An intraline of conical intersections for methylamine

An in aline o conical in e sec ions o me hylamine C. Le i,1G. J. Halász,2Á. Vibók,3I. Ba ,1Y. Zei i,4R. Koslo ,5and M. Bae 6,a兲 1Depa men o Physics, Ben-Gu ion Uni e si y, Bee She a 84105, Is ael 2Depa men o In o ma ion Technology, Uni e si y o Deb ecen, P.O. Box 12, H-4010 Deb ecen, Hunga y 3Depa men o Theo e ical Physics, Uni e si y o Deb ecen, P.O. Box 5, H-4010 Deb ecen, Hunga y 4Depa men o Biomedical Enginee ing, Ben-Gu ion Uni e si y, Bee She a 84105, Is ael 5Depa men o Physical Chemis y and he F i z Habe Cen e o Molecula Dynamics, The Heb ew Uni e si y o Je usalem, Je usalem 91904, Is ael 6The F i z Habe Cen e o Molecula Dynamics, The Heb ew Uni e si y o Je usalem, Je usalem 91904, Is ael 共Recei ed 20 Ma ch 2008; accep ed 19 May 2008; published online 23 June 2008兲 In his a icle a e conside ed he conical in e sec ions 共ci’s兲 ela ed o he N–H bond in he me hylamine, CH3NH2, molecule. The no el ea u e ha was e ealed is ha he wo lowes s a es 1A⬘and 1A⬙a e coupled by a line o cis loca ed in HC–NHH plane—a line ha is o med by mo ing a single hyd ogen on ha plane while ixing he 共six兲o he a oms. The alidi y o his line was p o en i s by s udying he singula i ies o he 共angula 兲nonadiaba ic coupling e ms and hen by e ealing he degene acy poin s o med by he wo in e ac ing adiaba ic po en ial ene gy su aces 共PESs兲. A heo e ical analysis indica ed ha he line has o be a ini e closed line. We also calcula ed he Be y phase o a con ou ha su ounds his line and ound i o be 3.127 ad, namely, a alue easonably close o ␲ . The exis ence o such lines o cis—ins ead o isola ed cis 共as exhibi ed by o he n-a omic 共n⬎3兲molecules such as HNCO o C2H2兲—may enhance signi ican ly he ansi ion a e om an uppe adiaba ic s a e o a lowe one. The e a e also nume ical ad an ages in such si ua ions, ha is, i such a line is p ope ly placed in ha plane 共like in he p esen case兲 he wa e-packe ea men o he nuclei can be ca ied ou employing a single diaba ic PES ins ead o ha ing o conside wo coupled PESs. © 2008 Ame ican Ins i u e o Physics. 关DOI: 10.1063/1.2943143兴 I. INTRODUCTION The s udy o he elec onic nonadiaba ic coupling e ms 共NACTs兲and conical in e sec ions 共ci兲concen a ed, o he pas wo decades, mainly on small molecules, namely, i- a omic and e a-a omic sys ems.1–17 In he p esen a icle we conside a much la ge sys em, he me hylamine molecule, CH3NH2, consis ing o se en a oms. The s uc u e o he molecule is gi en in Fig. 1共a兲whe e i can be seen ha he me hyl g oup, CH3, is sepa a ed om he amine g oup, NH2 by he CN bond. This molecule has long been a ques ion o conside able in e es since i comp ises wo s ongly coupled la ge ampli ude mo ions, speci ically, he o sion o he me- hyl op and he in e sion o he amine g oup.17 Se e al ex- pe imen al s udies p o ided e idence ha he dominan channel in i s pho odissocia ion in he i s abso p ion band co esponds o he N–H bond ission.18–20 Ve y ecen ly21–23 i was shown ha in he ib a ional media ed pho odissocia- ion o CD3NH2abou 90% o he o al obse ed p oduc s 共only hyd ogen iso opes we e measu ed兲a e H pho o ag- men s eleased om he amine g oup. Also, heo e ical ind- ings showed he cu s h ough he ab ini io po en ial ene gy su aces 共PESs兲 o he g ound 1A⬘and he i s exci ed 1A⬙ s a es o me hylamine. Pa icula ly, i was poin ed ou ha he 1A⬙s a e po en ial ha leads o he b eakup o he N–H bond is cha ac e ized by a small ba ie 共⬃3000 cm−1兲a sho ange and is ollowed, a la ge bond ex ensions, by a ci loca ed a an in e sec ion poin wi h he 1A⬘s a e whe e bo h s a es belong o he Cssymme y.24,25 In wha ollows we epo on a s udy o NACTs ha couple he abo e men ioned wo s a es and b ie ly e e o he highe exci ed s a e, 2A⬙. II. FORMING THE INTRALINE OF CONICAL INTERSECTIONS In wha ollows we epo on a s udy o NACTs ha couple he abo e men ioned h ee s a es. To ca y ou his s udy he ia omic amine g oup is assumed o o m a plane wi h HC dia om o he me hyl g oup 共see Fig. 1兲, and ou s udy concen a es on sea ching o Jahn–Telle cis 共Re s. 26–28兲be ween he h ee lowe s a es, namely, he g ound 1A⬘s a e and he nex wo exci ed s a es 1A⬙and 2A⬙共 he sea ch o hese cis is ca ied ou by allowing he amine hyd ogen o mo e in his plane only兲. To his end we employ he MOLPRO package29 wi h he mul i e e ence con igu a ion in e ac ion me hod and he 6-31+G*basis se . We used he ac i e space, including wo alence elec ons dis ibu ed on i e o bi als 共 h ee o hem belonging o he i educible ep esen a ion o A⬘兲. Th ee elec onic s a es we e compu ed wi h equal weigh s. All se en a oms we e conside ed, bu he sea ch o cis is ca ied ou by ac i a ing only one a om—an amine hyd ogen—as a es pa icle. To ollow he mo ion o he hyd ogen we em- a兲Elec onic mail: [email p o ec ed]. THE JOURNAL OF CHEMICAL PHYSICS 128, 244302 共2008兲 0021-9606/2008/128共24兲/244302/6/$23.00 © 2008 Ame ican Ins i u e o Physics128, 244302-1 Downloaded 10 Jul 2008 o 193.174.59.126. Redis ibu ion subjec o AIP license o copy igh ; see h p://jcp.aip.o g/jcp/copy igh .jsp ploy wo 共pola 兲coo dina es, he adial coo dina e qand he angula coo dina e ␸ , de ined o a sys em o coo dina es loca ed a some poin in he plane 关see Fig. 1共b兲兴. The de ec- ion is done by calcula ing he angula NACT, 共1/q兲 ␶ ␸ jk共 ␸ ,q兩s兲, whe e ␶ ␸ jk共 ␸ ,q兩s兲is de ined as ␶ ␸ jk共 ␸ ,q兩s兲= 冓 ␨ j共se兩 ␸ ,q,s兲冏 ⳵ ⳵␸␨ k共se兩 ␸ ,q,s兲 冔 .共1兲 He e ␨ i共se兩 ␸ ,q,s兲;i=j,ka e he elec onic 共adiaba ic兲Bo n– Oppenheime 共BO兲eigen unc ions,30,31 ses ands o he se o elec onic coo dina es, and sp esen s he g oup o all nuclea coo dina es excluding ␸ and q. Ha ing de ined he sys em o coo dina es he sea ch s a s by o ming ci cula con ou s o di e en adii and calcula ing he angula NACTs along hese con ou s. The nex s ep is de i ing he adiaba ic- o-diaba ic ans o ma ion 共ADT兲angle 共known also as he mixing angle兲, ␥ 共 ␸ ,q兲共Re . 32兲共see also Re . 1, Chap. 3.1兲, ␥ 共 ␸ ,q兩s兲= 冕 0 ␸ d ␸ ⬘ ␶ ␸ 共 ␸ ⬘,q兩s兲.共2兲 In he case o a closed con ou we ge ␥ 共 ␸ =2 ␲ ,q兩s兲 = ␣ 共q兩s兲, whe e ␣ 共q兩s兲is ecognized as he Be y phase.33,34 I is impo an o emphasize ha o si ua ions whe e he NACT is o med, in a gi en egion, by wo isola ed s a es 共namely, s a es a ec ed, a mos , sligh ly by highe s a es in ha egion兲, he phase ␣ 共q兩s兲is expec ed o be close o an in ege mul iple o ␲ 共o ze o兲.33–36 Pa o he nume ical s udy was de o ed o his issue. We calcula ed NACTs be- ween he h ee lowes s a es o he Cssymme y, namely, 1A⬘,1A⬙, and 2A⬙共men ioned ea lie 兲and ound ha only he NACTs o med by he wo lowe s a es a e nonze o whe eas he o he wo ha in ol e he second exci ed s a e a e p ac ically ze o 共 hus gua an eeing he isola ion o he wo lowe s a es in he egion o in e es 兲. Figu e 2共a兲p esen s he angula NACT, along such a ci cle, as a unc ion o ␸ wi h q=0.1 Å, whe e he coo di- na es o he cen e a e a 共R1, ␪ 1兲=共1.83 Å,122°兲and 共R2, ␪ 2兲=共1.0 Å,120°兲关see Fig. 1共a兲兴. The igu e shows wo ela i ely sha p peaks wi h opposi e signs. I u ns ou ha he alue o he in eg al unde each peak is ⬃1.554 ad 共i.e., a alue close o ␲ /2兲and he alue o he Be y phase is ␣ 共q兩s兲⬃0. I is impo an o emphasize ha he s uc u e o he angula NACT di e s signi ican ly om all o he angula NACTs o which we usually encoun e wo1,4共b兲,4共c兲,5,6,7共a兲,8,28 共and some imes h ee4共a兲兲posi i e maximal alues and he Be y phase becomes, app oxima ely, an in ege mul iple o ␲ 共and no ze o兲. Thus he ques ion is wha makes his pa icula case so di e en ? Following addi ional calcula ions we ind ha he same plane hos s many mo e cis. All a e cha ac e ized by he same ea u e as discussed abo e; namely, he co esponding angula NACTs posses wo sha p peaks o opposi e signs 共see ano he one gi en in Fig. 2共b兲wi h q=0.1 Å and he cen e a 关共R1, ␪ 1兲=共1.70 Å,85°兲兴. Finally i was es ablished ha he jus men ioned cis o m a con inuous 共 ini e兲line 共see Fig. 3兲. This line is cons uc ed in p ac ice by connec - ing all he posi i e 共o nega i e兲peaks o NACTs o he ype p esen ed in Figs. 2共a兲and 2共b兲. I is in e es ing o no e ha one o he ci poin s, i.e., he one calcula ed a R1=1.83 Å and ␪ 1=122°, is close 共bu no iden ical兲 o he posi ion o a degene acy poin e ealed some ime ago by Dunn and Mo okuma.24 In Fig. 3is p esen ed such a line in a plane de e mined FIG. 1. 共Colo online兲The equilib ium s uc u e o me hylamine. In he p esen s udy i e a oms, namely, he ca bon, he ni ogen, wo amine hy- d ogens, and a me hyl hyd ogen a e assumed o o m a ixed plane— he HC–NH2plane—and hei ela i e mo ion is cons ained o his plane. 共a兲 The coo dina es ␪ 1and R1show he posi ion o he es hyd ogen 共wi h espec o he ni ogen兲 ha we e a ied du ing he ci sea ch 共 he coo dina es ␪ 2and R2a e held ixed du ing his p ocess兲.共b兲The pola coo dina es qand ␸ show he posi ion o es hyd ogen wi h espec o an assumed poin close oaci poin . FIG. 2. 共Colo online兲The angula NACT p esen ed as a unc ion o ␸ o q=0.1 Å as calcula ed a h ee di e en loca ions along he seam 共see Fig. 3兲:共a兲The NACT a poin 共R1, ␪ 1;R2, ␪ 2兲=共1.83 Å,122° ;1.0 Å,120°兲.共b兲 The NACT a poin 共R1, ␪ 1;R2, ␪ 2兲=共1.70 Å,85° ;1.0 Å,120°兲.共c兲The NACT a poin 共R1, ␪ 1;R2, ␪ 2兲=共1.70 Å,159° ;1.0 Å,120°兲. 244302-2 Le i e al. J. Chem. Phys. 128, 244302 共2008兲 Downloaded 10 Jul 2008 o 193.174.59.126. Redis ibu ion subjec o AIP license o copy igh ; see h p://jcp.aip.o g/jcp/copy igh .jsp by holding he second amine hyd ogen ixed a 共R2, ␪ 2兲 =共1.0 Å,120°兲关see Fig. 1共a兲兴: This line is p esen ed in e ms o Ca esian coo dina es 共x,y兲关=R1cos共 ␲ − ␪ 1兲, R1sin共 ␲ − ␪ 1兲兴, whe e he cen e o he coo dina es is loca ed a he ni ogen a om. This jus desc ibed line can be in e p e ed as a seam 共de ined as he line ha connec s all hese in ini e numbe o ci poin s兲. Like any o he seam his one is also expec ed o be a line ha connec s he degene acy poin s be ween he wo adia- ba ic su aces. In Fig. 3is p esen ed a second line ha ol- lows he degene acy poin s and closely o e laps wi h he seam 共 he de ia ions a e wi hin he e o limi o he calcu- la ions兲. Mo e de ails conce ning his line o degene acy poin s a e gi en in Fig. 4. In Fig. 4共a兲a e p esen ed he wo in e - ac ing adiaba ic PESs 1A⬘and 1A⬙, assigned as V1共x,y兲and V2共x,y兲. In Fig. 4共b兲is p esen ed ⌬V共x,y兲, he di e ence ⌬V共x,y兲=V2共x,y兲−V1共x,y兲be ween he wo adiaba ic PESs. The wo lines in he x-yplane in Fig. 3a e seen o be ini e and open. In ac addi ional calcula ions based on he NACTs o ind ou mo e abou hese lines ailed al oge he because he NACTs became spikie o he ex en ha inally we los hem al oge he . Howe e addi ional calcula ions based on degene acy poin s we e ca ied ou , and hey showed ha , undoub edly, he line is ex ending owa d he wo hyd ogens— he amine hyd ogen on one side and he co esponding me hyl hyd ogen on he o he . Thus, his ype o calcula ions seems o indica e ha he line is ini e. How- e e we could no es ablish whe he he line is open o closed mainly because i closed— he missing segmen 共s兲is 共a e兲a he high po en ial egion whe e he accu acy is no good enough o es ablish poin s o degene acy. These issues a e u he discussed in he nex chap e . Mo e abou he PESs o his sys em is shown in Fig. 5. In con as o he adiaba ic su aces in Fig. 4共a兲 hese a e he co esponding diaba ic ones. This igu e clea ly shows he impo an ad an age in ha ing a line o degene acy poin s in a ixed plane. This si ua ion causes he co esponding diaba- ic coupling e m o be ze o along his line,37,38 hus allowing a wa e packe o mo e, eely and undis u bed, om he uppe adiaba ic su ace o he lowe adiaba ic su ace 共mo e on his issue is discussed in he nex chap e 兲. III. DISCUSSION AND CONCLUSIONS This a icle concen a es on an excep ional con inuous line o cis e ealed in he me hylamine molecule. This line can be in e p e ed as a seam, bu i i is a seam i is less common because some o i s ea u es a e di e en . In wha ollows we concen a e on se e al issues: 共1兲In all p e ious s udies1–8,14,15,28,34 he seams a e loca ed ou side he planes 共e.g., he g-hplane, which is de ined as he pe pendicula plane o he seam7共a兲兲 ha con ain he mo ing es pa icle 共which in he p esen case is he HC–NHH plane兲. In o he wo ds he me hylamine c e- a es a phenomenon whe e he wo in e ac ing s a es i.e., 1A⬘and 1A⬙, a e coupled by a con inuous line o cis o med by one single mo ing a om whe e all o he 共six兲a oms o he molecule a e ixed in con igu a ion space. Since his hyd ogen is mo ing wi hin he HC– FIG. 3. 共Colo online兲The in a 共 ixed con igu a ion space兲seam o he ci’s loca ed in he HC–NH2plane as calcula ed o R2=1.0 Å and ␪ 2=120°: 共----兲The seam as calcula ed by connec ing he ci poin s; 共----兲 he seam as calcula ed by connec ing he degene acy poin s. The wo Ca esian coo di- na es 共x,y兲a e ela ed o he pola coo dina es 共R1, ␪ 1兲, namely, 共x,y兲 =关R1cos共 ␲ − ␪ 1兲,R1sin共 ␲ − ␪ 1兲兴. A h ee loca ions a e d awn small ci cles o indica e he posi ions o h ee con ou s men ioned in Fig. 2. FIG. 4. 共Colo online兲The PESs as a unc ion o xand yas calcula ed o R2=1.0 Å and ␪ 2=120°: 共a兲P esen s he wo adiaba ic 共1A⬘and 1A⬙兲PESs, V1共x,y兲and V2共x,y兲;共b兲p esen s he di e ence ⌬V共x,y兲=V2共x,y兲 −V1共x,y兲be ween he wo PESs. The wo igu es a e shown a di e en angles o emphasize he mo e signi ican ea u es in each case. 244302-3 Conical in e sec ions o me hylamine J. Chem. Phys. 128, 244302 共2008兲 Downloaded 10 Jul 2008 o 193.174.59.126. Redis ibu ion subjec o AIP license o copy igh ; see h p://jcp.aip.o g/jcp/copy igh .jsp NHH plane, he line, jus desc ibed, is loca ed in ha 共HC–NHH兲plane as well 共see Fig. 3兲. As i s ands his si ua ion canno be jus i ied. To unde s and hese ind- ings we ha e o conside in mo e de ail bo h he plane ha con ains he mo ing hyd ogen and he ci-line as well as he wo s a es ha o m he NACTs. I is well known ha in o de o wo s a es 共no coupled by any o he s a e兲 o o m degene acy a a gi en poin , wo condi ions ha e o be ul illed, namely,38 H11 −H22 =0, 共3a兲 H12共=H21兲=0, 共3b兲 whe e Hij,i,j=1,2 a e he ma ix elemen s o he elec- onic Hamil onian. Fo a andomly selec ed plane hese condi ions a e a mos sa is ied a isola ed poin s. Howe e i a s udy is ca ied ou wi h espec o a plane o symme y 共as in he p esen case兲and he wo s a es a e o di e en symme ies 共as indeed 1A⬘and 1A⬙ a e兲, hen condi ion 共3b兲is sa is ied e e ywhe e by symme y. So now we a e le wi h only one equi e- men o be sa is ied, namely, Eq. 共3a兲, and his one can be ul illed along a line in he abo e men ioned plane wi hou necessa ily c ea ing any imp obable si ua ions 共as indeed is shown by he nume ical ea men 兲. 共2兲Once we es ablished he exis ence o he abo e men- ioned ci-line— o be assigned as ⌳— wo ques ions ha e o be answe ed: 共i兲Isi a ini e o an in ini e line? 共ii兲I i is a ini e line is i an open o a closed line? As indica ed in he p e ious chap e we could no answe hese ques ions nume ically 共al hough i is e y likely ha he line is indeed ini e兲. In o de o analyze he a ious possibili ies we conside wo kinds o con ou s, namely, ⌫c, ha in e sec ⌳and ⌫s, which su ound i 共see Fig. 6兲. As long as one mo es along ⌫s he o de o he s a es 1A⬘and 1A⬙a poin s Aand B emains un- changed whe eas i one ollows a con ou o he kind ⌫c关see Fig. 6共a兲兴, each c ossing is accompanied wi h a change in he o de o he s a es 关see Re s. 37共a兲and 37共b兲whe e a simila si ua ion is analyzed while s udy- ing cha ge ans e o he 共H+H2兲+sys em兴. Thus, i a poin As a e 1A⬙is abo e 1A⬘we ind, ollowing con- ou ⌫c, ha a B1A⬙is below 1A⬘. Mo ing along a con ou ⌫c ha c osses ⌳ wice lea es he o de o he s a es unchanged 关see Fig. 6共c兲兴. We e u n now o he abo e men ioned ques ions and we s a by discussing 共i兲:I ⌳is an in ini e line hen he o de o s a es can be a ec ed e en in a si ua ion whe e he es hyd ogen is a an in ini e sepa a ion om he CH2NH adical. How- e e a la ge dis ances he hyd ogen has no in luence on he adical 共and ice e sa兲and he e o e canno a ec he s a es o he adical. In o he wo ds no con ou s o he kind ⌫ccan be ound a he in ini e sepa a ion e- gion, which implies ha ⌳is a ini e line. Nex we discuss 共ii兲. Since ⌳is a ini e line, i can be ei he an open o a closed line. In o de o be able o ollow he o hcoming analysis we e e again o Fig. 6in which a e p esen ed se e al possibili ies. In Fig. 6共a兲is shown an open line, ⌳, and wo con ou s ⌫cand ⌫s ha con- nec Aand B. F om he ea lie discussion we ind ha mo ing along hese wo con ou s is expec ed o yield, physically, unaccep able esul s; namely, mo ing along ⌫cchanges he o de o s a es bu mo ing along ⌫s lea es hem unchanged. A di e en si ua ion is encoun- e ed when ⌳is a closed line as gi en in Figs. 6共b兲and 6共c兲. In he case o Fig. 6共b兲any con ou om A o Bis FIG. 5. 共Colo online兲The wo diaba ic PESs as a unc ion o xand yas calcula ed o R2=1.0 Å and ␪ 2=120°. In he igu e is emphasized he 共physical兲diaba ic po en ial, which leads in a con inuous way om he uppe adiaba ic PES o he lowe one. FIG. 6. A schema ic pic u e showing con ou s ⌫in e sec ing ini e ci lines ⌳ o a ious si ua ions. Aand Ba e wo a bi a y poin s on he x-yplane. 共a兲 An open line ⌳, a con ou , ⌫c, ha in e sec s i , and a con ou , ⌫s, ha su ounds i . 共b兲A closed line, ⌳, poin Ainside he su ounded a ea, poin Bou side he su ounded a ea, and wo con ou s ⌫c ha in e sec ⌳.共c兲A closed line, ⌳, and poin s Aand Bou side he su ounded a ea. ⌫cis a con ou ha in e sec s ⌳ wo imes and ⌫sis a con ou ha su ounds i . 244302-4 Le i e al. J. Chem. Phys. 128, 244302 共2008兲 Downloaded 10 Jul 2008 o 193.174.59.126. Redis ibu ion subjec o AIP license o copy igh ; see h p://jcp.aip.o g/jcp/copy igh .jsp a⌫c-con ou and he e o e a ec s, once, he o de o he s a es. The opposi e is ue o he case desc ibed in Fig. 6共c兲; namely, any con ou om A o B, whe he in e sec ing, ⌳, o no lea es he o de o he s a es unchanged. To summa ize his analysis we say ha open lines, in con as o closed lines, lead o unphysi- cal si ua ions and he e o e canno exis in molecula sys ems. 共3兲In he p esen s udy we also encoun e an uncommon Be y- ype phase which is ⬃0 ins ead o being an in e- ge mul iple o ␲ . Such a esul is expec ed when a seam and he closed ci cula con ou a e loca ed in he same plane so ha he con ou in e sec s he seam in- s ead o su ounds i 共by lea ing he plane兲. In Re s. 37共a兲and 37共b兲we analyzed simila si ua ions and showed ha each in e sec ion be ween he con ou and such a seam o ms o he co esponding 共in ou p esen case, he angula 兲componen o he NACT, a na ow unc ion ha in he limi ing case becomes a Di ac ␦ - unc ion, namely, 共 ␲ /2兲 ␦ 共 ␸ − ␸ s兲, which yields upon in eg a ion 共o e ␸ 兲 he ADT angle ␥ = ␲ /2. He e ␸ sis he ␸ - alue a he in e sec ion. The co esponding nu- me ical in eg a ions along he closed con ou e ealed in ou case ha a he icini y o he i s in e sec ion, he alue o he ADT angle ␥ is app oxima ely + ␲ /2 and a he second i is app oxima ely − ␲ /2. As a esul , he inal Be y phase is app oxima ely ze o. This ou - come, al hough somewha unexpec ed, is in ag eemen wi h o he s udies o simila si ua ions.37,39,40 共4兲In o de o s eng hen he idea ha his line o cis is indeed a seam, we managed also o calcula e he angu- la NACTs along a con ou , which is sligh ly il ed away om he HC–NHH plane. In his way he con- ou , ins ead o in e sec ing he line o cis, su ounds i . The esul s a e p esen ed in Fig. 2共c兲. I is clea ly seen ha in con as o he wo p e ious unc ional shapes his one is cha ac e ized by wo posi i e peaks. Calcu- la ing he Be y phase o his case yields he alue o 3.124, a alue close o ␲ . This issue will be u he discussed in a o hcoming publica ion. Finally we would like o e e o wo aspec s due o his s udy, a physical aspec and a nume ical one: 共a兲F om he physical poin o iew his new ype o seams is expec ed o gua an ee a much mo e e icien decay p ocess om an uppe adiaba ic s a e o a lowe one. The exis ence o such lines o cis—ins ead o isola ed cis 共as exhibi ed by o he polya omic 共n⬎3兲molecules such as HNCO 关Re . 8共b兲兴o C2H2关Re s. 4共b兲and 4共c兲兴—may enhance signi ican ly he ansi ion a e be- ween he wo PESs. 共b兲F om a compu a ional poin o iew his inding may a ec u u e quan um mechanical dynamical s udies o sys ems o his ype. Thus, i such a degene acy line is loca ed in he pa h o a wa e packe which mo es om he uppe PES o he lowe one 共as seen in Fig. 5兲we may pe o m, ins ead o a 共diaba ic兲 wo-s a e calcula- ion, a diaba ic single-s a e calcula ion and s ill ob ain accu a e esul s. In ac such a calcula ion is on i s way, and we hope o epo he ou comes in he nea u u e. In his espec we would like o men ion again he i s heo e ical s udy on his sys em.24 In addi ion o de e - mining he posi ion o he poin o degene acy 共 o he same con igu a ion兲, Dunn and Mo okuma24 also cal- cula ed he wo adiaba ic po en ial ene gy cu es along he N–H dissocia ion channel, which a e p esen ed in hei Fig. 2 as dashed lines. A good quali a i e i is achie ed be ween hei po en ial cu es and ou PESs. ACKNOWLEDGMENTS M.B. and C.L. acknowledge he US-Is ael Bi-na ional Science Founda ion o pa ly suppo ing his s udy. Á.V. ac- knowledges OTKA G an No. T067923 and he compu a- ional esou ces p o ided by he John- on-Neumann Ins i- u e, Resea ch Cen e Juelich 共P ojec ID ehu01兲. I.B. and C.L. acknowledge he pa ial suppo o he James F anck Bina ional Ge man-Is aeli P og am in Lase -Ma e In e ac- ion and he Is ael Science Founda ion ounded by The Is ael Academy o Science and Humani ies. 1M. Bae , Beyond Bo n Oppenheime : Elec onic non-Adiaba ic Coupling Te ms and Conical In e sec ions 共Wiley, Hoboken, NJ, 2006兲. 2in pa icula , see 共a兲M. S. Child, Ad . Chem. Phys. 124,1共2002兲;共b兲S. Adhika i and G. D. Billing, ibid. 124, 143 共2002兲;共c兲R. Englman and A. Yahalom, ibid. 124, 197 共2002兲;共d兲A. Kuppe mann and R. Ab ol, ibid. 124, 323 共2002兲;共e兲G. A. Wo h and M. A. Robb, ibid. 124,355共2002兲. 3A. Kuppe mann, in Dynamics o Molecules and Chemical Reac ions, edi ed by R. E. Wya and Z. H. Zhang 共Ma cel, New Yo k, 1996兲,p.411; R. Bae , D. M. Cha u z, R. Koslo , and M. Bae , J. Chem. Phys. 105, 9141 共1996兲; S. 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