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Graph Theory in Molecular Modeling The Hückel method: using adjacency lists to determine electron density Graph Theory in Chemistry Fullerenes are complex polyhedra The number of pentagon-pentagon adjacencies help separate stable and unstable isomers.

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graph theory in molecular modeling

Graph Theory in Molecular Modeling

The Hückel method: using adjacency lists to determine electron density

graph theory in chemistry
Graph Theory in Chemistry
  • Fullerenes are complex polyhedra
    • The number of pentagon-pentagon adjacencies help separate stable and unstable isomers.
  • Other graph invariants often match chemical properties:
    • Kekule counts help to qualitatively determine stability
    • Independence numbers serve as an indication of limits to chemical reactivity.
    • Adjacency eigenvalue spectrums determine electronic structure.
molecular orbitals a really basic chemistry review
Molecular Orbitals(a really basic chemistry review)
  • Atoms are surrounded by electron clouds, represented by continuous density functions.
  • The smooth surfaces are really misleading
molecular orbitals ii
Molecular orbitals ii
  • Orbitals also extend over bonds between atoms—a [covalent] bond is fundamentally just shared electrons.
molecular orbitals iii
Molecular orbitals iii
  • Electron density over atoms and bonds is the driving force of chemical reactivity:
    • Determines charge & partial charge.
    • Each has an energy level, ergo thermodynamic reactions proceed.
  • Best tool for raw prediction
h ckel method
Hückelmethod
  • Use a weighted graph representation of a heteromolecule (or approximate one)
    • Atoms become vertices
    • Bonds become edges
    • We already draw them like this!
  • Include a number for the total edge weight
    • The number of electrons that make up our bonds
  • Add a set of functions we want to minimize
    • These functions will model electron repulsion
the hard part
The hard part
  • A lot of equations, which are agonizing and I’m glossing over.
  • Basically, we have orbitals φμ and undetermined coefficients Cµi, for a given atom, orbital energies ε for a given orbital, and we get a ψi for each atom.
  • We want find values of Cµi to minimize E. This is the final equation.
ugh it gets worse
Ugh, it gets worse
  • We can turn our previous constraints into a set of linear equations.*
  • New complexity requires approximations and simplification of the problem.

* This requires solving the Schrödinger equation for a specific Hamiltonian, so I’m not going to cover it. If you’re curious, I can direct you to instructional sources.

i h ckel
I Hückel

Hückel made it easier (an approximation):

Now, we don’t need to even understand the form of the Hamiltonian (h), much less how it works.

example demonstration
Example & Demonstration
  • Ethane, an example from Australian Computational Chemistry via the Internet Project (ACCVIP).
  • There are a lot of programs, both commercial and free, that do all of this. A free visual one is the Simple Hückel Molecular Orbital (SHMO) calculator
extensions
Extensions
  • We’ve only accounted for heteromolecules.
  • Hückel assumes equal length bonds.
  • Other methods augment or replace Hückel:
    • CNDO/2
    • Strict quantum chemical adherence
    • Fenske-Hall method