Appendix a Space Groups

Appendix A Space Groups

Table A.1 gives the list of 230 three-periodic space groups. The point group international symbols are underlined for the space groups that appear as the ﬁrst ones in the list of a given crystal class.

Table A.1 230 three-periodic space groups

NG G IG NG G IG NG G IG Triclinic 1 1 1 C1 P1 2 Ci P1 Monoclinic 1 2 3 3 C2 P2 4 C2 P21 5 C2 C2 1 2 3 6 Cs Pm 7 Cs Pc 8 Cs Cm 4 1 / 2 / 9 Cs Cc 10 C2h P2 m 11 C2h P21 m 3 / 4 / 5 / 12 C2h C2 m 13 C2h P2 c 14 C2h P21 c 6 / 15 C2h C2 c Orthorhombic 1 2 3 16 D2 P222 17 D2 P2221 18 D2 P21212 4 5 6 19 D2 P212121 20 D2 C2221 21 D2 C222 7 8 9 22 D2 F222 23 D2 I 222 24 D2 I 212121 1 2 3 25 C2v Pmm2 26 C2v Pmc21 27 C2v Pcc2 4 5 6 28 C2v Pma2 29 C2v Pca21 30 C2v Pnc2 7 8 9 31 C2v Pmn21 32 C2v Pba2 33 C2v Pna21 10 11 12 34 C2v Pnn2 35 C2v Cmm2 36 C2v Cmc21 13 14 15 37 C2v Ccc2 38 C2v Amm2 39 C2v Abm2 16 17 18 40 C2v Ama2 41 C2v Aba2 42 C2v Fmm2 19 20 21 43 C2v Fdd2 44 C2v Imm2 45 C2v Iba2 22 1 2 46 C2v Ima2 47 D2h Pmmm 48 D2h Pnnn 3 4 5 49 D2h Pccm 50 D2h Pban 51 D2h Pmma (continued)

Table A.1 (continued)

NG G IG NG G IG NG G IG 6 7 8 52 D2h Pnna 53 D2h Pmna 54 D2h Pcca 9 10 11 55 D2h Pbam 56 D2h Pccn 57 D2h Pbcm 12 13 14 58 D2h Pnnm 59 D2h Pmmn 60 D2h Pbcn 15 16 17 61 D2h Pbca 62 D2h Pnma 63 D2h Cmcm 18 19 20 64 D2h Cmca 65 D2h Cmmm 66 D2h Cccm 21 22 23 67 D2h Cmma 68 D2h Ccca 69 D2h Fmmm 24 25 26 70 D2h Fddd 71 D2h Immm 72 D2h Ibam 27 28 73 D2h Ibca 74 D2h Imma Tetragonal 1 2 3 75 C4 P4 76 C4 P41 77 C4 P42 4 5 6 78 C4 P43 79 C4 I 4 80 C4 I 41 1 2 1 / 81 S4 P4 82 S4 I 4 83 C4h P4 m 2 / 3 / 4 / 84 C4h P42 m 85 C4h P4 n 86 C4h P42 n 5 / 6 / 1 87 C4h I 4 m 88 C4h I 41 a 89 D4 P422 2 3 4 90 D4 P4212 91 D4 P4122 92 D4 P41212 5 6 7 93 D4 P4222 94 D4 P42212 95 D4 P4322 8 9 10 96 D4 P43212 97 D4 I 422 98 D4 I 4122 1 2 3 99 C4v P4mm 100 C4v P4bm 101 C4v P42cm 4 5 6 102 C4v P42nm 103 C4v P4cc 104 C4v P4nc 7 8 9 105 C4v P42mc 106 C4v P42bc 107 C4v I 4mm 10 11 12 108 C4v I 4cm 109 C4v I 41md 110 C4v I 41cd 1 2 3 111 D2d P42m 112 D2d P42c 113 D2d P421m 4 5 6 114 D2d P421c 115 D2d P4m2 116 D2d P4c2 7 8 9 117 D2d P4b2 118 D2d P4n2 119 D2d I 4m2 10 11 12 120 D2d I 4c2 121 D2d I 42m 122 D2d I 42d 1 / 2 / 3 / 123 D4h P4 mmm 124 D4h P4 mcc 125 D4h P4 nbm 4 / 5 / 6 / 126 D4h P4 nnc 127 D4h P4 mbm 128 D4h P4 mnc 7 / 8 / 9 / 129 D4h P4 nmm 130 D4h P4 ncc 131 D4h P42 mmc 10 / 11 / 12 / 132 D4h P42 mcm 133 D4h P42 nbc 134 D4h P42 nnm 13 / 14 / 15 / 135 D4h P42 mbc 136 D4h P42 mnm 137 D4h P42 nmc 16 / 17 / 18 / 138 D4h P42 ncm 139 D4h I 4 mmm 140 D4h I 4 mcm 19 / 20 / 141 D4h I 41 amd 142 D4h I 41 acd Rhombohedral 1 2 3 143 C3 P3 144 C3 P31 145 C3 P32 4 1 2 146 C3 R3 147 C3i P3 148 C3i R3 1 2 3 149 D3 P312 150 D3 P321 151 D3 P3112 4 5 6 152 D3 P3121 153 D3 P3212 154 D3 P3221 7 1 2 155 D3 R32 156 C3v P3m1 157 C3v P31m (continued) Appendix A: Space Groups 655

Table A.1 (continued)

NG G IG NG G IG NG G IG 3 4 5 158 C3v P3c1 159 C3v P31c 160 C3v R3m 6 1 2 161 C3v R3c 162 D3d P31m 163 D3d P31c 3 4 5 164 D3d P3m1 165 D3d P3c1 166 D3d R3m 6 167 D3d R3c Hexagonal 1 2 3 168 C6 P6 169 C6 P61 170 C6 P65 4 5 6 171 C6 P62 172 C6 P64 173 C6 P63 1 1 / 2 / 174 C3h P6 175 C6h P6 m 176 C6h P63 m 1 2 3 177 D6 P662 178 D6 P6122 179 D6 P6522 4 5 6 180 D6 P6222 181 D6 P6422 182 D6 P6322 1 2 3 183 C6v P6mm 184 C6v P6cc 185 C6v P63cm 4 1 2 186 C6v P63mc 187 D3h P6m2 188 D3h P6c2 3 4 1 / 189 D3h P62m 190 D3h P62c 191 D6h P6 mmm 2 / 3 / 4 / 192 D6h P6 mcc 193 D6h P63 mcm 194 D6h P63 mmc Cubic 195 T 1 P23 196 T 2 F23 197 T 3 I 23 4 5 1 198 T P213 199 T I 213 200 Th Pm3 2 3 4 201 Th Pn3 202 Th Fm3 203 Th Fd3 5 6 7 204 Th Im3 205 Th Pa3 206 Th Ia3 1 2 3 207 O P432 208 O P4232 209 O F432 4 5 6 210 O F4132 211 O I 432 212 O P4332 7 8 1 213 O P4132 214 O I 4132 215 Td P43m 2 3 4 216 Td F43m 217 Td I 43m 218 Td P43n 5 6 1 219 Td F43c 220 Td I 43d 221 Oh Pm3m 2 3 4 222 Oh Pn3n 223 Oh Pm3n 224 Oh Pn3m 5 6 7 225 Oh Fm3m 226 Oh Fm3c 227 Oh Fd3m 8 9 10 228 Oh Fd3c 229 Oh Im3m 230 Oh Ia3d Appendix B Layer Groups as Subroups of Space Groups

Table B.1 shows the correspondence DG ↔ G/T3 between three-dimensional diperiodic (DG) and three-periodic (G) space groups. This correspondence was at ﬁrst given by Wood [1] in the setting that differs for rectangular diperiodic groups from that given in Table B.1 (the setting in this Table corresponds to the standard setting of [2]). Table B.2 presents the non-standard factorization of the diperiodic space groups. The generalized translational group, Z describes the periodical arrangement of the el- ementary motifs along two independent directions (these two directions are assumed to be in the xy-plane).

Table B.1 Correspondence between three-dimensional two-periodic (DG) and three-periodic (G) space groups (DG ↔ G/T3), [2] DG G DG G DG G 1 2 1 1 P1 C1 1 28 Pm21b C2v 26 55 P4mm C4v 99 1 2 2 2 P1 Ci 2 29 Pb21m C2v 26 56 P4bm C4v 100 1 3 1 3 P112 C2 3 30 Pb2b C2v 27 57 P42m D2d 111 1 4 3 4 P11m Cs 6 31 Pm2a C2v 28 58 P421m D2d 113 2 7 5 5 P11a Cs 7 32 Pm21n C2v 31 59 P4m2 D2d 115 / 1 5 7 6 P112 m C2h 10 33 Pb21a C2v 29 60 P4b2 D2d 117 / 4 6 / 1 7 P112 a C2h 13 34 Pb2n C2v 30 61 P4 mmm D4h 123 1 14 / 3 8 P211 C2 3 35 Cm2m C2v 38 62 P4 nbm D4h 125 2 15 / 5 9 P2111 C2 4 36 Cm2a 4C2v 39 63 P4 mbm D4h 127 3 1 / 7 10 C211 C2 5 37 Pmmm D2h 47 64 P4 nmm D4h 129 1 3 1 11 Pm11 Cs 6 38 Pmaa D2h 49 65 P3 C3 143 2 4 1 12 Pb11 Cs 7 39 Pban D2h 50 66 P3 C3i 147 3 5 1 13 Cm11 Cs 8 40 Pmam D2h 51 67 P312 D3 149 / 1 5 2 14 P2 m11 C2h 10 41 Pmma 4D2h 51 68 P321 D3 150 (continued) © Springer-Verlag Berlin Heidelberg 2015 657 R.A. Evarestov, Theoretical Modeling of Inorganic Nanostructures, NanoScience and Technology, DOI 10.1007/978-3-662-44581-5 658 Appendix B: Layer Groups as Subroups of Space Groups

Table B.1 (continued) DG G DG G DG G / 2 7 1 15 P21 m11 C2h 11 42 Pman D2h 53 69 P3m1 C3v 156 / 4 8 2 16 P2 b11 C2h 13 43 Pbaa D2h 54 70 P31m C3v 157 / 5 9 / 1 17 P21 b11 C2h 14 44 Pbam D2h 55 71 P312 m D3d 162 / 3 11 / 3 18 C2 m11 C2h 12 45 Pbma D2h 57 72 P32 m1 D3d 164 1 13 1 19 P222 D2 16 46 Pmmn D2h 59 73 P6 C6 168 2 19 1 20 P2122 D2 17 47 Cmmm D2h 65 74 P6 C3h 174 3 21 / 1 21 P21212 D2 18 48 Cmma D2h 67 75 P6 m C6h 175 6 1 1 22 C222 D2 21 49 P4 C4 75 76 P622 D6 177 1 1 1 23 Pmm2 C2v 25 50 P4 S4 81 77 P6mm C6v 183 4 / 1 1 24 Pma2 C2v 28 51 P4 m C4h 83 78 P6m2 D3h 187 8 / 3 3 25 Pba2 C2v 32 52 P4 n C4h 85 79 P62m D3h 189 11 1 / 1 26 Cmm2 C2v 35 53 P422 D4 89 80 P6 mmm D6h 191 1 2 27 Pm2m C2v 25 54 P4212 D4 90 The numbers of 17 plane groups are underlined

Table B.2 The non-standard factorization of the diperiodic groups [3] DG H I PZ Int.symb. DG H I PZ Int.symb 2 21 2 ( 21 2 2 ) 1 C2h C1 T p1 41 D2h D2h C2vTh p a m m m m a ¯ 2 2 21 ( 2 21 2 ) 2 S2 S2T p1 42 D2h D1dTh p n m a m a n 2 2 21 ( 2 21 2 ) 3 C2 C2T p211(112) 43 D2h D121Th p a b a b a a 2 21 21 ( 21 21 2 ) 4 C1h C1hT pm11(11m) 44 D2h C2hT v p m b a b a m 2 21 21 ( 21 21 2 ) 5 C1h Th pb11(11a) 45 D2h C221Th p a b m b m a 2 ( 2 ) 2 21 21 ( 21 21 2 ) 6 C2h C2hT p m 11 11 m 46 D2h C2vT h p n m m m m n 2 ( 2 ) 2 2 2 7 C2h C2Th p b 11 11 a 47 D2h D2hT p m m m 2 2 2 ( 2 2 2 ) 8 D2h D1 D1T p112(211) 48 D2h C2vT h c a m m m m e 9 D1 21 p1121(2111) 49 D4h C4 C4T p4 ¯ 10 D1 D1T c112(211) 50 S4 S4T p4 11 C1v C1vT p11m(m11) 51 C4h C4hT p4/m 12 C1v Tv p11a(b11) 52 C4h C4T h p4/n 13 C1v C1vT c11m(m11) 53 D4 D4T p422 2 ( 2 ) 14 D1d D1dT p11 m m 11 54 D4 C421 p4212 21 ( 21 ) 15 D1d S221 p11 m m 11 55 C4v C4vT p4mm (continued) Appendix B: Layer Groups as Subroups of Space Groups 659

Table B.2 (continued) DG H I PZ Int.symb. DG H I PZ Int.symb 2 ( 2 ) 16(18) D1d D1dT c11 m m 11 56 C4v C4T v p4bm 2 ( 2 ) ¯ 17(16) D1d S2Tv p11 a b 11 57 D2d D2dT p42m 21 ( 21 ) ¯ 18(17) D1d S2T v p11 a b 11 58 D2d S421 p421m ¯ 19 D2 D2T p222 59 D2d D2dT p4m2 ¯ 20 D2 C221 p2221(2122) 60 D2d S4T v p4b2 ( ) 4 2 2 21 D2 C221 p22121 21212 61 D4h D4hT p m m m 4 2 2 22 D2 D2T c222 62 D4h D2dT h p n b m ( ) 4 21 2 23 C2v C2vT p2mm mm2 63 D4h C4hT v p m b m ( ) 4 21 2 24(27) D1h D1hT pmm2 m2m 64 D4h D2dTh p n m m 25(29) D1h C1h21 pm21a(b21m) 65 D6h C3 C3T p3 ¯ 26(28) D1h C1v21 pbm21(m21b) 66 S6 S6T p3 27(30) D1h D1Tv pbb2(b2b) 67 D3 D3T p312 28(24) C2v C2Tv p2ma(ma2) 68 D3 D3T p321 29(31) D1h D1Th pam2(m2a) 69 C3v C3vT p3m1 30(33) D1h 21Th pab21(b21a) 70 C3v C3vT p31m ( ) ¯ 2 31(34) D1h D1Th pnb2 b2n 71 D3d D3dT p31 m ( ) ¯ 2 32 D1h C1vT h pnm21 m21n 72 D3d D3dT p3 m 1 33(25) C2v C2T v p2ba(ba2) 73 C6 C6T p6 ¯ 34(26) C2v C2vT c2mm(mm2) 74 C3h C3hT p6 35 D1h D1hT cmm2(m2m) 75 C6h C6hT p6/m 36 D1h D1T h cam2(m2e) 76 D6 D6T p622 2 2 2 37 D2h D2hT p m m m 77 C6v C6vT p6mm 2 2 2 ( 2 2 2 ) ¯ 38 D2h D2Th p a m a m a a 78 D3h D3hT p6m2 2 2 2 ( 2 2 2 ) ¯ 39 D2h D2T h p n b a b a n 79 D3h D3hT p62m 2 21 2 ( 21 2 2 ) 6 2 2 40 D2h C2hTv p m m a m a m 80 D6h D6hT p m m m The sequental numbers of diperiodic groups and their International symbols are given according to [1]. The sequental numbers and the International symbols from [2] are given in brackets if they differ from those in [1]. For each diperiodic group DG, the holohedry H, the isogonal point group I, the factorization PZ and the international symbol according to [1], are given Appendix C Line Groups

Table C.1 is taken from [4]. All line groups are divided into 13 families, each family includes inﬁnitely many line groups. Among these families there are 5 symmorphic (families 2, 3, 6, 9, 11) for which factorization (2.22) coincides with the standard one. The families 1 and 5 contain both incommensurate and commensurate (symmorphic and nonsymmorphic) line groups.

Table C.1 Line Groups. For each family F different factorizations, the roto-helical subgroup L(1), generators, and the isogonal point group PI, are given in the ﬁrst line (1) F qeven Factorizations q odd L Coset representatives PI + (F) NF International F i 1 TQ × Cn TQ × Cn (CQ | f ), Cn Cq

1 Lqp 1 2 T ∧ S2q T × Cq (E|a), S2q S2q 2 L2q Lq 1 S2q 3 T ∧ Cqh T × Cq (E|a), Cq ,σh Cqh 2 Lq/m L2q 1 σh 1 = 1 1 × ( | / ), ,σ 4 T2nCnh T2nS2n T2n Cn C2n a 2 Cn h C2nh 2 L(2n)n/m 1 σh 5 TQ ∧ Dn TQ × Cn (CQ | f ), Cn, U Dq 2 Lqp22 Lqp2 1 U 6 T × Cqv = Cqv ∧ T T × Cq (E|a), Cq ,σv Cqv 2 Lqmm Lqm 6 σv 7 Cq ∧ T T × Cq (σv|a/2), Cq Cqv (continued)

Table C.1 (continued) (1) F qeven Factorizations q odd L Coset representatives PI + (F) NF International F i 2 Lqcc Lqc 7 (σv|a/2) ∧ 1 = ∧ 1 × ( | / ), ,σ 8 Cnv T2n Cnv T d T2n Cn C2n a 2 Cn v C2nv 2 L(2n)nmc 8 σv 9 T ∧ Dqd = T ∧ Dqd T × Cq (E|a), Cq , Ud ,σv Dqd 4 L2q2m Lqm 6 σv, Ud , S2q 10 T S2q = T d Dq T × Cq (σv|a/2), S2q Dqd 4 L2q2c Lqc 7 (σv|a/2), S2q ,(Ud |a/2) 11 T ∧ Dqh = T Dqh T × Cq (E|a), Cq , U,σv Dqh 4 Lq/mmm L2qm2 7 σv, U,σh 12 T Cqh = T Dq T × Cq (σv|a/2), Cq ,σh Dqh 4 Lq/mcm L2q2c 7 (σv|a/2), U,(S2q |a/2) 1 = 1 = 1 × ( | / ), , ,σ 13 T2nDnh T2nDnd T2n Cn C2n a 2 Cn U v D2nh T d Dnh = T d Dnd 4 L(2n)n/mcm 8 σv, U,σh (F) (1) + NF =|L |L |, the international symbol (of commensurate groups only), family F of the ( ) positive subgroup L+ (for positive F = F+), and the coset representatives F for i > 1of(2.25) + i (when NF = 4, the ﬁrst one gives L ). T d and Ud are glide plane and horizontal axes bisecting vertical mirror planes, while S2q =Cq σh. For families 1 and 5, the order q of the isogonal principle axis is given by Q = q/r (for commensurate groups), otherwise q is inﬁnite [4]. In families 4,8,13 q = 2n is even Appendix D Rod Groups as Subgroups of Space Groups

Table D.1 gives the correspondence between rod and space groups—their international notation, the number of the corresponding SG and its Schönﬂis symbol. 75 rod groups can be distributed over 13 families, as is shown in Table D.2. Table D.3 is taken from [4] and presents rod groups in the form accepted for a line group classiﬁcation.

Table D.1 Rod Groups (RG) as subperiodic subgroups of Space Groups (SG) RG SG RG SG RG SG 1 4 3 1 P1 1 C1 26 P43 78 C4 51 P3m1 164 D3d 1 1 1 2 P1 2 Ci 27 P4 81 S4 P31m 162 D3d 1 / 1 4 3 P211 (3) C2 28 P4 m 83 C4h 52 P3c1 165 D3d 1 P121 (3) C2 1 / 2 2 4 Pm11 (6) Cs 29 P42 m 84 C4h P31c 163 D3d 1 P1m1 (6) Cs 2 1 1 5 P1c1 (7) Cs 30 P422 89 D4 53 P6 168 C6 2 3 2 Pc11 (7) Cs 31 P4122 91 D4 54 P61 169 C6 / 1 5 4 6 P12 m1 (10) C2h 32 P4222 93 D4 55 P62 171 C6 / 1 7 6 P2 m11 (10) C2h 33 P4322 95 D4 56 P63 173 C6 / 4 1 5 7 P12 c1 (13) C2h 34 P4mm 99 C4v 57 P64 172 C6 / 4 3 3 P2 c11 (13) C2h 35 P42cm 101 C4v 58 P65 170 C6 1 7 1 8 P112 (3) C2 P42mc 105 C4v 59 P6 174 C3h 5 / 1 36 P4cc 103 C4v 60 P6 m 175 C6h 2 1 / 2 9 P1121 4 C2 37 P42m 111 D2d 61 P63 m 176 C6h 1 5 1 10 P11m (6) Cs P4m2 115 D2d 62 P622 177 D6 2 2 38 P42c 112 D2d 63 P6122 178 D6 (continued) © Springer-Verlag Berlin Heidelberg 2015 663 R.A. Evarestov, Theoretical Modeling of Inorganic Nanostructures, NanoScience and Technology, DOI 10.1007/978-3-662-44581-5 664 Appendix D: Rod Groups as Subgroups of Space Groups

Table D.1 (continued) RG SG RG SG RG SG / 1 6 4 11 P112 m (10) C2h P4c2 116 D2d 64 P6222 180 D6 / 2 / 1 6 12 P1121 m 11 C2h 39 P4 mmm 123 D4h 65 P6322 182 D6 1 / 2 5 13 P222 16 D2 40 P4 mcc 124 D4h 66 P6422 181 D6 2 / 9 3 14 P2221 17 D2 41 P42 mmc 131 D4h 67 P6522 179 D6 1 / 10 1 15 Pmm2 (25) C2v P42 mcm 132 D4h 68 P6mm 183 C6v 3 1 2 16 Pcc2 27 C2v 42 P3 143 C3 69 P6cc 184 C6v 2 2 4 17 Pmc21 (26) C2v 43 P31 144 C3 70 P63mc 186 C6v 2 3 3 Pcm21 (26) C2v 44 P32 145 C3 P63cm 185 C6v 1 1 1 18 Pm2m (25) C2v 45 P3 147 C3i 71 P6m2 187 D3h 1 1 3 P2mm (25) C2v 46 P312 149 D3 P62m 189 D3h 4 2 2 19 P2cm (28) C2v P321 150 D3 72 P6c2 188 D3h 4 3 4 Pc2m (28) C2v 47 P3112 151 D3 P62c 190 D3h 1 4 / 1 20 Pmmm 47 D2h P3121 152 D3 73 P6 mmm 191 D6h 3 5 / 2 21 Pccm 49 D2h 48 P3212 153 D3 74 P6 mcc 192 D6h 5 6 / 4 22 Pcmm (51) D2h P3221 154 D3 75 P63 mmc 194 D6h 5 1 / 3 Pmcm (51) D2h 49 P3m1 156 C3v P63 mcm 193 D6h 1 2 23 P4 75 C4 P31m 157 C3v 2 3 24 P41 76 C4 50 P3c1 158 C3v 3 4 25 P42 77 C4 P31c 159 C3v The numbers of SG generating different RG or RG with different settings are given in brackets

Table D.2 Rod Groups Distribution over Line Group Families (q—order of the rotation or screw axis), F—family number, N, Nc—number of groups and classes in family F RGF Class Internat. N, q = 1 q = 2 q = 3 q = 4 q = 6 Symbol Nc (odd q,even q)

1 Cq q, qp 16 1(1) 8(112), 42(3) 23(4), 24(41) 53(6), 54(61) 9((1121) 43(31) 25(42), 26(43) 55(62), 56(63) 44(32) 57(64), 58(65) Classes 5 C1 C2 C3 C4 C6 5 Dq q2, q22 16 3(211) 13(222), 46(312) 30(422), 31(4122) 62(622), 63(6122) qq 2, qp22 14((2221) 47(3112) 32(4222), 33(4322) 64(6222), 65(6322) 48(3212) 66(6422), 67(6522) Classes 5 D1 D2 D3 D4 D6 2 S2q q, 2q 3 2(1) 27(4) 45(3) Classes 3 S2(Ci ) S4 S6(C3i) 3 Cqh 2q, q/m 5 10(11m) 11(112/m) 59(6) 28(4/m) 60(6/m) (continued) Appendix D: Rod Groups as Subgroups of Space Groups 665

Table D.2 (continued) RGF Class Internat. N, q = 1 q = 2 q = 3 q = 4 q = 6 Symbol Nc (odd q,even q)

4 Cqh (2n)n/m 3 12(1121/m) 29(42/m) 61(63/m) Classes 5 C1h C2h C3h C4h C6h 6 Cqv qm, qmm 6 4(m11) 15(mm2) 49(3m1) 34(4mm) 68(6mm) 7 Cqv qc, qcc 5 5(c11) 16(cc2) 50(3c1) 36(4cc) 69(6cc)

8 Cqv (2n)nmc 3 17(mc21) 35(42mc) 70(63mc) Classes 5 C1v C2v C3v C4v C6v 9 Dqd qm, 2q 2m 2 6(2/m11) 37(42m) 51(3 m1) 10 Dqd qc, 2q 2c 3 7(2/c11) 38(42c) 52(3 c1) Classes 3 D1d D2d D3d 11 Dqh 2qm2, q/mmm 5 18(2mm) 20(mmm) 71(6 m2) 39(4/mmm) 73(6/mmm) 12 Dqh 2q 2c, q/mcc 5 19(2cm) 21(ccm) 72(6 c2) 40(4/mcc) 74(6/mcc) 13 Dqh (2n)n/mmc 3 22(mcm) 41(42/mmc) 75(63/mmc) Classes 5 D1h D2h D3h D4h D6h total 75 rod groups, 31 crystal classes In families 4, 8, 13 q = 2n is even

Table D.3 Rod Groups (international and factorized notation) listed by families F of line groups according to the order q of the principal axis of the isogonal group, [4] F q = 1 q = 2 q = 3 q = 4 q = 6

1 1L1= TC1 8L2= TC2 42 L3 = TC3 23 L4 = TC4 53 L6 = TC6

9L21 = T2C1 43 L31 = T3C1 24 L41 = T4C1 54 L61 = T6C1 = 2 = = 44 L32 T3C1 25 L42 T4C2 55 L62 T6C2 = 3 = 26 L43 T4C1 56 L63 T6C3 = 2 57 L64 T6C2 = 5 58 L65 T6C1 2 2L1 = TS2 27 L4 = TS4 45 L3 = TS6

3 10 L2 = TC1h 11 L2/m = TC2h 59 L6 = TC3h 28 L4/m = TC4h 60 L6/m = TC6h / = / = 1 / = 1 4 12 L21 m T2C1h 29 L42 m T4C2h 61 L63 m T6C3h 5 3L12= TD1 13 L222 = TD2 46 L32 = TD3 30 L422 = TD4 62 L622 = TD6

14 L2122 = T2C1 47 L312 = T3C1 31 L4122 = T4D1 63 L6122 = T6D1 = 2 = = 47 L322 T3D1 32 L4222 T4D2 64 L6222 T6D3 = 1 = 1 33 L4322 T4D2 65 L6322 T6D3 = 2 66 L6422 T6D2 = 5 67 L6522 T6D1 6 4L1m = TC1v 15 L2mm = TC2v 49 L3m = TC3v 34 L4mm = TC4v 68 L6mm = TC6v 7 5L1c = T C1 16 L2cc = T C2 50 L3c = T C3 36 L4cc = T C4 69 L6cc = T C6

8 17 L21mc = T2C1v 35 L42mc = T4C2v 70 L63mc = T6C3v

9 6L1m = TD1d 37 L42m = TD2d 51 L3m = TD3d 10 7L1c = T S2 38 L42c = T S4 52 L3c = T S6

11 18 L22m = TD1h 20 L2/mmm = TD2h 71 L62m = TD3h 39 L4/mmm = TD4h 73 L6/mmm = TD6h 12 19 L22c = T C1h 21 L2/mcc = T C2h 72 L62c = T C3h 40 L4/mcc = T C4h 74 L6/mcc = T C6h

13 22 L21/mcm = T2D1d 41 L42/mcm = T4D2d 75 L63/mcm = T6D3d 666 Appendix D: Rod Groups as Subgroups of Space Groups

References

1. E.A. Wood, 80 Diperiodic Groups in Three Dimensions (Bell System Monograph No 4680, New York, 1964) 2. V. Kopsky, D.B. Litvin (eds.), International Tables for Crystallography. Subperiodic Groups, vol. E (Kluwer Academic Publishers, Dodrecht/Boston/London, 2002) 3. I. Milosević, B. Nikolić, M. Damnjanović, M. Kr˜cmar. J. Phys. A: Math. Gen. 31, 3625 (1998) 4. M. Damnjanović, I. Milosˇević, Line Groups in Physics. Lecture Notes in Physics, vol. 801 (Springer, Berlin, 2010) Index

A tetragonal, 17 Adiabatic approximation, 188 triclinic, 16, 17 type, 14 Brillouin zone B special points Band representation, 93 Monkhorst–Pack, 152 MgO and Si crystals, 95 symmetry line, 89 SrZrO3 crystal, 95 symmetry point, 89 Basic translation vectors, 13 Bulk crystals Basis sets in molecular calculations AlN, GaN localized atomic-like orbitals calculations, 384 correlation-consistent basis, 125 structure, 383 Gaussian-basis notations, 124 band structure Gaussian-type orbitals, 120 MgO, TiO2, 191 polarization functions, 122 BaTiO3 Slater-type orbitals, 118 calculations, 580, 581 Basis sets in periodic calculations structure, 579, 581 localized atomic-like orbitals, 118 symmetry, 579 basis set superposition error, 132 BN Gaussian-type basis, 128 structure, 349, 350 molecular-basis adaptation, 125 Boron, 218 plane waves, 115 electronic properties, 219 portable basis structure, 219 TZVP, 132 density of states Boundless crystal, 13 SrTiO3, SrZrO3, 194 Bravais lattice, 13 projected density of states, 194 2D (plane), 48 electron-density maps base-centered, 15 Cu2O, 195 body-centered, 15 Germanium-structure, 254 cubic, 17 group-III metal nitrides face-centered, 15 structure, 383 hexagonal, 17 MeB2 monoclinic, 16 structure, 239, 240 orthorhombic, 17 MoS2 parameters, 15 density of states, 613 point symmetry, 14 MoS2,WS2 primitive, 15 band structure, 612 © Springer-Verlag Berlin Heidelberg 2015 667 R.A. Evarestov, Theoretical Modeling of Inorganic Nanostructures, NanoScience and Technology, DOI 10.1007/978-3-662-44581-5 668 Index

calculations, 612 rutile, 34 structure, 611 ZrO2, 37 phonon symmetry type, 9, 28 TiO2, 204 Crystal system Silicon cubic, 14 structure, 254 hexagonal, 14 SrTiO3 monoclinic, 14 band structure, 547, 550 orthorhombic, 14 calculations, 546 rhombohedral, 14 Density of states, 550 syngony, 14, 15 phonon calculations, 551 tetragonal, 14 SrZrO3 triclinic, 14 band structure, 547 Crystallographic point groups Density of states, 550 international notation, 17 TiS2 Schönﬂies notation, 17 structure, 631 TiS2,ZrS2 D calculations, 635 Density-functional theory (crystals) structure, 634 LCAO approximation, 117 vibrational frequences HSE03 hybrid functional, 176 TiO2, 206 Kohn–Sham equations, 171 calculation, 202 linear-scaling method, 173 Bulk properties screened Coulomb hybrid SiC, GeC functional, 178 structure, 306 Density-functional theory (molecules) generalized-gradient approximation GGA C exchange-correlation Conventional unit cell, 17 functionals, 164 Crystal structure PBE functional, 165 cubic Hohenberg–Kohn theorems, 154 diamond, 32 Kohn–Sham equations, 156, 158 ﬂuorite, 32 local density approximation LDA perovskite, 33 exchange-correlation zincblende, 33 functionals, 161 database, 30 local spin density approximation LSDA, description, 29 162 hexagonal orbital-dependent exchange-correlation functionals AlB2,TiB2, 44 BN, 43 hybrid functionals, 167, 169 graphite, 42 Thomas-Fermi model, 153 graphite, boron nitride, Density Functional Tight Binding (DFTB) molybdenite, 40 bulk crystals, 114 parameters, 114 MoS2, 41 TiS2,TiB2, 41 wurtzite ZnS, 44 E orthorhombic Effective core potentials BaTiO3,V2O5, lepidocrocite, 39 nonrelativistic tetragonal energy-consistent, 137 BaTiO3,PbTiO3, 39 generation, 138 SrTiO3, 37 Hay–Wadt potentials, 140 anatase, 34 separable embedding potentials, 141 BaTiO3, 39 valence basis-sets, 139 Index 669

relativistic, 142 family 8, 101 Stevens–Basch–Krauss ﬁrst family, 100 potentials, 141 orbits, 66, 67 Stuttgart-Dresden potentials, 141 stabilizer, 67 Electron correlation spin correlation, 145 M Monoatomic crystal, 14 F Fullerene-like N SiC Nanochains structure, 325 BN calculations, 379 G structure, 378, 379 Group Nanolayers plane, 48 AlN, GaN space, 9, 11 calculations, 385 subperiodic, 44 structure, 385 BaTiO3 calculations, 582–585, 587 H stability, 585 Hartree-Fock method structure, 582 LCAO approximation Boron crystals, 150, 151 structure, 219–221 restricted for open shells ROHF, 148 symmetry, 222 restricted RHF, 147 Germanium unrestricted UHF, 148 calculations, 260 MeB2 structure, 241 I MoS Induced representations 2 density of states, 618 of space group MoS ,WS q-basis, 92 2 2 phonon calculations, 616 Irreducible representations, 9, 11 structure, 615, 616 of space group SiC full representation, 90 calculations, 309 of translation group structure, 256, 307 k vector, 84 SiC, GeSi Bloch functions, 85 calculations, 307 Brillouin zone, 85 Silicon, Germanium calculations, 256, 257 L structure, 254 Layer group SrTiO3, SrZrO3 Brillouin zone, 97 (0 0 1), density of states, 556 element, 49 surface (0 0 1), 553 irreducible representations, 97 surface (0 0 1) energy, 555 setting, 45 TiS2,ZrS2 site-symmetry, 50 calculations, 636 table, 657 structure, 635 Line groups, 63 Nanoribbons factorization, 61, 62 MoS2 families, 60 band gap, 630 irreducible representations, 100 calculations, 629 family 4, 100 properties, 629 670 Index

WS2 calculations, 223, 225–227 calculations, 631 double-wall, 232 Nanoscrolls double-wall, calculations, 233, 235 BN electronic structure, 232 structure, 371, 373 structure, 224 GeSi structure, 320 double-wall Nanotubes symmetry, 75 AlB2 GaN calculations, 246, 247 band gap, 391 AlN band structure, 390 band structure, 398 Germanium faceted structure, 397 calculations, 261 stability, 389 GeSi strain energy, 388, 397, 398 structure, 386 binding energy, 322 AlN calculations, 387 calculations, 322 AlN, GaN MeB2 calculations, 387 calculations, 241, 242, 245, 246 AlN/GaN structure, 241 band gap, 394 MgB2 band structure, 393 calculations, 245 calculations, 395 structure, 393 MoS2 bundles, 626 BaTiO3 calculations, 589, 596, 597 defective, 625 properties, 590 strain energy, 622 strain energy, 594 symmetry, 620 structure, 590, 591 MoS2,WS2 BeB2,MgB2 calculations, 623 calculations, 244 DFTB calculations, 624 BeB2,MgB2,AlB2 calculations, 242 structure, 620 BN multiwall band structure, 366 symmetry, 76 bundle calculations, 364 rolled up bundle structure, 363 single-wall, 68 double-wall, band structure, 369 symmetry, 67, 69 double-wall, binding energy, 367 symmetry implementation, 107, 108 double-wall, electronic structure, 368 SiC double-wall, structure, 366 growth mechanism, 360, 361 band structure, 311 phonon calculations, 356, 357 bundles, density of states, 314 polyhedral, calculations, 352 bundles, structure, 313, 314 pressure inﬂuence, 364 calculations, 258, 313 Raman spectra, 359 density of states, 318 rolled up, calculations, 353 double-wall, calculations, 316 double-wall, structure, 315 structure, 351 thermodynamics, 362 faceted, structure, 317 Boron properties, 319 bundle calculations, 228, 230, 231 stability, 259 bundle structure, 228 structure, 307 Index 671

SiC, GeC BaTiO3 structure, 310 calculations, 600–603 structure calculations, 310 ferroelectricity, 604 Silicon MD simulations, 607 calculations, 258, 266 properties, 604 double-wall, 260, 263, 264 structure, 598 phonon calculations, 266 BN stability, 259 DFTB calculations, 377 structure, 261, 262, 265 MD simulations, 377 SrTiO3 stability calculations, 374, 375 calculations, 566 structure, 373 double wall, symmetry, 559, 560 structure calculations, 373 single wall, calculations, 561–564 Boron single wall, symmetry, 559 calculations, 238 symmetry, 545 calculations, electronic SrTiO3,BaTiO3 properties, 238 comparison, 594 structure, 236, 237 TiB2 GaN calculations, 248–250 band gap, 419 TiO2 double wall band structure, 415, 416 calculations, 466 formation energy, 419 TiO2 single wall stability, 420 calculations, 459 structure, 415 rectangular, 470, 473 surface energy, 420, 421 rectangular, calculations, 473 Young’s modulus, 422 symmetry, 458 Ge/Si and Si/Ge TiS2,ZrS2 band structure, 340 calculations, 640 calculations, 337 comparison with dioxides, 642 Germanium double-wall, calculations, 645 band gap calculations, 305 double-wall, stability, 644 calculations, 298, 299, 301, 302 double-wall, structure, 646, 647 stability calculations, 302, 304 double-wall, symmetry, 643 structure, 298 strain energies, 640 GeSi structure, 638, 640 band structure, 338 WS2 calculations, 333–335, 337 structure, 619 structure, 331, 333, 336 ZnO-symmetry, 436 MoS2 ZrO2-calculations, 533, 538 calculations, 628 ZrO2-symmetry, 530 MoS2,WS2 Nanowires models of the structure, 627 AlN structure, 626 band structure, 398 SiC calculations, 401 band gap, 329 density of states, 402 band structure, 329 elastic constants, 411, 413 calculations, 323, 324 elastic properties, 410, 414 electronic structure, 328, 329 formation energy, 405 structure, 323, 324 phase transitions, 406 Silicon phonon calculations, 410 calculations, 269, 271, 273, 275–279, structure, 400, 403 281, 282 thermal properties, 407, 409 calculations of mechanical properties, 295 672 Index

elastic properties, 293 rectangular lattices, 74 MD calculations, 285 square lattice, 73 mechanical properties calculations, Site-symmetry group, 24 296–298 oriented, 28 phonon calculations, 289–293 Space group, 20 phonon properties, 283, 286, 288 designations, 21 structure, 269, 271, 283 elements, 20 thermal properties, 283, 284, 286 nonsymmorphic, 21 vibrational spectra, 292 symmorphic, 21 Wulff construction, 271, 273 table, 21 Young’s modulus, 294 Supercell Silicon, Germanium for centered lattices, 19 comparison, 301 transformation, 18 structure, 268 Surfaces of crystals SrTiO3 models calculations, 566, 568 single slab, 45 ZnO Surfaces types symmetry, 436 type-2 TiO2 (110), 53 Symmetry operations, 12 O Symmetry properties Orbits of points, 9, 12, 28 of crystalline orbitals, 82 crystallographic, 24 of molecular orbitals, 82 time-reversal transformation, 83 P Periodic boundary conditions, 13 T Primitive unit cell, 13 Translation improper or fractional, 20 proper, 20 R Reference unit cell, 14 Relativistic theory molecules, 142 U Rod groups U2N3-crystal as line groups, 64, 663 calculations, 144 axial point groups, 63 UF6 molecule, 143 correspondence to space groups, 63, 663 elements, 63 families, 64, 663 W family 4, 64 Wavevector ﬁrst family, 64 little group, 90, 99 realization, 65 point-symmetry group, 88 star, 90, 99 Wyckoff positions, 9, 28 S notation, 24 Single wall nanotubes, 70 parameter-dependent, 24 hexagonal lattice, 71 parameter-free, 24