Draft:Archive/2024/Hückel method/Hückel annotated results for ethene

Annotated Hückel calculation results for ethene
Below is the annotated output of the Hückel calculator results for ethene:

BOND       BETA 1    2     1.00000


 * This summarises the data you put in, indicating that the connectivity is 1.0. We call that here the BETA value. It will be different from 1.0 when we introduce heteratoms.

ENERGIES IN ASCENDING ORDER MO NO.  1          2 1.00000  -1.00000

[Equation #1]
 * The energy values as $$x_i$$ are given; i.e.

COEFFICIENTS IN COLUMNS AO NO.  1   0.70711    0.70711 2  0.70711   -0.70711

[Equation #2] The 2 molecular orbitals are: [Equation #3]
 * The coefficients of the atomic orbitals appear in columns.

BOND ORDERS BOND      ORDER 1   2     1.00000


 * Bond orders, in general, are a measure of the numbers of electrons in the bond. This is certainly how it is used in the general qualitative MO picture of diatomic molecules. In Huckel theory, however, it is more a measure of bond strength. Twice the sum of the bonds does not always give the number of electrons. Here it does, but for aromatic compounds twice the sum of the bond orders is greater than the number of electrons and this reflects the extra stability of these compounds. In this case it is saying that the [pi] bond is of order 1.0. Adding the [sigma] bond gives the double bond we expect. Bond orders are calculated from the coefficients of the occupied molecular orbitals.

CHARGE DENSITY, SPIN DENSITY AND FREE VALENCE ATOM   DENSITY        SPIN DENSITY      FREE VALENCE POSITIVE   NEGATIVE ION        ION 1    1.00000     0.50000     0.50000     0.73205   2     1.00000     0.50000     0.50000     0.73205


 * Charge density is, like bond orders, calculated from the coefficients. It gives an estimate of how the [pi] electrons are distributed between the atoms. In this case the two [pi] electrons are equally distributed between the atoms. Spin density is a measure of what the odd electron is doing in an open shell system - in this case the cation and anion derived from ethene by removing or adding an electron. There will be more about this later. Free valence is a measure of how strongly radical substitution will occur at the designated atom.


 * For more detail on bond orders, charge densities, spin densities, free valence etc., click here.

NO OF ATOMS          2 NO OF BONDS          1 NO OF ELECTRONS      2 TOTAL PI ENERGY      2.00000 RESONANCE ENERGY     0.00000


 * The total π electron energy is just the sum of the orbital energies. In this case it is 2 [alpha] + 2 [beta]. The output just sums the xi values. The resonance energy compares the total energy with the energy of the required number of double bonds (using the ethene energy) in a Kekule structure. In this case, therefore, it is not surprising that it is zero. For benzene, it would compare the energy of benzene from the Hückel calculation with the the energy of three ethenes since there are three double bonds in the Kekule structure of benzene.


 * Thus the resonance energy is a measure of how the structure is stabilised by resonance or delocalisation.

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