H. Li, Gregory S. Ezra and Laura A. Philips- Vibrationally induced rotational axis switching: A...

8/3/2019 H. Li, Gregory S. Ezra and Laura A. Philips- Vibrationally induced rotational axis switching: A novel mechanism for vi

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Vibrationally induced rotational axis switching: A novel mechanismfor vibrational mode couplingH. Li, Gregory S. Ezra, and Laura A. PhilipsDepartment of Chemistty, CorneN University, Ithaca, New York 14853-1301(Received 23 September 1991; accepted 20 July 1992)High resolution IR spectra of small- to medium-sized molecules such as 2-fluoroethanol(2FE) show that the effective density of coupled states is often greater than that obtained bya direct count of vibrational states. A novel mechanism for rotation-vibration interaction,vibrationally induced rotational axis switching (VIRAS), is proposed as a possible explanationfor these discrepancies. VIRAS has its origin in centrifugal distortion, and is physicallydistinct from Coriolis coupling. In the case of 2FE, we explicitly treat the coupling of overallrotation with large-amplitude internal rotation about the C-C bond. Assuming auniform coupling of all dark vibration-torsion states to the bright state, we predict a densityof coupled states in good agreement with that observed in the C-H stretching region at 2980cm-.

I. INTRODUCTIONIntramolecular vibrational energy redistribution(IVR) has been the focus of intensive work, both theoret-ical and experimental, over the past decade.lm21 he idea ofstate-selective laser excitation to control chemical reactionrates has been the motivation for a variety of experiments.One would like to mode-selectively excite a molecule toobtain reaction rates or product distributions that are func-tions of the particular mode excited, rather than just thetotal energy and angular momentum. Rapid IVR is, how-ever, a major obstacle to such dreams. Factors influencingrates and mechanisms of IVR are varied and extensive, andnot as yet completely understood. As a result, both exper-

imentalists and theorists continue striving to understandthe phenomenon of IVR.The consequences of IVR are manifest in both time-domain and frequency-domain experiments.-I4 Fre-quently, IVR is viewed as an explicitly time-dependentphenomenon, in which a nonstationary state is initiallyprepared and evolves in time. Energy flows out of the ini-tially excited mode, which may be localized in one part ofthe molecule, to other modes and, consequently, otherparts of the molecule. The nonstationary state initially pre-pared is often referred to as the bright state, as it carriesoscillator strength for the spectroscopic transition of inter-est, and IVR results in the flow of amplitude into the man-ifold of so-called dark states that are not excited directly.It is of interest to understand what physical interactionscouple different modes, allowing energy to flow betweenthem. In contrast to time-domain measurements, experi-ments in the frequency domain excite eigenstates of themolecule, which do not evolve with time. Splittings andperturbations observable in high resolution spectra canyield detailed information on the amount of mode couplingin a given molecule. The molecular eigenstates can be de-scribed as a superposition of zeroth-order states that wouldbe excited in the corresponding time-domain experiments.Conversely, time-domain experiments may be analyzed interms of the preparation of superposition states of the mo-

lecular eigenstates. Once again, we would like to predictwhich bright and dark states couple and what physicalmechanism causes such coupling.Various mechanisms that have been invoked to explainvibrational mode coupling include anharmonic, centrifu-gal, and Coriolis coupling. Each coupling mechanism hasbeen successfully employed to account for mode couplingin different circumstances. For example, anharmoniccoupling is independent of rotational angular momentumand therefore only directly couples states with the sameangular momentum quantum numbers. Both centrifugaland Coriolis coupling terms are dependent on angular mo-mentum and will therefore depend on the rotational levelof the molecule. For example, Kommandeur et al. analyzeboth time and frequency resolved data and discuss in detailthe application of these mechanisms to mode coupling inthe mid-sized molecule pyrizine. From these as well asother experiments,4-3 it is clear that rotational angularmomentum can play an important role in state mixing, andour understanding of the physical mechanism of rotationdependent vibrational mode coupling is far from complete.High resolution vibrational spectroscopy has been usedto resolve individual rovibrational eigenstates in moleculesof intermediate size. Perturbations in the spectra ofmany molecules lead to the appearance of clusters of peakswhere one would expect to see a single rovibrational tran-sition. These clusters or clumps of peaks are direct evi-dence of state coupling. Each additional peak in such aclump of peaks represents mixing of the bright state with adark state. Such spectral features have been observed in anumber of studies of mid-sized organic molecules in mo-lecular beams.* In some cases, there is a rotational de-pendence to the mode coupling,ti while in others themode coupling is independent of rotation.56 In either case,there is often a discrepancy between experiment and theoryin the number of coupled states.One approach used to determine the mechanism ofvibrational coupling is to compare the theoretically pre-dicted density of coupled states to the density of coupledstates measured experimentally. Using a zeroth-order basis

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set of bright and dark states, the total number of vibra-tional states available to couple to the bright state is esti-mated from the calculated density of dark states in thespectral region of interest. In the absence of rotation-vibration coupling, this total density of vibrational statesshould give an upper bound for the number of coupledstates, since all dark states will not necessari ly couple tothe bright state. Experimentally, the density of coupledstates is inferred from the number of transitions in a clumpof peaks in the spectrum. The spacing of the dark states ascalculated from the spacing of the transitions in the spec-trum is a measure of the density of states actually coupledto the bright state.5-8 In all of the experiments cited abovethe experimental density of coupled states is larger than thecalculated density of states. This discrepancy can be ex-plained if, for example, a single rovibrational bright state isable to couple to more than one rotational state associatedwith each of the vibrational dark states. Such coupling mayoccur via the Coriolis mechanism.22Y23Coriolis couplingalone, however, does not always account for the observeddiscrepancy between experiment and theory. As a result,other variations and combinations of Coriolis coupling andanharmonic coupling have been proposed.569*15We ana-lyze here an alternate mechanism, physically distinct fromCoriolis coupling, whereby a single rovibrational brightstate can couple to more than one rovibrational dark state.The mechanism considered is a consequence of centrifugalcoupling, and has a straightforward physical interpretationin terms of vibrationally induced changes in the orientationof the principal axes.Recent work in our laboratory on the high resolutionspectroscopy of 24luoroethanol (2FE) has motivated ou rinterest in coupling mechanisms.** In the C-H stretchingregion of 2FE at 2980 cm-, extensive mode-mode cou-pling has been observed. The experimental density of cou-pled states is determined to be 200-250 states/cm-. Thedensity of vibrational states in the neighborhood of 2980cm- estimated on the basis of a simple rigid rotor/harmonic oscillator/hindered rotor model is between 35and 58 states/cm-, depending on the potential functionsused in the calculation. In this calculation the fundamentalvibrational frequencies for all but the torsional modes wereused from experimental measurements,25 and a 2% anhar-monicity was assumed. The torsional modes were treatedas hindered rotors.26 An additional coupling mechanism isapparently required to account for the discrepancy be-tween experiment and calculation.One possible coupling mechani sm that will lead torovibrational mixing is standard Coriolis coupling. TheCoriolis term in the Watson Hamiltonian directly coupleszeroth-order vibrational states differing by one quantum ineach of two modes. As dark states in 2FE at approximately3000 cm- contain as many as 80 quanta, it is necessary toinvoke very high order combinations of anharmonic andCoriolis perturbations to explain the observed level density.The observed J dependence of the density of coupled statesis, however, relatively weak (see Fig. 7), suggesting thatCoriolis coupling is significant only to low order, if at all.We are therefore led to examine additional rotation-

vibration coupling mechanisms to explain the perturba-tions observed in the high resolution infrared spectrum of2FE. In this paper, we analyze a mechani sm for rovibra-tional mixing based on the phenomenon of vibrationallyinduced rotational axis switching (VIRAS). VIRAS ismost pronounced for dark states containing torsionalmodes, where large-amplitude internal rotation occurs.A summary of the paper follows: In Sec. II we intro-duce the key physical idea of VIRAS in the context of theconventional description of rotation-vibration interactionin semirigid molecules. The theory is then developed inmore detail for a molecular model in which one large-amplitude internal motion is treated explicitly. In Sec. IIIwe apply the theory to 2FE. The VIRAS mechanism isable to provide a very reasonable estimate for the density ofstates coupled to a given bright state for the example of2FE. Conclusions are given in Sec. IV.

II. THEORETICAL APPROACHES TOROTATION-VIBRATION COUPLINGIn this section we first introduce the essential physicalidea of vibrationally induced rotational axis switching inthe simple context of the usual rotation-vibration Hamil-tonian for semirigid molecules. To apply the theory to theparticular case of 2FE, however, it is necessary to considerexplicitly large-amplitude torsional motion.

A. Bright and dark statesThe standard spectroscopic approach to the problemof vibration-rotation coupling in semirigid molecules in-volves calculation of the eigenvalues and eigenvectors ofthe Watson Hamiltonian for small amplitude motion.22

The zeroth-order solution to the vibration-rotation prob-lem is based on the harmonic-oscillator/rigid rotor separa-tion of vibration and rotation. The vibrational modes aretreated as a set of independent harmonic oscillators and therotational states are taken to be eigenstates of a rigid asym-metric top. The corresponding zeroth-order Hamiltonianhas the usual form for nonlinear semirigid molecules:

where it is assumed that the Eckart frame i s aligned wi ththe principal axes of the inertia tensor at the equilibriumconfiguration (zero vibrational displacement). Pk is themomentum conjugate to the kth normal vibrati on andU(Q) is the potential energy for (3N-6) independentharmonic oscillators. Eigenfunctions of Bs are products1 ni)> 1 ,A) of the 3N-6 harmonic oscillator wave func-tions 1 nil), where {nj} is a set of vibrational quantumnumbers for the 3N-6 modes of the molecule, and asym-metric rotor wavefuncti ons 1 ,A), where A is a bookkeep-ing label that serves to distinguish between the 2J+ 1 statesof the same total rotational angular momentum J. Thesezeroth-order wave functions are often taken to describe thebright state and dark states in time-resolved experiments atlow levels of excitation. Transition probabilities may bederived based on properties of these states.

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Coupling terms are obtained by subtracting Hc fromthe full Watson Hamiltonian. Anharmonic coupling termsdepend only on vibrational variables, while Coriolis andcentrifugal coupling terms depend on both vibrational androtational variables. Due to the presence of these couplingterms, the functions 1 ni}) IJ,L) are no longer eigenstatesof the full rovibrational Hamiltonian. Nevertheless, thezeroth-order states 1 nj)) j J,A) serve as a complete basisset in which to expand the true rotation-vibration eigen-states, I Y,J) :

1Js4 ( F, CC~~},A~{nil> IJA)*JA bright state is a particular zeroth-order state/ (nj}) I J,A) that carries oscill ator strength for a givenspectroscopic transition in the harmonic-oscillator/rigid-rotor approximation. If the bright state contributes signif-icantly to several molecular eigenstates j Y,J), one will seemore transitions in the experimental spectrum than pre-dicted by a calculation based on the rigid rotor/harmonicoscillator model . The bright state/dark state picture be-comes less useful when the molecular eigenstates have sig-nificant admixtures of several bright states, producing aspectrum that is unassignable in terms of the zeroth-orderquantum numbers.Different coupling mechanisms will mix different typesof zeroth-order states in 1q,J). For example, anharmoniccoupling terms, which involve only vibrational variables,will only mix directly different {nj}s, while the rotational

part (J,A) remains unaffected.Both centrifugal and Coriolis coupling al so mix differ-ent / J,A)s, but total J must, of course, be conserved. Co-riolis coupling or centrifugal coupling pl us anharmoniccoupling can mix, i n principle, all of the rotational statesassociated with any two vibrational states.The physical origin of Coriolis coupling is the interac-tion of molecular rotation with the vibrational angular mo-mentum induced by the simultaneous excitation of a pairof vibrational modes. For example, in the case of a lineartriatomic molecule, Coriolis interactions can couple theasymmetric stretch and the bend. As the vibrating mole-cule rotates, there is a Coriolis force on each nucleus equalto the vector product of its velocity with the angular ve-locity vector. For the asymmetric stretch mode, the com-bination of forces on the individual nuclei results in exci-tation of a componen t of bending motion. The effects ofCoriolis coupling are most pronounced when the twomodes have nearly equal frequencies.Centrifugal coupling arises from the vibrational coor-dinate dependence of the molecular moments of inertia. Inthe VIRAS mechanism we focus attention on a rotation-vibration coupling phenomenon that is a consequence ofcentrifugal interaction, and so is physically distinct fromCoriolis coupling. This VIRAS phenomenon is analogousto axis switching in electronic transitions.24

5958 Li, Ezra, and Philips: Vibrational mode coupl ingB. Vibrationally induced rotational axis switching insemirigid molecules

To introduce the VIRAS phenomenon for semirigidmolecules, consider an adiabatic separation of (3N-6)fast vibrations from slow rotation. Each vibrationalstate then has an effective rotational Hamiltonian obtainedby averaging over the vibrational coordinates, and anhar-manic couplings will couple different zeroth-order vibra-tional states. The bright vibrational state is defined as theproduct of an appropriate excited vibrational state with aneigenstate of the associated rotational Hamiltonian. In gen-eral, rotational constants and, most importantly, the orien-tation of principal inertial axes (with respect to the Eckartframe) will be different for different vibrational states. Thechange in orientation of the inertial axes with vibrationalstate is called vibrationally induced rotational axis switch-ing (VIRAS).Suppose for the moment that the molecular Hamil-tonian contains only anharmonic perturbations, whichserve to mix vibrational states. The dependence of rota-tional constants and principal axis orientation on vibra-tional state then implies that the bright state is coupled toa dark s tate that has principal axes rotated with respect tothe principal axes of the bright state. Since the asymmetrictop quantum numbers refer to a specific orientation of theprincipal axes, if the axes are rotated, conservation of theasymmetric top quantum numbers breaks down. The resultis that a given rotational bright state, with given angularmomentum quantum numbers, can in fact couple via an-harmonic perturbation to several rotational states associ-ated with a particular dark vibrational state. That is,VIRAS results in a breakdown of rotational selectionrules, just as in the analogous phenomenon for electronictransitions.24The amount of rotational mixing is of course a func-tion of vibrational state. Vibrational modes which inducethe largest changes in both rotational constants and orien-tation of the principal axes are likely to have the mostdramatic rotational state mixing via VIRAS. In particular,vibrational modes involving large-amplitude low frequencytorsional motion will cause considerable changes in molec-ular geometry and concomitant changes in rotational con-stants and principal axis orientation. For example, we ex-pect the torsional modes in 2FE to have the largest VIRASeffect (see next section). As an example, shown in Fig. 1 isa diagrammatic representation of the reorientation of theprincipal axes for two different vibrational dark states of2FE, one containing no torsion and the other with signif-icant amounts of torsion.The above qualitative discussion has presented a phys-ical description of the VIRAS effect. Within the discussion,we have implicitly assumed the use of the Watsonrotation-vibration Hamiltonian for a semirigid molecule,based upon the Eckart frame. Before present ing the formaltreatment of the VIRAS effect for 2FE another aspec t ofangular momentum coupling is considered. The internalangular momentum generated by the large-amplitude tor-sional motion of the hindered rotor can couple to the over-all rotation of the molecule. In order to develop a quanti-



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A = 0.5318cm-B=O.l816cm~ A = 0.5621cm -C= 0.1513 cm- B = 0.1708cm-C = 0.1485 cm -Ia b

C-H StretchA = 0.53 17 cm-*B = 0.1816cm-C = 0.1513cm-

FIG. 1. Two examples of the relative orientation of the principal axes ofthe bright state and a particular dark state are shown. In (a), the darkstate contains no torsion and there is little difference in the axes or therotational constants for the two vibrational states. The dark state is com-posed of 1 quanta of I, and 1 quanta of I,,,. In contrast, (b), the darkstate contains torsion and there is substantial reorientation of theprincipal axes and concomitant change in the rotational constants. Thereare 74 quanta of torsion in this dark state, well as 1 quanta of Vi, and 1quanta of Vi,.

tative theory for 2FE, it is necessary o go beyond theWatson Hamiltonian and treat explicitly the coupling oflarge-amplitude torsional motion with overall rotation.

C. Coupling of internal rotation and overall rotationin 2-fluoroethanolIn this subsection, we derive a Hamiltonian describinginteraction of overall rotation with a single large-amplitudeinternal rotation coordinate. 26*27 he model Hamiltonian isthen applied to describe the interaction of rotation withinternal rotation about the C-C bond in 2FE. Internal ro-tation about the C-O bond involves solely the motion of ahydrogen atom. The perturbation caused by this internalrotation is small in terms of the VIRAS effect and is as-sumed to be only weakly coupled to the overall rotation.

Therefore, the C-O torsion is not considered explicitly, butis treated as an uncoupled hindered rotor. All remaining3N-8 internal coordinates are treated as high frequencysmall amplitude vibrati onal coordinates. Although weshall not need to specify the precise choice of embeddingfor the molecule-fixed frame, the Eckart-Sayvetz frame22 isa natural choice.The coordinate system used is shown in Fig. 2. Note inparticular that the internal rotor angle x is half the FCC0torsion angle, so that internal rotation generates only asmall amount of angular momentum about the C-C axis.In the treatment of the FCC0 torsion, the OH group is asingle point mass.Following standard procedures,27 he quantum me-

Li, Ezra, and Philips: Vibrational mode coupling 5959

FIG. 2. The axis system used to define the geometry of 2-fluoroethanol ispresented. The angle x is half of the FCC0 dihedral angle.

chanical rotation-torsion Hamiltonian for the model ofFig. 2 is found to beH=J~+GP+m- V(x) --2q.44

P,,

G=;.Here, V(x) is the torsional potential, ma and J are massand position vector of crth atom. I is the inertia tensor,which varies with torsional angle x.H can be partitioned into three parts, corresponding tooverall rotation, internal rotation, and the interaction be-tween internal rotation and overall rotation. The first termon the right-hand side of Eq. (3) represents overall rota-tion, the second and fourth terms represent internal rota-tion, and the third and fifth terms are interaction terms.Note that both the p tensor and the G coefficient depend onx. The above Hamiltonian can be expanded and simplifiedto the form

R~~=&P+,L@+~#). (4)In this form of the Hamiltonian we have made one furtherapproximation: the x dependenceof G in the internal ro-



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tation part of the Hamiltonian is ignored. This modifica-tion is expected to have only a minor effect on the resultsand greatly simplifies the calculation. The eigenfunctionsof HT are denoted jm), m=O,l;**. For a given J,rotation-torsion eigenfunctions are expanded as sums ofproducts of symmetric rotor wave functions with torsionalwave functions 1m),

I$)=? ~CK,IJWm). (5)The rotation-torsion component of the bright state associ-ated with the vibrational transition of interest is taken to bea product of a rotational eigenfuncti on for an asymmetri ctop with appropria te rotational constants and the torsionalground state function 1m = 0). The bright state can also beexpanded in the same product basis set used aboveI#)= ~&JJ,K)lm=O). (6)

The bright and dark states are now written as products ofrotation-torsion states, expanded as above, with vibrationalstates for the 3N-I remaining degrees of freedom. Toestimate the number of dark states coupled to a givenbright state, it is necessary to evaluate matrix elements ofthe form

($1 (Cnjwwk-H=1) Id); (7)where H; is the term in the Hamiltonian representing theeffect (direct and via higher order perturbations) of resid-ual anharmonic interactions between the torsion-vibrational modes. The set of dark states contain vibra-tional states of very different character. At the energy ofinterest, a given dark state may correspond to excitation ofonly a few quanta of torsion, or as many as 70 or 80quanta. The precise nature of the anharmonic perturba-tions which couple states with such disparate numbers ofquanta is unclear. Fr om experiment, it is apparent that thedensity of coupled states far exceeds the number of avail-able vibrational states. We therefore make the drastic as-sumption that all avai lable dark states in the vicinity of thebright state will couple, via an unknown anharmonic per-turbation, with the bright state. We further assume that thematrix elements of the anharmonic perturbation connect-ing the bright state Im=O) I {nj}> with any dark state( m) I {nf}) are equal. Using this approximation, the rela-tive magnitudes of the coupling matrix elements become

- CDK 1 C~(mlH$lm=O)

- &:)9 (8)mwhere CC,,,,} are the expansion coefficients of thetorsional-rotational eigenstates in the I J,K) I m) basis, andDk are expansion coefficients of the bright state in the same1 ,K) basis. Equation (8) therefore exp resses the magni-tude of the coupling matrix element as a sum of overlaps ofrotational components.

5960 Li, Ezra, and Philips: Vibrational mode coupli ng

-1 ooo--- C-C TorsionE -1500-5 -2ooo-

-2500-C-O Torsion

0 1 2 3 4 5Torsional Angle(Radians)

FIG. 3. The torsional potential functions for the C-C torsion and theC-O torsion are shown above. The C-O potential function was calculatedat the MP3/6-31lf +G** level by Wiberg and Murcko (Ref. 29). Forthe C-O torsion, the geometry about the C-C bond was held at the transconformation. The C-C torsional potential was generated by a sum ofcosines to correspond to experimentall y determined values of the barrierheights and relative well depths.

When the principal axes of a dark state do not coincidewith those of the bright state, the bright state can, in prin-ciple, couple to the entire manifold of rotational states as-sociated with the dark state as long as J is conserved. Inpractice, if the coupling matrix element is sufficientlysmall, a transition to that state would not be observedexperimentally. Based on the experimental signal-to-noiseratio, we set the threshold overlap of dark states withbright states to be 1% of the maximum overlap, and fur-ther assume that any dark state having an overlap with thebright state above the threshold is detectable experimen-tally.This procedure is demonstrated in detail for the case of2FE in the next section.

III. VIRAS AND P-FLUOROETHANOLUsing the procedure outlined in the previous section,we now calculate the density of states available for cou-pling in 2FE.There are 21 normal modes in 2FE, of which two aretorsional modes, corresponding to rotation about the C-Cbond and the C-O bond, respectively. The fundamentalfrequencies of these modes have been measured and as-signed experimentally.25 In order to make a crude estimate

of the vibrational density of states, we treat the 2 torsionalmodes separately as hindered rotors, while the other 19modes are treated as uncoupled anharmonic oscillators.6*7A 2 % anharmonicity is assumed for all modes. The poten-tial functions for the two torsional modes are shown in Fig.3. The C-O torsional potential is a calculated functionfrom the work of Wiberg and Murcko.28 The C-C tor-sional potential was generated to correspond to experimen-tally determined values of the barrier heights and relativewell depths. Direct counting of the vibrational states in-cluding a 2% anharmonicity then gives an average calcu-lated density of states of 56 states/cm- at 2980 cm-.With anharmonic coupling alone, the maximum number ofJ. Chem. Phys., Vol. 97, No. 9, 1 November 1992

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TABLE I. The vibrational modes in 2FE. The calculated frequencies arefrom the normal mode calculations. The assignments are from the exper-iments of Azrak and Buckton (Ref. 25).Observed CalculatedI (cm-) (cm-) Assignment

1 3626 3998 vOH2 2980 3256 v&H3 2961 3248 v,CH4 2926 3197 v$H5 2896 3185 v$H6 1460 1597 &CH7 1458 1544 oCH8 1400 1513 oCH9 1373 1452 &ZH10 1348 1430 fCH11 1252 142512 1203 1400 6$C13 1103 1315 v,occ14 1080 1241 v,occ15 1026 1124 VCF16 885 1010 rCH,v$CF17 850 948 ICH18 513 581 GOCC,CCF in phase19 342 345 GOCC,CCF, out phase20 292 289 rCOH21 161 122 rFCC0

available states in this region is therefore 56/cm-. BothCoriolis coupling and the VIRAS mechanism can result inrovibrational mixing, thereby increasing the number ofstates available for coupli ng in this spectral region. Wenow calculate the number of coupled states using theVIRAS model of the previous section.A normal mode analysis of 2FE was performed usingMOPAC. The input geomet ry of 2FE was determined fromexperimental data.2 MOPAC executed a MND03 calcula-tion on the input geometry until a stable geometry wasfound; only minor modifications to the input geometry re-sulted. The MNDO calculation also produced force con-stants, vibrational frequencies, normal modes, and thetransformation matrix from Cartesian coordinates to nor-mal coordinates. The calculated frequencies from the nor-mal mode analysis are in good agreement with experimentfor the high frequency modes, but some discrepancies arefound for the low frequency modes (see Table I). Thenormalized normal coordinate eigenvectors were used tocalculate classical rms displacements of each atom at avibrational energy equal to the zero-point energy. Assum-ing an adiabatic separat ion of vibration and rotation, thegeometry change upon excitation of the 19 small-amplitudeharmonic modes is not sufficient to significantly rotate theprincipal axes in the molecule, so that VIRAS will not beimportant for these modes. Excitation of the torsionalmodes, however, results in much larger average displace-ments of the atoms in the molecule and possibly concom-itant rotati on of the principal axes. Torsion about the C-Ogroup has only a small effect on the inertial parameters ofthe molecule. The mass of the hydrogen is so small thatchanges in the inertia tensor as a function of C-O torsionangle are negligible.When calculatingmolecular geome-

C-C Torsion

-2500 -I I I \ ,I\7 \ I \ II I I I I I0 1 2 3 4 5 6

Torsional Angle(Radians)


FIG. 4. The energy levels for the C-C torsional potential are shown forenergy levels up to u=50. There are three kinds of energy levels. Thereare symmetric and antisymmetric levels that are generated from the twoidentical gauchepotential wells, and levels corresponding to wave func-tions that have their maximum amplitude in the rrans potential well.

tries we therefore fix the torsional angle of the OH groupalong the C-O bond at 55.5, based on the geometry frommicrowave experiments.25Since there are such small changes in geometry uponexcitation of vibrations other than the torsional modes,changes in inertial parameters due to excitation of theseother modes are henceforth neglected. Excitation of thelarge-amplitude C-C torsion mode leads to the largestVIRAS effect, and we now consider the interaction of thismode with overall rotation.The Hamiltonian describing interaction of rotationwith l arge-amplitude C-C torsion was given in the previ-ous section. The torsional basis used consists of eigenstatesof the torsional Hamiltonian HP The torsional potentialfor the C-C torsion was modeled as a periodic sum ofcosines,

where the Vn s are derived f rom experimental values forthe barrier heights and relative well depths (see Fig.3). 28P29 he C-C torsional potential is shown in Fig. 4,together with the energy levels determined for this poten-tial. Torsional eigenfunctions were expanded as a sum ofcosine and sine functions, and the expansion coefficientscalculated based on the algorithm of Lewis et al. 28The sizeof the expansion basis was increased until convergencewithin 0.2 cm- was attained. A representative selection ofwave functions is shown in Fig. 5. Note that Vc-c is aperiodic funct ion with two identical minima at the twogauche forms and a single local minimum at the Tt form.The low energy torsional levels are nearly doubly degener-ate with maximum probability density at the gauche forms.With increasing quantum number, new states appear withprobability density at the trans form at approximately v=20. At higher quantum levels, the wave function beginsto delocalize ver all angularspace U> 48).



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n=50

n=20n=lO

\f-06 ; h i 4. k 6Torsional angle(Radians)FIG. 5. The corresponding wave functions for a representative selectionof the energy levels from Fig. 4 are shown.

A total of 120 torsional states are used in the expansionof the rotation-torsion eigenstates of Hamiltonian Eq. (4).From the coordinate system of Fig. 2, it can be seen thatthe transformation x-+x + 2n results in an overall rotationof the molecule by r about the C-C axis. The rotation-torsion basis functions must therefore satisfy the boundarycondition

1cI(e+a,x+27T) =$(&xl (10)where 8 is an Euler angle describing overall rotation aboutthe C-C bond. Note that the full range of x is 0-4~.Requiring the rotation-torsion eigenfunction to be singlevalued then implies that for an odd number of torsionalquanta the K quantum number be odd, and for an evennumber of torsional quanta K must be even. For given J, abasis of 240 torsional states was used, half of which havethe right symmetry to match even K states 1 ,K), the otherhalf odd K. The set of basis functions consisting of theproduct of the 1 ,K) rotational states and the 1m) tor-

Vs= 1

FIG. 6. Shown schematically is an example of the relative energies of thebright state and a dark state containing 1 quanta of V, and 1 quanta ofV,e, and the manifold of the states t+$ etermined from the diagonalizationof the rotation-torsion Hamiltonian given in Eq. (4) and described in thetext.

5962 Li, Ezra, and Philips: Vibrational mode coupling

I I 1 ,0 1 2 3 4 5JFIG. 7. Shown above is a plot of the density of coupled states vs thequantum number J for the experimental data from the spectrum of 2FEand VIRAS: Squares: experimental data (dotted lines represent the ex-perimental uncertainties); diamonds: the prediction of the VIRAS cou-pling model.

sional states is used to construct the rotation-torsionHamiltonian matrix.The Hamiltonian matrix is block diagonal in J, andeach J block is diagonalized independently (e.g., the J=5block is 1320x 1320) to give rotation-torsion eigenvaluesand eigenstates. The rotation-torsion states are combinedwith various other vibrations to form states of energynearly resonant with the bright state. Such a combinationstate is shown schematically in Fig. 6. The rotation-torsioncomponent of the total eigenvectors can be projected ontothe rotational bright states using Eq. (8). As discussed inthe previous section, to compare theory with experiment itis necessary to set a cutoff value for the overlap Eq. (8))below which the associated transition is deemed unobserv-able. This threshold is set at 1% of the maximum overlap.It should be emphasized that the 1% value is determinedby the dynamic range of the experiment and is not anarbitrarily adjustable parameter.The results of our quantum mechanical treatment ofVIRAS in 2FE are shown in Fig. 7 and presented in TableII. Many more rotational states associated with a singlevibrational dark state are available to couple to a givenrotational level in the bright state than wou ld be predictedfrom anharmonic coupling alone. For example, for J=4,the effective density of coupled states is increased from 56states/cm- with anharmonic coupling alone to 276TABLE II. The density of states as predicted by the VIRAS couplingmodel, as a function of the rotational quantum number, J.J Density of states (per wave number)0 51.701 91.402 142.93 198.44 276.15 310.6



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states/cm-, as compared to the experimental measuredvalue of 278 states/cm-. For the available experimentalresults, where J=O-5, our calculations are in very goodagreement with the experimental data and well within theexperimenta l uncertainties. The uncertainties in the exper-imental data are intrinsic to the physical quant ity beingmeasured , and therefore are not limited by the noise in thedata.7*8 The data suggests that there is mode selectivity inthe vibrational coupling that is not accounted for in thecalculation. Mode-select ive coupling resul ts in differencesin the magnitude of the coupling matrix elements for dif-ferent vibrational dark states, and therefore different effec-tive densities of states. The calculations presented here donot account for these differences because uniform couplingfor different vibrational states is assumed.The agreement between our model and the experimenthas been achieved without any adjustable parameters. Theprecise nature of the dark states in the vicinity of a givenbright state, however, is expected to be a sensitive functionof the details of the torsional potential function. Althoughour potential function was constructed to agree with ex-perimentally measured parameters, the form of the poten-tial chosen is necessarily an approximation. To evaluatethe effect of our choice of potential on the VIRAS predic-tions, we changed the barrier heights and relative welldepths by 10% and recalculated the predicted density ofcoupled states. These changes in the potential surface hadonly a small effect on the theoretical predictions, and ourresults remained in good agreement with the experimentaldata.IV. CONCLUSION

The phenomenon of vibrationally induced rotationalaxis switching (VIRAS) has been proposed as a mecha-nism for explaining enhanced densities of coupled states inhigh-resolution spectra. VIRAS is a direct consequence ofcentrifugal distortion, and is best understood in terms of anadiabatic separation of vibrational and rotational motion.Each vibrational state has an effective rotational Hamil-tonian obtained by averaging over the vibrational coordi-nates. The rotational constants and orientation of the prin-cipal axes will in general be different for each vibrationalstate, so that rotational eigenstates associated with differentvibrational states are no longer orthonormal. The numberof dark states to which a given bright state can be coupledby an anharmonic coupling term in the Hamiltonian isthen increased due to nonzero rotational Franck-Condonfactors.Application of the theory to 2FE showed that the C-Ctorsional mode exhibited the most pronounced VIRAS ef-fect. An exact torsional-rotational Hamiltoni an for 2FEwas diagonali zed to obtain rotation-torsion eigenstates for2FE. Zeroth-order states were taken to be products oftorsion-rotation states with vibrati onal states for the 3N- 7 remaining modes. Assuming that anharmonic terms inthe Hamilton ian couple every dark torsion-vibration stateequally strongly to the bright torsi on-vibration state, thecoupling matrix element can be expressed as a sum of over-laps of rotational states. Using a reasonable threshold in-

tensity for detection, we calculate the density of rovibra-tional states available for coupling. The theoreticalpredictions are found to be in good agreement with exper-iment.

ACKNOWLEDGMENTSThis work is supported by: The National Institute of

Health under Grant No. 08-R9N527039A, The Office ofNaval Research under Grant No. NOOO14-90-J-197 , andProctor and Gamble (to L.A.P.); and NSF Grant No.CHE-9101357 (to G.S.E.). G.S.E. is a Camille and HenryDreyfus Foundation Teacher-Scholar. The authors grate-fully acknowledge S. W. Mork, C. L. Brummel, and M.Shen for generous help and stimulating discussion. We areparticularly grateful to E. L. Sibert for his penetratingcomments on an earlier version of this paper.

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J. Chem. Phys., Vol. 97, No. 9, 1 November 1992

H. Li, Gregory S. Ezra and Laura A. Philips- Vibrationally induced rotational axis switching: A...

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