C. Examples of geometries using symmetry

Below are examples of the use of the SYMMETRY directive in the compound GEOMETRY directive (Section 6). 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 print may be added to the SYMMETRY directive to obtain this information.

C.1 $C_{s}$ methanol

The $\sigma_h$ plane is the xy plane.

  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

C.2 $C_{2v}$ water

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

  geometry units au
    O     0.00000000    0.00000000    0.00000000
    H     0.00000000    1.43042809   -1.10715266
    symmetry group c2v
  end

C.3 $D_{2h}$ 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.

  geometry units au
    symmetry group d2h
    C      0.000000000    0.000000000   -1.115108538
    H      0.000000000    0.000000000   -3.106737425
  end

C.4 $D_{2h}$ 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.

  geometry units angstroms
    C 0 0 0.659250 
    H 0 0.916366 1.224352 
    symmetry d2h
  end

C.5 $T_d$ 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.

  geometry units au
    c   0.0000000      0.0000000      0.0000000
    h   1.1828637      1.1828637      1.1828637
    symmetry group Td
  end

C.6 $I_h$ buckminsterfullerene

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

  geometry units angstroms # Bonds = 1.4445, 1.3945
    symmetry group Ih
    c   -1.2287651   0.0   3.3143121
  end

C.7 $S_4$ 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.

  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

C.8 $D_{3h}$ 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.)

  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

C.9 $D_{3d}$ 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.

  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

C.10 $C_{3v}$ 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.

  geometry units angstroms
    N 0     0     -0.055 
    H 0.665 0.665 -0.481 
    symmetry c3v
  end

C.11 $D_{6h}$ 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.

  geometry units au
    C 1.855 1.855 0 
    H 3.289 3.289 0 
    symmetry D6h
  end

C.12 $C_{3h}$ $BO_3H_3$

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

  geometry units au
    b 0 0 0 
    o 2.27238285 1.19464491 0.00000000 
    h 2.10895420 2.97347707 0.00000000 
   symmetry C3h
  end

C.13 $D_{5d}$ 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.

  geometry units angstroms
    symmetry d5d

    fe 0 0     0 
    c  0 1.194 1.789 
    h  0 2.256 1.789 
  end

C.14 $C_{4v}$ $SF_5Cl$

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.

  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

C.15 $C_{2h}$ trans-dichloroethylene

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

  geometry units angstroms
    C  0.65051239 -0.08305064 0 
    Cl 1.75249381  1.30491767 0 
    H  1.14820954 -1.04789741 0 
    symmetry C2h
  end

C.16 $D_{2d}$ $CH_2CCH_2$

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.

  geometry units angstroms
    symmetry d2d
    c 0     0     0 
    c 0     0     1.300 
    h 0.656 0.656 1.857 
  end

C.17 $D_{5h}$ 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.

  charge -1
  geometry units angstroms
    symmetry d5h
    c 0 1.1853 0 
    h 0 2.2654 0 
  end

C.18 $D_{4h}$ 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.

  geometry units au
    Au 0 0     0 
    Cl 0 4.033 0
    symmetry D4h
  end