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Subsections

- C.1 methanol
- C.2 water
- C.3 acetylene
- C.4 ethylene
- C.5 methane
- C.6 buckminsterfullerene
- C.7 porphyrin
- C.8 iron penta-carbonyl
- C.9 sodium crown ether
- C.10 ammonia
- C.11 benzene
- C.12
- C.13 ferrocene
- C.14
- C.15 trans-dichloroethylene
- C.16
- C.17 cyclopentadiene anion
- C.18 gold tetrachloride

# 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 methanol

The 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 water

The z axis is the axis and the 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 acetylene

Although acetylene has symmetry the subgroup 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 axes are the x, y, and z axes. The 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 plane.

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

# C.4 ethylene

The axes are the x, y, and z axes. The 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 plane.

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

# C.5 methane

For ease of use, the primary axis should be the x=y=z axis. The 3 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 buckminsterfullerene

One of the 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 porphyrin

The and rotation axis is the z axis. The reflection plane for the 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 iron penta-carbonyl

The axis is the z axis. The plane is the xy plane. One of the perpendicular axes is the x=y axis. One of the planes is the plane containing the x=y axis and the z axis. (The other axes and planes are generated by the 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 sodium crown ether

The axis is the z axis. The point of inversion is the origin. One of the perpendicular axes is the x=y axis. One of the 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 ammonia

The axis is the z axis. One of the 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 benzene

The axis is the z axis. The point of inversion is the origin. One of the 6 perpendicular axes is the x=y axis. (-x=y works as a axis.) The plane is the xy plane. The planes contain the axis and the z axis. The planes contain the 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

The axis is the z axis. The 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 ferrocene

The axis is the z axis. The center of inversion is the origin. One of the perpendicular axes is the x axis. One of the 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

The axis is the z axis. The planes are the yz and the xz planes. The 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 trans-dichloroethylene

The axis is the z axis. The origin is the inversion center. The 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

The axis is the z axis (z is also the axis). The x and y axes are the perpendicular s. The 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 cyclopentadiene anion

The axis is the z axis (z is also the axis). The y axis is one of the perpendicular axes. The plane is the xy plane and one of the 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 gold tetrachloride

The axis is the z axis (z is also the axis). The axes are the x and y axes and the axes are the x=y axis and the x=-y axis. The inversion center is the origin. The plane is the xy plane. The planes are the xz and yz planes and the 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

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