几类含碳分子体系的对称性和模糊对称性理论研究
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摘要
关于体系对称性的研究，研究人员已经建立了一套系统的理论和科学的方法。但是，当人们处理一些不具备完整对称性的体系时，意见就产生了分歧。对于某一体系，其对称性可能会由于某种原因（比如发生取代反应）致使原有的对称性下降，就产生了不完整对称性。例如，苯分子发生卤代反应以后，其原有的D6h的对称性会失去，而产生不完整对称性。过往，物理、化学领域对这类不完整对称性的描述，往往采取两种极端的方式：一种观点是否认不完整对称性的存在，也就是认为不完整对称性即不具有对称性，而不予以考虑；另外一种观点是忽略不完整性，即认为体系仍近似看作具有原来的对称性。这两种观点都有一定的道理，但又具有片面性。怎样描述这类不完整对称性才更为科学呢？随着上世纪60年代Zadeh建立模糊数学后，化学工作者将模糊数学方法引入到化学领域，成功建立了模糊对称性理论，诸如上述问题便迎刃而解了。
模糊对称性是理论化学中一个非常重要的议题，按照模糊对称性方法对一些分子进行研究，得到了不少重要的结果，一些研究分子以及分子轨道(MO)模糊对称性特征的方法也被提出。此外，静态和动态分子体系模糊对称性特征的研究也能得以开展。例如，分子整体骨架的模糊对称性；由重复结构单元组成的分子的模糊对称性；分子反应时的模糊对称性；平面及非平面分子的模糊对称性；线型分子的模糊对称性以及分子轨道的模糊对称性和不可约表示成分等。
在改进计算方法和提高数据处理效率的基础上，使得研究更为深入，相关的模糊空间对称性的研究也能进行了。一般说来，对于n维空间里仅在某一个特定方向具有周期对称性的体系，其对称性研究可借助于G1n群。因此，我们对具有一维模糊周期性的线状分子，比如聚炔、累积多烯以及全碳环分子，按照模糊G11对称性进行探讨；具有二维模糊周期性的平面分子，比如石墨烯分子，可以按照模糊G12对称性进行探讨；对于在三维空间中，仅在某一个方向具有周期性的体系，比如碳纳米管，其对称性可借助于柱面群G13和柱面正交坐标系进行探讨。对更为复杂的体系，例如M buis环带分子，此类分子的对称性属于分子环面群，对其称性的研究需借助于环面正交曲线坐标系进行探讨。
基于上述方法和理论，本论文主要研究了几类含碳分子体系的对称性和模糊对称性，具体包括：线状的聚炔、累积多烯分子和全碳环分子，平面的石墨烯分子，具有立体结构的M buis环带分子以及碳纳米管。分别从分子骨架的对称性及模糊对称性，π-MO的能量，分子轨道的隶属函数以及不可约表示成分等方面进行了系统研究。具体的研究对象是：
(1)含有不同碳原子数的线状聚炔分子(包括C_(10)H_2, C_(20)H_2, C_(30)H_2和C_(40)H_2分子)，属于D_(2h)点群的累积多烯C_(2n)H_4和属于D2d点群的累积多烯C_(2n-1)H_4分子，全碳环C_6和C_(18)分子。探讨了它们分子骨架的对称性和模糊对称性，π-MO的模糊对称性，π-MO关于平移对称变化的隶属函数以及相应的不可约表示成分。
(2)具有D_(2h)点群对称性的一组锯齿型石墨烯分子(包括C_(100)H_(32)、C_(84)H_(28)、C_(68)H_(24)和C_(52)H_(20))和一组扶手椅型石墨烯分子(包括C_(108)H_(32)、C_(72)H_(24)和C_(36)H_(16))，两个分别具有D_(2h)点群的子群对称性的石墨烯分子C_(94)H_(30)(C_(2h))和C_(94)H_(30)(C_(2v))，和一个以D_(2h)点群作为子群，属于D_(6h)点群的石墨烯C_(96)H_(24)(D_(6h))分子。分别讨论了它们分子骨架的对称性和模糊对称性，π-MO的能量分布特点，分子硬度，π-MO关于平移对称变化的隶属函数以及相应的不可约表示成分。
(3)分别计算了Hückel型环并苯分子和M buis型环并苯分子。前者属于柱面群，需用柱面正交曲线坐标系讨论；后者属于一种不同于以往点群或空间群的新的分子环面群，具有环面螺旋旋转变换(TSR)的对称性，需利用环面正交曲线坐标系对其进行对称性研究。此外，还对多次扭转的M buis带环的对称性进行了分析和讨论。
(4)初步探讨了一系列的扶手椅型碳纳米管和锯齿型碳纳米管，对它们的对称性，分子轨道的能量。这类体系中存在着螺旋对称性，而这种螺旋状分布结构对于某些生物大分子（如DNA与RNA等）是重要的，当然它们比碳纳米管还要复杂得多。由于这些体系较大，分子轨道的成分复杂，数据信息量很大，对我们的工作提出了更高的要求，目前的研究工作还处于尝试和探索阶段，是今后工作的重要内容。我们希望通过努力，使对体系模糊对称性的研究从简单的一维线状分子发展到二维的平面分子以及三维的立体结构，得到更多重要信息，使模糊对称性理论得到更为广泛的应用。
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 π-MOs, the membership functions ofπ-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 C_(10)H_2, C_(20)H_2,C_(30)H_2and C_(40)H_2); cumulative polyene C_(2n)H_4with the D_(2h)group symmetry andcumulative polyene C_(2n-1)H_4with the D2dgroup symmetry; full carbon ringmolecules (including C_6and C_(18)). We investigated their symmetry of themolecular skeleton, fuzzy symmetry of their π-MOs, the membership functions ofπ-MOs about the translating symmetry transformation and the irreduciblerepresentation components.
(2) A set of zigzag graphenes with the D_(2h)group symmetry (including C_(100)H_(32),C_(84)H_(28),C_(68)H_(24)and C_(52)H_(20)) and a set of armchair graphenes (C_(108)H_(32), C_(72)H_(24)andC_(36)H_(16)); two graphenes C_(94)H_(30)(C_(2h)) and C_(94)H_(30)(C_(2v)) which possesses thesubgroup symmetry of D_(2h)group; graphenes C_(96)H_(24)(D_(6h)) which takes the D_(2h)group as its subgroup. We investigated their symmetry of the molecular skeleton,energies of the π-MOs, molecular hardness, the membership functions of π-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 π-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.
模糊对称性是理论化学中一个非常重要的议题，按照模糊对称性方法对一些分子进行研究，得到了不少重要的结果，一些研究分子以及分子轨道(MO)模糊对称性特征的方法也被提出。此外，静态和动态分子体系模糊对称性特征的研究也能得以开展。例如，分子整体骨架的模糊对称性；由重复结构单元组成的分子的模糊对称性；分子反应时的模糊对称性；平面及非平面分子的模糊对称性；线型分子的模糊对称性以及分子轨道的模糊对称性和不可约表示成分等。
在改进计算方法和提高数据处理效率的基础上，使得研究更为深入，相关的模糊空间对称性的研究也能进行了。一般说来，对于n维空间里仅在某一个特定方向具有周期对称性的体系，其对称性研究可借助于G1n群。因此，我们对具有一维模糊周期性的线状分子，比如聚炔、累积多烯以及全碳环分子，按照模糊G11对称性进行探讨；具有二维模糊周期性的平面分子，比如石墨烯分子，可以按照模糊G12对称性进行探讨；对于在三维空间中，仅在某一个方向具有周期性的体系，比如碳纳米管，其对称性可借助于柱面群G13和柱面正交坐标系进行探讨。对更为复杂的体系，例如M buis环带分子，此类分子的对称性属于分子环面群，对其称性的研究需借助于环面正交曲线坐标系进行探讨。
基于上述方法和理论，本论文主要研究了几类含碳分子体系的对称性和模糊对称性，具体包括：线状的聚炔、累积多烯分子和全碳环分子，平面的石墨烯分子，具有立体结构的M buis环带分子以及碳纳米管。分别从分子骨架的对称性及模糊对称性，π-MO的能量，分子轨道的隶属函数以及不可约表示成分等方面进行了系统研究。具体的研究对象是：
(1)含有不同碳原子数的线状聚炔分子(包括C_(10)H_2, C_(20)H_2, C_(30)H_2和C_(40)H_2分子)，属于D_(2h)点群的累积多烯C_(2n)H_4和属于D2d点群的累积多烯C_(2n-1)H_4分子，全碳环C_6和C_(18)分子。探讨了它们分子骨架的对称性和模糊对称性，π-MO的模糊对称性，π-MO关于平移对称变化的隶属函数以及相应的不可约表示成分。
(2)具有D_(2h)点群对称性的一组锯齿型石墨烯分子(包括C_(100)H_(32)、C_(84)H_(28)、C_(68)H_(24)和C_(52)H_(20))和一组扶手椅型石墨烯分子(包括C_(108)H_(32)、C_(72)H_(24)和C_(36)H_(16))，两个分别具有D_(2h)点群的子群对称性的石墨烯分子C_(94)H_(30)(C_(2h))和C_(94)H_(30)(C_(2v))，和一个以D_(2h)点群作为子群，属于D_(6h)点群的石墨烯C_(96)H_(24)(D_(6h))分子。分别讨论了它们分子骨架的对称性和模糊对称性，π-MO的能量分布特点，分子硬度，π-MO关于平移对称变化的隶属函数以及相应的不可约表示成分。
(3)分别计算了Hückel型环并苯分子和M buis型环并苯分子。前者属于柱面群，需用柱面正交曲线坐标系讨论；后者属于一种不同于以往点群或空间群的新的分子环面群，具有环面螺旋旋转变换(TSR)的对称性，需利用环面正交曲线坐标系对其进行对称性研究。此外，还对多次扭转的M buis带环的对称性进行了分析和讨论。
(4)初步探讨了一系列的扶手椅型碳纳米管和锯齿型碳纳米管，对它们的对称性，分子轨道的能量。这类体系中存在着螺旋对称性，而这种螺旋状分布结构对于某些生物大分子（如DNA与RNA等）是重要的，当然它们比碳纳米管还要复杂得多。由于这些体系较大，分子轨道的成分复杂，数据信息量很大，对我们的工作提出了更高的要求，目前的研究工作还处于尝试和探索阶段，是今后工作的重要内容。我们希望通过努力，使对体系模糊对称性的研究从简单的一维线状分子发展到二维的平面分子以及三维的立体结构，得到更多重要信息，使模糊对称性理论得到更为广泛的应用。
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 π-MOs, the membership functions ofπ-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 C_(10)H_2, C_(20)H_2,C_(30)H_2and C_(40)H_2); cumulative polyene C_(2n)H_4with the D_(2h)group symmetry andcumulative polyene C_(2n-1)H_4with the D2dgroup symmetry; full carbon ringmolecules (including C_6and C_(18)). We investigated their symmetry of themolecular skeleton, fuzzy symmetry of their π-MOs, the membership functions ofπ-MOs about the translating symmetry transformation and the irreduciblerepresentation components.
(2) A set of zigzag graphenes with the D_(2h)group symmetry (including C_(100)H_(32),C_(84)H_(28),C_(68)H_(24)and C_(52)H_(20)) and a set of armchair graphenes (C_(108)H_(32), C_(72)H_(24)andC_(36)H_(16)); two graphenes C_(94)H_(30)(C_(2h)) and C_(94)H_(30)(C_(2v)) which possesses thesubgroup symmetry of D_(2h)group; graphenes C_(96)H_(24)(D_(6h)) which takes the D_(2h)group as its subgroup. We investigated their symmetry of the molecular skeleton,energies of the π-MOs, molecular hardness, the membership functions of π-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 π-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.
引文
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