晶體的對稱性與群論

1、高等無機(jī)化學(xué)高等無機(jī)化學(xué)Advanced Inorganic Chemistry高等無機(jī)化學(xué)高等無機(jī)化學(xué)課程簡介課程簡介: 無機(jī)化學(xué)在近代化學(xué)史上占有極為重要的地位, 在化學(xué)基本理論研究及實(shí)際應(yīng)用方面起著越來越重要的作用，研究范圍越來越大。近年來它已滲透到生物、分離分析、醫(yī)藥、催化冶金、材料科學(xué)、環(huán)境科學(xué)等領(lǐng)域,與各學(xué)科有著日益廣泛的聯(lián)系。學(xué)習(xí)本課程，掌握無機(jī)化合物及無機(jī)材料方面的知識，著重提高相關(guān)化學(xué)理論水平,了解現(xiàn)代無機(jī)化學(xué)的主要研究方向、研究方法、應(yīng)用及其發(fā)展趨勢，并培養(yǎng)把握學(xué)科前沿的能力，為研究生論文工作及今后從事相關(guān)研究工作打下堅(jiān)實(shí)的基礎(chǔ)。31. 1. 對稱操作對稱操作2. 2. 群

2、論基本概念群論基本概念3. 3. 分子的點(diǎn)群分子的點(diǎn)群4.4.群的表示和特征表標(biāo)群的表示和特征表標(biāo)5.5.波函數(shù)和對稱性波函數(shù)和對稱性6.6.群論的應(yīng)用群論的應(yīng)用第一章第一章: 對稱性和群論對稱性和群論 參考書目：參考書目： 1. Advanced Inorganic Chemistry F. Albert Cotton, Geoffrey, Wilkinsion, Carlos A. Murillo, Manfred Bochmann, John. Wiley. New York, 1999. 6th. Ed. 2. 中級無機(jī)化學(xué)中級無機(jī)化學(xué). 朱文祥朱文祥 編編, 高等教育出版社高等教育出

3、版社, 2004年年7月月. 3.無機(jī)化學(xué)無機(jī)化學(xué)D. F. Shriver, P. W. Atkins, C. H. Langford 著著, 高憶慈高憶慈 史啟禎史啟禎 曾克慰曾克慰 李丙瑞李丙瑞 等譯等譯 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.無機(jī)化學(xué)新興領(lǐng)域?qū)д摕o機(jī)化學(xué)新興領(lǐng)域?qū)д? 項(xiàng)斯芬編著項(xiàng)斯芬編著. 北京大學(xué)出版社北京大學(xué)出版社, 1988年年11月。月。教材：教材： 高等無機(jī)化學(xué)，高等無機(jī)化學(xué)， 陳蕙蘭主編，北京：高等教育出陳蕙蘭主編，北京：高等教育出版社，版社，2005.6第一章：對稱性與群論第一章：對稱性與群論要求：要求：1、確定簡單分子所屬

4、點(diǎn)群、確定簡單分子所屬點(diǎn)群2、解讀特征標(biāo)表、解讀特征標(biāo)表3、群論在無機(jī)化學(xué)中的應(yīng)用、群論在無機(jī)化學(xué)中的應(yīng)用 a. 對稱性與分子極性對稱性與分子極性 b. 分子的振動與分子的振動與IR、Raman光譜光譜 c. 化學(xué)鍵與分子軌道等化學(xué)鍵與分子軌道等 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對稱性：對稱性： 物體或圖像中所具有的相似性或勻稱感。物體或圖像中所具有的相似性或勻稱感。對稱操作：對稱操作： 經(jīng)過某一不改變物體中任何兩點(diǎn)間距離的操作經(jīng)過某一不改變物體中任何兩點(diǎn)間距離的操作使圖形復(fù)原。使圖形復(fù)原。對稱元素：對稱元素： 實(shí)現(xiàn)對稱操作所依賴的點(diǎn)、線或面。實(shí)現(xiàn)對稱操作所依賴的點(diǎn)、線或面

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 對稱操作和對稱元素l分子的對稱性分子的對稱性, 對稱操作及對稱元素對稱操作及對稱元素定義: 分子的對稱性分子的對稱性是指存在一定的操作，它在保持任意兩點(diǎn)間距離不變?nèi)我鈨牲c(diǎn)間距離不

9、變的條件下，使分子內(nèi)部各部分變換位置，而且變換后的分子整體又恢復(fù)恢復(fù)原狀原狀，這種操作稱為對稱操作對稱操作(symmetry operation)對稱操作據(jù)以進(jìn)行的元素稱為對對稱元素稱元素(symmetry element).Symmetry operations are:The corresponding symmetry elements are:點(diǎn)對稱性點(diǎn)對稱性 點(diǎn)對稱元素：對稱操作過程中點(diǎn)對稱元素：對稱操作過程中至少有一點(diǎn)保持不至少有一點(diǎn)保持不變。變。 一個物體一個物體所有的點(diǎn)對稱元素必通過一個所有的點(diǎn)對稱元素必通過一個共同點(diǎn)共同點(diǎn)。 一個分子的對稱性完全一個分子的對稱性完全可以用一套

10、點(diǎn)對稱元素描可以用一套點(diǎn)對稱元素描述。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為主軸。為主軸。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旋轉(zhuǎn)l定義:圍繞通過分子的某一根軸轉(zhuǎn)動 2/n 度能使分子復(fù)原的操作稱為旋轉(zhuǎn)(proper rotation)對稱操作，簡稱旋轉(zhuǎn).l符號:Cnl對稱元素:旋轉(zhuǎn)軸(rotation axis)l分子中常出現(xiàn)的旋轉(zhuǎn)軸： 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反映與對稱面反映與對稱面 s s對稱面對稱面 s s：分子在通過該平面反映后，產(chǎn)生一個不分子在通過該平面反映后，產(chǎn)生一個不可分辨的結(jié)構(gòu)取向?？煞直娴慕Y(jié)構(gòu)取向。水平對稱面水平對稱面 s sv ： 通過分子主軸通過分子主軸垂直對稱面垂直對稱面 s sh ： 垂直于分子主軸垂直于分子主軸平分對稱面平分對稱面 s sd ： 平分與主軸垂直的平分與主軸垂直的C2軸的夾角軸的夾角反映 定義: 通過某一鏡面將分子的各點(diǎn)反映到鏡面另一側(cè)位置相當(dāng)處，結(jié)果使分子又恢復(fù)

19、原狀的操作稱為反映(reflection)對稱操作，簡稱反映. 符號: 對稱元素:鏡面(mirror plane) 鏡面類型： v 通過主軸 h 和主軸垂直 d 通過主軸并平分兩個副軸間夾角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反演與對稱中心反演與對稱中心 i對稱中心對稱中心 i：每一個原子沿每一個原子沿直線通過該中心達(dá)到另一直線通過該中心達(dá)到另一邊相等距離時都能遇到相邊相等距離時都能遇到相同的原子。同的原子。正方形的正方形的PtCl42離子有

22、對離子有對稱中心，四面體的稱中心，四面體的SiF4分子分子沒有對稱中心。沒有對稱中心。 平面正方形平面正方形PtCl42 四面體四面體SiF4 有對稱中心有對稱中心 無對稱中心無對稱中心反演 定義: 將分子的各點(diǎn)移到和反演中心連線的延長線上，且兩邊的距離相等. 若分子能恢復(fù)原狀，即反演(inversion)對稱操作，簡稱反演. 符號:i 對稱元素:對稱中心(center of symmetry) 例: 平面正方形的 PtCl42- 或八面體的PtCl62- 離子中，鉑原子核的位置即為相應(yīng)離子的對稱中心.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*二重軸與反演中

24、心的區(qū)別二重軸與反演中心的區(qū)別C21i46旋轉(zhuǎn)旋轉(zhuǎn)-反映與映軸反映與映軸 Sn 如果繞一根軸旋轉(zhuǎn)如果繞一根軸旋轉(zhuǎn)2 /n角度后再對垂直于這根軸的一平面角度后再對垂直于這根軸的一平面進(jìn)行反映，產(chǎn)生一個不可分辨的構(gòu)型，那么這個軸就是進(jìn)行反映，產(chǎn)生一個不可分辨的構(gòu)型，那么這個軸就是n重重旋轉(zhuǎn)旋轉(zhuǎn)-反映軸反映軸，稱作稱作映軸映軸 Sn。交錯構(gòu)型的乙烷分子交錯構(gòu)型的乙烷分子與與C3軸重合的軸重合的S6軸軸n重非真旋轉(zhuǎn)軸重非真旋轉(zhuǎn)軸(improper rotation) Sn旋轉(zhuǎn)-反映 定義: 旋轉(zhuǎn)和反映的聯(lián)合操作稱為旋轉(zhuǎn)-反映(rotation-reflection)對稱操作，簡稱旋轉(zhuǎn)-反映. 符號:S

25、n 對稱元素:旋轉(zhuǎn)-反映軸(rotation-reflection axis) 旋轉(zhuǎn)-反映對稱操作: 先繞一根軸旋轉(zhuǎn)2/n度，接著按垂直該軸的鏡面進(jìn)行反映，使分子復(fù)原.Sn操作操作CH4分子的四重非真旋轉(zhuǎn)軸分子的四重非真旋轉(zhuǎn)軸S4CH4 三根與平分三根與平分HCH角的角的三根三根C2軸相重合的軸相重合的S4軸軸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對稱操作小結(jié)對稱操作小結(jié)對稱元素對稱元素 對稱操作對稱操作 對稱符號對稱符號 恒等操作恒等操作 En重對稱軸重對稱軸 旋轉(zhuǎn)旋轉(zhuǎn)2/n Cn鏡面鏡面 反映反映 反演中心反演中心 反演反演 in重非真旋轉(zhuǎn)軸重非真旋轉(zhuǎn)軸 先旋轉(zhuǎn)先旋轉(zhuǎn)2/n 或旋轉(zhuǎn)反映或旋轉(zhuǎn)反映 再對垂直于旋轉(zhuǎn)軸的再對垂直于旋轉(zhuǎn)軸的 Sn 鏡面進(jìn)行反映鏡面進(jìn)行反映 進(jìn)行這些操作時，分子中至少有一個點(diǎn)保持不動進(jìn)行這些操作時，分子中至少有一個點(diǎn)保持不動 “點(diǎn)群對稱點(diǎn)群對稱”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群的定義 定義：在元素的集合G上定義一種結(jié)合法(稱為乘法乘法)，若G對于給定的乘法乘法滿足下述四條公設(shè)(postulate)，則集合G稱為給定的乘法乘法的一個群(group)： 1封閉性封閉性。G中任何兩個(不同的或相同的)元素 A 和 B，它們的乘積 AB 仍是G中的元素。 2結(jié)合律結(jié)合律(associative law)成立。G中任意元素A，B，C，有(AB)C=A(BC)。 3單位元單位元E(unit element)存在。對于G中任何元素A，有EA=AE=A. 4逆元素逆元素(inverse element)存在。對于G中每一元素A，都有G中的一個元素B=A-1-1, 稱為

34、A的逆元，使得AB=BA=E 群論中的乘法不必然等于代數(shù)乘法1.1. 3 點(diǎn)對稱操作點(diǎn)對稱操作群群2.1 群的定義群的定義GCCabGbGa,. 1則必有若封閉性,22ECCyzxyzvssyzxzzCCzyxzyxzyxzyxzyxyzzxzssss2,22. 2. 群論基本概念群論基本概念cabbcaGcba)()(,. 2則若結(jié)合律yzxzyzxzyzyzyzxzyzxzCCECECCCssssssssss)()()()(222222為恒等元素則若存在一恒等元素EaaEEaGEGa,. 3baabEbaabGbGa1,. 4的逆元素，記作為且則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.一個分子所具有的對稱操作的完全集合構(gòu)成一個點(diǎn)群一個分子所具有的對稱操作的完全集合構(gòu)成一個點(diǎn)群每個點(diǎn)群有一個特定的符號每個點(diǎn)群有一個特定的符號C C2v2v 點(diǎn)群點(diǎn)群,22ECCxzyzvss封閉性： 元素相乘符合結(jié)合律 ：ECCCyzxz 222)(s ss syzxzCss2ECCCyzyzyzxz s ss s)(2yzxzyzxzCCs ss ss ss s)()(22 點(diǎn)群中有一恒等操作E：222CECEC ECCCC 122212每個元素都有其逆元素：1 xzzxs ss sShriver/AtkinsC21C31C21C21=C22=EC31C31xC32=C

37、32=C31C31C32C31=C33=E68 分子可以按分子可以按“對稱群對稱群”或或“點(diǎn)群點(diǎn)群”加以分類。加以分類。 其中，任何具有一條其中，任何具有一條C2軸，軸，2個對稱面和恒個對稱面和恒等操作等操作E這四種對稱操作組合的分子屬于這四種對稱操作組合的分子屬于C2v“點(diǎn)群點(diǎn)群”。H2O分子就屬于分子就屬于C2v點(diǎn)群點(diǎn)群 在一個分子上所進(jìn)行的對稱操作的完全組合在一個分子上所進(jìn)行的對稱操作的完全組合構(gòu)成一個構(gòu)成一個“對稱群對稱群”或或“點(diǎn)群點(diǎn)群”。 點(diǎn)群符號點(diǎn)群符號:如如C2、C2v、D3h、Oh、Td等。等。2.2 主要點(diǎn)群主要點(diǎn)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 (二十面體二十面體)Napex = 12, Nface= 20, Nedge= 30 2CC20H20B12H122（hydrogen omitted）Ih group: two long-known examples a. B12H122- (icosahedral borane dianion二十面硼烷二負(fù)離子)Chem.

55、 Rev. 2005, 105, 3643 and references therein. b. C20H20 (dodecahedrane正十二面體烷) 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)幾種主要分子點(diǎn)群(復(fù)習(xí)與小結(jié)）(1) C1點(diǎn)群點(diǎn)群(2) Cn 點(diǎn)群點(diǎn)群 非對稱化合物非對稱化合物 除除C1外，無任何對稱元素外，無任何對稱元素 僅含有一個僅含有一個Cn軸軸 幾種主要分子點(diǎn)群(3) Cs點(diǎn)群點(diǎn)群(4) Cnv 點(diǎn)群點(diǎn)群僅含有一個鏡面僅含有一個鏡面s s 含有一個含有一個Cn軸和軸和 n個豎直對稱面?zhèn)€豎直對稱面 (5) Cnh 點(diǎn)群點(diǎn)群(6) Dn 點(diǎn)群點(diǎn)群含有一個Cn軸和一個垂直于Cn軸的面s sh C2h點(diǎn)群點(diǎn)群 一個Cn軸和n個垂直于Cn軸的C2 軸 (8) Dnd 點(diǎn)群點(diǎn)群(7)

58、 Dnh 點(diǎn)群點(diǎn)群具有一個Cn軸, n個垂直于Cn軸的C2軸 和一個 s sh 具有一個Cn軸, n個垂直于Cn軸的C2 軸和n個分角對稱面 s sd D4h 點(diǎn)群點(diǎn)群D5d點(diǎn)群點(diǎn)群(9) Sn 點(diǎn)群點(diǎn)群只具有一個只具有一個Sn軸軸 S4 點(diǎn)群點(diǎn)群 (10) Td點(diǎn)群點(diǎn)群4C3,3C2, 3S4 , 6s sd (11) Oh點(diǎn)群點(diǎn)群3C4, 4C3, 3C2, 6C2, 4S6, 3S4, 3s sh, 6s sd, iTd點(diǎn)群點(diǎn)群Oh點(diǎn)群點(diǎn)群(12) Dh點(diǎn)群點(diǎn)群C , Sn, s sv, i(13) Cv點(diǎn)群點(diǎn)群Cv, s sv Dh點(diǎn)群點(diǎn)群Cv點(diǎn)群點(diǎn)群一些化學(xué)中重要的點(diǎn)群一些化學(xué)中重要

59、的點(diǎn)群點(diǎn)群點(diǎn)群 對對 稱稱 元元 素素(未包括恒等元素未包括恒等元素) 實(shí)例實(shí)例Cs 僅有一個對稱面僅有一個對稱面 ONCl, HOClC1 無對稱性無對稱性 SiFClBrICn 僅有一根僅有一根n重旋轉(zhuǎn)軸重旋轉(zhuǎn)軸 H2O2, PPh3Cnv n重旋轉(zhuǎn)軸和通過該軸的鏡面重旋轉(zhuǎn)軸和通過該軸的鏡面 H2O, NH3Cnh n重旋轉(zhuǎn)軸和一個水平鏡面重旋轉(zhuǎn)軸和一個水平鏡面 反反N2F2Cv 無對稱中心的線性分子無對稱中心的線性分子 CO，HCNDn n重旋轉(zhuǎn)軸和垂直該軸的重旋轉(zhuǎn)軸和垂直該軸的n根根C2軸軸 Cr(C2O4)33Dnh Dn的對稱元素、再加一個水平鏡面的對稱元素、再加一個水平鏡面 BF

60、3，PtCl42Dh 有對稱中心的線性分子有對稱中心的線性分子 H2, Cl2, CO2Dnd Dn的對稱元素、再加一套平分每一的對稱元素、再加一套平分每一C2軸的垂直鏡面軸的垂直鏡面 B2Cl4,交錯交錯C2H6Sn 有唯一對稱元素有唯一對稱元素(Sn映軸映軸) S4N4F4Td 正四面體分子或離子正四面體分子或離子，4C3、3C2、3S4和和6s sd CH4, ClO4Oh 正八面體分子或離子正八面體分子或離子，3C4、4C3、6C2、6s sd、3s sh、i SF6Ih 正二十面體，正二十面體，6C5、10C3、15C2及及15 B12H122Linear?i ?D hC vTwo