We are going to consider the helium atom from the point of view of electron spin and the Pauli principle. We know that the ground state has a wave function 1s(1)1s(2), where 1s represents the hydrogenoid orbital and the number in brackets refers to the electron. To take spin into account, we must multiply this spatial function by a proper function of spin. Let's look at the possible eigenfunctions of spin:

\begin{equation} \alpha(1)\alpha(2);\;\;\beta(1)\beta(2);\;\;\alpha(1)\beta (2);\;\;\beta(1)\alpha(2) \end{equation}

The first two functions are symmetric with respect to the exchange of electrons and do not distinguish between them, therefore being valid to construct the wave function of helium. The third and fourth functions are not valid because they violate the principle of indistinguishability. Furthermore, they are neither symmetric nor antisymmetric towards the exchange of electrons. To solve this problem, we build linear combinations of the form:

\begin{equation} \frac{1}{\sqrt{2}}[\alpha(1)\beta(2)+\beta(1)\alpha(2 )]\\ \frac{1}{\sqrt{2}}[\alpha(1)\beta(2)-\beta(1)\alpha(2)] \end{equation}

The first of the combinations is symmetric with respect to exchange, while the second is antisymmetric. To build the total wave function (including spin) of the helium atom, we must multiply the spatial (symmetric) part by an antisymmetric spin function, as indicated by the Pauli principle.

\begin{equation} \psi^{0}1s(1)1s(2)\cdot\frac{1}{\sqrt{2}}[\alpha(1)\beta(2)-\beta(1) \alpha(2)] \end{equation}

Let us now consider the excited states of helium. The one with the lowest energy has an antisymmetric spatial wavefunction, which multiplied by each of the three symmetric spin functions gives rise to three quantum states. This energy level is therefore triply degenerate.

\begin{equation} \frac{1}{\sqrt{2}}[1s(1)2s(2)-2s(1)1s(2)]\alpha(1)\alpha(2) \end{equation}

\begin{equation} \frac{1}{\sqrt{2}}[1s(1)2s(2)-2s(1)1s(2)]\beta(1)\beta(2) \end{equation}

\begin{equation} \frac{1}{\sqrt{2}}[1s(1)2s(2)-2s(1)1s(2)]\frac{1}{\sqrt{2}} [\alpha(1)\beta(2)+\beta(1)\alpha(2)] \end{equation}

The next energy level is built with the spatially symmetric part, multiplied by the single antisymmetric spin function.

\begin{equation} \frac{1}{\sqrt{2}}[1s(1)2s(2)+2s(1)1s(2)]\frac{1}{\sqrt{2}}[\ alpha(1)\beta(2)-\beta(1)\alpha(2)] \end{equation}