Theoretical and Experimental Studies for theInterpretation of Vibrational Circular DichroismSpectra in the CH-Stretching Overtone Region

SERGIO ABBATE,* GIOVANNA LONGHI, AND CLAUDIO SANTINADipartimento di Scienze Biomediche e Biotecnologie, Universita di Brescia, Brescia, Italy, and

INFM, UdR Brescia

ABSTRACT Two theoretical models for the interpretation of the existing data ofCH-stretching overtones’ vibrational circular dichroism data are presented. The firstmodel is based on the quantum mechanical Van Vleck contact transformation theory andis applied to the full vibrational problem, the second is based on classical trajectoriescalculations, by which we treat a simplified three-degrees of freedom Hamiltonian. Thelatter allows one to derive a qualitative but efficacious picture of the behavior of coupledanharmonic oscillators. In this framework, we analyze the Poincare Surfaces of Section,and calculate the Fourier Cross Spectra of coupled CH-stretchings. Values for the har-monic frequencies and anharmonicities are derived from absorption spectra in the nearinfrared on partially deuterated compounds. The effect of large amplitude, low-frequencypuckering or twisting modes on the ensemble of coupled CH-stretching is taken intoaccount. Chirality 12:180–190, 2000. © 2000 Wiley-Liss, Inc.

KEY WORDS: vibrational circular dichroism; near infrared; local modes; classical tra-jectories; Fourier cross spectrum; Van Vleck contact transformations

Vibrational circular dichroism (VCD) has become a well-established technique for monitoring the absolute configu-ration as well as the possible conformations of chiral mol-ecules from low to mid to large size.1–6 However, due inpart to difficulties in the interpretation, data from the CH-stretching fundamental region have become somewhat ne-glected lately, not to mention the CH-stretching overtoneregion, in the near infrared (NIR) range. Curiously, someresearch groups have started their investigations in VCDfirst at the NIR region;7 presently, the group of Sugeta8 isworking routinely in the 2,500–5,000 cm−1 range. The ex-isting VCD data in the NIR range have been reviewed inRef. 9: two main categories of spectra emerged from thatanalysis, the prototypical cases being those of (R)-limonene and (3R)-methylcyclopentanone. (R)-Limoneneexhibits a (+, −) couplet in order of increasing frequency atall orders of overtone up to Dv = 6. The couplets correlatethrough a Birge-Sponer plot with the strongest couplet ofVCD bands observed in the fundamental region Dv = 1.10,11

(3R)-Methylcyclopentanone instead shows a broad nega-tive feature from Dv = 1 to Dv = 6.9

Three observations have been made in Refs. 9–11, look-ing at NIR-VCD data:

1) The persistence in the overtone regions of featuresequal in sign and with very similar D«/« ratio with respectto what is observed in the fundamental, and especially ofbisignate features, is strongly suggestive of normal modebehavior, i.e., of coupled CH-stretchings; this is particularlyat odds with the usual belief that in the overtone region

CH-stretchings should behave as local oscillators, that is tosay, as decoupled.10,12,13.

2) The two distinct behaviors for (R)-limonene and ana-logs, and for (3R)-methylcyclopentanone and analogs havebeen correlated to the presence of a C=C as a cause of the(+, −) couplet in the first case and of a C=O as a cause forthe (−) band in the second. Also, the distance of the chiralcarbon atom C* from the C=C or C=O bonds (and thus theabsolute configuration of C*) has been considered to be ofsome relevance.

3) The most intense NIR-VCD spectra have been ob-served for cyclic molecules possessing large amplitude,low barrier ring deformation (or, to be more specific, ringinversion) modes.

The purpose of this article is to find some theoreticaljustification for observations 1), 2), and 3). In the nextsection we will present a model based on a variant of theperturbation theory, called Van Vleck contact transforma-tion theory.14–19 This allows us to verify that, subject to anumber of approximations, only persisting normal modesexhibit consistent with NIR-VCD spectra (as claimed inpoint 1). However, in order to get rid of the numerousapproximations made in the next section for highly anhar-monic motions, and yet to achieve a qualitative picture of

*Correspondence to: Sergio Abbate, Dipartimento di Scienze Biomedichee Biotecnologie, Universita di Brescia, via Valsabbina 19, 25123 Brescia,Italy, and INFM, UdR Brescia.Received for publication 1 October 1999; Accepted 27 February 2000

CHIRALITY 12:180–190 (2000)

© 2000 Wiley-Liss, Inc.

how circular dichroism is generated, we decided to con-sider the classical dynamics of the chiral HCC*H fragment,which is found (with the same chirality and approximategeometry) in our two prototype molecules. Indeed, classi-cal dynamics is recognized to be quite informative for theinterpretation of spectra in cases of highly anharmonicstates, all lying within narrow energy ranges with highdensity.20,21 The results of the calculations will be pre-sented in the fourth section, after the Morse potential char-acteristics of the CH-bonds in the two molecules are spec-troscopically determined in the third section by means ofNIR absorption spectroscopy. The work in section threeallows substantiation of the empirical observation made inpoint 2) of this introduction. Finally, it has been possible toincorporate, to some extent, the effect of large amplitudetorsional motions for the HCC*H fragment, and thus tocritically evaluate observation 3) above.

CONTACT TRANSFORMATION THEORY

The theoretical models dealing with anharmonic effectsin VCD are few; in the past only Fermi resonance has beenconsidered to some extent by more than one group ofworkers22,23 since Fermi resonance makes its appearancein many regions of the fundamental IR spectroscopic re-gion. More recently, Polavarapu18 reconsidered the resultsof the Minnesota group,14,15 with the purpose of analyzingthe optical activity of a single “chiral” Morse oscillator forthe transitions Dv = 1,2,3. Also, Bak et al.19 incorporatedanharmonic effects into ab initio calculations for fundamen-tal transitions. Our approach in this section makes use ofcontact transformation theory and is quite similar to thatpresented in Refs. 18 and 19; however, we repeat the deri-vation of the results here, in order to verify whether deal-ing with more than one oscillator brings in complicationswith respect to what is anticipated in Ref. 18. As pointed outin Ref. 18, the first systematic use of contact transformationtheory for VCD calculations was made by the Minnesotaschool.14,15,24 We adopt here the approach and notationintroduced in Refs. 14 and 15, and we assume that thedynamics of the system is described by the followingsimple Hamiltonian operator:

H = H0 + «H1 + «2H2 (1)

where:

H0 = ~hc/2!(ivi@~pi/"!2 + qi2# (1)

«H1 = hc(ikiiiqi3 (19)

«2H2 = hc(ikiiiiqi4 (1-)

In Eq. (1) and ff., h is Planck’s constant, c is the speed oflight, qi are n normal mode coordinates (for example, thoseresulting from n linear combinations of the n CH-stretchings), in dimensionless units, pi are their conjugatemomenta, vi are their frequencies, in cm−1 units, and kiiiand kiiii are the only anharmonic cubic and quartic normalmodel force constants, in cm−1, that have been assumed tobe nonzero in the present model. The validity of such anassumption will be critically discussed at the end of thissection.

The method of evaluating the effects of kiii and kiiii con-sists in finding a first contact transformation (a contacttransformation is infinitesimally close to the unit transfor-mation), defined as:

T1 = exp(i«S1) = 1 + i«S1 − (1/2)«2S12 + . . . [28]

such that the modified Hamiltonian H8 has the form:

H8 = T1HT1−1 = H80 + «H81 + «2H82 + . . . (29)

where:

H80 = H0 (29)

H81 = H1 + i@S1,H0# (2iv)

H82 = H2 + i@S1,H1# − ~1/2!@S1,@S1,H0## (2v)

where i is here the imaginary unit, not to be confused withthe running index of Eq. (1) and ff., and:

[S1,H0] = S1H0 − H0S1 [2vi]

From the above equations one sees that the Van Vleckcontact transformation is the quantum analog of the Lieseries canonical transformation.25 At first order one re-quires that the term S1 be such that H81 in Eq. (2iv) is zero,i.e.:

H1 = −i[S1,H0] (3)

We subject then the new Hamiltonian H8 to a second con-tact transformation T2, of the form:

T2 = exp(i«2S2) (4)

so that the further modified Hamiltonian H+ has the form:

H+ = T2H8T2−1 = T2T1HT1

−1T2−1 (48)

With similar requirements as those in Eqs. (28–2vi) and (3),we have a transformation T2T1 for the Hamiltonian as wellas all other operators f, in such a way that one does nothave to transform wavefunctions, but take the transitionmoments and expectation values of the transformed opera-tor on the basis of zeroth order wavefunction. For example,for an operator f of the form:

f = f0 + «f1 + «2f2 + «3f3 + . . . (5)

one can obtain the transformed operator f* up to order 3, aslong as the zeroth order approximation f0 is independent ofq’s and p’s, so that it commutes with the generating opera-tor S3 (and also S2 and S1):15

f* = T3T2T1fT1−1T2

−1T3−1 = f*0 + «f*1 + «2f*2 + «3f*3+....

(6)

It can be shown that:15–17

f*0 = f0 (7)

f*1 = f1 (79)

f*2 = f2 + i@S1,f1# (79)

f*3 = f3 + i@S1,f2# − ~1/2!@S1,@S1,f1## + i@S2,f1# (7-)

INTERPRETATION OF VIBRATIONAL CIRCULAR DICHROISM 181

For the Hamiltonian operator written in Eq. (1) and ff., thefirst order contact transfromation is:

S1 = Si @Siiipi3 + ~1/2!Si

ii~piqi2 + qi

2pi!# (8)

where:

Siii = −~2/3!@kiii/vi"3# (88)

Siii = −@kiii/vi"# (89)

The second contact transformation is instead:

S2 = Si @~1/2!Siiii~qipi

3 + pi3qi! + ~1/2!Siii

i~piqi3 + qi

3pi!# (9)

where:

Siiii = −~3/8!@kiiii/vi"

3# (98)

Siiii = − ~5/8!@kiiii/vi"# (99)

The two operators, the transformation properties of whichwe are interested in, are the electric dipole moment opera-tor µ and magnetic dipole moment operator m. FollowingRefs. 14 and 15, we can rewrite them as follows:

m = m0 + «Si~­m/­qi!0qi = m0 + «m1 (10)

m = «Sijipi + «2Sijijqipj = «m1 + «2m2 (11)

where:

ji = aiSA~ZAe/2c!RA0 x tAi (118)

jij = ~ai/aj!SA~ZAe/2c!tAi x tAj (119)

The last two equations have been written in the fixedpartial charge (FPC) approximation: we notice that in Refs.18 and 19 the authors have gone beyond this approxima-tion, but we will see below that sticking to the FPC approxi-mation will still allow one to grasp some insight into theproblem of overtones VCD. In the equations, index A runsover the atoms, the partial charge of which is ZA times theelementary charge e; their position at equilibrium is RA

0,tAi is the contribution of the ith normal mode to the Carte-sian displacement of atom A. Thus one has for the instan-taneous position RA and linear momentum pA of atom A:

RA = RA0 + SitAi~1/ai!qi

pA = mASitAiaipi

mA is the mass of atom A, and ai is the appropriate factorconverting from the normal coordinates in the usual units,Qi, to dimensionless normal coordinates qi, i.e.:

ai = [2pcvi/"]1/2

At this point, we apply transformations [7] to [7-] (takinginto account the explicit forms of the transformations S1and S2 in [8] to [89] and [9] to [99]) to the electric dipolemoment operator µ and magnetic dipole moment operatorm. We report these expressions in the Appendix. Finally,one can take the following transition moments:

^ vS = 0|m|vs = n & and ⟨ vs = 0|m|vs = n & n = 1,2,3,4,…

Here |vs = n⟩ is the (n − 1)th overtone harmonic wavefunc-tion relative to the s-th normal mode. One then substitutes

Eqs. (88), (89) and (98), (99) into Eqs. (A1) and (A2); finally,recalling the matrix elements of operators qi, pi, and theirlowest powers on the basis of the harmonic wavefunc-tions,15 one can calculate dipole and rotational strengths$0n(s) and 50n(s), for the fundamental (n = 1) and over-tone (n > 1) transitions relative to the normal mode s fromthe definitions:

$01~s! = | ^ vS = 0|m|vs = 1 & |2

= ~1/2!~­m/­qs!02@1 + ~ksss

2/vs2! − ~3/4!~kssss/vs!#

2

(12)

501~s! = Im ^ vs = 0|m|vs = 1 & ? ^ vs = 1|m|vs = 0 &= ~"/2!~­m/­qs)0 ? js ? @1 + ~ksss

2/vs2! − ~3/4!

~kssss/vs!# ? @1 − ~5/2!~ksss2/vs

2!+ ~3/2~kssss/vs!# (128)

Analogously, one obtains, for overtone transitions:

$02~s! = ~1/2!~­m/­qs!02~ksss

2/vs2! (13)

502~s! = "~­m/­qs!0 ? js~ksss2/vs

2! (138)

$03~s! = ~3/4!~­m/­qs!02@3~ksss

2/vs2! + ~1/2!~kssss/vs!#

2

(14)

503~s! = ~9"/4!~­m/­qs!0 ? js@3~ksss2/vs

2!

+ 1⁄2~kssss/vs# ? @~ksss2/vs

2! + 1⁄2~kssss/vs!# (148)

$0n~s! = 0 ; 50n~s! = 0 n $ 4 (15)

Within the assumptions made in the present model, allcombination bands are predicted to have zero dipole aswell as rotational strengths (in both cases this is ultimatelydue to the assumption of electrical harmonic approxima-tion made in Eq. (10)). The conclusions that can be drawnfrom Eqs. (12) through (15) can be synthesized as follows:if one makes the assumption that quartic force constantsare much smaller than cubic ones, or alternatively if oneassumes that they are positive, one sees that overtone ro-tational strengths are of the same sign as the fundamentalup to n = 3. For n $ 4 they are zero at this order oftruncation of the perturbative treatment. In essence, theovertone VCD spectrum is a replica of the VCD spectrumof the fundamental region, at much lower intensity, butwith comparable dissymmetry factors. Comparing Ref. 18,we can state that, within the approximations made, over-tones of normal modes behave like overtones of isolated“chiral” anharmonic oscillators, no matter how complicatedthe linear combinations of individual oscillators giving riseto normal modes may be. The comparison with the resultsof Polavarapu18 is quantitative for transitions Dv = 1, 2, 3(the comparison should be carried out taking into accountthat we have assumed that the only nonzero magnetic di-pole moment derivative, "j, corresponds to (­m/­p) inRef. 18, and that the only nonzero electric dipole momentderivative is (­µ/­q)0. Finally, from (12) through (14) wehave 502(s)/$02(s) ≈ 2501(s)/$01(s), and 503(s)/$03(s) ≈ 3501(s)/$01(s) ? [(ksss

2/vs2) + 1⁄2(kssss/vs)]/

[3(ksss2/vs

2) + 1⁄2(kssss/vs)].The result just obtained has the virtue of extreme sim-

plicity and cogency and allows us to explain observation (1)

182 ABBATE ET AL.

of the introduction made on the NIR-VCD data. However,once more we draw the attention of the reader to the Dra-conian hypotheses made in deriving Eqs. (12) through(15): (1) all anharmonic force constants beyond the quarticterms have been neglected, and Eq. (15) is a direct conse-quence of such an assumption; (2) all off-diagonal anhar-monic force constants have been neglected, while we knowthat at least Darling-Dennison type terms kiijj are importantto explain the transition from normal to local modes;25–27

(3) we assumed no degeneracy, neither permanent noraccidental, between normal mode combination or overtonefrequencies. In conjunction with the existence of anhar-monic interaction force constants kiijj, degeneracies havebeen shown to be quite important in Darling-Dennisonresonance phenomena.27 (4) Finally, all the above expres-sions were derived in the FPC approximation; however,this should not be that big an inconvenience, as long as thesymbols ji and jij in Eqs. (118) and (119) can be replaced byless specified parameters (­m/­pi) and (­m/­qi­pj). Com-paring again with Ref. 18, where higher order derivatives ofthe electric and magnetic dipole moments were consid-ered, one can say that when the dissymmetry factors50n(s)/$0n(s) are not increasing with n, this is not simplydue to a failure of the FPC approximation but rather of theelectrical/magnetic harmonic approximation: indeed, theimportance of electrical anharmonicity has been known fora while from overtone absorption intensities.28,29 We willnot pursue the latter complications any further, and we willconcentrate on the dynamics, attempting to grasp a quali-tative picture from classical dynamics calculations that al-low us to incorporate all anharmonic effects nonperturba-tively. Besides, the semiclassical approach will save us thetremendous algebraic effort needed to carry on to furtherorders the perturbation treatment of this section with theinclusion of higher-order terms.

DETERMINATION OF THE SPECTROSCOPICPARAMETERS FOR (R)-LIMONENE

AND (3R)-METHYLCYCLOPENTANONE

A correlation of the VCD data analogous to that evokedin the introduction of this article for the NIR region hasbeen put forward for the IR fundamental region in Refs. 30and 31. Therein, the rationale of the data was sought in theconformation rather than in the configuration of an ex-tended CH2CH2C*H fragment. The correlation was fol-lowed by a normal mode analysis: it was proposed that acharacteristic VCD triplet (+, −, +) observed for over 20cyclic compounds containing the CH2CH2C*H fragmentcame from the couplings of the two locally antisymmetricnormal modes “resident” in the two CH2 units with theC*H local mode.31 Other articles have critically dealt onthis point.23,32,33 Besides, all authors agree that the lowestfrequency negative VCD fundamental feature is due tosymmetric CH2 normal modes.

However, based on the experience of this work in theCH-stretching fundamental region, we find that it is im-perative to define first the dynamical parameters, since thenormal mode analysis is a consequence of the values forthe CH-stretching principal and interaction force constants:in the articles referenced at the beginning of this section

most of these values came from two IR and Raman studieson cyclohexanone and cyclopentanone and deuterated iso-topomers.34,35 For the overtone region, the most relevantparameters for which we need to determine the values are:v (mechanical frequency), x (anharmonicity) of the CHbonds in the various methylene units, and harmonic inter-action force constants K between various CH bonds. Thesedata come from NIR absorption spectroscopy and ab initiocalculations.

For (R)-limonene the most pertinent NIR studies arethose from the Bordeaux group on cyclohexene and cyclo-hexene-2,2,5,5-d4,36–39 performed on gaseous samples byphotoacoustic spectrometry, or by traditional spectrometryutilizing multipass cells. In the first column of Table 1 wereport the results of these studies: the values for v0 and xfor the CH’s far away from the C=C bond (b and g posi-tions) have been determined in Refs. 36 and 38 by a thor-ough study encompassing the explanation of the numerousFermi resonances present in various overtone regions ofcyclohexene-2,2,5,5-d4 (see Table III of Ref. 36). Such astudy allowed us to also determine harmonic interactionforce constants, also given in Table 1. Further constantsfrom Refs. 36 and 38 are not given here. The values for vand x for the CH’s close to the C=C bond (a positions) havebeen determined from a simple Birge-Sponer plot of ourown, utilizing the data of undeuterated cyclohexene fortransitions Dv = 3, 5, and 6 of Refs. 37–39; for completeness,we have drawn such Birge-Sponer plots also for cyclo-hexene-2,2,5,5-d4 regarding the b and g positions and re-port in Table 1 the results of such plots. One may see thattwo effects of comparable magnitude are affecting the val-ues of v and x, namely, the CH-bonds v’s depend both onthe distance from the C=C bond, and on their conformationbeing pseudo-axial or pseudo-equatorial (the latter bondsare indicated as “ax” or “eq” in the table). The axial/equatorial conformational dependence has been known forsome time and has found different checks by independenttechniques, e.g., ab initio calculations providing bondlengths, the latter quantities being directly connected tothe “strengths” of the CH-bonds, and consequently to v0.For comparison, we report the results of an analogousstudy on cyclopentene, by the same group.40 In the lattercase, a slightly greater dependence on the position in thering, is noticed rather than from the conformation, whilefor cyclohexene it is the other way around. If one assumesthat the vibrational optical activity observed in the NIR isgenerated by the interaction of the ensemble of ring CH-stretchings, one can perceive that the absolute configura-tion and the distance from the C=C bond may influence thegeometry of the normal modes of the ensemble, since anoticeable dependence of the v’s in the ensemble is ex-perimentally observed: this may be parallel to the correla-tion pointed out in point (2) of the introduction betweenVCD signals and configuration/distance from the C=C.

In order to obtain values for the same parameters v, andx for (3R)-methylcyclopentanone, we took the NIR absorp-tion spectra of cyclopentanone and cyclopentanone-2,2,5,5,-d4. The spectra were recorded on a Cary-2300 instrumentusing the neat liquids in cells of 1 to 100 mm with a reso-lution of 2 nm. In order to parallel the information given in

INTERPRETATION OF VIBRATIONAL CIRCULAR DICHROISM 183

Table 1, we recorded the spectra of cyclohexanone andcyclohexanone-2,2,6,6-d4, under the same experimentalconditions. The deuterated compounds were purchasedfrom Merck, Sharp & Dohme (West Point, PA) and theperhydro-compounds from Aldrich (Milwaukee, WI) andnot further purified. The spectra are reported in Figures1–3, for overtones Dv = 2, 3, and 4 and the observed fre-quencies were simply analyzed through Birge-Sponerplots, the results of which are also reported in Table 1,together with the data for cyclopentene and cyclohexene.The analysis of the data and the assignment of the spectrawas conducted as follows. At Dv = 4 (Fig. 3), where thelocal mode behavior is established and the values for v, x,and electric dipole transition moment of the individual CH-bonds are well determined,12,13 two signals are present forcyclopentanone, and, with lower resolution, for cyclohexa-none. Since the same signals are present for cyclopenta-none-2,2,5,5-d4 and cyclohexanone-2,2,6,6-d4, with thesame frequencies and with an intensity decrease equal tothe number of deuterium atoms replacing the hydro-gens,12,13 we assign the two signals to axial and equatorialCH-bond stretchings in either a, b, and g (for cyclohexa-none) positions. The equatorial CH-bond stretchings havehigher frequencies and intrinsic intensities, as has beenobserved in cyclohexene,36 and as has been recognizedand rationalized to some extent in a variety of ring mol-ecules, comprising cyclohexane,41 dioxane, tetrahydropy-rane, and piperidine.42 The previous overtone regions Dv =2 and Dv = 3 (Figs. 2 and 3, respectively) allow us to con-firm this assignment and to identify some shoulders, asslightly distinguishing the a from b positions for equatorial(or pseudo-equatorial) CH-bonds stretching in cyclopenta-none. What is of most relevance here is the behavior of v’s:the conformational effect (namely the dependence of v onthe CH bond being axial or equatorial) is bigger than thepositional (or configurational) effect (namely, the distance

of the CH bond from the C=O). This is at odds with theinfrared data of Ref. 35, but agrees with an ab initio studyby Polavarapu et al.,43 by which it was found that the cal-culated CH-bond lengths are more dependent from theconformation than from the distance from the C=O; in pass-ing, we observe that a discrepancy between the findings ofinfrared and NIR data was pointed out in Ref. 36, in com-menting the results for cyclohexene and cyclopentene. Thesame can be said for cyclohexanone, where the experimen-tal accuracy in the determination of v and x is not as highas for cyclopentanone, since the assignment and deconvo-lution of individual bands is poorer: indeed, the values forthe parameters at the a-positions reported in Table 1 aredetermined from weak features and shoulders of perhydro-cyclohexanone spectra. However, we point out that a re-cent ab initio study32 reached similar conclusions. Ouranalysis of NIR data allows us to conclude that the depen-dence of the harmonic force constants from the distance ofthe double bond, hypothesized earlier on the basis of IRdata,35 is not very important. Qualitatively we conclude thatfor cyclopentanone-type molecules the interacting CH-stretchings giving rise to VCD are quite similar in regard totheir dynamic characteristics; whether the difference in thebehavior of v’s between C=C and C=O containing mol-ecules is in accord with observation (2) of the introductionof this work, much work still needs to be done.

In conclusion, we observed distinct configurational andconformational effects in the values of mechanical frequen-cies v, and, as a consequence, in the local mode charac-teristics of the CH bonds. The first effect is much morepronounced in (R)-limonene than in (3R)-methylcyclo-pentanone. Of course, in order to verify how all this affectsthe dynamics of coupled CH stretchings on the ring, oneshould perform calculations of some sort, e.g., like those ofRef. 36 or classical dynamics calculations, explicitly incor-porating the data of Table 1.

TABLE 1. Experimental values of the mechanical frequencies v (cm−1), of the anharmonicities x (cm−1), of the CH-stretching/CH-stretching interaction force constant K (mdyne/Å), and for the interconversion barrier heights (cm−1) for

cyclopentene, cyclohexene, cyclopentanone, and cyclohexanone

Cyclopentene Cyclohexene Cyclopentanone Cyclohexanone

v x v x v x v x

CHax(a) 3,008.5 66a 3,013 67.5b 3,065 67c 3,000 62e

CHeq(a) 3,057.5 64.5a 3,045 66b 3,085 61c 3,070 63e

CHax(b,g) 3,049 62.5a 3,026 63b 3,065 67c,d 3,062 67f

3,033 67a

CHeq(b,g) 3,077.5 61.5a 3,056 63b 3,097 65d 3,062 67f

3,061 64.5a 3,085 61c

K(CHax − CHeq) 0.078a

K(CHax − CHax) −0.032a

K(CHeq − CHeq) 0.005a

Barrier 3600g 750h

aFrom Refs. 36 and 38.bFrom Birge Sponer plots on NIR data of Refs. 37–39 on perhydrocyclohexene and cyclohexene-2,2,4,4-d4.cThis work, using NIR data of perhydrocyclopentanone.dThis work, using NIR data of cyclopentanone-2,2,5,5-d4.eThis work, using NIR data of perhydrocyclohexanone.fThis work, using NIR data of cyclohexanone-2,2,6,6-d4.gRef. 48.hRef. 49.

184 ABBATE ET AL.

In Table 1 we report also the values of the harmonicinteraction force constants between CH bonds. These val-ues are dependent on the relative conformations and rela-tive distances of the CH bonds: this was concluded in Refs.36 and 38 from a parameter fitting overtone NIR data. Assuch, these values are not 100% sure; however, similar val-ues were obtained also for n-paraffins by overtone-NIR andfundamental-IR spectroscopy.44,45 Also, ab initio studieshave confirmed and signs and orders of magnitude of thesevalues.46 For these reasons we think that the values foundin Ref. 36 for cyclohexene apply to (R)-limonene and to(3R)-methylcyclopentanone.

The last entry of Table 1 is the barrier height for cyclo-hexene and cyclopentanone, which were taken from Refs.47 and 48, respectively. We will use these values for (R)-limonene and (3R)-methylcyclopentanone in the calcula-tions of the following section, even though we know that

substituent groups may influence significantly interconver-sion barriers, as verified by molecular mechanics calcula-tions:49,50 what will turn out to be important, though, is thefact that there is approximately one order of magnituderatio in the values of the barrier of cyclopentanone-typemolecules with respect to the values of the barrier of cy-clohexene-type molecules.

CLASSICAL DYNAMICS CALCULATIONS

In this section, we evaluate whether or not the torsionalor twisting motion of the ring in the two prototype ringmolecules of this work may play a role in VCD. We do thisby examining the classical dynamics of a three degrees offreedom model, with the aim of describing the couplings ofthe two CH-stretchings at the ends of an HCC*H fragment,which is allowed to torque around the CC* bond, as a

Fig. 1. Top: superimposed absorption spectra for cyclopentanone(dashed line) and cyclopentanone-2,2,5,5-d4 (solid line) in the first over-tone region (Dv = 2); (bottom) superimposed absorption spectra for cyclo-hexanone (dashed line) and cyclohexanone-2,2,6,6-d4 (solid line) in thefirst overtone region (Dv = 2). In both cases a 1 mm pathlength cell wasemployed, filled with neat liquid.

Fig. 2. Top: superimposed absorption spectra for cyclopentanone(dashed line) and cyclopentanone-2,2,5,5-d4 (solid line) in the second over-tone region (Dv = 3); (bottom) superimposed absorption spectra for cyclo-hexanone (dashed line) and cyclohexanone-2,2,6,6-d4 (solid line) in thesecond overtone region (Dv = 3). In both cases a 1 cm pathlength cell wasemployed, filled with neat liquid.

INTERPRETATION OF VIBRATIONAL CIRCULAR DICHROISM 185

consequence of ring twisting.47,49 The fragment HCC*Hunder study is defined as the chiral fragment in both (R)-limonene and (3R)-methylcyclopentanone, located at theCC* bond antipodal to C=C or C=O: respectively, the chiralcarbon C* corresponds to carbon atom #4 in (R)-limoneneand #3 in (3R)-methylcyclopentanone, and the other car-bon atom corresponds to carbon atom #5 in (R)-limoneneand #4 in (3R)-methylcyclopentanone, the C*4H (C*3H)being axial and C5H (C4H) equatorial. The fragment isassumed to have a perfectly tetrahedral geometry. Thefragment has the same chirality for the two molecules andis schematically drawn in Figure 4. The vibrational Hamil-tonian function that we assume to describe the vibrationaldynamics of such a fragment is:

H = ~p12/2m! + ~p2

2/2m! + ~pw2/27~l1,l2!!

+ D1~w!@1 − exp~−a1~l1 − l01!#2

+ D2~w!@1 − exp~−a2~l2 − l02!#2 + K~l1 − l01!~l2 − l02!

+ V@1 − cos~w + p/3! − cos~3w + p!#/@1 − ~w/p!2# [16]

In Eq. [16], l1 and l2 are the instantaneous lengths of thetwo CH bonds (#1 being the one originating from the chiralcarbon, see Fig. 4), the equilibrium bond lengths being l01and l02 (here l01 = l02 = 1Å). The third degree of freedom isw, which is defined as the dihedral angle between theplanes H1C*1C2 and H2C2C*1, being positive for a clock-wise rotation of H1C*1 onto H2C2 (see Fig. 4). The corre-sponding conjugate momenta are p1, p2, and pw. m is themass of the hydrogen atom (here 1 a.m.u.). 7(l1,l2) is thereduced moment of inertia of the fragment HCC*H, de-fined as follows:

(7(l1,l2))−1 = (71 + ml12sin2u1)−1 + (72 + ml22sin2u1)−1 [168]

where 71 and 72 are ad hoc values, so as to represent themoments of inertia of the molecular moieties bonded to C*and C (except H1 and H2), respectively, in the two mol-ecules: for (R)-limonene we took 71 = 101.2 a.m.u.Å2, and72 = 26.35 a.m.u.Å2; for (3R)-methylcyclopentanone wetook 71 = 50.6 a.m.u.Å2, and 72 = 26.35 a.m.u.Å2. Group 1in (R)-limonene has a higher moment of inertia 71 withrespect to (3R)-methylcyclopentanone, since it comprisesan isopropylene group in addition to a CH2CH2 fragment,whereas the corresponding fragment in (3R)-methyl-cyclopentanone comprises a methyl in addition to CH2CH2.u1 and u2 are tetrahedral. In Eq. (168) we use an approxi-mation of the proper kinetic term: indeed, in reducing thefull vibrational dynamics of either molecules to a modelcomprised of just large amplitude modes, the kinetic en-ergy should be evaluated following one of the laboriousmethods proposed in the literature,51–53 since the Wilson Gmatrix formulas are valid only for the complete vibrationalproblem. In particular, for six-membered rings one shouldconsider the ring bending and the ring twisting (the defi-nitions are those of Laane et al.53,54). In the case of (R)-limonene and cyclohexene, it appears that the ring inter-conversion is mostly driven by just one coordinate, namelythe torsion around the CC bond antipodal to the endocyclicC=C bond.50 Considering the latter coordinate should beenough to see if there is an effect of the ring flexibility onthe CH-stretching vibrations. In this respect, we adopt aneffective potential in the torsional coordinate, which, fol-lowing Ref. 54, corresponds to twice the twisting coordi-

Fig. 3. Top: superimposed absorption spectra for cyclopentanone(dashed line) and cyclopentanone-2,2,5,5-d4 (solid line) in the third over-tone region (Dv = 4); (bottom) superimposed absorption spectra for cyclo-hexanone (dashed line) and cyclohexanone-2,2,6,6-d4 (solid line) in thethird overtone region (Dv = 4). The spectra of the perhydro compoundswere taken with a 10 cm pathlength cell filled with neat liquid; the deuter-ated compounds were measured in a 1 cm pathlength cell filled with neatliquid. In order to have both spectra on the same absorbance scale, wemultiplied those of the deuterated compounds by 10.

Fig. 4. Schematics and Newman projection of the fragment HCC*Htreated in the present work, and definition of the coordinates l1, l2, and wemployed in the Hamiltonian function of Eq. (16) (l10 = l20 = 1 Å, u1 = u2 =109.47°, r = 1.54 Å).

186 ABBATE ET AL.

nate. We extend our model to (3R)-methylcyclopentanone,where the large amplitude ring modes are less separatedthan in (R)-limonene. The expression we use for the tor-sional potential of both molecules corresponds to the lastterm in Eq. (16) and gives two inequivalent minima sepa-rated by an interconversion barrier; the denominator issuch as to forbid a whole rotation of one CH with respectto the other. The absolute minimum is at w = −61.5° for themost stable conformation of the HCC*H such that bondC1H1 is axial and bond C2H2 is equatorial, as described inFigure 4. The parameter V is in relation to the barrierheight, DU, between the minimum conformation and thetransition state (which is the cis, at 0°): we have taken DU= 3600 cm−1 for (R)-limonene, as found for cyclohexene,47

and DU = 750 cm−1 for (3R)-methylcyclopentanone, as forcyclopentanone48 (compare with Table 1).

The two CH-stretchings in Eq. (16) are Morse oscilla-tors, the dissociation energy Di of which is assumed todepend on the dihedral angle w. We recall that:

Di = vi2/4xi , ai = =8p2mcx/h ~i = 1,2! [169]

where the parameters vi and xi (i = 1,2) are the mechanicalfrequencies and anharmonicities for the two CH bonds.The dependence of Di on w is prescribed through the de-pendence of vi on w, that had been proposed by McKeanand Watt some time ago55 and by Cavagnat and Las-combe,56 to account for the spectroscopic behavior of CH-stretching vibration in almost freely rotating methyls. Fol-lowing Ref. 55 we take, for the two CH bonds:

v1~w! = v0 + Bcos~w − 2p/3!

v2~w! = v0 + Bcos~w + 2p/3![16-]

By substituting w = −p/3 (namely, sitting very close to theminimum of the torsional well) in Eq. [16-], one obtains forbonds 1 and 2, respectively: v1 = v0 − B, and v2 = v0 + B/2.The values of v0 and B, which allow us to account for theaxial and equatorial CH mechanical frequencies of Table 1are: v0 = 3046 cm−1 and B = 20 cm−1 for cyclohexane and v0= 3086 cm−1 and B = 21 cm−1 for cyclopentanone: we usedthe former values for (R)-limonene and the latter for (3R)-methylcyclopentanone.

Finally, in Eq. [16] we allowed for a direct interaction ofthe two bonds via a force constant K, the value of which wehave taken for both molecules from Table 1 and Ref. 36 toreproduce the axial/equatorial interaction in cyclohexene,namely K = +0.078 mdyne/Å.

After defining all the parameters in Eq. [16], let us de-scribe the results of classical trajectory calculations. Weused a computer program based on the Runge-Kutta algo-rithm at the fourth order. The integration step-size was inall cases 0.05 femtosec, the total integration times rangedfrom 2 × 103 to 106 femtosec. For the latter time intervalsthe total energy was found to be stable to 1 part over10,000. The initial conditions were chosen with the aim ofcovering all possible spectroscopically interesting semi-classical trajectories. We proceeded as follows. First, thetotal energy values ET were taken so as to fulfill the semi-classical correspondence principle in the total stretching

quantum number for representing transitions 0 → v (v = 1,2, 3, 4). We achieved this objective approximately by ne-glecting the torsional contribution to ET and by making thehypothesis that the two Morse oscillators be uncoupledand identical to a mean oscillator M, with vM and xM givenby:

vM = ~v1 + v2!/2 xM = ~x1 + x2!/2 [17](v1 = vax; v2 = veq, e.g.). Hence, we defined the total en-ergy ET assuming that all vibrational quanta, appropriate torepresent the transition, be localized on one of the twooscillators. Following Heisenberg,57 we recall that thequantum matrix elements of an operator can be put incorrespondence with the Fourier coefficients of the relatedclassical function calculated on the proper classical trajec-tory. The trajectory to be used for a transition 0 → v is theone corresponding to the mean action between the initialand final states.58 This has been, in particular, verified fora Morse oscillator.59 In conclusion we take:

ET = vM[(v/2) + (1/2)] 1 xM[(v/2)

+ (1/2)]2 + vM/2 1 xM/4 [178]

In this way ET = 4483, 5925, 7335, and 8713 cm−1 for (R)-limonene and ET = 4538, 5996, 7420, and 8812 cm−1 for(3R)-methylcyclopentanone, to represent transitions Dv =1, 2, 3, and 4, respectively. Needless to say, ET is a constantof the motion; the other initial conditions were chosen sothat, by the method of Poincare surfaces of section (PSS),60

one could get a qualitative picture of all possible transitionswithin the manifold described by the total stretching tran-sition number Dv. In all cases it was taken: l1(t = 0) = l10, l2(t= 0) = l20, and w(t = 0) = −61.5°. The initial torsional energywas given a single value ETORS, which was taken as 0, at thezero-point energy (ZPE), and at 350 cm−1 in three distinctsets of calculations. The difference DE in the initial ener-gies E1 and E2 of the stretchings was allowed to take anumber of values from −ESTR to +ESTR (ESTR = ET − ETORS),each separated by the next by a constant value DE. Thetorsional momentum pw(t = 0) was always taken positive,while the CH-stretching conjugated moments p1(t =0) andp2(t = 0) were taken both positive and negative.

By first assuming that the effect on the stretching fre-quencies claimed by McKean and Watt55 is negligible (i.e.,assuming w = −p/3 in Eq. [16-] at all times), we obtainedthe results of Figure 5 for Dv = 1 and Dv = 4, where in bothcases the initial torsional energy ETORS was given the quan-tum zero-point value (ZPE), i.e., 39 cm−1 for (3R)-methylcyclopentanone and 78 cm−1 for (R)-limonene. Thelatter values were calculated as (1/2)vSA, where vSA is thevibrational frequency of the torsional coordinate w at smallamplitudes. The PSS is defined as the loci of the points inthe (l2, p2)-plane at the instants when l1 = l10, p1 > 0, irre-spective of what happens to w and pw.60 All curves areregular and thus multiperiodic.60–63 Three families ofcurves are topologically distinguished in Figure 5, the ex-tent of each being different in going from Dv = 1 to Dv = 4.The first family is composed of deformed ellipses centeredaround a fixed point at l2 = l20 and p2 > 0: we call this familynormal modes comprising purely symmetric normal modesclose to the fixed point and deformed symmetric normal

INTERPRETATION OF VIBRATIONAL CIRCULAR DICHROISM 187

modes with large antisymmetric contributions, the latterbeing those curves crossing the line p2 = 0.63,64 The secondand third families are the inner and outer circles, respec-tively, both centered approximately at l2 = l20 and p2 = 0: wecall both of them local modes. The general aspect of Figure5 is determined by the harmonic interaction force constantK between C1H1 and C2H2 being positive.63 The torsion isseen to have no appreciable influence, due to the smallnessof its interaction with the CH-stretchings, represented bythe kinetic terms of Eq. [168]. When Eq. [16-] is not valid,even by giving initially higher torsional energy valuesETORS, the general aspect of Figure 5 does not change.Moreover, it is interesting to notice that the differencebetween the two molecules and between Dv = 1 and Dv = 4is not very big. The latter fact is due in part to the assump-tion of Eq. [178] for the correspondence principle.

In another set of calculations, we incorporated the effectcaused by Eq. [16-]. In Figure 6 we compare the results ofFigure 5 for Dv = 1 that we repeat in the two squares of thefirst column65 with the PSS obtained allowing for w depen-dent v’s, with values ETORS at ZPE (second column), orETORS = 350 cm−1 (third column). The torsional motionbrings in definitely some perturbation, which is in the formof a “blurring” of the initial trajectories, the general aspectremaining unchanged. The Fourier analysis below will al-low better definition of the torsional perturbation. It is in-teresting to notice that some local modes are not at allaffected by the torsion. The effects are bigger for (3R)-methylcyclopentanone, since the same torsional excitationenergy implies larger amplitude w-motions in this moleculewith respect to (R)-limonene and consequently biggerchanges in v’s.

In order to generate spectra to compare with the over-tone and fundamental VCD experiments, we employed thecross-spectrum algorithm in the FPC approximation of Ref.64. We used Eq. #26 of Ref. 64 and employed analysis timesof 13,072 femtosec, with a value of 0.05 e.s.u. for the chargeof both hydrogen atoms, and with the CC equilibrium bond

length R = 1.54 Å. The “choice” of the trajectories from theensemble of Figure 6 has to come from semiclassical quan-tization considerations.58,59,64,66). We will not employ pre-cise semiclassical quantization here, but just make the ob-servation that two trajectories have to be chosen for thetransition Dv = 1, three for Dv = 2, four for Dv = 3, and fivefor Dv = 4. This is due to the existence of a second quantumnumber 1 that varies from 0 to v. This second quantumnumber describes some special characteristics of the tra-jectories of Figure 6 (area in special rectified phase spacecoordinates). Approximately, low values for 1 correspondto small DE values of the two stretchings energies and thusto banana-shaped normal modes curves of increasing area;high values of 1 to higher DE values and thus to localmodes, close to v being assigned wholly to one bond.63.For this reason, we report in Figure 7 the Fourier cross-spectra64 of an almost pure symmetric normal mode A, ofa more perturbed normal mode B, and of a local mode C,respectively, for (3R)-methylcyclopentanone (see Figure6), the torsional excitation being at ZPE, with the McKean-Watt effect of Eq. [16-] (for this torsional excitation (R)-limonene and (3R)-methylcyclopentanone behave simi-larly). In all cases, the Fourier spectrum is a line spectrum,meaning that the motions are still multiperiodic for thislevel of torsional excitation. The symmetric normal mode(A) has one positive feature at high frequency in the fun-damental and at all overtones (the square around 3000cm−1 and ff., respectively). The deformed normal mode (B)has two components of opposite sign in the fundamental(first square), the positive one at high frequency, and thenegative one of smaller magnitude at low frequency, a trip-let in the first overtone (second square) and so on, withsign alternation as described in Ref. 64. The local mode (C)has one VCD line which alternates in sign in going fromthe fundamental (first square), to the first overtone (sec-

Fig. 5. Poincare surfaces of section for (R)-limonene (left) and (3R)-methylcyclopentanone (right). HCC*H fragments for total energies ETcorresponding to transitions Dv = 1 and Dv = 4 in the stretching totalquantum number v. In both cases no change in v0’s was allowed with wmotions. An initial value ETORS = 78 cm−1 and ETORS = 39 cm−1 was assignedto the torsional energy of (R)-limonene and (3R)-methylcyclopentanone,respectively (see text).

Fig. 6. Comparison of the Poincare surfaces of section of (3R)-methylcyclopentanone (top), and (R)-limonene (bottom) HCC*H frag-ments representing the transition Dv = 1 in the stretching total quantumnumber, when no variation is allowed in v0 with w motions (left) and whenthe dependencies of Eq. (16-) are prescribed to v0’s (center and right). Forthe cases of the center squares, the initial value for the torsional energyETORS was prescribed at ZPE (ETORS = 78 cm−1 for (R)-limonene and ETORS= 39 cm−1 for (3R)-methylcyclopentanone). For the cases of the rightsquares, ETORS = 350 cm−1. A, B, and C indicate a symmetric normal mode,a deformed normal mode, and a local mode, respectively (see text andFig. 7).

188 ABBATE ET AL.

ond square), to the second overtone (third square), etc.64

The magnitude of the VCD local mode signal, though, ismuch smaller than that from normal modes. For this rea-son we conclude that local modes do not contribute, withinthe FPC approximation, to the VCD spectrum. It is stillpremature to see whether the theoretical VCD spectra ofthe normal modes are in accord with the experiments bothfor (R)-limonene and for (3R)-methylcyclopentanone. In-deed, in the first case a positive line is observed at lowfrequency and a negative one at high frequency,9,10 whilethe predictions of the present calculations are that the posi-tive line be at high frequency in the fundamental and at allovertones, at lower frequency there being a line with anegative sign and lower absolute value in the fundamental,with possible sign changes in the overtone. Since the ge-ometry of the fragment is known and close to that of Figure4,47,48 we think that calculations and experiments would bereconciled, on one hand, by introducing more degrees free-dom in the calculations, since more than two CH stretch-ings are likely to be responsible for the observed signal, asindicated in the fundamental region.30,31,33 On the otherhand, in order to prove the validity of our classical dynam-ics model VCD data are needed for selectively deuteratedmolecules, containing just the HCC*H fragment. In fact, ifone obtained nonzero experimental rotational strengths atall Dv’s, this would be a test of the validity of the threedegrees of freedom model of this section, together with thepossibility of getting better estimates for the parametersemployed in the expressions of the kinetic and potentialenergies in Eq. [16]. In any case, we feel that for (R)-limonene the torsion does not have a dramatic influence onthe VCD spectrum, as it does for (3R)-methylcyclopenta-none. Indeed, for a trajectory corresponding even to a Dv =

1 transition, if one increases ETORS to 350 cm−1, the VCDspectrum is markedly affected and starts to show chaotic-type nonline Fourier features (see Fig. 8).

In conclusion, we feel that some hints to explaining ob-servation (3) of the introduction have been made, eventhough much remains to be done, both theoretically andexperimentally.

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Fig. 7. Fourier cross-spectra for the symmetric normal mode A (topline), for the deformed normal mode B (center line), and for the local modeC (bottom line) of Figure 6 in the case of the (3R)-methylcyclopentanoneHCC*H fragment. The w-dependence of the v0’s of Eq. (16-) was includedand the relevant initial conditions were ET = 4538 cm−1 (Dv = 1) and DE =374 cm−1 (difference in the energies of the two bonds, E1–E2), p1 > 0, p2 >0, ETORS = 39 cm−1 (ZPE) for A, ET = 4538 cm−1, and DE = 1874 cm−1, p1 >0, p2 > 0, ETORS = 39 cm−1 (ZPE) for B, and ET = 4538 cm−1, and DE = −4124cm−1, p1 > 0, p2 > 0, ETORS = 39 cm−1 (ZPE) for C. In the three cases, thefirst squares from the left correspond to the fundamental region, the sec-ond to the first overtone, the third to the second overtone, the fourth to thethird overtone. The ordinate scale of the fundamental and overtones arearbitrary units; the relative units between A, B, and C, and between thevarious overtones are correct.

Fig. 8. Poincare surfaces of section and Fourier cross-spectra for achaotic mode in the (3R)-methylcyclopentanone HCC*H fragment with thew-dependence of the v0’s of Eq. (16-). The relevant initial conditions wereET = 4546 cm−1 (Dv = 1) and DE = −3839 cm−1 (difference in the energiesof the two bonds, E2–E2), p1 > 0, p2 > 0, ETORS = 350 cm−1 (see Fig. 7). Thefour squares of the Fourier cross-spectra report data from left to right fromthe fundamental to the third overtone region. The ordinate scale of thefundamental and overtones are arbitrary units; the relative units betweenthe various overtones are correct.

INTERPRETATION OF VIBRATIONAL CIRCULAR DICHROISM 189

21. Frederick JH, Heller EJ. Comp Phys Comm 1988;51:83.22. Abbate S, Havel HA, Laux L, Pultz V, Moscowitz A. J Phys Chem

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Journal de Physique, Col. 7, 1991;517.38. Rodin S. Redistribution intramoleculaire de l’ energie vibrationelle de

molecules flexibles dans les domaines du proche infrarouge et duvisible. PhD Thesis, Universite de Bordeaux I; 1993.

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APPENDIX

The operator µ of Eq. (10) has contributions up to thefirst order in « (the perturbation order is established by thehighest power of qi and pi). Applying Eqs. (6) through (6-)and (7) through (7-) with the appropriate transformationoperators S1 and S2 as given by Eqs. (8) and (9), one ob-tains:

m+ = m0 + (i~­m/­qi!0qi + "(i~­m/­qi!0~3Siiipi2 + Sii

iqi2!

− ~"2/2!(i~­m/­qi!0@3SiiiSiii~pi

2qi + qipi2! − 2~Sii

i!2qi3#

+ "(i~­m/­qi!0@~3/2!Siiii~pi

2qi + qip12! + Siii

iqi3# (A1)

The operator m of Eq. (11) has contributions from first andsecond order terms in «. Applying again Eq. (6) through(9), one obtains:

m+ = (ijipi + (ijijpipj − "(ijiSiii~piqi + qipi!

+ ~"2/2!(iji@~Siii!2~piqi

2 + qi2pi! − 6Sii

iSiiipi3#

− "(iji@~3/2!Siiii~piqi

2 + qi2pi! + Si

iiipi3# (A2)

190 ABBATE ET AL.