A symmetry adapted approach to molecular spectroscopy: the anharmonic oscillator symmetry model

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].