1Group Theory and Molecular Symmetry Molecular Symmetry Symmetry Elements and Operations Identity element – E - Apply E to object and nothing happens. Object is unmoved. Rotation axis – Cn - Rotation of object through 360°/n yields an indistinguishable object. C axis2 C axis3 C axis4 - 22CC E⋅=, 333CCC E⋅⋅=, etc … - The Cn axis with the highest n is an object’s principal axis. Reflection plane - σ - vertical - σv - plane is parallel to principal axis - 1,1,1 trichloroethane (eclipsed or staggered) is molecule with only σv Cl HCl HCl HClHClHClH - horizontal - σh - plane is perpendicular to principal axis - ethane (eclipsed) is molecule with σh - dihedral - σd - plane is parallel to principal axis and bisects angle between two C2 axes. - staggered ethane is molecule with σd - Examples of composite operations - Eσ⋅σ= , d2CEσ⋅ = , 4V 2CC⊥⋅σ= Center of inversion – i - turns molecule “inside out” - make following replacements xx→− , yy→− and zz→− - staggered ethane has i, eclipsed ethane does not.2Improper rotation axes – Sn - reflection followed by rotation through 360°/n - object may have Sn without having σh or Cn separately. - Allene is an example. CCCHHHH Symmetry Groups - all symmetry operations on a molecule form a closed group called a symmetry group. Example: C2v C2v E C2 σv σv' E E C2 σv σv' C2 C2 E σv' σv σv σv σv' E C2 σv' σv' σv C2 E OHHσvσv’C2 Special Groups C1 – has no symmetry NCCHClHHHOO Ci – has only inversion symmetry BrClBrCl Cs – has only mirror plane ClBrBrCl3Cn Groups Cn – has E and (n-1) Cn symmetry elements CCHOHHOOHHH C3 Cnv – has E, (n-1) Cn and n σv symmetry elements NHHH C3v Cnh – has E, (n-1) Cn, (n-1) Sn, σh, et. al. symmetry elements HClClH C2h Dn Groups – similar to Cn, but has n C2 axes perpendicular to Cn axis Dn – has E and (n-1) Cn, n C2 axes ⊥ principal axis, et. al. symmetry elements HHHHHH D3 Dnh – has E, (n-1) Cn, n σv, σh et. al. symmetry elements Ru D5h Dnd – has E, (n-1) Cn, σv, σd, et. al. symmetry elements HHHHHH D3d neither staggered nor eclipsed4Sn groups Sn – has E, (n-1) Sn symmetry elements Cubic groups T, Td, Th – tetrahedral groups SiFFFF Td O, Oh – octahedral groups CoClCl ClClClCl Oh Icosahedral group Ih – icosahedral group Buckminsterfullerene Linear groups C∞v - Heteronuclear diatomic molecules are most important example. D∞h - Homonuclear diatomic molecules are most important example. Immediate Consequences of Symmetry 1. Polarity - Molecule cannot have dipole moment perpendicular to Cn axis. - Molecule may have dipole moment parallel to principal axis if no C2 axes are perpendicular to principal axis - Molecule with Sn axis may not have dipole moment. - Thus only C1, Cs, Cn and Cnv molecules can have dipole moment. 2. Chirality - Molecule without Sn axis may be chiral - A chiral molecule cannot have center of inversion.5Representations of Groups Recall that the symmetry elements for a C2v molecule form a group that, by definition, is closed under multiplication. C2v E C2 σv σv' E E C2 σv σv' C2 C2 E σv' σv σv σv σv' E C2 σv' σv' σv C2 E OHHσvσv’C2 Rather than relying on our ability to visualize the spatial changes that occur during a pair of symmetry operator, we would like to use numerical representations for the group. Consider the following vector to represent the nine coordinates that describe the position of a water molecule. 111222333HxHyHzOxvOyOzHxHyHz⎛⎞⎜⎟⎜⎟⎜⎟⎜⎟⎜⎟⎜⎟=⎜⎟⎜⎟⎜⎟⎜⎟⎜⎟⎜⎟⎝⎠ If we wanted to represent the identity element with a matrix, the following matrix E would be a sensible choice. Note the effect of multiplying our coordinate vector v by the matrix E. Indeed, E is an identity element. 100000000010000000001000000000100000E000010000000001000000000100000000010000000001⎛⎞⎜⎟⎜⎟⎜⎟⎜⎟⎜⎟⎜⎟=⎜⎟⎜⎟⎜⎟⎜⎟⎜⎟⎜⎟⎝⎠ 111111222222333333Hx Hx100000000Hy Hy010000000Hz Hz001000000Ox Ox000100000E v Oy Oy000010000Oz Oz000001000Hx Hx000000100Hy Hy000000010Hz Hz000000001⎛⎞⎛⎛⎞⎜⎟⎜⎜⎟⎜⎟⎜⎟⎜⎟⎜⎟⎜⎟⎜⎟⎜⎟⎜⎟⎜⎟⎜⎟⋅= ⋅ =⎜⎟⎜⎟⎜⎟⎜⎟⎜⎟⎜⎟⎜⎟⎜⎟⎜⎟⎜⎟⎜⎟⎜⎟⎝⎠⎝⎠⎝⎞⎟⎜⎟⎜⎟⎜⎟⎜⎟⎜⎟⎜⎟⎜⎟⎜⎟⎜⎟⎜⎟⎜⎟⎠6With the C2 rotation, H1 and H3 switch places and the x and y directions change sign. The following matrix represents the C2 rotation. 20 000 00 1000 000 000 100 000 000 01000100000C0000 100000000010001000 000 000 100 000 00001000000−⎛⎞⎜⎟−⎜⎟⎜⎟⎜⎟−⎜⎟⎜⎟=−⎜⎟⎜⎟⎜⎟−⎜⎟−⎜⎟⎜⎟⎝⎠131313222 2222333HxHx0 000 00 100HyHy0 000 000 10HzHz0 000 000 01OxOx0 00 1000 00Cv OyOy0000 10000OzOz000001000HxH1000 000 00Hy0 100 000 00Hz0 010 000 00−−⎛⎞⎛⎞⎜⎟⎜⎟−−⎜⎟⎜⎟⎜⎟⎜⎟⎜⎟⎜⎟−−⎜⎟⎜⎟⎜⎟⎜⎟⋅= =−−⎜⎟⎜⎟⎜⎟⎜⎟⎜⎟⎜⎟−−⎜⎟⎜⎟−⎜⎟⎜⎟⎜⎟⎜⎟⎝⎠⎝⎠111xHyHz⎛⎞⎜⎟⎜⎟⎜⎟⎜⎟⎜⎟⎜⎟⎜⎟⎜⎟⎜⎟⎜⎟−⎜⎟⎜⎟⎝⎠ Thus vσ and v′σ also have 9 × 9 matrix representations. v0000001000000000 100000000010001000000000 100000000010001000000000 10000000001000000⎛⎞⎜⎟−⎜⎟⎜⎟⎜⎟⎜⎟⎜⎟σ=−⎜⎟⎜⎟⎜⎟⎜⎟−⎜⎟⎜⎟⎝⎠ 13131322v 22223133HxHx000000100HyHy0000000 10HzHz000000001OxOx000100000vOyOy0000 10000OzOz000001000HxHx100000000HyHy0 10000000Hz001000000⎛⎞⎛⎞⎜⎟⎜⎟−−⎜⎟⎜⎟⎜⎟⎜⎟⎜⎟⎜⎟⎜⎟⎜⎟⎜⎟⎜⎟σ⋅ = ⋅ =−−⎜⎟⎜⎟⎜⎟⎜⎟⎜⎟⎜⎟⎜⎟⎜⎟−−⎜⎟⎜⎟⎜⎟⎜⎟⎝⎠⎝⎠11Hz⎛⎞⎜⎟⎜⎟⎜⎟⎜⎟⎜⎟⎜⎟⎜⎟⎜⎟⎜⎟⎜⎟⎜⎟⎜⎟⎝⎠ v100 0 00 0 00010000000001000000000 100000000010000000001000000000 100000000010000000001−⎛⎞⎜⎟⎜⎟⎜⎟⎜⎟−⎜⎟⎜⎟′σ=⎜⎟⎜⎟⎜⎟−⎜⎟⎜⎟⎜⎟⎝⎠ 11111122v 22223333Hx Hx100000000Hy Hy010000000Hz Hz001000000Ox Ox000 100000vOyOy000010000Oz Oz000001000Hx Hx000000 100Hy H000000010Hz000000001−−⎛⎞⎛⎞⎜⎟⎜⎟⎜⎟⎜⎟⎜⎟⎜⎟⎜⎟⎜⎟−−⎜⎟⎜⎟⎜⎟⎜⎟′σ⋅ = ⋅