Chemistry · Lesson
Coordination geometries and isomerism
Distinguish a donor count from a spatial arrangement. Use ideal donor positions and symmetry to understand geometric isomers and a coarse model of a chelated mirror pair.
Back to the Chemistry mapBy the end, you can
- Compare square-planar, tetrahedral and octahedral donor arrangements.
- Distinguish cis/trans and fac/mer in their stated geometries.
- Count equivalent placements rather than treating every labelled position choice as a new isomer.
- Explain why a chelated mirror pair can be non-superposable without assigning optical rotation from a drawing.
A donor count leaves a geometry question
Coordination number 4 can accompany square-planar or tetrahedral arrangements, among other actual distortions. Coordination number 6 commonly accompanies octahedral arrangements but does not uniquely require them. Metal identity, electron configuration, ligand interactions, steric effects and phase all matter. Structural evidence distinguishes an actual geometry from a drawing chosen for explanation.
Here ideal donor coordinates have unit metal–donor radius. Square-planar sites are +x,+y,−x,−y in one plane. Tetrahedral sites are the four selected (±1,±1,±1)/√3 combinations with an even number of negative signs. Octahedral sites are ±x,±y,±z. These are dimensionless ideal directions, not measured distances or a proposed real complex’s coordinates.
Equivalent placements depend on the geometry
In ideal square-planar MA₂B₂, adjacent B sites are cis with donor angle 90°, and opposite B sites are trans with angle 180°. Choosing two of four labelled sites gives six position sets: four adjacent and two opposite. Proper spatial rotations of the square identify all members within each class, leaving two geometric isomers. Rotation of a complex does not create another isomer.
In ideal tetrahedral MA₂B₂, all six choices of a B pair are equivalent under the twelve proper tetrahedral rotations. Every distinct pair of donor directions has dot product −1/3 and angle arccos(−1/3)≈ 109.471°. There is no separate cis/trans pair of this tetrahedral composition. That statement applies to the stipulated two-A/two-B donor model, not to every tetrahedral ligand set.
An octahedron supports two useful geometric distinctions
For ideal octahedral MA₄B₂, the fifteen B-pair choices divide into twelve perpendicular cis pairs and three opposite trans pairs. Its twenty-four proper rotations again give two classes. The ninety-degree versus one-hundred-eighty-degree relation refers to donor directions at M, not angles measured from an arbitrary flat projection.
For octahedral MA₃B₃, a facial arrangement places the three B donors on one triangular face: all three B–M–B pairs are mutually cis. A meridional arrangement contains an opposite B pair and a third B perpendicular to that axis. The twenty ways to choose three B sites divide into eight fac and twelve mer placements, leaving two geometric classes. Distortion or changing the donor model can require a more detailed analysis.
A connected ligand framework can make a handed arrangement
Three bidentate ligands in the donor-only model M(en)₃ connect the six octahedral sites into three adjacent pairs. There are eight perfect matchings using adjacent sites, where a perfect matching pairs each donor once. Proper octahedral rotations split these into two classes of four. Reflecting x→−x exchanges the two classes; no proper rotation superposes one class on the other. This coarse connectivity model demonstrates a mirror pair.
For an explicit index convention let sites 0…5 be +x,−x,+y,−y,+z,−z. One matching is (0, 2),(1, 4),(3, 5); its reflected partner is (1, 2),(0, 4),(3, 5). These links indicate which donor sites belong to one bidentate ligand. They do not place the N–C–C–N chains, specify chelate-ring conformation or provide atomistic bond lengths.
The conventional Δ and Λ descriptors require a correctly oriented helicity assignment. We leave the absolute labels unassigned in this coarse model. Neither Δ/Λ nor a mirror sketch predicts the sign or magnitude of measured optical rotation. Geometry, absolute configuration, conformational detail and the conditions of an optical measurement are distinct information.
Original schematic projections of unit donor directions. Solid M–donor lines indicate the specified direct attachments; circles A and squares B distinguish identical ligand types in the first six panels. Displayed angles are the three-dimensional ideal donor relations, not angles measured from this projection. In the last two panels, labels N0…N5 index the octahedral donors; dashed coloured curves group the two donors belonging to one chelate. They are connection guides, not located ligand chains, extra donor–donor bonds or reaction arrows. Reflection x→−x exchanges the two matchings. Absolute Δ/Λ and optical-rotation sign are unassigned. No measured geometry or distances are supplied.
Square-planar B pairs are adjacent cis or opposite trans. All tetrahedral B-pair choices are equivalent. An octahedral trans pair is opposite; fac triples are mutually cis, while mer triples contain an opposite pair. The two connected adjacent-pair matchings are exchanged by reflection and are not superposable by a proper rotation. The coordinate table preserves the full three-dimensional donor directions and connections.
Show original donor coordinates and connectivity
| Panel | Indexed unit donor directions | Marked B sites or paired N indices | Ideal relation |
|---|---|---|---|
| Square planar: cis | 0:(1,0,0); 1:(0,1,0); 2:(-1,0,0); 3:(0,-1,0) | 0, 1 | B–M–B angle 90°; Four donor sites in one plane |
| Square planar: trans | 0:(1,0,0); 1:(0,1,0); 2:(-1,0,0); 3:(0,-1,0) | 0, 2 | B–M–B angle 180°; Same composition; opposite B pair |
| Tetrahedral: one pair class | 0:(0.57735,0.57735,0.57735); 1:(0.57735,-0.57735,-0.57735); 2:(-0.57735,0.57735,-0.57735); 3:(-0.57735,-0.57735,0.57735) | 0, 1 | B–M–B angle ≈109.471°; All six B-pair choices equivalent |
| Octahedral: trans | 0:(1,0,0); 1:(-1,0,0); 2:(0,1,0); 3:(0,-1,0); 4:(0,0,1); 5:(0,0,-1) | 0, 1 | B–M–B angle 180°; Six donors; opposite B sites |
| Octahedral: fac | 0:(1,0,0); 1:(-1,0,0); 2:(0,1,0); 3:(0,-1,0); 4:(0,0,1); 5:(0,0,-1) | 0, 2, 4 | All three B–M–B pairs 90°; B donors occupy one triangular face |
| Octahedral: mer | 0:(1,0,0); 1:(-1,0,0); 2:(0,1,0); 3:(0,-1,0); 4:(0,0,1); 5:(0,0,-1) | 0, 1, 2 | One B pair 180°; other pairs 90°; One opposite pair among the B sites |
| Three chelates: matching A | 0:(1,0,0); 1:(-1,0,0); 2:(0,1,0); 3:(0,-1,0); 4:(0,0,1); 5:(0,0,-1) | (0,2); (1,4); (3,5) | Dashed curves pair N donor sites; Absolute Δ/Λ unassigned |
| Three chelates: reflected B | 0:(1,0,0); 1:(-1,0,0); 2:(0,1,0); 3:(0,-1,0); 4:(0,0,1); 5:(0,0,-1) | (1,2); (0,4); (3,5) | Reflection x→−x exchanges 0 and 1; Absolute Δ/Λ unassigned |
| Ideal arrangement and composition | Position choices before rotation | Classes after proper rotations | Relation distinguishing the classes |
|---|---|---|---|
| Square-planar MA₂B₂ | 6 B pairs: adjacent 4, opposite 2 | 2 | cis 90°; trans 180° |
| Tetrahedral MA₂B₂ | 6 B pairs | 1 | All pair angles arccos(−1/3)≈ 109.471° |
| Octahedral MA₄B₂ | 15 B pairs: perpendicular 12, opposite 3 | 2 | cis 90°; trans 180° |
| Octahedral MA₃B₃ | 20 B triples: fac 8, mer 12 | 2 | fac: all mutual cis; mer: one opposite pair |
| Octahedral three adjacent chelates | 8 perfect matchings; rotation classes 4+4 | 2, exchanged by reflection | Non-superposable donor-only mirror pair; absolute labels unassigned |
Keep connectivity fixed when testing superposition
To compare proposed isomers, first specify the same composition and which atoms are connected. Apply translations and proper rotations when testing superposition; a mirror reflection is a separate operation. Permuting chemically identical ligands through a symmetry operation can remove a false distinction, while breaking a ligand into independent donors can remove a real one.
A geometric or optical isomer may have different physical properties, but these symmetry counts do not measure those properties or establish isolation under every condition. Rearrangement pathways and their rates determine whether a particular arrangement persists on an experimental timescale.
Worked examples
Remove the repeated square placements
Use the four ideal square-planar sites and select two identical B donors.
- There are 4× 3/2=6 unordered pairs.
- Two are opposite and four adjacent.
- The eight proper spatial symmetries of the square identify placements within each group.
Two geometric classes: cis and trans.
Use the direction dot product
Use normalized tetrahedral donor directions (1, 1, 1)/√3 and (1,−1,−1)/√3.
- Dot product is (1−1−1)/3=−1/3.
- Angle is arccos(−1/3)≈ 109.471°.
- The other pairs have the same relation and one symmetry orbit.
One ideal MA₂B₂ class, without a separate cis/trans distinction.
Count fac and mer before identifying equivalence
Use six octahedral sites and three identical B donors.
- Choose three of six:20 labelled subsets.
- Eight triples occupy triangular faces.
- The twelve remaining triples include one opposite pair.
Two classes, fac and mer, after proper octahedral rotations.
Reflect connected donor pairs rather than independent points
Use matching (0, 2),(1, 4),(3, 5) in the stated octahedral index convention.
- Reflection x→−x exchanges site 0 with 1 and fixes sites 2…5.
- The reflected pairs are (1, 2),(0, 4),(3, 5).
- The independently enumerated proper-rotation classes are disjoint.
A coarse non-superposable mirror pair; no absolute Δ/Λ or optical-rotation sign has been assigned.
Try it yourself
In the ideal square-planar donor model, are B sites +x and −y cis or trans, and what donor angle separates them?
Show the answer and reasoning
cis;90°.
Their unit direction dot product is zero. The sites are adjacent rather than opposite.
Classify ideal octahedral B triples(+x,+y,+z) and(+x,−x,+z) in MA₃B₃.
Show the answer and reasoning
The first is fac; the second mer.
The first triple is mutually perpendicular and occupies a triangular face. The second includes an opposite pair.
Can the three B donors of ideal fac-MA₃B₃ include an opposite pair?
Show the answer and reasoning
No.
They occupy one triangular face and are mutually cis.
Does assigning an absolute helicity descriptor determine the sign of measured optical rotation?
Show the answer and reasoning
No.
The descriptor and a wavelength- and condition-dependent optical measurement answer different questions.
What to carry forward
- A donor count does not uniquely give a geometry.
- Apply proper rotations before counting different placements.
- Use cis/trans and fac/mer only in their stated geometry.
- Preserve a chelate’s connected donor pairs.
- Separate absolute configuration from optical-rotation sign.
References
Teaching resources and reference material supporting this lesson’s explanations and examples.
- Brief Guide — stereodescriptorsRichard M. Hartshorn; Karl-Heinz Hellwich; Andrey Yerin; Ture Damhus; Alan T. HuttonSource
- Chemistry, edition 2, section 19.2: Coordination Chemistry of Transition MetalsPaul Flowers; Klaus Theopold; Richard Langley; William R. RobinsonSource
- ClassXII Chemistry PartI Unit5: Coordination CompoundsNational Council of Educational Research and Training; collective textbook authorshipSource · PDF