äÓíÍÇ ÉLÉÉÉâÉNÉ^Å[ÉeÅ[ÉuÉã.pdf
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1 1 V 2 D 3 Y 4 W (a, b, c D SHG SFG Y CARS, CSRS, THG W SFG SHG CARS 4 2 D W D abba D ab * SFG (page point group) (page point group) 2 C s *, C i, C 2 * 12 D 4h 3 C 2v, * C 2h * 13 D 6h 4 D 2, D 2h 14 C 3 *,C 4 * 5 C 3v *, D 3 * 15 C 6 *,S 4 * 6 C 5v *, C 4v * 16 S 6,C 3h * 7 D 4 *, D 2d * 17 C 4h, C 6h 8 C 6v *, D 6 * 18 T d *, T h 9 D 3d *, D 4d 19 T*, O 10 D 5d, D 6d 20 O h 11 D 3h *, D 5h 21 C v * 22 D h - 1
2 C s E σ xy A' A" +1-1 T x, T y R x, R y x 2, y 2, z 2, xy xz, yz C s HN 3 A' V 1b, V 1a, D (0) 0, D (1) 0, D (2) 0, D (2) 2b, D (2) 2a, Y (1) 1b, Y (1) 1a, Y (2) 1b, Y (2) 1a, Y (3) 1b, Y (3) 1a, Y (3) 3b, Y (3) 3a W (0) 0, W (1) 0, W (2) 0, W (2) 2b, W (2) 2a, W (3) 0, W (3) 2b, W (3) 2a, W (4) 0, W (4) 2b, W (4) 2a, W (4) 4b, W (4) 4a A" V 0, D (1) 1b, D (1) 1a, D (2) 1b, D (2) 1a, Y (0) 0, Y (1) 0, Y (2) 0, Y (3) 0, Y (2) 2b, Y (2) 2a, Y (3) 2b, Y (3) 2a W (1) 1b, W (1) 1a, W (2) 1b, W (2) 1a, W (3) 1b, W (3) 1a, W (3) 3b, W (3) 3a, W (4) 1b, W (4) 1a, W (4) 3b, W (4) 3a β aaa, β bbb, β acc, β bcc, β abb, β aab, β cac, β cca, β cbc, β ccb, β bab, β bba, β aba, β baa, β aaa, β aaa, γ aaaa, γ bbbb, γ cccc, γ aabb, γ bbcc, γ ccaa, γ bbaa, γ ccbb, γ aacc, γ abba, γ bccb, γ caab, γ baab, γ cbbc, γ acca, γ abab, γ baba, γ bcbc, γ cbcb, γ caca, γ acac, γ aaab, γ bbab, γ ccab, γ abaa, γ abbb, γ abcc, γ aaba, γ bbba, γ ccba, γ baaa, γ babb, γ bacc, γ accb, γ cabc, γ bcca, γ cbac, γ acbc, γ cacb, γ bcac, γ cbcac, γ caac, γ bbba R, R, R all components C i E I A g A u +1-1 T x, T y, x y z C i trans-cfh 2 -CFH 2 without internal rotation A g D (0) 0, D (1) 0, D (1) 1b, D (1) 1a, D (2) 0, D (2) 1b, D (2) 1a, D (2) 2b, D (2) 2a, W (0) 0, W (1) 0, W (1) 1b, W (1) 1a, W (2) 0,W (2) 1b, W (2) 1a, W (2) 2b, W (2) 2a, W (3) 0, W (3) 1b, W (3) 1a, W (3) 2b, W (3) 2a, W (3) 3b, W (3) 3a, W (4) 0, W (4) 1b, W (4) 1a, W (4) 2b, W (4) 2a, W (4) 3b, W (4) 3a, W (4) 4b, W (4) 4a A u V 1b, V 1a, V 0, Y (0) 0, Y (1) 0, Y (1) 1b, Y (1) 1a, Y (2) 0, Y (2) 1b, Y (2) 1a, Y (2) 2b, Y (2) 2a, Y (3) 0, Y (3) 1b, Y (3) 1a, Y (3) 2b, Y (3) 2a, Y (3) 3b, Y (3) 3a, ( γ') C 2 E C 2z A B +1-1 T x, T y R x, R y x 2, y 2, z 2, xy xz, yz C 2 H 2 O 2, H 2 S 2 (Two HOO/HSS planes are perpendicular.) A V 1a, D (0) 0, D (1) 1a, D (2) 0, D (2) 1b, D (2) 2b, Y (0) 0, Y (1) 1a, Y (2) 0, Y (2) 1b, Y (2) 2b, Y (3) 1a, Y (3) 2a, Y (3) 3a, W (0) 0, W (1) 1a, W (2) 0, W (2) 1b, W (2) 2b, W (3) 1a, W (3) 2a, W (3) 3a, W (4) 0, W (4) 1b, W (4) 2b, W (4) 3b, W (4) 4b B V 1b, V 0, D (1) 0, D (1) 1b, D (2) 1a, D (2) 2a, Y (1) 0, Y (1) 1b, Y (2) 1a, Y (2) 2a, Y (3) 0, Y (3) 1b, Y (3) 2b, Y (3) 3b W (1) 0, W (1) 1b, W (2) 1a, W (2) 2a, W (3) 0, W (3) 1b, W (3) 2b, W (3) 3b, W (4) 1a, W (4) 2a, W (4) 3a, W (4) 4a β bbb, β baa, β aab, β aba, β bcc, β ccb, β cbc, β abc, β bac, β bca, β cba, β cab, β acb, γ aaaa, γ bbbb, γ cccc, γ aabb, γ bbcc, γ ccaa, γ bbaa, γ ccbb, γ aacc, γ abba, γ bccb, γ caac, γ baab, γ cbbc, γ acca, γ abab, γ baba, γ bcbc, γ cbcb, γ caca, γ acac, γ aaca, γ bbca, γ ccca, γ caaa, γ cabb, γ cacc, γ aaac, γ bbac, γ ccac, γ acaa, γ acbb, γ accc, γ cbba, γ bcab, γ abbc, γ bacb, γ cbab, γ bcba, γ abcb, γ babc - 2
3 C 2v E C 2z σ v (yz) σ v (xz) A A B B T y T x R x R y x 2, y 2, z 2 xy yz xz C 2v H 2 O, A 1 V 1a, D (0) 0, D (2) 0, D (2) 2b, Y (1) 1a, Y (2) 1b, Y (3) 1a, Y (3) 3a, W (0) 0, W (2) 0, W (2) 2b, W (3) 2a, W (4) 0, W (4) 2b, W (4) 4b A 2 D (1) 1a, D (2) 1b, Y (0) 0, Y (2) 0, Y (2) 2b, Y (3) 2a, W (1) 1a, W (2) 1b, W (3) 1a, W (3) 3a, W (4) 1b, W (4) 3b B 1 V 0, D (1) 1b, D (2) 1a, Y (1) 0, Y (2) 2a, Y (3) 0, Y (3) 2b, W (1) 1b, W (2) 1a, W (3) 1b, W (3) 3b, W (4) 1a, W (4) 3a B 2 V 1b, D (1) 0, D (2) 2a, Y (1) 1b, Y (2) 1a, Y (3) 1b, Y (3) 3b, W (1) 0, W (2) 2a, W (3) 0, W (3) 2b, W (4) 2a, W (4) 4a β bbb, β baa, β bcc, β aab, β ccb, β aba, β cbc γ aaaa, γ bbbb, γ cccc, γ aabb, γ bbcc, γ ccaa, γ bbaa, γ ccbb, γ aacc, γ abba, γ bccb, γ caac, γ baab, γ cbbc, γ acca, γ abab, γ baba, γ bcbc, γ cbcb, γ caca, γ acac C 2h E C 2z σ h (xy) I A g A u B g B u T x, T y R x, R y x 2, y 2, z 2, xy xz, yz C 2h trans-dichloroethylene without internal rotation A g D (0) 0, D (1) 0, D (2) 0, D (2) 2a, D (2) 2b, W (0) 0, W (1) 0, W (2) 0, W (2) 2a, W (2) 2b, W (3) 0, W (3) 2b, W (3) 2a, W (4) 0, W (4) 2b, W (4) 2a, W (4) 4b, W (4) 4a A u V 0, Y (0) 0, Y (1) 0, Y (2) 0, Y (2) 2b, Y (2) 2a, Y (3) 0, Y (3) 2b, Y (3) 2a B g D (1) 1b, D (1) 1a, D (2) 1b, D (2) 1a, W (1) 1b, W (1) 1a, W (2) 1b, W (2) 1a, W (3) 1b, W (3) 1a, W (3) 3b, W (3) 3a, W (4) 1b, W (4) 1a, W (4) 3b, W (4) 3a B u V 1b, V 1a, Y (1) 1b, Y (1) 1a, Y (2) 1b, Y (2) 1a, Y (3) 1b, Y (3) 1a, Y (3) 3b, Y (3) 3a γ aaaa, γ bbbb, γ cccc, γ aabb, γ bbcc, γ ccaa, γ bbaa, γ ccbb, γ aacc, γ abba, γ bccb, γ caac, γ baab, γ cbbc, γ acca, γ abab, γ baba, γ bcbc, γ cbcb, γ caca, γ acac, γ aaab, γ bbab, γ ccab, γ abaa, γ abbb, γ abcc, γ aaba, γ bbba, γ ccba, γ baaa, γ babb, γ bacc, γ accb, γ cabc, γ bcca, γ cbac, γ acbc, γ cacb, γ bcac, γ cbca - 3
4 D 2 E C 2z C 2y C 2x A B B B T y T x R y R x x 2, y 2, z 2 xy xz yz D 2 A D (0) 0, D (2) 0, D (2) 2b, Y (0) 0, Y (2) 0, Y (2) 2b, Y (3) 2a, W (0) 0, W (2) 0, W (2) 2b, W (3) 2a, W (4) 0, W (4) 2b, W (4) 4b B 1 V 0, D (1) 0, D (2) 2a, Y (1) 0, Y (2) 2a, Y (3) 0, Y (3) 2b, W (1) 0, W (2) 2a, W (3) 0, W (3) 2b, W (4) 2a, W (4) 4a B 2 V 1a, D (1) 1a, D (2) 1b, Y (1) 1a, Y (2) 1b, Y (3) 1a, Y (3) 3a, W (1) 1a, W (2) 1b, W (3) 1a, W (3) 3a, W (4) 1b, W (4) 3b B 3 V 1b, D (1) 1b, D (2) 1a, Y (1) 1b, Y (2) 1a, Y (3) 1b, Y (3) 3b, W (1) 1b, W (2) 1a, W (3) 1b, W (3) 3b, W (4) 1a, W (4) 3a β abc, β bca β cab, β cba, β cab, β acb γ aaaa, γ bbbb, γ cccc, γ aabb, γ bbcc, γ ccaa, γ bbaa, γ ccbb, γ aacc, γ abba, γ bccb, γ caac, γ baab, γ cbbc, γ acca, γ abab, γ baba, γ bcbc, γ cbcb, γ caca, γ acac D 2h E C 2z C 2y C 2x I σ xy σ xz σ yz A g A u B 1g B 1u B 2g B 2u B 3g B 3u T y T x R y R x x 2, y 2, z 2 xy xz yz D 2h ethylene without internal rotation A g D (0) 0, D (2) 0, D (2) 2b, W (0) 0, W (2) 0, W (2) 2b, W (3) 2a, W (4) 0, W (4) 2b, W (4) 4b A u Y (0) 0, Y (2) 0, Y (2) 2b, Y (3) 2a B 1g D (1) 1b, D (2) 1a, W (1) 1b, W (2) 1a, W (3) 1b, W (3) 3b, W (4) 1a, W (4) 3a B 1u V 1b, Y (1) 1b, Y (2) 1a, Y (3) 1b, Y (3) 3b B 2g D (1) 1a, D (2) 1b, W (1) 1a, W (2) 1b, W (3) 1a, W (3) 3a, W (4) 1b, W (4) 3b B 2u V 1a, Y (1) 1a, Y (2) 1b, Y (3) 1a, Y (3) 3a B 3g D (1) 0, D (2) 2a, W (1) 0, W (2) 2a, W (3) 0, W (3) 2b, W (4) 2a, W (4) 4a B 3u V 0, Y (1) 0, Y (2) 2a, Y (3) 0, Y (3) 2b γ aaaa, γ bbbb, γ cccc, γ aabb, γ bbcc, γ ccaa, γ bbaa, γ ccbb, γ aacc, γ abba, γ bccb, γ caac, γ baab, γ cbbc, γ acca, γ abab, γ baba, γ bcbc, γ cbcb, γ caca, γ acac - 4
5 C 3v E 2C 3z 3σ v A A E ; C 3v CH 3 X A 1 V 0, D (0) 0, D (2) 0, Y (1) 0, Y (3) 0, Y (3) 3b, W (0) 0, W (2) 0, W (3) 3a, W (4) 0, W (4) 3b A 2 D (1) 0, Y (0) 0, Y (2) 0, Y (3) 3a, W (1) 0, W (3) 0, W (3) 3b, W (4) 3a E (V 1b,V 1a ), (D (1) 1b, D (1) 1a), (D (2) 1b, D (2) 1a), (D (2) 2b, D (2) 2a), (Y (1) 1b, Y (1) 1a), (Y (2) 1b, Y (2) 1a), (Y (2) 2b, Y (2) 2a), (Y (3) 1b, Y (3) 1a), (Y (3) 2b, Y (3) 2a), (W (1) 1b, W (1) 1a), (W (2) 1b, W (2) 1a), (W (2) 2b, W (2) 2a), (W (3) 1b, W (3) 1a), (W (3) 2b, W (3) 2a), (W (4) 1b, W (4) 1a), (W (4) 2b, W (4) 2a), (W (4) 4b, W (4) 4a) β ccc, β aac = β bbc, β aca = β bcb, β caa = β cbb, β aaa = -β abb = -β bba = -β bab, γ abba = γ baab, γ acca = γ bccb, γ caac = γ cbbc, γ aaca = -γ bbcb, γ caaa = -γ cabb = -γ cbab = -γ cbba, γ aaac = -γ bbac = -γ abbc = -γ babc, γ acaa = -γ acbb = -γ bcab = -γ bcba, D 3 E 2C 3z 3 C 2 A A E ; D 3 A 1 D (0) 0, D (2) 0, Y (0) 0, Y (2) 0, Y (3) 3b, W (0) 0, W (2) 0, W (3) 3b, W (4) 0, W (4) 3a A 2 V 0, D (1) 0, Y (1) 0, Y (3) 0, Y (3) 3a, W (1) 0, W (3) 0, W (3) 3a, W (4) 3b E (V 1b,V 1a ), (D (1) 1b, D (1) 1a), (D (2) 1b, D (2) 1a), (D (2) 2b, D (2) 2a), (Y (1) 1b, Y (1) 1a), (Y (2) 1b, Y (2) 1a), (Y (2) 2b, Y (2) 2a), (Y (3) 1b, Y (3) 1a), (Y (3) 2b, Y (3) 2a), (W (1) 1b, W (1) 1a), (W (2) 1b, W (2) 1a), (W (2) 2b, W (2) 2a), (W (3) 1b, W (3) 1a), (W (3) 2b, W (3) 2a), (W (4) 1b, W (4) 1a), (W (4) 2b, W (4) 2a), (W (4) 4b, W (4) 4a) β aaa = -β abb = -β bba = -β bab, -γ bbbc = γ aabc = γ baac = γ abac, -γ bcbb = γ bcaa = γ acba = γ acab, -γ bbcb = γ aacb = γ abca = γ baca, -γ cbbb = γ cbaa = γ caab = γ caba, - 5
6 C 5v E 2C 5z 2 2C 5 5σ v A A E 1 E cos72 2cos cos144 2cos72 0 C 5v A 1 V 0, D (0) 0, D (2) 0, Y (1) 0, Y (3) 0, W (0) 0, W (2) 0, W (4) 0 A 2 D (1) 0, Y (0) 0, Y (2) 0, W (1) 0, W (3) 0 E 1 (V 1b,V 1a ), (D (1) 1b, D (1) 1a), (D (2) 1b, D (2) 1a), (Y (1) 1b, Y (1) 1a), (Y (2) 1b, Y (2) 1a), (Y (3) 1b, Y (3) 1a), (W (1) 1b, W (1) 1a), (W (2) 1b, W (2) 1a), (W (3) 1b, W (3) 1a), (W (4) 1b, W (4) 1a), (W (4) 4b, W (4) 4a) E 2 (D (2) 2b, D (2) 2a), (Y (2) 2b, Y (2) 2a), (Y (3) 2b, Y (3) 2a), (Y (3) 3b, Y (3) 3a), (W (2) 2b, W (2) 2a), (W (3) 2b, W (3) 2a), (W (3) 3b, W (3) 3a), (W (4) 2b, W (4) 2a), (W (4) 3b, W (4) 3a) β ccc, β aac = β bbc, β aca = β bcb, β caa = β cbb γ abba = γ baab, γ acca = γ bccb, γ caac = γ cbbc C 4v E 2C 4z C 2 4 C 2 " 2σ v 2σ d A A B B E x 2 y 2 xy C 4v A 1 V 0, D (0) 0, D (2) 0, Y (1) 0, Y (3) 0, W (0) 0, W (2) 0, W (4) 0, W (4) 4b A 2 D (1) 0, Y (0) 0, Y (2) 0, W (1) 0, W (3) 0, W (4) 4a B 1 D (2) 2b, Y (2) 2a, Y (3) 2b, W (2) 2b, W (3) 2a, W(4)2b B2 D (2) 2a, Y (2) 2b, Y (3) 2a, W (2) 2a, W (3) 2b, W (4) 2a E (V 1b, V 1a ), (D (1) 1b, D (1) 1a), (D (2) 1b, D (2) 1a), (Y (1) 1b, Y (1) 1a), (Y (2) 1b, Y (2) 1a), (Y (3) 1b, Y (3) 1a), (Y (3) 3b, Y (3) 3a), (W (1) 1b, W(1) 1a ), (W(2) 1b, W(2) 1a ), (W(3) 1b, W(3) 1a ), (W(3) 3b, W(3) 3a ), (W(4) 1b, W(4) 1a ), (W(4) 3b, W(4) 3a ) β ccc, β aac = β bbc, β aca = β bcb, β caa = β cbb γ cccc, γ aaaa = γ bbbb, γ aabb = γ bbaa, γ aacc = γ bbcc = γ ccaa = γ ccbb, γ abba = γ baab, γ abab = γ baba, γ acca = γ bccb = γ caac = γ cbbc, γ acac = γ bcbc = γ caca = γ cbcb - 6
7 D 4 E 2C 4z C 2 4 C 2 " 2C 2 2C 2 ' A A B B E x 2 y 2 xy D 4 A 1 D (0) 0, D (2) 0, Y (0) 0, Y (2) 0, W (0) 0, W (2) 0, W (4) 0, W (4) 4b A 2 V 0, D (1) 0, Y (1) 0, Y (3) 0, W (1) 0, W (3) 0, W (4) 4a B 1 D (2) 2b, Y (2) 2b, Y (3) 2a, W (2) 2b, W (3) 2a, W (4) 2b B 2 D (2) 2a, Y (2) 2a, Y (3) 2b, W (2) 2a, W (3) 2b, W (4) 2a E (V 1b,V 1a ), (D (1) 1b, D (1) 1a), (D (2) 1b, D (2) 1a), (Y (1) 1b, Y (1) 1a), (Y (2) 1b, Y (2) 1a), (Y (3) 1b, Y (3) 1a), (Y (3) 3b, Y (3) 3a), (W (1) 1b, W (1) 1a), (W (2) 1b, W (2) 1a), (W (3) 1b, W (3) 1a), ((W (3) 3b, W (3) 3a), W (4) 1b, W (4) 1a), (W (4) 3b, W (4) 3a) β abc = -β bac, β bca = -β acb, β cab = -β cba γ cccc, γ aaaa = γ bbbb, γ aabb = γ bbaa, γ aacc = γ bbcc = γ ccaa = γ ccbb, γ abba = γ baab, γ abab = γ baba, γ acca = γ bccb = γ caac = γ cbbc, γ acac = γ bcbc = γ caca = γ cbcb D 2d E 2S 4z S 2 4 C 2 " 2C 2 2σ d A A B B E x 2 y 2 xy D 2d A 1 D (0) 0, D (2) 0, Y (2) 2a, Y (3) 2b, W (0) 0, W (2) 0, W (4) 0, W (4) 4b A 2 D (1) 0, Y (2) 2b, Y (3) 2a, W (1) 0, W (3) 0, W (4) 4a B 1 D (2) 2a, Y (0) 0, Y (2) 0, W (2) 2a, W (3) 2b, W (4) 2a B 2 V 0, D (2) 2b, Y (1) 0, Y (3) 0, W (2) 2b, W (3) 2a, W (4) 2b E (V 1b,V 1a ), (D (1) 1b, D (1) 1a), (D (2) 1b, D (2) 1a), (Y (1) 1b, Y (1) 1a), (Y (2) 1b, Y (2) 1a), (Y (3) 1b, Y (3) 1a), (Y (3) 3b, Y (3) 3a), (W (1) 1b, W (1) 1a), (W (2) 1b, W (2) 1a), (W (3) 1b, W (3) 1a), ((W (3) 3b, W (3) 3a), W (4) 1b, W (4) 1a), (W (4) 3b, W (4) 3a) β aac = -β bbc, β aca = -β bcb, β caa = -β cbb γ cccc, γ aaaa = γ bbbb, γ aacc = γ bbcc = γ ccaa = γ ccbb, γ aabb = γ bbaa, γ abba = γ baab, γ abab = γ baba, γ bcbc = γ acac, γ caca = γ cbcb, γ bccb = γ acca, γ caac = γ cbbc - 7
8 C 6v E 2C 6z 2C 2 6 2C 3 C 3 6 C 2 " 3σ v 3σ d A A B B E E C 6v A 1 V 0, D (0) 0, D (2) 0, Y (1) 0, Y (3) 0, W (0) 0, W (2) 0, W (4) 0 A 2 D (1) 0, Y (0) 0, Y (2) 0, W (1) 0, W (3) 0 B 1 Y (3) 3b, W (3) 3a, W (4) 3b B 2 Y (3) 3a, W (3) 3b, W (4) 3a E 1 (V 1b,V 1a ), (D (1) 1b, D (1) 1a), (D (2) 1b, D (2) 1a), (Y (1) 1b, Y (1) 1a), (Y (2) 1b, Y (2) 1a), (Y (3) 1b, Y (3) 1a), (W (1) 1b, W (1) 1a), (W (2) 1b, W (2) 1a), (W (3) 1b, W (3) 1a), W (4) 1b, W (4) 1a) E 2 (D (2) 2b, D (2) 2a), (Y (2) 2b, Y (2) 2a), (Y (3) 2b, Y (3) 2a), (W (2) 2b, W (2) 2a), (W (3) 2b, W (3) 2a), (W (4) 2b, W (4) 2a), (W (4) 4b, W (4) 4a) β ccc, β aac = β bbc, β aca = β bcb, β caa = β cbb γ abba = γ baab, γ acca = γ bccb, γ caac = γ cbbc D 6 E 2C 6z 2C 2 6 2C 3 C 3 6 C 2 " 3C 2 3C 2 ' A A B B E E D 6 A 1 D (0) 0, D (2) 0, Y (0) 0, Y (2) 0, W (0) 0, W (2) 0, W (4) 0 A 2 V 0, D (1) 0, Y (1) 0, Y (3) 0, W (1) 0, W (3) 0 B 1 Y (3) 3a, W (3) 3a, W (4) 3b B 2 Y (3) 3b, W (3) 3b, W (4) 3a E 1 (V 1b,V 1a ), (D (1) 1b, D (1) 1a), (D (2) 1b, D (2) 1a), (Y (1) 1b, Y (1) 1a), (Y (2) 1b, Y (2) 1a), (Y (3) 1b, Y (3) 1a), (W (1) 1b, W (1) 1a), (W (2) 1b, W (2) 1a), (W (3) 1b, W (3) 1a), W (4) 1b, W (4) 1a) E 2 (D (2) 2b, D (2) 2a), (Y (2) 2b, Y (2) 2a), (Y (3) 2b, Y (3) 2a), (W (2) 2b, W (2) 2a), (W (3) 2b, W (3) 2a), (W (4) 2b, W (4) 2a), (W (4) 4b, W (4) 4a) β abc = -β bac, β bca = -β acb, β cab = -β cba γ abba = γ baab, γ acca = γ bccb, γ caac = γ cbbc - 8
9 D 3d E 2S 6z 2S 2 6 2C 3z I 3C 2 3σ d A 1g A 1u A 2g A 2u E g E u ; D 3d A 1g D (0) 0, D (2) 0, W (0) 0, W (2) 0, W (3) 3a, W (4) 0, W (4) 3b A 1u Y (0) 0, Y (2) 0, Y (3) 3a A 2g D (1) 0, W (1) 0, W (3) 0, W (3) 3b, W (4) 3a A 2u V 0, Y (1) 0, Y (3) 0 Y (3) 3b E g (D (1) 1b, D (1) 1a), (D (2) 1b, D (2) 1a), (D (2) 2b, D (2) 2a), (W (1) 1b, W (1) 1a), (W (2) 1b, W (2) 1a), (W (2) 2b, W (2) 2a), (W (3) 1b, W (3) 1a), (W (3) 2b, W (3) 2a), (W (4) 1b, W (4) 1a)(W (4) 2b, W (4) 2a), (W (4) 4b, W (4) 4a) E u (V 1b,V 1a ), (Y (1) 1b, Y (1) 1a), (Y (2) 1b, Y (2) 1a), (Y (2) 2b, Y (2) 2a), (Y (3) 1b, Y (3) 1a), (Y (3) 2b, Y (3) 2a) β ccc, β aac = β bbc, β aca = β bcb, β caa = β cbb, β aaa = -β abb = -β bba = -β bab, γ abba = γ baab, γ acca = γ bccb, γ caac = γ cbbc, γ aaca = -γ bbcb, γ caaa = -γ cabb = -γ cbab = -γ cbba, γ aaac = -γ bbac = -γ abbc = -γ babc, γ acaa = -γ acbb = -γ bcab = -γ bcba D 4d E 2S 8z 2S C 4 2S 8 S 4 8 C 2 " 4C 2 4σ d A A B B E E E D 4d A 1 D (0) 0, D (2) 0, W (0) 0, W (2) 0, W (4) 0 A 2 D (1) 0, W (1) 0, W (3) 0 B 1 Y (0) 0, Y (2) 0, W (4) 4b B 2 V 0, Y (1) 0, Y (3) 0, W (4) 4a E 1 (V 1b,V 1a ), (Y (1) 1b, Y (1) 1a), (Y (2) 1b, Y (2) 1a), (Y (3) 1b, Y (3) 1a), (W (3) 3b, W (3) 3a), (W (4) 3b, W (4) 3a) E 2 (D (2) 2b, D (2) 2a), (Y (2) 2b, Y (2) 2a), (Y (3) 2b, Y (3) 2a), (W (2) 2b, W (2) 2a), (W (3) 2b, W (3) 2a), (W (4) 2b, W (4) 2a) E 3 (D (1) 1b, D (1) 1a), (D (2) 1b, D (2) 1a), (Y (3) 3b, Y (3) 3a), (W (1) 1b, W (1) 1a), (W (2) 1b, W (2) 1a), (W (3) 1b, W (3) 1a), (W (4) 1b, W (4) 1a) γ abba = γ baab, γ acca = γ bccb, γ caac = γ cbbc - 9
10 D 5d E 2C 5 2 2C 5 I 5C 2 3 5σ d 2S 10 2S 10 A 1g A 1u A 2g A 2u E 1g +2 +2cos72 +2cos cos72 +2cos144 E 1u +2 +2cos72 +2cos cos72-2cos144 E 2g +2 +2cos144 +2cos cos144 +2cos72 E 2u +2 +2cos144 +2cos cos144-2cos72 D 5d A 1g D (0) 0, D (2) 0, W (0) 0, W (2) 0, W (4) 0 A 1u Y (0) 0, Y (2) 0 A 2g D (1) 0, W (1) 0, W (3) 0 A 2u V 0, Y (1) 0, Y (3) 0 E 1g (D (1) 1b, D (1) 1a), (D (2) 1b, D (2) 1a), (W (1) 1b, W (1) 1a), (W (2) 1b, W (2) 1a), (W (3) 1b, W (3) 1a), (W (4) 1b, W (4) 1a), (W (4) 4b, W (4) 4a) E 1u (V 1b,V 1a ), (Y (1) 1b, Y (1) 1a), (Y (2) 1b, Y (2) 1a), (Y (3) 1b, Y (3) 1a) E 2g (D (2) 2b, D (2) 2a), (W (2) 2b, W (2) 2a), (W (3) 2b, W (3) 2a), (W (3) 3b, W (3) 3a), (W (4) 2b, W (4) 2a), (W (4) 3b, W (4) 3a) E 2u (Y (2) 2b, Y (2) 2a), (Y (3) 2b, Y (3) 2a), (Y (3) 3b, Y (3) 3a) γ abba = γ baab, γ acca = γ bccb, γ caac = γ cbbc D 6d E 2S 12z 2C 6 2S 4 2C 3 5 2S 12 C 2 6C 2 ' 6σ d A A B B E E E E E D 6d A 1 D (0) 0, D (2) 0, W (0) 0, W (2) 0, W (4) 0 A 2 D (1) 0, W (1) 0, W (3) 0 B 1 Y (0) 0, Y (2) 0 B 2 V 0, Y (1) 0, Y (3) 0 E 1 (V 1b,V 1a ), (Y (1) 1b, Y (1) 1a), (Y (2) 1b, Y (2) 1a), (Y (3) 1b, Y (3) 1a), E 2 (D (2) 2b, D (2) 2a), (W (2) 2b, W (2) 2a), (W (3) 2b, W (3) 2a), (W (4) 2b, W (4) 2a) E 3 (Y (3) 3b, Y (3) 3a), (W (3) 3b, W (3) 3a), (W (4) 3b, W (4) 3a) E 4 (Y (2) 2b, Y (2) 2a), (Y (3) 2b, Y (3) 2a), (W (4) 4b, W (4) 4a) E 5 (D (1) 1b, D (1) 1a), (D (2) 1b, D (2) 1a), (W (1) 1b, W (1) 1a), (W (2) 1b, W (2) 1a), (W (3) 1b, W (3) 1a), W (4) 1b, W (4) 1a) γ abba = γ baab, γ acca = γ bccb, γ caac = γ cbbc - 10
11 D 3h E 2C 3z 3C 2 σ h 2S 3z 3σ v A 1 ' A 1 " A 2 ' A 2 " E' E" ; D 3h A 1 ' D (0) 0, D (2) 0, Y (3) 3b, W (0) 0, W (2) 0, W (4) 0 A 1 " Y (0) 0, Y (2) 0,, W (3) 3b, W (4) 3a A 2 ' D (1) 0,, Y (3) 3a, W (1) 0, W (3) 0 A 2 " V 0, Y (1) 0, Y (3) 0 W (3) 3a, W (4) 3b E' (V 1b,V 1a ), (D (2) 2b, D (2) 2a), (Y (1) 1b, Y (1) 1a), (Y (2) 1b, Y (2) 1a), (Y (3) 1b, Y (3) 1a), (W (2) 2b, W (2) 2a), (W (3) 2b, W (3) 2a), (W (4) 2b, W (4) 2a), (W (4) 4b, W (4) 4a) E" (D (1) 1b, D (1) 1a), (D (2) 1b, D (2) 1a), (Y (2) 2b, Y (2) 2a), (Y (3) 2b, Y (3) 2a), (W (1) 1b, W (1) 1a), (W (2) 1b, W (2) 1a), (W (3) 1b, W (3) 1a), (W (4) 1b, W (4) 1a) β aaa = -β abb = -β bba = -β bab, γ abba = γ baab, γ acca = γ bccb, γ caac = γ cbbc, D 5h E 2C 5z 2 2C 5z σ h 5C 2 5σ v 2S 5 3 2S 5 A 1 ' A 1 " A 2 ' A 2 " E 1 ' +2 +2cos72 +2cos cos72 +2cos144 E 1 " +2 +2cos72 +2cos cos72-2cos144 E 2 ' +2 +2cos144 +2cos cos144 +2cos72 E 2 " +2 +2cos144 +2cos cos144-2cos72 D 5h A 1 ' D (0) 0, D (2) 0, W (0) 0, W (2) 0, W (4) 0 A 1 " Y (0) 0, Y (2) 0 A 2 ' D (1) 0, W (1) 0, W (3) 0 A 2 " V 0, Y (1) 0, Y (3) 0 E 1 ' (V 1b,V 1a ), (Y (1) 1b, Y (1) 1a), (Y (2) 1b, Y (2) 1a), (Y (3) 1b, Y (3) 1a), (W (4) 4b, W (4) 4a) E 1 " (D (1) 1b, D (1) 1a), (D (2) 1b, D (2) 1a), (W (1) 1b, W (1) 1a), (W (2) 1b, W (2) 1a), (W (3) 1b, W (3) 1a), (W (4) 1b, W (4) 1a) E 2 ' (D (2) 2b, D (2) 2a), (Y (3) 3b, Y (3) 3a), (W (2) 2b, W (2) 2a), (W (3) 2b, W (3) 2a), (W (4) 2b, W (4) 2a) E 2 " (Y (2) 2b, Y (2) 2a), (Y (3) 2b, Y (3) 2a), (W (3) 3b, W (3) 3a), (W (4) 3b, W (4) 3a) γ abba = γ baab, γ acca = γ bccb, γ caac = γ cbbc - 11
12 D 4h E 2C 4z C 2 4z C 2 " 2C 2 2C 2 ' σ h 2σ v 2σ d 2S 4z S 2 I A 1g A 1u A 2g A 2u B 1g B 1u B 2g B 2u E g E u x 2 y 2 xy D 4h A 1g D (0) 0, D (2) 0, W (0) 0, W (2) 0, W (4) 0, W (4) 4b A 1u Y (0) 0, Y (2) 0 A 2g D (1) 0, W (1) 0, W (3) 0, W (4) 4a A 2u V 0, Y (1) 0, Y (3) 0 B 1g D (2) 2b, W (2) 2b, W (3) 2a, W (4) 2b B 1u Y (2) 2b, Y (3) 2a B 2g D (2) 2a, W (2) 2a, W (3) 2b, W (4) 2a B 2u Y (2) 2a, Y (3) 2b E g (D (1) 1b, D (1) 1a), (D (2) 1b, D (2) 1a), (W (1) 1b, W (1) 1a), (W (2) 1b, W (2) 1a), (W (3) 1b, W (3) 1a), (W (3) 3b, W (3) 3a), (W (4) 1b, W (4) 1a), (W (4) 3b, W (4) 3a) E u (V 1b,V 1a ), (Y (1) 1b, Y (1) 1a), (Y (2) 1b, Y (2) 1a), (Y (3) 1b, Y (3) 1a), (Y (3) 3b, Y (3) 3a) γ abba = γ baab, γ acca = γ bccb, γ caac = γ cbbc - 12
13 D 6h E 2C 6z 2C 6 2 C 6 3 3C 2 3C 2 ' σ h 3σ v 3σ d 2S 6 2S 3 S 6 3 A 1g A 1u A 2g A 2u B 1g B 1u B 2g B 2u E 1g E 1u E 2g E 2u 2C 3 C 2 " I D 6h A 1g D (0) 0, D (2) 0, W (0) 0, W (2) 0, W (4) 0 A 1u Y (0) 0, Y (2) 0 A 2g D (1) 0, W (1) 0, W (3) 0 A 2u V 0, Y (1) 0, Y (3) 0 B 1g W (3) 3a, W (4) 3b B 1u Y (3) 3a B 2g W (3) 3b, W (4) 3a B 2u Y (3) 3b E 1g (D (1) 1b, D (1) 1a), (D (2) 1b, D (2) 1a), (W (1) 1b, W (1) 1a), (W (2) 1b, W (2) 1a), (W (3) 1b, W (3) 1a), (W (4) 1b, W (4) 1a) E 1u (V 1b,V 1a ), (Y (1) 1b, Y (1) 1a), (Y (2) 1b, Y (2) 1a), (Y (3) 1b, Y (3) 1a) E 2g (D (2) 2b, D (2) 2a), (W (2) 2b, W (2) 2a), (W (3) 2b, W (3) 2a), (W (4) 2b, W (4) 2a), (W (4) 4b, W (4) 4a), E 2u (Y (2) 2b, Y (2) 2a), (Y (3) 2b, Y (3) 2a) γ abba = γ baab, γ acca = γ bccb, γ caac = γ cbbc - 13
14 C 3 E 2C 3z A E +2-1 ; C 3 A V 0, D (0) 0, D (1) 0, D (2) 0, Y (0) 0, Y (1) 0, Y (2) 0, Y (3) 0, Y (3) 3a, Y (3) 3b, W (0) 0, W (1) 0, W (2) 0, W (3) 0, W (3) 3a, W (3) 3b, W (4) 0, W (4) 3a, W (4) 3b E (V 1b, V 1a ), (D (1) 1b, D (1) 1a), (D (2) 1b, D (2) 1a), (D (2) 2b, D (2) 2a), (Y (1) 1b, Y (1) 1a), (Y (2) 1b, Y (2) 1a), (Y (2) 2b, Y (2) 2a), (Y (3) 1b, Y (3) 1a), (Y (3) 2b, Y (3) 2a), (W (1) 1b, W (1) 1a), (W (2) 1b, W (2) 1a), (W (2) 2b, Y (2) 2a), (W (3) 1b, W (3) 1a), (W (3) 2b, W (3) 2a), (W (4) 1b, W (4) 1a), (W (4) 2b, W (4) 2a), (W (4) 4b, W (4) 4a) β ccc, β aac = β bbc, β caa = β cbb, β aca = β bcb, β abb = β bba = β bab = -β aaa, β baa = β aab = β aba = -β bbb, β abc = -β bac, β bca = -β acb, β cab = -β cba, γ cccc, γ aaaa = γ bbbb, γ aabb = γ bbaa, γ abba = γ baab, γ abab = γ baba, γ aacc = γ bbcc, γ ccaa = γ ccbb, γ acca = γ bccb, γ caac = γ cbbc, γ acac = γ bcbc, γ caca = γ cbcb, γ bcaa = γ acba = γ acab = -γ bcbb, γ aabc = γ baac = γ abac = -γ bbbc, γ bcaa = γ acba = γ acab = -γ bcbb, γ aacb = γ abca = γ baca = -γ bbcb, γ bbca = γ bacb = γ abcb = -γ aaca, γ cabb = γ cbab = γ cbba = -γ caaa, γ bbac = γ abbc = γ babc = -γ aaac, γ aaab = -γ bbba, γ aaba = -γ bbab, γ abaa = -γ baaa, γ baaa = -γ abbb, γ ccab = -γ ccba, γ abcc = -γ bacc, γ accb = -γ bcca, γ cabc = -γ cbac, γ acbc = -γ bcac, γ cacb = -γ cbca C 4 E 2C 4z C 2 4 C 2 A B E x 2 y 2, xy C 4 A V 0, D (0) 0, D (1) 0, D (2) 0, Y (0) 0, Y (1) 0, Y (2) 0, Y (3) 0, W (0) 0, W (1) 0, W (2) 0, W (3) 0, W (4) 0, W (4) 4a, W (4) 4b B D (2) 2b, D (2) 2a, Y (2) 2b, Y (2) 2a, Y (3) 2b, Y (3) 2a, W (2) 2b, W (2) 2a, W (3) 2b, W (3) 2a, W (4) 2b, W (4) 2a E (V 1b, V 1a ), (D (1) 1b, D (1) 1a), (D (2) 1b, D (2) 1a), (Y (1) 1b, Y (1) 1a), (Y (2) 1b, Y (2) 1a), (Y (3) 1b, Y (3) 1a), (Y (3) 3b, Y (3) 3a), (W (1) 1b, W (1) 1a), (W (2) 1b, W (2) 1a), (W (3) 1b, W (3) 1a), (W (3) 3b, W (3) 3a), (W (4) 1b, W (4) 1a), (W (4) 3b, W (4) 3a) β ccc, β aac = β bbc, β caa = β cbb, β aca = β bcb, γ cccc, γ aaaa = γ bbbb, γ aabb = γ bbaa, γ abba = γ baab, γ abab = γ baba, γ aacc = γ bbcc, γ ccaa = γ ccbb, γ acca = γ bccb, γ caac = γ cbbc, γ acac = γ bcbc, γ caca = γ cbcb, γ aaab = -γ bbba, γ aaba = -γ bbab, γ abaa = -γ babb, γ baaa = -γ abbb, γ ccab = -γ ccba, γ abcc = -γ bacc, γ accb = -γ bcca, γ cabc = -γ cbac, γ acbc = -γ bcac, γ cacb = -γ cbca - 14
15 C 6 E 2C 6z 2C 2 6 2C 3 C 3 6 C 2 " A B E E C 6 A V 0, D (0) 0, D (1) 0, D (2) 0, Y (0) 0, Y (1) 0, Y (2) 0, Y (3) 0, W (0) 0, W (1) 0, W (2) 0, W (3) 0, W (4) 0 B Y (3) 3b, Y (3) 3a, W (3) 3b, W (3) 3a, W (4) 3b, W (4) 3a E 1 (V 1b, V 1a ), (D (1) 1b, D (1) 1a), (D (2) 1b, D (2) 1a), (Y (1) 1b, Y (1) 1a), (Y (2) 1b, Y (2) 1a), (Y (3) 1b, Y (3) 1a), (W (1) 1b, W (1) 1a), (W (2) 1b, W (2) 1a), (W (3) 1b, W (3) 1a), (W (4) 1b, W (4) 1a) E 2 (D (2) 2b, D (2) 2a), (Y (2) 2b, Y (2) 2a), (Y (3) 2b, Y (3) 2a), (W (2) 2b, W (2) 2a), (W (3) 2b, W (3) 2a), (W (4) 2b, W (4) 2a), (W (4) 4b, W (4) 4a) β ccc, β aac = β bbc, β caa = β cbb, β aca = β bcb, γ cccc, γ aaaa = γ bbbb, γ aabb = γ bbaa, γ abba = γ baab, γ abab = γ baba, γ aacc = γ bbcc, γ ccaa = γ ccbb, γ acca = γ bccb, γ caac = γ cbbc, γ acac = γ bcbc, γ caca = γ cbcb, γ aaab = -γ bbba, γ aaba = -γ bbab, γ abaa = -γ babb, γ baaa = -γ abbb, γ ccab = -γ ccba, γ abcc = -γ bacc, γ accb = -γ bcca, γ cabc = -γ cbac, γ acbc = -γ bcac, γ cacb = -γ cbca S 4 E 2S 4z S 2 4 S 2 A B E x y 2, xy S 4 A D (0) 0, D (1) 0, D (2) 0, Y (2) 2b, Y (2) 2a, Y (3) 2b, Y (3) 2a, W (0) 0, W (1) 0, W (2) 0, W (3) 0, W (4) 0, W (4) 4b, W (4) 4a B V 0, D (2) 2b, D (2) 2a, Y (0) 0, Y (1) 0, Y (2) 0, Y (3) 0, W (2) 2b, W (2) 2a, W (3) 2b, W (3) 2a, W (4) 2b, W (4) 2a, E (V 1b, V 1a ), (D (1) 1b, D (1) 1a), (D (2) 1b, D (2) 1a), (Y (1) 1b, Y (1) 1a), (Y (2) 1b, Y (2) 1a), (Y (3) 1b, Y (3) 1a), (Y (3) 3b, Y (3) 3a), (W (1) 1b, W (1) 1a), (W (2) 1b, W (2) 1a), (W (3) 1b, W (3) 1a), (W (3) 3b, W (3) 3a), (W (4) 1b, W (4) 1a), (W (4) 3b, W (4) 3a) β aac = -β bbc, β caa = -β cbb, -β aca = β bcb, β abc = β bac, β bca = β acb, β cab = β cba, γ cccc, γ aaaa = γ bbbb, γ aabb = γ bbaa, γ abba = γ baab, γ abab = γ baba, γ aacc = γ bbcc, γ ccaa = γ ccbb, γ acca = γ bccb, γ caac = γ cbbc, γ acac = γ bcbc, γ caca = γ cbcb, γ aaab = -γ bbba, γ aaba = -γ bbab, γ abaa = -γ babb, γ baaa = -γ abbb, γ ccab = -γ ccba, γ abcc = -γ bacc, γ accb = -γ bcca, γ cabc = -γ cbac, γ acbc = -γ bcac, γ cacb = -γ cbca - 15
16 S 6 E 2S 6z 2S 2 6 2C 3 S 3 6 S 2 A g A g E g E u , S 6 A g D (0) 0, D (1) 0, D (2) 0, W (0) 0, W (1) 0, W (2) 0, W (3) 0, W (3) 3b, W (3) 3a, W (4) 0, W (4) 3b, W (4) 3a A u V 0, Y (0) 0, Y (1) 0, Y (2) 0, Y (3) 0, Y (3) 3b, Y (3) 3a E g (D (1) 1b, D (1) 1a), (D (2) 1b, D (2) 1a), (D (2) 2b, D (2) 2a), (W (1) 1b, W (1) 1a), (W (2) 1b, W (2) 1a), (W (2) 2b, W (2) 2a), (W (3) 1b, W (3) 1a), (W (3) 2b, W (3) 2a), (W (4) 1b, W (4) 1a), (W (4) 2b, W (4) 2a), (W (4) 4b, W (4) 4a) E u (V 1b, V 1a ), (Y (1) 1b, Y (1) 1a), (Y (2) 1b, Y (2) 1a), (Y (2) 2b, Y (2) 2a), (Y (3) 1b, Y (3) 1a), (Y (3) 2b, Y (3) 2a) γ cccc, γ aaaa = γ bbbb, γ aabb = γ bbaa, γ abba = γ baab, γ abab = γ baba, γ aacc = γ bbcc, γ ccaa = γ ccbb, γ acca = γ bccb, γ caac = γ cbbc, γ acac = γ bcbc, γ caca = γ cbcb, γ bcaa = γ acba = γ acab = -γ bcbb, γ aabc = γ baac = γ abac = -γ bbbc, γ bcaa = γ acba = γ acab = -γ bcbb, γ aacb = γ abca = γ baca = -γ bbcb, γ bbca = γ bacb = γ abcb = -γ aaca, γ cabb = γ cbab = γ cbba = -γ caaa, γ bbac = γ abbc = γ babc = -γ aaac, γ aaab = -γ bbba, γ aaba = -γ bbab, γ abaa = -γ baaa, γ baaa = -γ abbb, γ ccab = -γ ccba, γ abcc = -γ bacc, γ accb = -γ bcca, γ cabc = -γ cbac, γ acbc = -γ bcac, γ cacb = -γ cbca C 3h E 2C 3z σ h 2S 6 A' A" E' E" C 3h A' D (0) 0, D (1) 0, D (2) 0, Y (3) 3b, Y (3) 3a, W (0) 0, W (1) 0, W (2) 0, W (3) 0, W (4) 0 A" V 0, Y (0) 0, Y (1) 0, Y (2) 0, Y (3) 0, W (3) 3b, W (3) 3a, W (4) 3b, W (4) 3a E' (V 1b, V 1a ), (D (2) 2b, D (2) 2a), (Y (1) 1b, Y (1) 1a), (Y (2) 1b, Y (2) 1a), (Y (3) 1b, Y (3) 1a), (W (2) 2b, W (2) 2a), (W (3) 2b, W (3) 2a), (W (4) 2b, W (4) 2a), (W (4) 4b, W (4) 4a) E" (D (1) 1b, D (1) 1a), (D (2) 1b, D (2) 1a), (Y (2) 2b, Y (2) 2a), (Y (3) 2b, Y (3) 2a), (W (1) 1b, W (1) 1a), (W (2) 1b, W (2) 1a), (W (3) 1b, W (3) 1a), (W (4) 1b, W (4) 1a) β aaa = -β abb = -β bba = -β bab, β bbb = -β baab = -β aab = -β aba γ cccc, γ aaaa = γ bbbb, γ aabb = γ bbaa, γ abba = γ baab, γ abab = γ baba, γ aacc = γ bbcc, γ ccaa = γ ccbb, γ acca = γ bccb, γ caac = γ cbbc, γ acac = γ bcbc, γ caca = γ cbcb, γ aaab = -γ bbba, γ aaba = -γ bbab, γ abaa = -γ babb, γ baaa = -γ abbb, γ ccab = -γ ccba, γ abcc = -γ bacc, γ accb = -γ bcca, γ cabc = -γ cbac, γ acbc = -γ bcac, γ cacb = -γ cbca - 16
17 C 4h E 2C 4z C 2 4z C 2 " σ h 2S 4z S 2 I A g A u B g B u E g E u x 2 y 2, xy C 4h A g D (0) 0, D (1) 0, D (2) 0, W (0) 0, W (1) 0, W (2) 0, W (3) 0, W (4) 0, W (4) 4b, W (4) 4a A u V 0, Y (0) 0, Y (1) 0, Y (2) 0, Y (3) 0 B g D (2) 2b, D (2) 2a, W (2) 2b, W (2) 2a, W (3) 2b, W (3) 2a, W (4) 2b, W (4) 2a, B u Y (2) 2b, Y (2) 2a, Y (3) 2b, Y (3) 2a E g (D (1) 1b, D (1) 1a), (D (2) 1b, D (2) 1a), (W (1) 1b, W (1) 1a), (W (2) 1b, W (2) 1a), (W (3) 1b, W (3) 1a), (W (3) 3b, W (3) 3a), (W (4) 1b, W (4) 1a), (W (4) 3b, W (4) 3a) E u (V 1b, V 1a ), (Y (1) 1b, Y (1) 1a), (Y (2) 1b, Y (2) 1a), (Y (3) 1b, Y (3) 1a), (Y (3) 3b, Y (3) 3a) γ cccc, γ aaaa = γ bbbb, γ aabb = γ bbaa, γ abba = γ baab, γ abab = γ baba, γ aacc = γ bbcc, γ ccaa = γ ccbb, γ acca = γ bccb, γ caac = γ cbbc, γ acac = γ bcbc, γ caca = γ cbcb, γ aaab = -γ bbba, γ aaba = -γ bbab, γ abaa = -γ babb, γ baaa = -γ abbb, γ ccab = -γ ccba, γ abcc = -γ bacc, γ accb = -γ bcca, γ cabc = -γ cbac, γ acbc = -γ bcac, γ cacb = -γ cbca C 6h E 2C 6z 2C 6 2 2C 3 C 6 3 C 2 " σ h 2S 6 2S 3 S 6 3 I A g A u B g B u E 1g E 1u E 2g E 2u C 6h A g D (0) 0, D (1) 0, D (2) 0, W (0) 0, W (1) 0, W (2) 0, W (3) 0, W (4) 0 A u V 0, Y (0) 0, Y (1) 0, Y (2) 0, Y (3) 0 B g W (3) 3b, W (3) 3a, W (4) 3b, W (4) 3a B u Y (3) 3b, Y (3) 3a E 1g (D (1) 1b, D (1) 1a), (D (2) 1b, D (2) 1a), (W (1) 1b, W (1) 1a), (W (2) 1b, W (2) 1a), (W (3) 1b, W (3) 1a), (W (4) 1b, W (4) 1a) E 1u (V 1b, V 1a ), (Y (1) 1b, Y (1) 1a), (Y (2) 1b, Y (2) 1a), (Y (3) 1b, Y (3) 1a) E 2g (D (2) 2b, D (2) 2a), (W (2) 2b, W (2) 2a), (W (3) 2b, W (3) 2a), (W (4) 2b, W (4) 2a), (W (4) 4b, W (4) 4a) E 2u (Y (2) 2b, Y (2) 2a), (Y (3) 2b, Y (3) 2a) γ cccc, γ aaaa = γ bbbb, γ aabb = γ bbaa, γ abba = γ baab, γ abab = γ baba, γ aacc = γ bbcc, γ ccaa = γ ccbb, γ acca = γ bccb, γ caac = γ cbbc, γ acac = γ bcbc, γ caca = γ cbcb, γ aaab = -γ bbba, γ aaba = -γ bbab, γ abaa = -γ babb, γ baaa = -γ abbb, γ ccab = -γ ccba, γ abcc = -γ bacc, γ accb = -γ bcca, γ cabc = -γ cbac, γ acbc = -γ bcac, γ cacb = -γ cbca - 17
18 T d E 8C 3 6σ d 6S 4 2 3S 4 = 3C 2 A A E F F (T x, T y, ) (R x, R y, ) x 2 +y 2 +z 2 (x 2 +y 2-2z 2, x 2 y 2 ) (xy, xz, yz) T d A 1 D (0) 0, Y (3) 2b, W (0) 0 A 2 Y (0) 0, W (3) 2b E (D (2) 0, D (2) 2a), (Y (2) 0, Y (2) 2a), (W (2) 0, W (2) 2a) F 1 (D (1) 1b, D (1) 1a, D (1) 0), (Y (2) 2b, Y (2) 1b,Y (2) 1a), (W (1) 1b, W (1) 1a, W (1) 0) F 2 (V 1b, V 1a, V 0 ), (D (2) 2b, D (2) 1b, D (2) 1a), (Y (1) 1b, Y (1) 1a,Y (1) 0), (W (2) 2b, W (2) 1b, W (2) 1a) [Y (3) 0, Y (3) 1b, Y (3) 1a, Y (3) 2a, Y (3) 3b, Y (3) 3a], [W (3) 0, W (3) 1b, W (3) 1a, W (3) 2a, W (3) 3b, W (3) 3a], [W (4) 1b, W (4) 1a, W (4) 2b, W (4) 3b, W (4) 3a, W (4) 4a]: distributed over F 1 + F 2, [W (4) 0, W (4) 2a, W (4) 4b]: distributed over A 1 + E. (without consideration of the above "distributed terms,) β aac = β caa = β aca = -β bbc = -β cbb = -β cbc γ aaaa = γ bbbb = γ cccc, γ aabb = γ bbaa = γ aacc = γ bbcc = γ ccaa = γ ccbb, γ abba = γ baab = γ acca = γ bccb = γ caac = γ cbbc, γ abab = γ baba = γ acac = γ bcbc = γ caca = γ cbcb T h E 8C 3 3C 2 " I 8S 6 3σ h A g A u E g E u F g F u (T x, T y, ) (R x, R y, ) x 2 +y 2 +z 2 (x 2 +y 2-2z 2, x 2 y 2 ) (xy, xz, yz) T h A g D (0) 0, W (0) 0, W (3) 2b A u Y (0) 0, Y (3) 2b E g (D (2) 2a, D (2) 0), (W (2) 2a, W (2) 0) E u (Y (2) 2a, Y (2) 0) F g (D (1) 1b, D (1) 1a, D (1) 0), (D (2) 2b, D (2) 1b, D (2) 1a), (W (1) 1b, W (1) 1a, W (1) 0), (W (2) 2b, W (2) 1b,W (2) 1a) F u (V 1b, V 1a, V 0 ), (Y (1) 1b, Y (1) 1a, Y (1) 0), (Y (2) 2b, Y (2) 1b, Y (2) 1a) [Y (3) 0, Y (3) 1b, Y (3) 1a, Y (3) 2a, Y (3) 3b, Y (3) 3a]: distributed over 2F u, [W (3) 0, W (3) 1b, W (3) 1a, W (3) 2a, W (3) 3b, W (3) 3a], [W (4) 1b, W (4) 1a, W (4) 2b, W (4) 3b, W (4) 3a, W (4) 4a]: distributed over 2F g, [W (4) 0, W (4) 2a, W (4) 4b]: distributed over A g + E g. (without consideration of the above "distributed terms,) γ aaaa = γ bbbb = γ cccc, γ aabb = γ bbaa = γ aacc = γ bbcc = γ ccaa = γ ccbb, γ abba = γ baab = γ acca = γ bccb = γ caac = γ cbbc, γ abab = γ baba = γ acac = γ bcbc = γ caca = γ cbcb, γ aaab = γ bbba, γ bbab = γ aaba, γ abaa = γ babb, γ abbb = γ baaa, γ ccab = γ ccba = -γ abcc = -γ bacc, γ accb = γ bcca = -γ cabc = -γ cbac, γ acbc = γ bcac = -γ cacb = -γ cbca, - 18
19 T E 8C 3 3C 2 A E F (T x, T y, ) (R x, R y, ) x 2 +y 2 +z 2 (x 2 +y 2-2z 2, x 2 y 2 ) (xy, xz, yz) T A D (0) 0, Y (0) 0, Y (3) 2b, W (0) 0, W (3) 2b E (D (2) 2a, D (2) 0), (Y (2) 2a, Y (2) 0), (W (2) 2a, W (2) 0) F (V 1b, V 1a, V 0 ), (D (1) 1b, D (1) 1a, D (1) 0), (D (2) 2b, D (2) 1b, D (2) 1a), (Y (1) 1b, Y (1) 1a, Y (1) 0), (Y (2) 2b, Y (2) 1b, Y (2) 1a), (W (1) 1b, W (1) 1a, W (1) 0), (W (2) 2b, W (2) 1b,W (2) 1a) [Y (3) 0, Y (3) 1b, Y (3) 1a, Y (3) 2a, Y (3) 3b, Y (3) 3a], [W (3) 0, W (3) 1b, W (3) 1a, W (3) 2a, W (3) 3b, W (3) 3a], [W (4) 1b, W (4) 1a, W (4) 2b, W (4) 3b, W (4) 3a, W (4) 4a]: distributed over 2F, [W (4) 0, W (4) 2a, W (4) 4b]: distributed over A + E. (without consideration of the above "distributed terms,) β aac = β caa = β aca = -β bbc = -β cbb = -β bcb, β abc = β bca = β cab = -β bac = -β acb = -β cba γ aaaa = γ bbbb = γ cccc, γ aabb = γ bbaa = γ aacc = γ bbcc = γ ccaa = γ ccbb, γ abba = γ baab = γ acca = γ bccb = γ caac = γ cbbc, γ abab = γ baba = γ acac = γ bcbc = γ caca = γ cbcb, γ aaab = γ bbba, γ bbab = γ aaba, γ abaa = γ babb, γ abbb = γ baaa, γ ccab = γ ccba = -γ abcc = -γ bacc, γ accb = γ bcca = -γ cabc = -γ cbac, γ acbc = γ bcac = -γ cacb = -γ cbca O E 8C 3 6 C 2 6C 4 3C 4 2 3C 2 " A 1 A 2 E F 1 F (T x, T y, ) (R x, R y, ) x 2 +y 2 +z 2 (x 2 +y 2-2z 2, x 2 y 2 ) (xy, xz, yz) O A 1 D (0) 0, Y (0) 0, W (0) 0 A 2 Y (3) 2b, W (3) 2b E (D (2) 2a, D (2) 0), (Y (2) 2a, Y (2) 0), (W (2) 2a, W (2) 0) F 1 (V 1b, V 1a, V 0 ), (D (1) 1b, D (1) 1a, D (1) 0), (Y (1) 1b, Y (1) 1a, Y (1) 0), (W (1) 1b, W (1) 1a, W (1) 0) F 2 (D (2) 2b, D (2) 1b, D (2) 1a), (Y (2) 2b, Y (2) 1b, Y (2) 1a), (W (2) 2b, W (2) 1b, W (2) 1a) [Y (3) 0, Y (3) 1b, Y (3) 1a, Y (3) 2a, Y (3) 3b, Y (3) 3a], [W (3) 0, W (3) 1b, W (3) 1a, W (3) 2a, W (3) 3b, W (3) 3a], [W (4) 1b, W (4) 1a, W (4) 2b, W (4) 3b, W (4) 3a, W (4) 4a]: distributed over F 1 + F 2, [W (4) 0, W (4) 2a, W (4) 4b]: distributed over A 1 + E. (without consideration of the above "distributed terms,) β abc = β bca = β cab = -β bac = -β acb = -β cba γ aaaa = γ bbbb = γ cccc, γ aabb = γ bbaa = γ aacc = γ bbcc = γ ccaa = γ ccbb, γ abba = γ baab = γ acca = γ bccb = γ caac = γ cbbc, γ abab = γ baba = γ acac = γ bcbc = γ caca = γ cbcb - 19
20 O h 2 E 8C 3 6C 2 6C 4 3C 4 S 2 6S 4 8S 6 3σ h 6σ d 3C 2 " I A 1g A 1u A 2g A 2u E g E u F 1g F 1u F 2g F 2u (T x, T y, ) (R x, R y, ) x 2 +y 2 +z 2 (x 2 +y 2-2z 2, x 2 y 2 ) (xy, xz, yz) O h A 1g D (0) 0, W (0) 0 A 1u Y (0) 0 A 2g W (3) 2b A 2u Y (3) 2b E g (D (2) 2a, D (2) 0), (W (2) 2a, W (2) 0) E u (Y (2) 2a, Y (2) 0) F 1g (D (1) 1b, D (1) 1a, D (1) 0), (W (1) 1b, W (1) 1a, W (1) 0) F 1u (V 1b, V 1a, V 0 ), (Y (1) 1b, Y (1) 1a, Y (1) 0) F 2g (D (2) 2b, D (2) 1b, D (2) 1a), (W (2) 2b, W (2) 1b, W (2) 1a) F 2u (Y (2) 2b, Y (2) 1b, Y (2) 1a) [Y (3) 0, Y (3) 1b, Y (3) 1a, Y (3) 2a, Y (3) 3b, Y (3) 3a]: distributed over F 1u + F 2u, [W (3) 0, W (3) 1b, W (3) 1a, W (3) 2a, W (3) 3b, W (3) 3a], [W (4) 1b, W (4) 1a, W (4) 2b, W (4) 3b, W (4) 3a, W (4) 4a]: distributed over F 1g + F 2g, [W (4) 0, W (4) 2a, W (4) 4b]: distributed over A 1g + E g. (without consideration of the above "distributed terms,) γ aaaa = γ bbbb = γ cccc, γ aabb = γ bbaa = γ aacc = γ bbcc = γ ccaa = γ ccbb, γ abba = γ baab = γ acca = γ bccb = γ caac = γ cbbc, γ abab = γ baba = γ acac = γ bcbc = γ caca = γ cbcb - 20
21 C v E 2C φ 2C 2φ 2C 3φ σ v Σ + Σ - Π Φ cosφ 2cos2φ 2cos3φ cos2φ 2cos4φ 2cos6φ cos3φ 2cos6φ 2cos9φ 0 C v Σ + V 0, D (0) 0, D (2) 0, Y (1) 0, Y (3) 0, W (0) 0, W (2) 0, W (4) 0 Σ - D (1) 0, Y (0 )0, Y (2) 0, W (1) 0, W (3) 0 P (V 1b, V 1a ), (D (1) 1b, D (1) 1a), (D (2) 1b, D (2) 1a), (Y (1) 1b, Y (1) 1a), (Y (2) 1b, Y (2) 1a), (Y (3) 1b, Y (3) 1a), (W (1) 1b, W (1) 1a), (W (2) 1b, W (2) 1a), (W (3) 1b, W (3) 1a), (W (4) 1b, W (4) 1a), D (D (2) 2b, D (2) 2a), (Y (2) 2b, Y (2) 2a), (Y (3) 2b, Y (3) 2a), (Y (3) 2b, Y (3) 2a), (W (1) 2b, W (1) 2a), (W (2) 2b, W (2) 2a), (W (3) 2b, W (3) 2a), (W (4) 2b, W (4) 2a) Φ (Y (3) 3b, Y (3) 3a), (W (3) 3b, W (3) 3a), (W (4) 3b, W (4) 3a) Γ (W (4) 4b, W (4) 4a) β ccc, β aac = β bbc, β aca = β bcb, β caa = β cbb γ abba = γ baab, γ acca = γ bccb, γ caac = γ cbbc - 21
22 D h E 2C φ 2C 2φ 2C 3φ σ h C 2 σ v 2S φ 2S 2φ S 2 Σ g + Σ u + Σ g - Σ u - Π g Π u g u Φ g Φ u cosφ +2cos2φ +2cos3φ cosφ -2cos2φ cosφ +2cos2φ +2cos3φ cosφ +2cos2φ cos2φ +2cos4φ +2cos6φ cos2φ -2cos4φ cos2φ +2cos4φ +2cos6φ cos2φ +2cos4φ cos3φ +2cos6φ +2cos9φ cos3φ -2cos6φ cos3φ +2cos6φ +2cos9φ cos3φ +2cos6φ -2 Γ Γ u D h Σ g + D (0) 0, D (2) 0, W (0) 0, W (2) 0, W (4) 0 Σ u + V 0, Y (1) 0, Y (3) 0 Σ g - D (1) 0, W (1) 0, W (3) 0 - Σ u Y (0) 0, Y (2) 0 Π g (D (1) 1b, D (1) 1a), (D (2) 1b, D (2) 1a), (W (1) 1b, W (1) 1a), (W (2) 1b, W (2) 1a), (W (3) 1b, W (3) 1a), (W (4) 1b, W (4) 1a), Π u (V 1b, V 1a ), (Y (1) 1b, Y (1) 1a), (Y (2) 1b, Y (2) 1a), (Y (3) 1b, Y (3) 1a) g (D (2) 2b, D (2) 2a), (W (2) 2b, W (2 2a), (W (3) 2b, W (3) 2a), (W (4) 2b, W (4) 2a) u (Y (2) 2b,Y (2) 2a), (Y (3) 2b,Y (3) 2a) Φ g (W (3) 3b,W (3) 3a), (W (4) 3b,W (4) 3a) Φ u (Y (3) 3b,Y (3) 3a) Γ Γ u (W (4) 4b,W (4) 4a) γ abba = γ baab, γ acca = γ bccb, γ caac = γ cbbc - 22
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離散数学
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知能科学：ニューラルネットワーク
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