Topological effects for nonsymmetrical configurations: The C2H2+ as a case study

Topological e ec s o nonsymme ical con igu a ions: The C2H2 +as a case s udy G. J. Halász Depa men o In o ma ion Technology, Uni e si y o Deb ecen, P.O. Box 12, H-4010 Deb ecen, Hunga y Á. Vibók Depa men o Theo e ical Physics, Uni e si y o Deb ecen, P.O. Box 5, H-40410 Deb ecen, Hunga y M. Bae a兲 The F i z Habe Resea ch Cen e o Molecula Dynamics, The Heb ew Uni e si y o Je usalem, Je usalem 91904, Is ael 共Recei ed 25 June 2007; accep ed 10 Augus 2007; published online 12 Oc obe 2007兲 Du ing he las decade he s udy o opological e ec s o med by molecula sys ems became a ou ine bu i was always ca ied ou o con igu a ions ha we e limi ed by symme y condi ions. To be mo e speci ic his applied o he Jahn-Telle 共JT兲e ec o med by molecula con igu a ions o plana symme y 关see, e.g., Bae e al., Fa aday Discuss. 127, 337 共2004兲兴 and he Renne -Telle e ec o med by con igu a ions o axial symme y 关see, e.g., Halász e al., J. Chem. Phys. 126, 154309 共2007兲兴. In his a icle we conside o he i s ime molecula con igu a ions ha a oid any symme y condi ions o , in o he wo ds, a e cha ac e ized by he C1poin g oup. We epo on a de ailed s udy o opological e ec s o med by such a molecula sys em. The s udy concen a es on bo h, he wo-s a e 共Abelian兲case and he mul is a e 共non-Abelian兲case. I is shown ha he heo y ha was o iginally de eloped o ea opological e ec s due he JT in e sec ion and also applies o he s udy o opological e ec s in he mos gene al case. The s udy is accompanied wi h nume ical esul s. © 2007 Ame ican Ins i u e o Physics.关DOI: 10.1063/1.2779035兴 I. INTRODUCTION The dynamics igge ed in a molecule a e abso bing a pho on is usually discussed in e ms o he Bo n- Oppenheime 共BO兲 heo y,1whe e he as elec onic deg ees o eedom a e ea ed sepa a ely om he slow nuclei. In his pic u e, elec ons and nuclei do no easily exchange en- e gy. Ye , in some nuclea con igu a ions gene ally e med conical in e sec ions 共CIs兲, ene gy exchange can become sig- ni ican . I is widely ecognized oday ha we e i no o CIs impo an pho obiochemical p ocesses such as ision and pho osyn hesis o i amin D could no ake place. The BO ea men may yield a ious kinds o in e sec ions1bu in he li e a u e mainly wo a e discussed: 共i兲 he Renne -Telle 共RT兲in e sec ion usually cha ac e ized as a pa abolic in e sec ion;2–16 共ii兲 he Jahn-Telle 共JT兲in e - sec ion which, in gene al, is cha ac e ized as a CI.17–35 The wo ypes di e also in wo addi ional ways: 共a兲 he BO-RT nonadiaba ic coupling e ms 共NACTs兲a e o med by s a es o opposi e symme y and he esul ing opological 共Be y兲 phase ␣ is equal o 2 ␲ .2–4,7共b兲The BO-JT NACTs a e o med by s a es o same symme y and he esul ing JT phase ␣ is equal o ␲ 共Re s. 25 and 27兲共see also Re . 33, Chap. 3兲. I is impo an o emphasize ha he wo kinds o in e sec ions a e encoun e ed in hose cases whe e he sys em o a oms main ains plana o axial symme y and whe e i is cha ac e ized by he ollowing poin g oups: C2h,C2 ,o C2. The na u al ques ion o be asked is wha happens in case he molecula sys em lacks he abo e men ioned symme ies. Recen ly we aced such a si ua ion while s udying he in e - sec ions o he C2H2 +ion.17 In his case he wo ca bons o m he linea axis and he wo, o -axis, hyd ogens 共 oge he wi h he wo ca bons兲 o m a plane. In case we le he wo hyd ogens, simul aneously, su ound he CC axis, he plana symme y is p ese ed and consequen ly we encoun e he o dina y RT case 共 ␣ =2 ␲ 兲. Howe e , i one o he hyd ogens is clamped while he o he hyd ogen is allowed o su ound he axis, a nonsymme ic C1poin g oup is c ea ed, which s ill leads o opological e ec s bu seem o be mo e cha ac- e is ic o he JT e ec because he alue o ␣ is ound o be ␲ 共Re . 17兲and no he 共 ypical兲RT phase ␣ =2 ␲ . Al hough his ou come is somewha o a su p ise, i is s aigh o wa d o show ha , indeed, his is he co ec esul .17共a兲 De ini ion: In wha ollows we e e o he o dina y JT/RT in e sec ions as he symme ical in e sec ions and he C1con igu a ion 共 o which he plana /axial symme y is no main ained兲as he nonsymme ical in e sec ion. The p esen a icle is de o ed o he 共 i s -o -i s-kind兲 s udy o opological e ec s a in e sec ions o med by non- symme ical con igu a ions. This s udy is ca ied ou along wo lines. 共1兲In he i s pa we concen a e on he wo-s a e opo- logical ea u es as o med by he lowe s a es 1 2A⬙and he 1 2A⬘共which in he linea , poin g oup D⬁hcon igu- a ion co espond o he wo degene a e X2⌸u s a es15共b兲兲and he wo uppe s a es 2 2A⬙and 2 2A⬘ a兲Au ho o whom co espondence should be add essed. Elec onic mail: [email p o ec ed] THE JOURNAL OF CHEMICAL PHYSICS 127, 144108 共2007兲 0021-9606/2007/127共14兲/144108/7/$23.00 © 2007 Ame ican Ins i u e o Physics127, 144108-1 Downloaded 12 Ma 2008 o 129.187.254.46. Redis ibu ion subjec o AIP license o copy igh ; see h p://jcp.aip.o g/jcp/copy igh .jsp 共which in he linea , poin g oup D⬁hcon igu a ion, co espond o he wo degene a e X2⌸Gs a es15共b兲兲.We in end o show ha hese wo pai s o s a es yield, o he p esen nonsymme ical case, he cha ac e is ic JT phase, namely, ␣ = ␲ . 共2兲I is well known ha opological e ec s a e o med in he icini y o a poin o degene acy be ween wo ad- jacen s a es 共poin s o degene acy a e be e known as he poin s o conical/pa abolic in e sec ions兲. Howe e , wo s a es may no be enough o o m opological e - ec s o la ge egions in con igu a ion space17,25,27,34,35 共see also Re . 33, Chaps. 1 and 4兲. The e o e, i we a e in e es ed in s udying he numbe o in e sec ions and hei cha ac e in an ex ended egion we ha e o con- side nume ous s a es simul aneously. Fo his pu pose, some ime ago, we in oduced he adiaba ic- o-diaba ic T ans o ma ion 共ADT兲ma ix A共s兩⌫兲共Re . 36兲共he e, s s ands o he collec ion o nuclea coo dina es and ⌫is he con ou ha de ines he bounda ies o he egion兲 and he co esponding opological ma ix D共⌫兲共Re . 25兲which yields he equi ed in o ma ion. As is known bo h ma ices a e calcula ed along he con ou ⌫bu whe eas A共s兩⌫兲is calcula ed along an open con ou , he Dma ix is calcula ed along closed con ou s 共as is elabo a ed below兲. The need o ea he opological e ec s employing ma- ices o ms he ansi ion om an Abelian phenomenon 共cha ac e ized by wo s a es兲 o a non-Abelian phenomenon o med by se e al 共⬎2兲s a es37 共see also Re . 33, Chap. 1兲. We in end o show ha he analy ical ools de eloped o s udying he JT mul is a e in e sec ions25,27 apply also o he nonsymme ical mul is a e in e sec ions. II. THEORETICAL COMMENTS The analy ical ools ha we e men ioned ea lie a e all based on he BO-NACT, ␶ jk共s兲, a e m ha couples s a es j and kand is gi en in he o m1共see also Re . 33, Chap. 1兲 ␶ jk共s兲=具 ␨ j共se兩s兲兩ⵜ ␨ k共se兩s兲典.共1兲 He e ␨ i共se兩s兲;i=j,ka e he elec onic 共adiaba ic兲BO eigen- unc ions, seand ss and o he collec ion o elec onic co- o dina es and nuclea coo dina es, espec i ely, and ⵜis he 共nuclea 兲g ad ope a o . A his s age we ha e o be mo e speci ic ega ding he nuclea coo dina e s. Ou sys em is de ined in e ms o ou a oms: wo ixed ca bons ha o m he axis and wo o -axis hyd ogens H1and H2. He e H1is clamped a a dis ance q1 om he CC axis and H2, which is loca ed a a dis ance q2 om his axis, is ee o mo e a ound ha axis 共see sche- ma ic d awing in Fig. 1兲. In wha ollows we concen a e on he pola coo dina es o his, ee, a om, namely, 共 ␸ ,q2兲.Asa esul sis w i en in he ollowing o m: s⬅共 ␸ ,q2兩q1,R兲, whe e Ris he CC dis ance 共which is ixed h oughou his a icle兲. Ou o he a ious possible componen s o ␶ jk共 ␸ ,q2兩q1,R兲, he only one o in e es o us is he compo- nen angen ial o he con ou along which he in eg a ion is pe o med.27共a兲As men ioned ea lie , in ou case he con ou is a ci cle a ound a cen e loca ed a he CC axis, and con- sequen ly he componen o in e es is he ele an angula ␸ componen , which can be w i en as 共1/q兲 ␶ ␸ jk共 ␸ ,q2兩q1,R兲, whe e ␶ ␸ jk共 ␸ ,q2兩q1,R兲is ␶ ␸ jk共 ␸ ,q2兩q1,R兲= 冓 ␨ j共se兩 ␸ ,q2兩q1,R兲 冔 ⫻冏 ⳵ ⳵␸␨ k共se兩 ␸ ,q2兩q1,R兲 冔 .共2兲 In o de o simpli y ou no a ion we d op he a iable Rbu we s ill e ain he wo dis ances 共 adii兲共q1,q2兲. Ha ing in oduced he angula NACT he co esponding Ama ix, o med along his ci cula con ou , can be shown o be o he o m 27共a兲共see also Re . 33, Chap. 4兲 A共 ␸ ,q2兩q1兲=㜷exp 冉 − 冕 0 ␸ d ␸ ␶ ␸ 共 ␸ ,q2兩q1兲 冊 ,共3兲 whe e ␶ ␸ 共 ␸ ,q2兩q1兲is he ma ix o he elemen s ␶ ␸ jk共 ␸ ,q2兩q1兲,㜷is he o de ing ope a o ha eminds us ha in o de o e alua e he exponen ia ed ma ix i has o be done in a ce ain o de because o di icul ies due o he commu a ion ela ions. I is impo an o pay a en ion ha he dimension o he Ama ix 共and la e , o he Dma ix兲is iden ical o he dimension o he ␶ ma ix. Commen : In wha ollows we conside only he angula componen o ␶ . Consequen ly we d op he subsc ip ␸ so FIG. 1. 共Colo online兲Ene gy cu es o he i e lowe s a es o he C2H2 + ion p esen ed as a unc ion ␸ o he case 共q1,q2兲=共0.1,0.8兲Å. This calcu- la ion is done o he C1poin g oup whe e a om H1is clamped a a poin 共namely, a q1=0.1 Å兲and H2is allowed o o a e a ound he CC axis 共see schema ic d awing兲, whe e he adius o he ci cula con ou is q2=0.8 Å. The lowe po en ial cu es a e o he 1 2A⬙and 1 2A⬘s a es 共which in he linea poin g oup, D⬁hcon igu a ion, co espond o he wo degene a e X2⌸us a es兲, he uppe cu es a e o he 2 2A⬙and 2 2A⬘s a es 共which in he linea , poin g oup D⬁hcon igu a ion, co espond o he wo degene a e X2⌸gs a es兲and he hi d 共middle兲cu e is o an A⬘s a e ha has i s o igin in he isola ed D⬁h,12⌺g +con igu a ion. 144108-2 Halász, Vibók, and Bae J. Chem. Phys. 127, 144108 共2007兲 Downloaded 12 Ma 2008 o 129.187.254.46. Redis ibu ion subjec o AIP license o copy igh ; see h p://jcp.aip.o g/jcp/copy igh .jsp ha ␶ and ␶ jk s and, om now on, o ␶ ␸ and ␶ ␸ jk, espec- i ely. Nex is p esen ed he opological ma ix D共again de ails a e gi en in Re . 27共a兲兲, D共q2兩q1兲=㜷exp 冉 − 冕 0 2 ␲ d ␸ ␶ 共 ␸ ,q2兩q1兲 冊 .共4兲 Ou in ensi e in e es in he Dma ix is due o he ac ha in case he abo e men ioned Ns a es o m a quasi-Hilbe sub- space 共in he egion o in e es 兲, he ma ix Dis diagonal. In o he wo ds whene e a N⫻NDma ix becomes diagonal 共wi h ±1 along he diagonal in he case o eal eigen unc- ions兲, his ac implies ha he Ns a es o m, app oxima ely, an isola ed 共namely, decoupled兲g oup o s a es 共app oxi- ma ely兲independen o he es o he s a es ha o m he Hilbe space25,27共a兲共see also Re . 33, Chap. 2.1.3兲.I isim - po an o ealize ha he dimension o he Dma ix 共as well as o he Ama ix兲is de e mined by he dimension o he ␶ ma ix. In o he wo ds, all h ee o hem a e o dimension N⫻N. In he case ha N=2, Eqs. 共3兲and 共4兲simpli y signi i- can ly because any 2⫻2 o hogonal ma ix can be w i en in e ms o 36共a兲 A共2兲共 ␸ ,q2兩q1兲 = 冉 cos共 ␥ 12共 ␸ ,q2兩q1兲兲 sin共 ␥ 12共 ␸ ,q2兩q1兲兲 − sin共 ␥ 12共 ␸ ,q2兩q1兲兲 cos共 ␥ 12共 ␸ ,q2兩q1兲兲 冊 .共5兲 In his p esen a ion, ␥ 12共 ␸ ,q2兩q1兲is e med as he ADT angle and is gi en as an 共line兲in eg al,36共a兲 ␥ 12共 ␸ ,q2兩q1兲= 冕 0 ␸ ␶ 12共 ␸ ⬘,q2兩q1兲d ␸ ⬘.共6兲 A simila exp ession applies o ␣ 12共q2兩q1兲, he opological 共Be y兲phase. Thus,25,27共a兲 ␣ 12共q2兩q1兲= 冕 0 2 ␲ ␶ 12共 ␸ ,q2兩q1兲d ␸ .共7兲 The co esponding Dma ix is simila o he Ama ix as gi en in Eq. 共5兲bu whe e ␣ 12共q2兩q1兲 eplaces ␥ 12共 ␸ ,q2兩q1兲, namely, D共2兲共q2兩q1兲= 冉 cos共 ␣ 12共q2兩q1兲兲 sin共 ␣ 12共q2兩q1兲兲 − sin共 ␣ 12共q2兩q1兲兲 cos共 ␣ 12共q2兩q1兲兲 冊 .共8兲 I is well no iced ha he condi ion o he Dma ix o be diagonal is ha ␣ 12共q2兩q1兲=2 ␲ n,共9兲 whe e nis an in ege o hal an in ege . Commen : F om p e ious s udies we know ha a g oup o Ns a es yields a single- alued diaba ic po en ial i and only i his g oup is app oxima ely isola ed om he es o he s a es ha o m he comple e Hilbe space. This happens when he NACTs o he ype ␶ jk, ha couple any s a e j wi hin he g oup o Ns a es wi h any s a e kou side ha g oup, a e negligibly small, i.e., 兩 ␶ jk兩⬃O共␧兲; o j艋Nand k艌N. In such a case his g oup o Ns a es o ms a Hilbe subspace 共mo e abou his issue can be ound in Re s. 27 and 33, see Chap. 1兲. III. RESULTS In acco dance wi h he goals p esen ed in he in oduc- ion, we conside wo-s a e, h ee-s a e, and i e-s a e esul s. 共1兲As o he wo-s a e case we concen a e on he wo lowe s a es 1 2A⬙and 1 2A⬘共 ha o igina e om he wo lowe degene a e X2⌸us a es15共b兲and on he wo uppe s a es 共 he ou h and he i h s a es兲, namely, he s a es ha a e labeled as 2 2A⬙and 2 2A⬘and ha e hei o igin in he wo 共highe 兲 degene a e X2⌸gs a es.15共b兲We in end o show ha bo h he wo lowe s a es and he wo uppe s a es yield he opologi- cal phase ␣ 共q兲⬃ ␲ 关see Eq. 共9兲兴.共2兲As o he h ee-s a e sys em 共which is a non-Abelian sys em兲we concen a e on he ele an Dma ix 关see Eq. 共4兲兴and show ha in hose cases whe e he 2⫻2Dma ix ails o be diagonal, he 3 ⫻3Dma ix akes o e and sa is ies he diagonali y equi e- men . 共3兲Finally, we also p esen he diagonal elemen s o he 5⫻5Dma ix jus o show ha hey a e consis en wi h he diagonal elemen s o he ele an 3⫻3Dma ices. The calcula ion o he ene gy cu es and he 共angula 兲 NACTs is ca ied ou a he s a e-a e age comple e ac i e space sel -consis en ield 共CASSCF兲le el, employing he ollowing basis unc ions: Fo he ca bons as well as o he hyd ogens, we applied s,p, and d unc ions. We used he ac i e space, including all nine alence elec ons dis ibu ed on en o bi als. Fi e elec onic s a es we e compu ed by he s a e-a e age CASSCF le el using he 6-311G** basis se wi h equal weigh s. In ce ain cases hese calcula ions we e epea ed wi h h ee/ ou s a es o check o con e gence. All calcula ions we e done o he ollowing collinea con igu a- ion: he C–C dis ance: R=RCC=1.254 Å and he 共 wo兲H–C dis ances: RHC=1.080 Å. The nume ical ea men is ca ied ou employing he MOLPRO p og am.38 Figu es 1and 2p esen ed he i e lowe po en ial ene gy cu es ela ed o he nonsymme ical s a es o med by wo o -axis hyd ogens, H1and H2. He e H1is clamped a a dis ance q1 om he CC axis and H2, which is loca ed a a dis ance q2 om his axis, is ee o mo e a ound ha axis. I is well no iced ha he wo lowe cu es a e a ibu ed o 12A⬙and 1 2A⬘s a es; he hi d cu e is he po en ial ene gy cu e o he s a e A⬘ ha has i s o igin in he isola ed D⬁h, 12⌺g +con igu a ion and he wo uppe cu es a e ela ed o he 2 2A⬙and 2 2A⬘s a es. The po en ial cu es in Fig. 1we e de i ed o he case ha 共q1,q2兲=共0.1,0.8兲Å, and hose o 共q1,q2兲 =共1.0,1.0兲Å a e p esen ed in Fig. 2. In wha ollows he i s se o 共nonsymme ical兲q’s is e e ed o as case 1 and he second as case 2. Fo easons o con enience we label he i e s a es ac- co ding o hei ene gies a ␸ =0, as s a e 1, s a e 2, e c. Thus, o ins ance, ␶ 23共 ␸ 兩q1,q2兲is he NACT be ween s a e 2 共in ou case 1 2A⬙兲and s a e 3 共in ou case A⬘兲, e c. A. The opological „Be y…phases Figu e 3p esen s h ee ␸ -dependen NACTs o ␶ 45共 ␸ 兩q1,q2兲, as calcula ed o h ee di e en alues o q 144108-3 Topological e ec s: C2H2 +case s udy J. Chem. Phys. 127, 144108 共2007兲 Downloaded 12 Ma 2008 o 129.187.254.46. Redis ibu ion subjec o AIP license o copy igh ; see h p://jcp.aip.o g/jcp/copy igh .jsp whe e q=q1=q2. A simila igu e bu o ␶ 12共 ␸ 兩q1,q2兲is gi en in Fig. 2共a兲, as published in Re . 17共a兲. The calcula- ions a e ca ied ou o q=0.25,0.5,0.8 Å. I is well no iced ha he h ee cu es a e s ongly oscilla ing unc ions 共in pa icula , he one o q=0.8 Å兲in he icini y o he alue ␶ 45共 ␸ 兩q1,q2兲=0.5 ad−1 共a ac ha implies ha he co e- sponding opological phases a e expec ed o be ␣ 45共 ␸ 兩q1,q2兲⬃ ␲ e en in ex eme cases兲. The ele an Be y phases o bo h ␶ 12 and ␶ 45 a e summa ized in Table I.I is no iced ha he la ge he qis, he mo e signi ican is he de ia ion o he co esponding phase, ␣ 共 ␣ 12 o ␣ 45兲 om ␲ . This dependence on q, which is ypical o he symme ical JT and RT in e sec ions, is due o inc easing dis u bances caused by nea by s a es. I is no iced ha in case o q =0.8 Å, he dis u bances a e so signi ican ha NACTs, due o h ee s a es, ha e o be included in he calcula ions in o de o es o e he diagonali y o he Dma ix 共see nex sec ion兲. Ano he in e es ing ac ha esul s om his s udy is ha in bo h cases, he Be y phases 共namely, ␣ 12 and ␣ 45兲a e equal o ␲ 共like in he JT case兲and no o 2 ␲ 共like in he RT case which is cha ac e ized by an axial symme y兲. B. Abelian e sus non-Abelian sys ems o nonsymme ic con igu a ions space As al eady discussed in he In oduc ion he main em- phasis in his a icle is on he abili y o he gene al heo y 共 ha was success ully applied in he symme ical JT and RT si ua ions兲 o emedy o he cases whe e he wo-s a e sys- em ails o yield opological phases close enough o an in- ege mul iple o ␲ 共 hus 2⫻2 diagonal Dma ices兲.We in end o show ha in such a si ua ion adding a hi d s a e, jus like in he symme ical cases, imp o es signi ican ly he opological ea u es, as e ealed by he Dma ices. We s a by p esen ing he NACTs o he ype ␶ ij共 ␸ 兩q1,q2兲as a unc ion o ␸ o di e en pai s o 共q1,q2兲. Figu es 4–6p esen he NACTs ␶ 12, ␶ 23, and ␶ 13 共namely, NACTS ela ed o he lowe s a es兲and in Fig. 7a e p e- sen ed NACTs o ␶ 45, ␶ 53, and ␶ 43 共namely, NACTS o he uppe s a es兲. The alues o q1and q2a e inse ed in each igu e. I is no iced ha in all ou cases bo h ␶ 12 and ␶ 45 ha e alues ha oscilla e in he icini y o ␶ ⬃0.5 ad−1, whe eas he o he ␶ -ma ix elemen s, namely, ␶ 13, ␶ 23, ␶ 34, and ␶ 35 ha e, on he a e age, much smalle alues. Among hose we see ha he NACTs ela ed o wo uppe s a es, i.e., ␶ 34 and ␶ 35, a e much smalle han he co esponding ones ela ed o he lowe s a es, i.e., ␶ 23 and ␶ 34. This ac has implica ions ega ding he possibili y o he h ee-s a e Dma ix o em- edy he nondiagonali y o he wo-dimensional Dma ix as discussed nex . Ha ing he NACTs we a e now in a posi ion o calcula e he diagonal elemen s o he a ious D-ma ix elemen s. In Table II a e compa ed esul s as calcula ed o he wo-s a e case 共 hi d column兲and he h ee-s a e case 共columns 4, 5, and 6兲. I is no iced ha when he elemen s D11 共2兲共q1,q2兲 共⬅D22 共2兲共q1,q2兲兲 a e signi ican ly di e en om 共−1兲,some- imes bu no always, he ele an elemen s o he h ee-s a e FIG. 2. 共Colo online兲Ene gy cu es o he i e lowe s a es o he C2H2 + ion p esen ed as a unc ion ␸ o he case 共q1,q2兲=共0.8,0.8兲Å. The es , see Fig. 1. FIG. 3. 共Colo online兲Ab ini io C1 ␸ -dependen nonadiaba ic coupling e m, ␶ ␸ 45共 ␸ 兩q1,q2兲. Resul s o he case q1=q2共=q兲;共---兲q=0.25 Å; 共---兲q =0.5 Å; 共¯¯兲q=0.8 Å. TABLE I. The opological 共Be y兲phases ␣ 12, as calcula ed o he s a es 12A⬙and 1 2A⬘, and ␣ 45, as calcula ed o he s a es 2 2A⬙and 2 2A⬘along ci cles wi h di e en adii 共q兲. q共Å兲 ␣ 12 共 ad兲cos ␣ 12 ␣ 45 共 ad兲cos ␣ 45 0.25 3.105 −0.9993 3.109 −0.9995 0.50 3.010 −0.991 3.103 −0.9992 0.80 2.858 −0.960 2.812 −0.946 144108-4 Halász, Vibók, and Bae J. Chem. Phys. 127, 144108 共2007兲 Downloaded 12 Ma 2008 o 129.187.254.46. Redis ibu ion subjec o AIP license o copy igh ; see h p://jcp.aip.o g/jcp/copy igh .jsp Dma ix, namely, Djj 共3兲共q1,q2兲;j=1,2,3 a eclose o ±1. We emphasize he wo d some imes because his is ue o he lowe s a es bu no o he uppe ones. In he case o he lowe s a es we see ha he ex ension o h ee s a es, indeed, imp o ed signi ican ly he diagonali y o he Dma ix 共namely, lead o diagonal alues close o ±1兲. The ac ha in he case o he uppe s a es no imp o emen is achie ed is due o wo easons. 共a兲The wo addi ional NACTs, ␶ 34 and ␶ 35, ha e, on he a e age, alues oo small o be able o a ec he ele an Dma ix. 共b兲I is mos likely ha NACTs o med by highe s a es 共i.e., j⬎5兲a ec he wo- dimensional NACTs, bu hey a e no included in he calcu- la ion. Nex we commen on he posi ions o he minus signs along he diagonal o he h ee-s a e Dma ix. I is seen ha in he case o he lowe s a es, he minus signs a e a ached o he i s wo elemen s, namely, Djj,j=1,2 which implies ha he ci cula con ou su ounds a degene acy poin be- ween s a es 1 and 2 共 he lowe s a es ou o i e兲. In con as FIG. 4. 共Colo online兲Ab ini io C1 ␸ -dependen nonadiaba ic coupling e m, ␶ ␸ ij共 ␸ 兩q1,q2兲. Resul s o he case 共q1,q2兲=共0.1,0.8兲Å: 共---兲 ␶ ␸ 12;共---兲 ␶ ␸ 13 共·····兲 ␶ ␸ 23. FIG. 5. 共Colo online兲Ab ini io C1 ␸ -dependen nonadiaba ic coupling e m, ␶ ␸ ij共 ␸ 兩q1,q2兲. Resul s o he case 共q1,q2兲=共0.8,0.8兲Å: 共---兲 ␶ ␸ 12;共---兲 ␶ ␸ 13 共·····兲 ␶ ␸ 23. FIG. 6. 共Colo online兲Ab ini io C1 ␸ -dependen nonadiaba ic coupling e m, ␶ ␸ ij共 ␸ 兩q1,q2兲. Resul s o he case 共q1,q2兲=共1.0,1.0兲Å: 共---兲 ␶ ␸ 12;共---兲 ␶ ␸ 13 共·····兲 ␶ ␸ 23. FIG. 7. 共Colo online兲Ab ini io C1 ␸ -dependen nonadiaba ic coupling e m, ␶ ␸ ij共 ␸ 兩q1,q2兲. Resul s o he case 共q1,q2兲=共0.8,0.8兲Å: 共---兲 ␶ ␸ 45;共---兲 ␶ ␸ 35 共·····兲 ␶ ␸ 23. 144108-5 Topological e ec s: C2H2 +case s udy J. Chem. Phys. 127, 144108 共2007兲 Downloaded 12 Ma 2008 o 129.187.254.46. Redis ibu ion subjec o AIP license o copy igh ; see h p://jcp.aip.o g/jcp/copy igh .jsp o his, he minus sign in he case o he uppe s a es is a ached o he wo las elemen s Djj,j=4,5 which implies ha he ci cula con ou su ounds a degene acy poin be- ween s a es 4 and 5 共 he uppe s a es ou o i e兲. Fo comple eness we also calcula ed he diagonal ele- men s o one co esponding 5⫻5Dma ix 关a calcula ion based on all elemen s o he ype ␶ jk共 ␸ 兩q1,q2兲;j,k=兵1,5其兴, and hese a e p esen ed in Table III. I is no iced ha he minus signs a e a ached o he i s wo elemen s and he las wo elemen s consis en wi h posi ions o he minus signs in he h ee-s a e D-ma ix elemen s as discussed in he p e i- ous pa ag aph. IV. DISCUSSION AND CONCLUSIONS In his a icle we s udied, o he i s ime, a molecula sys em which is no es ic ed by any symme y condi ions. This ac leads o a si ua ion whe e any s a e in he elec onic mani old is coupled o any o he s a e. Limi ing ou sel es o a gi en egion in con igu a ion space, he ques ion is whe he he app oach ha o iginally de eloped o ea he JT e ec and ecen ly, success ully, ex ended o ea he RT e ec is applicable also o ea C1poin g oups, namely, con igu a- ions ha a e no con olled by any symme y condi ions. To ge an answe o his ques ion we conside ed he e a-a omic sys em C2H2 +, whe e he wo ca bons o m he axis o he molecule and he wo o -axis hyd ogens beha e as ollows. One hyd ogen is clamped o a ixed poin , while he second hyd ogen is allowed o su ound he CC axis, which se es as an axis o o a ion 共in o he wo ds he mo - ing hyd ogen ollows a ci cula con ou 兲. A any poin along he con ou —excep a ␸ =0, ␲ — he ou a oms a e no e- s ic ed by any symme y condi ions. Ou s udies concen a e on opological e ec s o med by ollowing he jus men- ioned closed con ou s. Ou main indings a e as ollows. 共1兲Al oge he we s udied he i e lowe s a es o he C2H2 + ion, all o hem ela ed o he C1poin g oup: Fo he lowe pai o s a es, 1 2A⬙and 1 2A⬘, and he uppe pai o s a es, 2 2A⬙and 2 2A⬘, we e ealed, in each case, a poin o degene acy, mos likely loca ed on he CC axis. In o he wo ds, we es ablished ha he CC axis which is known o con ain he poin s o degene acy ha o m he RT e ec con ains poin s o degene acy ha a e o med by s a es belonging o he C1poin g oup. The main di e ence be ween he wo ypes is ha whe eas he Be y phase o he RT e ec is 2 ␲ , he Be y phase o his C1poin g oup is ␲ . 共2兲I is well known ha o hose si ua ions whe e a wo- s a e ea men 共which yields he opological-Be y phase兲b eaks down, we a e o ced o employ he non- Abelian app oach 共 he ex ended mul is a e e sion兲 which is based on he opological Dma ix as in o- duced in Eq. 共4兲and which was success ully applied whene e we s udied he 共symme ical兲JT o he RT e ec s. In he p esen a icle we show ha he same non-Abelian app oach applies also o he p esen , non- symme ical, C1poin g oup. We ind ha whene e he adius o he con ou ge s oo la ge 共i.e., ⬎0.6 Å兲, he Abelian wo-s a e e sion has o be eplaced by he non-Abelian Dma ix. The impo ance o his s udy is in e ealing he ac ha a heo y ha was de ised o a symme ical si ua ion 共a si u- a ion ha is only a ely encoun e ed in eali y兲applies well o he mos gene al molecula con igu a ion. ACKNOWLEDGMENTS The au ho s acknowledge he US-Is ael Bi-na ional Sci- ence Founda ion o pa ly suppo ing his s udy. One o he au ho s 共A.V.兲acknowledges he 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. Ano he au ho 共M.B.兲would like o hank P o esso A. Mebel o aluable discussions ega ding he elec onic s uc u e o he C2H2 +ion. 1M. Bo n and J. R. Oppenheime , Ann. Phys. 84, 457 共1927兲; M. Bo n, Fes sch i Gö ingen Nach. Ma h. Phys. K1,1共1951兲; M. 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