We report an investigation of the translation-rotation (TR) level structure of H_{2} entrapped in C_{60}, in the rigid-monomer approximation, by means of a low-order perturbation theory (PT). We focus in particular on the degree to which PT can accurately account for that level structure, by comparison with the variational quantum five-dimensional calculations. To apply PT to the system, the interaction potential of H_{2}@C_{60} is decomposed into a sum over bipolar spherical tensors. A zeroth-order Hamiltonian, $H\u02c60$, is then constructed as the sum of the TR kinetic-energy operator and the one term in the tensor decomposition of the potential that depends solely on the radial displacement of the H_{2} center of mass (c.m.) from the cage center. The remaining terms in the potential are treated as perturbations. The eigenstates of $H\u02c60$, constructed to also account for the coupling of the angular momentum of the H_{2} c.m. about the cage center with the rotational angular momentum of the H_{2} about the c.m., are taken as the PT zeroth-order states. This zeroth-order level structure is shown to be an excellent approximation to the true one except for two types of TR-level splittings present in the latter. We then show that first-order PT accounts very well for these splittings, with respect to both their patterns and magnitudes. This allows one to connect specific features of the level structure with specific features of the potential-energy surface, and provides important new physical insight into the characteristics of the TR level structure.

## I. INTRODUCTION

The entrapment of H_{2} molecule (and its isotopologues HD and D_{2}) inside the fullerene C_{60} gives rise to the quantization of the translational motions of the center of mass (c.m.) of H_{2}. These are coupled to the quantized rotations of the guest molecule by a high-symmetry (*I _{h}*), nearly spherical confining potential of the C

_{60}cage.

^{1}The intermolecular translation-rotation (TR) motions couple very weakly to the intramolecular vibrations of both H

_{2}and C

_{60}, allowing the two monomers to be treated as rigid to a very good approximation. Moreover, the interaction between H

_{2}molecules occupying different C

_{60}cages is apparently negligible. As a result, H

_{2}@C

_{60}represents an isolated, highly symmetric, and yet challenging system whose coupled quantum TR dynamics one can aspire to understand in detail at a fundamental level and describe quantitatively from first principles.

Since the preparation of H_{2}@C_{60} in macroscopic quantities^{2,3} by the organic synthetic approach known as molecular surgery, this paradigmatic endohedral complex has been the subject of numerous and diverse experimental and theoretical investigations. The fully coupled quantum five-dimensional (5D) rigid-monomer calculations have accurately characterized the TR eigenstates of H_{2} and isotopologues in C_{60},^{4–6} and elucidated the salient features of the TR dynamics summarized below. The TR energy level structure has been also probed directly by means of infrared absorption (IR)^{7–10} and inelastic neutron scattering (INS) spectroscopy.^{11–13} Through close interaction between theoretical and IR spectroscopic investigations, intermolecular potential energy surfaces (PESs) for the interaction between the entrapped H_{2} and the interior of C_{60} have been developed,^{6,7,9} allowing accurate modeling of the experimental spectra. The synergy between theory and experiment, INS in particular, was strengthened significantly by the recent development of the quantum methodology for rigorous calculation of the INS spectra of a hydrogen molecule confined inside an arbitrarily shaped nanoscale cavity,^{14–16} which incorporates the computed 5D TR wave functions. Using this methodology, detailed and highly realistic simulations of the INS spectra of H_{2} and HD in C_{60} have been performed,^{17–19} leading to their interpretation and assignment, and further refinement of the 5D intermolecular PES of H_{2}@C_{60}.^{17} The culmination of this line of research was the unexpected discovery of a selection rule in the INS spectroscopy of H_{2}^{17} and HD^{18} inside C_{60}, subsequently confirmed experimentally^{19} and generalized.^{20,21}

Theoretical studies^{4–6} have established that the TR eigenstates can be well-described in terms of two elementary excitations. One of them is that of an isotropic 3D oscillator corresponding to the translational motion of the H_{2} c.m.; it can be assigned with the principal quantum number *n* = 0, 1, 2, …, of the 3D isotropic harmonic oscillator (HO) and its orbital angular momentum quantum number *l* = *n*, *n* − 2, …, 1 or 0, for odd and even *n*, respectively. The other is the excitation of a free linear rotor corresponding to the rotation of the H_{2} about its c.m.; the quantum numbers *j* = 0, 1, 2, …, of a rigid linear rotor can be used for its assignment. The angular momenta associated with these two excitations, the orbital angular momentum ** l** and the rotational angular momentum

**, couple vectorially to give the total angular momentum**

*j***λ**having the values

*λ*=

*l*+

*j*,

*l*+

*j*− 1, …, |

*l*−

*j*|, with the degeneracy of 2

*λ*+ 1.

This coupling of the two angular momenta is responsible for the most prominent feature of the TR energy-level structure of H_{2}@C_{60}:^{4} the TR eigenstates with the same (nonzero) quantum numbers *n* and *j* are split into distinct closely spaced energy levels, each labeled with one of the possible values of the quantum number *λ*, and exhibiting the degeneracy of 2*λ* + 1 (e.g., Fig. 1 of Ref. 17). Hereafter, we refer to these splittings as “*λ* splittings.” These splittings are of the order of several wavenumbers.^{4–6,17}

The second conspicuous feature of the TR levels of H_{2}@C_{60} is the “crystal field” splitting of their 2*λ* + 1 degeneracy, for *λ* > 2, by the icosahedral (*I _{h}*) environment of C

_{60}.

^{4,21}For example,

*λ*= 3 levels split into near-degenerate pairs of levels typically less than 1 cm

^{−1}apart. We will denote these splittings as “

*m*

_{λ}splittings.” Their small magnitude reflects the very weak

*I*corrugation, i.e., deviation from spherical symmetry, of the interaction potential for H

_{h}_{2}in C

_{60}. Similar “crystal field” splittings were observed for CO@C

_{60}

^{22}and H

_{2}O@C

_{60}.

^{23}

It is important to note that both *λ* splittings and *m*_{λ} splittings of a given (*n*, *l*, *j*) multiplet (for nonzero *l*, *j*) are much smaller in magnitude than the energy separation between the multiplets having different (*n*, *l*, *j*) values (see Fig. 1 of Ref. 19). The implication is that these and other important aspects of the quantum TR dynamics of H_{2} in C_{60} could be described accurately, and additional insight gained, by means of a perturbation theory (PT) treatment, starting from a natural, physically transparent set of states, “zeroth-order states,” that are eigenstates of a large portion of the TR Hamiltonian. The remaining small terms, responsible for *λ* and *m*_{λ} splittings, can be viewed as perturbations. Indeed, this general scheme has previously been applied successfully to a study of the TR states of H_{2} molecule confined in the *interstitial* octahedral site of solid C_{60}.^{24}

This approach is adopted in the present study. In this paper we report the results of a PT treatment of TR states of H_{2}@C_{60}. A particular objective of the study is to help clarify the connection between the *λ* and *m*_{λ} splittings present in the TR level structure of the species and specific parts of the TR Hamiltonian. In this effort our study borrows from those of CO@C_{60} by Olthof *et al.*^{22} and H_{2}@C_{60} by Mamone *et al.*,^{7,8} in that, like them, we are able to make such connections by decomposing the interaction PES into a sum over bipolar spherical tensors. However, we make an important step beyond these earlier studies, and use the tensor components identified as dominant in the first-order PT calculations of *λ* and *m*_{λ} splittings. The aims of these PT calculations are twofold. The first one is to gain physical insight into the details of the TR level structure revealed by the quantum 5D calculations,^{4–6,17} such as the energy ordering of the *λ* components in a given (*n*, *l*, *j*) multiplet, for which a simple explanation has been lacking. This aim is significantly facilitated by the simplicity of the PT approach. The second aim is to assess just how accurate PT can be in treating the system. Such assessment may prove valuable in future work that aspires to go beyond the rigid-monomer approximation.

The paper is organized as follows. In Section II we describe the division of the TR Hamiltonian ($H\u02c6$) into the sum of a zeroth-order term ($H\u02c60$) and perturbation terms. This is accomplished via the aforementioned tensor decomposition of the intermolecular PES of H_{2}@C_{60}. The $H\u02c60$ so obtained has commutation properties that significantly facilitate the determination of its eigenfunctions and eigenvalues. Section III pertains to the calculation of those $H\u02c60$ eigenfunctions and eigenvalues, a calculation that involves the straightforward solution of an ordinary differential equation. From these eigenfunctions we then construct zeroth-order states that incorporate the coupling of the angular momenta associated with H_{2} translation and rotation, respectively, mentioned above. By comparison with previously reported 5D calculations^{6} we show that these zeroth-order states are indeed excellent first approximations to the low-energy TR states of the species. Section IV describes variational calculations utilizing the zeroth-order basis. We first outline the calculation of matrix elements in that basis. We then describe the diagonalization of $H\u02c6$ and show that the 5D results so-obtained match those computed elsewhere^{6} for the same PES but with a different basis. This agreement provides a check on our methodology, and the results serve to illustrate the specific features of the level structure that we wish to describe with PT. Last in this section we undertake a variational calculation in which the Hamiltonian is $H\u02c60$ plus a single term—the leading isotropic^{25} perturbation term in the spherical-tensor expansion of the PES. The results are instructive in that they show that this limited calculation can account for all but the finest details of the TR level structure. Section V describes the PT calculations. Two features in the level structure are addressed. The first, *λ* splittings, arise from isotropic perturbations in the PES. These splittings represent the bulk of the deviation of the true level structure away from the zeroth-order one. The first-order PT treatment is shown to lead to excellent agreement with the 5D results in respect to both the patterns of *λ* splittings and the splitting magnitudes. For those instances in which the first-order results are not so good, the deviations can be attributed to specific second-order couplings. We also show that one can understand qualitatively the energy ordering of the *λ* components in a given (*n*, *l*, *j*) multiplet by a surprisingly simple physical picture involving the degree of alignment between the c.m. displacement of the H_{2} and its internuclear vector. The second feature we address arises from anisotropic terms in the PES. These much smaller *m*_{λ} splittings, which occur only for levels having sufficient total angular momentum (*λ* ≥ 3), must be treated with degenerate PT, and generally a matrix must be diagonalized. We show that such diagonalization leads to closed-form expressions for those *m*_{λ} splittings that occur in the low-energy part of the TR level structure and that, moreover, the evaluation of those first-order PT expressions produces excellent agreement with the 5D results. Section VI concludes.

## II. DECOMPOSITION OF THE TRANSLATION-ROTATION HAMILTONIAN

In the rigid monomer approximation with C_{60} fixed, the TR Hamiltonian of H_{2}@C_{60} is given by

Here ∇^{2} is the Laplacian associated with the vector **R** that points from the center of the C_{60} cage to the c.m. of the H_{2}, *M* is the mass of the H_{2}, $J\u02c62$ is the operator associated with the square of the rotational angular momentum of the H_{2} rotor, *B* is the rotational constant of that rotor, *V* is the 5D PES for the interaction between the H_{2} and C_{60} interior, *R* ≡ |**R**|, Ω ≡ (Θ, Φ) represents the polar and azimuthal angles that **R** makes with respect to a Cartesian axis system fixed to the C_{60} cage (hereafter the “SF” axis system), and *ω* ≡ (*θ*, *ϕ*) represents the polar and azimuthal angles that **r**, the internuclear vector of H_{2}, makes with respect to the SF system. For the sake of specificity, in all that follows we shall utilize the particular Hamiltonian parameters, including the 5D PES, used previously by Xu *et al.*,^{6} summarized in the supplementary material.

Four relevant operators almost commute with $H\u02c6$ of Eq. (1). These are $L\u02c62$ (corresponding to the square of the angular momentum of the c.m. about the SF origin), $J\u02c62$, $\Lambda \u02c62\u2261|L\u02c6+J\u02c6|2$, and $\Lambda \u02c6Z\u2261L\u02c6Z+J\u02c6Z$ (the SF *Z* component of $\Lambda \u02c6$). They do not fully commute with $H\u02c6$ because of *V*, specifically because of its dependence on Ω and *ω*. We seek to divide $H\u02c6$ into a term, $H\u02c60$, with which these operators do commute, plus terms with which they do not. To accomplish this we decompose *V*(*R*, Ω, *ω*) in Eq. (1) into a sum over bipolar spherical tensors analogous to the procedure employed by Olthof and co-workers^{22}

where the $GL,JK,Q(R)$ are scalar functions of *R*, the $TL,J(K,Q)(\Omega ,\omega )$ are the bipolar spherical tensors [for example, Ref. 26, Eq. (5.63), p. 192]

and the $Yqk$ are spherical harmonics. The determination of the $GL,JK,Q(R)$ expansion coefficients for a given potential is straightforward

which follows from Eq. (2) and the orthonormality of the $TL,J(K,Q)$. Equation (4) can be evaluated by using Eq. (3) and computing the resulting integrals numerically. We have done this by using Gauss-Legendre quadrature for the Θ and *θ* integrals, and quadrature on equally spaced (Fourier) grids for the Φ and *ϕ* integrals.

The decomposition produces directly the part of *V* that depends only on *R*, as the *K* = *Q* = *L* = *J* = 0 term of the expansion in Eq. (2), and that, therefore, commutes with $L\u02c62$, $J\u02c62$, $\Lambda \u02c62$, and $\Lambda \u02c6Z$. From this, one defines $H\u02c60$ as

The remaining terms in the expansion of *V* can be treated as perturbations to the zeroth-order states that are the eigenstates of $H\u02c60$.

Examining the properties of the expansion terms one notes several points. First, many of them are zero by symmetry. For example, because of the *I _{h}* symmetry of the PES, only terms with

*K*= 0, 6, 10, … are nonzero.

^{22}Further, because of our choice of SF axes (supplementary material) the only nonzero

*K*= 6 terms correspond to

*Q*= 0, ±5 and these are such that

Similarly, symmetry limits the nonzero $GL,JK,Q$ to those that have both even *L* and even *J*. Second, only certain combinations of *L*, *J*, *K* produce nonzero terms. This follows from Eq. (3) and the fact that the 3 − *j* symbol therein is nonzero only when *L*, *J*, *K* obey the triangle rule. Third, the magnitudes of the $GL,JK,Q(R)$ fall off markedly with *K* and with *L* and *J*. Figure 1 shows the plots of several $GL,JK,Q(R)$ vs. *R* that illustrate this point. In fact, the magnitude fall-off is such that quantitative results from the diagonalization of $H\u02c6$ can be obtained even with all terms having *K* > 6 discarded, as we show below. Finally, the perturbation terms are of two distinct types based on whether they commute with $\Lambda \u02c62$ and $\Lambda \u02c6Z$ or not. The *K* = 0 (or isotropic^{25}) perturbation terms, like $T2,2(0,0)$, do commute with these operators, whereas the *K*≠0 (anisotropic) terms do not. These two types of terms give rise to qualitatively different manifestations in the TR level structure.

## III. CALCULATION OF ZEROTH-ORDER STATES

The Schrödinger equation that involves $H\u02c60$ of Eq. (5) as Hamiltonian corresponds to that of a 3D oscillator/linear rotor in a radial potential

where *E*_{0} is the relevant energy eigenvalue. Given the commutation properties of $H\u02c60$, it is easy to see that this equation can be simplified by separation of variables

and subsequent division by $Ymll(\Omega )Ymjj(\omega )$ to yield the ordinary differential equation for *F _{nl}*(

*R*),

where *E _{nl}* ≡

*E*

_{0}−

*Bj*(

*j*+ 1) is the c.m. translational energy of the state and

*n*=

*l*,

*l*+ 2, … is a state-labeling index that in the limit of harmonic $G0,00,0(R)$ is the principal quantum number of the oscillator. Equation (9) can be solved variationally for any given value of

*l*. We have done this for

*l*values from 0 to 8 by using a basis composed of the

*R*-dependent parts of the 3D isotropic-HO eigenfunctions corresponding to each

*l*(supplementary material). For the HO functions we have chosen 2

*πMν*, where

*ν*is the linear frequency of the oscillator, to be 2.995 a.u. (which equates to a c.m. translational-mode harmonic frequency of about 180 cm

^{−1}) and have included all functions up to

*v*= 8, where

*v*is the principal quantum number of the oscillator. In this basis the kinetic-energy matrix elements are easily obtained analytically. To evaluate the matrix elements of $G0,00,0(R)$, we used 12-point Gauss-associated-Laguerre quadrature based on normalized associated-Laguerre polynomials (supplementary material). The

*E*obtained from solution of Eq. (9) for

_{nl}*n*= 0 to 5 are given in Table I. Included also in the table are the energies of

*j*= 0 states obtained from the 5D calculations of Xu

*et al.*,

^{6}that employ the full potential. One notes the excellent agreement between the two sets of results, indicating that the gross TR level structure of H

_{2}@C

_{60}for this potential is largely determined by our $H\u02c60$.

n
. | l
. | E
. _{nl} | E_{5D}
. |
---|---|---|---|

0 | 0 | −1257.01 | −1257.16 |

1 | 1 | −1073.23 | −1073.69 |

2 | 2 | −873.80 | −874.29 |

2 | 0 | −846.42 | −847.78 |

3 | 3 | −660.10 | −661.07/−660.24^{a} |

3 | 1 | −617.02 | −618.95 |

4 | 4 | −433.14 | −434.47/−433.56^{a} |

4 | 2 | −375.55 | −378.11 |

4 | 0 | −351.06 | −354.35 |

5 | 5 | −193.70 | |

5 | 3 | −122.37 | |

5 | 1 | −83.12 |

n
. | l
. | E
. _{nl} | E_{5D}
. |
---|---|---|---|

0 | 0 | −1257.01 | −1257.16 |

1 | 1 | −1073.23 | −1073.69 |

2 | 2 | −873.80 | −874.29 |

2 | 0 | −846.42 | −847.78 |

3 | 3 | −660.10 | −661.07/−660.24^{a} |

3 | 1 | −617.02 | −618.95 |

4 | 4 | −433.14 | −434.47/−433.56^{a} |

4 | 2 | −375.55 | −378.11 |

4 | 0 | −351.06 | −354.35 |

5 | 5 | −193.70 | |

5 | 3 | −122.37 | |

5 | 1 | −83.12 |

^{a}

These *j* = 0 levels are *m*_{λ}-split by the *I _{h}* crystal field.

The (2*l* + 1) × (2*j* + 1)-fold degenerate eigenfunctions of $H\u02c60$,

are not optimal as zeroth-order states. Ones that anticipate the coupling between **L** and **J** to produce **Λ** are better. The latter are given by the usual angular-momentum coupling approach as

These are not only eigenfunctions of $H\u02c60$, $L\u02c62$, and $J\u02c62$, but also of $\Lambda \u02c62$ [eigenvalue *λ*(*λ* + 1)] and $\Lambda \u02c6Z$ (eigenvalue *m*_{λ}). They are the states that we shall use as our zeroth-order basis.

## IV. VARIATIONAL CALCULATIONS

### A. Matrix elements in the zeroth-order basis

We wish to compute the TR level structure of the H_{2}@C_{60} species by diagonalizing $H\u02c6$ of Eq. (1) in the basis of Eq. (11). To accomplish this, we need expressions for the matrix elements of the terms of Eq. (2) in that basis. These take the form of a product of a radial and an angular matrix element

where the |*n*, *l*〉 are the *F _{nl}*(

*R*) obtained from the solution of Eq. (9). The radial matrix element that appears on the rhs of Eq. (12) can be evaluated by Gauss-associated-Laguerre quadrature (supplementary material). Table II presents values for some of these radial factors for the

*V*of Xu

*et al.*,

^{6}and the states listed in Table I. These values give one a good sense of the most important terms in the tensor decomposition of

*V*. One notes in particular the dominance of the

*L*=

*J*= 2 isotropic factors relative to all others.

n, l
. | $G2,20,0$ . | $G4,40,0$ . | $G6,60,0$ . | $G0,66,0$ . | $G2,46,0$ . | $G6,06,0$ . | $G4,26,0$ . |
---|---|---|---|---|---|---|---|

0,0 | 93.32 | 0.63 | 0.0022 | 0.0996 | 0.932 | 0.90 | 1.38 |

1,1 | 159.64 | 1.38 | 0.0057 | 0.130 | 1.61 | 2.26 | 2.95 |

2,2 | 229.39 | 2.35 | 0.0112 | 0.163 | 2.33 | 4.27 | 4.94 |

2,0 | 234.86 | 2.79 | 0.0151 | 0.165 | 2.40 | 5.56 | 5.76 |

3,3 | 302.63 | 3.54 | 0.0189 | 0.197 | 3.09 | 6.96 | 7.32 |

3,1 | 312.17 | 4.27 | 0.0266 | 0.202 | 3.21 | 9.30 | 8.61 |

4,4 | 379.28 | 4.94 | 0.0291 | 0.233 | 3.89 | 10.43 | 10.05 |

4,2 | 393.05 | 5.97 | 0.0413 | 0.240 | 4.07 | 13.84 | 11.78 |

4,0 | 398.92 | 6.40 | 0.0465 | 0.243 | 4.15 | 15.26 | 12.51 |

n, l
. | $G2,20,0$ . | $G4,40,0$ . | $G6,60,0$ . | $G0,66,0$ . | $G2,46,0$ . | $G6,06,0$ . | $G4,26,0$ . |
---|---|---|---|---|---|---|---|

0,0 | 93.32 | 0.63 | 0.0022 | 0.0996 | 0.932 | 0.90 | 1.38 |

1,1 | 159.64 | 1.38 | 0.0057 | 0.130 | 1.61 | 2.26 | 2.95 |

2,2 | 229.39 | 2.35 | 0.0112 | 0.163 | 2.33 | 4.27 | 4.94 |

2,0 | 234.86 | 2.79 | 0.0151 | 0.165 | 2.40 | 5.56 | 5.76 |

3,3 | 302.63 | 3.54 | 0.0189 | 0.197 | 3.09 | 6.96 | 7.32 |

3,1 | 312.17 | 4.27 | 0.0266 | 0.202 | 3.21 | 9.30 | 8.61 |

4,4 | 379.28 | 4.94 | 0.0291 | 0.233 | 3.89 | 10.43 | 10.05 |

4,2 | 393.05 | 5.97 | 0.0413 | 0.240 | 4.07 | 13.84 | 11.78 |

4,0 | 398.92 | 6.40 | 0.0465 | 0.243 | 4.15 | 15.26 | 12.51 |

The angular matrix element in Eq. (12) is given by the Wigner-Eckart theorem [for example, Ref. 26, Eq. (5.14), p. 181] as

where the reduced matrix element $\u3008l\u2032,j\u2032,\lambda \u2032\Vert TL,J(K)\Vert l,j,\lambda \u3009$ is [Ref. 22 and Ref. 26, Eq. (5.68), p. 194, and Eq. (5.28), p. 183]

In Eq. (14) the term in curly brackets is a 9 − *j* symbol, and $K\u0304\u2261(2K+1)$, $\lambda \u0304\u2261(2\lambda +1)$, etc.

Equations (12)–(14) can be used to evaluate any spherical-tensor matrix element in the zeroth-order basis. In the specific case of *K* = 0, Eqs. (13) and (14) reduce to

To derive the above, we have used Eqs. (5.71) and (5.28) of Ref. 26, as well as the fact that *L* = *J* for *K* = 0. One sees that with respect to isotropic terms in the PES both *λ* and *m*_{λ} are good quantum numbers. This is in keeping with the fact, mentioned in Section II, that both $\Lambda \u02c62$ and $\Lambda \u02c6Z$ commute with these terms. In contrast, neither $L\u02c62$ nor $J\u02c62$ commutes with isotropic terms for which *L*≠0. Consistent with this, Eq. (15) shows that states with different *l*, different *j*, or both, are coupled by them.

Referring back to Eqs. (13) and (14), one sees that anisotropic (*K*≠0) terms can couple states with different *λ* and different *m*_{λ} values. To the extent that such terms contribute to the PES, those quantum numbers become “less good.” That said, the couplings between states with different *λ* are constrained due to the triangle rule that applies to *λ*′, *λ*, and *K* from the 3 − *j* symbol in Eq. (13). The couplings between states with different *l* and different *j* are also constrained by the values of *L* and *J*, respectively, due to the triangle rules that must be obeyed in Eq. (14).

### B. Diagonalization of $H\u02c6$ in the zeroth-order basis

With Eqs. (12)–(14) and the *E*_{0} from Eq. (9) we have constructed the matrix of $H\u02c6$ of Eq. (1) in the zeroth-order basis keeping only those nonzero spherical-tensor terms up to *K* = 6. The matrix is block diagonal in two blocks, one corresponding to even *j* (*para*-H_{2}) and the other to odd *j* (*ortho*-H_{2}). The *para* basis includes all zeroth-order states having *n* = 0 to 8 and even *j* = 0 to 8 (total basis size equal to 4769 states). The *ortho* basis is the same except that odd *j* = 1 to 7 states are included (total basis size equal to 4204 states). We diagonalized the two blocks separately, both by the Chebyshev variant^{27} of filter diagonalization.^{28}

Results from the calculation are summarized in the first two columns of Tables III and IV as Δ*E*_{full} (energy-level shifts from the ground state) and *g*_{full} (level degeneracies). The results agree quantitatively with those from the 5D calculations by Xu *et al.*^{6} The assignments of the levels in terms of the quantum numbers (*n*, *l*, *j*) and *λ* of the basis functions that dominate in contributing to them are given in columns 5 and 6 of the tables. These also agree with Xu *et al.*.^{6} Besides being a check on our procedure, this agreement indicates that the truncation of the tensor expansion of this particular PES at *K* = 6 has a negligible effect on the low-energy level structure of the species.

ΔE_{full}
. | g_{full}
. | ΔE_{iso}
. | g_{iso}
. | (n, l, j)
. | λ
. | ΔE_{0}
. | g_{0}
. |
---|---|---|---|---|---|---|---|

0 | 1 | 0 | 1 | (0, 0, 0) | 0 | 0.15 | 1 |

183.47 | 3 | 183.47 | 3 | (1, 1, 0) | 1 | 183.94 | 3 |

328.62 | 5 | 328.62 | 5 | (0, 0, 2) | 2 | 328.98 | 5 |

382.87 | 5 | 382.87 | 5 | (2, 2, 0) | 2 | 383.37 | 5 |

409.37 | 1 | 409.38 | 1 | (2, 0, 0) | 0 | 410.75 | 1 |

507.00 | 5 | 507.00 | 5 | (1, 1, 2) | 2 | 512.92 | 15 |

513.39 | 4 | 513.44 | 7 | 3(a) | |||

513.50 | 3 | 3(b) | |||||

518.21 | 3 | 518.22 | 3 | 1 | |||

596.09 | 4 | 596.45 | 7 | (3, 3, 0) | 3(a) | 597.07 | 7 |

596.92 | 3 | 3(b) | |||||

638.22 | 3 | 638.22 | 3 | (3, 1, 0) | 1 | 640.14 | 3 |

705.05 | 3 | 705.21 | 7 | (2, 2, 2) | 3(a) | 712.35 | 25 |

705.31 | 4 | 3(b) | |||||

707.13 | 5 | 707.09 | 5 | 2 | |||

713.66 | 4 | 713.82 | 9 | 4(a) | |||

713.95 | 5 | 4(b) | |||||

717.77 | 3 | 717.88 | 3 | 1 | |||

724.06 | 1 | 723.91 | 1 | 0 | |||

741.22 | 5 | 741.22 | 5 | (2, 0, 2) | 2 | 739.73 | 5 |

822.70 | 4 | 823.21 | 9 | (4, 4, 0) | 4(a) | 824.03 | 9 |

823.61 | 5 | 4(b) | |||||

879.08 | 5 | 879.07 | 5 | (4, 2, 0) | 2 | 881.62 | 5 |

903.0 | 1 | 903.89 | 1 | (4, 0, 0) | 0 | 906.11 | 1 |

ΔE_{full}
. | g_{full}
. | ΔE_{iso}
. | g_{iso}
. | (n, l, j)
. | λ
. | ΔE_{0}
. | g_{0}
. |
---|---|---|---|---|---|---|---|

0 | 1 | 0 | 1 | (0, 0, 0) | 0 | 0.15 | 1 |

183.47 | 3 | 183.47 | 3 | (1, 1, 0) | 1 | 183.94 | 3 |

328.62 | 5 | 328.62 | 5 | (0, 0, 2) | 2 | 328.98 | 5 |

382.87 | 5 | 382.87 | 5 | (2, 2, 0) | 2 | 383.37 | 5 |

409.37 | 1 | 409.38 | 1 | (2, 0, 0) | 0 | 410.75 | 1 |

507.00 | 5 | 507.00 | 5 | (1, 1, 2) | 2 | 512.92 | 15 |

513.39 | 4 | 513.44 | 7 | 3(a) | |||

513.50 | 3 | 3(b) | |||||

518.21 | 3 | 518.22 | 3 | 1 | |||

596.09 | 4 | 596.45 | 7 | (3, 3, 0) | 3(a) | 597.07 | 7 |

596.92 | 3 | 3(b) | |||||

638.22 | 3 | 638.22 | 3 | (3, 1, 0) | 1 | 640.14 | 3 |

705.05 | 3 | 705.21 | 7 | (2, 2, 2) | 3(a) | 712.35 | 25 |

705.31 | 4 | 3(b) | |||||

707.13 | 5 | 707.09 | 5 | 2 | |||

713.66 | 4 | 713.82 | 9 | 4(a) | |||

713.95 | 5 | 4(b) | |||||

717.77 | 3 | 717.88 | 3 | 1 | |||

724.06 | 1 | 723.91 | 1 | 0 | |||

741.22 | 5 | 741.22 | 5 | (2, 0, 2) | 2 | 739.73 | 5 |

822.70 | 4 | 823.21 | 9 | (4, 4, 0) | 4(a) | 824.03 | 9 |

823.61 | 5 | 4(b) | |||||

879.08 | 5 | 879.07 | 5 | (4, 2, 0) | 2 | 881.62 | 5 |

903.0 | 1 | 903.89 | 1 | (4, 0, 0) | 0 | 906.11 | 1 |

ΔE_{full}
. | g_{full}
. | ΔE_{iso}
. | g_{iso}
. | (n, l, j)
. | λ
. | ΔE_{0}
. | g_{0}
. |
---|---|---|---|---|---|---|---|

109.63 | 3 | 109.63 | 3 | (0, 0, 1) | 1 | 109.66 | 3 |

287.71 | 3 | 287.71 | 3 | (1, 1, 1) | 1 | 293.59 | 9 |

294.23 | 5 | 294.23 | 5 | 2 | |||

304.54 | 1 | 304.55 | 5 | 0 | |||

484.47 | 5 | 484.47 | 5 | (2, 2, 1) | 2 | 493.03 | 15 |

494.06 | 4 | 494.28 | 7 | 3(a) | |||

494.57 | 3 | 3(b) | |||||

496.10 | 3 | 496.10 | 3 | 1 | |||

524.20 | 3 | 524.21 | 3 | (2, 0, 1) | 1 | 520.41 | 3 |

658.15 | 4 | 658.154 | 7 | (0, 0, 3) | 3(a) | 657.96 | 7 |

658.16 | 3 | 3(b) | |||||

694.87 | 3 | 695.26 | 7 | (3, 3, 1) | 3(a) | 706.73 | 21 |

695.57 | 4 | 3(b) | |||||

707.92 | 4 | 708.30 | 9 | 4(a) | |||

708.55 | 5 | 4(b) | |||||

710.38 | 5 | 710.34 | 5 | 2 | |||

737.80 | 3 | 737.78 | 3 | (3, 1, 1) | 1 | 749.80 | 9 |

754.50 | 5 | 754.52 | 5 | 2 | |||

770.39 | 1 | 770.44 | 1 | 0 | |||

836.02 | 3 | 836.046 | 7 | (1, 1, 3) | 3(a) | 841.90 | 21 |

836.06 | 4 | 3(b) | |||||

844.02 | 4 | 844.107 | 9 | 4(a) | |||

844.17 | 5 | 4(b) | |||||

846.86 | 5 | 846.86 | 5 | 2 |

ΔE_{full}
. | g_{full}
. | ΔE_{iso}
. | g_{iso}
. | (n, l, j)
. | λ
. | ΔE_{0}
. | g_{0}
. |
---|---|---|---|---|---|---|---|

109.63 | 3 | 109.63 | 3 | (0, 0, 1) | 1 | 109.66 | 3 |

287.71 | 3 | 287.71 | 3 | (1, 1, 1) | 1 | 293.59 | 9 |

294.23 | 5 | 294.23 | 5 | 2 | |||

304.54 | 1 | 304.55 | 5 | 0 | |||

484.47 | 5 | 484.47 | 5 | (2, 2, 1) | 2 | 493.03 | 15 |

494.06 | 4 | 494.28 | 7 | 3(a) | |||

494.57 | 3 | 3(b) | |||||

496.10 | 3 | 496.10 | 3 | 1 | |||

524.20 | 3 | 524.21 | 3 | (2, 0, 1) | 1 | 520.41 | 3 |

658.15 | 4 | 658.154 | 7 | (0, 0, 3) | 3(a) | 657.96 | 7 |

658.16 | 3 | 3(b) | |||||

694.87 | 3 | 695.26 | 7 | (3, 3, 1) | 3(a) | 706.73 | 21 |

695.57 | 4 | 3(b) | |||||

707.92 | 4 | 708.30 | 9 | 4(a) | |||

708.55 | 5 | 4(b) | |||||

710.38 | 5 | 710.34 | 5 | 2 | |||

737.80 | 3 | 737.78 | 3 | (3, 1, 1) | 1 | 749.80 | 9 |

754.50 | 5 | 754.52 | 5 | 2 | |||

770.39 | 1 | 770.44 | 1 | 0 | |||

836.02 | 3 | 836.046 | 7 | (1, 1, 3) | 3(a) | 841.90 | 21 |

836.06 | 4 | 3(b) | |||||

844.02 | 4 | 844.107 | 9 | 4(a) | |||

844.17 | 5 | 4(b) | |||||

846.86 | 5 | 846.86 | 5 | 2 |

The levels listed in Tables III and IV are grouped into sets, each characterized by the same set of values of (*n*, *l*, *j*). The zeroth-order energy shifts and degeneracy of each set are given in columns 7 and 8 of the tables. The states in a given (*n*, *l*, *j*) set do not all have the same energy because of the angle-dependent terms in *V*. Instead states within a set that have different values of *λ* also have different energies. These are what we refer to as “*λ*-split” states. In addition, some states within a (*n*, *l*, *j*) set have the same value of *λ* but nevertheless are also split. These are what we call “*m*_{λ}-split” states. In Tables III and IV the levels involved in *m*_{λ} splitting are denoted with “(a)” or “(b),” next to the value of *λ* to which they correspond. Note that *m*_{λ}-split states occur for *λ* ≥ 3.

### C. An isotropic approximation to the PES

We consider now a calculation like that of Sec. IV B but with all the perturbation terms in the Hamiltonian except the *K* = 0, *L* = *J* = 2 excluded. That is, we set all the anisotropic terms in the decomposed PES to zero and keep only the leading isotropic perturbation term. This approximation renders the Hamiltonian matrix very sparse (less than 0.35% of the matrix elements in the *para* and *ortho* matrices are nonzero) and the calculation much faster than that of Subsection IV B.

The results of this “isotropic” calculation are given in columns 3 and 4 of Tables III and IV as energy-level shifts from the ground state (Δ*E*_{iso}) and degeneracies (*g*_{iso}). From them one notes, first, that there is near-quantitative agreement between the isotropic results and those from the essentially full treatment of Sec. IV B. In fact, the *λ* splittings are fully accounted for in the former. Clearly, those splittings result entirely, or almost entirely, from the $T2,2(0,0)$ contributions to the PES. This is consistent with the work of Mamone *et al.*^{7,8} who interpreted the *λ* splittings of the (*n*, *l*, *j*) = (1, 1, 2) and (1, 1, 3) sets of states in vibrationally excited H_{2} by reference to the same tensor. Second, those which are not reproduced in the isotropic results are the *m*_{λ} splittings. Instead, the computed Δ*E*_{iso} value corresponding to a pair of *m*_{λ}-split levels falls between the Δ*E*_{full} values of that pair and has a degeneracy that is the sum of the *g*_{full} values of the pair. Of course, since *λ* and *m*_{λ} are conserved in an isotropic potential, the (2*λ* + 1) degeneracy of a level of given *λ* must persist, and there are no *m*_{λ} splittings. It is only *K*≠0 contributions to the PES that can produce those splittings.

## V. PERTURBATION-THEORY ANALYSIS OF *λ* AND *m*_{λ} SPLITTINGS

### A. *λ* splittings

The results of Subsection IV C show that the $T2,2(0,0)$ part of the PES by Xu *et al.*^{6} essentially determines the *λ* splittings in the relevant low-energy level structure. One can understand this as follows. First, for most of the (*n*, *l*, *j*) *λ*-split multiplets in Tables III and IV the *only* tensor term that can contribute to diagonal matrix elements between the zeroth-order states is the $T2,2(0,0)$ term. This is because of the selection rules that apply to *l*, *j*, and *λ* from Eqs. (13) and (14). Second, the only group of states in the tables for which diagonal coupling by a second tensor term is possible is (*n*, *l*, *j*) = (2, 2, 2). But that tensor—$T4,4(0,0)$—contributes to the PES at a level that is about two orders of magnitude smaller than the $T2,2(0,0)$ contribution (e.g., see Table II and Fig. 1). Since diagonal matrix elements are the relevant quantities in first-order PT corrections to energies, it is not surprising that the $T2,2(0,0)$ term dominates in determining *λ*-splitting characteristics.

From the foregoing we can immediately write the first-order *λ*-splitting corrections for a given (*n*, *l*, *j*) multiplet as

where we have neglected the small $T4,4(0,0)$ contribution to the (*n*, *l*, *j*) = (2, 2, 2) case. Relevant radial factors on the rhs of Eq. (16) have been evaluated above (Sec. IV A). The angular factor in Eq. (16) is given by Eq. (15) evaluated for *L* = 2,

Equations (16) and (17) give the first-order energy shift for each of the possible *λ* components associated with a given (*n*, *l*, *j*). Note that within each (*n*, *l*, *j*) multiplet the *pattern* of shifts is entirely determined by the angular factor and depends only on *l* and *j*. In turn, the overall spread in energy of the *λ*-split states is determined by the radial factor and depends on *n* and *l*.^{29} Table V summarizes calculated first-order energy shifts for the relevant (*n*, *l*, *j*) multiplets in Tables III and IV. Also given in Table V are the relevant radial factors for each set of states, the angular factor for each *λ* component from Eq. (17), the first-order energy shifts from Eq. (16), and the corresponding energy shifts from the full variational calculations *δ*_{λ,var} ≡ Δ*E*_{full} − Δ*E*_{0} (or for levels split by *m*_{λ} splitting *δ*_{λ,var} ≡ Δ*E*_{iso} − Δ*E*_{0}). One sees that the pattern of splittings for each (*n*, *l*, *j*) is correctly predicted by the perturbative approach. Moreover, the splitting magnitudes are reproduced to within about 1 cm^{−1} in most cases. The upshot is that lowest-order PT in the basis of Eq. (11) generally provides an excellent first approximation to *λ* splitting in the species.

n, l, j
. | $\u3008n,l|G2,20,0|n,l\u3009$ . | λ
. | $\u3008l,j,\lambda |T2,2(0,0)|l,j,\lambda \u3009$ . | δ_{λ,PT}
. | δ_{λ,var}
. |
---|---|---|---|---|---|

1, 1, 1 | 159.64 | 0 | +0.071 176 3 | +11.36 | +10.95 |

1 | −0.035 588 1 | −5.68 | −5.68 | ||

2 | +0.007 117 6 | +1.14 | +0.64 | ||

1, 1, 2 | 159.64 | 1 | +0.035 588 1 | +5.68 | +5.29 |

2 | −0.035 588 1 | −5.68 | −5.92 | ||

3 | +0.010 168 0 | +1.62 | +0.52 | ||

1, 1, 3 | 159.64 | 2 | +0.028 470 5 | +4.54 | +4.96 |

3 | −0.035 588 1 | −5.68 | −5.85 | ||

4 | +0.011 862 7 | +1.89 | +2.21 | ||

2, 2, 1 | 229.39 | 1 | +0.035 588 1 | +8.16 | +2.07 |

2 | −0.035 588 1 | −8.16 | −8.56 | ||

3 | +0.010 168 0 | +2.33 | +1.25 | ||

2, 2, 2 | 229.39 | 0 | +0.050 840 2 | +11.66 | +11.71 |

1 | +0.025 420 1 | +5.83 | +5.42 | ||

2 | −0.010 894 3 | −2.50 | −5.22 | ||

3 | −0.029 051 5 | −6.66 | −7.14 | ||

4 | +0.014 525 8 | +3.33 | +1.47 | ||

3, 1, 1 | 312.17 | 0 | +0.071 176 3 | +22.22 | +20.59 |

1 | −0.035 588 1 | −11.11 | −12.00 | ||

2 | +0.007 117 6 | +2.22 | +4.70 | ||

3, 3, 1 | 302.63 | 2 | +0.028 470 5 | +8.62 | +3.65 |

3 | −0.035 588 1 | −10.77 | −11.47 | ||

4 | +0.011 862 7 | +3.59 | +1.57 |

n, l, j
. | $\u3008n,l|G2,20,0|n,l\u3009$ . | λ
. | $\u3008l,j,\lambda |T2,2(0,0)|l,j,\lambda \u3009$ . | δ_{λ,PT}
. | δ_{λ,var}
. |
---|---|---|---|---|---|

1, 1, 1 | 159.64 | 0 | +0.071 176 3 | +11.36 | +10.95 |

1 | −0.035 588 1 | −5.68 | −5.68 | ||

2 | +0.007 117 6 | +1.14 | +0.64 | ||

1, 1, 2 | 159.64 | 1 | +0.035 588 1 | +5.68 | +5.29 |

2 | −0.035 588 1 | −5.68 | −5.92 | ||

3 | +0.010 168 0 | +1.62 | +0.52 | ||

1, 1, 3 | 159.64 | 2 | +0.028 470 5 | +4.54 | +4.96 |

3 | −0.035 588 1 | −5.68 | −5.85 | ||

4 | +0.011 862 7 | +1.89 | +2.21 | ||

2, 2, 1 | 229.39 | 1 | +0.035 588 1 | +8.16 | +2.07 |

2 | −0.035 588 1 | −8.16 | −8.56 | ||

3 | +0.010 168 0 | +2.33 | +1.25 | ||

2, 2, 2 | 229.39 | 0 | +0.050 840 2 | +11.66 | +11.71 |

1 | +0.025 420 1 | +5.83 | +5.42 | ||

2 | −0.010 894 3 | −2.50 | −5.22 | ||

3 | −0.029 051 5 | −6.66 | −7.14 | ||

4 | +0.014 525 8 | +3.33 | +1.47 | ||

3, 1, 1 | 312.17 | 0 | +0.071 176 3 | +22.22 | +20.59 |

1 | −0.035 588 1 | −11.11 | −12.00 | ||

2 | +0.007 117 6 | +2.22 | +4.70 | ||

3, 3, 1 | 302.63 | 2 | +0.028 470 5 | +8.62 | +3.65 |

3 | −0.035 588 1 | −10.77 | −11.47 | ||

4 | +0.011 862 7 | +3.59 | +1.57 |

Notwithstanding the preceding, one does note from Table V four particular instances where the *δ*_{λ,PT} are anomalous, being several cm^{−1} off from the variational results. These values, highlighted in boldface in the table, are the *δ*_{λ,PT} for |*n*, *l*, *j*, *λ*〉 = |2, 2, 1, 1〉, |2, 2, 2, 2〉, |3, 3, 1, 2〉, and |3, 1, 1, 2〉. It is instructive to examine the reason for these anomalies. In short, second-order, off-diagonal couplings play a significant role in these cases, and these second-order energy shifts push the levels away from their first-order positions. Specifically, each of these levels is able to couple to a nearby level via the $T2,2(0,0)$ tensor. |2, 2, 1, 1〉 can also couple to |2, 0, 1, 1〉, and the latter is only about 27.4 cm^{−1} higher in zeroth-order energy. Similarly, |2, 2, 2, 2〉 can couple to the higher-energy |2, 0, 2, 2〉 (also a 27.4 cm^{−1} difference). |3, 3, 1, 2〉 and |3, 1, 1, 2〉 can couple with each other and are separated by a zeroth-order energy difference of only 43 cm^{−1}. The matrix elements connecting these level pairs can be readily evaluated by using Eqs. (12) and (15). With these matrix elements, one can then compute the second-order energy shifts to which the interactions give rise. One finds shifts of −4.2, −3.0, −4.3, and +4.3 cm^{−1}, respectively, for |2, 2, 1, 1〉, |2, 2, 2, 2〉, |3, 3, 1, 2〉, and |3, 1, 1, 2〉. With these second-order corrections, the PT splittings come much closer into coincidence with the variational ones.

### B. Physical insight into the energy ordering of *λ* components

In Subsection V A we have demonstrated that first-order PT, with the $T2,2(0,0)$ term as the perturbation, in general provides an accurate account of both the *λ* splitting patterns and the magnitudes of the splittings. However, this still leaves one wondering whether there is a simple physical explanation for the energy ordering of the *λ* components within a given (*n*, *l*, *j*) multiplet. For example, in the *λ* triplet with *n* = *l* = 1, *j* = 1, the components appear as *λ* = 1, 2, 0 in the order of increasing energies (see Table IV). *Why* this particular *λ* ordering? The same question can be asked for any *λ* multiplet.

It turns out that the answer to this question can be obtained by examining in some detail the $T2,2(0,0)$ tensor, shown to have the dominant role in producing the *λ*-splitting patterns exhibited by H_{2}@C_{60}. It is straightforward to show [for example, by using Ref. 26, Eq. (5.55), p. 190] that

where *α* is the angle between **R** and **r**. The tensor is most positive when these two vectors are aligned and most negative when they are perpendicular. In the H_{2}@C_{60} PES the radial factor multiplying $T2,2(0,0)$ is everywhere positive (see Fig. 1). Thus, when H_{2} is displaced from the cage center, the PES has a lower energy at a given *R* when the H_{2} orientation is perpendicular to the displacement direction than when it is parallel to it. (Incidentally, this is consistent with *ab initio* calculations of the interaction for H_{2} in C_{60}.^{30})

An understanding of the energy ordering of the states in a *λ*-split group should thus be obtainable by looking at the expectation value of 3 cos^{2} *α* − 1 for each of the states involved. The potential energy and hence the energy of the states should increase with increasing values of 〈3 cos^{2} *α* − 1〉. For *l* = 1, *j* = 1 these expectation values are 0.8, −0.4, and 0.08, for *λ* = 0, 1, and 2, respectively. **R** and **r** tend to be more parallel for *λ* = 0, more perpendicular for *λ* = 1, and in-between for *λ* = 2. Hence, one predicts the *λ* energy ordering from lowest to highest to be 1, 2, 0, in agreement with that from the variational calculations. For *l* = 2, *j* = 1 the 〈3 cos^{2} *α* − 1〉 values are 0.4, −0.4, and 0.114 286, respectively, for *λ* = 1, 2, and 3. The corresponding *λ* energy ordering should be 2, 3, 1. Table VI shows such results for all *λ*-split groups in Tables III and IV. In all instances, the predicted orderings are just those found from the variational calculations.

l, j or j, l
. | λ
. | 〈3 cos^{2} α − 1〉
. | Energy ordering . |
---|---|---|---|

1,1 | 0 | +0.8 | 1 → 2 → 0 |

1 | −0.4 | ||

2 | +0.08 | ||

2,1 | 1 | +0.4 | 2 → 3 → 1 |

2 | −0.4 | ||

3 | +0.114 286 | ||

2,2 | 0 | +0.5714 | 3 → 2 → 4 → 1 → 0 |

1 | +0.2857 | ||

2 | −0.1224 | ||

3 | −0.3265 | ||

4 | +0.1633 | ||

3,1 | 2 | +0.32 | 3 → 4 → 2 |

3 | −0.4 | ||

4 | +0.133 33 |

l, j or j, l
. | λ
. | 〈3 cos^{2} α − 1〉
. | Energy ordering . |
---|---|---|---|

1,1 | 0 | +0.8 | 1 → 2 → 0 |

1 | −0.4 | ||

2 | +0.08 | ||

2,1 | 1 | +0.4 | 2 → 3 → 1 |

2 | −0.4 | ||

3 | +0.114 286 | ||

2,2 | 0 | +0.5714 | 3 → 2 → 4 → 1 → 0 |

1 | +0.2857 | ||

2 | −0.1224 | ||

3 | −0.3265 | ||

4 | +0.1633 | ||

3,1 | 2 | +0.32 | 3 → 4 → 2 |

3 | −0.4 | ||

4 | +0.133 33 |

In summary, the energy ordering of the *λ* components within a multiplet reflects the degree to which **R** and **r** are aligned in a state, such alignment being energetically unfavorable due to the nature of the PES.

### C. *m*_{λ} splittings

Levels with *λ* ≥ 3 must necessarily be split for a PES of *I _{h}* symmetry since the highest possible degeneracy is five-fold. One can show

^{21,22}from Eq. (11) and the transformation properties of spherical harmonics that for

*λ*= 3 the splitting results in a triply degenerate level (

*I*irrep

_{h}*T*

_{2g}when

*l*+

*j*is even,

*T*

_{2u}for odd

*l*+

*j*) and a quadruply degenerate one (

*G*for even

_{g}*l*+

*j*and

*G*for odd

_{u}*l*+

*j*). A doublet also arises for

*λ*= 4, with one level quadruply degenerate (

*G*for even

_{g}*l*+

*j*,

*G*for odd) and the other quintuply degenerate (

_{u}*H*for even

_{g}*l*+

*j*,

*H*for odd). As noted above, there are several examples of such

_{u}*m*

_{λ}splitting in Tables III and IV.

In order to compute these splittings with first-order PT, one first identifies the relevant perturbation—the PES tensor term(s) that can give rise to nonzero off-diagonal matrix elements and/or *m*_{λ}-dependent diagonal matrix elements between the (2*λ* + 1) states of a given |*n*, *l*, *j*, *λ*〉 level. Since only terms of nonzero rank can do so [e.g., see Eq. (15)], and since the only appreciable contributors to the PES considered here are of rank 6, we focus only on those terms. Generally, only a small number of these tensors is relevant to the *m*_{λ} splitting of a given level due to the selection rules on *L* and *J* in Eq. (14). Further—see Section II and Eq. (6)—of the set of 13 $TL,J(6,Q)$ tensor components for a given *L*, *J* only *Q* = 0 and *Q* = ± 5 contribute to the PES decomposition for our choice of the SF axis system, and these components have amplitudes that are in a fixed relation to one another. Second, one then evaluates the (2*λ* + 1)^{2} matrix elements (which may be contributed to by more than one tensor) for a given level. These are given by Eqs. (12)–(14). Finally, one diagonalizes the matrix. It turns out that for both *λ* = 3 and *λ* = 4 one can derive (supplementary material) closed-form expressions for the eigenvalues of these matrices, and therefore for the *m*_{λ} splittings. One finds for the *λ* = 3 levels

and for the *λ* = 4 levels

where the summations are over those values of *L* and *J* corresponding to nonzero values of the reduced matrix elements $\u3008l,j,\lambda \Vert TL,J(6)\Vert l,j,\lambda \u3009$. Note that we have defined (supplementary material) the *m*_{λ} splittings, $\delta m\lambda (\lambda )$, in such a way that the quantity is positive if the level with higher degeneracy is at a higher energy than the one with lower degeneracy.

We have used Eqs. (19) and (20) to calculate the first-order splittings for each of the *m*_{λ}-split levels listed in Tables III and IV. The results are given in Table VII. Included in the table are the tensor element(s) and radial factor(s) associated with the splitting of a given level, the splittings from perturbation theory, and the splittings from the variational results of Tables III and IV. One sees that not only are the orderings of the split levels correctly predicted by PT, but that the latter also does quite well in predicting the magnitudes.

n, l, j, λ
. | Tensors . | $\u3008nl|GL,J6,0|nl\u3009$ . | $\delta m\lambda ,PT(\lambda )$ . | $\delta m\lambda ,var(\lambda )$ . |
---|---|---|---|---|

1,1,2,3 | $T2,4(6,Q)$ | 1.61 | −0.19 | −0.11 |

2,2,2,3 | $T2,4(6,Q)/T4,2(6,Q)/T4,4(6,Q)$ | 2.33/4.94/−0.26 | +0.29 | +0.26 |

2,2,2,4 | $T2,4(6,Q)/T4,2(6,Q)/T4,4(6,Q)$ | 2.33/4.94/−0.26 | +0.40 | +0.29 |

3,3,0,3 | $T6,0(6,Q)$ | 6.96 | −0.81 | −0.81 |

4,4,0,4 | $T6,0(6,Q)$ | 10.34 | +0.93 | +0.91 |

2,2,1,3 | $T4,2(6,Q)$ | 4.94 | −0.59 | −0.51 |

3,3,1,3 | $T4,2(6,Q)/T6,0(6,Q)$ | 7.32/6.96 | +0.75 | +0.70 |

3,3,1,4 | $T4,2(6,Q)/T6,0(6,Q)$ | 7.32/6.96 | +0.80 | +0.63 |

0,0,3,3 | $T0,6(6,Q)$ | 0.10 | −0.01 | −0.01 |

1,1,3,3 | $T2,4(6,Q)/T0,6(6,Q)$ | 1.61/0.13 | +0.04 | +0.04 |

1,1,3,4 | $T2,4(6,Q)/T0,6(6,Q)$ | 1.61/0.13 | +0.12 | +0.15 |

n, l, j, λ
. | Tensors . | $\u3008nl|GL,J6,0|nl\u3009$ . | $\delta m\lambda ,PT(\lambda )$ . | $\delta m\lambda ,var(\lambda )$ . |
---|---|---|---|---|

1,1,2,3 | $T2,4(6,Q)$ | 1.61 | −0.19 | −0.11 |

2,2,2,3 | $T2,4(6,Q)/T4,2(6,Q)/T4,4(6,Q)$ | 2.33/4.94/−0.26 | +0.29 | +0.26 |

2,2,2,4 | $T2,4(6,Q)/T4,2(6,Q)/T4,4(6,Q)$ | 2.33/4.94/−0.26 | +0.40 | +0.29 |

3,3,0,3 | $T6,0(6,Q)$ | 6.96 | −0.81 | −0.81 |

4,4,0,4 | $T6,0(6,Q)$ | 10.34 | +0.93 | +0.91 |

2,2,1,3 | $T4,2(6,Q)$ | 4.94 | −0.59 | −0.51 |

3,3,1,3 | $T4,2(6,Q)/T6,0(6,Q)$ | 7.32/6.96 | +0.75 | +0.70 |

3,3,1,4 | $T4,2(6,Q)/T6,0(6,Q)$ | 7.32/6.96 | +0.80 | +0.63 |

0,0,3,3 | $T0,6(6,Q)$ | 0.10 | −0.01 | −0.01 |

1,1,3,3 | $T2,4(6,Q)/T0,6(6,Q)$ | 1.61/0.13 | +0.04 | +0.04 |

1,1,3,4 | $T2,4(6,Q)/T0,6(6,Q)$ | 1.61/0.13 | +0.12 | +0.15 |

## VI. DISCUSSION AND CONCLUSION

We have shown that low-order PT produces a very good approximation to the low-energy TR level structure of H_{2}@C_{60}. This includes the *λ* and *m*_{λ} splitting patterns and magnitudes calculated using first-order PT, that are generally in good agreement with those from variational quantum 5D calculations. One reason for this is that the interaction potential is dominated by the purely radial term $G0,00,0(R)T0,0(0,0)$. Therefore, a natural choice for the zeroth-order Hamiltonian is the one that includes only this portion of the PES. Second, in part because of the small inertial parameters associated with H_{2}, the zeroth-order states that follow from this choice of $H\u02c60$ have level spacings that tend to be large relative to the perturbation terms not included in $H\u02c60$. Hence, the perturbation-induced mixing between nondegenerate zeroth-order states tends to be weak. In consequence, first-order PT provides a good description of the level structure. While we have demonstrated this for one particular PES,^{6} we expect it to be true for other reasonable PESs of this system as well.

This is not to say that the approach presented here would necessarily be equally successful in applications to the endohedral complexes of C_{60} with other diatomics. For example, one interaction PES for CO@C_{60}^{22} has contributions from isotropic terms other than $G0,00,0(R)T0,0(0,0)$, that are comparable in magnitude to the latter. In this situation, a choice of $H\u02c60$ (and, hence, zeroth-order states) analogous to ours would not be useful. On the other hand, Olthof *et al.*,^{22} have shown that a variational calculation similar to that of Sec. IV C, in which only the contributions of isotropic tensors are retained in the TR Hamiltonian, produces a level structure that is close to that of the full Hamiltonian. The small deviations are due to *m*_{λ} splittings. The point is that for this species a PT calculation of the *m*_{λ} splittings, using the eigenstates of the isotropic Hamiltonian as the zeroth-order states, might have some utility.

Fully coupled 5D variational calculations are now commonplace, and for any given PES, essentially exact solutions for the TR states of H_{2}@C_{60} in the rigid-body approximation can be obtained.^{4–6,17} Nevertheless, the connection between particular pieces of the Hamiltonian and specific features of the level structure is not necessarily clear from those solutions. The accurate low-order PT results presented herein extend the work of Mamone *et al.*,^{7,8} in helping to clarify such connections. Moreover, they facilitate the development of physical insight into the nature of the level structure. A particularly good example of this is the simple and intuitive explanation of the energy ordering of the *λ* components within the multiplets presented in this paper, that has eluded the variational treatments. Finally, any extensions of the variational approach to incorporate additional H_{2}@C_{60} degrees of freedom (the H_{2} stretch, vibrational modes of the C_{60} cage) will involve significant increases in computational cost and difficulty of interpretation. Our results provide encouragement that these difficulties might be reasonably mitigated without marked loss in accuracy by judicious application of perturbation theory.

## SUPPLEMENTARY MATERIAL

See supplementary material section S1 for the details of the specific $H\u02c6$ we have employed including the H_{2} geometry and the SF coordinates of the carbon nuclei. Section S2 presents details of the 3D isotropic harmonic-oscillator basis and the Gauss-associated-Laguerre quadrature scheme. Section S3 presents a derivation of the closed-form expressions for the *m*_{λ} splitting of *λ* = 3 and *λ* = 4 levels from first-order perturbation theory.

## Acknowledgments

Z.B. is grateful to the National Science Foundation for its partial support of this research through Grant CHE-1112292. P.M.F. thanks Professor Daniel Neuhauser for generously allowing access to his computational resources and Professor Christian Müller for hosting him at Ruhr Universität Bochum while some of this research was done.

## REFERENCES

Throughout this paper we use “isotropic” to refer to any tensor invariant to rotation about any axis fixed to the C_{60} cage and passing through the center of that cage including all spherical tensors of rank 0.

This matches the *λ*-splitting behavior characterized for (*n*, *l*, *j*) = (1, 1, 1) of H_{2} in a spherically symmetric cavity. See Appendix B of Ref. 24.