Solution of the Hartree-Fock-Roothaan equations for helium

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

whereĀ (assuming no contribution from the exchange integral) andĀ .

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

 

Question

Show that .

Answer

Since and are real,


Using the Hermitian property of the KE operator,

 

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:

  1. Evaluate the integrals , , and in eq161 either analytically or numerically by letting and , and using the initial guess values of .
  2. 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 .
  3. 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.
  4. Repeat steps 1 through 4 until , and are invariant up to 6 decimal places.

The results are as follows:

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