Molecular Geometry & Bonding
what the shape decides
Concept

Partial charge — where it appears

A charge assigned to one atom of a molecule, which no measurement returns. Every scheme for dividing the density between atoms is a convention, and the conventions disagree by amounts comparable with the charges themselves.

Named by 29 essays across 6 fields — each of them below, with the objects they name alongside it.

What the group settles. For each molecule, the point group found from its coordinates and the two properties that follow from the group alone. Neither column required knowing anything about the bonds.

The dipole is not a sum of bonds

Adding bond dipoles as vectors gets the easy cases right and rests on a quantity with several incompatible definitions. The symmetry argument is exact, needs no electronegativities, and says when the answer must be zero.

wrong · Dipole Four scales, four orderings. Each column ranks the elements by one electronegativity scale, most electronegative at the top, with a line joining each element across the columns. Every crossing is a pair of elements that two scales order differently.

Electronegativity is not one quantity

Four scales, four definitions, four sets of units, and a defence — "they correlate well" — that answers a question nobody asked. Two scales can correlate at 0.99 and still put hydrogen on the wrong side of carbon.

wrong · Electronegativity pyrrole against cyclopentadienyl anion. The Hückel levels of pyrrole beside those of cyclopentadienyl anion, which is the same graph with one diagonal entry and the bonds touching it changed. The parent's levels are symmetric about α because its matrix has nothing on the diagonal; the substituted system's are not, and the asymmetry is the size of the fitted parameter rather than a result.

Six electrons in a ring that is not all carbon

Pyrrole and the cyclopentadienyl anion hold the same six π electrons in the same five-membered ring. The anion spreads them perfectly evenly; pyrrole's nitrogen keeps 1.720 of them and furan's oxygen keeps 1.791 — the more electronegative atom donates less, and the bond orders to it fall with it.

beyond · Delocalisation The moment that does not depend on where the origin is. For each molecule, the dipole and the largest quadrupole component about the centre, and the same two about an origin moved 1.6 bohr away. The lowest non-vanishing moment is unchanged in every row and the one above it moves in every row. The 4 non-polar molecules here have a quadrupole that is a property of the molecule; the polar ones have one that is a property of a choice.

What a dipole cannot tell apart

A dipole moment is three numbers extracted from a whole charge distribution, and enormously many distributions give the same three. The moment above it is not even a property of the molecule unless the one below it vanishes — which is why a quadrupole is quoted with an origin and a dipole is not.

shape · Dipole Three orbitals in a line, four electrons in them. The three levels of a linear three-centre system, with the two lowest filled. The lower one is bonding across all three centres; the second has exactly zero amplitude on the middle atom, by symmetry rather than by arithmetic, so the two electrons in it sit entirely on the ends. That is where the charges come from: -0.5 on each end and +1 in the middle. Each of the two bonds has an order of 0.7071, which is 1/√2 and not the half the electron count suggests.

Hypervalency is about the ligands

The three-centre four-electron bond is offered as the reason sulfur hexafluoride needs no d orbitals. Read it forwards instead of backwards and it is a requirement rather than a permission — the arrangement puts half an electron onto each ligand before any electronegativity difference is applied, which is why the hypervalent compounds are fluorides and SH₆ is not a compound.

beyond · Hypervalency The bond sum, the measurement, and what is left over. For each pyramid: the vector sum of three bond moments estimated from the electronegativity difference and the measured geometry, the measured dipole, and the difference between them. Positive is towards the lone pair. Ammonia's bonds point that way and nitrogen trifluoride's point the other, which is why the trifluoride's far more polar bonds give it a dipole six times smaller. The leftover is 0.58 D for ammonia and at least 1.48 D for the trifluoride, so it is not one lone pair's property.

The lone pair is not the missing term

Ammonia's dipole is 1.47 debye and nitrogen trifluoride's is 0.235, although the N–F bonds are far more polar than the N–H ones. The bond sums explain the reversal exactly and point in opposite directions — and the lone pair that is supposed to make up the difference has to be worth 0.58 debye in one molecule and at least 1.48 in the other.

shape · Dipole Bonds the scales disagree about. For each bond, which atom each scale calls the more electronegative. Every row is a bond whose polarity would be drawn in opposite directions depending on which of these four tables in common use was consulted: Pauling, Mulliken, Allred–Rochow, Allen.

A ranking is not a difference

Four electronegativity scales agree about order to a Spearman coefficient of 0.99 and disagree about size by a factor of sixty-two. Put each on its own range and the O–H bond is 39 per cent of the way across on three scales and 4.5 on the fourth.

wrong · Electronegativity Strong outside, weak inside. The bond orders along each chain. A three-centre system has two equal bonds of 0.707 — not the one half the electron count suggests — and every longer chain alternates, strong at the ends and weak in the middle. The spread grows with the chain: 3:0.000, 5:0.211, 7:0.271, 9:0.296. Nothing here is about iodine.

Hypervalency does not stop at three centres

The three-centre four-electron bond is written up everywhere as an arrangement peculiar to hypervalent molecules. It is the first member of a family — five centres and six electrons, seven and eight — and the family predicts alternating bond strengths that the polyiodide crystal structures have.

beyond · Hypervalency The dipole moment, and how many bands there are. For each of five molecules: the dipole moment of the point-charge model, the number of modes whose dipole derivative does not vanish, and the largest derivative. The molecules with no dipole at all have the most active bands, which is the whole of the argument.

A dipole is not what an infrared spectrum sees

Carbon dioxide has no dipole moment at all and three of its four modes are infrared active. Methane has none and six of nine; boron trifluoride none and five of six. Water, which has the largest dipole of the five, has three modes and three bands — and its dipole predicted neither number.

shape · Dipole The ionic weight against how much ionic structure is in the wavefunction. The percentage each convention calls ionic, at a fixed structure overlap, as the amount of ionic structure in the wavefunction is raised from none to the molecular orbital value. They meet at both ends of the sweep and disagree everywhere between. Every curve is a weight and every set sums to one.

A weight that depends on how it is weighed

The ionic character of a two-electron bond is quoted as a percentage. For one wavefunction at hydrogen's bond length, three conventions in the literature give 18.73, 34.74 and 5.88 per cent — a factor of six — and on a wavefunction with no ionic structure in it at all, one of them still reports a quarter.

bonding · Models Two derivatives of the same energy, and only one is tabulated. The 18 elements of the electronegativity tables, placed by their chemical potential — half the sum of the ionisation energy and the electron affinity, which is the Mulliken electronegativity — against their hardness, half the difference of the same two numbers. Hardness runs from 1.92 to 7.3 electronvolts, a factor of 3.8, and does not follow the horizontal axis. Ringed points are the three elements whose anion is not bound, so whose affinity is not a measurement.

The quantity no scale prints

Every electronegativity table is half of a calculation. The other half is the hardness — half the difference of the same two measurements the Mulliken scale is half the sum of — and it varies by a factor of 3.8 across eighteen elements. Put both halves in and B–F, with an electronegativity difference of 6.12 electronvolts, moves less charge than lithium iodide, whose difference is 3.75.

bonding · Electronegativity What boron trifluoride's bands are strong in, and what they move. Every infrared-active mode of boron trifluoride, with its band strength and the root-mean-square displacement of its atoms in the zero point, each scaled to its own largest. The two do not order the modes the same way — the rank correlation between them is 0.2 — and the strongest band belongs to the mode at 719 cm⁻¹, in which 89.93 per cent of the motion is the lightest atom's. Every mode moves the same weighted amount of mass, exactly, so that is not what separates them either.

The mode that moves least radiates most

Boron trifluoride's strongest infrared band is the one in which the fluorines barely move: ninety per cent of the motion belongs to the boron, which is a fifth of the molecule's mass. The mode that moves the most mass is nine and a half times weaker. Across five molecules the rank correlation between band strength and how far the atoms actually go runs from +1 to −0.66, and every normal mode carries exactly the same weighted motion by construction.

shape · Dipole Which numbers survive a change of frame, and which are coordinates. Every quantity this field quotes, against the four things that can be changed without changing the molecule: the zero of energy, the unit, the reference state a stabilisation is measured from, and what a "per" quantity is divided by. A filled mark is a quantity that moves. three of the 9 survive all four, and every one of them is a property of the eigenvectors rather than of the energies. The first two changes are exact symmetries of the model, so a quantity that moves under either is a coordinate and not a quantity at all.

Which numbers carry a frame

A Hückel calculation is written in two numbers nobody computes and quoted against reference states nobody measures, so every quantity it prints is a quantity in a frame. Nine of them, tested against four changes of frame: three survive all four, and the one that survives both exact symmetries and looks safest — a dimensionless ratio between two molecules — is the most fragile of all, because against one reference its denominator is exactly zero.

bonding · Delocalisation The fraction a table encloses is not one number. For each ion, the fraction of its own electron density that lies inside its tabulated radius. Along each isoelectronic series the answer runs over eight percentage points, where three neutral atoms at their contact distances spanned three tenths of one. And it peaks at the neutral rather than falling through it, because a van der Waals radius is fitted to the distance between two atoms that are not bonded and an ionic radius is one term of a sum fitted to the distance between two that are.

The surface a table draws

Three noble gases stop at a surface enclosing between 99.38 and 99.75 per cent of their density — a near-constant, and an argument that a contour is a real boundary. Charge the atoms and it collapses. Across ten electrons the tabulated radius encloses anything from 94.4 to 99.98 per cent, it peaks at the neutral rather than trending through it, and radii built at a fixed enclosure do not add up to a single measured separation.

orbitals · Contour Where the orphan pair actually sits. A σ-only Hückel model of each molecule, built from its own coordinates: the central atom's four valence orbitals, one σ orbital on each ligand, and the coupling between them the direction cosine of that ligand. Lone pairs need no special handling — they come out of the diagonalisation as the central orbitals no ligand combination transforms like. The charge is measured rather than assigned, and PF₅ comes out with two kinds of fluorine at 3 at -0.13 and 2 at -0.3, the more charged pair being the axial one that carries the orphan.

The count is the population

The census counted how many ligand combinations have no partner on the central atom and called the count n + L − 4. A σ-only model built from each molecule's own coordinates says what that count is worth: with no electronegativity difference anywhere, the mean charge on a ligand is minus the orphan count divided by the ligand count, exactly, in all ten cases. And the prediction the census made — that the charge grows with the orphan count — is refused by the divisor.

beyond · Hypervalency Carbon is not one number. The charge equalisation puts on carbon in the four fluoromethanes, and on hydrogen in the three that have one. Carbon's runs from 0.07 to 0.31, and hydrogen's changes sign between methane and fluoromethane, so the same C–H bond is polarised one way in one molecule and the other way in the next. A table with one number for carbon is printing the value an iteration starts from.

The value that only exists in the bond

A difference in electronegativity moves charge, and moving charge closes the difference — until every atom in the molecule has the same chemical potential. Solving that gives one number per molecule and a charge per atom, and carbon's runs from +0.0692 in methane to +0.3074 in tetrafluoromethane. Hydrogen's changes sign between the first and the second, so the same C–H bond is polarised one way in one compound and the other way in the next.

bonding · Electronegativity Four unweighted rules against the answer the hardnesses give. Twelve molecules, each drawn with the exact equalised electronegativity as a filled mark and the four unweighted means of the free atoms' values as open ones. The geometric mean — Sanderson's rule — is the closest on average, at 0.0706 eV against the arithmetic mean's 0.1666, and it is the closest on only 4 of the twelve. Every rule below the arithmetic mean is constrained to sit below it, and 2 of the exact answers do not.

A mean that is low rather than right

Sanderson's rule takes the geometric mean of the atoms' electronegativities, and it works better than the plain average. Computed against the exact equalised answer for twelve molecules it is better by erring downwards — and two of the twelve have exact answers above the plain average, where no geometric mean can go at any parameter.

wrong · Electronegativity The metal's charge is a coordinate, and the count is not. The metal's charge in an octahedral d6 complex, against how much of each shared pair the ligand is given. Half each is Mulliken's rule and the whole to the ligand is the assumption an oxidation state makes; the answer runs over 2.06 electrons between them. The oxidation state itself is 0, which is off the end of the range, and the electron count is the same number at every point on it.

An integer nobody measured

The oxidation state of chromium in the hexacarbonyl is zero. Its charge, computed from the same wavefunction, is anywhere between −3.04 and −0.98 depending on how the shared electrons are divided — and the integer sits outside that whole range. The electron count, meanwhile, is eighteen at every point on it.

applied · Electron count Four tables, one answer, and a reason it could not be otherwise. The intensity–motion correlation — infrared intensity against how far the atoms move — computed with charges from four published electronegativity tables. Every molecule gives the same number on all four, to machine precision, because each is made of two elements: its charges are one number times a fixed pattern, a change of table changes only that number, and a rank correlation does not notice a rescaling. The dipole moments beside them do notice, which is the check that the tables are genuinely different.

The table that could not have mattered

Charges taken from one of four electronegativity tables invite a worry, because the tables disagree with each other. They do disagree — hydrogen cyanide's dipole runs over a factor of eleven between them — and for the molecules in question the worry could not have applied, because a molecule of two elements has charges that are one number times a fixed pattern and a rank correlation does not notice a rescaling.

shape · Dipole The even sharer is the one the repulsion likes least. trigonal bipyramid: 2 kinds of ligand, spread 0.1137, repulsion 6.4747; square pyramid: 2 kinds of ligand, spread 0.1658, repulsion 6.4844; pentagonal planar: 1 kind of ligand, spread 0.0000, repulsion 6.8819. The planar arrangement gives all five ligands exactly the same charge and costs 6.3 per cent more in repulsion than the bipyramid, which is the arrangement chemistry actually adopts — so the two models disagree about which arrangement is preferred, and about how much charge is moved.

Two models that disagree about the shape

The identity says how much charge a hypervalent molecule's ligands must share and nothing about how. Working out which arrangement shares it most evenly puts the σ model and the repulsion model on one axis for the first time: the even sharer is the pentagonal plane, which is the arrangement the repulsion likes least — and along the interchange chemistry actually uses, one model sees 0.15 per cent of a change and the other sees 54.

beyond · Hypervalency However hard the π channel is driven, the counted orbital stays the metal's. The metal's share of the filled T₂g orbital against the π coupling, with the eighteen-electron count drawn beside it. The share falls from 1 to 0.5467 across a coupling range of 80,000 cm⁻¹ and approaches a half from above without reaching it: the lower eigenvector of a two-level problem always carries more of the lower basis function, whatever the coupling. The count is eighteen at every point.

The count that cannot be broken by strength

Back-donation puts electrons into orbitals that are not the metal's, and the eighteen-electron rule counts the metal's nine. Turning the π channel up as far as it will go never breaks it: the counted orbital's metal share falls from 100 per cent to 54.67 and approaches a half from above without reaching it. What does flip it is not strength but order.

applied · Electron count Twelve hydrogen bonds, four tables. The bond dipole of hydrogen against each partner, in debye, on each of the four tables after all four are anchored to the same hydrogen–fluorine separation. Positive is hydrogen at the positive end. A bond whose marks straddle the axis is one the tables disagree about the direction of, and there are 3 of them.

Four tables and one molecule to disagree about

A molecule of two elements is provably safe from the choice of electronegativity table, and a series down a group should land where the tables do disagree. It does. All four agree that hydrogen iodide is the exception — and they disagree about what its dipole is by 1.245 debye, which is two and a half times the 0.448 that was measured.

shape · Dipole A fourth data point, and 3 negative electronegativities. Each element's electronegativity from three points on its energy curve, the cubic coefficient a fourth point adds, and what the fourth point leaves. The cubic coefficient is one sixth of a second difference of the ionisation series, so it is largest where that series has a kink — and the alkali metals, whose second electron comes out of a closed shell, are pushed to Li -7.91, Na -3.42, K -1.49 eV. The last column is where the two roots of the fixed-point equation collide, beyond which the atom has no solution at all.

Where a closed form stops being one

What happens when the quadratic energy is not enough? A cubic makes the equalisation condition a quadratic with two roots, and something has to choose between them. The choice is easy and the finding is somewhere else — a cubic fitted through the dication gives lithium an electronegativity of −7.907 eV, a capacity of 0.073 of an electron, and a molecule of two alkali metals no solution at all.

wrong · Electronegativity The capacity, against whether the anion exists. Each atom at its electron affinity and the capacity the cubic model gives it — the largest amount of electron the model says it can accept. A negative affinity is an anion that is not bound, which is the statement that the true capacity at the integer is zero. The three atoms with negative affinities are beryllium, magnesium and nitrogen, and they have the largest capacities in the set: two of them infinite and the third 31.5. The three smallest capacities all belong to atoms whose anions are bound.

A capacity that is largest where there is none

A cubic through four charge states implies a largest amount of electron an atom can accept, and it is natural to ask whether the idea survives its own model. It does not. The three atoms whose anions are not bound — beryllium, magnesium and nitrogen — have the largest capacities in the set, two of them infinite; the three smallest capacities all belong to atoms whose anions are bound. And among the fourteen where the test cannot bite, the ordering is sensible.

wrong · Electronegativity One geometry, four electron counts, four answers. The bond order between two orbitals with exactly no overlap and no resonance integral, as the third orbital's energy is swept, at every count the trio can hold. With none it is identically zero. With two it is positive and rises past one. With four it is negative and reaches -0.954. With six it is a horizontal line — the third orbital's energy stops mattering entirely.

A filled shell is not an empty statement

A bond order of −0.954 between two orbitals with no overlap and no resonance integral invites the prediction that at six electrons — every level occupied, the sum over a complete set — it would be exactly zero. It is exactly one seventh, and the reason is that a complete set in a non-orthogonal basis sums to the inverse of the overlap matrix, which has entries where the overlap has none.

wrong · Overlap The two overlaps, squared, at the measured bond lengths. For each chromium(III) donor: the σ overlap squared, a metal 3d(z²) against the donor's p(z), and the π overlap squared, a 3d(xz) against its p(x). Everything is computed — the radial functions from Slater's rules, the separation from the measured bond length, the integral by quadrature. Chloride's π overlap is 3.5 times fluoride's, which is the opposite of what the overlap argument required of it.

The overlap the model is not proportional to

Without a computed π overlap, the natural argument reasons about one instead: the denominator over-predicts the halide trend, so the overlap must shrink down the group to cancel part of it. Computed, it grows — 3.5 times from fluoride to chloride. And the fitted parameter changes sign across the series, which no ratio of squared overlaps can do.

applied · Ligand field Two charges, and three ligands no charge reaches. The metal effective charge each ligand would need on its own for the model's ratio to equal the fitted one. The band is the range Slater's rules allow chromium. Two ligands have an answer, both far outside it and 2.30 apart from each other. Chloride needs more than the overlap rule can be trusted to compute. Ammonia's fitted parameter is exactly zero and cyanide's is negative, and a quotient of squares is neither.

A contraction that cannot reach three of them

The angular overlap model's own derivation gives a π/σ ratio that disagrees with the fitted parameters by up to sixfold, and the metal's contraction is the obvious candidate to account for it. The whole range Slater's rules allow moves the ratio by a factor of two. Two ligands need charges far outside it, one needs a charge past where the overlap rule can be trusted at all, and two are unreachable at any charge because a quotient of squared overlaps cannot be zero or negative.

applied · Ligand field Every window sits above the value it was meant to reach. For each of the five chromium(III) complexes, the whole range of π/σ ratios the model can produce as the ligand's donor atom is taken through every oxidation state it has — from its bare nucleus to its closed-shell anion — drawn as a bar, with the fitted parameter marked beneath it. The three ligands whose fitted parameter is positive have bars that begin above it and never come down. The other two have fitted parameters of zero and of a negative number, which a quotient of squared overlaps cannot be at any charge.

The correction that moves three of them backwards

Every ligand radial function in the angular overlap sweeps was a neutral atom's, while three of the five donors carry a formal charge. Giving each one the charge it actually has moves three of the five computed ratios — and moves all three away from the fitted parameter, none towards it. The whole window each donor's own oxidation states allow sits above the value it was meant to reach.

applied · Ligand field Straight segments, and the curve fitted through their ends. Cl's energy against the charge it carries. The exact theory says the energy is straight between integers, with a kink at each one: the slope below the neutral atom is the electron affinity and the slope above it is the first ionisation energy, and the two are different numbers. Every quantity in this argument comes from the smooth curve fitted through those points instead — and the curve's second derivative, which is the hardness, is a property the segments do not have at all.

Four quantities go and one question stays

The exact theory says an atom's energy against electron count is straight segments between integers, and that model gets the alkali metals right for free — which is exactly where the fitted curves give lithium a negative electronegativity. What it costs is the hardness, the capacity, the electronegativity and equalisation, and what it leaves is a question whose answer is an integer.

wrong · Electronegativity

Named alongside it

The objects these essays reach for when they reach for this one.

Model limitConventionElectronegativityDipole momentPolarityMulliken scalePauling scaleBond dipoleElectron affinityIonisation energyOverlap integralRank correlation

All concepts