Why energy changes when an electron is added to a neutral atom?
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If the electron statistical distribution around the atomic kernel had been perfectly spherically symmetric, and if the electron occurance distribution had not mutually overlapped, than by the Gauss law of electrostatics, the net force between a neutral atom and an electron would have been zero.
But as neither of above conditions is true, a kernel charge is not fully screened off by electrons, acting as having a residual, "effective charge", what allows releasing energy by bounding an extra electron. See Slater rules.
An extra electron puts among other electrons some extra stress=mutual repulsion, what somewhat decreases this released energy.
As effective kernel charge and electron mutual repulsion ( classical and Pauli ones ) are 2 major factors affecting energies of electron orbitals in multi-electron atoms.
When the former factor is stronger, energy is released by an extra electron bounding, like for fluorine. And vice versa, like for helium.
Not every atom releases energy; it is all about the stability. For your understanding, stability is inversely proportional to energy, so as the stability increases atom try to de-excite itself. If you look at the periodic table, metals generally have no affinity towards the electron because they are e donor, so they release energy by losing an electron. In non-metals like chlorine, which has highest electron affinity, release much high energy and gain the stability. And why it happens and how it happens, you need something more than normal Bohr and other atomic model, which is not the real model of an atom.