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why does one choose in Hartree Fock theory the following type of spin orbitals?

spinorbital(r,s) = spacialOrbital(r)*alpha(s)

or

spinorbital(r,s) = spacialOrbital(r)*beta(s)

where alpha(s) is the spin up function and beta(s) the spin down function.

Why does one not choose the spin part of the spin orbitals as a linear combination of spin up and down, that is:

spinorbital(r,s) = spacialOrbital(r)*(a*alpha(s) + b*beta(s) )

--derivator

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# Hartree Fock spin orbitals

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