The fuzzy D<Subscript>6h</Subscript>-symmetries of azines molecules and their molecular orbitals

Based on our previous study on the elementary characterization of fuzzy symmetry, we inquire into the fuzzy symmetries of some simple linear and plane molecules. These systems belong to point groups that include the identity and twofold symmetry elements, but not include higher multi-fold symmetry ones, and their molecular orbitals (MOs) only belong to one-dimension irreducible representations. In this paper, we take the azines as a typical model to examine the fuzzy symmetry in relation to the D_6h point group. As this group includes multi-fold symmetry elements such as a sixfold rotation axis, some of the MOs may belong to two-dimensional irreducible representations. We inquire into the fuzzy symmetry of these molecules and their MOs in terms of membership functions, representation components and correlation diagrams. In addition to these neutral closed shell molecules, pyridine hydride radical, anion, and cation are also analyzed.     

分子的模糊对称性

将模糊数学方法引入对分子对称点群的研究,建立描述具有不完整分子对称性的模糊点对称群(集合).建立具有模糊对称性分子轨道的模糊表示及其模糊特征标(模糊广义宇称).通过对典型的线状分子、平面分子以及非平面的立体分子等进行分析,展示了一个新的理论化学园地.初步探讨了具有模糊对称性的动态反应体系.从模糊对称性出发,探讨了分子轨道对称守恒原理的半定量特征.     

直链共轭多烯的模糊ta/2对称性

近年来关于分子模糊对称性的工作多属于模糊点对称性的研究．关于模糊空间对称性探讨较少．只曾对线状一维模糊周期分子进行过一些分析．本文在此基础上进一步对于较复杂的平面一维模糊周期分子——直链共轭多烯（简称为共轭多烯）分子进行了较仔细的探讨．除模糊平移变换外,这里还将涉及模糊的螺旋旋转和滑移反映等空间变换．此外,还讨论了存在其中的其他模糊点对称变换．对于点对称元素的变动导致的模糊对称性特征,往往和某种空间对称变换的模糊对称性特征相关．对于分子轨道,除模糊对称变换的隶属函数外,分析了所属不可约表示成分．对这些分子的某些性质和其模糊对称性特征之间的相关性进行探讨．     

聚炔、累积多烯与全碳环分子的模糊对称性

近年来我们关于分子模糊对称性的工作多属于模糊点对称性的研究, 关于模糊空间对称性探讨较少. 聚炔作为线状一维模糊周期分子, 我们曾对其进行了初步分析. 虽然对于聚炔分子骨架的分析比较全面, 但由于繁冗的计算使我们对分子轨道(MO)模糊对称性的分析只限于少数典型分子. 本文将对不同的聚炔分子MO模糊对称性特征进行较为系统的分析. 结果表明包含不同碳原子数目的分子轨道模糊对称性参数值之间有一定相关性. 此外我们还对一些相关体系分子的MO进行分析, 累积多烯分子虽然并非线型分子, 但其π-MO相关的碳原子处于线性位置, 可依模糊一维周期的G11体系处理. 按Born-Karman近似, 即n个单元的一维周期对称群与Cn点群同构, 本文还分析了相关的全碳环分子的MO的对称性和模糊对称性. 努力寻求与一维周期性相关的模糊对称性规律性特征.     

丙二烯分子的模糊对称性

利用作者提出的探讨分子及其轨道模糊对称性的方法分析了丙二烯在内旋转过程中模糊对称性特征. 考虑到此过程中经历不同状态所属的对称点群, 即D2h、D2d与D2. 利用包含这些点群中所有元素的最小点群D4h进行分析. 将D4h中的元素分为四组: (i) G0——包含在D2点群中的元素, 也是所有上述点群中都存在的元素; (ii) G1——包含在D2h点群中, 但不包含在D2d点群中的元素; (iii) G2——包含在D2d点群中, 但不包含在D2h点群中的元素; (iv) G3——包含在D4h点群中, 但不包含在D2h与D2d点群中的元素. 分别分析在内旋转过程中各个分子轨道(MO)相应每一组元素的隶属函数的共性变化规律性.     

Fuzzy ta/2 symmetries of straight chain conjugate polyene molecules

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.     

Approximate symmetry characteristics using fuzzy-subset theory study for chiral transitions of allene-1,3,-dihalides

Based on our study of the application of fuzzy-subset theory to the characterization of imperfect symmetry in some stable molecular systems and simple dynamic molecular systems, we analyze the internal rotation process of allene-1,3- dihalides. Allene-1,3-dihalides (CHX=C=CHY, where X and Y may be the same or different halogen atoms) are optically chiral nonplanar molecules. The two end-groups may internally rotate about the near straight linear C=C=C axis, and the molecule may change its chirality. The internal rotation process may pass through two different planar transition state (TS): cis-TS and trans-TS, which belong to C_2v and C_2h point groups (as X and Y to be same), respectively. The intrinsic reaction coordinate (IRC) corresponding to the two TS processes is denoted as cis-IRC and trans-IRC. However, for the whole IRC reaction process, only their subgroup C₂ well-defined symmetry remains. Other symmetry transformations in C_2v and C_2h point groups can only be examined in terms of imperfect symmetry, although there appear certain reaction reversal joint point group G(R_cC_2v) and G(R_tC_2h) well-defined symmetry in the dynamics through the IRC processes. If X and Y are different, the stable molecule has no conventional nontrivial point group symmetry. The internal rotation processes may pass through two different planar TS’s (cis-and trans-TS). The TS will still be a planar molecule belonging to C_S point group with the molecule plane as its symmetry plane. Other states in the IRC may belong to certain reaction reversal joint point groups, G(RM)_C and G(RM)_T. We have thus examined the approximate symmetry of MO’s related to C₂ point group. Moreover, we have also analyzed the membership functions, representation components, and their relationships shown in the MO fuzzy main representation correlation diagrams.     

Continuous symmetry measures of irreducible representations: application to molecular orbitals

The formalism of continuous symmetry measures is extended to describe the extent to which a function, such as a molecular orbital, transforms under the symmetry operations of a given point group according to each irreducible representation of this group. For distorted molecules we are able to calculate the degree to which any molecular orbital transforms with respect to the irreducible representations of the pseudosymmetry group that is valid for a higher symmetry, idealized geometry, showing which irreducible representations participate in each molecular orbital upon symmetry loss in the geometry of the nuclear framework.     

Fuzzy space periodic symmetries for polyynes and their cyano-compounds

In our previous papers on the molecular fuzzy symmetry, we analyzed the basic characterization in connection with the fuzzy point group symmetry. In this paper, polyynes and their cyano-derivatives are chosen as a prototype of linear molecules to probe the one-dimensional fuzzy space group of parallel translation. It is notable that the space group is an infinite group whereas the point group is a finite group. For the fuzzy point group, we focus on considering the fuzzy characterization introduced due to the difference of atomic types in the monomer through point symmetry transformation in the beginning; and then we consider the difference between the infinity of space group and the finite size of real molecules. The difference between the point group and the space group lies in the translation symmetry transformation. This is the theme of this work. Starting with a simple case, we will only analyze the one-dimensional translation transformation and space fuzzy inversion symmetry transformation in this paper. The theory of the space group is often used in solid state physics; and some of its conclusions will be referred to. More complicated fuzzy space groups will be discussed in our future papers.     

Molecular torus group

Generally speaking, the highest symmetry of M?bius cyclacene molecule possesses the C₂ symmetry based on the theory of point group according to the previous works. However, based on the topology principle, the fundamental group of M?buis strip is an infinite continuous cyclic group and its border is made up of twice of the generator. Of course, the M?bius strip-like molecule is associated with a finite discrete cyclic symmetry group. For the cyclacene isomers constructed by several (n) benzene rings, these isomers include: the common cylinder Hückel cyclacene (HC-[n]) molecules, the M?bius cyclacene (MC-[n]) molecules by twisting the linear precursor one time (180°), and the multi-twisted M?bius strip-like cyclacene (M ^m C?[n]) molecules by twisting the linear precursor m times (m × 180°). The relevant results suggest that in addition to the point symmetry, there is a new kind of symmetry called the torus screw rotation (denoted as TSR). In this article, we take the M ^m C?[n] molecules as examples to discuss their TSR group and point group symmetry, and the relative symmetry adaptive atom sets as well as their atomic orbital (AO) sets. Here, the Cartesian coordinates is not quite fit for the investigation of these AOs, so it is replaced by either the torus orthogonal curvilinear coordinates (for M ^m C?[n] molecule) or the cylinder orthogonal curvilinear coordinates (for HC-[n] molecule). According to the features of cyclic group, the character table of the irreducible representation of the TSR group could be constructed easily. Some other relevant point-group symmetries maybe also belong to the molecule, so its symmetry maybe called as the torus group symmetry. For multi-twisted M?bius strip-like molecule, there are some correlations in topology characteristics.     

Maximum Interaction Model Force Constant Calculation of the Mlx (Co)6-x Type Molecule

C—O stretching frequency of inorganic carbonyl molecules is quite characteristic in the vibrational spectroscopy (the IR and Raman). Since such stretching modes are quite different from the skeletal modes, a non-mechanical coupling model has been proposed by Cotton¹⁾. The C—O stretching force constants and their interaction constants may be calculated semiquantitatively from their characteristic frequencies. The trouble is that the number of vibrational frequencies is always less than the number of force constants, whenever a symmetry irreducible representation contains more than one frequency. Cotton suggest that the interaction constants for octahedral type molecule from the dpπ bonding point of view^1-2). Instead of this, a model of maximum interaction between M—C—O bonds is proposed in this paper. Such model not only gives better remedy to the problem of octahedral complexes, but also is useful to solve the normal modes in molecules of trigonal bipyramidal and many other symmetries.     

Group theory and normal coordinate calculations

A method is proposed to obtain a set of computer generated symmetry adapted basis vectors which completely factorize a matrix associated with any operator which commutes with all the symmetry operations of a molecule. The cartesian coordinates of all atoms are used to derive the irreducible representations of the appropriate molecular point group and the transformation properties of any starting set of coordinates. Symmetry coordinates are then projected out each irreducible representation.     

M(o)bius环并苯的分子对称性

一般来说,点群理论认为M(o)bius带环分子最高的对称性只能是C2.本文讨论了由18个苯环组成的环并苯的异构体分子,包括柱面的Hückel型分子(HC-[18])和扭转180°的M(o)bius带环分子(MC-[18]).结果表明除了点对称性外,M(o)bius带环分子还存在一种可称为环面螺旋旋转(TSR)变换的对称性,为此还引用了环面正交曲线坐标系.此外,还讨论了这些分子关于TSR对称性匹配的原子集和原子轨道(AO)集.根据TSR对称性的循环群特征,可以建立此类群的不可约表示及有关特征标.这类分子的分子轨道(MO)关于TSR群的不可约表示是纯的,然而所含的相应的原子轨道对称性匹配的线性组合(SALC-AO)成分可以是多种的.     

The symmetrized random matrix approach, an efficient method to obtain relativistic molecular symmetry adapted functions

In relativistic quantum chemical calculation of molecules, where the spin-orbit interaction is included, the electron orbitals possess both the double point group symmetry and the time-reversal symmetry. If symmetry adapted functions are employed as the basis functions of electron orbitals, it would allow a significant reduction of the computational expense. The point group symmetry adapted functions can be obtained by the group projection operators via its actions on the atomic orbital functions. We have proposed an efficient and simple method to obtain all irreducible representation matrices, which are the basis of the group projection operators, of any finite double point group. Both double point group symmetry and time-reversal symmetry are automatically imposed on the representation matrices. This is achieved by the symmetrized random matrix (SRM) approach, where the SRM is constructed in the regular representation space of a finite group and the eigenfunctions of SRM provide all irreducible representation matrices of the given point group.     

Möbius环并苯的分子对称性

一般来说, 点群理论认为Möbius带环分子最高的对称性只能是C₂. 本文讨论了由18个苯环组成的环并苯的异构体分子, 包括柱面的Hückel型分子(HC-[18])和扭转180°的Möbius带环分子(MC-[18]). 结果表明除了点对称性外, Möbius带环分子还存在一种可称为环面螺旋旋转(TSR)变换的对称性, 为此还引用了环面正交曲线坐标系. 此外, 还讨论了这些分子关于TSR对称性匹配的原子集和原子轨道(AO)集. 根据TSR对称性的循环群特征, 可以建立此类群的不可约表示及有关特征标. 这类分子的分子轨道(MO)关于TSR群的不可约表示是纯的, 然而所含的相应的原子轨道对称性匹配的线性组合(SALC-AO)成分可以是多种的.     

前过渡金属四核原子簇簇骼对称性变化的分子轨道理论研究

本文用半经验分子轨道理论方法计算了前过渡金属四核原子簇的T_d、C_(2v)、D_(4b)、D_(2h)等簇骼对称性的化合物。在表示对称性变化的Walsh图上,适当地确定HOMOLUMO分界线而得到的簇核电子(CCE)数能说明多数具有上述对称性的前过渡金属四核簇合物结构特征及其二级Jahn-Teller畸变。比较了各类配体的作用。     

Fuzzy symmetries of two classes of linear polyacene molecules

The fuzzy symmetries of two kinds of linear polyacene molecules are probed into in the paper. In these molecules, any one of the benzene rings abreast connects at most other two rings in two ways: either its two opposite C–C bonds combine with two other rings, respectively, or its two meta-position C–C bonds connect two rings in cis- and trans-form, respectively. The former is called p-polyacenes (or straight polyacenes), and the latter is m-polyacenes (or kinked ones). It can be thought as the planar molecule with approximate one-dimensional space periodic transformation (parallel translation) symmetry, namely, group G₁²{{\rm G}_{1}^{2}} symmetry, when the number of its benzene ring is very large; on the other hand, it can be considered as the fuzzy group G₁²{{\rm G}_{1}^{2}} symmetry, if the benzene ring number is not large enough. The p-polyacene and m-polyacene with 20 benzene rings are analyzed as typical examples, and the energies of the π-molecular orbital (MO) and the fuzzy symmetry characters related to the space symmetry transformations are carefully examined. Moreover, the π-MOs of the p-polyacenes and m-polyacenes with different numbers of benzene ring are investigated to obtain the related rules.     

Fuzzy Symmetries for Linear Molecules and their Molecular Orbitals

In this paper, the fuzzy symmetry of some prototypical linear molecules has been analyzed. The results show that some molecular orbitals (MOs) are less symmetrical but some others are more symmetrical than the molecular skeleton, which the MOs correspond to. The membership functions of space inversion for MOs are closely related to the chemical characteristics of the MOs. Sometimes, although the symmetry of a molecular skeleton is not obvious, however that of some MO is quite obvious. The membership functions of the fuzzy inversion symmetry depend on the choice of the position of the center of inversion. As compared to those of diatomic molecules and linear tri-atomic molecules, the linear polyatomic molecules in which a distinctive fuzzy symmetry of space translation may exist, and thus a significant effect on their properties can be expected.     

环型硫分子的从头算及其应用

通过对环型硫分子S6～S20同素异构体的实验结构分析,用从头算(ab initio)RHF/6-311G*(包括BLYP/6-311G* 和MP4/6-311G*)方法进行基态几何结构优化,获得了与实验结构相吻合的新的理论稳定构型.其中S18的两种变体S18(α)和S18(β)的能量相近,都是相互稳定的分子构型.通过计算,从理论上推测S20有D4点群的物相.还就对称性、偶极矩和红外振动光谱与分子构象的相互关系,以及环分子结构与化学活性之间的关系进行了讨论,获得了具有实验意义的结论.     

Symmetries and fuzzy symmetries of graphene molecules

In this paper, we will investigate the fuzzy layer group symmetries of two-dimensional (2D) periodic molecules. Here, we select several graphene molecules as typical examples to discuss. For these two-dimensional graphene molecules, their MO energies, symmetries and fuzzy symmetries are preliminarily studied. In addition, we especially make a detailed comparison between the zigzag and armchair graphene molecules. These studies will develop a theoretical framework that will help us to investigate the fuzzy symmetries of various layer group molecules as well as molecules with 3D periodic structure.