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 = 1sA + 0 sB sB = 0sA + 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