Iodine pentafluoride,

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Group theory and symmetry · Group theory and symmetry

Invariance, Ch Representations, Schur's lemma and characters

Problem

Iodine pentafluoride, IF5\mathrm{IF_5}, is a square-pyramidal molecule of point group C4vC_{4v}: the iodine sits above a square of four equivalent basal fluorines, with one apical fluorine on the fourfold axis. The classes of C4vC_{4v}, with sizes, are E(1)E(1), 2C4(2)2C_4(2), C2(1)C_2(1), 2σv(2)2\sigma_v(2) (mirror planes each containing the axis and two opposite basal fluorines) and 2σd(2)2\sigma_d(2) (mirror planes bisecting the angle between neighbouring basal fluorines). Its irreducible representations have characters on (E,2C4,C2,2σv,2σd)(E,2C_4,C_2,2\sigma_v,2\sigma_d): A1A_1: (1,1,1,1,1)(1,1,1,1,1) A2A_2: (1,1,1,−1,−1)(1,1,1,-1,-1) B1B_1: (1,−1,1,1,−1)(1,-1,1,1,-1) B2B_2: (1,−1,1,−1,1)(1,-1,1,-1,1) EE: (2,0,−2,0,0)(2,0,-2,0,0) Let Γ\Gamma be the representation generated by the permutation of the four basal fluorine labels. ((a)) Find χ(g)\chi(g) on each class for Γ\Gamma: C4C_4 and C2C_2 leave no basal fluorine fixed; each σv\sigma_v fixes the two basal fluorines it contains and swaps the other two; each σd\sigma_d leaves none fixed. ((b)) Decompose Γ\Gamma.

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