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315 lines
8.8 KiB
TeX
315 lines
8.8 KiB
TeX
%
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% $Id$
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%
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\label{symexamples}
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Below are examples of the use of the \verb+SYMMETRY+ directive in
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the compound \verb+GEOMETRY+ directive (Section \ref{sec:geom}).
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The z axis is always the primary rotation axis.
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When in doubt about which axes and planes are used for the group
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elements, the keyword \verb+print+ may be added to the \verb+SYMMETRY+
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directive to obtain this information.
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\section{\protect$C_{s}$ methanol}
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The $\sigma_h$ plane is the xy plane.
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\begin{verbatim}
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geometry units angstroms
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C 0.11931097 -0.66334875 0.00000000
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H 1.20599017 -0.87824237 0.00000000
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H -0.32267592 -1.15740001 0.89812652
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O -0.01716588 0.78143468 0.00000000
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H -1.04379735 0.88169812 0.00000000
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symmetry cs
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end
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\end{verbatim}
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\section{\protect$C_{2v}$ water}
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The z axis is the $C_2$ axis and the $\sigma_v$ may be either the xz or the
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yz planes.
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\begin{verbatim}
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geometry units au
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O 0.00000000 0.00000000 0.00000000
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H 0.00000000 1.43042809 -1.10715266
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symmetry group c2v
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end
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\end{verbatim}
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\section{\protect$D_{2h}$ acetylene}
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Although acetylene has symmetry $D_{\infty h}$ the subgroup
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$D_{2h}$ includes all operations that interchange equivalent atoms
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which is what determines how much speedup you gain from using symmetry
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in building a Fock matrix.
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The $C_2$ axes are the x, y, and z axes. The $\sigma$ planes are the xy,
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xz and yz planes. Generally, the unique atoms are placed to use the z
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as the primary rotational axis and use the xz or yz planes as the $\sigma$
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plane.
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\begin{verbatim}
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geometry units au
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symmetry group d2h
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C 0.000000000 0.000000000 -1.115108538
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H 0.000000000 0.000000000 -3.106737425
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end
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\end{verbatim}
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\section{\protect$D_{2h}$ ethylene}
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The $C_2$ axes are the x, y, and z axes. The $\sigma$ planes are the xy,
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xz and yz planes. Generally, the unique atoms are placed to use the z
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as the primary rotational axis and use the xz or yz planes as the $\sigma$
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plane.
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\begin{verbatim}
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geometry units angstroms
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C 0 0 0.659250
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H 0 0.916366 1.224352
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symmetry d2h
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end
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\end{verbatim}
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\section{\protect$T_d$ methane}
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For ease of use, the primary $C_3$ axis should be the x=y=z axis.
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The 3 $C_2$ axes are the x, y, and z.
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\begin{verbatim}
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geometry units au
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c 0.0000000 0.0000000 0.0000000
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h 1.1828637 1.1828637 1.1828637
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symmetry group Td
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end
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\end{verbatim}
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\section{\protect$I_h$ buckminsterfullerene}
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One of the $C_5$ axes is the z axis and the point of inversion is the origin.
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\begin{verbatim}
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geometry units angstroms # Bonds = 1.4445, 1.3945
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symmetry group Ih
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c -1.2287651 0.0 3.3143121
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end
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\end{verbatim}
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\section{\protect$S_4$ porphyrin}
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The $S_4$ and $C_2$ rotation axis is the z axis. The reflection plane
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for the $S_4$ operation is the xy plane.
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\begin{verbatim}
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geometry units angstroms
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symmetry group s4
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fe 0.000 0.000 0.000
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h 2.242 6.496 -3.320
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h 1.542 4.304 -2.811
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c 1.947 6.284 -2.433
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c 1.568 4.987 -2.084
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h 2.252 8.213 -1.695
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c 1.993 7.278 -1.458
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h 5.474 -1.041 -1.143
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c 1.234 4.676 -0.765
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h 7.738 -1.714 -0.606
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c 0.857 3.276 -0.417
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h 1.380 -4.889 -0.413
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c 1.875 2.341 -0.234
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h 3.629 3.659 -0.234
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c 0.493 -2.964 -0.229
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c 1.551 -3.933 -0.221
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c 5.678 -1.273 -0.198
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c 1.656 6.974 -0.144
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c 3.261 2.696 -0.100
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n 1.702 0.990 -0.035
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end
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\end{verbatim}
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\section{\protect$D_{3h}$ iron penta-carbonyl}
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The $C_3$ axis is the z axis. The $\sigma_h$ plane is the xy plane. One of
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the perpendicular $C_2$ axes is the x=y axis. One of the $\sigma_v$
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planes is the plane containing the x=y axis and the z axis. (The
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other axes and planes are generated by the $C_3$ operation.)
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\begin{verbatim}
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geometry units au
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symmetry group d3h
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fe 0.0 0.0 0.0
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c 0.0 0.0 3.414358
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o 0.0 0.0 5.591323
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c 2.4417087 2.4417087 0.0
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o 3.9810552 3.9810552 0.0
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end
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\end{verbatim}
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\section{\protect$D_{3d}$ sodium crown ether}
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The $C_3$ axis is the z axis. The point of inversion is the origin. One of
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the perpendicular $C_2$ axes is the x=y axis. One of the $\sigma_d$ planes
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is the plane containing the -x=y axis and the z axis.
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Note that the oxygen atom is rotated in the x-y plane 15
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degrees away from the y-axis so that it lies in a mirror
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plane. There is a total of six atoms generated from the
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unique oxygen, in contrast to twelve from each of the carbon
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and hydrogen atoms.
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\begin{verbatim}
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geometry units au
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symmetry D3d
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NA .0000000000 .0000000000 .0000000000
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O 1.3384771885 4.9952647969 .1544089284
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H 6.7342048019 -0.6723850379 2.6581562148
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C 6.7599180056 -0.4844977035 .6136583870
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H 8.6497577017 0.0709194071 .0345361934
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end
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\end{verbatim}
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\section{\protect$C_{3v}$ ammonia}
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The $C_3$ axis is the z axis. One of the $\sigma_v$ planes is the
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plane containing the x=y axis and the z axis.
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\begin{verbatim}
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geometry units angstroms
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N 0 0 -0.055
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H 0.665 0.665 -0.481
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symmetry c3v
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end
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\end{verbatim}
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\section{\protect$D_{6h}$ benzene}
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The $C_6$ axis is the z axis. The point of inversion is the origin. One
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of the 6 perpendicular $C_2'$ axes is the x=y axis. (-x=y works as a
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$C_2''$ axis.) The $\sigma_h$ plane is the xy plane. The $\sigma_d$
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planes contain the $C_2''$ axis and the z axis. The $\sigma_v$ planes
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contain the $C_2'$ axis and the z axis.
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\begin{verbatim}
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geometry units au
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C 1.855 1.855 0
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H 3.289 3.289 0
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symmetry D6h
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end
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\end{verbatim}
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\section{\protect$C_{3h}$ \protect$BO_3H_3$}
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The $C_3$ axis is the z axis. The $\sigma_h$ plane is the xy plane.
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\begin{verbatim}
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geometry units au
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b 0 0 0
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o 2.27238285 1.19464491 0.00000000
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h 2.10895420 2.97347707 0.00000000
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symmetry C3h
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end
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\end{verbatim}
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\section{\protect$D_{5d}$ ferrocene}
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The $C_5$ axis is the z axis. The center of inversion is the origin.
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One of the perpendicular $C_2$ axes is the x axis. One of the $\sigma_d$
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planes is the yz plane.
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\begin{verbatim}
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geometry units angstroms
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symmetry d5d
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fe 0 0 0
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c 0 1.194 1.789
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h 0 2.256 1.789
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end
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\end{verbatim}
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\section{\protect$C_{4v}$ \protect$SF_5Cl$}
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The $C_4$ axis is the z axis. The $\sigma_v$ planes are the yz and the xz
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planes. The $\sigma_d$ planes are: 1) the plane containing the x=y axis and
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the z axis and 2) the plane containing the -x=y axis and the z axis.
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\begin{verbatim}
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geometry units au
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S 0.00000000 0.00000000 -0.14917600
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Cl 0.00000000 0.00000000 4.03279700
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F 3.13694200 0.00000000 -0.15321800
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F 0.00000000 0.00000000 -3.27074500
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symmetry C4v
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end
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\end{verbatim}
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\section{\protect$C_{2h}$ trans-dichloroethylene}
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The $C_2$ axis is the z axis. The origin is the inversion center.
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The $\sigma_h$ plane is the xy plane.
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\begin{verbatim}
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geometry units angstroms
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C 0.65051239 -0.08305064 0
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Cl 1.75249381 1.30491767 0
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H 1.14820954 -1.04789741 0
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symmetry C2h
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end
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\end{verbatim}
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\section{\protect$D_{2d}$ \protect$CH_2CCH_2$}
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The $C_2$ axis is the z axis (z is also the $S_4$ axis). The x and y axes
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are the perpendicular $C_2'$s. The $\sigma_d$ planes are:
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1) the plane containing the x=y axis and the z axis and 2) the plane
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containing the -x=y axis and the z axis.
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\begin{verbatim}
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geometry units angstroms
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symmetry d2d
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c 0 0 0
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c 0 0 1.300
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h 0.656 0.656 1.857
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end
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\end{verbatim}
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\section{\protect$D_{5h}$ cyclopentadiene anion}
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The $C_5$ axis is the z axis (z is also the $S_5$ axis). The y axis is one
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of the perpendicular $C_2$ axes. The $\sigma_h$ plane is the xy plane and
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one of the $\sigma_d$ planes is the yz plane.
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\begin{verbatim}
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charge -1
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geometry units angstroms
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symmetry d5h
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c 0 1.1853 0
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h 0 2.2654 0
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end
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\end{verbatim}
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\section{\protect$D_{4h}$ gold tetrachloride}
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The $C_4$ axis is the z axis (z is also the $S_4$ axis).
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The $C_2'$ axes are the x and y axes and the $C_2''$ axes are the x=y axis
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and the x=-y axis. The inversion center is the origin. The $\sigma_h$
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plane is the xy plane. The $\sigma_v$ planes are the xz and yz planes and
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the $\sigma_d$ planes are 1) the plane containing the x=-y axis and the z axis
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and 2) the plane containing the x=y axis and the z axis.
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\begin{verbatim}
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geometry units au
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Au 0 0 0
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Cl 0 4.033 0
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symmetry D4h
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end
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\end{verbatim}
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