15 Molecular Theory of Graphene
251
planar sheets of sp 2 -bonded carbon atoms that are densely packed in a honeycomb
crystal lattice [9]’. This definition clearly exhibits a molecular-crystal duality of
this extraordinary substance. From the molecular viewpoint, the extraordinariness
is provided with the availability of odd electrons that are responsible for the sp 2
configuration of valence electrons of carbon atoms. The 2D-dimensionality, on the
other hand, dictates peculiar properties of a regularly packed honeycomb pattern.
Due to this, the graphene properties are similar to those of both polycondensed benzenoid molecules and 2D-dimensional crystals. Obviously, fundamental characteristics of the two forms are tightly interconnected. Thus, as will be discussed below,
such seemingly solid state properties as magnetism and mechanics of graphene are
of molecular origin.
The above mentioned peculiar duality is embodied in the computational strategy
of graphene, as well. On one hand, the solid state microscopic theory of quasiparticles in a 2D space forms the ground for the description of the graphene crystal.
On the other hand, quantum molecular theory creates the concept of the graphene
molecule. Seemingly, the two theoretical approaches, obviously different from the
computational viewpoint, have nevertheless much in common. Thus, the solid state
quasiparticles are usually described in the approach based on a unit cell and/or supercell followed by periodic boundary conditions; besides, the unit cell is described
at the molecular theory level thus presenting the molecular object in the same way as
in the case of the molecular theory. However, the very molecular object provides a
crucial difference between the two approaches. In the case of a correct solid state formulation, the cell and/or supercell should be strictly chosen as a known crystalline
motive. Accordingly, the two-atomic cell of graphene crystal finds its exhibition
in the peculiarities of the crystal electron band structure. However, nowadays, the
solid state approach is explored in the graphene science in regard to practically all
the phenomena including graphene chemical modification, graphene deformation
and magnetization. The two-atomic unit cell of the crystal does not meet conditions
needed for examining these complicated events, particularly, related to the chemical
modification. The cell is substituted by a supercell, whose structure is taken at one’s
own choosing, once in the preponderance of cases just ‘drawn’ in stead off attributed
to a reality. Moreover, regular structure of the graphene object is fastened by the periodical boundary conditions. The two features of the solid-state approach, namely,
the arbitrarily chosen supercell and the fastened periodicity make clear the Hoffmann answer “Not much” to the question “What you can trust about theory?” [1].
Then Hoffmann continues: “Aside from the natural prejudice for simplicity, people really want translational periodicity in their calculations, for then the quantum
mechanical problem reduces to one of the size of the unit cell. But the real world
refuses to abide by our prejudices. And it is often an aperiodic, maximally defectridden, amorphous world, where emergent function is found in matter that it is as far
from periodic as possible”. The reality of the graphene science, particularly, related
to the chemical modification, strongly witnesses the domination of aperiodic structures. In view of this, the molecular theory of graphene has a convincing preference
since its molecular object is created in the course of computations without structural
restrictions introduced in advance.
251
planar sheets of sp 2 -bonded carbon atoms that are densely packed in a honeycomb
crystal lattice [9]’. This definition clearly exhibits a molecular-crystal duality of
this extraordinary substance. From the molecular viewpoint, the extraordinariness
is provided with the availability of odd electrons that are responsible for the sp 2
configuration of valence electrons of carbon atoms. The 2D-dimensionality, on the
other hand, dictates peculiar properties of a regularly packed honeycomb pattern.
Due to this, the graphene properties are similar to those of both polycondensed benzenoid molecules and 2D-dimensional crystals. Obviously, fundamental characteristics of the two forms are tightly interconnected. Thus, as will be discussed below,
such seemingly solid state properties as magnetism and mechanics of graphene are
of molecular origin.
The above mentioned peculiar duality is embodied in the computational strategy
of graphene, as well. On one hand, the solid state microscopic theory of quasiparticles in a 2D space forms the ground for the description of the graphene crystal.
On the other hand, quantum molecular theory creates the concept of the graphene
molecule. Seemingly, the two theoretical approaches, obviously different from the
computational viewpoint, have nevertheless much in common. Thus, the solid state
quasiparticles are usually described in the approach based on a unit cell and/or supercell followed by periodic boundary conditions; besides, the unit cell is described
at the molecular theory level thus presenting the molecular object in the same way as
in the case of the molecular theory. However, the very molecular object provides a
crucial difference between the two approaches. In the case of a correct solid state formulation, the cell and/or supercell should be strictly chosen as a known crystalline
motive. Accordingly, the two-atomic cell of graphene crystal finds its exhibition
in the peculiarities of the crystal electron band structure. However, nowadays, the
solid state approach is explored in the graphene science in regard to practically all
the phenomena including graphene chemical modification, graphene deformation
and magnetization. The two-atomic unit cell of the crystal does not meet conditions
needed for examining these complicated events, particularly, related to the chemical
modification. The cell is substituted by a supercell, whose structure is taken at one’s
own choosing, once in the preponderance of cases just ‘drawn’ in stead off attributed
to a reality. Moreover, regular structure of the graphene object is fastened by the periodical boundary conditions. The two features of the solid-state approach, namely,
the arbitrarily chosen supercell and the fastened periodicity make clear the Hoffmann answer “Not much” to the question “What you can trust about theory?” [1].
Then Hoffmann continues: “Aside from the natural prejudice for simplicity, people really want translational periodicity in their calculations, for then the quantum
mechanical problem reduces to one of the size of the unit cell. But the real world
refuses to abide by our prejudices. And it is often an aperiodic, maximally defectridden, amorphous world, where emergent function is found in matter that it is as far
from periodic as possible”. The reality of the graphene science, particularly, related
to the chemical modification, strongly witnesses the domination of aperiodic structures. In view of this, the molecular theory of graphene has a convincing preference
since its molecular object is created in the course of computations without structural
restrictions introduced in advance.