# Direction of spins in the singlet vs. triplet state

Suppose we consider the first excited state of the helium atom. We know that the first excited state of helium can exist as a triplet or singlet. The possible functions related to the spin of the two electrons in the triplet state are

$$\alpha(1)\alpha(2)$$ $$\beta(1)\beta(2)$$ $$\dfrac{1}{\sqrt{2}} [\alpha(1)\beta(2) + \beta(1)\alpha(2) ]$$

while the one for the singlet state is

$$\dfrac{1}{\sqrt{2}} [\alpha(1)\beta(2) - \beta(1)\alpha(2) ]$$

The triplet state predicts that the spins of the two electrons are parallel, but according to this equation

$$\dfrac{1}{\sqrt{2}} [\alpha(1)\beta(2) + \beta(1)\alpha(2) ]$$

there is a 50% probability that electron 1 is in the alpha state and a 50% probability that it is in the beta state: the same goes for electron 2. So, if this function predicts that the two spins are antiparallel, why is it part of one of the triplet states?

• I think your confusion stems from the fact that "up" and "down" refer to the sign of the z component of the spin of individual electrons, whereas "triplet" and "singlet" describe the magnitude of the total spin of the combined system, not just the z components. Thus, one of the three triplet states has zero spin in the z-direction (one electron up and one down), but still has non-zero total spin, ie. the combined spin vector is within the x-y plane. The singlet state has zero overall spin, ie. the two spin vectors completely cancel each other, rather than just canceling in the z direction. Aug 18, 2021 at 20:19
• @Andrew That fully answers the question I believe... I hope you invest the extra effort to add it as one? That would be appreciated by many, if you find the time. Aug 19, 2021 at 14:01
• @Andrew can you see this image?upload.wikimedia.org/wikipedia/commons/4/49/… If one of the triplet excited states has one alpha and one beta electron, shouldn't this be the configuration? 1s $\uparrow$ 2s $\downarrow$ Aug 19, 2021 at 19:58
• This answer has a diagram that you should find informative. chemistry.stackexchange.com/questions/50277/… Aug 20, 2021 at 6:56
• @stian yttervik I didn’t write it as an answer because it seemed like a good answer should have images like in the answer porphyrin linked, and I didn’t have time to make them. I think that linked answer covers it though Aug 20, 2021 at 14:58