Bound He Rydberg states

The Hamiltonian for two spinless electrons at points r, and r2 relative to a He2+ core is given by3 

To describe an atom with one electron in a Rydberg state a reasonable approach, termed Heisenberg s method by Bethe and Salpeter,3 is to separate the Hamiltonian of Eq (18.1) into two parts. Explicitly, 

For consistency with other bound Rydberg states we shall measure the energies relative to the ground state of He+. Expressed in this way, the energy of the state given in Eq. (18.5) is 

Following the Pauli exclusion principle we must antisymmetrize the wavefunctions when we include the spins of the two electrons. The wavefunction is given by a product of spatial and spin wave functions, i.e. 

Here 0+ is the spin wavefunction, and the + and - signs refer to the orthogonal singlet, 5 = 0 and triplet, 5=1 states, where 5 is the total electron spin. Using the wavefunctions of Eq. (18.7) we can calculate the first order corrections to Eq. (18.6) using the Hamiltonian of Eq. (18.2c). Explicitly, 

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