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