Calculation of the Herman Wallis effect in vibrational overtone transitions in a linear molecule: Comparison with HCN experimental results

Transcription

1 Calculaton of the Herman Walls effect n vbratonal overtone transtons n a lnear molecule: Comparson wth HCN expermental results Danele Romann a) and Kevn K. Lehmann Department of Chemstry, Prnceton Unversty, Prnceton, New Jersey Receved 6 May 995; accepted 28 March 996 The hgh senstvty of cavty rng-down spectroscopy has allowed us to observe a few perpendcular vbratonal overtone transtons of HCN n the vsble. These transtons dsplay a szable Herman Walls effect, that s an asymmetry n the relatve ntenstes of the R and P branch lnes. We have developed a theory for the frst-order Herman Walls effect based upon usng varatonal vbratonal wave functons but treatng the vbraton rotaton nteracton by frst-order perturbaton theory. In the specfc case of perpendcular transtons, the frst-order effect s domnated by Corols mxng of and overtone states. We used the emprcal energy surface by Carter, Mlls, and Handy J. Chem. Phys. 99, restrcted to the stretchng degrees of freedom. Bendng was ncluded by multplcaton of these stretchng wave functons by harmonc wave functons of the bend. Vbratonal transton moments were calculated usng a polynomal surface ft to ab nto CCSD T dpole moment ponts by Botschwna et al. Chem. Phys. 90, and prvate communcaton. We expected that ths treatment would be accurate but the calculated Herman Walls effect s about one order of magntude too large. To gan further nsght nto the poor agreement between theory and experment, we have calculated the senstvty of the Herman Walls coeffcent and of the transton moment to the dpole and energy surface parameters. From ths, t appears that the dpole surface, whle producng accurate band ntenstes, could at the same tme be nadequate to account for the Herman Walls effect. A smlar possblty stands for the energy surface, whch however s hghly constraned by the requrement to ft the observed band orgns. 996 Amercan Insttute of Physcs. S I. INTRODUCTION Vbratonal spectroscopy has long been used to study the potental energy and dpole moment functons of polyatomc molecules. Tradtonal IR studes, however, yeld nformaton only about the potental curvature and dpole dervatves near the potental mnmum. Study of the overtone spectrum expands the regon sampled by the vbratonal wave functons and thus gves more global nformaton on the potental and dpole surfaces. However, at the hgher energes, the tradtonal perturbatve treatments for dealng wth anharmonc effects become suspect, and precse comparsons between theory and experment requres varatonal calculatons for the vbratonal wave functons. Because of the exponental ncrease of the computatonal cost of such calculatons wth the number of vbratonal degrees of freedom, tratomc molecules provde the best systems for comparsons. The most careful comparsons between theory and experment have been made for the smple hydrdes, H 2 O and HCN. Work n our laboratory has focused on the vsble and near-ir spectrum of HCN. Recently, we reported,2 on the extenson of the overtone spectrum usng the technque of cavty rng down spectroscopy. 3 5, The observed band orgns, rotatonal constants and sotopc shfts were compared wth the varatonal predctons from an emprcal potental a Current address: Laboratore de Spectrométre Physque-CNRS URA 08, Unversté J. Fourer/Grenoble, B.P Sant Martn d Hères Cedex, France. energy surface PES by Carter, Mlls, and Handy CMH 6 that was optmzed to the prevously observed lower energy bands of HCN. The agreement was excellent, wth the emprcal potental even correctly predctng weak anharmonc resonances of a type not prevously observed. In addton, we used the varatonal wave functons and the ab nto dpole moment surface DMS by Botschwna et al. 7 to calculate vbratonal transton moments and thus ntegrated band ntenstes. Agan, excellent agreement between theory and experment was found. Based upon ths work, we beleved that both the potental and dpole moment surfaces of HCN were well known, at least over a range of stretchng coordnates correspondng to over half of the C H bond dssocaton energy. In addton to the ntegrated band ntensty, one can examne the rotatonal dependence of the observed lne ntenstes. In the approxmaton of neglgble ro-vbratonal couplng, the ntegrated ntensty of ndvdual rotatonal lnes n a molecular vbratonal transton can be wrtten n terms of the thermal populaton dstrbuton and the rotatonal Hönl London factor. Ths factor arses from the transformaton of the dpole moment operator between the laboratory frame and the molecule-fxed frame Ref. 8, p Ths transformaton results n the electrc dpole operator beng wrtten as a product of the dpole moment as a functon of the nternal molecular coordnates alone tmes a drecton cosne relatng the two reference systems. When one neglects rotaton vbraton nteractons, the overall wave functon can be wrtten as a product of a vbratonal part and a rotatonal part. 68 J. Chem. Phys. 05 (), July /96/05()/68/3/$ Amercan Insttute of Physcs

2 D. Romann and K. K. Lehmann: Herman Walls effect n transtons 69 Ths gves the transton dpole matrx element as a product of a vbratonal transton moment ndependent of the rotatonal state, and a drecton cosne matrx element, whch depends upon the rotatonal quantum numbers and the drecton of the dpole moment alone. If rotaton vbraton nteracton s not neglgble, such a smple factorng of the transton dpole s not possble. The correlatons now present between vbratonal and rotatonal motons are reflected n the fact that the ro-vbratonal wave functon must be wrtten as sums over product terms of zeroorder wave functons for the rotatons and the vbratons. These dfferent terms n the expanson of the ro-vbratonal wave functon can lead to nterference between the correspondng contrbutons to the overall transton moment. The lowest-order effect of ths type was orgnally ntroduced for datomc molecules by Herman and Walls HW. 9 They found that for the fundamental bands, centrfugal dstorton resulted n an nterference between terms arsng from the permanent dpole as well as the dpole dervatve. Ths effect allows spectroscopc determnaton of the relatve drectons of these two components, whch s not possble from the vbratonal band ntensty alone. Subsequently, the theory of the HW effect has been extended to polyatomc molecules usng the contact transformaton method. These treatments use perturbaton theory to account for both the anharmonc and the rotaton vbraton nteractons. As such, they have been restrcted to vbratonal bands allowed n low order,.e. fundamentals and low order combnatons and ther assocated hot bands. 0, In our spectroscopc study of the vsble overtone spectrum of HCN, we found a szable HW effect n each of the three observed perpendcular bands. These are the and the transtons n H 3 CN and the n the common sotopomer. Only these perpendcular bands where measured manly because they are approxmately two orders of magntude weaker than the correspondng parallel bands havng the same change n stretchng quantum numbers. Based upon the above mentoned success of the PES and DMS n predctng the band ntenstes, we fully expected that these should be able to predct the observed HW effects as well. We hoped that the effect would be domnated by the nteractons of at most a few nearby states, and thus we could use the sgn of the observed effect as a test for the relatve drectons of the vbratonal transton moments. In addton to the perpendcular HCN bands we detected n the vsble, several low-energy combnaton bands were recently nvestgated by Mak et al. 2,3 who determned both frst- and second-order HW correctons to the rgd rotor lne strengths. Among these, the results for the fundamental bendng transton and another combnaton band, wll also be consdered here for comparson to our theoretcal values. Note that due to a dfference n the defnton of the transton moment for a degenerate state, the reported dpole moments values from Mak et al. have been dvded by 2. In ths paper we use perturbaton theory to derve expressons for the HW coeffcents n a lnear molecule n terms of the matrx elements of the dpole moment and of the ro-vbratonal couplng Hamltonan between the vbratonal egenfunctons. Numercal results for the observed perpendcular overtone transtons n HCN are then calculated usng varatonal wave functons determned by dagonalzaton of the vbratonal Hamltonan over a large product bass set expanson. Terms n the Hamltonan are ordered accordng to the power of the rotatonal operators, so the Corols nteractons are part of the frst-order perturbaton and centrfugal dstorton gves correctons n the second order. The physcal motvaton for ths approach s based upon the emprcal observaton, evdent from the spectroscopy of HCN, that whle vbratonal anharmonc effects are qute strong, each vbratonal state has a regular rotatonal spectrum, wth effectve rotatonal constants that vary smoothly wth vbratonal quantum numbers. Ths clearly ndcates that vbraton rotaton nteractons lead to energy correctons that have the form predcted by perturbaton theory, where such nteractons gve contrbutons to the vbratonal dependence of the effectve rotatonal constants. We have worked out the general perturbaton formulas for both parallel and perpendcular bands of lnear molecules. In keepng wth earler work, we fnd that the second order for the ntenstes of lnes n the R and P branches are gven by S ( A m A 2 m 2 ) 2, 9 where m s J or Jfor the R and P branch, respectvely. Here, S s connected to the zero-order band ntensty S 0 by a factor f whch s close to unty, S fs 0. Our experments and calculatons ndcate that the lnear HW coeffcent A to be wrtten smply as A from now on domnates for thermally populated rotatonal levels. The A 2 coeffcent only has contrbutons from second and hgher orders n our perturbaton expanson. In order not to make a long techncal dgresson, the dervaton of the frst order formulas of both A and f s gven n the Appendx. It s nterestng to pont out that the frst-order treatment may stll be vald even when the HW effect s qute szable. Ths can arse when consderng a vbratonal band wth a partcularly weak transton moment, such as one of the perpendcular bands n the vsble overtone spectrum of HCN. The reason s smply that a very small mxng-n of excted states wth much larger transton moments can produce a very large effect on the ntensty. A partcularly evdent and nterestng example s the parallel HCN band, whch s anomalously weak due to almost perfect accdental cancellaton of transton dpole moment and dsplays a strong HW effect. 4 Our calculated values of A for the bendng fundamental and the other low-lyng combnaton band reported by Mak et al. are n good agreement wth the expermental ones. However, for the perpendcular bands n the vsble overtone spectrum that motvated our calculaton, the calculated effect s about one order of magntude too large. Ths result was extremely surprsng gven the already proven predctve power of both PES and DMS. In order to carefully check our results, the perturbaton formulas were derved ndependently by both authors, and each also wrote ndependent computer programs to perform the calculatons. We found that, n the calculaton of the HW effect, there are two contrbutons that systematcally almost quanttatvely cancel.

3 70 D. Romann and K. K. Lehmann: Herman Walls effect n transtons FIG a and b overtone transtons n H 3 CN. The apparent weakness of the P branches s due to the HW effect. The sample pressure was 00 torr and the effectve absorpton path length was 42 km for and 60 km for The nstrumental resoluton s on the order of 0. cm full wdth, so that the observed lne wdths are domnated by pressure broadenng. The P branch lnes are marked to dstngush them from the overlappng Q lnes. In the R branch of the band, some lnes belongng to the P branch have also been marked out. Ths suggests that a better bass set for the perturbaton theory should exst, though we do not know what that bass could be. Ths cancellaton makes t possble, f not lkely, that the HW effect be much more senstve to features of the PES and DMS than the band energes and the ntegrated ntenstes. Thus, despte the fact that the PES and DMS of HCN are certanly among the most accurately known, they appear not to gve even an order of magntude estmate for the effect of vbraton rotaton nteractons on the observed spectroscopc lne ntenstes. In order to gan further nsght nto ths paradox, we have calculated the senstvty of both the Herman Walls coeffcent A and the transton moment to the DMS parameters. We fnd that the senstvty of these quanttes s qute dfferent, so that t s possble for a DMS to accurately predct the known band ntenstes whle beng nadequate to account for the Herman Walls effect. We have also calculated the senstvty to the PES parameters, whch smlarly shows that the HW effect and the transton ntenstes are senstve to dfferent aspects of the potental. However, the PES s subjected to the rather strngent requrement of accurately fttng the band orgns and rotatonal constants. On the other hand, the stuaton may be consdered smlar for the DMS, snce the polynomal expanson we use has been ft to accurately determned ab nto ponts. The problem mght then le wth the polynomal ft, not wth the accuracy of the ab nto calculatons themselves. In fact, we wll see that our senstvty analyss suggests that the order of the dpole polynomal surface s not completely converged wth respect to HW coeffcents and ntenstes. A smlar observaton holds for the PES polynomal expanson. An addtonal source of error s our approxmate treatment of bend stretch nteractons n the varatonal calculatons. Whle we expected ths to be an excellent approxmaton for the low bendng exctaton states consdered here, the exstence of near cancellatons mples that even a small error could be dramatcally amplfed n the fnal result. In the future we hope to elmnate ths source of error by repeatng our calculatons usng a full three-dmensonal varatonal calculaton that explctly accounts for bend stretch anharmonc nteractons. In any case, the present results clearly ndcate that even subtle effects can provde a radcally dfferent nsght nto our understandng of a well studed molecule. II. EXPERIMENT Our data were obtaned by cavty rng-down spectroscopy, as descrbed n our prevous papers.,2 Ths s a hghsenstvty, lnear, and absolute absorpton technque, 4 where the decay rate of pulsed laser lght njected nto a hgh-q optcal cavty s montored whle the laser wavelength s tuned. The gaseous sample s kept n the cavty and ts weak absorpton features produce changes n the lght decay rate. To quantfy the senstvty of the technque, one can ntroduce an effectve absorpton path length, proportonal to the decay tme constant. The effectve absorpton lengths for the and the spectra n the H 3 CN sotopomer were 42 and 60 km, respectvely we follow the l HCN normal mode labelng conventon n CN -n bend -n CH ). It can be seen n Fg. that as a result of the HW effect, lne ntenstes n these bands are substantally weaker n the P

4 D. Romann and K. K. Lehmann: Herman Walls effect n transtons 7 FIG a and b lne ntenstes as a functon of m, after dvdng out populaton and Hönl London factors. The HW rotatonal dependence of the R and P branch ntenstes s ftted to S ( Am) 2. Full dots are used for the R and P branches, empty squares for the Q branch, and contnuous lnes for the fts. branch, whch s partly obscured by the strong Q branch progresson. In fact, snce the Hönl London factors are twce as large n the Q branch as n the P and R branches whch are nearly the same sze, the R branch s clearly enhanced relatve to the Q branch. Unfortunately, we have observed only one other perpendcular band, that of the transton of the common -2-4 sotopomer. In Fg. 2 we plot the lne ntenstes obtaned by fttng wth ndependent Vogt profles each lne n the and H 3 CN bands, usng a procedure that works well even for strongly blended lnes. 2 The populaton and Hönl London factors have been dvded out, and the lne ntenstes are plotted as a functon of m ( J or Jn the R or P branch, respectvely. Notce that, due to the large B of these hgh overtones, many of the Q branch lnes could be resolved and ftted separately, yeldng accurate ntenstes also for the Q branch. A ft wth S ( Am) 2 s shown n the same fgure, whch reproduces the HW J-dependence to wthn expermental errors. Accordng to the HW theory, ths fttng form s accurate to frst order n the vbraton rotaton nteracton and n the case of fundamental nfrared bands of lnear molecules, the A coeffcent has been gven n terms of molecular parameters. 5,2 The A values from these fts are reported n Table I, where they are compared wth our theoretcal values. In the same table we also compare the transton moments for the same bands. The values by Mak et al. have been dvded by 2 due to dfferent defntons of the transton moment for a degenerate state. In fact, to relate the observed absorpton band ntensty I to the transton dpole moment, we use the defnton gven n Ref. 6. In ths defnton the degeneracy g of the upper level s explctly elmnated: (Debye) 0.63 I(km/mol)/ 0 (cm )/g, where 0 s the transton band orgn. On the other hand, the defnton used by Mak et al. n Ref. 2 mplctly ncludes the degeneracy factor nto the transton moment. The dfference smply reflect whether one s nterested n the vbratonal transton moment to one of the (x,y) polarzed bendng states as we are here, or to one of the states of e or f symmetry, where for a gven rotatonal branch only one s allowed. III. THEORY In the followng we wll consder the zero-order separable Hamltonan H 0 of the vbratng rgd-rotor, perturbed by vbraton rotaton nteracton, gven by the Corols couplng H c and the centrfugal dstorton H d. The mxng of the zero-order egenstates nduced by H d s on the order of the square of that nduced by H c, so t can be neglected. TABLE I. Comparson of expermental and calculated transton moments and HW factors. Expermental values for the and transtons are from Mak et al. Refs. 2 and 3 after dvson by 2 due to a dfferent defnton of the transton moment for a degenerate state. Observed Calculated Transton S Debye A S Debye f A HCN: H 3 CN: HC 5 N:

5 72 D. Romann and K. K. Lehmann: Herman Walls effect n transtons Whle ths s expected on the bass of general arguments, 7 the expressons n the Appendx have been derved keepng the H d frst-order contrbuton. We verfed numercally that all of the H d mxng coeffcents are ndeed on the order of the square of those due to H c. An addtonal reason for neglectng the centrfugal dstorton s that the contrbuton t gves to the transton moment s of perpendcular type, whle that due to H c s parallel and therefore has much larger magntude. As dscussed n the Appendx however, n the case of parallel transtons ths last stuaton s reversed, and the mxng due to H d cannot be so easly neglected. The effect of Corols couplng s to mx a small amount of and character nto the zero-order states, and some character nto the zero-order states. When calculatng the ntensty of a transton proportonal to 2 ), the and dpole moment matrx elements from the mxng nterfere wth the zeroorder term. Mxng of character, on the other hand, produces no frst-order effect on the ntenstes because s zero. Snce the mxng depends on the rotatonal quantum number, the ntenstes are J dependent, as requred by the HW effect. The and transton moments can be very large relatve to the zero-order moment whch s obvous f one consders that the overtone spectrum of HCN s domnated by parallel bands therefore even a small mxng can produce szable effects. The largest mxng coeffcents that we obtan are on the order of 0 4 J for Corols couplng (0 8 J 2 for the centrfugal dstorton, confrmng that a frst-order treatment s approprate for thermally populated levels (J 0). By symmetry, the Corols mxng wth vbratonal states whch for tratomc molecules must be of symmetry and the assocated ntensty effect can only nvolve e sublevels of the upper state, whch are those reached by R and P branches n transtons. Ths gves the above ntensty dependence I RP (J) fs 0 ( A m A 2 m 2 ) 2. On the other hand, the Q branch contans transtons to the f sublevels of the state, and frst-order Corols nteracton can mx these sublevels only wth states, wth no frstorder change n the lne ntenstes. However, mxng of the lower state wth e sublevels of other states gves a contrbuton to the Q branch transton ntenstes, wth the same global multplcatve factor f as the R and P branches, and a lowest-order squared J dependence I Q (J) fs 0 ( A Q J(J )) 2, where A Q depends only on H d and s therefore much smaller than A. For our calculatons, the vbratonal HCN egenfunctons of H 0 were obtaned usng the very accurate emprcal PES by Carter, Mlls, and Handy 6 see Table II. Due to computatonal lmtatons, we treated varatonally only the stretchng degrees of freedom. The CMH PES s gven as a truncated seres n Morse coordnates exp( r) for the bond stretches, and n even powers of the angle for the bend. Therefore, as convenent bass functons for dagonalzaton of the stretch-only problem, we used products of bound Morse egenfunctons for the CH and CN bond coordnate. All matrx elements of the poston and momentum operators n the Morse problem can be evaluated analytcally. 8 Ths s TABLE II. Force constants for the expanson n Morse coordnates of the emprcal HCN PES by Carter, Handy, and Mlls Ref. 6. The equlbrum structure s gven by r e (CH) Å and r e (CN) Å. All values are n unts of aj 0 8 J except for the Morse range parameters CH and CN whch are n unts of Å. CH V(,3) CN V(0,4) bend X x 8.37 V(5,0) bend X y V(4,) bend X xy 2.45 V(3,2) V(2,0) V(2,3) V(,) V(,4) V(0,2) V(0,5) V(3,0) V(6,0) V(2,) V(5,) V(,2) V(4,2) 0 V(0,3) V(3,3) V(4,0) V(2,4) 0 V(3,) V(,5) V(2,2) V(0,6) very convenent n settng up both the matrx H 0 to be dagonalzed and to determne the matrces of all other operators that enter the calculaton. Only product bass states wth energy expectaton value below a maxmum of cm were used, gvng a bass set of about 40 states. We checked that all the quanttes of nterest are fully converged by comparng wth results obtaned usng a bass that contans all bound Morse product states wth energy expectaton value below cm ( 400 wave functons. A frstorder anharmonc correcton accountng for exctaton of the bend s ncluded n the calculaton, followng the method ntroduced by Botschwna 9 and modfed for a local mode calculaton by Smth et al. 20 Ths results n slghtly dfferent effectve PESs for the and states, and thus slghtly dfferent vbratonal wave functons for the stretchng coordnates. The energes that we obtan are very close to the expermental values: wthn 5 cm for the and 0 cm for the states. The software s a modfed verson of the code wrtten by Dr. Alce M. Smth. 2 As s shown later below see Eqs. A7, A26, and A27, calculaton of the HW effect requres the matrx elements of two operators (K and U n the notaton of the Appendx, wth H c KU) whch control the mxng coeffcents among the zero-order states. These are smple functons of the poston and momentum operators for the molecular degrees of freedom. The same holds for the dpole moment, whch s a power seres Table III n the bond coordnates and bend angle, obtaned by ft to ab nto CCSD T ponts by Botschwna et al. 7,22,23 The parallel and perpendcular components of the dpole operator are referenced to the Eckart axs system. They are thus not sotopcally nvarant, but we wll gnore the small changes n these axes produced by sotopc change of the heavy atoms. It may be the case that ths approxmaton s responsble for our underestmate of the expermentally observed sotopc dependence of the HW effects. We use a harmonc approxmaton for the bendng wave functons, and thus snce the Corols couplng s lnear n the

6 D. Romann and K. K. Lehmann: Herman Walls effect n transtons 73 TABLE III. Botschwna s CCSD T dpole moment surface Refs. 7 and 23 expanded as a power seres of the form C(m,n,kz)r m CH r n k CN bend. Terms wth k 0 gve z whle those wth k gve mu x. C(0,0,0) C(6,,0) C(0,0,) C(,0,0) C(0,2,0) C(,0,) C(2,0,0) C(,2,0) C(2,0,) C(3,0,0) C(2,2,0) C(3,0,) C(4,0,0) C(3,2,0) C(4,0,) C(5,0,0) C(4,2,0) C(0,,) C(6,0,0) C(0,3,0) C(,,) C(7,0,0) C(,3,0) C(2,,) C(8,0,0) C(2,3,0) C(3,,) C(0,,0) C(3,3,0) C(0,2,) C(,,0) C(0,4,0) C(,2,) C(2,,0) C(,4,0) C(2,2,) C(3,,0) C(2,4,0) C(0,3,) C(4,,0) C(0,5,0) C(,3,) C(5,,0) C(0,6,0) C(0,4,) bendng coordnate and momentum, we get the selecton rule that states are coupled that dffer by only quanta n the bendng mode. Snce the states we are consderng all have 0 or quanta of bendng exctaton, the only bendng matrx elements we need are those between the ground and the frst excted bendng states. These matrx elements are needed for the matrx of the perpendcular dpole moment x and the K matrx, as n Eq. A27. As mentoned earler, we use analytcal expressons for Morse oscllator matrx elements of operators nvolvng the coordnates and momenta of the CN and CH bond local modes. By usng matrx nverson, product, and addton, we can calculate the K, U, x, and z matrces. Then, to transform these to the dagonal representaton of H 0, we use the matrx expressng the egenfunctons of H 0 n terms of the product wave functons. Ths orthogonal transformaton matrx s obtaned from the dagonalzaton of the vbratonal Hamltonan. For mproved accuracy, we use two transformaton matrces where they are needed as for K j ): one for the egenfunctons, calculated from the effectve PES wth no bend exctaton, and one for the bass, from the effectve PES wth one quantum of bendng exctaton. It s qute nformatve to consder the sze of the transton moment nduced by the Corols mxng. In Fg. 3 we plot the terms a s k z j x k x j, 0 state wth, and has an assocated transton moment x 3 z x. We also plot the runnng sum of these terms, whch converges to the A coeffcent apart from the factor / f of Eq. A22. The nterestng pont s that the largest values of a and b, due to the permanent dpole moment from the matrx elements 0 z 0 and x 3 z x 3 ), almost cancel each other. The same holds for the other terms: each a has a correspondent b almost opposte n sze. Ths cancellaton effect s due to the smlarty of s k to x k z x, and that of s j to z j. The frst par of terms s characterzed by peaks for the same values of, clustered around k, whle the second par has the same knd of peak structure around j. For j 3 ( 5 state, we fnd that the largest values correspond to the states 04, 05, and 06. The smlarty of these terms must be due to the operator U( I 0 ) beng almost dagonal, so that s j s manly determned by K, whch lke s b k x z x s j x, k x j s j JMe H c j JM J J E j E, wth j 0 and k 3 for the transton. These appear n the defnton of A and f n Eq. A22. Corols mxng of the upper k 3 state wth produces the a term, whch s therefore assocated wth the z j 0 contrbuton to the transton moment. Symmetrcally, the b term comes from mxng of the lower FIG. 3. Plot of the terms a crosses, b empty squares, and ther runnng sums full dots, for the case of the transton (k 3). These quanttes are defned explctly n Eq.. They appear n the defnton of f and ther sum n practce gves A, as one can see n Eq. A22. The ndex n the abscssa runs over the frst 50 and states that contrbute to the Corols mxng of the ntal and fnal levels n the transton.

7 74 D. Romann and K. K. Lehmann: Herman Walls effect n transtons nearly lnear n the P and Q operators of the stretches. In summary, we observe that systematc cancellaton of lower and upper state contrbutons to the frst order HW coeffcent s a general feature, whch makes the HW effect qute senstve to the PES and DMS surfaces. Ths effect s counter to our ntal expectatons n that we physcally expect that the Corols effect on the hgh energy, large ampltude upper state would be much more mportant than on the ground zero pont level. IV. COMPARISON OF THEORETICAL AND EXPERIMENTAL RESULTS TABLE IV. Partal logarthmc dervatves of transton dpole moment and HW A coeffcent for the transton. Dervatves are taken wth respect to the coeffcents of the DMS lsted n Table III, and wth respect to the parameters of the emprcal PES of Table II. Only dervatves wth magntude larger than 0. are lsted. Parameter ln A ln(parameter) ln ln(parameter) C(0,0,0) C(,0,0) C(3,0,0) C(4,0,0) C(,0,) C(2,0,) C(3,0,) C(4,0,) C(0,,) CH CN V(2,0) V(0,2) V(4,0) V(6,0) In Table I we report the calculated HW A and f factors for all perpendcular transtons that could be compared wth avalable expermental data. Besdes the large dsagreement for the hgher overtones, another major pont to be notced s that the expermental values by Mak et al. 2,3 show a much larger varaton wth sotopc substtuton than the calculated ones ( and transtons. In fact, ths s also n contrast to our two expermental values for the transton n HCN and H 3 CN, whch are very close. Ths dsagreement could result from our neglect of the sotopc effect on the DMS, due to a rotaton of the Eckart axes. Whle we dd not check the consequences of ths effect on our calculatons, t seems unlkely that ths effect could gve rse to such a large sotopc varaton of the HW coeffcents. Therefore, at ths moment we do not have a plausble explanaton for the large sotopc varaton of the HW coeffcents observed by Mak et al. We expect the frst-order perturbaton treatment to be an excellent approxmaton. Thus, n order to ratonalze the poor agreement between our theoretcal estmates and the observed value of the HW coeffcents, we wll consder our approxmate treatment of the bendng dynamcs and possble nadequacy of the PES and DMS. The harmonc approxmaton should descrbe the bendng moton accurately because the states mportant to our calculatons nvolve only 0 or bendng quanta. In addton, stretch bend nteracton s qute elusve n HCN: almost no perturbatons are found, even n the hgher stretchng overtones, and two-dmensonal ab nto calculatons of ths molecule perform unusually well. Ths s probably the combned effect of a small stretch bend anharmonc couplng and of effectve frequences whose ratos are not favorable to the occurrence of quas-resonances. The lowest order near resonances are that twce the CN harmonc frequency s almost 6 tmes that of the bend, whle twce the CH frequency s somewhat smaller than 0 tmes the bend the HCN harmonc wave numbers are 227, 727, and 3440 cm 24.In addton, the effect of anharmoncty s to tune the effectve stretchng frequences away from these near resonances when the stretches are excted as n the observed overtone transtons. Notce that symmetry allows only anharmonc resonances wth even number of quanta n the bendng mode. In the regon of nterest, the excellent ft of the observed frequences by the CMH PES seems a good ndcaton of ts accuracy. Therefore by excluson, the DMS s the least certan component of the present calculaton. It has been shown 25 that the overtone ntenstes from the ground state are especally senstve to the repulsve part of the PES. The fact that the CCSD T DMS predcts accurate overtone ntenstes lkely ndcates that ths surface s relable n the regon of the repulsve wall, but t cannot assure a global accuracy. On the other hand, the HW effect appears to be more senstve to global features of the DMS. In fact, we have shown that t s gven by the mxng of several zeroorder transton moments, some of whch connect overtone wave functons that are relatvely close n energy. There s no reason to thnk that the dpole matrx element between two such wave functons should be partcularly senstve to the repulsve wall regon of the PES. It should be noted that the predcted perpendcular band ntenstes are n sgnfcantly worse agreement wth experment than are those of the parallel bands. Ths suggests that the perpendcular, x, dpole moment surface s less accurate than the parallel, z, surface, at least as long as the molecule remans lnear. Gven the obvous loss of symmetry when the molecule bends, ths s perhaps not an unexpected result. To examne how dfferent features of the DMS nfluence the predctons, we determned numercally the senstvty of the HW A coeffcent and of the transton moment to varaton of the coeffcents of the DMS polynomal surface, whch are gven n Table III. In Table IV we lst the dmensonless partal logarthmc dervatves of transton dpole moment and HW A coeffcent for the transton, taken wth respect to the coeffcents of the polynomal surface ft to the ab nto CCSD T dpole moment ponts by Botschwna et al. Only the dervatves wth magntude larger than 0. are reported. Our defnton of logarthmc dervatve s ln f(x)/ ln x (x/f) f(x)/ x, whose meanng s qute smple. For example, ln A/ ln C(0,0,0) 0.57 mples that

8 D. Romann and K. K. Lehmann: Herman Walls effect n transtons 75 a varaton of % n the C(0,0,0) wll produce a varaton of 0.57% n the HW A coeffcent of the transton. To obtan these dervatves, we terated the calculaton of the ntenstes and the HW coeffcents after systematc small relatve varaton of each of the DMS coeffcents. To check aganst problems of numercal stablty, the whole procedure was repeated wth the relatve varatons equal to 0.%, 0 4, and 0 5. The values of the dervatves where found to be ndependent of the varaton sze, except for a few whch dropped to zero n the 0 5 case, due to numercal roundoff. The DMS senstvty pattern dsplayed n Table III s consstent wth the HW effect as descrbed before n terms of Corols mxng of and states and nterference of the nduced transton matrx elements. In fact, the A coeffcent are senstve to both the C(m,n,0) and C(m,n,) coeffcents, whle the ntenstes are dependent only on the C(m,n,) coeffcents. Note that n parallel transtons the ntenstes would depend only on the C(m,n,0). Wth respect to the values themselves, t s evdent that the senstvty of A and of the transton moment to the C(m,n,) coeffcents are very well correlated by a factor of about Another evdent feature, whch follows from the large dpole component of the CH bond, s that the CH stretch DMS coeffcents C(n,0,0) and C(n,0,) are the most relevant, whle for the CN stretch only C(0,,0) produces a dervatve larger than 0.. These senstvty patterns are generally vald for the transtons we are consderng, therefore the values gven n the table for a specfc transton are typcal for other perpendcular bands too. Ths suggests that naccuraces n the DMS can have lttle effect on the band ntenstes whle leadng to large errors n the HW coeffcents. In ths respect, we remark that many parallel bands have been observed, but only a few hgh-lyng perpendcular bands, and the accuracy of the predcted band ntenstes for the latter are not nearly as good as for the former. However, t s also possble that the ab nto CCSD T ponts be very accurate, but that the polynomal ft to these ponts be not a proper representaton. Thus even though the ab nto ponts used for fttng the polynomal cover a range of stretch and bend values that should amply suffce for the present calculatons, 23 the large value of the dervatve ln A/ ln C(4,0,) 0.40 s cause for concern, snce C(4,0,) s the hghest degree CH coeffcent for the perpendcular DMS component. Ths suggests that hgher-order coeffcents are needed and therefore the order of the polynomal expanson s not converged wth respect to the HW coeffcents and transton moments. As we dd for the DMS, we performed calculated the logarthmc dervatves of transton moments and HW A coeffcents wth respect to varaton of the PES parameters. The PES dervatves larger than 0. are also lsted n the same Table IV. Whle t s more dffcult to physcally nterpret these results, we note that there s agan a correlaton between the dervatves for A and the transton moment, but not as well defned as for the DMS. Somewhat unexpectedly, the senstvty s only about two tmes larger for the HW effect, wth the notable excepton of ln A/ ln V(2,0), whch s 8 tmes smaller and has the same sgn! than the correspondng dervatve for the transton moment. Smlarly to the DMS, here t also appears that the order of the PES expanson n powers of the Morse coordnate s not fully converged wth respect to the HW coeffcent, snce the dervatve respect to V(6,0) s stll qute szable compared wth the maxmum values assumed by the other dervatves. We note however that the PES polynomal expanson s not as free as that for the DMS snce t s the result of a successful ft to the very large data set composed by most of the observed HCN band orgns and rotatonal constants. 6 V. CONCLUSION At the orgn of ths work, s the szable HW dstorton of the R/P branch ntenstes affectng the few perpendcular transtons we observed n the lnear molecule HCN. We have used perturbaton theory to predct the effect of rovbratonal nteractons on the overtone ntenstes, and n the Appendx we gve explct expressons for the HW parameters n terms of transton dpole moments among zero-order vbratonal wave functons. As t turns out, the treatment s rather more complex than n the case of fundamental vbratonal transtons, and numercal evaluaton of these expressons s requred. Calculatons have been performed usng an approxmate varatonal bass that results from the dagonalzaton of the best emprcal PES today avalable for HCN, and a polynomal ft to ab nto CCSD T DMS ponts, whch reproduces rather accurately all the known overtone ntenstes. As ths model completely fals to gve the correct sze of the HW effect, we have argued that ether the PES or the DMS mght be not suffcently accurate. We have calculated the senstvty of the HW parameters and of the band ntenstes to the DMS coeffcents. We have found the physcally smple result that the HW parameter depends upon both the parallel and the perpendcular components of the DMS, whle the ntenstes depend only on one or the other, dependng on the type of transton. Ths ndcates that the good performance of the DMS n predctng the ntenstes s not ncompatble wth a poor predcton of the HW effect. In partcular, ths mght ndcate that the perpendcular DMS component s less accurate than the parallel component, as t s also ndcated by the reduced accuracy of the perpendcular band ntenstes. We have also observed that the order of the DMS polynomal expanson seems not to be suffcently converged wth respect to the ntenstes and the HW coeffcents. Ths suggests that the problem s wth the polynomal expanson and not wth the ab nto DMS ponts t was ft to. Wth respect to the analyss of the senstvty to the PES expanson parameters, t smlarly shows that the HW effect and the transton ntenstes are senstve to dfferent aspects of the potental. However, the PES expanson s subjected to the strngent requrement of accurately reproducng the vast data set of HCN band orgns and rotatonal constants. But, because of the extremely small sze of overtone transton moments, very small changes n the vbratonal wave func-

9 76 D. Romann and K. K. Lehmann: Herman Walls effect n transtons tons, whch would have mnor effects on the vbratonal energy levels, could have a large effect on the ntenstes. 25 We conclude that the HW effect can be a valuable probe of the accuracy of the DMS of PES avalable for a gven molecule. Gven the progress n hgh senstvty and hgh resoluton technques, ths effect s lkely observable n several systems, especally n perpendcular overtone transtons. Too often n the past, the possblty of HW perturbatons of the ntensty profle has been neglected. Besdes the loss of valuable nformaton, ths can lead to errors n the evaluaton of vbratonal ntenstes. In the present case of HCN, more nvestgatons, both expermental and computatonal, are needed to shed more lght over ths problem, whch remans partally unresolved. ACKNOWLEDGMENTS We wsh to thank Professor W. Quapp and Dr. A. Mak for sharng wth us ther data pror to publcaton and Professor P. Botschwna for provdng a copy of P. Sebald s thess and hs CCSD T dpole results pror to publcaton. Dr. J. K. G. Watson s acknowledged for a careful readng of a draft verson of ths paper. Ths work was supported by a grant from the Natonal Scence Foundaton. APPENDIX A: RO-VIBRATIONAL PERTURBATION EXPANSION OF THE TRANSITION STRENGTH IN A TRIATOMIC LINEAR MOLECULE In ths Appendx we derve a general expresson for the J-dependent ro-vbratonal transton ntensty from ( bend 0) to ( bend,l ) overtone states of a lnear tratomc molecule. Ths expresson s good to the frst perturbatve order n the rotaton vbraton couplng, wth the zero order gven by the separable Hamltonan of vbratons and rgd rotatons. We derve the smple form HL(m) fs 0 ( Am) 2, where m J n the R branch and mj n the P branch, HL(m) s the Hönl London rotatonal coeffcent, A s the HW coeffcent and f gves a correcton to S 0, whch s the zero-order vbratonal ntensty of the transton. For the Q branch lne ntenstes we obtan HL Q (J) fs 0 ( A Q J(J )) 2. In the followng framework, t s relatvely easy to derve correspondng expressons for transtons other than, ncludng second-order perturbaton correctons, whch descrbe the HW effect to powers hgher than J 2. In our case, the expermental error bars allow to determne only a lnear J dependence. We wll show that hgher perturbatve correctons to the coeffcents of J and J 2 (2A and A 2 ) are qute neglgble n transtons. Two mportant ponts should be underlned from the begnnng. The frst s that even very small state mxng, well descrbed by frst-order perturbaton theory, can produce a szable HW effect, due to the strong transton moment carred by some of the mxed components. The second pont, whch can be source of confuson, s that the correspondence between perturbatve orders and powers of J s not trval. The ro-vbratonal Hamltonan of a lnear molecule n the Born Oppenhemer approxmaton was rgorously derved by Watson 26 only n 970: 3N 5 P k 2 H k 2 2I Q J x x 2 J y y 2 V Q, A where Q k and P k are the normal-coordnates poston and momentum operators, I(Q) s the effectve moment of nerta, J s the total angular momentum, s the vbratonal angular momentum assocated wth the twofold degenerate bendng moton, and V(Q) s the nuclear PES. The z axs of the molecule-fxed Cartesan coordnates s along the C symmetry axs, whle x can be assgned to the molecule rotaton axs, so that the projecton of J along y s zero. The space-fxed Cartesan axes wll be labeled by X,Y,Z. Greek ndces wll be reserved for x,y,z. The vbratonal angular momentum s defned by wth 3N 5 k,l x,y,z kl Q k P l N kl, l,k l,l, A2 where the coeffcent l,k gves the component of the massweghted Cartesan dsplacement m dr of the atom for the normal coordnate Q k : k l,k Q k. The symmetry of the normal modes allows to decde mmedately whch of the Corols kl coeffcents must be zero. For the nonzero coeffcents n HCN and ts sotopomers, the followng values were calculated usng the force constants by Carter, Mlls, and Handy 6 : Isotopomer a x y 4 3 b x y HCN H3CN HC5N A3 Here, the normal mode numberng s: CN, 2 CH, 3 x bend, 4 y bend. Ths labelng s not standard and wll be used only n ths Appendx. The sgns of the Corols coeffcents depends on the choce of the coordnate systems for the dsplacements. Here, the z axs ponts from the C to the N atom, and the sgns of the z dsplacements of all 3 atoms are relatve to the drecton of the z axs. Notce that n HCN the stretchng normal modes can be easly dentfed and labeled by the bond mode wth predomnant ampltude. The Corols and centrfugal-dstorton terms (H c and H d ) couplng rotatons and vbratons wll be treated as perturbatons to the lnear-molecule rgd-rotor Hamltonan H 0. They are obtaned by expandng the mddle term of the full Hamltonan gven above, subtractng the rgd-rotor term contanng the equlbrum moment of nerta I 0, and neglectng ( 2 x 2 y )/2I, whch corresponds to a rotatonndependent frequency shft:

10 D. Romann and K. K. Lehmann: Herman Walls effect n transtons 77 H c I Q J x x J y y 2I Q J J, A4 H d 2 I Q I 0 J x 2 J y 2 2 I Q I 0 J 2 J z 2. A5 Notce that the satsfy the usual commutaton rules for the angular momentum so that ( x y )/2 behave as rasng/lowerng operators for z. On the other hand, the J have anomalous commutaton J,J J wth cyclc values of,, and ). We wll adopt the conventon of defnng J (J x J y )/2, whch behave as lowerng/rasng notce the nverson operators for J z. Ths subtle anomaly depends on the fact that the total angular momentum J s defned as the generator of rotatons n the space-fxed frame, where t obeys normal commutaton rules. The anomalous rules are a consequence of transformng the operator to the rotatng body-fxed frame Ref.8p.84. The ro-vbratonal wave functons that dagonalze H 0 are formed by the products of each symmetrc-top rotatonal egenfuncton JMK wth M and K quantum numbers of the J projectons along Z and z) wth the vbratonal egenfunctons havng the same value of K as an egenvalue of z. Ths ensures that the Sayvetz condton s satsfed. As an example, the zero-order wave functons for the overtone state 0 6 standard notaton, are 06 JMK JM, A6 2 where we have used the crcularly polarzed bendng states ( 0 0 )/ 2. By defnton, 0 and 0 are the bendng states lnearly polarzed along x and y. One of the effects of the Corols couplng H c s to remove the degeneracy n K. It has been shown 27 that the lnear combnatons of the states wth K partally dagonalze H c : Wth respect to the prevous example, we are here referrng to the states 06 JM 06 JM. Snce the party of the combnatons s ( ) J, these states can be mxed by H c whch commutes wth party wth and 2 ( bend 2,l 0) states of the same J, whle the combnatons do not mx at all to frst order. On the other hand, snce the transton dpole moment changes the party, the combnatons partcpate only to P and R branch transtons to and 2 states, whle the combnatons partcpate to the Q branch. In a perpendcular transton, the Corols mxng of a small amount of character to the state carres wth t a much larger parallel transton moment, and determnes the observed HW anomaly of R/P-branch J-dependent ntenstes. Smlarly, the mxng of some character to the state also gves a contrbuton to ths effect, through a szable transton moment. Mxng wth 2 states has a neglgble effect, snce the transton moment for a change of bend 2 s small. ( bend 2,l 2) states also mx to the states, but the - transton moment s zero ( l 2). The accepted conventon 28 s to label as e the states wth party ( ) J, and as f the others. In our case, the generc e/ f states are defned as JM f e 2 JM JM ). A7 At a frst glance, the ntensty J dependence of a overtone transton would appear to be domnated by mxng of the e sublevels wth the closest states only. On the contrary, all the states wth energy comprsed between the and state gve a relevant contrbuton. The mxng coeffcents are proportonal to e H c /(E E ), whle the contrbuton of each state to the transton moment s proportonal to. In a harmonc approxmaton of the vbratonal potental V(Q), the matrx elements would be zero for changes of more than one vbratonal quanta, but due to the substantal anharmoncty of the CH chromophore, the matrx element decreases almost exponentally wth the dfference n vbratonal quanta. On the other hand, and H c are both nearly lnearly dependent on the Q s; the decrease of one matrx element s thus almost exactly compensated by the ncrease of the other, for all states between the ntal and the fnal state. The remanng slow / E dependence wll gve a larger weght to the states closer to the state, but wll not completely elmnate the contrbuton of the more dstant states. A smlar argument holds for states mxng wth the state. Whle H c s nearly lnear n the bendng momentum and poston operators, H d does not depend on these operators at all, whch can be deduced from the defnton of z n Eq. A2 and I(Q) n Eq. A26. As we have seen, H c has matrx elements between states whch dffer by one quantum of bendng exctaton, whle H d must preserve the bendng quantum number. Therefore, H d mxes wth and wth states only and the transton moment t ntroduces n a transton s perpendcular n character, and thus s not expected to gve a large contrbuton to the HW coeffcent. However, we wll nclude the H d mxng terms whch allowed us to confrm numercally that ther effect on the ntenstes of the perpendcular transtons of nterest s small. We now proceed by wrtng the ro-vbratonal wave functons to the frst perturbatve order, j JM j 0 JM k J M e k 0 J M e l s j 0 JMe j c j 0 JM, l k d lk l 0 J M e, p lk l 0 J M where the mxng coeffcents are gven by s j 0 JMe H c j 0 JM / E j 0 E 0, A8

11 78 D. Romann and K. K. Lehmann: Herman Walls effect n transtons c j 0 JM H d j 0 JM / E j 0 E 0, p lk l 0 J MH c k 0 J M e / E k 0 E l 0, d lk l 0 J M e H d k 0 J M e / E k 0 E l 0. A9 Notce that the rotatonal dependence of the energy can be neglected. Apart from proportonalty factors, the transton ntensty s Ref. 8, p. 284 : S J J e k j 3 k J M e Z j JM 2. M,M A0 For smplcty we dropped the perturbatve-order superscrpts. The factor 3 has been ntroduced by assumng spatal sotropy and summng over the space-fxed components of. If we substtute Eq. A8 nto Eq. A0 and neglect second-order terms lke,l s j p lk...,weseethat we need 3 knds of Z matrx elements: S J J e k j 3 k J M e Z j JM M,M l s j k J M e Z JMe p lk l J M Z j JM j c j k J M e Z JM l k d lk l J M e Z j JM 2. A To smplfy these matrx elements, we transform Z to the rotatng frame, Z q,0 D 0q *,,, q, A2 where the tensoral components of are (, )( x y )/ 2 and (,0) z, and the J D MK (,, ) are the rotaton matrces, whch are functon of the Euler angles Ref. 8, pp. 78 and 85. Then, we need Eq. A7, plus the defnton of the symmetrc top egenfunctons Ref. 8, p. 05 JMK 2J 8 2 D J MK *,, 2J M K 8 2 J D M K, A3 and the followng ntegral Ref. 8, p. 03 J D 3 J M 3 K D 2 J 3 M 2 K D 2 M K d 8 2 J M J 2 M 2 J 3 M 3 J J 2 J 3 K K 2 K. 3 A4 By also usng the cyclc propertes of the 3-J symbols, and ther explct expressons from Zare s book Ref. 8, Table 2.5, the requred matrx elements for the R and P branch (JJ ) smplfy to k J M e Z j JMF J,J,M,M k J M e Z JMe k x x j m, F J,J,M,M x k z x sgn m m m, A5 m l J M Z j JM F J,J,M,M l z j sgn m m, F J,J,M,M J J M J MM 0, where we have used the lnearly-polarzed bendng states x n n 2 0, whch are more convenent for drect numercal evaluaton. We also lst the other nonzero matrx elements among and states, whch are needed to carry out the calculatons for the Q branches or partly for the second order: l J M Z JMe F J,J,M,M l x x m, k J M f Z JMf k J M e Z JMe, k JM e Z JMf k JM f Z JMe F J,J,M,M k x z x 2J J J, l JM Z JMf JM f Z l JM F J,J,M,M l x x 2J. A6 Notce that for the second-order perturbaton, matrx elements nvolvng states wth bend 2 are also needed, and can be calculated followng the same procedure. Wth respect to the H c mxng coeffcents, ther J dependence can be made explct by decomposng H c nto the product of the operators U 2I Q,Q 2 and K J J. A7 These operators are real and Hermtan therefore symmetrc, and they commute: H c UK KU. 26 In addton, U s dagonal n the rotatonal and bendng degrees of freedom. Therefore,