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# Geometry examples

(Difference between revisions)
 Revision as of 14:52, 24 May 2010 (view source) (Created page with '=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 …')← Older edit Revision as of 14:54, 24 May 2010 (view source)Newer edit → Line 5: Line 5: ==$C_{s}$ methanol== ==$C_{s}$ methanol== - The $\sigma_h<\math> plane is the xy plane. + The [itex]\sigma_h plane is the xy plane. geometry units angstroms geometry units angstroms Line 16: Line 16: end end - ==[itex]C_{2v}$ water + ==$C_{2v}$ water== The z axis is the $C_2$ axis and the $\sigma_v$ may be either the xz or the yz planes. The z axis is the $C_2$ axis and the $\sigma_v$ may be either the xz or the yz planes. Line 26: Line 26: end end - ==$D_{2h}$ acetylene + ==$D_{2h}$ 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. 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. Line 38: Line 38: end end - ==$D_{2h}$ ethylene + ==$D_{2h}$ ethylene== The $C_2$ 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. The $C_2$ 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. Line 48: Line 48: end end - ==$T_d$ methane + ==$T_d$ 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. 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. Line 58: Line 58: end end - ==$I_h$ buckminsterfullerene + ==$I_h$ buckminsterfullerene== One of the $C_5$ axes is the z axis and the point of inversion is the origin. One of the $C_5$ axes is the z axis and the point of inversion is the origin. Line 67: Line 67: end end - ==$S_4$ porphyrin + ==$S_4$ 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. The $S_4$ and $C_2$ rotation axis is the z axis. The reflection plane for the $S_4$ operation is the xy plane. Line 96: Line 96: end end - ==$D_{3h}$ iron penta-carbonyl + ==$D_{3h}$ 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.) 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.) Line 112: Line 112: end end - ==$D_{3d}$ sodium crown ether + ==$D_{3d}$ 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. 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. Line 129: Line 129: end end - ==$C_{3v}$ ammonia + ==$C_{3v}$ 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. 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. Line 139: Line 139: end end - ==$D_{6h}$ benzene + ==$D_{6h}$ 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 $C_{2}'[itex] axis and the z axis. The [itex]C_6$ 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 $C_{2}'[itex] axis and the z axis. Line 149: Line 149: end end - ==[itex]C_{3h}$ $BO_{3}H_{3}$ + ==$C_{3h}$ $BO_{3}H_{3}$== The $C_3$ axis is the z axis. The $\sigma_h$ plane is the xy plane. The $C_3$ axis is the z axis. The $\sigma_h$ plane is the xy plane. Line 160: Line 160: end end - ==$D_{5d}$ ferrocene + ==$D_{5d}$ 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. 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. Line 172: Line 172: end end - ==$C_{4v}$ $SF_{5}Cl$ + ==$C_{4v}$ $SF_{5}Cl$== 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. 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. Line 185: Line 185: end end - ==$C_{2h}$ trans-dichloroethylene + ==$C_{2h}$ trans-dichloroethylene== The $C_2$ axis is the z axis. The origin is the inversion center. The $\sigma_h$ plane is the xy plane. The $C_2$ axis is the z axis. The origin is the inversion center. The $\sigma_h$ plane is the xy plane. Line 196: Line 196: end end - ==$D_{2d}$ $CH_{2}CCH_{2}$ + ==$D_{2d}$ $CH_{2}CCH_{2}$== 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. 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. Line 207: Line 207: end end - ==$D_{5h}$ cyclopentadiene anion + ==$D_{5h}$ 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. 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. Line 218: Line 218: end end - ==$D_{4h}$ gold tetrachloride + ==$D_{4h}$ 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. 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.

# 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.

## Cs methanol

The σh 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


## C2v water

The z axis is the C2 axis and the σv 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


## D2h acetylene

Although acetylene has symmetry $D_{\infty h}$ the subgroup D2h 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


## D2h 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


## Td 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


## Ih 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


## D3h iron penta-carbonyl

The C3 axis is the z axis. The σh plane is the xy plane. One of the perpendicular C2 axes is the x=y axis. One of the σv 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


## D3d 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 σd 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


## C3v ammonia

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


## D6h 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 σh plane is the xy plane. The σd planes contain the C2'' axis and the z axis. The σv planes contain the C2' < math > axisandthezaxis.geometryunitsauC1.8551.8550H3.2893.2890symmetryD6hend = = < math > C3h BO3H3==

The C3 axis is the z axis. The σh 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


## D5d 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 σd 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


## C4v SF5Cl

The C4 axis is the z axis. The σv planes are the yz and the xz planes. The σd 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


## C2h trans-dichloroethylene

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


## D2d 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 σd 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


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

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


## D4h 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 σh plane is the xy plane. The σv planes are the xz and yz planes and the σd 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