Chiral Rotational Spectroscopy
CChiral Rotational Spectroscopy
Robert P. Cameron ∗ and J¨org B. G¨otte Max Planck Institute for the Physics of Complex Systems, Dresden D-01187, Germany andSchool of Physics and Astronomy, University of Glasgow, Glasgow G12 8QQ, U.K.
Stephen M. Barnett
School of Physics and Astronomy, University of Glasgow, Glasgow G12 8QQ, U.K. (Dated: January 16, 2018)We introduce chiral rotational spectroscopy: a new technique that enables the determination ofthe orientated optical activity pseudotensor components B XX , B Y Y and B ZZ of chiral molecules,in a manner that reveals the enantiomeric constitution of a sample and provides an incisive signaleven for a racemate. Chiral rotational spectroscopy could find particular use in the analysis ofmolecules that are chiral solely by virtue of their isotopic constitution and molecules with multiplechiral centres. A basic design for a chiral rotational spectrometer together with a model of itsfunctionality is given. Our proposed technique offers the more familiar polarisability components α XX , α Y Y and α ZZ as by-products, which could see it find use even for achiral molecules. I. INTRODUCTION
Chirality pervades the natural world and is of particu-lar importance to life, as the molecules that comprise liv-ing things are chiral and their chirality is crucial to theirbiological function [1–4]. Our ability to probe and har-ness molecular chirality remains incomplete in many re-spects, however, and new techniques for chiral moleculesare, therefore, highly sought after.The basic property of a chiral molecule that is probedin typical optical rotation experiments using fluid sam-ples [3, 6–9] is the isotropic sum ( B XX + B Y Y + B ZZ ) (1)with B XX , B Y Y and B ZZ components of the opticalactivity pseudotensor [10, 11] referred to molecule-fixedaxes X , Y and Z . These experiments yield no informa-tion about B XX , B Y Y or B ZZ individually. Other well-established chiroptical techniques such as circular dichro-ism [2, 3, 8, 12–15] and Raman optical activity [3, 8, 15–19] yield other chirally sensitive molecular properties butthe fact remains that it is the isotropically averaged formsof these that are usually observed in practice.The ability to determine orientated rather thanisotropically averaged chiroptical information, in partic-ular the individual, orientated components B XX , B Y Y and B ZZ [20] is highly attractive, as these offer a wealthof information about molecular chirality that is only par-tially embodied by the isotropic sum (1). At present suchinformation can only be obtained, however, using an ori-entated sample as in a crystalline phase [21, 22]. Thepreparation of such samples is not always feasible andeven when it can be achieved, signatures of chirality areusually very difficult to distinguish from other effects,in particular those due to linear birefringence. Indeed, ∗ [email protected] it was noted in 2012 that “ we have a shockingly smallamount of data on the chiroptical responses of orientatedmolecules, a vast chasm in the science of molecular chi-rality ” [22].In the present paper we introduce chiral rotationalspectroscopy: a new technique with the ability to (i) determine B XX , B Y Y and B ZZ individually, thuspromising to fill the “ vast chasm ” described above; (ii) measure the enantiomeric excess of a sample andprovide an incisive signal even for a racemate, thusnegating the need for dissymmetric synthesis or res-olution, which are instead required by traditionaltechniques; (iii) probe the chirality of molecules that are chiralsolely by virtue of their isotopic constitution, theimportance of which is becoming increasingly ap-parent whilst traditional techniques remain some-what lacking in their sensitivities; (iv) distinguish clearly and in a chirally sensitive man-ner between subtly different molecular forms, mak-ing it particularly useful for molecules with multi-ple chiral centres, the analysis of which using tra-ditional techniques represents a serious challenge.Chiral rotational spectroscopy is distinct from anotherclass of techniques introduced recently, in which thephase of a microwave signal is used to discriminate be-tween opposite enantiomers [23–32]. We refer to thesecollectively as ‘chiral microwave three wave mixing’. Inwhat follows we will compare and contrast chiral rota-tional spectroscopy and chiral microwave three wave mix-ing. It is our hope that these techniques will one daycomplement each other.Let us emphasise that chiral rotational spectroscopyboasts the abilities (i) - (iv) simultaneously . Variousother techniques boast some of these abilities but noneoffer them all , together. For example: vibrational circu-lar dichroism and Raman optical activity are inherently a r X i v : . [ phy s i c s . a t m - c l u s ] J a n sensitive to isotopic molecular chirality to leading orderas per (iii) [14, 33, 34] but are blind to racemic samplesand so cannot fully realise (ii) , nor are they particularlywell suited for (iv) ; chiral microwave three-wave mixingcan realise (iv) [27, 28, 31, 32] but is also blind to racemicsamples and so cannot fully realise (ii) ; Coloumb explo-sion imaging can probe the molecular chirality of race-mates as per (ii) [35, 36] and is inherently sensitive toisotopic molecular chirality to leading order as per (iii) [36], but is seemingly restricted in its application to rela-tively simple structures under well-controlled conditionsand is not particularly well suited for realising (iv) .We work in a laboratory frame of reference x , y and z with time t and ˆ x , ˆ y and ˆ z unit vectors in the + x , + y and + z directions. A lower-case index a can take on thevalues x , y or z and upper-case indices A , B and C cantake on the values X , Y or Z . The summation conventionis to be understood throughout. II. CHIRAL ROTATIONAL SPECTROSCOPY
In the present section we elucidate the basic premiseof chiral rotational spectroscopy: chiral molecules illumi-nated by circularly polarised light yield orientated chirop-tical information via their rotational spectrum .To enable the determination of orientated chiropticalinformation we recognise the need to (i) prepare a chiral sample of orientated character, (ii) evoke a chiroptical response from the sample and (iii) observe and interpret the response so as to obtainorientated chiroptical information.We envisage fulfilling these objectives as follows.
FIG. 1. An un-ionised L - α -alanine molecule. L - α -alanine isan amino acid found in abundance in living things [2, 4, 40].Produced using data from [43]. A. Chiral sample of orientated character
A chiral molecule with unimpeded rotational degrees offreedom, as in the gas phase or a molecular beam [37, 38],can already be regarded as a chiral sample of orientatedcharacter, as we will now demonstrate. Let us assumethat the molecule is at rest or moving slowly and that itoccupies its vibronic ground state, in which it is small,polar and non-paramagnetic [9, 39–42]. We model therotation of the molecule as that of an asymmetric rigidrotor, with equilibrium rotational constants A > B > C associated with rotations about the molecule-fixed, prin-cipal axes of inertia X , Y and Z , as depicted in FIG.1. The rotational and nuclear-spin degrees of freedom ofthe molecule should be well described [44] then by theeffective Hamiltonianˆ H = ˆ H rotor + δ ˆ H (2)with ˆ H rotor = ̵ h ( A ˆ J X + B ˆ J Y + C ˆ J Z ) (3)the rotor Hamiltonian [37, 40, 41, 45] and δ ˆ H account-ing for nuclear spin [37–39, 46–50] and perhaps also cor-rections to the rigid rotor model such as those due tocentrifugal distortion [9, 37, 40]. The components ˆ J A of the rotor angular momentum account for the entiretyof the molecule’s intrinsic angular momentum except fornuclear spin [40]. Let us neglect δ ˆ H for the moment andfocus our attention upon the rotor states ∣ J τ,m ⟩ and rotorenergies w J τ , which satisfyˆ H rotor ∣ J τ,m ⟩ = w J τ ∣ J τ,m ⟩ (4)with J ∈ { , , . . . } determining the magnitude of the ro-tor angular momentum, τ ∈ { , . . . , ± J } labeling the rotorenergy and m ∈ { , . . . , ± J } determining the z componentof the rotor angular momentum [37, 40, 41, 45]. Some ofthese are depicted in FIG. 2. In the J τ,m = , rotorstate the molecule possesses a vanishing rotor energy of w =
0, as it is not rotating. All orientations of X , Y and Z relative to x , y and z are, therefore, equally likely to befound. In the 1 − ,m rotor states, however, the moleculepossesses a rotor energy of w − = B + C , as it will neverbe found rotating about the X axis but is equally likelyto be found rotating about the Y or Z axis. The con-ceivable motions of the rotor then conspire such that for m = X axis is most likely to be found perpendicu-lar to the x - y plane whereas for m = ± x - y plane. Analogous observationshold for the 1 ,m rotor states, in which it is the Y axisthat is treated preferentially, and the 1 ,m rotor states, inwhich it is the Z axis. They can be extended, moreover,to the J ∈ { , . . . } manifolds, although the analysis be-comes increasingly complicated with increasing J . Theimportant point here is that the rotation and hence ori-entation of the molecule in any particular rotor state is not of isotropic character, in general. Indeed, the mostprobable orientations of the molecule differ for differentrotor states. The isotropic character usually ascribed tothe gas phase or a molecular beam emerges only whenthese states are suitably averaged over, in accord withthe principal of spectroscopic stability [51]. Such obser-vations are complicated by the inclusion of δ ˆ H , but onlysuperficially. FIG. 2. The blue regions here depict equally probable valuesof the rotor angular momentum J relative to the molecule-fixed axes X , Y and Z and to the laboratory-fixed axes x , y and z for some of the molecule’s rotor states whilst the greenregions indicate the most probable orientations of X , Y and Z relative to x , y and z . B. Evoking a chiroptical response from the sample
Suppose now that the molecule is illuminated by faroff-resonance visible or perhaps near infrared circularlypolarised light of moderate intensity I and wavevector k pointing in the z direction, with the ellipticity parame-ter − ≤ σ ≤ ± H ′ = ˆ H rotor + ˆ H light + δ ˆ H (5) with ˆ H light = − I (cid:15) c ( ˆ (cid:96) xA ˆ (cid:96) xB + ˆ (cid:96) yA ˆ (cid:96) yB ) (6) × ( α AB + σ ∣ k ∣ B AB ) accounting for the energy associated with the oscillations[55]. The ˆ (cid:96) aA are direction cosines [3, 40], which quan-tify the orientation of the molecule relative to the light.The α AB are components of the electronic electric-dipole/ electric-dipole polarisability [3, 8, 51], which quantifythe susceptibility of the charge and current distributionsof the molecule to be distorted by the light in a chi-rally insensitive manner: α XX , α Y Y and α ZZ in partic-ular are identical for opposite enantiomers. The B AB are components of the electronic optical activity pseu-dotensor [10, 11], which quantify the susceptibility of thecharge and current distributions of the molecule to bedistorted in a chirally sensitive manner: B XX , B Y Y and B ZZ in particular each possess equal magnitudes but op-posite signs for opposite enantiomers and are the molecu-lar properties upon which chiral rotational spectroscopyis based. Let us focus our attention now upon a moleculewith nuclear spins of 0 or 1 / H rotor ≫ ˆ H light ≫ δ ˆ H with no accidental degeneracies ofimportance whilst neglecting the possibility of any effectsdue to the spin statistics of similar nuclei [9, 37, 38, 56].The energy of the perturbed 0 , rotor state together witha nuclear-spin state ∣ n ⟩ is then essentially w + ∆ w , + δw , (7)with w = , ∆ w , = ⟨ , ∣ ˆ H light ∣ , ⟩ + . . . (8) = − I(cid:15) c ⎡⎢⎢⎢⎢⎣ ( α XX + σ ∣ k ∣ B XX )+ ( α Y Y + σ ∣ k ∣ B Y Y )+ ( α ZZ + σ ∣ k ∣ B ZZ ) ⎤⎥⎥⎥⎥⎦ + . . .δw , = ⟨ n ∣⟨ , ∣ δ ˆ H ∣ , ⟩∣ n ⟩ (9)the unperturbed rotor energy, an energy shift due tothe light and a further energy shift due to nuclear in-tramolecular interactions and perhaps also corrections tothe rigid rotor model, where we assume ∣ n ⟩ to be di-agonal in ⟨ , ∣ δ ˆ H ∣ , ⟩ . The components α XX , α Y Y , α ZZ , B XX , B Y Y and B ZZ make isotropically weightedcontributions, reflecting the idea that all orientations ofthe molecule relative to the light are equally likely to befound in the 0 , rotor state: the electric and magneticfield vectors of the light can be said to drive oscillationsequally along the X , Y and Z axes. In contrast, the en-ergy of the perturbed 1 − , rotor state together with anuclear-spin state ∣ n ′ ⟩ is essentially w − + ∆ w − , + δw − , (10)with w − = B + C, ∆ w − , = ⟨ − , ∣ ˆ H light ∣ − , ⟩ + . . . (11) = − I(cid:15) c ⎡⎢⎢⎢⎢⎣ ( α XX + σ ∣ k ∣ B XX )+ ( α Y Y + σ ∣ k ∣ B Y Y )+ ( α ZZ + σ ∣ k ∣ B ZZ ) ⎤⎥⎥⎥⎥⎦ + . . .δw − , = ⟨ n ′ ∣⟨ − , ∣ δ ˆ H ∣ − , ⟩∣ n ′ ⟩ , (12)where we assume ∣ n ′ ⟩ to be diagonal in ⟨ − , ∣ δ ˆ H ∣ − , ⟩ .The components α XX , α Y Y , α ZZ , B XX , B Y Y and B ZZ now make anisotropically weighted contributions reflect-ing the idea that the X axis is most likely to be foundperpendicular to the x - y plane in the 1 − , rotor state:the electric and magnetic field vectors of the light can besaid to drive oscillations less frequently along the X axisand more frequently along the Y and Z axes. Such ob-servations can be extended, of course, to other rotor andnuclear-spin states. The important point here is thatthe energy shifts due to the light exhibit different de-pendencies upon B XX , B Y Y and B ZZ for different rotorstates whilst differing for opposite circular polarisations:the rotation and hence orientation of the molecule rel-ative to the light differs for different rotor states whilstone enantiomorphic form of the helically twisting elec-tric and magnetic field vectors that comprise circularlypolarised light [57] is more competent at driving chiraloscillations in the charge and current distributions of themolecule than the other, much as one enantiomorphicform of a glove is a better fit for a human hand thanthe other. Similarly for a fixed circular polarisation andopposite enantiomers. In contrast the chirally sensitivephase that underpins chiral microwave three wave mixingderives from the sign of the product of three orthogonalelectric-dipole moment components, which is opposite foropposite enantiomers [23–32]. The diagonalisation of ˆ H ′ is discussed in more detail in Appendix A. C. Observing and interpreting the response
We envisage having a large number of molecules inpractice, occupying many rotational and nuclear-spinstates in accord with some thermal distribution, say. Werecognise the need, therefore, to observe and interprettheir chiroptical response in a manner that distinguishesbetween different rotational states, lest we lose the ori-entated character that is inherent to these states indi-vidually but absent from them collectively [51]. We pro-pose simply measuring the rotational spectrum of the molecules in the microwave domain [58], which will ap-pear modified due to the light. For example, the mi-crowave energy required to induce a 1 − , ← , rota-tional transition in a molecule follows from the differencebetween (10) and (7) as B + C − I(cid:15) c ⎡⎢⎢⎢⎢⎣ − ( α XX + σ ∣ k ∣ B XX )+ ( α Y Y + σ ∣ k ∣ B Y Y ) + ( α ZZ + σ ∣ k ∣ B ZZ ) ⎤⎥⎥⎥⎥⎦ + . . . plus a small correction moreover that is particular to thenuclear-spin states involved. B XX , B Y Y and B ZZ canbe determined individually by recording such energies fortwo distinct rotor transitions and both circular polarisa-tions of the light and making use of the measured value ofthe isotropic sum (1). This is the essence of chiral rota-tional spectroscopy. Let us emphasise here, however, thatchiral rotational spectroscopy also has abilities reachingbeyond this particular task, as we will see in what follows. D. Additional remarks
Knowledge of B XX , B Y Y and B ZZ might assist in theassignment of absolute configuration, as the measuredsigns of these should be easier to correlate with those pre-dicted by quantum chemical calculations than in the caseof the isotropic sum (1), which is often somewhat smallerin magnitude than its constituents B XX / B Y Y / B ZZ / B XX , B Y Y and B ZZ might also serve asprobes of isotopic molecular chirality and cryptochiral-ity in general, where the isotropic sum (1) fails ratherdramatically, as we will elucidate in § III B. Although ourfocus in the present paper is upon the chirality of individ-ual molecules, we observe that knowledge of B XX , B Y Y and B ZZ might in some cases facilitate the explorationand exploitation of the myriad contributions to the opti-cal properties of crystals [21, 22] comprised, wholly or inpart, of such molecules. We recognise moreover that ourproposed technique offers α XX , α Y Y and α ZZ and po-tentially even the distortion of such quantities by staticfields (see Appendix B) as by-products, which is in itselfan attractive feature that could see our proposed tech-nique find use even for achiral molecules.It is interesting to note that ˆ H light is, in fact, the a.c.Stark Hamiltonian, but calculated here to higher orderthan is usual [55]. The associated energy shifts are thesame as those that govern the refraction of light propa-gating through a medium [55], with circular birefringencedue to B XX , B Y Y and B ZZ giving rise to natural opticalrotation [55]. Spatial gradients in such shifts give rise,moreover, to forces, including the dipole optical forceused to trap atoms in optical lattices [60] and the dis-criminatory optical force [55, 61–70]: a viable manifesta-tion of chirality in the translational degrees of freedomof chiral molecules.Let us conclude the present section now with a discus-sion of other phenomena and techniques centred upon therotational degrees of freedom of chiral molecules, by wayof comparison with chiral rotational spectroscopy. Mi-crowave optical rotation and circular dichroism have beenconsidered in theory [71–81]. These phenomena promisechirally sensitive information about a molecule’s perma-nent electric-dipole moment and rotational g tensor butare anticipated to be weak, owing primarily to the small-ness of molecules relative to the twist inherent to cir-cularly polarised microwaves. Rotational Raman opticalactivity has also been considered in theory [72, 82]. Thisphenomenon promises certain combinations of orientatedpolarisability components. A difficulty with rotationalRaman optical activity is the anticipated proximity ofthe relevant Stokes and anti-Stokes lines to the Rayleighline [83]. In light of these challenges it is little surpriseperhaps that “ no experimental observations ... of opticalactivity associated with pure rotational transitions of chi-ral molecules ... (had) been reported ” by 2004 [3]. Thesuccessful implementation in 2013 of chiral microwavethree wave mixing [23–32], however, demonstrated thatthe exploitation of rotational degrees of freedom is, infact, viable. Two additional works of interest came toour attention whilst preparing the present paper for sub-mission. The first of these is a theoretical proposal for ori-entating chiral molecules using multi-coloured light [84].The second is a theoretical proposal, published on thearXiv, for the use of “ near-resonant AC Stark shifts ” todetect molecular chirality in the microwave domain via a“ five wave mixing ” process [85]. Let us emphasise thatchiral rotational spectroscopy is quite distinct from thesetechniques, including chiral microwave three wave mix-ing, and that it offers fundamentally different informationabout molecular chirality. III. CHIRAL ROTATIONAL SPECTRA
In the present section our goal is to illustrate, simply,some of the features that might be seen in chiral rota-tional spectra for various different types of sample. Toproduce FIG. 3, FIG. 5, FIG. 6 and FIG. 7 we plottedLorentzians, centred at the relevant rotational transitionfrequencies as given by the leading-order perturbative re-sults described in Appendix A but with δ ˆ H neglectedhere. Each Lorentzian was ascribed a frequency full-width at half-maximum of 1 . × s − and taken to beproportional in amplitude to the number of contributingmolecules. The same features persist when higher-ordercorrections and the effects of δ ˆ H are included and forlarger rotational linewidths: it is acceptable to have ro-tational lines overlap significantly if their centres, say,can still be distinguished with sufficient resolution. Theforms of the rotational lines seen in a real chiral rota-tional spectrum will depend, of course, upon the natureand functionality of the chiral rotational spectrometerused to obtain the spectrum, but should nevertheless of- fer the same information. The calculated molecular prop-erties used to produce FIG. 3, FIG. 5, FIG. 6 and FIG. 7are reported in Appendix C. The reader will observe thehigh precision with which I and 2 π /∣ k ∣ are quoted in thepresent section. In principle this represents no difficultyand ensures that FIG. 3, FIG. 5, FIG. 6 and FIG. 7 aredrawn accurately to a frequency resolution of 10 s − . Inpractice it should be possible in many cases to reducestringent requirements on the uniformity and stability ofthe intensity of the light by exploiting certain, ‘magic’rotational transitions, as discussed in § III E.
A. Orientated chiroptical information
FIG. 3. A rotational line for an enantiopure sample of the low-est energy conformer of (S)-propylene glycol in the absence oflight (a), illuminated by left-handed light (b) and illuminatedby right-handed light (c). The separation between rotationallines (b) and (c) in particular reveals orientationally and chi-rally sensitive information about the response of the moleculesto the light.
Consider first an enantiopure sample of the lowest en-ergy conformer of (S)-propylene glycol [86]. Racemicpropylene glycol is employed as an antifreeze and is akey ingredient in electronic cigarettes. Depicted in FIG.3 is: (a) the 2 − ← − rotational line in the absence oflight; (b) the 2 − , ← − , rotational line in the presenceof light with I = . × kg.s − , 2 π /∣ k ∣ = . × − mand σ =
1; (c) the same as in (b) but with σ = −
1. Theseparation between rotational line (a) and the centroid ofrotational lines (b) and (c) yields a certain combinationof α XX , α Y Y and α ZZ whilst that between rotationallines (b) and (c) yields a certain combination of B XX , B Y Y and B ZZ , as described in § II.
B. Isotopic molecular chirality
FIG. 4. Singly deuterated chlorofluoromethane derives chi-rality from the arrangement of its neutrons. These moleculesshould exist in small quantities in some refrigerators.
Chirality is more widespread at the molecular levelthan is sometimes appreciated, for even a molecule withan achiral arrangement of atoms may in fact be chiralsolely by virtue of its isotopic constitution, as illustratedin FIG. 4. Isotopically chiral molecules might have beenamongst the very first chiral molecules, formed perhapsin primordial molecular clouds [87]. They might evenhave given rise to biological homochirality, by trigger-ing dissymmetric autocatalysis reactions [87–91]. At amore fundamental level still, isotopically chiral moleculeshave been put forward [92] as promising candidatesfor the measurement of minuscule differences believedto exist between the energies of opposite enantiomers[2, 3, 40, 93–96]. It is well established that isotopic sub-stitution in certain achiral molecules can significantlymodify their interaction with living things. Heavy wa-ter can change the phase and period of circadian oscil-lations [97], for example. In spite of this there “ havebeen very few studies on isotope-generated chirality inbiochemistry ” [88].Isotopic molecular chirality can already be probed us-ing various techniques, in particular vibrational circulardichroism and Raman optical activity, which are inher-ently sensitive to chiral mass distributions [14, 33, 34, 98].A difficulty, however, is that enantiopurified samples ofisotopically chiral molecules can often only be synthe-sised in small quantities [34] whilst resolution of race-mates is extremely challenging [99]. Chiral rotationalspectroscopy may prove particularly useful here as it is,like vibrational circular dichroism and Raman optical ac-tivity, inherently sensitive to isotopic molecular chirality
FIG. 5. A rotational line for a 60 ∶
40 mixture (a), a 50 ∶ ∶
60 mixture (c) of opposite enantiomersof isotopically chiral housane in the presence of left-handedlight. The chiral splitting is apparent even for the racematewhilst the relative heights of the constituent lines reveal theenantiomeric excesses of the samples. and, in addition, gives an incisive signal even for a race-mate, thus negating the need for dissymmetric synthesisor resolutionWe find in electronic calculations within the Born-Oppenheimer approximation for a rigid nuclear skele-ton [40, 42] that the isotropic sum (1) vanishes for anisotopically chiral molecule, as it is rotationally invari-ant and the electronic charge and current distributionsof the molecule are achiral. Chirally sensitive vibra-tional corrections to this picture do exist but are usuallysmall at visible or near infrared frequencies as consid-ered here. The individual components B XX , B Y Y and B ZZ , and therefore chiral splittings in chiral rotationalspectroscopy, can nevertheless attain appreciable magni-tudes for an isotopically chiral molecule as each of theseis dependent upon the orientation of the principal axesof inertia relative to the molecule and is, therefore, sensi-tive to the distribution of mass throughout the molecule,which is where the molecule’s chirality resides. Chi-ral rotational spectroscopy might be similarly useful forother molecules exhibiting cryptochirality [100] where theisotropic sum (1) is essentially zero whilst two or three ofits constituents B XX / B Y Y / B ZZ / C atomto give the opposite enantiomers of an isotopically chi-ral molecule. Depicted in FIG. 5 is the 1 − , ← , rotational line for light with I = . × kg.s − ,2 π /∣ k ∣ = . × − m and σ = ∶
40 mixture of opposite enan-tiomers; (b) a 50 ∶
50 mixture; (c) a 40 ∶
60 mixture.In all three cases the chiral splitting is apparent whilstthe relative heights of the constituent lines reveal theenantiomeric excess of the sample and so enable its de-termination. Let us highlight the significance of panel(b) in particular. We have here an obvious and revealingsignature of chirality from a racemate of isotopically chi-ral molecules, as claimed. The chirality of each of thesemolecules derives solely from the placement of a singleneutron, which constitutes but 1% of the total mass of themolecule. Techniques such as electronic optical rotationand electronic circular dichroism in contrast are nearlydouble blind under such circumstances and even vibra-tional circular dichroism, Raman optical activity and chi-ral microwave three wave mixing would yield vanishingsignals.
C. Molecules with multiple chiral centres
Standard rotational spectroscopy can often distinguishwell between different isomers, provided they are not op-posite enantiomers [86]. Chiral rotational spectroscopycan distinguish well between different isomers including opposite enantiomers. It may find particular use, there-fore, in the analysis of molecules with multiple chiral cen-tres, which permit an exponentially large number of dif-ferent stereoisomers, many of which are opposite enan-tiomers. This in turn could see chiral rotational spec-troscopy find particular use in the food and pharmaceu-tical industries, where different isomers must be individ-ually justified [101] and molecules with multiple chiralcentres are recognised as being “ challenging ” [102].Consider now then a sample of tartaric acid. The twochiral centres permit three different stereoisomers. Oneof these; mesotartaric acid, is achiral whilst the othertwo; L -tartaric acid and D -tartaric acid, are oppositeenantiomers. L -tartaric acid is found in grapes and wasone of the first molecules recognised as being opticallyactive [3]. The racemate of L - and D -tartaric acid, also FIG. 6. A rotational line for a sample of the three dif-ferent stereoisomers of tartaric acid as it might appear instandard rotational spectroscopy (a) and in chiral rotationalspectroscopy using left-handed light (b). The standard ro-tational spectrum fails to distinguish between the oppositeenantiomers whilst the chiral rotational spectrum instead dis-tinguishes between all three molecular forms. Here ∆ =√ A + B + C − AB − AC − BC . known as paratartaric acid [3] or racemic acid [103], wasthe subject of Pasteur’s original chiral separation [2, 3].Depicted in FIG. 6 (a) is the 2 − ← − rotational line fora 50 ∶ n ∶ ( − n ) mixture of mesotartaric acid, L -tartaricacid and D -tartaric acid in the absence of light [104].The contribution due to mesotartatic acid appears wellseparated from that due to L -tartaric acid and D -tartaricacid. The spectrum gives no information, however, aboutthe relative abundances of L -tartaric acid and D -tartaricacid, only their combination. Depicted in FIG. 6 (b) isthe 2 − , ← − , ± rotational line for a 50 ∶ ∶
30 mix-ture in the presence of light with I = . × kg.s − ,2 π /∣ k ∣ = . × − m and σ =
1. Contributions due to all three stereoisomers now appear well distinguished whilstyielding a wealth of new information, as claimed.Rotational spectra are sufficiently sparse that the anal-ysis of molecules with significantly more chiral centres inthis way should not be met with any fundamental difficul-ties. This ability to distinguish well and in a chirally sen-sitive manner between subtly different molecular formspersists moreover for more general mixtures containingmultiple types of molecule. The chirally sensitive analy-sis of complicated mixtures using traditional techniquesrepresents a serious challenge. Indeed, it was suggested in2014 that “ only one mixture analysis (based upon circulardichroism, vibrational circular dichroism or Raman opti-cal activity) was reported so far ” [27], although the useof chiral microwave three wave mixing to analyse variousmixtures has now been well demonstrated [27, 28, 31, 32].
D. Scaling
FIG. 7. A chiral splitting induced in a rotational line of aracemate of ibuprofen using a relatively low intensity of left-handed light.
Polarisabilities tend to increase with the size of amolecule. The light intensity required to induce observ-able shifts in a rotational spectrum therefore tends todecrease with the size of a molecule, as is evident in theexamples above. This favourable scaling is ultimatelycounteracted in that larger molecules are usually moredifficult to sample appropriately, tend to exhibit lowerrotational transition frequencies, are often more likely toabsorb light and might require higher levels of theoryto accurately describe. It seems then that there shouldbe a certain molecular size range for which chiral rota-tional spectroscopy is particularly well suited. To illus-trate these ideas let us consider a racemate of a par-ticular conformer of ibuprofen, which is somewhat moremassive than the molecules considered in the other ex-amples above. Such a sample would yield no informationabout the chirality of the molecules when analysed us-ing traditional techniques, in spite of the fact that it isonly the (S)-enantiomeric form of ibuprofen that acts asthe anti-inflammatory agent whilst the (R)-enantiomericform is ineffective in this context [105]. Enantiopureibuprofen is sometimes sold under a different name suchas Seractil R ○ . Depicted in FIG. 7 is the chiral split- ting of the 1 − , ← , rotational line due to light with I = . × kg.s − , 2 π /∣ k ∣ = . × − m and σ = E. Practical considerations
Requirements on the monochromaticity and stabilityof the wavelength of the light are stringent but are easedsomewhat by the fact that the α AB vary slowly withwavelength far off-resonance. For most rotational transi-tions requirements on the uniformity and stability of theintensity of the light are very stringent, as small varia-tions in the intensity can easily overwhelm chiral split-tings. In many cases rotational transitions can be found,however, for which the chirally insensitive piece of the ro-tational transition frequency shift due to the light is con-siderably smaller than is typical whilst the chirality sen-sitive piece remains appreciable. These magic rotationaltransitions should be particularly well suited to chiral ro-tational spectroscopy as they reduce requirements on theuniformity and stability of the intensity of the light. Itshould be possible moreover to significantly refine somemagic transitions by fine-tuning the polarisation proper-ties of the light or even the strength and direction of anapplied static field. IV. CHIRAL ROTATIONAL SPECTROMETER
In the present section we discuss a basic design for achiral rotational spectrometer [106]. This represents butone of many conceivable possibilities for the implemen-tation of chiral rotational spectroscopy: the ideas intro-duced in § II and § III have a generality reaching beyondthe present discussions. The design certainly has its lim-itations, but should nevertheless permit high-precisionmeasurements based upon α XX , α Y Y and α ZZ for manytypes of molecule, be they chiral or achiral, as well asmeasurements based upon B XX , B Y Y and B ZZ for sometypes of chiral molecule under favourable circumstances.The key components of the spectrometer are depictedin FIG. 8 (a), with an expanded view of the active regionin FIG. 8 (b). We summarise their functionality as fol-lows. A quantitative model of the spectrometer is givenin Appendix D. FIG. 8. Key components of a chiral rotational spectrome-ter. Drawn approximately to scale but with portions of theHelmholtz coils and the vacuum chamber removed, for thesake of clarity. (i)
A pulsed supersonic expansion nozzle together witha collimation stage is employed to generate nar-row pulses of internally cold chiral molecules withunimpeded rotational degrees of freedom. A noz- zle of the piezoelectric variety permits a high rateof measurement [107]. The collimation stage mightinclude a skimmer augmented by an aperture [108–110]. (ii)
An optical cavity houses far off-resonance visibleor perhaps near infrared circularly polarised lightof moderate intensity, to shift the rotational ener-gies of the molecules in a chirally sensitive man-ner. Fine-tuning the polarisation properties of thelight in the active region enables the refinement ofmagic transitions, to help overcome stringent re-quirements on the intensity of the light and thusobtain a clean chiral rotational spectrum. The op-tical cavity might be of the skew-square ring va-riety, comprised of low-loss, ultra-high-reflectivity,low-anisotropy mirrors [111, 112] whilst the lightmight originate from an external cavity diode laser,the output of which is fibre amplified and modematched into the ring with stability actively en-forced [113, 114]. We envisage the light to be con-tinuous wave here, with a central intensity of atleast 10 kg.s − (10 W.cm − ). Each molecule takessome 10 − s to traverse the light; a time intervallarge enough to facilitate a notional microwave fre-quency linewidth of around 10 s − . Variants of ourdesign that use pulsed light rather than continuouswave light are also conceivable and may prove eas-ier to implement in practice. We will discuss thesein more detail elsewhere. (iii) A microwave cavity and associated componentsgenerate and detect microwaves as in the well-established technique of cavity enhanced Fouriertransform microwave spectroscopy [39, 86, 115–123]but here with the aim of measuring chirally sensi-tive distortions of the rotational spectrum of themolecules due to the light. The microwave cavitymight be of the Fabry-P´erot variety, comprised ofspherical mirrors with microwaves coupled in andout of the microwave cavity via waveguide or per-haps via antennas [39, 86, 115, 116, 118–123]. (iv)
An evacuated chamber encompasses the key com-ponents described above to eliminate atmosphericinterference with the molecular pulses and facilitatethe removal of molecules between measurements.The absence of air, dust and other such influencesshould assist moreover in maintaining the stabilityof the optical cavity [114]. (v)
A static magnetic field of moderate strength andhigh uniformity defines a quantisation axis parallelto that defined by the direction of propagation ofthe light in the active region whilst enabling addi-tional refinement of magic transitions if necessary.The static magnetic field might be produced by apair of superconducting Helmholtz coils [125] orperhaps even an appropriate arrangement of per-manent magnets with some degree of tunability.0Note that the static magnetic field plays no directrole in probing the chirality of the molecules. Itsinfluence is discussed in more detail in AppendixB.A chiral rotational spectrum is recorded as the averageof many measurements, each of which proceeds as fol-lows. The nozzle is opened at some initial time, allowinga molecular pulse to begin expanding towards the activeregion. In the initial stage of this expansion the inter-nal temperature of the molecules decreases dramatically,as collisions convert enthalpy into directed translationalenergy. The molecules thus occupy their electronic andvibrational ground states and a small collection of rota-tional and nuclear-spin states, with their internal angularmomenta preferentially quantised parallel to the staticmagnetic field. Following this initial stage the moleculesproceed largely collision free. A subset of the moleculesselected by the collimation stage eventually permeate thelight in the active region, which shifts their rotational en-ergies in a chirally sensitive manner. When the overlapbetween the molecules, the light and the microwave modeis optimum a microwave pulse permeates the microwavecavity and induces coherence in those (light-shifted) rota-tional transitions that lie near the chosen microwave cav-ity frequency and within the microwave cavity frequencybandwidth. The molecules then radiate back into themicrowave cavity over a longer time, with the signal di-minishing primarily as a result of residual collisions. Thisfree induction decay signal is monitored and the real partof its Fourier transform, say, calculated and regarded asthe measurement.In Appendix E we estimate the signal-to-noise ratioand find that a very agreeable chiral rotational spectrumcould be obtained for a recording time of a few hours un-der favourable operating conditions. This is approachingthe time usually taken to record a complete standard ro-tational spectrum [86], but here with the effort focusedentirely upon a single rotational line. This is acceptableas four spectra spread over two lines for opposite cir-cular polarisations might already permit the extractionof all of the chirally sensitive information on offer herefor a particular enantiomer. We are reminded of earlyRaman optical activity spectrometers, which demandedrecording times of several hours [19]. Even now, “ tra-ditional chiroptical spectroscopy techniques take minutesto hours ” [24]. Chiral microwave three wave mixing incontrast exhibits an excellent signal-to-noise ratio, withmeasurement times “ as fast as tens of seconds ” havingbeen claimed in one of the earliest publications [24]. A linearly polarised standing wave of light with a sig-nificantly lower intensity, housed simply in a two-mirroroptical cavity perhaps, might already suffice if measure-ments based upon α XX , α Y Y and α ZZ , for either chiralor achiral molecules, are all that is sought. Note addedpost-publication : measurements of individual, orientatedcomponents of α AB using a combination of optically in-duced ac Stark shifts and microwaves have recently beenreported for heteronuclear molecules [124]. This work,performed independently of ours, confirms the validity ofthe basic theory first presented by us for chiral rotationalspectroscopy. V. SUMMARY AND OUTLOOK
In the present paper we have introduced chiral rota-tional spectroscopy: a new technique for chiral moleculesthat combines the chiral sensitivity inherent to naturaloptical activity with the orientational sensitivity and highprecision inherent to standard rotational spectroscopy.Chiral rotational spectroscopy enables the determinationof the orientated optical activity pseudotensor compo-nents B XX , B Y Y and B ZZ of chiral molecules, in a man-ner that reveals the enantiomeric constitution of a sampleand provides an incisive signal even for a racemate. Itcould find use in the analysis of molecules that are chiralsolely by virtue of their isotopic constitution, moleculeswith multiple chiral centres and more besides.There is much to be done: our formalism and calcula-tions can and should be refined; the nature of the infor-mation offered by B XX , B Y Y and B ZZ requires furtherattention; the use of our proposed technique to determineother polarisability components remains to be exploredin more detail; designs for chiral rotational spectrometersand their functionality demand further investigation. Wewill return to these and related tasks elsewhere. VI. ACKNOWLEDGEMENTS
This work was supported by the Engineering and Phys-ical Sciences Research Council grants EP/M004694/1,EP/101245/1 and EP/M01326X/1; the alumnus pro-gramme of the Newton International Fellowship and theMax Planck Institute for the Physics of Complex Sys-tems. We thank Melanie Schnell, Laurence D. Barronand Fiona C. Speirits for helpful correspondences. [1] The word ‘chiral’ was introduced by Lord Kelvin [5].[2] W. J. Lough and I. W. Wainer 2002
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Adv. Chem. Phys. T and T in the J → J. Chem. Phys. J. Chem. Phys. [133] It has been shown that mirrors of the type we envis-age here can sustain continuous illumination with noapparent damage by light with an intensity of at least1 . × kg.s − [113][134] B. Deppe, G. Huber, C. Kr¨ankel and J. K¨upper 2015High-intracavity-power thin-disk laser for the alignmentof molecules Opt. Express Appendix A: Diagonalisation of ˆ H ′ In the present appendix we discuss the diagonalisationof ˆ H ′ in more detail. We again focus our attention upona molecule with nuclear spins of 0 or 1 /
2, assume thatˆ H rotor ≫ ˆ H light ≫ δ ˆ H with no accidental degeneracies ofimportance and neglect the possibility of effects due tothe spin statistics of similar nuclei [9, 37, 38, 56]. For amolecule with nuclear spins of 1 or greater the interactionbetween nuclear electric-quadrupole moments and in-tramolecular electric field gradients [37, 38, 49, 50] mightgive rise to a large δ ˆ H such that ˆ H rotor ≫ ˆ H light ≫ δ ˆ H is not a valid assumption and a more involved approachtowards diagonalisation than that described here is re-quired.We begin by considering ˆ H rotor in isolation. Letus introduce here the familiar symmetric rotor states ∣ J, K, m ⟩ , with K ∈ { , . . . , ± J } determining the Z com-ponent of the (oblate) rotor’s angular momentum, say[9, 37, 40, 41]. We expand the ∣ J τ,m ⟩ in terms of these as ∣ J τ,m ⟩ = J ∑ K =− J ˜ a J,τ ( K )∣ J, K, m ⟩ . (A1)Closed forms for the ˜ a J,τ ( K ) are not known at present.It has been established [40, 45], however, that the ( J + ) × ( J + ) matrix of ˆ H rotor for given values of J and m can be partitioned into smaller blocks, referred to as the E + , E − , O + and O − blocks with associated basis states E + ∶ ∣ J, , m ⟩ together with1 √ (∣ J, K, m ⟩ + ∣ J, − K, m ⟩) ( K even , ≥ ) ,E − ∶ √ (∣ J, K, m ⟩ − ∣ J, − K, m ⟩) ( K even , ≥ ) ,O + ∶ √ (∣ J, K, m ⟩ + ∣ J, − K, m ⟩) ( K odd , ≥ ) O − ∶ √ (∣ J, K, m ⟩ − ∣ J, − K, m ⟩) ( K odd , ≥ ) . The ˜ a J,τ ( K ) can then be found by diagonalising theseblocks individually, the associated eigenvalues being the w J τ with τ running from − J to J with increasing en-ergy. For the lowest values of J this procedure can beperformed analytically. For higher values of J the E + , E − , O + and O − blocks must themselves be diagonalisednumerically. In what follows we focus our attention upona particular pair of values of J and τ . We assume theassociated ˜ a J,τ ( K ) to be known and that these satisfy ∑ JK =− J ∣ ˜ a J,τ ( K )∣ =
1, thus ensuring normalisation of the ∣ J τ,m ⟩ .Next, we consider the perturbation of ˆ H rotor by ˆ H light to first order. The (2 J +1)-fold m rotational degeneracyinherent to ˆ H rotor is partially broken by ˆ H light , as ⟨ J τ,m ∣ ˆ H light ∣ J τ,m ′ ⟩ = (A2) − δ mm ′ I(cid:15) c ⎡⎢⎢⎢⎢⎣ a J,τ (∣ m ∣) ( α XX + σ ∣ k ∣ B XX )+ b J,τ (∣ m ∣) ( α Y Y + σ ∣ k ∣ B Y Y )+ c J,τ (∣ m ∣) ( α ZZ + σ ∣ k ∣ B ZZ ) ⎤⎥⎥⎥⎥⎦ with a J,τ (∣ m ∣) = J ∑ K =− J J ∑ K ′ =− J ˜ a ∗ J,τ ( K ) ˜ a J,τ ( K ′ ) (A3) ⟨ J, K, ∣ m ∣∣ ( ˆ (cid:96) xX + ˆ (cid:96) yX ) ∣ J, K ′ , ∣ m ∣⟩ , b J,τ (∣ m ∣) = J ∑ K =− J J ∑ K ′ =− J ˜ a ∗ J,τ ( K ) ˜ a J,τ ( K ′ ) (A4) ⟨ J, K, ∣ m ∣∣ ( ˆ (cid:96) xY + ˆ (cid:96) yY ) ∣ J, K ′ , ∣ m ∣⟩ c J,τ (∣ m ∣) = J ∑ K =− J J ∑ K ′ =− J ˜ a ∗ J,τ ( K ) ˜ a J,τ ( K ′ ) (A5) ⟨ J, K, ∣ m ∣∣ ( ˆ (cid:96) xZ + ˆ (cid:96) yZ ) ∣ J, K ′ , ∣ m ∣⟩ being numbers that quantify the average orientation ofthe molecule. The independence upon the sign of m indi-cated here leaves a ∣ m ∣ rotational degeneracy, of course,and may be appreciated by noting that a parity inver-sion of the system changes the sign of the component ofthe molecule’s angular momentum along the direction ofpropagation of the light whilst leaving the energy of thesystem unchanged. The absence at this order of certaincomponents such as α XY may be appreciated by notingthat these are not uniquely defined in the present context:a rotation of the molecular axes by π about the original X axis without changing the molecule leaves ˆ H rotor unaf-fected whilst nevertheless changing the sign of α XY , forexample. It is tedious but straightforward to evaluatethe matrix elements appearing in (A3), (A4) and (A5),by performing angular integrations over direction cosinesand symmetric rotor wavefunctions explicitly [40, 41] orby multiplying well-established expressions for directioncosine matrix elements in the symmetric rotor basis per-haps [37, 126]. We refrain from reproducing here thesomewhat lengthy expressions thus obtained. We note,5however, that the summations a J,τ (∣ m ∣) + b J,τ (∣ m ∣) + c J,τ (∣ m ∣) = , (A6)1 ( J + ) J ∑ m =− J a J,τ (∣ m ∣) = , (A7)1 ( J + ) J ∑ m =− J b J,τ (∣ m ∣) =
13 (A8)1 ( J + ) J ∑ m =− J c J,τ (∣ m ∣) =
13 (A9)yield isotropic values as indicated, in accord with theprinciple of spectroscopic stablity [51]. Higher-order cor-rections in the α AB can be significant, but are chirallyinsensitive. We refrain, therefore, from including themexplicitly in the present paper. Their presence is indi-cated in § II by dots and is neglected in § III.Finally, we consider the additional perturbation ofˆ H rotor + ˆ H light by δ ˆ H to first order. Let us introduce herethe nuclear-spin states ∣ I j , m j ⟩ , with I j ∈ { , / } deter-mining the magnitude of the spin and m j ∈ { , . . . , I j } determining the z component of the spin for the j th nu-cleus [9, 37, 38, 40]. Our approach is to consider eachdistinct value of ∣ m ∣ = ∣ m ′ ∣ ∈ { , . . . , J } in turn and diag-onalise the matrix with elements of the form ⎛⎝∏ j ⟨ I j , m j ∣⎞⎠ ⟨ J τ,m ∣ δ ˆ H ∣ J τ,m ′ ⟩ ⎛⎝∏ j ′ ∣ I j ′ , m ′ j ′ ⟩⎞⎠ . The energy shifts thus obtained give rise in particularto hyperfine structure in the rotational spectrum of themolecule in the presence of the light.The leading-order perturbative results described abovesuffice to illustrate the basic features of chiral rotationalspectroscopy and are the ones upon which we base ourexplicit discussions and calculations in the present paper.We note here, however, that near degeneracies of impor-tance are, in fact, rather common. In general then, ˆ H ′ should be diagonalised numerically. Appendix B: Influence of an applied static magneticfield
In the present appendix we briefly discuss the influenceof an applied static magnetic field. We consider the situ-ation described by ˆ H ′ as in Appendix A but augmentedhere by a uniform, static magnetic field B of moderatestrength pointing in the z direction. The rotational andnuclear-spin degrees of freedom of the molecule shouldnow be well described by the effective Hamiltonianˆ H ′′ = ˆ H rotor + ˆ H light + ˆ H B nucl + ˆ H B rotor (B1) + ˆ H B + ˆ H B light + δ ˆ H + δ ˆ H B with ˆ H B nucl = − µ N ̵ h ∑ j g j ˆ I jz B z (B2) accounting for the interaction energy between B and thenuclear magnetic-dipole moments [37, 38],ˆ H B rotor = − µ N ̵ h ( ˆ (cid:96) zA ˆ J B + ˆ J B ˆ (cid:96) zA ) g AB B z (B3)accounting for the interaction energy between B and therotational magnetic-dipole moment [48, 126, 127],ˆ H B = −
12 ˆ (cid:96) zA ˆ (cid:96) zB χ AB B z (B4)accounting for the interaction energy between B and themagnetic-dipole moment induced by B ,ˆ H B light = − I (cid:15) c σk z ∣ k ∣ ˆ (cid:96) xA ˆ (cid:96) yB ˆ (cid:96) zC α ′ AB,C B z (B5)accounting for the distortion by B of the electronicelectric-dipole / electric-dipole polarisability [55] and δ ˆ H B accounting for additional effects associated with B such as nuclear-spin shielding. µ N is the nuclear mag-neton; g j is the g factor of the j th nucleus; ˆ I jz is the z component of the spin of the j th nucleus; B z is the z com-ponent of B ; the g AB are components of the rotational g tensor, which has nuclear and electronic contributions[48, 126, 127]; the χ AB are components of the electronicstatic magnetic susceptibility tensor, which has diamag-netic and temperature-independent paramagnetic contri-butions [3, 48, 51], and the α ′ AB,C are components of theelectronic Faraday-B polarisability [3].The α ′ AB,C might in some cases give a σ -dependentcontribution to ˆ H ′′ comparable to that from the B AB .Note, however, that the α ′ AB,C are chirally insensitive: α ′ Y Z,X , α ′ ZX,Y and α ′ XY,Z in particular are identical foropposite enantiomers. In principle the effects of the α ′ AB,C can be distinguished from those of the B AB bycomparing spectra obtained with k and B parallel andantiparallel. Indeed, the contribution made to ˆ H ′′ by the α ′ AB,C is to magnetic or Faraday optical rotation and thespin of light what the contribution made by the B AB isto natural optical rotation and the helicity of light [55].We begin by considering the perturbation of ˆ H rotor byˆ H light + ˆ H B nucl + ˆ H B rotor + ˆ H B + ˆ H B light to first order. Thenuclear-spin degeneracy is at least partially broken byˆ H B nucl , as ⎛⎝∏ j ⟨ I j , m j ∣⎞⎠ ˆ H B nucl ⎛⎝∏ j ′ ∣ I j ′ , m ′ j ′ ⟩⎞⎠ (B6) = − ⎛⎝∏ j δ m j m ′ j ⎞⎠ µ N ⎛⎝∑ j ′ g j ′ m j ′ ⎞⎠ B z with nuclear-spin degeneracies remaining when multiplenuclei of the same type with spins of 1 / ( J + ) -fold m ro-tational degeneracy inherent to ˆ H rotor is fully broken byˆ H B rotor , as ⟨ J τ,m ∣ ˆ H B rotor ∣ J τ,m ′ ⟩ = − δ mm ′ µ N g J,τ mB z (B7)6with g J,τ m = ̵ h ⟨ J τ,m ∣ ( ˆ (cid:96) zX ˆ J X + ˆ J X ˆ (cid:96) zX ) ∣ J τ,m ⟩ g XX (B8) + ̵ h ⟨ J τ,m ∣ ( ˆ (cid:96) zY ˆ J Y + ˆ J Y ˆ (cid:96) zY ) ∣ J τ,m ⟩ g Y Y + ̵ h ⟨ J τ,m ∣ ( ˆ (cid:96) zZ ˆ J Z + ˆ J Z ˆ (cid:96) zZ ) ∣ J τ,m ⟩ g ZZ defining the effective rotational g factor g J,τ [126, 127].Further ∣ m ∣ -dependent energy shifts arise through ˆ H B ,as ⟨ J τ,m ∣ ˆ H B ∣ J τ,m ′ ⟩= − δ mm ′ { [ − a J,τ (∣ m ∣)] χ XX (B9) + [ − b J,τ (∣ m ∣)] χ Y Y + [ − c J,τ (∣ m ∣)] χ ZZ } B z , and through ˆ H B light , as ⟨ J τ,m ∣ ˆ H B light ∣ J τ,m ′ ⟩= − δ mm ′ I (cid:15) c σk z ∣ k ∣ { [ b J,τ (∣ m ∣) + c J,τ (∣ m ∣) − ] α ′ Y Z,X + [ c J,τ (∣ m ∣) + a J,τ (∣ m ∣) − ] α ′ ZX,Y (B10) + [ a J,τ (∣ m ∣) + b J,τ (∣ m ∣) − ] α ′ XY,Z } B z , although the magnitudes of these are not necessarilylarger than those of the energy shifts that arise through δ ˆ H + δ ˆ H B .We conclude by considering the perturbation ofˆ H rotor + ˆ H light + ˆ H B nucl + ˆ H B rotor + ˆ H B + ˆ H B light by δ ˆ H + δ ˆ H B to first order. Our approach is to consider each distinctpair of values of ∑ j g j m j = ∑ j ′ g j ′ m ′ j ′ and m ∈ {− J, . . . , J } in turn and diagonalise the matrix with elements of theform ⎛⎝∏ j ⟨ I j , m j ∣⎞⎠ ⟨ J τ,m ∣ ( δ ˆ H + δ ˆ H B ) ∣ J τ,m ⟩ ⎛⎝∏ j ′ ∣ I j ′ , m ′ j ′ ⟩⎞⎠ . The energy shifts thus obtained give rise in particularto hyperfine structure in the rotational spectrum of themolecule in the presence of the light and B .Again, the perturbative results described above suf-fice to illustrate the basic features introduced by B butshould not be used in lieu of a numerical diagonalisationof ˆ H ′′ in general. Appendix C: Calculated molecular properties
In the present appendix we report the calculatedmolecular properties upon which FIG. 3, FIG. 5, FIG.6 and FIG. 7 are based. We evaluated A = ̵ h ∑ j M j ( Y j + Z j ) , (C1) B = ̵ h ∑ j M j ( Z j + X j ) (C2) C = ̵ h ∑ j M j ( X j + Y j ) (C3)using the nuclear coordinates X j , Y j and Z j tabulatedbelow together with mass / 10 − kg H 0.1673533 C 1.9926468 C 2.1642716 O 2.6560180for the masses M j . The NWChem computational chem-istry program [11, 128] was employed to calculate theelectronic energy eigenstates ∣ k ⟩ and associated electronicenergy eigenvalues ̵ hω k asˆ H elec ∣ k ⟩ = ̵ hω k ∣ k ⟩ (C4)withˆ H elec = ⎡⎢⎢⎢⎢⎣ ∑ i ˆ P iA ˆ P iA m e (C5) + ∑ i ∑ i ′ > i e π(cid:15) √( ˆ X i − ˆ X i ′ ) + ( ˆ Y i − ˆ Y i ′ ) + ( ˆ Z i − ˆ Z i ′ ) − ∑ i ∑ j Z j e π(cid:15) √( ˆ X i − X j ) + ( ˆ Y i − Y j ) + ( ˆ Z i − Z j ) ⎤⎥⎥⎥⎥⎦ the electronic Hamiltonian [3, 9, 40, 42], where the ˆ P iA are components of the canonical linear momentum of the i th electron; m e is the mass of the electron; e is themagnitude of the electronic charge; ˆ X i , ˆ Y i and ˆ Z i arethe coordinates of the i th electron and Z j is the atomicnumber of the j th nucleus. These gave [3, 7, 8] α XX = ̵ h ∑ k ω k ω k − c ∣ k ∣ R (⟨ ∣ ˆ µ X ∣ k ⟩⟨ k ∣ ˆ µ X ∣ ⟩) , (C6) G ′ XX = − ̵ h ∑ k c ∣ k ∣ ω k − c ∣ k ∣ I (⟨ ∣ ˆ µ X ∣ k ⟩⟨ k ∣ ˆ m X ∣ ⟩) (C7) A X,Y Z = ̵ h ∑ k ω k ω k − c ∣ k ∣ R (⟨ ∣ ˆ µ X ∣ k ⟩⟨ k ∣ ˆΘ Y Z ∣ ⟩) (C8)with µ X = − e ∑ i ˆ X i , (C9)ˆ m X = − e m e ∑ i ( ˆ Y i ˆ P iZ − ˆ Z i ˆ P iY ) (C10)ˆΘ Y Z = − e ∑ i ˆ Y i ˆ Z i , (C11)7for example, where ∣ ⟩ and ̵ hω pertain to the groundstate in particular. Then [10, 11] B XX = − c ∣ k ∣ G ′ XX + ( A Y,ZX − A Z,XY ) , (C12)for example. Note that the nuclei are held here in thesame, rigid constellation for different electronic stateswith the nuclear and electronic centres of mass regardedas one and the same [40, 48]. Myriad corrections to thismodel, not least the inclusion of the vibrational degreesof freedom of the molecule, might be entertained in morerefined calculations. We found the b3lyp exchange func-tionals to be more reliable for the smaller molecules hereand the xcamb88 exchange functionals to be more reli-able for the larger ones.For the lowest energy conformer of (S)-propylene glycol(upper signs) or (R)-propylene glycol (lower signs) X / 10 − m Y / 10 − m Z / 10 − m H ± ± ∓ H ∓ ∓ ∓ H ∓ ∓ ± H ∓ ∓ ± H ± ∓ ± H ± ∓ ∓ H ± ∓ ± H ∓ ∓ ± C ± ∓ ∓ C ∓ ∓ ± C ∓ ∓ ∓ O ∓ ± ∓ O ± ∓ ± s − A / π ̵ h B / π ̵ h C / π ̵ h − kg − .s .A α XX / α Y Y / α ZZ / ∣ k ∣ B XX ± ∣ k ∣ B Y Y ∓ ∣ k ∣ B ZZ ± π /∣ k ∣ = . × − m using DFT with the aug-cc-pVDZ basis set and the b3lyp exchange functionals.For isotopically chiral housane, with the upper andlower signs referring to the enantiomers obtained by re-placing the usual C atom at the bottom-left or bottom-right of the ‘house’ with a C atom, X / 10 − m Y / 10 − m Z / 10 − m H ∓ ∓ ± H ∓ ± ± H ± ∓ ∓ H ± ± ∓ H ± ∓ ± H ± ± ± H ∓ ∓ ∓ H ∓ ∓ ∓ C ∓ ∓ ± C ∓ ± ± C ± ± ∓ C ∓ ∓ ∓ C ± ∓ ∓ s − A / π ̵ h B / π ̵ h C / π ̵ h − kg − .s .A α XX / α Y Y / α ZZ / ∣ k ∣ B XX ∓ ∣ k ∣ B Y Y ± ∣ k ∣ B ZZ ± π /∣ k ∣ = . × − m using DFT with the aug-cc-pVDZ basis set and the b3lyp exchange functionals.For mesotartaric acid X / 10 − m Y / 10 − m Z / 10 − m H 3.2759269 -1.4121684 0.0008288 H 0.3010403 0.0938862 1.4929986 H 0.4320884 1.9990179 -0.6188789 H -0.3010403 -0.0938862 -1.4929986 H -0.4320884 -1.9990184 0.6188789 H -3.2759269 1.4121684 -0.0008288 C 1.9010044 0.0508956 0.0829577 C 0.4800303 0.3819448 0.4570136 C -0.4800303 -0.3819448 -0.4570136 C -1.9010044 -0.0508956 -0.0829577 O 0.2630994 1.7859661 0.3091107 O -0.2630994 -1.7859661 -0.3091107 O -2.6250383 -0.910888 0.3600024 O -2.3629433 1.2001377 -0.2408646 O 2.3629433 -1.2001377 0.2408646 O 2.6250383 0.91088792 -0.3600024from [129]. These gave value / 10 s − A / π ̵ h B / π ̵ h C / π ̵ h − kg − .s .A α XX /2 7.107610 α Y Y /2 6.572130 α ZZ /2 4.668277at 2 π /∣ k ∣ = . × − m using DFT with the aug-cc-pVDZ basis set and the xcamb88 exchange functionals.For L -tartaric acid (upper signs) or D -tartaric acid(lower signs) X / 10 − m Y / 10 − m Z / 10 − m H ∓ ∓ ∓ H ∓ ∓ ± H ∓ ± ± H ± ∓ ∓ H ± ± ∓ H ± ∓ ± C ∓ ∓ ± C ∓ ∓ ± C ± ∓ ∓ C ± ∓ ∓ O ± ± ∓ O ∓ ± ± O ∓ ± ± O ∓ ∓ ∓ O ± ± ∓ O ± ∓ ± s − A / π ̵ h B / π ̵ h C / π ̵ h − kg − .s .A α XX /2 7.047660 α Y Y /2 5.985290 α ZZ /2 5.183870 ∣ k ∣ B XX ∓ ∣ k ∣ B Y Y ∓ ∣ k ∣ B ZZ ± π /∣ k ∣ = . × − m using DFT with the aug-cc-pVDZ basis set and the xcamb88 exchange functionals.For a particular conformer of (S)-ibuprofen (uppersigns) or (R)-ibuprofen (lower signs) X / 10 − m Y / 10 − m Z / 10 − m H ∓ ∓ ± H ∓ ± ± H ∓ ± ∓ H ± ± ± H ∓ ∓ ± H ∓ ∓ ∓ H ∓ ∓ ∓ H ∓ ± ± H ∓ ∓ ∓ H ∓ ∓ ± H ∓ ± ± H ∓ ∓ ∓ H ± ± ± H ± ∓ ∓ H ± ± ∓ H ± ± ∓ H ± ± ∓ H ± ∓ ± C ∓ ∓ ± C ∓ ± ∓ C ∓ ± ∓ C ± ± ± C ± ± ± C ∓ ∓ ∓ C ∓ ∓ ± C ∓ ± ± C ∓ ± ∓ C ± ± ± C ± ± ∓ C ± ± ∓ C ± ∓ ± O ± ∓ ± O ± ∓ ± s − A / π ̵ h B / π ̵ h C / π ̵ h − kg − .s .A α XX / α Y Y / α ZZ / ∣ k ∣ B XX ± ∣ k ∣ B Y Y ∓ ∣ k ∣ B ZZ ∓ π /∣ k ∣ = . × − m using DFT with the 6-311+G ∗ basis set and the xcamb88 exchange functionals. Appendix D: Functionality of the chiral rotationalspectrometer
In the present appendix we give a quantitative modelof the chiral rotational spectrometer discussed in § IV.9Our derivation borrows heavily from [116, 118–120, 130].Let us focus our attention here upon a particular de-sign in which the molecular pulses are assumed to havethe usual form but with a sharp angular collimation, theoptical cavity is of the skew-square ring variety, the mi-crowave cavity is of the Fabry-P´erot variety with spher-ical mirrors and the static magnetic field is produced byHelmholtz coils, as seen in FIG. 8 and also FIG. 9. Weconsider a single form of molecule, taking t = x , y , z at the centre of the microwave cavity, withthe y axis parallel to the axis of the microwave cavity andthe direction of propagation of the light in the active re-gion defining the + z direction. In addition we introducea secondary set of axes x ′ , y ′ , z ′ which are parallel to x , y , z but have their origin at the centre of the activeregion, located at r = y ˆ y with respect to x , y , z . TheHelmholtz coils are centred upon r where they producea static magnetic field in the z direction. We imagine per-fect vacuum save, of course, for the molecules themselvesand any atoms that accompany them. Stray fields andradiation, including the earth’s gravitational and mag-netic fields and background blackbody radiation, are ne-glected, as is noise. A number of possible interactions,including the formation of clusters and other such com-plications within the molecular pulses, adhesion of themolecules to the light mirrors, changes in the resonantfrequencies of the optical cavity due to heating by thelight or refraction by the molecules and perturbation ofthe operation of the microwave cavity by the light mir-rors, are neglected. These may need to be consideredmore carefully in some circumstances.Our model should be well suited to values such as [39, 111, 112, 115, 116, 118–123, 130–132] y ∈ [ l − d / , d / ] ,I = . × kg.s − ,γ = . × − m , π /∣ k ∣ = . × − m , ∣ σ ∣ = . ,l = . × − m , ∣ β ∣ = . × − ,χ = . × − ,δ = × − , R = . × − m , µ N ∣I∣/ √ R ∈ [ . × − kg.s − .A − , . × kg.s − .A − ] , Γ / π = . × s − ,v = m.s − ,h = . × − m ,µ = . × − ,N = m − ,D = × − m ,p ∈ [− × − , × ] ,θ = . × − ,T = K ,τ = − s ,E µ βαz τ /̵ h ≲ π / ,τ c = − s ,ω c / π ∈ [ . × s − , . × s − ] ,R = . × − m d ∈ [ . × − m , . × − m ] , for example, with the symbols as defined below and indi-cated in FIG. 9. Note in particular the implied circulat-ing light power of 1 . × kg.m .s − in the optical cavity,which should be achievable using an input light power of1 . × kg.m .s − or less, assuming a transmittance of2 . × − or less for each light mirror and neglectingloss [112–114, 133]. Significantly higher circulating lightpowers than this have certainly been demonstrated, alsoin the context of molecular alignment [134].Let ˆ H ′′ ( r ) be the effective Hamiltonian describingthe rotational and nuclear-spin degrees of freedom of amolecule the centre of mass of which is notionally heldfixed at some position r = x ˆ x + y ˆ y + z ˆ z in the active re-gion. The energy eigenstates ∣ r ( r )⟩ and associated energyeigenvalues ̵ hω r ( r ) of ˆ H ′′ ( r ) satisfyˆ H ′′ ( r )∣ r ( r )⟩ = ̵ hω r ( r )∣ r ( r )⟩ . (D1)It is convenient to partition these as ∣ r ( r )⟩ = ∣ r ⟩ + ∆ ∣ r ( r )⟩ (D2) ̵ hω r ( r ) = ̵ hω r + ̵ h ∆ ω r ( r ) (D3)0 FIG. 9. Some of the parameters of our model and an indica-tion of their significance, for quick reference: Γ, v , N , D , p , θ and T pertain to the pulsed supersonic expansion nozzleand the molecular pulses; l , β , χ and δ pertain to the opticalcavity; I , γ , k and σ pertain to the light in the active region; τ c , ω c , R and d pertain to the microwave cavity; τ and E pertain to the polarising microwave pulses and R , N and I pertain to Helmholtz coils which produce the static magneticfield, as described in the text. with ∣ r ⟩ and ̵ hω r denoting the particular forms takenat r = r in the absence of the light. We assume thatthe ∆ ∣ r ( r )⟩ constitute but small corrections to the ∣ r ⟩ of interest, as, by construction throughout the active re-gion, the static magnetic field is highly uniform and di-rected essentially parallel to the direction of propagationof the light, so that essentially the same quantisation axisfor the rotational degrees of freedom of the molecules isfavoured by both. We therefore neglect the ∆ ∣ r ( r )⟩ andtake ∣ r ( r )⟩ = ∣ r ⟩ in what follows. We also assume thatthe ̵ h ∆ ω r ( r ) constitute but small corrections to the ̵ hω r of interest. We nevertheless retain the ̵ h ∆ ω r ( r ) in whatfollows unless otherwise stated, as they appear in the ar-guments of sensitive mathematical functions and are theessence of chiral rotational spectroscopy. The followingexplicit forms might be employed:ˆ H ′′ ( r ) = ̵ h ( A ˆ J X + B ˆ J Y + C ˆ J Z )− I ( r ) (cid:15) c {[ ( cos η cos ι + sin η sin ι ) ˆ (cid:96) xA ˆ (cid:96) xB + cos 2 η sin 2 ι ˆ (cid:96) xA ˆ (cid:96) yB + ( cos η sin ι + sin η cos ι ) ˆ (cid:96) yA ˆ (cid:96) yB ] α AB + σ ∣ k ∣ ( ˆ (cid:96) xA ˆ (cid:96) xB + ˆ (cid:96) yA ˆ (cid:96) yB ) B AB }− µ N ̵ h ∑ j g j ˆ I jz B z ( r )− µ N ̵ h ( ˆ (cid:96) zA ˆ J B + ˆ J B ˆ (cid:96) zA ) g AB B z ( r )−
12 ˆ (cid:96) zA ˆ (cid:96) zB χ AB B z ( r ) (D4) − I ( r ) (cid:15) c σ ˆ (cid:96) xA ˆ (cid:96) yB ˆ (cid:96) zC α ′ AB,C B z ( r )− δ ˆ H − δ ˆ H B ( r ) is the effective Hamiltonian indicated in (B1) but herevarying through the active region as a function of r witha more general pure polarisation state for the light, where I ( r ) is the intensity profile of the light, k = ∣ k ∣ ˆ z is thecentral wavevector of the light, − η is the ellipticity angleof the light as in [3] (with σ = sin 2 η here), − ι is thepolarisation azimuth of the light as in [3] and B z ( r ) is the z component of the static magnetic field B ( r ) , assumingthe light to propagate in a fundamental mode that is nottightly focussed. This might be augmented with I ( r ′ ) = I exp (− x ′ + y ′ γ ) , (D5)which is a Gaussian transverse intensity profile, with γ the 1 / e width. The precise resonance frequencies c ∣ k ∣/ π of the remaining longitudinal modes supported by theoptical cavity depend, of course, upon the precise length l of each side of the ring as well as the fold angle β , as c ∣ k ∣ π = c l ( g + ± √ βπ ) , (D6)say, with g the longitudinal mode number and where theupper and lower signs distinguish opposite circular polar-isations [111, 112]. Tacit in (D6) is the assumption that1 >> ∣ β ∣ >> ∣ χ ∣ , δ , with χ and δ reflection anisotropies of1the light mirrors as in [112]. Furthermore, we might take B z ( r ′ ) = µ RN I π ∫ π ⎧⎪⎪⎨⎪⎪⎩ (D7) + (R − x ′ cos φ − y ′ sin φ )[( x ′ − R cos φ ) + ( y ′ − R sin φ ) + ( z ′ − R ) ] / + (R − x ′ cos φ − y ′ sin φ )[( x ′ − R cos φ ) + ( y ′ − R sin φ ) + ( z ′ + R ) ] / ⎫⎪⎪⎬⎪⎪⎭ d φ, which is the usual expression for the z component of thestatic magnetic field produced by Helmholtz coils, with R and N the radius and number of turns of each coil and I the current running through each turn. Our neglect ofthe x and y components of the light’s wavevectors and B ( r ) in extrapolating the spatially-dependent form (D4)from the more idealised form (B1) should introduce lit-tle error as, again by construction throughout the activeregion, the light propagates in a near-planar fashion and B ( r ) is well directed. Similarly for our neglect of diffrac-tion in (D5).Let us assume now that the energies of molecules inmotion follow the forms ̵ hω r ( r ) adiabatically and takethe master equation describing the molecules to bei ̵ h ( ∂∂t + v ⋅ ∇ ) σ rs ( r , v , t ) = ̵ hω rs ( r ) σ rs ( r , v , t )+ ∑ w [ σ rw ( r , v , t ) µ wsa − µ rwa σ ws ( r , v , t )] E a ( r , t )− i ̵ h Γ [ σ rs ( r , v , t ) − σ rs eq ( r , v , t )] (D8)with v a velocity in phase space; ∇ the gradient operatorwith respect to r ; the σ rs ( r , v , t ) = ⟨ r ∣ ˆ σ ( r , v , t )∣ s ⟩ matrixelements of the density operator ˆ σ ( r , v , t ) ; the ̵ hω rs ( r ) =̵ hω r ( r ) − ̵ hω s ( r ) energy differences; the µ rs = ⟨ r ∣ ˆ µ ∣ s ⟩ ma-trix elements of the single-molecule electric-dipole mo-ment operator ˆ µ ; E ( r , t ) the polarising microwave elec-tric field; Γ a decay rate which models decoherence dueprimarily to residual molecular collisions and σ rs eq ( r , v , t ) the equilibrium density matrix [118–120]. Our neglect ofabsorption, Raman scattering and other such processesshould be justified as the molecules are only illuminatedby the light for a short time and molecular collisions oc-cur infrequently during this time. Our use of a single decay rate ( i.e. Γ ≈ / T ≈ / T ) is in accord with em-pirical observations [130–132]. Let us imagine that themolecules proceed from the nozzle in straight lines, inwhich case we can assign a unique velocity v ( r ) to each r .Our neglect of forces including those due to light shouldbe well justified. It is convenient then to introduce theparametrisations σ rs ( r , v , t ) = N ( r , t ) δ [ v − v ( r )] ρ rs ( r , t ) (D9) σ rs eq ( r , v , t ) = N ( r , t ) δ [ v − v ( r )] ρ rs eq (D10)with N ( r , t ) the number density distribution of themolecules, ρ rs ( r , t ) elements of the reduced density ma-trix appropriate to single molecules following the tra-jectory defined by v ( r ) and ρ rs eq elements of the re-duced equilibrium density matrix appropriate to single molecules. Our failure to acknowledge the spatial varia-tion of ρ rs eq should be of little consequence as the ̵ h ∆ ω r ( r ) are small relative to the ̵ hω r . Let us assume further that v ( r ) ⋅ ∇ v a ( r ) = ∂N ( r , t ) ∂t + ∇ ⋅ [ v ( r ) N ( r , t )] = v andmaking use of (D9)-(D12) we obtain the reduced masterequationi ̵ h [ ∂∂t + v ( r ) ⋅ ∇ ] ρ rs ( r , t ) = ̵ hω rs ( r ) ρ rs ( r , t )+ ∑ w [ ρ rw ( r , t ) µ wsa − µ rwa ρ ws ( r , t )] E a ( r , t )− i ̵ h Γ [ ρ rs ( r , t ) − ρ rs eq ] (D13)which we will begin making use of shortly. The followingexplicit forms might be employed: v ( r ′ ) = v ( x ′ − h sin µ ) ˆ x + y ′ ˆ y + ( z ′ − h cos µ ) ˆ z √( x ′ − h sin µ ) + y ′ + ( z ′ − h cos µ ) (D14)is a velocity field describing molecules emanating radiallyfrom the nozzle orifice, with v the speed of the molecules, h the distance from r to the centre of the nozzle orificeand µ the angle from the + z axis to the line joining thesetwo points [120]. N ( r ′ , t ) = C( r ′ ) N D (D15) × ( h − x ′ sin µ − z ′ cos µ ) p [( x ′ − h sin µ ) + y ′ + ( z ′ − h cos µ ) ] + p / describes the shape of the expanding gas pulse far fromthe nozzle orifice as being proportional to the usual‘cos p θ / r ’ form, with N is the number density ofmolecules in the nozzle reservoir, D the nozzle diame-ter, p quantifying the angular spread of a (hypothetical)freely expanding pulse [116, 118, 120] and C( r ′ ) a modu-lating function included by us to account for the effects ofcollimation and perhaps other effects due to short pulsetimes and the off-centre position of the nozzle with re-spect to the microwave cavity mirrors. The value of p isspecific to the time elapsed since the opening of the noz-zle and to the mixture of molecules and atoms present inthe expansion, but can be regarded as being essentiallyconstant over the course of a measurement as the shapeof the molecular pulse varies relatively slowly. In oneparticular experiment (with no collimation) it was deter-mined that, for example, p = − × − a time 3 . × − safter the opening of a nozzle and p ≥ × at the latertime of 5 . × − s, describing an initial depletion fromthe beam axis and increased directivity at later times, aseemingly general trend [120]. For a nozzle with no colli-mation, C( r ′ ) =
1. A simple approach therefore might beto take C ( r ′ ) = ⎧⎪⎪⎪⎨⎪⎪⎪⎩ , cos − { ( h − x ′ sin µ − z ′ cos µ )√( x ′ − h sin µ ) + y ′ +( z ′ − h cos µ ) } ≤ θ , otherwise2which describes a sharp angular collimation, with halfangle θ . This disregards any effects due to collimationupon the evolution of the molecular pulse shape as em-bodied by p , but might nevertheless prove valid for θ suitably small, as p has little effect upon N ( r ′ , t ) close tothe axis of the molecular pulse, where ‘ cos p θ ≈ ′ . ρ rs eq = δ rs exp (− ̵ hω r k B T )∑ w exp (− ̵ hω w k B T ) (D16)is an equilibrium density matrix pertaining to a thermaldistribution, with T the temperature of the distribution.First, consider times t <
0, during which there are es-sentially no microwaves present in the microwave cavity.Thus, we take E ( r , t ) = ρ rs ( r , t ) = ρ rs eq (D17)which is the equilibrium solution of the reduced masterequation (D13).Next, consider the time interval 0 ≤ t ≤ τ , during whicha microwave pulse polarises the molecules. We model themicrowave pulse temporally as being of duration τ withconstant amplitude E and spatially as being in a singleTEM mnq mode of the microwave cavity. Our neglect ofthe finite rise and decay times of the microwave pulseshould introduce little error as the polarisation of themolecules is an integrated quantity and the microwavecavity rise / decay time τ c is much shorter than τ [119,120]. Thus, we take E ( r , t ) = [ ˜ E ( r ) exp (− i ω c t ) + ˜ E ∗ ( r ) exp ( i ω c t )] (D18)for ˜ E ( r ) = ˆ z E u ( r ) exp [ i ( ϑ + ω c τ )] (D19)with E the microwave pulse amplitude, u ( r ) = w w ( y ) H m [ √ xw ( y ) ] H n [ √ zw ( y ) ]× exp [− x + z w ( y ) ] (D20) × cos [ ω c yc + ω c ( x + z ) Rc − Φ ( y ) − πq ] the microwave mode shape, ω c = πcd [ q + + π ( m + n + ) cos − ( − dR )] (D21)the microwave mode angular frequency and ϑ a tunablephase angle, where w = [ cd ( R − d ) ω c ] / (D22) w ( y ) = w [ + cyω c w ] / (D23) are beam waists andΦ ( y ) = tan − ( cyω c w ) (D24)is a phase factor [116, 118–120]. The H m ( x ) are Her-mite polynomials, with H ( x ) = R is the radius ofcurvature of each microwave mirror and d is the separa-tion between the microwave mirrors. Our focus upon a single microwave mode should be acceptable as the fre-quency spacings between these modes are considerablylarger than the frequency bandwith of the microwave cav-ity. Note that we have focussed our attention explicitlyhere upon microwaves that are linearly polarised parallelto z , which are appropriate for probing ∆ m = x say, which areappropriate for probing ∆ m = ± v = = ρ rs ( r , t ) = ρ rs eq + E i ̵ h ( ρ rr eq − ρ ss eq ) µ rsz u ( r )× ⎡⎢⎢⎢⎢⎣ exp ( i { ϑ + [ ω c − ω rs ( r )] t })× sinc { [ ω c − ω rs ( r )] t } (D25) + exp (− i { ϑ + [ ω c + ω rs ( r )] t })× sinc { [ ω c + ω rs ( r )] t } ⎤⎥⎥⎥⎥⎦ to first order in E ( r , t ) , with sinc ( x ) = sin ( x )/ x . Ourneglect of higher-order contributions in E ( r , t ) should in-troduce little error, assuming E and τ to be such thatthe polarisation of the molecules is a little less than wouldresult from a π / t > τ , during which themolecules exhibit a free induction decay. Taking E ( r , t ) = ρ rs ( r , t ) = ρ rs eq + exp [− Γ ( t − τ )] (D26) × exp {− i ∫ tτ ω rs [ r − v ( r )( t − t ′ )] d t ′ }× { ρ rs [ r − v ( r )( t − τ ) , τ ] − ρ rs eq } . The polarisation P ( r , t ) = N ( r , t ) ∑ r ∑ s µ rs ρ sr ( r , t ) (D27)3associated with the molecules generates a microwave elec-tric field E s ( r , t ) in the microwave cavity, which we taketo satisfy ( ∇ − c ∂ ∂t ) E s ( r , t ) = c τ c ∂∂t E s ( r , t ) + µ ∂ ∂t P ( r , t ) (D28)with ∇ = ∇ ⋅ ∇ and E s ( r , t ) = ˆ z s ( t ) u ( r ) (D29)a single-mode form echoing that of the polarising mi-crowave pulse [119]. The electric field variation s ( t ) di-minishes slowly in average magnitude over time, primar-ily as a result of residual molecular collisions. Let ussuppose simply here that the detected lineshape S ( t ) is0 for t ≤ τ and proportional to s ( t ) for t > τ [119, 120].We then calculate the Fourier transform˜ S ( ω ) = exp (− i ωτ ) ∫ ∞ τ S ( t ) exp ( i ωt ) d t. (D30)and regard the real part R [ ˜ S ( ω )] of this as being themeasurement, given an appropriate choice of ϑ . Using(D28)-(D30) and taking ∇ u ( r ) = − ω c c u ( r ) (D31)we obtain˜ S ( ω ) ∝ ω ω c − ω + i ωτ c ( ω c − ω ) + ω τ c E ̵ h × ∑ r ∑ s ∣ µ srz ∣ ( ρ ss eq − ρ rr eq )× ∫∫∫∫ ∞ N ( r , t ) u ( r ) u [ r − v ( r ) t ] exp [( i ω − Γ ) t ]× exp {− i ∫ t ω sr [ r − v ( r )( t − t ′ )] d t ′ }× ⎧⎪⎪⎨⎪⎪⎩ exp [ i ( ϑ + { ω c − ω sr [ r − v ( r ) t ]} τ )] (D32) × sinc ( { ω c − ω sr [ r − v ( r ) t ]} τ )+ exp [− i ( ϑ + { ω c + ω sr [ r − v ( r ) t ]} τ )]× sinc ( { ω c + ω sr [ r − v ( r ) t ]} τ ) ⎫⎪⎪⎬⎪⎪⎭ d t d r where we have made the replacement N ( r , t + τ ) → N ( r , t ) , which is justified as the shape of the molecu-lar pulse varies slowly relative to τ , and have assumedthat µ rs = r = s , which is justified as asymmetricrotors do not exhibit first-order Stark shifts.Let us focus our attention now upon a particular, well-isolated rotational transition. We label the nuclear-spinstate manifold of the lower rotational state as α and thenuclear-spin state manifold of the upper rotational state as β . For ω ∼ ω c ∼ ω βα ≫ Γ the dominant contribution to(D32) then comes when r sums over α whilst s sums over β . If we assume moreover that ∣ ω βα − ω c ∣ ≪ ∣ ω βα + ω c ∣ , wecan safely neglect the term proportional to exp (− i ϑ ) . Inaddition, we will consider only those frequencies that liewell within the microwave cavity frequency bandwidth,as ∣ ω − ω c ∣ ≪ / τ c , in which case we can safely make thereplacement ω ( ω c − ω + i ω / τ c )/[( ω c − ω ) + ω / τ c ] → i Q ,with Q = ω c τ c the quality factor of the microwave cavity.We are left then with˜ S ( ω ) ∝ QE ̵ h exp ( i ϑ )× ∑ α ∑ β ∣ µ βαz ∣ ( ρ ββ eq − ρ αα eq )× ∫∫∫∫ ∞ N ( r , t ) u ( r ) u [ r − v ( r ) t ] exp [( i ω − Γ ) t ]× exp {− i ∫ t ω βα [ r − v ( r )( t − t ′ )] d t ′ } (D33) × exp ( i2 { ω c − ω βα [ r − v ( r ) t ]} τ )× sinc ( { ω c − ω βα [ r − v ( r ) t ]} τ ) d t d r . This form may be appreciated by considering the ide-alised limit v ( r ) → − v ˆ z ,N ( r ′ , t ) → n δ ( x ′ ) δ ( y ′ ) ,u ( x , y , z + v t ) → u ( x , y , z ) ,ω βα ( x , y , z + v t ) → ω , exp ( i2 { ω c − ω βα [ r − v ( r ) t ]} τ ) → , sinc ( { ω c − ω βα [ r − v ( r ) t ]} τ ) → ρ ββ eq − ρ αα eq → ∆ ρ pertaining to well-collimated molecular pulses movingparallel to the axis of the active region with n the num-ber of molecules per unit length on axis; motion throughthe microwave mode during the course of a measurementneglected; variations of I ( r ) and B ( r ) on axis as well ashyperfine splittings neglected, leaving a unique rotationaltransition angular frequency ω ; the microwave cavitymode chosen to be on resonance as ω c = ω and a singlevalue ∆ ρ taken to be a fair representation of the differ-ences in the equilibrium Boltzmann factors for the upperand lower manifolds. The rotational spectrum itself thentends towards R [ ˜ S ( ω )] ∝ QE κ ∆ ρn ζ π ΓΓ + ( ω − ω ) (D34)for ϑ =
0, with ζ = π ∫ u ( x , y , z ) d z / ̵ h a geomet-rical factor that accounts for the overlap between themicrowave mode and the molecular beam and κ =∑ α ∑ β ∣ µ βαz ∣ a measure of the strength of the rotationaltransition. This is a Lorentzian, like those plotted in § III.4To approach this limit in practice would require in par-ticular that the molecules pass through a light mirror,which might be difficult to realise without significantlycompromising the optical cavity.
FIG. 10. Lineshapes for different molecular pulse geometriesand opposite circular polarisations of the light.
In general (D33) must be integrated numerically. Ourpreliminary investigations here reveal that under morerealistic operating conditions the spectrometer yields alineshape resembling a Lorentzian but with a significant broadening on one side; that closest to the microwaverotational transition frequency as it would appear in theabsence of the light. This asymmetric broadening occursbecause those molecules removed from the most intenseregions of the light experience weaker (but not stronger)energy shifts due to the light. Chirally sensitive informa-tion can be extracted from these lineshapes in spite oftheir unusual forms, although for some tasks such as thedetermination of enantiomeric excess a fitting proceduremight be required.To illustrate these ideas let us consider ω βα ( r ′ )/ π ={ . + ( . + . σ ) exp [−( x ′ + y ′ )/ γ ]} × s − , which is representative of a rea-sonably magic rotational transition, be it naturallyoccurring or refined, for a chiral molecule with asignificant but not exceptional chiroptical response,comparable to that of ibuprofen say. Depicted in FIG.10 are fair numerical approximations to R [ ˜ S ( ω )] from(D33) for: (a) µ = . θ = . × − , which wouldrequire in particular that the molecules pass through alight mirror; (b) µ = . × − and θ = . × − , whichshould be viable without compromising the stability ofthe optical cavity; (c) µ = . × − and θ = . × − ,which should also be viable without compromising thestability of the optical cavity but corresponds to aweaker angular collimation than is desirable. In all threecases y = . × − m, γ = . × − m, σ = ± . / π = . × s − , v = . × m.s − , h = . × − m, p = . τ = . × − s, ω c / π = . × s − , d = . × − m and ϑ = .
00 in particular with op-eration in the TEM microwave mode. The lineshapesare strongly dependent upon both the tilt ( µ ) andangular collimation ( θ ) of the molecular pulses, as wemight expect. The σ -dependent splitting is neverthelessapparent in all three cases, however.We note finally that calculating the real part of theFourier transform ˜ S ( ω ) might not be the most transpar-ent way of interpreting the free induction decays recordedin chiral rotational spectroscopy. A different function inwhich the unusual geometry inherent to the spectrometeris compensated for might be calculated instead, perhapsyielding clearer chiral rotational spectra without furtherwork. Appendix E: Signal-to-noise ratio
In the present appendix we estimate the signal-to-noiseratio that might be attained for the chiral rotational spec-trometer discussed in § IV and Appendix D.The precise value of the signal-to-noise ratio will de-pend, of course, upon the quality of the components usedand the care with which the spectrometer is built andmeasurements are taken, as well as the nature of the sam-ple and the rotational line under consideration. We cannevertheless give some idea here of what should be pos-sible by recalling that a comparable standard cavity en-hanced Fourier transform microwave spectrometer gave5a signal-to-noise ratio of 5 × − s − / √ ∆ t with ∆ t thetotal recording time for a measurement rate of 5 × s − of the J = ← O C S moleculesthat exist with a natural abundance of 0 . ∶
20 Ne:He carrier gas[123]. We extrapolate from this a signal-to-noise ratio ofSNR ′ = ( . × − ) − ( × − s − / √ ∆ t )= × s − / √ ∆ t (E1)for a pure sample, containing a single form of molecule( O C S). Our approach now is to estimate the signal-to-noise ratio SNR of the chiral rotational spectrometerby scaling the signal-to-noise ratio SNR ′ of the standardrotational spectrometer in accord with the following con-siderations. (i) There will be fewer molecules in the chiral rota-tional spectrometer than in the standard rotationalspectrometer, due in particular to the sharp an-gular collimation of the molecular pulses. This isperhaps the single most significant cause of signalreduction. Let us introduce f = ∫ θ cos θ sin θ d θ ∫ π / cos θ sin θ d θ = − cos θ (E2)as the ratio of the ‘solid angle’ occupied by themolecular pulses in the chiral rotational spectrom-eter to the analogous quantity in the standard ro-tational spectrometer, with each of these ‘solid an-gles’ weighted by the number density distribution ofthe molecular pulses, assuming p =
2. For θ = − we obtain f = − . Further reductions in thenumber of molecules seem likely, for example if themolecules are of lower volatility than O C Sand so cannot be sampled at the same density orif it not possible to prepare a pure sample due tothe existence of multiple stereoisomers. Let us in-troduce f = N N ′ (E3)as the ratio of the number density N of moleculesin the chiral rotational spectrometer to the analo-gous quantity N ′ = m − for the standard rota-tional spectrometer. We might hope that 10 − ≤ f ≤ for first demonstrations of chiral rota-tional spectroscopy but recognise that significantlysmaller values of f may be encountered undermany circumstances of practical interest. (ii) The fraction of molecules in the rotational statesof interest and also the coupling of the rotationaltransition of interest to the microwave pulses will ingeneral differ, of course, from the particular values described above for the standard rotational spec-trometer. Let us introduce f = ∆ ρ ∆ ρ ′ (E4)as the ratio of the difference ∆ ρ in the equilib-rium populations of the rotational states of interestin chiral rotational spectroscopy to the analogousquantity ∆ ρ ′ for the standard rotational spectrom-eter. The molecules of interest in chiral rotationalspectroscopy will be larger and more complicatedthan O C S, with lower rotational energies andmore rotational states such that 10 − ≤ f ≤ − ,perhaps. Let us introduce f = ( µµ ′ ) (E5)as the square of the ratio of the transition electricdipole-moment moment µ for the rotational statesof interest in the chiral rotational spectrometer tothe analogous quantity µ ′ = − m.s.A for thestandard rotational spectrometer. The moleculesof interest in chiral rotational spectroscopy willlikely have larger permanent electric-dipole mo-ments than O C S such that 10 ≤ f ≤ ,perhaps. Let f account for additional reductionsor perhaps enhancements in signal strength suchas those due to the choice of carrier gas, which cansee the signal strength vary by orders of magnitude[123]. We would hope that f = under wellchosen operating conditions. (iii) The rate at which measurements are taken in astandard cavity enhanced Fourier transform mi-crowave spectrometer is limited physically by thetime taken to evacuate molecules between succes-sive measurements [116, 118, 122, 123]. In a chi-ral rotational spectrometer measurements might betaken instead at an increased rate owing to thesmaller number of molecules present, perhaps upto 7 . × s − [107]. This is approaching con-tinuous operation of the pulsed supersonic expan-sion nozzle, with measurements made around onceevery ten free induction decay times, assumingΓ / π = . × s − say. Let us introduce f = √ ΛΛ ′ (E6)as the square root of the ratio of the measurementrate Λ in the chiral rotational spectrometer to theanalogous quantity Λ ′ = × s − in the standardrotational spectrometer. We might hope for an en-hancement of 10 ≤ f ≤ × here.Finally, we take SNRSNR ′ = f f f f f f . (E7)6For f = − , 10 − ≤ f ≤ , 10 − ≤ f ≤ − ,10 ≤ f ≤ , f = and 10 ≤ f ≤ we obtain10 − ≤ SNR / SNR ′ ≤ − , or 10 − s − / √ ∆ t ≤ SNR ≤ − s − / √ ∆ t . This suggests in turn that a very agree-able chiral rotational spectrum could be obtained for arecording time of a few hours under favourable operatingconditions, as discussed in §§