Numerical calculations transformational invariant

To calculate the profiles and the differential capacitance of the interface numerically we have to choose a differential equation solver. However, the usual packages require that the problem is posed on a finite interval rather than on a semi-infinite interval as in our problem. In principle, we can transform the semi-infinite interval into a finite one, but the price to pay is a loss of translational invariance of the equations and the point mapped from that at infinity is singular, which may pose a problem on the solver. Most of the solvers are designed for initial-value problems while in our case we deal with a boundary-value problem. To circumvent these inconveniences we follow a procedure strongly influenced by the Lie group description. 

In Section 5.1 we describe the independent electron pair approximation (lEPA). We use an approach that leads quickly to the computational formalism but which may give the misleading impression that lEPA is an approximation to DCI. After showing what is involved in performing pair calculations, we will return to the physical basis of the formalism and show that in fact both lEPA and DCI are different approximations to full Cl. In Subsection 5.1.1 we describe a deficiency of the lEPA, not shared by DCI or the perturbation theory of Chapter 6 namely, that the lEPA is not invariant under unitary transformations of degenerate molecular orbitals. In Subsection 5.1.2 we present some numerical results which show that while the lEPA is reasonably accurate for small atoms, it has serious deficiencies when applied to larger molecules. 

---