SYMMETRY CONCEPT APPLIED IN MOLECULAR SPECTROSCOPY

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1 SYMMETRY CONCEPT APPLED N MOLECULAR SPECTROSCOPY 1 D.O. DOROHO, C.E. HRETCANU, 3 M.C. CRASMAREANU 1 Fculty of Physcs, Al.. Cuz Unversty, ş, Romn Food Engneerng Fculty, Ştefn cel Mre Unversty, Sucev, Romn 3 Fculty of Mthemtcs, Al.. Cuz Unversty, ş, Romn Abstrct: Ths pper dels wth estblshment of the permtted trnstons n the rottonl spectr of molecule belongng to Abeln pont group C v nd hvng the two-fold s prllel to the mn s correspondng to the smllest vlue of the nert moment ( O CH - formldehyde). The molecule s consdered s beng rgd top n ths study. Keywords: pont groups of low symmetry, formldehyde, pure rotton spectrum 1. NTRODUCTON The mthemtcl concept of symmetry hs multple prctcl pplctons n scence nd technology. One epresson of symmetry s the lgebr of the pont groups. Moleculr physcs uses the pont group theory for evluton of some electro-optcl prmeters. The pont group theory s lso ppled n moleculr spectroscopy n order to estblsh the permtted trnstons when the molecule ntercts wth optcl rdtons. So, for the molecules belongng to the pont groups wth smll symmetry, the permtted trnstons n UV, VS nd R cn be esly estblshed. The AB, C 6 H 5 R, or R AB nd so fr, belong to the low symmetry Cv pont group. There re four symmetry opertons tht trnsform these molecules nto tself (Vncent-1977, Lndu-1965): (1) () reflecton n the moleculr plne σ v, reflecton n plne σ v perpendculr to the moleculr plne nd rotton wth round two-fold s C tht s the ntersecton of the two plnes of symmetry, nd dentty, operton tht does not chnge nythng. These opertons do not chnge the sptl dstrbuton of the nucle. They

2 must no chnge the electro-optc prmeters of the molecule. t results tht the electrc dpole moment of the molecules belongng to C v, must be prllel wth the two-fold s, representng the unchnged drecton by ll opertons of ths group of symmetry. When only the rotton movement s tken nto consderton, the molecules ABC nd BC, re clssfed dependng on the orentton of the moleculr electrc dpole moment reltve to the prncpl es of the nert (Lndu , Herzberg-1945, Elşevc-1966). ). molecules wth ther two-fold s prllel to the mn s correspondng to the ntermedte vlue of the nert moment ( OH ;SH ; DH ); b). molecules wth ther two-fold s prllel to the mn s correspondng to the smllest vlue of the nert moment ( O CH - formldehyde). f the energy levels nd ther symmetry for the two cses re known, the trnstons permtted n the rottonl spectrum (n fr R or/nd n mcrowve rnge) cn be esly estblshed. The molecules re consdered s symmetrc rgd tops n ther electronc ground stte (Elşevc ). The cse ) of molecules representng rgd tops wth ther symmetry s orented long the s of the ntermedte moment of nert, such s wter, sulphurous hydrogen, ) ws studed n (Herzberg , Doroho 004). The molecules OCH, OCD or OCCl (Elşevc -1966, Doroho - 004) hve the two-fold s orentted prllel to the C=O chemcl bond nd bsectng the ngle between the dentcl chemcl bonds. Ths s lso corresponds to the prncpl s of the smllest moment of nert. Formldehyde s molecule hvng ts two-fold s long the prncpl s of nert correspondng to the smllest vlue of the nert moment. The dpole moment of ths molecule s lso prllel to the symmetry s. The symmetry selecton rules for the pure rottons requre the sme symmetry for the dpole moment vrton nd the product of the egenfunctons. t results tht the followng trnstons between the pure rotton levels of formldehyde re permtted by the symmetry selecton rules:

3 (1). ( +, + ) ( +, ) nd (, ) (, + ) The dpolr molecules nterct wth dpolr rdtons. So, fr R nd mcrowve bsorpton nduces rottonl motons of these molecules. The permtted trnstons for the dpolr rdton re gven by the selecton rule (Herzberg , Elşevc -1966): () J = 0, ± 1 where J s the quntum number of the knetc moleculr moment, M p. h (3) M p = J ( J + 1); J = 0, 1,, 3,.. 4π The pure rotton energy levels of n symmetrc rgd top cn be estmted by usng the formul: 1 M p M pb M pc (4) E = + + b c As one cn see, rotton energy s n nvrnt n rpport wth the symmetry opertons of C v group. Usully, one mkes the followng nottons (Elşevc -1966, Doroho - 004): (5) A h h h = ; B = ; C = 8π 8π b 8π c The rotton constnts cn be epressed n cm -1. n reltons (3-5), h s Plnck constnt, M p, M pb M pc re the components of the knetc moment on the mn es o, ob nd oc of nert., b, c sgnfy the mn moments of nert. ESTMATON OF THE MAN MOMENTS OF NERTA: The estmton of the mn moments of nert for molecule cn be mde by usng the followng reltons (Elşevc -1966, Doroho - 004): 4 (6) m ( y + z ) = = 1 4 (7) m ( + z ) y = = 1

4 4 (8) m ( + y ) z = = 1 To eser compre the energy levels of the prolte nd oblte tops, the followng conventon s mde: the mn es of the nert moment re renmed wth o, ob nd oc n order of ncresng vlues of the nert moments. From relton (11) t results tht y nd y z. So, the prncpl s oc must be tken prllel wth oy s. The es o nd ob re tken prllel wth o, respectve wth oz, f the nequlty z s stsfed. Contrrly, f z, the es o nd ob re tken prllel wth oz, respectvely wth o. ENERGY LEVELS OF A MOLECULAR ASYMMETRC RGD TOP: n order to estmte the energy for the pure rotton moton of molecule consdered s rgd symmetrc top, the results of Wng (199) hve been used. The centrfugl dstorton of the molecule becomes mportnt for the rpd motons nd the supposton bout moleculr rgdty cn not be ppled to rottonl levels wth bg J vlues. Molecules cn be consdered s rgd tops only for the smll vlues of J. Only rottons correspondng to J 3 were tken nto consderton n ths study. For the hevy molecules wth gret vlues of, b, nd c (smll vlues of the A, B, C constnts), ths correcton could ttend vlues comprble wth the trnston wvenumber. The pure rotton energy levels were computed by usng the formul (Doroho, 004): 1 1 (9) E( J ) = ( B + C ) J ( J + 1) + A ( B + C ) W where W re the solutons of the equtons: (10) J=0: W 0 =0; (11) J=1: W = 0 nd W W + ( 1 b ) = 0 (1) J=: W 1 + 3b = 0 ; W 1 3b = 0 ; 4 = 0 W 4 W 1b (13) J=3: W 4 = 0 ; W 4 W 60b = 0 = 0 W ;

5 W ( 10 6b) W + ( 9 54b 15b ) = 0 ( b) W + ( b 15b ) = 0 W n reltons (9)-(13), b sgnfes: C B b = (14) 1 A ( B + C ) n order to obtn the energy vlues for gven J, the (J+1) equtons from (6-11) n unknown term W were solved nd the obtned solutons W used n the energy epresson (5). The lowest vlue of W for gven J s noted wthw J, the followng lowest, wth W, nd so on (Elşevc , Jones -1990). J + 1. COMPUTATONAL PROGRAM A computtonl progrm ws mde (Doroho 004). t permts to: 1. estmte the moleculr mn vlues of the moments of nert;. estmte the energy of n symmetrc rgd top, hvng the computed vlues for the prncpl moments of nert; c 3. ttch the symmetry speces for rottons round C nd C prncpl es of moments of nert; 4. decde wht re the permtted trnstons n the fr R nd mcrowve spectrl rnges, correspondng to pure rottonl moton of the molecule; 5. Attch the sttstcl weghts of rottonl levels, permttng the estmton of the ntensty of the pure rottonl spectrl lnes. Ths progrm could help spectroscopsts to ttrbute the lnes n the pure rotton spectr to gven trnston or t cn be lso used for demonstrtve courses, n ddctcl purposes. 3. RESULTS AND DSCUSSONS Formldehyde hs ts two-fold s (electrc dpole moment drecton) prllel to the s of the smllest moment of nert. ts structurl prmeters re (Rton-1997): the lengths of the chemcl

6 bonds CO=1.3A; CH =1.06A; the ngle HCH= 0 13 ; the tomc msses 4 mo = g, 4 mc = g 4 mh = g. Addtonlly the constnt h/8 π 40 = cm 1 ws consdered (Herzberg -1945, Elşevc -1966). Tble 1 Dennson s notton nd symmetry of the formldehyde energy levels J=1 J= J=3 1-1(-,+) - (+, +) 3-3 (-, +) 10 (-,-) -1 (+, -) 3- (-, -) 11 (+,-) 0 (-, -) 3-1 (+, -) 1 (-, +) 30 (+, +) (+, +) 31 (-, +) 3 (-, -) 33 (+, -) Tble Pure rotton energy levels of formldehyde J=1 J= J= The notton nd symmetry of the energy levels of formldehyde re shown n Tble 1. The pure rotton energy levels of formldehyde were computed (Tble ) nd ther symmetry hs been ssgned The trnstons permtted by reltons (1), cn be esly estblshed by usng symmetry selecton rules (Jones, 1990) for the pure rotton spectrum of ths molecule, consdered s rgd top. Tkng nto consderton the symmetry selecton rules, epressed by relton (1), the permtted trnstons gve the wvenumbers of the

7 mcrowve rnge whch re n good greement wth the epermentl dt (Wng-199, Nelsen , Rndll -1937, Dennson , Plczek-1933). 4. CONCLUSONS The pure rotton spectr of molecules clssfed to C v pont group of symmetry, hvng ther electrc dpolr moment prllel wth the s correspondng to the smllest moment of nert, cn be smulted. Errors of the results occur n the hypothess tht the molecule s rgd top, n whch the centrfugl dstorton s neglected. Epermentl evluton of the mgntude of the term descrbng centrfugl deformtons shll gve the possblty to use correcton dependent on the quntum number J nd on the structurl fetures of the studed molecule. BBLOGRAPHY A. Vncent, 1977, Moleculr Symmetry nd Group Theory, John Wley nd Sons, London, New York, Sdney, Toronto, cpt., p.. L.D. Lndu, E. M. Lfşţ, 1965, Mecnc Cuntcă, Teor nereltvstă, Ed. Tehncă, Bucureşt. G. Herzberg, 1945, Moleculr spectr nd moleculr structure, nfrred nd Rmn spectr of polytomc molecules, Vn Nostrnd Renhold Compny, New York, Cncnnt, Toronto, London, Melbourne, cpt. 4, p.4. M. A. Elşevc, Spectroscop Atomcă ş Moleculră, Ed. Acd. Romne, Bucureşt, 1966, cpt. 19.6, p D. Doroho, G. Amrnde, C. Stn, 004, Rev. Chm. (Bucurest), 55, 1, p D. Doroho, Tensor, 004, Vol. 65, no.3, p.66-7,. H. F. Jones, 1990, Groups, Representtons nd Physcs, Adm Hlher, Brstol nd New York, cpt.5, p. 78 Boc Rton, , Hndbook of Chemstry nd Physcs, CRC Press nc., New York, London, Tokyo. S.C. Wng, 199, Phys. Rev., 34, H.N. Nelsen, 1931, Phys. Rev., 38, H. M. Rndll, D.M. Dennson, N. Gnsburg, L.R. Weber, 1937, Phys. Rev. 5, D.M. Dennson, 1931, Rev. Mod. Phys., 3, G. Plczek nd E. Teller, 1933, Z. Physk, 81,