V. ROTATIONAL (MICROWAVE) SPECTROSCOPY

Transcription

1 V. ROTATONAL (MCROWAVE) SPECTROSCOPY n 934 Cleeton and Wllams observed absorptons at mcrowave frequences by NH 3, and thus MW spectroscopy began. World War proved to be a boon to the development of technology for mcrowave spectroscopy due to the mltary need for radar (RADAR, rado detecton and rangng) capabltes. The mcrowave spectrum of H O was dscovered when the US Navy found that unusually hgh attenuaton of radar sgnals occurred at certan frequences due to atmospherc water vapor. Mcrowave (MW) regon of the electromagnetc (EM) spectrum (MW oven:.45 GHz): wavelength: (0., 00) cm wavenumber: ~ (0.0, 0) cm frequency: (0.3, 300) GHz Terahertz (THz) regon: prncpally above THz, up to 3-4 THz H:\Attla\Osszestett\WORD\Otatas\Eloadaso\ElméletKéma-Angol\Wee.docx

2 n these regons of the EM spectrum pure atonal transtons domnate. Nevertheless, there are alternate causes of absorpton n the mcrowave regon: Tunnelng splttng of the nverson (vbratonal) mode of NH 3 (at 4 GHz 0.80 cm ) Nuclear hyperfne splttng of the S state of the H atom ( cm - ), leadng to MW emsson from outer space Splttng of degenerate electronc states due to the nteracton of electronc orbtal angular momentum and molecular aton: Λ-doublng of the Π / state of the OH radcal for = ½ ( cm ) H:\Attla\Osszestett\WORD\Otatas\Eloadaso\ElméletKéma-Angol\Wee.docx

3 3 Hallmars of atonal spectroscopy A) Hgh-precson data Wavenumbers and frequences of some measured atonal transtons of CO ~ (cm - ) (GHz) (GHz) B) Determnaton of geometrcal structures Hgh precson atonal constants (A 0,B 0,C 0 ) moments accurate bond lengths for parent molecule and sotopologues of nerta and bond angles [Useful readng: Nemes László: A moleulageometra meghatározása forgás spetroszópával (Kéma Újabb Eredménye, 5. ötet, 98)] H:\Attla\Osszestett\WORD\Otatas\Eloadaso\ElméletKéma-Angol\Wee.docx

4 4 C) Determnaton of permanent dpole moments n the Star effect, dscovered by Hughes and Wlson n 947, an external electrc feld s appled n the absorpton cell. Ths effect ncreases senstvty and also causes splttng of atonal energy levels n accordance wth the magntude of the dpole moment. D) Radofrequency astronomy Detecton and dentfcaton of molecules n nterstellar space from emsson and absorpton spectroscopy. Bacground energy denstes characterstc of a blac body at.7 K. Bacground radaton n the unversty s supposedly left over from the Bg Bang. Unqueness of hgh precson transton frequences s complcated by the Doppler effect. Many atonal lnes from nterstellar space reman to be characterzed. A challengng queston: are there molecules characterstc of lfe on Earth (e.g., amno acds and sugars) n space? H:\Attla\Osszestett\WORD\Otatas\Eloadaso\ElméletKéma-Angol\Wee.docx

5 5 Lot more than 00 unque neutral molecules, as well as several catons and a few anons, have been observed va ther mcrowave and mllmeterwave (emsson) spectra. Some of these chemcal speces are as follows ( 0 s a requrement, so the most mportant molecule leadng the gas-phase chemstry of cold places, H 3+, has only been dentfed recently, ts propertes could be nvestgated only n the laboratory): datomcs: OH, CO, CN, CS, SO, SS, NO, NS, CH, CH + tratomcs: H O, HCN, HNC, OCS, H S, N H +, SO, HNO, C H, etc. 0-atoms: NH CH COOH(?) 3-atoms: NC CC CC CC CC CC H H:\Attla\Osszestett\WORD\Otatas\Eloadaso\ElméletKéma-Angol\Wee.docx

6 6 Classcal mechancs of rgd aton Prncpal reference: H. Goldsten et al., Classcal Mechancs, 3 rd ed., 00 Assume that we have a rgd body whch s freely atng n space,.e, ts moton s unconstraned and no torque s appled to t. Such s the model for the atons of polyatomc molecules, and the model s called rgd or. We have the followng equatons of moton: L = = const. (conservaton of angular momentum) T T T ω ω L L const. (conservaton of energy) where L s the atonal angular momentum (a constant vector n the lab-fxed frame) s the angular velocty of aton, drected along the nstantaneous axs of aton (a tme-dependent vector n the lab-fxed frame) s the nerta tensor (a 3 3 tme-dependent matrx n the lab-fxed frame) T s the netc energy of the aton (a constant snce energy s conserved and there s no potental energy nvolved) H:\Attla\Osszestett\WORD\Otatas\Eloadaso\ElméletKéma-Angol\Wee.docx

7 7 Translatonal moton along the x axs Analoges between translaton and aton Rotatonal moton around atonal axs z coordnate: x angle (angular dstorton): velocty: d x v x angular velocty of aton: z d t d vx d x d acceleraton: ax angular acceleraton: z d z d t d t d t d t mass: m moment of nerta: m d x d equaton of moton: F x m equaton of moton: M z d t d t (lnear) momentum: px mvx angular momentum: Lz z netc energy: T mv x netc energy: T z d d t H:\Attla\Osszestett\WORD\Otatas\Eloadaso\ElméletKéma-Angol\Wee.docx

8 8 Lab-fxed and molecule-fxed axs systems H:\Attla\Osszestett\WORD\Otatas\Eloadaso\ElméletKéma-Angol\Wee.docx

9 9 n each nstant n tme one can fnd a untary matrx U(t) whch relates the body-fxed (x, y, z) coordnate system to the lab-fxed (X, Y, Z) system. The equatons of moton n the body-fxed frame are L ( t) ω( t) and T T ω( t) ω( t ) constant, where L ( t) U( t) L (L constant), ω ( t) U( t) ω( t), and T. U( t ) ( t) U( t) The convenence of choosng the body-fxed frame s that s a constant wth respect to tme. H:\Attla\Osszestett\WORD\Otatas\Eloadaso\ElméletKéma-Angol\Wee.docx

10 0 Equatons for the moment of nerta tensor For a freely atng object, the orgn of the body-fxed frame s the center of mass (abbrevated as CM or sometmes COM). For a rgd body consstng of N partcles wth masses m, elements of the moments of nerta are as follows (comng from the defnton ): of the angular momentum as H:\Attla\Osszestett\WORD\Otatas\Eloadaso\ElméletKéma-Angol\Wee.docx

11 H:\Attla\Osszestett\WORD\Otatas\Eloadaso\ElméletKéma-Angol\Wee.docx moments of nerta: N xx z y m ) ( N yy z x m ) ( N zz y x m 33 ) ( products of nerta: N yz z y m 3 3 N xy y x m N xz z x m 3 3

12 To vsualze the nerta tensor we can do the followng. For every possble axs through the COM compute the moment of nerta N m r (r s the perpendcular dstance from the partcle of mass m to the axs ). Lay off on each sde of the center of mass a dstance numercally equal to ellpsod of nerta. /. The resultng surface n three dmensons s the H:\Attla\Osszestett\WORD\Otatas\Eloadaso\ElméletKéma-Angol\Wee.docx

13 H:\Attla\Osszestett\WORD\Otatas\Eloadaso\ElméletKéma-Angol\Wee.docx 3 By choosng the body-fxed frame (embeddng) approprately, we can mae dagonal: c b a zz zy zx yz yy yx xz xy xx n ths case, the netc energy expresson grossly smplfes, ) ( ) ( ) ( ) ( z y yz z x xz y x xy z zz y yy x xx T t t T ω ω becomes c c b b a a T. The axes whch dagonalze the nerta tensor n ths manner are called the prncpal axes of aton. The resultng a, b, and c values are the prncpal moments of nerta.

14 4 Algorthm to determne the prncpal axes and prncpal moments () Attach an arbtrary coordnate system to the (rgd) body and compute the coordnates of the center of mass: R COM N m M r () Translate the coordnate system to places the COM at the orgn. Recompute all coordnates of the partcles n ths reference frame. (3) Compute the nerta tensor n ths body-fxed, COM reference frame. (4) Fnd the egenvalues and egenvectors of : v v, a,b,c The v are the prncpal axes and the are the prncpal moments. H:\Attla\Osszestett\WORD\Otatas\Eloadaso\ElméletKéma-Angol\Wee.docx

15 5 Facts whch ad the rapd determnaton of the prncpal axes: () The prncpal axes are mutually orthogonal. (The nerta tensor s symmetrc.) () Any plane of symmetry contans two prncpal axes and s perpendcular to a thrd prncpal axs. (3) Any axs of aton s a prncpal axs. f t s a C n axs for n >, the plane perpendcular to the axs s a prncpal plane correspondng to degenerate moments of nerta. Examples:. H O placed n the yz plane yz symmetry plane x must be a prncpal axs (based on ()) xz symmetry plane y s than also a prncpal axs (based on ()) (x,y) prncpal axes z s a prncpal axs (based on ()). Ammona C n (z) z s a prncpal axs (based on (3)) n = 3 (x,y) prncpal plane. Then any two orthogonal vectors of the (x,y) planecan be chosen as prncpal axes. Axes x and y s one of the possble choces. H:\Attla\Osszestett\WORD\Otatas\Eloadaso\ElméletKéma-Angol\Wee.docx

16 6 Spectroscopc atonal constants and prncpal mooments of nerta t s a generally accepted conventon that the prncpal moments of nerta correspondng to (a,b,c) must satsfy the followng relatonshp: c b a The atonal constants, whch wll be used to help to nterpret atonal spectra, are defned accordng to the same conventon as: ~ A hc a ~ B hc b ~ A ~ B ~ C ~ C hc c H:\Attla\Osszestett\WORD\Otatas\Eloadaso\ElméletKéma-Angol\Wee.docx

17 7 Example: Rotatonal constants of H O (C v, placed n the yz plane) O : r = {0, 0, 0} (natural choce) H : r = {0, r sn(/), r cos(/)} (r = r OH and = HOH) H 3 : r 3 = {0, r sn(/), r cos(/)} Step : Based on R r determne the COM: tp N m M M, the total mass: M = m O + m H R COM m m O H (0, 0, m H r cos( / ) Step : ntroduce the new coordnates: O : r = r {0, 0, ( mh / M )cos( / ) } H : r = r {0, sn(/), (m O /M) cos(/)} H 3 : r 3 = r {0, sn(/), (m O /M) cos(/)} H:\Attla\Osszestett\WORD\Otatas\Eloadaso\ElméletKéma-Angol\Wee.docx

18 H:\Attla\Osszestett\WORD\Otatas\Eloadaso\ElméletKéma-Angol\Wee.docx 8 Step (3): Determnaton of the elements of the nerta tensor: yy = z m z x m ) ( = O H O H ) / ( cos 4 ) / ( cos M m m r M m m r = ) / ( cos ) (4 ) / ( cos O H O H O H r M m m m m m m M r zz = ) / ( sn ) ( H r m m y y x m xx = zz yy xx z y m ) (, whch holds, as can be easly shown, for all planar molecules.

19 9 m x y 0 and m x z 0, xy yz r sn( / )( m m y z 0 O xz / M )cos( / ) r sn( / )( m O / M )cos( / ) 0 Step (4): Note that the moment of nerta tensor s already dagonal, so the egenvalues are just the dagonal elements and the egenvectors are smply the vectors (, 0, 0), (0,, 0) and (0, 0, ). Therefore, the prncpal axes are (x, y, z), as deduced above usng the rules ()-(3) concernng prncpal axes. Recommendaton: t s hghly advsable to use Facts ()-(3) concernng prncpal axes n settng up the ntal coordnate system. f there s any pont-group symmetry operaton characterzng the molecule, an approprate choce of the ntal axs system smplfes the tas of dagonalzng the nerta tensor. H:\Attla\Osszestett\WORD\Otatas\Eloadaso\ElméletKéma-Angol\Wee.docx

20 0 We have the followng data for the water molecule [ u = m( C)/]: m O = u, m H = u, r e (O H) = Å, e HOH = 04.5 (not exactly the correct equlbrum structure of H 6 O, but close) Based on the relaton just obtaned: yy = u Å = g m zz =.549 u Å = g m = u Å = g m xx yy zz Note the assocaton y a, z b, x c. H:\Attla\Osszestett\WORD\Otatas\Eloadaso\ElméletKéma-Angol\Wee.docx

21 The assocated equlbrum atonal constants ( cm = MHz and MHz = cm ): ~ e A = MHz = cm - ~ B e= MHz = cm - ~ C e = MHz = cm - ~ The equlbrum atonal constants e ~ ~ (measurable) atonal constants 0 ~ ~ A, B e, and C e are to be dstngushed from the ~ A, B 0, and C 0, whch nclude zero-pont vbratonal effects and thus are slghtly dfferent from ther equlbrum counterparts (quantum chemstry provdes outstandng estmates for the small but sgnfcant dfferences). H:\Attla\Osszestett\WORD\Otatas\Eloadaso\ElméletKéma-Angol\Wee.docx

22 (A) Lnear molecules Molecular Rotatonal Types c = b a = 0 There exsts only one unque atonal constant, whch s labelled B and s equal to C. (B) Sphercal tops c = b = a = A = B = C There exsts only one unque atonal constant, whch s labelled as B. Examples: CH 4, SF 6, basetball. H:\Attla\Osszestett\WORD\Otatas\Eloadaso\ElméletKéma-Angol\Wee.docx

23 3 (C) Symmetrc tops Case (): Prolate symmetrc top c = b a 0 A B = C There exsts two unque atonal constants, A and B, where A B. Examples: CH 3, CH 3 Cl, allene, Amercan football. Case (): Oblate symmetrc top c b = a 0 A = B C There exsts two unque atonal constants, B and C, where B C. Examples: NH 3, chloroform (CHCl 3 ), benzene, frsbee. Planar oblate symmetrc tops, f they are truly symmetrc (thus possess at least one atonal axs C n wth n > ) do not have a dpole moment and thus have no pure atonal spectra. (D) Asymmetrc tops c b a 0 A B C There exst three unque atonal constants, A, B, and C, where A B C. Most molecules belong to ths top type. Examples: H O, H CO, C H 4. f c b a, the molecule s a prolate near symmetrc top (e.g., HNCO), whle f c b a, the molecule s an oblate near symmetrc top (e.g., furan, C 4 H 4 O). H:\Attla\Osszestett\WORD\Otatas\Eloadaso\ElméletKéma-Angol\Wee.docx

24 4 Group theory and atonal types The character table for the pont group of the molecule ndcates whether t s an asymmetrc top, a symmetrc top, or a sphercal top by showng how the (a, b, c) axes, whch correspond to some choces of (x, y, z), transform. Asymmetrc top: a, b, and c are unque and transform as a one-dmensonal rreducble representaton (e.g., H O, H CO). Symmetrc top: (a, b) or (b, c) par transforms as a two-dmensonal rreducble representaton. The other axs s unque and transforms as a one-dmensonal rrep (e.g., C 6 H 6, CH 3 Br). Sphercal top: (a, b, c) trplet transforms as a three-dmensonal rrep (e.g., CH 4, SF 6 ). H:\Attla\Osszestett\WORD\Otatas\Eloadaso\ElméletKéma-Angol\Wee.docx

25 5 Quantum mechancs of rgd aton Consder a molecule atng n sotropc, feld-free space wth no appled torque. After nvong the Born Oppenhemer (BO) approxmaton (wthout whch electronc and nuclear motons would be scrambled n complcated molecular Hamltonans, and extensve numercal computatons would be necessary to extract even the most qualtatve features of vbratonal and atonal structure) we wsh to solve the nuclear Schrödnger equaton: where H N T trans H E, N T N. T N vb-rovb N T V Vbratonal moton s typcally much faster than atonal moton. Thus, the vbratonaton nteracton term, T vb-, can be averaged over the gven vbratonal state, whence H, v T vb-ro T v H:\Attla\Osszestett\WORD\Otatas\Eloadaso\ElméletKéma-Angol\Wee.docx

26 6 s the atonal Hamltonan for the gven vbratonal state (for atonal moton V = 0). Consequently, N trans vb,v and the tme-ndependent Schrödnger equaton for the atonal moton s Classcally, or T H T, v E, a, v a a a, v b b b b, v c, where a,b,c are components of the atonal angular momentum n the body-fxed (molecule-fxed) frame of reference. c c c H:\Attla\Osszestett\WORD\Otatas\Eloadaso\ElméletKéma-Angol\Wee.docx

27 H:\Attla\Osszestett\WORD\Otatas\Eloadaso\ElméletKéma-Angol\Wee.docx 7 Use quantum mechancal operators: v c c v b b v a a v H,,,,, where the moments of nerta depend (slghtly) on the vbratonal state of concern characterzed by the quantum number v. Understandng ths dependence, we henceforth drop the subscrpt v. One could, n prncple, use operators for the components of the angular momentum about the space-fxed (lab-fxed) axes X Ĵ, Y Ĵ, és Z Ĵ. We now that Z Y X c b a. Note the dfferences n commutaton relatons: Z Y X, c b a, X Z Y, a c b, Y X Z, b a c,

28 8 Solutons to the atonal Schrödnger equaton H Regardless of the values of a, b, and c, Thus, a a b b c H, 0 and, 0 s also an egenfuncton of c H. Ĵ and Ĵ Z. We now from the theory of angular momenta (see also the outstandng textboo R. N. Zare: Angular Momentum, Wley-nterscence: New Yor, 988) that and H E, ( ) and = 0,,, 3,, Z Z M and M = 0,,,,, H:\Attla\Osszestett\WORD\Otatas\Eloadaso\ElméletKéma-Angol\Wee.docx

29 9 where s the quantum number for total atonal angular momentum, M s the quantum number for the projecton of on the space-fxed Z axs. Thus, for any or F(, )exp( M), where the three atonal coordnates are the three Euleran angles (,, ) ( 0 ;0 ; 0 ), defnng the orentaton of a rgd body n space. Wth the help of the Euleran angles, and some algebra, the angular momentum operator can be expressed both n the space- and body-fxed frames: cos csc cos cot sn a, sn csc sn cot cos b, c, H:\Attla\Osszestett\WORD\Otatas\Eloadaso\ElméletKéma-Angol\Wee.docx

30 30 and t can also be shown that cos cot cos csc sn X, sn cot sn csc cos Y, Z. csc csc cot csc cot. n what follows let us nvestgate the dfferent tops separately. H:\Attla\Osszestett\WORD\Otatas\Eloadaso\ElméletKéma-Angol\Wee.docx

31 3 Lnear molecules We learned the followng about rgd lnear molecules ( pencls ): c = b a = 0, thus, there s only one atonal constant, the conventon s to call t B and ts value concdes wth C. and The Hamlton operator depend only on the Euler angles (, ) but not on. Thus, H, b cot, sn YM (, ) ( ) Y (, ),. Y M (, ) M H:\Attla\Osszestett\WORD\Otatas\Eloadaso\ElméletKéma-Angol\Wee.docx

32 3 We get the followng expresson for the atonal energy levels of a lnear molecule: furthermore, E ( ), 0,,, and M 0,,,,. b All atonal levels depend only on the quantum number but not on M; thus, the levels are ( + )-fold degenerate (M can tae + values for each ). n spectroscopy, t s usual to tal about term values (unt: cm -, the same as that of B ~ ): E ~ F B ( ). hc H:\Attla\Osszestett\WORD\Otatas\Eloadaso\ElméletKéma-Angol\Wee.docx

33 33 Sphercal top We learned the followng about rgd sphercal tops ( footballs ): c = b = a = A = B = C, thus, there s only one atonal constant, usually denoted as B. The atonal Hamltonan employed n the Schrödnger equaton s thus a b c H and the tme-ndependent Schrödnger equaton becomes The egenfunctons can be wrtten as M where cos( ) ( / ) E. cos( ) exp( M) P M, P are assocated Legendre polynomals. The correspondng egenvalues are E ( ), 0,,,. H:\Attla\Osszestett\WORD\Otatas\Eloadaso\ElméletKéma-Angol\Wee.docx

34 34 Whle the atonal Hamltonan does not generally commute wth Ĵ a; for sphercal tops H, a 0. The egenvalues of Ĵ a are K, wth K 0,,,,. The quantum number K gves the atonal angular-momentum component along a molecule-fxed axs of the sphercal top. Naturally, the correspondng egenfunctons have the form exp( ) K. Hence the overall sphercal-top egenfunctons have the form H KM ( ) exp( M)exp( K). There are three quantum numbers characterzng ths top:, K, and M. Accordng to the defnton of the quantum numbers (n the present case M 0,,,, ), each atonal level s ( ) -fold degenerate. The atonal term values are E ~ F B ( ). hc H:\Attla\Osszestett\WORD\Otatas\Eloadaso\ElméletKéma-Angol\Wee.docx

35 35 Symmetrc top molecules Case : Prolate symmetrc top c = b a 0 A B = C There are two unque atonal constants, A and B, and accordng to ur conventon A B. Examples: CH 3, CH 3 Cl, allene, amercan football. H a a b b c a a b a b a a b The latter form was chosen so that we can tae advantage of the followng commutaton relatons durng soluton of the Schrödnger equaton: H, 0 and, 0 Based on our nowledge we can wrte H E ( ) a K K 0,,, Z M M 0,,, H. a H:\Attla\Osszestett\WORD\Otatas\Eloadaso\ElméletKéma-Angol\Wee.docx

36 H:\Attla\Osszestett\WORD\Otatas\Eloadaso\ElméletKéma-Angol\Wee.docx 36 The tme-ndependent Schrödnger equaton s ) ( K H b a b b a a b. Thus, we obtan the followng equaton for the energy levels of a prolate symmetrc top: b a b K E ) (. The correspondng term value expresson s ) ~ ~ ( ) ( ~ hc K B A B E F.

37 37 Case : Oblate symmetrc top c b = a 0 A = B C There are two unque atonal constants, B and C, accordng to the usual conventon B C. Followng a small and trval modfcaton of the formulas we learned for prolate symmetrc tops, we obtan F E hc ~ B ( ) ( C B) K. Rotatonal wavefunctons of symmetrc tops: KM (,, ) ~ ~ G KM ( )e M K Based on the defnton of the Euleran angles, descrbes the aton about the space-fxed Z axs (thus, M s assocated wth the projecton of on Z), whle s a aton about a body-fxed axs (K s assocated wth the projecton of on molecule-fxed z, whch s dentcal, n the case of a prolate symmetrc top, to a, the at least three-fold prncpal axs). e H:\Attla\Osszestett\WORD\Otatas\Eloadaso\ElméletKéma-Angol\Wee.docx

38 38 Degeneracy of E (,K): K = 0 (+)-fold degeneracy n M K 0 (+)-fold degeneracy n M, -fold n K The degeneracy n M can be lfted f an external electrc feld s appled on the molecule. Fnally, t s noted parenthetcally that the tme-ndependent Schrödnger equaton of rgd ors can be solved analytcally for symmetrc top molecules, as well as for lnear and sphercal-top molecules. The analytcal solutons for the atonal wavefunctons contan the three-dmensonal Wgner aton matrces, generalzaton of the onedmensonal orthogonal polynomals and the two-dmensonal sphercal harmoncs. H:\Attla\Osszestett\WORD\Otatas\Eloadaso\ElméletKéma-Angol\Wee.docx

39 39 H:\Attla\Osszestett\WORD\Otatas\Eloadaso\ElméletKéma-Angol\Wee.docx

40 40 Asymmetrc top molecules c b a 0 A B C There are three unque atonal constants, A, B, and C, wth A B C. a b c H, 0 H ( ) H, Z Z M H, a 0 0 K s not a good quantum number a Only the and M quantum numbers reman good quantum numbers. The tme-ndependent Schrödnger equaton for an asymmetrc top cannot be solved n closed form for arbtrary values of. Analytc solutons can be obtaned for low values, however, and a prescrpton for obtanng numercal results for arbtrary s straghtforward. b c H:\Attla\Osszestett\WORD\Otatas\Eloadaso\ElméletKéma-Angol\Wee.docx

41 4 Wthout further elaboraton, the fnal results, n the two lmtng cases, are: prolate symmetrc top lmt E(, K) ~ F(, K) B ( ) ( A B) K hc ~ ~ B B C ) ( oblate symmetrc top lmt E(, K) ~ F(, K) B ( ) ( C B) K hc ~ ~ B A B ) ( H:\Attla\Osszestett\WORD\Otatas\Eloadaso\ElméletKéma-Angol\Wee.docx

42 4 where Selecton rules for polyatomc atons Transton dpole matrx elements: R nm * n d q r s the dpole moment operator (sum runs over electrons and nucle). For pure atonal transtons Fnal result: (,, ) vb ( Q;,, ) el m ( r ;,,, Q) m, m e, (,, ) vb ( Q;,, ) el ( r ;,,, Q) n, n e. R nm * e ed e vb, mdq d * *, n vb R nm * vb( Q ) ( Q) vb ( Q dq, m d *, n ) The ntegral n the bracet s the expectaton value of the dpole moment. To conclude, the transton dpole ntegral wll be zero unless the molecule possesses a permanent dpole moment 0. Thus, only polar molecule exhbt pure atonal spectra. H:\Attla\Osszestett\WORD\Otatas\Eloadaso\ElméletKéma-Angol\Wee.docx

43 43 Lnear molecules Selecton rules: =, M = 0,, and 0 0. The transton wth = + corresponds to absorpton, the transton wth = corresponds to emsson (here and later on). Symmetrc top Selecton rules: =, K = 0, M = 0, and 0 0. The K = 0 restrcton arses from the fact that the molecule has no component of electrc dpole moment perpendcular to ts symmetry axs no torque by the feld. n other words, quantum number K determnes the speed of aton around the prncpal axs and ths does not have an effect on the permanent dpole. H:\Attla\Osszestett\WORD\Otatas\Eloadaso\ElméletKéma-Angol\Wee.docx

44 44 Sphercal top A specal case of the symmetrc top; thus, the same selecton rules pertan. However, by symmetry 0 = 0 holds for sphercal tops. Therefore, pure atonal spectra cannot be observed for (rgd) sphercal tops. n practce, extremely wea transtons may be observed due to centrfugal dstorton effects (devatons from a rgd body). Asymmetrc top Selecton rules: = 0,, M = 0,, and 0 0. Because K s not a good quantum number, there s no general restrcton on K. Consder the dpole moment n the molecule-fxed axs system: = ( a, b, c ). For molecules of certan symmetry not all the dpole components have a non-zero value. Based on these values, we can tal about a-type ( a 0), b-type ( b 0), and c-type ( c 0) transtons. For example, for the man water sotopologue H 6 O only b-type transtons can be observed. H:\Attla\Osszestett\WORD\Otatas\Eloadaso\ElméletKéma-Angol\Wee.docx

45 45 Appearance of atonal spectra When we substtute the selecton rules nto the energy expressons governng atons of rgd ors, we can obtan the transtons whch can be observed. For example, for the + transton and based on the energy expresson E = B ( + ) or E B( )( ) B ( ) B( ) ~ B( ). f we also tae centrfugal dstorton nto account ~ 3 B( ) 4D ( ). The spectroscopc constant D (called quartc centrfugal dstorton constant) s about - orders of magntude smaller than the atonal constant B; thus, ts contrbuton s mnuscule for small values. Thus, n the rgd-or approxmaton atonal spectra contan lnes separated by B. H:\Attla\Osszestett\WORD\Otatas\Eloadaso\ElméletKéma-Angol\Wee.docx

46 46 Example: atonal spectrum of NH 3 For ammona B = cm - ; thus, for the + transtons: = 0 3 ~ / cm Thus, measured atonal lnes are unformly separated by about 9.95 cm -. H:\Attla\Osszestett\WORD\Otatas\Eloadaso\ElméletKéma-Angol\Wee.docx

47 47 ntensty of atonal lnes We now that the observed absorpton s the dfference of stmulated absorpton and stmulated emsson. ntensty of the lnes correspondng to the gven transtons thus bascally depend upon the relatve populaton of the two levels. Accordng to the Boltzmann dstrbuton populaton of the atonal states decreases exponentally when ncreases. However, as we learned from the atonal energy expressons, almost all atonal levels are degenerate and the degree of degeneracy ncreases as ncreases. Due to ths degeneracy and the small dfference between the atonal energes t s not the level wth the smallest whch wll have the largest populaton. ntensty of absorpton lnes (assumng thermal equlbrum and the valdty of the Boltzmann dstrbuton) s determned by q exp( E / T ) ( ) exp( hcb ( ) / T ), where s the quantum number of the lower (absorbng) state. For typcal molecules at room temperature the max quantum number wth maxmum populaton can reach max 0. H:\Attla\Osszestett\WORD\Otatas\Eloadaso\ElméletKéma-Angol\Wee.docx

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49 49 A) Structure determnaton Applcatons of atonal spectroscopy Assgnment of spectral lnes to transtons between specfc atonal levels A 0, B 0, C 0 atonal constants moments of nerts. Not enough data to determne complete structure (e.g., for a symmetrc top we get only B 0 ) MW spectra of sotopcally substtuted speces are examned. sotopc substtuton wll cause a neglgble change n the equlbrum geometrc parameters (nuclear masses do not appear n the electronc Schrödnger equaton, thus the usual Born-Oppenhemer PESs are sotope ndependent). Example: structure of PCl PCl 3 : P 35 Cl3 + P Cl Cl + P Cl Cl + P 37 Cl3 3 P=00%, 35 Cl=75%. symm. top asymm. top symm. top B 0 = 67. MHz B 0 = MHz H:\Attla\Osszestett\WORD\Otatas\Eloadaso\ElméletKéma-Angol\Wee.docx

50 50 Two atonal constants are suffcent to determne the structure: r(p Cl) =.04 Å és Cl P Cl = 00. The largest source of error n structure determnatons based on mcrowave spectroscopy s related to zero-pont vbratons. For example, for the lnear OCS molecule f we choose four dfferent pars of atonal constants avalable for the dfferent sotopologues, the CO dstance wll change between.55 and.65 Å, whch s clearly unacceptable. B) Determnaton of dpole moment (Star effect) H:\Attla\Osszestett\WORD\Otatas\Eloadaso\ElméletKéma-Angol\Wee.docx

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