Molecular Geometry & Bonding
what the shape decides
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Series

A field says what an essay is about. A series follows one idea essay by essay — from the question that introduces it to the one that assumes all the others.
The Td character table. The irreducible representations of the molecule's point group. The class headings carry the number of operations found by generating the group from the coordinates, and those counts were produced before this table was opened.

Representation

  1. 1 Character tables and reduction
  2. 2 Selection rules are one theorem
  3. 3 Site symmetry, and what it constrains
  4. 4 Degeneracy is a group theorem
  5. 5 Why a d–d band is weak
  6. +16 more
21 essays · symmetry Choosing a contour for 3s. The fraction of the density enclosed by a contour, against the contour's level. Picking a level is picking a point on this curve, and the usual practice of picking one that looks right is picking a point without knowing which. Contours drawn: 3s at 50% of its density, |ψ| = 6.24e-3; 3s at 90% of its density, |ψ| = 2.64e-3; 3s at 99% of its density, |ψ| = 7.15e-4.

Contour

  1. 1 Say what it encloses
  2. 2 How big is an orbital
  3. 3 One level is not one comparison
  4. 4 A filled shell has no shape
  5. 5 The isovalue nobody chose
  6. +14 more
19 essays · orbitals ammonia — C3v. The molecule with the point group found from its coordinates rather than looked up: every candidate operation was applied and kept when it permuted the atoms among themselves.

Point group

  1. 1 Point groups from coordinates
  2. 2 Symmetry forbids a dipole
  3. 3 Chirality is a symmetry statement
  4. 4 Descent in symmetry
  5. 5 An infinite group, worked in a finite one
  6. +13 more
18 essays · symmetry The wrong shape, fitted as well as it can be. The exact hydrogen 1s orbital and the best sums of one, two, three and six Gaussians, each with its exponents optimised for the energy. Three of them already reproduce the exact function to 99.94 per cent by overlap, which is why the method works at all — and the two places it goes wrong, at the nucleus and far out, are exactly where the other faces of this figure look.

Basis

  1. 1 A Gaussian is the wrong shape
  2. 2 A contraction is a decision made once
  3. 3 A function that is already there
  4. 4 The measurement a basis was not fitted to
  5. 5 The property that gets worse
  6. +11 more
16 essays · orbitals sulfur hexafluoride — Oh. The molecule with the point group found from its coordinates rather than looked up: every candidate operation was applied and kept when it permuted the atoms among themselves.

Hypervalency

  1. 1 Hypervalency without d orbitals
  2. 2 Three-centre bonding, computed
  3. 3 Six bonds and four orbitals
  4. 4 Hypervalency is about the ligands
  5. 5 Hypervalency does not stop at three centres
  6. +11 more
16 essays · beyond The spin-only count against nine measured moments. Each ion's magnetic moment computed from the number of unpaired electrons alone, √(n(n+2)) Bohr magnetons, beside the measured value. The two agree to a hundredth for the first five and the measurement exceeds the count by up to 0.93 for Co²⁺ — always in the same direction, which is what an omission looks like rather than noise.

Magnetism

  1. 1 A moment counts electrons, not orbitals
  2. 2 The pairing energy decides the moment
  3. 3 What couples two spins
  4. 4 An orbital carries no angular momentum
  5. 5 The g-value is the orbital coming back
  6. +11 more
16 essays · applied Which rings close a shell. Each ring is filled with its own number of pi electrons and asked whether the highest occupied shell came out full. Of the rings drawn here, C6 and C10 close — at 6 and 10 electrons — which is Hückel's 4n+2, produced here rather than recalled.

Multicentre

  1. 1 Where two-centre bonding stops
  2. 2 The band limit
  3. 3 What one pair can hold together
  4. 4 A cage needs one pair more than it has corners
  5. 5 Four centres, and the pair that will not localise
  6. +11 more
16 essays · beyond benzene — D6h. The molecule with the point group found from its coordinates rather than looked up: every candidate operation was applied and kept when it permuted the atoms among themselves.

Delocalisation

  1. 1 Delocalisation
  2. 2 Hückel theory and what it gets right
  3. 3 Bond order from the eigenvectors
  4. 4 Conjugation, and its limits
  5. 5 Hückel with a heteroatom
  6. +10 more
15 essays · beyond Eighteen electrons, from a reduction. The ligand σ orbitals of an octahedral complex reduced in Oh (A₁g ⊕ Eg ⊕ T₁u), matched against the metal's nine valence orbitals by species, and counted. 6 bonding and 3 non-bonding orbitals hold 18 electrons.

Electron count

  1. 1 Eighteen is a count
  2. 2 Back-bonding is two interactions
  3. 3 Sixteen is also a count
  4. 4 The bonds are what is left over
  5. 5 The count that is not always eighteen
  6. +10 more
15 essays · applied The 2pz orbital. The 2pz orbital at the contour enclosing 90 per cent of its density — a level solved for by integration rather than chosen. The two colours are the two signs of the wavefunction, which is what distinguishes a bonding interaction from an antibonding one. Contours drawn: 2pz at 90% of its density, |ψ| = 9.48e-3.

Orbital

  1. 1 What an orbital is
  2. 2 Nodes
  3. 3 Where the electron is
  4. 4 The radial distribution across the periodic table
  5. 5 Complex harmonics against real ones
  6. +10 more
15 essays · orbitals 1s with 1s at 2.8 bohr. The two orbitals in the plane containing both nuclei, with the regions where their product is positive and negative shown faintly. The overlap integral is the signed volume of that product, and where symmetry makes the two regions mirror images it comes out exactly zero. Contours drawn: 1s at 50% of its density, |ψ| = 1.48e-1; 1s at 50% of its density, |ψ| = 1.48e-1.

Overlap

  1. 1 Overlap decides
  2. 2 Exactly zero
  3. 3 The antibonding level goes up more
  4. 4 A double bond is not two single bonds
  5. 5 The trans influence is an overlap argument
  6. +10 more
15 essays · bonding carbonyl sulfide: a linear. The principal axes of carbonyl sulfide drawn at its centre of mass, with the moment of inertia about each. Which kind of top this makes it is decided by comparing three numbers, and the point group forces the same answer independently: an axis of order infinity means a linear molecule, with one moment at zero forbids an asymmetric top.

Rotation

  1. 1 The rotational spectrum is a moment of inertia
  2. 2 A bond length out of a spectrum
  3. 3 The moment that is the sum of the other two
  4. 4 The rotor that stretches
  5. 5 The constant a spectrum cannot see
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15 essays · spectra 6 molecules, counted. How many vibrations each molecule has, how many symmetry species they fall into, and how many frequencies are infrared active, Raman active, both, or neither. Every column after the first counts frequencies rather than modes, because a degenerate pair is one line in a spectrum. The molecules with a centre of inversion are the ones with nothing in the both column — mutual exclusion as a computed count. Nothing here uses a force constant.

Spectrum

  1. 1 How many frequencies, not how many modes
  2. 2 Two structures, two spectra
  3. 3 What an absence proves
  4. 4 A spectrum counts environments, not atoms
  5. 5 A spectrum that changes when only a mass does
  6. +10 more
15 essays · spectra Chains of 2, 4, 8, 16, 40: the levels crowd, the edges do not move. Every level of a chain of 2, 4, 8, 16, 40 sites, drawn at its computed energy on one axis. Adding sites adds levels inside a fixed interval rather than widening it, which is what makes the large system a band rather than a wider molecule.

Bands in a solid

  1. 1 A solid is a molecule that did not stop
  2. 2 The width of a band is a count of neighbours
  3. 3 A density of states is not a spectrum
  4. 4 A band with no structure in it
  5. 5 Where a molecule stops being one
  6. +9 more
14 essays · solids 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.

Dipole

  1. 1 The dipole is not a sum of bonds
  2. 2 One coordinate, three point groups
  3. 3 What a dipole cannot tell apart
  4. 4 The lone pair is not the missing term
  5. 5 Zero dipole is not no interaction
  6. +9 more
14 essays · wrong A d shell in an octahedral field. The five d energies in octahedral coordination, computed and drawn in units of the octahedral splitting. The dashed line is the barycentre, which the five levels sum to whatever the arrangement.

Ligand field

  1. 1 The splitting is a symmetry statement
  2. 2 Two models, one ratio
  3. 3 The spectrochemical series is not electrostatics
  4. 4 Where a d–d band falls
  5. 5 The double hump and what removes it
  6. +9 more
14 essays · applied Splitting goes with overlap. For each pair, the atomic levels on the outside and the combinations they form in the middle, with the splitting drawn in proportion to the computed overlap. A pair that symmetry forbids does not split at all, because its overlap is exactly zero.

Models

  1. 1 Molecular orbital and valence bond
  2. 2 Where molecular orbital theory dissociates
  3. 3 Two pictures, one plane
  4. 4 A weight that depends on how it is weighed
  5. 5 One spectrum, a line of models
  6. +9 more
14 essays · bonding H₂O: 3 distinct modes. The displacement of every atom in 3 normal modes of H₂O, drawn from the eigenvectors of the mass-weighted Hessian. Under each is the internal coordinate with the largest share of the motion and how large that share is; where no coordinate holds nine tenths, the mode is not a motion of one bond or one angle and is labelled so.

Normal mode

  1. 1 Normal modes are not bond stretches
  2. 2 The force field is not in the spectrum
  3. 3 The isotope shift is arithmetic
  4. 4 The frequency is not the bond strength
  5. 5 Group frequencies, and where they stop
  6. +9 more
14 essays · spectra methane: 2 valence bands. The measured valence photoelectron bands of methane, each labelled with the symmetry species of the orbital it comes from, and beside them the species the valence basis spans — the central atom's s and p functions and one s on each ligand, reduced in the molecule's own group. A band carrying a species the reduction does not produce would stop this figure being drawn.

Photoelectron

  1. 1 What a photoelectron spectrum measures
  2. 2 Water's lone pairs are not a pair
  3. 3 Fewer bands than electrons
  4. 4 Koopmans' theorem is exact for nothing
  5. 5 The width of a band is a bond length
  6. +9 more
14 essays · wrong The largest angle each ring size can have. The ceiling on a bond angle in a closed ring of equal bonds, 180°(n−2)/n, which is the interior angle of the regular planar polygon and follows from a closed curve having to turn through a full circle. The line at 109.47° is the tetrahedral angle: rings of 3, 4, 5 atoms cannot reach it at any geometry whatever, and every larger ring can, by leaving the plane.

Strain

  1. 1 The angle a ring cannot have
  2. 2 The ring that cannot hold still
  3. 3 The double bond a ring cannot hold
  4. 4 The strain that is not in the angles
  5. 5 The atoms that meet across a ring
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14 essays · shape

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