Drake G.W.F. (editor) Handbook of Atomic, Molecular, and Optical Physics

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566 Part C Molecules

ming/integrating over rotational/vibrational degrees of

freedom have been given [37.39]. As can be seen by

writing out the expression for P



, the phase space for-

mula for the state-to-state cross section in (37.25) obeys

microscopic reversibility:

2µT

σ(J

, J

→ J

, J

)

= 2µ



σ(J

, J

← J

, J

) (37.26)

where the reduced mass and asymptotic translational

energy of the products are denoted by primes. Thus,

thermalization of the forward and backward rates leads

to detailed balancing.

As a detailed theory for prediction of reaction prod-

ucts and their state distributions, phase space theory and

its variants have had mixed success. It is true that the

theory correctly predicts that exoer gic reactions occur-

ring on barrierless potential surfaces proceed on every

strong collision. It is also true, however, that the theory

is generally not useful in predicting the branching ratios

among several sets of exoergic products because poten-

tial surfaces do not often show barrierless pathways for

more than one set of products. When applied to product

state distributions, the theory yields statistical results, so

that population inversions in nondegenerate degrees of

freedom (such as the vibrations of a diatomic molecule)

are not replicated. With respect to more thermalized en-

tities, such as total cross sections vs. collision energy for

endoergic reactions [37.40], the theory can be quite suc-

cessful, especially when the potential surface involves

a deep minimum or intermediate complex. In this in-

stance, the strong coupling hypothesis comes closest to

actuality. A deep well is also associated with a high dens-

ity of vibrational states in the quasicontinuum above the

dissociation limit of the molecular structure deﬁned by

the potential minimum [37.1, 41,42]. Such a high dens-

ity renders direct (speciﬁc) dynamical processess less

likely.

A useful variant of phase space theory if complex

lifetimes are needed [37.43] (see Sects.37.1.4 and 37.2)

is based on a unimolecular decay theory of Klots[37.44].

The reaction is considered to proceed via a capture cross

section to form the intermediate complex, which then

can dissociate into all available reactant and product

states consistent with conservation of energy and an-

gular momentum. The complex dissociation rate k

uni

into a speciﬁc state can be obtained via the principle of

microscopic reversibility in terms of the capture cross

section from that state, obtained with P



= 1in(37.25),

to form the complex. In particular, if a complex with to-

tal angular momentum J can dissociate into one state of

reactants A and B separating with translational energy

,then

uni

= ρ

−1

vib

g(J)

−1

g(J

)g(J

)2µT

σ(J

, J

→ J),

(37.27)

where ρ

vib

is the density of complex vibrational states

obtained via a variety of prescriptions [37.1,41,42], and

g is the rotational degeneracy. A cross section analogous

to that in (37.25) can be formulated in terms of capture

to form the complex multiplied by the complex disso-

ciation rate into a particular state divided by the total

(summed) dissociation rate. Note that the dissociation

rate of the complex is proportional to ρ

−1

vib

.Sinceρ

vib

a strong function of the well depth, long-lived com-

plexes are associated with large well depths (>1eV).

The Klots form for k

uni

is especially useful for ion–

molecule systems, where the cross section for complex

formation can be assumed to be Langevin or ion–dipole.

The concept of a long-range potential is less useful for

most neutral–neutral systems, so that unimolecular rate

coefﬁcients for unstable entities are normally obtained

quite differently in terms of ρ

vib

at the transition state

(Sect. 37.1.4).

For smaller reaction systems, especially those in-

volving ions, it is quite common for the electronic states

of reactants to correlate with more than one potential

surface of the combined system, although typically only

one surface leads to energetically accessible products.

In such instances, it is normal to assume statistical par-

titioning, although ﬁne structure effects can complicate

this picture. A generalization of phase space theory to ac-

count for multiple potential surfaces has been proposed

for the C

+ H

reaction [37.45].

Phase space theory has been used to predict prod-

uct branching ratios for dissociative recombination

reactions between polyatomic positive ions and elec-

trons [37.46]; its success here has been limited at

best.

37.1.3 Short-Range Barriers

Most reactions involving neutral molecules, as well as

a minority of ion–molecule reactions, possess potential

surfaces with short-range barriers. The experimen-

tal rate coefﬁcients of these systems over selected

temperature ranges are normally ﬁt to the Arrhenius

expression (37.17).

The simplest theoretical method of taking short-

range potential barriers into account is the line-of-

centers approach, which resembles the capture theories

previously discussed [37.7]. In this crude approxima-

Part C 37.1

Gas Phase Reactions 37.1 Normal Bimolecular Reactions 567

tion, it is assumed that structureless reactants colliding

with impact parameter b along a repulsive potential

must reach some minimum distance, d, for reaction to

occur. If the potential energy in the absence of angu-

lar momentum at d is E

, this condition implies that

the asymptotic translational energy of reactants T

must exceed the sum of E

and the centrifugal energy

, which in turn yields a maximum impact par-

ameter b

max

. Thermal averaging of the rate coefﬁcient

k = vπb

max

over a Maxwell–Boltzmann distribution

then yields

k(T) = πd

(8k

T/πµ)

1/2

exp(−E

T),

(37.28)

which bears some resemblance to the Arrhenius expres-

sion if one equates the parameter E

with the activation

energy. Assuming T = 300 K, µ = 10 u, and d = 1Å,

the pre-exponential factor is 2.5×10

−11

−1

,which

lies in a typical range.

The simple line-of-centers approach reduces the

problem to that of structureless particles. The standard

method of including all degrees of freedom is to use

canonical ensemble statistical mechanics and to imagine

that the transition state is in equilibrium with reactants.

In the transition state, one of the vibrational degrees of

freedom of a normal polyatomic molecule is replaced

by a degree of freedom (the reaction coordinate) along

which the potential is a maximum (with a corresponding

imaginary frequency of vibration). The reaction coordi-

nate is treated as a separable translation, so that the

reaction rate coefﬁcient can be envisaged as the equi-

librium coefﬁcient between transition state (minus one

coordinate) and reactants multiplied by the (average)

speed of the transition state structure over the saddle

point in the potential energy surface. The canonical

result for k(T ) is given by [37.1,7]

k(T) =

†

exp(−E

T) (37.29)

for reactants A and B,whereE

is the energy dif-

ference between the transition state and the reactants

referred to zero-point levels, the

† refers to the tran-

sition state, and the q are partition functions per unit

volume. The partition functions can be factored into

products representing electronic, vibrational, rotational,

and translational degrees of freedom [37.47]. This for-

mulation for k(T ) is known as the activated complex

theory (ACT); a more appropriate name would be tran-

sition state theory since the term activated complex is

also used to refer to an unstable state of a molecule in

a deep potential well. The rate coefﬁcient can also be

written in terms of the thermodynamic parameters ∆H

†

and ∆S

†

[37.1].

Both the size and temperature dependence of the

pre-exponential factor depend critically on the char-

acteristics of transition state and reactants. Originally,

ACT theory was used mainly to ﬁt transition state char-

acteristics to rate data. Increasingly accurate ab initio

calculations of potential surfaces now allow purely the-

oretical determinations of k(T ) [37.48].

In addition to the assumption of a separable re-

action coordinate that can be treated as a translation,

several other assumptions have gone into the deriva-

tion of the ACT rate coefﬁcient. First, molecules at

the saddle point conﬁguration (the transition state) have

been arbitrarily chosen to be in equilibrium with re-

actants. More recently, the transition state assumption

has been generalized to refer to that portion of the po-

tential surface in which the reaction ﬂux or number of

states is a minimum [37.49], whether or not this oc-

curs at a well-deﬁned saddle point. Loose transition

states can thus be deﬁned even if there is no barrier

along the reaction coordinate [37.50]. The procedure

can be undertaken for transition states as a function of

temperature, energy, or, in its most detailed version, en-

ergy and angular momentum. The variational theorem

shows rate coefﬁcients determined in this way to be up-

per bounds to the true rate coefﬁcients, the more detailed

procedures leading to the better bounds [37.23, 49, 50].

Secondly, the implicit assumption is made that trans-

lation along the reaction coordinate at the transition

state structure invariably leads to products. Sometimes

a transmission coefﬁcient κ is used as an unknown factor

in the expression for k(T) to account for the possi-

bility that translation leads back to reactants instead.

Thirdly, the assumption is made that two-body colli-

sions can lead to canonical statistical equilibrium. This

assumption becomes worse as temperature declines, be-

cause at low temperatures the long-range centrifugal

constraints on angular momentum become more signiﬁ-

cant [37.8]. The inﬂuence of the long-range potential on

ACT is contained in the statistical adiabatic theory dis-

cussed earlier [37.23]. Here loose transition states are

deﬁned as the maxima of effective adiabatic potential

curves, and (37.29) must be modiﬁed since, as in de-

tailed variational transition state theory, the transition

states are themselves dependent on energy and angular

momentum. A related problem is how to treat angular

momentum constraints at low temperatures for potential

surfaces which also contain a large short-range barrier

or tight transition state. One approach is discussed in

Sect. 37.1.4.

Part C 37.1

568 Part C Molecules

Rate coefﬁcients derived from ACT do not indicate

what the reaction products or sets of products might be.

This information can be obtained from ab initio studies

of the potential surface if different transition states lead

to different products.

37.1.4 Complexes Followed by Barriers

Since long-range forces are always attractive, it makes

sense to consider theories in which attraction and re-

pulsion occur sequentially. For ion–molecule systems,

this is especially important because many potential sur-

faces are monotonically attractive from long range to

formation of a deep minimum at short range, but possess

transition state barriers in their exit channels which are

not large enough to prevent reaction but which affect the

reaction dynamics. In addition, there are ion–molecule

systems with potential surfaces closer to the norm for

neutral–neutral species; in these systems there is only

a weak long-range minimum followed by a short-range

transition state barrier with energy above that of reac-

tants. For the former type and, more arguably, for the

latter type of potential surface, one can assume that the

reaction proceeds through initial formation of an ergodic

complex, followed by dissociation of the complex back

into reactants or over the transition state barrier. If the

reactants are labeled A and B,thecomplexAB

∗

,and

the transition state AB

†

, the mechanism is

A+ B

 AB

∗

−→ AB

†

−→ Product , (37.30)

which leads to the steady-state rate law

d[A]/ dt =−k[ A][B] ,

(37.31)

k =

+ k



, (37.32)

where k

, k

,andk



are the rate coefﬁcients for

complex formation, redissociation into reactants, and

dissociation into products over the transition state, re-

spectively. One statistical approach to such systems is

to use a capture theory for complex formation, to use

the Klots formulation of complex redissociation into

reactants, and to use a different theory for the uni-

molecular dissociation of the complex into products.

Transition state theories of unimolecular complex decay

have been studied for many years [37.1]. The current

version is called RRKM theory, after the four authors

Rice, Ramsperger, Kassel,andMarcus [37.10–13]. This

theory builds upon the earlier RRK approach, in which

random intramolecular vibrational energy transfer leads

to large amounts of energy in the bond to be broken.

An alternative hypothesis, in which amplitudes of well-

deﬁned normal modes add up to extend the bond to be

broken past a certain amount, is now discredited [37.51].

In the RRKM approach, which is perfectly analogous to

ACT theory, an equilibrium is envisaged between the

activated complex and transition state species. The main

use of the theory is in thermal unimolecular decay where

it explains complex activation, deactivation, and random

unimolecular dissociation as a function of gas density.

At high density in a thermal environment, an equilib-

rium among complex, activated complex, and transition

state leads to a thermal unimolecular decay rate (37.13)

RRKM

(T ) =

†

exp(−E

T), (37.33)

which is perfectly analogous to the ACT result. At

lower pressures, the activated complex is not in ther-

mal equilibrium with the stable complex and the details

of complex activation and deactivation are impor-

tant [37.12,13].

In the microcanonical formulation of RRKM theory,

the dissociation rate coefﬁcient k

RRKM

as a function of

(activated) complex total energy E and angular momen-

tum J is given by

RRKM

(E, J) =

†

[E − E

− E

rot

(J)]

hρ

∗

[E − E

rot

(J)]

(37.34)

where N

†

refers to the total number of vibrational states

of the transition state from its minimum allowable saddle

point energy E

through E,andρ

∗

refers to the density

of vibrational states of the complex at energy E.For

both the transition state and the complex, the available

vibrational energy is the total available energy minus the

rotational energy E

rot

(J), which is a function of the an-

gular momentum. The energy not used for vibration and

rotation in the transition state is considered to belong to

the separable reaction coordinate. The most common ex-

pressions for the number and density of vibrational states

are the semiclassical empirical values [37.1,41, 42]; di-

rect counting schemes also exist [37.52]. Both empirical

and direct counting refer to states representing a bath

of harmonic oscillators; anharmonic effects are rarely

treated.

A theory of reaction rates for the mechanism

in (37.30) incorporating a capture theory, and the Klots

and RRKM unimolecular decays has been applied suc-

cessfully to a variety of ion–molecule reactions in

competition with association channels for which the re-

action proceeds via a deep well and an exit channel

barrier (see Sect. 37.2.3) [37.53–55]. Some authors pre-

fer to use RRKM theory to describe redissociation of the

Part C 37.1

Gas Phase Reactions 37.1 Normal Bimolecular Reactions 569

complex into reactants [37.13, 56] by means of a loose

transition state [37.57], which can be deﬁned via a va-

riety of variational and other methods [37.50]. Some

recent work on unimolecular decay of polyatomic ions

and neutrals following excitation indicates the superior-

ity of the variational RRKM approach to unimolecular

decay [37.58], especially if there is a signiﬁcant amount

of energy. Both the Klots and RRKM approaches to

complex redissociation have problems associated with

them: the Klots expression obeys microscopic reversibil-

ity but requires both a capture cross section and the

assumption that the phase space formulation of the

probability of individual quantum states is accurate;

the RRKM expression does not obviously obey micros-

copic reversibility and, unless a variational calculation

is performed, can create a somewhat artiﬁcial transition

state.

Reaction mechanisms with more than one potential

barrier can also be treated via a combination of ACT,

capture, and unimolecular approaches [37.59,60].

37.1.5 The Role of Tunneling

The role of tunneling in bimolecular and unimolecu-

lar reactions grows in importance as the temperature is

lowered and hopping over potential barriers becomes

more difﬁcult. A simple one-dimensional correction for

tunneling in the ACT expression for bimolecular rate co-

efﬁcients, obtained by Wigner [37.1, 61], appears to be

reasonable if the tunneling mechanism is not dominant.

This correction Γ>1 is simply the quantum correction

to the partition function for the reaction coordinate:

Γ ≈ 1 −

(hν

)

24(k

+ ··· , (37.35)

where ν

is the (imaginary) harmonic frequency at

the saddle point. Improved corrections have also been

developed [37.62, 63] and, for the H + H

reaction,

tested versus accurate quantum calculations [37.64]. At

very low temperatures, such as those found in the in-

terstellar medium, tunneling corrections are likely to

be very large and to require proper multidimensional

treatments.

A one-dimensional tunneling correction to the

RRKM expression for the microcanonical unimolecular

decay rate coefﬁcient also exists [37.65]. The effec-

tive potential representing the reaction coordinate at the

saddle point is assumed to be an Eckart barrier. The

probability of tunneling for each vibrational state of the

transition state is computed, and this probability takes

the place of simply counting the state in the standard

formula for k

RRKM

. In particular, N

†

(E − E

− E

rot

) in

the equation for k

RRKM

(E, J) is replaced by

†

(E − E

− E

rot

) =



P(E − E

− E

rot

− 

(37.36)

where 

is the vibrational energy of state n of the tran-

sition state with respect to its zero point energy, and

the sum is over all states n for which the energy in the

reaction coordinate (the energy in parentheses on the

right-hand-side) is negative, but not so negative that the

classical energy in the reaction coordinate lies below

the minimum of the complex potential well. In general,

only a few vibrational states of the transition state need

be considered for the tunneling correction.

This tunneling correction has been incorporated into

statistical theories for the rate of reactions that proceed

via complexes and transition state barriers [37.60]. It

has been used recently in a calculation of the slow ion–

molecule reaction

+ H

−→ NH

+ H , (37.37)

which successfully reproduces the observation that,

despite initially decreasing as the temperature is re-

duced below 300 K, the reaction rate coefﬁcient begins

to increase as the temperature is decreased below

100 K [37.66]. The parameters for the calculation were

obtained from an ab initio calculation of the potential

surface, which shows the system to proceed through

a weakly-bound long-range complex before encoun-

tering a transition state with a rather small barrier of

≈ 0.25 eV. Although the dynamical theory is not quan-

titative (presumably because of the one-dimensional

tunneling approximation), it does reproduce the iso-

tope effects seen when the reactants NH

+ D

and

+ H

are used, deﬁnitively showing that tunnel-

ing is the cause. The actual increase in rate at very low

temperature comes from the fact that the tunneling rate

is less dependent on temperature than the dissociation

rate of the complex into reactants. Similar calculations

have been performed for the analogous ion–molecule

reaction

+ H

−→ C

+ H (37.38)

to explain a similar observation, although in this latter

instance there is still a controversy concerning whether

or not the reaction is truly exoergic [37.67]. It is in-

teresting to speculate on whether similar effects can be

detected for analogous neutral radical–H

reactants with

moderate activation energy barriers, such as

CCH + H

−→ C

+ H . (37.39)

Part C 37.1

570 Part C Molecules

This system possesses a transition state of rather moder-

ate energy [37.48], but the only long-range complex is

presumably the very weakly-bound van der Waals struc-

ture. Unlike the ion–molecule case, there are very few

experimental studies of low temperature neutral–neutral

reactions. Studies of pressure broadening at very low

temperatures do reveal, however, the strong inﬂuence of

the van der Waals bond [37.68].

37.2 Association Reactions

Association reactions have been studied for both

ion–molecule and neutral–neutral systems. For the ion–

molecule case, association processes can be investigated

in great detail at low densities near or in the radiative

association regime because ions can be stored in low

pressure traps. Review articles [37.69,70] describe such

experiments for small and large ions, respectively. The

results of many higher pressure studies of termolecular

ion–molecule association, especially those undertaken

in the SIFT apparatus, are also available [37.71]. In

general, below the high pressure limit, the rate of as-

sociation reactions without activation energy is known

to increase with (a) an increase in the size of the reac-

tants, (b) an increase in the bond energy of the product

species, and (c) a decrease in temperature. These trends

are all reproduced by statistical theories in which associ-

ation proceeds by the formation of a complex followed

by radiative and/or collisional stabilization [37.8]. More

limited data for termolecular neutral–neutral reactions,

with and without short-range potential barriers, havealso

been reviewed [37.72]. Although radiative stabilization

can proceed via a single photon, collisional stabilization

proceeds most probably in a stepwise fashion rather than

in a single strong collision. The most detailed statistical

theories incorporating multistep collisional stabiliza-

tion use master equation techniques; in general it is

preferable to solve the inverse problem of unimolecular

dissociation via detailed RRKM theory and then invoke

detailed balance [37.13,56]. Such theories are more suc-

cessful than strong collision approaches, especially in

considering the dependence of ternary association rates

on pressure. This dependence can be especially difﬁcult

to treat when association competes with an exoergic

channel which does not necessarily dominate because

of a barrier in the exit channel.

37.2.1 Radiative Stabilization

The problem of radiative association in the absence of

competitive exoergic channels is in principle much more

simple. In addition to the rates of complex formation and

dissociation, one needs only the rate of stabilization via

spontaneous emission. If the complex abundance is at

steady state, the rate law for radiative association of

species A and B is simply

d[A]/ dt =−k

[A][B] , (37.40)

+ k

rad

, (37.41)

where k

represents the rate coefﬁcient for radiative

association and k

rad

refers to the rate coefﬁcient for ra-

diative stabilization. If ternary association is considered

in addition, the rate coefﬁcient for radiative associa-

tion is the low pressure limit of a more complex rate

expression.

Radiative stabilization, which is normally consid-

ered to proceed via emission of a single infrared photon,

has been studied in some detail [37.73–75]. If the

unstable complex is imagined to be a collection of har-

monic oscillators, each vibrational state i, which is best

regarded as a Feshbach resonance, can be expressed

as a set of occupation numbers n

, n

,... ,





for the

assorted modes. If it is further assumed that upon forma-

tion, the complex has a probability P

of being formed

in state i, the rate of single-photon vibrational emission

rad



−1



is then given by

rad



i, j

i→ j



],[n

]

]→[n

]

(37.42)

where the sum is overinitial states i and ﬁnal states j,and

A is the Einstein coefﬁcient for spontaneous emission.

With some algebraic manipulation and the assumption

that dipole selection rules apply, the result simpliﬁes to

rad



k=1



1→0

, (37.43)

where the index k refers to each of the s normal modes,

the index n

to the occupation number of mode k,and

the Einstein A coefﬁcient, which here refers to the fun-

damental transition of mode k, can be expressed in terms

of the absorption intensity (see Chapt. 10). The proba-

Part C 37.2

Gas Phase Reactions 37.2 Association Reactions 571

bility P

that mode k is excited to state n

is given by

the statistical formula

= ρ



vib

− n

hν

)/ρ

vib

), (37.44)

where ρ

vib

is the vibrational density of states and ρ



vib

the reduced density of states when n

quanta are as-

signed to mode k. The role of overtone and combination

contributions to the radiative emission rate has also

been investigated [37.74], as have a canonical approx-

imation to the microcanonical formulation discussed

here [37.75] and the small additional effect of stabiliza-

tion by sequential emission [37.73]. With the use of the

classical approximation to the standard vibrational dens-

ity of state functions [37.41,42], (37.43)fork

rad

reduces

to a linear function of the vibrational energy:

rad

vib



k=1

1→0

hν

. (37.45)

The constant of proportionality between k

rad

and E

vib

depends strongly on the fundamental intensities; many

of these can be obtained from infrared absorption spec-

tra, although there is normally insufﬁcient information

for all of the active modes in polyatomic molecules,

especially ions. Reasonable estimates, as well as lim-

ited experimental data, show that k

rad

varies between

10 and 1000 s

−1

for vibrational energies up to a few

eV [37.69, 70, 73–75]. The radiative stabilization rate

typically decreases slightly with increasing molecular

size due to the inclusion of infrared inactive modes of

vibration. There also appears to be a distinction between

ions and neutral species, with rates for ions apparently

faster .

Since spontaneous emission rates between elec-

tronic states are often far more rapid than infrared

vibrational rates, the possibility that radiative stabi-

lization can proceed via electronic emission has been

examined [37.9]. The well-studied association between

and H

appears to proceed via electronic relax-

ation, with a radiative stabilization rate one to two orders

of magnitude larger than would be the case for vibra-

tional emission. Whether or not the association between

and H

proceeds via an electronically excited com-

plex has been debated for some time; the answer is

apparently negative [37.69].

37.2.2 Complex Formation and Dissociation

The formation and dissociation of the complex can be

studied with a variety of statistical approximations other

than what is discussed in this section. In particular, com-

plex dissociation has been treated by a thermal RRKM

approach [37.70, 75], an energy but not angular mo-

mentum speciﬁc RRKM approach [37.73], variational

transition state theory, statistical adiabatic theory, and

ﬂexible transition state theory [37.72]. We ﬁrst consider

a simple thermal approximation for systems without

activation energy incorporating microscopic reversibil-

ity [37.8]. In the limit that the complex redissociation

into reactants is much more rapid than radiative stabi-

lization, the rate coefﬁcient k

for radiative association

in the thermal model is

= K(T )k

rad

, (37.46)

where K(T ) is the ratio between k

and k

in a ca-

nonical ensemble, and the radiative stabilization rate

has been assumed to be independent of temperature.

The equilibrium coefﬁcient K(T ) between complex

and reactants can be rewritten in terms of partition

functions in the standard way [37.47]. However, the par-

tition function of the complex is best calculated via the

equation

∗

∞



(E + D

) exp(−E/k

T) dE ,

≈ ρ

T , D

 k

T , (37.47)

where ρ

is the density of vibrational-rotational states

[37.41, 42], E is the energy of reactants, and D

is the

bond energy of the complex. One immediate prediction

of this approach (deriving mainly from the rotational

partition functions of the reactants) is that k

pos-

sesses a strong inverse dependence on temperature

since

K(T) ∝ T

−(r

+1)/2

, (37.48)

wherer refers to the number of rotational degrees of free-

dom (two for a linear molecule and three for a nonlinear

one). It also predicts a strong dependence on well depth

and size of the complex since the density of complex

states is a strong function of both these parameters. Al-

though the thermal model agrees with the strong inverse

temperature dependence of the coefﬁcients for both

radiative and, more commonly, ternary ion–molecule

association [37.8,71] (in which case, the thermal model

with strong collisions replaces k

rad

with a collisional

rate coefﬁcient), the absolute rate coefﬁcients calculated

tend to range up to an order of magnitude too high. Given

the large range of values exhibited by radiative associa-

tion rate coefﬁcients



−20

–10

−9

−1



, this might

not seem too large a problem. It has been shown, how-

Part C 37.2

572 Part C Molecules

ever, that the thermal model is deﬁcient, especially at

low temperatures because thermal equilibrium cannot

be achieved by two-body collisions [37.8]. A revised

approach, called the modiﬁed thermal theory, replaces

the rovibrational density of complex states in the ther-

mal model with a vibrational density of states coupled

with sums over the allowable ranges of complex angular

momentum achievable from the speciﬁc capture model

assumed [37.8]. The result is a somewhat lessened in-

verse temperature dependence and somewhat smaller

rate coefﬁcients, especially at low temperatures. Both

modiﬁcations result in better agreement with experi-

ment [37.8].

The modiﬁed thermal approach considers structure-

less reactants; full consideration of the internal states of

the reactants is achieved via the Klots version of phase

space theory, in which k

and k

in (37.41) are the cap-

ture and unimolecular rates discussed in Sect. 37.1.2.

The phase space treatment reduces to the modiﬁed ther-

mal treatment for small reactant angular momentum if

the possibility of saturation [k

(E, J)<k

rad

] may be

ignored. Both conditions are normally met. The phase

space approach has also been used for ion–molecule

ternary association reactions, but here is coupled with

the strong collision hypothesis and must be regarded

as inferior to the more detailed RRKM calculations

with master equation treatments for collisional stabi-

lization [37.13, 56].

37.2.3 Competition with Exoergic Channels

There have been numerous reactions reported (es-

pecially ion–molecule reactions) in which exoergic

reaction channels compete with association channels,

both radiative (at low pressure) and ternary (at higher

pressures). One view of such competition is that it

occurs via a sequential mechanism in which the reac-

tants form a long-lived collision complex which either

redissociates, dissociates into exoergic products, or is

stabilized [37.53, 54]. The rate coefﬁcient for competi-

tive radiative association is then given by the steady-state

expression

+ k



+ k

rad

, (37.49)

where k



refers to complex dissociation into products.

The additional and normally large k



term in the de-

nominator means that the thermal and modiﬁed thermal

theories cannot, in general, be used. The phase space

treatment for this mechanism [37.53, 54, 76, 77]shows

that association cannot compete with exoergic channels

unless there is a signiﬁcant barrier in the exit chan-

nel which considerably slows the dissociation rate of

the complex into products, especially for partial waves

of high angular momentum. Such barriers tend to be

large enough to slow dissociation down, but not large

enough to require tunneling. Although the results of

phase space calculations are in good agreement with ex-

periment for a variety of competitive systems [37.69, 76],

they are once again inferior to RRKM calculations

with master equation collisional deactivation when the

competition involves collisional stabilization of the com-

plex [37.13, 53,54, 56].

Another mechanism exists for competition between

association and normal exoergic channels [37.69]. Ab

initio studies show that there is a parallel type of com-

petition in which the product and association channels

occur on different portions of the potential surface, with

a branching at long range in the entrance channel. For

example, the competing reactions [37.69,78]

+ H

−→ C

+ H , (37.50)

+ H

−→ C

, (37.51)

occur via distinct pathways. The former reaction is a pos-

sibly endoergic direct process in which the molecular

hydrogen attacks perpendicularly, leading to the cyclic

form of the C

ion, whereas the association reac-

tion occurs via the deep well of the ethylene ion. The

competition in the analogous C

+ H

system is not

currently well understood, with both parallel and series

mechanisms suggested [37.69,79].

37.3 Concluding Remarks

While the study of chemical reaction dynamics and ki-

netics is a relatively mature area of investigation, many

challenges remain. A central one is the development

of an efﬁcient, fully quantum mechanical method for

evaluating k(T) that can be applied to a range of sys-

tems [37.80]. A second set of challenges comes in the

intrinsically non-adiabatic nature of chemical reactions.

As a system moves from reactants to products, there

is a dramatic change in the electronic wave function.

In fact the very existence of a transition state reﬂects

the avoided crossing of two potential energy surfaces.

There is experimental and theoretical evidence that the

Part C 37.3

Gas Phase Reactions References 573

fact that the dynamics takes place on multiple potential

surfaces has a large effect on the gas-phase state-to-state

dynamics, and eventually the rate coefﬁcients. Finally,

as the evaluation of the electronic structure of atoms and

molecules becomes more easily accomplished through

the use of sophisticated computer packages, researchers

will be able to ask more detailed questions about the re-

lationships between the topology of potential surfaces

and the corresponding rate coefﬁcients than was possi-

ble as recently as ten years ago [37.81].

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Part C 37

574 Part C Molecules

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Part C 37

575

Gas Phase Ion

38. Gas Phase Ionic Reactions

Ionic reactions in the gas phase is a broad ﬁeld

encompassing a multitude of interactions between

ions (both positively and negatively charged),

electrons and neutrals. These can be as simple as

the transfer of an electron between molecules, or

can be complex with considerable bond breaking,

reforming and rearrangement. Reactions of the

general type

+‚−

+ B

−‚n

→ products , (38.1)

illustrating this diversity, are given in Table 38.1.

The reaction process can be considered as

consisting of three parts: (a) the initial interaction,

in which the colliding particles, A

+‚−

and B

−‚n

are drawn together by an attractive potential, (b)

the reaction intermediate and transition state, in

which reactants are transformed into products

38.1 Overview............................................. 575

38.2 Reaction Energetics ............................. 576

38.3 Chemical Kinetics ................................ 578

38.4 Reaction Processes............................... 578

38.4.1 Binary Ion–Neutral Reactions ..... 579

38.4.2 Ternary Ion–Molecule Reactions.. 581

38.5 Electron Attachment ............................ 582

38.6 Recombination.................................... 583

38.6.1 Electron–Ion Recombination ...... 583

38.6.2 Ion–Ion Recombination

(Mutual Neutralization).............. 584

References .................................................. 585

and (c) the weakening interaction as the product

particles separate. Interactions can also occur

where only elastic scattering is involved and these

are considered in Chapt. 67.

38.1 Overview

Part (a) of the interaction is readily studied theoretically

by classical mechanics in terms of reactant particle mo-

tions controlled by the interaction potential u(r) between

the particles, where r is the interparticle separation.

For ion–neutral interactions, this can take the form

(i) of an attractive ion-polarization or induced dipole

potential [38.1] u(r) =−α

/2r

,whereα

is the po-

larizability of the neutral and q is the charge on the

ion; anisotropy in the polarizability can also be taken

into account [38.2], (ii) of an ion-permanent dipole po-

tential [38.3] with u(r) =−qµ

cos θ/r

,whereµ

is the permanent dipole of the neutral and θ is the

angle that the dipole makes with r, and (iii) of an ion-

quadrupole potential [38.4] with u(r) =−Qq(3cos

θ −

1)/2r

,whereQ is the quadrupole moment and θ

is the angle the quadrupole axis makes with r.Such

capture theories are considered in Chapt. 37. Other in-

teraction potentials can be considered, but these are

of lesser signiﬁcance. Coulombic interaction poten-

tials u(r) =−q

/r,whereq

and q

are the charges

on the interacting particles, are appropriate for pos-

itive ion–negative ion recombination (more correctly

termed mutual neutralization) and electron–ion re-

combination. For processes involving electrons, i. e.,

electron attachment and electron–ion recombination,

the wave nature of the electron also has to be consid-

ered [38.5,6].

In addition to providing a means of bringing the

reactant particles together, the attractive interaction po-

tential has another function. In part (b) of the reaction

mechanism, where considerable rearrangement of the

atoms in the colliding species may occur (i. e., in

the intermediate complex and transition state), there

may be barriers to reaction. Here, the kinetic energy

gained from the interaction potential is available in

the intermediate complex for overcoming such energy

barriers (very much less interaction energy is avail-

able in neutral-neutral reactions and the effects of

energy barriers are very much more evident [38.7]).

Of course, this energy has to be reconverted to po-

tential energy as the product particles separate, and is

therefore not available to drive the overall reaction.

However, the amount of energy returned may differ

from that initially converted to kinetic energy since

Part C 38