Fundamental symmetry aspects of optical activity

rnterestmg rug~estron \VZXT made recently m thts fourna~ that every motecuie 5s opt~cafly active when 11 1s free to rotate [I ] and thnr a normal sample of meknte gas, for example, should properly be regarded as J. racemrc mixture, with molecules m rotattonaf. states 1-IA0 and I.! -AI) bemg enantlomers (21. lhs source of optmf actmty was correctly ascribed to In& of time reversal mvartance (reversahty), as m the Faraday effect. but it WASfurther suggested that lack of reversahty IS also the source of rtarurol opt1caI activity ?ks Comment points out that fundrimentnt symmetr) arguments show that a molecule m a rotanonal state U M> or )J K &I) IS not a truly choral &~ect tn the same sertse as a chxral molecule, so taik of “‘ch~ai dlscrunmation” m spher& tops f2] should not be taken too hteralfy Furthermore, one can see from several dofferem clewpomts that, wh&2 the magnets optimal rotatmn observable 1s certamly tunesodd, the natural optIcal rotation observable IS tme-even, so lack of reversabty of not the source of nnturai optical act1v3ty One way of demonstrative that natural and ma~etic optical actwty are tlmezven and tzme-odd phenomena. respectively, is to ‘took at the e\presslon far rhe opttcat rotation angle in terms of components of molecular property tensors ft fs found that alt the contnbuhons 10 natural optical rotation are generated by trme-even tensclrs and that all the ~o~tr~butio~s to magnetic optical r~tatlon are generated by trme-odd tensors [3] Thrs can be expressed m a more ‘“physrcat’” manner by reahzmg that the symmetry of natural optical actlvlty under ttme reversal (that IS, a q&t-handed enantiomer, The

0 009-26

say, rematns rtght-handed tf all the momenta and spms arc reversed) reqmres the appraprrate molecular prOpfXty tensors to be symmernc under time reversal [4] _The most fundamens~~ approach 1s to m~ent operators Mose e\pectatlon values generate the optIcal act~ty abservnbles, and to ascertam the behavrour of the operators under time reversal T (and Indeed under space mversfon P and charge conjuganon C) f5] . It 1s found that the natural optlcaf rotatton observable, bemg a tame-even pseudoscalar, IS generated by a tn-ne-even odd-parity operator, and the magnetic optrcd rutattun observable, berg a tune-odd avlal vector, 1s generated by a tnne-odd even-parity operator [S 1. Measurements on a system m 3 state of defkute panty can reveai only observabies wtth even panty, so oddparity observabtes can only be shown by a system m a state of mlxed parxty It;] - Srnce the nafural optrcal rotation observable has odd parity, a resolved choral molecute must east m a state of mrxed panty, and the nature of these mtxed-panty states has been dIscussed ET’] Smce the magnetic opticaX rota&on observable has even par&y. we can understand why a rotational state IJiM), which has defimte panty, can show nagnettc optical rotatlan. But iJM> does not have definite reversahty because TfJMl IS a new state or~~~~~~~ tu iJN>, which is conslstent with the result that a time-odd observable such as magnetic optIcal rotation can anfy be supported by a system in a state of mwed reversal&y [6] . Of course the degeneracy between \J M) and TIJN) must be lifted by a time-odd external mfluence such as a mqnetlc Geld (or mdeed a rota&on of the

14/S l/OO~O-00~0~s 02 SO Q ~or~-Ho~~~d

~bhs~~g

Company

Volume 79. number 2

15

CHWIICAL PHYSICS LETTERS

bulk sample [ 11). Consequently, while it is certamly true that a methane molecuIe, say, in a pure IJM) state wlU generate optical rotation II-Ia hght beam propagatmg parallel to the quantlzatlon axis, it IS not strictly correct to call IJM) and 1.J --Ai) enantlomers smce each is an elgenstate of the panty operator and so they 3re not mterconverted by space mverslon. However, it might be thought that It 1s still correct to refer to symmetrictop elgenstates IJKAI) and IJ -KM) 3s enantlomenc because these do have mixed parity, but In fact it can be shown usmg tune-reversal arguments that they are sttil not true enantlomers because they cannot support the natural optical rotation observable [S] What seems to characterrze a chual ObJeCt IS that It IS found m two dlstmct enantlomeric states that cannot be mterconverted by t!me reversal combined with any proper spatial rotation improper operations are always required. We therefore have confidence in the postulated small energy &fference between optlcal enantlomers mduced by the parity-vlolatmg weak neutral current [8.9] because this would not vlolate reversahty. This leads us to a new defmltlon of the enantiomer of a clural molecule it IS the moiecule mth the mverted spatial configuration and composed of antlpartlcles. Space mverslon alone does not generate a true enantlomer because of the shght energy difference, where3s the combmed CP operation generates a molecule m the anrnvorld wth the opposite chlrahty and e.~act& the same energy. This follows from the CPT theorem (mvanance of any process under the combined operatlon of charge coqugation, space inverslon and time reversal IlO]) and the asszutzptzotz that T IS not violated. Smce P vlolatlon automatlcally unphes C vlolatlon here, It also follows that there 1s a small energy &fference between a ctiral molecule m the real world and the correspondmg choral molecule wth the same absolute configuration m the antnvorld This more general definition of enanflomers IS consistent with the “chlrahty” of free atoms [8,9]. The panty-vlolatmg weak neutral current generates only one type of clural atom m the real world- the conventional enantiomer of 3 chiral atom obtamed by space mverslon alone does not exist. Clearly, the enantiomer of a chiral atom IS generated by the combmed CP operation. Thus the corresponding atom composed of antiparticles will of necessity have the opposite “absolute configuration” and wll show an opposite sense of optical rotation. Chiral atoms are therefore analogous to

Apd

1981

can also exist m only two states which namely, 3 neutrmo with a posltlve heliclty and an antmeutrmo with a negative hehclty. A cucularly polarized photon is a choral ObJect by wtue of Its motion (smce tune reversal simply reverses the dmxtlon of motion wlthout changmg the ciurahty) but IS a special case because, smce a photon IS its own antiparticle, the enantlomer IS generated by P alone It might be objected that, since CP, and hence T, are known to be weakly violated In certam high-energy elementary particle processes [ 101 , we should allow for thus posslblbty LII the realm of atoms and molecules and so define the true enantiomer of 3 choral molecule 3s that generated from the orlgmal by the combined CPT operatton To my knowIedge, there IS no cogent theory of CP violation m atoms and molecules, and certamly no evperlmental evidence for It. Should such evidence arlse, we can easliy fall back on tlus more general definition. But for the moment, we only have good reasons for suspectmg P-vlolatlon (although there hns been speculation that T-vlolatlon leads to opposite “mternal tlmmgs” m optlcal enantiomers [11] ) These constderatrons appear to confhct with the assertron that “optlcal actlvlty has to be understood rn a macroscopic context as a loss of inverslon symmetry of the whole materml medium. and that chlrahty IS not a property that can be related to isolated molecules” [I?-141. If physlcal laws permit the existence of truly choral smgle objects such as circularly poIarlzed photons, neutrmos and chual atoms, why do we need a mac:o. scope context to understand ttzoleadar chuahty 7 This question seems pertment irrespective of whether or not we take account of the dlstmctlon between systems havrng a P-mvanant harmltoman (such as chual molecules, neglecting the weak neutral current) and systems havmg 3 P-vlolatmg hamlltonian (such as atoms subject to the weak neutral current) In the P-mvarlant case the chlrahty does, m a sense, anse from the enwonment neutrmos,

which

are enantiomeric,

because a chual molecule

can only be formed

m a ctural

production process- its optlcal actlvlty only remams observable so long as the observation tune IS short compared wrth the mterconverslon tune between enantlomers, which IS proportional to the mverse of the frequency characterlstlc of the energy sphttmg between the true eigenstates which have defiiite parity and zero chzrahty [7,9 J . In the P-vlolatmg case the eigenstates themselves are panty mixed \nth resultant chrrallty for a free system. A clrcularIy polarized photon 1s analogolls 393

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79, number

2

CHEWCAL

PHYSKS

to a ctura1 molecule WI that It requtres a chunl productlon process, but differs m that mterconverslon between photon enanttomers zs forb&kx2 by ixn-nxvat~on of anguhr 3w3menwm hwwxxtvers~~n between the enantwmenc newtno and antmeutruto stares 1s forbIdden both by conservation of angular momentum and consecvatlon of lepton number. But despite these differences in internal structure. all can eust as smgle choral objects exhlbltmg pseudoscalar observables

LETTERS [31 A D Buckmg.bsm.C

(41 lSj 161 [7j [S]

References

[I] P W AtLlns.md J 4 N I- Comes, Chem Phys Letters 39t1976t319 [ 11 P W 4thmh. Chem Phls Lcttcrs 7-t (1980) 358

394

Graham and R E Raab. Chem Phys Letters 8 (1971) 622 A D Buckmgham, Phd Trans Roy Sot A293 (1979) 239 L D Bnrron. to be published F.A. Kaempffer, Concepts m quantum mechamcs (Academrc Tress New York, 1965) L D Bxron. J 4m Chem Sot 101 (1979) 769 P G H Sandars, m Fundamental mteractlons and structure of matter. cds K Crowe J Duclos. G Florentlnl

and G Tore111 (Plenum 19

I thank Drs J K Tyler and R G. Woolley for stlmulatlng dtscussrons, and a referee for helpfuui comments

15 Aprti 1981

Press, Nerb Yorh,

1980)

p 57

[ R A Harris, m Quantum

dynnmlcs of molecules, ed R G WoolIcy (Plenum Press, New York, 1980) p 357 ( lOl W hl Gibson and B R PoIlard, Symmete pnncrples tn eIemcntar_v partrcle physics (CambrIdge Unrv Press. London. 1976) [ 11 I A S CaraL m Orlgms of optlcal xtlwty m nature ed DC Walker (Elsevler, Amsterdam, 1979) p 245 [ 121 R G Woolley, Advan Phys 25 (1976) 27

f 131 R G WooUey, J. 4m Chem (141

R G Woolley,

Israel J Chem

Sot.

100 (1978) 30

I9 (i980)

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