i am looking for a derivation that can prove that the shape of an atom is spherical . I experimentally proved this statement but i need a theoretical way.
Answer
If you want to prove an isolated atom is spherically symmetrical, you could proceed by showing the sum of the probabilty functions (wave functions squared) of each orbital results is a spherically symmetrical distribution.
Certainly s-orbitals are spherically symmetrical.
The sum for an entire subshell of orbitals (such as all three 2p orbitals) is also spherically symmetrical (despite wikipedia cartoons that may look otherwise).
According to Shapes of Atoms, J. Chem. Educ., 1965, 42 (3), pages 145-146, all isolated atoms with half-filled valence shells or filled valence shells are spherical, while the remainder are not. For example, the articles says boron and oxygen are prolate and halogens are oblate.
The article concludes the non-spherical shapes based upon a particular subset of orbitals being occupied in the valence subshell; however, personally I would think that for an isolated atom all the valence orbitals would be degenerate and the wave function would be a linear combination of all the degenerate orbitals. Hopefully someone will say how my thinking is misguided.
In fact it is the above article that is incorrect, and J. Chem. Education published a retraction a few months later J. Chem. Educ., 1965, 42, pages 397-398 stating "all isolated atoms are spherical" (emphasis in original) and that the previous article failed to consider degeneracy of valence orbitals.
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