= Solution
A <helium atom> has two <electrons> bound to nuclear charge $+2e$. In the nonrelativistic fixed-nucleus approximation, using <atomic units>,
$$
H=-\frac12\nabla_1^2-\frac12\nabla_2^2-\frac2{r_1}-\frac2{r_2}+\frac1{r_{12}}.
$$
The last term couples the <electrons>, so a product of independent hydrogenic orbitals is not an exact solution. The dominant ground configuration is $1s^2$. Its spatial <wavefunction> is symmetric, so Fermi antisymmetry requires the <spin> singlet
$$
\chi_{00}=\frac{|\uparrow\downarrow\rangle-|\downarrow\uparrow\rangle}{\sqrt2},\qquad
\boxed{\text{helium ground level }1s^2\;{}^1S_0.}
$$
A triplet cannot put both <electrons> in the same $1s$ spatial orbital. Correlation changes the exact spatial function, including its dependence on $r_{12}$, while preserving this ground-state symmetry.
A useful <helium variational effective charge> takes normalized orbitals $\phi_\zeta(\mathbf r)=(\zeta^3/\pi)^{1/2}e^{-\zeta r}$ for both <electrons> and multiplies their product by $\chi_{00}$. The <energy> in Hartree units is
$$
E(\zeta)=\zeta^2-4\zeta+\frac58\zeta,\qquad
\boxed{\zeta_{\rm opt}=\frac{27}{16},\quad E_{\rm var}=-\frac{729}{256}E_h\simeq-77.5\,\mathrm{eV}.}
$$
The reduced effective charge expresses screening. Neglecting repulsion gives $-4E_h$, while its expectation in the unscreened $\zeta=2$ product gives $-2.75E_h$; optimizing the orbital improves that trial estimate. The exact ground <energy> is lower than the optimized product <energy>, since the product omits explicit <electron correlation>. This calculation keeps nuclear charge two in $H$; only the trial orbital parameter is varied.
For a <singly excited helium state> with distinct orthonormal orbitals $a=1s$ and $b=n\ell$, the spatial combinations are
$$
\Phi_\pm=\frac{a(1)b(2)\pm b(1)a(2)}{\sqrt2}.
$$
The symmetric $\Phi_+$ pairs with the antisymmetric <spin> singlet, while antisymmetric $\Phi_-$ pairs with the symmetric <spin> triplet. With the same orbitals, their leading energies are
$$
E_\pm=\epsilon_a+\epsilon_b+J_{ab}\pm K_{ab},\qquad
\boxed{E_{\rm singlet}-E_{\rm triplet}=2K_{ab}.}
$$
Here $J_{ab}$ is the direct <Coulomb integral> and $K_{ab}$ is the <exchange integral>. The latter is positive for the <Coulomb interaction>, so the triplet is lower for this fixed-orbital comparison. <singlet-triplet exchange splitting in helium> arises from antisymmetry and spatial correlation in the <Coulomb interaction>, not from inserting a separate classical spin-spin force. Spin-singlet <helium> states are <parahelium>; spin-triplet <helium> states are <orthohelium>. The two spectra approximately form separate families because the leading electric dipole operator does not change total <spin>.<Topics in Quantum Mechanics by David Tong>[§3.2.2, Helium and the Exchange Energy, printed pp. 91–96: spatial/spin symmetry and exchange splitting]{ref}
The lowest $1s2s$ terms are $ {}^3S_1$ and $ {}^1S_0$, with the triplet lower. The $1s2p$ configuration gives $ {}^3P_{0,1,2}^{\rm o}$ and $ {}^1P_1^{\rm o}$, where the odd <parity> allows singlet $P$ to connect to the even-parity ground singlet by an <electric dipole transition>. Both $1s2s$ levels lack an allowed direct electric dipole decay to $1s^2$: the triplet is <spin> forbidden, and singlet $S\to S$ is forbidden by <orbital angular momentum> and <parity>. This explains <helium metastability> without implying an infinite lifetime once weaker radiative processes or collisions are allowed.
\Image[/past-exam-of-the-advanced-quantum-physics-course-of-the-university-of-cambridge/2015/paper-1-helium.png]
{title=Low neutral-helium singlet and triplet levels, plotted from independently cited tabulated energies}
The observed first $1s2s$ excitation energies are approximately $19.82$ and $20.62\,\mathrm{eV}$ for the triplet and singlet, respectively. Higher singly excited Rydberg series approach the He II ground-state ionization threshold; doubly excited states above that threshold can autoionize and are not all stable bound excited levels.<Energy Levels of Neutral Helium by National Institute of Standards and Technology>[1s2s and 1s2p configuration rows; He II ground-state limit at 198310.6691 inverse centimetres]{ref}
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