The trans influence is an overlap argument

Take a metal with two identical ligands on opposite sides of it and replace one of them with a better donor. Something happens to the bond on the other side, and what happens is that it gets longer.

The effect is large enough to be unmissable in a crystal structure. In platinum(II) complexes a chloride trans to a hydride or a methyl group sits some 0.05 to 0.10 ångströms further from the metal than the same chloride trans to another chloride — a change comparable with the difference between a single and a partial double bond, produced by altering something on the far side of the metal.

The explanation is one sentence long and the rest of this essay is about making it computable: there is only one metal orbital pointing along that axis, and the two ligands are sharing it.

Three orbitals in a line #

The smallest system that can show the effect has three orbitals: a ligand on the left, one metal orbital, a ligand on the right. Two electrons occupy the lowest combination, which is the σ bonding pair holding the fragment together.

Solve it exactly — a 3×3 matrix, diagonalised — and take the metal–ligand bond orders from the filled eigenvector in the usual way, as in bond order from the eigenvectors: twice the product of the two coefficients.

With both ligands interacting equally the two bond orders are equal, at 0.6155 each. Strengthen the left-hand interaction by sixty per cent and its own bond order rises to 0.7807 while the right-hand one falls to 0.4880 — a drop of a fifth, produced by changing nothing on the right.

The total does not fall. The fragment as a whole is more strongly bound after the change, at −5.70 against −4.85 in units of the interaction parameter. What has happened is a redistribution: the filled orbital has more amplitude on the left and correspondingly less on the right, and the bond order follows the amplitude.

Why one orbital and not two #

The claim that there is “only one metal orbital pointing along that axis” is the load-bearing one, and it is not an approximation.

A metal has nine valence orbitals. Along a given axis — call it z — exactly one of them is a p orbital pointing that way, and one d orbital, dz², has amplitude along it as well. The s orbital points everywhere and so is shared by all six positions rather than by one trans pair.

So the resource being competed for is small and specific: a p orbital, plus a share of dz² and of s. Two ligands at ±z both need it and there is one of each to go round. Two ligands at ±x need px instead, and are not in competition with the first pair at all except through the s orbital, whose contribution is spread over everything.

That asymmetry is the prediction. The influence should be transmitted strongly along an axis and weakly around it — trans, not cis — and the observed effect is emphatically a trans one. Cis bond lengths in the same complexes move by a few thousandths of an ångström where trans lengths move by tens.

It also explains why the effect is so much clearer in square-planar complexes than in octahedral ones. A square plane has four ligands and two in-plane p orbitals: two clean, separate competitions. An octahedron has six ligands and three p orbitals, and the d and s contributions are shared more widely, so a change at one position is diluted among more partners.

The refusal that isolates the mechanism #

An explanation of this kind needs a test that can fail, because “they share an orbital” is the sort of statement that sounds explanatory whether or not it is doing any work.

The test is to remove the sharing. Set the right-hand interaction to zero, so the two ligands are no longer connected by anything, and strengthen the left-hand one exactly as before. The right-hand bond order changes by $10^{-16}$ — arithmetic noise, and not a small number that might be argued about.

That is the control, and it is the one that makes the other two mean something. Two ligands with no shared orbital do not influence each other in this model, so when they do influence each other the shared orbital is what carried it. There is no other channel in the calculation.

There is a second control available, and it is the one that says what sets the size of the effect rather than whether there is one. Move the ligand level closer to the metal’s and the two mix more thoroughly; move it further away and they mix less. The scan range is held identical for both so that only the denominator differs.

The same three orbitals as a hypervalent bond #

The three-level system in this essay is not new to the site. It is the same system three-centre bonding computed uses, one interaction weaker.

That essay puts four electrons into three orbitals in a line — two in the bonding combination and two in the non-bonding one — and argues that the result is a genuine chemical bond across three centres with no d orbitals required, which is what hypervalency actually is. The present essay puts two electrons in and asks what happens to the bond orders when the two ends are made unequal.

The connection is worth making explicit because it says what kind of claim the trans influence is. It is not a special effect of transition metals; it is what happens whenever two ligands bond through one orbital on a shared centre, and the same arithmetic describes the linear I–I–I of a triiodide ion, the F–Xe–F of xenon difluoride, and the Cl–Pt–Cl of a platinum complex.

What is special about square-planar platinum is that the geometry makes the competition visible. Sixteen-electron d⁸ complexes are square planar, as eighteen is a count works out, and a square plane has two independent axes with one metal p orbital along each — so the two ligands on one axis compete with each other and not with the pair on the other axis. The effect is confined to trans pairs, which is exactly what is observed.

The symmetry says the same thing in one line. In the square-planar σ framework the metal’s two in-plane p orbitals span Eu, and each of them lies along one axis — so each is shared by exactly one trans pair, and eighteen is a count builds that framework and labels it. That is the structural reason the influence is transmitted along an axis rather than around the plane, and it is why the two competitions in a square plane are independent of each other.

The measured version #

The model produces bond orders and a crystal structure produces bond lengths, so the comparison is qualitative in form and quantitative in size.

Ordering ligands by how much they lengthen the bond trans to them gives a series that is reproducible across many complexes:

H⁻ ≈ CH₃⁻ > PR₃ > CO ≈ Cl⁻ > NH₃ > H₂O

A hydride or a methyl group is at the top, and both are pure σ donors with no π chemistry available to them at all. That is the fact that identifies the mechanism: if the effect ran through π interactions, carbon monoxide would be at the top; it runs through the σ framework, and the strongest σ donors win.

The magnitudes are worth stating. In the anion of Zeise’s salt the Pt–Cl trans to the ethene ligand is about 2.34 Å against 2.30 for the two cis chlorides. In hydride complexes the difference reaches 0.1 Å. A tenth of an ångström is roughly four per cent of a bond length, which for a quantity determined by X-ray diffraction to three decimal places is enormous.

The reference compound is the one with no comparison in it: tetrachloridoplatinate, four identical chlorides round platinum(II) in D4h, every bond the same length because every ligand has the same trans partner. Replacing one chloride is what makes the comparison, and it is the bond opposite the replacement that moves. So the question the model can be asked is how far the trans bond order goes when the substituting ligand is at the top of that series rather than the middle of it.

Influence and effect are two claims #

The vocabulary here is unusually careful and the care is worth respecting, because two different things are being named.

The trans influence is a ground-state property: a bond length, or a bond order, or a coupling constant, measured on a molecule that is sitting still. It is what this essay computes.

The trans effect is kinetic: the observation that substitution at a position trans to certain ligands is faster, by factors that run to millions. It is a statement about a transition state, and it therefore depends on what stabilises the transition state as well as on what destabilises the ground state.

The two correlate and their orderings are not identical. Ligands high in the trans effect series include some — carbon monoxide, cyanide, ethene — that are good π acceptors, and their advantage is largely in stabilising the five-coordinate intermediate rather than in weakening the ground-state bond.

Nothing here is a rate. No activation energy, transition state or mechanism is computed. What is computed is a bond order in a ground state, which is a structural claim, and the kinetic half is named and left alone.

The energy the redistribution costs #

One more number is worth extracting, because it answers an objection.

If strengthening one bond weakens its trans partner, why does any complex bother with a strong donor? The answer is that the total falls: the fragment’s energy goes from −4.85 to −5.70 as the left-hand interaction is raised from 1.0 to 1.6, so the gain on one side outweighs the loss on the other.

That is the general shape of a competition for a shared resource, and the same shape appears in what a lone pair is worth, where a lone pair takes more than its share of space around a central atom and the angles adjust. The lone pair does not make the molecule less stable; it redistributes.

What the pair is competing for is easiest to name by taking one ligand away. One ligand orbital and one metal orbital, two electrons in the lower combination, is the ordinary two-level problem overlap decides is built on — and the trans influence is that problem with a second claimant on the same metal orbital. What the two ligands compete for is amplitude in one filled orbital, and a bond order is how much of it each of them got.

What the model cannot say #

It has one orbital per centre. A real ligand brings σ and π orbitals and a real metal brings nine, so the three-level picture is a caricature of a σ framework that the complete calculation of eighteen is a count treats properly. What it captures is the competition, which is what the effect is about.

It has no π in it. Ligands that influence their trans partner through π interactions — and there are some — are outside this model entirely. That is a real limitation and it is also why the σ-donor ordering of the observed series is evidence for this mechanism rather than for another.

Bond order is not bond length. The two are related monotonically and not by any formula computed here. What is claimed is the direction and the comparison, not a number in ångströms.

There is no electron repulsion. The two electrons in the filled orbital are treated as independent, which overestimates how much amplitude will move to one side. The size of that overestimate on a system small enough to solve exactly is in the smallest many-electron calculation.

The two-ligand limit #

Reducing the problem to two ligands and nothing else makes the shared orbital as visible as it can be made.

In a linear two-coordinate complex the sharing is as bare as it gets. Of the five d orbitals only dz² points along the axis, and in an angular overlap treatment it takes the whole of the σ interaction at 2eσ while the other four take none — the ligand field of a linear complex computes that split. One orbital, two ligands, and nothing else in the coordination sphere to dilute it.

Two-coordinate complexes are rare and instructive. Gold(I) and silver(I) form them, they are linear, and the linearity itself is a consequence of the same competition: with only one σ orbital available along an axis, two ligands get more from being opposite each other, where they share one orbital fully, than from any bent arrangement.

What repulsion alone would do with four ligands is put them at the corners of a tetrahedron, which has no trans pairs at all and therefore no axis along which an influence could be transmitted — what repulsion alone predicts draws that arrangement and says where it stops applying. Square-planar complexes have trans pairs because a ligand field made them square planar, which is why a phenomenon that follows from shared orbitals in general is, in practice, a transition-metal one.

That last point is worth holding on to. The trans influence needs two ligands on one axis, and a four-coordinate complex only has such pairs if it is square planar rather than tetrahedral — so a phenomenon that is an ordinary consequence of shared orbitals is confined, in practice, to the compounds whose geometry the ligand field decides.

What a bond order is doing here #

A word about the quantity being computed, since bond order is a term used loosely in chemistry and precisely here.

The Mulliken bond order between two centres is twice the product of their coefficients in a filled orbital, summed over filled orbitals. It is a property of the wavefunction and not an observable: nobody measures a bond order, and two different partitionings of the same density give different values, which is the standing caution of a basis is not a thing and of the localisation transformation.

What makes it usable here is that only differences are being claimed, and the same partitioning is applied to both bonds of the same molecule in the same calculation. When the left-hand bond order rises and the right-hand one falls, the two numbers were produced by one formula from one eigenvector, so the comparison survives the arbitrariness that would sink an absolute claim.

That is a general habit worth stating, because it applies to several quantities: a delocalisation energy, a bond order, a hybridisation ratio and an atomic charge are all partitioning-dependent, and all of them can carry an argument provided the argument is about a difference computed the same way twice.

Two series with one name, and this argument explains one of them #

The competition computed here has a name in coordination chemistry and it has a second name that is very nearly the same word, for a quantity that is different — and separating them says exactly what the overlap argument covers.

The trans influence is a ground-state property. It is measured as a bond length, a stretching force constant or a nuclear spin–spin coupling: a ligand’s ability to weaken the bond opposite it, in a molecule sitting still. That is the quantity computed here, and the mechanism is the one the arithmetic gives — two ligands competing for the same metal orbital, with the stronger σ donor winning and the loser’s bond order falling.

The trans effect is a rate. It is measured by how fast a ligand opposite a given group is replaced, and it is a statement about a reaction rather than about a structure.

The two orderings are not the same, and where they differ is informative.

Hydride and methyl sit at the top of the structural series, because they are excellent σ donors — exactly what the overlap competition predicts. Carbon monoxide and ethene sit near the bottom of that series, because they are poor σ donors; they push very little.

In the kinetic series carbon monoxide and ethene are at the top, above almost everything. The mechanism there is not σ competition at all: substitution at a square-planar centre goes through a five-coordinate intermediate, and a π-accepting ligand stabilises that intermediate by taking density off the metal as it becomes crowded. That is a statement about a transition state, and nothing in a ground-state bond order reaches it.

So the two series measure two mechanisms, and one word is routinely used for both.

The overlap argument covers the first and not the second. It explains why a bond trans to a hydride is a tenth of an ångström longer, and it has nothing to say about why a ligand trans to a carbonyl is replaced a thousand times faster — which is a σ argument and a π argument, on a structure and on a barrier, and the overlap competition is only one of the four.

That division has a practical use as well as a tidying one. A ligand high in both series — a phosphine, or hydride — is one whose σ donation does the work in each, and a prediction about it from either series will hold. A ligand high in one and low in the other is a ligand whose two mechanisms disagree, and predictions about it have to say which quantity is meant. Carbon monoxide is the standing example: opposite it, bonds are ordinary and substitutions are fast, which is exactly what a poor σ donor and a strong π acceptor should produce and is inexplicable if the two words name one property.

Who found it, and when #

The kinetic version came first: Ilya Chernyaev established the trans effect series in Russia in 1926, from preparative work on platinum ammines, and it was used as a synthetic tool for decades before anybody knew why it worked — the standard preparations of cis- and trans-platin both rely on it.

The structural version was separated out later, as crystallography became routine enough to measure bond lengths in related complexes reliably, and the name “trans influence” was coined in the 1960s specifically to distinguish the ground-state phenomenon from the kinetic one. The molecular orbital account — that the two ligands share a metal orbital — belongs to the same period.

The order of discovery explains a persistent confusion. A single term was in use for forty years covering two effects with different orderings and different explanations, and the literature from that period has to be read with the distinction supplied by the reader. Naming the two separately was most of the work of understanding them.