MCSCF linear response study of the three-bodydissociative recombination CHþ

2 þ e! Cþ 2HBoris F. Minaev, Mats Larsson*

Department of Physics, SCFAB Stockholms University, Stockholm SE-10691, Sweden

Received 9 December 2001; in final form 9 April 2002

Abstract

A number of excited states of the CH2 molecule and of its ion are optimized by MCSCF method. Potential energy

curves (PEC) along the linear and bent reaction coordinates for 36 excited states obtained by linear response calculation

are presented. Account of these data permit us to simulate the linear and bent model for the three-body dissociative

recombination (DR) CHþ2 þ e� ! Cð3P; 1D; 1SÞ þ 2H. From a simple linear reaction coordinate a principal possibility

of the direct mechanism for the three-body DR by the electron capture to the 13R�u state is qualitatively understood. It is

stressed that the DR is governed by exchange interaction. For the more realistic bent structure the analysis is more

complicated. The bending potentials for the upper excited states (in the close energetic proximity with the ion 12A1state) are not very sensitive to the angle variation in the region 180�–130�. Nevertheless the bent structure of the ionfacilitates the three-body dissociation: the valence excited 33A2 state fits the energy of the ion and dissociates without

barrier to the carbon atom and triplet spin pairing of the hydrogen radical pair Cð3PÞ þ 2Hð3Rþu Þ. The hydrogen radical

pair spin pattern is important for preventing the H2 molecule production. The role of spin functions and exchange

coupling are crucial for the DR process. � 2002 Published by Elsevier Science B.V.

1. Introduction

The methylene molecule (CH2) and its ion playan important role in models of interstellar molec-ular clouds [1–5]. The ions reactions are generallyimportant in this environment because chemicalprocesses between neutral molecules are frozen atlow temperature (needing heating to overcomeactivation barrier). The ions can recombine withelectrons by slow collisions and gain a lot of en-ergy by this recombination: the process is opposite

to ionization in terms of thermodynamic balance.The excessive electronic energy usually exceeds thedissociation barrier and the dissociative recombi-nation (DR) occurs [2,6]. Thus numerous neutralproduct can be produced.The hydrocarbon chemistry in the interstellar

medium is initiated by the UV ionization of car-bon atoms and by the radiative association

H2 þ Cþ ! CHþ2 þ hm ð1Þ

The CHþ2 ion can be also formed by the hydrogen

abstraction reaction

H2 þ CHþ ! CHþ2 þH ð2Þ

Chemical Physics 280 (2002) 15–30

www.elsevier.com/locate/chemphys

*Corresponding author.

0301-0104/02/$ - see front matter � 2002 Published by Elsevier Science B.V.

PII: S0301-0104 (02 )00509-8

Continuation of the chain leads to larger ionsproduction [5]. In 1941 Douglas and Herzbergidentified the ion CHþ in the diffuse molecularclouds and since then the problem to explain theabundance of the CHþ ions has arisen [7]. Thisabundance has recently been tentatively explainedby the turbulence model [5,8]. As follows fromEqs. (1) and (2) the CHþ

2 ion should play an im-portant role in the hydrocarbon chemistry of in-terstellar clouds, but it has never been observed inthis environment [5]. CHþ

2 ion is thought to beremoved by DR processes. The DR cross-sectionhas been measured first time in a single-passmerged electron–ion beam experiment [9]. Larsonet al. [5] have measured recently in the CRYRINGthe branching ratios for DR of CHþ

2 ion with slowelectrons. Contrary to what has been believedearlier [2,3,10,11], the DR is dominated by thethree-body channel CþHþH (63%), Eq. (3),whereas the other, more exothermic processes,Eqs. (4) and (5), occur with branching ratios of25% and 12%, respectively. The exothermicity ofthe reactions

CHþ2 þ e! CþHþHþ 2:3 eV ð3Þ

CHþ2 þ e! CHþHþ 5:8 eV ð4Þ

CHþ2 þ e! CþH2 þ 6:8 eV ð5Þ

determines the excess kinetic energies for theground state products; the CHþ

2 ion is consideredin its lowest vibrational level [5]. On the ground ofthe valence bond method, Bates [2] andMillar et al.[10] argued that the reaction (4) should be thedominant. The reaction (3) has been neglected inthe recent models of the interstellar clouds [3,11].This disagreement between theoretical predictions[2,3,10,11] and the direct experimental measure-ment [5] prompted the authors to calculate po-tential energy curves (PEC) of the three-bodychannel, reaction (3), in order to see the possibilityof such unexpected process.The basic principles of DR have been proposed

by Bates in 1950 [6]. The DR process can be di-vided into two steps. First, the free electron entersinto an unoccupied bound molecular orbital (MO)and simultaneously excites a bound electron. In-coming electron can not just occupy the lowest

unoccupied bound MO in the ion, because of thebig energy gap between ion and the neutral mol-ecule (ionization potential). The energy resonancecondition for electron attachment to the ion needsto overcome additional electron excitation. Thisproduces a neutral, doubly excited species, whichcan rapidly dissociate. This is the so-called directmechanism of DR. The non-direct mechanism,proposed by Bardsley [12], accounts that theelectron loses its energy entering a Rydberg or-bital. Then a radiationless transition from theRydberg state to the doubly excited state occurswith a rapid dissociation.Applying these arguments for specific ions, one

has to consider two types of species, depending onthe type of the ground state molecule, produced byion+ electron recombination. The first (and moregeneral) case corresponds to the ground statemolecule with a closed shell electronic structure.The second case corresponds to the open-shellground state species. The ions like H2O

þ, COþ,C2H

þ2 refer to the first case, while the ions H

þ3 ,

CHþ2 , CH

þ3 , NH

þ2 , NH

þ4 , CH

þ5 , HCNH

þ, studiedso far [13], correspond to the second case. Almostall ions of the last case (besides CHþ

2 ) have theeven number of electrons and the singlet closedshell ground state; the corresponding neutral spe-cies are the radicals with the doublet ground state.The CHþ

2 ion is rather particular, since its neutralcounterpart is an open shell biradical with thetriplet ground state. This classification is impor-tant mostly from the point of view of computa-tional details, since different types of electroncorrelation are involved in the species. The spinpatterns are also different in these DR processes.In the first case the doublet electron spin of the ioninteracts with the incoming electron spin: threetriplet and one singlet states are available in ratio3:1 by statistics for each 4 collisions. The tripletand singlet channels of the DR process are differ-ent. The triplet–singlet (T–S) transitions can beinduced by internal (spin–orbit coupling (SOC)) orexternal magnetic field, which can in principle in-fluence the rate and yield (products ratio) of theDR. The T–S transitions in the separated radicalpair (ion+ electron) are well known to be inducedby the difference of g-factors and by hyperfine in-teractions with nuclear spins in the ion [14] in the

16 B.F. Minaev, M. Larsson / Chemical Physics 280 (2002) 15–30

presence of the external magnetic field. The elec-tronic g-factor of the ion is determined by the in-ternal SOC and depends on the ion electronicstructure, while the free-electron g factor is con-stant (ge ¼ 2:0023). For a very low energy of freeelectron collision with ion the T–S transitions canoccur depending on the external magnetic fieldorientation to the radical pair axes at large dis-tances between particles (before the recombinationcapture). In experiments, like in CRYRINGmeasurements, the external magnetic field is pre-sent in order to confine the electron beam. Thusone can not exclude in advance the possibility ofthe external magnetic field effect on the DR forsuch type of ions. Radical recombination reactionsare influenced by magnetic field in solvents (this isalso the reason for chemically induced dynamicnuclear spin polarization, CIDNP) and also in gasphase [14].For the second case species the ion has a singlet

ground state and in the DR process only thedoublet states are involved. (There are no startingquartet states in the ion+ electron recombinationprocess.) In this case the external magnetic fieldeffect on the DR process is definitely excluded.Besides this tentative classification with respect

to the possible complication from the magnet ofthe apparatus, the different ion cases have differenttype of qualitative analysis of the electronicstructure for the capture process. We shall con-sider this for the case of CHþ

2 ion. One has to studyfirst the neutral CH2 species and the ion structurein a variety of states.The methylene molecule possesses a triplet

ground state, X3B1 and low-lying excited singletstate a1A1 [15]. Methylene was first observed byHerzberg [7], who detected both of these states inabsorption. Herzberg [7] concluded that the tripletis the lowest state and determined quite accuratelythe angle in the singlet state (103�) and the bondlength (re ¼ 1:116 �AA). The structure of the tripletground state has been established later by meansof laser magnetic resonance (LMR) spectroscopyin the far-infrared (IR) region [16] and subsequentmicrowave studies (re ¼ 1:075 �AA, \HCH ¼133:9�)[15,17–19]. Assignment of the far-IR LMR spectrato the rotational–vibrational transitions, includingthe singlet–triplet (S–T) a1A1–X

3B1 transitions,

leads to a direct determination of the S–T splitting(’0.4 eV) [15]. Calculation of vibronic SOC per-turbations between the S and T levels obtainedwith the nonrigid bender Hamiltonian (includingthe proper Franck–Condon factors) [15] can beconsidered as a great achievement of molecularspectroscopy and ab initio theory. Much less isknown about higher excited states of the methyl-ene. For the b1B1; c

1A1 and Rydberg states alimited number of studies have been reported[1,20–24]. Readers should refer to the explicit re-views of previous theoretical studies presented bySchaefer and co-workers [21,22] and by Be€aardaet al. [1].Dissociation in two-body channels, Eqs. (4) and

(5), have been considered in connection withphotodissociation problem using multireferenceconfiguration-interaction (MRCI) method [1]. Theionic states and close-lying neutral excited levelsimportant for the DR processes have not beencalculated by Be€aarda et al. [1]. We have repro-duced qualitatively the results of MRCI calcula-tions for these reaction paths [1] in our linearresponse (LR) method and added some higherexcited states whose energy is close to the ioniza-tion potential. Thus we shall not present all PECfor the two-body channels, Eqs. (4) and (5); in-stead we shall discuss qualitatively the new PECsin addition to the data presented in [1]. The three-body process ðCþ 2HÞ has not been considered sofar and the corresponding PECs are the mainsubject of the present study.

2. Method of calculations

Multi-configuration self-consisted field(MCSCF) method in a complete active space(CAS) [25] is a general basis for the present ap-proach. Ten excited states of each symmetry havebeen calculated by LR method [25] along the DRcoordinates. The MCSCF LR calculations of theCH2 molecule dissociation reactions, Eqs. (3)–(5),have been performed for linear and bent struc-tures. Since the CHþ

2 ion and the CH2 molecule arevery close to linear form in terms of potential en-ergy expenses in their ground states, we have usedfirst the linear reaction coordinate for symmetric

B.F. Minaev, M. Larsson / Chemical Physics 280 (2002) 15–30 17

three-body dissociation (with the D2h point group)as the simplest model for the DR process, Eq. (3).Many (but not all) excited states studied have alinear structure, thus this model is a good startingapproximation for the qualitative study of themain electronic interactions, responsible for theDR process and its driving forces.Different complete active spaces (CAS) have

been used for testing our calculations. For linearCH2 molecule the ground configuration is

3w0 X3R�

g

h i¼ Að1rgÞ2ð2rgÞ2ð1ruÞ2ðpuÞ2; ð6Þ

where A indicates the antisymmetrization prod-uct. The 1s inner MO on carbon atom, ð1rgÞ, withthe energy )11.3 a.u. was excluded from CASquite naturally. All other occupied MOs of theground state constitute the ground of all CASsstudied; the 2, 3ru, 3–6rg and 2pu;g empty orbitalswere also included in some CAS. After numeroustesting those CAS has been adopted for massivecalculations of the DR process, for which the 2pgempty orbitals were excluded from the previousset. Their account do not influence the LR calcu-lations of all important excited states. Thus theCAS includes 6 electrons in 10 orbitals ð6e �10 MOÞ The total number of configuration statefunctions (CSFs) is about two thousands CSFs inthis CAS for different symmetries of irreduciblerepresentations in the D2h point group.For the more realistic nonlinear structure the

C2v point group has also been implemented. Theground state of the CH2 molecule is known to bebent with the HCH angle equal 134� (the C–Hdistance rCH ¼ 1:076 �AA) [15]. The dominant CSFis

3w0½X3B1 ¼ Að1a1Þ2ð2a1Þ2ð1b2Þ2ð3a1Þð1b1Þ: ð7ÞIn that case the number of active orbitals in the

A1;B1;B2;A2 irreducible representations are equalto (6, 2, 3, 1), respectively; the total number ofconfigurations is about six thousands CSFs in thisCAS ð6e� 12 MOÞ. The larger CAS in the C2vgroup is used quite naturally, since the pg arestrongly splited in the bent form.The geometry of the ground and of all excited

states of each symmetry for the CH2 molecule andfor its ion have been optimized by MCSCF pro-

cedure in the C2v and D2h point groups. These dataare used for qualitative discussion of the DRmechanism. The ground state optimized HCHangle in the CH2 molecule (132�) and in the ion(138.3�) have been used for simulation of thesymmetric dissociation reaction coordinate modelof the DR process, Eq. (3), in nonlinear geometryof the C2v point group.The two-body channels, Eqs. (4) and (5), are

calculated for the neutral species with the CAS(9,3) in the Cs point group for 6 active electrons,which is compatible with the C2v CAS. Calcula-tions of PECs for 24 states are performed for twobond angles (132� and 138�) as functions of one C–H bond length, keeping the other C–H distance atthe equilibrium value in the ion (rCH ¼ 1:101 �AA).Since the results for 10 lowest states are very closeto the previous MRCI calculations [1] we shall notpresent these PEC and only discuss qualitatively anumber of new features.The standard choice of axes is used in the pre-

sent work. The CH2 molecule is placed in the yzplane and the z axis is the C2 rotation axis. Thecorrelation consistent basis of Peterson and Dun-ning [19] augmented by polarization functions, theso-called ‘‘aug-cc-pVTZ’’ basis, which contains108 primitives and 92 contracted atomic orbitalsfor CH2 molecular is used. The MCSCF LR cal-culations were performed with the DALTON [26]program packages.

3. Results and discussion

The incoming electron (when it collides with theCHþ

2 ion) can be described by wave function ofcontinuous spectrum of any symmetry, thus thereis no symmetry selection rule for DR process (allsymmetry channels are allowed, but have differentprobabilities). The cross-section of DR obtainedfrom the direct mechanism is assumed to be largeat very small electron collision energies [2,6,27].Whenever there is a crossing between the repulsiveneutral state and the ionic CHþ

2 PEC, this repul-sive potential will also cross the Rydberg statePEC situated below the ionic state potential.The crossing between the Rydberg state PEC

and the dissociative curve of the same symmetry

18 B.F. Minaev, M. Larsson / Chemical Physics 280 (2002) 15–30

take place in the framework of a quasi-diabaticrepresentations [28]. In our adiabatic treatment(the usual standard of quantum chemistry) thesestates avoid crossing and a qualitative conclusionabout the indirect mechanism can be withdrawn.The indirect mechanism is indeed possible as soonas the ionisation channel to which the Rydbergstate pertains is closed, and this can happen even ifthe dissociative state and the Rydberg state do notcross (as in the case of non-adiabatic couplingvalid for DR for Hþ

3 and HeHþ ions [28,29]). The

only efficient method used up to now for quanti-tative account of indirect process is based on thework of Giusti [28], where multichannel quantumdefect theory was applied to dissociative processes.The DR cross-section of CHþ

2 measured byCRYRING [5] indicates an inverse energy depen-dence for collision energies up to 0.7 eV. The factthat the 1=E dependence is observed indicates thedominance of the direct recombination process.Resonance structures in the DR cross-sectionCHþ

2 þ e cause deviation from the 1=E behaviorabove 1 eV and a small wide peak is observed at anenergy of 2 eV [5]. In order to explain these DRfeatures we shall consider the excited states PEC ofthe ionic and neutral CH2 species in respect to thepossible DR reaction channels, Eqs. (3)–(5). Fewvalues of the bond angle \HCH will be fixed foreach reaction coordinate. The corresponding PECsare presented in Figs. 1–5. Fig. 6 represents thebond angle dependence of the excited states ener-gies.

3.1. The ground and excited states of methylenediradical

The first information needed for understandingof the DR process are the electronic states. Sec-ond, we need to learn the coupling between states.This refers especially to the coupling between theionization continuum, Rydbers states PEC and thedissociative potential curves. We shall address tothis issue latter by analysis of configuration statemixing.The ground state of CH2 molecule is the triplet

state with zero orbital angular momentum, Eq. (7)[7,30,31]. Its properties and the S–T energy gaphave been calculated by many theorists; a review

can be found elsewhere [21]. Our MCSCF re-sults with the aug-cc-pVTZ basis set (Table 1) arein a good agreement with experimental data[7,30,31] and with the recent calculations ofSchaefer and co-workers [21]. The predicted

Fig. 1. Potential energy curves for symmetric three-body dis-

sociation of the linear CH2 molecule to the Cþ 2H products

calculated by MCSCF CAS (3rg, 3ru, 2pu) linear responsemethod with the aug-cc-pVTZ basis set. The ionic states are

obtained by MCSCF calculation (thin solid line) and by linear

response calculation with the 2Rþg state as a reference (dashed

line).

Fig. 2. Potential energy curves for symmetric three-body dis-

sociation CH2 ! Cþ 2H at the ground state equilibrium bondangle (\HCH ¼ 132�). Excited triplet states of A1 and A2symmetry.

B.F. Minaev, M. Larsson / Chemical Physics 280 (2002) 15–30 19

equilibrium geometry of the ground state(rCH ¼ 1:083 �AA, \HCH ¼ 132�) agrees quite wellwith the spectroscopically derived parametersðrCH ¼ 1:078 �AA, \HCH ¼ 133:9�Þ. The bendingpotential (Fig. 6) is very flat, thus a small devia-tion in the equilibrium angle (1.9�) is not surpris-ing. The calculated dipole moment (0.592 debye)corresponds to the C�Hþ polarization and agreeswith that calculated by Schaefer and co-workers(0.594 debye) [21]. For the lowest singlet a1A1

Fig. 3. Potential energy curves for symmetric three-body dis-

sociation CH2 ! Cþ 2H at the ground state equilibrium bondangle (\HCH ¼ 132�). Triplet states of B1 and B2 symmetry.

Fig. 4. Potential energy curves for symmetric three-body dis-

sociation CH2 ! Cþ 2H at the ground state equilibrium bondangle (\HCH ¼ 132�). Excited singlet states.

Fig. 5. Potential energy curves for symmetric three-body dis-

sociation CH2 ! Cþ 2H at the ionic ground state equilibriumbond angle (\HCH ¼ 138:3�) for few important states.

Fig. 6. Angular dependence of potential energy curves for

various states of methylene and its ion (bold lines). Only the

lowest 3A1 state (dot-dashed line) is shown. The four lowest

state of other symmetries for methylene are presented by dotted

(3A2 states), solid (3B1 states) and by long dashed lines (

3B2states).

20 B.F. Minaev, M. Larsson / Chemical Physics 280 (2002) 15–30

state we get 1.74 debye in agreement with the big-basis-set result of Schaefer et al. (1.76 debye) [21].Equilibrium geometry of the singlet a1A1 state isalso well known [7,21,30,31] and our MCSCFapproximation reproduces it perfectly (Table 1).The gas-phase IR spectrum for the triplet

methylene has been observed directly only for x2

bending vibration [21,31]. The two stretchingmodes have not been detected so far in the IRspectrum, since intensity of these fundamentaltransitions is negligible [21,31]. Our calculationreproduce this trend (Table 1). In contrast the IRintensities of the two stretching modes in the sin-glet a1A1 state are much larger (Table 1): all threevibrational frequencies have been observed in theIR spectrum of the singlet methylene [21,31]. Thecalculated IR frequencies in our MCSCF methodare quite reasonable.The excited states of CH2 molecule have been

intensively studied by ab initio [1,20–22,32] and byexperimental methods [7,15,23,24,30,31]. Herzbergand Johns [30] have interpreted the vacuum UVabsorption spectrum (141.5 nm) of CH2 moleculein terms of its bent structure and assigned theabsorption lines to the (K00 ¼ 0! K0 ¼ 0) sub-band of the X3B1 ! 23A2 transition with Rydbergd character. Previously these lines were attributedto a X3R�

g ! 3R�u absorption of the linear CH2

molecule [7]. Excited triplet states of CH2 andCD2 species have been studied later by REMPI(resonance-enhanced multiphoton ionization)technique [24]. The REMPI spectra occur from3-photon resonances with the B3A2ð3dÞ and otherð3d; 4dÞ Rydberg states between 8.45 and 9.8 eV[24]. The C(3d), D(3d) and 3A2ð4dÞ have beendetected. Combination of the three nd Rydbergseries with the group of lines at 141.5 nmproduces the first ionization potential of methyl-ene to be 10.2–10.4 eV [7,24]. Experimental dataon other excited triplet states of valence nature areabsent.Results of geometry optimization for a number

of low-lying states of the methylene molecule andof its ion are presented in Tables 1 and 2, respec-tively. These results are in a qualitative agreementwith the angle dependent PECs for a number ofstates obtained by MRCI studies [1]. Unfortu-nately the adiabatic ionization potential of meth-ylene is underestimated in our MCSCFcalculations (9.65 eV instead of experimental resultof 10.4 eV). At the same time the LR calculationsof the excited states of methylene are in muchbetter agreement with very accurate MRCI results[1,32]. In order to overcome this defect we havecalculated the ionic energy by LR technique usingthe 2Rþ

g state as a reference. The corresponding

Table 1

Equilibrium properties of a number of states of CH2 molecule calculated by MCSCF method with the ‘‘aug-cc-pVTZ’’ basis sets and

complete active space 6e� 12 MOsa

State re HCH� Te x3 asðb2Þ I3 x1 sða1Þ I1 x2ða1Þbend I2

X3B1 1.083 132.0 )39.02850 3282.4 0.7 3093.9 0.04 1117.3 5.11

X3B1([21])b 1.075 132.9 )39.07505 3399 0.1 3174 0.3 1134 6.9

X3B1ðexpÞc 1.078 133.9 – 3213 0.7 2992 – 963 –

a1A1 1.110 101.9 )39.02253 2987.1 61.1 2913.2 67.2 1434.0 1.49

a1A1([21])b 1.105 102.0 )39.05570 3036 69.0 2968 69.7 1434 1.3

a1A1ðexpÞc 1.111 102.4 – 2864.5 – 2805.9 – 1352.6 –

13A2 1.300 41.1 )38.90971 1578.2 2.5 2620.0 64.5 1001.7 69.7

13Puð13A1Þd 1.373 180.0 )38.83247 2077.7 i – 1411.9 0.00 961.6 0.95

13B2 1.169 78.1 )38.78917 2163.9 139.6 2536.2 27.1 1074.8 2.25

15Pgð15A2Þd 1.762 180.0 )38.76266 1912 i – 740.0 0.001 495.8 17.8

a Equilibrium internuclear distances, re (�AA), Te is the total energy (a.u.), xn is vibrational frequency (cm�1), the asymmetric C–H

stretch (as) of b2 type is the most intensive in the IR absorption spectrum; bend means bending vibration of the a1 type. In means IRintensity (km/mol).bCISD calculation with the largest basis set from [21].c Experimental data are from [7,21,30].d Imaginary frequency corresponds to unstable state in respect to CHþH dissociation.

B.F. Minaev, M. Larsson / Chemical Physics 280 (2002) 15–30 21

PEC of the ion is shown in Fig. 1 by the thin da-shed line.The CHþ

2 ion has a larger angle (138.3�) thanthe ground triplet state of methylene (132�) and alonger bond length (rCH ¼ 1:1 and 1.083 �AA, re-spectively). But still the differences in both equi-libria are not very big, thus the dissociationreaction can be studied with both starting geom-etries, or with one of them.The vertical excitation spectrum of the CH2

molecule is given in Table 3. Our MCSCF LRresults are in a good agreement with the MRCIstudies [1,32]. The ground ionic state energy iscalculated by MCSCF method and this result isworse in comparison with the LR data for excitedstates of methylene. The number of particles isconserved in the response methods, thus we cannot use the same approximation for the neutralspecies and for the ion. One has to keep in mindthis shortcoming in a qualitative discussion of theDR process.The 3A2 states are proven to be the most im-

portant in this study. The first excited triplet stateof this symmetry, 13A2, is the valence state pro-duced by 1b2 ! 3a1 excitation from the groundterm, Eq. (7):

3w0½13A2 ¼ Að1a1Þ2ð2a1Þ2ð1b2Þð3a1Þ2ð1b1Þ ð8ÞIt can easily dissociate along the symmetric three-body channel (Fig. 2), but it is too low in energy incomparison with the ion state. Thus this state isnot accessible by direct relaxation after the elec-tron capture by the ion. Besides that, the 13A2valence state is strongly stabilized by diminishing

the HCH angle until 41.1� (Table 1, Fig. 6) inagreement with previous studies [1,32]. Onebonding electron (1b2) is excited to the partly an-tibonding MO (3a1) upon X

3B1 ! 13A2 excitation.(The 3a1 is partly antibonding in respect to s-atomic orbitals, but it is bonding in respect to pzAO.) Thus the 13A2 valence state has longer C–Hdistance (rCH ¼ 1:3 �AA). The C–H bond prolonga-tion is also seen from the Fig. 2 at the fixed HCHangle (132�). The 13A2 can dissociate toCð3PÞ þH2ðX1RgÞ limit, Eq. (5), since the state isgetting bound in respect to H–H coupling. Thesmall yield of this channel, Eq. (5), can be quali-tatively explained by a long nonradiative cascadefrom the ion to the low-lying 13A2 valence state ofthe neutral species.The second triplet state of this symmetry, 23A2,

is a Rydberg state with diffuse s orbitals on H at-oms. We have not obtained large contributionfrom the ð3a1Þð1a2Þ CSF (where 1a2 orbital is apure 3d�2 AO) as it was obtained in [22]. Butnevertheless the calculated vertical transition en-ergy of the 23A2 Rydberg state is in a goodagreement with experimental data on theX3B1 ! 23A2 transition [23,24,30]. The observedabsorption wave length (141.5 nm) corresponds totransition energy 8.74 eV in agreement with ourvalue 8.83 eV. From the angle dependence of the3A2 PECs (dotted lines in Fig. 6) one can see thatthe observed Rydberg transition at 141 nm [30]comes to the upper 23A2 state produced by avoi-ded crossing with the 13A2 state. This is the reasonwhy the upper state, 23A2, has a minimum at lar-ger bond angle (\HCH ¼ 144�) than in the

Table 2

Equilibrium properties of a number of states of CHþ2 ion calculated by MCSCF method with the ‘‘aug-cc-pVTZ’’ basis sets and

complete active space 5e� 10 MOsa

State re HCH� Te x3 asðb2Þ I3 x1 sða1Þ I1 x2ða1Þbend I2

X2A1 1.101 138.4 )38.64238 3203.3 144.3 2965.4 21.3 998.4 30.8

X2A1([33]) 1.094 140.8 )38.64238 3324 – 2934 – 1033 –

X2A1([37]) 1.094 139.8 )38.64238 3226.4 – 2938 – 838.8 –2Pu 1.097 180.0 )38.6368 3256.2 279.6 2964.3 0.00 1200.4 5.02Pu([33]) 1.092 180.0 )38.6368 3416 2964 – 2450 –2B1 1.101 138.4 )38.60246b – – – – – –4R�

g 1.268 180.0 )38.30495 2802.1 – 1784.4 – 901.5 –

a Equilibrium internuclear distances, re ð�AAÞ, Te is the total energy (a.u.), xn is vibrational frequency (cm�1), the asymmetric C–H

stretch (as) of b2 type is denoted as x3; bend means bending vibration of the a1 type. In means IR intensity (km/mol).b This state corresponds to the vertical excitation.

22 B.F. Minaev, M. Larsson / Chemical Physics 280 (2002) 15–30

ground state. Our result is in agreement with cal-culations of Yamaguchi and Schaefer [22]; theypredicted the optimized geometry of the 23A2ð3dÞRydberg state to be \HCH ¼ 141:3� andrCH ¼ 1:093 �AA. The bond length is not muchchanged, as it follows also from our data for the23A2 state (Fig. 2). The bending-induced avoidedcrossing of two lowest 3A2 states was not takeninto account in [32], but this is a very importantresult for interpretation of the classical experiment[30].The next 33A2 state, which correlates with the

13R�u state of linear structure, is the main subject of

the three-body DR, as it will be shown in the nextsections. This state has not been calculated inprevious ab initio studies.The first excited triplet state of methylene is the

Rydberg 13A1ð3sÞ state which has the dominantCSF of the form:

3w0½13A1 ¼ Að1a1Þ2ð2a1Þ2ð1b2Þ2ð3a1Þð4a1Þ ð9Þ

Its vertical excitation energy is about 6.1–6.3 eV inaccordance with calculations presented in Table 3.This is 1b1 ! 4a1 excitation from nonbonding(1b1) to the antibonding 4a1ðr�

CHÞ orbital withlarge 3s Rydberg character. It is strange that thistransition has not been observed; the verticaltransition moment is very large (Table 3). Thisstate is found to be unstable in respect to dissoci-ation into the CHðX2PÞ þH limit in agreementwith [1,32]. Contrary to the MCSCF geometryoptimization, the LR method predicts a small de-viation from the linear structure of the 13A1 state(Fig. 6), which coincides with the result of [1]. Thisstate has a strong polarization: the carbon atomiccharge is equal to +1.4 e.The lowest quintet state 15A2 is repulsive in

respect to all three dissociation channels, CHþH,

Table 3

The vertical excitation spectrum of the CH2 molecule calculated by MCSCF linear response method with the ‘‘aug-cc-pVTZ’’ basis sets

and complete active space 6e� 12 MOs at the calculated ground state equilibrium (re ¼ 1:0825 �AA, \HCH ¼ 132�)a

State DEb DEc DEd CSF Db Dc fb fc fd

X3B1 0.00 0.00 0.00 ð1b2Þ2ð3a1Þð1b1Þ – – – – –

13A1 6.11 6.28 6.38 ð1b2Þ2ð3a1Þð4a1Þ 0.65 0.54 0.064 0.045 0.056

13A2 7.22 7.34 7.56 ð1b2Þð3a1Þ2ð1b1Þ 0.04 0.05 0.0003 0.0005 0.001

13B1 7.51 7.42 7.52 ð1b2Þ2ð4a1Þð1b1Þ 0.57 0.53 0.059 0.051 0.045

13B2 7.63 7.61 7.60 ð1b2Þ2ð3a1Þð2b2Þ 0.00 – – – –

23A1 7.67 7.80 7.81 ð1b2Þ2ð3a1Þð5a1Þ 0.19 0.03 0.007 0.0001 0.0006

23B2 7.92 8.01 8.15 ð1b2Þð3a1Þð1b1Þ2 0.00 – – – –

23B1 8.19 7.85 7.82 ð1b2Þ2ð4a1Þð2b1Þ 0.08 0.07 0.001 0.001 –

33A1 8.67 8.45 8.59 ð1b2Þ2ð3a1Þð6a1Þ 0.46 0.12 0.016 0.003 0.0001

23A2 8.83 8.73 8.71 ð1b2Þ2ð2b2Þð1b1Þ 0.22 0.05 0.011 0.0006 0.001

43A1 9.07 – – ð1b2Þ2ð3a1Þð6a1Þ 0.18 – 0.009 – –

33B1 9.33 – – ð1b2Þ2ð5a1Þð1b1Þ 0.10 – 0.003 – –

33B2 9.70 – – ð1b2Þð4a1Þð1b1Þ2 0.00 – – – –

43B1 9.97 – – ð1b2Þ2ð3a1Þð2b1Þ 0.24 – 0.014 – –

15A2 10.98 11.06 – ð1b2Þð3a1Þð1b1Þð4a1Þ – – – – –

15B1 12.33 12.51 – ð1b2Þð3a1Þð1b1Þð2b2Þ – – – – –

a1A1 1.21 1.01 1.09 ð1b2Þ2ð3a1Þ2 – – – – –

b1B1 1.49 1.56 1.64 ð1b2Þð3a1Þð1b1Þ – – – – –

c1A1 3.52 – – ð1b2Þð1b1Þ2 – – – – –

12A1 9.67e 10.22 10.21 ð1b2Þ2ð3a1Þ – – – – –

12B1 10.80e 11.29 11.29 ð1b2Þ2ð1b1Þ – – – – –

aDE is excitation energy (eV), D is the electric dipole transition moment (ea0), f is the oscillator strength, CSF is the dominant

configuration state function. The ground X3B1 state energy is )39.0285 a.u.b This work.cRef. [1]. (E½X3B1 ¼ �39:0539 a:u:)dRef. [32]. The experimental ground state geometry.eMCSCF result.

B.F. Minaev, M. Larsson / Chemical Physics 280 (2002) 15–30 23

CþH2 and Cþ 2H, in a wide range of the internalcoordinates for the CH2 system. MCSCF geome-try optimization with the C2v point group con-strain predicts a weakly bound van der Waalscomplex between Cð3PÞ and H2ð3Rþ

u Þ pair (Table1) near the dissociation limit. This intermediatestate is unstable in respect to dissociation to thelower products CHþH.The singlet states of methylene are also well

reproduced in our MCSCF LR method (Table 3).Adiabatic energy gap between the ground tripletand the lowest singlet a1A1 states in our MCSCFgeometry optimization is equal to 0.18 eV in areasonable agreement with the experimentally de-rived value 0.39 eV [31] and the best theoreticalestimation [21]. The vertical ST energy gap ob-tained by the LR method (1.2 eV) is in a goodagreement with the MRCI calculations (1.1 eV)[1,32]. (The MCSCF method gives 0.65 eV). The T0value for the second singlet state, b1B1, is equal1.47 eV, while experiment gives 1.41 eV [31]. Thusthe chosen CAS and bases set are adequate foranalysis of potential energy hypersurface of theexcited CH2 system.Not all Rydberg states are reproduced in our

basis set, but this is not the aim of our studies. Thecomprehensive description of the Rydberg series isgiven in [32]. All important Rydberg states whichinteract with valence excitation and are relevant tothe DR problem are included.

3.2. Linear reaction coordinate model

The ground state of the CH2 molecule is onlyslightly stabilized (0.25 eV) by the bending dis-tortion of the HCH angle from 180� to 132�. TheCHþ

2 ion is more complicated case, since the linearground state 2Pu is the subject of the Renner–Teller distortion and is splited to the 2A1 and

2B1states [33]; the lowest one, 2A1, is stabilized by 0.15eV with the bending distortion from 180� to 138.4�HCH angle (Table 2). Neglecting this small energystabilization upon bending mode, we get a veryuseful approximation for the qualitative analysisof potential energy curves of excited states beingresponsible for the DR of the three-body channel.At linear structure some states of the CH2 mole-cule are getting degenerate. For example, the 13A1

and 13B1 states become the degenerate 13Pu state

at 180� angle, which illustrates the Renner–Tellercoupling [1,32]. One can see that the 13A1 statealmost does not depend on the angle in the range180–140�, but the 13B1 states energy increases withbending [1,32]. (Compare Figs. 2 and 3.) At thelinear reaction coordinate the lowest Rydberg13Pu state has a deep minimum near the C–Hdistance equal 1.4 �AA (Fig. 1). Prolongation of theC–H bond length leads to a barrier. R€oomelt et al.[32] have shown that the lower component (13A1)of the 13Pu Rydberg state (pu ! 3s) have an ex-tremely shallow minimum away from the linearstructure (around 150�), while the higher compo-nent 13B1 prefers the linear arrangement. In ourLR calculation the 13B1 state is a little bit shiftedfrom linear structure (Fig. 6). This state has the3A00 symmetry along one-bond distortion and dis-sociates to the CHþH limit [32].The 3A2 and

3B2 states become the degenerate3Pg state at the linear structure. Behaviour ofPECs of all these states in respect to the three-body dissociation is qualitatively similar for thelinear and bent reaction paths, although there aresome differences in positions of minima andmaxima.The behaviour of the states which energy is

close to the ion is the most important for ourDR analysis. In linear structure such states aredivided into the groups of close-lying states withdifferent orbital angular momentum projection.A qualitative analysis of their PECs in the linearreaction coordinate prompt us a general ideaabout the possibility of the three-body DR, Eq.(3). The linear model stresses the most essentialfeatures of the three body dissociation channel.From Fig. 1 we see that there are at least twopossible candidates to be the intermediate statein DR process: these are the 13R�

u and 13Rþu

states, which have close-lying energies near theequilibrium C–H distance (rCH ¼ 1:1 �AA), wherethey are Rydberg states by nature. Both stateshave very similar orbital configuration of thetype

ð1rgÞ2ð2rgÞ2ð3rgÞ1ðpgÞ2ð1ruÞ1 ð10Þbut their spin-pairing patterns are different. Forthe 13Rþ

u state the wave function is

24 B.F. Minaev, M. Larsson / Chemical Physics 280 (2002) 15–30

W½13Rþu ¼ 0:69 jð3rgÞ1aðpg;xÞ2ð1ruÞ1aj

h

� jð3rgÞ1aðpg;yÞ2ð1ruÞ1aji; ð11Þ

where the closed shell part ð1rgÞ2ð2rgÞ2 is omitted.For the 13R�

u state the spin-pairing patterns lookslike the following:

W½13R�u ¼ 0:48ð½aaab � ½aaba þ ½abaa � ½baaaÞ;

ð12Þ

where orbital configuration is ð3rgÞðpg;xÞðpg;yÞð1ruÞ. It is qualitatively important that bothstates have different behaviour in respect to thethree-body dissociation, in spite the fact that thesame bonding and antibonding MO are occupied.The 13R�

u state has a very small barrier (goingfrom the Rydberg character to the valence stateof the same configuration), but the 13Rþ

u stateovercomes a big barrier (Fig. 1). Energy differ-ence between these two states is determined byexchange interaction. The 3Du state (Fig. 1) con-sists of two degenerate patterns which are similarto the wave functions given by Eqs. (11) and (12),but with different signs combinations. The 3Dustate also has a big barrier at the starting point ofthe three-body DR reaction (Fig. 1). The 13R�

u

state is the only plausible candidate for the DRprocess: its potential energy curve (PEC) crossesthe ionic PEC near the zero vibrational energylevel and leads to the proper dissociation limitCð3PÞ þ 2Hð2SÞ. Though the electric dipole tran-sition from the ground state X3R�

g � 13R�u is al-

lowed by selection rule, the transition moment isclose to zero. Thus this channel is not accessibleby optical photodissociation. The 13R�

u statecorrelates with the 33A2 state of the bent CH2

molecules, which is known to have an equilibriumangle close to the linear structure (about 172�,Fig. 6). Thus we come to a qualitative conclusionthat the three-body dissociation process (Eq. (3))is governed by electronic exchange interaction andstrongly depends on the orbital angular momentumprojection and spin pairing pattern. Close-lyingRydberg states with different orbital angularmomentum projection properties are character-ised by different barriers and dissociation prod-ucts.

Almost all other triplet states (besides the 13Pu

low-lying Rydberg term) have large barriers forthe three-body DR process (Fig. 1). Thus they arenot interesting for our study. The low-lying 13Pu

state correlates with the 13A1 and 13B1 Rydberg

states and all these terms have no relevance to themain purpose of our study, as was discussed be-fore.The symmetric and antisymmetric stretching

potentials for the almost linear ion (179�, as amodel for the 2B1 state) have been studied byReuter et al. [33]. They have shown that the linear(and bent) ion is pretty stable in respect to bothstretchings, symmetric and antisymmetric, whichsimulate the reaction coordinate for the tree-bodyand two-body dissociation processes, respectively.Thus far no experimental information on the CHþ

2

spectrum has been published. But a number oftheoretical researches are available (the review isgiven in [33].) The vibronic and rovibronic struc-ture of the CHþ

2 spectrum have been based on thethree-dimensional hypersurface [33]. This studyhas presented an effective one-dimensional bend-ing potential for the X2A1 and 1

2B1 states of theion obtained from MRCI electronic structure cal-culations taking into account the bond lengthchanges during bending. The reason for such ap-proach is that the large-amplitude bending vibra-tion accompanying the Renner–Teller effect is themost important aspect of the CHþ

2 spectrum [33].For the DR processes, Eqs. (3)–(5), the Renner–Teller effect is not so important, since the groundstate of ion interacts with the colliding electron.Nevertheless it is useful for our purposes to ana-lyse the results of [33]. The symmetric stretchingpotential for the C–H bonds prolongation is muchmore soft, than the antisymmetric one [33]. This isin agreement with our results for the PECs alongthe Cþ 2H and CHþH reactions in the ionic andin a number of Rydberg states for the linear andbent reaction coordinates.

3.3. Nonlinear reaction coordinate models

Since the DR process is determined by PEC ofthe neutral species, the natural nonlinear reactioncoordinate model includes the ground state HCHangle of the triplet methylene which is equal 132�.

B.F. Minaev, M. Larsson / Chemical Physics 280 (2002) 15–30 25

Potential energy curves relative for the three-bodyDR process are presented in Figs. 2–4. We alsostudied the PECs along the reaction coordinatewith the fixed HCH angle, which is equal 138.3� asin the ground 2A1 state of the ion (Table 2). Thesecurves behave in a qualitatively similar mannerand only a small part of these PECs is presented inthe paper (Fig. 5).There are no possible candidates among the

singlet states (Fig. 4) for the three-body DR re-action governed by direct mechanism. The 31A2state (which is shown in Fig. 4 by heavy line) leadsto the excited (C1D) atom after a barrier. Thissinglet state has much higher barrier at any HCHangle than its triplet counterpart. The 31B1 state(dotted line in Fig. 4) lies about 1 eV higher thanthe ion state. This state can be responsible for theresonant structure in the DR cross-section. Sincethe 31B1 state has a minimum at rCH ’ 1:8 �AA, thisweakly bound intermediate exciplex can predisso-ciate through the repulsive 31A2 state (Fig. 4).Similar speculations could be possible for the 61A1state, which has the first minimum at long C–Hdistance (1.33 �AA) and can predissociate. The low-lying singlet states all have big barriers for thethree-body dissociation and can not be accessiblein the DR process (Fig. 4). It is remarkable thatthe singlet and triplet states of methylene are ra-ther different in respect to Cþ 2H dissociation.The triplet states are more interesting for the

three-body DR channels. The 3A1,3A2 and

3B1,3B2 states PECs for the three-body DR reactionare presented in Figs. 2 and 3, respectively. Thelast group of states do not contain plausible can-didates for such DR reaction. Many of these stateshave a big barrier in respect to the three-bodydissociation Cþ 2H channel (Fig. 3). Exception isthe lowest excited 13B1 rate, which has a Rydberg(3s) character and can easily loose two hydrogenatoms. It has a shallow minimum at rCH ¼ 1:34 �AAand a small barrier (0.45 eV) at long C–H distance(2.1 �AA). Anyway this state is not accessible in theDR process since its energy is too low in com-parison with the ionization potential. Similar ar-guments can be applied to another low-lying states(13A1 and 1

3A2) with relatively small barriers (Fig.2.) Only the 33A2 state, which correlates with the13R�

u state of the linear coordinate, can be respon-

sible for the three-body DR reaction (Fig. 2). It isdifficult to obtain proper dissociation limits innonlinear reaction coordinate. Convergence ofMCSCF and of the LR solutions at long rCH dis-tances is bad in the C2v point group (probablybecause of a big variety of close-lying state of onesymmetry and because of the H–H interactionwith different spin pairing). Correlation betweenthe D2h and C2v point groups indicates that the33A2 state will dissociate to the Cð3PÞ þ 2H limit atlinear structure (Fig. 1) but at the bond angle\HCH ¼ 132� it dissociates to the Cð1DÞ þ 2Hlimit (Fig. 2). This is in agreement with recentfinding of the angle distribution for theCð1DÞ þ 2H and Cð3PÞ þ 2H products in DRCHþ

2 þ e� [34].The incoming electron (when it collides with the

CHþ2 ion) can be described by wave function of

continuous spectrum of any symmetry, thus thereis no symmetry selection rule for DR process. Thecross-section of DR obtained from the directmechanism is assumed to be large at very smallelectron collision energies [2,6,27].In the vicinity of the ion ground state equilib-

rium the 33A2 dissociative state energy is very closeto the ionization energy; its PEC is very flat andalmost has no barrier for three-body dissociation(Fig. 2). The nearest 43A2 state has a barrier ofabout 6 kcal/mol at rCH ¼ 1:3 �AA (at the HCHangle 132�). This barrier is getting higher at largerbond angle (Fig. 5). Other states of the 3A2 sym-metry are not repulsive in respect to the symmetricthree-body dissociation process. The fact that onlythe 33A2 fits the DR process, Eq. (3), is supportedby direct MCSCF calculations with larger CASand basis set (‘‘aug-cc-pVQZ’’). The ionic energyin Figs. 2–4 (dot-dashed line) is obtained from theLR calculation with the excited ion, thus it is veryapproximate. The MCSCF energy of the X2A1ionic state is about 1 eV lower and goes in parallel.At the ion equilibrium bond angle \HCH equal

to 138.3� the energy gap between the ion groundstate and the 33A2 dissociative state is the largestone in comparison with other calculated angles. InFig. 5 the ionic ground state calculated at theMCSCF level is shown by bold solid line. In thesame Figure the zero vibrational level is shown(calculated in harmonic approximation). By

26 B.F. Minaev, M. Larsson / Chemical Physics 280 (2002) 15–30

dashed lines the two vibrational levels for the x2

bending mode are given and one symmetric stretchquantum is presented by the dotted line. The lastlevel is almost in resonance with the crossing pointbetween ion and the 33A2 dissociative state (Fig.5). The symmetric stretch mode is absent in the33A2 state. Irrespective of our error in the ionicenergy calculation the following qualitative con-clusion is drawn: easily excited bending vibrationscan assist the electron capture and the three-bodyDR process, Eq. (3).Another probable explanation of this DR

mechanism can account the electron capture withnonradiative transition to the 33B2 state. This va-lence state is bound in respect to the Cþ 2H dis-sociation reaction, but its minimum is shifted incomparison with the ionic X2A1 equilibrium (Fig.5). The 33B2 state can predissociate through the33A2 potential (Fig. 5): both states are mixed byrotational interaction (rotation around y-axis).The 33A2 state is analogous to the 1

3R�u state of

linear configuration: at short distance (rCH ¼1:1 �AA) it has the main orbital configuration

W½33A2 ¼ Að1a1Þ2ð2a1Þ2ð1b2Þ1ð3a1Þð4a1Þð1b1Þ;ð13Þ

which has to be augmented by the spin function ofthe same type as that given in Eq. (12). Fromanalysis of the MCSCF wave function of this statethe 4a1-MO is not a Rydberg orbital, thus the statecan be treated as a valence state. This is doublyexcited CSF in respect to the closed shell singletstate of the CH2 species. But this is the singly ex-cited configuration in respect to the triplet groundstate methylene. For the open-shell molecules thetransition to the doubly excited state in the DRprocess (in terms of Bates model [6,2]) is not nec-essary.Movement along the symmetric prolongation of

the both C–H bonds leads to a fast change of themain electronic configuration: at rCH ¼ 1:4 �AA theprevious wave function is changed to a simple CSFwith two nonpaired electrons:

W½33A2 ¼ Að1a1Þ2ð2a1Þ2ð1b2Þ2ð1b1Þð2b2Þ: ð14Þ

This is just the 3a1 ! 2b2 excitation from theground X3B1 state of methylene. The 2b2-MO is

antibonding in respect to both C–H chemicalbonds.The quintet 15A2 state with the main orbital

configuration of the same type, like that shown inEq. (13), is also dissociative. Its PEC crosses theionic state potential near the equilibrium point andthen goes lower than the triplet 33A2 state PEC.The quintet PEC slope is much steeper in the ionicequilibrium region. Similar behavior of the quintet15A2 state potential has been obtained before forthe two-body dissociation channel CHþH [1] andwe have reproduced this potential. For this reac-tion the dissociation limit is much lower and thequintet PEC slope is more steeper than in theCþ 2H channel. The direct DR mechanism withaccount of the quintet 15A2 state is spin forbidden.A nondirect DR process with SOC account be-tween the quintet and possible Rydberg states(33B2, 4

3B1) is of very low probability, because theSOC matrix elements are negligible. Thus thequalitative analysis of three possible channels, Eqs.(3)–(5), can be done by comparison of the corre-sponding triplet state potentials.

4. Comparison of the dissociation channels

The ion CHþ2 represents a system with strong

Renner–Teller type of interaction [7]. That is whyit is nonlinear in the ground state 2A1. OurMCSCF calculation with the ‘‘aug-cc-pVTZ’’ ba-sis gives the HCH angle 138.36�, the distancerCH ¼ 1:101 �AA. Geometry optimization in the 2B1state of the C2v group leads to linear structure withthe shorter distance rCH ¼ 1:097 �AA. Starting withthe ion ground state equilibrium we have calcu-lated by MCSCF LR technique the neutral CH2

PECs of 36 singlet and 37 triplet states for all threechannels of the DR process, Eqs. (3)–(5). The samecalculations have been repeated with the bondangles 132� (ground state equilibrium) and 180�(simple model). We do not present our PECs forthe CHþH channel, since they are very similar tothe potentials obtained in [1]. The only new findingis that the higher energetic states, which are closeto the ionic state energy, (6; 7; 8A00) are attractivein respect to the C–H bond cleavage and all havehigh barriers.

B.F. Minaev, M. Larsson / Chemical Physics 280 (2002) 15–30 27

We have not found repulsive states for theCHþH channel which cross the ion state 12A1 inthe vicinity of its minimum (with energy close tothe ionic zero-vibrational level). Only the lowestexcited 13A1 and 1

3A2 states of the methylene arerepulsive states in respect to the CHþH dissoci-ation as it was obtained in [1]. The first one is theRydberg state which is strongly mixed with thevalence 13A0 state upon the asymmetric stretchvibration. This state can not contribute to the di-rect two-body DR, Eq. (4), because of the largeexcess energy between the ionic and this Rydbergstate (4.2 eV).The valence 13A2 state transforms adiabatically

smoothly to the repulsive PEC which correlateswith the excited product CHða4R�Þ þH [1]. As itdiscussed in the following the 13A2 state could beinvolved as an intermediate step in the cascadeleading to the many-step dissociation, Eq. (5), witha low probability, since it is unstable in respect tothe bending mode (Fig. 6).Many of the excited 3A00 states undergo avoided

crossing along the C–H bond dissociation reactionaround rCH ¼ 1:6 �AA [1]. The one-bond dissociationprocess starts at this bond length and seems to becompleted at rCH ¼ 2:2 �AA. Between these pointsthere is a change in the orbital character of 4a0 and5a0-MOs [1]. This is the reason why many stateshave a barrier in this region [1]. There are also fewconical intersections (CIS) along the C–H bonddissociation channel [1,35,36].As far as the CþH2 DR channel concerns we

have found only one possible mechanism: it shouldstart with simultaneous prolongation of two C–Hbonds and come to the intermediate state withshifted minimum along rCH coordinate (symmetricstretching prolongation). There are few candidatesfor such intermediate: the most probable is the33B2 state, shown in Figs. 5, 6 and 3. Electroncapture by the ion can easily lead to such a neutraltrap. The C–H bond lengths are equal 1.18 �AA,which are longer than in the ion (1.10 �AA).This intermediate trap is pretty stable in respect

to the further C–H bonds prolongation in thethree-body DR channel (Fig. 3), but it is unstablein respect to the bending (Fig. 6). The 33B2 statehas a deep minimum at the bond angle\HCH ¼ 97�. Thus the highly excited bending

vibrations occur. This vibronically excited 33B2state can dissociate to the Cð3PÞ þH2 product. Atthe small bond angle (\HCH ’ 60�, the leftturning point during bending vibration) there is aprobability for such dissociation, as it followsfrom our PEC calculations at different bond an-gles. It includes the nonradiative transition to the13B2 state through the symmetric stretch vibra-tions (the latter oscillator has longer C–H bonds,but short H–H distance). Finally the nonradiativetransition from the 13B2 state to the 1

3A2 statethrough the bending vibrations (Fig. 6) and rota-tion can occur. The low-lying 13A2 state has a deepminimum at the HCH bond angle equal 41� (Table1). This vibronically excited 13A2 state will disso-ciate to the Cð3PÞ þH2 product. The energytransfer from symmetric stretch to bending vibra-tion occurs in the first two steps and finally thebending energy is transferred to the kinetic energyof the dissociation products CþH2. The proba-bility of such three-step process could be definitelysmall in comparison with the Cþ 2H channel,which follows immediately the electron captureand dissociation along one state (33A2) potential.One can stress that importance of the two-di-

mensional (stretching + bending) movements inthe Cþ 2H channel of the DR process is easilyseen from comparison of Figs. 2–5 and Fig. 6.Besides those intermediate traps mentioned above,some other stepwise processes could be considered.But for proper cross-section calculation one needsto use a quasi-diabatic representation with accountof nuclear kinetic energy.As follows from our adiabatic PECs calcula-

tions, the DR channel CHþH, Eq. (4), can occuralso through the intermediate trap state. We shallprovide a qualitative analysis for this channel onlyin brief. In contrast to the DR reactionCð3PÞ þH2, Eq. (5), this channel is a two-stepprocess. After trapping the excitation in the 33B2state, the system would come to the 13A2 statemoving along the bending coordinate. Again thesequence of stretching and the following bendingmodes determines the reaction coordinate. Butnow the 13A2 state can dissociate at any angle tothe CHða4RÞ þH products. This result has beenobtained first time by R€oomelt et al. [32] and hasbeen supported in [1] and by our calculations.

28 B.F. Minaev, M. Larsson / Chemical Physics 280 (2002) 15–30

Other channels are also possible with combinationof antisymmetric stretch and bending vibrationsthrough the 3B2 and

3B1 states including numerousCISs obtained in our LR studies.Yarkony has considered the CIS between low-

lying states 13A2 and 13B1 states near the equilib-

rium geometry of the ground X3B1 methylene andstressed that this CIS plays key role in nonadia-batic decomposition originated on the 13B1ð33A00Þsurface during photodissociation to CHþHproducts [35,36]. All these states have 3A00 sym-metry along the reaction coordinate when thebond lengths are different. This CIS in the Csgroup corresponds to 23A00–33A00 PECs avoidedcrossing; the CIS can lead to the geometric phaseeffect [35,36]. The nonadiabatic transition33A00 ! 23A00 owing to the funneling effect of CIScan influence the photodissociation process in theCHþH channel [35]. Similar nonadiabatic tran-sitions can occur in the final step of the DR reac-tion, Eq. (4). More detailed consideration will bethe object of our future work.

5. Conclusions

Simultaneous cleavage of two chemical bonds isa very unlikely process in ordinary chemistry. Thereason why it occurs in DR of molecular ionCHþ

2 þ e� ! Cð3PÞ þ 2Hð2SÞ is the following.This is a very particular way of chemical activa-tion, when the active state is prepared by electroncapture and a big internal electronic energy storedin the separated particles can be released. Heat offormation of each particle is more than 300 kcal/mol thus their sum exceeds the heat of formationof the products. The problem how this excess en-ergy can be transformed to the kinetic energy ofthe dissociative particles has been addressed in thepresent study in terms of adiabatic potential curvesanalysis.We looked for dissociative PECs of the neutral

species which cross the ionic state PEC. SuchPECs have been found only for the three-body DRprocess, Eq. (3). From the linear model we haveobtained one essential feature of the three bodydissociation channel. From two possible candi-dates for the intermediate state in DR process,

13R�u and 1

3Rþu states, we can see (Fig. 1) that only

the first state has a very small barrier. Its potentialenergy curve crosses the ionic PEC near the zerovibrational energy level and leads to the properdissociation limit Cð3PÞ þ 2Hð2SÞ. After electroncapture by the ion this neutral species statetransforms easily from Rydberg character to thevalence state nature and change the orbital con-figuration. At the same time the 13Rþ

u state over-comes a big barrier (Fig. 1). Energy differencebetween these two states is determined by ex-change interaction, since the states have the sameorbital configuration ð3rgÞ1ðpgÞ2ð1ruÞ1 at thestarting point and differs by spin pairing. Thelinear model simulation of the three-body disso-ciation process (Eq. (1)) produces a qualitativeconclusion that this DR process is governed in agreat extent by electronic exchange interaction.The three-body channel, Eq. (3), is much more

efficient because it follows one PEC (33A2) alongthe symmetric stretching mode. The initially cre-ated wavepacket after an electron capture by theion can evolve on the 33A2 PEC and smoothlytransform electronic excitation energy into kineticenergy of the three body products. The two-bodychannels include at least one more additionalmovement – the bending mode and an additionalintermediate state. The direct mechanism of elec-tron attachment leads to the 33B2 state with longerC–H bond lengths than in the ionic state. Thecreated wavepacket may encompass a closed patharound the CIS. Energy transfer to the bendingvibration can induce nonadiabatic transition to the13A2ð23A00Þ state. This state can dissociate alongthe asymmetric stretching movement and lead tothe CHþH dissociation channel at any bondangle, or can dissociate to the CþH2 product atthe particular small bond angle. The last process isless probable. Thus qualitative arguments agreewith the observed product ratio in the DRCHþ

2 þ e� reaction.Most of the calculations performed in this study

are one-dimensional, either in the linear or in theC2v symmetry. Previous DR studies (on Hþ

3 forexample [5,13,29]) have shown that three-dimen-sional effects could be very important. In this studywe also come to importance of two- and three-dimensional effects in crossings and interactions of

B.F. Minaev, M. Larsson / Chemical Physics 280 (2002) 15–30 29

potential energy hypersurfaces. Thus for CHþHDR process the CIS between the 23A00 and 33A00

PECs can be responsible for efficient nonradiativeescape through this funnel. Simultaneous consid-eration of the symmetric and asymmetric stretch-ing movement together with bending vibration isclearly seen as an important origin for few DRchannels, which could be accounted for properanalysis of the branching ratios, Eqs. (3)–(5).

Acknowledgements

This work was supported by the Royal SwedishAcademy of Sciences and the Swedish ResearchCouncil.

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