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336 10MOLECULAR SYMMETRY
P10B.8 Performing the matrix multiplications gives
σxσx = (0 11 0)(0 11 0) = (1 00 1) = σ0 σxσy = (0 11 0)(0 −ii 0 ) = ( i 00 −i) = iσz
σxσz = (0 11 0)(1 00 −1) = (0 −11 0 ) = −iσy σxσ0 = (0 11 0)(1 00 1) = (0 11 0) = σx
σyσx = (0 −ii 0 )(0 11 0) = (−i 00 i ) = −iσz σyσy = (0 −ii 0 )(0 −ii 0 ) = (1 00 1) = σ0
σyσz = (0 −ii 0 )(1 00 −1) = (0 ii 0) = iσx σyσ0 = (0 −ii 0 )(1 00 1) = (0 −ii 0 ) = σy
σzσx = (1 00 −1)(0 11 0) = ( 0 1−1 0) = iσy σzσy = (1 00 −1)(0 −ii 0 ) = ( 0 −i−i 0 ) = −iσx
σzσz = (1 00 −1)(1 00 −1) = (1 00 1) = σ0 σzσ0 = (1 00 −1)(1 00 1) = (1 00 −1) = σz
σ0σx = (1 00 1)(0 11 0) = (0 11 0) = σx σ0σy = (1 00 1)(0 −ii 0 ) = (0 −ii 0 ) = σy
σ0σz = (1 00 1)(1 00 −1) = (1 00 −1) = σz σ0σ0 = (1 00 1)(1 00 1) = (1 00 1) = σ0
�e RR′ multiplication table is therefore
R ↓ R′ → σx σy σz σ0
σx σ0 iσz −iσy σx
σy −iσz σ0 iσx σy
σz iσy −iσx σ0 σz
σ0 σx σy σz σ0
�e four matrices do not form a group under multiplication. �is is because
they do not satisfy criterion 3 in Section 10B.1 on page 397, that the combination
RR′ must be equivalent to a single member of the collection. For example,
σxσy = iσz , but iσz is not one of the four matrices.
P10B.10 �e twowavefunctions are shown schematically in Fig. 10.16, where the shading
indicates the sign of the wavefunction.
It is clear from Fig. 10.16 that a C+4 rotation, taken to be 90○ anticlockwise,
converts ψ2,3 into ψ3,2 and ψ3,2 into −ψ2,3: (ψ3,2 −ψ2,3 )← (ψ2,3 ψ3,2 ). Sim-
ilarly C−4 transforms (−ψ3,2 ψ2,3 )← (ψ2,3 ψ3,2 ), C2 changes the sign of both
wavefunctions, while E leaves both unchanged. Writing these using matrix
multiplication gives