15.3. THEPERIODICTABLE 247
manyotherparticles,ignoringthesesubtlecorrellationsisoftenjustified. Wewon’t
actuallydoanycalculationsofmulti-electronatomicwavefunctionsusingthismethod
(thecomputationsarequiteinvolved),butitsstillinterestingandusefultoseehow
onegetsstarted.
Thefirst part of the Hartreeapproximation isto imagine that eachelectron,
numbered 1 , 2 ,..,Zisinanindividualstate,denotedφ 1 ,φ 2 ,...,φZ,sothetotalwave-
functionwouldbe
Φ(1, 2 ,...,Z)=φ 1 (1)φ 2 (2)...φZ(Z) (15.61)
wherethearguments 1 , 2 ,..refertothecoordinatesandspinstateoftheindicated
particle. Thisisofcoursealreadyinconflictwiththespin-statisticstheorembecause
Φ shouldbeantisymmetric. Infact, theapproximationcan beimprovedbyanti-
symmetrizingΦ(itscalledthe”Hartree-Fock”approximation),buttheimprovement
isonthe orderof 10 −20%, sowe willignorethisadditional complication. The
PauliExclusionprinciple,however,mustberespected!Thisisimposedbyrequiring
thattheφnareallorthogonal,ornearlyso,sothatnotwoelectronsareinthesame
quantumstate. Wewillalsosuppose,fornow,thattheφnareoftheform
φn(n)=φn(xn)χn± (15.62)Theessence oftheHartreeapproximationisthat theelectroninthek-thstate
”sees”theelectroninthej-thstateasbeingacloudofcharge,withchargedensity
givenby
ρ(x)=−e|φj(x)|^2 (15.63)
Inthatcase,wecanwritedownaSchrodingerequationforthewavefunctionofthe
k-thelectron,treatingtheotherelectronsasthoughtheyweresimplyclassicalcharge
distributionsoftheform(15.63),i.e.
− ̄h^2
2 m∇^2 −
Ze^2
r+
∑n(=k∫
d^3 ye^2 |φn(y)|^2
|%x−%y|
φk(x)=Eφk(x) (15.64)Tomakethingsevensimpler,thelasttermisapproximatedbyitsangularaverage,
i.e. {
−h ̄^2
2 m
∇^2 −
Ze^2
r+Vk(r)}
φk(x)=Eφk(x) (15.65)where^2
Vk(r)=1
4 π∫
dΩ∑n(=k∫
d^3 ye^2 |φn(y)|^2
|%x−%y|(15.66)
Therearetwoimportantobservationstomakeabouteq. (15.65). First,itisan
equationwithasphericallysymmetricpotential. ThismeansthattheHamiltonian
(^2) ThisistheCoulombterm.TheHartreeapproximation,becauseithasnotbeenproperlyanti-
symmetrized,missestheexchangeterm.