We present a methodology that, for the first time, allows rigorous quantum calculation of the inelastic neutron scattering (INS) spectra of a triatomic molecule in a nanoscale cavity, in this case, H2O inside the fullerene C60. Both moieties are taken to be rigid. Our treatment incorporates the quantum six-dimensional translation–rotation (TR) wave functions of the encapsulated H2O, which serve as the spatial parts of the initial and final states of the INS transitions. As a result, the simulated INS spectra reflect the coupled TR dynamics of the nanoconfined guest molecule. They also exhibit the features arising from symmetry breaking observed for solid H2O@C60 at low temperatures. Utilizing this methodology, we compute the INS spectra of H2O@C60 for two incident neutron wavelengths and compare them with the corresponding experimental spectra. Good overall agreement is found, and the calculated spectra provide valuable additional insights.

The fullerene C60 has fascinated scientists with its exceptionally high symmetry, and the potential for encapsulating atoms and small molecules in its cavity, from the moment it was serendipitously discovered by Kroto et al.1 This tantalizing potential became a reality with the introduction of the “molecular surgery” procedure.2–4 The approach proved to be remarkably successful and versatile, and within a decade, it led to the synthesis of the light-molecule endofullerenes (LMEFs) H2@C60,5,6 H2O@C60,7 HF@C60,8 and, most recently, CH4@C60.9 These endofullerenes are now available with high purity and in macroscopic quantities, making possible a wide range of spectroscopic studies of their fundamental properties and potential practical applications.10 

A distinctive feature of H2O@C60 and other LMEFs that has prompted a large number of experimental10 and theoretical studies11,12 are strong nuclear quantum effects (NQEs), which dominate the dynamics and spectroscopy of the guest molecules, particularly for low temperatures (typically ranging from 1.5 K to about 30 K), at which the spectroscopic measurements are typically performed. These species are the embodiments of many of the fundamental principles of quantum mechanics to the degree and with a clarity that are unrivaled in molecular systems amenable to experimental investigations. NQEs have multiple sources. One of them is the quantization of the translational center-of-mass (c.m.) degrees of freedom (DOFs) of the encapsulated molecules arising from their confinement inside the fullerene cavity (the textbook particle-in-a-box effect). Because of the tight confinement and low molecular masses of the guest molecules, the energy differences between their translational eigenstates are large relative to kT (k is the Boltzmann constant). The same holds for the quantized rotational states of these light molecules having one or more hydrogen atoms, due to their large rotational constants. Furthermore, the quantized translational and rotational DOFs of the guest molecule are coupled by the confining potential of the fullerene interior, resulting in a sparse and intricate translation–rotation (TR) energy level structure.11,12

When a molecule, such as H2O and H2, has two symmetrically equivalent H atoms, it exhibits the phenomenon of nuclear spin isomerism that introduces additional prominent NQEs in its TR dynamics inside the fullerene cages. The two identical 1H nuclei are fermions since their nuclear spin is 1/2. Satisfying the Pauli principle requires that the total molecular wave function of the caged molecule, its spatial and spin components, must be antisymmetric with respect to the exchange of the two fermions. A consequence of this is the particular entanglement of the spin and spatial quantum states, which results in nuclear spin isomers, para and ortho, of H2O (and H2), having total nuclear spins I = 0 and 1, respectively. The rotational states of H2O are conventionally labeled with the asymmetric top quantum numbers jkakc; for para-H2O, ka + kc has even parity, while for ortho-H2O, ka + kc has odd parity.13 The constraints that the Pauli principle places on the rotational quantum numbers of the spin isomers of H2O make its already sparse TR level structure inside the fullerenes even sparser, pushing the TR dynamics even deeper in the quantum regime.

Theoretical studies have thoroughly characterized the TR level structure of H2O@C60.11,12 The TR eigenstates of para- and ortho-H2O in an (isolated) C60 cage with Ih symmetry have been obtained by means of fully coupled quantum calculations in 6D, treating both H2O and C60 as rigid,14 and also in 9D, for flexible H2O and its excited inter- and intramolecular vibrational states inside C6015 (the 9D calculations in Ref. 16 were performed only for H2O in the ground intramolecular vibrational state). These calculations14,15 demonstrated that the TR level structure of H2O@C60 exhibits all of the key qualitative features previously identified for H2@C60.11,12,17,18 The purely translational eigenstates can be assigned with the quantum numbers of the 3D isotropic harmonic oscillator (HO), the principal quantum number n = 0, 1, 2, …, and the orbital angular momentum quantum number l = n, n − 2, …, 1 or 0, for odd and even n, respectively. For the H2O rotational excitations, the asymmetric top quantum numbers jkakc are employed. When the TR eigenstates are excited both translationally and rotationally, the orbital angular momentum l of the c.m. of H2O and the rotational angular momentum j of H2O couple to give the total angular momentum λ = l + j, with λ = l + j, l + j − 1, …, |lj|. The values of l are those allowed for the quantum number n of the 3D isotropic HO. The TR states having the same quantum numbers n and jkakc are split into as many distinct levels as there are different values of λ, each with the degeneracy 2λ + 1.14 A remarkable and comprehensive infrared (IR) spectroscopic study probing these and other features of intra- and intermolecular vibrations of H2O in C60 was published recently.19 

Much of what is known experimentally about the quantum TR dynamics of molecules having one or more hydrogen atoms, such as H2, HD, HF, and H2O, encapsulated inside C60 has come from the inelastic neutron scattering (INS) spectroscopy of these endohedral complexes.8,20–25 The exceptional power and utility of the INS stem from two unique characteristics. One of them is the unusually large cross section for the incoherent neutron scattering from the hydrogen (1H) nucleus; it is ∼15 times greater than for the nucleus of any other elements, including the isotope deuterium (D). As a result, INS is an exquisitely selective probe of the dynamics of the confined H-containing molecules. The peaks present in the measured INS spectra arise from the transitions between the TR eigenstates of the caged molecules induced by the incident neutrons. The second important advantage of the INS is that neutrons, unlike photons, can change the total nuclear spin of the guest molecule, allowing the observation of rotational Δj = 1 transitions that are forbidden in optical, IR and Raman, spectroscopy, since they involve the orthopara conversion.

The rich information content of the INS spectra of endohedral complexes has been underutilized for a long time. Its quantitative extraction is possible only with the help of high-level theory, given the intricacy of the TR dynamics. Complete analysis and assignment of the experimental INS spectra of the entrapped light molecules require theory that is capable of computing rigorously not only the TR excitation energies but also the intensities of the INS transitions. Together, they constitute a unique spectroscopic fingerprint of the system considered, enabling direct and more conclusive comparison with the experimental INS data. However, rigorous methodology for such calculations simply did not exist until rather recently.

In the past decade, we have developed an approach that enables rigorous and efficient quantum calculation of the INS spectra, i.e., the energies and the INS intensities of the TR transitions, of a diatomic molecule, initially homonuclear, such as H2, confined in a nanocavity of an arbitrary shape, assuming that the guest molecule and the cavity are rigid.26,27 What sets this approach apart from all others in the literature is that it uses the quantum 5D TR wave functions of the entrapped diatomic molecule, which fully couple its translational and rotational DOFs, as the spatial parts of the initial and final states of the INS transitions. As a result, the simulated INS spectra exhibit an unprecedented degree of realism and reflect the complexity of the TR dynamics of the guest molecule in nanoconfinement. The first demonstration of this novel methodology involved the calculation of the INS spectra of para- and ortho-H2 in the small cage of the sII clathrate hydrate,26 in remarkably good agreement with the experimental data.28 Subsequent application involved the computation of the INS spectra of H2@C60,29 for a range of temperatures. In the next step, this approach was extended to a heteronuclear diatomic molecule entrapped inside a nanoscale cavity.30 It was implemented to compute the INS spectra of HD in the small cage of the sII clathrate hydrate, which agreed well with the measured INS spectra of this system,28,31 and also in C60.32 The most recent application was to the quantum simulation of the INS spectra of HF@C60,33 which was instrumental in the assignment of the measured spectrum.8 An alternative approach for calculating the INS spectrum of a homonuclear diatomic molecule inside a spherical nanocavity was reported by Mamone et al.34 based on the expansion of the confining potential into multipoles of the coupled rotational and translational angular variables. It was applied to an impressive simulation of the INS spectrum of H2@C60.

In addition to being indispensable for the analysis of the experimental INS spectra, the new methodology has led to some genuine conceptual advances. The most important among them is the surprising discovery of the INS selection rule, in the field where it has long been taken for granted, as an article of faith, that the incoherent INS spectroscopy of the vibrations of discrete molecular systems is not subject to any selection rules.35–38 However, rigorous quantum calculations revealed that certain transitions in the INS spectra of H2@C6029 and HD@C6032 have zero-intensity (to 4–5 significant figures), strongly suggesting that they are forbidden according to some as yet unknown, and unsuspected, INS selection rule. This motivated the analytical derivation of the first ever selection rule in the INS vibrational spectroscopy, applicable to transitions originating from the ground TR state of para-H229 or HD32 confined in a near-spherical nanocavity, such as that of C60. Within a year of its publication, this precedent-setting selection rule was validated by a joint experimental and theoretical INS investigation of H2@C60.39 Shortly afterward, the generalized selection rule was derived, which defines all possible forbidden INS transitions, originating from a variety of TR states, ground and excited, of ortho-H2, para-H2, and HD, inside a near-spherical nanocavity.40 Almost simultaneously, the same generalized selection rule was derived for H2@C60, but not HD@C60, using group-theory arguments.41 A recent comprehensive review of the methodology for accurate quantum calculations of the INS spectra of diatomic molecules inside nanocavities and its applications is available.42 

Once the quantum methodology for computing the INS spectra of diatomic molecules in nanoconfinement was developed, it was natural to take the next step and extend the formalism to the INS spectra of triatomic molecules inside nanocavities, for which such a treatment is not available. Additional strong motivation for embarking on this endeavor came from the INS spectroscopic measurements of H2O@C60 by Horsewill and co-workers.24,25 In particular, Ref. 25 presented the INS spectra of H2O@C60 taken for several incident neutron wavelengths and a range of temperatures, as well as their analysis. These INS measurements showed that the three-fold degenerate 101 ground state of ortho-H2O is split by 4.19 cm−1 into a non-degenerate lower-energy state and a doubly degenerate higher-energy state, clear evidence that the symmetry of the environment felt by the caged H2O must be lower than Ih. Similar symmetry breaking was observed for solid H2@C6043 and HF@C60.8 Quantitative first-principles explanation of this phenomenon in terms of an electrostatic interaction model was developed by us in Refs. 11, 44, and 45.

In this paper, the methodology is introduced, which makes possible rigorous quantum computation of the INS spectra of a triatomic molecule in a nanocavity, specifically H2O inside the C60 cage. The present approach builds on our previous treatment of the INS spectra of nanoconfined diatomic molecules.26,27,30,33 However, going from a diatomic molecule to a (nonlinear) triatomic molecule inside a cavity increases the complexity of the treatment significantly, which also accounts for the effects of symmetry breaking. With this methodology, the INS spectra of H2O@C60 are calculated for two incident neutron wavelengths and compared to their experimental counterparts in Ref. 25. There is good overall agreement between the theory and experiment, and the former sheds additional light on the measured INS spectra.

This paper is organized as follows. The theory is described in Sec. II. In Sec. III, we present and discuss the results. Section IV contains the conclusions.

As in our previous treatments of the INS spectra of nanoconfined diatomic molecules (H2, HD, and HF),26,27,30,33 the starting point of the methodology for calculating the INS spectrum of H2O@C60 in this work is the standard expression for the neutron scattering double differential cross section in the first Born approximation,46,47

d2σdΩdω=kkS(κ,ω),
(1)

where

S(κ,ω)=ipif|Mfi|2δ[ω(ϵfϵi)/]
(2)

and

Mfi=nf|B̂nexp(iκrn)|i.
(3)

In Eqs. (1)(3), |i⟩ stands for the initial state of the scattering molecular system with the energy ϵi, pi is its statistical weight, |f⟩ is the final state with the energy ϵf, κ=kk, with k and k being the wave vectors of the incident and the scattered neutrons, respectively, ℏω = EE′ = 2(k2k2)/(2m), with m being the neutron mass, B̂n is the scattering length operator, and rn is the position of nucleus n. For a system of protons,

Bn̂=b1+b2σ2Sn,
(4)

where σ2 represents the neutron spin and Sn is the spin of proton n. The coefficients b1 and b2 can be found in Ref. 47.

The coordinates used in the treatment of the INS spectrum of H2O@C60 described below are depicted in Fig. 1. They are as follows:

rO=R+ρRO,rH1=R+ρRH1,rH2=R+ρRH2.
(5)

In Eq. (5), R fixes the position of the H2O c.m. relative to a space-fixed Cartesian system with the origin at the center of the C60 cage, while ρRO, ρRH1, and ρRH2 connect the c.m. of H2O to O, H1, and H2 atoms, respectively. In addition, the vector pointing from H1 to H2 is denoted as ρH2, and the vector from the c.m. of H2O to the midpoint of ρH2 is denoted as ρRHH. It is easy to see that

ρH2=rH2rH1,ρRHn=ρRHH+(1)n2ρH2.
(6)

It should be noted that ρRO, ρRH1, ρRH2, ρH2, and ρRHH all lie in the plane defined by the three atoms of H2O.

FIG. 1.

Vectors defining the position and orientation of H2O: R is the H2O c.m. position vector. ρRO, ρRH1, and ρRH2 connect the c.m. of H2O to O, H1, and H2, respectively. ρH2 is the vector pointing from H1 to H2. ρRHH is the vector from the H2O c.m. to the midpoint of ρH2.

FIG. 1.

Vectors defining the position and orientation of H2O: R is the H2O c.m. position vector. ρRO, ρRH1, and ρRH2 connect the c.m. of H2O to O, H1, and H2, respectively. ρH2 is the vector pointing from H1 to H2. ρRHH is the vector from the H2O c.m. to the midpoint of ρH2.

Close modal

With the above, Mfi in Eq. (3) can be written as

Mfi=n=13f|Bn̂exp(iκrn)|i=f|B̂Oexp(iκrO)+B̂Hexp(iκrH1)+B̂Hexp(iκrH2)|i=f|expiκRb1O+b2Oσ2SOexpiκρRO+n=12b1H+b2Hσ2SHnexpiκρRHn|i=f|expiκRV0+V1+V2+V3|i,
(7)

where

V0=b1OexpiκρRO,
(8)
V1=2b1Hexp(iκρRHH)cosκρH22,
(9)
V2=b2H2σ(SH1+SH2)exp(iκρRHH)cosκρH22,
(10)
V3=ib2H2σ(SH1SH2)exp(iκρRHH)sinκρH22.
(11)

The above equations already reflect that for 16O atoms, b2O=0.47 Moreover, the 17O and 18O isotopes have negligible abundance.

For H2O confined inside the C60 cavity, the total wave functions of the initial (|i⟩) and final states (|f⟩) in Eqs. (1)(3) can be written in the following product form:

|i=|Îi|Ψi6D(R,ϕ,θ,χ),|f=|Îf|Ψf6D(R,ϕ,θ,χ).
(12)

In Eq. (12), ϕ, θ, and χ are the Euler angles that specify the orientation of the body-fixed (BF) axes of H2O relative to the axes fixed to the C60 frame.14 |Îτ and |Ψτ6D(R,ϕ,θ,χ)(τ=i,f) are the nuclear-spin and 6D spatial wave functions, respectively, of the caged H2O molecule, respectively. |Ψτ6D(R,ϕ,θ,χ) are the eigenstates of the 6D TR Hamiltonian for rigid H2O molecules in (rigid) C60 obtained as described below. Hereafter, |Ψτ6D(R,ϕ,θ,χ) are denoted as |Ψτ6D(τ=i,f).

Taking advantage of the product form of the initial and final states in Eq. (12), Eq. (7) can be written as

Mfi=b1OδIfIiΨf6D|exp(iκR)exp(iκρRO)|Ψi6D+2b1HδIfIiΨf6D|exp(iκR)exp(iκρRHH)cosκρH22|Ψi6D+b2H2If|σ(SH1+SH2)|IiΨf6D|exp(iκR)exp(iκρRHH)×cosκρH22|Ψi6D+ib2H2If|σ(SH1SH2)|Ii×Ψf6D|exp(iκR)exp(iκρRHH)sinκρH22|Ψi6D.
(13)

The above equation for Mfi contains two distinct types of matrix elements: (i) those involving the nuclear-spin wave functions |Iτ⟩ (τ = i, f) and (ii) those involving the 6D spatial (TR) wave functions |Ψτ6D(τ=i,f). In the following, they are referred to as the spin and spatial matrix elements, respectively, and are evaluated separately below.

For H2O, the nuclear-spin state |ÎH2O, either para- (I = 0) or ortho- (I = 1), can be written as

|ÎH2O=|ÎO×|ÎH2=|0O×|ÎH2.
(14)

The nuclear-spin state |0⟩O of the oxygen atom is a singlet, while the nuclear-spin wave functions |ÎH2 are the same as those of the H2 molecule, singlet para- (I = 0) or triplet ortho- (I = 1) states. Consequently, for simplicity, |ÎH2O is hereafter referred to as |IH2.

In Eq. (13), it is necessary to compute two spin matrix elements that describe the coupling between the neutron spin and the nuclear spins of the H1 and H2 protons,

If|σ(SH1+SH2)|IiIfMIf|σI|IiMIi,If|σ(SH1SH2)|IiIfMIf|σ(i1i2)|IiMIi.
(15)

In the above equation, Sn is replaced by in(n=1,2), and the total spin ISH1+SH2 is defined. In addition, the nuclear-spin wave function |Iτ⟩ is written explicitly as |IτMIτ(τ=i,f).

Following closely the treatment of the spin matrix elements for H2 in a nanocavity,27 one obtains the equations for the nuclear-spin transition IiIf of the caged H2O molecule,

0014π2σcohHcosif(κ)H2+σcohOexpif(κ)O2,1114π2σcohHcosif(κ)H2+σcohOexpif(κ)O2+83σincHcosif(κ)H22,0114π4σincHsinif(κ)H22,1014π43σincHsinif(κ)H22,
(16)

where

cosif(κ)H2Ψf6D|exp(iκR)exp(iκρRHH)cosκρH22|Ψi6D,sinif(κ)H2Ψf6D|exp(iκR)exp(iκρRHH)sinκρH22|Ψi6D,expif(κ)OΨf6D|exp(iκR)exp(iκρRO)|Ψi6D.
(17)

Equation (16) represents the final expressions for the neutron scattering cross sections for the transitions between different nuclear-spin states of H2O. What remains is the evaluation of the spatial matrix elements in Eq. (17).

In Eq. (16), the coherent and incoherent scattering cross sections for the proton, σcohH and σincH, respectively, and the coherent scattering cross section for the O atom, σcohO, are48 

σcohH=4πb1H2=1.7587×1028m2,σincH=34πb2H2=80.2769×1028m2,σcohO=4πb1O2=4.2346×1028m2.
(18)

The spatial matrix elements in Eq. (17) are evaluated in Sec. II E. However, in order to set the stage for this, in Sec. II D, we first describe the computational approach used to calculate the 6D TR eigenstates |Ψτ6D(τ=i,f) of H2O inside the C60 cage. They are the spatial components of the states |i⟩ and |f⟩ in Eq. (12), which are the initial and final states, respectively, of the INS transitions.

The 6D TR eigenstates |Ψτ6D(τ=i,f) of H2O@C60 employed in our calculations of the INS spectra of this system incorporate the effects of symmetry breaking. The electrostatic interaction model, which we have developed to accomplish this and account quantitatively for the magnitude of the 1:2 splitting of the 101 ground state of ortho-H2O observed in the INS spectra of solid H2O@C60 at low temperatures,24,25 is described in detail in Refs. 11, 44, and 45, and only its salient features are summarized here.

At temperatures below 90 K, the fullerene molecules in the solid C60 are locked in, and coexist as, two orientationally ordered configurations of neighboring units.49 In the dominant one, referred to as the P orientation, the electron-rich double bonds shared by two hexagons (denoted 6:6) of one C60 unit face directly the electron-poor pentagons of the neighboring cages.49,50 The other, H orientation has the electron-rich 6:6 bonds of one C60 immediately adjacent to the electron-poor hexagonal faces of the neighboring units.49 Below 90 K, and at ambient pressure, the relative proportion of molecules in the P and H orientations is 5:1.49 The P and H orientations are relevant for symmetry breaking since for both the point-group symmetry of the environment at the center of a C60 cage is S6,43,51 capable of splitting the three-fold degeneracy of the 101 ground state of ortho-H2O.

Our approach11,44,45 focuses on the smallest fragment of solid C60 that can give rise to S6 symmetry in its center, comprised of 12 nearest-neighbor (NN) cages around the central cage, in either P or H crystal orientation. The cages are treated as rigid and assumed to have Ih symmetry. Only the central cage is occupied by the H2O molecule, while the 12 NN cages are left empty. At the heart of this approach is the proposition that the predominant source of the symmetry breaking in M@C60(s) (M = H2O, H2, HF) at low temperatures is the electrostatic interaction between the charge densities on the NN C60 cages and that on M in the central cage. The electron density on the NN cages is obtained from a first-principles density functional theory (DFT) calculation, and the charge density on M is related to its body-fixed quadrupole moments reported in the literature. Thus, the M–NN electrostatic interaction, denoted as VES, contains no adjustable parameters.

For this electrostatic interaction model, the 6D TR Hamiltonian of (rigid) H2O inside the central cage of the (rigid) 13-cage fragment of H2O@C60(s) can be written as

Ĥ=T̂+VH2OC60+VES,
(19)

where T̂ is the operator associated with the TR kinetic energy of H2O, VH2OC60 is the part of the 6D intermolecular PES arising from the H2O–central cage interaction, and VES is the H2O–NN electrostatic interaction above. VH2OC60 is defined in our previous study of H2O@C60;14 it is constructed as a sum over the pairwise interactions, modeled with the Lennard-Jones 12-6 potentials, of each atom of H2O with each atom of C60. In principle, one could try to represent VH2OC60 by a polynomial/multipole expansion, as was done for He@C60.52 However, it remains to be seen how well would this approach work for a 6D PES. Only the leading term in the multipole expansion of the H2O–NN electrostatic interaction is retained in VES.44,45 Owing to the S6 point-group symmetry of the 13-cage (central + NN) crystal fragment,53,54 that term is the quadrupolar one,

VESVquad=m=22(1)mQm(2)Im(2),
(20)

where Qm(2) constitute the electric-quadrupole spherical tensor of H2O@C60 and Im(2) constitute the electric-field-gradient tensor arising from the charges on the NN cages. It has been shown that retaining just the leading quadrupole terms in Vquad suffices to quantitatively account for the manifestations of symmetry breaking observed in all the three M@C60(s) (M = H2O, H2, and HF) species considered thus far.44,45 We demonstrated that the P crystal orientation, besides being dominant at low temperatures, causes much larger TR level splittings than the H crystal orientation44,45 and is therefore the one employed in the present calculations. The geometry of the 13-cage fragment of H2O@C60(s) is fully defined in the supplementary information of Ref. 44.

The only, but computationally significant, modification in the above procedure made in this work is that R, the position vector of the c.m. of H2O, is expressed in terms of Cartesian coordinates {x, y, z}, instead of the spherical polar coordinates {R, β, α} (where R|R|) used in our past work on H2O@C60.14,15 This allows us to use the 3D direct-product discrete variable representation (DVR)55 basis {|Xα⟩|Yβ⟩|Zγ⟩} in the x, y, z coordinates (the sinc-DVR56) as the basis for the translational c.m. degrees of freedom (DOFs) of H2O; it is labeled by the grid points {Xα}, {Yβ}, and {Zγ}, at which the respective DVR basis functions are localized. We showed previously27 that the use of this 3D DVR is crucial for the efficient computation of the spatial matrix elements, as it greatly simplifies the otherwise rather forbidding equations and drastically reduces the computational effort involved in their evaluation. The symmetric-top eigenfunctions, i.e., the normalized Wigner rotation functions {|j,m,k(2j+1)/8π212Dmkj(ϕ,θ,χ)*}, serve as the basis for the angular DOFs of H2O. The Euler angles ϕ, θ, and χ specify the orientation of the body-fixed (BF) axes of H2O relative to the axes fixed to the C60 frame. The BF z axis of H2O is taken to be the symmetry axis of the molecule, the x axis is taken to be the other principal axis of the molecule lying in the molecular plane, and the y axis is perpendicular to the molecular plane and is such that the axis system is right-handed.

Together, the above translational and rotational bases constitute the 6D direct-product basis set {|Xα⟩|Yβ⟩|Zγ⟩|jmk⟩}. The eigenstates |Ψi6D obtained by diagonalizing in this basis the 6D TR Hamiltonian for H2O@C60 in Eq. (19) are

|Ψi6D=αβγjmkAαβγjmk,i6D|Xα|Yβ|Zγ|jmk=αβγjmk2j+18π2Aαβγjmk,i6D|Xα|Yβ|ZγDmkj(Ω)*.
(21)

Let the direction of κ in Eq. (13) be defined by the angles (θκ,ϕκ). On the other hand, the directions of the three vectors ρη (η = RO, RH1, and RH2) in Fig. 1 are defined by the angles (θη, ϕη), respectively, in the cage-fixed Cartesian axis system. Therefore, we make use of the well known expression as follows:57 

exp(iκρη)=lmil4πjl(κρη)Ylm*(θκ,ϕκ)Ylm(θη,ϕη),
(22)

where ρη|ρη| and jl(κρη) are the spherical Bessel functions.58 

We now consider explicitly the term in Eq. (13) associated with the O atom. Then, following Eq. (22), we can write

exp(iκρRO)=lmil4πjl(κρ)Ylm*(θκ,ϕκ)Ylm(θρRO,ϕρRO).
(23)

Taking the initial angles of ρRO to be (θρRO0,ϕρRO0)=(π,π), under the rotation Ω ≡ (ϕ, θ, χ), we have

Ylm(θρRO,ϕρRO)=m=llDmml(Ω)*Ylm(π,π),
(24)

where Dmml(R)* is the complex conjugate of an element of the Wigner D-matrix. With this, Eq. (23) can be written as

exp(iκρRO)=lmmil4πjl(κρRO)Ylm*(θκ,ϕκ)Dmml(Ω)*×Ylm(π,π)LMMiL4πjL(κρRO)YLM*(θκ,ϕκ)×DMML(Ω)*YLM(π,π).
(25)

The change to upper-case indices L, M, and M″ in the above equation is made in order to avoid possible confusion with the indices used in the further derivation.

In the same vein, expressions analogous to that in Eq. (25) for ρRO can be obtained for ρRH1 and ρRH2. From the information in Table I, one can deduce that their initial angles are (θρRH10,ϕρRH10)=(θρRH10,0) and (θρRH20,ϕρRH20)=(θρRH10,ϕρRH10+π)=(θρRH10,π), respectively. With this, one can obtain the following expressions:

exp(iκρRH1)=LMMiL4πjL(κρRH1)YLM*(θκ,ϕκ)×DMML(Ω)*YLM(θρRH10,0)
(26)

and

exp(iκρRH2)=LMMiL4πjL(κρRH2)YLM*(θκ,ϕκ)×DMML(Ω)*YLM(θρRH10,π)=LMMiL4πjL(κρRH2)YLM*(θκ,ϕκ)×DMML(Ω)*(1)MYLM(θρRH10,0).
(27)

Based on Eqs. (26) and (27), and noticing that ρRH1=ρRH2, the following factors in Eq. (13) can be expressed as

TABLE I.

Cartesian coordinates (in Å), relative to the body-fixed axis system, of the H2O nuclei for the water geometry employed in the H2O@C60 calculations.

LabelXYZ
0.0000 0.0000 −0.065 638 07 
0.7575 0.0000 0.520 861 93 
−0.7575 0.0000 0.520 861 93 
LabelXYZ
0.0000 0.0000 −0.065 638 07 
0.7575 0.0000 0.520 861 93 
−0.7575 0.0000 0.520 861 93 
exp(iκρRHH)cosκρH22=LMMiL4πjL(κρRH1)YLM*(θκ,ϕκ)DMML(Ω)*1+(1)M2YLM(θρRH10,0),exp(iκρRHH)sinκρH22=LMMiL4πjL(κρRH1)YLM*(θκ,ϕκ)DMML(Ω)*(1)M12iYLM(θρRH10,0).
(28)

In the following, we make use of the equation

dΩDm,kj(Ω)DM,ML*(Ω)Dm,kj*(Ω)=8π2(1)m+kLjjMkkLjjMmm,
(29)

which follows from the well-known expression for the integral over the product of three rotation matrices.57,59 This allows us to obtain the matrix elements involving the expressions in Eqs. (25) and (28) for the initial and final angular states |jmk⟩ and |jmk′⟩, respectively,

jkm|exp(iκρRO)|jkm=(2j+1)(2j+1)(1)m+kLiL4πjL(κρRO)MLjjMmmYLM*(θκ,ϕκ)MLjjMkkYLM(π,π),jkm|exp(iκρRHH)cosκρH22|jkm=(2j+1)(2j+1)(1)m+kLiL4πjL(κρRH1)MLjjMmmYLM*(θκ,ϕκ)MLjjMkk1+(1)M2YLM(θρRH10,0),jkm|exp(iκρRHH)sinκρH22|jkm=(2j+1)(2j+1)(1)m+kLiL4πjL(κρRH1)MLjjMmmYLM*(θκ,ϕκ)MLjjMkk(1)M12iYLM(θρRH10,0).
(30)

In order to make the notation more compact, we now define the following quantities:

expjkm,jkmiκρRO(θκ,ϕκ)jkm|exp(iκρRO)|jkm,cosjkm,jkmiκρRHH(θκ,ϕκ)jkm|exp(iκρRHH)cosκρH22|jkm,sinjkm,jkmiκρRHH(θκ,ϕκ)jkm|exp(iκρRHH)sinκρH22|jkm,
(31)

where the matrix elements on the right-hand side are given in Eq. (30).

From the properties of the spherical harmonics,

Ylm(π,π)=(1)l2l+14πδm,0,
(32)
Ylm(θ,0)=(1)m2πPlm(θ).
(33)

With this and Eq. (30), for the matrix elements in Eq. (31), we obtain

expjkm,jkmiκρRO(θκ,ϕκ)=LiL4πjL(κρRO)(1)L(2j+1)(2j+1)(1)m+k2L+14πLjj0kk×MLjjMmmYLM*(θκ,ϕκ),cosjkm,jkmiκρRHH(θκ,ϕκ)=LiL4πjL(κρRH1)(2j+1)(2j+1)(1)m+k×MLjjMmmYLM*(θκ,ϕκ)MLjjMkk1+(1)M22πPLM(θρRH10),sinjkm,jkmiκρRHH(θκ,ϕκ)=LiL4πjL(κρRH1)(2j+1)(2j+1)(1)m+k×MLjjMmmYLM*(θκ,ϕκ)MLjjMkk1(1)M22πiPLM(θρRH10).
(34)

Taking into account Eqs. (21) and (34), and the DVR points Xα,Yβ,Zγ, the following final expressions can be derived for the entities on the left-hand side of Eq. (17):

expif(κ)O=αβγjmkjmkAαβγjmk,f6D*Aαβγjmk,i6Dexpjkm,jkmiκρRO(θκ,ϕκ)×expiκsinθκcosϕκXα+sinθκsinϕκYβ+cosθκZγ,cosif(κ)H2=αβγjmkjmkAαβγjmk,f6D*Aαβγjmk,i6Dcosjkm,jkmiκρRHH(θκ,ϕκ)×expiκsinθκcosϕκXα+sinθκsinϕκYβ+cosθκZγ,sinif(κ)H2=αβγjmkjmkAαβγjmk,f6D*Aαβγjmk,i6Dsinjkm,jkmiκρRHH(θκ,ϕκ)×expiκsinθκcosϕκXα+sinθκsinϕκYβ+cosθκZγ.
(35)

Equation (35) represents the key result of this derivation, as it allows the evaluation of the matrix elements in Eq. (17). These in turn enter Eq. (16), which gives the final equations for the neutron scattering cross sections for the transitions between different nuclear-spin states of the encapsulated H2O.

The experimental INS spectra of H2O@C60, with which in Sec. III we compare our computed INS spectra, were taken from a polycrystalline sample,25 in which the fullerene cages are randomly oriented with respect to the incoming neutron beam. Therefore, to achieve a more realistic comparison with the measured spectra, the INS spectra computed by the approach described above are averaged over all possible orientations of C60 by the procedure described previously27 and implemented in our quantum simulations of the INS spectra of H2/HD@C6029,32 and HF@C60.33 The numerical integration involved in the powder averaging of S(κ,ω) in Eq. (2) over the directions of κ, defined by the angles (θκ,ϕκ), to yield S(κ, ω), is carried out using the ten-point Gauss–Legendre quadrature in θκ and the 20-point Gauss–Chebyshev quadrature in ϕκ. Moreover, these spectra were measured on the IN4C spectrometer, which allows the spectra to be recorded for a broad range of κ|κ| values. This means that S(κ, ω) needs to be integrated also over κ in order to obtain the desired S(ω), as described in Ref. 32.

The above integration of the computed INS spectrum over both the angles (θκ,ϕκ) and κ is performed for each INS transition in the range of excitation energies considered. The procedure is computationally time consuming but necessary and results in a realistic simulation of the experimental INS spectra.

In the calculations of the 6D TR eigenstates of H2O@C60 described in Sec. II D, the mass of the H2O molecule equal to 18.010 56 amu is used, together with the moments of inertia of H2O about the BF axes, Ix = 0.604 72, Iy = 1.8156, and Iz = 1.161 64 amu Å2 (taken to be those associated with the measured rotational constants of H2O in the ground vibrational state:60A = 27.877, C = 9.285, and B = 14.512 cm−1, respectively). The dimension of the sinc-DVR basis is 14 for each of the three Cartesian coordinates x, y, and z, spanning the range from −1 to +1 Å in each coordinate. In addition, all angular |j, m, k⟩ basis functions for 0 ≤ j ≤ 6 are included in the 6D basis.

The energies of the transitions in the INS spectra discussed here do not exceed 10 meV, while the first translationally excited state of para-H2O on the PES used in this work is at 21.4 meV. Therefore, all of the TR states involved are purely rotational, in the ground translational state. For them, the translational quantum numbers n and l of the 3D isotropic HO14,15 are equal to zero and are not included in the assignments. Hence, hereafter, the TR states are assigned as (jkakc,λ,|mλ|), where |mλ| is the absolute value of the component of λ along the C3 axis of the S6 point group.45 Both λ and |mλ| remain very good quantum numbers in the presence of the quadrupolar H2O–NN interactions, and the latter labels the sublevels arising from the symmetry breaking.45 

In the analysis of the INS spectra, below, we will often refer to Table 3 of Ref. 45, which gives the energies and other properties of the calculated TR eigenstates of H2O@C60 in the presence of symmetry breaking, computed as described in Sec. II D.

The INS spectra of H2O@C60(s) computed for the incident neutron wavelengths λn = 3.0 Å and λn = 2.3 Å, at 0 K, are shown in Figs. 2 and 3, respectively. They correspond to the experimental INS spectra of Goh and co-workers25 recorded for the same neutron wavelengths and at 1.6 K, which are displayed in Figs. 2 and 5, respectively, of Ref. 25. The benefits of using different incident neutron wavelengths are apparent from the comparison of the INS spectra, calculated or measured, for these two incident wavelengths. Decreasing the wavelength, i.e., increasing the energy of the incoming neutrons, enables excitation of higher-energy TR states of the guest H2O. However, this also lowers the energy resolution of the INS spectrum, which, as discussed below, leaves some important features of the TR energy level structure unresolved—which are resolved for longer incident wavelengths.

FIG. 2.

INS spectrum of H2O@C60(s) computed for 0 K and the incident neutron wavelength λn = 3.0 Å. The quantity on the horizontal axis is the neutron energy transfer ΔE = EE′, where E and E′ are the energies of the incident and scattered neutrons, respectively. ΔE is positive in NE loss and negative in NE gain. The stick spectra shown are calculated assuming the 3:1 ratio of ortho-H2O to para-H2O. The transitions originating from the ground state of para-H2O are shown with black vertical lines, while the transitions from the ground state of ortho-H2O are represented with red vertical lines. The full red line depicts the stick spectrum for the 3:1 ratio of ortho-H2O to para-H2O convolved with the instrumental resolution function. The dashed blue line depicts the stick spectrum for the 1:1 ratio of ortho-H2O to para-H2O (not shown) convolved with the instrumental resolution function. The INS transitions associated with labels 1–6 are listed in Table II.

FIG. 2.

INS spectrum of H2O@C60(s) computed for 0 K and the incident neutron wavelength λn = 3.0 Å. The quantity on the horizontal axis is the neutron energy transfer ΔE = EE′, where E and E′ are the energies of the incident and scattered neutrons, respectively. ΔE is positive in NE loss and negative in NE gain. The stick spectra shown are calculated assuming the 3:1 ratio of ortho-H2O to para-H2O. The transitions originating from the ground state of para-H2O are shown with black vertical lines, while the transitions from the ground state of ortho-H2O are represented with red vertical lines. The full red line depicts the stick spectrum for the 3:1 ratio of ortho-H2O to para-H2O convolved with the instrumental resolution function. The dashed blue line depicts the stick spectrum for the 1:1 ratio of ortho-H2O to para-H2O (not shown) convolved with the instrumental resolution function. The INS transitions associated with labels 1–6 are listed in Table II.

Close modal
FIG. 3.

INS spectrum of H2O@C60(s) computed for 0 K and the incident neutron wavelength λn = 2.3 Å. The quantity on the horizontal axis is the neutron energy transfer ΔE = EE′, where E and E′ are the energies of the incident and scattered neutrons, respectively. ΔE is positive in NE loss and negative in NE gain. The stick spectra shown are calculated assuming the 3:1 ratio of ortho-H2O to para-H2O. The transitions originating from the ground state of para-H2O are shown with black vertical lines, while the transitions from the ground state of ortho-H2O are represented with red vertical lines. The full red line depicts the stick spectrum for the 3:1 ratio of ortho-H2O to para-H2O convolved with the instrumental resolution function. The dashed blue line depicts the stick spectrum for the 1:1 ratio of ortho-H2O to para-H2O (not shown) convolved with the instrumental resolution function. The INS transitions associated with labels 1–6 are the same as in Fig. 2 and are listed in Table II. The three transitions associated with label 7, at 9.53, 9.66, and 10.01 meV, originate from the (000, 0, 0) ground state of para-H2O. Their final states are (212, 2, |mλ|), |mλ| = 0, 1, and 2, respectively. The quantum numbers (jkakc,λ,|mλ|) are explained in the text.

FIG. 3.

INS spectrum of H2O@C60(s) computed for 0 K and the incident neutron wavelength λn = 2.3 Å. The quantity on the horizontal axis is the neutron energy transfer ΔE = EE′, where E and E′ are the energies of the incident and scattered neutrons, respectively. ΔE is positive in NE loss and negative in NE gain. The stick spectra shown are calculated assuming the 3:1 ratio of ortho-H2O to para-H2O. The transitions originating from the ground state of para-H2O are shown with black vertical lines, while the transitions from the ground state of ortho-H2O are represented with red vertical lines. The full red line depicts the stick spectrum for the 3:1 ratio of ortho-H2O to para-H2O convolved with the instrumental resolution function. The dashed blue line depicts the stick spectrum for the 1:1 ratio of ortho-H2O to para-H2O (not shown) convolved with the instrumental resolution function. The INS transitions associated with labels 1–6 are the same as in Fig. 2 and are listed in Table II. The three transitions associated with label 7, at 9.53, 9.66, and 10.01 meV, originate from the (000, 0, 0) ground state of para-H2O. Their final states are (212, 2, |mλ|), |mλ| = 0, 1, and 2, respectively. The quantum numbers (jkakc,λ,|mλ|) are explained in the text.

Close modal

The stick spectra appearing in Figs. 2 and 3 are calculated for the 3:1 ratio of ortho-H2O to para-H2O. In order to facilitate visual comparison with the measured INS spectra, the stick spectra in both figures are convolved with the instrumental resolution function, resulting in smooth spectra depicted by the full red line. Also shown in Figs. 2 and 3 are smooth INS spectra depicted by the dashed blue line; they correspond to the stick spectra simulated for the 1:1 ratio of ortho-H2O to para-H2O (not shown for clarity) convolved with the instrumental resolution function. The purpose of computing the INS spectra for 3:1 and 1:1 ratios of ortho-H2O to para-H2O is to roughly mimic the corresponding experimental INS spectra recorded in the first hour and eighth hour after cooling, respectively.25 Spontaneous interconversion from ortho-H2O to para-H2O over several hours observed in H2O@C60 decreases the initial 3:1 ortho- to para-H2O ratio.25 

The quantity appearing on the abscissa of Figs. 2 and 3 is the neutron energy transfer ΔE = EE′, where E and E′ are the energies of the incident and scattered neutron, respectively. ΔE is positive in neutron-energy (NE) loss, when the incoming neutron transfers a part of its energy to the sample and excites the caged H2O from a lower to a higher TR energy level. In NE gain, when ΔE is negative, the scattered neutron gains energy when H2O makes a downward transition from a higher to a lower TR energy level.

The peaks in Figs. 2 and 3 have labels 1–6 and 1–7, respectively. The corresponding transitions 1–6 are given in Table II.

TABLE II.

INS transitions associated with each of the labels 1–6 in Fig. 2. They are out of the (101, 1, 0) ground state of ortho-H2O or the (000, 0, 0) ground state of para-H2O. ΔE is the neutron energy transfer. The intensities calculated for the incident neutron wavelength λn = 3.0 Å are in the units of barn (b); 1 b = 1 × 10−24 cm2. The quantum numbers (jkakc,λ,|mλ|) of the initial and final states of the INS transitions listed are explained in the text; g is the degeneracy of the final state.

LabelΔE (meV)Intensity (b)Initial (jkakc,λ,|mλ|)Final (jkakc,λ,|mλ|)Final g
0.52 12.11 (101, 1, 0) (101, 1, 1) 
1.91 7.67 (101, 1, 0) (111, 1, 1) 
1.96 4.67 (101, 1, 0) (111, 1, 0) 
2.42 38.94 (101, 1, 0) (110, 1, 1) 
2.94 0.000 (101, 1, 0) (110, 1, 0) 
2.61 97.10 (000, 0, 0) (101, 1, 0) 
3.13 171.07 (000, 0, 0) (101, 1, 1) 
5.81 19.37 (101, 1, 0) (202, 2, 0) 
5.93 29.52 (101, 1, 0) (202, 2, 1) 
6.35 0.04 (101, 1, 0) (202, 2, 2) 
6.92 11.37 (101, 1, 0) (212, 2, 0) 
7.05 15.51 (101, 1, 0) (212, 2, 1) 
7.40 0.76 (101, 1, 0) (212, 2, 2) 
LabelΔE (meV)Intensity (b)Initial (jkakc,λ,|mλ|)Final (jkakc,λ,|mλ|)Final g
0.52 12.11 (101, 1, 0) (101, 1, 1) 
1.91 7.67 (101, 1, 0) (111, 1, 1) 
1.96 4.67 (101, 1, 0) (111, 1, 0) 
2.42 38.94 (101, 1, 0) (110, 1, 1) 
2.94 0.000 (101, 1, 0) (110, 1, 0) 
2.61 97.10 (000, 0, 0) (101, 1, 0) 
3.13 171.07 (000, 0, 0) (101, 1, 1) 
5.81 19.37 (101, 1, 0) (202, 2, 0) 
5.93 29.52 (101, 1, 0) (202, 2, 1) 
6.35 0.04 (101, 1, 0) (202, 2, 2) 
6.92 11.37 (101, 1, 0) (212, 2, 0) 
7.05 15.51 (101, 1, 0) (212, 2, 1) 
7.40 0.76 (101, 1, 0) (212, 2, 2) 

The only states populated at 0 K in the calculations (and 1.6 K in the experiments25) are the (000, 0, 0) ground state of para-H2O and the (101, 1, 0) ground state of ortho-H2O. The latter is the lower-energy component of the three-fold degenerate 101 ground state of ortho-H2O split in the 1:2 pattern by the symmetry breaking.25,44 Therefore, all INS transition present in the calculated spectra in Figs. 2 and 3, as well as in the measured INS spectra shown in Figs. 2 and 5 of Ref. 25, must originate from one of these two states.

We now discuss the INS spectrum in Fig. 2 calculated for the incident neutron wavelength λn = 3.0 Å. Only one peak is present in the NE gain part of the spectrum, at −2.61 meV; its origin is in the transition from the (101, 1, 0) ground state of ortho-H2O to the (000, 0, 0) ground state of para-H2O. This provides clear evidence for the existence of two nuclear spin isomers of water, ortho-H2O, and para-H2O. The same peak is present in the measured INS spectrum25 at −2.56 meV. Its intensity decays with time, evident in Fig. 2 of Ref. 25, as a result of slow conversion from ortho- to para-H2O. This agrees with the fact that the height of the peak at −2.61 meV in Fig. 2 computed for the 3:1 ratio of ortho-H2O to para-H2O is greater than that obtained for the 1:1 ratio of these two spin isomers.

In the NE loss, the lowest-energy calculated peak at 0.52 meV, labeled 1 in Fig. 2, arises from the transition (101, 1, 0) → (101, 1, 1) between the lower- and higher-energy components of the 1:2 split ground state of ortho-H2O. This peak is not identified in the measured INS spectrum since it appears as barely a shoulder of the massive peak centered at 0 meV.25 

Label 2 applies to the peak, which according to Table II corresponds to the pair of orthopara transitions, (101, 1, 0) → (111, 1, 1) and (101, 1, 0) → (111, 1, 0), at 1.91 and 1.96 meV, respectively.

The peak at 2.42 meV labeled 3 is associated with the orthoortho transition (101, 1, 0) → (110, 1, 1) (Table II). However, it is evident from Table 3 of Ref. 45 that the final state 110 is actually split by the symmetry breaking into two components, (110, 1, 1) at 5.04 meV and (110, 1, 0) at 5.55 meV. Therefore, another peak should be present in the spectrum at 2.94 meV due to the transition (101, 1, 0) → (110, 1, 0), but it is absent since its calculated intensity is zero (0.000000). This is reminiscent of the situation for H2@C60, where our calculations revealed that certain INS transitions have zero intensity as well. This led us to postulate and derive the first ever, and entirely unexpected, selection rule in the INS spectroscopy of discrete molecular compounds.29 Soon thereafter, the predicted precedent-setting selection rule was validated by a joint experimental and theoretical INS investigation of H2@C60.39 It remains to be seen whether a similar INS selection rule exists for H2O@C60.

In the corresponding measured INS spectrum in Fig. 2 of Ref. 25, this orthoortho transition, marked as 110, is assigned to a shoulder just below 2 meV. However, as discussed above, according to our calculations, this is where the two orthopara transitions (101, 1, 0) → (111, 1, 1) and (101, 1, 0) → (111, 1, 0) appear (label 2).

Label 4 refers to the two peaks at 2.61 and 3.13 meV, respectively, associated with the transitions out of the (000, 0, 0) ground state of para-H2O to the two components of the ground state of ortho-H2O, (101, 1, 0) and (101, 1, 1), resulting from the symmetry breaking. In the experimental INS spectrum, these two transitions are observed at slightly different energies, 2.50 and 3.02 meV, respectively.25 However, the measured and calculated splittings of the ground state of ortho-H2O are the same, 0.52 meV.44,45

Label 5 applies to three orthopara transitions at 5.81, 5.93, and 6.35 meV, respectively: (101, 1, 0) → (202, 2, 0), (101, 1, 0) → (202, 2, 1), and (101, 1, 0) → (202, 2, 2), respectively (Table II). They arise from the symmetry breaking of the final rotational state 202 of para-H2O into three components (Table 3 of Ref. 45). Only two of these INS transitions, at 5.81 and 5.93 meV, are visible in the calculated spectrum in Fig. 2 since the intensity of the third at 6.35 meV is extremely small (Table II). In the INS spectrum measured for λn = 2.3 Å, this orthopara transition is assigned to the peak at 5.68 meV (Table 1 in Ref. 25).

Finally, label 6 denotes the three orthoortho transitions at 6.92, 7.05, and 7.40 meV, respectively: (101, 1, 0) → (212, 2, 0), (101, 1, 0) → (212, 2, 1), and (101, 1, 0) → (212, 2, 2), respectively (Table II). They are due to the symmetry breaking of the final rotational state 212 of ortho-H2O into three components (Table 3 of Ref. 45). Of these three INS transitions, only two, at 6.92 and 7.05 meV, are visible in the calculated spectrum in Fig. 2 since the intensity of the third at 7.40 meV is very small (Table II). In the INS spectrum measured for λn = 2.3 Å, this orthoortho transition is assigned to the peak at 6.51 meV (Table 1 in Ref. 25).

In the following, we discuss briefly the INS spectrum in Fig. 3 calculated for the incident neutron wavelength λn = 2.3 Å. In its main features, including the transitions with labels 1–6, it bears close resemblance to the spectrum in Fig. 2 computed for λn = 3.0 Å. An additional band appears, labeled 7, which is centered at 9.6 meV. It is associated with the peaks at 9.53, 9.66, and 10.01 meV, respectively, corresponding to the three transitions out of the (000, 0, 0) ground state of para-H2O to the final states (212, 2, |mλ|), |mλ| = 0, 1, and 2 of ortho-H2O. In the INS spectrum measured for λn = 2.3 Å, this paraortho transition is assigned to the peak at 9.20 meV (Table 1 in Ref. 25).

As mentioned earlier, the ability to access higher-energy TR states comes at price of a lower resolution of the INS spectrum. In particular, the feature labeled 4 arising from the transitions out of the (000, 0, 0) ground state of para-H2O to the (101, 1, 0) and (101, 1, 1) components of the ground state of ortho-H2O, which appears as two well-resolved peaks in the λn = 3.0 Å spectrum in Fig. 2, is unresolved in the λn = 2.3 Å spectrum. The same is true for the corresponding experimental INS spectra shown in Figs. 2 and 5 of Ref. 25.

Overall, there is good correspondence between the INS spectra calculated in this work and the measured INS spectra reported in Ref. 25. The energies of the calculated transitions are typically higher by a fraction of a wavenumber than the corresponding ones in the experimental INS spectra. This undoubtedly reflects the shortcomings of the PES employed in the calculations, not surprising given its rather crude pairwise-additive character. On the other hand, the calculated INS spectra provide valuable insights that are obscured in the measured spectra. Thus, they reveal that most bands in the measured spectra in fact comprise 2–3 unresolved individual transitions, largely caused by symmetry breaking, whose intensity can vary greatly. They also reveal transitions, such as (101, 1, 0) → (101, 1, 1) at 0.52 meV, that are hidden as shoulders of much more intense transitions and are therefore difficult to identify with confidence in the measured INS spectra. Moreover, one calculated transition, (101, 1, 0) → (110, 1, 0), is shown to have zero intensity, hinting at the possible existence of an INS selection rule in this system, akin to the one we uncovered for H2@C60.29,39

We have presented a methodology that allows rigorous quantum calculation of the INS spectra of H2O inside C60, with both moieties taken to be rigid. With relatively straightforward modifications, the method can be applied to simulating the INS spectra of other triatomic molecules confined inside arbitrarily shaped nanocavities. This methodology represents a natural extension of the quantum treatment pioneered by us previously to compute the INS spectra of diatomic molecules in nanocavities,26,27,30,33 which was applied successfully to diatomics nanoconfined inside the cages of C60 and clathrate hydrates.11,42 The treatment in this paper incorporates the quantum 6D TR wave functions of the entrapped water molecule, which couple its translational and rotational DOFs, as the spatial parts of the initial and final states of the INS transitions. This allows the simulated INS spectra to reflect in full the complexity of the TR dynamics of the guest molecule in nanoconfinement. Moreover, the 6D TR eigenstates are calculated in a way that accounts quantitatively for the effects of the symmetry breaking.44,45 As a result, the computed INS spectra exhibit characteristic features present in the INS spectra measured for solid H2O@C60 at low temperatures,25 which arise from symmetry breaking.

In this paper, the newly developed methodology is utilized to compute the INS spectra of H2O@C60 for two incident neutron wavelengths, λn = 3.0 Å and λn = 2.3 Å, at 0 K. Their comparison with the experimental INS spectra25 recorded for the same incident neutron wavelengths at 1.6 K reveals very good overall match between the theory and experiment. However, the calculated INS spectra lead to deeper understanding and interpretation of their experimental counterparts. What emerges from the former is that underneath most of the bands present in the measured spectra, there are 2–3 individual transitions, mostly stemming from symmetry breaking, which can differ greatly in intensity, that the experiments do not resolve. Moreover, the simulated spectra are able to reveal transitions that in the measured spectra are hidden in the shoulders of more intense transitions, and whose existence, and assignment, therefore cannot be established unambiguously. An intriguing result of our calculations is that one calculated orthoortho transition, (101, 1, 0) → (110, 1, 0), has zero intensity. This suggests the possibility that an INS selection rule exists in this system that remains to be identified, analogous to the one we uncovered for H2@C60.29,39 Future research will explore this fundamental issue in detail.

The accuracy of the spectra computed by the methodology developed in this work is limited largely by the quality of the intermolecular PES employed. In the case of H2O@C60, clearly, there is room for improvement, and we hope that the present results will add to the motivation for computing a high-quality endohedral PES of this system. Moreover, the recent synthesis of CH4@C609 suggests further methodological development that would enable quantum simulations of the INS spectra of this endohedral complex as well.

Z.B. and P.M.F. acknowledge the National Science Foundation for its partial support of this research through Grant Nos. CHE-2054616 and CHE-2054604, respectively. P.M.F. is grateful to Professor Daniel Neuhauser for his support.

The authors have no conflicts to disclose.

The data that support the findings of this study are available within the article.

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