For months now, I've been hoping to figure out what exactly Lucas did to derive the exchange functional. Well, as fate would have it, I just did.
The trick involved taking a double integral in one dimension and noting that the integrand only contained the difference of the two coordinates. The replacement is obvious if you wave your hands! I get the feeling that this is a common trick in the ABCs but it must've flown completely over my head! Li recognized the switch as from a particular class of problems (the Hooke's atom) but man...the math step was not obvious!
In my defense, I did ask Lucas several times for how he did this and nowhere in the notes was this trick. I kept getting an exponential integral. Playing Devil's advocate, however, I should have learned it during the now fifth ABC class I'm in! Again in my defense, somehow this should have been exposed, you'd think. Again playing Devil's advocate: go and do your work! Stop blogging!