Coulomb and Exchange Integrals: In preparation


         (most of these integrals have been derived in G.F. Neumark's thesis, loc.cit. p.A1-A4, with errors on p.A4)
  1. Potential energy of overlapping uniformly charged one electron spheres with same radius

    Combining the different terms gives the following integral:

    This is not difficult to evaluate and leads to 12 terms. Simplification gives the final expression:
    For r -> 0 we get the old expression 6/5R for the interaction of two electrons in one cloud. for r = 2R we have separated spheres with V = 1/2R.


  2. Potential energy of overlapping uniformly charged one electron spheres with different radii

    Evaluation and collection of terms gives the result:


  3. Small sphere inside larger sphere

    Potential energy of concentric spheres, R=0, with same size, P=Q, yield again 6/5R. Concentric spheres with different sizes (e.g. 1s-2s overlap) have the potential energy:


  4. Potential energy of overlaps of s- and p-type spheres


  5. Potential of overlaps of σ- and π-spheres
  6. Exchange integrals
    He(1s2s), singlet, no exchange but Coulomb repulsion:

    He(1s2s), triplet, with exchange and Coulomb repulsion:

    He(1s2p), singlet, no exchange but Coulomb repulsion:

    He(1s2p), triplet, with exchange and Coulomb repulsion:

    Exchange lenses:
    lens 1
    lens 2
    lens 3