On the Non-Orthogonality Problem Connected with the Use of Atomic Wave Functions in the Theory of Molecules and Crystals

The treatment of molecules and crystals by the Heitler-London method or by the collective electron model can be based on the atomic orbitals φμ of the system. These orbitals are in general overlapping, and the corresponding overlap integrals Sμν, given by (1), have almost universally been neglected in the literature as causing undesirable complications. Here we will take these overlap integrals into consideration and show that they, instead of being negligible, are of essential importance in molecules and in crystals. The problem is simply solved by considering the orthonormalized functions open phiμ, given by (21), as the real atomic orbitals. The solution is worked out in detail for (I) the molecular orbital method of treating molecules, (II) the Bloch orbital method of treating crystals, and (III) the Heitler-London method of treating both these systems in some simple spin cases. Some numerical applications are given for ionic crystals, showing that the overlap effects are responsible for all the repulsive forces in these solids. It is also shown that the overlapping adds interesting new features to the properties of molecules and crystals, namely the ``many-orbital-effects,'' corresponding to certain ``many-body-forces'' in ionic solids.