a Xi :physics/9611029 1 [physics.chem-ph] 29 No 1996
A Symme y-Adap ed Algeb aic App oach o Molecula
Spec oscopy
A. F ank1,2), R. Lemus1), R. Bijke 1), F. P´e ez-Be nal3) and J.M. A ias3)
1) Ins i u o de Ciencias Nuclea es, U.N.A.M.,
A.P. 70-543, 04510 M´exico D.F., M´exico
2) Ins i u o de F´ısica, Labo a o io de Cue na aca,
A.P. 139-B, Cue na aca, Mo elos, M´exico
3) Depa amen o de F´ısica A ´omica, Molecula y Nuclea ,
Facul ad de F´ısica, Uni e sidad de Se illa,
Apdo. 1065, 41080 Se illa, Espa˜na
1 In oduc ion
The s udy o molecula ib a ional spec a [1] equi es heo e ical models in o de o analyze and in e p e
he measu emen s. These models ange om simple pa ame iza ions o he ene gy le els, such as he
Dunham expansion [2], o ab ini io calcula ions, whe e solu ions o he Sch ¨odinge equa ion in di e en
app oxima ions a e sough [3, 4, 5, 6]. In gene al, he la e in ol e he use o in e nal coo dina es and he
e alua ion o o ce ield cons an s associa ed o de i a i es a he po en ial minima. While his me hod
can be eliably applied o small molecules [7], i quickly becomes a o midable p oblem in he case o
la ge molecules, due o he size o hei con igu a ion spaces. New calcula ional ools o desc ibe complex
molecules a e hus needed.
In 1981 an algeb aic app oach was p oposed o desc ibe he o o- ib a ional s uc u e o dia omic
molecules [8], subsequen ly ex ended o linea i- and ou - a omic molecules [9] and ce ain non-linea
ia omic molecules [10]. Al hough hese we e encou aging esul s, he model could no be ex ended
o polya omic molecules, due o he impossibili y o inco po a ing he unde lying disc e e symme ies.
This di icul y could be su moun ed by ea ing he ib a ional deg ees o eedom sepa a ely om he
o a ions. In 1984 Van Roosmalen e al. p oposed a U(2) based model o desc ibe he s e ching
ib a ional modes in ABA molecules [11] which was la e ex ended o desc ibe he s e ching ib a ions
o polya omic molecules such as oc ahed al and benzene like molecules [12]. Recen ly he bending modes
ha e also been included in he amewo k, which was subsequen ly applied o desc ibe C2 - ia omic
molecules [13] and he lowe exci a ions o e ahed al molecules [14], using a scheme which combines
Lie-algeb aic and poin g oup me hods. In a di e en app oach, i has also been sugges ed [15] o use a
U(k+ 1) model o he k= 3n−3 o a ional and ib a ional deg ees o eedom o a n-a omic molecule.
This model has he ad an age ha i inco po a es all o a ions and ib a ions and akes in o accoun he
ele an poin g oup symme y, bu o la ge molecules he numbe o possible in e ac ions and he size
o he Hamil onian ma ices inc ease e y apidly, making i imp ac ical o apply.
Al hough he algeb aic o mula ions ha e p o ed use ul, se e al p oblems emained, mos impo an
o which is he absence o a clea connec ion o adi ional me hods. On he o he hand, a ela ed
p oblem is he lack o a sys ema ic p ocedu e o cons uc all physically meaning ul in e ac ions in he
algeb aic space. In his pape we show ha bo h hese issues can be esol ed by conside ing a symme y-
adap ed e sion o he U(2) algeb aic model o he analysis o molecula ib a ional spec a. In his
app oach i is possible o cons uc algeb aic ope a o s wi h well de ined physical meaning, in pa icula
in e ac ions undamen al o he desc ip ion o he degene a e modes p esen in sys ems exhibi ing high
1
deg ee o symme y. The p ocedu e o cons uc hem akes ull ad an age o he disc e e symme y o
he molecule and gi es ise o all possible e ms in a sys ema ic ashion, p o iding a clea -cu connec ion
be ween he algeb aic scheme and he adi ional analyses based on in e nal coo dina es, which co espond
o he ha monic limi o he model [16].
As a es o he symme y-adap ed app oach we discuss an applica ion o h ee D3h- ia omic molecu-
la sys ems, namely H+
3, Be3and Na+
3, and o wo e ahed al molecules, he Be4clus e and he me hane
molecule. Since small molecules can in gene al be well desc ibed by means o ab ini io calcula ions [17, 18],
we emphasize he basic pu pose o his wo k. We es ablish an exac co espondence be ween con igu a-
ion space and algeb aic in e ac ions by s udying he ha monic limi o he U(2) algeb a. This gene al
p ocedu e no only allows o de i e algeb aic in e ac ions om in e ac ions in con igu a ion space, bu
can also be applied o cases o which no con igu a ion space in e ac ions a e a ailable. The D3h- ia omic
molecules cons i u e he simples sys ems whe e degene a e modes appea and whe e he new in e ac ions
in he model become signi ican . In he case o Be4we p esen a compa ison wi h ab ini io calcula ions,
while o CH4we p esen a de ailed compa ison wi h expe imen . The applica ion o hese echniques o
o he molecules, as well as a mo e comple e p esen a ion can be ound in e e ences [16, 19, 20, 21].
2 The U(2) ib on model
The model is based on he isomo phism o he U(2) Lie algeb a and he one dimensional Mo se oscilla o
H=−¯h2
2µ
d2
dx2+D(e−2x/d −2e−x/d),(1)
whose eigens a es Ecan be associa ed wi h U(2) ⊃SO(2) s a es [22]. In o de o see how his isomo phism
comes abou , conside he adial equa ion
1
2−1
d
d d
d +σ2
2+ 2φ( ) = (N+ 1)φ( ),(2)
which co esponds o a wo-dimensional ha monic oscilla o (in uni s whe e ¯h=µ=e= 1) associa ed
o a U(2) symme y algeb a [23]. By ca ying ou a change o a iable
2= (N+ 1)e−ρ,
Eq. (2) ans o ms in o
−d2
dρ2+N+ 1
22
(e−2ρ−2e−ρ)φ(ρ) = −σ
22
φ(ρ).(3)
This can be iden i ied wi h Eq. (1) a e de ining x=ρd and mul iplying by ¯h2/2µd2, p o ided ha
D=¯h2
8µd2(N+ 1)2,(4)
E=−¯h2
2µd2m2,(5)
whe e we ha e de ined m=σ/2. In he amewo k o he U(2) algeb a, he ope a o ˆ
Nco esponds o he
o al numbe o bosons and is ixed by he po en ial shape acco ding o Eq. (4), while m, he eigen alue o
he SO(2) gene a o Jz, akes he alues m=±N/2, ±(N−2)/2,.... The Mo se spec um is ep oduced
wice and consequen ly o hese applica ions he m- alues mus be es ic ed o be posi i e. In e ms o
he U(2) algeb a, i is clea om Eqs. (3-5) ha he Mo se Hamil onian has he algeb aic ealiza ion
ˆ
H=−¯h2
2µd2ˆ
J2
z=−Aˆ
J2
z.(6)
2
In addi ion, he U(2) algeb a includes he aising and lowe ing ope a o s ˆ
J+and ˆ
J−, which connec
di e en ene gy s a es, while he angula momen um ope a o is gi en by ˆ
J2=ˆ
N(ˆ
N+ 2)/4, as can be
eadily shown.
The Mo se Hamil onian o Eq. (6) can be ew i en in he mo e con enien o m
ˆ
H′=ˆ
H+Aˆ
N2
4=A
2[( ˆ
J+ˆ
J−+ˆ
J−ˆ
J+)−ˆ
N],(7)
whe e we ha e used he ela ion ˆ
J2
z=ˆ
J2−(ˆ
J+ˆ
J−+ˆ
J−ˆ
J+)/2 and added a cons an e m Aˆ
N2/4 in o de
o place he g ound s a e a ze o ene gy. The pa ame e s Nand Aa e ela ed o he usual ha monic
and anha monic cons an s ωeand xeωeused in spec oscopy. To ob ain his ela ion i is con enien o
in oduce he quan um numbe
=N
2−m , (8)
which co esponds o he numbe o quan a in he oscilla o . In e ms o , he co esponding ene gy
exp ession akes he o m
E′=−A(m2−N2
4) = −A
2(N+ 1/2) + A(N+ 1)( + 1/2) −A( + 1/2)2,(9)
om which we immedia ely ob ain
ωe=A(N+ 1) ,
xeωe=A . (10)
Thus, in a dia omic molecule he pa ame e s Aand Ncan be de e mined by he spec oscopic cons an s
ωeand xeωe.
We now conside he Ui(2) ⊃SUi(2) ⊃SOi(2) algeb a, which is gene a ed by he se {ˆ
Gi} ≡
{ˆ
Ni,ˆ
J+,i,ˆ
J−,i,ˆ
J0,i}, sa is ying he commu a ion ela ions
[ˆ
J0,i,ˆ
J±,i] = ±ˆ
J±,i ,[ˆ
J+,i,ˆ
J−,i] = 2 ˆ
J0,i ,[ˆ
Ni,ˆ
Jµ,i] = 0 ,(11)
wi h µ=±,0. As men ioned be o e, o he symme ic i educible ep esen a ion [Ni,0] o Ui(2) one can
show ha he Casimi ope a o is gi en by ~
J2
i=ˆ
Ni(ˆ
Ni+ 2)/4 [23], om which ollows he iden i ica ion
ji=Ni/2. The SOi(2) label is deno ed by mi.
3 The Be4clus e
As a speci ic example, we conside he Be4clus e , which has a e ahed al shape. D3hmolecules can
be simila ly ea ed. In he Be4case he e a e six Ui(2) algeb as in ol ed (i= 1,...,6). In he p esen
app oach each ele an in e a omic in e ac ion is associa ed wi h a Ui(2) algeb a. The ope a o s in he
model a e exp essed in e ms o he gene a o s o hese algeb as, and he symme y equi emen s o he
e ahed al g oup Tdcan be eadily imposed [14, 24]. The local ope a o s {ˆ
Gi}ac ing on bond ican be
p ojec ed o any o he undamen al i eps Γ = A1,Eand F2. Using he ˆ
Jµ,i gene a o s we ob ain he
Td enso s
ˆ
TΓ
µ,γ =
6
X
i=1
αΓ
γ,i ˆ
Jµ,i ,(12)
whe e µ=±,0 and γdeno es he componen o Γ. The explici exp essions a e gi en by
ˆ
TA1
µ,1=1
√6
6
X
i=1
ˆ
Jµ,i ,
3
ˆ
TE
µ,1=1
2√3ˆ
Jµ,1+ˆ
Jµ,2−2ˆ
Jµ,3+ˆ
Jµ,4−2ˆ
Jµ,5+ˆ
Jµ,6,
ˆ
TE
µ,2=1
2ˆ
Jµ,1−ˆ
Jµ,2−ˆ
Jµ,4+ˆ
Jµ,6,
ˆ
TF2
µ,1=1
√2ˆ
Jµ,1−ˆ
Jµ,6,
ˆ
TF2
µ,2=1
√2ˆ
Jµ,2−ˆ
Jµ,4,
ˆ
TF2
µ,3=1
√2ˆ
Jµ,3−ˆ
Jµ,5.(13)
The Hamil onian ope a o can be cons uc ed by epea ed couplings o hese enso s o a o al symme y
A1, since i mus commu e wi h all ope a ions in Td[14].
All calcula ions a e ca ied ou in a symme y-adap ed basis, which is p ojec ed om he local basis
U1(2) ⊗ ··· ⊗ U6(2) ⊃SO1(2) ⊗ ··· ⊗ SO6(2) ⊃SO(2)
↓ ↓ ↓ ↓ ↓
|[N1], . . . , [N6] ; 1, . . . , 6;Vi
(14)
in which each anha monic oscilla o is well de ined. By symme y conside a ions, Ni=N o he six
oscilla o s, i=Ni/2−mideno es he numbe o quan a in bond iand V=Pi iis he o al numbe o
quan a. The local basis s a es o each oscilla o a e usually w i en as |Ni, ii, whe e i= (Ni−2mi)/2 =
0,1,...[Ni/2] deno es he numbe o oscilla o quan a in he i- h oscilla o . The s a es wi h one quan um
V= 1 a e deno ed by |iiwi h i= 1 and j6=i= 0. Using he same p ojec ion echnique as o he
gene a o s (13), we ind he six undamen al modes
|1φΓ
γi=
6
X
i=1
αΓ
γ,i |ii.(15)
The expansion coe icien s a e he same as in Eq. (13). The s a es wi h a highe numbe o quan a |VφΓ
γi
can be cons uc ed using he Clebsch-Go dan coe icien s o Td[14, 24]. Since all ope a o s a e exp essed
in e ms o powe s o he Ui(2) gene a o s, hei ma ix elemen s can be easily e alua ed in closed o m.
The symme y-adap ed ope a o s o Eq. (13) and symme y-adap ed basis s a es a e he building blocks
o he model.
We now p oceed o expici ly cons uc he Be4Hamil onian. Fo in e ac ions ha a e a mos
quad a ic in he gene a o s he p ocedu e yields
ˆ
H0=ω1ˆ
HA1+ω2ˆ
HE+ω3ˆ
HF2+α2ˆ
VE+α3ˆ
VF2,(16)
wi h
ˆ
HΓ=1
2NX
γˆ
TΓ
−,γ ˆ
TΓ
+,γ +ˆ
TΓ
+,γ ˆ
TΓ
−,γ,
ˆ
VΓ=1
NX
γ
ˆ
TΓ
0,γ ˆ
TΓ
0,γ .(17)
No e ha we ha e no included ˆ
VA1in ˆ
H0, since he combina ion
X
Γˆ
HΓ+ˆ
VΓ=1
4N
6
X
i=1
ˆ
Ni(ˆ
Ni+ 2) ,(18)
is a cons an 3(N+ 2)/2 . The i e in e ac ion e ms in Eq. (16) co espond o linea combina ions o
he Casimi ope a o s o [14]. Howe e , o a good desc ip ion o he ib a ional ene gies o Be4i is
4
necessa y o include in e ac ions which a e ela ed o he ib a ional angula momen a associa ed wi h
he degene a e modes Eand F2. These kind o e ms is absen in he o me e sions o he model
[12, 14]. We now p oceed o show how hey can be ob ained in he p esen o malism. In con igu a ion
space he ib a ional angula momen um ope a o o he Emode is gi en by [25]
ˆ
lA2=−iqE
1
∂
∂qE
2−qE
2
∂
∂qE
1,(19)
whe e qE
1and qE
2a e he no mal coo dina es associa ed o he Emode. This ela ion can be ans o med
o he algeb aic space by means o he ha monic oscilla o ope a o s
bÆ
γ=1
√2qΓ
γ−∂
∂qΓ
γ, bΓ
γ=1
√2qΓ
γ+∂
∂qΓ
γ,(20)
o ob ain
ˆ
lA2=−ibE†
1bE
2−bE†
2bE
1.(21)
He e bE
γ=PiαE
γ,i bi, wi h a simila o m o bΓ†
γ, while he αE
γ i can be ead om Eqs. (12,13). In o de
o ind he algeb aic exp ession o ˆ
lA2we i s in oduce a scale ans o ma ion
¯
b†
i≡ˆ
J−,i/pNi,¯
bi≡ˆ
J+,i/pNi.(22)
The ele an commu a o can hen be exp essed as
[¯
bi,¯
b†
i] = 1
Ni
[ˆ
J+,i,ˆ
J−,i] = 1
Ni
2ˆ
J0,i = 1 −2ˆ i
Ni
,(23)
whe e
ˆ i=ˆ
Ni
2−ˆ
J0,i .(24)
The o he wo commu a ion ela ions o Eq. (11) a e no modi ied by he scale ans o ma ion o Eq. (22).
In he ha monic limi , which is de ined by Ni→ ∞, Eq. (23) educes o he s anda d boson commu a o
[¯
bi,¯
b†
i] = 1. This limi co esponds o a con ac ion o SU(2) o he Weyl algeb a and can be used o
ob ain a geome ic in e p e a ion o algeb aic ope a o s in e ms o hose in con igu a ion space. In he
opposi e sense, Eq. (22) p o ides a p ocedu e o cons uc he anha monic ep esen a ion o ha monic
ope a o s h ough he co espondence b†
i→¯
b†
i=ˆ
J−,i/√Niand bi→¯
bi=ˆ
J+,i/√Ni. Applying his
me hod o he ib a ional angula momen um we ind
ˆ
lA2=−i
Nˆ
TE
−,1ˆ
TE
+,2−ˆ
TE
−,2ˆ
TE
+,1.(25)
Fo he ib a ional angula momen um ˆ
lF1
γassocia ed wi h he F2mode we ind a simila exp ession.
The co esponding in e ac ions a e
ˆ
H1=g22 ˆ
lA2ˆ
lA2+g33 X
γ
ˆ
lF1
γˆ
lF1
γ.(26)
Wi h his me hod we ob ain an algeb aic ealiza ion o a bi a y con igu a ion space in e ac ions. As a
simple example, a one-dimensional ha monic oscilla o Hamil onian ˆ
Hi= (b†
ibi+bib†
i)/2, ans o ms in o
1
2N(ˆ
J−,i ˆ
J+,i +ˆ
J+,i ˆ
J−,i) = 1
N(ˆ
J2
i−ˆ
J2
0,i) = ˆ i+ 1/2−ˆ 2
i
N,(27)
5

whe e in he las s ep we used Eq. (24). The spec um o Eq. (27) has an anha monic co ec ion, analogous
o he quad a ic e m in he Mo se po en ial spec um. We a e hus subs i u ing ha monic oscilla o s by
Mo se oscilla o s.
A mo e in e es ing applica ion is o use ou model o i he spec oscopic da a o se e al polya omic
molecules. In he case o Be4 he ene gy spec um was analyzed by ab ini io me hods in [17], whe e
o ce- ield cons an s co esponding o an expansion o he po en ial up o ou h o de in he no mal
coo dina es and momen a we e e alua ed. We ha e gene a ed he ab ini io spec um up o h ee quan a
using he analysis in [25]. Fo he algeb aic Hamil onian we ake [16]
ˆ
H=ω1ˆ
HA1+ω2ˆ
HE+ω3ˆ
HF2+X12 ˆ
HA1ˆ
HE+X13 ˆ
HA1ˆ
HF2+X33 ˆ
HF22
+g33 X
γ
ˆ
lF1
γˆ
lF1
γ+ 33 ˆ
O33 + 23 ˆ
O23 ,(28)
The e ms ˆ
O33 and ˆ
O23 ep esen he algeb aic o m o he co esponding in e ac ions in [25] which a e
esponsible o he spli ing o he ib a ional le els in he (ν1, νm
2, νl
3) = (0,00,22) and he (0,11,11)
o e ones [16].
In Table I we show he he esul s o a leas -squa e i o he ib a ional ene gies o Be4wi h he
Hamil onian o Eq. (28). The .m.s. de ia ion ob ained is 2.6 cm−1, which can be conside ed o spec o-
scopic quali y. We poin ou ha in [25, 26] se e al highe o de in e ac ions a e p esen which we ha e
neglec ed. Since ou model can be pu in o a one o one co espondence wi h he con igu a ion space
calcula ions, i is in ac possible o imp o e he accu acy o he i conside ably, bu we ha e used a
simple Hamil onian han he one o [25, 26]. When no ab ini io calcula ions a e a ailable (o easible)
he p esen app oach can be used empi ically, achie ing inc easingly good i s by he inclusion o highe
o de in e ac ions [16].
We no e ha he Be4Hamil onian o Eq. (28) p ese es he o al numbe o quan a V. This is a good
app oxima ion o his case acco ding o he analysis o [25, 26], bu i is known ha Fe mi esonances
can occu o ce ain molecules when he undamen al mode equencies a e such ha (V, V ′) s a es wi h
V6=V′a e close in ene gy. These in e ac ions can be in oduced in he Hamil onian bu he size o he
ene gy ma ices g ows e y apidly, so he bes way o deal wi h his p oblem is h ough pe u ba ion
heo y.
4D3h ia omic molecules
Fo D3hmolecules we ollow a simila p ocedu e, namely, we cons uc he D3hsymme y-adap ed op-
e a o s and s a es analogous o Eq. (13,15) and ca y ou he building up p ocedu e o cons uc he
Hamil onian and s a es wi h a highe numbe o quan a wi h he app op ia e p ojec ion ope a o s and
Clebsch-Go dan coe icien s [19].
In Table II we p esen he i s o he spec a o Be3, Na+
3and H+
3up o h ee quan a. While ema kably
accu a e desc ip ions o he i s wo molecules can be achie ed using a ou -pa ame e Hamil onian, we
we e o ced o include ou addi ional highe o de e ms in he H+
3Hamil onian in o de o p ope ly
desc ibe his molecule. This is in acco dance wi h he wo k o Ca e and Meye [18], who we e o ced
o include wice as many e ms in he po en ial ene gy su ace o H+
3 han o he Na+
3molecule. The
H+
3ion is a e y “so ” molecule which, due o he ligh mass o i s a omic cons i uen s ca ies ou la ge
ampli ude oscilla ions om i s equilib ium posi ions [18].
5 The me hane molecule
We now u n ou a en ion o he CH4molecule, o which we shall make a de ailed desc ip ion. Fo
me hane we ha e ou U(2) algeb as co esponding o he C-H in e ac ions and six mo e ep esen ing he
6
H-H couplings. The assignmen s and he choice o he Ca esian coo dina e sys em a e he same as in
[14]. The molecula dynamical g oup is hen gi en by he p oduc
G=U1(2) ⊗U2(2) ⊗...⊗U10(2) .(29)
The labeling is such ha i= 1,...,4 co espond o he C-H couplings while he o he alues o i
a e associa ed wi h H-H in e ac ions. Consequen ly he e a e wo di e en boson numbe s, Ns o he
C-H couplings and Nb o he H-H couplings, which co espond o he s e ching and bending modes,
espec i ely. The e ahed al symme y o me hane is aken in o accoun by p ojec ing he local ope a o s
{ˆ
Gi}, which ac on bond i, on he i educible ep esen a ions Γ o he e ahed al g oup Td. The explici
exp essions o he Td enso s o he s e ching modes a e
ˆ
TA1,s
µ,1=1
2
4
X
i=1
ˆ
Jµ,i ,
ˆ
TF2,s
µ,1=1
2ˆ
Jµ,1−ˆ
Jµ,2+ˆ
Jµ,3−ˆ
Jµ,4,
ˆ
TF2,s
µ,2=1
2ˆ
Jµ,1−ˆ
Jµ,2−ˆ
Jµ,3+ˆ
Jµ,4,
ˆ
TF2,s
µ,3=1
2ˆ
Jµ,1+ˆ
Jµ,2−ˆ
Jµ,3−ˆ
Jµ,4,(30)
while o he bending modes we ha e
ˆ
TA1,b
µ,1=1
√6
10
X
i=5
ˆ
Jµ,i ,
ˆ
TEb
µ,1=1
2√3ˆ
Jµ,5+ˆ
Jµ,6−2ˆ
Jµ,7+ˆ
Jµ,8−2ˆ
Jµ,9+ˆ
Jµ,10,
ˆ
TEb
µ,2=1
2ˆ
Jµ,5−ˆ
Jµ,6−ˆ
Jµ,8+ˆ
Jµ,10,
ˆ
TF2,b
µ,1=1
√2ˆ
Jµ,5−ˆ
Jµ,10,
ˆ
TF2,b
µ,2=1
√2ˆ
Jµ,6−ˆ
Jµ,8,
ˆ
TF2,b
µ,3=1
√2ˆ
Jµ,7−ˆ
Jµ,9.(31)
As be o e, he algeb aic Hamil onian can be cons uc ed by epea ed couplings o hese enso s o a o al
symme y A1.
The me hane molecule has nine ib a ional deg ees o eedom. Fou o hem co espond o he
undamen al s e ching modes (A1⊕F2) and he o he i e o he undamen al bending modes (E⊕F2)
[27]. The p ojec ed enso s o Eqs. (30) and (31) co espond o en deg ees o eedom, ou o which
(A1⊕F2) a e ela ed o s e ching modes and six (A1⊕E⊕F2) o he bendings. Consequen ly we
can iden i y he enso ˆ
TA1,b
µ,1as he ope a o associa ed o a spu ious mode. This iden i ica ion makes
i possible o elimina e he spu ious s a es exac ly. This is achie ed by (i) igno ing he ˆ
TA1,b
µ,1 enso
in he cons uc ion o he Hamil onian, and (ii) diagonalizing his Hamil onian in a symme y-adap ed
basis om which he spu ious mode has been emo ed ollowing he p ocedu e o [14]. We no e ha
he condi ion on he Hamil onian ha was used in [14] o exclude he spu ious con ibu ions, does no
au oma ically hold o s a es wi h highe numbe o quan a.
Acco ding o he abo e p ocedu e, we now cons uc he Tdin a ian in e ac ions ha a e a mos
quad a ic in he gene a o s and conse e he o al numbe o quan a
ˆ
HΓx=1
2NxX
γˆ
TΓx
−,γ ˆ
TΓx
+,γ +ˆ
TΓx
+,γ ˆ
TΓx
−,γ,
7
ˆ
VΓx=1
NxX
γ
ˆ
TΓx
0,γ ˆ
TΓx
0,γ .(32)
He e Γ = A1,F2 o he s e ching ib a ions x=sand Γ = E,F2 o he bending ib a ions x=b. In
addi ion he e a e wo s e ching-bending in e ac ions
ˆ
Hsb =1
2√NsNbX
γˆ
TF2,s
−,γ ˆ
TF2,b
+,γ +ˆ
TF2,s
+,γ ˆ
TF2,b
−,γ ,
ˆ
Vsb =1
√NsNbX
γ
ˆ
TF2,s
0,γ ˆ
TF2,b
0,γ .(33)
The ze o h o de ib a ional Hamil onian is now w i en as
ˆ
H0=ω1ˆ
HA1,s +ω2ˆ
HEb+ω3ˆ
HF2,s +ω4ˆ
HF2,b +ω34 ˆ
Hsb
+α2ˆ
VEb+α3ˆ
VF2,s +α4ˆ
VF2,b +α34 ˆ
Vsb .(34)
The in e ac ion ˆ
VA1,s has no been included since, in analogy o Eq. (18), he combina ion
X
Γˆ
HΓs+ˆ
VΓs=1
4Ns
4
X
i=1
ˆ
Ni(ˆ
Ni+ 2) ,(35)
co esponds o a cons an Ns+ 2. A simila ela ion holds o he bending in e ac ions, bu in his case
he in e ac ion ˆ
VA1,b has al eady been excluded in o de o emo e he spu ious A1bending mode. The
subsc ip s o he pa ame e s co espond o he (ν1, νl2
2, νl3
3, νl4
4) labeling o a se o basis s a es o he
ib a ional le els o CH4. He e ν1,ν2,ν3and ν4deno e he numbe o quan a in he A1,s,Eb,F2,s and
F2,b modes, espec i ely. The labels lia e ela ed o he ib a ional angula momen um associa ed wi h
degene a e ib a ions. The allowed alues a e li=νi, νi−2,...,1 o 0 o νiodd o e en [27].
In he ha monic limi he in e ac ions o Eqs. (32) and (33) again a ain a pa icula ly simple o m,
which can be di ec ly ela ed o con igu a ion space in e ac ions. This limi is ob ained, as be o e, by
escaling ˆ
J+,i and ˆ
J−,i by √Niand aking Ni→ ∞, so ha
lim
Ni→∞
ˆ
J+,i
√Ni
=bi,
lim
Ni→∞
ˆ
J−,i
√Ni
=b†
i,
lim
Ni→∞
1
Ni
[ˆ
J+,i,ˆ
J−,i] = lim
Ni→∞
2ˆ
J0,i
Ni
= 1 .(36)
whe e he ope a o s biand b†
jsa is y he s anda d boson commu a ion ela ion [bi, b†
j] = δij. Applying
he ha monic limi o he in e ac ions o Eqs. (32) and (33) we ob ain
lim
Nx→∞
ˆ
HΓx=1
2X
γbΓx†
γbΓx
γ+bΓx
γbΓx†
γ,
lim
Nx→∞
ˆ
VΓx= 0 ,
lim
Ns,Nb→∞
ˆ
Hsb =1
2X
γbF2,s †
γbF2,b
γ+bF2,s
γbF2,b †
γ,
lim
Ns,Nb→∞
ˆ
Vsb = 0 .(37)
8
He e he ope a o s bΓx†
γa e gi en in e ms o he local boson ope a o s b†
i h ough he coe icien s αΓx
γ,i
gi en in Eqs. (30,31)
bΓx†
γ=
10
X
i=1
αΓx
γ,i b†
i,(38)
wi h a simila ela ion o he annihila ion ope a o s. F om Eq. (37) he physical in e p e a ion o
he in e ac ions is immedia e. The ˆ
HΓx e ms ep esen he anha monic coun e pa o he ha monic
in e ac ions, while he ˆ
VΓx e ms a e pu ely anha monic con ibu ions which anish in he ha monic
limi . In an applica ion o he ozone molecule i was ound ha hese e ms can accoun o he s ong
anha monici ies and inco po a e he e ec o Da ling-Dennison ype couplings [20].
The ze o h o de Hamil onian o Eq. (34) is no su icien o ob ain a high-quali y i o he ib a ions
o me hane. Se e al physically meaning ul in e ac ion e ms ha a e essen ial o such a i a e no
p esen in Eq. (34). They a ise in ou model as highe o de in e ac ions. I is an ad an age o he
symme y-adap ed model ha he a ious in e ac ion e ms ha e a di ec physical in e p e a ion and a
speci ic ac ion on he a ious modes [16]. Hence he addi ion o highe o de e ms and anha monici ies
can be done in a sys ema ic way. Fo he s udy o he ib a ional exci a ions o me hane we use he Td
in a ian Hamil onian [21]
ˆ
H=ω1ˆ
HA1,s +ω2ˆ
HEb+ω3ˆ
HF2,s +ω4ˆ
HF2,b +α3ˆ
VF2,s
+X11 ˆ
HA1,s 2
+X22 ˆ
HEb2
+X33 ˆ
HF2,s 2
+X44 ˆ
HF2,b 2
+X12 ˆ
HA1,s ˆ
HEb+X14 ˆ
HA1,s ˆ
HF2,b 
+X23 ˆ
HEbˆ
HF2,s +X24 ˆ
HEbˆ
HF2,b +X34 ˆ
HF2,s ˆ
HF2,b 
+g22 ˆ
lA22
+g33 X
γ
ˆ
lF1
s,γ ˆ
lF1
s,γ +g44 X
γ
ˆ
lF1
b,γ ˆ
lF1
b,γ +g34 X
γ
ˆ
lF1
s,γ ˆ
lF1
b,γ
+ 33 ˆ
Oss + 44 ˆ
Obb + 34 ˆ
Osb + 23 ˆ
O2s+ 24 ˆ
O2b.(39)
The in e p e a ion o he ωiand α3 e ms ollows om Eq. (37). The Xij e ms a e quad a ic in he
ope a o s ˆ
HΓxand hence ep esen anha monic ib a ional in e ac ions. The gij e ms a e ela ed o
he ib a ional angula momen a associa ed wi h he degene a e ib a ions. As men ioned be o e, hese
in e ac ions, which a e undamen al o desc ibe molecula sys ems wi h a high deg ee o symme y, a e
absen in p e ious e sions o he ib on model in which he in e ac ion e ms a e exp essed in e ms o
Casimi ope a o s and p oduc s he eo [12, 14]. They gi e ise o a spli ing o ib a ional le els wi h
he same alues o (ν1, ν2, ν3, ν4) bu wi h di e en l2,l3and/o l4. Thei algeb aic ealiza ion is gi en
by
ˆ
lA2=−i√21
Nb
[ˆ
TEb
−׈
TEb
+]A2,
ˆ
lF1
x,γ = +i√21
Nx
[ˆ
TF2,x
−׈
TF2,x
+]F1
γ.(40)
The squa e b acke s in Eq. (40) deno e he enso coupling unde he poin g oup Td
[ˆ
TΓ1׈
TΓ2]Γ
γ=X
γ1,γ2
C(Γ1,Γ2,Γ; γ1, γ2, γ)ˆ
TΓ1
γ1
ˆ
TΓ2
γ2,(41)
whe e he expansion coe icien s a e he Clebsch-Go dan coe icien s o Td[14, 24]. In he ha monic limi
he expec a ion alue o he diagonal e ms in Eq. (39) leads o he amilia Dunham expansion [27]
X
i
ωi( i+di
2) + X
j≥iX
i
Xij ( i+di
2)( j+dj
2) + X
j≥iX
i
gij lilj.(42)
9
Table IV: Fi o ib a ional exci a ions o CH4. The alues o he pa ame e s a e gi en in he second column o
TableIII. He e ∆E=Ecal −Eexp. The expe imen al ene gies a e aken om [28]. The wa e numbe s a e gi en
in cm−1.
Γ (ν1, ν2, ν3, ν4)Ecal Eexp ∆EΓ (ν1, ν2, ν3, ν4)Ecal Eexp ∆E
A1(1000) 2916.32 2916.48 –0.16 (0111) 5844.98
E(0100) 1533.46 1533.33 0.13 (1200) 5974.81
F2(0001) 1309.86 1310.76 –0.90 (1011) 7147.49
(0010) 3018.09 3019.49 –1.40 (0021) 7303.38
(2100) 7315.60
A1(0002) 2587.77 2587.04 0.73 (0120) 7479.48
(0200) 3063.66 3063.65 0.01 (0120) 7557.17
(0011) 4323.81 4322.72 1.09 (1020) 8833.05
(2000) 5790.13 5790 0.13 F1(0003) 3920.46 3920.50 –0.04
(0020) 5966.57 5968.1 –1.53 (0102) 4128.38 4128.57 –0.19
E(0002) 2624.14 2624.62 –0.48 (0201) 4364.39 4363.31 1.08
(0200) 3065.22 3065.14 0.08 (0012) 5620.08
(0011) 4323.09 4322.15 0.94 (0012) 5630.76
(1100) 4446.41 4446.41 0.00 (1101) 5755.58
(0020) 6045.03 6043.8 1.23 (0111) 5829.79
F1(0101) 2845.35 2846.08 –0.73 (0111) 5848.94
(0011) 4323.15 4322.58 0.57 (0210) 6061.57
(0110) 4537.57 4537.57 0.00 (1011) 7147.53
F2(0002) 2612.93 2614.26 –1.33 (0021) 7303.29
(0101) 2830.61 2830.32 0.29 (0021) 7343.21
(1001) 4223.46 4223.46 0.00 (1110) 7361.79
(0011) 4321.02 4319.21 1.81 (0120) 7518.70
(0110) 4543.76 4543.76 0.00 (0030) 8947.65 8947.95 –0.30
(1010) 5845.53 F2(0003) 3871.29 3870.49 0.80
(0020) 6003.65 6004.65 –1.00 (0003) 3931.36 3930.92 0.44
(0102) 4143.09 4142.86 0.23
A1(0003) 3909.20 3909.18 0.02 (0201) 4349.01 4348.77 0.24
(0102) 4131.92 4132.99 –1.07 (0201) 4378.38 4379.10 –0.72
(0300) 4595.26 4595.55 –0.29 (1002) 5523.80
(1002) 5498.66 (0012) 5594.92 5597.14 –2.22
(0012) 5617.16 (0012) 5620.68
(0111) 5836.11 (0012) 5632.36
(1200) 5973.26 (1101) 5740.86
(1011) 7147.56 (0111) 5830.28
(0021) 7300.85 (0111) 5848.46
(0120) 7562.91 (0210) 6054.58
(3000) 8583.81 (0210) 6067.03
(1020) 8727.97 (2001) 7094.16
(0030) 8975.64 8975.34 0.30 (1011) 7145.84
A2(0102) 4161.52 4161.87 –0.35 (0021) 7266.11
(0300) 4595.28 4595.32 –0.04 (0021) 7303.38
(0111) 5844.61 (0021) 7344.87
(0120) 7550.53 (1110) 7365.83
E(0102) 4105.22 4105.15 0.07 (0120) 7514.67
(0102) 4152.15 4151.22 0.93 (2010) 8594.90
(0300) 4592.13 4592.03 0.10 (1020) 8786.05
(1002) 5535.04 (0030) 8907.91 8906.78 1.13
(0012) 5620.36 (0030) 9045.36 9045.92 –0.56
(0111) 5836.45
16