So a positive and a positive wave function create a bonding orbital where the probability of finding an electron is summed while a positive and a negative create an anti-bonding orbital with a lower electron probability in the region between them leading to a repulsion. My confusion stems from not having any idea as to what a negative wave function is representing – can anyone give me some physical intuition on how this negative wave function could be correlated to something in reality?
[Physics] Physical intuition behind negative values for wave function
orbitalsquantum mechanicswavefunction
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You are quite correct that atoms don't have a precise size. When defining the size of atoms we tend to use either bond lengths, if the atoms are reactive, or interatomic potentials for atoms that aren't reactive.
For example take Argon atoms, which are unreactive. The force between two argon atoms is well described by the London dispersion force, which in the case of Argon looks like:
Typically we get a minimum in the energy (at about 380pm in this case) then a hard core repulsion i.e. the energy rises steeply as you push the atoms past the minimum. We can take this as a measure of the size of the atom.
Where atoms react we can use the spacing in the molecules as a guide. For example the O to O distance in the O$_2$ molecule is 121pm giving us a radius for the Oxygen atom of a bit over 60pm.
However we will get different values for the atomic radii depending on how exactly we define it. For example Carbon forms single, double and triple C-C bonds, and they all have different lengths giving us different radii for the Carbon atom. This means the figures for the sizes of atoms are a guide rather than a precise value.
Wikipedia has a list of atomic sizes here that shows the different values for the radii obtained using different measures.
Yes, atomic orbitals are very significant.
An electron being in a particular orbital corresponds to a specific energy. If an electron transitions between two orbitals, the energy of the photon absorbed or emitted is the difference between the energy levels of the orbitals.
The probablilty density function is also important. For example, an s-orbital electron has a probabilty of being within the nucleus. The gives rise to Fermi Contact Interaction, which has observable effects in NMR and ESR spectroscopy and electron capture.
Best Answer
The wavefunction contains all the information about the system, but it is not itself observable. Since our intuition tends to be linked to what we have observed in the past, it's not surprising that the wavefunction seems unintuitive. As you say it can be negative, but it can also be a complex number making it even less intuitive (if that's possible!).
Anything we can observe is obtained from the wavefunction using a function called an operator. In your example of calculating the probability of finding the electron in some infinitesimal volume $dV$, this probability is given by:
$$ P = \psi^*\psi dV $$
where $\psi^*$ is the complex conjugate of $\psi$. The product $\psi^*\psi$ is always real and positive, so the probability of finding an electron is always real and positive. We don't need to worry that the wavefunction can be negative or complex.