Part III Symmetry and BondingChapter 21Representations第二章(群)表示Prof.Dr.XinLu（吕鑫）Email:xinlu@xmu.edu.cnhttp:/ /pcoss.xmu.edu.cn/xlv/index.htmlhttp://pcoss.xmu.edu.cn/xlv/courses/theochem/index.html
Part III Symmetry and Bonding Chapter 2 Representations 第二章 (群)表示 Prof. Dr. Xin Lu （吕鑫） Email: xinlu@xmu.edu.cn http://pcoss.xmu.edu.cn/xlv/index.html http://pcoss.xmu.edu.cn/xlv/courses/theochem/index.html
2.Representations(群的表示)>The key thing about a symmetry operation is that it leaves the molecule in anindistinguishable orientation to the starting position, e.g., Ox, over H,O.>What effect do these symmetry operations have on functionswithin' the molecule, such as the atomic orbitals?e.g, the 2s, 2p-, 2px, 2p, valence atomic orbitals (VAOs) of O in H,O>What we will see in this section is that it is very convenient to arrange for the orbitals tobehave inawaywhichreflectsthe symmetryofthemolecule>This discussionwillleadus tointroducerepresentations(表示)andtheall-importantirreduciblerepresentations（不可约表示）ofthepointgroups
2. Representations (群的表示) What we will see in this section is that it is very convenient to arrange for the orbitals to behave in a way which reflects the symmetry of the molecule. This discussion will lead us to introduce representations (表示) and the all-important irreducible representations (不可约表示) of the point groups. The key thing about a symmetry operation is that it leaves the molecule in an indistinguishable orientation to the starting position, e.g., xz over H2O. e.g, the 2s, 2pz , 2px , 2py valence atomic orbitals (VAOs) of O in H2O. What effect do these symmetry operations have on functions ‘within’ the molecule, such as the atomic orbitals?
2.1 Introducing representations. The idea of a representation is best introduced using an example: H,O (C2,)Symmetry elementsforH,O(C):theidentity(E),atwo-fold axis ofrotation(theprincipal axisC)and two (vertical) mirror planes (o)By convention thez-axis is coincident with theprincipal axis, but we are at libertyto put thex-andy-axes wherewe like. (e.g,right handed coordinates!)Vycomingoutofplaneofpaperalternative way of indicating axes·Allowed symmetryoperations forH,o (C,):E, C2, o", ".(Thesefouroperations areof coursetheelementsoftheC2,pointgroup!)
2.1 Introducing representations • The idea of a representation is best introduced using an example: H2O (C2v) Symmetry elements for H2O (C2v): the identity (E), a two-fold axis of rotation (the principal axis, C2 ) and two (vertical) mirror planes (v ). • By convention the z-axis is coincident with the principal axis, but we are at liberty to put the x- and y-axes where we like. (e.g., right handed coordinates!) • Allowed symmetry operations for H2O (C2v): E, 𝑪𝟐 𝒛 , xz , yz . (These four operations are of course the elements of the C2v point group!)
2.1.1 Behavior of the oxygen AOs in H,OWhitefor+and black/red for-value ofthe wavefunctions..Howaretheoxygen atomic orbitals (AOs)affected by the symmetry operations of theStarteffect of C2effect of oxzeffectofyz&effectofECpoint group: C2, arz and arz.. Under the symmetry operations theseAOseitherremainthesameorsimplyEs= (+1)sgrzs= (+1)sas= (+1)sC2s= (+1)schange sign; they neither move to anotherPxposition nor become other orbital.In each case, the effect of a symmetryapx= (+1)pxEpr= (+1)PxC2Pr= (-1)Pxg%px=(-1)pxoperationRcanbeexpressed intheformofR =A(A=+1 or-1).In GroupTheory these AOs are an exampleof a set of basis functions,they are simplyEp,= (+1)PzC2P,= (+1)Pz"P,=(+1)Pzα"p,= (+1)Pzreferredtoasabasisp.The effectofthesymmetry operations onp.C2p,= (-1)PyEp,= (+1)Pyozp,= (-1)Pyo"p,= (+1)P)can be summarized as (+1, -1, +1, -1)
2.1.1 Behavior of the oxygen AOs in H2O • How are the oxygen atomic orbitals (AOs) affected by the symmetry operations of the point group: 𝑪𝟐 𝒛 , σ xz and σ yz . White for + and black/red for value of the wavefunctions. Start & effect of E • Under the symmetry operations these AOs either remain the same or simply change sign; they neither move to another position nor become other orbital. • In each case, the effect of a symmetry operation 𝑹 can be expressed in the form of 𝑹 𝝍 = 𝑨𝝍 (A = +1 or –1). Es = 𝑪𝟐 𝒛 s = σ yz s s σ xzs = s s = s Epx = 𝑪𝟐 𝒛 px = σ yz px −px σ xzpx = px px = −px Epz = 𝑪𝟐 𝒛 pz = σ yz pz pz σ xzpz = pz pz = pz Epy = 𝑪𝟐 𝒛 py = σ yz py −py σ xzpy = py py = py • In Group Theory these AOs are an example of a set of basis functions; they are simply referred to as a basis. (1)px (+1)s (+1)s (+1)s (+1)s (+1)px (1)px (+1)px (+1)pz (+1)pz (+1)pz (+1)pz (1)py (+1)py (+1)py (1)py • The effect of the symmetry operations on px can be summarized as (+1, −𝟏, +1, −𝟏). effect of 𝑪𝟐 𝒛 effect of 𝝈 𝒙𝒛 effect of 𝝈 𝒚𝒛
2.1.1 Behaviour of the oxygen AOs in H,O? Taking the O p, orbital as the basis, the effect of the symmetry operations can be summarizedby grouping together as follows : (+1, -1, +1, -1).:In Group Theory this is said to be a representation of the operations of the group in a basisconsisting of just theprAO, and can be found as a row in the character table.EoJzC2axzC2v1111x2:VA12:72Z11-1A2-1Rzxy1-11B1-1R,(+1,-1, +1,-1) in the basis p,xXZ1B2-11-1Rxyyz. In the character table the rows are a very special set of representations called the irreduciblerepresentations(IRs)
2.1.1 Behaviour of the oxygen AOs in H2O • Taking the O px orbital as the basis, the effect of the symmetry operations can be summarized by grouping together as follows : (+1, −𝟏, +1, −𝟏). • In Group Theory this is said to be a representation of the operations of the group in a basis consisting of just the pxAO, and can be found as a row in the character table. (+1,−1, +1,–1) in the basis px • In the character table the rows are a very special set of representations called the irreducible representations (IRs)