It is instructive to introduce the formalism of SARFs for cylindrical confinement by elaborating a concrete example. We consider a C₆₀ molecule encapsulated in a CNT with its center of mass on the tube’s long axis (Figure 1). We treat the molecule as a rigid cluster of 60 carbon atoms, labeled Λ_a = 1, . . . , 60, and the nanotube as a homogeneous cylindrical density distribution n(r⃗) of carbon atoms (the subscript _a stands for atom). The interaction energy then reads

where r⃗_Λa = (x_Λa, y_Λa, z_Λa) is the position vector of atom Λ_a of the C₆₀ molecule. The function v(d) is the pair potential function giving the energy of two interacting centers (a carbon atom of the molecule and a carbon atom of the tube) at a distance d apart. Its precise form is not essential at this moment. For a tube of radius R, n(r⃗)dr⃗ = σδ(ρ − R)ρdρdΦdZ, and V becomes

Here, cylindrical coordinates (ρ,Φ,Z) have been introduced so that r⃗ = (ρ cos Φ, ρ sin Φ, Z), and r⃗_R = (Rcos Φ, R sin Φ, Z). The quantity σ is the tube’s surface density, the value for rolled-up graphene sheets (CNTs) is 0.372 Å⁻².

We have not yet specified the molecule’s orientation. Let us introduce a reference orientation for the C₆₀ molecule, we choose it to be the orientation where two-fold symmetry axes coincide with the coordinate axes—the so-called standard orientation (Figure 1). The molecule’s center of mass coincides with the origin and the z-axis is chosen to coincide with the tube’s long axis. Molecular rotations can now be specified with respect to this standard orientation. Using the Euler angle convention of Ref. [11], any rotation can be described as the succession of three Euler rotations: (i) a rotation ℜ_z(α) over 0 ≤ α < 2π about the z-axis, followed by (ii) a rotation ℜ_y(β) over 0 ≤ β ≤ π about the y-axis, and finally (iii) a rotation ℜ_z(γ) over 0 ≤ γ < 2π about the z-axis again. The x-, y- and z-axes are kept fixed. Note that α = β = γ = 0 then corresponds to the standard (reference) orientation. With the convention of Ref. [11], any coordinate function f(r⃗) is transformed as ℜ (α, β, γ)f(r⃗ ) = f(ℜ⁻¹(α, β, γ) r⃗ ), where ℜ (α, β, γ) = ℜ_z(γ) ℜ_y(β) ℜ_z(α). Applying this to the pair potentials v, essentially functions of the atomic positions r⃗_Λa, results in the following explicit expression for the molecule-tube interaction energy V (α, β, γ) for a rotated molecule:

Expression (3) does not make use of the molecule’s symmetry; its numerical implementation requires the use of explicit Euler rotation matrices and is computationally heavy. To exploit the symmetries of both the molecule and the surrounding tube, we proceed as follows. First, we introduce spherical coordinates for the atoms of the molecule, r⃗_Λa = (r_Λa sin θ_Λa cos φ_Λa, r_Λa sin θ_Λa sin φ_Λa, r_Λa cos θ_Λa), and rewrite the interaction energy for a molecule in the standard orientation as

with

The distance |r⃗_R − r⃗_Λa| reads

Only the difference of Φ and φ_Λa enters expression (6), as the argument of the function cos. Therefore, the quantity w is independent of φ_Λa, since a change of variables Φ′ = Φ − φ_Λa eliminates φ_Λa from the expression for |r⃗_R − r⃗_Λa| and since ∫_φΛa^2π+φΛ_a dΦ′f(Φ′) = ∫₀^2π dΦ′f(Φ′) for a function f with periodicity 2π. For a C₆₀ molecule, all C atoms have the same radial coordinate r_Λa ≡ r_a, which can therefore be considered a constant rather than a variable in expressions (4) – (6). Hence, we end up with the equations

The quantity w(R; θ_Λa), taken as a function of θ_Λa, can be expanded into m = 0 spherical harmonics:

Here, we use the Bradley and Cracknell spherical harmonics (Ref. [11]). Upon rotation, spherical harmonics transform into linear combinations of spherical harmonics. One has

The rotation operator ℜ(α, β, γ) has been introduced before; the quantities (α, β, γ) are the Wigner -functions. In the present case, cylindrical symmetry implies m = 0, for which the α-independent Wigner -functions reduce to

Collecting the previous equations results in the following potential energy expression for a rotated molecule:

The potential energy V (β, γ) depends on the molecule’s orientation and on the tube’s radius; it can therefore be considered as a potential energy field (nanotube field) set up by the surrounding tube and experienced by the molecule. Note that there is no α-dependence since the initial rotation ℜ_z(α) about the z-axis has no effect on the interaction energy—a consequence of the “smooth-tube approximation”.

So far, only the cylindrical symmetry—the site symmetry—has been used. The molecular symmetry is accounted for by the distribution of C atoms. We introduce atomic form factors c_lⁿ,

molecular shape factors g_l,

and normalised atomic form factors α_lⁿ,

Icosahedral molecular symmetry implies that c_lⁿ differs from zero only for n even and l = 0, 6, 10, 12, . . .. The non-vanishing g_l and α_lⁿ values are tabulated in Table 1 up to l = 12. We can now rewrite V (β, γ) as

where

are molecular-and-site-symmetry-adapted rotator functions (SARFs). Rotator functions, originally introduced by James and Keenan [2], are the appropriate variables for the description of orientationaldependent properties of molecules in crystals [3,4]. They account for the symmetry of the molecule and the symmetry of the crystal site point group. In the present case the crystal site symmetry is the D_∞h symmetry of the (smooth) nanotube. The cylindrical site symmetry has the consequence that the Wigner -functions are linear combinations of spherical harmonics [Equation (13)], and that the rotator functions (β, γ) can be easily implemented:

The symmetry of a C₆₀ molecule implies some restrictions on the atomic form factors. The combination of a center of inversion and the (x, z)- and (x, y)-planes being mirror planes results in the follow properties:

The same relations hold for the normalized molecular form factors α_lⁿ. In particular, it follows that the rotator functions are real functions since α_l⁻ⁿ = α_lⁿ and [Y_lⁿ (β, γ)]^* = Y_l⁻ⁿ (β, γ).

In summary, as a numerically much more efficient alternative to Equation (3) involving explicit coordinate transformations (Euler rotations), one can maximally exploit the symmetry of both the molecule and its environment by first calculating the atomic form factors c_lⁿ [Equation (15)], molecular shape factors g_l [Equation (16)], and normalised atomic form factors α_lⁿ [Equation (17)]. The latter then serve as coefficients in linear combinations of spherical harmonics defined as SARFs [Equation (20)]. These coefficients have only to be calculated once. For a given tube radius R, the expansion coefficients v_l(R) are calculated by numerical integration [Equation (11)]. The evaluation of a few leading terms of [Equation (18)] then serves as an excellent approximation for the molecule’s potential energy V (β, γ).

For actual calculations, a potential function and potential parameters have to be specified. In Refs. [12] and [13], a Born–Mayer–van der Waals C-C pair potential,

was introduced for studying C₆₀-C₆₀ interactions in C₆₀-fullerite (solid buckminsterfullerene); it led to a crystal field potential and a structural phase transition temperature [14,15] in good agreement with experiments. Using the potential constants C₁ = 3.24 × 10⁷ K × k_B, C₂ = 3.6 Å⁻¹ and B = 4.579 × 10⁵ K × k_B · Å⁶ of Refs. [6] and [7] results in the v_l(R) coefficients, obtained via numerical integration of expression (11), shown in Table 2. The amplitude of these coefficients decreases with increasing l. More indicative are the weighted coefficients g_lv_l(R), also given in Table 2. They reveal the relative importance of the contributing l terms. The l = 12 terms clearly contribute much less than the l = 6 terms— the lowest-order terms introducing (β, γ)-dependence, but the l = 10 contribution obviously plays an important role for the R = 6.0 and R = 7.0 cases.

Having calculated the quantities g_l, v_l(R) and α_lⁿ allows to construct the rotator functions (β, γ) and to evaluate expression (18). In Figure 2, we show the results of V (β, γ) for R = 6.0 Å, R = 7.0 Å and R = 8.0 Å for a calculation up to l = 12. For R = 6.0 Å, there are 12 local minima, corresponding to the twelve equivalent molecular orientations where two opposing pentagons of the C₆₀ molecule are perpendicular to the z-axis. The 20 maxima correspond to orientations where hexagons are perpendicular to the z-axis. (The “soccer-ball structure” indeed features 12 pentagons and 20 hexagons.) At R = 7.0 Å, there are 30 local minima: they correspond to the molecular configurations where opposing double bonds are perpendicular to the tube’s long axis (z-axis). The potential energy is maximal when a hexagon is perpendicular to the z-axis. Finally, at R = 8.0 Å, minima and maxima correspond to hexagons and pentagons perpendicular to the z-axis, respectively.

When comparing the nanotube fields shown in Figure 2 to the (β, γ)-maps obtained via the exact formula with explicit coordinate transforms [Equation (3)], there is no visual difference: the SARFs expansion up to l = 12 is an excellent approximation. Using the same integration routines, the direct calculations took a few hours each, however, while the evaluation of the v_l(R) coefficients and the SARFs expansion was a matter of seconds. Therefore, in addition to providing physical and mathematical understanding of the symmetries present in the system, SARFs have a considerable computational advantage.

The important feature in the construction of an expansion into SARFs is the vanishing of several atomic form factors c_lⁿ (α_lⁿ). This is a direct consequence of the symmetry of the considered molecule. In terms of group theory, only certain linear combinations of spherical harmonics Y_lⁿ (β, γ) belong to the unit representation (A_1g irreducible representation) of the molecular symmetry group and contribute to the expansion of V (β, γ) into SARFs. For a cylindrical tube there is no α-dependence, and the expansion into SARFs is in fact nothing but an expansion into linear combinations of spherical harmonics Y_lⁿ (β, γ) belonging to the unit representation.

Often, several types of molecular sites are treated as interaction centers (ICs). In the case of a molecule consisting of different types of atoms, every atomic type interacts differently with the surrounding nanotube, which can be accounted for by using different potential constants (or even different potential functions). The pair potential v(d) and the expansion coefficients v_l(R) then become type-dependent:

Here, the superscript ^t stands for the IC type. The ICs need not only be atoms; in the case of C₆₀ molecules, it is customary to place ICs on bonds. For a C₆₀ molecule, double bonds (fusing hexagons) and single bonds (fusing hexagons and pentagons) are considered, and labeled t = db and t = sb, respectively. (By bonds, the midpoints of bonds are understood.) These additional ICs were originally introduced to account for variations in the charge distribution of a C₆₀ molecule [13,16]. For each of the IC types, t = a, db, sb, the atomic form factors

the molecular shape factors

and the normalised atomic form factors

have to be calculated. Here, θ_Λt and φ_Λt stand for the polar and azimuthal angles of the ICs of type t, labeled Λ_t = 1, . . . ,N_t, with N_t the number of ICs of type t (N_a = 60, N_db = 30 and N_sb = 60). Remarkably, it turns out that

with ξ_l^t = ±1. It is therefore useful to introduce modified molecular shape factors

so that

where α_lⁿ ≡ α_l^a,n. This allows to write the following generalised expression for V (β, γ):

Note that the same SARFs as before [Equation (19)] can be used.

The nanotube fields of a C₆₀ molecule arising from the extended interaction model described in the present subsection do not differ qualitatively from the ones shown in Figure 2. For full details, we refer to Ref. [8].

The manifestation of potential energy landscapes as in Figure 2 as a consequence of molecular symmetry and cylindrical confinement and their dependence on the tube radius R has important implications on the physical properties of a “peapod”—a chain of several molecules encapsulated in a CNT. In particular, the peapod system of C₆₀ molecules encapsulated in a CNT displays unusual dynamical behavior as demonstrated by different experimental techniques: inelastic neutron scattering [17], nuclear magnetic resonance [18,19] and high-pressure near-infrared Raman scattering [20]. Obviously, for a realistic description of peapods, molecule-molecule interactions have to be taken into account as well. For typical tube radii (R ~ 7 Å) of C₆₀ peapod samples, though, the intermolecular interactions are several orders of magnitude smaller than the molecule-tube interaction [8] and do therefore not significantly influence the molecules’ optimal orientation with respect to the surrounding CNT. A discussion of the dynamics of the molecules in a C₆₀-peapod is beyond the scope of the present paper, however — we recall that our purpose is to provide the mathematical framework for the effective exploitation of the molecular and the environmental symmetry for calculating potential energies. For details, we refer to the relevant experimental [17,18,19,20] and theoretical [6,7,8] literature.

In the foregoing we have introduced SARFs for a C₆₀ molecule, displaying icosahedral symmetry, with atoms, double and single bonds considered as three different types of ICs. A special feature of the C₆₀ molecule is that for each IC type, the radial coordinates of the ICs are equal (dependent on t, not on Λ_t): r_Λa ≡ r_a, r_Λdb ≡ r_db, r_{Λ sb} ≡ r_sb. This does not hold for all symmetries, however. The general formulation of a molecule’s nanotube field V (β, γ) has to take this into account.

To fix ideas, we take the example of a C₇₀ molecule, which has an ellipsoidal shape andD_5h symmetry. (All formulas of this subsection will hold in general, though.) A popular IC cluster model for C₇₀ features the 70 carbon atoms (t = a), 20 so-called D-centers on bonds near the top and bottom of the molecule (t = D) and 30 so-called I-centers in the “equatorial zone” of the molecule as ICs (Figure 3). The essential step to develop appropriate SARFs is to group ICs with the same value of the radial coordinate r_Λt. In the case of a C₇₀ molecule, ICs with the same |z_Λt| value have the same r_Λt value. Therefore, we can think of layers of ICs having the absolute value of their z-coordinate in common. We use the term ‘layers’ in an abstract way and let it refer to a group of ICs with equal r_Λt values. We label the layers by an index λ_t, and the ICs within layer λ_t by an index ν_λt. This results in a compound index

to address IC Λ_t. Introducing the layer-dependent analogues of Equations (15) – (17),

and the layer-dependent SARFs

results in the following expression for the molecule’s nanotube field:

Equation (34) is the most general form of the SARFs expansion for a molecule placed on the long axis of a CNT. It takes into account different IC types and the non-spherical distribution of ICs. Note that there is no proportionality rule like Equation (29) in the case of “layered” structures (different r_Λt values for IC type t). Results for the C₇₀ molecule’s nanotube fields and their physical implications can be found in Ref. [9].

We now apply the SARFs procedure to an example with cubic molecular symmetry. Cubane, C₈H₈, has eight carbon atoms arranged on the corners of a cube to each of which a hydrogen atom is bound (Figure 4). While its chemical synthesis dates back to 1964 [21], it has gained renewed interest after the successful synthesis of fullerene-cubane, C₆₀.C₈H₈, a remarkable molecular crystal consisting of icosahedral (I_h) and cubic (O_h) molecules with stoichiometry 1:1 [22].

We consider a cubane molecule encapsulated in a CNT with radius R; it is intended as a generic example of a cubic molecule inserted into a nanotube. An example of an actual molecule with O_h symmetry that has successfully been encapsulated in a nanotube is octasilesquioxane, Si₈H₈O₁₂ [23,24].

We model the cubane molecule as a simple cubic cluster of 8 ICs placed on the H atoms and define the standard orientation (α = β = γ = 0) as the orientation where the cube’s faces are parallel to the coordinate planes. The ICs then have coordinates (±a,±a,±a), (±a,±a, a), (±a, a,±a) and (a,±a,±a) with a = 1.4139 Å. First, the atomic form factors c_lⁿ and the derived quantities g_l and α_lⁿ have to be determined [Equations (15) – (17)]. In Table 3 we show the non-zero g_l and α_lⁿ coefficients up to l = 12. The symmetry relations (21a) – (21d) are also valid for cubic symmetry. In addition, c_lⁿ (α_lⁿ) coefficients vanish if n is not a multiple of 4:

The lowest non-zero l-value yielding non-vanishing c_lⁿ coefficients is l = 4. This is a well-known result from group theory; the l = 4 rotator function

is proportional to the cubic harmonic K₄(β, γ). The next non-zero terms in the SARFs expansion have l = 6, 8, 10, 12, . . ..

For the pair interaction potential v(d) we take the Lennard-Jones potential used for modeling cubanefullerene interactions in C₆₀.C₈H₈ [25]:

with ɛ = 16.733 K × k_B and σ = 2.895 Å. For two radii, R = 5.0 Å and R = 7.0 Å, we first calculate the expansion coefficients v_l(R) [Equation (11)]. They are shown in Table 4 up to l = 12, together with the weighted expansion coefficients g_lv_l(R). The magnitude of g_lv_l(R) decreases rapidly with increasing l. The R = 5.0 Å and R = 7.0 Å nanotube fields, calculated via the SARFs expansion up to l = 12, Equation (18), are shown in Figure (5). Both display cubic symmetry, but the local minima and maxima are located differently. For R = 5.0 Å, 6 equivalent minima can be distinguished. They correspond to the 6 realisations of the standard orientation where the cubane molecule’s faces are aligned with the coordinate planes. There are 12 maxima, they correspond to the orientations where the z-axis (long axis of the tube) intersects the midpoints of two opposing edges of the cube. At R = 7.0 Å, the local minima have become local maxima. There are 8 local minima, corresponding to orientations where the tube’s long axis intersects two opposing vertices of the cube. These findings are relevant for explaining actual experimental results; Si₈H₈O₁₂ molecules (of cubic symmetry) inserted in CNTs with radii R ≈ 6 Å - 7 Å self-assemble into Si_4nH₈O_8n−4 ladder-like structures [26]. We argue that the optimal orientations found here are a necessary prerequisite for the formation of the experimentally observed chemical bonds between neighboring octasilesquioxane monomers.