Fuzzy ta/2 symmetries of straight chain conjugate polyene molecules

Abstract

On the basis of our recent studies on the molecular fuzzy point group symmetry, we further probe into the more complicated planar one-dimensional fuzzy periodic molecules−straight chain conjugate polyene. Except for the fuzzy translation transformation, the space transformation of the fuzzy screw rotation and the glide plane will be referred to. In addition, other fuzzy point symmetry transformation lain in the space transformation is discussed. Usually there is a correlation between the fuzzy symmetry characterization caused by the transition of the point symmetry elements and by certain space symmetry transformation. For the molecular orbital, the irreducible representation component is analyzed besides the membership function of the fuzzy symmetry transformation. Also, we inquire into the relativity between some molecular property and the fuzzy symmetry characterization.conjugate polyene, one-dimensional fuzzy periodic transformation, the membership function, irreducible representation component

Symmetries and fuzzy symmetries of graphene molecules

“…In addition, only small energy gaps exist between the OMO and VMO for all the molecules. We have pointed out that for some conjugated molecules, such kind of π -OMO and π -VMO branches are corresponding to the full band and the conduction band of one-dimensional crystals, and the gap is corresponding to the forbidden band gap [21,22]. So, it can be expected that a zigzag graphene molecule containing a large quantity of C-atom should have electrical conductivity, which is consistent with the known results [39].…”

“…The reason can be found in the following presentation. In investigating the fuzzy one-dimensional periodic systems with limited size [17,21], we have indicated that when the characteristic size of one-dimensional system is ten times more than the translation length, the corresponding translational symmetry membership function is relative large (>0.8 ∼ 0.9) and its symmetry can be approximately disposed by a perfect space group. For graphene molecule belongs to the fuzzy 2D system, we need to analyze the translational membership function along two different periodic directions.…”

“…8, we can see that for one appointed armchair graphene, the energy dots corresponding to OMO and VMO are segregated in two sets but close to each other, forming approximately two separated branches. We have mentioned that for some conjugated molecules, such π -OMO or π -VMO branches correspond to the full band or the conduction band of the one-dimensional crystal, and the gap corresponds to the forbidden band gap [21,22]. Thus, it can be expected that an armchair graphene molecule containing a large quantity of C-atoms should also have electrical conductivity.…”

“…Because there is a certain dependence between the membership function of π -MO and the irreducible representation components [13,20,21], we can plot the membership functions of the π -MOs in C 94 H 30 (C 2h ) and C 94 H 30 (C 2V ) versus the corresponding irreducible representations components, as shown in Fig. 15a and b.…”

“…and we have done such studies on the molecules of polyyne [17], cumulative polyene [23], conjugate polyene [21,22], and their derivatives. These studies are about fuzzy point symmetry and fuzzy space periodic symmetry which is only involved for systems with 1D directional periodicity.…”