晶体的对称性与群论

1、高等无机化学高等无机化学Advanced Inorganic Chemistry高等无机化学高等无机化学课程简介课程简介: 无机化学在近代化学史上占有极为重要的地位, 在化学基本理论研究及实际应用方面起着越来越重要的作用，研究范围越来越大。近年来它已渗透到生物、分离分析、医药、催化冶金、材料科学、环境科学等领域,与各学科有着日益广泛的联系。学习本课程，掌握无机化合物及无机材料方面的知识，着重提高相关化学理论水平,了解现代无机化学的主要研究方向、研究方法、应用及其发展趋势，并培养把握学科前沿的能力，为研究生论文工作及今后从事相关研究工作打下坚实的基础。31. 1. 对称操作对称操作2. 2. 群

2、论基本概念群论基本概念3. 3. 分子的点群分子的点群4.4.群的表示和特征表标群的表示和特征表标5.5.波函数和对称性波函数和对称性6.6.群论的应用群论的应用第一章第一章: 对称性和群论对称性和群论 参考书目：参考书目： 1. Advanced Inorganic Chemistry F. Albert Cotton, Geoffrey, Wilkinsion, Carlos A. Murillo, Manfred Bochmann, John. Wiley. New York, 1999. 6th. Ed. 2. 中级无机化学中级无机化学. 朱文祥朱文祥 编编, 高等教育出版社高等教育出

3、版社, 2004年年7月月. 3.无机化学无机化学D. F. Shriver, P. W. Atkins, C. H. Langford 著著, 高忆慈高忆慈 史启祯史启祯 曾克慰曾克慰 李丙瑞李丙瑞 等译等译 高等教育出版社高等教育出版社 1997年年7月月 第二版第二版 4.无机化学新兴领域导论无机化学新兴领域导论, 项斯芬编著项斯芬编著. 北京大学出版社北京大学出版社, 1988年年11月。月。教材：教材： 高等无机化学，高等无机化学， 陈蕙兰主编，北京：高等教育出陈蕙兰主编，北京：高等教育出版社，版社，2005.6第一章：对称性与群论第一章：对称性与群论要求：要求：1、确定简单分子所属

4、点群、确定简单分子所属点群2、解读特征标表、解读特征标表3、群论在无机化学中的应用、群论在无机化学中的应用 a. 对称性与分子极性对称性与分子极性 b. 分子的振动与分子的振动与IR、Raman光谱光谱 c. 化学键与分子轨道等化学键与分子轨道等 Chapter 1.1 Molecular symmetry and symmetry point group1.1.1 Symmetry elements and symmetry operationsSymmetry exists all around us and many people see it as being a thing of b

5、eauty, e.g., the snow flakes. A symmetrical object contains within itself some parts which are equivalent to one another. What are the key symmetry elements pertaining to these objects? The molecular configuration can be expressed more simply and distinctly. The determination of molecular configurat

6、ion is greatly simplified. It assists giving a better understanding of the properties of molecules. To direct chemical syntheses; the compatibility in symmetry is a factor to be considered in the formation and reconstruction of chemical bonds.Why do we study the symmetry concept?symmetry operationAn

7、 action that leaves an object the same after it has been carried out is called symmetry operation.Example:1) Symmetry elements and symmetry operations对称性：对称性： 物体或图像中所具有的相似性或匀称感。物体或图像中所具有的相似性或匀称感。对称操作：对称操作： 经过某一不改变物体中任何两点间距离的操作经过某一不改变物体中任何两点间距离的操作使图形复原。使图形复原。对称元素：对称元素： 实现对称操作所依赖的点、线或面。实现对称操作所依赖的点、线或面

8、。Symmetry operations are carried out with respect to points, lines, or planes called symmetry elements.symmetry elements(b) an H2O molecule has a twofold (C2) axis. (a) An NH3 molecule has a threefold (C3) axis 对称操作和对称元素l分子的对称性分子的对称性, 对称操作及对称元素对称操作及对称元素定义: 分子的对称性分子的对称性是指存在一定的操作，它在保持任意两点间距离不变任意两点间距离不

9、变的条件下，使分子内部各部分变换位置，而且变换后的分子整体又恢复恢复原状原状，这种操作称为对称操作对称操作(symmetry operation)对称操作据以进行的元素称为对对称元素称元素(symmetry element).Symmetry operations are:The corresponding symmetry elements are:点对称性点对称性 点对称元素：对称操作过程中点对称元素：对称操作过程中至少有一点保持不至少有一点保持不变。变。 一个物体一个物体所有的点对称元素必通过一个所有的点对称元素必通过一个共同点共同点。 一个分子的对称性完全一个分子的对称性完全可以用一套

10、点对称元素描可以用一套点对称元素描述。述。15IFClBr Operation by the identity operator leaves the molecule unchanged. All objects can be operated upon by the identity operation.a) The identity (E)16matrix representation of an operator222111zyxzyxC333231232221131211cccccccccC133132131212312212121131121112zcycxczzcycxcyzcy

11、cxcx222111333231232221131211zyxzyxccccccccc100010001EzyxzyxzyxE100010001matrix representation of EzzyyxxEEE;Ezxy(x,y,z)(x,y,z)rrzyxzzyxyzyxxEEE100;010;001lExample: Rotation of trigonal planer BF3C31C32Rotation about an n-fold axis (rotation through 360o/n) is denoted by the symbol Cn.b) Rotation and

12、 the n-fold rotation axis (Cn)The principle rotation axis is the axis of the highest fold.One three-fold (C3) rotation axes. (=2/3)C3为主轴。为主轴。BF3分子有分子有1C3、3C2？：？：H2O，PtCl42+， C5H5-，C6H6C5The principle rotation axis is the axis of the highest fold.C6The matrix representations:30（x,y）（x1,y1）C31xyx1 = -

13、rsin(30) = -rsin30cos - rcos30sin = (-1/2)x + (-3/2)yy1 = rcos(30) = rcos30cos - rsin30sin = (3/2)x + (-1/2)yzyxyxzyxzyxzyxCzyx2232321000212302321100034cos34sin034sin34cos23222zyxyxzyxzyxzyxCzyx2232321000212302321100032cos32sin032sin32cos132221000cossin0sincosknCnk2The matrix representations of rota

14、tions around a Cn axis:CnConditions:Principle axis is aligned with the z-axis23旋转l定义:围绕通过分子的某一根轴转动 2/n 度能使分子复原的操作称为旋转(proper rotation)对称操作，简称旋转.l符号:Cnl对称元素:旋转轴(rotation axis)l分子中常出现的旋转轴： C2 C3 C4 C5 C6 C If reflection of an object through a plane produces an indistinguishable configuration, that pla

15、ne is a plane of symmetry (mirror plane, s s).c) Reflection and the Mirror plane (s s)100010001xyszyxzyxzyxxy100010001s（x, y,-z）（x,y,z）zyxThere are three types of mirror planes:If the plane is perpendicular to the vertical principle axis, it is labeled sh.If the plane contains the principle axis, it

16、 is labeled sv.If a s plane contains the principle axis and bisects the angle between two adjacent 2-fold axes, it is labeled sd.If the plane is perpendicular to the vertical principle axis then it is labeled sh.lExample: BF3 has a sh plane of symmetry.If the plane contains the principle axis then i

17、t is labeled sv.lExample: H2OuHas a C2 principle axis.uHas two planes that contain the principle axis, sv and sv.HHO sv sv Example: H2C=C=CH2 C2 C2 sd sd If a s plane contains the principle axis and bisects the angle between two adjacent 2-fold axes then it is labeled sd.(Dihedral mirror planes )The

18、 angle between the two sd planes is actually the dihedral angle H-CC-H !29反映与对称面反映与对称面 s s对称面对称面 s s：分子在通过该平面反映后，产生一个不分子在通过该平面反映后，产生一个不可分辨的结构取向。可分辨的结构取向。水平对称面水平对称面 s sv ： 通过分子主轴通过分子主轴垂直对称面垂直对称面 s sh ： 垂直于分子主轴垂直于分子主轴平分对称面平分对称面 s sd ： 平分与主轴垂直的平分与主轴垂直的C2轴的夹角轴的夹角反映 定义: 通过某一镜面将分子的各点反映到镜面另一侧位置相当处，结果使分子又恢复

19、原状的操作称为反映(reflection)对称操作，简称反映. 符号: 对称元素:镜面(mirror plane) 镜面类型： v 通过主轴 h 和主轴垂直 d 通过主轴并平分两个副轴间夹角H2ONH33233d) Inversion and the inversion center (i)An object has a center of inversion, i, if it can be reflected through a center to produce an indistinguishable configuration.34反演中心不在原子上35These do not ha

20、ve a center of inversion.These objects have a center of inversion i.For example100010001zyxzyxzyxi100010001Its matrix representationInverts all atoms through the centre of the object（-x,- y,-z）（x,y,z）zyxIzxy(-x,-y,-z)(x,y,z)rrEzxy(x,y,z)(x,y,z)rrzyxzyxIzyxzyxIzyxzyxI2EI1000100012EI100010001100010001

21、1000100012333231232221131211333231232221131211333231232221131211cccccccccbbbbbbbbbaaaaaaaaa213112213123212211211231211232132212121111311111311321121111cbacbababacbacbababacbacbababaiiiiiiiii反演与对称中心反演与对称中心 i对称中心对称中心 i：每一个原子沿每一个原子沿直线通过该中心达到另一直线通过该中心达到另一边相等距离时都能遇到相边相等距离时都能遇到相同的原子。同的原子。正方形的正方形的PtCl42离子有

22、对离子有对称中心，四面体的称中心，四面体的SiF4分子分子没有对称中心。没有对称中心。 平面正方形平面正方形PtCl42 四面体四面体SiF4 有对称中心有对称中心 无对称中心无对称中心反演 定义: 将分子的各点移到和反演中心连线的延长线上，且两边的距离相等. 若分子能恢复原状，即反演(inversion)对称操作，简称反演. 符号:i 对称元素:对称中心(center of symmetry) 例: 平面正方形的 PtCl42- 或八面体的PtCl62- 离子中，铂原子核的位置即为相应离子的对称中心.42Rotate 360/n followed by reflection in mirro

23、r plane perpendicular to axis of rotationa. n-fold rotation + reflection, Rotary-reflection axis (Sn)e) The improper rotation axisS4The staggered form of ethane has an S6 axis composed of a 60 rotation followed by a reflection. S6 sh S6Example: H3C-CH344Special Cases: S1 andS2hhCSss11iCSh22s*二重轴与反演中

24、心的区别二重轴与反演中心的区别C21i46旋转旋转-反映与映轴反映与映轴 Sn 如果绕一根轴旋转如果绕一根轴旋转2 /n角度后再对垂直于这根轴的一平面角度后再对垂直于这根轴的一平面进行反映，产生一个不可分辨的构型，那么这个轴就是进行反映，产生一个不可分辨的构型，那么这个轴就是n重重旋转旋转-反映轴反映轴，称作称作映轴映轴 Sn。交错构型的乙烷分子交错构型的乙烷分子与与C3轴重合的轴重合的S6轴轴n重非真旋转轴重非真旋转轴(improper rotation) Sn旋转-反映 定义: 旋转和反映的联合操作称为旋转-反映(rotation-reflection)对称操作，简称旋转-反映. 符号:S

25、n 对称元素:旋转-反映轴(rotation-reflection axis) 旋转-反映对称操作: 先绕一根轴旋转2/n度，接着按垂直该轴的镜面进行反映，使分子复原.Sn操作操作CH4分子的四重非真旋转轴分子的四重非真旋转轴S4CH4 三根与平分三根与平分HCH角的角的三根三根C2轴相重合的轴相重合的S4轴轴49iS450Stereographic ProjectionsoxoxWe will use stereographic projections to plot the perpendicular to a general face and its symmetry equivalen

26、ces, to display crystal morphology o for upper hemisphere; x for lower oESCSCSSCSCS63235313433323231313;sssS3 hhCCSss333It is not an independent symmetry element.ESCSCSCS44343412241414;ssESCSCSCSCSSCSCSCSCS1054595358525751565554545353525251515;sssssxS43xS41S4244CShshhCCSss555iCCSh366Only S4 and S8 a

27、re independent symmetry elements.Other Sn axes can be expressed by i, s sh, or Cn+ i, Cn+s sh.1313CiI 2323C IE C C C C63235313433323231313IiIIiIIiIIniCnI1iC1i, I2iC2shI3C3ixxb. n-fold rotation + inversion, Rotary-inversion axis (In)Rotation of Cn followed by inversion through the center of the axisL

28、ikewise, only I4 and I8 are independent point groups of symmetry! Summary对称操作小结对称操作小结对称元素对称元素 对称操作对称操作 对称符号对称符号 恒等操作恒等操作 En重对称轴重对称轴 旋转旋转2/n Cn镜面镜面 反映反映 反演中心反演中心 反演反演 in重非真旋转轴重非真旋转轴 先旋转先旋转2/n 或旋转反映或旋转反映 再对垂直于旋转轴的再对垂直于旋转轴的 Sn 镜面进行反映镜面进行反映 进行这些操作时，分子中至少有一个点保持不动进行这些操作时，分子中至少有一个点保持不动 “点群对称点群对称”操作。操作。 = C

29、3 + sh+S3S6 C2H6+.= C6 + sh= C3 + i36 = C6 + iCrystallographers Herman Maugin - RotoinversionSchoenflies - Rotoreflection571.1.2 Combination rules of symmetry elementsA. Combination of two axes of symmetry The combination of two C2 axes intersecting at angle of 2/2n, will create a Cn axis at the poi

30、nt of intersection which is perpendicular to both the C2 axes and there are nC2 axes in the plane perpendicular to the C2 axis.Cn + C2 () nC2 ()B. Combination of two planes of symmetry.If two mirrors planes intersect at an angle of 2/2n, there will be a Cn axis of order n on the line of intersection

31、. Similarly, the combination of an axis Cn with a mirror plane parallel to and passing through the axis will produce n mirror planes intersecting at angles of 2/2n.Cn + v n v vvvvCCssss3232 Ex. H2O, NH3C2C. Combination of an even-order rotation axis with a mirror plane perpendicular to it. Combinati

32、on of an even-order rotation axis with a mirror plane perpendicular to it will generate a center of symmetry at the point intersection.Each of the three operations xy, C2n and i is the product of the other two operations.10001000110001000112xyCs1 (x,y,z)3 (-x,-y,-z)2 (-x,-y,z)C2yxiCxy 10001000112sTd

33、OhIh61群的定义 定义：在元素的集合G上定义一种结合法(称为乘法乘法)，若G对于给定的乘法乘法满足下述四条公设(postulate)，则集合G称为给定的乘法乘法的一个群(group)： 1封闭性封闭性。G中任何两个(不同的或相同的)元素 A 和 B，它们的乘积 AB 仍是G中的元素。 2结合律结合律(associative law)成立。G中任意元素A，B，C，有(AB)C=A(BC)。 3单位元单位元E(unit element)存在。对于G中任何元素A，有EA=AE=A. 4逆元素逆元素(inverse element)存在。对于G中每一元素A，都有G中的一个元素B=A-1-1, 称为

34、A的逆元，使得AB=BA=E 群论中的乘法不必然等于代数乘法1.1. 3 点对称操作点对称操作群群2.1 群的定义群的定义GCCabGbGa,. 1则必有若封闭性,22ECCyzxyzvssyzxzzCCzyxzyxzyxzyxzyxyzzxzssss2,22. 2. 群论基本概念群论基本概念cabbcaGcba)()(,. 2则若结合律yzxzyzxzyzyzyzxzyzxzCCECECCCssssssssss)()()()(222222为恒等元素则若存在一恒等元素EaaEEaGEGa,. 3baabEbaabGbGa1,. 4的逆元素，记作为且则必存在若存在逆元素EECCCCxzxz112

35、121212ss , ,C,CE, 2313sss ECCC332313231313CCC132323CCC)()(132313132313CCCCCCECC2313symmetry elements:Closure.Identity operation.Associative rule.Inversion.EExample: NH3 Therefore, all symmetry elements of this molecule constitute a group, namely C3V. (group order = 6) E, C31,C32 is a subgroup of the

36、 C3V group.一个分子所具有的对称操作的完全集合构成一个点群一个分子所具有的对称操作的完全集合构成一个点群每个点群有一个特定的符号每个点群有一个特定的符号C C2v2v 点群点群,22ECCxzyzvss封闭性： 元素相乘符合结合律 ：ECCCyzxz 222)(s ss syzxzCss2ECCCyzyzyzxz s ss s)(2yzxzyzxzCCs ss ss ss s)()(22 点群中有一恒等操作E：222CECEC ECCCC 122212每个元素都有其逆元素：1 xzzxs ss sShriver/AtkinsC21C31C21C21=C22=EC31C31xC32=C

37、32=C31C31C32C31=C33=E68 分子可以按分子可以按“对称群对称群”或或“点群点群”加以分类。加以分类。 其中，任何具有一条其中，任何具有一条C2轴，轴，2个对称面和恒个对称面和恒等操作等操作E这四种对称操作组合的分子属于这四种对称操作组合的分子属于C2v“点群点群”。H2O分子就属于分子就属于C2v点群点群 在一个分子上所进行的对称操作的完全组合在一个分子上所进行的对称操作的完全组合构成一个构成一个“对称群对称群”或或“点群点群”。 点群符号点群符号:如如C2、C2v、D3h、Oh、Td等。等。2.2 主要点群主要点群2.2 Point Groups, the symmetr

38、y classification of moleculesPoint group:All symmetry operations arising from all symmetry elements of a given molecule constitute a group of symmetry operations. All possible symmetry operations corresponding to the symmetry elements have at least one common point unchanged.Such group of symmetry i

39、s thus called point group.It is convenient to represent the symmetry of a molecule by point group. 70The group C1 A molecule belongs to the group C1 if it has no element of symmetry other than the identity.1. The groups C1, Ci, and CsA molecule belongs to C1 if it has only the identity E.The group C

40、i = E, i It belongs to Ci if it has the identity and inversion alone. Example: meso-tartaric acid, HClBrC-CHClBrA molecule belongs to Ci if it has only the identity E and i.The group Cs It belongs to Cs if it has the identity and a mirror plane alone.A molecule belongs to Cs if it has only the ident

41、ity E and s s. E, s s The group Cn = E, Cn1, , Cnn-1 A molecule belongs to the group Cn if it has only an n-fold axis. Example: H2O22. The groups Cn, Cnv, Cnh and SnGroup order of Cn is ?.C3H2Cl2C2C2H3Cl3C3The group Cnv If in addition to a Cn axis it also has n vertical mirror planes sv, then it it

42、belongs to the Cnv group.CvC3vC2vC=OThe order of Cnv point group is 2n!C2vC3vC4vThe group Cnh Objects having a Cn axis and a horizontal mirror plane belong to Cnh.trans-CHCl=CHClC2H2Cl2I7-C10H6Cl2C2h? H-N=N-H, HCN, I3-, N3-81C3hC4hBOOOHHHB(OH)3C3hA Cnh point group is 2n-order, consisting of operatio

43、ns Cnm (m=1,.n), Snm (m=1, n-1) and s sh! Note: In such a case, Snn/2 = i (for even n) . 222424)(CCSnhnnnnsiCSnhnnnn)(1212241224siCh2s The group Sn Objects having a Sn improper rotation axis belong to Sn.Group S1=CsGroup S2=Ci S4S4S6hnhnnnnCSss)(12121212122. Objects having a S4n+2 axis have symmetry

44、 elements S4n+2, C2n+1 (=S4n+22) and i (=S4n+22n+1) ! (n1)S81. Objects having an odd-fold S2n+1 axis should also have a s sh mirror plane, thus having actually a C2n+1 axis (n=1,2,), thus belonging to C(2n+1)h. 1122124124224)()(nhnnnCCCSsiCSnhnnnn)(1212241224sMono-axis groups Such point groups Cn, C

45、nv, Cnh, Sn etc having only one rotary axis are called mono-axis groups. In short, there exist Sn groups only when n =2m (m1)! Sn point group is n-order.3. Only S4n axes are independent symmetry elements! Objects having an S4n axis also have a C2n axis. (n=1,2,)3. The group Dn, Dnh, DndDn: A molecul

46、e that has an n-fold principle axis and n twofold axes perpendicular to Cn belongs to Dn.Dn is 2n-order.(Ethane in a non-equilibrium state)Co(dien)2D2D3D4D5D6 A molecule with a mirror plane (h) perpendicular to a Cn axis, and with n two-fold axes (C2) in the plane, belongs to the group Dnh. The grou

47、ps DnhCCHHHHC4C2C2shD2hClClClAuClC4C2C2C2C2shD4hDnh is 4n-order.C2DnhD3hC2H4C10H10 SiF4(C5H5N)2BF3PCl5Tc6Cl6D2hD3hNi(CN)42-M2(COOR)4X2Re2Cl8Cr2(C6H5)2C6H6O=C=OD4hD6hThe group Dnd A molecule that has an n-fold principle axis and n twofold axes perpendicular to Cn belongs to Dnd if it posses n dihedra

48、l mirror planes.The order of group=4nD2dDndD3dTiCl62-TaF83-S8D3dD4dD5d4. High order point groups The aforementioned point groups have one axis or one n-fold axis plus several C2 axes. Molecules having three or more high symmetry elements (several n-fold axes, n2) may belong to one of the following:

49、T: 4 C3, 3 C2 (Th: +3h) (Td: +3S4) O: 4 C3, 3 C4 (Oh: +3h) I: 6 C5, 10 C3 (Ih: +i) Cubic groupCubic groupsShapes corresponding to the point groups (a) T. The presence of the windmill-like structures reduces the symmetry of the object from Td to T. C3C2T: 4 C3, 3 C2 (Th: +3h) (Td: +3S4) The tetrahedr

50、on-shaped object belongs to Td point group.T= =E，4C31，4C32，3C2 order =12Pure rotation group!Cubic groupsThT Th h = =E，4C31，4C32，3C2，i，4S6，4S65，3hOrder = 24Order = 24 T + 3s sh ( C2)s sh C2 iC3 + i S6TdTd =E，3C2，8C3，6S4，6dOrder =24C3C3C3C2S4s sd,vT + 3sdC(CH3)4 P8C12 CH4 dTThTdTdCo4(CO)12 P4O6 Ti8C12

51、 (Td is more stable than Th)Chem. Rev. 2005, 105, 3643.TdCubic groupsShapes corresponding to the point groups O. The presence of the windmill-like structures reduces the symmetry of the object from Oh to O. OO: 4C3, 3C4 (Oh: +3h)C3C4C2O= E, 4C31, 4C32, 3C41 , 3C43 , 3C2 , 6C2Order =24C2 Pure rotatio

52、n group!FSFFFFFOhs sd/vC4S6C2C2 Oh = E, 6C4, 3C2, 8C3, 6C2 , i, 8S6, 6S4, 3s sh, 6s sv(3C41,3C43) (3C42)(4C31,4C32)(3S41,3S43)(4S61,4S65)Group order= 48S4C3s shSF6 C8H8OhRh13 O = lacks i, S4, S6, sh and sd and is called the pure rotation subgroup of Oh.Td = this group lacks C4, i and sh and is the g

53、roup of tetrahedral molecules, e.g., CH4.Th = this uncommon group is derived from Td by removing S4 and sd elements.T = the pure rotation subgroup of Td contains only C3 and C2 axes.Typical subgroups of OhC5I and Ih groupsI: 6 C5, 10C3 (Ih: +i) C3 I = E, 12C5, 12C52, 20C3, 15C2) order =60 Pure rotat

54、ion group!Ih: (I + i)C5 + i = S10 C3 + i = S6 C2 + i = h E, 12C5, 12C52, 20C3, 15C2，i, 12S10, 12S103, 20S6, 15h order = 120 Icosahedron (二十面体二十面体)Napex = 12, Nface= 20, Nedge= 30 2CC20H20B12H122（hydrogen omitted）Ih group: two long-known examples a. B12H122- (icosahedral borane dianion二十面硼烷二负离子)Chem.

55、 Rev. 2005, 105, 3643 and references therein. b. C20H20 (dodecahedrane正十二面体烷) First synthesized by Paquette in 1982, three years before the discovery of C60. It is indeed the first fullerene derivative synthesized by mankind. IhC60, bird-views from the 5-fold axis and 6-fold axis.Ih= E，12C5，12C52，20

56、C3，15C2， i，12S10，12S103，20S6，15h Order =120 12 pentagons and 20 hexagons; C60, a molecule starring in the past two decades.tetrahedral symmetry groupIcosahedral symmetry groupIh Icosahedral symmetry (Buckminster-fullerene, C60)Td Species with tetrahedral symmetryoctahedral symmetry groupOh Species w

57、ith octahedral symmetry (many metal complexes)几种主要分子点群(复习与小结）(1) C1点群点群(2) Cn 点群点群 非对称化合物非对称化合物 除除C1外，无任何对称元素外，无任何对称元素 仅含有一个仅含有一个Cn轴轴 几种主要分子点群(3) Cs点群点群(4) Cnv 点群点群仅含有一个镜面仅含有一个镜面s s 含有一个含有一个Cn轴和轴和 n个竖直对称面个竖直对称面 (5) Cnh 点群点群(6) Dn 点群点群含有一个Cn轴和一个垂直于Cn轴的面s sh C2h点群点群 一个Cn轴和n个垂直于Cn轴的C2 轴 (8) Dnd 点群点群(7)

58、 Dnh 点群点群具有一个Cn轴, n个垂直于Cn轴的C2轴 和一个 s sh 具有一个Cn轴, n个垂直于Cn轴的C2 轴和n个分角对称面 s sd D4h 点群点群D5d点群点群(9) Sn 点群点群只具有一个只具有一个Sn轴轴 S4 点群点群 (10) Td点群点群4C3,3C2, 3S4 , 6s sd (11) Oh点群点群3C4, 4C3, 3C2, 6C2, 4S6, 3S4, 3s sh, 6s sd, iTd点群点群Oh点群点群(12) Dh点群点群C , Sn, s sv, i(13) Cv点群点群Cv, s sv Dh点群点群Cv点群点群一些化学中重要的点群一些化学中重要

59、的点群点群点群 对对 称称 元元 素素(未包括恒等元素未包括恒等元素) 实例实例Cs 仅有一个对称面仅有一个对称面 ONCl, HOClC1 无对称性无对称性 SiFClBrICn 仅有一根仅有一根n重旋转轴重旋转轴 H2O2, PPh3Cnv n重旋转轴和通过该轴的镜面重旋转轴和通过该轴的镜面 H2O, NH3Cnh n重旋转轴和一个水平镜面重旋转轴和一个水平镜面 反反N2F2Cv 无对称中心的线性分子无对称中心的线性分子 CO，HCNDn n重旋转轴和垂直该轴的重旋转轴和垂直该轴的n根根C2轴轴 Cr(C2O4)33Dnh Dn的对称元素、再加一个水平镜面的对称元素、再加一个水平镜面 BF

60、3，PtCl42Dh 有对称中心的线性分子有对称中心的线性分子 H2, Cl2, CO2Dnd Dn的对称元素、再加一套平分每一的对称元素、再加一套平分每一C2轴的垂直镜面轴的垂直镜面 B2Cl4,交错交错C2H6Sn 有唯一对称元素有唯一对称元素(Sn映轴映轴) S4N4F4Td 正四面体分子或离子正四面体分子或离子，4C3、3C2、3S4和和6s sd CH4, ClO4Oh 正八面体分子或离子正八面体分子或离子，3C4、4C3、6C2、6s sd、3s sh、i SF6Ih 正二十面体，正二十面体，6C5、10C3、15C2及及15 B12H122Linear?i ?D hC vTwo