Chapter 3 - Molecular Symmetry

tion, for the moregeneral case (complicated molecule with many bonds),we can use the reduction formula.29

Reducible RepresentationsThe reduction can be achieved using the reduction formula. It is amathematical way of reducing that will always work when the answercannot be spotted by eye. It is particularly useful when there arelarge numbers of bonds involved.The vibrational modes of the molecule are reduced to produce areducible representation into the irreducible representations. Thismethod uses the following formula reduction formula:N 1χ rx χ ix n x h xN is the number of times a symmetry species occurs in the reduciblerepresentation,h is the ‘order of the group’: simply the total number of symmetry operations inthe group.The summation is over all of the symmetry operations. For each symmetryoperation, three numbers are multiplied together. These are:Χr is the character for a particular class of operation in the irreduciblerepresentationΧi is the character of the irreducible representation.n is the number of symmetry operations in the classThe characters of the reducible representation can be determinedby considering the combined effect of each symmetry operation onzthe atomic vectors.zAtomic contributions, by symmetry operations,to the reducible representation for the 3Ndegrees of freedom for a molecule.OperationContributionper yyxxzxiyxyz*Cn 1 2cos(360/n); Sn -1 2cos(360/n)30

zDerivation of reducible representation for degrees of freedom in H2OUnshifted Oσ(yz)3HaHbHaHbObtain the reducible representation (for H2O)by multiplying the number of unshifted atomstimes the contribution per atom.EUnshifted Atoms3Contribution per atom 39ΓtotC21-1-1σv(xz)111σv(yz)31331

Reducible RepresentationsThe reduction can be achieved using the reduction formula. It is amathematical way of reducing that will always work when the answercannot be spotted by inspection. It is particularly useful when thereare large numbers of atoms and bonds involved.The vibrational modes of the molecule are reduced to produce areducible representation into the irreducible representations. Thismethod uses the following formula reduction formula:N 1χ rx χ ix n x h xN is the number of times a symmetry species occurs in the reduciblerepresentation,h is the ‘order of the group’: simply the total number of symmetry operations inthe group.The summation is over all of the symmetry operations. For each symmetryoperation, three numbers are multiplied together. These are:Χr is the character for a particular class of operation in the reduciblerepresentationΧi is the character of the irreducible representation.n is the number of symmetry operations in the classTabulate our known information.Reducible Representation (for H2O)1)ΓrEC2σv(xz)σv(yz)9-113Character Table2)irrepC2vE C2 σv(xz) σv(yz)A1111A211–1B11-1B21 –11 zh 4x2, y2, z2–1 Rzxy1–1 x, Ryxz–11 y, RxyzN 1χ rx χ ix n x h xA1: (1/h)[(χrΕ)(χiΕ)(nΕ) (χrC2)(χiC2)(nC2) (χr σv(xz))(χiσv(xz))(nσv(xz)) (χr σv(yz))(χiσv(yz))(nσv(yz))A : (1/4)[(9)(χ Ε)(nΕ) (-1)(χ C2)(nC2) (1)(χ σv(xz))(nσv(xz)) 1i(3)(χiσv(yz))(nσv(yz))iiA1: (1/4)[(9)(1)(nΕ) (-1)(1)(nC2) (1)(1)(nσv(xz)) (3)(1)(nσv(yz))A1: (1/4)[(9)(1)(1) (-1)(1)(1) (1)(1)(1) (3)(1)(1) 332

Calculate irreducible representation A2A2: (1/h)[(χrΕ)(χiΕ)(nΕ) (χrC2)(χiC2)(nC2) (χr σv(xz))(χiσv(xz))(nσv(xz)) (χ σv(yz))(χ σv(yz))(nσv(yz))riA2: (1/4)[(9)(1)(1) (-1)(1)(1) (1)(-1)(1) (3)(-1)(1) 1Calculate irreducible representation B1B1: (1/4)[(9)(1)(1) (-1)(-1)(1) (1)(1)(1) (3)(-1)(1) 2Calculate irreducible representation B2B2: (1/4)[(9)(1)(1) (-1)(-1)(1) (1)(-1)(1) (3)(1)(1) 3The reducible representation ΓrEC2σv(xz)σv(yz)9-113 is resolved into three A1, one A2, two B1, and three B2 species.C2vE C2 σv(xz) σv(yz)A1111A211–1B11-1B21 –1Γtot1 zx2, y2, z2–1 Rzxy1–1 x, Ryxz–11 y, Rxyz 3A1 A2 2B1 3B2-[Γtrans A1 B1 B2]-[Γrot A2 Γvib 2A1 B2B1 B2]Notice this is the same result weobtained by analyzing the symmetriesof the vibrational modes.-each mode is IR active33

Consider only the OH stretches in H2O-consider the number of unchanged O-H bonds under the symmetryoperations of the point groupC2vE C2 σv(xz) σv(yz)Unchanged OH bonds 2CharacterTable00ReducibleRepresentation2C2vE C2 σv(xz) σv(yz)A1111A211–1B11-1B21 –1x2, y2, z21 z–1 Rzxy1–1 x, Ryxz–11 y, RxyzBy inspection, the reducible representation is composed of the A1and B2 representation.E C2 σv(xz) σv(yz)A1 1111B2 1-1-11Sum of rows 2002General Method: Determine OH stretching in H2O usingreducible representations and reduction formulaC2vEC2σv(xz)σv(yz)111002Coefficient1Order of group4Unchanged bonds (OH)2C2vE C2 σv(xz) σv(yz)A1111A211–1B11-1B21 –11 zx2, y2, z2–1 Rzxy1–1 x, Ryxz–11 y, RxyzUsing reducible representations and the reductionformula, one obtains A1 B2 modes.34

Derive Γtot for BCl3 given the character table for D3hDerive the number of vibrational modes and assign modes for BCl3.E2C33C2σh2S33σvUnshifted atoms412412Contribution per atom30-11-21Γtot120-24-22D 3hE2C 33C 2σΞhh2S 3A -1-1-2-1-11-110E′A 1′′A′′2E ′′x 2 y 2, z 2Rz(x , y )(x 2 – y 2, 2xy )z(R x , R y ) (xy , yz )Results of using the reduction formula.xxxΧr *Χi *nA 1′A ′2E′A 1′′A ′′2E ′′121224121224000000-6 46 40 8-6 -46 -40 fore, we have determinedΓtot A1′ A ′2 3 E′ 2 A ′′2 E ′′but, subtract off the translational representations.E′ A ′′2 ]-[Γtrans and subtract off the rotational representations.-[Γrot Γvib E ′′ 2 E′ A ′′2 ]A ′2A1′]Each E’representationdescribes twovibrationalmodes of equalenergy.35

Symmetricalstretching.Raman active.Out-of-planebending mode.IR active.Unsymmetricalstretching.In-planebending mode.Raman and IR active.We can use isotopic substitution to interpret spectra, since thecharacteristic frequency of the mode will depend on the masses ofthe atoms moving in that mode.Review: What do I do when I need to ?Assign symmetry labels to vibrational modes? If the vibrational mode is known and illustrated, sketch the resultingvibrational mode before and after each symmetry operation of thepoint group. Using the character table, assign the symmetry label andidentify if the mode is IR and/or Raman active.Determine the symmetries of all vibrational modes and if the modesare IR and/or Raman active? Determine how many atoms are left unchanged by each symmetryoperation. Find the reducible representation and reduce into theirreps. Subtract translational and rotational modes Identify whichmodes are IR and/or Raman active. Determine the symmetries of only the stretching modes and if themodes are IR and/or Raman active? Determine how many bonds are left unchanged by each symmetryoperation. Find the reducible representation and reduce into theirreps. Identify which are IR and/or Raman active. Develop a character table? Determine the effect of each symmetry operation on the x, y, ztranslation and the rotation Rx, Ry, and Rz. The resulting set ofcharacters correspond to an irrep in the character table.36

Determine the number of and assign the vibrational modes of the following:How many peaks in the (1) IR spectra and (2) Raman spectra1. NH32. CH43. [PtCl4]24. SF65. SF5ClDetermine number of CO stretching modes in1. Mn(CO)62. Mn(CO)5Cl3. trans-Mn(CO)4Cl24. cis-Mn(CO)4Cl25. fac-Mn(CO)3Cl36. mer-Mn(CO)3Cl337

Symmetry Point Groups Symmetry of a molecule located on symmetry axes, cut by planes of symmetry, or centered at an inversion center is known as point symmetry . Collections of symmetry operations constitute mathematical groups . Each symmetry point group has a particular designation. Cn, C nh, C nv Dn, D nh, D nd S2n C v ,D h