Re is a Mexicana de Física
42,
Suplemen o
1 (1996) 73-83
A symme y adap ed app oach o molecula
spec oscopy: he anha monic oscilla o symme y
model
A. FRANK',
R.
LEMUS,
R.
BIJKER
Ins i u o de Ciencias Nuclea es, Uni e sidad Nacional Au ónoma de México
Apa ado pos al 70-543, 04510 México, D.F., México
F. PÉREZ-BERNAL AND
J.M.
ARIAS
Depa amen o de Física A ómica, Molecula
y
Nuclea
Facul ad de Física, Uni e sidad de Se illa
Apa ado 1065, 41080 Se illa, España
ABSTRACT.
Ve apply he anha monic oscilla o symme y model o he desc ip ion o ib a ional
exci a ions in
V
3h
and
Td
molecules. A sys ema ic p ocedu e can be used
lO
es ablish he ela ion
be ween he algeb aic and con igu a ion space o mula ions,
by
means o which new in e ac ions
a e ound in he algeb aic model, leading o eHable spec oscopic p edic ions. Ve illus a e he
me hod o he case o ia omic
D
3h
molecules and he
T
d
ne,-clus e .
RESUMEN.
U ilizamos el modelo de sime ía de oscilado es ana mónicos pa a desc ibi las exci a-
ciones ib acionales en moléculas con sime ía
V
3h
yTd.
Un p ocedimien o sis emá ico pe mi e
es ablece la eladón en e la o mulación algeb aica y la de! espado de con igu ación. Median e
es a conexión se encuen an nue as in e acciones en el modelo algeb aico que dan luga a p edic-
ciones espec oscópicas con iables. Ilus amos el mé odo pa a el caso de moleculas ia ómicas
V3h
y pa a el cúmulo de be ilio Be4 con sime ía c aéd ica.
PACS: 33.20.Tp; 33.15.M ; 03.65.Fd
The s udy o molecula ib a ional spec a [1] equi es heo e ical models in o de o ana-
Iyze and in e p e he measu emen s [2]. These models ange om simple pa ame iza ions
o he ene gy le els, such as he Dunham expansion [21, o ab ini io calcula ions, whe e
solu ions o he Sch iidinge equa ion in di e en app oxima ions a e sough [3-5). In gen-
e 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 assoda ed o de i a i es a he po en ial minima. While his me hod can be
eliably applied o small molecules [6]' i quickly becomes a o midable p oblem in he
case o la ge Illolecules, due o he size o hei con igu a ion spaces. New calcula ional
ools o desc ibe cOlllplex molecules a e hus needed.
In 1981 an algeb aic app oach was p oposed o desc ibe he o o- ib a ionaI s uc u e o
dia omic molecules [7]' subsequen ly ex en<!cd o linea i- and Oll -a omic molecules
[81
• Also a Ins i u o de Física, Labo a o io de Cuc na aca, Apa ado pos al
139-D.
Cuc lla 'aca,
Mo e!os, México.
73
74 A.
FRANK ET AL.
and ee ain non-linea ia omie moleeules [9]. Al hough hese we e eneou aging esul s,
he model eould no be ex ended o polya omie moleeules, due o he impossibili y o
ineo po a ing he unde lying dise e e symme ies. This di ieul y eould 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) ba.sed model o desc ibe he s e ehing ib a ional modes
in ABA moleeules [10], la e ex ended o desc ibe he s e ehing ib a ions o polya omie
moleeules sueh as oe ahed al and benzene like moleeules [11]. Reeen ly he bending modes
ha e a!so been included in he amewo k, whieh was subsequen ly applied o desc ibe
C2 - ia omie moleeules [12J and he lowe exci a ions o e ahed al moleeules [13], using
a seheme whieh combines Lie-algeb aie and poin g oup me hods. In a di e en app oach,
i has also been sugges ed 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 omie moleeule. This model has he ad an age ha
i ineo po a es all o a ions and ib a ions and akes in o aeeoun he ele an poin g oup
symme y [14J, bu o la ge moleeules he numbe o possible in e ae ions and he size
o he Hami1 onian ma ices ine ease e y apidly, making i imp ae ieal o apply.
A1 hough he algeb aie o mula ions ha e p o ed use ul, se e al p oblems emained,
mos impo an o whieh is he absenee o a clea eonnee ion o adi ional me hods. On
he o he hand, a ela ed p oblem is he laek o a sys ema ie p oeedu e o eons ue all
physically meaning ul in e ae ions in he algeb aie spaee. In his pape we show ha bo h
hese issues can be esol ed by means o a gene al model o he analysis o molecula i-
b a ional spee a, he anha monie oscilla o symme y model (AOSl l). In his app oaeh
i is possible o cons ne algeb aie ope a o s wi h well de ined physieal meaning, in pa -
icula in e ae ions undamen al o he dese ip ion o he degene a e modes p esen in
sys ems exhibi ing high deg ee o symme y. The p oeedu e o cons ue hem akes ull
ad an age o he dise e e symme y o he moleeule and gi es ise o all possible e ms
in a sys ema ie ashion. The ha monic limi o he model p o ides a clea -eu connec ion
be ween he algeb aic scheme and he adi ional analyses based on in e nal coo dina es.
As a es o his app oach we apply he AOSM o he Be4 clus e [15J and o h ee
D
3h
ia omic molecula sys ems, namely Hj, I3e3 and Naj [16]. 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. Ve ha e es ablished an exac co espondence be ween
con igu a ion space and algeb aic in e ac ions by s ndying he ha monic limi o he U(2)
algeb a. This gene al p ocedu e no only allows o de i e he in e ac ions in he AOSM
om in e ac ions in eon igu a ion space, bu can also be applied o cases o which no
con igu a ion space in e ae ions a e a ailable. The 'D3h- ia omie moleeules cons i n e he
simples sys ems whe e degene a e modes appea and whe e he new in e ae ions in he
model become signi ican . In he case o I3e4, a di ec compa ison wi h ab ini io ealcula ions
will be p esen ed. The applica ion o hese echuiqnes o mo e complex sys ems, such as
he me hane moleenles, is p esen ly unde in es iga ion [19].
The model is based on he isomo phism o he U(2) Lie algeb a and he one dimensional
Mo se oscilla o
¡
2
d
2
'h
11
= ---
+D(e-
d -
2e-;1),
(1)
2" dx
2
whose eigens a es [ can be a.,"o<:ia ed wi h U(2)
:>
SO(2) s a es [20]. In o de o see how
A
SYMMETRY ADAPTED APPROACH TO MOLECULAR.. .
75
his isomo phism comes abou , conside he adial equa ion
I ( I
d d
a
22)
- ---d
-
d
+
2"
+
<,i>( )
=
(N
+
I)<,i>( ),
2
(2)
which co esponds o a wo-dimensional ha monic oscilla o (in uni s whe e h
=
J1,
=
e
=
1)
associa ed o a
U(2)
symme y algeb a [21]. By ca ying ou a change o a iable
Eq. (2) ans o ms in o
[d2 (N+I)2 ] (a)2
- d
p
2
+
-2-
(e-
2p -
2e-
P)
<,i>(p)
= -
"2
<,i>(p),
(3)
which can be iden i ied wi h (1) a e de ining x
=
pd
and mul iplying by
h
2
/2J1,d
2,
p o ided
ha
(4)
[=
h
2
2
- 2J1,d
2
m ,
(5)
whe e we ha e de ined m
=
a
/2. In he amewo k o he
U(2)
algeb a, he ope a o
IV
co esponds o he o al numbe o bosons and is ixed by he po en ial shape acco ding
o (4), while m, he eigen alue o he 50(2) gene a o
i"
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 n- alues mus be es ic ed o be posi i e. In e ms o he U(2) algeb a,
i is elea om (3-5) ha he Mo se Hamil onian has he algeb aic ealiza ion
(6)
In addi ion, he U(2) algeb a ineludes he aising and lowe ing ope a o s
.i+, .i_,
which
connec di e en ene gy s a .es in (3), while he angula momen um ope a o is gi en by
j2
=
Ñ(Ñ
+
2)/4, as can be eadily shown.
The Mo se Hamil onian (6) can be ew i en in lle mo e con enien onn
(i)
wlle e we ha e used he ela ion
.i¡
=
.i
2-
(';+.L +';_';+)/2 and add"d a cons an
e m AÑ2/4 in o de o place he g ound s a e a ze o ene gy. The pa ame e s Nand A
appea ing in
(i)
a e ela ed o he usual ha monic and anha monic cons an s
We
and
XeWe
76 A.
FRANK ET AL.
used in spec oscopy [7). To ob ain his ela ion i is con enien o in oduce he quan um
numbe
N
V=--ln
2 ' (8)
which co esponds o he numbe o quan a in he oscilla o [211. In e ms o u, he
co esponding ene gy exp ession akes he o m
(N2) A
E'
=
-A
m2-
4
=
-2(N
+1/2) +
A(N
+l)(u +1/2) -
A(u
+1/2)2,
om which we immedia ely ob ain
W
e
=
A(N
+1),
XeWe=A.
(9)
(10)
Thus, in a dia omic molecule he pa ame e s
A
and
N
can be de e mined by he spec n-
scopic cons an s W
e
and XeWe-
We now conside he U,(2) :J 5U,(2) :J 50,(2) algeb a, which is gene a eo by he se
{G;}
==
{Ñ"
j+,;,
L",
jo,;},
sa is ying he commu a ion ela ions
[Ñ
i,
j~,d
=
O,
(11)
wi h
J1
=
:l:, O. As men ioned be o e, o he symme ic i educible ep esen a ion
[N;,
O]
o U;(2) one can show ha he Casimi ope a o is gi en by [2111;2
=
Ñ,UV,
+
2)/4, om
which ollows he iden i ica ion
j,
=
N;j2.
The 50,(2) label is deno ed by
m,.
In he algeb aic app oach eaeh ele an in e a omic in e ac ion is associa ed wi h a Ui(2)
algeb a [11]. As a speci ic example, we eonside he ne. clus e , whieh has a e ahed al
shape. 1)3. molecules can be simila ly ea ed. In he ne. case he e a e six Ui(2) algeb as
in ol ed (i
=
1, ... ,6). 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
T
dcan be
eadily imposed [13,22]. The local ope a o s
{G;}
ae ing on bond
i
can be p ojec ed o
any o he undamen al i eps
=
Al, E
and
F2.
Using he
j~"
gene a o s (11) we ob ain
he
T,¡
enso s
(12)
whe e
JI
=
:l:, O and
"1
deno es he componen u
.
The explici . exp essions a e gi en by
1
6 •
Al '"
J
~,1
=
.j6
L. ~,"
1=1
;'1
=
2~
(j",l
+
j",2 - 2j,.,3
+
.J",. -
2.J~,5
+
.J"
,6) ,
A
SYMMETRY ADAPTED APPROACH TO MOLECULAR •..
77
(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
Al,
since i mus commu e wi h aH ope a ions in
T. .
This is accomplished
by means o he T. -Clebsch-Go dan coe icien s [13,22,231.
AH 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) (9 ... (9 U6(2)
::>
501(2) (9 ... (9 506(2)
::>
50(2)
! !! !!
IINd, ... ,
[N6]; VI, ... , V6;
V) (14)
in which each anha monic osciHa o is weH de ined. I3y symme y conside a ions,
Ni
=
N
o he six oscilla o s,
Vi
=
N;J2 -
mi
deno es he phonon numbe in bond
i
and V
=
Li
Vi
is he o al numbe o phonons [13,21]. The one-phonon s a es
V
=
1 a e deno ed by
I
i)
wi h
Vi
=
1 and
Vj¡ i
=
O. Using he same p ojec ion echnique as o he gene a o s (13),
we ind he six undamen al modes
6
14>;
=
L
,';';1
i).
i=1
(15)
The expansion coe icien s a e he same as in (13). The highe phonon s a es
4>;
can also
be cons uc ed using he Clebsch-Go dan coe icien s o
T.
[13,22]. 5ince aH ope a o s a e
exp essed in e ms o powe s o he U,(2) gene a o s, h,ei ma ix elemen s can be easily
e alua ed in closed o mo The symme y-adap ed ope a o s (13) and s a es (15) a e he
building blocks o he model. l 'ole ha o mo e complex molecules, he me hod aHows
he exac elimina ion o spu ious s a es [1U].
Ve now p oceed o expici ly consl uc he I3e4 Hamil onian. Fo in e ac ions lha a e
a mos quad a ic in he gene alo s he p ocedu e yields
(16)
wi h
(17)

78
A.
FRANK ET AL.
No e ha we ha e no included
V
Al
in (16), since he combina ion
¿(i +V )
=
NI
¿Ñ;(Ñ
i
+2),
4 ; (18)
is a cons an 3(N +2)/4. The i e in e ac ion e ms in Eq. (16) co espond o linea
combina ions o he ones ob ained in lowes o de in Re s.
[11,131.
Howe e , i is 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
E
and
F
2•
These kind o e ms is absen in he o me e sions
o he model
[11,131.
We now p oceed o show how hey can be ob ained in he AOSM. In
con igu a ion space he ib a ional angula momen um ope a o o he Emode is gi en
by [24]
i
A, _ . (E
a
E
a )
- -1
q¡ aq - q2 aq ' (19)
whe e q and q a 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
o ob ain
_
1
( a)
b~ -
J2 q~ - aq~ '
l( a)
b~
=
J2 q~
+
aq~ ' (20)
(21)
He e
b
=
L;
u ,;
b
i,
wi h a simila o m o b~ , while he
u ;
can be ead om (13).
In o de o ind he algeb aic exp ession o
i
A,
we i s in oduce a scale ans o ma ion
in (11)
(22)
The ele an commu a o can be exp essed as
whe e
• Ñ
i•
Vi
=
2-
Jo,¡o
(23)
(24)
The o he wo commu a o s in (11) a e no lIlodi ied by (22). In he ha lIlonic limi , which
is de ined by
Ni ~
00,
Eq. (23) educes o he s anda d boson commu a o
[b
i,
bl)
=
1.
This lilIli co esponds o a con ac ioll o SU(2) o he Weyl algeb a and can be used o
A
SYMMETRY ADAPTED APPROACH TO MOLECULAR...
79
ob ain a geome ic in e p e a ion o AOSM 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
b1 -. i1
=
j
-,;/.., liTi
and
b. -.
i.
=
j+.;/.., liTi.
Applying his me hod o he ib a ional angula momen um
(21) we ind
(25)
Fo he ib a ional angula momen um
i '
associa ed wi h he
F2
mode we ind a simila
exp ession. The AOSM o m o he co esponding Hamil onian in e ac ions is
NI
=
922
i
A,
i
A,
+
933
L
i ' i '.
1
(26)
Vi 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
Ni
=
(b1b.
+
b
i
b1)/2,
ans o ms in o
(27)
whe e in he !as s ep we used ela ion
(24).
The spec um o
(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. Ve a e hus
subs i u ing ha monic oscilla o s by Mo se oscilla o s in he AOSM.
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 I3e4 he ene gy spec um was analyzed by
ab
ini io
me hods in
[171,
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 phonons using he analysis in
[24).
Fo
he algeb aic Hamil onian we ake [15]
N
=
Wl
HA,
+
W2
HE
+
W3
HF,
+
X33
(HF,)
2
+
X
12
(HA, HE)
+X
I3
(HA, HF,)
+
933
L
i ' i ' +
33
0
33
+
13
0
13.
(28)
1
The e ms
0
33
and
0
23
ep esen he algeb aic o m o he co esponding in e ac ions
in
[24)
which a e esponsible o he spli ing o he ib a ional le els in he
(VI,
2' !)
=
(0,0°,2
2)
and he (0,1
1
,1
1)
o e ones [15).
In
Table I we show he i o Be4 using he Hamil onian (28). The i includes all
le els up o
V
=
4
phonon s a es and gi es a .m.s. de ia ion o 2.6 cm-
1,
which can
be conside ed o spec oscopic quali y.
In
Table 1 we only show he esul s o
V
:5 3
le els. We poin ou ha in
[17,241
se e al highe o de in e ac ions a e p esen which we
ha e neglec ed. Since ou mode! can be pu in o a one o one co espondence wi h he
80
A. FRANKET AL.
TABLE I. Vib a ional exci a ions o Be, using he algeb aic Hamil onian wi h pa ame e s gi en
in he ex o The ab ini io
(N -
00) spec um is gene a ed wi h he pa ame e s om
[17J.
The
ene gies a e gi en in cm-l.
V
(VI,
2", !)
Ab ini io P esen
V
(Vh
;',
~)
Ab ini io P esen
N-oo N =44 N-oo N =44
1 (1,0°,0°)
A,
638.6 637.0 3(1,0°,2°)
A,
2106.8 2105.6
(0,1',0°)
E
453.6 455.0 (1,0°,2')
E
2000.1 1999.8
(0,0°,1')
F,
681.9 678.2
F,
2056.8 2052.8
2 (2,0°,0°)
A,
1271.0 1269.2 (0,3',0°)
E
1341.3 1343.7
(1,1',0°)
E
1087.1 1087.0 (0,33,0°)
A,
1355.5 1352.5
(1,0°,1')
F,
1312.6 1308.3
A,
1355.5 1354.4
(0,2°,0°)
Al
898.3 901.4 (0,2°.',1
1)
F,
1565.5 1565.7
(0,2',0°)
E
905.4 906.1
F,
1584.4 1583.1
(0,1',1')
F,
1126.7 1125.1 (0,2',1')
F,
1578.5 1578.0
F,
1135.5 1134.1 (O,1',2°.')
E
1821.4 1821.6
(0,0°,2°)
A,
1484.0 1483.0
E
1929.5 1929.0
(0,0°,2')
E
1377.3 1373.9 (O,1',2')
A,
1813.3 1813.1
F,
1434.1 1429.6
A,
1830.8 1831.7
3 (3,0°,0°)
A,
1897.0 1896.7
F,
1874.4 1873.2
(2,1',0°)
E
1714.3 1714.3
F,
1883.2 1883.0
(2,0°,1')
F,
1937.0 1933.7 (0,0°,3,,3)
F,
2136.5 2134.2
(1,2°,0°)
A,
1526.6 1529.2
F,
2327.3 2326.9
(1,2',0°)
E
1533.7 1532.8 (0,0° ,3')
F,
2199.8 2197.1
(1,1',1')
F,
1752.2 1749.7
A,
2256.5 2254.4
F,
1761.0 1759.8
con igu a ion space calcllla ions, i is in ac possible o imp o e he accll acy o he i
conside ably, bu we ha e used a simple Hamil onian han he one o
[17,24].
Vhen no
ab
ini io ca1cllla ions a e a ailable (o easible) he AOS~I 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
[191.
The Be, Hamil onian (28) p ese ws he o al phonon-numbe
V.
This is a good ap-
p oxima ion o his case accOlding o he analysis o
117,24],
bu i is known ha Fe mi
esonances can occu o ce ain molecu!es when he undamen al mode equencies a e
such ha
(V, V')
s a es wi h
V
i
V'
a e close in ene gy. These in e ac ions can be in o-
duced in he lIamil 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 heOlY.
Fo
D3h
molecules we ollow an analogous p oced u e, nillnely, we cons uc he
D3h
symme y-adap ed ope a o s and s a es cOl esponding o (13) and (15) and ca y ou he
building-up p ocedu e o cons uc he Hamil onian amI highe phonon s a es [16]. He e
we omi he de ails Ol lack o space and only p esen he i o he ene gy spec um [16,25].
A SYMMETRYADAPTEDAPPROACHTO MOLECULAR...
81
TABLE11.Leas -squa e ene gy li o he ib a ional exci a ions o Hj, Be3 and Naj. The ene gy
di e ences
I::: ..E
=
E
h -
E
exp
a e gi en in
cm-l.
H+ Be3 Naj
3
V
( " ~)
J.E J.E J.E
1
(O,
1')
E
-1.55 0.51 0.93
(1,0°)
A,
0.42 0.02 1.95
2(0,2°)
A,
7.48 -0.74 0.37
(0,2
2)
E
-5.69 0.17 0.84
(U')
E
-0.61 0.82 1.68
(2,0°)
A,
-0.11 -0.04 1.26
3 (0,3' )
E
-4.46 -2.05 -1.19
(0,33)
A,
3.18 -1.23 -0.34
(0,33)
A
2
2.44 0.61 -0.33
(1,2°)
Al
0.66 1.90 -0.01
(1,2
2)
E
-5.00 -1.36 0.34
(2, 1')
E
4.07 0.79 -0.19
(3,0°)
A,
-1.23 -1.66 -2.06
.m.s. 5.84 1.35 1.33
Pa ame e s 84 4
In Table II we p esen AOSM li s o he spee a o 13e3,Na and H up o h ee phonons.
While ema kably aeeu a e dese ip ions o he i s wo moleeules can be aehie ed using
a ou -pa ame e Hamil onian, we had o inelude ou addi ional highe o de e ms in
he H Hamil onian in o de o p ope ly desc ibe his moleeule. This is in aeeo danee
wi h he wo k o Ca e and Meye [18], who we e o eed o inelude wiee as many e ms
in he po en ial ene gy su a'Ce o H han o he Na moleeule. The H ion is a e y
"so " moleeule whieh, due o he ligh mass o i s a omie eons i uen s ea ies ou la ge
ampli ude oseilla ions om i s equilib ium posi ions [18].
In ano he es o he model we s udied he ib a ional ene gies o wo ozone iso opes,
16
0
3 and
18
0
3 [26]. A leas -squa e i o all published expe imen al le els (up o en
quan a) yields a .m.S. de ia ion o 2.5 and 1.0 em-', espee i ely.
A s ill ine es o he model is o use he wa e une ions o e alua e in a ed and
Raman ansi ions. The algeb aie ealiza ion o he ansi ion ope a o s can be ob ained
om hei exp ession in eonligu a ion spaee using he la ge
N
eonnee ion, o pu ely al-
geb aieally by hei enso ial p ope ies unde he ele an poin g oup [19]. Ve ema k
ha he nadel can also be ex ended o include
i e
o a ional deg ees o ee Iom,
by
cou-
pling he ib a ional wa e une ions o o a ional s a es p ope ly symme ized o ca y
he poin g oup ep esen a ions [24].