5.10 Pauli Re­pul­sion

Be­fore pro­ceed­ing to a de­scrip­tion of chem­i­cal bonds, one im­por­tant point must first be made. While the ear­lier de­scrip­tions of the hy­dro­gen mol­e­c­u­lar ion and hy­dro­gen mol­e­cule pro­duced many im­por­tant ob­ser­va­tions about chem­i­cal bonds, they are highly mis­lead­ing in one as­pect.

In the hy­dro­gen mol­e­cule cases, the re­pul­sive force that even­tu­ally stops the atoms from get­ting to­gether any closer than they do is the elec­tro­sta­tic re­pul­sion be­tween the nu­clei. It is im­por­tant to rec­og­nize that this is the ex­cep­tion, rather than the norm. Nor­mally, the main re­pul­sion be­tween atoms is not due to re­pul­sion be­tween the nu­clei, but due to the Pauli ex­clu­sion prin­ci­ple for their elec­trons. Such re­pul­sion is called ex­clu­sion-prin­ci­ple re­pul­sion or Pauli re­pul­sion.

To un­der­stand why the re­pul­sion arises, con­sider two he­lium ions, and as­sume that you put them right on top of each other. Of course, with the nu­clei right on top of each other, the nu­clear re­pul­sion will be in­fi­nite, but ig­nore that for now. There is an­other ef­fect, and that is the in­ter­est­ing one here. There are now 4 elec­trons in the 1s shell.

With­out the Pauli ex­clu­sion prin­ci­ple, that would not be a big deal. The re­pul­sion be­tween the elec­trons would go up, but so would the com­bined nu­clear strength dou­ble. How­ever, Pauli says that only two elec­trons may go into the 1s shell. The other two 1s elec­trons will have to di­vert to the 2s shell, and that re­quires a lot of en­ergy.

Next con­sider what hap­pens when two he­lium atoms are not on top of each other, but are merely start­ing to in­trude on each other’s 1s shell space. Re­call that the Pauli prin­ci­ple is just the an­ti­sym­metriza­tion re­quire­ment of the elec­tron wave func­tion ap­plied to a de­scrip­tion in terms of given en­ergy states. When the atoms get closer to­gether, the en­ergy states get con­fused, but the an­ti­sym­metriza­tion re­quire­ment stays in full force. When the filled shells start to in­trude on each other’s space, the elec­trons start to di­vert to in­creas­ingly higher en­ergy to con­tinue to sat­isfy the an­ti­sym­metriza­tion re­quire­ment. This process ramps up much more quickly than the nu­clear re­pul­sions and dom­i­nates the net re­pul­sion in al­most all cir­cum­stances.

In every­day terms, the stan­dard ex­am­ple of re­pul­sion forces that ramp up very quickly is bil­liard balls. If bil­liard balls are a mil­lime­ter away from touch­ing, there is no re­pul­sion be­tween them, but move them closer a mil­lime­ter, and sud­denly there is this big re­pul­sive force. The re­pul­sion be­tween filled atom shells does not ramp up that quickly in rel­a­tive terms, of course, but it does ramp up quickly. So de­scrib­ing atoms with closed shells as bil­liard balls is quite rea­son­able if you are just look­ing for a gen­eral idea.

Key Points

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If elec­tron wave func­tions in­trude on each oth­ers space, it can cause re­pul­sion due to the an­ti­sym­metriza­tion re­quire­ment.

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This is called Pauli re­pul­sion or ex­clu­sion prin­ci­ple re­pul­sion.

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It is the dom­i­nant re­pul­sion in al­most all cases.