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Most of the books I've seen they say that the first excited state of Helium (with two electrons, one in orbital 1s and other in 2s) can have the two electrons with parallel spin (orthohelium) or anti-parallel spin (parahelium).

If ##\operatorname{X_{↑}}{\left (n \right )}## represent the state of particle n with spin up, there are the following possible states:

Anti-parallel spin:

##

A \left(\operatorname{X_{↑}}{\left (1 \right )} \operatorname{X_{↓}}{\left (2 \right )} - \operatorname{X_{↑}}{\left (2 \right )} \operatorname{X_{↓}}{\left (1 \right )}\right) \left(\operatorname{Ψ_{1s}}{\left (r_{1} \right )} \operatorname{Ψ_{2s}}{\left (r_{2} \right )} + \operatorname{Ψ_{1s}}{\left (r_{2} \right )} \operatorname{Ψ_{2s}}{\left (r_{1} \right )}\right)

##

Parallel spin:

##

A \left(\operatorname{Ψ_{1s}}{\left (r_{1} \right )} \operatorname{Ψ_{2s}}{\left (r_{2} \right )} - \operatorname{Ψ_{1s}}{\left (r_{2} \right )} \operatorname{Ψ_{2s}}{\left (r_{1} \right )}\right) \operatorname{X_{↑}}{\left (1 \right )} \operatorname{X_{↑}}{\left (2 \right )}

##

##

A \left(\operatorname{Ψ_{1s}}{\left (r_{1} \right )} \operatorname{Ψ_{2s}}{\left (r_{2} \right )} - \operatorname{Ψ_{1s}}{\left (r_{2} \right )} \operatorname{Ψ_{2s}}{\left (r_{1} \right )}\right) \operatorname{X_{↓}}{\left (1 \right )} \operatorname{X_{↓}}{\left (2 \right )}

##

##

A \left(\operatorname{X_{↑}}{\left (1 \right )} \operatorname{X_{↓}}{\left (2 \right )} + \operatorname{X_{↑}}{\left (2 \right )} \operatorname{X_{↓}}{\left (1 \right )}\right) \left(\operatorname{Ψ_{1s}}{\left (r_{1} \right )} \operatorname{Ψ_{2s}}{\left (r_{2} \right )} - \operatorname{Ψ_{1s}}{\left (r_{2} \right )} \operatorname{Ψ_{2s}}{\left (r_{1} \right )}\right)

##

From what I understand, any linear combinations of the above functions also represent two electrons in 1s and 2s orbitals, but if for example i add the first and the four function I get:

##

A \left(\operatorname{X_{↑}}{\left (1 \right )} \operatorname{X_{↓}}{\left (2 \right )} \operatorname{Ψ_{1s}}{\left (r_{1} \right )} \operatorname{Ψ_{2s}}{\left (r_{2} \right )} - \operatorname{X_{↑}}{\left (2 \right )} \operatorname{X_{↓}}{\left (1 \right )} \operatorname{Ψ_{1s}}{\left (r_{2} \right )} \operatorname{Ψ_{2s}}{\left (r_{1} \right )}\right)

##

Becasue I add the Anti-parallel spin function and one of the Parallel spin functions, now I get a function that also represent two electrons in 1s and 2s orbitals, but it is neither parallel or antiparallel.(it's a superposition of both possibilities)

Is this correct or I'm missing something?

If ##\operatorname{X_{↑}}{\left (n \right )}## represent the state of particle n with spin up, there are the following possible states:

Anti-parallel spin:

##

A \left(\operatorname{X_{↑}}{\left (1 \right )} \operatorname{X_{↓}}{\left (2 \right )} - \operatorname{X_{↑}}{\left (2 \right )} \operatorname{X_{↓}}{\left (1 \right )}\right) \left(\operatorname{Ψ_{1s}}{\left (r_{1} \right )} \operatorname{Ψ_{2s}}{\left (r_{2} \right )} + \operatorname{Ψ_{1s}}{\left (r_{2} \right )} \operatorname{Ψ_{2s}}{\left (r_{1} \right )}\right)

##

Parallel spin:

##

A \left(\operatorname{Ψ_{1s}}{\left (r_{1} \right )} \operatorname{Ψ_{2s}}{\left (r_{2} \right )} - \operatorname{Ψ_{1s}}{\left (r_{2} \right )} \operatorname{Ψ_{2s}}{\left (r_{1} \right )}\right) \operatorname{X_{↑}}{\left (1 \right )} \operatorname{X_{↑}}{\left (2 \right )}

##

##

A \left(\operatorname{Ψ_{1s}}{\left (r_{1} \right )} \operatorname{Ψ_{2s}}{\left (r_{2} \right )} - \operatorname{Ψ_{1s}}{\left (r_{2} \right )} \operatorname{Ψ_{2s}}{\left (r_{1} \right )}\right) \operatorname{X_{↓}}{\left (1 \right )} \operatorname{X_{↓}}{\left (2 \right )}

##

##

A \left(\operatorname{X_{↑}}{\left (1 \right )} \operatorname{X_{↓}}{\left (2 \right )} + \operatorname{X_{↑}}{\left (2 \right )} \operatorname{X_{↓}}{\left (1 \right )}\right) \left(\operatorname{Ψ_{1s}}{\left (r_{1} \right )} \operatorname{Ψ_{2s}}{\left (r_{2} \right )} - \operatorname{Ψ_{1s}}{\left (r_{2} \right )} \operatorname{Ψ_{2s}}{\left (r_{1} \right )}\right)

##

From what I understand, any linear combinations of the above functions also represent two electrons in 1s and 2s orbitals, but if for example i add the first and the four function I get:

##

A \left(\operatorname{X_{↑}}{\left (1 \right )} \operatorname{X_{↓}}{\left (2 \right )} \operatorname{Ψ_{1s}}{\left (r_{1} \right )} \operatorname{Ψ_{2s}}{\left (r_{2} \right )} - \operatorname{X_{↑}}{\left (2 \right )} \operatorname{X_{↓}}{\left (1 \right )} \operatorname{Ψ_{1s}}{\left (r_{2} \right )} \operatorname{Ψ_{2s}}{\left (r_{1} \right )}\right)

##

Becasue I add the Anti-parallel spin function and one of the Parallel spin functions, now I get a function that also represent two electrons in 1s and 2s orbitals, but it is neither parallel or antiparallel.(it's a superposition of both possibilities)

Is this correct or I'm missing something?

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