Greens function and tip wavefunctions

The Green s function for the Schrodinger equation, Eq. (3.2), is defined by the differential equation (Arfken, 1968) 

With the boundary condition that it is regular at r — ro — the explicit 

Denoting p = r — ro, the Green s function can be written in terms of a spherical modified Bessel function of the second kind. 

Therefore, the. r-wave tip wavefunction is equal to the Green s function up to a constant, with the center of the apex atom taken as ro, 

By taking the derivative of both sides of Eq. (3.10) with respect tozo, and using the relation (see Appendix C)  

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