8.3 Infinite groupsI. Non-centrosymmetric linear molecules (e.g. OCS, NNO) ~ Csymmetrylabel&angularmomentumECoov2C (α)80yabout thez-axis(2)x2 +y2;21+(Ar)1ZZ~1-DIR2=01-1N11Rz(A2)II~2-DIR.2=±1II20(Er)2cosα(xz, yz)(x,y)(Rx,R,)△~2-DIR.2=±22△0(E2)(x2 - y2,2xy)2cos2a20@(E3)2cos3α@~ 2-DIR, 2=±3+/-~ symmetry under o,.The long axis of such molecules is the principal axis (z)and a rotation through any angle a about this axis is asymmetry operation. Thereare thus an infinite numberof such rotation axes, identifiedas C(a). There are an infinite number of mirror planes (coo,) containing the internuclear axis.:A state possessinga certain amount of angular momentumabout theprincipal axis transforms as aparticularIR
8.3 Infinite groups I. Non-centrosymmetric linear molecules (e.g. OCS, NNO) ~ C∞v. • The long axis of such molecules is the principal axis (z) and a rotation through any angle α about this axis is a symmetry operation. There are thus an infinite number of such rotation axes, identified as Cz (α). • There are an infinite number of mirror planes (∞σv ) containing the internuclear axis. • A state possessing a certain amount of angular momentum about the principal axis transforms as a particular IR. symmetry label & angular momentum about the z-axis () ~ 1-D IR, = 0 ~ 2-D IR, = 1 ~ 2-D IR, = 2 ~ 2-D IR, = 3 +/– ~ symmetry under v
8.3 Infinite groupsII. Centrosymmetric linear molecules (e.g. CO2, BeH2) ~ Doh:Infinitenumberofoperations!Q:ForCO2,i)determinethesymmetriesofthenormalmodesDohEi2C(α)252(α)00C2C0Vi) considering onlyx2+y2;22111112t1(Alg)fundamentaltransitionofeach2g1111Rz-1-1(A2g)Ig2020(Eig)2cosa2cosα(Rx,Ry)(xz, yz)normalmode,determineeach0202Ag2cos2a(x2 -y2,2xy)(E2g)2cos2anormalmodeisactiveinthe2200dg(E3g)2cos3a-2cos3αIR/RAMANspectrum..Nt111-1(Alu)-1-1z111Eu-1-1-1(A2u)...0Ilu20-2(Elu)2cosQ2cosα(x,y)..200-2Au2cos2(E2u)-2cos2a0du20-22cos3a2cos3a(E3u)t..++.....:Asaresultof theinfinitenumber of operations contained bythesegroups it is notquitestraightforwardtoapplythevariousmethodsthathavebeendescribedaboveforfinitegroups
8.3 Infinite groups II. Centrosymmetric linear molecules (e.g. CO2 , BeH2 ) ~ D∞h. • As a result of the infinite number of operations contained by these groups it is not quite straightforward to apply the various methods that have been described above for finite groups. Infinite number of operations! Q: For CO2 , i) determine the symmetries of the normal modes; ii) considering only fundamental transition of each normal mode, determine each normal mode is active in the IR/RAMAN spectrum
8.3 Infinite groupsWe can enumerate the particular properties of the IRs and the significance of their labels1. One-dimensional IRs are labelled Z.2. For IRs, the superscript + or - indicates the behavior under any one of the o, planes+ ~symmetric under , (i.e. the character is +1),- ~antisymmetric under o, (i.e. the character is -1)3. In Dh, the g or u subscript indicates the symmetry under the inversion operation:g ~ symmetric under i(i.e. the character is positive),u~ antisymmetric under i (i.e. the character is negative)
8.3 Infinite groups We can enumerate the particular properties of the IRs and the significance of their labels. 1. One-dimensional IRs are labelled Σ. 2. For Σ IRs, the superscript + or – indicates the behavior under any one of the σv planes: + ~symmetric under σv (i.e. the character is +1), – ~antisymmetric under σv (i.e. the character is –1). 3. In D∞h , the g or u subscript indicates the symmetry under the inversion operation: g ~ symmetric under i (i.e. the character is positive), u ~ antisymmetric under i (i.e. the character is negative)
H8.3 Infinite groups4.A Z IR indicates that there is no angular momentum about the principal axis5. I, and @ IRs are all two-dimensional; they correspond to ±1, ±2, ±3 units, respectivelyof angular momentum about theprincipal axis.e.g, In Hz, the MOs formed from the overlap of two 1s AOs are labelled og and ot. They transform as the IRs g and t, respectively: Both are symmetric with respect to o,. They differ in their symmetry with respect to i: Neither orbital has any angular momentum about theogoutprincipal axis
8.3 Infinite groups 4. A Σ IR indicates that there is no angular momentum about the principal axis. 5. Π, ∆ and Φ IRs are all two-dimensional; they correspond to ±1, ±2, ±3 units, respectively, of angular momentum about the principal axis. e.g, In H2 +, the MOs formed from the overlap of two 1s AOs are labelled σ𝐠 + and σ𝐮 +. • They transform as the IRs Σ𝐠 + and Σ𝐮 + , respectively. • Both are symmetric with respect to σv . • They differ in their symmetry with respect to i. • Neither orbital has any angular momentum about the principal axis
8.3 Infinite groups: Two 2p AOs overlap head on' to form two MOs with symmetry labels o and ot. If the 2p AOs overlap 'sideways on' the resulting MOs have symmetry labels u, and ng.1) Each is doubly degenerate since there are infact two pairs of p orbitals (two 2pxand two 2py)Tuoverlapping.2)EachMOhas±lunitof angularmomentumabout the principal axisTg
8.3 Infinite groups • Two 2p AOs overlap ‘head on’ to form two MOs with symmetry labels σ𝒈 + and σ𝐮 +. • If the 2p AOs overlap ‘sideways on’ the resulting MOs have symmetry labels u and g . 1) Each is doubly degenerate since there are in fact two pairs of p orbitals (two 2px and two 2py ) overlapping. 2) Each MO has ±1 unit of angular momentum about the principal axis. πu πg