinetic energy of the ith compound
Non-statistical part of the energy partitioning	ENS	J		Non-statistical part of the energy partitioning for reaction Ai → Aj
Rotational barrier	ERB	J		Rotational barrier for reaction Ai → Aj used in RRKM calculations
Rotational correction	ERC	J		Rotational correction for reaction Ai → Aj used in energy partitioning
Rotational energy	Erot	J		Rotational energy of the ith compound
				Sum of rotational energies of reaction products j and x
Heat of reaction at 0 K	ΔERXN	J		Also known as energy balance. It is the difference in the electronic energy (including zero-point vibrational energy) of products and reactants for reaction Ai → Aj
Thermal energy	Etherm	J	Etherm(T)	Thermal internal energy at temperature T
‘Unavailable’ energy for reaction	EUA	J		‘Unavailable’ energy for reaction Ai → Aj
Error function	Erf	–	Erf	Mathematical error function
Collision inelasticity	η	–	η	Collision inelasticity (fraction of com collision energy transferred to internal energy)
Super-collision energy transfer	ηsuper	–	ηsuper	Collision inelasticity of a super-collision
Planck constant	h	J s	h	6.6256 × 10^−34
Integrated IR intensity	I0	Practical units	I0,i,m	Integrated IR intensity (of 0 → 1 quantum state transition) of the mth oscillator of the ith reactant
			I0(scaled)	Scaled integrated IR intensity
Unimolecular rate constant	k	s^−1	k^i,j(E)	Rate constant for reaction Ai → Aj at internal energy E
Boltzmann constant	kb	J K^−1	kb	1.3805 × 10^−23
Radiative cooling rate	kcool	s^−1	kcool	Radiative cooling rate
Flight length/distance	l	m	l	Obtained from the geometry of the mass spectrometer
Mass	m	kg	mi	Mass of ith compound
			mG	Mass of the collision gas
Density of states	N	–	Ni(E)	Density of states at E energy for the ith compound
				Density of states at energy E for a complementary product x in the reaction Ai → Aj + Ax, including translational degree of freedom formed in the reaction
			NRT,G(E)	Density of states of the collision gas including translational degree of freedom of the two collision partners grouped together
Number of compounds	nA	–	nA	Number of compounds considered in the reaction scheme
Super-collision probability	πsuper	–	πsuper	Probability of super-collisions relative to all energy transfer collisions
Gas pressure	P	Pa	PG	Pressure of the collision gas
Internal and kinetic energy distribution	P	–	Pi(Eint,Ekin)	Probability of the ith molecule has internal energy Eint and kinetic energy Ekin
Accelerative state transfer probability function	P^AST	–		Probability of transition in unit time from a state defined by E′int and E′kin to a state defined by Eint and Ekin for the ith compound due to external (electric) fields
Collisional state transfer probability function	P^CST	–		Probability of transition in unit time from a state defined by E′int and E′kin to a state defined by Eint and Ekin for the ith compound due to collision(s)
Single collisional state transfer probability function	P^CST,SC	–		Probability of transition in a single collision from a state defined by E′int and E′kin to a state defined by Eint and Ekin for the ith compound
Electrostatic state transfer probability function	P^EST	–		Probability of kinetic energy transition in unit time from E′kin to Ekin kinetic energy for the ith compound due to external (electrostatic) field
Single collisional internal energy transfer probability function	P^IET,SC	–		Probability of internal energy transition in a single collision from a state defined by E′int and E′kin to a state defined only by Eint for the ith compound in the case of the long-lived collision complex model
				Probability of internal energy transition in a single collision from a state defined by E′int and E′kin to a state defined only by Eint for the ith compound in the case of the partially elastic collision model. The shape of the function is indicated by either exp (exponential), hG (half-Gaussian) or G (Gaussian)
			× (Eint,E′int,E′kin)	Probability of internal energy transition in a single collision from a state defined by E′int and E′kin to a state defined only by Eint for the ith compound in the case of the partially elastic with cooling collision model. The ‘up’ refers to the excitation and ‘down’ to the de-excitation part of the function
			× (Eint,E′int,E′kin)
Internal energy distribution	P	–		Internal energy distribution of the ith molecule
Single collisional kinetic energy transfer probability function	P^KET,SC	–		Probability of kinetic energy transfer in a single collision from kinetic energy E′kin to Ekin
Kinetic energy distribution	P	–		Kinetic energy distribution of the ith molecule
Microcanonical occupation probability	P^occ	–		Probability that mth oscillator of ith molecule (at internal energy E′) contains energy E
Internal and kinetic energy partitioning probability function	PPART	–		Probability that in a reaction Aj → Ai, from the jth compound being in state E′int,E′kin the ith compound is formed in the Eint,Ekin state
Internal energy partitioning probability function	PPART,INT	–		Probability that in a reaction Aj → Ai, from the jth compound being in state E′int, the ith compound is formed in the Eint state
Kinetic energy partitioning probability function	PPART,KIN	–		Probability that in a reaction Aj → Ai, from the jth compound being in state E′kin the ith compound is formed in the Ekin state
Radiative energy transfer probability function	P^RET	–		The probability of radiative internal energy transfer in unit time from E′int to Eint (due to photon emission/adsorption)
Radiative state transition probability	P^RST	–	P^RST(Eint,Ekin,E′int,E′kin)	Probability of transition in unit time from a state defined by E′int and E′kin to a state defined by Eint and Ekin for the ith compound due to IR radiation. Note that P^RST does not depend on Ekin, so (according to Eqn (64) P^RST≡P^RET)
State transfer probability function	P^ST	–		Probability of transition in unit time from a state defined by E′int and E′kin to a state defined by Eint and Ekin for the ith compound due to collisions, radiative energy transfer and external fields
Unit state transfer probability function	P^UNIT	–		Unity function, describing the case when Ekin and Eint do not change
Charge	q	C	qi	Charge state of the ith compound
Radiation density	ρ	J m^−3 Hz^−1	ρ(ν)	Radiation density at frequency ν
Collision cross-section	σ	m^2	σi	Collision cross section of the ith compound
				Collision cross-section of the ith compound, resulting in ion loss due to scattering, neutralization or charge transfer in a single collision
Degeneracy	σ^d	–	σ^d	Reaction path degeneracy used in the RRKM expression
Full width at half-height	σ^hw	J	σ^hw	Full width at half-height of a (Gaussian) distribution
IR scaling factor	SfIR	–	SfIR	Scaling factor of integrated IR intensities, used to determine absorption/emission coefficients of IR radiation
Com frame scattering angle	Θ	rad	Θ
Temperature	T	K	TG	Temperature of collision gas
			TMS	Temperature of the mass spectrometer (external temperature)
Collision temperature	Tcoll	K	Tcoll	Collision temperature, related to Ecom
Translational energy loss	TEL	J		Translational energy loss of the ith compound, inelastic collision
				Translational energy loss of the ith compound, elastic collision
				Translational energy loss of the ith compound, partially inelastic collision
Excitation time	tex	s	tex	Excitation time
Internal temperature	Tint	K	Tint	Internal temperature, related to the internal energy
Potential energy difference	ΔU	V	ΔU
Electrostatic potential	U	V	U	Electrostatic potential
			U0	Amplitude of electric field
Vibrational frequency	ν	Hz	νi,m	mth vibrational frequency of the ith reactant, m = 1,…,Di
Scalar velocity	v	m s^−1	vi	Scalar velocity of the ith compound
			vG	Scalar velocity of the collision gas
			vi,G	Relative velocity of the ith compound and collision gas
Collision frequency	ω	s^−1	ωi	Collision frequency of the ith compound
Cyclotron frequency	ωc	s^−1	ωc	Cyclotron frequency in FT-ICR cell
Frequency of the excitation field	ωex	s^−1	ωex	Frequency of the excitation field in FT-ICR SORI experiment
Sum of states	W	–	Wi(E)	Sum of states at energy E for the ith compound

Table 5. Abbreviations used

AST	Accelerative state transfer
BIRD	Black body infrared radiation-induced decomposition
CID	Collision-induced dissociation
CST	Collisional state transfer
EL	Elastic (collision model)
EST	Electrostatic state transfer
FT-ICR	Fourier transform ion cyclotron resonance
IED	Internal energy distribution
IET	Internal energy transfer
INEL	Inelastic (collision model)
IR	Infrared
IRMPD	Infrared multiphoton dissociation
KER	Kinetic energy release
KET	Kinetic energy transfer
LLC	Long-lived collision (complex)
MIKE	Mass-analyzed ion kinetic energy (spectroscopy)
PEL	Partially elastic collision (model)
PELC	Partially elastic collision with cooling (model)
QET	Quasi-equilibrium theory
RET	Radiative energy transfer
RRK	Rice–Ramsperger–Kassel theory
RRKM	Rice–Ramsperger–Kassal–Marcus theory
RST	Radiative state transfer
SORI	Sustained off-resonance irradiation
TEL	Translational energy loss

Acceleration

The kinetic energy of ions (which determines their scalar velocity) is changed in electric fields (e.g. in electrostatic fields or electrodynamic fields established by applying radiofrequency pulses). Assuming the ions to be point changes, their motion is described by well-known physical equations. Changes in the direction of motion (e.g. as a result of electromagnetic fields) do not influence the kinetic energy, and will not be considered here. Acceleration has no direct effect on product distribution, or on internal energy distribution. Note, however, that a change in kinetic energy will change the collision conditions, and so will have an indirect influence on the internal energy and product distributions.

In a constant electrostatic field, ions are accelerated due to the potential difference, ΔU:

• (1)

To describe the effect of such fields on ions traveling a given distance, it is often advantageous to express it as a function of electrostatic field strength, ε = ΔU/l:

• (2)

The effect of complex (non-homogenous) electrostatic fields can be approximated by a series of homogenous fields. In addition to electrostatic fields, ions can also be accelerated by radiofrequency pulses. Two important cases are resonant excitation and SORI used in FT-ICR instruments. In the case of resonant excitation:

• (3)

where U₀ is the amplitude of the electric field and, t_ex is the excitation time. In the case of SORI, the kinetic energy of the ions changes in time according to the equation

• (4)

Reactions and reaction rates

Interconversion among n_A species can be described by () elementary reactions. Elementary reactions can be treated individually (although there are some common features, e.g. the transition state for A_i → A_j is the same as that for A_j → A_i). Typically many of the () reactions do not occur (have zero rate constant), so simplifying the reaction scheme. An example of a fairly complex reaction scheme, including consecutive, competitive (also called parallel) and reversible reactions is the following:

• (5)

Description of reaction rates is straightforward if all reactions are unimolecular, i.e. if the reaction rate depends only on the concentration of the precursor compound and the rate constant (which, in turn, depends on the structure and the state of the precursor). For a unimolecular reaction the concentration change is given by the following:

• (6)

For a set of reactions, the product distribution can be expressed by the following set of differential equations:

• (7)

Note that in this equation the first term indicates decomposition, the second formation of the ith compound (in the present context isomers can be regarded as different compounds). Note that the reaction rate constant depends on the internal energy, so the Eqn (7) has to be integrated over the whole internal energy range. Note also that the internal energy may change in time (e.g. due to excitation). If so, the change of internal energy (distribution) has to be followed in time, using the master equation approach.

Note that in this reaction scheme only one precursor and one product is considered for each elementary reaction. This situation is that which relates to mass spectrometric experiments, in which (usually) only one product (the charged product) is observed. If both products are charged, or if neutrals are of interest (e.g. in non-mass spectrometric applications) the reaction scheme should display the other products (and reactants) as well. Taking this into account, the first reaction would be more accurately given as A₁ → A₂ + A_x; A₁ → A_2a + A_2b; or A_1a + A_1b → A_2a + A_2b. For bi- and trimolecular reactions, the unimolecular k^i,jc_j(t) rate expressions in Eqn (7) should be modified also. For example, the rate expression for the bimolecular A₁ + A₂ → A₃ reaction is

• (8)

When necessary, these extensions can be incorporated into Eqn (7). To simplify discussion, this complication will not be explicitly mentioned in the following presentation. As a further simplification, instead of compounds, usually ions will be mentioned in the following, mainly because in mass spectrometry ions are nearly always considered.

For unimolecular reactions, the reaction rate depends on the molecular parameters of one molecule only, and the (electronic, vibrational and rotational) state of this (the precursor) compound. Most commonly, it is assumed that the precursor converts rapidly (compared with the reaction rate) into the ground electronic state. (If not, the electronic states should be considered separately and their interconversions can be treated as if they represented isomerizations.) Furthermore, it is commonly assumed that the internal energy distribution within a compound is statistically distributed among vibrations and internal rotations, and that the energy flow among these is fast compared with the reaction rate. These assumptions are used in most reaction rate theories, and are usually termed ‘statistical’ behavior. Note that in this paper, as is often done elsewhere, the internal energy (E_int) is defined as the sum of energies in vibrations and internal rotations of a molecule (or ion) above the zero point energy level. The translation and overall rotation of the molecule (and the associated energies) are treated separately.

At present there are two main theories describing internal energy-dependent reaction rates; the more common is the transition state theory5 and the other is the phase-space theory.52 In MassKinetics we use the transition state theory to calculate rate constants. In its early form, the transition state theory uses the RRK expression

• (9)

This is a good approximation only if distance between the energy levels of the oscillators (hν) is small compared with the mean internal energy per oscillator. As this condition is typically not satisfied, the RRK theory is good only for qualitative studies. To correct the gross mathematical error, the number of oscillators (D above) is often reduced by an arbitrary factor of 3–5 (and the term ‘effective oscillators’ is introduced).

Development of RRK theory (mainly the use of an appropriate mathematical treatment) resulted in the RRKM theory.12, 13 The latter is capable of accurately and quantitatively describing both microcanonical and canonical rate constants without the need for an empirical constant. The internal energy-dependent microcanonical rate expression is

• (10)

One of the main assumptions of the RRKM theory is that the reaction rate is determined by the nature of the transition state. In cases where there is no well-defined transition state (i.e. when the reverse reaction has no activation energy), the reaction rate is better described by phase space theory16, 52, 53 or by variational transition state theory.16, 54, 55 However, it has been found that RRKM can often describe reaction rates accurately, even if there is no well-defined transition state.

A complicating factor in rate constant calculations is that it is also influenced by the overall rotation of the molecule: the total angular momentum of a system should be conserved in the course of its reactions.56 Possibly the simplest way to treat this problem is to express it as a rotational barrier, E_RB,57 and modify the critical energy (E₀) by adding a term E_RB. (Note that the term ‘rotational barrier’ is used most often in the context of bimolecular reactions. In the case of a typical unimolecular fragmentation, part of the overall rotational energy of the molecule will be converted into translational energy of the products, so the ‘rotational barrier’ will be of negative sign, and will actually decrease the critical energy. Note also that the importance of this effect usually decreases with increasing molecular size.) Reaction rate theories will not be discussed in more detail here; they can be found in textbooks and in excellent reviews.12, 13

Chemical reactions influence not only the product distribution, but also the internal and kinetic energy distributions. Decomposition depletes the concentration of the precursor ion of a given internal energy, so the internal energy distribution of the precursor ion changes as a result of fragmentation. The product ion resulting from a given elementary process will be formed with a certain internal and kinetic energy distribution, which are determined by internal and kinetic energy partitioning functions. Note that energy distributions related to the product have to be calculated and followed in time only if the product is itself the precursor of a subsequent (consecutive) reaction.

Partitioning of kinetic energy is simple: it is directly proportional to the relative mass of the products. In the case of the reaction A_i → A_j:

• (11)

where refers to kinetic energy of the precursor and to that of the product ion. Note that the product kinetic energy will be slightly modified by the ‘kinetic energy release’ of a reaction, but this effect is usually so small that it is neglected.

Internal energy partitioning in the course of unimolecular dissociation may become complicated, as several effects have to be considered. In a simple case a given fraction of the internal energy of the activated reactant is not available (E_UA) since it represents the heat of reaction at 0 K (ΔE_RXN, also called the energy balance of a reaction), the difference of electronic and zero point vibrational energy level between products and reactants. The rest (E_int − ΔE_RXN) is statistically distributed among the degrees of freedom of the products, including relative translational motion (appearing as kinetic energy release, E_KER).

There are two effects modifying this simple picture which may be neglected in many cases, but have to be considered for completeness. One of these is the constraint of rotational momentum conservation and the other is the occurrence of non-statistical effects. The different energy types encountered in the discussion are illustrated in Fig. 1. Owing to angular momentum conservation, there are restrictions on rotational energy partitioning, so the rotational energy of precursor and products will be different.58–60 This difference can be expressed as the rotational correction (E_RC), and this energy will also be unavailable for internal energy partitioning:

• (12)

Note that in unimolecular dissociation reactions, part of the rotational energy is converted into internal or kinetic energy, so E_RC usually has a negative value. In the context of bimolecular reactions and collisions, analogous restrictions are usually expressed as a rotational or centrifugal barrier.

Figure 1. Energy profile of a reaction, indicating various types of energies involved in energy partitioning

Another complicating factor is that part of the available internal energy may be distributed non-statistically. In most cases it is due to the non-statistical part of the kinetic energy release distribution.61 In other words, this means that more energy will go to relative translational motion than that determined statistically:

• (13)

The non-statistical part of the kinetic energy release is important only in those cases when the corresponding metastable peak is wide and/or flat topped, which is easy to check experimentally.61 This usually happens when the reverse reaction has a significant critical energy (). Note also that in rate calculations the internal energy has far more importance than the kinetic energy. Partly for this reason (and partly because the relative velocity of reaction products is usually isotropic) it is reasonable to neglect the KER contribution to the kinetic energy [Eqn (11)], while it is important to consider its effect on the internal energy. Most of the assumptions used in the present treatment of energy partitioning relate to the statistical approach chosen, and it may not be important to consider other non-statistical effects. If necessary, a certain amount of energy may be deducted from before energy partitioning and added to or after statistical energy partitioning is performed (X referring to the other, usually neutral, reaction product). If the correction is small, a similar non-statistical correction could be achieved by increasing or decreasing in Eqn (13).

The statistically distributed part of the internal energy is divided among the degrees of freedom, including the relative translational degrees of freedom of the reaction products. This may be regarded as a consequence of the fast flow of internal energy inside the molecule. In the reaction A_i → A_j + A_x, the precursor being in the level, the product A_j being formed in the level, the internal energy partitioning probability function can be expressed as follows:

• (14)

Translational degrees of freedom of the relative motion of reaction products (i.e. the kinetic energy release) and internal degrees of freedom of A_x are grouped together, indicated by N_T,x above (note that relative rotational energies were taken into account using E_RC). Note also that the calculation requires vibrational frequencies for the complementary product A_x as well, which is usually neutral and often not considered explicitly.

A far simpler, but less accurate version of the same idea for energy partitioning is equipartitioning of the available internal energy among vibrations, internal and external rotations and translational motion:

• (15)

In several cases, especially for small molecules and in the case of competitive and consecutive reactions, equipartitioning [Eqn (15)] may lead to significant errors.58, 62, 63 Note that in relation to reaction kinetics, the internal energy is of prime importance, whereas the kinetic energy plays only a secondary role, mainly as an energy reservoir. For this reason it is important to consider small corrections (such as E_KER,NS and E_UA) related to the internal energy distribution, while similar corrections may be neglected in relation to kinetic energy distributions. This is done here (compare Eqns (11–15)), and will also be done in several cases later. These approximations become much better with increasing molecular size.

Collisional processes

Collisional processes depend on two main factors: the probability of a collision and the outcome of a single collision. The outcome of multiple collisions and collision cascades is determined by these two parameters using the master equation approach, to be described later.

The number of collisions is determined by the collision frequency (ω) and the time period over which collisions may occur. In FT-ICR and ion-trap instruments, the latter is determined explicitly. In the case of other instruments, the ions have to traverse a certain distance (e.g. the length of the collision cell), such that the collision time is determined from the geometry of the instrument and from the ion velocity (which may change in time, as collisions can slow the ions). The collision frequency depends on the collision cross-section, the relative velocity of collision partners (which often changes over time), and the number density of the collision gas. Assuming ideal gas behavior:

• (16)

Note that to simplify indices, dependence on the collision gas is usually not indicated (viz. use of ω_i instead of ω_i,G). An exception is v_i,G, used to emphasize that it is the relative velocity of the two collision partners. The collision cross-section always depends on both collision partners, and on the relative velocity (alternatively expressed as E_kin) also. In a simple case, the collision radius may be approximated by the sum of the radii of the two collision partners. For a fast ion, when the collision (or target) gas can be regarded as stationary, the relative velocity is

• (17)

When the fast ion is slowed in a collision cascade, the mean relative velocity will be determined by the thermal velocities:

• (18)

where µ_i,G is the reduced mass of the ith ion and the collision gas:

• (19)

Note that in such a case the collision partners will move in random directions. When the ions are slowing so that the collision gas cannot be regarded as stationary, exact calculation of the relative velocity distribution or of the mean relative velocity is complex. For most purposes, it is sufficiently accurate to use Eqn (17) to determine the relative velocity until it drops to the thermal relative velocity determined by Eqn (18). When the ions are moving with lower velocity (with respect to the laboratory) than that determined by Eqn (18), the latter equation should be used to approximate the mean relative velocity of the ion and the collision gas.

The duration of a single collision is considered, in most cases, to be short compared with the time taken for fragmentation. (Note that for a small organic ion, accelerated to keV energies, the collision time is ∼10⁻¹⁵ s, for a large ion at a few eV kinetic energy it is ∼10⁻¹² s, whereas a vibrational period is ∼10⁻¹³ s.) In a single collision, energy transfer may take place, and this is the main topic of this section. Fragmentation in such a case occurs only after the breakup of the collision complex and after energy randomization within the scattered ion. It is assumed that collisions will not influence the temperature of the collision gas (i.e. the heat capacity of the collision gas is much larger than that of the ions). In addition to energy transfer, charge permutation reactions (notably neutralization), scattering, chemical reactions etc. may also occur.64 These processes often result in the ‘loss’ of ions from the mass spectrum. When necessary, the probability of these processes can adequately be described by a cross-section and collision frequency, e.g. ion ‘loss’ is given by and , and their outcome (ion loss in this case) is taken into account in the master equations. Ion loss is usually considered important in keV collisions, where it has a high probability—the main reason why in sector instruments only one or few collisions are used. In low-energy collisions, ion ‘loss’ processes may or may not be neglected, and each case has to be considered separately. In the following (if not mentioned otherwise), only energy transfer collisions are discussed.

Kinetic to internal energy conversion in collisions is one of the major processes affecting internal energy in the course of CID. Note that in the following, energy transfer in a single collision is considered. Multiple collisions or collision cascades are built up of individual collision events, as discussed. Several models have been developed to describe collisional energy transfer,28–37, 65 but details of this process are still under discussion. The most important parameter influencing energy transfer in a collision is the center of mass (com) collision energy, E_com. It is the maximum amount of kinetic energy that may be converted into internal energy in a collision in a completely inelastic scattering. In the case of a fast projectile (ion) colliding with an approximately stationary gas molecule, E_com is determined from the (laboratory frame) kinetic energy:

• (20)

When the projectile is slowed (its laboratory frame kinetic energy is close to zero), the com collision energy will be determined by the temperature of the collision gas:

• (21)

As in the case of determining the relative velocities, E_com is calculated by Eqn (20) until E_com resulting from the directional motion (and calculated by Eqn (20)) falls below that due to thermal motion. When it happens, the value determined by Eqn (21) is used.

The collision energy has a distribution and may change in time—either because of collisions or due to external electrostatic fields, as in the case of SORI or resonant excitation. Beside the collision energy, molecular properties of the collision partners (internal energy, degrees of freedom, polarizability, charge state, dipole moment, etc.), collision trajectory (characterized usually by the impact parameter and the scattering angle), lifetime of the collision complex (which is related to E_kin), and orientation of the reaction partners in the collision also have a significant influence.30, 37, 42, 66 In most experiments, the collision trajectory, lifetime of the collision complex and orientation of the reaction partners cannot be selected, so the experimental result represents a particular average over these parameters. In model calculations, accordingly, the outcome of a collision should reflect a representative average over these parameters. When such an averaging is not feasible, Monte Carlo-type selection of initial conditions may also be used. The change in the kinetic energy of the fast ion (in the laboratory frame) resulting from kinetic energy transferred to the collision gas depends on the scattering angle, and can be calculated based on well-known kinematic equations.42 Collisional energy transfer can be described using various models; the following are the most important and most often used.

Long-lived collision complex model

This is also called the orbiting transition state model67 and abbreviated LLC. ‘Long-lived’ in reaction dynamics often means longer than a rotational period, sufficient to lead to random scattering angles. ‘Long-lived’ in the present context indicates a lifetime sufficiently long to lead to energy randomization in the collision complex. The latter is usually shorter than the rotational period, in the range 10⁻¹⁰–10⁻¹³ s. (Note that energy randomization does not necessarily require more time for large compared with small molecules.) In the long-lived collision complex model, the total energy of the system (sum of internal energies of the two collision partners and the com collision energy) is distributed statistically. Energy partitioning in an inelastic collision can be expressed in an analogous way to energy partitioning in a reaction, as described above. Owing to the necessity of angular momentum conservation, only part of the rotational energies of the collision partners will be available for energy partitioning. (‘Rotational energy’ relates to the overall rotation of a molecule; ‘internal rotations’ are included in the internal energy and hence in energy partitioning.) In theory, and within the framework of the classical mechanical model, angular momentum conservation is a straightforward way to take into account rotations in energy partitioning. However, in practice it is difficult to determine the rotational energy of a molecule flying through a mass spectrometer, because it depends on a number of variables and on experimental conditions (collisions, acceleration, etc.). Especially for larger molecules, it may not have a large effect. For simplicity in the present case we assume that the rotational energies are conserved in the collision. Another simple option is to add a certain amount of energy correction (e.g. 3/2 k_bT_G) to the internal energy in Eqn (22) below.

As will be discussed in more detail in the section on master equations, the probability of internal energy transfer in a single collision is indicated by a ‘collisional internal energy transfer’ probability function (P^IET), (often called simply collisional energy transfer). This relates to a single collision (P^IET,SC), based on the long-lived collision complex model (P^IET,SC,LLC), which has the arguments E_int, E_kin, E′_int and E′_kin, and which relates to the ith compound indicated by . This describes the probability that the ith compound, having E′_int internal and E′_kin kinetic energy will have E_int internal and E_kin kinetic energy after a single collision. To simplify abbreviations, the index related to the compound (i) and identification of the arguments (or some of the arguments) are sometimes omitted. When a particular collision model is discussed (such as LLC now), the index relating it to single collisions (SC) may be left out without loss of meaning. Taking these simplifications into account, will often be abbreviated as P^IET,LLC:

• (22)

Note that E′_kin characterizes the initial state of the compound and it determines E_com [Eqn (20)], which is used in the expression above. N_T,G(E) is the density of states of internal oscillators of the collision gas plus the relative translational motions of the two collision partners grouped together. The total internal energy of the collision complex is E_int + E_{int, G} + E_com, where E_com is the relative translational energy of the two colliding partners. Note that the calculation requires, among other data, the vibrational frequencies of the collision gas. The kinetic (translational) energy loss of the fast ion (in the laboratory frame) will be very close to that determined for inelastic collisions (see below), so that Eqn (27) can also be used for this collision model:

• (23)

As in the case of energy partitioning in the course of chemical reactions, statistical energy partitioning in a collision complex (within the LLC model) can be approximated by equipartitioning, with some (occasionally substantial) loss of accuracy. In such a case, the internal energy after a collision (E_int) is

• (24)

As discussed before, (part of) the rotational energy may be involved in equipartitioning; in such a case E_rot and D_rot contributions have to be included in Eqn (24). Analogously to Eqn (22), equipartitioning can also be expressed in the form of an internal energy transfer probability function, which has the following form:

• (25)

where P^IET,LLC,equi(E_int,E_kin,E′_int,E′_kin) is the probability of an internal energy transition in a single collision from a state defined by E′_int and E′_kin (note that E_com is defined by E′_kin) to a state defined by E_int and E_kin. This expression means that if the energy transfer is equal to the value determined by Eqn (24) its probability is one, otherwise no energy transfer can occur, so its probability is zero.

The long-lived collision complex model is usually thought to be a good assumption when the relative translational energy is smaller than or comparable to the binding energy of the collision complex. For ion–molecule complexes, the binding energy is in the order of 1 eV, so long-lived collision complexes may be expected to occur for energies below 1 eV com collision energy. Note, however, that satisfying this condition does not necessarily mean that this collision model will describe the system adequately.

Partially inelastic collision model

This model has two limiting cases: completely inelastic and completely elastic collisions. In the case of completely inelastic collisions, all of the com collision energy is converted into internal energy and (in principle) the two collision partners stick together:

• (26)

As the center of mass is moving with respect to the laboratory, this means that the collision gas will gain kinetic energy in the laboratory frame of reference. The kinetic energy of the fast ion (in the laboratory frame) will decrease by the sum of the amount of E_com, (converted into internal energy) and the kinetic energy imparted to the collision gas, this sum is also called the translational energy loss (TEL):

• (27)

In the case of completely elastic collisions, simple scattering only takes place in the com frame. In the laboratory frame, kinetic energy is transferred to the target, so elastic collisions result in translational energy loss of the projectile. The amount of energy transferred depends on the scattering angle, Θ, measured in the com frame:

• (28)

For random com scattering angles (integrating Eqn (28) over all possible scattering angles), the average translational energy loss of the fast ion becomes

• (29)

In reality, partially inelastic collisions occur, when a certain fraction (η) of the com collision energy is converted into internal energy, the rest remains as (center of mass) translational energy:

• (30)

The degree of inelasticity of a collision can be defined by η, which is often expressed as a percentage. While η defines the mean value of collisional internal energy transfer (IET), it is not a fixed value, but has a distribution. (Note that, in principle, the same also applies to the translational energy loss, TEL. Using the mean TEL value instead of the distribution is likely to have a very small effect on the results, and this approximation is used here. The effect is so small that this complication is often not even mentioned.) Note that the energy transfer in this model depends only on the com collision energy (for a given ion and a given collision gas), but does not depend on the internal energy. The translational energy loss (change of laboratory frame collision energy) depends strongly on the scattering angle, similarly to that discussed for elastic collisions. For a given (com) scattering angle, it is given as

• (31)

For random com scattering angles, the average TEL is

• (32)

In the case of partially inelastic collisions, the fraction of (com) translational energy converted into internal energy of the fast ion (η) can be regarded as a parameter. This parameter is likely to depend of the com collision energy, so it may be more accurate to indicate it as η(E_com). Our present knowledge suggests that with increasing collision energy, the fraction of com collision energy converted into internal energy decreases.30, 64, 66 Alternatively expressed, with increasing collision energy the interaction time between the colliding partners decreases, and with this the inelasticity also decreases.

At a given collision energy, the amount of internal energy transferred has a distribution. This distribution can be calculated, measured or estimated but, unfortunately, there is little direct information on the precise form, or of fine details, of the collisional energy transfer function. Fortunately, most mass spectrometric experiments are not very sensitive to the shape of the P^IET function. Using trajectory calculations,68–70 the internal energy transfer function is easily available from the calculation output. Measurement of P^IET requires the determination of either the vibrational energy content of the molecule16 or the full three-dimensional velocity map of the products (F. Muntean and P. B. Armentrout, work in progress). Usually such data are not available, and estimates are used. It seems both desirable and feasible to use only a few parameters in describing the shape of the P^IET function. Commonly it is assumed that the probability of energy transfer decreases with the amount of energy transferred, and usually an exponential form is assumed.30, 41 Note that, in the partially inelastic collision model, excitation always takes place while de-excitation is not considered, so E_int > E′_int. The probability of an internal energy change can be expressed as

• (33)

Note that, the parameter ξ would equal η if the function were not truncated at E_com. The parameter ξ can be determined by iteration using the following equation:

• (34)

The exponential function does not require parameters apart from the average inelasticity, η. In addition to the simple exponential, other distributions, such as half-Gaussian distributions, are occasionally used to describe collisional internal energy transfer probabilities:41

• (35)

where Erf is the mathematical error function (necessary for normalization to unit area). Note that η is also sufficient to describe this function without introducing further parameters. As an example, the shapes of these distributions are shown in Fig. 2(a), in the case of η = 0.1 (i.e. 10% efficiency). Regardless of the form of the P^IET distribution, the maximum amount of energy transfer cannot be larger than the com collision energy (i.e. the exponential or half-Gaussian distributions have to be truncated at E_com, as indicated in Eqns (33–35)).

Figure 2. Typical types of P^IET functions used in the ‘partially inelastic collision’ model: (a) exponential (solid line) and half-Gaussian (dashed line), with η = 0.1; (b) Gaussian, with η = 0.8 and δ = 0.2; and (c) super-collision, described by superposition of an exponential (η = 0.1) and a Gaussian (η = 0.8 and δ = 0.2), the latter having 5% relative probability

Note that it is not possible to describe efficiencies higher than 50% with such distributions (or with any other monotonically decreasing function) and that for over ∼30% energy transfer efficiency, both the exponential and half-Gaussian distributions will start to have unreasonable shapes. It is known that in some experiments high energy transfer does occur and in some collisions very close to 100% inelasticity must occur (up to a few eV com collision energy), otherwise threshold CID experiments could not provide accurate thermochemical results.38, 39, 71, 72 We suggest that such cases could be modeled by Gaussian distributions. (Note that this distribution also has to be truncated at E_com). To describe this case the mean energy transfer efficiency (η) and the width of the Gaussian distribution (δ) can be used as parameters:

• (36)

where σ^hw is the full width at half-height of the Gaussian distribution and δ is its width relative to E_com. Note that owing to the necessary truncation at E_com, the mean energy transfer will not coincide with the maximum of the Gaussian curve. The shape of this distribution is expressed by Eqns (36–38), and is shown in Fig. 2(b) (in the case of η = 0.8, δ = 0.2):

• (37)

The parameter E_cent is the center of this Gaussian distribution. Note that the average amount of energy transferred in a collision (ηE_com) is not equal to the center value of the Gaussian distribution (E_cent) because of the necessary truncation at zero and at E_com. The value of E_cent is related to ηE_com, but can be determined only by numerical methods to satisfy Eqn (38):

• (38)

where the left-hand side is the calculated average energy transfer, which is defined as ηE_com.

It is usually assumed that most collisions are glancing collisions, in which only a small amount of energy transfer takes place. There is experimental evidence41 that in a small fraction of collisions far more kinetic energy is transferred to internal energy than that deduced from the mean inelasticity of collisions. (According to some studies, this fraction may be significant.38, 39, 71, 72 These are usually called super-collisions (not to be confused with superelastic collisions). We suggest that in addition to the commonly accepted collisional energy transfer functions (described above), super-collisions may also be relevant to mass spectrometry, and it should be possible to take them into account. Super-collisions may have particular relevance when describing keV collisions, where it is commonly assumed that energy transfer has a long high-energy ‘tail’. We suggest that super-collisions should be described by their probability (π_super, expressed as a fraction of all collisions), the average amount of energy transferred (η_super, the inelasticity of the super-collision), and the shape of the energy transfer distribution. The latter can be exponential, half-Gaussian or Gaussian, the same way as for ‘normal’ collisions described above [Eqns (33–38)]. In other words, superposition of two distributions can be used, one to describe ‘normal’ and the other to describe super-collisions. The ‘total’ inelasticity of a collision will be

• (39)

Such an example is shown in Fig. 2(c), when ‘normal’ collisions have an exponential IET function with η = 0.1 (and have 90% probability), whereas super collisions have 10% probability (π_super = 0.1) and a Gaussian shape with η_super = 0.8, δ_super = 0.2.

Partially inelastic collisions including cooling effects

Both the ‘long-lived collision complex’ and the ‘partially inelastic collision’ models have their own advantages. The first adequately describes very low-energy collisions (although the definition of what can be regarded as a ‘very low’ collision energy is fairly vague), but fails completely at higher energies. A favorable aspect of this model is that after many collisions a statistical (i.e. thermal) energy distribution is obtained—the model in fact, assumes energy equilibration in each and every collision, which may be too strong an assumption. The ‘partially inelastic collision’ model can describe energy transfer in higher energy collisions, but fails for collision cascades. It cannot describe de-excitation, which is important when the ion is slowed close to thermal kinetic energies. In the case of a physically reasonable model, relatively low inelasticity should occur at high collision energies, but after a very large number of collisions a thermal internal energy distribution (corresponding to the temperature of the collision gas), should be obtained. Note that when the system is ‘driven’ (ions are accelerated externally, as in an ion trap experiment) a Boltzmann-like distribution is obtained, but the temperature characterizing it will be much higher than ambient.35

To combine the favorable features of the two models we have developed an improved collision model based on the partially inelastic collision model but incorporating cooling effects. This is indicated as the ‘partially inelastic collisions with cooling’ (PELC) model. The IET function in this case (P^IET,PELC) consists of an excitation term, P^IET,PELC,up, which is equal to the internal energy transfer defined for partially elastic collisions:

• (40)

A de-excitation term (P^{IET,PELC,down}) makes it possible to account for cooling effects. In the suggested collision model we do not use empirical estimates for de-excitation, but we have derived the P^{IET,PELC,down} function by modifying the concept of detailed balance, and applying it to isolated systems studied in a mass spectrometer. Note that the relationship between P^IET,PELC,up and P^{IET,PELC,down} may be derived from the concept of microscopic reversibility, but its exact derivation is not trivial. The model discussed below is an approximation, based on model calculations, and we believe that it is fairly accurate.

The concept of detailed balance40 is derived from the concept of microscopic reversibility, and is used in thermal systems. It describes how the probability of transition from E′_int to E_int relates to the probability of transition from E_int to E′_int. The relationship between these two probabilities is derived from the requirement of a thermal energy distribution at equilibrium in a thermal system. (Note that the first argument of the internal energy transfer function indicates the final, the second the initial, internal energy level):

• (41)

The mean collision energy in a thermal system can be indicated as 〈E_coll,T〉, which is equal to 3/2k_bT. (Note that in a thermal system, both the mean kinetic energy of a particle in the laboratory frame, and the collision energy in the com frame, will be equal to 3/2k_bT.) In the course of a collision, there will be a certain amount of energy exchange between translational and internal energy modes, which also depends on the internal energy (E′_int). The energy flow will depend only on the internal and the (com) kinetic energies, but does not depend on the origin of kinetic (or internal) energy. In other words, if the com collision energy in a given collision (with given E_com) is equal to 〈E_coll,T〉, the same energy exchange will take place regardless of whether the kinetic energy was obtained in a thermal system or by ion acceleration. Substituting

• (42)

into Eqn (41), we may obtain the expression defining the ratio of P^up and P^down for collisions. As this relates to internal energy transfer probabilities using the ‘partially inelastic collisions with cooling effects’ model, the distribution functions are indicated as P^IET,PELC,up and P^{IET,PELC,down}. As the kinetic energy (related to E_com) is of critical importance, it is also indicated among the arguments of the probability function:

• (43)

Knowing the form of the excitation side of the collisional energy transfer probability function [Eqns (33–38)], the de-excitation side can be determined using Eqn (43) by means of the Gilbert–King recursion equation.40

Note that in this model, the average energy transfer in a collision will not be determined by ηE_com only (a term which relates to the activation steps), but also by the de-activation process. In other words, the average energy transfer () will be less than ηE_com, and will depend both on the internal and the kinetic energy before the collision:

• (44)

In the case of a cooling collision, will be a negative value. In this case internal energy will be converted to kinetic energy (this process is also called a ‘super-elastic collision’, not to be confused with the ‘super-collisions’ described above). Note that the term P^IET,PELC in this expression is the internal energy transfer probability function, defined by Eqns (40) and (43). As a consequence, and in principle, equations describing translational energy loss of the projectile [Eqns (31) and (32)] should also be modified (substituting η with ). This correction of TEL is, however, so insignificant in most cases that it is neglected.

The direction of energy flow in a collision (i.e. the sign of in Eqn (44)) can be understood, in a semi-quantitative way, by comparing the com collision energy with the mean amount of internal energy in an oscillator of the molecule. Discussion is simple within the framework of the RRK approximation (non-quantized oscillators). In this case is zero, if 2/3 E_com is equal to E_int/D. If E_com is larger than this, heating occurs, if smaller then cooling collisions occur.

In the case of quantized oscillators the idea is the same, but the discussion is much more difficult. The mean internal energy of a molecule in thermal equilibrium at temperature T is 〈E_int(T)〉; in the harmonic oscillator model the relationship between temperature and internal energy can be described using the following equation:13, 15

• (45)

where k_bT is the measure of energy available in a thermal system; in non-thermal collisions, this can be substituted by 2/3 E_com, as described by Eqn (42). The amount of internal energy in equilibrium with the collision energy, indicated by , is

• (46)

Using this expression the energy flow in single collisions can be described qualitatively. When E_int exceeds , collisional excitation occurs. In the next collision both E_com and will be somewhat smaller (because of translational energy loss), whereas E_int will be larger. When E_int equals , there will be no net energy flow between translational and internal energies in the collision. However, even in such a case, kinetic energy will be transferred from the projectule to the collision gas, which lowers the collision energy (and therefore ) for the next collision. From this point on, will be lower than E_int, so collisional cooling will occur. Eventually both and E_int will decrease until (after many collisions) they will be in equilibrium with their environment (i.e. with the temperature of the collision gas, T_G). In this model, the more different are E_int and , the faster is the energy flow (and the more asymmetric is the P^IET,PELC function). As already noted, the ‘long-lived collision complex’ model assumes that collisional and internal energies equilibrate in each collision. In the ‘partially inelastic collision with cooling’ model the same equilibration occurs, but over the course of several collisions. The speed that this occurs depends (indirectly) on the form of the P^IET,PELC function. A nice feature of the ‘partially inelastic collision with cooling’ model developed here is that the de-excitation side does not have to be assumed, optimized or parameterized, but can be determined directly from the principle of detailed balance.

The same idea can be stated more expressively (and less accurately) by characterizing translational and internal energy levels by collisional and internal ‘temperatures’, and estimating energy flow based on the temperature difference. Note that these ‘temperatures’ are used only to facilitate discussion and to characterize the mean energy levels. They are not used to derive the equations above, nor do they imply a thermal-like (Maxwell–Boltzmann) population of ions.

The mean internal energy of a molecule in thermal equilibrium at temperature T is 〈E_int(T)〉. If an isolated molecule has exactly this amount of internal energy {E_int(A) = 〈E_int(T)〉}, E_int(A) may be characterized by this temperature, which may be called its ‘internal’ temperature, T_int. In an analogous way, the actual com collision energy can be related to the mean thermal kinetic energy (as described by Eqn (42)) and may be characterized by a ‘collision temperature’ (T_coll). We reiterate that T_coll and T_int (which change from collision to collision) are not ‘real’ temperatures, as they do not describe systems in thermal equilibrium, but are used to characterize single collisions with specific internal and kinetic energies. Using the concept of these fictitious temperature values, the energy flow in single collisions can be described qualitatively. When T_coll exceeds T_int, collisional excitation occurs. In the course of a collision cascade, in the next collision T_coll will be somewhat smaller (owing to translational energy loss), whereas T_int will be larger. When T_coll is equal to T_int, there will be no net energy flow from translational to internal energy in the collision. However, even in such a case, kinetic energy will be transferred to the collision gas, which lowers the collision energy (and therefore T_coll) for the next collision. From this point on, T_coll will be lower than T_int, so collisional cooling will occur. Eventually, both T_coll and T_int will decrease until (after many collisions) they will be in equilibrium with their environment (i.e. with the real temperature of the collision gas, T_G). In this model, the farther from each other the two temperatures are, the faster is the energy flow (the more asymmetric is the P^IET,PELC function).

To summarize the various collision models, their essential features are shown in Table 3. The data all relate to single collisions, so index SC is omitted. The table lists the mean internal and kinetic energy change according to the collision model, and refers to the relevant equation. The main features, advantages and approximations of the models are also given there.

Radiative transitions

In addition to collissions, radiative energy transfer (RET) is the main physical process influencing the internal energy (distribution) of compounds. The probability of spontaneous and induced emission and absorption can be modeled using a theoretical framework that is accurate and well known.19, 50 In any ambient environment there is a certain photon density; in the absence of external radiation it is usually in equilibrium with the surroundings. The ambient environment is typically approximated by a ‘black body’. Radiation density in such a case is described by the Planck equation:

• (47)

where T_MS is the temperature of the surrounding black body, i.e. the temperature of the mass spectrometer, while ρ(ν) indicates the radiation intensity at frequency ν. Photon emission and absorption are described using the so-called Einstein A_Ein and B_Ein coefficients:

• (48)

• (49)

where I₀ is the integrated IR intensity (of 0 → 1 quantum state transition) of an oscillator at frequency ν in the nth quantum state. Up-pumping (E_int > E′_int) occurs by photon absorption, and is described by the absorption term of the radiative energy transfer probability:

• (50)

where δ is the Dirac delta function. Down-pumping (E_int < E′_int) occurs by photon emission (induced and spontaneous), also expressed by probabilities:

• (51)

where P^RET(E_int,E′_int) indicates the probability of radiative energy transfer from E′_int to E_int energy. The microcanonical occupation probability (P^occ) can be calculated by exact state counting.46 (Note that owing to the frequency distribution of photons, nearly always IR photon emission and absorption are considered but, if necessary, visible and UV photon absorption/emission can also be taken into account, using equations analogous to Eqns (47–51), but related to electronic state transfer.)

Modeling requires knowledge of the ambient temperature, the internal energy, molecular frequencies (as for RRKM) and IR intensities for each oscillator. The latter, in turn, are calculated from the transition dipole moments of vibrational modes. IR intensities can be calculated or estimated, but usually only with large uncertainty. For this reason, theoretically calculated or estimated IR emission/absorption coefficients are often scaled by an adjustable scaling factor,18 usually in the range 0.1–10. Expediently, one ‘average’ scaling factor (Sf_IR) can be used for all oscillators of a given molecule or ion. As all components of P^RET depend linearly on will also be scaled linearly with this factor:

• (52)

An alternative to the theoretically accurate approach described above is the ‘standard hydrocarbon model’ of Dunbar and conclusions thereof.18 The philosophy of this approach is that characteristics of various kinetic processes, in particular radiative transitions and unimolecular dissociations, are approximately the same for all hydrocarbon-type molecules when they are appropriately scaled by size, dissociation energetics and perhaps a few other molecule-specific parameters. In its original form the ‘standard hydrocarbon model’ described de-excitation by IR photons. A main feature of the model is that the internal energy, in excess of the value in equilibrium with ambient photons (in the case of black body radiation with ambient temperature), will decay exponentially in time. The decay rate (expressed in energy units) is assumed to be proportional to the excess internal energy per oscillator. The decay rate was calculated for a small hydrocarbon, and thus is called the ‘standard hydrocarbon model.’ The conclusions of the ‘standard hydrocarbon model’ can be reformulated in the following form, which is more convenient to use in calculations:

• (53)

where E_therm(T_MS) indicates the mean (thermal) internal energy at the temperature of the mass spectrometer. The value of k_cool describes the radiative decay rate, and its value is eV⁻¹ s⁻¹ in the case of the ‘standard hydrocarbon model,’ determined from Dunbar's original data.18 To take into account molecular parameters of a given compound and to account for inaccuracies of the model, the value of k_cool can be scaled empirically for a given molecule or ion: . This approach is less elegant but, with respect to the mean internal energy, yields very similar results to the ‘exact’ procedure described by Eqns (47–51). One of the main drawbacks of using Eqn (53) is that the internal energy distribution will collapse eventually into a single value (corresponding to the mean thermal energy) and not to an energy distribution.

Activation by IR photons (BIRD21–23 or IRMPD47–49) can also be described using the same approaches, either Eqns (47–51) or Eqn (53). Note that in the case when E_therm(T_MS) is larger that E_int, Eqn (53) will describe activation by IR photon absorption.

The master equation model

The individual physical processes, which have a direct or indirect effect on reaction kinetics, have been discussed above. The ‘master equation’ (which is in fact a set of differential equations) is the proper mathematical model to describe situations in which a chemical reaction occurs in parallel with the change of internal energy. In master equations, the probability of a compound being in a given state is followed in time, and the probabilities of different processes are considered, usually through so-called state transfer (ST) probability functions. Conventionally, a given ‘state’ is characterized by its internal energy, which is correct if energy equilibration within a molecule is fast compared to the reaction rate (this is the statistical assumption, used e.g. in the RRKM theory). The conventional form of the master equation used to describe collisional energy transfer and unimolecular decomposition in a thermodynamic system is the following:41

• (54)

where P(E_int) is the internal energy distribution, ω is the collision frequency and P^CST,SC(E_int,E′_int) is the probability of transition from one (internal energy) state (E′_int) to the other state (E_int) in one collision (‘single collision’, SC). Note that P^CST is called the collisional state transfer probability function. This equation describes the probability of a compound being in a given state (dP(E_int)/dt), which is equal to the probability of energy transfer from all other states to this state (the first integral) minus energy transfer from this state to any other state (second integral) minus ‘loss’ of the ion in this state due to fragmentation (the third term in Eqn (54)). In Eqn (54), the single collision state transfer multiplied by the collision frequency (ωP^CST,SC) is related to the transition probability in unit time (in contrast to a single collision), and is indicated by P^CST. This terminology will be used below, as it simplifies the following discussion, making it easier to connect collisional and radiative energy transfers.

To model fairly complex chemical reactions occurring in mass spectrometry, a more general form of the master equation should be considered. First, the given state of an ion should be defined not only by its internal energy but also by its kinetic energy—the latter is of prime importance for determining the outcome of collisions. Unfortunately, both have a distribution, and to model fragmentation processes accurately both distributions have to be considered. Second, in a complex reaction scheme dissociation may occur (depleting the amount of the ion being studied), but they may also be accompanied by reactions leading to this ion (e.g. consecutive or reversible reactions). Third, energy transfer may occur both by collisional and by radiative energy transfer. Fourth, ion ‘loss’ in collisions, as described in the section on collisional processes, may also have to be considered. To take into account these effects, we have developed the following ‘extended’ master equation:

• (55)

where P_i(E_int,E_kin) is the probability of the ith compound having E_int internal and E_kin kinetic energy. is the probability of transition from a state defined by E′_int and E′_kin to a state defined by E_int and E_kin for the ith compound, in unit time (these process are called state transfer (ST) processes, while P^ST is called the state transfer probability function, as mentioned before). State transfer may occur by three different physical processes (collisional, radiative or accelerative ST), which influence either the internal or the kinetic energy (or both).

The probability of the compound being in state (E_int,E_kin) is increased by kinetic and/or internal energy transfer from all other states to this state (the first integral in Eqn (55)). Note that the double integrals are necessary, as the internal and kinetic energies are independent variables. The given state (E_int,E_kin) is depleted by kinetic and/or internal energy transfer from this state to any other state (the second integral in Eqn (55)) and by fragmentation (reactions leading to products, the third integral in Eqn (55)). The ion losses due to scattering, neutralization and charge transfer, which are particular effects in some mass spectrometric experiments, are considered in the fourth term. The last term in Eqn (55) describes the probability of a chemical reaction leading to this (the ith) compound in this selected (E_int,E_kin) state from any other compound in any state. To describe this last process, a partition function has to be used: describes the probability that in a reaction A_j → A_i, from the jth compound being in state E′_int,E′_kin the ith compound is formed in the E_int,E_kin state.

To solve Eqn (55), the state transfer and partition functions have to be defined. Starting with the simpler problem, the partition function in Eqn (55) is fairly straightforward. As described in the previous sections, internal energy partitioning and kinetic energy partitioning are independent processes (or at least can be considered as independent with negligible loss of accuracy), so can be separated into independent kinetic energy and internal energy partitioning functions:

• (56)

The internal energy partition function was described before in the statistical case [Eqn (14)]. In the case of equipartitioning, the following equation can be used on the basis of Eqn (15):

• (57)

The kinetic energy partitioning probability function can be expressed on the basis of Eqn (11) in the following form:

• (58)

State transfer is described by {P^ST(E_int,E_kin,E′_int,E′_kin)}, and this occurs by three independent, simultaneous processes: collisional and radiative state transitions and kinetic energy transition due to acceleration in external (electrostatic or radiofrequency) fields. As these processes are independent, their respective probabilities can be separately expressed (as P^CST,P^RST and P^AST, respectively) and then convoluted (indicated by ⊗). (To simplify discussion, identification of the compound (i) and of states (i.e. E_int,E_kin,E′_int,E′_kin) in the probability functions is often omitted in the following equations):

• (59)

The collisional state transfer, P^CST(E_int,E_kin,E′_int,E′_kin), depends on the collision frequency (ω), described by Eqns (16–18)), and on the single collision state transfer, P^CST,SC:

• (60)

Here P^CST,SC(E_int,E_kin,E′_int,E′_kin) is the single collision state transfer function, describing kinetic and internal energy changes in a single collision, and ω(E′_kin) is the collision frequency at E′_kin energy (described by Eqn (16)). Note that the unity function (P^UNIT) describing the case when nothing happens is necessary to avoid normalization problems:

• (61)

In all collision models described above, the internal and kinetic energy changes can be separated. The change in kinetic energy is independent of the internal energy; while for describing internal energy transfer the final kinetic energy is irrelevant. Taking these into account, the P^CST,SC(E_int,E_kin,E′_int,E′_kin) function can be partially separated as expressed in the following equation:

• (62)

For other collision models, in which this assumption is not valid, P^CST,SC should be appropriately defined and used in Eqn (59). P^IET,SC has been discussed before [Eqns (22), (25), (40) and (43)]; P^KET,SC is the kinetic energy transfer probability function for a single collision:

• (63)

where the translational energy loss (TEL) depends on the collision model and is expressed by Eqns (27–32). These are relatively simple functions, but (depending on the collision model) other expressions for P^KET,SC may also be used within the general framework described here. The internal energy transfer functions P^IET,SC were discussed before and expressed by Eqns (22), (25) and (33–38).

The radiative state transition probability, P^RST (E_int,E_kin, E′_int,E′_kin) in Eqn (59) is independent of the kinetic energy, and is equal to the radiative energy transfer function, P^RET(E_int,E′_int), described before [Eqns (50) and (51)]:

• (64)

Acceleration in electric fields has been described by Eqns (1–4). To accommodate the nature of the master equations, which describe changes in time, Eqns (1–4) have to be converted into a similar form, which may be called ‘accelerative state transfer’, P^AST. In the case of acceleration due to electrostatic fields (‘electrostatic state transfer’, P^EST) it depends on the field strength (ε) and on the mass and charge of the ion. Taking these into account:

• (65)

In the case of resonant excitation, kinetic energy change is fast compared with the collision rate, and the kinetic energy is determined by Eqn (3). In the case of SORI, the kinetic energy of the ion of interest depends predominantly on the external electromagnetic fields and the effect of collisions on the kinetic energy (P^KET,SC) can be neglected. The kinetic and internal energy distributions become independent variables and (as excitation time is much longer than the periodicity of velocity change), a time-averaged kinetic energy distribution can be used. This is calculated by the following equation, derived from Eqn (4):

• (66)

where

• (67)

MODELING MASS SPECTROMETRIC EXPERIMENTS: THE MASSKINETICS ALGORITHM

In the previous section the physical processes occurring in mass spectrometry were summarized and described mathematically. Based on this description, a computer algorithm and software (MassKinetics) were created to describe reaction kinetics under such conditions. In MassKinetics, the properties of an ion population are followed in time. Most important among these is the product distribution (relative concentration of various interconverting species). Each compound (usually an ion) in this distribution is characterized by a certain (time-dependent) internal energy distribution. Each internal energy level of each ion is further characterized by a kinetic energy distribution. These distributions change as a result of ion acceleration in electrostatic/radiofrequency fields, and due to collisions, photon emission/absorption, and chemical reactions. Most mass spectrometric experiments can be modeled by a combination of these processes; the mass spectrum can be regarded as a ‘snapshot’ of the product distribution at the time of ion detection. In Table 1, the main parameters characterizing the ion population and the main physical processes considered in this model are summarized.

In any mass spectrometric experiment the conditions influencing ions change in time and/or in space. A given experiment can be modeled by a series of ‘events’; such events are listed in Table 2. For example, a mass-analyzed ion kinetic energy (MIKE) experiment on a sector instrument can be built up from the following ‘events:’ (1) formation of ions (in the ion source); (2) field-free flight (from the source to the accelerating lens); (3) acceleration; (4) field-free flight (from the accelerating lens to the magnet); (5) ion selection (field-free flight through the magnet with loss of ions formed during this period, completed by setting all ion abundances to zero with the exception of the selected ion); (6) field-free flight through the second field-free region; (7) ion selection by scanning (field-free flight through the electrostatic analyzer with loss of ions formed during this period); and (8) detection (registering the product distribution). In this example the various processes are separated ‘in space.’ In such experiments, which require the transfer of ions for a given distance, the movement of ions through the instrument is followed in time by MassKinetics. In other instruments (ion traps and FT-ICR), experimental events are defined by time directly, which is used by MassKinetics.

Table 2 lists experimental events that may freely be combined. One advantage of using such simple ‘events’ is that nearly all mass spectrometric experiments can be built up by a combination of them. Another advantage is that ‘external conditions’ do not change during an ‘event’, so the master equations [Eqn (55)] can be used in a straightforward manner. As discussed, the master equations take into account acceleration, chemical reactions, collisions and radiative energy transfer. Depending on the experimental event sequence some of these may not occur, or can be neglected, which simplifies solution of the master equations significantly. ‘Ion formation’ means the setting of initial conditions (e.g. concentrations, energy distributions, etc.). One of the simplest ‘events’ is ‘field-free flight without collisions’—this means that neither external acceleration nor collisions take place, which also means that the kinetic energy is constant. In the master equation for this event, only chemical processes and radiative energy transfer have to be considered, so the form of Eqn (55) will become

• (68)

and the solution becomes relatively simple. If the time of field-free flight is short (<1 ms), there is practically no time for radiative energy transfer, which makes the solution of Eqn (55) very easy. When the duration of field-free flight is long (often called the ‘residence time’ in this case), radiative energy transfer processes may have a predominant role (as in the case of BIRD experiments). ‘Single ion selection’ and ‘ion selection by scanning’ are a combination of field-free flight without collisions, with a subsequent ‘arbitrary’ change or detection of product distribution. ‘Acceleration’ is also a simple event, a combination of collision-free flight with a change in kinetic energy.

Collisional processes are more complex to model, as kinetic and internal energies are interrelated, so the solution of Eqn (55) becomes far more complicated. Among these, ‘acceleration with collisions’ is most complex, when all three state transfer functions (P^CST,P^RST and P^AST) have to be considered in Eqn (59). The typical collisional process is the ‘field-free flight with collisions’ event. There are four important cases with particular characteristics, which have to be considered when solving Eqn (55). In the case of ‘keV collisions in a collision cell,’ there are usually only a few collisions, the kinetic energy can be regarded as a constant value and the flight time is short. In this case, ion loss processes are important and should be taken into account. In the case of ‘eV collisions in a collision cell’ there may be many collisions, which may slow the ion significantly, but not bring it close to thermal energy. In this case ion loss processes may need to be taken into account, depending on the system studied. In ‘collision cascades’ the kinetic energy of the ions is reduced close to zero, and cooling collisions may have a large influence. The residence time is usually fairly long (of the order of seconds), so collisional and radiative cooling occur in parallel. The last collisional experiment discussed here is ‘collisions occurring in parallel with rapid, periodical changes in ion velocity,’ as occurs in SORI in FT-ICR or ‘resonance excitation’ in ion traps. In this case the ion kinetic energy will depend on the external field, and is independent of the internal energy. This time dependence can be converted into a time-averaged kinetic energy distribution, simplifying solution of Eqn (55). Note that, usually, the external field determines the kinetic energy of only one ion; other ions (such as those formed in a reaction) will behave like those in a collision cascade.

On the basis of the physical model and its mathematical formulation developed in the previous section, and on the description of mass spectrometric experiments described above, the computer program was written (MassKinetics 1.0) in C++ programming language (at present it is over 10 000 lines long). MassKinetics runs under Windows, and a simplified version of the program will soon be available on our server for general use (www.chemres.hu/ms/masskinetics). The most difficult part of the calculations is the solution of Eqn (55). These differential equations are homogenous and linear, and so can be solved analytically using the matrix form of the equation. Unfortunately, full diagonalization of a matrix dependent on six parameters (i, E_int,E_kin, j, E′_int,E′_kin), seems infeasible for a PC at present, so in MassKinetics numerical methods are used: the three (product, internal and kinetic energy) distributions are followed in time. This solution is mathematically more complex, but can be done without the use of very large matrices.

Determination of reaction rates (as a function of internal energy) is a critical step in the calculations. At present only unimolecular reactions are considered [Eqn (7)]. Calculation of the rate expression within the framework of RRKM [Eqn (10)] always requires assumptions or approximations. In MassKinetics the harmonic oscillator model is used with internal rotations are approximated by low frequency vibrations, an often-used approach.12, 73 Including internal rotations explicitly and/or using anharmonic oscillator models is a possible extension of the present program, but it was not deemed necessary at present. Angular momentum effects are taken into by the formalism of the ‘rotational barrier,’ which may be estimated or calculated separately, and the results inserted into MassKinetics. On the other hand, it is important to calculate the density and sum of states by a direct count, which was done using the Beyer–Swinehart algorithm.74

PARAMETERS DESCRIBING MASS SPECTROMETRIC REACTION KINETICS

Parameters describing the experiment

These parameters include definition of the sequence of ‘events’ according to Table 2; the reaction scheme (list of compounds and reactions with non-zero rate constants); initial concentrations (for most mass spectrometric applications only the precursor ion will have a non-zero concentration); initial internal energy distributions (which is not trivial, and will be discussed below); and initial laboratory-frame kinetic energies (in most applications these will be thermal (or zero) for all ions). Definition of the experiment (of the individual ‘events’) should include the time-scale (flight and/or residence time), defined either explicitly or indirectly using flight distances (ion velocities are calculated from the kinetic energy, if necessary taking into account deceleration due to collisions). Potentials or other parameters defining ion acceleration should also be given. To convert laboratory to com frame kinetic energies and to calculate ion velocities, the masses of the ions involved and of the collision gas has to be defined. To take into account cooling effects, the temperature of the collision gas and of the mass spectrometer have to be specified—these are nearly always the same.

Among the distributions needed, only the initial internal energy distribution is non-trivial. This has to be specified for all compounds having non-zero concentration—in practice only for the precursor ion. In many experiments, the internal energy of ions leaving the ion source has a close to thermal (or thermal-like) distribution (e.g. in the case of electrospray ionization,75 in chemical ionization or in most FT-ICR experiments). In other cases, the internal energy distribution should be determined experimentally or should be estimated. Luckily, the initial internal energy distribution has only a small effect on the results in most experiments.

Parameters describing chemical reactions (reaction rates and energy partitioning)

Internal energy dependent reaction rates are determined in MassKinetics by the RRK or RRKM formalisms. For RRK calculations the critical energy (E₀), the pre-exponential (or frequency) factor and the degrees of freedom of the reactants should be specified.

The far more accurate RRKM calculations require the critical energy (E₀) and frequency models for the ground and for the transition states (the ‘frequency model’ is the list of molecular normal frequencies). The ‘looseness’ of the transition state can be characterized by the Arrhenius-type pre-exponential factor, which is a very useful derived parameter in RRKM. The same property can also be expressed by the entropy of activation. Note that in the case of a loose transition state, the entropy of activation is closely related to the reaction entropy (determined mainly by the change in the number of translational and external rotational degrees of freedom in a reaction).

Normal frequencies for the ground and for the transition states can be calculated by molecular orbital (quantum chemical) calculations. Significantly, RRKM calculations are not very sensitive to the accuracy of frequency models, only the (derived) pre-exponential factor characterizing the transition state is important. (Formally, the choice of the reaction coordinate frequency in the transition state may be regarded as an empirical parameter for such a purpose.) Advanced level RRKM calculations can also take into account non-harmonic oscillators, rotations and hindered rotations. However, parameters describing hindered rotations, for example, are difficult to obtain, and in most cases (with the exception of small molecules) these effects are believed to influence the results to a very minor degree. In consequence, most RRKM calculations use the harmonic oscillator model and internal rotations are often substituted by low-frequency vibrations—an approach also used in MassKinetics. Rotational barriers are typically of much more importance. Rotational barriers are most often estimated (2–4 kcal mol⁻¹ (1 kcal = 4.184 kJ)),57 although they can also be calculated knowing molecular parameters and the rotational energy of the molecule.

In conclusion, to calculate energy-dependent reaction rates the important parameters are the critical energy and the pre-exponential factor for each reaction considered. Molecular frequencies can be obtained from simple quantum chemical calculations with reasonable accuracy. When the reaction mechanism is known (e.g. H-rearrangement through a five-membered ring), the pre-exponential factor may be estimated57 or calculated using medium-level quantum chemical calculations, usually with sufficient accuracy. The reaction mechanism (the likely transition state) can also be determined by quantum chemistry, but that usually requires many calculations. When the reaction mechanism is not known, the frequency factor has to be regarded as an adjustable parameter. The critical energy (E₀) can be determined experimentally, estimated based on thermochemical data or calculated using high-quality ab initio methods. When neither option is available, critical energy (E₀) can be regarded as an empirical (adjustable) parameter.

When consecutive reactions are included in the reaction scheme, the internal energy distribution of the intermediate product has to be determined by energy partitioning. Equipartitioning is simple and requires only the number of degrees of freedom of the reactants. Statistical energy partitioning requires the frequency model of the reactants (including the neutrals).

Collisional processes

Modeling collisional processes is one of the prime features of MassKinetics. These are described by the probability of a collision, the duration while collisions may happen, and the information on what happens in a single collision. In the present case our main purpose is to describe reaction kinetics, so that most important aspect of collisions is energy transfer. Other processes may also occur, usually resulting in ion ‘loss,’ as described above, which is also characterized by a certain probability. Energy transfer in a single collision is described by the collision models discussed above, each of them requiring different parameters (although there is a lot of overlap among the various models).

The probability of collisions is determined according to Eqns (16–18). (Note that it is different for each component in the reaction scheme.) This probability depends on the kinetic energy (which is followed in MassKinetics), on the pressure of the collision gas (which can be measured76) and on the collision cross-section. The latter can be measured or estimated. The cross-section may change with the collision energy, but this effect is usually neglected. In some cases, e.g. in some ion trap experiments when ion velocities change a great deal, the use of collision energy dependent cross sections is preferable (W. Plass and R. G. Cooks, unpublished work). The cross-section (σ) is defined here to relate only to energy transfer collisions, which is a usual approach. The only other process considered here, ion loss, is characterized by the appropriate σ^loss value. Note that the collision number is particularly important in those cases, when the collision energy changes relatively little in the experiment. In such a case, the mean collisional energy transfer is proportional to the product of the collision number (ωt) and the efficiency of collisional energy transfer (η). In other experiments (such as collision cascades), the collision frequency has a much smaller influence, because the collision energy will decrease to thermal value anyway.

Energy transfer in a single collision depends on the collision model used. In the case of the long-lived collision complex model, energy partitioning is used, requiring the frequency model (as used for rate calculations). Note, however, that in this case the internal degrees of freedom or the vibrational frequencies of the collision gas also have to be specified.

In the ‘partially inelastic collision’ model, the most important parameter is the average degree of inelasticity (η), determining the fraction of com collision energy converted into internal energy. This may change from zero (completely elastic collision) to unity (completely inelastic collision). Note that the degree of inelasticity may change with the kinetic energy. If the collision energy varies to a large extent during the experiment, this change has to be specified by a user-determined function, η(E_com). The shape of the energy transfer probability function has to be specified as well. This distribution has often been assumed to be exponential; occasionally half-Gaussian or Gaussian curves can also be used. The latter requires one more parameter, δ, signifying the relative width of the Gaussian curve. Luckily, most mass spectrometric experiments are not sensitive to the exact shape of the energy distribution function. If super-collisions are also assumed to take place, these have to be characterized by their relative probability (π_super), their mean energy transfer efficiency (η_super) and the shape of the energy transfer function.

The com scattering angle has a large effect on the amount of translational energy loss, which is used mainly to increase the kinetic energy of the collision gas in the laboratory frame [Eqns (27–32)]. This effect is significant only if there is a large number of collisions, and if there is no external acceleration. This is the case in collision cascades, where the fast ion loses nearly all of its kinetic energy—converted either into internal energy or into the kinetic energy of the collision gas. In many cases the assumption of random angle scattering (in the com frame) seems reasonable, especially at low collision energies,77, 78 which is the region of prime importance in collision cascades.

As this discussion has shown, there are a fairly large number of parameters that influence collisional energy transfer. Some are easy to determine accurately, some are usually only estimated empirically. A number of these parameters influence the shape of the resulting internal energy distribution. In most multiple collision mass spectrometric experiments, variations in the shape of the single collision internal energy transfer distribution (P^IET,SC) do not have a major influence on ion ratios. The mean amount of kinetic energy transferred into internal energy in the whole collision process, on the other hand, has a very large influence on the results. In many cases it is possible to incorporate uncertainties into one adjustable parameter, usually the efficiency of energy transfer (η), and fix all other parameters. In this way collisional processes can be described in MassKinetics with reasonable accuracy, using only one or occasionally two adjustable parameters.

Radiative transitions

To describe radiative heating and cooling effects by Eqns (47–51) one needs to know the external photon density (and its frequency distribution, ρ(ν)), the internal energy distribution and the infrared intensities (I₀) for each oscillator of the molecule. From the IR intensities Einstein A and B coefficients are determined using Eqns (48) and (49). If the external photons are due to thermal (black body) radiation, their frequency distribution is described by the Planck equation [Eqn (47)], which (apart from the external temperature) does not need further parameters for its description. IR intensities can be calculated (using quantum chemistry) or estimated, and may be scaled by a common empirical factor (Sf_IR) to compensate for errors in the calculation (or estimation). If radiative transitions are to be described by Dunbar's model [Eqn (53)], one empirical factor is also required to scale the radiative decay rate of the molecule (or ion) to that of a ‘standard hydrocarbon.’

As described above, in MassKinetics many parameters are used. Some describe the experiment, and represent trivial information. Some describe the molecular system studied, most of which are straightforward to calculate. Further parameters are used to describe the (collisional and radiative) energy transfer processes. Among these parameters only relatively few influence the results significantly. These have to be known, measured, calculated or scaled accurately, if calculations using MassKinetics are to be compared directly with experimental results. The critically important parameters are the following: the critical energy (E₀) and pre-experimental factor (A_PE), a parameter (usually the inelasticity, η) describing the efficiency of collisional energy transfer, and a parameter scaling the radiative heating (or cooling) rate. In most applications it seems sufficient to use a common scaling factor for all ions. Furthermore, in most experiments, either collisional or radiative energy transfer is predominant, in which case a standard (i.e. not scaled or optimized) parameter can be used for the less important process.

APPLICATIONS AND OUTLOOK

Initial results obtained with the program have been published recently,75, 79, 80 and several other studies are in progress. The results are very encouraging; close to experimental accuracy has been obtained in several systems with the use of no or only one adjustable parameter.

Studying parallel metastable reactions is very attractive (both experimentally and theoretically), as the pre-exponential factors often cancel, and the difference in critical energies of the two parallel processes is the only really important parameter. For this reason, the first applications of MassKinetics were related to modeling of the kinetic method.79 Probably the best illustration is the case of protonated alcohol clusters,80 studied experimentally by Holmes et al.,81 using metastable ions. In our study, the critical energy and transition state model was taken from Holmes et al.81 and vibrational frequencies were calculated by the (very simple) AM182 molecular orbital method. The internal energy distribution of ions leaving the chemical ionization ion source was assumed to be thermal, and the time-scale was calculated from straightforward instrument parameters. Because of the short, microsecond time-scale, photon exchange was neglected. The calculated ion abundances were in surprisingly good agreement with the experimental data, as shown in Fig. 3. Similar excellent agreement was also found between experimental and calculated ion abundances in the case of other metastable ion studies.79 Note that in these calculations no adjustable parameters or scaling factors were used. The very good agreement between experiments and calculations is a very strong indication that MassKinetics is able to calculate ion abundances accurately without the need for adjustable parameters, scaling factors or a fitting procedure—at least when there is no energy exchange with the environment. It is also an important starting point for studying collisional and IR activation: one can be certain that only parameters related to energy transfer have to be evaluated, as the rest of the calculation was shown to work well. Note that modeling collisional energy transfer is still fairly tricky.

Figure 3. Comparison of ln([R₁OH₂⁺]/[R₂OH₂⁺]) vs proton affinity plots of obtained experimentally (Holmes et al.,81 shown by +) and calculated by the MassKinetics program (Thomas et al.,80 shown by •)

Very encouraging results have also been obtained on applying MassKinetics to collisional activation. Fragmentation of small salt clusters was evaluated recently using the kinetic method.83 The data indicated a surprisingly high effective temperature (in 3000–4000 K range), which was successfully explained using simulations with MassKinetics (L. Drahos, K. Vékey and R. G. Cooks, in preparation). In this study, only a single parameter (the efficiency of energy transfer, η) was optimized and used for a series of small alkali metal halide clusters; relative fragment abundances and effective temperatures were accurately described. Note that such small systems (with only three atoms) are not ideally suited to statistical treatment, but in spite of this the results are very encouraging. In another study,84 very low (< 100 K) effective temperatures were accurately determined. The reason for these was also understood (and accurately calculated) using MassKinetics; details will be published in the near future.

Collision cascades, cooling collisions and the effect of IR cooling have also been studied, simulating conditions on an FT-ICR instrument. Radiative transitions were modeled in this case using Dunbar's original ‘standard hydrocarbon model.’ Using only one parameter only (η), changes in the experimentally observed molecular ion survival yields were successfully explained.85 Initial success with MassKinetics suggests that mass spectra can be calculated with surprising accuracy knowing molecular properties and using an adequate (fairly complex) theoretical framework but very few adjustable parameters. We believe that MassKinetics can be used to obtain valuable information on energy transfer in collisional processes, at least an average η value can be determined. MassKinetics can also be used to obtain critical energies or other molecular parameters from mass spectrometric experiments, and can also be used to explain unexpected experimental results or to pinpoint possible error sources in special experiments.

In this paper we have summarized the main physical effects determining product abundances in reactions under non-equilibrium conditions, i.e. those applicable to mass spectrometry. The physical processes themselves are well known, but we believe that this is the first case in which all of them (reactions, energy distributions, collisional and radiative energy exchange and energy partitioning) have been incorporated into one model, which also takes into account their simultaneous occurrence. We have found that in several, seemingly detailed, theoretical descriptions of mass spectrometric processes, key elements are often omitted. We have tried not to commit this error, and to evaluate all parameters and assumptions used. We have also tried to formulate all of the relevant equations: the collection provided here could be a very useful reference (or starting) point in future studies. Some of these equations are fairly trivial and relatively straightforward to derive, but are not always easy to find in the literature. Some others we believe to be new, or at least generalized versions of well-known concepts. Such an example is the master equation: conventionally only a relatively simple form [Eqn (54)] is used, but we have extended it to account for a reaction scheme (and not only for a single elementary reaction), to include the influence of kinetic energy and to include collisional ‘loss’ processes [Eqn (55)]—the latter are of prime importance in mass spectrometry. We have also developed a model to account for cooling collisions.

We believe that the theoretical framework presented here is correct and accurate but, of course, its implementation involves a number of approximations, as discussed above. The correspondence between theoretically calculated and experimentally observed ion abundances (see, e.g., Fig. 3) seems better than the estimated accuracy of individual elements (e.g. rate constant calculations by RRKM) used in the MassKinetics algorithm. We believe that this is because there are a number of ‘negative feedback’ mechanisms operating in mass spectrometry (just as in other natural phenomena), and also because of the ‘smoothing’ effect of distributions influencing these reactions. An example of ‘negative feedback’ is in the case of collisional excitation. If it is overestimated, the internal energy will be higher, causing faster IR cooling. The latter process will partially offset the effect of the erroneous degree of collisional excitation. A good example of the effect of distributions is in the case of metastable fragmentations. If erroneously high rate constants were determined by RRKM, this will cause a relatively small upward shift in the metastable energy window, but the width of this window (which is limited on both sides by the degree of fragmentation) will not change significantly.79 As a consequence, the abundance ratio of metastable and stable ions will change only very little. We also believe that the analogous effects contributed to the tremendous success of organic mass spectrometry in the 1960s and 1970s, which made interpretation of mass spectra possible based on relatively few qualitative concepts.1–3

Acknowledgements

We are particularly grateful to R. G. Cooks (Purdue University, USA) for his help and encouragement. We also thank P. B. Armentrout (University of Utah, USA), J. R. Christie (School of Chemistry, Bundoora, Australia), W. R. Plass (Purdue University, USA), R. M. A. Heeren (FOM AMOLF, The Netherlands), J. H. Futrell (Pacific Northwest National Laboratory, USA) and Gy. Lendvay (Hungarian Academy of Sciences, Hungary) for suggestions and helpful discussions.