The Calculation of Second Order Polarization Energies in Atoms. The Molecular Structures of the Bromoethanes, Boron Trimethyl, and Three Boron Trihalides

The development of the quantum mechanics made possible a systematic attack on many problems of atomic energy. The problem of polarization in an external field is one to which the approximation methods of quantum mechanics are applicable, since in cases of interest the external field is small compared to internal atomic forces. Indeed, the first use to which Schrod1nger1 put his newly developed first order perturbation theory was the treatment of the first order Stark effect in atomic hydrogen. A second order perturbation theory was developed independently by Epstein2 , Wentzel3 , and Waller4 and applied to the second order Stark effect in hydrogen. Other treatments of hydrogen and more complex atoms in a uniform field followed, notably by Buckingham5 , who applied the variation method to the problem. The more complex problem of polarization in a nonhomogeneous field has received relatively little attention. Th. Neugebauer6 has given an approximate treatment of ion deformation in binary crystals, but he was not concerned with the energy effect of the deformation. Inasmuch as an ion in an ionic crystal is subject to a complex electric field arising from the distribution of positive and negative charges in its environment, knowledge of the theory of atomic polarization in non-uniform fields is essential to a complete understanding of crystal lattice energies. It 1s with this application primarily in mind that the following treatment is undertaken. The method to be followed in the subsequent calculation is first to carry through as rigorous as possible a treatment for an electron in an hydrogen-like orbital, then to generalize the result by substituting an effective nuclear charge for the true nuclear charge. This method is, to be sure, an approximate one, since it assumes that the field about one electron due to the rest is central, a condition obviously not fulfilled. The expression for the hydrogen-like 'atom will be obtained by application of a general perturbation theory7 which is capable of giving results which are exact (for the hydrogen-like atom) to second-order terms in the parameters describing the field. However, since filled-shell ions are of primary interest, simplifications will be made which, while not affecting the results for filled-shell ions, will destroy the rigor of the treatment for hydrogen.