513.001 Molecular and Solid State Physics

Singlet-triplet spliting

A helium atom has two electrons. The ground state electronic configuration is 1s². The first excited state is 1s¹2s¹. There is a singlet-triplet splitting of the first excited state. The goal of this problem is to calculate the singlet-triplet splitting.

Construct an approximate two electron wavefunction for the 1s¹2s¹ state by neglecting the electron-electron interactions. The properly antisymmetrized two electron wavefunctions can either have a symmetric spatial component and an antisymmetric spin component or an antisymmetric spin component and a symetric spin component.

The singlet state is,

and the triplet states are,

The difference in energies between the singlet and triplet states can be written as,

Here H_atom is the Hamiltonian without the electron-electron interaction term and H_e-e is the electron-electron interaction term. Show that the difference in energy between the 1s¹2s¹ singlet state and the 1s¹2s¹ triplet state is,

The integrals that need to be performed are difficult to do. Try to calculate these integrals using a computer algebra program. Explain where the difficulty comes from.