Molecular Symmetry: Elements, Operations and Point Groups
Symmetry operations return a fixed molecular geometry to an indistinguishable arrangement. Their full set defines its point group. This group constrains orbital mixing, chirality and the directions in which a permanent dipole is allowed.
What a learner can do afterwards
- Tests rotations, reflections, inversion and improper rotations by comparing atom positions
- Uses a stated decision branch to distinguish the C2v, C3v and D3h molecular examples
- Applies the absence of all improper rotations as the chirality test for a fixed geometry
- Distinguishes a symmetry-forbidden permanent dipole from a symmetry-allowed dipole
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You have used symmetry to test orbital mixing. A symmetry operation returns a fixed geometry to an indistinguishable arrangement: each atom reaches a position occupied by the same kind of atom. Identical atoms may exchange positions. E is the identity operation, which leaves every position unchanged. An element is an operation's axis, plane or point. A Cn rotation turns through 360/n degrees about an axis.
Reflection, σ, reverses the coordinate perpendicular to a mirror plane. Inversion, i, sends (x,y,z) to (-x,-y,-z) through a centre. An improper rotation, Sn, combines a 360/n-degree rotation with reflection through the plane perpendicular to its axis. Test the combined result against the original atom positions. S1 is reflection; S2 is inversion.
For the three examples here, find the highest-order rotation axis. Look for C2 axes perpendicular to it. If present, a horizontal mirror plane leads to Dnh. If absent, vertical mirror planes lead to Cnv. Horizontal means perpendicular to the main axis; vertical means containing it. Other point groups need other branches.
Water is bent. Its C2 axis passes through oxygen along the H-O-H angle bisector. A 180-degree turn exchanges the hydrogens. One mirror plane contains the molecule; another contains C2 and is perpendicular to the molecular plane. There are no perpendicular C2 axes. The operations are E, C2 and two vertical reflections: C2v. Water has no C3 axis.
Ammonia is trigonal pyramidal. Its C3 axis passes through nitrogen and the centre of the hydrogen triangle. Turns of 120 and 240 degrees cycle the hydrogens. Three mirror planes contain this axis, each containing one N-H bond. No perpendicular C2 axes or horizontal mirror plane exist. Its operations are E, two C3 turns and three reflections: C3v.
BF3 has three equal B-F bonds at 120 degrees in one plane. C3 is perpendicular to that plane; three C2 axes lie along the bonds. The molecular plane is σh. Its operations are E, two C3 turns, three C2 turns, σh, two S3 operations and three vertical reflections. This is D3h. The three equal bond dipoles cancel, making BF3 nonpolar.
A fixed geometry is chiral when its mirror image cannot be superimposed by rotations and translations. This happens exactly when no Sn operation of any order exists. Mirror planes, S1, and inversion centres, S2, therefore forbid chirality. A geometry with only proper rotations besides E is chiral. Flexible molecules may interconvert between mirror-related geometries; this test describes one geometry.
A permanent dipole must survive every symmetry operation. Ammonia has C3 and vertical mirrors but no perpendicular C2 axes: the Cnv branch with n=3 gives C3v. C2v and C3v allow a dipole along the main axis. Water's and ammonia's bond dipoles do not cancel, so both are polar. In D3h, C3 forbids an in-plane component and σh forbids a perpendicular component. Permission alone does not prove a nonzero dipole.
Compare atom positions after each operation. Use the point group to test chirality and find which dipole directions symmetry allows.
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Operations and their elements
A symmetry operation is a coordinate transformation that reproduces the molecular arrangement. It need not change any coordinates: E, the identity, leaves every position unchanged. An element is the axis, plane or point used by an operation.
A Cn turn has angle 360/n degrees. A C3 axis permits both 120-degree and 240-degree turns. The atom positions after each turn must match equivalent positions before it.
Reflection through the xy plane sends (x,y,z) to (x,y,-z). Inversion through the origin sends it to (-x,-y,-z). An Sn operation combines a rotation with reflection through the plane perpendicular to its axis. S1 is reflection, and S2 is inversion.
These operations test a fixed geometry. They do not instruct you to bend bonds or move individual atoms into a different shape.
Checking three molecular point groups
Use the three-dimensional atom arrangement. First find the highest-order rotation axis, then look for perpendicular C2 axes and mirror planes.
Water's main axis bisects its H-O-H angle. A half turn exchanges the hydrogens. Its molecular plane is a mirror plane, and a second mirror plane exchanges the hydrogens. Both contain the C2 axis. Together with E, these operations give C2v.
Ammonia's C3 axis passes through nitrogen and the centre of the hydrogen triangle. Its three mirror planes contain that axis and one N-H bond each. It has no perpendicular C2 axes, giving C3v.
In BF3, the main C3 axis is perpendicular to the molecular plane. Three perpendicular C2 axes lie along the B-F bonds. The molecular plane is a horizontal mirror plane. This selects D3h. These checks form one branch of point-group classification; they do not classify every possible molecule.
Testing possible dipole directions
A molecule's permanent dipole must remain unchanged under every operation of its point group. Any component reversed by a symmetry operation must be zero.
Water allows a component along its C2 axis. Ammonia allows one along its C3 axis. Their charge distributions produce nonzero values in these directions.
For BF3, a C3 turn prevents a nonzero component in the molecular plane. Reflection in that plane reverses any perpendicular component. No nonzero direction survives both tests. The bond-vector check agrees: three equal B-F dipoles separated by 120 degrees have zero sum.
A symmetry-allowed direction does not establish the dipole's value. It tells you where a nonzero dipole could point. You still need information about the charge distribution to decide whether a nonzero value exists.
Atom arrangements and chirality
Molecular geometry describes atom positions. Electron-pair geometry also includes lone-pair regions. Ammonia has a tetrahedral electron-pair arrangement but a trigonal pyramidal atom arrangement. Water's atom arrangement is bent. Use the atoms when assigning these molecular point groups.
Atom identity matters. A valid operation can exchange identical hydrogens, but it cannot exchange hydrogen and oxygen. Distinguishable isotopes must also be treated as different.
A fixed geometry is chiral if its mirror image cannot be superimposed by rotations and translations. Its point group then contains no improper rotation of any order. Check mirror reflection and inversion as well, since these are S1 and S2.
A flexible molecule may pass between mirror-related geometries. The symmetry test describes the specified geometry; it does not calculate the rate of that interconversion.
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