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Two electrons in different orbitals can exchange with each other and if this happens they release a certain amount of exchange energy. But where does this energy come from? And wouldn't the atom collapse if exchange energy is continuously released?

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The electrons are not being continually exchanged. Rather, the fact that they could be exchanged without affecting the quantum state alters their wavefunctions in a way that impacts their mutual repulsion energy. As Wikipedia explains, the distance between electrons in identical subshell and spin states is effectively increased, so they naturally have less repulsion between them.

The effect is due to the wave function of indistinguishable particles being subject to exchange symmetry, that is, either remaining unchanged (symmetric) or changing sign (antisymmetric) when two particles are exchanged. Both bosons and fermions can experience the exchange interaction. For fermions [including electrons], this interaction is sometimes called Pauli repulsion and is related to the Pauli exclusion principle. For bosons, the exchange interaction takes the form of an effective attraction that causes identical particles to be found closer together, as in Bose–Einstein condensation.

The exchange interaction alters the expectation value of the distance when the wave functions of two or more indistinguishable particles overlap. This interaction increases (for fermions) or decreases (for bosons) the expectation value of the distance between identical particles (compared to distinguishable particles).[1] Among other consequences, the exchange interaction is responsible for ferromagnetism and the volume of matter. It has no classical analogue.

The number of electron-electron repulsions that are affected in this way grows quadratically as the number of electrons in identical subshell and spin states increases: one repulsion between two electrons, three repulsions between three electrons, six between four electrons, and so on. So when we have a lot of electrons that could be unpaired with identical spins in different orbitals of a subshell, there is a growing tendency for them to favor this unpairing, leading the the effect we see in $d^5$ transition metal complexes (e.g. $\ce{Mn(II),Fe(III)}$).

Cited Reference

  1. David J. Griffiths: Introduction to Quantum Mechanics, Second Edition, pp. 207–210.
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