1% 2% $Id$ 3% 4\label{symexamples} 5 6Below are examples of the use of the \verb+SYMMETRY+ directive in 7the compound \verb+GEOMETRY+ directive (Section \ref{sec:geom}). 8The z axis is always the primary rotation axis. 9When in doubt about which axes and planes are used for the group 10elements, the keyword \verb+print+ may be added to the \verb+SYMMETRY+ 11directive to obtain this information. 12 13 \section{\protect$C_{s}$ methanol} 14 15The $\sigma_h$ plane is the xy plane. 16 17\begin{verbatim} 18 geometry units angstroms 19 C 0.11931097 -0.66334875 0.00000000 20 H 1.20599017 -0.87824237 0.00000000 21 H -0.32267592 -1.15740001 0.89812652 22 O -0.01716588 0.78143468 0.00000000 23 H -1.04379735 0.88169812 0.00000000 24 symmetry cs 25 end 26\end{verbatim} 27 28 \section{\protect$C_{2v}$ water} 29 30The z axis is the $C_2$ axis and the $\sigma_v$ may be either the xz or the 31yz planes. 32 33\begin{verbatim} 34 geometry units au 35 O 0.00000000 0.00000000 0.00000000 36 H 0.00000000 1.43042809 -1.10715266 37 symmetry group c2v 38 end 39\end{verbatim} 40 41 \section{\protect$D_{2h}$ acetylene} 42 43Although acetylene has symmetry $D_{\infty h}$ the subgroup 44$D_{2h}$ includes all operations that interchange equivalent atoms 45which is what determines how much speedup you gain from using symmetry 46in building a Fock matrix. 47 48The $C_2$ axes are the x, y, and z axes. The $\sigma$ planes are the xy, 49xz and yz planes. Generally, the unique atoms are placed to use the z 50as the primary rotational axis and use the xz or yz planes as the $\sigma$ 51plane. 52 53\begin{verbatim} 54 geometry units au 55 symmetry group d2h 56 C 0.000000000 0.000000000 -1.115108538 57 H 0.000000000 0.000000000 -3.106737425 58 end 59\end{verbatim} 60 61 \section{\protect$D_{2h}$ ethylene} 62 63The $C_2$ axes are the x, y, and z axes. The $\sigma$ planes are the xy, 64xz and yz planes. Generally, the unique atoms are placed to use the z 65as the primary rotational axis and use the xz or yz planes as the $\sigma$ 66plane. 67 68\begin{verbatim} 69 geometry units angstroms 70 C 0 0 0.659250 71 H 0 0.916366 1.224352 72 symmetry d2h 73 end 74\end{verbatim} 75 76 77 \section{\protect$T_d$ methane} 78 79For ease of use, the primary $C_3$ axis should be the x=y=z axis. 80The 3 $C_2$ axes are the x, y, and z. 81 82\begin{verbatim} 83 geometry units au 84 c 0.0000000 0.0000000 0.0000000 85 h 1.1828637 1.1828637 1.1828637 86 symmetry group Td 87 end 88\end{verbatim} 89 90 \section{\protect$I_h$ buckminsterfullerene} 91 92One of the $C_5$ axes is the z axis and the point of inversion is the origin. 93 94\begin{verbatim} 95 geometry units angstroms # Bonds = 1.4445, 1.3945 96 symmetry group Ih 97 c -1.2287651 0.0 3.3143121 98 end 99\end{verbatim} 100 101 \section{\protect$S_4$ porphyrin} 102 103The $S_4$ and $C_2$ rotation axis is the z axis. The reflection plane 104for the $S_4$ operation is the xy plane. 105 106\begin{verbatim} 107 geometry units angstroms 108 symmetry group s4 109 110 fe 0.000 0.000 0.000 111 h 2.242 6.496 -3.320 112 h 1.542 4.304 -2.811 113 c 1.947 6.284 -2.433 114 c 1.568 4.987 -2.084 115 h 2.252 8.213 -1.695 116 c 1.993 7.278 -1.458 117 h 5.474 -1.041 -1.143 118 c 1.234 4.676 -0.765 119 h 7.738 -1.714 -0.606 120 c 0.857 3.276 -0.417 121 h 1.380 -4.889 -0.413 122 c 1.875 2.341 -0.234 123 h 3.629 3.659 -0.234 124 c 0.493 -2.964 -0.229 125 c 1.551 -3.933 -0.221 126 c 5.678 -1.273 -0.198 127 c 1.656 6.974 -0.144 128 c 3.261 2.696 -0.100 129 n 1.702 0.990 -0.035 130 end 131\end{verbatim} 132 133 \section{\protect$D_{3h}$ iron penta-carbonyl} 134 135The $C_3$ axis is the z axis. The $\sigma_h$ plane is the xy plane. One of 136the perpendicular $C_2$ axes is the x=y axis. One of the $\sigma_v$ 137planes is the plane containing the x=y axis and the z axis. (The 138other axes and planes are generated by the $C_3$ operation.) 139 140\begin{verbatim} 141 geometry units au 142 symmetry group d3h 143 144 fe 0.0 0.0 0.0 145 146 c 0.0 0.0 3.414358 147 o 0.0 0.0 5.591323 148 149 c 2.4417087 2.4417087 0.0 150 o 3.9810552 3.9810552 0.0 151 end 152\end{verbatim} 153 154 \section{\protect$D_{3d}$ sodium crown ether} 155 156The $C_3$ axis is the z axis. The point of inversion is the origin. One of 157the perpendicular $C_2$ axes is the x=y axis. One of the $\sigma_d$ planes 158is the plane containing the -x=y axis and the z axis. 159 160 Note that the oxygen atom is rotated in the x-y plane 15 161 degrees away from the y-axis so that it lies in a mirror 162 plane. There is a total of six atoms generated from the 163 unique oxygen, in contrast to twelve from each of the carbon 164 and hydrogen atoms. 165 166\begin{verbatim} 167 geometry units au 168 symmetry D3d 169 170 NA .0000000000 .0000000000 .0000000000 171 O 1.3384771885 4.9952647969 .1544089284 172 H 6.7342048019 -0.6723850379 2.6581562148 173 C 6.7599180056 -0.4844977035 .6136583870 174 H 8.6497577017 0.0709194071 .0345361934 175 176 end 177\end{verbatim} 178 179 \section{\protect$C_{3v}$ ammonia} 180 181The $C_3$ axis is the z axis. One of the $\sigma_v$ planes is the 182plane containing the x=y axis and the z axis. 183 184\begin{verbatim} 185 geometry units angstroms 186 N 0 0 -0.055 187 H 0.665 0.665 -0.481 188 symmetry c3v 189 end 190\end{verbatim} 191 192 \section{\protect$D_{6h}$ benzene} 193 194The $C_6$ axis is the z axis. The point of inversion is the origin. One 195of the 6 perpendicular $C_2'$ axes is the x=y axis. (-x=y works as a 196$C_2''$ axis.) The $\sigma_h$ plane is the xy plane. The $\sigma_d$ 197planes contain the $C_2''$ axis and the z axis. The $\sigma_v$ planes 198contain the $C_2'$ axis and the z axis. 199 200\begin{verbatim} 201 geometry units au 202 C 1.855 1.855 0 203 H 3.289 3.289 0 204 symmetry D6h 205 end 206\end{verbatim} 207 208 \section{\protect$C_{3h}$ \protect$BO_3H_3$} 209 210The $C_3$ axis is the z axis. The $\sigma_h$ plane is the xy plane. 211 212\begin{verbatim} 213 geometry units au 214 b 0 0 0 215 o 2.27238285 1.19464491 0.00000000 216 h 2.10895420 2.97347707 0.00000000 217 symmetry C3h 218 end 219\end{verbatim} 220 221 \section{\protect$D_{5d}$ ferrocene} 222 223The $C_5$ axis is the z axis. The center of inversion is the origin. 224One of the perpendicular $C_2$ axes is the x axis. One of the $\sigma_d$ 225planes is the yz plane. 226 227\begin{verbatim} 228 geometry units angstroms 229 symmetry d5d 230 231 fe 0 0 0 232 c 0 1.194 1.789 233 h 0 2.256 1.789 234 end 235\end{verbatim} 236 237 \section{\protect$C_{4v}$ \protect$SF_5Cl$} 238 239The $C_4$ axis is the z axis. The $\sigma_v$ planes are the yz and the xz 240planes. The $\sigma_d$ planes are: 1) the plane containing the x=y axis and 241the z axis and 2) the plane containing the -x=y axis and the z axis. 242 243\begin{verbatim} 244 geometry units au 245 S 0.00000000 0.00000000 -0.14917600 246 Cl 0.00000000 0.00000000 4.03279700 247 F 3.13694200 0.00000000 -0.15321800 248 F 0.00000000 0.00000000 -3.27074500 249 250 symmetry C4v 251 end 252\end{verbatim} 253 254 \section{\protect$C_{2h}$ trans-dichloroethylene} 255 256The $C_2$ axis is the z axis. The origin is the inversion center. 257The $\sigma_h$ plane is the xy plane. 258 259\begin{verbatim} 260 geometry units angstroms 261 C 0.65051239 -0.08305064 0 262 Cl 1.75249381 1.30491767 0 263 H 1.14820954 -1.04789741 0 264 symmetry C2h 265 end 266\end{verbatim} 267 268 \section{\protect$D_{2d}$ \protect$CH_2CCH_2$} 269 270The $C_2$ axis is the z axis (z is also the $S_4$ axis). The x and y axes 271are the perpendicular $C_2'$s. The $\sigma_d$ planes are: 2721) the plane containing the x=y axis and the z axis and 2) the plane 273containing the -x=y axis and the z axis. 274 275\begin{verbatim} 276 geometry units angstroms 277 symmetry d2d 278 c 0 0 0 279 c 0 0 1.300 280 h 0.656 0.656 1.857 281 end 282\end{verbatim} 283 284 \section{\protect$D_{5h}$ cyclopentadiene anion} 285 286The $C_5$ axis is the z axis (z is also the $S_5$ axis). The y axis is one 287of the perpendicular $C_2$ axes. The $\sigma_h$ plane is the xy plane and 288one of the $\sigma_d$ planes is the yz plane. 289 290\begin{verbatim} 291 charge -1 292 geometry units angstroms 293 symmetry d5h 294 c 0 1.1853 0 295 h 0 2.2654 0 296 end 297\end{verbatim} 298 299 300 \section{\protect$D_{4h}$ gold tetrachloride} 301 302The $C_4$ axis is the z axis (z is also the $S_4$ axis). 303The $C_2'$ axes are the x and y axes and the $C_2''$ axes are the x=y axis 304and the x=-y axis. The inversion center is the origin. The $\sigma_h$ 305plane is the xy plane. The $\sigma_v$ planes are the xz and yz planes and 306the $\sigma_d$ planes are 1) the plane containing the x=-y axis and the z axis 307and 2) the plane containing the x=y axis and the z axis. 308 309\begin{verbatim} 310 geometry units au 311 Au 0 0 0 312 Cl 0 4.033 0 313 symmetry D4h 314 end 315\end{verbatim} 316