Infrared spectroscopy of endohedral water in C_{60}
A. Shugai, U. Nagel, Y. Murata, Yongjun Li, S. Mamone, A. Krachmalnicoff, S. Alom, R. J. Whitby, M. H. Levitt, T. Room
IInfrared spectroscopy of endohedral H O in C . A. Shugai, U. Nagel, Y. Murata, Yongjun Li, S. Mamone, A. Krachmalnicoff, S. Alom, R. J. Whitby, M.H. Levitt, and T. Rõõm a) National Institute of Chemical Physics and Biophysics, Akadeemia tee 23, 12618 Tallinn,Estonia Institute for Chemical Research, Kyoto University, Kyoto 611-0011, Japan Department of Chemistry, Columbia University, New York, New York 10027,USA b)4)
School of Chemistry, Southampton University, Southampton SO17 1BJ, United Kingdom (Dated: 15 February 2021)
Infrared absorption spectroscopy study of endohedral water molecule in a solid mixture of H O@C and C wascarried out at liquid helium temperature. From the evolution of the spectra during the ortho - para conversion process,the spectral lines were identified as para - and ortho -H O transitions. Eight vibrational transitions with rotational sidepeaks were observed in the mid-infrared: ω , ω , ω , 2 ω , 2 ω , ω + ω , ω + ω , and 2 ω + ω . The vibrationalfrequencies ω and 2 ω are lower by 1.6% and the rest by 2.4%, as compared to free H O. A model consisting of arovibrational Hamiltonian with the dipole and quadrupole moments of H O interacting with the crystal field was usedto fit the infrared absorption spectra. The electric quadrupole interaction with the crystal field lifts the degeneracy of therotational levels. The finite amplitudes of the pure v and v vibrational transitions are consistent with the interaction ofthe water molecule dipole moment with a lattice-induced electric field. The permanent dipole moment of encapsulatedH O is found to be 0 . ± . O in the molecular cage of C was observed at 110 cm − (13.6 meV). I. INTRODUCTION
Endohedral fullerenes consist of atoms or molecules fullyencapsulated in closed carbon cages. The remarkable syn-thetic route known as “molecular surgery" has led to the syn-thesis of atomic endofullerenes He@C and Ar@C , andseveral molecular endofullerene species, including H @C ,H O@C , HF@C , CH @C , and their isotopologs .It is well established that these endohedral molecules donot form chemical bonds with the carbon cage and rotatefreely . The rotation is further facilitated by the nearlyspherical symmetry of the C cage. The thermal andchemical stability of A@C opens up the unique possibil-ity of studying the dynamics and the interactions of a smallmolecule with carbon nano-surfaces. The trapping potentialof dihydrogen H has been described with high accuracy us-ing infrared (IR) spectroscopy , inelastic neutron scatter-ing (INS) and theoretical calculations . The non-sphericalshape of the small molecule and the quantized translationalmotion of its center of mass leads to coupled rotational andtranslational dynamics .The water-endofullerene H O@C is of particular interest.The encapsulated water molecule possesses rich spatial quan-tum dynamics. It is an asymmetric-top rotor, supports threevibrational modes, displays nuclear spin isomerism ( para - and ortho -water) and has an electric dipole and quadrupole mo-ment.Low-temperature dielectric measurements on solidH O@C show that the electric dipole moment of the encap-sulated water is reduced to 0 . ± .
05 D from the free water a) Electronic mail: toomas.room@kbfi.ee b) Present address Merck & Co., 126 E Lincoln Ave, Rahway, New Jersey,07065, USA. value 1.85 D. The C carbon cage responds to the endohedralwater molecule with a counteracting induced dipole, resultingin the lower total dipole moment .The dynamics of isolated or encapsulated single watermolecules have been studied before in other environments,such as noble gas matrices, solid hydrogen and liquid heliumdroplets. Although the trapping sites in these matrices havehigh symmetry and allow water rotation, these systems ex-ist only at low temperature or for a very short time .Water has also been studied in crystalline environments withnano-size cavities. However, in this case, the interactionswith the trapping sites inhibit the free rotation of the watermolecules .Several spectroscopic techniques have been used to studyH O@C , including nuclear magnetic resonance (NMR),INS and IR and time-domain THz spectroscopy .The low-lying rotational states of the encapsulated moleculeare found to be very similar to those of an isolated watermolecule, with the notable exception of a 0.6 meV splittingin the J = . This indicates that the localenvironment of the water molecule in H O@C has a lowersymmetry than the icosahedral point group of the encapsulat-ing C cage. The splitting has been attributed to the interac-tion between the electric quadrupole moment of H O and theelectric field gradients generated by the electronic charge dis-tribution of neighbouring C molecules . The merohedraldisorder present in solid C leads to two C sites with dif-ferent quadrupolar interactions. This merohedral disorder alsoleads to splittings of the IR phonons in solid C
60 34 . There isalso evidence from dielectric measurements that merohedraldisorder leads to electric dipolar activity in solid C .Confined molecules exhibit quantization of their transla-tional motion (“particle in a box"), in addition to their quan-tized rotational and vibrational modes. Quantized transla-tional modes have been observed at 60 to 70 cm − for wa-ter in noble gas matrices . However, comparatively lit- a r X i v : . [ phy s i c s . c h e m - ph ] F e b tle is known about the centre-of-mass translational mode ofH O@C . The fundamental frequency of the water trans-lation mode in H O@C has been predicted to occur at ∼
160 cm − . This relatively high frequency reflects therather tight confinement of the water molecule in the C cage.The energy level separation between the ground para rota-tional state and the lowest ortho rotational state in H O@C has been determined to be 2.6 meV (21 cm − ) by INS . Thisenergy level separation corresponds to a temperature of 28 K.The full thermal equilibration of H O@C at temperaturesbelow 30K therefore requires the conversion of ortho into para water. This conversion process takes between tens ofminutes to several hours below 20K in H O@C .The spin-isomer conversion is much faster at ambient tem-perature, with a time constant of about 30 s reported forH O@C dissolved in toluene .In this paper, we report on a detailed low-temperature far-and mid-IR spectroscopic study of H O@C and C solidmixtures. The IR technique allows us to measure the fre-quencies of rotational, vibrational and translational modes andfrom the line intensities to determine the dipole moment of en-capsulated water. In addition, the IR spectra reveal the inter-action of endohedral water with the electrostatic fields presentin solid C .The rest of this paper is organized as follows. Section II dis-cusses the sample preparation, the recording procedures of theIR spectra, and the determination of the IR absorption cross-sections for different filling factors, temperatures and ortho - para ratios. The quantum mechanical vibrating rotor modelfor the encapsulated water molecules is introduced in SectionIII A. We include in this model the interactions between theH O electric dipole and quadrupole moments with the elec-trostatic fields present in solid C . The theory of the IR lineintensities is presented in Section III B. Section IV presentsthe measured IR spectra and the fitting of this data by thequantum-mechanical model. The results are discussed in Sec-tion V, followed by a summary in Section VI. The Appendixcontains more detailed theory for the interaction of the watermolecules with the electrostatic fields and the infrared radia-tion, and more details on the fitting of the experimental databy the quantum-mechanical model. II. METHODSA. Sample preparation H O@C was prepared by multi-step synthetic routeknown as “molecular surgery" . The H O-filled (numberdensity N • ) and empty (number density N ◦ ) C were mixedand co-sublimed to produce small solvent-free crystals with afilling factor f = N • / ( N • + N ◦ ) . Five samples with filling fac-tors f = . , . , . , .
18, and 0.80 were studied. Thepowdered samples were pressed into pellets under vacuum.The diameter of sample pellets was 3 mm and the thickness d varied from 0.2 mm to 2 mm. The thinner samples were usedin the mid-IR because of light scattering in the powder sam-ple. Samples were thicker for lower filling factors and thinner for higher filling factors to avoid the saturation of absorptionlines in the far-IR. B. Measurement techniques
The far-IR measurements were done with a Martin-Pupletttype interferometer and He cooled bolometer from 5 to200 cm − as described in Ref. 11. The IR measurementsbetween 600 cm − and 12000 cm − were performed with aninterferometer Vertex 80v (Bruker Optics) as described inRef. 12.Two methods were used to record the H O@C absorptionspectra. Method 1.
The intensity through the sample, I s , was ref-erenced to the intensity through a 3 mm diameter hole, I .The sample was allowed to reach ortho - para thermal equilib-rium at a temperature of 30 or 45 K, and the temperature wasrapidly reduced to 10 or to 5 K. The sample spectrum I s ( t = ) was recorded immediately after the temperature jump. Sincethe ortho - para conversion process is slow, the ortho frac-tion was assumed to be preserved during the T jump, corre-sponding to the high temperature ortho fraction n o ≈ . α was calculated from the ratio Tr = I s ( t = ) / I as α ( t = ) = − d − ln [( − R ) − Tr ] wherefactor ( − R ) with R = ( η − ) ( η + ) − corrects for thelosses of radiation, one reflection from the sample front andone from the back face. The refraction index of solid C wasassumed to be given by η = . To identify para - and or-tho -water absorption peaks the difference of two spectra wascalculated, ∆ α = α ( t = ) − α ( ∆ t ) , where α ( ∆ t ) is the ab-sorption spectrum measured after the waiting time ∆ t . Onlythe para - and ortho -H O@C peaks show up in the differen-tial absorption spectra, with the para - and ortho -H O@C peak amplitudes having different signs, ortho positive and para negative. This method was used for the far- and mid-IR part of the spectrum. Method 2.
The sample was allowed to reach ortho - para thermal equilibrium at a temperature of 30 or 45 K, leadingto an ortho -rich state, as in the first method. The temper-ature was rapidly reduced to 10 or to 5 K and a series ofspectra recorded at intervals of a few minutes starting im-mediately after the T jump, and continued until the ortho - para equilibrium was reached. The differential absorption ∆ α = − d − ln [ I s ( ) / I s ] was calculated, where I s ( ) is thespectrum recorded immediately after the T jump and I s is thespectrum recorded when the low temperature equilibrium wasreached. The equilibrium ortho fraction is approximately 0 . C. Line areas and absorption cross-sections
The absorption line area A ( k ) ji was determined by fitting themeasured absorption α ( k ) ji ( ω ) with Gaussian line shape. | i (cid:105) and | j (cid:105) are the initial and final states of the transition and k denotes para ( k = p) or ortho ( k = o) species. From theseexperimental line areas, A ( k ) ji , a temperature and para ( ortho )fraction independent line area A ( k ) ji was calculated, A ( k ) ji ( f ) = A ( k ) ji ( f ) n k ( p ( k ) i − p ( k ) j ) . (1)The population difference of initial and final states, p ( k ) i − p ( k ) f ,is given by the sample temperature T , while the para ( ortho )fraction n k depends on the history of the sample because of the ortho - para conversion process. The normalized absorptionline area (cid:104) A ( k ) ji (cid:105) was determined from the linear fit of A ( k ) ji ( f ) for each absorption line, A ( k ) ji ( f ) = f (cid:104) A ( k ) ji (cid:105) . (2)Thus, the normalized absorption line area (cid:104) A ( k ) ji (cid:105) is the absorp-tion line area of a sample with a filling factor f = n k = p ( k ) i =
1. We used (cid:104) A ( k ) ji (cid:105) to calculate a syntheticexperimental spectrum for the spectral fit with the quantummechanical model, Section IV B.Furthermore, to compare the absorption cross-sections ofH O@C in solid C and free H O, a normalized absorptioncross-section (cid:104) σ ( k ) ji (cid:105) was obtained as (cid:104) σ ( k ) ji (cid:105) = (cid:18) √ ηη + (cid:19) σ ( k ) ji n k ( p ( k ) i − p ( k ) j )= (cid:18) √ ηη + (cid:19) (cid:104) A ( k ) ji (cid:105) N C , (3)where η is the index of refraction of solid C and σ ( k ) ji isthe absorption cross-section of an endohedral water molecule,Eq. (20) in Section III B. This H O@C absorption cross-section can be compared to the free water normalized cross-section (cid:104) σ ( k ) ji (cid:105) = σ ( k ) ji n − k ( p ( k ) i − p ( k ) j ) − , where p ( k ) i − p ( k ) j and n k ( T ) are given by the temperature of the water vapour in theexperiment reporting σ ( k ) ji . III. THEORYA. Quantum mechanical model of H O@C : Confinedvibrating rotor in an electrostatic field We use the following Hamiltonian to model endohedral wa-ter molecule in solid C : H = H M + H ES + H T , (4)where H M = H v + H rot is the free-molecule rovibrationalHamiltonian and H ES is the electrostatic interaction of H Owith the surrounding electric charges. The translationalHamiltonian H T consists of water center of mass kinetic andpotential energy in the molecular cavity of C molecule. pppooo pppooo ppooopo0 p a)b) c) + + + FIG. 1. (a) Molecule-fixed coordinate frame M = { x , y , z } and theaxes of principal moments of inertia, { a , b , c } . (b) Vibrations: v -symmetric stretch, v - symmetric bend and v - asymmetric stretch.(c) para - and ortho -water rotational energy levels in the ground andexcited vibrational states, see Section III A 5, and the rovibrationalIR transitions (arrows) between the levels. The rotational states arelabelled by J K a K c and p ( para ) and o ( ortho ). The IR transitions arebetween the para or between the ortho states. 1, 3, and 5 are ortho -H O transitions and 2, and 4 are para -H O transitions. Transitions 1and 2 are forbidden for a free water molecule. The transitions wherethe one-quantum excitation of the asymmetric stretch vibration v is involved, are numbered 6 ( para -H O), and 7 and 8 ( ortho -H Otransitions). The rotational and translational far-IR transitions in theground vibrational states are shown in the inset to Fig. 2.
We neglect couplings between vibrational modes and be-tween vibrational and rotational modes. Also, the couplingbetween translational motion and rotations is neglected. In H ES , terms describing the coupling of the solid C crystalfield to the electric dipole and quadrupole moment of H O areincluded.The fitting of IR absorption spectra, Section IV B, is donewith the Hamiltonian where the translational part is excluded: H = H M + H ES . (5)We employ three coordinate frames. The space-fixed coor-dinate frame is denoted A . M = { x , y , z } is the molecule-fixedcoordinate frame, Fig. 1. The Euler angles Ω A → M trans-form A to M . The crystal coordinate frame is C = { x (cid:48) , y (cid:48) , z (cid:48) } with the z (cid:48) axis along the three-fold symmetry axis of the S point group, the symmetry group of C site in solid C . TheEuler angles Ω A → C transform A to C and Ω C → M transform C to M . The coordinate systems A and C are used because theradiation interacting with the molecule is defined in the space-fixed coordinate frame A while the local electrostatic fields aredefined by the crystal coordinate frame C , which in a powdersample has a uniform distribution of orientations relative tothe space-fixed frame A .
1. Vibrations H O has three normal vibrations: the symmetric stretch ofO–H bonds, quantum number v , the bending motion of theH–O–H bond angle, v , and the asymmetric stretch of O–Hbonds, v , as sketched in Fig. 1(b). The vibrational stateis denoted | v (cid:105) where the symbol v denotes the three vibra-tional quantum numbers, v ≡ v v v , each of which takes val-ues v i ∈ { , , , . . . } . The vibrational energy for a harmonicvibrational potential is E v = ∑ i = ω i ( v i + ) , v i = , , , . . . , (6)where ω i is the vibrational frequency of the i th vibrationmode, i ∈ { , , } and [ ω i ] = cm − .
2. Rotations H O has the rotational properties of an asymmetric top withprincipal moments of inertia I a < I b < I c . The rotationalstates are indexed by three quantum numbers J K a K c where J = , , . . . is the rotational angular momentum quantumnumber. K a and K c are the absolute values of the projectionof J onto the a and c axes, in the limits of a prolate ( I a = I b )and an oblate ( I b = I c ) top respectively; K a , K c ≤ J . Eachrotational state | J K a K c , m (cid:105) is ( J + ) -fold degenerate, where m ∈ {− J , − J + , . . . J } is the projection of J onto the z (cid:48) axisof the crystal coordinate frame C .The energies and the wavefunctions, | J K a K c , m (cid:105) , of the freerotor Hamiltonian H rot depend on the rotational constants ofan asymmetric top, A v > B v > C v , H rot = A v ˆ J a + B v ˆ J b + C v ˆ J c , (7)where ˆ J i are the components of the angular momentum oper-ator ˆ J along the principal directions a , b , and c . The index v labels the rotational constants in vibrational state | v (cid:105) .A molecule-fixed coordinate system M (axes x , y and z ,Fig. 1) with its origin at the nuclear centre of mass is definedwith the following orientations relative to the principal axesof the inertial tensor: x (cid:107) b , y (cid:107) c , and z (cid:107) a where y is perpen-dicular to the H–O–H plane and x points towards the oxygenatom. The rotational wavefunctions of an asymmetric top inthat basis are : | J , k , m , ±(cid:105) = ( | J , k , m (cid:105) ± | J , − k , m (cid:105) ) / √ , (8) | J , , m , ±(cid:105) ≡ | J , , m (cid:105) if k = , where k is the projection of J on the a axis, axis z of M ; for k > k ∈ { , , . . . , J } and each state is doubly degenerate.Here, | J , k , m (cid:105) denotes a normalized rotational function | J , k , m (cid:105) = (cid:114) J + π (cid:2) D Jmk ( Ω C → M ) (cid:3) ∗ , (9)where the Euler angles Ω C → M transform the crystal-fixed co-ordinate frame C into the molecule-fixed coordinate frame M and D Jmk ( Ω ) is the Wigner rotation matrix element or theWigner D -function .The correspondence between the asymmetric top wave-functions | J , k , m , ±(cid:105) , Eq. (8), and asymmetric top wavefunc-tions | J K a K c , m (cid:105) , is given in . The latter notation of wavefunc-tions is useful for the symmetry analysis and can be used torelate the wavefunction to para and ortho states of the watermolecule, see Section III A 5.
3. Electrostatic interactions
We assume two contributions to the electrostatic interac-tion: H ES = H Q + H ed , (10)denoting the coupling of the quadrupole and dipole momentsof H O to the corresponding multipole fields created by thesurrounding charges.
Quadrupolar interaction.
It was shown by Felker et al. that C molecules in neighbouring lattice sites generate anelectric field gradient at the centre of a given C molecule.In H O@C , the electric field gradient couples to the electricquadrupole moment of the water molecule, lifting the three-fold degeneracy of the J = ortho -H O rotational groundstate .The quadrupolar Hamiltonian may be expanded in rank-2spherical tensors as follows : H Q = ∑ m = − ( − ) m V ( ) − m Q Cm , (11)where { V ( ) m } are the spherical components of the electric fieldgradient tensor and { Q Cm } is the quadrupole moment of thewater molecule, both expressed in the crystal-fixed coordinateframe C .The experimental value of the H O quadrupole mo-ment in the molecule-fixed coordinate frame M is given by { Q xx , Q yy , Q zz } = {− . , − . , . } e.s.u × cm . Since | Q xx | << | Q yy | , | Q zz | , it holds that Q zz ≈ − Q yy , and we mayapproximate the water quadrupole moment in spherical coor-dinates as follows: { Q Mm } = { ( Q xx − Q yy ) , , (cid:114) Q zz , , ( Q xx − Q yy ) }≈ Q zz { , , (cid:114) , , } , m = − , . . . , + . (12)The site symmetry of the C molecule in solid C is S ,with the three-fold symmetry axis along the cubic [ ] axis,which is chosen here to be the z (cid:48) axis of the crystal coordinateframe C . The spherical tensor component in the frame C is { V ( ) Cm } = V Q { , , , , } , (13)where m ∈ {− , − , , , } , transforms like the fully sym-metric A g irreducible representation of the point group S .After the transformation of the quadrupole moment (12) fromthe H O-fixed molecular frame M to the crystal frame C (seeEq. A5), the quadrupolar Hamiltonian (11) is given by: H Q = V Q Q zz (cid:34)(cid:114) (cid:2) D ( Ω C → M ) (cid:3) ∗ (14) + (cid:2) D , − ( Ω C → M ) + D ( Ω C → M ) (cid:3) ∗ (cid:21) . Dipolar interaction.
Dielectric measurements of solid C have provided evidence for the existence of electric dipoles insolid C . We assume these electric dipoles can be sourceof an electric field in the C cage center.Consider a crystal electric field E with spherical coor-dinates { E , φ E , θ E } in the crystal-fixed frame C , see Ap-pendix A. For simplicity, we assume a homogeneous crystalfield with uniform orientation in the crystal-fixed frame. Theinteraction of the electric dipole moment with the electric fieldis given by H ed = − ∑ m = − ( − ) m E E − m µ Em (15) = − ∑ m (cid:48) = − E (cid:2) D m (cid:48) ( Ω E → C ) (cid:3) ∗ × (cid:34) ∑ m (cid:48)(cid:48) = − (cid:2) D m (cid:48) m (cid:48)(cid:48) ( Ω C → M ) (cid:3) ∗ µ Mm (cid:48)(cid:48) (cid:35) , where the dipole moment in the molecule-fixed frame M isgiven by { µ Mm } = µ x √ {− , , } , m ∈ {− , , } (16)where µ x is the permanent dipole moment of water in theCartesian coordinates of frame M , Fig. 1(a). Since thereare no other anisotropies than the axially symmetric electricfield gradient tensor, the angle φ E is arbitrary and we choose φ E = J ≤ | (cid:105) and for the threeexcited vibrational states | (cid:105) , | (cid:105) , and | (cid:105) . The groundstate and the three excited vibrational states are assumed tohave independent rotational constants A v , B v , and C v , where v = , ,
010 or 001.After separation of coordinates, see Appendix B 2, thequadrupole and dipole moments in equations (14) and (15) arereplaced by their expectation values, (cid:104) v | Q M | v (cid:105) , (cid:104) v | Q M ± | v (cid:105) and (cid:104) v | µ M ± | v (cid:105) , in the ground and in the three excited vibra-tional states. We assume for simplicity that the dipole andquadrupole moments of H O are independent of the vibra-tional state | v (cid:105) .
4. Confined water translations: spherical oscillator
The translational motion is the center of mass motion and isquantized for a confined molecule. The high icosahedral sym-metry of the C cavity is close to spherical symmetry and therefore the translational motion of trapped molecule can bedescribed by the three-dimensional isotropic spherical oscil-lator model . For simplicity we write the potential in theharmonic approximation : V ( R ) = V R , (17)where R is the displacement of H O center of mass from theC cage center. The C cage is assumed rigid and its centerof mass is fixed.The frequency of the spherical harmonic oscillator is: ω = (cid:112) V / m , (18)where m is the mass of a molecule moving in the potential and [ ω ] = rad s − . The energy of spherical harmonic oscillator isquantized, E N = ¯ h ω ( N + ) , (19)where N is the translational quantum number, N ∈{ , , , . . . } . The orbital quantum number L takes values L = N , N − , . . . ( ) for N odd (even). Energy of the harmonicspherical oscillator does not depend on L and in isotropicapproximation there is an additional degeneracy of each E N level in quantum number M L , taking 2 L + M L ∈{− L , − L + , . . . L } .
5. Nuclear spin isomers: para and ortho water
The Pauli principle requires that the total quantum state isantisymmetric with respect to exchange of the two protonsin water, which as spin-1/2 particles are fermions. This con-straint leads to the existence of two nuclear spin isomers, withtotal nuclear spin I = para -H O) and I = ortho -H O),and different sets of rovibrational states. The antisymmet-ric nature of the quantum state has consequences on the IRspectra: only para to para and ortho to ortho transitions areallowed.The ortho -H O states have odd values of K a + K c while the para -H O states have even values of K a + K c in the groundvibrational state | (cid:105) , see Fig. 1(c). The allowed rotationaltransitions are depicted in the inset to Fig. 2(a). The samerule applies to the excited vibrational states | (cid:105) and | (cid:105) ,Fig. 1(c), upper left part. However, the rules are inverted forthe states | (cid:105) , | (cid:105) and | (cid:105) , which involve one-quantumexcitation of the asymmetric stretch mode v . In these cases , para -H O has odd values of K a + K c , while ortho -H O haseven values of K a + K c , Fig. 1(c), upper right part.The energy difference between the lowest para rotationalstate | (cid:105) and the lowest ortho rotational state | (cid:105) is2.6 meV (28 K) . Above 30 K the ratio of ortho and para molecules is n o / n p ≈
3. Hence, if the sample is cooledrapidly to 4 K, the number of para molecules slowly growsin the subsequent time interval, while the number of the ortho molecules slowly decreases to the thermal equilibrium value n o ≈
0. The full conversion takes several hours . B. Absorption cross-section of H O@C The strengths of the transitions between the rotational statesof a polar molecule are determined by the permanent elec-tric dipole moment of the molecule and by the electric fieldof the infrared radiation, corrected by the polarizability of themedium. In principle, the polarizability χ of the solid dependson the fraction f of C cages which contain a water molecule, χ ( f ) = χ C + f χ H O@C . However, we found that withinthe studied range of filling factors, f = . O@C was independent of f . Hence,only the polarizability of solid C is relevant, χ ≈ χ C , andthe problem is similar to the optical absorption of an isolatedimpurity atom in a crystal .Following Ref. 46, the electric field at the molecule em-bedded into medium with an index of refraction η is E eff = E ( η + ) /
3, where E is the electric field of radiation in thevacuum. The refractive index of solid C is η = , andhence E eff = E .In the following discussion, we use the index k ∈ { o , p } toindicate the ortho or para nuclear spin isomers. The absorp-tion cross-section for a given nuclear spin isomer k , includ-ing the effective field correction, is given by σ ( k ) ji = N − k (cid:90) Line α ( k ) ji ( ω ) d ω (20) = π h ε c η (cid:18) η + (cid:19) ω ( k ) ji (cid:16) p ( k ) i − p ( k ) j (cid:17) S ( k ) ji , where c is the speed of light in vacuum and ε the permittivityof vacuum; SI units are used and the frequency ω is in numberof waves per meter, [ ω ] = m − . The integral in (20) is the areaof the absorption line of the transition from the state | i (cid:105) to | j (cid:105) .The square of the electric dipole matrix element is given by S ( k ) ji = ∑ σ = − | (cid:104) j | µ C σ | i (cid:105) | , (21)where | i (cid:105) and | j (cid:105) are the eigenstates with corresponding ener-gies E ( k ) i and E ( k ) j . The symbol µ C σ denotes the dipole momentcomponents of a water molecule in the crystal-fixed frame C .This form of S ( k ) ji is valid for random orientation of crystals inthe powder sample and does not depend on the polarization oflight, see Appendix B.The absorption cross-section σ ( k ) ji was evaluated separatelyfor para and ortho water. The concentration of moleculesis N k = f N C n k , where n k is the ortho (or para ) fractionand n p + n o =
1. The filling factor is denoted by f and thenumber density of C molecules in solid C is given by N C = . × cm − . p ( k ) i and p ( k ) j are the probabilities that the initial and finalstates are thermally populated, p ( k ) n = (cid:16) Z ( k ) (cid:17) − exp (cid:32) − E ( k ) n − E ( k ) k B T (cid:33) , (22) where the statistical sum is Z ( k ) = ∑ n exp (cid:32) − E ( k ) n − E ( k ) k B T (cid:33) (23)and E ( k ) is the ground state energy of para ( ortho ) molecules.At the low temperatures considered in this work, only the vi-brational ground state is significantly populated. Thus, thesignificantly thermally populated states are rotational states inthe ground vibrational states, which can be written as linearcombinations of basis states in Eq. (8). Since we expect thatthe water molecule is not in a spherically symmetric environ-ment in H O@C , the degeneracy in quantum number m islifted, in general. Therefore, p ( k ) n is the thermal population ofa non-degenerate rotational state and the eigenstates | i (cid:105) and | j (cid:105) in Eq. (21) include all possible m values for a given J .When thermal equilibrium is reached between the para and ortho water, the fraction of nuclear spin isomer k is n k ( T ) = g ( k ) Z ( k ) g ( p ) Z ( p ) + g ( o ) Z ( o ) . (24)For spin isomer k , the nuclear spin degeneracy g ( k ) = I + I = para and I = ortho .The absorption line areas are calculated from Eq. (20),where the matrix elements in Eq. (21) are between the eigen-states | Φ v rot (cid:105) of the Hamiltonian given by Eq. (5). After sepa-ration of coordinates (see Appendix B 2), the matrix elementsin the crystal-fixed coordinate frame C are (cid:104) j | µ C σ | i (cid:105) = (25) = ∑ σ (cid:48) = − (cid:68) Φ v (cid:48) rot (cid:12)(cid:12)(cid:12) D σσ (cid:48) ( Ω C → M ) ∗ (cid:12)(cid:12) Φ (cid:11) (cid:10) v (cid:48) (cid:12)(cid:12) µ M σ (cid:48) | (cid:105) , where the initial state is | i (cid:105) = | (cid:105) (cid:12)(cid:12) Φ (cid:11) and | Φ v rot (cid:105) arethe linear combinations of states (8). The dipole moments (cid:104) v (cid:48) | µ M σ (cid:48) | (cid:105) are given by Eq. (B13) and (B14) in Cartesiancoordinates. IV. RESULTS AND INTERPRETATION OF SPECTRAA. Spectra
The water IR absorption lines were identified unambigu-ously by taking advantage of the slow ortho - para conversionat low temperature. After rapid cooling from 30 K to 5 K, theslow ortho - para conversion causes the intensity of the para lines to slowly increase, while the intensity of the ortho linesdecreases. The water absorption lines are readily identified,and assigned to one of the two spin isomers, by taking thedifference between spectra acquired shortly after cooling andspectra acquired after an equilibration time at the lower tem-perature.A group of H O lines, numbered 3, 4, and 5, is seen below60 cm − , Fig. 2(a). These far-IR absorption lines have beenreported earlier and correspond to the rotational transitions of
05 01 0 01 5 0 0 T p T o 1 T o 2 b ) T = 5 K5 % a (cm-1) W a v e n u m b e r ( c m - 1 ) T o 2 T p a ) T o 1 N = 1 N = 0 T = 5 K1 8 % FIG. 2. Far-IR absorption spectra of H O@C at 5 K. (a) Spec-trum α ( ) was measured after the temperature jump from 30 K to5 K (black) and the difference ∆ α = α ( ) − α ( ∆ t ) was measured ∆ t =
44 h later (blue). The sample filling factor f = .
05. Water ro-tational transitions corresponding to the absorption lines numbered 3and 5 ( ortho -water) and 4 ( para -water), are shown in the inset. (b)Spectrum α ( ) was measured after the temperature jump from 30 Kto 5 K (black) and the difference ∆ α = α ( ) − α ( ∆ t ) was measured ∆ t = f = .
18. The trans-lational transitions N = → N = para (T p ) and ortho (T o1 , T o2 ) H O@C are shown in the inset to panel (a). N is the quantumnumber of the spherical harmonic oscillator, Eq. (19). H O in the C cage . The rotational energy levels involvedare shown in the inset to Fig. 2 (a). Lines 3 and 5 are ortho wa-ter rotational transitions starting from the ortho water groundstate | (cid:105) . Line 4 is the para water transition from the groundrotational state | (cid:105) . No other rotational transitions were ob-served at 5 K which is consistent with the selection rules forthe electric dipole allowed rotational transitions from states | (cid:105) and | (cid:105) .Further H O lines are observed around 110 cm − , Fig. 2(b),and in six spectral regions above 600 cm − as shown in Fig-ures 3 to 6. Below, we address each wavenumber range sepa-rately and assign the spectral lines to the transitions shown inthe energy schemes of Fig. 1(c) and Fig. 2 (a). The absorptionlines associated with transitions of free water, labelled 1 to 8,are listed in Table I. Line assignments are supported by the re-sults of the spectral fitting using the model of a vibrating rotorin a crystal field.
1. Translational transitions
A group of absorption lines around 110 cm − is shown inFig. 2 (b). These lines do not correspond to any known wa-ter rotational transitions. We assign these peaks to the trans-lational transitions ( N = → N =
1) of para - and ortho -H O, corresponding to the quantized centre-of-mass vibra-tional motions of the water molecules in the encapsulating C cages. Here N denotes the quantum number of a spherical har- monic oscillator .The assignment of these peaks to water centre-of-masstranslational oscillations is supported by the presence of linesat 1680 cm − , visible in the difference spectrum shown in theright-hand inset to Fig. 3. These lines are 110 cm − higherthan the vibrational transitions 1 and 2 of the v mode andcorrespond to the simultaneous excitation of the v vibrationand the translational modes. A similar combination has beenobserved in H @C where a group of lines between 4240and 4270 cm − is the translational sideband to H stretchingvibration .The translational side peak of the v vibrational mode isexpected at about 3683 cm − . However, this frequency coin-cides with a strong rovibrational absorption line 6 of the v mode (Fig. 4), which probably obscures the 3683 cm − trans-lational side peak of the v vibration.The translational ortho transitions display a splitting of2.9 cm − in the ground vibrational state, see Fig.2(b), and 2.7cm − in the excited vibrational state | (cid:105) , see Fig. 4. Thesesplittings may be attributed to the coupling between the watertranslation and rotation, associated with the interaction of thenon-spherical rotating water molecule with the interior of theC cage. Spectral structure of this type has been analysedin detail for the case of H @C . The simplified theo-retical model used here does not include translation-rotationcoupling and cannot explain these splittings. A theoreticalanalysis of the translational peaks will be given in a later pa-per.
2. Vibrational and rovibrational transitions
The vibrational and rovibrational transitions are shown inFig. 3. The three major features that distinguish the spectrumof H O@C from the spectrum of free water are as follows:
1. Pure vibrational transitions.
Absorptions correspond-ing to pure vibrational transitions, i.e. without simultaneousrotational excitation, are present around ω = − and ω = − . Both features are split into two components,labeled 1 and 2, identified from the difference spectra, Fig. 3and 4, as para (1) and ortho (2) transitions.The transition 1 is a transition from the ground vibrationalstate to the excited vibrational state without a change of ro-tational state | (cid:105) . Transition 2 is a vibrational excitationwithout change in the rotational state | (cid:105) . The correspond-ing transitions are forbidden for an isolated water moleculesince the corresponding matrix element is zero . As dis-cussed below, the presence of pure vibrational transitions 1and 2 is consistent with the presence of an electric field insolid H O@C .The ortho - para splitting, i.e. the separation of lines 1 and2, is 0.5 cm − for the v vibrational mode, and 1.8 cm − forthe v vibrational mode,
2. Spectral splittings.
The rovibrational transitions 3 to 8are split into two or more components (see Figs. 3 and 4).These splittings, as in the case of the rotational transitions,are absent for a water molecule in the gas phase. Moreover,transitions 3, 4, and 5 have the same splitting pattern as the
TABLE I. The rotational and rovibrational transition frequencies ω ji and the normalized absorption cross-sections (cid:104) σ ( k ) ji (cid:105) , Eq. (3), from theground para , | (cid:105) , and ground ortho , | (cid:105) , rotational state of H O@C and of free H O. The initial vibrational state is | (cid:105) for alltransitions. The spectral lines are labelled by ω ji of lines 3 to 8 in H O@C is the intensity-weighted average of line sub-component frequencies. Gas phase ω ji are fromRef. 48. | v v v (cid:105) | J K a K c (cid:105) ω ji / cm − (cid:104) σ ( k ) ji (cid:105) / ( cm / molecule ) (cid:104) σ ( k ) ji (cid:105) ( gas ) / (cid:104) σ ( k ) ji (cid:105) ( C ) j i j gas C gas Ref. C
000 3 1 ( . ± . ) −
17 49 ( . ± . ) − ( . ± . ) − ( . ± . ) − ( . ± . ) − ( . ± . ) − ( . ± . ) − ( . ± . ) −
19 50 ( . ± . ) − ( . ± . ) − ( . ± . ) − ( . ± . ) − ( . ± . ) − ( . ± . ) − ( . ± . ) −
17 51 ( . ± . ) − ( . ± . ) − ( . ± . ) − ( . ± . ) − ( . ± . ) − ( . ± . ) −
18 50 ( . ± . ) − ( . ± . ) − ( . ± . ) − ( . ± . ) − ( . ± . ) − ( . ± . ) −
19 52 ( . ± . ) − ( . ± . ) − ( . ± . ) − ( . ± . ) − ( . ± . ) − ( . ± . ) −
19 53 ( . ± . ) − ( . ± . ) − ( . ± . ) − ( . ± . ) − ( . ± . ) − ( . ± . ) −
20 54 ( . ± . ) − rotational transitions in the ground vibrational state, also la-beled 3, 4, and 5, see Fig. 2. Transitions 6 and 7 are specialsince they are between the | (cid:105) and | (cid:105) rotational states andthus reflect directly the splitting of the triply degenerate | (cid:105) state either in the ground vibrational state, transition 7, or inthe excited vibrational state, transition 6, Fig. 4.We assign a weak ortho line at 3654 cm − , marked by *in Fig. 4, to the v rovibrational transition from the thermallyexcited rovibrational state | (cid:105) | (cid:105) to | (cid:105) | (cid:105) . This as-signment is further confirmed by calculating the transitionfrequency with the parameters from Table III: ω + E − E = ω + A + C − ( A + B ) = − .
3. Red shifts.
The frequencies of vibrations are red-shiftedrelative to free H O. The stretching mode frequencies are red-shifted by about 2.4%, while the bending mode frequenciesare red-shifted by about 1.6%, see Table II.
3. Overtone and combination rovibrational transitions
Overtone and combination vibrational transitions wheretwo vibrational quanta are excited are presented for the 2 ω transition in Fig. 5 (a) and for the ω + ω and ω + ω tran-sitions in Fig. 6 (a) and (b). A three-quantum transition,2 ω + ω , is shown in Fig. 5 (b). Rotational levels involved are sketched in Fig. 1 (c). Again, the splitting pattern of eachhigher order rovibrational transition is similar to the splittingof rotational transition with ∆ v i = ∆ v i = + + " at 7059 cm − to 2 ω plus para -H O rotational transition, 0 → , Fig. 6(b). An-other two rotational side peaks of 2 ω are ortho transitions3 and 5 expected at 2 ω + E − E = − and at2 ω + E − E = − , where E − E = A − C and E − E = A + C , with the approximation A = A ≈ A , C = C ≈ C . The numerical values of A and C are taken from Table III. The first line overlaps withline 7, rovibrational transition of ω + ω . The second line isnot observed but this could be due to the low intensity of ortho line 5 relative to the para line 4, see for example Fig. 5 (a).All two- and three-quantum vibrational transitions are red-shifted approximately by 2.4% except the 2 ω where the redshift is 1.5%, see Table II. B. Spectral fitting with a quantum mechanical model
A synthetic spectrum consisting of Gaussian lines with fullwidth at half maximum 1.5 cm − was calculated from the ex-perimental normalized line areas (cid:104) A ( k ) ji (cid:105) , Eq. (2), using f = - 2 002 04 0 W a v e n u m b e r ( c m - 1 ) a (cm-1) T = 5 K T o 1 T o 2 T p FIG. 3. Absorption spectra of H O@C vibrational, rovibrationaland vibration-translational transitions of bending vibration v at 5 K.Spectrum α ( ) was measured after the temperature jump from 30 Kto 5 K (black) and the difference ∆ α = α ( ) − α ( ∆ t ) was measured ∆ t = f = .
1. Lines numbered 1and 2 are pure vibrational transitions and 3, 4, and 5 are rovibrationaltransitions, Fig. 1 (c). Left inset show the para , line 1, and ortho , line2, components of the pure vibrational transitions difference spectrumwith time delay 1 h of the f = . v vibration-translational transitions T p , T o1 and T o2 , at 110cm − higher frequency from 1 and 2, is in the right inset.TABLE II. Frequencies of symmetric stretching ( ω ), symmetricbending ( ω ), and asymmetric stretching ( ω ) modes and of theircombinations 2 ω , ω + ω , 2 ω + ω , 2 ω , and ω + ω measuredfrom the ground vibrational state | (cid:105) for H O@C (this work)and for free H O . ortho and para components of pure vibrationaltransitions v and v are indicated by superscripyts o and p. Thefrequencies of transitions involving v in H O@C are the averageof frequencies of transition 6 and 7, see Fig. 4, 5 (b) and 6. Theovertone 2 ω is estimated from the frequency of line 4, Fig. 5(a),2 ω + E = − , where E = A + C is the energy of ro-tational state | (cid:105) , and A = A , C = C from Table III. The2 ω overtone frequency is estimated from the frequency of para line2 ω + E = − , Fig. 6(b), where E = A + C . Shift ∆ ω = ω C − ω gas . | v v v (cid:105) H O@C free H O ∆ ω ∆ ω / ω gas cm − cm − cm − | (cid:105) . o . − . − . . p | (cid:105) . o . − . − . . p | (cid:105) . . − . − . | (cid:105) . . − . − . | (cid:105) . . − . − . | (cid:105) . . − . − . | (cid:105) . . − . − . | (cid:105) . . − . − . ( · a (cm-1) W a v e n u m b e r ( c m - 1 ) * T = 5 K 3
0- 33 8 0 % 12
FIG. 4. Absorption spectra of H O@C vibrational, rovibrationaland vibration-translational transitions of symmetric stretching, v ,and anti-symmetric stretching vibration v at 5 K. Spectrum α ( ) was measured after the temperature jump from 30 K to 5 K (black)and the difference ∆ α = α ( ) − α ( ∆ t ) was measured ∆ t = f = . v and lines 6,7, 8 are rovibrational transitions of v , Fig. 1 (c). (*) marks the v rovibrational transition at 3654cm − from the thermally excited rota-tional state | (cid:105)| (cid:105) to rovibrational state | (cid:105)| (cid:105) . Inset showsthe para , line 1, and the ortho , line 2, components of the pure vibra-tional transition of mode v of the f = . - 0 . 50 . 00 . 51 . 0 a ) a (cm-1) T = 1 0 K - 0 . 10 . 00 . 10 . 20 . 3 b )
87 6 T = 1 0 K W a v e n u m b e r ( c m - 1 ) FIG. 5. Absorption spectra of H O@C at 10 K. (a) Overtone, 2 ω ,and (b) combination, 2 ω + ω , rovibrational transitions. Spectrum α ( ) was measured after the temperature jump from 45 K to 10 K(black) and the difference ∆ α = α ( ) − α ( ∆ t ) was measured ∆ t = .
75 h later (blue). The numbers 3 – 8 label the transitions shown inFig. 1 (c). Sample filling factor f = .
051 01 5 * T = 5 K a ) W a v e n u m b e r ( c m - 1 ) a (cm-1) - 0 . 50 . 00 . 51 . 01 . 52 . 0 + T = 5 K b ) FIG. 6. Absorption spectra of H O@C at 5 K. (a) Rovibrationalcombination ω + ω and (b) ω + ω transitions. Spectrum α ( ) was measured after the temperature jump from 30 K to 5 K (black)and the difference ∆ α = α ( ) − α ( ∆ t ) was measured ∆ t = .
83 hlater (blue). The numbers 6, 7, 8 label the transitions shown inFig. 1 (c). (*) in (a) at 5202cm − marks the ω + ω rovibrationaltransition from the thermally excited rovibrational state | (cid:105)| (cid:105) to | (cid:105)| (cid:105) . The 2 ω rovibrational transition at 7059cm − , marked(+) in (b), is from | (cid:105)| (cid:105) to | (cid:105)| (cid:105) . Sample filling factor f = . n o = . n p = . T = + δ a ν and − δ a ν in Eq. (C9), where the parameter variation ofthe ν -th parameter at its best value a ν min is | δ a ν | = . a ν min .The fit was applied to the rotational transitions in the groundvibrational state | (cid:105) and to the rovibrational transitions fromthe ground state to the vibrational states | (cid:105) , | (cid:105) , and | (cid:105) . In total, 45 absorption lines were fitted.The synthetic experimental spectra and the best fit spectraare shown in Fig. 7, with the best fit parameters given in Ta-ble III. The result of the fit overlaps well with the syntheticspectrum, except for the transition 5 as seen in the first threepanels of Fig. 7. While for other transitions one or two Gaus-sian components were sufficient, the experimental transitionlineshape required three components to get a reliable fit of itsline area. Also, transitions 6 and 7 were represented by twocomponents in the synthetic spectrum, although four peaksare seen in the experimental spectrum, Fig. 4. The additionalstructure of experimental peaks may originate from the mero-hedral disorder as discussed in Section V E.Tables IV and V list energies and the main componentsof rotational states in the ground vibrational state. The rota-tional energies are in qualitative agreement with recent com-putational estimates . The 2 J + O. TABLE III. Parameters obtained from the quantum mechanicalmodel fit of IR spectra of H O@C at T = ω i , rotational constants A v , B v , C v , and quadrupolarenergy V Q Q zz are in units of cm − ; electric field E in 10 Vm − , θ E in radians and the dipole moment µ x (B13) and the transition dipolemoments µ x , z i (B14) in D. It is assumed that µ x and Q zz do not de-pend on the vibrational state. The parameters with zero error werenot fitted.Parameter Value Error n o . f ω . . ω . . ω . . A .
15 0 . B . . C .
48 0 . A . . B . . C .
50 0 . A . . B . . C .
81 0 . A
26 6 B
15 3 C . . µ x .
474 0 . µ x . × − . × − µ x . × − . × − µ z . × − . × − E
110 5 θ E . . V Q Q zz − . . From our fit the permanent dipole moment µ x of the encap-sulated water is given by the absorption cross-section of theIR rotational transitions 3, 4, and 5 in the ground vibrationalstate. With the value of µ x in hand and by using the intensitiesof transitions 1 and 2 we were able to determine the internalstatic electric field in solid C . The interaction of µ x with thecrystal electric field mixes rotational states within ground andexcited vibrational states. For example, in case of ortho waterthe components of state | (cid:105) are mixed into the ground state | (cid:105) , Table V. This mixing gives the oscillator strength to thepure vibrational transitions 1 and 2. As shown in table III,the fitted value of the electric field at the C cage centres is ( ± ) Vm − .A splitting of 4 cm − is observed for transition 7 and is dueto the splitting of the ortho ground state | (cid:105) , Fig. 1(c). Inprinciple, a splitting could be caused by the interaction of thewater electric dipole with an electric field, or by the interactionof the water electric quadrupole moment with an electric fieldgradient. The electric field 110 × Vm − is too small tocause splitting of this magnitude. This electric field lifts thedegeneracy of m = ± | (cid:105) , but the gap between m = m = ± V Q and Q zz , itis not possible to have an estimate of how much is the water1quadrupole moment Q zz screened in C . V. DISCUSSIONA. Vibrations of confined H O All eight frequencies of the encapsulated H O vibrationsfound in this work are red-shifted relative to those of free wa-ter, see Table II. The red-shift of the vibrational frequency hasbeen observed for other endofullerenes, H @C , HD andD @C , and HF@C . Six water modes have a relativeshift between − .
3% and − . ω and its overtone fre-quency 2 ω are shifted by − .
6% and − . ω and ω is only partially consistent with previous DFT cal-culations. The DFT-based calculations published by Varad-waj et al. do predict vibrational redshifts, while some of thecalculations reported by Farimani et al. predict blue shiftsrather than red shifts.The calculation predicts a blue shift of the bending mode ω although ten times less in absolute value than the predictedshift of stretching modes . The experimental shift of ω isless than that of stretching modes but it is still red-shifted.The other method, fully coupled nine-dimensional calcula-tion, predicts blue-shifts for all three vibrational modes . B. Translations of H O Table VI lists the measured translational energies from theground to the first excited state, ω t , of small-molecule end-ofullerenes. It is known that the potential of di-hydrogen inC is anharmonic while the degree of anharmonicity ofHF@C and H O@C potentials is not known. For simplic-ity, we assume that the potential is harmonic for the currentcase of endohedral molecules, ω t ≈ ω , and show its scal-ing relative to H in Table VI. In this approximation the har-monic potential parameter is similar among the hydrogen iso-topologs but for HF and H O V is larger by a factor of 1.9 and3.4, respectively. The steeper translational potential for HFand H O, relative to dihydrogen, is consistent with the largersize of these molecules, and hence their tighter confinement.The last line of Table VI is the frequency and the harmonicpotential of H O@C derived by Bacic and co-workers using Lennard-Jones potentials. The calculated potential ismore steep than the experimentally determined potential.As seen in Fig. 2(b), the absorption line of the ortho -H Otranslational mode is split by 2.9 cm − . This splitting may beattributed to the coupling between the endohedral moleculetranslation and rotation, associated with the interaction of thenon-spherical rotating molecule with the interior of the C cage, as seen in H @C . For the particular transition,shown in Fig. 2(b), it is the coupling between translationalstate with N = , L = | (cid:105) of ortho wa-ter. Within spherical symmetry a good quantum number is Λ = J + L . Thus, the translation-rotation coupled translational state L = J = Λ -values 0, 1, and 2. The calculated energy difference of or-tho -water Λ = Λ = − as compared tothe experimental value 2.9 cm − . C. Rotations of H O There are two possibilities why the rotational constants ofwater change when it is encapsulated. First is that the bondlength and angles of H O change. The second is that H O,because of confinement and being non-centrosymmetric, isforced to rotate about the “center of interaction” which doesnot coincide with its nuclear center of mass .The rotational constant relates to the moment of inertia I aa as A = h ( π c I aa ) − where c is the speed of light. Similarly, B = h ( π c I bb ) − for the b -axis and C = h ( π c I cc ) − forthe c -axis rotation ( [ A ] = m − in SI units and 0 . [ A ] = cm − ).The moments of inertia are I αα = ∑ i m i ( β i + γ i ) , where { α i , β i , γ i } are the Cartesian coordinates of the i -th nucleuswith mass m i with the origin at the nuclear center of mass.For non-centrosymmetric molecules, the translation-rotation coupling shifts the rotational energy levels. Inquantum mechanical terms, the shift of rotational states iscaused by the mixing of rotational and translational statesby translation-rotation coupling, example is HD@C .Translation-rotation coupling was not included in our quan-tum mechanical model of H O@C . The rotational con-stants of the model were free parameters to capture the ef-fect of translation-rotation coupling and the change of theH O molecule geometry caused by the C cage. The ro-tational constants of the free water are A = .
88 cm − , B = .
52 cm − and C = .
28 cm − in the ground vibra-tional state . The rotational constants of endohedral waterhave relative shifts − , . , − .
7% for A , B and C , inthe ground vibrational state, Table III. In the following discus-sion, we use classical arguments to assess whether the shift ofrotational levels is caused by the change of water moleculegeometry or by the translation-rotation coupling.From the symmetry of the H O molecule, the nuclear cen-ter of mass is on the b axis, Fig. 1. The shift of H O center ofrotation in the negative direction of b axis decreases A and C while it does not affect B . If B changes it must be due to thechange of H-O-H bond angle and O-H bond length. The cal-culation predicts the lengthening of the H-O bond by 0.0026Åand decrease of the H-O-H bond angle by 0 . ◦ of caged wa-ter with respect to free water . With these parameters therelative change of rotational constants A , B and C is -2.5%,0.65%, -0.46%, an order of magnitude smaller than derivedfrom the IR spectra of H O@C . However, by shifting therotation center by -0.07Å in the b direction (further away fromthe oxygen) gives relative changes − , .
65% and − . A and C is not very differentfrom the values derived from the IR spectra while the relativechange of B is within the error limits, Table III. Thus, it islikely that the dominant contribution to the observed changesof rotational constants in the ground translational state comesnot from the change of H O molecule bond lengths and bond2
TABLE IV. para -H O@C rotational energies and wavefunctions in the ground vibrational state calculated with the best fit parameters fromTable III. Wavefunction components with amplitude absolute value less than 0.1 are omitted; | J , − k , − m (cid:105) ≡ (cid:12)(cid:12) J , ¯ k , ¯ m (cid:11) . J K a K c Energy/cm − Wavefunction in symmetric top basis | J , k , m (cid:105) − . | , , (cid:105) .
07 0 . (cid:0)(cid:12)(cid:12) , ¯1 , ¯1 (cid:11) + (cid:12)(cid:12) , ¯1 , (cid:11) − (cid:12)(cid:12) , , ¯1 (cid:11) − | , , (cid:105) (cid:1) . − . (cid:0)(cid:12)(cid:12) , ¯1 , (cid:11) − | , , (cid:105) (cid:1) .
75 0 . (cid:0)(cid:12)(cid:12) , ¯1 , ¯1 (cid:11) − (cid:12)(cid:12) , ¯1 , (cid:11) − (cid:12)(cid:12) , , ¯1 (cid:11) + | , , (cid:105) (cid:1) . − . | , , (cid:105) + . (cid:0)(cid:12)(cid:12) , ¯2 , (cid:11) + | , , (cid:105) (cid:1) .
73 0 . (cid:0)(cid:12)(cid:12) , , ¯1 (cid:11) + | , , (cid:105) (cid:1) .
35 0 . (cid:0)(cid:12)(cid:12) , , ¯1 (cid:11) − | , , (cid:105) (cid:1) .
27 0 . (cid:0)(cid:12)(cid:12) , , ¯2 (cid:11) − | , , (cid:105) (cid:1) . − . (cid:0)(cid:12)(cid:12) , , ¯2 (cid:11) + | , , (cid:105) (cid:1) . − . (cid:0)(cid:12)(cid:12) , ¯1 , (cid:11) + | , , (cid:105) (cid:1) + . (cid:0)(cid:12)(cid:12) , ¯1 , ¯2 (cid:11) + (cid:12)(cid:12) , ¯1 , (cid:11) + (cid:12)(cid:12) , , ¯2 (cid:11) + | , , (cid:105) (cid:1) . − . (cid:0)(cid:12)(cid:12) , ¯1 , ¯2 (cid:11) − (cid:12)(cid:12) , ¯1 , (cid:11) + (cid:12)(cid:12) , , ¯2 (cid:11) − | , , (cid:105) (cid:1) . − . (cid:0)(cid:12)(cid:12) , ¯1 , (cid:11) + | , , (cid:105) (cid:1) − . (cid:0)(cid:12)(cid:12) , ¯1 , ¯2 (cid:11) + (cid:12)(cid:12) , ¯1 , (cid:11) + (cid:12)(cid:12) , , ¯2 (cid:11) + | , , (cid:105) (cid:1) − . (cid:0) − (cid:12)(cid:12) , ¯1 , ¯1 (cid:11) + (cid:12)(cid:12) , ¯1 , (cid:11) − (cid:12)(cid:12) , , ¯1 (cid:11) + | , , (cid:105) (cid:1) .
91 0 . (cid:0) − (cid:12)(cid:12) , ¯1 , ¯1 (cid:11) + (cid:12)(cid:12) , ¯1 , (cid:11) − (cid:12)(cid:12) , , ¯1 (cid:11) + | , , (cid:105) (cid:1) − . (cid:0)(cid:12)(cid:12) , ¯1 , (cid:11) + | , , (cid:105) (cid:1) − . (cid:0)(cid:12)(cid:12) , ¯1 , ¯2 (cid:11) + (cid:12)(cid:12) , ¯1 , (cid:11) + (cid:12)(cid:12) , , ¯2 (cid:11) + | , , (cid:105) (cid:1) .
01 0 . (cid:0)(cid:12)(cid:12) , ¯1 , ¯1 (cid:11) + (cid:12)(cid:12) , ¯1 , (cid:11) + (cid:12)(cid:12) , , ¯1 (cid:11) + | , , (cid:105) (cid:1) TABLE V. ortho -H O@C rotational energies, zero energy at para ground state | (cid:105) , and wavefunctions in the ground vibrational statecalculated with the best fit parameters from Table III. Wavefunction components with amplitude absolute value less than 0.1 are omitted; | J , − k , − m (cid:105) ≡ (cid:12)(cid:12) J , ¯ k , ¯ m (cid:11) . J K a K c Energy/cm − Wavefunction in symmetric top basis | J , k , m (cid:105) .
89 0 . | , , (cid:105) . − . (cid:0)(cid:12)(cid:12) , , ¯1 (cid:11) + | , , (cid:105) (cid:1) + . (cid:0)(cid:12)(cid:12) , ¯1 , (cid:11) + | , , (cid:105) (cid:1) . − . (cid:0)(cid:12)(cid:12) , , ¯1 (cid:11) − | , , (cid:105) (cid:1) .
88 0 . (cid:0)(cid:12)(cid:12) , ¯1 , ¯1 (cid:11) − (cid:12)(cid:12) , ¯1 , (cid:11) + (cid:12)(cid:12) , , ¯1 (cid:11) − | , , (cid:105) (cid:1) .
03 0 . (cid:0)(cid:12)(cid:12) , ¯1 , ¯1 (cid:11) + (cid:12)(cid:12) , ¯1 , (cid:11) + (cid:12)(cid:12) , , ¯1 (cid:11) + | , , (cid:105) (cid:1) + . | , , (cid:105) .
62 0 . (cid:0)(cid:12)(cid:12) , ¯1 , (cid:11) + | , , (cid:105) (cid:1) .
29 0 . (cid:0)(cid:12)(cid:12) , ¯1 , (cid:11) − | , , (cid:105) (cid:1) . − . (cid:0)(cid:12)(cid:12) , ¯1 , ¯1 (cid:11) + (cid:12)(cid:12) , ¯1 , (cid:11) − (cid:12)(cid:12) , , ¯1 (cid:11) − | , , (cid:105) (cid:1) . − . (cid:0)(cid:12)(cid:12) , ¯1 , ¯1 (cid:11) − (cid:12)(cid:12) , ¯1 , (cid:11) − (cid:12)(cid:12) , , ¯1 (cid:11) + | , , (cid:105) (cid:1) . − . (cid:0)(cid:12)(cid:12) , ¯1 , ¯2 (cid:11) − (cid:12)(cid:12) , ¯1 , (cid:11) − (cid:12)(cid:12) , , ¯2 (cid:11) + | , , (cid:105) (cid:1) . − . (cid:0)(cid:12)(cid:12) , ¯1 , ¯2 (cid:11) + (cid:12)(cid:12) , ¯1 , (cid:11) − (cid:12)(cid:12) , , ¯2 (cid:11) − | , , (cid:105) (cid:1) . − . (cid:0)(cid:12)(cid:12) , ¯2 , ¯2 (cid:11) + (cid:12)(cid:12) , ¯2 , (cid:11) − (cid:12)(cid:12) , , ¯2 (cid:11) − | , , (cid:105) (cid:1) . . (cid:0)(cid:12)(cid:12) , ¯2 , ¯2 (cid:11) − (cid:12)(cid:12) , ¯2 , (cid:11) − (cid:12)(cid:12) , , ¯2 (cid:11) + | , , (cid:105) (cid:1) . . (cid:0)(cid:12)(cid:12) , ¯2 , ¯1 (cid:11) − (cid:12)(cid:12) , ¯2 , (cid:11) − (cid:12)(cid:12) , , ¯1 (cid:11) + | , , (cid:105) (cid:1) . . (cid:0)(cid:12)(cid:12) , ¯2 , ¯1 (cid:11) + (cid:12)(cid:12) , ¯2 , (cid:11) − (cid:12)(cid:12) , , ¯1 (cid:11) − | , , (cid:105) (cid:1) . . (cid:0)(cid:12)(cid:12) , ¯2 , (cid:11) − | , , (cid:105) (cid:1) angle but from the shifting its center of rotation away from thenuclear center of mass of H O molecule.
D. Permanent and transition dipole moments
A comparison of the normalized absorption cross-sections (cid:104) σ ji (cid:105) , Eq. (3), for H O@C and for free water is given in the last column of Table I. In general, for all observed transitionsthe absorption cross-section of endohedral water is smallerthan that of free water. The v mode has the smallest relativechange while the largest relative change is for the combinationmode ω + ω .The comparison of H O@C and free H O absorptioncross-sections, Table I, enables us to estimate independentlyfrom the spectral fit the permanent and the transition dipole3
TABLE VI. Translational energies from the ground to the first ex-cited state, ω t , of small-molecule endofullerenes and the scaling ofthe harmonic spherical potential V for the translational motion of anendohedral molecule, mass m i , relative to the H @C potential V H .The anharmonicity is neglected, ω t ≈ ω , where ω is the frequencyof an harmonic oscillator, Eq. 18.Molecule m i ω t / cm − V / V H Ref.H . HD 3 157 . D . HF 20 78 . H O 18 110 3.4 This workH O 18 162 7.3 Theory
TABLE VII. Absolute values of the dipole moment, unit D, ma-trix elements of rotational ( µ x , Eq. (B13)) and rovibrational ( µ x , µ x and µ z , Eq. (B14)) transitions of gaseous H O, as published, and ofH O@C calculated with Eq. (26) from the rotational and rovibra-tional absorption cross-sections, Table I, and determined by the fit ofIR spectra, Table III. The cross-section-derived dipole moments arethe cross-section-error weighted averages of the three transition, 3, 4and 5 or 6, 7 and 8 in Table I.Gas Ref. Cross-section Fit µ x . . ± .
05 0 . ± . µ x . ( . ± . ) − ( . ± . ) − µ x . ( . ± . ) − ( . ± . ) − µ z . ( . ± . ) − ( . ± . ) − moments of encapsulated H O, µ C ji = µ gas ji (cid:118)(cid:117)(cid:117)(cid:116) (cid:104) σ C ji (cid:105) ω gas ji (cid:104) σ gas ji (cid:105) ω C ji . (26)The results are collected together with the dipole momentsobtained from the fit of IR spectra in Table VII. The per-manent dipole moment of encapsulated water is nearly fourtimes smaller than of a free H O. The results of the IR spec-troscopy study are consistent with the dipole moment deter-mined by the capacitance method, 0 . ± .
05 D . The re-duction of fullerene-encapsulated water dipole moment hasbeen predicted by several theoretical calculations . E. Effect of solid C crystal field The electric dipolar and quadrupolar interactions of theH O@C molecule with the electrostatic field in solid C explains the oscillator strength of the pure v and v vibra-tional transitions and the splitting of the rotational states with J > shows that the source ofthe quadrupole crystal field is the orientation of electron-richdouble bonds of 12 nearest-neighbour C relative to the cen-tral H O@C . When the solid C is cooled below 90 K amerohedral disorder is frozen in where approximately 85% of C have electron-rich 6:6 bond (bond between the twohexagonal rings) facing pentagonal ring of a neighbouringcage, the P-orientation . The rest are H-oriented where the6:6 bond faces neighbouring cages hexagonal ring. The cal-culated quadrupolar interaction for the P-oriented moleculesis ten times bigger than for the H-oriented molecules . Theelectric field gradient couples to the quadrupole moment ofwater and splits the | (cid:105) rotational state by 4 . − , wherethe m = ± m = . This the-oretically predicted splitting of | (cid:105) state for the P-orientedmolecules is remarkably close to the observed experimentalvalue, seen as a 4 cm − splitting of line 7, transition startingfrom the ortho ground state, Fig. 4. It is not possible to de-termine the crystal electric field gradient tensor and the en-capsulated water quadrupole moment separately from our IRspectra.Further splitting is possible if the symmetry is lower than S , but the maximum number of components for J = by IR spectroscopy and was attributed to the merohedraldisorder. Thus, our work and IR study of phonons clearlyshow that there are two different quadrupolar interactions insolid C . As was proposed by Felker et al. , the crystalfield has a different magnitude for P-oriented sites and for H-oriented sites. The small population of H-oriented sites (about15%) justifies our spectral fitting with a single quadrupolarinteraction.We assumed that there is an internal electric field in C andthis field is a possible reason why the pure vibrational transi-tions 1 and 2, see Fig. 1(c), become visible in the IR spec-trum. It is also plausible that 1 and 2 gain intensity throughthe translation-rotation coupling from the induced dipole mo-ment of translational motion. However, there is evidence thatlocal electric fields exist in solid C as a result of merohe-dral disorder C . The estimate of un-balanced charge byAlers et al. was q = × − e assuming a dipole moment µ = qd , where e is the electron charge and d = . molecule. Our estimate is that the electricfield E = . × Vm − at the center of C cage is createdby the dipole moment with charge q = . × − e . Thesetwo estimates are very close.C has six nearest-neighbor equatorial cages and three ax-ial cages above and three axial cages below the equatorialplane, following the notation of Ref. . The z (cid:48) axis of the crys-tal field coordinate frame is normal to the equatorial plane. Asour fit shows, the electric field is rotated away from the z (cid:48) axisby θ E ≈ ◦ , Table III, almost into the equatorial plane. It ispossible that one of the 6 nearest-neighbours in the equatorialplane does not have P-orientation and this mis-oriented cageis the source of the electric field. θ E has a large relative errorconsistent with the probability to have the mis-oriented cagein the equatorial or in the axial position.4 a (cm-1) W a v e n u m b e r ( c m - 1 ) 543a ) - 1 ) 543b ) a (cm-1) W a v e n u m b e r ( c m - 1 ) 543c )
W a v e n u m b e r ( c m - 1 ) 867d )
FIG. 7. The synthetic experimental spectra (black solid line) and the spectra calculated with the best fit parameters from Table III (blue dashedline) with filling factor f = ortho fraction n o = . v and (c) v . (d) Rovibrational transitions (6, 7, 8) of v . VI. SUMMARY
The infrared absorption spectra of solid H O@C sampleswere measured close to liquid He temperature and rotational,vibrational, rovibrational, overtone, and combination rovibra-tional transitions of H O were seen. The spectral lines wereidentified as para and ortho water transitions by following the para - ortho conversion process over the timescale of hours.The vibrational frequencies are shifted by -2.4% relative tofree water, except bending mode frequency ω and its over-tone 2 ω , where the shift is -1.6%. An absorption mode due tothe quantized center of mass motion of H O in the molecularcage of C , was observed at 110 cm − . The dipole momentof encapsulated water is 0 . ± .
05 D, approximately 4 timesless than for free water and agrees with previous estimates .The rotational and rovibrational spectra were fitted with aquantum mechanical model of a vibrating rotor in electro-static field with dipolar and quadrupolar interactions. Thequadrupolar interaction splits the J ≥ O. The source of quadrupolar interaction is the relativeorientation of electron-rich chemical bonds relative to pentag-onal and hexagonal motifs of C and its 12 nearest neigh-bours . Further IR study by using pressure to change theratio of P- and H-oriented motifs would provide more infor-mation on the quadrupolar crystal fields of these motifs. The finite oscillator strength of the fundamental vibrational tran-sitions is attributed to a finite electric field at the centres ofthe C cages due to merohedral disorder, as has been postu-lated in different contexts . Our results are consistent with aninternal electric field of 10 Vm − . However, it is also plausi-ble that the fundamental vibrational transitions gain intensitythrough the translation-rotation coupling from the dipole mo-ment induced by the translational motion, something that canbe addressed in further theoretical studies.To conclude, H O in the molecular cavity of C behavesas a vibrating asymmetric top, its dipole moment is reduced,and the translational motion is quantized. The splitting of ro-tational levels is caused by the quadrupolar interaction withthe crystal field of solid C . Evidence is found for the exis-tence of a finite electric field at the centres of the C cages inthe water-endofullerene, due to merohedral disorder.Two out of three components necessary for a rigorous, com-prehensive description of the water translations, rotations andvibrations inside C molecular cage are now in place: firstly,the infrared spectroscopy data reported here and secondly, the9-dimensional quantum bound-state methodology plus thetheory of symmetry breaking in solid C . What is miss-ing is a high-quality ab initio 9-dimensional potential energysurface for this system.5 ACKNOWLEDGMENTS
We thank Prof. Zlatko Baˇci´c for useful discussions. Thisresearch was supported by the Estonian Ministry of Educationand Research institutional reseach funding IUT23-3, personalresearch funding PRG736, and the European Regional Devel-opment Fund project TK134. We thank EPSRC (UK), grantnumbers EP/P009980/1 and EP/T004320/1, for support.
Appendix A: Interaction of dipole moment with local electricfield
The dipole moment in spherical components is µ + = − √ ( µ x + i µ y ) , µ − = √ ( µ x − i µ y ) , µ = µ z . (A1)The dipole moment of water in the Cartesian molecule co-ordinate frame, as shown in Fig. 1, is µ M = {− µ x , , } , wherewe use a convention that the dipole moment is directed fromthe negative charge to the positive charge. Then, from Eq. A1the dipole moment in spherical components is { µ Mm } = µ x √ {− , , } , m = − , , + . (A2)We consider the coupling of H O dipole moment to the lo-cal electric field, E , with spherical coordinates { E , φ E , θ E } in the crystal frame C (frame where the electric field gra-dient tensor is defined) and in the coordinate frame E ofelectric field { E Em } = E { , , } , i.e. along z E axis. Corre-sponding Euler angles are Ω C → E = { φ E , θ E , } and Ω E → C = { π , θ E , − π − φ E } . The dipole moment of the molecule in theframe E is µ Em = ∑ m (cid:48) = − (cid:2) D mm (cid:48) ( Ω E → C ) (cid:3) ∗ µ Cm (cid:48) . (A3)Here we used Wigner D -functions relating the components ofa spherical rank j irreducible tensor T jm in coordinate frames A and B : T Bjm = j ∑ m (cid:48) = − j D jm (cid:48) m ( Ω A → B ) T Ajm (cid:48) (A4)and T Ajm = j ∑ m (cid:48) = − j (cid:104) D jmm (cid:48) ( Ω A → B ) (cid:105) ∗ T Bjm (cid:48) , (A5)where Ω A → B = { φ , θ , χ } are Euler angles transforming co-ordinate frame A into frame B . The angles for the inversetransformation are Ω B → A = { π − χ , θ , − π − φ } . The interaction of molecular dipole moment µ Mm with elec-tric field is H ed = − ∑ m = − ( − ) m E E − m µ Em = − ∑ m = − ( − ) m E E − m (cid:34) ∑ m (cid:48) = − (cid:2) D mm (cid:48) ( Ω E → C ) (cid:3) ∗ µ Cm (cid:48) (cid:35) = − ∑ m , m (cid:48) = − ( − ) m E E − m (cid:2) D mm (cid:48) ( Ω E → C ) (cid:3) ∗ µ Cm (cid:48) = − ∑ m , m (cid:48) = − ( − ) m E E − m (cid:2) D mm (cid:48) ( Ω E → C ) (cid:3) ∗ × (cid:34) ∑ m (cid:48)(cid:48) = − (cid:2) D m (cid:48) m (cid:48)(cid:48) ( Ω C → M ) (cid:3) ∗ µ Mm (cid:48)(cid:48) (cid:35) . The minus sign in front of sum in the formula above is consis-tent with H ed = − E · µ if the spherical components of vectors E and µ are defined as in (A1). Appendix B: Interaction of dipole moment with electric fieldof radiation
Here we derive the electric dipole transition matrix ele-ments of a molecule, part (21) of absorption cross section (20).The derivation of (21) starts from S fi = e | (cid:104) f | H ed | i (cid:105) | , (B1)and H ed = − ∑ m = − ( − ) m e A − m µ Am , (B2)where the electric field of radiation, e = { e Am } and the dipolemoment of a molecule, µ Am , are in the space-fixed frame A ; e = ∑ m = − ( e Am ) .The Euler angles of transformation of crystal frame C tospace-fixed frame A and vice versa are Ω C → A = { φ R , θ R , } and Ω A → C = { π , θ R , − π − φ R } .The dipole moment in frame A is µ Am = ∑ m (cid:48) = − (cid:2) D mm (cid:48) ( Ω A → C ) (cid:3) ∗ µ Cm (cid:48) , (B3)where µ Cm is the dipole moment in the crystal frame C .6The absolute value of matrix element squared is S fi == e − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:104) j | ∑ m = − ( − ) m e A − m µ Am | i (cid:105) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = e − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∑ m , m (cid:48) = − ( − ) m e A − m (cid:2) D mm (cid:48) ( Ω A → C ) (cid:3) ∗ (cid:104) j | µ Cm (cid:48) | i (cid:105) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = e − ∑ m , m (cid:48) , m , m (cid:48) = − ( − ) m + m e A − m (cid:2) e A − m (cid:3) ∗ × (cid:104) D m m (cid:48) ( Ω A → C ) (cid:105) ∗ D m m (cid:48) ( Ω A → C ) × (cid:104) j | µ Cm (cid:48) | i (cid:105) (cid:104) j | µ Cm (cid:48) | i (cid:105) ∗ . (B4)
1. Random orientation of crystals
We assume all the crystals are identical and there are noother static fields outside the crystal what could brake the di-rectional isotropy. We average (B4) over the random orien-tation of crystal coordinate frames with respect to the space-fixed frame, Ω A → C , and get: (cid:10) S fi (cid:11) Ω A → C == e ∑ m , m (cid:48) = − (cid:12)(cid:12) e A − m (cid:12)(cid:12) (cid:12)(cid:12) (cid:104) j | µ Cm (cid:48) | i (cid:105) (cid:12)(cid:12) = e (cid:32) ∑ m = − (cid:12)(cid:12) e A − m (cid:12)(cid:12) (cid:33) (cid:32) ∑ m (cid:48) = − (cid:12)(cid:12) (cid:104) j | µ Cm (cid:48) | i (cid:105) (cid:12)(cid:12) (cid:33) = (cid:32) ∑ m (cid:48) = − (cid:12)(cid:12) (cid:104) j | µ Cm (cid:48) | i (cid:105) (cid:12)(cid:12) (cid:33) . (B5)where we used a property of rotation matrices , (cid:90) (cid:104) D j m m (cid:48) ( Ω ) (cid:105) ∗ D j m m (cid:48) ( Ω ) d Ω (B6) = π j + δ j j δ m m δ m (cid:48) m (cid:48) , and (cid:90) π sin θ R d θ R (cid:90) π d φ R (cid:90) π d χ R = π . (B7)If the sample is in the powder form then it follows fromEq. (B5) that the absorption is independent of light polariza-tion.
2. Transition matrix element and separation of coordinates
The absorption of radiation by a molecule, Eq. (20), de-pends on the matrix elements of an electric dipole momentbetween the initial and final states, ∑ σ = − (cid:12)(cid:12)(cid:10) Φ (cid:48) (cid:12)(cid:12) µ A σ | Φ (cid:105) (cid:12)(cid:12) , (B8) where µ A σ is the molecule diopole moment in the space-fixedcoordinate frame. Molecule wavefunction consists of nu-clear spin wavefunction | Im I (cid:105) , electron | Φ e (cid:105) and electron spinwavefunction | Sm S (cid:105) , vibration wavefunction | Φ v (cid:105) , and rota-tion wavefunction | Φ r (cid:105) : | Φ (cid:105) = | Im I (cid:105) | Sm S (cid:105) | Φ e Φ v Φ r (cid:105) . (B9)Using these separable wavefunctions the matrix element(B8) is : (cid:10) Φ (cid:48) (cid:12)(cid:12) µ A σ | Φ (cid:105) = (cid:10) I (cid:48) m (cid:48) I (cid:12)(cid:12) Im I (cid:105) (cid:10) S (cid:48) m (cid:48) S (cid:12)(cid:12) Sm S (cid:105) (B10) × ∑ σ (cid:48) = − (cid:68) Φ r (cid:48) (cid:12)(cid:12)(cid:12) D σσ (cid:48) ( Ω A → M ) ∗ | Φ r (cid:105) (cid:68) Φ v (cid:48) (cid:12)(cid:12)(cid:12) µ M ( e ) σ (cid:48) | Φ v (cid:105) , where µ M ( e ) σ = (cid:68) Φ e (cid:48) (cid:12)(cid:12)(cid:12) µ M σ | Φ e (cid:105) (B11)and we have taken into account that the electric dipole mo-ment does not depend on nuclear and electron spin coordi-nates. Furthermore, if the energies of initial and final statesof the transition are independent of spin projections m I and m s , the summation over initial and final states in the transi-tion probability leads to degeneracy factors g I = I + g S = S + O, thus g S =
1. Degeneracy of para molecules ( I =
0) is g ( p ) I = ortho molecules ( I =
1) is g ( o ) I = | Φ e (cid:105) , µ M ( eg ) σ ≡ (cid:104) Φ e | µ M σ | Φ e (cid:105) . (B12)For the rest of the discussion we use a shorthand notation µ M σ for the molecule dipole moment in the ground electronic state, µ M ( eg ) σ .Using Cartesian coordinates the dipole moment in theground vibrational state is µ x = (cid:104) | µ Mx | (cid:105) . (B13)In the quantum mechanical model of H O@C what we usedto fit the IR spectra we set the dipole moment equal to µ x inthree excited vibrational states | (cid:105) , | (cid:105) , and | (cid:105) .The vibrational transition dipole moments are µ x = (cid:104) | µ Mx | (cid:105) , µ x = (cid:104) | µ Mx | (cid:105) , (B14) µ z = (cid:104) | µ Mz | (cid:105) . The relation between spherical and Cartesian dipole momentcomponents is given by Eq. A1.7
Appendix C: Fitting of synthetic spectra with quantummechanical model and model parameter error estimation
We determined the Hamiltonian parameters and the dipolemoments, parameter set a = { a , . . . , a ν , . . . , a M } , by findingthe paramater set a min what gives the minimum value, χ ,of function χ = N ∑ i = [ S ( ν i ) − f ( ν i ; a )] (C1)where S ( ν i ) is the synthetic spectrum, argument frequency ν i ,generated from the fit of the experimental spectra, and f ( ν i ; a ) is the spectrum calculated from the model with M parame-ters a = { a , . . . , a ν , . . . , a M } ; N is the number of points in thespectrum. The goal is to minimize χ over parameters a . Theresult is χ and a min .Lets define matrix F i ν = ∂ f ( ν i ; a ) ∂ a ν (C2)and dispersion matrix ¯ V , Eq. 12 in , V µν = N ∑ i = F T µ i F i ν . (C3)The covariance matrix, Eq. 11 in ,¯ Θ = σ ¯ V − , (C4)where σ = χ min / ( N − M ) and ¯ V − is inverse matrix of ¯ V ,¯ V − ¯ V = ¯ .The estimated variance of parameter a ν is ∆ a ν = (cid:112) Θ νν = (cid:115) χ ( ¯ V − ) νν N − M . (C5)The correlation matrix ¯ C is C νµ = Θ νµ (cid:112) Θ νν Θ µµ . (C6)The element of ¯ F , Eq.(C2), is F i ν = f ( ν i ; a min + δ a ν ) − f ( ν i ; a min ) δ a ν , (C7)where a min minimizes χ and δ a ν is a small variation of pa-rameter a ν .We change the sum for an integral over ν in (C1), χ = ∑ k (cid:90) ν k ν k A k [ S ( ν ) − f ( ν ; a )] d ν , (C8)where the experimental spectrum is available in several spec-tral ranges { ν k , ν k } indexed by k . A k is the weight factor foreach spectral range k . A k is chosen so that the strongest linesfor each range k are equal. 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