To solve the Hartree-Fock-Roothaan equations for He, we begin by noting that in the spin orbitals and . If the basis wavefunctions and are real, becomes . With reference to eq157, we have

where (assuming no contribution from the exchange integral) and .

The pair of simultaneous equations can be represented by the following matrix equation:

Let the basis wavefunctions be and . Since these Slater-type orbitals are real, . Eq158 becomes

Therefore, eq159 becomes

Eq160 is a linear homogeneous equation, which has non-trivial solutions if

Expanding the determinant and noting that , we get the characteristic equation:

The computation process is as follows:

- Evaluate the integrals , , and in eq161 either analytically or numerically by letting and , and using the initial guess values of .
- Substitute the evaluated integrals in eq161, solve for and retain the lower root. Substitute back in eq159 to obtain an expression of in terms of .
- Use the expression derived in part 2, together with , to solve for and , which are then used as improved values of and for the next iteration.
- Repeat steps 1 through 4 until , and are invariant up to 6 decimal places.

The results are as follows: