Tail cancellation and the global matching function

As discussed by Andersen [9, 10] for muffin-tin orbitals, the locally regular components y defined in each muffin-tin sphere are cancelled exactly if expansion coefficients satisfy the MST equations (the tail-cancellation condition) [9, 384], The standard MST equations for space-filling cells can be derived by shrinking the interstitial volume to a honeycomb lattice surface that forms a common boundary for all cells. The wave function and its normal gradient evaluated on this honeycomb interface define a global matching function %(cr). 

In any cell r = r/(. with surface a = a/x, any solution ijr of the Lippmann-Schwinger equation dehnes auxiliary funchons xm = f fr Govf and = Jm3 r Govifr, which are equal by construction. After integration by parts, 

The notation Wtl defines a Wronskian integral over ct/2. By the surface matching theorem, xm = X the interior component of 4 Since 4r is a solution of the Lippmann-Schwinger equation, this implies xout = Xm when evaluated in the interior of Tjj. This is a particular statement of the tail-cancellation condition. To show this in detail, after integration by parts 

All Wronskian integrals of iA — should vanish on the honeycomb lathee. Given the local expansion 4/ = J4r. lYl in particular cell r/(, (.// WI, 4/ 4) — determines Sl.vYu- WiA4f — ) = 0 implies that 

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