# Geometry examples

### From NWChem

(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 …') |
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==<math>C_{s}</math> methanol== | ==<math>C_{s}</math> methanol== | ||

- | The <math>\sigma_h< | + | The <math>\sigma_h</math> plane is the xy plane. |

geometry units angstroms | geometry units angstroms | ||

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end | end | ||

- | ==<math>C_{2v}</math> water | + | ==<math>C_{2v}</math> water== |

The z axis is the <math>C_2</math> axis and the <math>\sigma_v</math> may be either the xz or the yz planes. | The z axis is the <math>C_2</math> axis and the <math>\sigma_v</math> may be either the xz or the yz planes. | ||

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end | end | ||

- | ==<math>D_{2h}</math> acetylene | + | ==<math>D_{2h}</math> acetylene== |

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

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end | end | ||

- | ==<math>D_{2h}</math> ethylene | + | ==<math>D_{2h}</math> ethylene== |

The <math>C_2</math> 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 <math>C_2</math> 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. | ||

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end | end | ||

- | ==<math>T_d</math> methane | + | ==<math>T_d</math> methane== |

For ease of use, the primary <math>C_3</math> axis should be the x=y=z axis. The 3 <math>C_2</math> axes are the x, y, and z. | For ease of use, the primary <math>C_3</math> axis should be the x=y=z axis. The 3 <math>C_2</math> axes are the x, y, and z. | ||

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end | end | ||

- | ==<math>I_h</math> buckminsterfullerene | + | ==<math>I_h</math> buckminsterfullerene== |

One of the <math>C_5</math> axes is the z axis and the point of inversion is the origin. | One of the <math>C_5</math> axes is the z axis and the point of inversion is the origin. | ||

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end | end | ||

- | ==<math>S_4</math> porphyrin | + | ==<math>S_4</math> porphyrin== |

The <math>S_4</math> and <math>C_2</math> rotation axis is the z axis. The reflection plane for the <math>S_4</math> operation is the xy plane. | The <math>S_4</math> and <math>C_2</math> rotation axis is the z axis. The reflection plane for the <math>S_4</math> operation is the xy plane. | ||

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end | end | ||

- | ==<math>D_{3h}</math> iron penta-carbonyl | + | ==<math>D_{3h}</math> iron penta-carbonyl== |

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

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end | end | ||

- | ==<math>D_{3d}</math> sodium crown ether | + | ==<math>D_{3d}</math> sodium crown ether== |

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

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end | end | ||

- | ==<math>C_{3v}</math> ammonia | + | ==<math>C_{3v}</math> ammonia== |

The <math>C_3</math> axis is the z axis. One of the <math>\sigma_v</math> planes is the plane containing the x=y axis and the z axis. | The <math>C_3</math> axis is the z axis. One of the <math>\sigma_v</math> planes is the plane containing the x=y axis and the z axis. | ||

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end | end | ||

- | ==<math>D_{6h}</math> benzene | + | ==<math>D_{6h}</math> benzene== |

The <math>C_6</math> axis is the z axis. The point of inversion is the origin. One of the 6 perpendicular <math>C_{2}'</math> axes is the x=y axis. (-x=y works as a <math>C_{2}''</math> axis.) The <math>\sigma_h</math> plane is the xy plane. The <math>\sigma_d</math> planes contain the <math>C_{2}''</math> axis and the z axis. The <math>\sigma_v</math> planes contain the <math>C_{2}'<math> axis and the z axis. | The <math>C_6</math> axis is the z axis. The point of inversion is the origin. One of the 6 perpendicular <math>C_{2}'</math> axes is the x=y axis. (-x=y works as a <math>C_{2}''</math> axis.) The <math>\sigma_h</math> plane is the xy plane. The <math>\sigma_d</math> planes contain the <math>C_{2}''</math> axis and the z axis. The <math>\sigma_v</math> planes contain the <math>C_{2}'<math> axis and the z axis. | ||

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end | end | ||

- | ==<math>C_{3h}</math> <math>BO_{3}H_{3}</math> | + | ==<math>C_{3h}</math> <math>BO_{3}H_{3}</math>== |

The <math>C_3</math> axis is the z axis. The <math>\sigma_h</math> plane is the xy plane. | The <math>C_3</math> axis is the z axis. The <math>\sigma_h</math> plane is the xy plane. | ||

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end | end | ||

- | ==<math>D_{5d}</math> ferrocene | + | ==<math>D_{5d}</math> ferrocene== |

The <math>C_5</math> axis is the z axis. The center of inversion is the origin. One of the perpendicular <math>C_2</math> axes is the x axis. One of the <math>\sigma_d</math> planes is the yz plane. | The <math>C_5</math> axis is the z axis. The center of inversion is the origin. One of the perpendicular <math>C_2</math> axes is the x axis. One of the <math>\sigma_d</math> planes is the yz plane. | ||

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end | end | ||

- | ==<math>C_{4v}</math> <math>SF_{5}Cl</math> | + | ==<math>C_{4v}</math> <math>SF_{5}Cl</math>== |

The <math>C_4</math> axis is the z axis. The <math>\sigma_v</math> planes are the yz and the xz planes. The <math>\sigma_d</math> 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 <math>C_4</math> axis is the z axis. The <math>\sigma_v</math> planes are the yz and the xz planes. The <math>\sigma_d</math> 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. | ||

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end | end | ||

- | ==<math>C_{2h}</math> trans-dichloroethylene | + | ==<math>C_{2h}</math> trans-dichloroethylene== |

The <math>C_2</math> axis is the z axis. The origin is the inversion center. The <math>\sigma_h</math> plane is the xy plane. | The <math>C_2</math> axis is the z axis. The origin is the inversion center. The <math>\sigma_h</math> plane is the xy plane. | ||

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end | end | ||

- | ==<math>D_{2d}</math> <math>CH_{2}CCH_{2}</math> | + | ==<math>D_{2d}</math> <math>CH_{2}CCH_{2}</math>== |

The <math>C_2</math> axis is the z axis (z is also the <math>S_4</math> axis). The x and y axes are the perpendicular <math>C_{2}'</math>s. The <math>\sigma_d</math> 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 <math>C_2</math> axis is the z axis (z is also the <math>S_4</math> axis). The x and y axes are the perpendicular <math>C_{2}'</math>s. The <math>\sigma_d</math> 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. | ||

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end | end | ||

- | ==<math>D_{5h}</math> cyclopentadiene anion | + | ==<math>D_{5h}</math> cyclopentadiene anion== |

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

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end | end | ||

- | ==<math>D_{4h}</math> gold tetrachloride | + | ==<math>D_{4h}</math> gold tetrachloride== |

The <math>C_4</math> axis is the z axis (z is also the <math>S_4</math> axis). The <math>C_{2}'</math> axes are the x and y axes and the <math>C_{2}''</math> axes are the x=y axis and the x=-y axis. The inversion center is the origin. The <math>\sigma_h</math> plane is the xy plane. The <math>\sigma_v</math> planes are the xz and yz planes and the <math>\sigma_d</math> 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 <math>C_4</math> axis is the z axis (z is also the <math>S_4</math> axis). The <math>C_{2}'</math> axes are the x and y axes and the <math>C_{2}''</math> axes are the x=y axis and the x=-y axis. The inversion center is the origin. The <math>\sigma_h</math> plane is the xy plane. The <math>\sigma_v</math> planes are the xz and yz planes and the <math>\sigma_d</math> 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. |

## Revision as of 13:54, 24 May 2010

# 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*_{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 *C*_{2} 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 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 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 *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.

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 *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

## *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

## *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

## *D*_{3h} iron penta-carbonyl

The *C*_{3} axis is the z axis. The σ_{h} plane is the xy plane. One of the perpendicular *C*_{2} 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 *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

## *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 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 *C*_{3} 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 *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 σ_{h} plane is the xy plane. The σ_{d} planes contain the *C*_{2}'' axis and the z axis. The σ_{v} planes contain the *C*_{2}' < *m**a**t**h* > *a**x**i**s**a**n**d**t**h**e**z**a**x**i**s*.*g**e**o**m**e**t**r**y**u**n**i**t**s**a**u**C*1.8551.8550*H*3.2893.2890*s**y**m**m**e**t**r**y**D*6*h**e**n**d* = = < *m**a**t**h* > *C*_{3h} *B**O*_{3}*H*_{3}==

The *C*_{3} 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 *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 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} *S**F*_{5}*C**l*

The *C*_{4} 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 *C*_{2} 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} *C**H*_{2}*C**C**H*_{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 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 *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 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 *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 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