When considering the orbital model of the atom it seems like the shape of each orbital corresponds the shape that contains a volume such that there is a 90% chance of an electron being there. I also assume that in the majority of the literature, the equation that defines the shape of the different orbitals corresponds to the solution of some kind of 3-dimensional version of the Schrödinger Equation the configuration of which corresponds to only one electron. Are we then Supposed to add up all the orbitals? Don't all those orbitals with different shapes intersect? (if they do, i would doubt its a good solution) Would the orbitals of atoms with multiple electrons get distorted instead of cutting trough each other?
Taking all of this into consideration, is there any way to represent the three dimensional probability density of the electrons in a carbon atom for example?
Answer
The classical pretty shapes of orbitals like this
are "hydrogen-like" orbitals. Essentially, they depict exact solutions of the Schrodinger equation for atoms with a point-nucleus and no other electrons.
Atoms with multiple electrons have orbitals that are similar to those hydrogen-like orbitals, but not quite the same because of electron-electron interactions. The wikipedia article puts it well:
For atoms with two or more electrons, the governing equations can only be solved with the use of methods of iterative approximation. Orbitals of multi-electron atoms are qualitatively similar to those of hydrogen, and in the simplest models, they are taken to have the same form. For more rigorous and precise analysis, the numerical approximations must be used.
So, in answer to your last question, you could represent the 3D probability density of electrons in a Carbon atom, but it would have to be solved numerically; there isn't a closed-form solution, and in the end the picture would look pretty similar to a bunch of hydrogen-like orbitals superimposed.
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