《无机化学 Chemical Theory》课程教学资源（讲稿）Part II Molecular Spectroscopy Chapter 2 Representations 群表示

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)

2.1.1 Behaviour of the oxygen AOs in H,OEx. 5Similarly,thes,P,andp,AOseachresultinarepresentation:representation in the basis s:(+1,+1, +1,+1)representation in the basis py: (+1,-1, -1,+1)representation in the basis pz: (+1,+1, +1,+1). These are all described as one-dimensional representations since in each case there is only onebasis function. They also can be found in the character table of CyEoJzC3ozC2v111x2;y2;221AlZ(+1,+1, +1,+1) in the basis s or pz1A21-1-1Rzxy-11-1(+1,-1, +1,-1) in the basis p.B1R,1xXzB21-1-11Rxyyz(+1,-1, -1,+1) in the basis p,·In the present example, we would say that p,transforms as the irreducible representationB,. Similarly, p,transforms as B, and p, transforms as A

Which row for the basis py ? 2.1.1 Behaviour of the oxygen AOs in H2O • Similarly, the s, py and pzAOs each result in a representation: representation in the basis s: (+1,+𝟏, +1,+𝟏) representation in the basis py : (+1,−𝟏, −1,+𝟏) representation in the basis pz : (+1,+𝟏, +1,+𝟏) • These are all described as one-dimensional representations since in each case there is only one basis function. They also can be found in the character table of C2v. (+1,−1, −1,+1) in the basis py (+1,−1, +1,–1) in the basis px Which row for the basis pz ? (+1,+1, +1,+1) in the basis s or pz • In the present example, we would say that ‘px transforms as the irreducible representation B1 ’. Similarly, py transforms as B2 and pz transforms as A1 . Ex. 5

o2.1.2 Behavior of the hydrogen AOs in H,OSA=1xSA+0XSB. Two hydrogen 1s AOs in water (labeled as S and sB)SB = 0xSA + 1 ×SBstart&effectofEeffectofC,zeffectofozeffectofoyzC3s=SBSB.The basis functions s and Sare interconvertedVzOSBbytheoperations ofthegroup.(writeeqs.!)SA1V2SB=SA:The effect of a particular operation on an orbitalfunction is no longer simply to multiply it by lEsSAbut can be expressed as a linear combination ofESB=SBthe two AOs

2.1.2 Behavior of the hydrogen AOs in H2O • The basis functions sA and sB are interconverted by the operations of the group. (write eqs.!) • The effect of a particular operation on an orbital function is no longer simply to multiply it by ±1, but can be expressed as a linear combination of the two AOs. • Two hydrogen 1s AOs in water (labeled as sA and sB ). 𝑪𝟐 𝒛 sA = 𝑪𝟐 𝒛 sB = 𝑪𝟐 𝒛 sA sB = sB sA = 0 1 1 0 sA sB  𝒙𝒛 sA =  𝒙𝒛 sB = 𝝈 𝒙𝒛 sA sB = sA sB = 1 0 0 1 sA sB  y𝒛 sA =  𝒚𝒛 sB = 𝝈 𝒚𝒛 sA sB = sB sA = 0 1 1 0 sA sB 𝑬sA = 𝑬sB = 𝑬 sA sB = sA sB = 1 0 0 1 sA sB effect of C2 z effect of xz start & effect of E effect of yz sA = 1sA + 0 sB sB = 0sA + 1 sB sB sA sA sB sB sA sA sB

2.1.2 Behaviour of the hydrogen AOs in H,OThesefourmatricestogetherformarepresentationoftheoperationsofthegroup:The character ( of a matrix:the sum of thediagonal elements (alsoknownasthetrace)EC3X2. This is a two-dimensional representation, which is a set of 2 × 2 matrices, generated in thebasis consisting of two orbitals (or basis functions), s and sB..Thecharactersofthematricesaremoreimportantthanthematricesthemselves.Fortheabove representation in the s and s basis,the characters are0202ThematrixrepresentativeofE (identity)mustEalwaysbeaunitmatrix,soitscharactermustbeequaltothenumberofbasisfunctionsthedimensionalityoftherepresentation!

2.1.2 Behaviour of the hydrogen AOs in H2O • These four matrices together form a representation of the operations of the group: { 1 0 0 1 0 1 1 0 1 0 0 1 0 1 1 0 } 𝑪𝟐 𝒛 𝑬 σ xz σ yz • This is a two-dimensional representation, which is a set of 2 × 2 matrices, generated in the basis consisting of two orbitals (or basis functions), sA and sB . • The characters of the matrices are more important than the matrices themselves. For the above representation in the sA and sB basis, the characters are ( 2 , 0 , 2 , 0 ) 𝑪𝟐 𝒛 𝑬 σ xz σ yz The character () of a matrix: the sum of the diagonal elements (also known as the trace) The matrix representative of E (identity) must always be a unit matrix, so its character must be equal to the number of basis functions. the dimensionality of the representation!

2.1.3 Characters and reducible representations.Therepresentationwith characters (2,0,2,0)is notone oftheIRs inthe charactertableyzEotC2C2v.However,this setofnumbers canbeX2;y;21111A1Zobtained by adding together theRzxycharactersoftheIRA,withthoseoftheB11-111R,xXZIR B, i.e., A, ④ B,: (2,0,2,0)Rx田yz.y2200i.e.,therepresentationwithcharacters(2.0.2.0)isreducible(可约的）andcanbereducedtothe sum of the two IRs A,and B,i.e.,A, ④ B. (④~直和): The two-dimensional representation formed by the two hydrogen s orbitals ‘spans the IRsA,and B.Inotherwords,thesetwo orbitals transformas A, @B

2.1.3 Characters and reducible representations • The representation with characters (2,0,2,0) is not one of the IRs in the character table. i.e., the representation with characters (2,0,2,0) is reducible (可约的) and can be reduced to the sum of the two IRs A1 and B1 , i.e., A1 ⊕ B1 . (⊕~直和) • The two-dimensional representation formed by the two hydrogen 1s orbitals ‘spans the IRs A1 and B1 ’. In other words, ‘these two orbitals transform as A1 ⊕ B1 ’. • However, this set of numbers can be obtained by adding together the characters of the IR A1 with those of the IR B1 , i.e., A1 ⊕ B1 : (2,0,2,0) ⊕ 2 0 2 0

2.1.4 A quick method of finding charactersSinceweareonlyinterestedinthe characters oftherepresentativematrices(i.e.the sumofthe diagonal elements), then we only need to work out their diagonal elements.? If a symmetry operation moves an orbital to a different position there will be a O on thediagonal of the matrix. e.g. for the effect of C2 on sA? If the symmetry operation leaves the orbital in the same place, there will be a + on thediagonal, e.g., for the effect of rz on sa.? Finally, if the orbital remains in the same place but just changes sign, a -l will appear onthe diagonal, e.g., for the effect of C2 on the O px.H

2.1.4 A quick method of finding characters • If a symmetry operation moves an orbital to a different position there will be a 0 on the diagonal of the matrix. e.g. for the effect of 𝑪𝟐 𝒛 on sA. Since we are only interested in the characters of the representative matrices (i.e. the sum of the diagonal elements), then we only need to work out their diagonal elements. • If the symmetry operation leaves the orbital in the same place, there will be a +1 on the diagonal, e.g., for the effect of σ xz on sA . • Finally, if the orbital remains in the same place but just changes sign, a 1 will appear on the diagonal, e.g., for the effect of 𝑪𝟐 𝒛 on the O px