Therefore, we are now able to calculate approximate energy curves for molecules and compare the results with experimental data.įurthermore, the curve for ψ - shows no minimum. This is a quite satisfying result from experiments we get 2.77 eV. The calculated bonding energy is 1.77 eV. Thus, H 2 + should exist as a stable molecule. The curve for ψ + refers to the ground state of the molecule where a minimum energy is found for a nuclear distance of approximately 2a o (i.e. The energy curves for ψ + and ψ - reveal the following properties of the ion H 2 + The sum of this energy and the energyĮmerging from the repulsion between the nuclei represents the total energy of the molecule.įig.1 Curves representing the total energy for the bonding (+) and the antibonding (-) MO as a function of the internuclear distance R.Ītomic units used: a o = Bohr's radius, energy unit au is twice the ionization energy of the hydrogen atom. Now we are able to calculate α, β and S for any distance R between the two nuclei. |ψ-|² for the ion H 2 + are depicted here.īefore proceeding our calculation, we substitute φ A and φ B with the atomic orbital 1s of the hydrogen atom. The obtained wave functions Ψ+ and Ψ- as well as the square of their absolute values With a coefficient c A that is determined in the usual way by normalization. For the respective molecular orbitals we get: By inserting both energy values E ± into the secular equation, we obtain the two solutions for c A = ± c B. To determine the wave function ψ = c Aφ A + c Bφ B for the energy levels, we need the coefficients c A and c B. The only situation where we get an asymmetric splitting is for values of S that are not neglectable when compared with 1. The results are depicted schematically in the following figure. The other MO with the higher energy is the antibonding MO which is higher in energy than the hydrogen atom. Its energy is below the energy of the hydrogen atom and represents the ground state of the molecule. The MO with the lowest energy is called the bonding MO. It is as well recognizable that the extent of the splitting is determined by the value of (β - αS). The equations show the way a formerly degenerated pair of energy values α splits symmetrically in two values below and above the original energy. For "slightly larger" values of S the equation E ± ≈ α ± (β − αS) represents a good approximation. c A and c B are not zero) to our problem ( derivation) we obtainįor values of S small in comparison to 1. = α B = α and obtain the following secular equations:Īs solutions relevant (i.e. Keep in mind that α A and α B in both secular equations have identical values as there is no difference between the two ends of the molecule and an electron hat the same energy in φ A (around nucleus A) as in φ B (around nucleus B). It is recommendable to begin with the most simple among those systems, the hydrogen molecule ion H 2 +.Īs this molecule has only one electron, this molecule is for a consideration of the chemical bond as fundamental as the hydrogen atom for the structur of the atoms of the periodic table of elements. molecules that consist of two identical atoms, e.g. The LCAO method adopts an especially simple form for homonuclear diatomic molecules, i.e. The Hydrogen Molecule Ion H2+ The Hydrogen Molecule Ion H 2 +
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