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So I am reading about bonding and antibonding molecular orbitals and i can't understand how does the probability diagram show that the electron density in the sigma 1s molecular orbital is greatest between the two positively charged nuclei.

As i read this diagram [(b) in the picture],the probability of finding the 2 electrons in a hydrogen molecule is higher around each hydrogen atom nucleus. But molecular theory says that the electron density is the highest between the 2 nuclei. The diagram shows that the highest ψ^2 values are on the hydrogens nuclei and not between them. So what am i getting wrong?

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    $\begingroup$ This is a consequence of plotting in 2D and focusing on the wave function to emphasize the additive nature at the expense of clarity. It might help to compare the atomic 1s wave function to its probability as a function of radius, which is the wavefunction multiplied by a spherical volume element, so that even though $\psi^2$ is maximum at $r=0$, the maximum probability is actually at a distance r>0 as shown here: hyperphysics.phy-astr.gsu.edu/hbase/quantum/hydr.html. $\endgroup$
    – Andrew
    Commented Mar 6, 2023 at 16:07
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    $\begingroup$ Perhaps what is meant is not that the maximum of the probability lies there but that the integrated probability over that region is a maximum. $\endgroup$
    – Buck Thorn
    Commented Mar 6, 2023 at 17:25
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    $\begingroup$ What should be plotted, perhaps, is the difference between the probability density of two isolated hydrogen atoms and two bonded hydrogen atoms. That would make the situation clearer. $\endgroup$
    – matt_black
    Commented Apr 3, 2023 at 9:04

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The shape of the orange blob is highly affected by the chosen contour level. In the image below, if you choose a contour level of 0.04, you get a oval much like the one shown. If you choose a contour level of 0.2, you get a dumbbell shape. If you choose a contour level of 0.4, you get two spheres around the atoms, which does not look connected at all.

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It turns out that the electron density of two bonded atoms is very similar to the electron density of the two individual atoms, especially for atoms with inner electrons. The difference, called the deformation density, gives you an idea how the electron density "shifted" with respect to that of the unbound atoms. Unfortunately, I did not find a deformation density for dihydrogen in a quick internet search. Maybe someone can calculate it, though.

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