Within the nearest-neighbor approximation, the tight-binding Hamiltonian matrix for the two coupled atomic rings has a 2x2 block structure

.

The eigenproblem for this Hamiltonian can be reduced to two eigenproblems of

by

matrices

, where

is the number of atoms in each ring. The tridiagonal structure of such matrices allows one to treat their eigenproblems by the transfer matrix method [1].

The resulting secular equation for the given system reads

,

where

is the number of atomic sites in each ring;

is the electron momentum, quantized by the secular equation;

and

are the tight-binding hopping integrals for electron hoppings within and between the rings, respectively; and

is a parameter specifying the symmetry of the electronic wavefunctions.

The energy levels of the system are given by

,

where

is the

solution of the above secular equation.

Note that for

(blue branch of the secular equation), the wavefunctions are always symmetric with respect to the reflection in the plane that is normal to the coupling bond and intersects it in the middle while for

(red branch of the secular equation), such symmetry is broken.

Snapshot 1: the solutions of the secular equation of two coupled atomic rings for

and

Snapshot 2: the energy levels of the system presented as a function of the coupling strength

Snapshot 3: the wavefunction (eigenvector) of the two coupled rings for

and

For

, we have benzene rings so that the system is an actual biphenyl molecule. The energy levels of the biphenyl (or diphenyl) molecule has been treated by an expansion of the determinant generating function [2, 3].

The transfer matrix method also applies to the problem of two coupled carbon nanotubes or graphene nanoribbons, which are promising materials for THz application [4]. An example of a single tube and ribbon treated by the transfer matrix method can be found in [5].

[2] J. E. Lennard-Jones, F. R. S. and J. Turkevich, "The Electronic Structure of Some Polyenes and Aromatic Molecules. II–The Nature of the Links of Some Aromatic Molecules,"

*Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences*,

**158**(894), 1937 pp. 297–305.

doi:10.1098/rspa.1937.0021.

[3] C. A. Coulson, "The Electronic Structure of Some Polyenes and Aromatic Molecules. IV. The Nature of the Links of Certain Free Radicals,"

*Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences*,

**164**(918), 1938 pp. 383–396.

doi:10.1098/rspa.1938.0024.

[4] M. E. Portnoi, V. A. Saroka, R. R. Hartmann and O. V. Kibis, "Terahertz Applications of Carbon Nanotubes and Graphene Nanoribbons," in

*2015 IEEE Computer Society Annual Symposium on VLSI*, Montpellier, France, IEEE 2015 pp. 456–459.

doi:10.1109/ISVLSI.2015.97.

[5] V. A. Saroka, M. V. Shuba and M. E. Portnoi, "Optical Selection Rules of Zigzag Graphene Nanoribbons,"

*Physical Review B*,

**95**(15), 2017 155438.

doi:10.1103/PhysRevB.95.155438.