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. Bo n and K.
Huang, Dynamical Theo y o C ys al La ices 共Ox o d Uni e si y, New
Yo k, 1954兲.
2E. Renne , Z. Phys. 92,172共1934兲.
3G. He zbe g, Molecula Spec a and Molecula S uc u e 共K iege , Mala-
ba , 1991兲, Vol. III.
4H. C. Longue -Higgins, Ad . Spec osc. 共N.Y.兲2, 429 共1961兲.
5K. D essle and D. A. Ramsay, J. Chem. Phys. 27,971共1957兲; Philos.
T ans. R. Soc. London, Se . A 251, 553 共1958兲.
6J. A. Pople and H. C. Longue -Higgins, Mol. Phys. 1,372共1958兲.
7M. Pe ic and S. Peye imho , Ad . Chem. Phys. 124,583共2002兲.
TABLE II. The diagonal D-ma ix elemen s as calcula ed o wo s a es 共D11
共2兲兲and o h ee s a es Djj
共3兲j
=1,2,3.
共q1,q2兲aD11
共2兲D11
共3兲D22
共3兲D33
共3兲
␣
共 ad兲b
Lowe
s a es
共0.1,0.8兲−0.959 −0.999 −0.999 +1.000 2.854
共0.8,0.8兲−0.960 −0.993 −0.993 +1.000 2.858
共1.0,1.0兲−0.927 −0.986 −0.982 +0.996 2.757
Uppe
s a es
共0.8,0.8兲−0.946 +0.998 −0.952 −0.951 2.812
aUni s in Å.
bThe wo-s a e opological 共Be y兲phase.
TABLE III. The diagonal D-ma ix elemen s as calcula ed o i e s a es
共Djj
共5兲;j=兵1,5其兲.
共q1,q2兲aD11
共5兲D22
共5兲D33
共5兲D44
共5兲D55
共5兲
共0.8,0.8兲−0.986 −0.982 +0.997 −0.944 −0.946
aUni s in Å.
144108-6 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
8Ch. Jungen and A. J. Me e , Mol. Phys. 40,1共1980兲;40,95共1980兲;in
Molecula Spec oscopy, Mode n Resea ch, edi ed by K. N. Rao 共Aca-
demic, New Yo k, 1977兲, Vol. 2, p. 127.
9J. M. B own and F. Jo genson, Ad . Chem. Phys. 52,117共1983兲;J.M.
B own, in Compu a ional Molecula Spec oscopy, edi ed by P. Jensen
and P. R. Bunke 共Wiley, New Yo k, 2000兲, p. 517.
10 S. Ca e and N. C. Handy, Mol. Phys. 47, 1445 共1982兲;52, 1367
共1984兲.
11 R. Ba ow, R. N. Dixon, and G. Duxbu y, Mol. Phys. 27, 1217 共1974兲.
12 A. Alijah and G. Duxbu y, Mol. Phys. 70, 605 共1990兲; G. Duxbu y, B.
McDonald, M. Van Gogh, A. Alijah, Ch. Jungen, and H. Pali an, J.
Chem. Phys. 108, 2336 共1998兲.
13 M. Pe ic, R. J. Buenke , and S. Peye imho , Mol. Phys. 59, 1283
共1986兲; M. Pe ic, S. Peye imho and R. J. Buenke , Z. Phys. D: A ., Mol.
Clus e s 24,177共1992兲.
14 P. R. Bunke and P. Jensen, J. Mol. Spec osc. 118,18共1986兲;Molecula
Symme y and Spec oscopy 共NRC Resea ch, O awa, 1998兲.
15 共a兲M. Pe ic and S. D. Peye imho , J. Chem. Phys. 102,3685共1995兲;共b兲
M. Pe ic, B. Engels and M. Han a h, Chem. Phys. 238,33共1998兲.
16 G. J. Halász, Á. Vibók, R. Bae , and M. Bae , J. Chem. Phys. 125,
094102 共2006兲.
17 共a兲G. J. Halász, Á. Vibók, D. K. Ho man, D. J. Kou i, and M. Bae , J.
Chem. Phys. 126, 154309 共2007兲;共b兲G. J. Halász, Á. Vibók, R. Bae ,
and M. Bae , J. Phys. A 40, F267 共2007兲.
18 H. A. Jahn and E. Telle , P oc. R. Soc. London, Se . A 161, 220 共1937兲.
19 M. S. Child and H. C. Longue Higgins, Philos. T ans. R. Soc. London,
Se . A 254,259共1961兲.
20 G. He zbe g and H. C. Longue -Higgins, Discuss. Fa aday Soc. 35,77
共1963兲; H. C. Longue Higgins, P oc. R. Soc. London, Se . A 344, 147
共1975兲.
21 E. R. Da idson, J. Am. Chem. Soc. 99, 397 共1977兲.
22 M. S. Child, Ad . Chem. Phys. 124,1共2002兲; S. Adhika i and G. D.
Billing, ibid. 124,143共2002兲; R. Englman and A. Yahalom, ibid. 124,
197 共2002兲; A. Kuppe mann and R. Ab ol, ibid. 124, 323 共2002兲;G.A.
Wo h and M. A. Robb, ibid. 124, 355 共2002兲.
23 M. Chajia and R. D. Le ine, Phys. Chem. Chem. Phys. 1, 1205 共1999兲;
K. L. Kompa and R. D. Le ine, P oc. Na l. Acad. Sci. U.S.A. 98, 410
共2001兲.
24 A. Kuppe mann, in Dynamics o Molecules and Chemical Reac ions,
edi ed by R. E. Wya and J. Z. H. Zhang 共Dekke , New Yo k, 1996兲,p.
411; R. Ab ol, A. Shaw, and A. Kuppe mann, J. Chem. Phys. 116, 1035
共2002兲.
25 M. Bae , S. H. Lin, A. Alijah, S. Adhika i, and G. D. Billing, Phys. Re .
A62, 032506 共2000兲.
26 A. L. Sobolewski and W. Domcke, Chem. Phys. 259, 181 共2000兲.
27 共a兲M. Bae , T. Ve si, G. J. Halász, Á. Vibók, and S. Suhai, Fa aday
Discuss. 127, 337 共2004兲;共b兲A. M. Mebel, G. J. Halász, Á. Vibók, A.
Alijah, and M. Bae , J. Chem. Phys. 117, 991 共2002兲.
28 R. Englman and T. Ve esi, Phys. Le . A 354, 196 共2006兲;E.S.K y
-
achko, Ad . Quan um Chem. 44,119共2003兲; P. Ba agan, L. F. E ea, A.
Macias, L. Mendez, A. Rie a, J. M. Lucas, and A. Aguila , J. Chem.
Phys. 121, 11629 共2004兲; S. Gomez-Ca asco, A. Aquado, M. Paniaqua,
and O. Ronce o, ibid. 125, 104105 共2006兲; M. B. Se yuk, L. Y. Rusin,
S. Ca alli, and V. Aquilan i, J. Phys. Chem. A 108, 8731 共2004兲;V.A.
Mozhayski , D. Babiko , and A. I. K ylo , J. Chem. Phys. 124, 224309
共2006兲; K. A. Kis le and S. Ma sika, J. Phys. Chem. A 111, 2650 共2007兲;
E. Ab ahamsson, G. C. G oenenboom, and R. V. K ems, J. Chem. Phys.
126, 184309 共2007兲.
29 G. D. Billing, M. Bae , and A. M. Mebel, Chem. Phys. Le . 1, 372
共2003兲.
30 J. C. Tully, Fa aday Discuss. 127,463共2004兲.
31 Conical In e sec ions: Elec onic S uc u e, Dynamics and Spec oscopy,
edi ed by W. Domcke, D. R. Ya kony, and H. Köppel 共Wo ld Scien i ic,
Singapo e, 2004兲.
32 M. Ben-Nun and T. J. Ma inez, Chem. Phys. 259, 237 共2000兲.
33 M. Bae , Beyond Bo n Oppenheime : Elec onic Non-Adiaba ic Coupling
Te ms and Conical In e sec ions 共Wiley, Hoboken, 2006兲.
34 B. Sa ka and S. Adhika i, J. Chem. Phys. 124, 074101 共2006兲; P. Puza i,
B. Sa ka , and S. Adhika i, ibid. 121,707共2004兲.
35 O. Godsi, C. R. E enhuis, and M. Collins, J. Chem. Phys. 125, 164321
共2006兲.
36 共a兲M. Bae , Chem. Phys. Le . 35,112共1975兲;共b兲Mol. Phys. 40, 1011
共1980兲.
37 C. N. Yang and R. L. Mills, Phys. Re . 96, 191 共1954兲.
38 H.-J. We ne and P. J. Knowles, J. Almlö e al.,MOLPRO, a package o
ab ini io p og ams.
144108-7 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
