486 A Point Group Character Tables Table A.30.Character table for group O(cubic) 0(432) E 8C3 3C2=3C 6C 6C4 (x2+y2+z2) A1 1 1 1 1 A2 1 1 -1 -1 (x2-y2,3z2-r2) E 2 -1 0 0 (R=;Ry;R:) T 3 0 -1 -1 1 (c,y,2) (ty,y2,2x) T2 0 -1 -1 Oh=O⑧i,(m3m)(cubic) Table A.31.Character table for the cubic group Oh(cubic) repr.basis functions E3C6C46C码8C3i3iC6iC46iC吗 8iCs A时 1 111111111 1 x(2-z2)+ A y(z2-x2)+ 11-1-111 1-1 -1 1 (z4(x2-y2) E+ ∫x2-2 222-x2-y2 2200-12 20 0 -1 T E,y,2 3-11-1 0-3 1-1 1 0 T (x2-y2). 3-1-1 1 0-3 1 1 -1 0 xyz[4(y2-22)+ A (22-x2)+ 11111-1 -1-1 -1 -1 z4(x2-y2】 A 工y2 11-1-11-1-111 -1 E xyz(x2-y2)... 2200-1-2-2 00 1 T xy(a2-y2)... 3-11-103-11-1 0 T xy,yz,zx 3-1-1103-1-11 0 tThe basis functions for T2 are z(2-y2),x(y2-22),y(22-22),for E-are xyz(x2-y2),xyz(322-s2)and for Tf are ry(r2-y2),yz(y2-22),zx(22-x2) Table A.32.Character table for group Ta(cubic)a Ta(④3m) E 8C3 3C2 6od 6S4 x2+y2+z2 A 1 1 1 1 1 A2 1 1 1 -1 -1 (x2-y2,322-r2) E 2 -1 2 0 0 (R=;Ry;R=) T 2 0 -1 -1 1 y22t,xy） (x,,2) T 3 -1 1 -1 a Note that (yz,z,ry)transforms as representation Ti486 A Point Group Character Tables Table A.30. Character table for group O (cubic) O (432) E 8C3 3C2 = 3C2 4 6C 2 6C4 (x2 + y2 + z2) A1 1 1 111 A2 11 1 −1 −1 (x2 − y2, 3z2 − r2) E 2 −1 200 (Rx, Ry, Rz) (x, y, z) " T1 3 0 −1 −1 1 (xy, yz, zx) T2 3 0 −1 1 −1 Oh = O ⊗ i, (m3m) (cubic) Table A.31. Character table for the cubic group Oh (cubic)† repr. basis functions E 3C2 4 6C4 6C 2 8C3 i 3iC2 4 6iC4 6iC 2 8iC3 A+ 1 1 1 1 1 1 11 1 1 1 1 A+ 2 ⎧ ⎨ ⎩ x4(y2 − z2)+ y4(z2 − x2)+ z4(x2 − y2) 1 1 −1 −1 11 1 −1 −1 1 E+ ( x2 − y2 2z2 − x2 − y2 2200 −12 2 0 0 −1 T − 1 x, y, z 3 −1 1 −1 0 −3 1 −11 0 T − 2 z(x2 − y2)... 3 −1 −110 −311 −1 0 A− 1 ⎧ ⎨ ⎩ xyz[x4(y2 − z2)+ y4(z2 − x2)+ z4(x2 − y2)] 11111 −1 −1 −1 −1 −1 A− 2 xyz 1 1 −1 −1 1 −1 −111 −1 E− xyz(x2 − y2). . . 2 2 0 0 −1 −2 −200 1 T + 1 xy(x2 − y2). . . 3 −1 1 −1 03 −1 1 −1 0 T + 2 xy, yz, zx 3 −1 −1 1 03 −1 −11 0 † The basis functions for T − 2 are z(x2 − y2), x(y2 − z2), y(z2 − x2), for E− are xyz(x2 − y2), xyz(3z2 − s2) and for T + 1 are xy(x2 − y2), yz(y2 − z2), zx(z2 − x2) Table A.32. Character table for group Td (cubic)a Td (43m) E 8C3 3C2 6σd 6S4 x2 + y2 + z2 A1 1 1111 A2 111 −1 −1 (x2 − y2, 3z2 − r2) E 2 −1 200 (Rx, Ry, Rz) yz, zx, xy) " T1 3 0 −1 −1 1 (x, y, z) T2 3 0 −1 1 −1 a Note that (yz, zx, xy) transforms as representation T1