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. +20 more
25 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. +16 more
21 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. +16 more
21 essays · beyond 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. +16 more
21 essays · symmetry 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. +15 more
20 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. +14 more
19 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. +14 more
19 essays · bonding 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. +13 more
18 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. +13 more
18 essays · beyond 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. +12 more
17 essays · spectra 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
  6. +12 more
17 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. +12 more
17 essays · spectra Orbitals at the 90 per cent contour. Several orbitals drawn at the same enclosed fraction and, unless stated otherwise, at the same scale — so the sizes on the page are the sizes. Each contour was solved for separately by integrating that orbital's own density. Contours drawn: 1s at 90% of its density, |ψ| = 3.94e-2; 2s at 90% of its density, |ψ| = 7.40e-3; 2pz at 90% of its density, |ψ| = 9.48e-3.

Approximation

  1. 1 Orbitals are not where the electron is
  2. 2 The aufbau order is not a property of the atom
  3. 3 A better energy is not a better answer
  4. 4 The atoms are not at the points
  5. 5 The bond length that depends on the isotope
  6. +11 more
16 essays · wrong 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 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. +11 more
16 essays · beyond A chain of 100 is more stable alternating. The electronic energy gained by alternating the bonds of a half-filled chain, the elastic cost of doing it, and their sum, all per site and all against the alternation. The sum has its best value away from zero, so the evenly spaced chain is not the stable structure.

Peierls distortion

  1. 1 A chain cannot stay even
  2. 2 The gap is not the band width
  3. 3 Copper is never quite octahedral
  4. 4 The distortion the filling chooses
  5. 5 A distortion needs two states
  6. +11 more
16 essays · solids 4 sites, minimised. The arrangement of 4 points on a sphere that minimises their mutual repulsion. The angles printed were measured off the result rather than quoted, and the shape was not assumed.

VSEPR

  1. 1 VSEPR, computed
  2. 2 Five sites are not alike
  3. 3 Why water is bent
  4. 4 The shapes above six coordination
  5. 5 Bent's rule, against the substituent series
  6. +11 more
16 essays · shape 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

  1. 1 Electronegativity is not one quantity
  2. 2 A ranking is not a difference
  3. 3 A difference does not make a transfer
  4. 4 The quantity no scale prints
  5. 5 The value that only exists in the bond
  6. +10 more
15 essays · wrong 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. +10 more
15 essays · wrong 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 close — at 6 electrons — which is Hückel's 4n+2, produced here rather than recalled.

Aromaticity

  1. 1 Aromaticity as a computed shell closure
  2. 2 Delocalisation is stabilising, and other things that are false in general
  3. 3 The vibration that lowers the symmetry
  4. 4 The same ring, three charges
  5. 5 The ring with a twist in it
  6. +9 more
14 essays · bonding

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