THE JOURNAL OF CHEMICAL PHYSICS 141, 064312 (2014)

Theoretical investigation of intersystem crossing between the a 1A1and X 3B1 states of CH2 induced by collisions with helium

Lifang Ma(���),1 Millard H. Alexander,2,a) and Paul J. Dagdigian3,b)

1Department of Chemistry and Biochemistry, University of Maryland, College Park,Maryland 20742-2021, USA2Department of Chemistry and Biochemistry and Institute for Physical Science and Technology,University of Maryland, College Park, Maryland 20742-2021, USA3Department of Chemistry, The Johns Hopkins University, Baltimore, Maryland 21218-2685, USA

(Received 13 June 2014; accepted 25 July 2014; published online 13 August 2014)

Collisional energy transfer between the ground (X 3B1) and first excited (a 1A1) states of CH2 isfacilitated by strong mixing of the rare pairs of accidentally degenerate rotational levels in the groundvibrational manifold of the a state and the (020) and (030) excited bending vibrational manifolds ofthe X state. The simplest model for this process involves coherent mixing of the scattering T-matrixelements associated with collisional transitions within the unmixed a and X states. From previouscalculations in our group, we have determined cross sections and room-temperature rate constantsfor intersystem crossing of CH2 by collision with He. These are used in simulations of the timedependence of the energy flow, both within and between the X and a vibronic manifolds. Relaxationproceeds through three steps: (a) rapid equilibration of the two mixed-pair levels, (b) fast relaxationwithin the a state, and (c) slower relaxation among the levels of the X state. Collisional transferbetween the fine-structure levels of the triplet (X) state is very slow. © 2014 AIP Publishing LLC.[http://dx.doi.org/10.1063/1.4892377]

I. INTRODUCTION

Methylene (CH2), a small molecule with only six valenceelectrons, is an important intermediate in organic synthesis,1–3

chemical combustion,4, 5 and planetary photochemistry.6–8

The molecule is the simplest carbene, one of the most re-active intermediates in organic chemistry.1 The highest twofilled molecular orbitals of CH2 are, in linear geometry, thenon-bonding 2px and 2py atomic orbitals on the C atom (weassume that the linear molecule defines the z axis). In singletCH2 one of these orbitals is doubly occupied and the otherunoccupied, so that it will behave as either an electron donoror acceptor. In the ground triplet state each of these orbitals issingly occupied.

The difference in electronic structures of the a1A1 andX3B1 states CH2 leads to significantly different reactivities.The a state CH2 reacts rapidly with radicals, as well as withmany stable molecules. By contrast, the X state reacts withonly a few very reactive species, but at a slower rate comparedto the singlet.4, 5, 9

In addition, we have shown that rotationally inelastic re-laxation within the a state is 2–3 times faster than within theX state.10, 11 In these studies we investigated rotationally in-elastic collisions of He with CH2 in its first excited a stateand in its ground X state, separately. Collisions with othermolecules (H2, N2, H2O, CO2, CH4) or atoms (He, Ar) cancause electronic relaxation of the a to the ground X state.9

Because the two states lie close in energy (∼3000 cm−1), thiscollision-induced intersystem crossing (CIISC)12 is of key im-portance in the relaxation kinetics of CH2. Our investigations

a)Electronic mail: mha@umd.edu.b)Electronic mail: pjdagdigian@jhu.edu.

of rotational relaxation and preliminary work on collision-induced vibrational and electronic transitions have been re-viewed recently.13

Although CH2 contains only light atoms, there exists asmall spin-orbit coupling between the X and a states, on theorder of a few wavenumbers.14 This can lead to significantstate mixing whenever two ro-vibrational levels associatedwith the two electronic states lie within a few wavenumbersof each other, provided, further, that the states have the sametotal angular momentum j and parity.

Since CH2 is a hydride, the rotational levels are sparse.Additionally, only roughly half of the levels belong to a givennuclear spin modification. Thus, the occurrence of a pair ofa − X levels which are significantly mixed is rare. Bley andTemps14 conducted a meticulous investigation of the degreeof a − X mixing in CH2 for over 100 rotational levels in thea(0,0,0) manifold with rotational angular momentum n ≤ 10and its body-frame projection ka ≤ 6. They found (see Table5 of Ref. 14) that only 16 pairs have fractional mixings largerthan 0.01. Of these, only 5 pairs have fractional mixing largerthan 0.1; these are listed in Table I. Figure 1 shows the levelstructure in the vicinity of several of these nearly degeneratea − X pairs.

To a first approximation, collision-induced transitions be-tween the vibration-rotation manifolds of the two electronicstates will be governed by the small spin-orbit coupling be-tween these rare pairs of accidentally degenerate levels. Akinetic model based on this approximation was first devel-oped by Gelbart and Freed.15–18 The availability of spec-troscopic and kinetic data9, 19 makes the methylene radicalan ideal species to assess the accuracy of this mixed-statemodel.14, 20, 21

0021-9606/2014/141(6)/064312/10/$30.00 © 2014 AIP Publishing LLC141, 064312-1

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

162.129.250.14 On: Wed, 13 Aug 2014 13:35:28

064312-2 Ma, Alexander, and Dagdigian J. Chem. Phys. 141, 064312 (2014)

TABLE I. Energies, quantum numbers, and mixing coefficients sin 2θ (fromRef. 14) of the most strongly mixed CH2 a − X pairs. All a state levels are inthe vibrational ground state (0,0,0). The zero of energy is the lowest rotationallevel of a(000).

X

o/p a (v1v2v3) j, nkakc

Es(cm−1) Et(cm−1) sin 2θ

para 431 (0,3,0) 4, 312 285.366 287.376 0.149para 633 (0,3,0) 6, 616 502.132 501.302 0.138ortho 716 (0,3,0) 7, 615 537.160 537.560 0.463ortho 818 (0,2,0) 8, 937 567.028 568.728 0.256ortho 945 (0,2,0) 9, 946 1042.44 1031.40 0.240

In our previous quantum scattering studies of rotationalrelaxation in the a and X states,10, 11 we neglected electronicstate mixing. We compared calculated total removal rate con-stants for some ka = 1 rotational levels of CH2(a) with ex-perimental results from Hall, Sears, and their co-workers.20, 22

For most of the experimentally probed levels our predictedrate constants were about 20% larger than the experimentalresults (see Fig. 16 of Ref. 10 and Fig. 10 of Ref. 11).

In contrast, for the a(0,0,0) 818 level, which forms amixed pair with the X(0,2,0) 937 level, the calculated re-moval rate constant for the unmixed 818 state is about 70%larger than the experimental rate constant for the nominallya (singlet) component of the mixed pair. The calculated re-moval rate constant for the unmixed 937 level is about 10%smaller than the experimental rate constant for the nominallyX (triplet) component of the mixed pair. These differencesbetween our theoretical results, calculated under the assump-tion of no state mixing, and the experimental observations il-lustrates the importance of the mixing between the a and X

states.In this paper, we extend our previous work10, 11 to include

collision-induced intersystem crossing between the a and theX states of CH2, still in collision with He. Here, we includealso the fine-structure splitting in the ground triplet state. For

400

450

500

550

600141 14

1 2 3 41 2 3 4ka = 1

a (000)~X (020)~

14%

47%

26%

131 12

936937 542

541

E ro

t / c

m–1

112 101129

440441

836835

X (030)~

616

615

514514

515

541 542

642643

726716

818

633634

624

625615

717

FIG. 1. Energy level diagram showing three low-lying strongly mixed “gate-way” pairs of CH2 in the energy range of 400–600 cm−1, relative to the low-est rotational level of the a(0,0,0) vibronic manifold. The solid and dashedlines refer to levels of the ortho and para nuclear spin modifications, respec-tively. The percentage mixings determined in Ref. 14 are shown in light blue.The fine-structure splitting in the X state has been ignored in labeling thelevels in this state.

each mixed pair, only one of the triplet fine-structure levelswill be coupled with an a state rotational level. The mecha-nism of collisional singlet-triplet mixing in CH2 molecule isdiscussed in Sec. II. Section III summarizes the mixed-statequantum calculations of cross sections and rate constants, theresults of which are presented in Sec. IV. In Sec. V we intro-duce a simple kinetic model for the relaxation kinetics in thepresence of two coupled electronic states. We follow this witha full solution of the relaxation master equation for both statessimultaneously. In Sec. VI, we compare our mixed-state rateconstants with the experimental results of Hall and Sears.22

We finish with a brief summary.

II. MECHANISM OF SINGLET-TRIPLET STATEMIXING IN CH2

In the mixed-state model15–17 CIISC occurs only throughthe few pairs of levels which are mixed, and thus have par-tial singlet and partial triplet character. These pairs then act as“gateways” for the relaxation between spin-rotation-vibrationlevels associated with different electronic states. For nota-tional simplicity, we express the rovibronic levels of the a andX components of a mixed pair as

|ψs〉 = ∣∣χsv jks

aksc

⟩∣∣1A1

⟩, (1)

|ψt 〉 = ∣∣χtvjntkt

aktc

⟩∣∣3B1

⟩. (2)

Here, we have introduced the indices t and s for the tripletand singlet states and have subsumed the three vibrationalquantum numbers (symmetric stretch, antisymmetric stretch,bend) into a single index χv . In Eqs. (1) and (2) we haveexplicitly appended the electronic wave functions of the twostates. Note that although the total angular momentum j has tobe the same for the two levels, the rotational quantum numbern can be different since n can equal j = n and n ± 1 in the X

(triplet) state.In their analysis of a − X mixing, Bley and Temps14 pro-

vided estimates of the spin-orbit matrix elements Hst

Hst = 〈ψs |Hso|ψt 〉= ⟨

jkaskc

s∣∣⟨χs

v

∣∣⟨1A1

∣∣Hso

∣∣3B1

⟩∣∣χtv

⟩∣∣jntkat kc

t⟩

(3)

between the singlet and triplet levels of a mixed pair. The mix-ing between |ψ s〉 and |ψ t〉 is obtained by diagonalizing the2 × 2 matrix

H =(

Es Hst

Hst Et

), (4)

where Es and Et are the energies of the unmixed singlet andtriplet levels defined by Eqs. (1) and (2).

In terms of the pure (unmixed) singlet and triplet states,the wave functions for the mixed states can be written as

|ψA〉 = − cos θ |ψs〉 + sin θ |ψt 〉, (5)

|ψX〉 = sin θ |ψs〉 + cos θ |ψt 〉, (6)

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

162.129.250.14 On: Wed, 13 Aug 2014 13:35:28

064312-3 Ma, Alexander, and Dagdigian J. Chem. Phys. 141, 064312 (2014)

where θ , the so-called “mixing angle,” is

θ = 1

2tan−1

(2Hst

Et − Es

). (7)

The fractional mixing, is measured by the magnitude of sin 2θ

(see Table I), can rise as high as 50% (0.5) when the nominalsinglet and nominal triplet levels are completely degenerate.

Formally, the scattering amplitude between an initial |i〉and a final |f〉 state can be written as

〈i|T |f 〉 ≡ Tif , (8)

where T denotes a generalized transition operator. In themixed-state model,15, 17 the matrix elements of T are com-puted, separately, for the unmixed a and X levels. Then,Eq. (8) is extended to include the mixed character of the pair.

In the simplest approximation, we treat each pair ofmixed levels separately. The wave functions of the mixed pairare given by Eqs. (5) and (6). The transition amplitudes in-volving transitions to/from one or the other of the pair ofmixed states from/to any unmixed level will be straightfor-ward linear combinations of the T-matrix elements betweenthe unmixed states.

For example, from the mixed level with nominal a char-acter, whose wave function is given by Eq. (5), into an un-mixed level in the a state, which we designate |s′〉, we have

TAs ′ = 〈ψs |T |ψs ′ 〉= (− cos θ〈ψs | + sin θ〈ψt |)T |ψs ′ 〉= − cos θ〈ψs |T |ψs ′ 〉 + sin θ〈ψt |T |ψs ′ 〉= − cos θTss ′ . (9)

The cancellation of the 〈ψt |T |ψs ′ 〉 term in going from the 3rdto the 4th line of Eq. (9) occurs because we assume no col-lisional coupling between the singlet and triplet rovibronicmanifolds in the absence of spin-orbit mixing. In other words,in the mixed-state model it is only the weak spin-orbit cou-pling in the isolated CH2 molecules that allows for a mixingof the nearly degenerate pairs. Similarly, from the mixed levelwith nominal X character, whose wave function is given byEq. (6), into an unmixed level in the a state, which we desig-nate |s′〉, we have

TXs ′ = sin θTss ′ . (10)

Similar equations apply for transitions from the mixed pairinto unmixed levels in the X state.

The T-matrix elements for transitions between the pair ofgateway states are more complicated. There are two terms inboth Eqs. (5) and (6), giving rise to four terms contributing tothese T-matrix elements. Of these, two vanish because of theabsence of coupling between the unmixed rovibronic levels ofthe two states. We then find, taking as an example the state ofnominal a character,

TAA = cos2 θTss + sin2 θTtt . (11)

For the three other possible couplings (XX, AX, and XA)we find

TXX = sin2 θTss + cos2 θTtt , (12)

TXA = TAX = cos θ sin θ (−Tss + Ttt ). (13)

We note, in passing, that the T-matrix elements between thetwo different components of a mixed pair are linear combina-tions of the elastic Tss and Ttt matrix elements, which can beexpected to be both large in magnitude, but certainly not iden-tical. Thus, the T-matrix elements connecting the two mixedstates “borrow” from the T-matrix elements for elastic transi-tions, which are large. Thus we anticipate efficient collisionaltransfer between the components of a mixed pair.

III. SCATTERING CALCULATIONS

In the presence of the weak spin-orbit mixing betweenthe a and X states, the mixed-state model allows us to calcu-late elements of the two-state T matrix from the T matrix forcollisions of He with, separately, CH2(a) and CH2(X). Thesewe have already determined, and details of these calculationscan be found in the literature.10, 11

We further assume that the gateway process occursthrough only one mixed pair at a time. In other words, weignore interference between two different gateways. We haveconcentrated on pairs in which the mixing is >10%, and withrotational energies <2000 cm−1, but above the lowest rota-tional level of the X(0,2,0) vibronic manifold. There are only4 pairs (2 para and 2 ortho) that meet these criteria, all involv-ing levels within the a(0,0,0) vibronic manifold. These levelsand their X state partners are listed in Table I.

In our study of rotational relaxation of CH2(X),11 we ig-nored the fine-structure splitting of the rotational levels, treat-ing the molecule as a singlet electronic state. Since the a − X

mixing couples only one fine-structure level of a given n inthe X state, we must include the electron spin explicitly in theT matrix for the X state. Since the interaction potential doesnot explicitly depend on spin and the fine-structure splittingis only a fraction of a wavenumber,23 we can use the spin-free matrix elements to generate those for transitions betweenindividual fine-structure levels.

McCourt, Alexander, and their co-workers24, 25 haveshown how to obtain T matrix elements between fine-structurelevels with spin included from those computed in a spin-freecalculation:

T Jjsnk

akcl,j ′sn′k′

ak′c

= (−1)j′−j−l′+l[(2j + 1)(2j ′ + 1)]1/2

×∑

J

(2J +1)

{s n j

l J J

}{s n′ j ′

l′ J J

}T J

nkakcl,n′k′

ak′cl

′ .

(14)

Here, {:::} is a 6j symbol,26 and the electron spin s = 1 inthe X state. From the triangle relations on the angular mo-menta in the 6j symbols in Eq. (14), the angular momentum J

= J − 1,J ,J + 1 here.Scattering calculations at a collision energy of 300 cm−1

for these four pairs were carried out with the HIBRIDONsuite of programs.27 Additional scattering calculations at vari-ous collision energies (up to 1300 cm−1) for the 818 − 937 pairwere carried out to compute the rate constants for transitions

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

162.129.250.14 On: Wed, 13 Aug 2014 13:35:28

064312-4 Ma, Alexander, and Dagdigian J. Chem. Phys. 141, 064312 (2014)

involving this mixed pair, by numerical integration:28

kmm′(T ) =(

8

πμ(kBT )3

)1/2

×∫ ∞

0σm→m′(Ec) Ec exp (−Ec/kBT ) dEc. (15)

Here, we used repeated trapezoidal integration on a gridof collision energies (Ec) spaced every 20 cm−1 up to2500 cm−1.

IV. CROSS SECTIONS AND RATE CONSTANTS

A. Scattering out of the mixed-pair levels

It is interesting to investigate the changes in the inelas-tic and elastic cross sections for collision-induced rotationaltransitions within the a and X states caused by inclusion ofthe spin-orbit coupling in the scattering calculations.

We list in Table II several integral cross sections out ofthe four levels that constitute the two mixed ortho pairs inTable I. The upper three rows list cross sections computedwithout including the spin-orbit mixing between the two elec-tronic states. The lower part of the table presents cross sec-tions computed after including the coupling between the firstpair of states, and, in a separate calculation, the second pair.

It can be seen in Table II that the elastic cross sectionsare smaller for all four initial levels after the singlet-tripletmixing is included. On the other hand, the cross sections fortransitions to the other perturbed level(818↔937, 716↔615),are significantly larger than all the other inelastic cross sec-tions. In particular, for the 716 and 615 pair, where the mixingis nearly maximal (∼50%), the cross sections to the other per-turbed level are even larger than the elastic cross sections, dueto the maximized mixing angle. To a good approximation, in

the presence of mixing the non-mixed elastic flux is appor-tioned between the purely elastic transitions (A → A and X→ X) and the intra-pair transitions (X → A and A → X).

Also, as revealed by Eq. (9), the cross sections from themixed singlet level to other singlet levels is reduced by afactor of |cos θ |2. A similar consideration applies to transi-tions from the mixed triplet level to other triplet levels. Thisloss is partially compensated for by the addition of transitionsfrom the nominally singlet component of the mixed pair tothe triplet rotational levels and, vice versa, from the nomi-nally triplet component of the mixed pair to the singlet ro-tational levels, both of which appear with an intensity factorof |sin θ |2. Thus, since cos θ2 + sin θ2 = 1, the sum of theremoval cross sections out of the two unmixed levels of thenearly degenerate pair are very close in magnitude to the sumof the removal cross sections to the rotational manifolds ofboth the a and X states out of these same levels when they aremixed. This is seen by comparing the sum of the two bold-faced entries in the upper section of Table II with the sum ofthe four bold-faced entries in the lower section.

Figure 2 shows graphically the cross sections tabulatedin the lower section of Table II. Although the cross sectionsshown in Table II and Fig. 2 refer to a single collision energyof 300 cm−1, from them we can make some qualitative pre-dictions about the relaxation of a non-equilibrium sample ofCH2 in the two electronic states. Transfer of population be-tween the pair of mixed states will have the largest rate con-stant. Since the two states in a mixed pair have the same jand nearly the same energy, their thermal populations will beequal. Thus, the fastest relaxation process will be the equi-libration of the populations in the two mixed levels. Subse-quently, these two levels will relax to the other unperturbeda state levels. However, the largest state-to-state cross sectionfor this process is 3.1 Å2, an order of magnitude smaller thanthose for equilibration of population between the two mixedlevels. Finally, transitions to other unperturbed X state levels

TABLE II. Elastic, total removal, and the largest state-to-state cross sections for transitions from selected levelsof ortho-CH2 by collision with He at a collision energy of 300 cm−1.a

Cross section/Å2

Transitions a 818 X 937(0,2,0) a 716 X 615(0,3,0)

Without spin-orbit couplingElastic 65.3 73.0 65.3 73.9Total removal 16.0b 5.7b 18.4 4.7Largest state-to-state cross section 4.3c 1.3d 1.5e 2.3f

With spin-orbit couplingElastic 35.0 38.2 30.5 31.2To the other mixed-pair level 32.7 32.5 38.8 38.7Removal to all other singlet levels 11.3b 4.7b 9.9 8.5Removal to all other triplet levels 1.6b 4.0b 2.2 2.5Largest state-to-state cross section 3.1c 0.9d 0.8e 1.2f

aCross sections for two mixed-state pairs are presented: (1) a 818 and the j = 8 component of X(0,2,0) 937 and (2) a 716 and the j= 7 component of X(0,3,0) 615.bThese bold-faced entries will be discussed further in the main text.ca818 → a734.dX937(j = 8) → X 835(j = 7).ea716 → a707.fX615(j = 7) → X 413(j = 5).

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

162.129.250.14 On: Wed, 13 Aug 2014 13:35:28

064312-5 Ma, Alexander, and Dagdigian J. Chem. Phys. 141, 064312 (2014)

0

10

20

30

40

Cro

ss S

ectio

n / Å

2

818

to other perturbed levelto all other singlet levelsto all other triplet levels

937 716 615

FIG. 2. Bar plot of cross sections for transitions from both components ofthe two mixed pairs [a 818/X(0,2,0) 937 j = 8 and a 716/X(0,3,0) 615 j = 7]to the other component of the mixed level and to all other 1a and 3X levels,in collisions with He at a collision energy of 300 cm−1.

will be even slower since rotational relaxation in the X stateis slower than in the a state.10, 11

Since the (0,2,0) bending vibrational level of the X statelies more than 1300 cm−1 lower than the (0,0,0) vibrationallevel of the a state (see Fig. 2 of Ref. 11), for calculationsin which the collision energy in the a state is ∼ 1000 cm−1,the comparable collision energy for the levels of the X(0,2,0)state will be greater than 2000 cm−1. To simplify the compu-tational requirements, we carried out scattering calculationsfor collision energies only up to 1300 cm−1 and set cross sec-tions for collision energies greater than 1300 cm−1 equal tothe value at that energy. Because the Boltzmann weighting inEq. (15) [Ecexp ( − Ec/kBT)] decreases rapidly at high col-lision energy, test calculations indicate that the error in theroom temperature rate constants introduced by this extrapola-tion will be at most 1%.

B. Inclusion of the X state fine-structure levels

The spin-orbit interaction14 mixes an a-state level withonly one fine-structure component of a rotational level in theX state. As outlined in Sec. III, to describe collisions in whichthe X and a states are mixed, we expanded our previous spin-free calculation11 of rotational transitions in the X state toinclude the spin. Having done so, we can explore propensitiesin collision-induced rotational/fine-structure transitions in theX state.

Our earlier work13 has indicated that, except for low n(low j) there will be a propensity for transitions where �j= �n. Initially, �S and �n are coupled to form �j . The collisionaffects only �n. After the collision �n′ recouples with �S. Because�S is unaffected, in this recoupling the relative orientation of�S and �n′ is preserved. This propensity has been seen beforein studies of rotationally inelastic collisions of O2(3�−) andNH(3�−).29–31

To demonstrate this propensity, Table III lists some crosssections for transitions out of the nominally X component ofthe 818/937 mixed pair, the |jnkakc〉 = |8937〉 level, into (a) theother two n = 9 levels with different values of the total angularmomentum (j = 9 and j = 10) and (b) into the three fine-structure components of the n = 7 level. These calculations

TABLE III. Cross sections for transfer for transitions out of theX(0, 2, 0)937 level to other rotational/fine-structure levels in the X(0,2,0) vi-bronic manifold.a

Transitions Cross section/Å2

Spin-freeb Spin includedc

Elastic 73.09 73.04Total removal 5.61 5.66

j = 8 j = 9 j = 10To other 937 levels 73.09

73.04 4.23 × 10−2 5.55 × 10−4

j = 6 j = 7 j = 8To 735 levels 1.09

1.05 3.73 × 10−2 6.92 × 10−4

aRelaxation just within the X state, without spin-orbit coupling to the a state.bSpin-unresolved nk

akc

→ n′k′ak

′c transitions out of the 937 level.

cSpin-resolved jnkakc

→ j ′n′k′ak

′c transitions out of the 937(j = 8) fine-structure level.

address relaxation just within the X state, in the absence ofspin-orbit coupling with the a state.

The �j = �n propensity is seen in transitions both elasticand inelastic in n. Cross sections for transitions in which �j�= �n are 3 orders of magnitude smaller than for transitionsin which �j = �n.

When spin is included, inelastic flux for a single n → n′

transition becomes split over nine j, n → j′, n′ transitions. Wesee from Table III that to a very good approximation, the spin-resolved cross sections obey the simple conservation rule

σ(spin-free)n→n′ ≈ 1

2S + 1

n+1∑j=n−1

n′+1∑j ′=n′−1

σj,n→j ′,n′ . (16)

In our mixed-state model, we include only one pair ofmixed states in each separate calculation. We focus on the a

818/X(0,2,0) 937 pair, to compare with the available experi-mental data from the group of Hall and Sears at Brookhaven.

V. RELAXATION KINETICS

A. Simplistic model

We first consider a simplistic model for the relaxation ofa mixed pair, in which collisions can transfer population be-tween these levels as well as into two separate baths consist-ing of all the other a and X rotational levels. Population willnot be allowed to return from the baths to the mixed pair. Wedesignate the two mixed-state levels with nominal a and nom-inal X character as “A” and “X,” respectively. These levelswill be interconnected and coupled to the two baths by 6 rateconstants, as illustrated in Fig. 3.

With a realistic choice of rate constants, determination ofthe time dependence of the populations in the “X” and “A”levels will give us qualitative insight into the relaxation kinet-ics of a mixed level system. The master equation for relax-ation of the populations in these two levels can be written as

d

dtn = ρKn, (17)

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

162.129.250.14 On: Wed, 13 Aug 2014 13:35:28

064312-6 Ma, Alexander, and Dagdigian J. Chem. Phys. 141, 064312 (2014)

A X

BathSinglet

Bath Triplet

kASkAT kXS kXT

kXA

kAX

FIG. 3. A simplistic model for the relaxation of a pair of mixed levels andtwo baths made up of all singlet and, respectively, all triplet levels, exclusiveof the pair of levels which are mixed. Return of population from the baths tothe mixed levels is not allowed.

where n is a column vector of the populations nA and nX, ρ isthe number density of the buffer gas, and K is the rate constantmatrix

K =(−kA kAX

kAX −kX

). (18)

The diagonal element kii of K is the total removal rate of leveli, while the off-diagonal element kij is the rate for a transitionfrom level i to level j. The total removal rate constants equal

kA = kAX + kAS + kAT , (19)

kX = kXA + kXS + kXT . (20)

Since the gateway states have the same rotational quantumnumber and have nearly the same energy (see Table I), weassume kAX = kXA.

From our prior calculations on the unmixed a and X

states, and from the calculations described in Sec. IV, we es-timate the following values for the rate constants that appearin the rate matrix [Eq. (18)]: kA = −7.15, kX = −6.78, kAX= 5.26, all in units of 10−10 molecule cm−3 s−1. The mas-ter equation [Eq. (17)] may be solved, for example, by themethod described by Alexander et al.32

Figure 4 shows the time dependence of the populationsin the two mixed levels A and X when the initial populationis solely in either of the two levels. Under either initial condi-

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

initial state: A

Pop

ulat

ion

0 0.2 0.4 0.6 0.8 1

initial state: X

AX

t ρ t ρ

FIG. 4. Time dependence of the populations of the two mixed levels in Fig. 3with (left panel) initial population solely in the nominally singlet state (desig-nated “A”) and (right panel) solely in the nominally triplet state (designated“X”). The abscissa is time multiplied by the number density ρ in units of1010/cm3.

tions, the populations of the two levels equalize rapidly andthen both relax at the same rate. The population evolutionis thus governed by two rates, as expected for a 2 × 2 ratematrix [Eq. (18)]. Consistent with this analysis, Komissarovet al. observed in their study of the collisional thermalizationof CH2(a, X) that the populations of the mixed a 818 − X 937levels rapidly equalized and then decayed together at the samerate.20

B. Full relaxation master equation

A more accurate model for the relaxation includes tran-sitions into and out of all levels. For a system with p rota-tional levels in the a state and q rotational levels in the X state,relaxation is governed by a master equation of dimensional-ity p + q. The terms in the master equation, Eq. (17), mustbe augmented. The column vector of the populations in thep + q levels, namely (here we save space by giving the trans-pose of this column vector)

nT = (ns1· · · nA · · · ns

pnt1

· · · nX · · · ntq). (21)

In the absence of mixing, the rate constant matrix is blockdiagonal in the electronic state index, namely,

K =(

Ka 0

0 KX

), (22)

where Ka and KX are the p × p and q × q matrices of rate con-stants for transitions within the a and X states, respectively.We will treat, separately, each pair of mixed levels. These twolevels we designate, as in the simplistic model, “A” and “X.”As discussed formally in Sec. II, levels A and X will couplewith all levels, in both electronic states. Thus, the rate con-stant matrix will be modified from the block-diagonal form ofEq. (22) by changing the rows and columns corresponding tothe indices of the A and X levels into full columns and fullrows. This is illustrated, schematically, in Fig. 5.

Each rate constant is an integral over collision energy ofthe appropriate cross section [see Eq. (15)]. To simplify, theupper triangle of the rate matrix was determined by with ascattering calculation, while the lower triangle was obtainedby detailed balance.28

For transitions between a pair of unperturbed levels inthe a state, the rate constants are identical to those calcu-lated previously.10 In our study of rotational relaxation in theX state,11 we ignored the fine-structure splitting of the rota-tional levels due to the nonzero electron spin. Here, we ob-tain the cross sections, and hence rate constants, for transi-tions between the rotational/fine-structure levels of the X statethrough the use of the expression for the T matrix elementsgiven in Eq. (14). Rate constants in the two modified rows andcolumns of K, involving transitions to/from the mixed levels,were obtained through the treatment described in Sec. II.

The full master equation was solved by the method ofAlexander, Hall, and Dagdigian.32 Rather than integrate thedifferential equations describing the time evolution of thepopulations [Eq. (17)], here the rates governing the collisionalrelaxation are obtained by diagonalizing a modified rate ma-trix. Full details of this procedure are given in Ref. 32.

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

162.129.250.14 On: Wed, 13 Aug 2014 13:35:28

064312-7 Ma, Alexander, and Dagdigian J. Chem. Phys. 141, 064312 (2014)

FIG. 5. Schematic of the rate constant matrix for a pair of mixed levels. (Left matrix) The K matrix in the absence of mixing, block-diagonal in the spinmultiplicity [see Eq. (22)]. (Right matrix) In the presence of spin-orbit coupling the two rows and two columns (indicated by orange stripes) corresponding theA/X mixed pair, now extend across both the singlet and triplet levels.

C. Master equation simulation

Making use of our calculated rate constants, we wrotea MATLAB script to solve the relaxation master equation[Eq. (17) and Sec. V B]. This allows us to explore the timedependence of the populations of rotational levels in both thea and X states. In these simulations, we include 48 and 160rotational levels in the a(0,0,0) and X(0,2,0) vibronic man-ifolds, respectively. The three fine-structure components foreach triplet nk

akc

level [with j = n − 1, n, n + 1] are includedas distinct levels. The MATLAB script is available in the sup-plementary material.33

We ignore relaxation of the bending vibrational mode.We have recently determined that relaxation of this modeis at least two orders of magnitude smaller than rotationalrelaxation.34 Hence, for CIISC involving the a(0,0,0) 818 andX(0,2,0) 937 levels, we included only rotational levels in thea(0, 0, 0) and X(0, 2, 0) manifolds.

In our previous papers,10, 11 we have seen that the �ka= 0, �n = ±1 and �ka = 2, �n = 0 transitions make thedominant contribution to the total removal cross sections outof a single rotational level of the a state. Similarly the �ka= 0, �n = −1/ − 2 and �ka = −1, �n = 2 transitionsare the strongest for relaxation of the X(0,2,0) state. For the818/937(0, 2, 0) pair, these strong transitions lead to a state fi-nal levels 716, 918, and X state final levels 836 and 835, 735,and 112, 10.

The MATLAB script allows us to visualize the entire timeevolution, displaying as a function of time bar histograms ofthe populations in various levels. The first example is the evo-lution of the population subsequent to initial population in the818 (nominal a) component of the 818/937 mixed pair. We setboth the translation and rotational temperature to 300 K andthe bath pressure to 2 Torr. These values are consistent withthe ongoing experiment of Hall and Sears.22 We assume thatboth T and p do not change. The pressure affects the numberdensity of the collision partners ρ in Eq. (17), but not the ratematrix.

Figure 6 presents three snapshots, following initial popu-lation of the 818 level, the nominally a level of the particularortho-CH2 mixed-pair under study at Brookhaven.20, 35 In thetop panel, we plot the positions of some of the nearby rota-tion levels in both the a (blue) state and the X (green) states.

For the triplet levels we include in the bar plots both the othermixed pair (937) and the 735 level. The 937 → 735 transitionis one of the strongest relaxation pathways in the X state. Weshow explicitly each of the three fine-structure components ofthese two X rotational levels (these are degenerate to withinthe resolution of the figure). For the a state we include sev-eral additional final rotational states, covering transitions with�ka = [0, 1, 2], �n = [ − 3, −2, −1, 0, 1], and �E = −200∼ 200 cm−1.

0.1

0.2

0.3j = 8

t = 60 ns

0

0.01

0.02

0.03

Pop

ulat

ion

t = 900 ns

−200

0

200

j = 8

0

0.4

0.8

1.2 t = 0 ns

9 10

937

735

j = 6 7 8

918836

827818

616625

716550725

0

E–E

(818

) / c

m–1

j = 8 9 10937 735

6 7 8918 836 827 818616625716550725

FIG. 6. Rotational distribution of selected singlet and triplet levels at vari-ous times following initial population of the a(0, 0, 0) 818 component of themixed pair; p = 2 Torr and T = 300 K. The black rectangular empty boxesindicate the relative Boltzmann population including all a and X levels. Theblue (slightly narrower) empty boxes represent the relative Boltzmann popu-lation of the a levels in the absence of coupling to the X state.

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

162.129.250.14 On: Wed, 13 Aug 2014 13:35:28

064312-8 Ma, Alexander, and Dagdigian J. Chem. Phys. 141, 064312 (2014)

0 100 200 300 400 5000

0.2

0.4

0.6

0.8

1

Time / ns

Rel

ativ

e P

opul

atio

n 818(s) S(total)T(total)

0 2000 4000 6000 8000 10000

Time / ns

937(t)

FIG. 7. Time dependence of the (normalized) populations in the mixed singlet and the mixed triplet levels as well as the sum of the populations in all unperturbedsinglet and all unperturbed triplet levels. Initial population was solely in the mixed singlet 818 level. The left and right panels correspond to two different timeranges.

The three lower panels of Fig. 6 display bar histogramplots of the initially normalized populations at three differ-ent times (chosen on a logarithmic scale). Two square emptyboxes are plotted for each of the a state levels, of which theblack one represents the Boltzmann population including alla and X levels, while the blue empty box represents the rela-tive Boltzmann population among the a levels in the absenceof coupling to the X state.

By 60 ns (middle panel) the populations in the two mixedlevels, 818 and 937(j = 8) have already equilibrated, consider-ably before significant population is seen in any of the otherlevels. This immediate re-distribution between the mixed lev-els is consistent with the large size of the cross section fortransitions between the two mixed levels, which borrows in-tensity from the elastic scattering amplitudes in the two states[see Eq. (13)].

In the fourth panel of the figure, at t = 900 ns, the blueboxes for the a state levels are almost fully filled. By this timerotational relaxation within the a state has reached equilib-rium. From this point the population of all a state levels de-cay at the same rate. Since the 818 level lies ∼ 1900 cm−1

above the lowest level in the simulation [the a(0, 2, 0) 000(j= 1) level], at 300 K, eventually all the population will havedecayed out of the levels considered explicitly in this figure.

At t = 900 ns, the populations of the X state levels (ex-cept the X component in the mixed pair) are still low - theblack boxes indicating their Boltzmann populations are stillempty. This is true even for the 735 level, which is relativelystrongly coupled with the mixed-pair level 937. Note that evenafter progression from 60 ns (middle panel) to ∼ 900 ns(fourth panel), the fractional populations of the two mixedlevels remain equal, even though the fractional population inboth has decreased from ∼0.25 to ∼0.01. A Boltzmann pop-ulation distribution is attained at much longer times (t ≈ 2 ×105 ns).

We show in Fig. 7 the evolution of the rotational pop-ulations of several levels over two different time scales. Weinclude the two mixed states (818 and 937), as well as the totalpopulation in all singlet [designated S(total)] and all triplet[designated T(total)] unperturbed levels. This figure showsagain that the two mixed states merge quickly to the samedepletion rates within 50 ns. The a state population then max-imizes (at t ≈ 1000 ns) only to diminish at longer time, as

population recedes into the rotational levels of the energeti-cally lower X(0,2,0) vibronic manifold.

In conclusion, for initial population in the singlet com-ponent of one of the mixed pairs, the relaxation process iscomposed of three major steps: very fast energy transfer tothe mixed partner (818 → 937 in this case), relatively fast ro-tational re-distribution within the a state (900 ns), and slowerrotational re-distribution within the X state.

Population restricted initially to a nearby unperturbedlevel (rather than one of the mixed-pair levels), for exam-ple, a 616, provides an interesting contrast. This is shown inFig. 8. Here the initial relaxation is slower, corresponding toredistribution of population among the other rotational levelsof the a state. We see, though, evidence of the strong mixingbetween the 818 and 937 levels. As population appears in the818 level, it shows up simultaneously in the other mixed-pair.The prompt re-distribution between the two mixed-state lev-els is not as obvious in this simulation because both mixedlevels are initially empty. However, once population is trans-ferred into one, it appears in the other. Their populations arealready equal at t = 100 ns.

By t = 900 ns, the relative populations are very similarto those shown in Fig. 6. Once population has equilibrated inthe a state levels, the subsequent population evolution is in-dependent of which a state level was populated initially. Thisis true both for the relative and absolute populations.

VI. DISCUSSION

As discussed in the Introduction, for most of ka = 1 lev-els of ortho-CH2(a) that were studied experimentally,20, 35 thetotal removal rate constants calculated ignoring the mixingwith the X state are about 20% higher than the experimen-tal observations. Surprisingly, for the 818 level, the differencebetween the calculated and experimental total removal rateconstants, is much larger, about 70%. Here, we have appliedthe mixed-state model15, 17 to the CH2–He system, concentrat-ing on the 818 level as the a “gateway” level. Table II showsthat inclusion of the spin-orbit coupling leads to a significantdecrease in the 818 total removal cross section to all other lev-els (exclusive of the other mixed-pair state) from 16.0 Å2 inthe absence of spin-orbit coupling to 12.9 Å2. Similarly, ourestimate for the total removal rate out of the other compo-

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

162.129.250.14 On: Wed, 13 Aug 2014 13:35:28

064312-9 Ma, Alexander, and Dagdigian J. Chem. Phys. 141, 064312 (2014)

nent of the mixed pair [X(0, 2, 0)937(j = 8)] is larger (8.7 ascompared to 5.7 Å2) when spin-orbit mixing is included. Byignoring spin-orbit mixing, we overestimate the total removalcross sections for the 818 level but underestimate it for the 937level.

The comparison in Table II refers to cross sectionsat a particular collision energy. The total removal ther-mal rate constants at 300 K are 1.89 and 1.36 × 10−10

cm3 molecule−1 s for the 818 and 937 levels, respectively.These two numbers should be compared with the pointsmarked, respectively, “S” and “T” in Fig. 16(a) of Ref. 10.They are in considerably better agreement with experimentthan the rates calculated neglecting spin-orbit mixing [see Fig.16(a) of Ref. 10)] but are still ≈30% higher for the singlet and≈10% higher for the triplet level. The overall relaxation fromthe mixed 937 level is lower than that of the 818 level, sincethe latter possesses greater singlet character. The degree ofdisagreement with experiment is comparable to what we havefound for other ka = 1 levels. This confirms the accuracy ofthe mixed-state model for CH2.

In this paper, we have concentrated on a − X mixingin the CH2 ortho levels, in order to compare with the ex-periments of Hall, Sears, and co-workers.20, 22 Collisionalrelaxation of CH2 a/X para levels can be expected to besomewhat different since strong spin-mixing occurs for

0.1

0.4

0.5

t = 100 ns

0

0.01

0.02

0.03

Pop

ulat

ion

t = 900 ns

−200

0

200

j = 8

0

0.4

0.8

1.2 t = 0 ns

9 10

937

735

j = 6 7 8

918836

827818

616625

716550725

0

E–E

(818

) / c

m–1

j = 8 9 10937 735

6 7 8918 836 827 818616625716550725

0.3

0.2

FIG. 8. Rotational distribution of selected singlet and triplet levels at varioustimes following initial population of the a(0, 0, 0) 616 unperturbed level; p= 2 Torr and T = 300 K. The black rectangular empty boxes indicate therelative Boltzmann population including all a and X levels. The blue (slightlynarrower) empty boxes represent the relative Boltzmann population of the a

levels in the absence of coupling to the X state.

lower rotational levels of this nuclear spin modification (seeRef. 14 and Table I). Since rate of rotational relaxation in-creases with decreasing rotational quantum number n, ro-tational relaxation in the X state from the gateway levelsin para-CH2 should generally be faster than for the ortholevels.

Gannon et al.21 have measured rate constants as a func-tion of temperature for total removal of one unperturbed leveleach in the CH2 a(0,0,0) ortho and para manifolds in colli-sions with a number of nonreactive collision partners, includ-ing He. They observed for several collision partners, moststrongly for He, that the relaxation at low temperatures ofthe para level was faster than the ortho level and that thepara/ortho rate constant ratio decreased for modest increasesin temperature. This behavior is consistent with the effect oftemperature on the Boltzmann populations of the mixed levelsin the para and ortho manifolds. This effect is, however, notseen to persist above room temperature. In principle, calcu-lations such as those presented in this work could be used tounderstand the temperature dependence of the para and orthoremoval rates.

We have presented a simplistic and more complete simu-lation of relaxation involving the ground and the first excitedelectronic states of ortho-CH2 by collisions with He. In thesimplistic model, we treated all unperturbed singlet and tripletlevels as two large baths. The predicted time evolution of thepopulations of the two mixed levels was double exponential:a rapid equilibration of these two levels at short time followedby a slower overall relaxation. The complex simulation repro-duced this double-exponential behavior [Fig. 7] for the twomixed levels.

Here, we ignored vibrationally inelastic processes, whichrecent work shows to be quite unimportant.34 In addition, weonly included one mixed pair of levels. The next degree ofcomplexity would be to include more pairs of mixed lev-els for either of the nuclear-spin modifications of CH2, seeTable I. For ortho-CH2 we would need to add the 716/615 pair,which are nearly totally mixed. This more complete relaxationsimulation will involve two simultaneous gateways.

In work on collisional relaxation in the excited state ofthe Na2 dimer, Li et al.18 give an extensive discussion of thevalidity of the Gelbart-Freed gateway model. The spin-orbitcoupling in CH2 is so small, compared to the level spacing,that the spin-orbit induced mixing is effective only betweenthe isolated pairs of nearly degenerate levels listed in TableI. During the collision, since the interaction between He andCH2 in the X and a states is so different,10, 34 the relativepositions of the two sets of rotational manifolds will shifttransiently with respect to each other, by amounts on the or-der of 1–10 cm−1. Thus, transiently, other near degeneraciescan occur, through which intersystem crossing could occur.This effect could be included by adding in the quantum scat-tering calculations the rotational manifolds associated withboth electronic states and retaining all possible spin-orbit cou-plings with a geometry-dependent spin-orbit operator. Thiswould lead to a rate constant matrix in which, in comparisonto that of Fig. 5, the off-diagonal blocks are full.

We anticipate that the effect of this transient couplingwould be small. For much of the duration of the collision,

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

162.129.250.14 On: Wed, 13 Aug 2014 13:35:28

064312-10 Ma, Alexander, and Dagdigian J. Chem. Phys. 141, 064312 (2014)

the relative spacing of the a and X rotational manifolds re-mains little altered. Due to the sparseness of these manifoldsin the hydride CH2 (as compared to the Na2 dimer studied inRef. 18), the asymptotic mixed-pair description will remainlargely valid, except for the brief moment in which the differ-ence in the repulsive interactions are large enough to lead to atransient relative shift in the levels, which is large compared totheir asymptotic spacing. We have seen that the overwhelm-ing contribution to the large inter-state cross sections are dueto borrowing of amplitude from elastic transitions. These pro-cesses occur at long-range, where the relative shifting of thea with respect to X levels will be slight.

Eventually, when experiment becomes capable of de-termining intersystem rate constants at a state-to-state(a, n, ka → X, n′, k′

a) level, it will become worthwhile to in-vestigate the contribution of the transient relative shifting ofthe two rotational manifolds.

We now have a better understanding of a → X intersys-tem relaxation in CH2. Whatever the initial conditions, thepopulations in both components of the mixed pair rapidlyequilibrate, and then decay at the same rate. The next mostefficient process is rotational relaxation within the a state,which is ≈3 times faster than within the X state. The more ef-ficient relaxation within the a state is a consequence of severalfactors: First, the anisotropy of the CH2(a)He PES is larger.11

Second, because the CH2 molecule in the X state is less bent,the splitting between the k stacks is greater.11 Thus, there arefewer nearby rotation-projection states to participate in rota-tional relaxation. Eventually, though, the bulk of the popula-tion transfers to the X state, influenced not only by the higherdegeneracy of this state, but also because the zero-point levelsof both the (0,2,0) and (0,3,0) vibrational manifolds lie con-siderably below the origin of the a state, as seen in Fig. 2 ofRef. 11.

ACKNOWLEDGMENTS

This material is based upon work supported by the U.S.Department of Energy, Office of Science, Office of Basic En-ergy Sciences, under Grant No. DESC0002323. The authorsare most grateful to Greg Hall and Trevor Sears for their sug-gestions and comments. They are also indebted to one of thereferees for a helpful discussion about limitations in the sim-ple gateway model.

1M. J. Perkins, Radical Chemistry: The Fundamentals (Oxford UniversityPress, New York, 2000), p. 83.

2C. Wentrup, Reactive Molecules: The Neutral Reactive Intermedi-ates in Organic Chemistry (John Wiley & Sons, New York, 1984),p. 162.

3S. P. McManus, Organic Reactive Intermediates (Academic Press, NewYork, 1973), p. 61.

4T. J. Frankcombe and S. C. Smith, Faraday Discuss. 119, 159 (2001).5W. M. Shaub, and M. C. Lin, in Laser Probes for Combustion Chemistry,edited by D. R. Crosley (American Chemical Society, Washington, D.C.,1980), p. 408.

6D. F. Strobel, Space Sci. Rev. 116, 155 (2005).7Y. L. Yung and W. B. DeMore, Photochemistry of Planetary Atmospheres(Oxford University Press, New York, 1999), p. 144.

8D. F. Strobel, “The photochemistry of the atmospheres of the outer planetsand their satellites,” in The Photochemistry of Atmospheres, edited by J. S.Levine (Academic Press, New York, 1985), pp. 393–434.

9D. L. Baulch, C. T. Bowman, C. J. Cobos, R. A. Cox, T. Just, J. A. Kerr,M. J. Pilling, D. Stocker, J. Troe, W. Tsang, R. W. Walker, and J. Warnatz,J. Phys. Chem. Ref. Data 34, 757 (2005).

10L. Ma, M. H. Alexander, and P. J. Dagdigian, J. Chem. Phys. 134, 154307(2011).

11L. Ma, P. J. Dagdigian, and M. H. Alexander, J. Chem. Phys. 136, 224306(2012).

12J. I. Steinfeld, Molecules and Radiation: An Introduction to Modern Molec-ular Spectroscopy (Dover Publications, Cambridge, MA, 2005).

13P. J. Dagdigian, Int. Rev. Phys. Chem. 32, 229 (2013).14U. Bley and F. Temps, J. Chem. Phys. 98, 1058 (1993).15W. M. Gelbart and K. F. Freed, Chem. Phys. Lett. 18, 470 (1973).16K. F. Freed, in Potential Energy Surfaces, edited by K. P. Lawley (Wiley,

New York, 1980), p. 207.17B. Pouilly, J. M. Robbe, and M. H. Alexander, J. Phys. Chem. 88, 140

(1984).18L. Li, Q. Zhu, A. M. Lyyra, T.-J. Wang, W. C. Stwalley, R. W. Field, and

M. H. Alexander, J. Chem. Phys. 97, 8835 (1992).19M. E. Jacox, J. Phys. Chem. Ref. Data 32, 1 (2003).20A. V. Komissarov, A. Lin, T. J. Sears, and G. E. Hall, J. Chem. Phys. 125,

084308 (2006).21K. L. Gannon, M. A. Blitz, T. Kovacs, M. J. Pilling, and P. W. Seakins, J.

Chem. Phys. 132, 024302 (2010).22G. E. Hall and T. J. Sears, private communication (2010).23T. J. Sears, P. R. Bunker, A. R. W. McKellar, K. M. Evenson, D. A. Jen-

nings, and J. M. Brown, J. Chem. Phys. 77, 5348 (1982).24G. C. Corey and F. R. McCourt, J. Phys. Chem. 87, 2723 (1983).25M. Alexander, J. E. Smedley, and G. C. Corey, J. Chem. Phys. 84, 3049

(1986).26A. R. Edmonds, Angular Momentum in Quantum Mechanics, 2nd ed.

(Princeton University Press, Princeton, 1974).27HIBRIDON is a package of programs for the time-independent quan-

tum treatment of inelastic collisions and photodissociation written byM. H. Alexander, D. E. Manolopoulos, H.-J. Werner, B. Follmeg, Q.Ma, P. J. Dagdigian and others. More information and/or a copy of thecode can be obtained from the website http://www2.chem.umd.edu/groups/alexander/hibridon.

28I. W. M. Smith, Kinetics and Dynamics of Elementary Gas Reactions (But-terworths, London, 1980).

29M. H. Alexander and P. J. Dagdigian, J. Chem. Phys. 83, 2191 (1985).30J. L. Rinnenthal and K.-H. Gericke, J. Chem. Phys. 113, 6210 (2000).31J. L. Rinnenthal and K.-H. Gericke, J. Chem. Phys. 116, 9776 (2002).32M. H. Alexander, G. E. Hall, and P. J. Dagdigian, J. Chem. Educ. 88, 1538

(2011).33See supplementary material at http://dx.doi.org/10.1063/1.4892377 for the

MATLAB script to solve the master equation.34L. Ma, P. J. Dagdigian, and M. H. Alexander, J. Chem. Phys., “Theoretical

investigation of the relaxation of the bending mode of CH2(X) by collisionswith helium” (unpublished).

35G. E. Hall, A. V. Komissarov, and T. J. Sears, J. Phys. Chem. A 108, 7922(2004).

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

162.129.250.14 On: Wed, 13 Aug 2014 13:35:28