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I am trying to write the ground state wave function of a 10 electron atom as a product of space part and anti-symmetric spin part. $$1s\uparrow,1s\downarrow$$ $$2s\uparrow, 2s\downarrow$$ $$2p_{x}\uparrow,2p_{x}\downarrow$$ $$2p_{y}\uparrow, 2p_{y}\downarrow$$ $$2p_{z}\uparrow,2p_{z}\downarrow$$ There are totally ten electronic states and hence the slater determinant will be 10 x 10 .

Here, Is it possible to factor this slater determinant into a spin part and spatial part separately such that, the space part is just a product of wave functions? and,

In general what conditions, I need to check whether a slater determinant can be factored into space part and spin part?

Is there any general approach to find this out?

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