Calculation of the energy helium atom

To anticipate some of the problems involved in energy calculations, we use the helium atom functions already derived to obtain the actual energy levels. The 2-electron Hamiltonian (neglecting spin terms) is 

Before proceeding, we note that functions with different spin factors, when written in the factorized form used in Section 1.2, must give H/cl = Mkl = 0. For example, 

But this will vanish, owing to orthogonality of the spin functions (1.2.23), and hence H12 = 0. The only non-zero off-diagonal element is in fact W15, which we evaluate for illustration. The spin factors in and are both 

If we preserve the order of the subscripts 1, 2 throughout, this may be expanded and written in the abbreviated form 

The first and last terms are each unity, the others zero, and spin integration therefore gives only a factor unity, verifying the normalization of the spin function. Thus 

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