6.Many-electronatomsTheheliumatomTheHamiltonianfor He (with clamped nucleus)ishHHeV2mnV2meZe2(1)4元9斤Ze24元0/224元0:/12Because oftheterm in1/r1-this can't be separated into H(1)+H(2), sothe wavefunctioncan'tbewrittenas=(1)(2)
6. Many-electron atoms The helium atom The Hamiltonian for He (with clamped nucleus) is Because of the term in 1/𝑟12this can't be separated into 𝐻(1) + 𝐻(2), so the wavefunction can't be written as Ψ = 𝜓(1)𝜓(2). 2 2 He 1 2 2 2 2 0 1 2 0 2 2 0 12 2 2 4 4 4 e e H m m Ze r Ze r e r ò ò ò (1)
6.1 Central field approximationHoweverwecan writeapproximately(2)HHe ~ H(1)+H(2)withn_?+V(r)H:(3)2mwhere Vincludes a spherical average of therepulsion from theother electron.This is the central field approximation, and it allows us to treat the electrons as if theymoveindependentlyofeachotherThat is, = (1)(2), with H = EWriting out the kinetic energy term gives (in atomic units)121a22aH:or +2r2 +V(r)2r? Or
However we can write approximately (2) with (3) where V includes a spherical average of the repulsion from the other electron. This is the central field approximation, and it allows us to treat the electrons as if they move independently of each other. That is, Ψ = 𝜓(1)𝜓(2), with 𝐻𝜓 = 𝐸𝜓. Writing out the kinetic energy term gives (in atomic units) 6.1 Central field approximation H H H He 1 2 2 2 2 e H V r m 2 2 2 2 ˆ 1 2 2 H r V r r r r r l
The important feature is that V still depends only on r, not on and BecauseofthiswecanstillwriteV/nlm = R. (r) Ym(0, p)where Yim(, ) is a spherical harmonic, just as before, but Rn(r) now satisfies a differentradial equation:1(1+1)1 0,20RmR., + V(r) R., = E.m,R.l2r22r2 Orar
The important feature is that V still depends only on r, not on 𝜃 and 𝜑. Because of this we can still write where 𝑌𝑙𝑚(𝜃, 𝜑) is a spherical harmonic, just as before, but 𝑅𝑛𝑙(𝑟) now satisfies a different radial equation: ( , ) nlm nl lm R r Y 2 2 2 1 1 2 2 nl nl nl nl nl R l l r R V r R E R r r r r
TheSelf-ConsistentFieldmethodTheradialequation is1(1+1)1aaR,!m +V(r)Rm=EmRlR2r.22r2 arOrVis an average of the interactions with the other electron (or electrons, in general), so wecan't calculate it until we know where the electrons are. We have to start by guessing theform oftheorbitals.andthen(i) use the orbitals to evaluate V(r),(ii) solve the eq. to get new orbitals.andrepeattheprocessuntiltheneworbitalsagreewiththeoldonesThisiscalledtheSelf-ConsistentFieldorSCFmethod
The Self-Consistent Field method The radial equation is V is an average of the interactions with the other electron (or electrons, in general), so we can't calculate it until we know where the electrons are. We have to start by guessing the form of the orbitals, and then (i) use the orbitals to evaluate 𝑉(𝑟) , (ii) solve the eq. to get new orbitals, and repeat the process until the new orbitals agree with the old ones This is called the Self-Consistent Field or SCF method. 2 2 2 1 1 2 2 nl nl nl nl nl R l l r R V r R E R r r r r
Thepotential actingontheelectron is sphericalEach electron in an atom moves independently in a central potential due to the CoulombattractionofthenucleusandtheaverageeffectoftheotherelectronsintheatomHartree'sprocedureisasfollowss, = f(r)Yl(0,g)do = s(ri,0.9)s2(r,02.Φ)...s,(r,0n.n)eelectron 1Q.TP2P, = -e|s, P2dy. :dy4元804元80Y121Vi2 +Vi, +...+V.dyryi=2Ze12V(ri,0,g)=V(r)ririj=2n2V? +V(r) |t;(1) = s,t(1)2me
Each electron in an atom moves independently in a central potential due to the Coulomb attraction of the nucleus and the average effect of the other electrons in the atom. The potential acting on the electron is spherical electron 1 ( ) ( , ) l i m s f r Y Hartree’s procedure is as follows 0 1 1 1 1 2 2 2 2 ( , , ) ( , , ) ( , , ) n n n n s r s r s r 1 2 12 2 0 12 4 Q V dv r 2 2 2 2 12 | | ' s e dv r 0 ' 4 e e 2 2 2 e s| | 2 2 12 13 1 2 | | ' n j n j j ij s V V V e dv r 2 2 2 1 1 1 1 1 1 2 1 1 | | ' ( , , ) ' ( ) n j j j j s Ze V r e dr V r r r 2 2 1 1 1 1 1 ( ) (1) (1) 2 e V r t t m