What are coulombic and exchange energy and how are they determined?

A simplified way is to call them #Pi_c# and #Pi_e#, the coulombic and exchange components of the pairing energy #Pi#:

#Pi = Pi_c + Pi_e#

[This is gone into more detail here.]

For example, consider the ground state valence configuration of chromium:

#ul(uarr color(white)(darr))#
#4s#

#underbrace(ul(uarr color(white)(darr))" "ul(uarr color(white)(darr))" "ul(uarr color(white)(darr))" "ul(uarr color(white)(darr))" "ul(uarr color(white)(darr)))#
#" "" "" "" "" "" "color(white)(/)3d#

This has zero coulombic repulsion energy #Pi_c#, because no electrons are paired. However, if we focus on the #3d# electrons, there are #10# possible nonredundant exchanges.

Label each electron to get:

#uarr_1 uarr_2 uarr_3 uarr _4 uarr_5#

Now, the possible exchanges (that result in the same energy) involve all of the #3d# electrons:

#1harr{2, 3, 4, 5}# #-># #4# exchanges
#2harr{3, 4, 5}# #-># #3# exchanges
#3harr{4, 5}# #-># #2# exchanges
#4harr{5}# #-># #1# exchange

(Try not to double-count; so don't count #1harr2# and then #2harr1#; those are the same thing.)

This is true because all of the electrons are all indistinguishable. This results in an exchange energy stabilization of #10xxPi_e# (meaning #Pi_e < 0#), one for each exchange.

So the pairing energy in total is:

#Pi = 10Pi_e#

Now, suppose instead, the chromium atom was in this state:

#ul(uarr darr)#
#4s#

#underbrace(ul(uarr color(white)(darr))" "ul(uarr color(white)(darr))" "ul(uarr color(white)(darr))" "ul(uarr color(white)(darr))" "ul(color(white)(uarr darr)))#
#" "" "" "" "" "" "color(white)(/)3d#

Now there is one pair of electrons that ARE repelling each other, so the coulombic repulsion energy is #1xxPi_c#. However, the exchange energy stabilization now is only #6 xx Pi_e#...

So the pairing energy in total is higher than before:

#Pi = Pi_c + 6Pi_e#

This is less stable by #Pi_c - 4Pi_e#, and qualitatively shows that chromium should "prefer" the #3d^5 4s^1# configuration over the #3d^4 4s^2# configuration.

This approach also explains why tungsten prefers a #5d^4 6s^2# configuration.