## Approximation Methods

When one includes not just one electron, but many electrons, there is a large variety of effects that affect the energy levels of the various states. Quite generally, these additional effects cannot be treated exactly as was done for the one-electron atom. Even with the one-electron atom, we did not include the effects of the spin magnetic moments of either the electron or the nucleus nor their interactions with the orbital magnetic moment of the electron, nor did we include any relativistic corrections, and any one of these additional effects complicates the problem to such an extent that exact results can no longer be obtained with nonrelativistic quantum mechanics. In addition to these effects, the many-electron atom has all of the electron-electron effects. We may represent these effects, however, through a very complicated potential, so that the Hamiltonian becomes

H = T +Ven+Vee+Vso+Vss+Voo+Vesns+Vnseo+Vrel+Vexch+Vetc , (5.1)

where each term may be represented by the following expressions:

1. Kinetic energy:

T = p2n 2mn

+ N∑ i=1

. (5.2)

2. Electron-nucleus electrostatic interaction:

Ven = − N∑ i=1

Ze2

4pi²0ri . (5.3)

3. Electron-electron mutual electrostatic interaction (repulsion):

Vee = N∑ i=1

e2

4pi²0rij . (5.4)

and

4. Spin-orbit interaction (spin angular momentum — orbital angular momentum):

Vso = − N∑ i=1

σi · li m2ric2

dV dri

. (5.5)

5. Spin-spin interaction (electron spin-electron spin):

Vss = µ0 4pi

e2

m2

[ σi · σj r3ij

− 3(σi · rij)(σj · rij) r5ij

] . (5.6)

6. Orbit-orbit interaction (electron-electron orbital angular momentum):

Voo = N∑ i=1

Cijli · lj . (5.7)

7. Electron spin — nuclear magnetic moment:

Vesns = µ0 4pi

e

m

[ µn · σi r3i

− 3(µn · ri)(σi · ri) r5i

] . (5.8)

8. Nuclear spin — electron orbital angular momentum:

Vnseo = µ0 4pi

e

m

( µn · li 2pir3i

) . (5.9)

9. Relativistic correction (to the kinetic energy):

Vrel = − N∑ i=1

. (5.10)

10. “Exchange interaction.” (Due to the Pauli exclusion principle, which is spin-dependent, there is a tendency to align the spins and effectively cause the electrons to “repel” one another.)

11. Miscellaneous other effects, such as quadrupole interactions, finite nuclear size, etc.