Part III Symmetry and BondingChapter 8 MiscellanyProf.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 8 Miscellany 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
8.1 Dipole moments=rqPhys.Chem.: A (permanent) dipole moment is a physical property possessed by a moleculee.g.,for H,O, define μ (a vector). Does it change upon C3, axz and yz operations?R = E, C2, gxz, gyzRu = (+1)uThe characters are all +l.Invariant!The dipole moment must be invariant to symmetry operations possessed by that molecule!. That is, the dipole moment must transform as the totally symmetric IR The dipole moment itself is just the result of an uneven distribution of charge, and in generalit can only transform like x, y or z.A molecule possesses a permanent dipole moment only if x, y or ztransforms as the totallysymmetric IR. (Necessity for the presence of dipole moment!)? Molecules of such symmetry as C, Cn, and Cn, can have a permanent dipole
8.1 Dipole moments • The dipole moment must be invariant to symmetry operations possessed by that molecule! e.g.,for H2O, define 𝝁 (a vector). Does it change upon 𝑪𝟐 𝒛 , σ xz and σ yz operations? • That is, the dipole moment must transform as the totally symmetric IR. • The dipole moment itself is just the result of an uneven distribution of charge, and in general it can only transform like x, y or z. q– q+ r 𝝁 = 𝒓q • A (permanent) dipole moment is a physical property possessed by a molecule. The characters are all +1. Invariant! • A molecule possesses a permanent dipole moment only if x, y or z transforms as the totally symmetric IR. (Necessity for the presence of dipole moment!) Phys. Chem. 𝑹 = 𝑬, 𝑪𝟐 𝑹 𝝁 = ? 𝒛 , 𝝈 𝒙𝒛 , 𝝈 𝒚𝒛 (+𝟏)𝝁 • Molecules of such symmetry as Cs , Cn , and Cnv can have a permanent dipole
8.1 Dipole momentsC2O-y2Eo-tzC2v1Ai111ZA21-1-11Bi-11-1e.g., for H,O C2v, z transforms as A,x-1B21-11y it has a dipole along z.2CCa3C2DehE2C63C22S32S630d30vOh111111Aig-1111111/-1-1111R.A2g-1-1Blg-1-11-11-111-1-1-B2g-1一-11-1e.g., benzene(Dh),ztransforms as Azuand00Elg2-202~20(Rx,R,)20200E2g072-1(x,y) transform as Eiu-1Ain-1-111A21-1-1-1-1-112Bl-1-1-1-1I-11-111 no dipole at all!B211-1-1-11/1-1-1E20200(-1-20-2-1E2u2200-2-200-1-11Amolecule withax, y and z must all be anti-No dipole moment!symmetric IR with the label ucentre of symmetry: This discussion refers to the permanent dipole possessed by a molecule in its equilibrium geometry
8.1 Dipole moments no dipole at all! e.g., benzene (D6h), z transforms as A2u and (x,y) transform as E1u. • This discussion refers to the permanent dipole possessed by a molecule in its equilibrium geometry. A molecule with a centre of symmetry x, y and z must all be antisymmetric IR with the label u. No dipole moment! e.g., for H2O C2v, z transforms as A1 it has a dipole along z
8.2 Chirality: Chiral molecules have the physical property that they rotate the plane of polarized light. A molecule is chiral if it cannot be superimposed on its mirror image, or, in the language of grouptheory,ifthemoleculedoes notpossessany improperaxesofrotation,Sn: It is important to recall that a mirror plane is the same thing as S, and a centre of symmetry is thesame thing as S,.Therefore molecules possessing either mirror planes or a centre of symmetryarenotchiral.0: If in a molecule having no S, a carbon atom is attached to four differentgroups i.e. C(ABCD) then clearly the molecule is chiral. Such a carbonH2NOiscalledachiral centre.OHHRe.g., amino acids with the a carbon in a chiral centre
8.2 Chirality • Chiral molecules have the physical property that they rotate the plane of polarized light. • A molecule is chiral if it cannot be superimposed on its mirror image, or, in the language of group theory, if the molecule does not possess any improper axes of rotation, Sn . • It is important to recall that a mirror plane is the same thing as S1 and a centre of symmetry is the same thing as S2 . Therefore molecules possessing either mirror planes or a centre of symmetry are not chiral. • If in a molecule having no Sn a carbon atom is attached to four different groups i.e. C(ABCD) then clearly the molecule is chiral. Such a carbon is called a chiral centre. e.g., amino acids with the α carbon in a chiral centre
8.2ChiralityGeneratingchiralitywithoutuse of chiral centres.Neitherofthemolecules shownbelowhavechiralcentres.buttheyarenevertheless chiralas a result of restricted rotation about theC-C bond in the case ofthe molecule on the leftand the geometry of the fused four-membered rings on the right.NO2COOHNH2HaNHO2NCOOH
8.2 Chirality • Generating chirality without use of chiral centres. Neither of the molecules shown below have chiral centres, but they are nevertheless chiral as a result of restricted rotation about the C–C bond in the case of the molecule on the left, and the geometry of the fused four-membered rings on the right
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