# Symmetric molecular bearings can exhibit low energy barriers that are insensitive to details of the potential energy function

The potential energy of a single atom moving smoothly past a closely-spaced ring of other atoms is approximately sinusoidal, as shown in the upper diagram. The potential energy function for a *ring* of atoms moving smoothly past another ring is a sum of near-sinusoids, with the net potential having a spatial frequency that depends on the least common multiple of the symmetry-numbers of the two rings. Intuitively, the energy is at a maximum when (any) two atoms are at their point of closest approach, and from such a position, a small rotation places a different pair of atoms in the same situation (when the LCM is large, as in the example here).

This symmetry property of the potential energy function ensures that only high spatial frequency components of the interatomic potential contribute to rotational energy barriers. Explicit calculations and molecular mechanical models of detailed designs routinely yield rotational barriers of negligible magnitude (< 0.001zJ). These low barriers depend little on parameters such as bond lengths and bond angles, and hold true for any physically reasonable model of the interatomic potential. They do, however, require careful choice of symmetry and avoidance of designs in which elastic instabilities violate the smooth-interaction condition. Barriers for similar bearings of reduced symmetry (slightly deformed sleeves, displaced shafts) are larger, but can still be very low by the relevant mechanical standards.

#### What are symmetric sleeve bearings?

A shaft in a sleeve can form a rotary bearing

#### What are the conditions for smooth interatomic interactions?

*Stiffly* supported sliding atoms have a smooth interaction potential

*Softly* supported sliding atoms can undergo abrupt transitions in energy

#### Spatial frequencies and symmetry operations:

Merkle, R. C. (1993) “A Proof About Molecular Bearings” *Nanotechnology* **4:**86-90.

Drexler, K. E. (1992) *Nanosystems: Molecular Machinery, Manufacturing, and Computation.* Wiley/Interscience, pp.285–286.