Examples of geometries using symmetry¶

Below are examples of the use of the SYMMETRY directive in the compound GEOMETRY directive. 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_s methanol¶

The σ_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_2v water¶

The z axis is the C₂ axis and the σ_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

D_2h acetylene¶

Although acetylene has symmetry D_∞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 C₂ axes are the x, y, and z axes. The σ 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 σ plane.

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

D_2h ethylene¶

The C₂ axes are the x, y, and z axes. The σ 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 σ plane.

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

T_d methane¶

For ease of use, the primary C₃ axis should be the x=y=z axis. The 3 C₂ 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

I_h buckminsterfullerene¶

One of the C₅ 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

S₄ porphyrin¶

The S₄ and C₂ rotation axis is the z axis. The reflection plane for the S₄ 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

D_3h iron penta-carbonyl¶

The C₃ axis is the z axis. The σ_h plane is the xy plane. One of the perpendicular C₂ axes is the x=y axis. One of the σ_v planes is the plane containing the x=y axis and the z axis. (The other axes and planes are generated by the C₃ 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

D_3d sodium crown ether¶

The C₃ axis is the z axis. The point of inversion is the origin. One of the perpendicular C₂ axes is the x=y axis. One of the σ_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_3v ammonia¶

The C₃ axis is the z axis. One of the σ_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

D_6h benzene¶

The C₆ axis is the z axis. The point of inversion is the origin. One of the 6 perpendicular C₂‘ axes is the x=y axis. (-x=y works as a C₂‘’ axis.) The σ_h plane is the xy plane. The σ_d planes contain the C₂‘’ axis and the z axis. The σ_v planes contain the C₂‘ axis and the z axis.

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

C_3h H₃BO₃¶

The C₃ axis is the z axis. The σ_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

D_5d ferrocene¶

The C₅ axis is the z axis. The center of inversion is the origin. One of the perpendicular C₂ axes is the x axis. One of the σ_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_4v SF₅Cl¶

The C₄ axis is the z axis. The σ_v planes are the yz and the xz planes. The σ_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_2h trans-dichloroethylene¶

The C₂ axis is the z axis. The origin is the inversion center. The σ_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

D_2d CH₂CCH₂¶

The C₂ axis is the z axis (z is also the S₄ axis). The x and y axes are the perpendicular C₂’s. The σ_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

D_5h cyclopentadiene anion¶

The C₅ axis is the z axis (z is also the S₅ axis). The y axis is one of the perpendicular C₂ axes. The σ_h plane is the xy plane and one of the σ_d planes is the yz plane.

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

D_4h gold tetrachloride¶

The C₄ axis is the z axis (z is also the S₄ axis). The C₂‘ axes are the x and y axes and the C₂‘’ axes are the x=y axis and the x=-y axis. The inversion center is the origin. The σ_h plane is the xy plane. The σ_v planes are the xz and yz planes and the σ_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

Examples of geometries using symmetry¶

Cs methanol¶

C2v water¶

D2h acetylene¶

D2h ethylene¶

Td methane¶

Ih buckminsterfullerene¶

S4 porphyrin¶

D3h iron penta-carbonyl¶

D3d sodium crown ether¶

C3v ammonia¶

D6h benzene¶

C3h H3BO3¶

D5d ferrocene¶

C4v SF5Cl¶

C2h trans-dichloroethylene¶

D2d CH2CCH2¶

D5h cyclopentadiene anion¶

D4h gold tetrachloride¶