Self-consistent quantum kinetic theory of diatomic molecule formationRobert C. Forrey Citation: The Journal of Chemical Physics 143, 024101 (2015); doi: 10.1063/1.4926325 View online: http://dx.doi.org/10.1063/1.4926325 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/143/2?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Systematic study of first-row transition-metal diatomic molecules: A self-consistent DFT + U approach J. Chem. Phys. 133, 114103 (2010); 10.1063/1.3489110 Self-consistent field theory of polymer-ionic molecule complexation J. Chem. Phys. 132, 194103 (2010); 10.1063/1.3430745 Self-consistent theory of the gain linewidth for quantum-cascade lasers Appl. Phys. Lett. 86, 041108 (2005); 10.1063/1.1851004 Non-Born–Oppenheimer trajectories with self-consistent decay of mixing J. Chem. Phys. 120, 5543 (2004); 10.1063/1.1648306 A multiconfiguration self-consistent field/molecular dynamics study of the (n→π * ) 1 transition of carbonylcompounds in liquid water J. Chem. Phys. 113, 6308 (2000); 10.1063/1.1308283

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THE JOURNAL OF CHEMICAL PHYSICS 143, 024101 (2015)

Self-consistent quantum kinetic theory of diatomic molecule formationRobert C. ForreyDepartment of Physics, Penn State University, Berks Campus, Reading, Pennsylvania 19610-6009, USA

(Received 8 May 2015; accepted 24 June 2015; published online 8 July 2015)

A quantum kinetic theory of molecule formation is presented which includes three-body recombina-tion and radiative association for a thermodynamically closed system which may or may not exchangeenergy with its surrounding at a constant temperature. The theory uses a Sturmian representation ofa two-body continuum to achieve a steady-state solution of a governing master equation which isself-consistent in the sense that detailed balance between all bound and unbound states is rigorouslyenforced. The role of quasibound states in catalyzing the molecule formation is analyzed in completedetail. The theory is used to make three predictions which differ from conventional kinetic models.These predictions suggest significant modifications may be needed to phenomenological rate con-stants which are currently in wide use. Implications for models of low and high density systems arediscussed. C 2015 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4926325]

I. INTRODUCTION

Molecule formation via three-body recombination (TBR)and radiative association (RA) and their destructive inverses,collision induced dissociation (CID), and photodissociation,are fundamental processes in gas phase chemistry that havebeen extensively studied for more than fifty years. Whethera molecule forms by TBR or RA is largely determined bythe density of the gas; however, in both cases, the rate offormation is influenced by quasibound states trapped insidea centrifugal barrier. The widely used orbiting resonance the-ory (ORT) developed by Roberts, Bernstein, and Curtiss1 as-sumes that recombination is dominated by quasibound orbitingresonances which stabilize to a bound state upon collisionwith a third body. RA as a mechanism for molecule forma-tion is important at low densities such as occur in molecularclouds.2–10 The orbiting resonances are believed to contributeto RA in a similar manner to collisions but with stabilizationto a bound state occurring through spontaneous emission of aphoton. While quasibound states are central to both processes,the role of narrow resonances is problematic because it re-lies on a kinetic model to account for the population of thelong-lived states. Normally, a kinetic model which assumes anequilibrium distribution of unbound states yields the largestresonant contribution to the formation rate. The assumptionof equilibrium is removed in kinetic models which attempt todetermine the steady-state population of the resonant states. Asteady-state solution to a governing master equation should bemore accurate than an equilibrium approximation. However,steady-state approximations that are conventionally used inRA4–10 and TBR11 allow for the depletion of long-lived reso-nances but neglect mechanisms for their repopulation. In bothcases, the molecule formation rate γi(T) for the resonant state iis equal to an equilibrium constant weighted by a ratio of decaywidths and may be written in the form

γi(T) = Keqi

(Γ1Γ2

Γ1 + Γ2

)(1 + f ), (1)

where f is a kinetic feedback term that contains informationabout interactions with other bound and unbound states. ForRA, the decay widths account for tunneling and emission ofradiation.4–10 For TBR, the radiative width is replaced by acollision term which depends on the third-body concentra-tion.11 Because the feedback f generally depends on unknownconcentrations, it is usually neglected. This approach cansignificantly reduce the resonant contribution to the formationrate and has correctly been criticized in the case of TBR bySchwenke12 and Pack, Walker, and Kendrick.11,13 Their masterequation analysis demonstrated the importance of direct three-body collisions which are missing in kinetic models basedon ORT. It was shown13 that when three-body collisions areincluded in the feedback term in Equation (1), the long-livedresonances are not totally depleted. We further show in thispaper that the feedback term is essential for maintaining localthermodynamic equilibrium (LTE) when there is no energyexchange between the system of interest and its surround-ing environment. The possibility that radiative transitionscould play a similar repopulating role for resonant RA is alsoinvestigated. For astrophysical environments, it is desirable toconsider non-LTE systems where energy loss due to escapingphotons is balanced by gravitational contraction. For the pur-poses of this work, we assume that steady-state level popula-tions and particle abundances may be obtained at a constanttemperature.

For TBR, the relative importance of direct (D) versus indi-rect energy transfer (ET) and exchange (Ex) mechanisms hasbeen a subject of interest and some controversy. The quantumkinetic theory of Snider and Lowry14 was used by Wei, Alavi,and Snider15 to show that the D, ET, and Ex mechanismsshould each lead to the same TBR rate. This formal proof,however, has not been supported in practice. For example,Pack, Snow, and Smith16 calculated TBR rate coefficients forthe ET and Ex mechanisms for hydrogen recombination withH2, He, and Ar acting as the third-body. They selected onlythe longest lived quasibound states and obtained rate con-stants for the separate mechanisms which were not identical

0021-9606/2015/143(2)/024101/12/$30.00 143, 024101-1 © 2015 AIP Publishing LLC

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024101-2 Robert C. Forrey J. Chem. Phys. 143, 024101 (2015)

and showed different temperature dependencies. Esposito andCapitelli17 have performed the most complete classical trajec-tory study for hydrogen recombination. All three recombina-tion mechanisms were included in their calculations, and thecontributions from each mechanism were again found to bedistinctly different. Another recent study by Perez-Rios et al.18

provides a comparison of classical and quantal calculations forhelium TBR which contains interesting and important results.Although not central to the main issues tackled in their paper,it was suggested that their study provides an indirect test of theassertion15 that all alternative three-body mechanisms shouldproduce identical TBR rate coefficients. Because the 4He dimerhas only one bound state and no resonances, ORT predictsa vanishing TBR rate, whereas their quantal and classicalcalculations produced direct recombination rates that exceed10−30 cm6/s at low energy. Based on their results, they state“It remains unclear how this might be reconciled with theconclusions of Ref. 15.” This apparent disagreement between aformal proof and the results of practical computation is clearlyunsatisfactory.

While the proof15 has not been widely accepted, it hasbeen employed by the present author in a Sturmian theory19

which formulates TBR using the ET mechanism and withproper statistical accounting assumes the result representsthe entire TBR rate constant. The present work extends theSturmian formulation to include TBR and RA for closedthermodynamic systems which may or may not exchangeenergy with their surroundings. A steady-state solution to amaster equation is obtained which is self-consistent in thesense that detailed balance between bound and unbound statesis rigorously enforced in order to account for the feedback inEquation (1). An independent proof is given which clarifiesthe interpretation of the alternative recombination mechanismsand reconciles the apparent contradiction described above.The self-consistent solution also allows three predictions tobe made which suggest that significant modifications maybe needed to phenomenological rate constants which arecurrently in wide use. These predictions are (i) steady-state TBR and RA rate constants for an isolated systemcannot depend on concentration or tunneling widths, (ii)resonant TBR and RA rate constants should account for bothdepletion and repopulation of quasibound states, and (iii) forany closed system at constant temperature, the steady-stateresonant contribution should be included in its entirety (i.e.,without removal of contributions from long-lived quasiboundstates). The justification for these predictions is given inthe theoretical analysis of Section II and discussion whichfollows in Section III. The proof that the indirect ET andEx pathways yield the same recombination rate constantwhen formulated in a Sturmian basis set is given in theAppendix.

II. THEORY

The interaction of atoms A and B to form mole-cule AB may be viewed as a two-step process beginningwith

A + B ↔ A · · · B = AB(ui), (2)

where ui denotes an unbound state with energy Eui obtainedby diagonalizing a two-body Hamiltonian with potential vAB

in a Sturmian basis set. The energy eigenstates are assumedto provide a complete basis for both the dynamics and ki-netics. The unbound state may be a quasibound resonance ora discretized representative of the non-resonant background.19

The equilibrium constant between the free continuum and theinteracting unbound state at a temperature T is given by

Kequi =

gui exp(−Eui/kBT)QAQBQT

, (3)

where QA and QB are atomic partition functions, QT is thetranslational partition function, gui is the degeneracy of theunbound state, and kB is Boltzmann’s constant. The secondstep of the process stabilizes AB(ui) to form a bound stateAB(bj) through collision with a third body C or emission ofa photon ν,

AB(ui) + C → AB(bj) + C, (4)

AB(ui) → AB(bj) + ν. (5)

The rate constant for the two-step process is given by

γ(T) =i, j

Kequi γui→b j

, (6)

with

γui→b j=

Γui→b jRA

σui→b j· v TBR

, (7)

where v =

8kBT/πµ is the average velocity for reduced massµ. In Equation (6), there are no kinetic adjustments of thekind described in Equation (1) to account for depletion oflong-lived resonances or feedback mechanisms for their re-population. The connection between kinetic models and non-resonant states included in the summation of Equation (6) isaddressed in the master equation analysis given below. Partic-ular attention is given to quantum kinetic models which treatthe feedback and decay terms self-consistently.

The effective cross section for the TBR stabilization stepis given by

σui→b j= (kBT)−2

∞

0σui→b j

exp(−E/KBT) E dE. (8)

The radiative decay width Γui→b jmay be computed from

matrix elements of the lowest order terms of a multipole expan-sion. This formulation agrees with the result obtained previ-ously2,6 for resonant RA; however, the present theory alsoincludes non-resonant contributions in Equation (6). Likewise,the present formulation agrees with ORT but also includes non-resonant contributions. Different pathways for TBR may beconsidered within this formulation. In general, atoms A andB may recombine through a direct D pathway or through anindirect ET or Ex pathway as shown

A + B + C → AB + C, (9)A + B + C → A · · · B + C → AB + C, (10)A + B + C → A · · ·C + B → AB + C, (11)

with the corresponding transition operators

TD = (V − vAB)(1 + G+E V ), (12)

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024101-3 Robert C. Forrey J. Chem. Phys. 143, 024101 (2015)

TET = (V − vAB) 1 + G+E (V − vAB)ΩAB, (13)

TEx = (V − vAB) 1 + G+E (V − vAC)ΩAC, (14)

where G+E is the outgoing wave Green’s function for the fullHamiltonian consisting of the three-body potential V and thetotal kinetic energy operator. The Moller operators ΩAB andΩAC in Equations (13) and (14) show that the rate coefficientγui→b j

in Equation (7) may be identified as a collisional ratecoefficient kET

ui→b jor kEx

ui→b jwith the understanding that ui

refers to the AC subspace for the Ex pathway. The transitionoperator TD in Equation (12) cannot be considered in this typeof analysis because it does not use a Moller operator to converta free eigenstate to an interacting one. However, it has beenshown15 that TD may be expressed in terms of TET and TEx,so Equation (6) may be used in general for any of the kineticpathways leading to TBR.

Equation (6) suggests that the rate constant for moleculeformation may be written in terms of state-to-state rate coeffi-cients multiplied by the equilibrium constant between the inter-acting unbound state and the free continuum. It is commonlyassumed in ORT;1,20 however, that unbound states which arequasibound with negligible tunneling widths Γtun

uicannot reach

equilibrium and should be removed from the summation inEquation (6). More sophisticated versions of ORT17,21 haveused the criteria

Γtunui≫ [C]

j

kui→b j(15)

to selectively remove resonant contributions as a functionof third-body concentration [C]. This method produces asignificant pressure dependence in the TBR rate constant fora given temperature. For RA, the possibility that a quasi-bound molecular state can radiatively decay before it hastunneled is typically taken into account4–10 by calculatingan effective cross section as in Equation (8) using theopacity

P =ΓtunuiΓradui

(E − Eui)2 + (Γtunui + Γ

radui )2/4

, (16)

where Γradui

includes all possible downward transitions. As-suming Γtun

ui+ Γrad

ui≪ kBT allows the integral to be evaluated

analytically which leads to the modification

γ(T) =i

Kequi*,

ΓtunuiΓradui

Γtunui + Γ

radui

+-. (17)

It will be shown that Equation (17) corresponds to a steady-state kinetic model which has zero population for all stateswith Eu j

> Eui. As in the case of TBR, this modification pro-duces a significant decrease in the contribution from narrowresonances. In each of these commonly used kinetic models,the depletion of long-lived resonances is assumed withoutconsidering mechanisms for their repopulation. Pack, Walker,and Kendrick13 have shown that three-body collisions caneffectively keep long-lived resonances from being depleted.Their master equation analysis revealed pure third-order ki-netics with a weak dependence on concentration and tunnelingwidths. Here, we investigate these dependencies and the possi-

bility that radiative transitions could play a similar repopulat-ing role for resonant RA.

In the present theory, no a priori assumptions are madewith respect to the population of unbound states. The validityof Equation (6) is tested via a master equation analysis whichbalances the formation and destruction of AB via the effectiverate equation

ddt[AB] = Mr[A][B] − Md[AB], (18)

where the square brackets denote number density of the en-closed species, and Mr and Md are the respective rate constantsfor formation (recombination) and destruction (dissociation)of the molecule AB. The effective rate constants are definedby Mr = Γr + kr[C] and Md = Γd + kd[C]where Γr and Γd arethe respective rates for RA and photodissociation, and kr andkd are the respective rates for TBR and CID. Equation (6) maybe generalized using the definitions

Mr ≡i, j

Mui→b j

[AB(ui)][A][B] , (19)

Md ≡i, j

Mbi→u j

[AB(bi)][AB] , (20)

where Mi→ j = Γi→ j + ki→ j[C]. In order for the effective rateEquation (18) to be meaningful, it must be consistent witha set of state-specific rate equations. For the ET path-way,

ddt[AB(bi)] =

j

(Mu j→bi[AB(u j)] − Mbi→u j

[AB(bi)])

+j,i

(Mb j→bi[AB(bj)]

−Mbi→b j[AB(bi)]

)+ Φbi, (21)

ddt[AB(ui)] =

j

(Mb j→ui[AB(bj)] − Mui→b j

[AB(ui)])

+ kelasticf→ui

[A][B] − τ−1ui[AB(ui)]

+j,i

(Mu j→ui[AB(u j)]

−Mui→u j[AB(ui)]

)+ Φui, (22)

where τui is the lifetime of the unbound state ui, and kelasticf→ui

isthe two-body elastic scattering rate constant which is relatedto τ−1

ui= Γtun

uiby the equilibrium constant in Equation (3). The

functions Φbi and Φui represent source terms which allowenergy exchange with the environment. For the moment, weconsider a closed system where the source terms are zero.Steady-state solutions to rate equations (21) and (22) may beexpressed as

[AB(bi)] =

j,i[AB(bj)]Mb j→bi +

j[AB(u j)]Mu j→bij,i Mbi→b j

+

j Mbi→u j

(23)

and

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024101-4 Robert C. Forrey J. Chem. Phys. 143, 024101 (2015)

[AB(ui)] =Kequi[A][B] + τui

(j[AB(bj)]Mb j→ui +

j,i[AB(u j)]Mu j→ui

)1 + τui

(j Mui→b j

+

j,i Mui→u j

) . (24)

It is convenient to express the steady-state concentrations as

[AB(ui)][A][B] ≡ (1 + δui)

gui exp(−Eui/kBT)QAQBQT

, (25)

[AB(bi)][AB] ≡ (1 + δbi)

gbi exp(−Ebi/kBT)QAB

, (26)

where

QAB =i

(1 + δbi)gbi exp(−Ebi/kBT) (27)

is a molecular partition function, and δbi and δui are non-equilibrium concentration defects. Effective rate equation (18) will beconsistent with underlying master equations (21) and (22) if the following condition holds:

Mr

Md=

[AB][A][B] ≡

QAB

QAQBQT(1 + δ), (28)

with

1 + δ =

i, j(1 + δu j

)gbi exp(−Ebi/kBT)Mbi→u ji, j(1 + δbi)gbi exp(−Ebi/kBT)Mbi→u j

. (29)

Equations (25)-(28) may be substituted into Equations (23) and (24). Application of detailed balance yields

(1 + δbi) =

j,i(1 + δb j)Mbi→b j

+ (1 + δ)−1 j(1 + δu j

)Mbi→u jj,i Mbi→b j

+

j Mbi→u j

(30)

and

(1 + δui) =1 + τui

((1 + δ) j(1 + δb j)Mui→b j

+

j,i(1 + δu j)Mui→u j

)1 + τui

(j Mui→b j

+

j,i Mui→u j

) . (31)

After some algebra, we obtain the closed form solution

(1 + δui) =j

B−1uiu j

, (32)

(1 + δbi) = (1 + δ)−1xbi, (33)

Mr

Md= (QAQBQT)−1

i

xbigbi exp(−Ebi/kBT), (34)

where

xbi =j,k,l

A−1bib j

Mb j→ukB−1ukul

(35)

characterizes any deviations from equilibrium. Equation (34) reduces to the Saha relation when xbi = 1. The matrices A and Bwhich need to be inverted are defined by

Abib j= δi j *

,

k,i

Mbi→bk +k

Mbi→uk+-− (1 − δi j)Mbi→b j

, (36)

Buiu j= δi j

1 + τui *,

k,i

Mui→uk +k

Mui→bk+-

− τui

(1 − δi j)Mui→u j

+k,l

Mui→bk A−1bkbl

Mbl→u j

. (37)

Equations (32), (36), and (37) yield the set of homogeneousequations

j

Buiu jδu j= 0. (38)

Assuming |B| , 0, the solution set δui = 0 leads to xbi = 1 andδbi = (1 + δ)−1 − 1 where δ is an arbitrary constant. Therefore,when there is no energy exchange between the system andits surrounding environment, the steady-state solution of the

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024101-5 Robert C. Forrey J. Chem. Phys. 143, 024101 (2015)

master equation is a Boltzmann equilibrium distribution forall bound and unbound states. While this result is to be ex-pected, the derivation of Equation (38) requires that the ratioMr/Md and the rate constants kr and kd must be independent ofconcentration and tunneling widths. This requirement differsfrom kinetic models based on ORT and also conventionalkinetic studies of TBR based on the numerical solution of amaster equation.12,13 This point will be discussed further inSection III and be referred to as the first test of the theory.

The ratio Mr/Md may depend on concentration for sys-tems that allow energy exchange with their environment.Therefore, we now include the source terms Φb and Φu inmaster equations (21) and (22). For a closed system, these maybe defined as

Φbi =j,i

[AB(bj)]Φb j→bi +j

[AB(u j)]Φu j→bi

− [AB(bi)] *.,

j,i

Φbi→b j+

j

Φbi→u j

+/-, (39)

Φui =j

[AB(bj)]Φb j→ui +j,i

[AB(u j)]Φu j→ui

− [AB(ui)] *.,

j

Φui→b j+

j,i

Φui→u j

+/-, (40)

where the state-to-state terms Φi→ j are determined by theparticular application. If the source terms drive only upwardtransitions, then it is convenient to make the definitions

M INi→ j ≡ Mi→ j + Φi→ j Θ(Ei − E j), (41)

MOUTi→ j ≡ Mi→ j + Φi→ j Θ(E j − Ei), (42)

where Θ(x) is the Heaviside step function. Equations (32)-(35) will remain valid if the A and B matrices given by Equa-tions (36) and (37) are replaced by the following generaliza-tions:

Abib j= δi j *

,

k,i

MOUTbi→bk

+k

MOUTbi→uk

+-

− (1 − δi j)M INbi→b j

, (43)

Buiu j= δi j

1 + τui *,

k,i

MOUTui→uk

+k

MOUTui→bk

+-

− τui(1 − δi j)M IN

ui→u j

+k,l

M INui→bk

A−1bkbl

Mbl→u j

. (44)

These equations lead to the set of inhomogeneous equations

τ−1ui

j

Buiu jδu j=

j,k,l

M INui→bk

A−1bkbl

Mbl→u j−

j

Mui→b j

+j,i

(M IN

ui→u j− MOUT

ui→u j

). (45)

As a second test of the theory, we consider the case wherephotons from downward radiative transitions are able to exit

the system. This is equivalent to setting the state-to-state sourceterms Φi→ j = −β Γi→ j in order to remove upward radiativetransitions that are contained in the definition of Mi→ j. Foran optically thick gas, β = 0 and Equation (38) is recovered.For an optically thin gas, β = 1 and the set of inhomogeneousequations (45) becomes

τ−1ui

j

Buiu jδu j=

j,k,l

M INui→bk

A−1bkbl

Mbl→u j−

j

Mui→b j

+j>i

Γui→u j−

j<i

Γui→u j(46)

which may be solved to obtain the steady-state modificationfrom equilibrium. The solution may be expressed as

1 + δui =1 + τuiΨ

INi

1 + τuiΨOUTi

, (47)

with

ΨINi =

j,i

(1 + δu j) *.,

M INui→u j

+k,l

M INui→bk

A−1bkbl

Mbl→u j

+/-,

(48)

ΨOUTi =

j,i

MOUTui→u j

+k

*,

MOUTui→bk

−l

M INui→bk

A−1bkbl

Mbl→ui+-. (49)

In the low density [C] → 0 limit,

ΨINi =

j,i

(1 + δu j) *,Γu j→uiΘ(Eu j

− Eui) + ϵ ik

Γu j→bk+-

×gu j

guie−(Eu j−Eui

kBT

)(50)

ΨOUTi =

j<i

Γui→u j+ (1 − ϵ i)

k

Γui→bk, (51)

where

ϵ i =kb1→uij kb1→u j

(52)

represents a dissociative feedback pathway induced by colli-sions from the ground state b1. If this term is neglected, theresult

1 + δui =1 + τui

j>i Γui→u j

(1 + δu j)

1 + τui(

j<i Γui→u j+

j Γui→b j

) (53)

is obtained. This equation may be solved iteratively startingwith the quasibound state which has the largest energy andthe assumption that δu = 0 for all unbound states which arenot quasibound. It is noteworthy that Equation (53) reducesto

1 + δui =1

1 + τui(

j<i Γui→u j+

j Γui→b j

) , (54)

when δu j= −1 which corresponds to zero population for all

unbound states with j > i. This assumption leads to resonant

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024101-6 Robert C. Forrey J. Chem. Phys. 143, 024101 (2015)

rate coefficient (17) that is commonly used for RA. SubstitutingEquation (53) into Equation (25) yields

Γr =i

Kequi*,

ΓtunuiΓradui+ ∆ui

Γtunui + Γ

radui

+-, (55)

where

∆ui =j>i

Γui→u j(1 + δu j

)k

Γui→bk − τ−1ui

j<i

Γui→u j(56)

is an additional term which differs from Equation (17).The negative contribution removes downward transitions tounbound states which are included in conventional models.4–10

For narrow resonances, the negative term may be neglected,and the result is an enhancement in the resonant rate constantcompared to Equation (17). This enhancement would beeven larger when dissociative feedback (52) is included. Itis noteworthy that the assumption ϵ i = 0 cannot hold forall unbound states. If it did, rate equations (21) and (22)would require the steady-state concentrations to be zero for allstates other than the ground state. The dissociative feedbackpathway allows unbound states to be replenished throughcollisions at low density in a manner which is independentof third-body concentration. Normally, the population ofquasibound states is expected to be less than or equal to theequilibrium value so that δui ≤ 0. However, it is possible tohave unbound populations such that δui > 0. For example, atvery low temperatures, it may be possible to have ϵ i ≈ 1 forui equal to the lowest-lying quasibound state, if the energyof this state is close to zero. In this case, δui ≈ τuiΨIN

i whichcan be significantly larger than zero for a quasibound statewith a large lifetime. This would yield the resonant contri-bution

Γ(i)r = Keq

ui

1 + τuiΨ

INi

j

Γui→b j(57)

which would not only be an enhancement compared to Equa-tion (17) but would also be an enhancement compared toEquation (6). These types of resonant enhancements comparedto conventional kinetic models are the second prediction of thepresent theory.

Equations (43)-(49) may also be used to compute non-equilibrium coefficients (35). In the low density limit, theground state coefficient

xb1 =

i j(1 + δui)gui exp(−Eui/kBT)Mui→b j

gb1 exp(−Eb1/kBT) j kb1→u j[C] (58)

is obtained which dominates all other coefficients by O[C]−1.

The denominator is consistent with previous studies ofCID.22–24 Substitution into Equation (34) yields the result

Mr

Md=

i j(1 + δui)gui exp(−Eui/kBT)Mui→b j

QAQBQT

j kb1→u j[C] . (59)

Equation (59) reveals that the ratio Mr/Md has an inversedependence on the third-body concentration for a low densitynon-LTE system. This result follows from Equation (34);however, it may also be obtained directly from the defini-

tions

Mr =i j

(1 + δui)gui exp(−Eui/kBT)

QAQBQTMui→b j

, (60)

Md =i j

(1 + δbi)gbi exp(−Ebi/kBT)

QABMOUT

bi→u j(61)

which demonstrates the self-consistency of the theory. Equa-tion (59) applies to both RA and TBR at low densities. Inboth cases, the formula yields a large molecule-to-atom ratiodue to the small non-LTE dissociation rate in the denominator.The effect is amplified at low temperatures and goes againstexpectations for primordial gas which is purely atomic. Thereare two ways to reconcile this dilemma:

• The assumption that T is constant breaks down. Thisis consistent with hydrodynamic models of primor-dial star formation25,26 and suggests that steady-statepopulations may not be reachable within the coolingtimescale. In this case, it may be sufficient to assumethat rate equation (22) reaches a steady-state withoutimposing the same requirement for Equation (21). Thiswould allow Equations (53) and (60) to serve as a modelfor molecule formation for a given T before it changes.However, care should be used when attempting toinclude this model as part of a larger kinetic modelwith an associated master equation because the non-equilibrium defects δu may change.

• The assumption that [C] → 0 breaks down. Althoughthe system is assumed to be closed with respect to thenumber of atoms of a particular type, the steady-stateratio (59) allows for increasing gas density at a constantT so long as the density remains small enough thatcollisions are negligible compared to radiative decay.As the density increases, however, collisions becomeimportant and the non-LTE dissociation rate increasestowards its LTE value. In this case, Equation (59) wouldneed to be replaced by Equations (60) and (61) withupdated values of δu and δb to reflect the increaseddensity. For constant temperature, this would eventuallylead to high density.

In order to address this second possibility, we now consideran optically thin gas where radiative transition probabilitiesare small compared to collisional transition probabilities. If wemaintain the assumption of constant temperature, then Equa-tions (47)-(49) may be used, and the high density [C] → ∞limit yields the result

ΨINi = [C]

j,i

(1 + δu j)

× *.,kui→u j

+k,l

kui→bk A−1bkbl

Mbl→u j

+/-, (62)

ΨOUTi = [C]

j,i

kui→u j

+ [C]k

kui→bk*,1 −

l

A−1bkbl

Mbl→ui+-, (63)

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024101-7 Robert C. Forrey J. Chem. Phys. 143, 024101 (2015)

and

1 + δui =1 + τui[C] j,i(1 + δu j

) (kui→u j+

k,l kui→bk A−1

bkblMbl→u j

)1 + τui[C] j,i

(kui→u j

+

k,l kui→bk A−1bkbl

Mbl→u j

) . (64)

Equation (64) leads to a set of homogeneous equations forall δui independent of the value of τui. This shows that thesteady-state solution of the master equation is equal to a Boltz-mann equilibrium distribution to O([C]−1)when there is energyleaving the system at constant T . Therefore, we conclude thatthe usual practice of removing long-lived resonances from therecombination rate constant is not justified for an isothermalprocess. Because a low density gas described by Equation (59)contracts until high densities are reached, the steady-state solu-tion for an isothermal process is LTE. In this case, the reso-nant contribution should be included in its entirety, and Equa-tions (6) and (7) should be used without any modifications.This is the third prediction of the present theory.

III. DISCUSSION

As shown above, the self-consistent steady-state solutionof a Sturmian master equation for an isolated system at con-stant temperature is a Boltzmann equilibrium distribution forall bound and unbound states. This result differs from thevast majority of TBR studies which utilize a kinetic schemebased on ORT. The important works of Schwenke12 and Pack,Walker, and Kendrick11,13 noted that ORT yields an unphysicalpressure dependence. They provided a numerical solution toa master equation and a fitting procedure to obtain kr and kd.While this approach is valid, in principle, it is difficult to imple-ment and provides limited insight. It also yielded rate con-stants that were dependent on concentration12,13 although thedependence was far less than in ORT. Both of these studies12,13

considered isolated systems; however, the steady-state solu-tions of their master equations were not the same as a Boltz-mann distribution. In the calculations by Pack, Walker, andKendrick,13 the dependence on concentration may have beendue to their rate coefficients11 which did not exactly satisfydetailed balance. Rate coefficients must satisfy detailed bal-ance in order to get the kinetic equations to yield the cor-rect steady-state populations. It is noteworthy that Schwenke12

considered a mixture of third bodies whose concentrations maychange as the reaction proceeds. This greatly complicated hisanalysis and led to a breakdown of the “linear mixture rule”which assumes the rate constants are independent of concen-tration. In the present theory, there is no such breakdown of thelinear mixture rule for an isolated system because the ratio

Mr

Md=Γr

Γd=

krkd

(65)

must be independent of concentration. Equation (65) is the firsttest of the present theory. This test applies to mixtures of thirdbodies because the same analysis applies if we replace Γr and

Γd with collision terms such that Mr = kCr [C] + kD

r [D] andMd = kC

d[C] + kD

d[D]. The linear mixture rule is maintained

in this case due to the condition

Mr

Md=

kCr

kCd

=kDr

kDd

(66)

which is required in the derivation leading to Equation (38).The second test of the present theory considers a low-

density gas that loses energy to its surrounding due to exitingphotons, and it is assumed that a steady-state population of qua-sibound states may be obtained within the cooling timescale.The prediction here is that Equations (53) and (60) shouldreplace Equation (17) for computing the resonant contribu-tion. Considering that Equation (17) is normally used for verynarrow resonances that are difficult to resolve in a scatteringcalculation, the present prediction will necessarily increase therecombination rate constant compared to conventional calcu-lations4–10 that do not account for repopulation of quasiboundstates due to transitions from unbound states which are higherin energy. This enhancement may be significant at low temper-atures. For example, the resonant RA rate constant for CH+ ata temperature of 10 K decreased by two orders of magnitude8

when Equation (17) was used in place of Equation (6). If the∆ui term is included in Equation (55), the resonant rate constantwould likely increase by a comparable magnitude if the tran-sition matrix elements between the unbound and quasiboundstates are similar to those between unbound and bound states. Itwould be interesting to see whether this could help explain theobserved concentration of CH+ in the interstellar medium.27

A further increase would occur if a steady-state popula-tion of bound states may also be reached within the coolingtimescale. This corresponds to an isothermal process and leadsto the third test of the present theory. In this case, the den-sity increases as photons produced by spontaneous emissionexit the system, and the prediction is that all unbound states,including extremely long-lived quasibound states, will reach asteady-state population that is the same to leading order in anasymptotic expansion as would be obtained by a Boltzmannequilibrium distribution. As a result, the steady-state resonantcontribution should be included in its entirety (i.e., without theremoval of contributions from long-lived quasibound states)and Equations (6) and (7) should be used to calculate therecombination rate constants. This prediction applies to bothRA and TBR at constant temperature. However, Equation (59)shows that caution should be used when attempting to deter-mine Md from Mr . The usual rate quotient law given by xbi = 1in Equation (34) would provide a substantial overestimate ofMd for a given Mr at low densities.

The three predictions described above remain valid whenexchange processes are included in the master equation

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024101-8 Robert C. Forrey J. Chem. Phys. 143, 024101 (2015)

analysis (see Appendix). It is further demonstrated that theindirect ET and Ex pathways yield the same recombinationrate at equilibrium. Because the direct operator TD may beexpressed in terms of TET and TEx, the present theory is incomplete agreement with the analysis of Wei et al.15 Thereis some ambiguity associated with interpreting the variousmechanisms in the context of classical TBR. In previouswork,19 it was speculated that the direct contribution obtainedfrom classical trajectory calculations might correspond to thecomplete TBR rate in accordance with Ref. 15. It shouldbe noted, however, that the transition operator TD for directprocess (9) acts on free states, whereas the transition operatorsTET and TEx for the second step of indirect processes (10) and(11) act on continuum states of an interacting subsystem. Ifthe direct contribution is interpreted as corresponding to thenon-resonant part of a two-step mechanism, then the directand indirect contributions will clearly be independent and notidentical as they are both part of a single pathway. Thesediffering interpretations of the direct process are responsiblefor the apparent contradiction between the formal proof15 andthe practical calculations described in the Introduction. Whenthe resonant contribution is zero, as in the case of helium TBR,the non-resonant part of the ET or Ex pathway yields the fullrate coefficient. If the representation of interacting unboundstates in the two-body subspaces is complete, then the ratecoefficient computed in both pathways should be the same.

The present theory may be viewed as a practical imple-mentation and generalization of the quantum kinetic theoryof Snider and Lowry.14 A Sturmian representation provides anatural way to achieve this implementation because the en-ergy eigenstates form a complete basis for both dynamics andkinetics. A previous attempt to use an L2 representation ofbroad above barrier resonances within the ET mechanism11

was unsuccessful due to an apparent arbitrariness in the num-ber of nodes which were retained in the L2 representationbefore truncating and normalizing. This led the authors11 toconclude that non-resonant continuum states could not in prac-tice be included in an accurate quantum calculation of the ETmechanism. This issue was further investigated with a Stur-mian representation19 using the same energy sudden approxi-mation employed in Ref. 11, and it was found that at this levelof dynamical approximation, the broad resonances and nonres-onant continuum states could in practice be included within theET mechanism. This suggests that a practical demonstrationof the equivalence of alternative pathways may be possible;however, the formulation19 did not provide a numerical testbecause it considered only a single pathway. Likewise, therecent study of helium TBR18 calculated the rate constant forsingle direct pathway (9), so it does not provide a numericaltest of the equivalence with indirect pathways. Other studiesof TBR16,17,20 have considered multiple pathways, but thesealso do not provide a reliable test of the formal proof becausethey do not include a complete representation of interme-diate states along a given pathway. Furthermore, such calcu-lations sum the contributions from each pathway in an inco-herent manner which may introduce errors associated with thedouble-counting of transition probability.15

The completeness of a two-body Sturmian representationallows TBR to be formulated in any desired mechanism, or in

the case of identical particles, as a symmetrized combinationof ET and Ex mechanisms. Normally, it is most convenientto formulate the calculation using the ET mechanism. Forreactive systems which can easily exchange atoms, the conver-gence rate of the Sturmian basis set can be extremely slow.A similar situation arose in reactive scattering formulationswhich employed negative imaginary potentials to absorb thereactive flux.28 A rapid increase in the convergence rate wasachieved in these calculations by using a transformation fromthe asymptotic diabatic basis set to successive sectors of localadiabatic basis sets.28–35 A similar set of transformations wouldbe helpful to improve convergence rates using the Sturmianbasis sets implicit in the present theory. It is worth notingthat Sturmian representations have been very successful indescribing electron and photon collisions with atoms.36 There-fore, it is expected that the theory described here for moleculeformation should also be applicable to atomic ionization andrecombination.

A key advance here is the requirement that detailed bal-ance is enforced not just between bound state transitions,but also for transitions between bound and unbound states.This allows Equations (19) and (20) to be evaluated usingequilibrium concentrations for any thermodynamically iso-lated system. A complicated master equation analysis of thekind described in the classic TBR papers12,13 is not necessaryfor such systems because the equilibrium and steady-stateconcentrations are the same. For closed systems where thesteady-state concentrations differ from equilibrium, modifiedEquations (60) and (61) may be used to calculate the rateconstants. The non-equilibrium concentration defects δbi andδui may be obtained from basic Equations (32)-(35) usingmatrices (43) and (44). These equations apply to a non-LTEsystem where the source terms drive upward transitions. In theRA example considered in the present work, the source termswere negative (sinks) to remove upward radiative transitionscontained in the definitions of Mi→ j. Positive source termsdue to radiation from an external source such as a star couldbe handled in a similar manner. Because Equations (32)-(35)represent a formal solution to a governing master equation, aset of rate coefficients and widths may be used directly to obtainsteady-state association and dissociation rate constants withoutfurther numerical methods for handling the kinetics. Thismay be particularly useful for systems which have extremelylong-lived resonances that increase the stiffness of the masterequation.24

A final note concerns the utility of the present theoryfor dynamical formulations which do not employ Sturmianbasis sets. For RA, the conventional numerical techniques2–10

for computing matrix elements of the dipole operator neededin Equation (54) may also be employed in Equation (53).Therefore, it should be straight-forward to test prediction(ii). For TBR, the use of classical trajectories is the mostcommon method for computing the rate constant. For reasonsdiscussed above, this approach does not lead to identical rateconstants for each mechanism. Esposito and Capitelli17 haverecommended “carefully” summing up contributions from thevarious mechanisms in such a way that double-counting doesnot occur. For this approach to be consistent with prediction(iii) of the present theory, the strong pressure-variation of

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024101-9 Robert C. Forrey J. Chem. Phys. 143, 024101 (2015)

FIG. 1. TBR rate constant for H+H+H. The DEB + ORT curve correspondsto the low pressure limiting case reported by Esposito and Capitelli.17 Alsoshown is the quantum calculation performed by the present author.37

the ORT contribution arising from the criteria used to selectquasibound states [see Equation (15)] would need to be re-placed by the low pressure limiting result so that all resonancesare included independent of pressure. This point is illustratedin Figure 1 for the important case of hydrogen recombination.37

The so-called DEB result of Esposito and Capitelli17 refers todetailed balance applied to a quasiclassical CID calculationfor H + H2. As mentioned above, a previous interpretation19

suggested that the DEB result might correspond to the com-plete TBR rate constant in accordance with the operator TD.The alternative (and probably better) interpretation is that itcorresponds to the non-resonant part of the transition operatorsTET and TEx. When all of the resonant contributions are addedwithout removal of long-lived quasibound states, the DEB+ ORT result shown in Figure 1 is obtained. This combinedresult corresponds to the low pressure limiting case given byEsposito and Capitelli.17 For increasing pressure, the long-lived quasibound states were removed according to Equa-tion (15) which caused a strong variation in the reported

DEB + ORT results.17 This pressure dependence is incon-sistent with prediction (iii) of the present work for a closedsystem at constant temperature. In this case, the steady-stateresonant contribution should be included without removal ofcontributions from long-lived quasibound states. Therefore,the classical dynamical rate constant for all pressures shouldbe the DEB + ORT result shown in Figure 1. For the temper-atures shown in the figure, this classical rate constant is abouttwo times larger than the quantum mechanical rate constantcomputed by the present author37 using the energy suddenapproximation.

ACKNOWLEDGMENTS

This work was supported by NSF Grant No. PHY-1203228. Helpful communications with Professor ChrisGreene are gratefully acknowledged.

APPENDIX: PROOF THAT ET AND EX PATHWAYSYIELD IDENTICAL RATE CONSTANTS

TBR is considered for an isolated system which allowsboth the ET and Ex mechanisms given by Equations (10) and(11). As before, no a priori assumptions are made with respectto the population of bound or unbound states. The formationof molecule AB is described by the rate equation

ddt[AB] = kr[A][B][C] − kd[AB][C], (A1)

kr =i, j

kAB→ ABui→b j

[AB(ui)][A][B] =

i, j

kAC→ ABui→b j

[AC(ui)][A][C] , (A2)

kd =i, j

kAB→ ABbi→u j

[AB(bi)][AB] =

i, j

kAB→ ACbi→u j

[AB(bi)][AB] . (A3)

In order for the rate coefficients kr and kd in phenomenolog-ical rate equation (A1) to be meaningful, it is necessary thatthey be consistent with an underlying master equation whichproperly describes the system under consideration. The masterequations for the present system may be written as

ddt[AB(bi)] = [C] *.

,

j,i

kAB→ ABb j→bi

[AB(bj)] +j

kAB→ ABu j→bi

[AB(u j)]+/-

− [AB(bi)][C] *.,

j,i

kAB→ ABbi→b j

+j

kAB→ ABbi→u j

+j

kAB→ ACbi→b j

+j

kAB→ ACbi→u j

+/-

+ [B] *.,

j

kAC→ ABb j→bi

[AC(bj)] +j

kAC→ ABu j→bi

[AC(u j)]+/-, (A4)

ddt[AB(ui)] = [C] *.

,

j,i

kAB→ ABu j→ui

[AB(u j)] +j

kAB→ ABb j→ui

[AB(bj)]+/-+ kelastic

f→ui[A][B]

− [AB(ui)][C] *.,

j,i

kAB→ ABui→u j

+j

kAB→ ABui→b j

+j

kAB→ ACui→u j

+j

kAB→ ACui→b j

+/-

+ [B] *.,

j

kAC→ ABu j→ui

[AC(u j)] +j

kAC→ ABb j→ui

[AC(bj)]+/-− τ−1

ui[AB(ui)], (A5)

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024101-10 Robert C. Forrey J. Chem. Phys. 143, 024101 (2015)

with similar expressions for the bound and unbound concen-trations of AC. It is convenient to express the steady-stateconcentrations as

[AB(ui)][A][B] ≡ (1 + δAB

ui)g

ABui

exp(−EABui/kBT)

QAQBQT, (A6)

[AB(bi)][AB] ≡ (1 + δAB

bi)gABbi

exp(−EABbi/kBT)

QAB, (A7)

with

QAB =i

(1 + δABbi

)gABbi

exp(−EABbi/kBT) (A8)

and similar expressions for AC. For self-consistency, wedefine (

krkd

)AB

=[AB][A][B] ≡

QAB

QAQBQT(1 + δAB), (A9)

where

1 + δAB =

i, j(1 + δAB

u j)gAB

biexp(−EAB

bi/kBT)kAB→ AB

bi→u ji, j(1 + δAB

bi)gAB

biexp(−EAB

bi/kBT)kAB→ AB

bi→u j

.

(A10)

If we can prove that δABui= δACui

= 0 and (1 + δAB)(1 + δABbi

)= (1 + δAC)(1 + δACbi

) = 1, then phenomenological rate equa-tion (A1) will be consistent with the underlying master equa-tions (A4) and (A5). Substituting (A6)-(A9) into (A4) yieldsthe steady-state equation

(1 + δAC)(1 + δACbi)

=j

(AAC)−1bib j

k

kAC→ ACb j→uk

(1 + δACuk)

+k

kAC→ ABb j→uk

(1 + δABuk

) + (1 + δAB)

×k

kAC→ ABb j→bk

(1 + δABbk

), (A11)

where

AACbib j= δi j *

,

k,i

kAC→ ACbi→bk

+k

kAC→ ACbi→uk

+k

kAC→ ABbi→bk

+k

kAC→ ABbi→uk

+-− (1 − δi j)kAC→ AC

bi→b j. (A12)

A similar expression for (1 + δAB)(1 + δABbi

) may be obtainedand then substituted into the right-hand-side of Equation (A11)to yield the equation

(1 + δAC)(1 + δACbi) =

j,k

(AAC)−1bib j

M AC→ AC

b j→uk(1 + δACuk

)

+ M AC→ ABb j→uk

(1 + δABuk

), (A13)

where

AACbib j= AAC

bib j−

k,l

kAC→ ABbi→bk

(AAB)−1bkbl

kAB→ ACbl→b j

, (A14)

M AC→ ACbi→u j

= kAC→ ACbi→u j

+k,l

kAC→ ABbi→bk

(AAB)−1bkbl

kAB→ ACbl→u j

,

(A15)

M AC→ ABbi→u j

= kAC→ ABbi→u j

+k,l

kAC→ ABbi→bk

(AAB)−1bkbl

kAB→ ABbl→u j

.

(A16)

In order to evaluate the right-hand-side of Equation (A13), weneed to obtain δAB

uland δACul

. Substituting Equations (A6)-(A9)into Equation (A5) yields the steady-state equation

j

BABuiu j

(1 + δABu j

)

= 1 + τABui

[C]

j

kAB→ ACui→u j

(1 + δACu j)

+j

kAB→ ABui→b j

(1 + δAB)(1 + δABb j

)

+j

kAB→ ACui→b j

(1 + δAC)(1 + δACb j), (A17)

where

BABuiu j= δi j

(1 + τAB

uiψABui

[C]) − (1 − δi j)τABui

kAB→ ABui→u j

[C](A18)

with

ψABui=

k,i

kAB→ ABui→uk

+k

kAB→ ABui→bk

+k

kAB→ ACui→uk

+k

kAB→ ACui→bk

. (A19)

Expressing (1 + δAB)(1 + δABb j

) and (1 + δAC)(1 + δACb j) using

Equation (A13) and substituting into Equation (A17) yields theset of equations

j

Buiu j(1 + δu j

) = 1, (A20)

where δu jis a column vector comprised of δAB

u jand δACu j′

, and Bui,u jis an enlarged matrix defined by

Buiu j=*.,

BABuiu j− τAB

ui[C]N AB→ AB

ui→u j−τAB

ui[C]N AB→ AC

ui→u j

−τACui[B]N AC→ AB

ui→u jBACuiu j− τACui

[B]N AC→ ACui→u j

+/-, (A21)

with

N AB→ ABui→u j

=k,l

kAB→ ABui→bk

(AAB)−1bkbl

M AB→ ABbl→u j

+k,l

kAB→ ACui→bk

(AAC)−1bkbl

M AC→ ABbl→u j

, (A22)

N AB→ ACui→u j

= kAB→ ACui→u j

+k,l

kAB→ ABui→bk

(AAB)−1bkbl

M AB→ ACbl→u j

+k,l

kAB→ ACui→bk

(AAC)−1bkbl

M AC→ ACbl→u j

. (A23)

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024101-11 Robert C. Forrey J. Chem. Phys. 143, 024101 (2015)

Using the identities j

kAB→ ABui→b j

=j,k,l

kAB→ ABui→bk

(AAB)−1bkbl

(M AB→ AB

bl→u j+ M AB→ AC

bm→u j

), (A24)

j

kAB→ ACui→b j

=j,k,l

kAB→ ACui→bk

(AAC)−1bkbl

(M AC→ AB

bl→u j+ M AC→ AC

bm→u j

), (A25)

it is straight-forward to show that j

Buiu j= 1 (A26)

which yields in Equation (A20) a set of homogeneous equations for δu j. Assuming |B| , 0, the trivial solution δu j

= 0 isobtained which leads to (1 + δAB)(1 + δAB

bi) = (1 + δAC)(1 + δACbi

) = 1. If the arbitrary constants δAB and δAC are set to zero,Equations (A2) and (A3) reduce to

kr =i, j

kAB→ ABui→b j

gABui

exp(−EABui/kBT)

QAQBQT=

i, j

kAC→ ABui→b j

gACui

exp(−EACui/kBT)

QAQCQT, (A27)

kd =i, j

kAB→ ABbi→u j

gABbi

exp(−EABbi/kBT)

QAB=

i, j

kAB→ ACbi→u j

gABbi

exp(−EABbi/kBT)

QAB, (A28)

and it is easily seen that detailed balance may be applied to both the ET and Ex rate coefficients to show that(krkd

)AB

=QAB

QAQBQT. (A29)

This proof, which is based on representation theory and detailed balance, is in complete agreement with the operator theory proofgiven by Ref. 15. It shows that the rate constants kr and kd are independent of concentration and tunneling widths for an isolatedsystem and may be computed using either ET or Ex rate coefficients.

It is also noteworthy that the above notation may be generalized to include RA and any number of diatomic electronic statesγ using

Mγ1→γ2bi→u j

= Mγ1→γ2bi→u j

+k,l,γ

Mγ1→γ

bi→bk(Aγ)−1

bkblMγ→γ2

bl→u j, (A30)

Nγ1→γ2ui→u j

= (1 − δγ1,γ2)Mγ1→γ2ui→u j

+k,l,γ

Mγ1→γ

ui→bk(Aγ)−1

bkblMγ→γ2

bl→u j, (A31)

and

Buiu j=

*.......,

Bγ1uiu j− τγ1

ui Nγ1→γ1ui→u j

−τγ1ui Nγ1→γ2

ui→u j−τγ1

ui Nγ1→γ3ui→u j

...

−τγ2ui Nγ2→γ1

ui→u jBγ2uiu j− τγ2

ui Nγ2→γ2ui→u j

−τγ2ui Nγ2→γ3

ui→u j...

−τγ3ui Nγ3→γ1

ui→u j−τγ3

ui Nγ3→γ2ui→u j

Bγ3uiu j− τγ3

ui Nγ3→γ3ui→u j

...

... ... ... ...

+///////-

. (A32)

This leads to γ, j

Buiu jδγu j= 0 (A33)

which establishes a more general pathway independence at LTE. For a low-density closed system which radiates energy to itsenvironment, Equations (53) and (60) generalize to

1 + δγui =1 + τγui

γ′, j Γ

γ→γ′ui→u j

(1 + δγ′u j)Θ(Eγ′

u j− Eγ

ui)1 + τγui

(γ′, j Γ

γ→γ′ui→u j

Θ(Eγui − Eγ′

u j) +

γ′, j Γγ→γ′

ui→b j

) (A34)

and

Γγr =

i, j,γ′

Kequi (1 + δγui) Γγ→γ′

ui→b j, (A35)

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024101-12 Robert C. Forrey J. Chem. Phys. 143, 024101 (2015)

with

Kequi =

gγui exp(−Eγ

ui/kBT)QAQBQT

. (A36)

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