Correlation Between Time Reversal Symmetry andChirality in Certain Molecular Processes

TIANHU LI* AND ALAN NADIN*Department of Chemistry, The Scripps Research Institute, La Jolla, California

ABSTRACT If a molecule is identified not only by its static spatial constructions, butalso by the motions at the sub-molecular level, application of time reversal symmetryoperation to a certain molecule could lead to another distinguishable from the originalin the sense of sub-molecular motions, a phenomenon now defined as time reversalisomerism. Assessment of the consideration of certain enantiomers as distinguishabletime reversal isomers is suggested in order to evoke a comprehensive interpretation of alikely correlation between the two types of isomerisms. The conceptual basis of a con-nection between absolute asymmetric synthesis under the influence of external fieldsand the intrinsic time reversal symmetry violation at the molecular level is also estab-lished to encourage new experimental investigations on this theme. Chirality 10:289–293, 1998. © 1998 Wiley-Liss, Inc.

KEY WORDS: chirality; time reversal symmetry; asymmetric synthesis; enantiomerism;isomerism

Time reversal symmetry, one of the three fundamentaldiscrete symmetry operations (the other two being parityand charge conjugation) describing the symmetry of thelaws of nature, replaces the time coordinator (t) by (−t) inequations formulating physical laws.1 When the movingobjects and their paths are well defined in a system, timereversal symmetry can be taken as the operation that re-verses the motion of the constituent moving objects in thesystem along their original paths.2,3 In the past two de-cades, the correlation between time reversal symmetry andmolecular chirality has been the subject of many theoreti-cal investigations. A possible connection between the timereversal symmetry violation at the level of elementary par-ticles with the original driving force for chiral asymmetricemergence of biomolecules on earth, for example, hasbeen suggested.4 The implications of time reversal sym-metry in certain chemical reactions was also theoreticallyinvestigated in the search for asymmetric production ofchiral molecules assisted by external fields.5–8 In addition,on the basis of the basic principle of time reversal symme-try and elaborate theoretical analysis, the important con-cept of ‘‘true’’ and ‘‘false’’ chirality was deduced a decadeago, to help distinguish chirality from other types of struc-tural dissymmetry.9,10 Furthermore, the implications ofparity, time reversal, and charge conjugation in moleculardynamics were described recently, which provided a newview of the coherence of these symmetries at the molecu-lar level.11,12 Most recently, based on the consideration oftime reversal symmetry and analysis of characteristics ofmolecular structures, we presented a theory of intrinsictime reversal symmetry violation at the molecular level.3Based on this, we extend in this note the conception oftime reversal isomerism and describe the correlation be-

tween time reversal symmetry and molecular chirality froman alternative perspective. We hope that this discussionwill complement previous ideas4–12 and be constructive forexplaining the wider implications of chirality likely to beembedded in molecular processes.

AN EXTENDED VIEW OF TIMEREVERSAL ISOMERISM

Molecular structures are ordinarily described in terms oftheir static spatial constructions. However, certain types ofmotions at the sub-molecular level are commonly associ-ated with the static spatial constructions of the molecule,displaying a new dynamic transition of the molecule. Thesub-molecular motions in many molecular processes canbe readily defined and actually described. The p electronsin the aromatic molecule 3 (Fig. 1) under certain circum-stances can, for example, circulate along the conjugated pring, generating an electric current. The moving objects inthe aromatic molecule 3 (Fig. 1) can therefore be identifiedas the p electrons and their path is the conjugated p ring.Application of time reversal symmetry operation to the dy-namic transition 3 reverses the motions of the p electronsalong their original path,2,3 the conjugated p ring of 3,leading to a new dynamic transition 4 (Fig. 1). Leaving outthe motions at the sub-molecular level (the p electron cir-culating along the conjugated ring), 3 and 4 (Fig. 1) areidentical. However, if a molecule is identified not only by its

*Correspondence to: Dr. Tianhu Li, Department of Chemistry, The ScrippsResearch Institute, La Jolla, CA 92037. E-mail: tianhuli@scripps.edu or Dr.Alan Nadin, Merck, Sharp, and Dohme, Neuroscience Research Center,Terlings Park, Harlow, Essex CM20 2QR. UK. E-mail: alan nadin@merck.comReceived for publication 18 September 1996; Accepted 27 June 1997

CHIRALITY 10:289–293 (1998)

© 1998 Wiley-Liss, Inc.

static spatial constructions, but also the motions at thesub-molecular level, the two dynamic transitions 3 and 4(Fig. 1) can be designated as the pair of molecules relatedby time reversal symmetry, namely time reversal isomers.The pair of time reversal isomers are named indistinguish-able time reversal isomers (ITR isomers or ITRmers) ifthey are superimposable by rigid rotation, and named dis-tinguishable time reversal isomers (DTR isomers orDTRmers) if they are not superimposable. We suggest thatthe term time reversal isomerism3 be used to describe thegeneral existence of ITR isomers and DTR isomers in mo-lecular processes.

A specific example of time reversal isomers in aromaticring current molecules was identified previously.3 A fewnew examples of the isomers are illustrated in Figure 1.The pair of dynamic transitions 1 and 2, 5 and 6, 9 and10, and 13 and 14 are pairs of ITR isomers, and 3 and 4,7 and 8, 11 and 12, and 15 and 16 are pairs of DTRisomers. The time-dependent objects are p electrons in 1to 4, groups of molecules in 5 to 8, and atoms in 13 to 16.Sequential Cope rearrangements of hypostrophene arewell characterized chemical processes.13,14 Structures 9 to12 represent one series of structures singled out from thevarious consecutive rearrangements, in which the electronmigrations take place sequentially in a single directionalong the five-membered carbon ring. The time-dependentobjects in 9 to 12 are therefore p and s electrons associ-ated with the periodic variation of the molecular geometry.The structurally distinguishable nature of the pairs 3 and

4, 7 and 8, and 11 and 12 is independent of the timelength of observation; however 13 to 16 exhibit time-length-dependent time reversal isomerism. In 3-fluoropyridine 13, the F atom can vibrate above and belowthe plane of its aromatic ring in a simple harmonic fashion.The dynamic transition 13 in Figure 1 represents that ofthe time course of one period of vibration of the F atom(from position 1 to 2, and back to position 1) while 15represents the dynamic transition of a half period of vibra-tion (from position 1 to 2). The motion patterns of the twotime reversal transitions 13 and 14 are identical. How-ever, the motion patterns of 15 and 16 are distinguishablewith respect to the direction of the motion of F. The causeof the time-length-dependent time reversal isomerism in13 to 16 is that the pathway of vibration of the F atom inits second half period is identically equivalent to the timereversal of the motion pattern in its first half period. Manyother molecular motions, such as bond stretching, alsodisplay time-length-dependent time reversal isomerism.Unlike all previously categorized isomerisms derived inconsideration of space (e.g., cis-trans isomerism, enantiom-erism, and conformational isomerism), the newly intro-duced time reversal isomerism is a phenomenon with re-spect to time. The characteristic difference of time reversalisomerism from any other types of previously character-ized isomerism is that time reversal isomers possess mo-mentum vectors associated with the time-dependent ob-jects. The lack of a proper symmetry element within thestructure of DTR isomers with respect to these momentum

Fig. 1. Examples of time reversal molecular states in different types of molecular processes. The time reversal states 1 and 2, 5 and 6, 9 and 10,and 13 and 14 are pairs of ITR isomers, and 3 and 4, 7 and 8, 11 and 12, and 15 and 16 are pairs of DTR isomers.

290 LI AND NADIN

vectors makes the two time-dependent molecular statesstructurally distinguishable.

CONSIDERATION OF CERTAIN ENANTIOMERS ASTHE TIME REVERSAL ISOMERS

Since Pasteur’s first separation of a racemic organic com-pound into optically active crystals 140 years ago,15 severalelegant and comprehensive descriptions of the structuralfeatures of chiral molecules have been developed (for re-views, see refs.1,16,17 and also see refs.11,12,18). Amongthem, the recent interpretations of the implications of timereversal symmetry displayed a new perspective of the na-ture of molecular chirality.9–12,19 Inspired by the previousvisions of chirality from the viewpoint of time reversal sym-metry and the elucidation of the role of nuclear motion inmolecular chirality,20 we suggest that certain pairs ofenantiomers can be considered as time reversal isomers(Fig. 2).

In Figure 2, A represents one of a dynamic transition ofan isolated molecule (ABCDE) in which the nucleus E isconsidered to be in periodic circulation, and the other nu-clei A, B, and D are in translational motions relative to theframe of reference of the centered atom C as shown.Nucleus E has two different types of continuous motion inits complete cycle: (1) tunneling through the potential en-ergy barrier within the Coulombic equilibrium separationdistance (r0) (from position 1 to 2), as described by Hund’stheory18; and (2) translocation beyond r0 (from position 2to 1 via position 3). The circulation of nucleus E beyond r0in the nuclear motion model can be considered to be di-rectly analogous to the experimentally well-established[1,1] sigmatropic nuclear shift.21 One period of circulationof nucleus E is the sum of Dt1 and Dt2 where Dt1 is thetime taken for the translocation of nucleus E, and Dt2 is thetime taken for inversion through tunneling. The relation-ship Dt1 << Dt2 commonly holds because energy barriersto tunneling are known to be much higher than those ofthermal unimolecular nuclear translocation in the pro-cesses of most tetra-substituted tetrahedral molecules.22

Because one period of circulation of nucleus E is muchgreater than the time length of common observations, acomplete periodic motion of nucleus E is rarely observable.

Instead, the single nuclear configuration C in Figure 2 iscommonly observable because tunneling occurs far lessfrequently than nuclear translocation beyond r0 along thepathway (Dt1 << Dt2). Application of the time reversal sym-metry operator to A (Fig. 2) leads to B in Figure 2. The twodynamic transitions A and B (Fig. 2) are a pair of DTRisomers when the nuclei A, B, and D are structurally dif-ferent. The single configuration D is the most observableconfiguration of B owing to the same reasons given above.

According to the nuclear motion model, a certain pair ofenantiomers (e.g., C and D in Fig. 2) can be interpreted asthe two most observable nuclear configurations that occurin a pair of DTR isomers A and B, respectively. Becausethe time period is so long, observation of a complete periodis extremely unlikely and we perceive the DTRmers A andB as a pair of enantiomers (C and D). In addition, thismodel depicts unimolecular racemization as the processthat reverses the momentum vectors of the nuclei of DTRisomers. This time reversal symmetry consideration ap-pears to be only applicable to tetrahedral chiral molecules,and we are aware that molecules may be chiral other thanby having this feature.

CORRELATION BETWEEN ABSOLUTE ASYMMETRICSYNTHESIS AND THE INTRINSIC VIOLATION OF

TIME REVERSAL SYMMETRY AT THEMOLECULAR LEVEL

The possibility of chiral asymmetric synthesis under theinfluence of magnetic and/or electric fields has been thesubject of theoretical and experimental investigation foryears (for recent discussions on this topic, see refs.23–25).In addition, a new mode of interaction named ‘‘chiral inter-action’’ based on general symmetry considerations and de-signed experimental verification involving an external mag-netic field were proposed by Gilat.26 Most recently, basedon the consideration of a breakdown of microscopic revers-ibility and the CP violation, the theoretical grounds for theasymmetric production were carefully examined by Bar-ron, to solicit new specifically designed experiments onthis theme.27 Besides the conceptual basis established byGilat26 and Barron,27 it can be deduced on the basis of ourrecently established principle3 that there exists a correla-tion between asymmetric synthesis and the intrinsic timereversal symmetry violation at the molecular level. Timereversal symmetry conservation implies that a pair of DTRisomers exhibit time reversal degeneracy during any ex-perimental measurements. If time reversal symmetry isviolated at the molecular level, however, a pair of DTRisomers are not exactly degenerate and the non-degeneracy between them could be manifest in certainphysical and chemical processes. The conception of intrin-sic time reversal symmetry violation at the molecular leveland its correlation with chiral asymmetry production areillustrated in Figures 3 and 4.

If time reversal symmetry is conserved in molecular pro-cesses, the pair of time reversal isomers 1 and 2 (Fig. 3A)are degenerate. Under the examination of nuclear mag-netic resonance (NMR) spectroscopy, 1 and 2 should giveexactly the same NMR spectra. If time reversal symmetryis violated, one of the possible consequences of the sym-

Fig. 2. The nuclear motion model of chiral molecules.

TIME REVERSAL SYMMETRY AND CHIRALITY 291

metry violation could be that the electric currents in oppo-site directions along the aromatic ring in 1-fluoro-2-iodobenzene (1 and 2 in Fig. 3B) are not equally induced,giving rise to a slight energy difference, which could, inprinciple, be detected by a difference in their NMR spectra.

Figure 4 is an illustration of the addition reaction28 ofpositive species R+ [R+: e.g., Mn+(CO)3] to 1-fluoro-2-iodobenzene under paralleled electric and magnetic fields.The numbers of 1-fluoro-2-iodobenzene molecules withtheir molecular planes facing up and down with respect tothe direction of external fields are equal in the system dueto heat motion (1 and 2 in Fig. 4). Under the influence ofthe external electric field, positively charged species R+ are

moving upward along the direction of the external electricfield and therefore addition reactions are more likely totake place between 1-fluoro-2-iodobenzenes 1 and 2 andthose R+ underneath them, rather than those from above.If time reversal symmetry is conserved, a pair of enantio-mers 5 and 6 will be produced in equal amounts owing tothe degeneracy between 1 and 2 (Fig. 4A). If time reversalsymmetry is violated, the local magnetic flux densityaround 1 and 2 (Fig. 4B) is different. Therefore, the addi-tion reaction of positive species R+ to 1 and 2 could be akinetically non-degenerate process, and an excess of one ofthe enantiomers 5 or 6 (Fig. 4B) would be produced. Itshould be pointed out that the experiments shown in Fig-ure 4 differ from any of the previous attempts at absoluteasymmetric synthesis under the influence of magneticand/or electric fields. The necessary characteristics of thisexperiment are the co-existence of (1) a pair of DTR iso-mers possessing well-defined electric currents, (2) paral-leled external electric and magnetic fields, and (3) direc-tional motion of charged reactants (R+).

The interaction mediating molecular processes is elec-tromagnetic and there is at the present time no theoreticalreason why time reversal symmetry should be intrinsicallyeither conserved or violated in the electromagnetic inter-action.2,3 Instead, the determination of whether time rever-sal symmetry is strictly conserved in molecular processescould rely on future experimental examinations of theproperties of a pair of DTR isomers.3 From this point ofview, the designed experiments in Figures 3 and 4 could bethe proposed tests for the examinations of the limit of timereversal symmetry conservation in molecular processes.

CONCLUSIONSThe theoretical foundations of time reversal symmetry in

isolated molecular system have recently been establishedby Quack,19 and the involvement of nuclear motion in mo-lecular chirality and the idea of chirality beyond molecularshape has been noted by Cina and Harris.20 Inspired byprevious interpretations of the implication of time reversalsymmetry on the structural features of chiral mol-ecules,9–12,19,20 we suggest that certain pairs of enantio-mers should be considered as time reversal isomers andexpect that the embryonic model presented here will evokea new interpretation of molecular chirality. In addition, theexperimental designs in Figures 3 and 4 are the two ex-amples in which the time reversal symmetry violation, if itexists, could be manifest. The intrinsic magnitude of thiseffect could, however, be very small, as was found to be thecase of time reversal symmetry violation (implied by CPTTheorem) in the certain decay of K-meson,29 and its detec-tion could require elaborate experimentation.

ACKNOWLEDGMENTSThe authors thank Professor Rudolf Janoschek and a

reviewer for their valuable suggestions on the earlier ver-sion of the manuscript and Drs. K.C. Nicolaou, L.D. Bar-ron, and M. Quack for their interest in this work.

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Fig. 4. Diagrams of addition reactions of positively charged species toa pair of DTRmers under the influence of paralleled electric and magneticfields. A: Time reversal symmetry conservation B: Time reversal symme-try violation.

Fig. 3. Illustrations of energy profiles of the time reversal molecularstates of 1-fluoro-2-iodobenzene. A: Time reversal symmetry conservation.B: Time reversal symmetry violation.

292 LI AND NADIN

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