Neumann function

The metal cluster will be modeled as an infinitely deep spherical potential well with the represented by an infinitely high spherical barrier. Let us place this barrier in the center of the spherical cluster to simplify the calculations. The simple Schrodinger equation, containing only the interaction of the electrons with the static potential and the kinetic energy term and neglecting any electron-electron interaction, can then be solved analytically, the solutions for the radial wave functions being linear combinations of spherical Bessel and Neumann functions. 

The solution of the corresponding radial equation is a familiar problem -we follow Schiff (1968) outside the atomic sphere, there arc two independent solutions, the spherical Bessel function and Neumann function, 7i(fcr) and ni kr), both corresponding to energy For conceptual purposes, they maybe thought... 

The above equation has linearly independent solutions kxji(kx) and kxni(kx), where j kx), n kx) are the spherical Bessel and Neumann functions, respectively. Thus the solution of eqn (64) has the asymptotic form (when 00)... 

For an infinitely long cylinder the function F(r+) is determined by (2.166) with n = 1. This is the zero-order Bessel differential equation and its solutions are the Bessel function Jq and the Neumann function No both of zero-order ... 

For large x, the Bessel/Neumann functions behave like ... 

The expansion theorem means that the tail of the muffin-tin orbital (5.13), i.e. a spherical Neumann function including the angular part iVJ(r), as in (5.5), positioned at R may be expanded in terms of spherical Bessel functions centred at R1. The reason for the functional form (for simplicity R = 0)... 

It now remains to define the tail N icr) of the augmented muffin-tin orbital in a suitable form. Here we recall that the original spherical Bessel and Neumann functions obey the expansion theorem (5.14), and it is therefore natural to require that the augmented functions and also satisfy this theorem. Hence, we are led to the definition... 

The first of these, which connects the logarithmic derivatives of the spherical Bessel and Neumann functions, is a direct consequence of the Wronskian relation nj - jn = S [5.4], while the latter may be derived quite generally in analogy to (4.17), to which it reduces when the two boundary conditions are - i - 1 and . In addition, we renormalise the structure constants... 

In the k = 0 limit used in the atomic-sphere approximation, the wave equation (5.7) used to construct the tail of the partial wave (5.10) turns into the Laplace equation. Hence, in the definition of the muffin-tin orbitals (5.13,25) the spherical Bessel and Neumann functions should be substituted by the harmonic functions (r/S)z and (r/S)" "1, respectively. By means of the small kr limits (5.8) of the spherical Bessel and Neumann functions, the expansion theorem (5.14) becomes... 

In this approximation each individual atom of type t is surrounded by an atomic sphere of radius S., and the kinetic energy k in the region outside the spheres is zero. Hence, the spherical Bessel and Neumann functions which enter the theory become polynomials in (r/S) where S may be taken as a common radius different from S. The requirement of continuity and differentiability at the individual radii St determines the normalisation of and the function... 

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