5.3 Symmetry orbitalsAsymmetryorbitalisalinearcombinationofotherorbitals(usuallyAOs)whicharechosen in such a way that the symmetry orbital transforms as a single irreduciblerepresentation.In some texts these linear combinations are called symmetry adapted linear combinations,SALCS.Wewill describetwoapproachestotheconstructionof SOs:(1) by making use of the additional information presented in charactertables;CzG-xzZC2vE(2)by use oftheprojectionformula (投影公式)-111A11X2;y2;2ZInpractice,thefirst oneisbyfartheeasiest11-1-1A2Rxy1Bi1-1-1R,xz1B21-1-1Rxyyz
5.3 Symmetry orbitals • A symmetry orbital is a linear combination of other orbitals (usually AOs) which are chosen in such a way that the symmetry orbital transforms as a single irreducible representation. In some texts these linear combinations are called symmetry adapted linear combinations, SALCs. • We will describe two approaches to the construction of SOs: (1) by making use of the additional information presented in character tables; (2) by use of the projection formula (投影公式). In practice, the first one is by far the easiest
5.3.1 SOs in BH·Pointgroup:D3h·Firstconsidera basisconsistingof threeH lsAOs and countthecharactersE2S3D3h2C33C2Ch30v.Obviously,thecombinationof1111A'1x2+y2;z?1the hydrogen Is AOs whichA1Rz11-11Eitransforms as the totally202-10[(x, y)(x2 - y2,2xy)-1A"111-1-1-1symmetric IR A is1A211-1-1-1zi20-10-21(Rx,R,)(xz,yz)(sa+ $p+so)r301310=A'@E':Theremainingtwo SOstransformas E', similar to the basis (x,y)0
5.3.1 SOs in BH3 • First consider a basis consisting of three H 1s AOs and ‘count’ the characters. • Obviously, the combination of the hydrogen 1s AOs which transforms as the totally symmetric IR 𝑨𝟏 ′ is (sA+ sB+ sC ). 3 0 1 3 0 1 • The remaining two SOs transform as 𝑬 ′ , similar to the basis (x,y). = A1 E • Point group: ?D3h
一5.3.1 SOs in BHFortheSOthattransformslikethefunctionxL"likexsymmetryorbitalSO, = 0xSa +(+1)xSb +(-1)xScxcoord=0= SB - Scxcoord=Xcoord=+1Not normalized yet!·For the SO that transforms like the functiony"likey'symmetryorbitalSO2 = (+1)xsa +(-1/2)xSg + (-1/2)xScycoord=+1= SA - (Sp+sc)/2ycoord=-1/2ycoord=-1/2-1/2SB1/2Sc
• For the SO that transforms like the function ‘x’, 5.3.1 SOs in BH3 SO1 = 0sA + (+1)sB + (1)sC • For the SO that transforms like the function ‘y’, = sB sC SO2 = (+1)sA + (1/2)sB + (1/2)sC = sA – (sB+sC )/2 Not normalized yet!
5.3.1 SOs in BH:Hence the three H 1s AOs in BH, give the following three SOs,QE'x= SB - Sc, E'y = SA -(SB + Sc)/2OA = SA + SB + Sc ;· It is important to realise that S e',x and Se'y, together transform as the two-dimensional IR E':itisnotthateachalonetransformsasE'y"likex'solikey'soAT = SA + SB + SCQE.x=SB-SCQE.y=SA-1/2SB-1/2Sc
• Hence the three H 1s AOs in BH3 give the following three SOs， 𝜽𝑨𝟏 ′ = 𝒔𝑨 + 𝒔𝑩 + 𝒔𝑪 ; 𝜽𝑬 ′ ,𝒙 = 𝒔𝑩 − 𝒔𝑪 , 𝜽𝑬 ′ ,𝒚 = 𝒔𝑨 − (𝒔𝑩 + 𝒔𝑪)/𝟐 5.3.1 SOs in BH3 • It is important to realise that 𝜽𝑬 ′ ,𝒙 and 𝜽𝑬 ′ ,𝒚 together transform as the two-dimensional IR 𝑬 ′ : it is not that each alone transforms as 𝑬 ′
5.3.2 Normalization of symmetry orbitals*Φdt = 1·In quantummechanicsa wavefunctionyis normalized if.If a wavefunction y is not normalized, then defineNywar (normalization factor), and (Ny) is normalized.:A symmetry orbital is writtenas a linear combinationofatomic orbitals @;0 = C1Φ1 + C2Φ2 + C3Φ3 + ...IftheAOwavefunctions arethemselves normalized, and if weassumethattheAOsondifferent (but symmetrically equivalent) atoms do not overlap, i.e., J@,@,dt= , = 0 (itj)0 = C191 + c292 + C303+..theSOcanbenormalizedasci + c2 + c3 + ..or if the SO is normalized then c2+ c,2+ c2+...=1
5.3.2 Normalization of symmetry orbitals • In quantum mechanics a wavefunction ψ is normalized if 𝝍∗𝝍𝒅𝝉 = 𝟏 𝑵 = 𝟏 ∗𝒅 (normalization factor), and (Nψ) is normalized. 𝜽 = 𝒄𝟏𝚽𝟏 + 𝒄𝟐𝚽𝟐 + 𝒄𝟑𝚽𝟑 + ⋯ • If a wavefunction ψ is not normalized, then define • A symmetry orbital is written as a linear combination of atomic orbitals Φi : If the AO wavefunctions are themselves normalized, and if we assume that the AOs on different (but symmetrically equivalent) atoms do not overlap, i.e., ijd = ij = 0 (i≠j) the SO can be normalized as 𝜽 = 𝒄𝟏𝚽𝟏 + 𝒄𝟐𝚽𝟐 + 𝒄𝟑𝚽𝟑+ . . . 𝒄𝟏 𝟐 + 𝒄𝟐 𝟐 + 𝒄𝟑 𝟐 + ⋯ or if the SO is normalized then c1 2 + c2 2 + c3 2 + . = 1