An in aline o conical in e sec ions o me hylamine
C. Le i,1G. J. Halász,2Á. Vibók,3I. Ba ,1Y. Zei i,4R. Koslo ,5and M. Bae 6,a兲
1Depa men o Physics, Ben-Gu ion Uni e si y, Bee She a 84105, Is ael
2Depa men o In o ma ion Technology, Uni e si y o Deb ecen, P.O. Box 12, H-4010 Deb ecen, Hunga y
3Depa men o Theo e ical Physics, Uni e si y o Deb ecen, P.O. Box 5, H-4010 Deb ecen, Hunga y
4Depa men o Biomedical Enginee ing, Ben-Gu ion Uni e si y, Bee She a 84105, Is ael
5Depa men o Physical Chemis y and he F i z Habe Cen e o Molecula Dynamics, The Heb ew
Uni e si y o Je usalem, Je usalem 91904, Is ael
6The F i z Habe Cen e o Molecula Dynamics, The Heb ew Uni e si y o Je usalem,
Je usalem 91904, Is ael
共Recei ed 20 Ma ch 2008; accep ed 19 May 2008; published online 23 June 2008兲
In his a icle a e conside ed he conical in e sec ions 共ci’s兲 ela ed o he N–H bond in he
me hylamine, CH3NH2, molecule. The no el ea u e ha was e ealed is ha he wo lowes s a es
1A⬘and 1A⬙a e coupled by a line o cis loca ed in HC–NHH plane—a line ha is o med by
mo ing a single hyd ogen on ha plane while ixing he 共six兲o he a oms. The alidi y o his line
was p o en i s by s udying he singula i ies o he 共angula 兲nonadiaba ic coupling e ms and hen
by e ealing he degene acy poin s o med by he wo in e ac ing adiaba ic po en ial ene gy su aces
共PESs兲. A heo e ical analysis indica ed ha he line has o be a ini e closed line. We also calcula ed
he Be y phase o a con ou ha su ounds his line and ound i o be 3.127 ad, namely, a alue
easonably close o
. The exis ence o such lines o cis—ins ead o isola ed cis 共as exhibi ed by
o he n-a omic 共n⬎3兲molecules such as HNCO o C2H2兲—may enhance signi ican ly he ansi ion
a e om an uppe adiaba ic s a e o a lowe one. The e a e also nume ical ad an ages in such
si ua ions, ha is, i such a line is p ope ly placed in ha plane 共like in he p esen case兲 he
wa e-packe ea men o he nuclei can be ca ied ou employing a single diaba ic PES ins ead o
ha ing o conside wo coupled PESs. © 2008 Ame ican Ins i u e o Physics.
关DOI: 10.1063/1.2943143兴
I. INTRODUCTION
The s udy o he elec onic nonadiaba ic coupling e ms
共NACTs兲and conical in e sec ions 共ci兲concen a ed, o he
pas wo decades, mainly on small molecules, namely, i-
a omic and e a-a omic sys ems.1–17 In he p esen a icle we
conside a much la ge sys em, he me hylamine molecule,
CH3NH2, consis ing o se en a oms. The s uc u e o he
molecule is gi en in Fig. 1共a兲whe e i can be seen ha he
me hyl g oup, CH3, is sepa a ed om he amine g oup, NH2
by he CN bond. This molecule has long been a ques ion o
conside able in e es since i comp ises wo s ongly coupled
la ge ampli ude mo ions, speci ically, he o sion o he me-
hyl op and he in e sion o he amine g oup.17 Se e al ex-
pe imen al s udies p o ided e idence ha he dominan
channel in i s pho odissocia ion in he i s abso p ion band
co esponds o he N–H bond ission.18–20 Ve y ecen ly21–23
i was shown ha in he ib a ional media ed pho odissocia-
ion o CD3NH2abou 90% o he o al obse ed p oduc s
共only hyd ogen iso opes we e measu ed兲a e H pho o ag-
men s eleased om he amine g oup. Also, heo e ical ind-
ings showed he cu s h ough he ab ini io po en ial ene gy
su aces 共PESs兲 o he g ound 1A⬘and he i s exci ed 1A⬙
s a es o me hylamine. Pa icula ly, i was poin ed ou ha
he 1A⬙s a e po en ial ha leads o he b eakup o he N–H
bond is cha ac e ized by a small ba ie 共⬃3000 cm−1兲a
sho ange and is ollowed, a la ge bond ex ensions, by a ci
loca ed a an in e sec ion poin wi h he 1A⬘s a e whe e bo h
s a es belong o he Cssymme y.24,25
In wha ollows we epo on a s udy o NACTs ha
couple he abo e men ioned wo s a es and b ie ly e e o
he highe exci ed s a e, 2A⬙.
II. FORMING THE INTRALINE OF CONICAL
INTERSECTIONS
In wha ollows we epo on a s udy o NACTs ha
couple he abo e men ioned h ee s a es. To ca y ou his
s udy he ia omic amine g oup is assumed o o m a plane
wi h HC dia om o he me hyl g oup 共see Fig. 1兲, and ou
s udy concen a es on sea ching o Jahn–Telle cis 共Re s.
26–28兲be ween he h ee lowe s a es, namely, he g ound
1A⬘s a e and he nex wo exci ed s a es 1A⬙and 2A⬙共 he
sea ch o hese cis is ca ied ou by allowing he amine
hyd ogen o mo e in his plane only兲.
To his end we employ he MOLPRO package29 wi h he
mul i e e ence con igu a ion in e ac ion me hod and he
6-31+G*basis se . We used he ac i e space, including wo
alence elec ons dis ibu ed on i e o bi als 共 h ee o hem
belonging o he i educible ep esen a ion o A⬘兲. Th ee
elec onic s a es we e compu ed wi h equal weigh s. All
se en a oms we e conside ed, bu he sea ch o cis is ca ied
ou by ac i a ing only one a om—an amine hyd ogen—as a
es pa icle. To ollow he mo ion o he hyd ogen we em-
a兲Elec onic mail: [email p o ec ed].
THE JOURNAL OF CHEMICAL PHYSICS 128, 244302 共2008兲
0021-9606/2008/128共24兲/244302/6/$23.00 © 2008 Ame ican Ins i u e o Physics128, 244302-1
Downloaded 10 Jul 2008 o 193.174.59.126. Redis ibu ion subjec o AIP license o copy igh ; see h p://jcp.aip.o g/jcp/copy igh .jsp
ploy wo 共pola 兲coo dina es, he adial coo dina e qand he
angula coo dina e
, de ined o a sys em o coo dina es
loca ed a some poin in he plane 关see Fig. 1共b兲兴. The de ec-
ion is done by calcula ing he angula NACT,
共1/q兲
jk共
,q兩s兲, whe e
jk共
,q兩s兲is de ined as
jk共
,q兩s兲=
冓
j共se兩
,q,s兲冏
k共se兩
,q,s兲
冔
.共1兲
He e
i共se兩
,q,s兲;i=j,ka e he elec onic 共adiaba ic兲Bo n–
Oppenheime 共BO兲eigen unc ions,30,31 ses ands o he se
o elec onic coo dina es, and sp esen s he g oup o all
nuclea coo dina es excluding
and q. Ha ing de ined he
sys em o coo dina es he sea ch s a s by o ming ci cula
con ou s o di e en adii and calcula ing he angula NACTs
along hese con ou s. The nex s ep is de i ing he adiaba ic-
o-diaba ic ans o ma ion 共ADT兲angle 共known also as he
mixing angle兲,
␥
共
,q兲共Re . 32兲共see also Re . 1, Chap. 3.1兲,
␥
共
,q兩s兲=
冕
0
d
⬘
共
⬘,q兩s兲.共2兲
In he case o a closed con ou we ge
␥
共
=2
,q兩s兲
=
␣
共q兩s兲, whe e
␣
共q兩s兲is ecognized as he Be y phase.33,34
I is impo an o emphasize ha o si ua ions whe e he
NACT is o med, in a gi en egion, by wo isola ed s a es
共namely, s a es a ec ed, a mos , sligh ly by highe s a es in
ha egion兲, he phase
␣
共q兩s兲is expec ed o be close o an
in ege mul iple o
共o ze o兲.33–36 Pa o he nume ical
s udy was de o ed o his issue. We calcula ed NACTs be-
ween he h ee lowes s a es o he Cssymme y, namely,
1A⬘,1A⬙, and 2A⬙共men ioned ea lie 兲and ound ha only
he NACTs o med by he wo lowe s a es a e nonze o
whe eas he o he wo ha in ol e he second exci ed s a e
a e p ac ically ze o 共 hus gua an eeing he isola ion o he
wo lowe s a es in he egion o in e es 兲.
Figu e 2共a兲p esen s he angula NACT, along such a
ci cle, as a unc ion o
wi h q=0.1 Å, whe e he coo di-
na es o he cen e a e a 共R1,
1兲=共1.83 Å,122°兲and
共R2,
2兲=共1.0 Å,120°兲关see Fig. 1共a兲兴. The igu e shows wo
ela i ely sha p peaks wi h opposi e signs. I u ns ou ha
he alue o he in eg al unde each peak is ⬃1.554 ad 共i.e.,
a alue close o
/2兲and he alue o he Be y phase is
␣
共q兩s兲⬃0. I is impo an o emphasize ha he s uc u e o
he angula NACT di e s signi ican ly om all o he angula
NACTs o which we usually encoun e wo1,4共b兲,4共c兲,5,6,7共a兲,8,28
共and some imes h ee4共a兲兲posi i e maximal alues and he
Be y phase becomes, app oxima ely, an in ege mul iple o
共and no ze o兲.
Thus he ques ion is wha makes his pa icula case so
di e en ? Following addi ional calcula ions we ind ha he
same plane hos s many mo e cis. All a e cha ac e ized by he
same ea u e as discussed abo e; namely, he co esponding
angula NACTs posses wo sha p peaks o opposi e signs
共see ano he one gi en in Fig. 2共b兲wi h q=0.1 Å and he
cen e a 关共R1,
1兲=共1.70 Å,85°兲兴. Finally i was es ablished
ha he jus men ioned cis o m a con inuous 共 ini e兲line
共see Fig. 3兲. This line is cons uc ed in p ac ice by connec -
ing all he posi i e 共o nega i e兲peaks o NACTs o he ype
p esen ed in Figs. 2共a兲and 2共b兲. I is in e es ing o no e ha
one o he ci poin s, i.e., he one calcula ed a R1=1.83 Å and
1=122°, is close 共bu no iden ical兲 o he posi ion o a
degene acy poin e ealed some ime ago by Dunn and
Mo okuma.24
In Fig. 3is p esen ed such a line in a plane de e mined
FIG. 1. 共Colo online兲The equilib ium s uc u e o me hylamine. In he
p esen s udy i e a oms, namely, he ca bon, he ni ogen, wo amine hy-
d ogens, and a me hyl hyd ogen a e assumed o o m a ixed plane— he
HC–NH2plane—and hei ela i e mo ion is cons ained o his plane. 共a兲
The coo dina es
1and R1show he posi ion o he es hyd ogen 共wi h
espec o he ni ogen兲 ha we e a ied du ing he ci sea ch 共 he coo dina es
2and R2a e held ixed du ing his p ocess兲.共b兲The pola coo dina es qand
show he posi ion o es hyd ogen wi h espec o an assumed poin close
oaci poin .
FIG. 2. 共Colo online兲The angula NACT p esen ed as a unc ion o
o
q=0.1 Å as calcula ed a h ee di e en loca ions along he seam 共see Fig.
3兲:共a兲The NACT a poin 共R1,
1;R2,
2兲=共1.83 Å,122° ;1.0 Å,120°兲.共b兲
The NACT a poin 共R1,
1;R2,
2兲=共1.70 Å,85° ;1.0 Å,120°兲.共c兲The
NACT a poin 共R1,
1;R2,
2兲=共1.70 Å,159° ;1.0 Å,120°兲.
244302-2 Le i e al. J. Chem. Phys. 128, 244302 共2008兲
Downloaded 10 Jul 2008 o 193.174.59.126. Redis ibu ion subjec o AIP license o copy igh ; see h p://jcp.aip.o g/jcp/copy igh .jsp
by holding he second amine hyd ogen ixed a 共R2,
2兲
=共1.0 Å,120°兲关see Fig. 1共a兲兴: This line is p esen ed in
e ms o Ca esian coo dina es 共x,y兲关=R1cos共
−
1兲,
R1sin共
−
1兲兴, whe e he cen e o he coo dina es is loca ed
a he ni ogen a om.
This jus desc ibed line can be in e p e ed as a seam
共de ined as he line ha connec s all hese in ini e numbe o
ci poin s兲.
Like any o he seam his one is also expec ed o be a line
ha connec s he degene acy poin s be ween he wo adia-
ba ic su aces. In Fig. 3is p esen ed a second line ha ol-
lows he degene acy poin s and closely o e laps wi h he
seam 共 he de ia ions a e wi hin he e o limi o he calcu-
la ions兲.
Mo e de ails conce ning his line o degene acy poin s
a e gi en in Fig. 4. In Fig. 4共a兲a e p esen ed he wo in e -
ac ing adiaba ic PESs 1A⬘and 1A⬙, assigned as V1共x,y兲and
V2共x,y兲. In Fig. 4共b兲is p esen ed ⌬V共x,y兲, he di e ence
⌬V共x,y兲=V2共x,y兲−V1共x,y兲be ween he wo adiaba ic PESs.
The wo lines in he x-yplane in Fig. 3a e seen o be
ini e and open. In ac addi ional calcula ions based on he
NACTs o ind ou mo e abou hese lines ailed al oge he
because he NACTs became spikie o he ex en ha inally
we los hem al oge he . Howe e addi ional calcula ions
based on degene acy poin s we e ca ied ou , and hey
showed ha , undoub edly, he line is ex ending owa d he
wo hyd ogens— he amine hyd ogen on one side and he
co esponding me hyl hyd ogen on he o he . Thus, his ype
o calcula ions seems o indica e ha he line is ini e. How-
e e we could no es ablish whe he he line is open o closed
mainly because i closed— he missing segmen 共s兲is 共a e兲a
he high po en ial egion whe e he accu acy is no good
enough o es ablish poin s o degene acy. These issues a e
u he discussed in he nex chap e .
Mo e abou he PESs o his sys em is shown in Fig. 5.
In con as o he adiaba ic su aces in Fig. 4共a兲 hese a e he
co esponding diaba ic ones. This igu e clea ly shows he
impo an ad an age in ha ing a line o degene acy poin s in
a ixed plane. This si ua ion causes he co esponding diaba-
ic coupling e m o be ze o along his line,37,38 hus allowing
a wa e packe o mo e, eely and undis u bed, om he
uppe adiaba ic su ace o he lowe adiaba ic su ace 共mo e
on his issue is discussed in he nex chap e 兲.
III. DISCUSSION AND CONCLUSIONS
This a icle concen a es on an excep ional con inuous
line o cis e ealed in he me hylamine molecule. This line
can be in e p e ed as a seam, bu i i is a seam i is less
common because some o i s ea u es a e di e en . In wha
ollows we concen a e on se e al issues:
共1兲In all p e ious s udies1–8,14,15,28,34 he seams a e loca ed
ou side he planes 共e.g., he g-hplane, which is de ined
as he pe pendicula plane o he seam7共a兲兲 ha con ain
he mo ing es pa icle 共which in he p esen case is he
HC–NHH plane兲. In o he wo ds he me hylamine c e-
a es a phenomenon whe e he wo in e ac ing s a es i.e.,
1A⬘and 1A⬙, a e coupled by a con inuous line o cis
o med by one single mo ing a om whe e all o he
共six兲a oms o he molecule a e ixed in con igu a ion
space. Since his hyd ogen is mo ing wi hin he HC–
FIG. 3. 共Colo online兲The in a 共 ixed con igu a ion space兲seam o he ci’s
loca ed in he HC–NH2plane as calcula ed o R2=1.0 Å and
2=120°:
共----兲The seam as calcula ed by connec ing he ci poin s; 共----兲 he seam as
calcula ed by connec ing he degene acy poin s. The wo Ca esian coo di-
na es 共x,y兲a e ela ed o he pola coo dina es 共R1,
1兲, namely, 共x,y兲
=关R1cos共
−
1兲,R1sin共
−
1兲兴. A h ee loca ions a e d awn small ci cles
o indica e he posi ions o h ee con ou s men ioned in Fig. 2.
FIG. 4. 共Colo online兲The PESs as a unc ion o xand yas calcula ed o
R2=1.0 Å and
2=120°: 共a兲P esen s he wo adiaba ic 共1A⬘and 1A⬙兲PESs,
V1共x,y兲and V2共x,y兲;共b兲p esen s he di e ence ⌬V共x,y兲=V2共x,y兲
−V1共x,y兲be ween he wo PESs. The wo igu es a e shown a di e en
angles o emphasize he mo e signi ican ea u es in each case.
244302-3 Conical in e sec ions o me hylamine J. Chem. Phys. 128, 244302 共2008兲
Downloaded 10 Jul 2008 o 193.174.59.126. Redis ibu ion subjec o AIP license o copy igh ; see h p://jcp.aip.o g/jcp/copy igh .jsp
NHH plane, he line, jus desc ibed, is loca ed in ha
共HC–NHH兲plane as well 共see Fig. 3兲. As i s ands his
si ua ion canno be jus i ied. To unde s and hese ind-
ings we ha e o conside in mo e de ail bo h he plane
ha con ains he mo ing hyd ogen and he ci-line as
well as he wo s a es ha o m he NACTs. I is well
known ha in o de o wo s a es 共no coupled by any
o he s a e兲 o o m degene acy a a gi en poin , wo
condi ions ha e o be ul illed, namely,38
H11 −H22 =0, 共3a兲
H12共=H21兲=0, 共3b兲
whe e Hij,i,j=1,2 a e he ma ix elemen s o he elec-
onic Hamil onian. Fo a andomly selec ed plane
hese condi ions a e a mos sa is ied a isola ed poin s.
Howe e i a s udy is ca ied ou wi h espec o a plane
o symme y 共as in he p esen case兲and he wo s a es
a e o di e en symme ies 共as indeed 1A⬘and 1A⬙
a e兲, hen condi ion 共3b兲is sa is ied e e ywhe e by
symme y. So now we a e le wi h only one equi e-
men o be sa is ied, namely, Eq. 共3a兲, and his one can
be ul illed along a line in he abo e men ioned plane
wi hou necessa ily c ea ing any imp obable si ua ions
共as indeed is shown by he nume ical ea men 兲.
共2兲Once we es ablished he exis ence o he abo e men-
ioned ci-line— o be assigned as ⌳— wo ques ions
ha e o be answe ed: 共i兲Isi a ini e o an in ini e line?
共ii兲I i is a ini e line is i an open o a closed line? As
indica ed in he p e ious chap e we could no answe
hese ques ions nume ically 共al hough i is e y likely
ha he line is indeed ini e兲. In o de o analyze he
a ious possibili ies we conside wo kinds o con ou s,
namely, ⌫c, ha in e sec ⌳and ⌫s, which su ound i
共see Fig. 6兲. As long as one mo es along ⌫s he o de o
he s a es 1A⬘and 1A⬙a poin s Aand B emains un-
changed whe eas i one ollows a con ou o he kind
⌫c关see Fig. 6共a兲兴, each c ossing is accompanied wi h a
change in he o de o he s a es 关see Re s. 37共a兲and
37共b兲whe e a simila si ua ion is analyzed while s udy-
ing cha ge ans e o he 共H+H2兲+sys em兴. Thus, i a
poin As a e 1A⬙is abo e 1A⬘we ind, ollowing con-
ou ⌫c, ha a B1A⬙is below 1A⬘. Mo ing along a
con ou ⌫c ha c osses ⌳ wice lea es he o de o he
s a es unchanged 关see Fig. 6共c兲兴. We e u n now o he
abo e men ioned ques ions and we s a by discussing
共i兲:I ⌳is an in ini e line hen he o de o s a es can be
a ec ed e en in a si ua ion whe e he es hyd ogen is
a an in ini e sepa a ion om he CH2NH adical. How-
e e a la ge dis ances he hyd ogen has no in luence on
he adical 共and ice e sa兲and he e o e canno a ec
he s a es o he adical. In o he wo ds no con ou s o
he kind ⌫ccan be ound a he in ini e sepa a ion e-
gion, which implies ha ⌳is a ini e line. Nex we
discuss 共ii兲. Since ⌳is a ini e line, i can be ei he an
open o a closed line. In o de o be able o ollow he
o hcoming analysis we e e again o Fig. 6in which
a e p esen ed se e al possibili ies. In Fig. 6共a兲is shown
an open line, ⌳, and wo con ou s ⌫cand ⌫s ha con-
nec Aand B. F om he ea lie discussion we ind ha
mo ing along hese wo con ou s is expec ed o yield,
physically, unaccep able esul s; namely, mo ing along
⌫cchanges he o de o s a es bu mo ing along ⌫s
lea es hem unchanged. A di e en si ua ion is encoun-
e ed when ⌳is a closed line as gi en in Figs. 6共b兲and
6共c兲. In he case o Fig. 6共b兲any con ou om A o Bis
FIG. 5. 共Colo online兲The wo diaba ic PESs as a unc ion o xand yas
calcula ed o R2=1.0 Å and
2=120°. In he igu e is emphasized he
共physical兲diaba ic po en ial, which leads in a con inuous way om he
uppe adiaba ic PES o he lowe one.
FIG. 6. A schema ic pic u e showing con ou s ⌫in e sec ing ini e ci lines ⌳
o a ious si ua ions. Aand Ba e wo a bi a y poin s on he x-yplane. 共a兲
An open line ⌳, a con ou , ⌫c, ha in e sec s i , and a con ou , ⌫s, ha
su ounds i . 共b兲A closed line, ⌳, poin Ainside he su ounded a ea, poin
Bou side he su ounded a ea, and wo con ou s ⌫c ha in e sec ⌳.共c兲A
closed line, ⌳, and poin s Aand Bou side he su ounded a ea. ⌫cis a
con ou ha in e sec s ⌳ wo imes and ⌫sis a con ou ha su ounds i .
244302-4 Le i e al. J. Chem. Phys. 128, 244302 共2008兲
Downloaded 10 Jul 2008 o 193.174.59.126. Redis ibu ion subjec o AIP license o copy igh ; see h p://jcp.aip.o g/jcp/copy igh .jsp
a⌫c-con ou and he e o e a ec s, once, he o de o
he s a es. The opposi e is ue o he case desc ibed in
Fig. 6共c兲; namely, any con ou om A o B, whe he
in e sec ing, ⌳, o no lea es he o de o he s a es
unchanged. To summa ize his analysis we say ha
open lines, in con as o closed lines, lead o unphysi-
cal si ua ions and he e o e canno exis in molecula
sys ems.
共3兲In he p esen s udy we also encoun e an uncommon
Be y- ype phase which is ⬃0 ins ead o being an in e-
ge mul iple o
. Such a esul is expec ed when a
seam and he closed ci cula con ou a e loca ed in he
same plane so ha he con ou in e sec s he seam in-
s ead o su ounds i 共by lea ing he plane兲. In Re s.
37共a兲and 37共b兲we analyzed simila si ua ions and
showed ha each in e sec ion be ween he con ou and
such a seam o ms o he co esponding 共in ou p esen
case, he angula 兲componen o he NACT, a na ow
unc ion ha in he limi ing case becomes a Di ac
␦
- unc ion, namely, 共
/2兲
␦
共
−
s兲, which yields upon
in eg a ion 共o e
兲 he ADT angle
␥
=
/2. He e
sis
he
- alue a he in e sec ion. The co esponding nu-
me ical in eg a ions along he closed con ou e ealed
in ou case ha a he icini y o he i s in e sec ion,
he alue o he ADT angle
␥
is app oxima ely +
/2
and a he second i is app oxima ely −
/2. As a esul ,
he inal Be y phase is app oxima ely ze o. This ou -
come, al hough somewha unexpec ed, is in ag eemen
wi h o he s udies o simila si ua ions.37,39,40
共4兲In o de o s eng hen he idea ha his line o cis is
indeed a seam, we managed also o calcula e he angu-
la NACTs along a con ou , which is sligh ly il ed
away om he HC–NHH plane. In his way he con-
ou , ins ead o in e sec ing he line o cis, su ounds i .
The esul s a e p esen ed in Fig. 2共c兲. I is clea ly seen
ha in con as o he wo p e ious unc ional shapes
his one is cha ac e ized by wo posi i e peaks. Calcu-
la ing he Be y phase o his case yields he alue o
3.124, a alue close o
. This issue will be u he
discussed in a o hcoming publica ion.
Finally we would like o e e o wo aspec s due o his
s udy, a physical aspec and a nume ical one:
共a兲F om he physical poin o iew his new ype o seams
is expec ed o gua an ee a much mo e e icien decay
p ocess om an uppe adiaba ic s a e o a lowe one.
The exis ence o such lines o cis—ins ead o isola ed
cis 共as exhibi ed by o he polya omic 共n⬎3兲molecules
such as HNCO 关Re . 8共b兲兴o C2H2关Re s. 4共b兲and
4共c兲兴—may enhance signi ican ly he ansi ion a e be-
ween he wo PESs.
共b兲F om a compu a ional poin o iew his inding may
a ec u u e quan um mechanical dynamical s udies o
sys ems o his ype. Thus, i such a degene acy line is
loca ed in he pa h o a wa e packe which mo es om
he uppe PES o he lowe one 共as seen in Fig. 5兲we
may pe o m, ins ead o a 共diaba ic兲 wo-s a e calcula-
ion, a diaba ic single-s a e calcula ion and s ill ob ain
accu a e esul s. In ac such a calcula ion is on i s way,
and we hope o epo he ou comes in he nea u u e.
In his espec we would like o men ion again he i s
heo e ical s udy on his sys em.24 In addi ion o de e -
mining he posi ion o he poin o degene acy 共 o he
same con igu a ion兲, Dunn and Mo okuma24 also cal-
cula ed he wo adiaba ic po en ial ene gy cu es along
he N–H dissocia ion channel, which a e p esen ed in
hei Fig. 2 as dashed lines. A good quali a i e i is
achie ed be ween hei po en ial cu es and ou PESs.
ACKNOWLEDGMENTS
M.B. and C.L. acknowledge he US-Is ael Bi-na ional
Science Founda ion o pa ly suppo ing his s udy. Á.V. ac-
knowledges OTKA G an No. T067923 and he compu a-
ional esou ces p o ided by he John- on-Neumann Ins i-
u e, Resea ch Cen e Juelich 共P ojec ID ehu01兲. I.B. and
C.L. acknowledge he pa ial suppo o he James F anck
Bina ional Ge man-Is aeli P og am in Lase -Ma e In e ac-
ion and he Is ael Science Founda ion ounded by The Is ael
Academy o Science and Humani ies.
1M. Bae , Beyond Bo n Oppenheime : Elec onic non-Adiaba ic Coupling
Te ms and Conical In e sec ions 共Wiley, Hoboken, NJ, 2006兲.
2in pa icula , see 共a兲M. S. Child, Ad . Chem. Phys. 124,1共2002兲;共b兲S.
Adhika i and G. D. Billing, ibid. 124, 143 共2002兲;共c兲R. Englman and A.
Yahalom, ibid. 124, 197 共2002兲;共d兲A. Kuppe mann and R. Ab ol, ibid.
124, 323 共2002兲;共e兲G. A. Wo h and M. A. Robb, ibid. 124,355共2002兲.
3A. Kuppe mann, in Dynamics o Molecules and Chemical Reac ions,
edi ed by R. E. Wya and Z. H. Zhang 共Ma cel, New Yo k, 1996兲,p.411;
R. Bae , D. M. Cha u z, R. Koslo , and M. Bae , J. Chem. Phys. 105,
9141 共1996兲; S. Adhika i and G. D. Billing, ibid. 111,40共1999兲.
4共a兲G. J. Halász, Á. Vibók, A. M. Mebel, and M. Bae , Chem. Phys. Le .
358, 163 共2002兲;共b兲Á. Vibók, G. J. Halász, and M. Bae , ibid. 413,226
共2005兲;共c兲G. J. Halász, Á. Vibók, S. Suhai, and M. Bae , J. Chem. Phys.
127, 244101 共2007兲.
5G. J. Halász, Á. Vibók, R. Bae , and M. Bae , J. Phys. A: Ma h. Theo .
40, F267 共2007兲.
6G. J. Halász, Á. Vibók, D. K. Ho man, D. J. Kou i, and M. Bae , J.
Chem. Phys. 126, 154309 共2007兲.
7Conical 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. In pa icula see: 共a兲D. R. Ya kony, p. 41; 共b兲H..
Köppel, p. 175; 共c兲R. de Vi ie-Riedle and A. Ho mann, p. 803.
8共a兲R. G. Sadyko and D. R. Ya kony, J. Chem. Phys. 109,20共1998兲;共b兲
D. R. Ya kony, ibid. 114, 2614 共2001兲.
9E. R. Da idson, J. Am. Chem. Soc. 99,397共1977兲.
10 E. S. K yachko, Ad . Quan um Chem. 44,119共2003兲.
11 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; M. B. Se yuk, L. Y.
Rusin, S. Ca alli, and V. Aquilan i, J. Phys. Chem. A 108,8731共2004兲;
R. Englman and T. Ve esi, Phys. Le . A 354, 196 共2006兲.
12 S. Ama an and S. Kuma , J. Chem. Phys. 128, 154325 共2008兲;S.Ama
-
an, S. Kuma , and H. Koeppel, ibid. 128, 124305 共2008兲; S. Gomez-
Ca asco, A. Aquado, M. Paniaqua, and O. Ronce o, ibid. 125, 104105
共2006兲.
13 C. Hu, H. Hi ai, and O. Sugino, J. Chem. Phys. 128, 144111 共2008兲;
127, 064103 共2007兲.
14 E. Ab ahamsson, G. C. G oenenboom, and R. V. K ems, J. Chem. Phys.
126, 184309 共2007兲; V. A. Mozhayski , D. Babiko , and A. I. K ylo ,
ibid. 124, 224309 共2006兲.
15 M. Ba ba i, S. Belz, M. Leibsche , H. Lischka, and J. Manz, Chem.
Phys. 350, 145 共2008兲.
16 P. Puza i, B. Sa ka , and S. Adhika i, J. Chem. Phys. 121,707共2004兲;B.
Sa ka and S. Adhika i, ibid. 124, 074101 共2006兲; P. O. Godsi, C. R.
E enhuis, and M. Collins, ibid. 125, 164321 共2006兲.
17 M. K eglewski, J. Mol. Spec osc. 133,10共1989兲.
18 J. V. Michael and W. A. Noyes, J. Am. Chem. Soc. 85,1228共1963兲.
244302-5 Conical in e sec ions o me hylamine J. Chem. Phys. 128, 244302 共2008兲
Downloaded 10 Jul 2008 o 193.174.59.126. Redis ibu ion subjec o AIP license o copy igh ; see h p://jcp.aip.o g/jcp/copy igh .jsp
19 M. N. R. Ash old, R. N. Dixon, M. Kono, D. H. Mo daun , and C. V. L.
Reed, Philos. T ans. R. Soc. London, Se . A 355, 1659 共1997兲.
20 G. C. G. Waschewsky, D. C. Ki chen, P. W. B owning, and L. J. Bu le ,
J. Phys. Chem. 99,2635共1995兲.
21 R. Ma om, U. Zecha ia, S. Rosenwaks, and I. Ba , Chem. Phys. Le .
440,194共2007兲.
22 R. Ma om, U. Zecha ia, S. Rosenwaks, and I. Ba , Mol. Phys. 106,213
共2008兲.
23 R. Ma om, U. Zecha ia, S. Rosenwaks, and I. Ba , J. Chem. Phys. 128,
154319 共2008兲.
24 K. M. Dunn and K. Mo okuma, J. Phys. Chem. 100, 123 共1996兲.
25 E. Kassab, J. T. Glegho n, and E. M. E le h, J. Am. Chem. Soc. 105,
1746 共1983兲.
26 H. A. Jahn and E. Telle , P oc. R. Soc. London, Se . A 161, 220 共1937兲.
27 R. Englman, The Jahn-Telle E ec in Molecules and C ys als 共Wiley-
In e science, New Yo k, 1972兲.
28 M. Bae , A. M. Mebel, and R. Englman, Chem. Phys. Le . 354, 243
共2002兲.
29 MOLPRO, a package o ab ini io p og ams designed by H.-J. We ne and P.
J. Knowles, R. Lindh, F. R. Manby, M. Schü z e al.
30 M. Bo n and J. R. Oppenheime , Ann. Phys. 84, 457 共1927兲.
31 M. Bo n and K. Huang, Dynamical Theo y o C ys al La ices 共Ox o d
Uni e si y P ess, New Yo k, 1954兲, Chap. IV.
32 M. Bae , Chem. Phys. Le . 35,112共1975兲.
33 M. Bae and R. Englman, Mol. Phys. 75, 293 共1992兲.
34 D. R. Ya kony, J. Chem. Phys. 105, 10456 共1996兲.
35 M. Bae , J. Chem. Phys. 107, 2694 共1997兲.
36 M. Bae , S. H. Lin, A. Alijah, S. Adhika i, and G. D. Billing, Phys. Re .
A62, 032506 共2000兲.
37 共a兲Z. H. Top and M. Bae , Chem. Phys. 25,1共1977兲;共b兲M. Bae , G.
Niedne , and J. P. Toennies, J. Chem. Phys. 91, 4169 共1989兲;共c兲G. D.
Billing, M. Bae , and A. M. Mebel, Chem. Phys. Le . 372,1共2003兲.
38 C. A. Mead, J. Chem. Phys. 70, 2276 共1979兲.
39 Y. Lyanda-Gelle , Phys. Re . Le . 71, 657 共1993兲.
40 T. Ve esi and R. Englman, J. Phys. B 41, 025102 共2008兲.
244302-6 Le i e al. J. Chem. Phys. 128, 244302 共2008兲
Downloaded 10 Jul 2008 o 193.174.59.126. Redis ibu ion subjec o AIP license o copy igh ; see h p://jcp.aip.o g/jcp/copy igh .jsp
