A tri-atomic Renner-Teller system entangled with Jahn-Teller conical intersections

A i-a omic Renne -Telle sys em en angled wi h Jahn-Telle conical in e sec ions A. Csehi, A. Bende, G. J. Halász, Á. Vibók, A. Das e al. Ci a ion: J. Chem. Phys. 138, 024113 (2013); doi: 10.1063/1.4773352 View online: h p://dx.doi.o g/10.1063/1.4773352 View Table o Con en s: h p://jcp.aip.o g/ esou ce/1/JCPSA6/ 138/i2 Published by he Ame ican Ins i u e o Physics. Addi ional in o ma ion on J. Chem. Phys. Jou nal Homepage: h p://jcp.aip.o g/ Jou nal In o ma ion: h p://jcp.aip.o g/abou /abou _ he_jou nal Top downloads: h p://jcp.aip.o g/ ea u es/mos _downloaded In o ma ion o Au ho s: h p://jcp.aip.o g/au ho s Downloaded 14 Jan 2013 o 129.206.21.195. Redis ibu ion subjec o AIP license o copy igh ; see h p://jcp.aip.o g/abou / igh s_and_pe missions THE JOURNAL OF CHEMICAL PHYSICS 138, 024113 (2013) A i-a omic Renne -Telle sys em en angled wi h Jahn-Telle conical in e sec ions A. Csehi,1A. Bende,2G. J. Halász,1Á. Vibók,3A. Das,4D. Mukhopadhyay,4 and M. Bae 5,a) 1Depa men o In o ma ion Technology, Uni e si y o Deb ecen, H-4010 Deb ecen, P.O. Box 12, Hunga y 2Molecula and Biomolecula Physics Depa men , Na ional Ins i u e o Resea ch and De elopmen o Iso opic and Molecula Technologies, Cluj-Napoca, Romania 3Depa men o Theo e ical Physics, Uni e si y o Deb ecen, H-4010 Deb ecen, P.O. Box 5, Hunga y 4Depa men o Chemis y, Uni e si y o Calcu a, Kolka a 700 009, India 5The 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 (Recei ed 8 Sep embe 2012; accep ed 11 Decembe 2012; published online 14 Janua y 2013) The p esen s udy concen a es on a si ua ion whe e a Renne -Telle (RT) sys em is en angled wi h Jahn-Telle (JT) conical in e sec ions. S udies o his ype we e pe o med in he pas o con ou s ha su ound he RT seam loca ed along he collinea axis [see, o ins ance, G. J. Halász, Á. Vibók, R. Bae , and M. Bae , J. Chem. Phys. 125, 094102 (2006)]. The p esen s udy is cha ac e ized by plana con ou s ha in e sec he collinea axis, hus, o ming a unique ype o RT-non-adiaba ic cou- pling e ms (NACT) exp essed in e ms o Di ac-δ unc ions. Consequen ly, o calcula e he equi ed adiaba ic- o-diaba ic (mixing) angles, a new app oach is de eloped. Du ing his s udy we e ealed he exis ence o a no el molecula pa ame e , η, which yields he coupling be ween he RT and he JT NACTs. This pa ame e was ound o be a pu e numbe η=2√2/π (and he e o e independen o any pa icula molecula sys em) and is designa ed as Renne -Jahn coupling pa ame e . The p esen s udy also e eals an unexpec ed esul o he ollowing kind: I is well known ha each (comple e) g oup o s a es, esponsible o ei he he JT-e ec o he RT-e ec , o ms a Hilbe space o i s own. Howe e , he en anglemen be ween hese wo e ec s o ms a hi d e ec , namely, he RT/JT e ec and he s a es ha ake pa in i o m a di e en Hilbe space. © 2013 Ame ican Ins i u e o Physics. [h p://dx.doi.o g/10.1063/1.4773352] I. INTRODUCTION This a icle is one addi ional link in a se ies o a icles1–4 de o ed o he p oblem o e ealing igo ous, e icien , and ac- cu a e me hods o cons uc diaba ic po en ial ene gy su aces (PES) o mul i-s a e, poly-a omic molecula sys ems. In he pas , mos o such a icles we e de o ed o he calcula ion o diaba ic PESs o limi ed egions in con igu a ion space (CS), mainly, plana con ou s5–12 (see also a lis o s udies in Re . 4). Ex ending he a ailable app oaches o he equi ed chemical olume, so ha s udies o chemical p ocesses can be acili- a ed, was and s ill a o midable ask. The main di icul y is associa ed wi h he single- aluedness o hese diaba ic PESs, which is di icul o gua an ee e en o simple CSs such as planes once hey ex end o la ge sizes. The se e i y o his issue inc eases signi ican ly i one is in e es ed in s udying chemical exchange p ocesses ha equi e wo o mo e a - angemen channels. Recen ly, while s udying he HHF sys em, we managed o o e come, pa ially, his di icul y by in oducing an ap- p oxima e app oach ha ul ills wo basic equi emen s: (a) I yields single- alued diaba ic po en ials o an a bi a y la ge plane; (b) i is igo ous in he sense ha i o igina es om he Bo n-Oppenheime (BO) ea men .13,14 This app oxima- a)Email: [email p o ec ed]. ion is based on complemen a y con ou s and by ac i a ing i we calcula ed plana PESs o he HHF sys em o a (R, θ) g id (see Fig. 1) co e ing he ange 2.8 <R<10 a.u. and −π/2 <θ<+π/2, espec i ely.4The planes unde con- side a ion a e o med by he h ee a oms (a luo ine and wo hyd ogens in ou case) and a e pa ame e ized ia al- ues o , he in e -a omic dis ance o H2.The(R,θ)g id poin s desc ibe he pola coo dina es o he F-a om wi h espec o he cen e -o -mass o he H2molecule on ha plane. In he p esen a icle, we conside a mo e ad anced ap- p oach which is expec ed o yield diaba ic PESs wi h highe accu acy.3I is based on h ee (o mo e) in e ac ing s a es and he e o e equi es ea ing he adiaba ic- o-diaba ic ans- o ma ion (ADT) ma ices, A(s),5,12 a he han he o di- na y ADT (mixing) angles, γ(s) (see Eq. (1) in Re . 4). As is well known, he li e a u e con ains nume ous s ud- ies based on hese ma ices in connec ion wi h a ious di - e en , i-a omic, e a-a omic, and poly-a omic molecula sys ems.6(a),9,11,15,16(a),17,18 These s udies a e di ided in o wo gene al ca ego ies based on he ype o non-adiaba ic cou- pling e ms (NACT) included in he s udy: (a) ADT ma ices o med by Jahn-Telle (JT) NACTs;6(a),9,11,15 (b) ADT ma i- ces o med by a mix u e o bo h Renne -Telle (RT) NACTs and JT NACTs.16(a),17,18(b),18(c) This pa i ioning is somewha a i icial (in pa icula he exis ence o “pu e” JT NACTs) 0021-9606/2013/138(2)/024113/11/$30.00 © 2013 Ame ican Ins i u e o Physics138, 024113-1 Downloaded 14 Jan 2013 o 129.206.21.195. Redis ibu ion subjec o AIP license o copy igh ; see h p://jcp.aip.o g/abou / igh s_and_pe missions 024113-2 Csehi e al. J. Chem. Phys. 138, 024113 (2013) FIG. 1. A schema ic pic u e o he sys em o coo dina es: (a) Posi ions o a oms, poin o (1,2) ci and he cen e o all ci cula con ou s. (b) Sys em o coo dina es: (R,θ| ) s. (ϕ,q| ). In his igu e, a oms F, H1,andH 2s and o a oms A, B, and C, espec i ely, men ioned in he ex . because molecula sys ems always con ain bo h ypes o NACTs. Since ou aim is de i ing diaba ic PESs o he equi ed chemical olume we ha e o gua an ee ha he app oach o be de eloped ea s he wo kinds o NACTs. This was achie ed in he jus men ioned s udies16(a),17,18(b),18(c) in which he ADT ma ices a e calcula ed employing line in eg als along con- ou s ha su ound he collinea axis, whe e he NACTs ma- ices con ain bo h he RT and he JT NACTs (see also Re . 16(b)). The main sho coming o his app oach is ha he ADT ma ices a e calcula ed o g ids on a se ies o planes pe pen- dicula o he collinea axis ins ead o he equi ed g id on he i-a omic plane. (We emind he eade ha dynamical ea - men s a e ca ied ou on i-a omic planes.) In o he wo ds, his app oach demands in ica e ans o ma ions om hese nume ous g ids o he i-a omic g id—a p ocess ha se e ely complica es he dynamical ea men (may be makes i e en, al oge he , in easible). In wha ollows, we sugges calcula - ing he ADT ma ices o he i-a omic g id di ec ly. Ano he eason o his choice is ha , o each alue o he ib a- ional coo dina e , such a plane con ains all JT cisaswellas he collinea axis ha con ains all RT degene acy poin s and he e o e he co esponding plana g id poin s a e exposed o all he opological e ec s. Howe e , he e is s ill one hu dle o o e come. Since all con ou s a e assumed o be in he plane no con ou is capable o su ound his axis (which is in he plane) and he e o e a di e en way o include he RT e ec has o be ound. Based on ou pas expe ience19–21 a us ul way o include he RT e ec is o le he con ou s in e sec he degene acy line and in his way o enable he co esponding NACT ma ix ele- men s o pick up he esul ing RT e ec . The only p oblem en- coun e ed he e is ha hese calcula ed NACTs a e ex emely spiky— eminiscen o he Di ac δ- unc ion (see, e.g., Fig. 2 in Re . 21(a))—and he e o e hei co ec shape is equen ly missed. This si ua ion opened up he way o a heo e ical/ ma hema ical s udy acco ding o which hese NACTs a e indeed, up o a no maliza ion ac o , pu e Di ac δ- unc ions.19,21(b) The heo e ical indings o his s udy a e in- co po a ed in he p esen s udy (see Eqs. (11a) and (11b)). The a icle is a anged in he ollowing way: In Sec. II is gi en he heo e ical backg ound which concen a es on he NACT ma ices ( o he i-s a e and he e a-s a e cases), on he co esponding ex ended (p i ileged) wo-s a e ADT (mix- ing) angles and inally e e s o he de i a ion o he Renne - Jahn pa ame e η,3in Sec. III a e p esen ed he calcula ions and in Sec. IV is gi en he discussion and summa y o he esul s. We also men ion Appendix Awhich b ie ly discusses he connec ion be ween he o iginal Renne heo y and he p esen RT NACT. II. THEORETICAL BACKGROUND A. In oduc o y ema ks Ou app oach is based on sol ing he ollowing mul i- dimensional i s o de di e en ial equa ion:5,12 ∇A(s)+τ(s)A(s)=0,(1) whe e A(s), as p e iously men ioned, is he ADT ma ix, τ(s) is an an i-symme ic ma ix ha con ains he abo e men ioned ec o ial NACTs and sis a a iable ha p esen s he collec- ion o in e nal nuclea coo dina es. The ma ix τ(s), which equen ly con ains singula elemen s, appea s ( oge he wi h he adiaba ic, diagonal PES, u(s)) in he nuclea Sch ödinge equa ion (SE) ollowing he BO ea men .13,14 One way o a oid hese singula i ies is o elimina e he τ(s)-ma ix and o m a modi ied SE ee o all singula i ies bu go e ned by V(s), a ull po en ial ma ix, which eplaces he o iginal, di- agonal ma ix, u(s). The wo po en ial ma ices a e ela ed ia he ollowing ans o ma ion:5,12 V(s)=A(s)†u(s)A(s),(2) whe e A(s)†is he complex conjuga e ma ix o A(s). The ma- ix V(s) is known as he diaba ic PES—i s diagonal elemen s a e he co esponding diaba ic po en ials and i s o diagonal elemen s o m he diaba ic coupling e ms ( eminiscing o he NACTs). The common way o sol e Eq. (1) is o assume con ou s, , and o in eg a e i along such con ou s. Since V(s) has o be p esen ed a a gi en g id o poin s we mus gua an ee ha he chosen con ou s (along which A(s) and V(s), a e calcula ed) co e e icien ly he ull co esponding CS. While doing ha we ace again he oublesome singula i ies o he ma ix τ(s) which a e also known as poin s o conical in e sec ions (ci). These cis may cause he diaba ic po en ial V(s) o be mul i- alued (namely, non-single- alued) and he e o e, essen ially, o no physical use. To o e come his di icul y, he A-ma ix (which, acco ding o Eq. (2), is esponsible o he single- aluedness o V(s)) has o be calcula ed employing nume - ous (usually 3-4) s a es so ha upon comple ion o any closed con ou in ha CS i ends up as a diagonal ma ix.22 A di e en way o ea hese ma ices is as ollows: Since A(s) is an o hogonal ma ix i can be p esen ed in e ms o Downloaded 14 Jan 2013 o 129.206.21.195. Redis ibu ion subjec o AIP license o copy igh ; see h p://jcp.aip.o g/abou / igh s_and_pe missions 024113-3 Csehi e al. J. Chem. Phys. 138, 024113 (2013) 1A’ 1A’’ 2A’ 1A’ 1A’’ 2A’ 3A’ (a) T i-s a e sys em (b) Te a-s a e sys em RT(11’’) JT(12) RT(11’’) JT(12) JT(13) JT(23) FIG. 2. Schema ic igu es desc ibing he wo RT/JT models: (a) he i-s a e RT/JT model; (b) he e a-s a e RT/JT model. Eule kind o angles18,23 and consequen ly i s diagonali y is gua an eed i and only i hese angles, a he end o a closed con ou , become in ege mul iples o π.18(a),23(b) So a we e e ed o CSs in gene al bu he abo e men- ioned plana CSs a e he ones o be conside ed o ou pu - poses. (These plana CSs a e, a a la e s age, ex ended o o m he ull equi ed chemical CS.) We emind he eade ha in hese planes he JT cis appea as isola ed poin s and he RT de- gene acy is loca ed along he collinea i-a omic axis which, ob iously, is con ained in he same plane (see Fig. 1). B. Molecula s a es and he NACT ma ix Re u ning now o he las pa ag aph o he In oduc ion ou aim is o cons uc NACT-ma ices, τ(s), which con ain he wo ypes o NACTs: he JT NACTs and he RT NACTs. This will be done o wo molecula si ua ions: (i) A sys em o h ee s a es, wo A-s a e and one A-s a e (see Fig. 2(a)); (ii) a sys em o ou s a es, h ee Aand one A (see Fig. 2(b)). 1. T i-s a e NACT-ma ix He e, he coupling be ween 1Aand 2Ais o he JT- ype and designa ed as τ12 and he coupling be ween 1Aand 1A is o he RT- ype and designa ed as τ11 (see Fig. 2(a)). We emind he eade ha he wo s a es 1Aand 1A o m a RT degene acy line along he collinea HHF axis. The NACT-ma ix akes he o m τ(s)=⎛ ⎝ 0τ11 τ12 −τ11 00 −τ12 00 ⎞ ⎠,(3) whe e we assumed ha τ21≡0. Howe e , his o m is no well sui ed o he nume ical ea men as will be elabo a ed nex . Fo easons o con e- nience, we p e e o ha e τ12 a he (1,2) posi ion o τ(s). To achie e his a angemen , we pe mu e be ween he las wo ows and hen be ween he las wo columns so ha τ(s) be- comes τ(s)=⎛ ⎝ 0τ12 τ11 −τ12 00 −τ11 00 ⎞ ⎠(3) which is he NACT-ma ix o he desi ed o m. I is impo an o men ion ha he solu ion o Eq. (1) is no a ec ed by hese pe mu a ions. 2. Te a-s a e NACT ma ix He e, he NACTs be ween 1Aand 2A, be ween 1Aand 3Aand be ween 2Aand 3Aa e o a JT- ype, designa ed as τ12,τ13, and τ23, espec i ely, and he coupling be ween 1A and 1A is, like be o e, o he RT- ype and designa ed as τ11 (see Fig. 2(b)). The NACT-ma ix akes o m τ(s)=⎛ ⎜ ⎜ ⎝ 0τ11 τ12 τ13 −τ11 000 −τ12 00τ23 −τ13 0−τ23 0 ⎞ ⎟ ⎟ ⎠ .(4) As in he p e ious case, his ma ix is no well sui ed o ou nume ical ea men . Again o eason o con enience we p e e o ha e he six JT-NACTs o be concen a ed in he up- pe 3 ×3 diagonal co ne o he 4 ×4 ma ix and he wo RT-NACTs o be a he pe iphe al posi ions. To achie e his, we pe mu e be ween he second and he hi d ows and hen be ween he second and he hi d columns, so ha τ(s) be- comes τ(s)=⎛ ⎜ ⎜ ⎝ 0τ12 τ11 τ13 −τ12 00τ23 −τ11 000 −τ13 −τ23 00 ⎞ ⎟ ⎟ ⎠ .(4) Nex , we pe mu e be ween he wo las ows and hen be ween he wo las columns so ha τ(s) becomes τ(s)=⎛ ⎜ ⎜ ⎝ 0τ12 τ13 τ11 −τ12 0τ23 0 −τ13 −τ23 00 −τ11 000 ⎞ ⎟ ⎟ ⎠ (4) which is he NACT-ma ix o he equi ed o m. Downloaded 14 Jan 2013 o 129.206.21.195. Redis ibu ion subjec o AIP license o copy igh ; see h p://jcp.aip.o g/abou / igh s_and_pe missions 024113-4 Csehi e al. J. Chem. Phys. 138, 024113 (2013) C. ADT ma ices and p i ileged angles 1. Two-s a e case and he ADT angle In he ea ly days, we used o sol e Eq. (1) by ea ing he ull A-ma ix as i s ands wi hou paying much a en ion o i s in e nal s uc u e. The only excep ional case is he wo-s a e case whe e he A-ma ix which can be exp essed in e ms o one angle γ(s) ( o be e med as he ADT o mixing angle) and his leads o he ollowing simple line in eg al:12 γ12(s|)=s s0 ds·τ12(s|).(5) He e, τ12 was in oduced ea lie , designa es he con ou along which is ca ied ou he in eg a ion and he do p esen s he scala p oduc . Since we in end o conside ci cula con- ou s only he in eg a ion can be simpli ied o become o e an angle, ϕ γ12(ϕ,q) =ϕ 0 τϕ12(ϕ,q)dϕ,(6) whe e (ϕ, q) a e pola coo dina es: q is he adius, ϕis he angle associa ed wi h he (nuclea ) o a ion. Consequen ly, he co esponding angula componen o τ12 is, as usual, p e- sen ed in he o m: (1/q)τϕ12(ϕ,q), whe e τϕ12(ϕ,q)=ζ1(se|ϕ,q) ∂ ∂ϕζ2(se|ϕ,q).(7) He e ζj(ϕ|q); j =1, 2 a e he co esponding eigen unc ions ela ed o he wo lowe s a es (in ou case he s a es 12Aand 22A). In addi ion o he ADT angle, we a e also in e es ed in he angle α12(q)— he-end-o - he-con ou phase—de ined as α12(q) =2π 0 τϕ12(ϕ,q)dϕ(8) and is known as he opological/geome ical phase. Commen : Abou wo decades ago i was sugges ed24 o iden i y he opological phase wi h he Be y phase o a wo- s a e sys em.25 This connec ion was ound o be alid o all epo ed nume ical s udies o molecula sys ems wi h wo quasi-isola ed s a es.4,15–17,21(a),26–28 In case he wo-s a e sys em o ms a Hilbe subspace he opological phase becomes an in ege mul iple o π(o ze o). In he In oduc ion, we al eady men ioned ha he necessa y and su icien condi ion ha he diaba ic po en ials, o med by Eq. (2), a e o physical alue in a gi en egion in CS is ha he opological phase (see Eq. (8)) is equal o nπ(whe e n is an in ege ) o any closed con ou in ha egion. In case he wo-s a e opological phase is no equal o nπin he con- side ed egion, we a e o ced o include h ee s a es o some- imes mo e o gua an ee ha he ele an opological phase(s) is(a e) in ege mul iples o π. This will be done nex . 2. T i-s a e p i ileged angle As al eady men ioned in he In oduc ion, we ake ad- an age o he ac ha A(ϕ,q) is a 3 ×3 o hogonal ma- ix and he e o e i s nine elemen s can be p esen ed in e ms o he h ee quasi-Eule angles.18,23 This idea was al eady elabo a ed, applied, and analyzed in a se ies o a icles3,7,9,11,18,29,30 and, he e o e, is only b ie ly discussed he e. As in case o he o dina y Eule ma ix, he o hogonal A-ma ix is p esen ed as a p oduc o h ee o a ion ma ices Qij(γij)(i<j=2, 3) whe e he p oduc A=QklQmnQpq can be w i en in any o de . Subs i u ing his p oduc in Eq. (1) yields h ee coupled i s -o de di e en ial equa ions o he h ee co esponding quasi-Eule angles, γij. The inal se o equa ions as well as hei solu ion depends on he o de o he Q-ma ices.18(a) In an analysis ca ied ou se e al yea s ago,18(b),18(c),29 we a ibu ed physical meaning only o one o he h ee ADT an- gles, γij, p i ileged wi h an equa ion ha con ains he co e- sponding NACT, τij, as a ee isola ed e m ( he e is one equa- ion like ha in e e y g oup o h ee coupled equa ions). In wha ollows, we assume γ12 o be such an angle and conside o his pu pose he p oduc : A=Q12(γ12)Q13(γ13)Q23(γ23). Subs i u ing his p oduc in Eq. (1) yields h ee i s o de equa ions, o which wo equa ions ( o γ12 and γ13) o ma closed subg oup o wo coupled equa ions3 ∂ ∂ϕγ12 =−τ12 − an γ13(τ23cosγ12 +τ13 sinγ12),(9a) ∂ ∂ϕγ13 =τ23 sin γ12 −τ13 cos γ12.(9b) These wo equa ions a e sol ed wi h he aim o calcula - ing he p i ileged ADT angle γ12(ϕ, q). The in oduc ion o he p i ileged angle enables he ex ension o he ea lie de- ined wo-s a e opological phase, α12, o h ee-s a e sys ems. A s aigh o wa d choice is he end-o - he-con ou alue o he angle γ12. Thus, α12(q) =γ12(ϕ=2π,q). 3. Te a-s a e p i ileged angle To ea he ou -s a e case, we need o exp ess he 4×4A-ma ix in e ms o six quasi-Eule angles which implies—simila o he i-s a e case— ha Ahas o be p e- sen ed as a p oduc o six o a ional ma ices: Qij(γij)(i<j =2, 3, 4).18(c),30 In his con ex we e e , again, o he co esponding p i ileged angle, γ12 and employ o his pu pose he co esponding p oduc o elemen a y o a ional ma ices18(c) A=Q12Q13Q14Q23Q24Q34 ⇒Qij =Qij(γij). Subs i u ing his p oduc in Eq. (1) yields a g oup o six i s -o de di e en ial equa ions (see Eq. (20) in Re . 18(c)), o which he i s h ee equa ions ( o he angles γ12,γ13, and γ14) o m a closed subg oup o coupled equa ions ∇γ12 =−τ12 − an γ13(τ13 sin γ12 +τ23cosγ12) − an γ14 sec γ13(τ14 sinγ12 +τ24 cos γ12),(10a) ∇γ13 = anγ14 sinγ13(τ24 sinγ12 −τ14 cos γ12) +τ23 sinγ12 −τ13cosγ12 −τ34 anγ14 cos γ13,(10b) ∇γ14 =−τ14 cos γ12 cos γ13 +τ24 sin γ12 cos γ13 +τ34 sin γ13. (10c) Downloaded 14 Jan 2013 o 129.206.21.195. Redis ibu ion subjec o AIP license o copy igh ; see h p://jcp.aip.o g/abou / igh s_and_pe missions 024113-5 Csehi e al. J. Chem. Phys. 138, 024113 (2013) These h ee equa ions a e sol ed wi h aim o calcula - ing, γ12(ϕ,q), he p i ileged ADT angle and α12(q), he co - esponding opological phase, which, as be o e, is de ined as he end-o - he-con ou alue o he angle γ12, namely, α12(q) =γ12(ϕ=2π,q). D. The inclusion o he Renne -Telle e ec 1. P esen a ion o he in a-plana RT NACT Whe eas he calcula ions o wo-s a e JT-NACTs along ci cula con ou s, , in he i-a omic plane is well known we, mainly, concen a e on he RT-NACT, τ11, along he same con ou (see Sec. II B 1). As al eady men ioned ea lie , we concen a e on ci cles ha ha e hei cen e s on he collinea axis. Since he collinea axis is an in ini e long in e al −∞ <R<+∞ each such ci cle in e sec s his line a wo poin s, i.e., a ϕ=0 and a ϕ=π(see Fig. 1(b)). I is well known ha a each such in e sec ion poin , along a sho in- e al pe pendicula o he collinea axis, is o med a spiky non-ze o NACT (in his case an angula NACT) wi h ea u es eminiscen o a Di ac δ- unc ion.21(b),31 Thus, ou i s en- dency is o assume ha he angula RT-NACTs in he plana CS ake he o m3 τ11(ϕ|q,)=π 2δ(ϕ−ϑ) (11a) o any ci cle wi h a adius q (he e ϑdesigna es he in e - sec ion poin s and is ei he ze o o π—see Fig. 1(b)). Equa- ion (11a) has o be applied wi h some ca e because as i s ands i yields, o any ci cle, a quan ized opological phase (=π) o be expec ed o an undis u bed RT e ec along . Howe e , in Sec. II A we assumed he exis ence o a JT ci on he collinea axis and he e o e he RT e ec is mos likely weakened (o , e en ually, in ensi ied) by his ci.32 Con- sequen ly, he pu e RT quan iza ion is a ec ed and a way o inco po a e his possibili y is o ex end Eq. (11a) by adding a no maliza ion ac o , η, hus, τ11(ϕ|q,)=π 2ηδ(ϕ−ϑ),(11b) whe e η(mos likely ≤1) is a pa ame e o be de e mined he- o e ically. In wha ollows Eq. (11b) is e med as he quasi- Di ac δ- unc ion and ηis e med as he Renne -Jahn coupling pa ame e (RJCP). Mo e abou Eq. (11b) is gi en in Sec. IV. Al hough he ci cula con ou in e sec s he HHF axis a wo poin s, namely, a ϕ=0, πwe conside only wha hap- pens a ϕ=π. I can be shown ha he RT-NACT a ϕ=0 has no a ec on he esul s. 2. The i-s a e JT/RT coupled equa ions In o de o de i e he co esponding di e en ial equa- ions o he i-s a e RT/JT coupled sys em, we conside he NACT-ma ix gi en in Eq. (3)and subs i u e he ele an ma- ix elemen s in Eqs. (9a) and (9b). Thus, ∂ ∂ϕγ12(ϕ)=−τ12(ϕ)−τ11(ϕ) anγ13(ϕ)sinγ12(ϕ),(12a) ∂ ∂ϕγ13(ϕ)=−τ11(ϕ) cos γ12(ϕ),(12b) whe e he in eg a ion is done along a ci cula con ou , and he e o e he i s -o de di e en ia ion ope a o ∇, is eplaced by he angula de i a i e: (∂/∂ϕ).Pa so Eqs.(12) can be in- eg a ed analy ically aking ad an age o Eq. (11b). De ining he ollowing Hea iside s ep unc ion: (ϕ−π)=⎧ ⎨ ⎩ 0; 0 ≤ϕ≤π 1; π≤ϕ≤2π ,(13) i can be shown ha he solu ion o Eq. (12b) is p opo ional o his s ep unc ion γ13(ϕ)=(ϕ−π)γ(0) 13 (ϕ),(14a) whe e γ(0) 13 (ϕ)=−ηπ 2cos{γ12(ϕ=π)}(14b) and ha he solu ion o Eq. (12a) is also a s ep unc ion o a somewha mo e in ol ed o m γ12(ϕ)=−ϕ 0 dϕτ12(ϕ)+(ϕ−π)χ(ϕ=π),(15a) whe e χ(ϕ=π)isgi enin he o m χ(ϕ=π)=−ηπ 2 an γ(0) 13 (ϕ=π)sin{γ12(ϕ=π)}. (16) Equa ion (16) is cha ac e ized by he ollowing ea u es: (i) In case |γ12(ϕ=π)|>π/2 he sign o χ(ϕ=π) is opposi e o he sign o γ12(ϕ=π), e.g., when γ12(ϕ=π)<0, hen χ(ϕ=π)>0. (ii) In case |γ12(ϕ=π)|<π/2, he sign o χ(ϕ=π) is iden ical o he sign o γ12(ϕ=π), e.g., when γ12(ϕ=π)<0, hen χ(ϕ=π)<0. Fo he whole app oach o be meaning ul he up- wa d/downwa d e ical shi s, as exp essed in e ms o χ(ϕ=π), ha e o ul ill wo condi ions (see Appendix B o addi ional in o ma ion) (1) Since he diaba ic po en ials ha e o be single- alued a e e y poin in CS hey ha e o be so also a ϕ=π. Con- sequen ly, he alue o χ(ϕ=π) has o gua an ee he equali y: sin[γ12(ϕ=π)] =sin[γ12(ϕ=π)+χ] and a simila equali y (up o a sign) o he cosine unc ion. (2) Since he alue o γ12(ϕ) a he end o he closed ci cu- la con ou , namely, a ϕ=2πhas o be γ12(ϕ=2π) =±nπ he e ical shi , χ(ϕ=π), has o gua an ee he ollowing quan iza ion condi ion: α(q)=−2π 0 dϕτ12(ϕ)+χ(ϕ=π)=±nπ, (15b) whe e α(q) is ecognized as he ele an opological phase (see Eq. (8)). In Sec. II D 4, hese condi ions will be discussed in mo e de- ail o he case o in ini e small de ia ions. 3. The e a-s a e JT/RT coupled equa ions In wha ollows a e de i ed he di e en ial equa ions o he e a RT/JT coupled sys em. Fo his pu pose, we conside Downloaded 14 Jan 2013 o 129.206.21.195. Redis ibu ion subjec o AIP license o copy igh ; see h p://jcp.aip.o g/abou / igh s_and_pe missions 024113-6 Csehi e al. J. Chem. Phys. 138, 024113 (2013) he NACT-ma ix gi en in Eq. (4)( hus, τ24 and τ34 a e iden- ically ze o and τ14 is eplaced by τ11) and subs i u e he el- e an ma ix elemen s in Eqs. (10a)–(10c). Thus, ∂ ∂ϕγ12 =−τ12 − an γ13(τ13 sin γ12 +τ23cosγ12) −τ11 an γ14 sec γ13 sin γ12,(17a) ∂ ∂ϕγ13 =τ23 sinγ12 −τ13cosγ12 −τ11 anγ14 sinγ13 cos γ12, (17b) ∂ ∂ϕγ14 =−τ11 cos γ12 cos γ13.(17c) Nex , Eq. (17c) is in eg a ed analy ically aking ad an- age o Eq. (11b). Thus, γ14(ϕ)=(ϕ−π)γ(0) 14 (ϕ),(18a) whe e γ(0) 14 (ϕ)=−ηπ 2cos[γ12(ϕ=π)] cos[γ13(ϕ=π)].(18b) To con inue we dis inguish be ween he wo in e als: (i) 0 ≤ϕ<π. Along his in e al γ14(ϕ)≡0 and we a e le wi h ollowing wo equa ions ( o γ12,γ13): ∇γ12 =−τ12 − an γ13 (τ13 sin γ12 +τ23cosγ12), (19a) ∇γ13 =τ23 sin γ12 −τ13cosγ12.(19b) These equa ions a e sol ed o he ini ial condi ions γ12(ϕ=0) =γ13(ϕ=0) ≡0. I is in e es ing o emphasize ha Eqs. (19a) and (19b) a e iden ical o he se o equa ions which ha e o be sol ed o an o dina y sys em o h ee coupled JT s a es. (ii) π<ϕ≤2π. Along his in e al a e encoun e ed he (same) equa ions as gi en by Eqs. (19a) and (19b) (because τ11(ϕ)≡0) bu he e γ14(ϕ), al hough be- ing a cons an , di e s om ze o (see Eqs. (18a) and (18b)). Equa ions (19a) and (19b) a e, he e o e, sol ed along he in e al (ϕ≥π), o he ollowing ini ial condi ions: (I) γ(0) 12 (ϕ=π)=γ12(ϕ=π)+χ12(ϕ=π),(20a) whe e χ12(ϕ=π) — he (1,2) e ical shi — is gi en as χ12(ϕ=π)=−π 2η an γ(0) 14 (ϕ=π) ×sin {γ12(ϕ=π)}sec{γ13(ϕ=π)} (21a) and (II) γ(0) 13 (ϕ=π)=γ13(ϕ=π)+χ13(ϕ=π),(20b) whe e χ13(ϕ=π) — he co esponding (1, 3) e ical shi — is gi en in he o m χ13(ϕ=π)=−π 2η an γ(0) 14 (ϕ=π) ×cos{γ12(ϕ=π)}sin{γ13(ϕ=π)}. (21b) Co olla y: Equa ions (21a) and (21b) indica e, unam- biguously, ha no shi akes place whene e γ12(ϕ=π) =π/2. This conclusion is independen o he alue RJCP, η. 4. De i a ion o he Renne -Jahn coupling pa ame e η To de i e he RJCP, η, we conside he solu ion o γ12(ϕ) as gi en in Eq. (15a) o he case ha γ12(ϕ)a ϕ=πis sligh ly la ge han π/2, namely, γ12(ϕ=π)=π/2+ε. (22) He e, εis a cons an assumed o be small enough o gua an ee he ul illmen o he equi ed app oxima ions. Subs i u ing Eq. (22) in Eq. (16) yields o he co esponding e ical shi , χ(ϕ=π), he alue3 χ(ϕ=π)=ηπ 22 ε. (23) Based on con inui y we expec ha , o ε→0, wo e- qui emen s ha e o be ul illed by χ(ϕ)a ϕ=π.Wes a wi h he single- aluedness o he diaba ic po en ials (see Ap- pendix B o addi ional in o ma ion) which is ul illed when sin(π/2 +ε)=sin(π/2 +ε−χ){≡sin(π/2 −ε)}. In o he wo ds, he single- aluedness is ul illed when (π/2 +ε−χ) =(π/2 −ε)o χ(ϕ=π)=2ε. Recalling Eq. (23) we see ha his happens when ηis3 η=2√2 π=0.9003.(24) We con inue wi h he quan iza ion equi emen and o his pu pose we employ Eq. (15b).F omEq.(22) we ge , due o symme y, ha γ12(ϕ)a ϕ=2πbecomes π+2ε. Sub- s i u ing his ou come in (16), we ind ha he quan iza ion is ul illed whene e 2ε−χ=0 hus, as be o e, his equali y yields o η he esul gi en in Eq. (23) and consequen ly also Eq. (24). So a he η- alue in Eq. (24) was de e mined o he case ha ε→0 o when γ12(ϕ=π) is only sligh ly la ge han π/2 (see Eq. (22)). In Re . 3we conduc ed, o he F +H2 sys em, a nume ical s udy wi h he aim o inding ou o wha ex en his alue o ηapplies also o a bi a y la ge shi s. In- deed, o all conside ed cases (e en o e ical shi s up o ∼2 Rad.), we ind he di e ences be ween he heo e ical shi s and he equi ed nume ical ones o be negligibly small (see a compa ison along he wo las columns o Table 1 gi en in Re . 3). So a his de i a ion was ca ied ou o he i-s a e case. I can be shown ha an iden ical esul is ob ained o he e a-s a e case. In o he wo ds, he ansi ion om a i-s a e sys em o a e a-s a e sys em lea es RJCP una ec ed. Downloaded 14 Jan 2013 o 129.206.21.195. Redis ibu ion subjec o AIP license o copy igh ; see h p://jcp.aip.o g/abou / igh s_and_pe missions 024113-7 Csehi e al. J. Chem. Phys. 138, 024113 (2013) Commen : As al eady men ioned in he In oduc ion, we also s udied sys ems a ec ed by bo h, JT-NACTs and RT- NACTs16(a),17,18 whe e he con ou s su ound he RT seams (and he e o e do no in e sec hem) so ha he esul ing RT NACTs a e no o he Di ac δ- unc ion ype and he need o η- ype pa ame e s becomes edundan . III. NUMERICAL RESULTS A. In oduc o y commen s As men ioned ea lie we apply he nume ical ea men o he H2+F sys em ( he model desc ibed in Sec. II B is con- s uc ed o be sui able o his sys em). The F +H2sys em was ea ed by nume ous g oups du ing he las hal -cen u y bu we men ion he e only he s udies by We ne e al.33,34 In pa icula , we e e o he nume ical ea men by S a k and We ne 33(a) who no only p oduced he up- o-da e, g ound s a e (adiaba ic) po en ial o his sys em bu also e ealed he exis ence o a JT ci loca ed on he collinea axis in icini y o R∼5.5 a.u. (men ioned ea lie while cons uc ing he model) and de i ed he i s diaba ic po en ials o his sys em (mo e de ails a e gi en in Re s. 1,2, and 4). Ou s udy is ca ied ou o he plana CS as o med by assuming (=RHH) obe ixeda =1.4a.u.(seeFig.1). In his ea men a e calcula ed, employing MOLPRO,33(b) only JT-NACTs (see Appendix C o de ails) and he co e- sponding ADT angles. Fo his pu pose a e conside ed he h ee lowes s a es o ype Acoupled a h ee collinea (JT) ci-poin s: he (1,2) ci-poin is loca ed a he icini y o R =Rci∼5.5 a.u.;33(a) he (2,3) ci-poin loca ed a he icini y o R=Rci∼1.9 a.u. and he (3,4) ci-poin loca ed a he icini y o R =Rci∼1.8 a.u. (see Fig. 1). In his espec , we men ion ha he (1,2)ci is o med by a -s a e, assigned as 2Aand one o he wo -s a es (wi h he same symme y) assigned as 1A(see Fig. 2). As men ioned ea lie he RT degene acy is o med by wo -s a es, designa ed as 1Aand 1A (see Fig. 2) and, as usual, a e loca ed along he (collinea ) HHF axis. The co esponding RT-NACTs equi ed o he p esen s udy a e no calcula ed bu assumed o be quasi-Di ac-δ unc ions as discussed in Sec. II D. The calcula ions and he heo e ical s udy a e done (as equen ly men ioned) along closed ci cula con ou s. All ci - cles ha e hei cen e s a he same ixed poin on he collinea axis a R =Rc=6 a.u. This common cen e is chosen in such a way as o gua an ee ha he a ious ci cles (wi h he a ying adii) co e he whole plana CS o in e es . B. JT-NACTs along closed ci cles In Fig. 3a e p esen ed angula JT-NACTs as calcula ed along ou di e en (closed) ci cles wi h he adii: q =0.4, 3.0, 3,7, 4.0 a.u. In panel (a) is p esen ed one NACT, namely, τ12(ϕ|q =0.4 a.u.) whe eas in all o he panels we p esen h ee NACTs, namely, τ12(ϕ|q), τ13(ϕ|q), τ23(ϕ|q) ( ha cou- ple he h ee lowe As a es) calcula ed along ci cles wi h la ge adii: q =3.0, 3.7, 4.0 a.u. The ea u e ha cha ac e - izes he NACTs o q =0.4 a.u. is ha he ci cle does no su - 0π/2 π3π/2 2π -0.2 -0.1 0 0.1 0.2 τϕ [1/ adian] τ12 0π/2 π3π/2 2π -4 -3 -2 -1 0 1 2 τ ϕ [1/ adian] τ12 τ13 τ23 0π/2 π3π/2 2π -4 -3 -2 -1 0 1 2 τϕ [1/ adian] τ12 τ13 τ23 0π/2 π3π/2 2π ϕ [ adian] -4 -2 0 2 4 τϕ [1/ adian] τ12 τ13 τ23 (a) (b) (c) (d) q=0.4 au q=3.0 au q=3.7 au q=4.0 au FIG. 3. Angula NACTs, τϕ(ϕ|q), o ci cula con ou s a R =Rce =6a.u. along he in e al 0 ≤ϕ≤2π:(a)τ12ϕ(ϕ|q), o q =0.4 a.u.; (b) τ12ϕ(ϕ|q), τ13ϕ(ϕ|q) and τ23ϕ(ϕ|q), o q =3 a.u.; (c) he same as in (b) bu o q =3.7 a.u.; (d) he same as in (b) bu o q =4a.u. ound he poin o ci which is loca ed a a dis ance o 0.5 a.u. om he cen e o he ci cle. In all o he cases, he ci - cles su ound he ci-poin and he e o e he a ious τ12(ϕ|q >0.5 a.u.)’s exhibi a sligh ly mo e complica ed s uc u e. As o he wo NACTs ha couple he hi d s a e he ollowing can be said: (a) τ13(ϕ|q) ha dly changes as q inc eases; (b) Downloaded 14 Jan 2013 o 129.206.21.195. Redis ibu ion subjec o AIP license o copy igh ; see h p://jcp.aip.o g/abou / igh s_and_pe missions 024113-8 Csehi e al. J. Chem. Phys. 138, 024113 (2013) τ23(ϕ|q) changes signi ican ly and becomes spikie . The ea- son is associa ed wi h he ac ha a {R ∼1.8 a.u., θ=0} ≡{q ∼4.2 a.u., ϕ=π}(seeFig.1)) a e loca ed wo ad- di ional JT cis (as al eady men ioned ea lie ): a (2,3)ci and a (3,4)ci. As a esul , he mo e emo e is he cen e o he ci cle om hese cis he spikie (as a unc ion o ϕ)is he co esponding NACT, τ23(ϕ|q). These spiky NACTs may lead o inaccu acies in calcula ing he co esponding ADT angles. C. (1,2) ADT angles along closed ci cles 1. T i-s a e esul s In Fig. 4a e p esen ed he ( e ical) shi ed (1,2) ADT angles, γ12(ϕ|q), as calcula ed acco ding o he ecipe in Eqs. (15a) and (16). Cu es o nine q- alues in he ange 0.4 <q<5.0 a.u. a e gi en. We dis inguish be ween h ee ypes o cu es: (1) The cu e o q =0.4 a.u. is no shi ed (in o he wo ds, he shi is ze o) and i s opological phase, α12(q), is ze o which esul s om he ac ha he ci cle does no su - ound he (1,2)ci. (2) The cu e o q =1.0 a.u. is no shi ed (o he shi is negligible small) and is cha ac e is ic o he si ua ion ha he ci cle su ounds he (1,2)ci and he e o e yields: α12(q) =π. (3) In all o he cases, he ci cles su ound he (1,2) ci and a e shi ed downwa ds (a ϕ=π). These wo ac s a e he eason ha all he opological phases α12(q) a e co ec ly quan ized (namely, become equal o πwhen ϕ= 2π). A his s age, we emphasize again ha he downwa ds shi s o he a ious cases we e calcula ed acco ding o o - mula in Eq. (16) (see, also, Eq. (14b)) whe e ηis gi en by Eq. (24) and γ12(ϕ=π|q) is he co esponding p i ileged ADT angle. In o he wo ds, no a i icial i ing is done! Figu e 4 e eals one in e es ing (and impo an ) ea u e: The ϕ-dependence o he a ious cu es become simila and he cu es a e con e ging o each o he as he adius, q, o he ci cles inc eases. This phenomenon is gene al bu is en- hanced in he in e al o π/2 <ϕ<3π/4. Figu e 5shows, schema ically, he egion in CS — in he shape o a squa e — 0π/2 π3π/2 2π ϕ [ adian] 0 π/2 π 3-s a e γ12 q=0.4 au q=1.0 au q=2.0 au q=3.0 au q=3.7 au q=4.0 au q=4.6 au q=5.0 au q=4.2 au FIG. 4. The i-s a e ADT (mixing) angle, γ12(ϕ|q), o ci cula con ou s a R=Rce =6 a.u. along he in e al 0 ≤ϕ≤2πas calcula ed employing he RT/JT Eqs. (15a) and (16). Resul s a e shown o q =0.4, 1.0, 2.0, 3.0, 3.7, 4.0, 4.6, 5.0 a.u. FIG. 5. The ansi ion egion, p esen ed as a squa e, om eagen s channel o p oduc s channel. whe e he con e gence is mos e icien . As i happens his is he egion whe e he chemical eac ion akes place. In o he wo ds, i o e laps wi h he ansi ion egion om he eagen s a angemen o he p oduc s a angemen . 2. Te a-s a e esul s In Fig. 6is p esen ed a compa ison be ween p i ileged ADT angles, γ12(ϕ|q) as calcula ed, once employing h ee s a es (see Eqs. (15a) and (16)) and once employing ou s a es (see Eqs. (19)–(21)). These ADT angles a e calcu- la ed along ci cles wi h he ollowing adii: q =3.0, 3.7, 4.0 a.u. p esen ed in he ele an panels. As is no iced he e- sul s a e well con e ged o all h ee cases. The encou aged ac om his compa ison is ha al hough he e a-s a e ADT angles a e calcula ed using wo addi ional NACTs, namely, τ13(ϕ|q) and τ23(ϕ|q) (and he e o e a e based on mo e in- ol ed exp essions o calcula e he shi s a ϕ=π—see Eqs. (20) and (21) — s ill, he ADT angles, a e easonably well con e ged. IV. DISCUSSION AND CONCLUSIONS In his a icle is s udied he p i ileged, ADT angle γ12(ϕ|q) due o wo en angled NACTs— he JT-NACT and he RT-NACTs. This was no he si ua ion when we s a ed s udy- ing he FHH sys em.1,2A ha s age we employed one JT- NACT, namely, τ12(ϕ|q), and ollowing Eq. (6), we go quan- ized opological angles, α12(q), only along ci cles wi h small adii (q <2.0 a.u.). To imp o e he quan iza ion o ci cles wi h la ge adii (q >2.0 a.u.), we added ano he A-s a e as well as i s wo co esponding NACTs, τ13(ϕ|q) and τ23(ϕ|q), and sol ed Eqs. (19a) and (19b) — he ele an equa ions o he h ee-s a e JT si ua ion (sol ed wi h he ini ial condi- ions: γ12(ϕ=0) =γ13(ϕ=0) ≡0). In Fig. 7a e compa ed he wo JT-ADT angles γ12(ϕ|q), as calcula ed o q =4.0 a.u. (once by sol ing Eq. (6) and once by Eqs. (19a) and (19b)). I is well no iced ha he wo kinds o calcula ions yield sim- ila alues o he opological phases, α12(q) bu which di e signi ican ly om π. In o he wo ds, inc easing he JT sub- Hilbe space om wo s a es o h ee did no yield he ex- pec ed quan iza ion (unlike in nume ous o he cases27,29). Downloaded 14 Jan 2013 o 129.206.21.195. Redis ibu ion subjec o AIP license o copy igh ; see h p://jcp.aip.o g/abou / igh s_and_pe missions