A New Spin on Photoemission Spectroscopy Page: 42 of 259
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This Hamiltonian is separable, and so similar to equation 1.43, possible wavefunctions
include (with clear shorthand notation)
I (1, 2) = #b(1)#b(2) (1.49)
where 0,(1) is the solution to
hV12 + V (g)# = Ea# (1.50)
and #b(2) is the solution to the equivalent solution for electron 2 with energy Eb. Again, just
as above, the correct wavefunctions must be symmetrized/antisymmetrized and matched
with the correct spin singlet/triplet states. The two total wavefunctions are then:
qPs = [0a(1)Ob(2) + 0.(2)#b(1)] x8 singlet (1.51)
t = [0a(1)Ob(2) - #5(2)b(1)] Nt triplet. (1.52)
It should be clear from the example of the free electrons that these wavefunctions will
have position correlation. However, there is no energy difference between them yet; E =
f '*FHP8 dxidx2 = f '14H% dxidx2. Now let's introduce electron-electron interaction
through a perturbation, H12. This can be as simple as Coulomb repulsion between the
two electrons as H12 = - or something more complex, perhaps involving the additional
interaction between electron 1 and nucleus 2 (and vice versa) and the interaction between
the two nuclei in the case of the hydrogen molecule. The additional energies introduced by
H12 (assuming T1&11 remain good zero-order wavefunctions) can be calculated by E,t =
f P*,tH12t3,t giving:
Es = (K12 + J12) (1.53)
Et = (K12 - J12) (1.54)
K12 = #*f(1)#b(2)H120(1)b(2) dxl dx2 (1.55)
12 = f * (1)#[(2)H12a(2)#b(1) dxl dx2. (1.56)
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Jozwiak, Chris. A New Spin on Photoemission Spectroscopy, thesis or dissertation, December 1, 2008; United States. (https://digital.library.unt.edu/ark:/67531/metadc1014237/m1/42/: accessed April 18, 2019), University of North Texas Libraries, Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.