Molecular Geometry & Bonding
what the shape decides
Figure

The energy against how big the atomic orbital is

The total energy of a one-electron diatomic against the exponent of the 1s functions it is built from, at 4 separations. The free atom's value is one, marked; every minimum lies to the right of it, at 1.33, 1.24, 1.14, 1.06. Contracting costs kinetic energy, which goes as the square of the exponent, and buys attraction, which goes as the exponent — so where the balance falls is a property of the molecule and not of the atom.
The energy against how big the atomic orbital is. The total energy of a one-electron diatomic against the exponent of the 1s functions it is built from, at 4 separations. The free atom's value is one, marked; every minimum lies to the right of it, at 1.33, 1.24, 1.14, 1.06. Contracting costs kinetic energy, which goes as the square of the exponent, and buys attraction, which goes as the exponent — so where the balance falls is a property of the molecule and not of the atom.

One of the figures on radial functions: The one-dimensional half of a wavefunction: where the density is, what screening does to it, and how far out an orbital reaches.

Two essays draw this figure, each at the values its own argument needs rather than at the setting shown above. What each one uses it to show is below, in the words of its own caption.

In the essays

The atom does not bring its own orbital

The exponent the molecule chooses against the separation of the nuclei, with the binding curves at that exponent and at the free atom’s. One variational parameter moves the computed bond length from 2.49 bohr to 2.00 and the binding from 0.065 hartree to 0.087.

The energy against the exponent at four separations, with the minimum of each marked and the free atom’s value drawn across them. Every minimum is to the right of it, and the shorter the bond the further right it is.

The check the optimisation has to pass and could have failed. Twice the kinetic energy plus the potential, against the exponent, at the equilibrium separation — zero exactly where the energy is lowest and nowhere else, so the theorem locates the minimum rather than restating the arithmetic that found it. At the free atom’s exponent the residual is −0.168-0.168 hartree, and at the fixed-exponent calculation’s own minimum it is −0.182-0.182: the failure is not small, and no amount of lowering the energy would have produced the zero.

A contraction is a decision made once

The exponent the hydrogen molecule ion chooses at each separation, from a one-electron calculation with every integral in closed form. It rises to 1.39 at short range, passes through one near five bohr and returns to the free atom’s value at long range. The value at the equilibrium separation, 1.238, is the charge the basis test below is run at.

Every figure · Every orbital, by what it encloses · All essays