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We present a general description of the formalism of symmetry-adapted rotator functions (SARFs) for molecules in cylindrical confinement. Molecules are considered as clusters of interaction centers (ICs), can have any symmetry, and can display different types of ICs. Cylindrical confinement can be realized by encapsulation in a carbon nanotube (CNT). The potential energy of a molecule surrounded by a CNT can be calculated by evaluating a limited number of terms of an expansion into SARFs, which offers a significant reduction of the computation time. Optimal molecular orientations can be deduced from the resulting potential energy landscape. Examples, including the case of a molecule with cubic symmetry inside a CNT, are discussed.

Symmetry plays an extremely important role in nature. Accordingly, the mathematics of symmetry is embedded in many aspects of theoretical physics. In particular, many concepts from group theory have been applied to describe the crystal structure of solids.

Molecular crystals combine the symmetry of a crystal lattice with molecular symmetries (for a review, see Ref. [_{4} [

Traditionally, SARFs have been used to describe three-dimensional lattices. However, in recent years, molecules have been successfully inserted into carbon nanotubes (CNTs), the internal hollow space of which provides cylindrical confinement. The first reported synthesis of such a system (called “nanopeapod”) featured C_{60} molecules encapsulated in a CNT [_{60} molecule in cylindrical confinement were developed afterwards [_{60}, C_{70} and C_{80} peapods, each featuring different molecular symmetries (_{h}_{5}_{h}_{5}_{d}

The purpose of the present paper is to provide a general description of the construction of SARFs for molecules of any symmetry in cylindrical confinement. First, we present a pedestrian approach to the example of a C_{60} peapod: we show how the potential energy of the C_{60} molecule, positioned on the long axis of a CNT, can be expanded into a series of SARFs. We then discuss the resulting formulas, and extend the potential model used for calculating the interaction energy of a C_{60} molecule and the surrounding CNT. This is followed by the general construction of SARFs. While the main goal is to focus on the mathematical formalism behind SARFs, we will also show potential energy landscapes for various tube radii (“nanotube fields”) and point to the associated optimal molecular orientations. The practical advantage of SARFs expansions is discussed. In addition, we provide an original example with cubic molecular symmetry.

It is instructive to introduce the formalism of SARFs for cylindrical confinement by elaborating a concrete example. We consider a C_{60} molecule encapsulated in a CNT with its center of mass on the tube’s long axis (_{a} = 1_{a} stands for atom). The interaction energy then reads

where _{Λa} = (_{Λa}_{Λa}_{Λa}) is the position vector of atom Λ_{a} of the C_{60} molecule. The function

Here, cylindrical coordinates (_{R}^{−2}.

We have not yet specified the molecule’s orientation. Let us introduce a reference orientation for the C_{60} molecule, we choose it to be the orientation where two-fold symmetry axes coincide with the coordinate axes—the so-called standard orientation (_{z}_{y}_{z}^{−1}(_{z}_{y}_{z}_{Λa}, results in the following explicit expression for the molecule-tube interaction energy

Expression (_{Λa} = (_{Λa} sin _{Λa} cos _{Λa}_{Λa} sin _{Λa} sin _{Λa}_{Λa} cos _{Λa}), and rewrite the interaction energy for a molecule in the standard orientation as

with

The distance _{R}_{Λa}

Only the difference of Φ and _{Λa} enters expression (_{Λa}, since a change of variables Φ′ = Φ − _{Λa} eliminates _{Λa} from the expression for _{R}_{Λa}_{φ}_{Λa}^{2}^{π}^{+}^{φ}^{Λa} _{0}^{2}^{π}_{60} molecule, all C atoms have the same radial coordinate _{Λa} ≡ _{a}, which can therefore be considered a constant rather than a variable in expressions (

The quantity _{Λa}), taken as a function of _{Λa}, can be expanded into

Here, we use the Bradley and Cracknell spherical harmonics (Ref. [

The rotation operator ℜ(

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

The potential energy _{z}

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 _{l}^{n}

molecular shape factors _{l}

and normalised atomic form factors _{l}^{n}

Icosahedral molecular symmetry implies that _{l}^{n}_{l}_{l}^{n}

where

are molecular-and-site-symmetry-adapted rotator functions (SARFs). Rotator functions, originally introduced by James and Keenan [_{∞h}

The symmetry of a C_{60} molecule implies some restrictions on the atomic form factors. The combination of a center of inversion and the (

The same relations hold for the normalized molecular form factors _{l}^{n}_{l}^{−}^{n}_{l}^{n}_{l}^{n}^{*}_{l}^{−}^{n}

In summary, as a numerically much more efficient alternative to _{l}^{n}_{l}_{l}^{n}_{l}

For actual calculations, a potential function and potential parameters have to be specified. In Refs. [

was introduced for studying C_{60}-C_{60} interactions in C_{60}-fullerite (solid buckminsterfullerene); it led to a crystal field potential and a structural phase transition temperature [_{1} = 3.24 ^{7} K _{B}, _{2} = 3.6 Å^{−1} and ^{5} K _{B} · Å^{6} of Refs. [_{l}_{l}_{l}

Having calculated the quantities _{l}_{l}_{l}^{n}_{60} molecule are perpendicular to the

When comparing the nanotube fields shown in _{l}

The important feature in the construction of an expansion into SARFs is the vanishing of several atomic form factors _{l}^{n}_{l}^{n}_{l}^{n}_{1}_{g}_{l}^{n}

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 _{l}

Here, the superscript ^{t} stands for the IC type. The ICs need not only be atoms; in the case of C_{60} molecules, it is customary to place ICs on bonds. For a C_{60} 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 C_{60} molecule [

the molecular shape factors

and the normalised atomic form factors

have to be calculated. Here, _{Λt} and _{Λt} stand for the polar and azimuthal angles of the ICs of type t, labeled Λ_{t} = 1_{t}, with _{t} the number of ICs of type t (_{a} = 60, _{db} = 30 and _{sb} = 60). Remarkably, it turns out that

with _{l}^{t} =

so that

where _{l}^{n}_{l}^{a}^{,n}

Note that the same SARFs as before [

The nanotube fields of a C_{60} molecule arising from the extended interaction model described in the present subsection do not differ qualitatively from the ones shown in

The manifestation of potential energy landscapes as in _{60} molecules encapsulated in a CNT displays unusual dynamical behavior as demonstrated by different experimental techniques: inelastic neutron scattering [_{60} peapod samples, though, the intermolecular interactions are several orders of magnitude smaller than the molecule-tube interaction [_{60}-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 [

In the foregoing we have introduced SARFs for a C_{60} molecule, displaying icosahedral symmetry, with atoms, double and single bonds considered as three different types of ICs. A special feature of the C_{60} molecule is that for each IC type, the radial coordinates of the ICs are equal (dependent on t, not on Λ_{t}): _{Λa} ≡ _{a}, _{Λdb} ≡ _{db}, _{Λ sb} ≡ _{sb}. This does not hold for all symmetries, however. The general formulation of a molecule’s nanotube field

To fix ideas, we take the example of a C_{70} molecule, which has an ellipsoidal shape and_{5}_{h}_{70} 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 (_{Λt}. In the case of a C_{70} molecule, ICs with the same _{Λt}_{Λt} value. Therefore, we can think of layers of ICs having the absolute value of their _{Λt} values. We label the layers by an index _{t}, and the ICs within layer _{t} by an index _{λ}_{t}. This results in a compound index

to address IC Λ_{t}. Introducing the layer-dependent analogues of

and the layer-dependent SARFs

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

_{Λt} values for IC type t). Results for the C_{70} molecule’s nanotube fields and their physical implications can be found in Ref. [

We now apply the SARFs procedure to an example with cubic molecular symmetry. Cubane, C_{8}H_{8}, has eight carbon atoms arranged on the corners of a cube to each of which a hydrogen atom is bound (_{60}.C_{8}H_{8}, a remarkable molecular crystal consisting of icosahedral (_{h}_{h}

We consider a cubane molecule encapsulated in a CNT with radius _{h}_{8}H_{8}O_{12} [

We model the cubane molecule as a simple cubic cluster of 8 ICs placed on the H atoms and define the standard orientation (_{l}^{n}_{l}_{l}^{n}_{l}_{l}^{n}_{l}^{n}_{l}^{n}

The lowest non-zero _{l}^{n}

is proportional to the cubic harmonic _{4}(

For the pair interaction potential _{60}.C_{8}H_{8} [

with _{B} and _{l}_{l}_{l}_{l}_{l}_{8}H_{8}O_{12} molecules (of cubic symmetry) inserted in CNTs with radii _{4}_{n}_{8}O_{8}_{n}_{−4} ladder-like structures [

We have outlined the construction of SARFs for molecules of any symmetry in cylindrical confinement. The molecules are taken as discrete clusters of ICs, labeled Λ_{t}, of different types, labeled t. In general, SARFs _{t}; _{t} groups ICs having the same radial coordinate _{Λt} ≡ _{λ}_{t} (layers of ICs). The SARFs are type- and layer-dependent. In some special cases, e.g., for spherical clusters like C_{60}, type-independent SARFs can be constructed. The main consequence of the cylindrical site symmetry is the SARFs’ independence on the Euler angle _{l}^{t} (_{t}(_{60} molecule’s nanotube field take less than 1000 times the time for the direct calculation.

Knowledge of the nanotube field of a molecule encapsulated in a CNT immediately allows to identify stable molecular orientations. In the case of C_{60} molecules, depending on the tube radius _{8}H_{8}, we also find different regimes. For small radii, the cube’s faces are aligned to the crystal planes, while for large radii, two opposing edges are intersected halfway by the tube’s long axis of the tube.

The computational efficiency for nanotube field calculations is one of the main advantages of using SARFs. There are, however, many more situations in which SARFs are useful, especially in the context of orientational order-disorder phase transitions in molecular crystals (see e.g. Ref. [_{60}). The general theoretical framework of SARFs as described by Michel and Parlinski [

can play the role of order parameters of second-order orientational phase transitions and are also are quantities relevant for the interpretation of Raman and/or infra-red spectroscopic measurements.

Throughout the paper, we have worked under the smooth-tube approximation, neglecting the actual honeycomb network of carbon atoms of the CNT. As has been shown by comparing the results of both the smooth-tube approach and calculations taking the discrete structure of a CNT into account, this is a valid approximation [_{60} and higher (tubular) fullerene molecules, the energetically favorable position is off-axis from

_{60}in carbon nanotubes

_{60}Molecules in Carbon Nanotubes

_{60}molecules in carbon nanotubes

_{60}molecules in carbon nanotubes: Atomistic versus continuous approach

_{70}molecules in carbon nanotubes

_{70}and C

_{80}fullerenes in carbon nanotubes

_{60}

_{60}

_{60}

_{60}

_{60}

_{60}in Carbon Nanopeapods

_{60}dynamics inside single-walled carbon nanotubes: NMR observations

_{60}Encapsulated Single Wall Carbon Nanotubes

_{60}from nearinfrared Raman studies under high pressure

_{8}Si

_{8}O

_{12}in Carbon Nanotubes

_{60}fullerene-cubane

_{8}Si

_{8}O

_{12}

_{60}

A C_{60} molecule in a CNT with radius

Nanotube field _{60} molecule in a CNT with radius (a) _{B}. The absolute minima have been subtracted so that the local energy minima lie at zero.

Projection of a C_{70} molecule in the standard orientation on the (

A C_{8}H_{8} molecule.

Nanotube field _{8}H_{8} molecule in a CNT with radius (a) _{B}. The absolute minima have been subtracted so that the local energy minima lie at zero.

Atomic form factors
_{l}_{h}

_{l} |
| ||
---|---|---|---|

0 | 16.9257 | 0 | 1 |

6 | 2.6365 | 0 | −0.2073 |

6 | 2 | −0.4750 | |

6 | 4 | 0.3878 | |

6 | 6 | 0.3202 | |

10 | 19.2982 | 0 | 0.3545 |

10 | 2 | −0.2880 | |

10 | 4 | −0.3572 | |

10 | 6 | −0.0565 | |

10 | 8 | −0.4251 | |

10 | 10 | 0.2069 | |

12 | 9.0051 | 0 | −0.4145 |

12 | 2 | −0.1179 | |

12 | 4 | −0.1830 | |

12 | 6 | 0.4635 | |

12 | 8 | −0.0738 | |

12 | 10 | −0.2924 | |

12 | 12 | −0.2469 |

Expansion coefficients _{l}_{l}_{l}_{B} · Å.

_{0}( |
_{6}( |
_{10}( |
_{12}( | |
---|---|---|---|---|

6.0 Å | −2201.02 | −833.92 | −53.79 | 7.87 |

7.0 Å | −2151.95 | −7.81 | −1.99 | 0.36 |

8.0 Å | −886.63 | 4.23 | −0.04 | 0.01 |

_{0}_{0}( |
_{6}_{6}( |
_{10}_{10}( |
_{12}_{12}( | |

6.0 Å | −37253.82 | −2198.65 | −1038.13 | 62.97 |

7.0 Å | −36423.20 | −20.58 | −38.49 | 2.85 |

8.0 Å | −15006.86 | 11.14 | −0.71 | 0.09 |

Atomic form factors
_{l}_{h}

_{l} |
| ||
---|---|---|---|

0 | 2.2568 | 0 | 1 |

4 | 3.4473 | 0 | −0.7638 |

4 | 4 | −0.4564 | |

6 | 5.1143 | 0 | 0.3536 |

6 | 4 | −0.6614 | |

8 | 1.9797 | 0 | 0.7181 |

8 | 4 | 0.2700 | |

8 | 8 | 0.4114 | |

10 | 6.7237 | 0 | −0.4114 |

10 | 4 | 0.4146 | |

10 | 8 | 0.4934 | |

12 | 4.6866 | 0 | 0.0919 |

12 | 4 | −0.3625 | |

12 | 8 | 0.5977 | |

12 | 12 | −0.0849 |

Expansion coefficients _{l}_{l}_{l}_{B} · Å.

_{0}( |
_{4}( |
_{6}( |
_{8}( |
_{10}( |
_{12}( | |
---|---|---|---|---|---|---|

5.0 Å | −474.28 | 106.87 | −49.25 | 14.53 | −3.38 | 0.67 |

7.0 Å | −103.26 | −1.89 | 0.11 | 0.00 | 0.00 | 0.00 |

_{0}_{0}( |
_{4}_{4}( |
_{6}_{6}( |
_{8}_{8}( |
_{10}_{10}( |
_{12}_{12}( | |

5.0 Å | −1070.37 | 368.41 | −251.86 | 28.76 | −22.71 | 3.14 |

7.0 Å | −233.04 | −6.53 | 0.55 | −0.01 | 0.00 | 0.00 |