The Hamiltonian for
Helium
has the same terms as Hydrogen but has a large
perturbation due to the repulsion between the two electrons.

Note that the

The **Helium ground state has two electrons in the 1s level**.
Since the spatial state is symmetric, the spin part of the state must be antisymmetric so
(as it always is for closed shells).
For our zeroth order energy eigenstates, we will use
**product states of Hydrogen wavefunctions**

and ignore the perturbation. The energy for two electrons in the (1s) state for is then eV.

We can estimate the ground state energy in **first order perturbation theory**,
using the electron repulsion term as a (very large) perturbation.
This is not very accurate.

We can improve the estimate of the ground state energy using the **variational principle**.
The main problem with our estimate from perturbation theory is that we are not accounting for
**changes in the wave function of the electrons due to screening**.
We can do this in some reasonable approximation by reducing the charge of the nucleus in the wavefunction
(not in the Hamiltonian).
With the parameter
, we get a better estimate of the energy.

Calculation | Energy | |

Order | -108.8 | 2 |

Order perturbation theory | -74.8 | 2 |

Order Variational | -77.38 | |

Actual | -78.975 |

There is only one allowed
state and it is the ground state.
For **excited states**, the spatial states are (usually) different so
they can be either symmetric or antisymmetric (under interchange of the two electrons).
It turns out that the antisymmetric state has the electrons further apart so the repulsion is smaller
and the energy is lower.
If the spatial state is antisymmetric, then the spin state is symmetric, s=1.
So the triplet states are generally significantly lower in energy than the corresponding spin singlet states.
This
**appears to be a strong spin dependent interaction but is actually just the effect of the repulsion between the electrons**
having a big effect depending on the symmetry of the spatial state and hence on the symmetry of the spin state.

The **first exited state** has the hydrogenic state content of (1s)(2s) and has s=1.
We calculated the energy of this state.

We'll learn later that electromagnetic **transitions which change spin are strongly suppressed**causing the spin triplet (orthohelium) and the spin singlet states (parahelium) to have nearly separate decay chains.

Jim Branson 2013-04-22