Kirkwood-Buckingham relation

This is the Kirkwood-Buckingham relation useful for analysis of boundary value problems. For example, Equation (2.4) follows from Equation (3.17) and the variational inequality for the ground state wavefunctions for Dirichlet problems in regions ST and 12. One can find some generalizations of this relation in [54,55]. 

The idea of using functions of the form frjr for approximate solutions of the boundary value problems has been used in a large number of papers. For example, this idea was used in [70] for the one-dimensional Schrodinger equation to prove the Hull-Julius relation (4.21) and to study the confined IlJ molecule. The direct use of the Kirkwood-Buckingham relation for variational calculations of the hydrogen atom in a half space was... 

Note that one may use a pair of relations similar to Equations (3.5) and (3.9) to exclude the energy derivatives. This gives an integral relation, valid both for free and confined systems. One of the useful relations of such a kind was derived in an approximate way by Kirkwood [52] and Buckingham [53] to estimate the polarizability. For a simple proof and the history of these relations, widely used nowadays, see [54]. 

---