%
% $Id: symex.tex 19708 2010-10-29 18:04:21Z d3y133 $
%
\label{symexamples}

Below are examples of the use of the \verb+SYMMETRY+ directive in
the compound \verb+GEOMETRY+ directive (Section \ref{sec:geom}).
The z axis is always the primary rotation axis.
When in doubt about which axes and planes are used for the group
elements, the keyword \verb+print+ may be added to the \verb+SYMMETRY+ 
directive to obtain this information.

  \section{\protect$C_{s}$ methanol}

The $\sigma_h$ plane is the xy plane.

\begin{verbatim}
  geometry units angstroms
    C    0.11931097    -0.66334875     0.00000000
    H    1.20599017    -0.87824237     0.00000000
    H   -0.32267592    -1.15740001     0.89812652
    O   -0.01716588     0.78143468     0.00000000
    H   -1.04379735     0.88169812     0.00000000
    symmetry cs
  end
\end{verbatim}

  \section{\protect$C_{2v}$ water}

The z axis is the $C_2$ axis and the $\sigma_v$ may be either the xz or the
yz planes.

\begin{verbatim}
  geometry units au
    O     0.00000000    0.00000000    0.00000000
    H     0.00000000    1.43042809   -1.10715266
    symmetry group c2v
  end
\end{verbatim}

  \section{\protect$D_{2h}$ acetylene}

Although acetylene has symmetry $D_{\infty h}$ the subgroup
$D_{2h}$ includes all operations that interchange equivalent atoms
which is what determines how much speedup you gain from using symmetry
in building a Fock matrix.

The $C_2$ axes are the x, y, and z axes.  The $\sigma$ planes are the xy,
xz and yz planes.  Generally, the unique atoms are placed to use the z
as the primary rotational axis and use the xz or yz planes as the $\sigma$ 
plane.

\begin{verbatim}
  geometry units au
    symmetry group d2h
    C      0.000000000    0.000000000   -1.115108538
    H      0.000000000    0.000000000   -3.106737425
  end
\end{verbatim}

  \section{\protect$D_{2h}$ ethylene}

The $C_2$ axes are the x, y, and z axes.  The $\sigma$ planes are the xy,
xz and yz planes.  Generally, the unique atoms are placed to use the z
as the primary rotational axis and use the xz or yz planes as the $\sigma$ 
plane.

\begin{verbatim}
  geometry units angstroms
    C 0 0 0.659250 
    H 0 0.916366 1.224352 
    symmetry d2h
  end
\end{verbatim}


  \section{\protect$T_d$ methane}

For ease of use, the primary $C_3$ axis should be the x=y=z axis.
The 3 $C_2$ axes are the x, y, and z.

\begin{verbatim}
  geometry units au
    c   0.0000000      0.0000000      0.0000000
    h   1.1828637      1.1828637      1.1828637
    symmetry group Td
  end
\end{verbatim}

  \section{\protect$I_h$ buckminsterfullerene}

One of the $C_5$ axes is the z axis and the point of inversion is the origin.

\begin{verbatim}
  geometry units angstroms # Bonds = 1.4445, 1.3945
    symmetry group Ih
    c   -1.2287651   0.0   3.3143121
  end
\end{verbatim}

  \section{\protect$S_4$ porphyrin}

The $S_4$ and $C_2$ rotation axis is the z axis.  The reflection plane 
for the $S_4$ operation is the xy plane.

\begin{verbatim}
  geometry units angstroms
     symmetry group s4
  
     fe                0.000  0.000  0.000         
     h                 2.242  6.496 -3.320   
     h                 1.542  4.304 -2.811
     c                 1.947  6.284 -2.433
     c                 1.568  4.987 -2.084
     h                 2.252  8.213 -1.695
     c                 1.993  7.278 -1.458
     h                 5.474 -1.041 -1.143
     c                 1.234  4.676 -0.765
     h                 7.738 -1.714 -0.606
     c                 0.857  3.276 -0.417
     h                 1.380 -4.889 -0.413
     c                 1.875  2.341 -0.234
     h                 3.629  3.659 -0.234
     c                 0.493 -2.964 -0.229
     c                 1.551 -3.933 -0.221
     c                 5.678 -1.273 -0.198
     c                 1.656  6.974 -0.144
     c                 3.261  2.696 -0.100
     n                 1.702  0.990 -0.035
  end
\end{verbatim}

  \section{\protect$D_{3h}$ iron penta-carbonyl}

The $C_3$ axis is the z axis.  The $\sigma_h$ plane is the xy plane.  One of
the perpendicular $C_2$ axes is the x=y axis.  One of the $\sigma_v$ 
planes is the plane containing the x=y axis and the z axis.  (The 
other axes and planes are generated by the $C_3$ operation.)

\begin{verbatim}
  geometry units au
    symmetry group d3h

    fe        0.0         0.0         0.0

    c         0.0         0.0         3.414358
    o         0.0         0.0         5.591323

    c         2.4417087   2.4417087   0.0
    o         3.9810552   3.9810552   0.0
  end
\end{verbatim}

  \section{\protect$D_{3d}$ sodium crown ether}

The $C_3$ axis is the z axis. The point of inversion is the origin. One of
the perpendicular $C_2$ axes is the x=y axis. One of the $\sigma_d$ planes
is the plane containing the -x=y axis and the z axis.

  Note that the oxygen atom is rotated in the x-y plane 15
  degrees away from the y-axis so that it lies in a mirror
  plane.  There is a total of six atoms generated from the
  unique oxygen, in contrast to twelve from each of the carbon
  and hydrogen atoms.

\begin{verbatim}
  geometry units au
    symmetry D3d

   NA     .0000000000    .0000000000   .0000000000
   O     1.3384771885   4.9952647969   .1544089284
   H     6.7342048019  -0.6723850379  2.6581562148
   C     6.7599180056  -0.4844977035   .6136583870
   H     8.6497577017   0.0709194071   .0345361934

  end
\end{verbatim}

  \section{\protect$C_{3v}$ ammonia}

The $C_3$ axis is the z axis. One of the $\sigma_v$ planes is the 
plane containing the x=y axis and the z axis.

\begin{verbatim}
  geometry units angstroms
    N 0     0     -0.055 
    H 0.665 0.665 -0.481 
    symmetry c3v
  end
\end{verbatim}

  \section{\protect$D_{6h}$ benzene}

The $C_6$ axis is the z axis.  The point of inversion is the origin.  One
of the 6 perpendicular $C_2'$ axes is the x=y axis. (-x=y works as a 
$C_2''$ axis.)  The $\sigma_h$ plane is the xy plane.  The $\sigma_d$ 
planes contain the $C_2''$ axis and the z axis.  The $\sigma_v$ planes 
contain the $C_2'$ axis and the z axis.

\begin{verbatim}
  geometry units au
    C 1.855 1.855 0 
    H 3.289 3.289 0 
    symmetry D6h
  end
\end{verbatim}

  \section{\protect$C_{3h}$ \protect$BO_3H_3$}

The $C_3$ axis is the z axis.  The $\sigma_h$ plane is the xy plane.

\begin{verbatim}
  geometry units au
    b 0 0 0 
    o 2.27238285 1.19464491 0.00000000 
    h 2.10895420 2.97347707 0.00000000 
   symmetry C3h
  end
\end{verbatim}

  \section{\protect$D_{5d}$ ferrocene}

The $C_5$ axis is the z axis.  The center of inversion is the origin.
One of the perpendicular $C_2$ axes is the x axis.  One of the $\sigma_d$
planes is the yz plane.

\begin{verbatim}
  geometry units angstroms
    symmetry d5d

    fe 0 0     0 
    c  0 1.194 1.789 
    h  0 2.256 1.789 
  end
\end{verbatim}

  \section{\protect$C_{4v}$ \protect$SF_5Cl$}

The $C_4$ axis is the z axis.  The $\sigma_v$ planes are the yz and the xz
planes.  The $\sigma_d$ planes are: 1) the plane containing the x=y axis and
the z axis and 2) the plane containing the -x=y axis and the z axis.

\begin{verbatim}
  geometry units au
    S  0.00000000 0.00000000 -0.14917600 
    Cl 0.00000000 0.00000000  4.03279700 
    F  3.13694200 0.00000000 -0.15321800 
    F  0.00000000 0.00000000 -3.27074500 

    symmetry C4v
  end
\end{verbatim}

  \section{\protect$C_{2h}$ trans-dichloroethylene}

The $C_2$ axis is the z axis.  The origin is the inversion center.
The $\sigma_h$ plane is the xy plane.

\begin{verbatim}
  geometry units angstroms
    C  0.65051239 -0.08305064 0 
    Cl 1.75249381  1.30491767 0 
    H  1.14820954 -1.04789741 0 
    symmetry C2h
  end
\end{verbatim}

  \section{\protect$D_{2d}$ \protect$CH_2CCH_2$}

The $C_2$ axis is the z axis (z is also the $S_4$ axis).  The x and y axes
are the perpendicular $C_2'$s. The $\sigma_d$ planes are: 
1) the plane containing the x=y axis and the z axis and 2) the plane 
containing the -x=y axis and the z axis.

\begin{verbatim}
  geometry units angstroms
    symmetry d2d
    c 0     0     0 
    c 0     0     1.300 
    h 0.656 0.656 1.857 
  end
\end{verbatim}

  \section{\protect$D_{5h}$ cyclopentadiene anion}

The $C_5$ axis is the z axis (z is also the $S_5$ axis).  The y axis is one
of the perpendicular $C_2$ axes.  The $\sigma_h$ plane is the xy plane and
one of the $\sigma_d$ planes is the yz plane.

\begin{verbatim}
  charge -1
  geometry units angstroms
    symmetry d5h
    c 0 1.1853 0 
    h 0 2.2654 0 
  end
\end{verbatim}


  \section{\protect$D_{4h}$ gold tetrachloride}

The $C_4$ axis is the z axis (z is also the $S_4$ axis).  
The $C_2'$ axes are the x and y axes and the $C_2''$ axes are the x=y axis
and the x=-y axis.  The inversion center is the origin.  The $\sigma_h$
plane is the xy plane.  The $\sigma_v$ planes are the xz and yz planes and
the $\sigma_d$ planes are 1) the plane containing the x=-y axis and the z axis
and 2) the plane containing the x=y axis and the z axis.

\begin{verbatim}
  geometry units au
    Au 0 0     0 
    Cl 0 4.033 0
    symmetry D4h
  end
\end{verbatim}
