Intramolecular stretching vibrational states and frequency shifts of (H2)2 confined inside the large cage of clathrate hydrate from an eight-dimensional quantum treatment using small basis sets.

We report the results of calculations pertaining to the HH intramolecular stretching fundamentals of (p-H2)2 encapsulated in the large cage of structure II clathrate hydrate. The eight-dimensional (8D) quantum treatment assumes rotationless (j = 0) H2 moieties and a rigid clathrate structure but is otherwise fully coupled. The (H2)2-clathrate interaction is constructed in a pairwise-additive fashion, by combining the ab initio H2-H2O pair potential for flexible H2 and rigid H2O [D. Lauvergnat et al., J. Chem. Phys. 150, 154303 (2019)] and the six-dimensional (6D) H2-H2 potential energy surface [R. J. Hinde, J. Chem. Phys. 128, 154308 (2008)]. The calculations are performed by first solving for the eigenstates of a reduced-dimension 6D "intermolecular" Hamiltonian extracted from the full 8D Hamiltonian by taking the H2 moieties to be rigid. An 8D contracted product basis for the solution of the full problem is then constructed from a small number of the lowest-energy 6D intermolecular eigenstates and two discrete variable representations covering the H2-monomer internuclear distances. Converged results are obtained already by including just the two lowest intermolecular eigenstates in the final 8D basis of dimension 128. The two HH vibrational stretching fundamentals are computed for three hydrate domains having an increasing number of H2O molecules. For the largest domain, the two fundamentals are found to be site-split by ∼0.5 cm-1 and to be redshifted by about 24 cm-1 from the free-H2 monomer stretch frequency, in excellent agreement with the experimental value of 26 cm-1. A first-order perturbation theory treatment gives results that are nearly identical to those of the 8D quantum calculations.


I. INTRODUCTION
Hydrogen clathrate hydrates are inclusion compounds in which one or more hydrogen molecules are confined inside closely packed polyhedral cavities within the three-dimensional (3D) crystalline framework created by hydrogen-bonded water molecules. [1][2][3] Simple hydrogen clathrate hydrates, with only hydrogen molecules as guests, first identified by Dyadin et al., 4 and subsequently studied in more detail by Mao et al., 5 adopt the classical structure II (sII). 1,2,5 Its unit cell is cubic, comprised of two types of cages: (a) sixteen small cages, each consisting of 20 H 2 O molecules and denoted 5 12 due to its 12 pentagonal faces, and (b) eight large cages, each formed by 28 H 2 O molecules arranged in 12 pentagonal and 4 hexagonal faces, and therefore denoted as 5 12 6 4 . The small cage has been shown to accommodate only one H 2 molecule, while up to four H 2 molecules can be encapsulated in the large cage. 6 Hydrogen clathrate hydrates have been the subject of a great deal of research in recent years, owing to their potential as hydrogen storage materials that could be both economical and environmentally friendly. 1,2,[7][8][9][10] The dynamics and spectroscopy of hydrogen molecules entrapped inside the cages of the clathrate hydrate are dominated by strong quantum effects, to a degree seen only for light molecules inside fullerenes. 11 These quantum effects have multiple sources. Nanoscale confinement in the hydrate cages gives rise to the quantized translational center-of-mass (c.m.) degrees of freedom (DOFs) of the guest molecule(s) (particle-in-a-box effect). They are coupled by the confining potential of the hydrate cage to the also quantized rotational DOFs of the hydrogen molecule(s). The translation-rotation (TR) energy level structure that results is sparse, due to the the low molecular mass of H 2 /HD/D 2 , their large rotational constants, and the small size of the hydrate cavities. For a single hydrogen molecule in the cages of the sII clathrate hydrate, the salient features of its TR eigenstates, notably the splittings of both the translational fundamental and rotational levels, as well as their manifestations in the inelastic neutron scattering (INS) spectra, have been characterized by Bačić and coworkers through quantum 5D bound-state calculations [12][13][14][15] and rigorous computations of the corresponding INS spectra. [15][16][17][18][19] These features have also been observed experimentally, in the INS 20,21 and the Raman spectra 22-24 of the binary tetrahydrofuran (THF) + H 2 /HD/D 2 sII clathrate hydrate. The quantum TR dynamics of up to four H 2 molecules in the large hydrate cage has been studied at T = 0 K by means of the diffusion Monte Carlo (DMC) calculations, 25 and at elevated temperatures (T = 25 − 200 K) using path-integral molecular dynamics (PIMD) simulations. 26 . In addition, fully quantal calculations of the TR eigenstates have been performed for two 27,28 and four 29 H 2 molecules inside the large clathrate hydrate cage. In all these calculations, both the H 2 molecules and the hydrogen-bonded clathrate hydrate framework were treated as rigid. Recently, 30 this constraint was relaxed partially, by performing quantum 5D calculations of the TR levels of (rigid) H 2 in the small sII hydrate cage, while taking into account the quantum delocalization of the proton nuclei of the framework water molecules arising from their hindered rotations about the fixed positions of their O atoms.
Another spectroscopic manifestation of the encapsulation of hydrogen molecules in the cages of clathrate hydrates, particularly relevant for this study, is the shift in the frequency of the H 2 intramolecular stretching vibration away from that in the gas phase. It is readily observable in the Raman spectra of the binary tetrahydrofuran (THF) + H 2 sII hydrate, where the large cages are completely occupied by the THF while the small cages are singly occupied by H 2 , and also in those of simple sII hydrates in which H 2 molecules are the only guests. 10,22,23 The vibrational frequencies of H 2 molecules encapsulated in the sII hydrates are always lower than, i.e., redshifted relative to, the gas-phase H 2 . The largest redshift, −34 cm −1 , is observed in the Raman spectra of the THF + H 2 sII hydrate, and can be assigned unambiguously to the singly H 2 occupied small cage. 10,22,23 The same redshift of −34 cm −1 appearing in the Raman spectra of the simple sII hydrate is therefore also attributed to H 2 in the small cage.
Also observed in the Raman spectra of the simple sII hydrate are the bands redshifted by −26, −18, and −11 cm −1 , respectively, 10,22,23 that must represent contributions from the large cages whose H 2 occupancy ranges between two and four. Thus, the frequency shifts are very sensitive to the number of H 2 molecules confined in the cage. However, interpreting them in terms of a particular H 2 occupancy of the large cages has turned out to be nontrivial, and the initial attempts 22 proved to be incorrect. The subsequent elaborate experiments led to the assignment of these three redshifts to double, triple, and quadruple H 2 occupancies of the large cages, respectively. 23 In the case of sII hydrogen hydrates, it was possible to assign the observed frequency shifts to different H 2 occupancies of the small and large clathrate cages based on detailed experimental data alone. However, in general, e.g., molecular hydrogen in metal-organic frameworks (MOFs), 31,32 extracting the information encoded in the measured vibrational frequency shifts regarding the H 2 occupancies of the cavities of nanoporous materials, and other structural as well as dynamical aspects of the entrapped H 2 , demands theoretical methods capable of accurate calculation of the frequency shifts. This task is highly challenging, for two reasons. The first is posed by the high dimensionality of the problem. Even if the host cavity is treated as rigid, the dimensionality of the calculations is 6nD, n being the number of encapsulated H 2 molecules, when treated as flexible. Thus, for n = 1 − 4, one has to be able to deal with the problem whose dimensionality ranges from 6D to 24D. This requires having accurate high-dimensional potential energy surfaces (PESs), that incorporate the H 2 -clathrate interactions and, in the case of multiple occupancy, the interactions among the guest H 2 molecules. In addition, the interactions must include the dependence on the H 2 intramolecular stretch coordinate and its coupling to the intermolecular DOFs. Second, dynamical quantum effects and anharmonicities in both intra-and intermolecular DOFs are significant, particularly at the low temperatures of the Raman spectroscopy measurements.
Consequently, these key features have to be described correctly by any first-principles theoretical method whose goal is to determine accurate intramolecular vibrational frequency shifts of encapsulated hydrogen molecules.
A number of approaches, relying on different approximations, have been utilized to address this fundamental and difficult problem. In some of them, the H 2 molecules encapsulated in the isolated small or large hydrate cages were treated as static, frozen in the geometry corresponding to the minimum energy of the system. [33][34][35] This leaves out nuclear quantum effects, especially the averaging over the large-amplitude intermolecular vibrations of the guest H 2 molecules. This problem was also approached by combining classical molecular dynamics (MD) and PIMD simulations with electronic structure calculations at the DFT (B3LYP) and MP2 levels. 36 The H 2 vibrational frequencies calculated in 1D for the H 2 intermolecular coordinates taken from many snapshots of the MD simulations covered a broad distribution of frequencies that extended to that of the free H 2 at 4155 cm −1 . Their maxima agree reasonably well with experiment, after a scaling factor was introduced in the calculations. Finally, classical MD simulations within the DFT framework were performed for an sII hydrate unit cell, and the H 2 vibrational spectra were calculated by Fourier transforming the H-H bond length autocorrelation function. 37 This classical treatment does not account for the quantum effects. Moreover, it gives the vibrational spectra that are shifted by 100-150 cm −1 to higher frequencies relative to the experimental results, and above the stretch fundamental of free H 2 .
Recently, Powers et al. 38 have calculated the frequency shift of H 2 inside the small cage of the sII hydrate, isolated and surrounded by spherical hydrate domains of increasing size, allowing the investigation of the effects of the condensed-phase environment. The approach employed was that developed earlier by Bačić and co-workers for the purpose of computing the HF stretch frequency shift in Ar n HF clusters. [39][40][41][42] The H 2 frequency shift was obtained by means of the quantum 5D bound-state calculations of the coupled TR eigenstates on a pair of effective pairwise-additive intermolecular PESs for rigid H 2 in a (rigid) hydrate domain that depend on the vibrational state of H 2 , v = 0 or v = 1, respectively. These 5D PESs were constructed using the 5D (rigid-monomer) pair potential for the interaction of ground-state calculations, but already the first excited state poses a challenge arising from the need to locate the node in the wave function, which is generally unknown (unless it can be determined from symmetry considerations 45 ). Therefore, the calculations for the first excited vibrational state of the caged H 2 were done in the fixed-node approximation, applying the "adiabatic" method of McCoy and co-workers 46 to find the position of the node.
In our recent study, 47 we presented the results of the first fully coupled quantum 6D calculations of the vibration-translation-rotation (VTR) eigenstates of a single flexible H 2 , HD and D 2 molecule entrapped in the (rigid) small cage of the sII hydrate, that extended to the first excited (v = 1) vibrational state of H 2 . Prior to this work, it has been a widely held opinion that for molecular systems which have both high-frequency intramolecular vibrational mode(s) and low-frequency intermolecular vibrations, such as H 2 in hydrate cages and hydrogen-bonded and van der Waals (vdW) molecular complexes, rigorous calculation of fundamental (and overtone) excitation(s) of their intramolecular vibrational mode(s) would be an extremely difficult and prohibitively costly task. The main source of the difficulty was seen to be the very large number of highly excited intermolecular vibrational eigenstates in the manifold of the intramolecular ground state lying below the energy of the intramolecular vibrational excitation(s), and the assumption that they all have to be converged in order to accurately compute the latter.
However, we demonstrated that, contrary to the above expectation, accurate computation of the intramolecular stretch fundamental of the entrapped H 2 at ≈4100 cm −1 required having only a modest number of converged TR states in the v = 0 manifold up to at most 400-450 cm −1 above the ground state, and none within several thousand wave numbers of of the intramolecular fundamental. 47 Our explanation for this most surprising finding was that, although the number of highly excited intermolecular v = 0 TR states in the vicinity of the H 2 stretch fundamental is large, their coupling to it is extremely weak. Consequently, not including them in the calculations has a negligible effect on the accuracy with which the intramolecular vibrational excitation is calculated. Of course, this resulted in a major reduction of the basis size required, and transformed a computationally very demanding task to one that was readily tractable. 47 Exploiting the valuable insight gained in the above study, we recently developed a very efficient method for full-dimensional, and fully coupled, quantum calculations of intramolecular vibrational fundamentals and overtones of weakly bound molecular dimers. 48 Its application to the 6D problem of (HF) 2 produced results in excellent agreement with those in the literature, but with a fraction of the basis sets required by the other methods.
In this paper, we go one step further and report the results of the calculations of the intramolecular stretch fundamentals of (p-H 2 ) 2 confined inside the large cage of sII clathrate hydrate. These challenging calculations, like those for (HF) 2 , 48 are made possible by what we learned in our investigation of a single H 2 in the small clathrate hydrate cage. 47 They also borrow from the computational methodology implemented in the (HF) 2 calculations. 48 The quantum treatment is in 8D (instead of 12D), since it assumes that the H 2 moieties are rotationless (j = 0), and that the hydrate framework is rigid. Apart from that, the approach is fully coupled. The 8D partially contracted product basis includes a very small They yield the vibrational stretching fundamentals of the caged (p-H 2 ) 2 and their redshifts from the free-H 2 stretch frequency, that for the largest domain considered are in excellent agreement with the measured values. We also formulate a first-order perturbation theory treatment, whose results are extremely close to those from the quantum 8D calculations. This paper is organized as follows: The general approach is described in Sec. II, and the computational details in Sec. III. In Sec. IV, we present and discuss the results. Sec. V summarizes the work and outlines possible directions of further research.

II. GENERAL APPROACH
As stated in the Introduction, for n (flexible) H 2 molecules inside a nanocavity, taken to be rigid, the dimensionality of the quantum bound-state treatment is 6nD. Therefore, computing the coupled VTR eigenstates of (H 2 ) 2 in a cage represents a 12D problem. This poses a serious computational challenge, especially when one is interested in the excited intramolecular stretching vibrational eigenstates of (H 2 ) 2 . In order to alleviate this problem, in the following, the two caged p-H 2 molecules are treated as rotationless, i.e., in ground (j = 0) rotational state, and thus effectively as spherical particles. This reduces the dimensionality of the problem to be solved from 12D for two rotating H 2 molecules to 8D for two rotationless (j = 0) H 2 monomers. This approximation is made rather commonly, e.g., in the well known Silvera-Goldman isotropic effective pair potential for H 2 . 50 Moreover, as demonstrated in Sec. IV A for the vibrational calculations on a single p-H 2 in the large clathrate hydrate cage, the error introduced by the j = 0 approximation is not large, and it decreases further as the size of the hydrate domain increases.
The 8D Hamiltonian for the coupled VRT motions of (H 2 ) 2 inside a large cage, isolated or within a larger sII hydrate domain, assumed to be rigid, and for rotationless (j = 0) H 2 -monomer moieties, can be written aŝ Here R i is the position vector of the center of mass (c.m.) of the ith H 2 moiety measured with respect to a cage-fixed axis system with origin at cage center, ∇ 2 i is the Laplacian associated with that vector, V H 2 −domain is the interaction energy between a rotationless H 2 moiety and the hydrate domain, r i is the distance between the two H nuclei in the ith H 2 , V HH is the intramolecular PES of monomer H 2 in its ground electronic state, R 12 ≡ |R 1 −R 2 |, V H 2 H 2 is the isotropic interaction energy between the two H 2 moieties, m = 2.01565 amu is the mass of H 2 and µ = 0.5039125 amu is the reduced mass of H 2 .
The approach we take in order to solve for the eigenstates and eigenvalues of the 8D H in Eq. (1) adopts some of the key elements of our recent full-dimensional treatment of the intramolecular vibrational eigenstates of weakly bound molecular dimers, 48 specifically (HF) 2 . We first re-writeĤ by separating out its six-dimensional (6D) "intermolecular" portion from the remainder where the constantr is an intramolecular HH distance close to the bottom of the V HH (r i ) and Second, as in Ref. 48, we diagonalize the 6DĤ inter in Eq. (3) to obtain the intermolecular eigenfunctions |κ and associated eigenvalues E inter κ . Third, we construct a partially contracted 8D basis, consisting of states |κ, γ 1 , γ 2 , as the direct product of the N inter lowestenergy |κ and two one-dimensional (1D) Morse discrete variable representations (DVRs) covering the r 1 and r 2 coordinates, respectively, and consisting of N Morse functions apiece: Finally, we compute the matrix ofĤ in this basis and diagonalize.
TheĤ inter matrix in this basis is diagonal with elements given by The matrix elements of the other pieces ofĤ are easily written as follows: and The efficiency of this approach in calculating accurate intramolecular fundamental frequencies for the (H 2 ) 2 @cage species depends crucially on the assumption of weak coupling between the intramolecular vibrational fundamentals of the confined (H 2 ) 2 and its inter-

III. COMPUTATIONAL DETAILS
A. Potential Energy Surfaces PES describing the interaction of j = 0 H 2 moiety #i with the clathrate-water moieties is taken as where the sum is over the water moieties in the sII hydrate domain chosen, whose number is coordinates of water #k, and ω i denotes the two angles that fix the orientation of H 2 #i with respect to the cage-fixed axis system. The 6D pair potential h,w k has the same meaning as in our recent work: 47  we substitute the V H 2 H 2 value corresponding to R 12 = 4.25 bohr. We anticipate minimal errors due to this approximation since the potential at these small R 12 values sufficiently exceeds that at the potential minimum that low-energy intermolecular wave functions should have negligible amplitude at these distances.

B. Diagonalization ofĤ inter
ForĤ inter we use Eq. (3) withr = 0.74Å, a value close to that corresponding to the minimum of V HH . We diagonalize the operator in a 6D product basis consisting of 1D sine DVRs 53 covering the six Cartesian coordinates associated with R 1 and R 2 . These 1D DVRs are all of identical form, with each derived from the 15 lowest-energy eigenfunctions of a particle in a box whose interior ranges from −2.7 to +2.7Å. Diagonalization was effected by using the Chebyshev version 54 of filter diagonalization. 55 This procedure involves the repeated application of the Hamiltonian on a random initial state vector. We symmetrized the initial state vector such that it was either symmetric or antisymmetric with respect to H 2 interchange prior to implementing the filter diagonalization algorithm. Thus, the complete diagonalization ofĤ inter consisted of separate calculations yielding "even" (|κ + ) and then the corresponding "odd" (|κ − ) intermolecular eigenfunctions. (While the odd intermolecular states are not physical on their own, they can combine with (H 2 ) 2 intramolecular states that are also antisymmetric with respect to H 2 interchange to produce physically realizable states. Hence they must be included in the construction of the full 8D basis.) Operation withĤ inter on the state vector was accomplished by matrix-on-vector multiplication. The matrix elements of the pieces of the kinetic-energy operator inĤ inter were easily obtained by first evaluating them analytically in the basis of 1D particle-in-box eigenfunctions, and then transforming the results to the DVR representation. Potential-energy matrix elements are diagonal in the 6D DVR basis and are given by the value of the potential at the relevant DVR point. Table I lists Table I are all very small, and the properties of the two states in each tunneling pair are almost identical. Apparently the barrier to H 2 interchange is large compared to the excitation energies of these low-energy states.
Further insight into the low-energy intermolecular level structure can be gained by considering the single-H 2 c.m. probability densities (PDs) associated with the states. (Note that, though ρ κ in Eq. (12) is for H 2 #1, the same density function pertains to H 2 #2 owing to interchange symmetry.) Isosurface plots of ρ κ are presented in Fig. 1 for several of the low-energy states listed in Table I. From the Figure   one notes that the intermolecular ground state, |0 + , has its H 2 moieties localized in two regions ( Fig. 1(a)). This ground-state behavior is different from that reported in previous studies 25,28 in which a different PES was assumed, and in which delocalization of the H 2 c.m. positions over four sites was found. As the other plots in Fig. 1 show, however, the states at higher energies show increasing degrees of angular delocalization, consistent with excitations involving hindered rotational motion.

C. Diagonalization ofĤ
Per Eq. (6), the 8D basis for the diagonalization ofĤ was constructed from states of Table   I  More can now be said about the calculation of the matrix elements that appear in Eqs. In all cases, if one intermolecular state of an interchange-tunneling pair was included in the basis, then the other one was as well. Erroneous results obtain if this is not done.

D. Intramolecular stretch fundamental of a single H 2 in the large hydrate cage
While our main concern here is with the intramolecular vibrational fundamentals of ( where R is the c.m. position vector of the H 2 moiety, ∇ 2 is the Laplacian associated with R, r is the HH internuclear distance, ω denotes the two angles that fix the H 2 axis with respect to the cage axis system, and This potential involves the same V HH and the same H 2 - h,w k (R, ω, r), that we use for the two-H 2 system [see Eq. (11)] and that we used in Ref. 47, except that there is no averaging over H 2 orientation in Eq. (15). Clathrate hydrate domains identical to those defined in Sec. III A and employed in the (H 2 ) 2 calculations were also used in evaluating Eq. (15).
For the 6D bases we used (a) the product of three 1D sine DVRs identical to those described in Sec. III B to cover the R degrees of freedom times (b) a 1D Morse DVR identical to that described in Sec. III C to cover the r degree of freedom times (c) a spherical harmonic Y m j (ω) to cover the H 2 rotational degrees of freedom. In one 6D basis we only included the j = 0 spherical harmonic. In a second one we included all spherical harmonics with j = 0, 2 and 4.

IV. RESULTS AND DISCUSSION
A. Single H 2 in the large hydrate cage Table II lists results from the vibrational calculations on a single p-H 2 in the large cage.
The results pertain to all three clathrate hydrate domains, and are given for both the rotationless (j max = 0) basis and for the basis that allows for H 2 rotational motion (j max = 4).
There are three significant trends to note from these results. First, both the ground-state energy and the frequency of the intramolecular stretch fundamental decrease with increasing N H 2 O , although the decrease for the latter is very small. This behavior matches that which has previously been observed in computational studies 38,47 of one H 2 in the small cage of the sII clathrate hydrate. It is readily rationalized in that the H 2 -H 2 O interactions relevant to the hydrate inclusion compounds tend to be predominantly attractive, and such as to lower the H 2 intramolecular fundamental frequency. Hence the more such interactions, the lower the ground-state energy and the smaller the frequency of the H 2 stretch fundamental(s).
Of course, one expects these effects to saturate as N H 2 O → ∞, given that the long-range splitting) was also found to hold for H 2 in the small cage. 38,47 The trend is easily understood once one recognizes that the angular anisotropy in the H 2 -cage interaction is due to the asymmetry in the spatial distribution of H-atoms in the hydrogen-bond framework of the hydrate. The effects of such asymmetry tend to get averaged away as more layers of water moieties are included in the larger hydrate domains. In any case, the single-H 2 results indicate that the j = 0 approximation that we have made in the (H 2 ) 2 calculations causes convergence errors of small and acceptable magnitude.
Finally, we note that the magnitudes of the HH-frequency shifts in Table II are significantly less than those that we have computed (∼ −42 cm −1 ) for H 2 in the small hydrate cage. 47 This is in keeping with the relative ground-state energies in the two cases (the smallcage H 2 ground state is stabilized by approximately 200 cm −1 more than the large-cage one relative to monomer H 2 ), and the fact that the redshift of the stretch fundamental tracks with ground-state stabilization. Indeed, the ratio of redshifts for the two cases closely matches the ratio of the ground-state stabilization energies. Table III lists results from the diagonalization ofĤ for the j = 0, (p-H 2 ) 2 system for two 8D bases whose dimensions differ by a factor of ten -one corresponding to N inter = 2 (the two lowest-energy states in Table I), and N tot = 128, and the other to N inter = 20 (all the states in Table I In this regard it is instructive to examine the intermolecular basis-state composition of the pertinent 8D eigenfunctions computed by using the N inter = 20 6D intermolecular basis.

B. (H 2 ) 2 in the large hydrate cage
To do so we calculate for 8D eigenstate |ψ n the quantities For |ψ 0 , the only 6D intermolecular state that contributes appreciably is |0 + , and P 0 + (ψ 0 ) = 0.9986. For the intramolecular stretch fundamental states, |ψ νa and |ψ ν b , just two lowestenergy 6D intermolecular states in Table I -|0 + and |0 − -dominate in their contributions The implications of this are two-fold: (1) The remaining nine higher-energy even/odd intermolecular eigenfunction pairs in Table I mix negligibly with |ψ νa and |ψ ν b . (2) The 8D basis that we have used very efficiently overlaps the 8D eigenstates that have small degrees of intermolecular excitation.
These observations help explain a most striking result of this study, that converged values of the intramolecular stretch fundamentals of the caged (p-H 2 ) 2 are obtained already with the smallest 6D intermolecular basis (N inter = 2), and N tot = 128, which is incapable of describing any excited intermolecular vibrational states of (p-H 2 ) 2 . Even the largest inter-molecular basis employed, with N inter = 20, extends to only ∼ 64 cm −1 above the ground state, which is far below the energies of the (p-H 2 ) 2 intramolecular stretch fundamentals, around 4100 cm −1 . At this energy, the density of the intermolecular vibrational states in the v 1 + v 2 = 0 (ground-state) intramolecular vibrational manifold of (p-H 2 ) 2 must be very high, but their coupling to the intramolecular vibrational excitations is negligible, so that leaving them out of the treatment has no effect whatsoever on the calculated intramolecular One also sees from Table III that  It is of some interest to get a more complete picture of the nature of the |ψ νa and |ψ ν b intramolecular excitations than is available from their energies alone. In particular, are the excitations local within the dimer or delocalized? Equivalently, what portion of the ν a /ν b splitting is due to site splitting as opposed to excitation exchange? We have addressed these questions by computing the center-of-mass probability density corresponding to the vibrationally excited H 2 moiety in both the |ψ νa and the |ψ ν b states. To do this, we projected out of each 8D eigenstate |ψ νx , x = a, b, the 6D function where |v 1 , v 2 corresponds to an intramolecular excitation in which H 2 moiety #1 has vibrational quantum number v 1 and H 2 moiety #2 has vibrational quantum number v 2 . The v 1 , v 2 |γ 1 , γ 2 were obtained by solving for the free H 2 -monomer vibrational states |v i in the |γ i basis. We then computed the single-particle c.m. PD νx (R 2 ).) Table IV gives the dominant coefficients κ, 1, 0|ψ νx and κ, 0, 1|ψ νx in Eq. (17) computed for x = a/b for the N inter = 20 basis. From the sum of squares of these values (the right-most column of Table IV) it is clear that the two eigenstates correspond overwhelmingly to v 1 + v 2 = 1 intramolecular excitations, as assigned. Calculation of ρ (1,0) νx (R 1 ) for the two states from such coefficients yields the plots shown in Fig. 2. The striking feature of these plots is the marked degree of spatial localization associated with each intramolecular excitation. Further, if one compares these PDs with the ground state PD, ρ (0,0) ν 0 (R 1 ), which is essentially ρ 0 + (R 1 ) of Fig. 1(a), one sees that the "excitation locale" for |ψ νa matches one of the high-density regions found for |ψ 0 , whereas that for |ψ ν b matches the other such ground-state region. One concludes that, for the PES we have employed, the splitting of the intramolecular vibrational fundamentals in the encaged (H 2 ) 2 is dominated by effects due to inequivalent H 2 sites within the clathrate cage.
C. First-order perturbation theory treatment is accurate for caged (H 2 ) 2 Given the results in Table IV, it is pertinent to make one last point in closing this section.
Namely, if we were to choose an 8D basis consisting of just the four "zeroth-order" states, |κ, v 1 , v 2 = |0 ± , 1, 0 and |0 ± , 0, 1 , and diagonalizeĤ in that basis, we would capture the bulk of the eigenvector content of both intramolecular vibrational fundamentals |ψ νa and |ψ ν b . Similarly, the character of the 8D ground state is almost entirely captured by a single state in the |κ, v 1 , v 2 basis: | 0 + , 0, 0|ψ 0 | 2 = 0.9984. The upshot is that computing the energies of these states, and hence the intramolecular stretch fundamental frequencies, would seem to fall well within the reach of first-order perturbation theory (PT). We have tested this as follows. We take the unperturbed Hamiltonian to bê This has |0 + , 0, 0 , |0 ± , 1, 0 and |0 ± , 0, 1 as eigenstates. The perturbation then iŝ We then compute (a) the ground-state energy by means of nondegenerate first-order PT and (b) the energies of the |ψ νx (x = a, b) vibrationally excited states by using degenerate first-order PT. The results for all three hydrate domains are given in Table V. Comparison with the quantum 8D results in Table III shows  For the (H 2 ) 2 system the two intramolecular fundamentals are computed to be sitesplit by about 0.5 cm −1 and to be red-shifted from the free-monomer stretch frequency by