The Symmetries And Fuzzy Symmetries Of Several Carbon-involved Molecular Systems: A Theory Study

On the symmetry investigation of system, reseachers have established systemictheories and scientific methods. However, when we deal with some systems withoutcomplete symmetry, our views are quiet different. For a system, its original symmetryprobably declined due to some reasons （such as substitution reaction） and producedthe incomplete symmetry. For example, after the halogenated reaction, benzene lostits original D6hsymmetry, and produced the incomplete symmetry. In the past, thedescription of such incomplete symmetry in the physical and chemical fields, peoplehave often taken two extreme methods: one view is to deny the existence ofincomplete symmetry and considered that incomplete symmetry is non-symmetry;another view is neglect the incomplete symmetry and considered that the system stillapproximately possessed its original symmetry. These two views are all have somecorrectness, but both are one-sided. How to describe such incomplete symmetrywould be more scientific? In1960s, mathematician Zadeh established the FuzzyMathematics. Then, chemical workers introduced it into the field of chemistry andsuccessfully established the fuzzy symmetry theory. So, the above problems are allsolved.Fuzzy symmetry is a very interesting topic in theoretical chemistry and a fewimportant results have been obtained. In our previous papers, some research methodshave been established to study the fuzzy symmetry characteristics of the moleculestructures and molecular orbitals （MOs） for the static and dynamic molecular systems.For example, the fuzzy symmetry of the whole molecule skeleton, the fuzzysymmetry of molecule with repetitive unit, the molecule reactions, the nonplanarmolecules and the planar molecules, linear molecules, as well as the fuzzyrepresentation and fuzzy parity of MO.After the improvement of our calculation level and efficiency of data processing,we can study the fuzzy space symmetry. Generally speaking, for moleculespossessing periodicity in one-dimensional direction, they are usually analyzed by using the cylinder group G1n. Therefore, we investigate the linear moleculespossessing periodicity in one-dimensional direction （such as polyyne, cumulativepolyene and full carbon ring molecules） by the G11symmetry; planar molecules withtwo-dimensional fuzzy periodicity （e.g. graphene molecules） by the G12symmetry;and for spacial molecules possessing periodicity in one-dimensional direction, theirsymmetry can be investigated by the G13group and the orthogonal cylindricalcoordinate system is introduced to study these molecules.Basing on the above theory and method, we mainly studied severalcarbon-involved molecular systems，including: polyyne, cumulative polyene and fullcarbon ring molecules, graphene molecules, M buis cyclacenes and carbon nanotubein this dissertation. We investigated their symmetry and fuzzy symmetry of themolecular skeleton, fuzzy symmetry of their Ï€-MOs, the membership functions ofÏ€-MOs about the translating symmetry transformation and the irreduciblerepresentation components. The research systems are as follows:（1） Linear polyyne molecules with different carbon atoms (including C₁₀H₂, C₂₀H₂,C₃₀H₂and C₄₀H₂); cumulative polyene C_2nH₄with the D_2hgroup symmetry andcumulative polyene C_2n-1H₄with the D2dgroup symmetry; full carbon ringmolecules (including C₆and C₁₈). We investigated their symmetry of themolecular skeleton, fuzzy symmetry of their Ï€-MOs, the membership functions ofÏ€-MOs about the translating symmetry transformation and the irreduciblerepresentation components.（2） A set of zigzag graphenes with the D_2hgroup symmetry (including C₁₀₀H₃₂,C₈₄H₂₈,C₆₈H₂₄and C₅₂H₂₀) and a set of armchair graphenes (C₁₀₈H₃₂, C₇₂H₂₄andC₃₆H₁₆); two graphenes C₉₄H₃₀(C_2h) and C₉₄H₃₀(C_2v) which possesses thesubgroup symmetry of D_2hgroup; graphenes C₉₆H₂₄(D_6h) which takes the D_2hgroup as its subgroup. We investigated their symmetry of the molecular skeleton,energies of the Ï€-MOs, molecular hardness, the membership functions of Ï€-MOsabout the translating symmetry transformation and the irreducible representationcomponents.（3） Calculated the Hückel cyclacenes and M buis cyclacenes. The former belongs tocylindrical group and need to use the orthogonal cylindrical coordinate system to study them. The latter belongs to a new kind of molecular torus group which isdifferent with the general point group and space group. They possess the torusscrew rotation （TSR） symmetry and need to use the torus orthogonal curvilinearcoordinate system to study them. In addition, we also investigated themulti-twisted M buis cyclacenes.（4） Primarily probed some armchair canbon nanotubes and zigzag canbon nanotubes,analyzed their molecular skeleton symmetry and energies of the Ï€-MOs. Thosesystems exists screw symmetry and this spiral structure in some biologicalmacromolecules （such as DNA and RNA, etc.） is very important. Of course, thebiological macromolecules are much more complex than carbon nantubes. For theabove systems, their compositions of the molecular orbital are complex and dataprocess tedium. So, the current research is still to try and exploratory stage and itis the focus of the future.We hope that the symmetry and fuzzy symmetry study can be from the simpleone-dimensional linear molecular system exploring to two-dimensional planar andthree-dimensional structure systems. Through our efforts, the fuzzy symmetry theorycan be more widely applied in chemical field.