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
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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
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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
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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.
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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)
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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
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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.
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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)
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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).
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