The derivative rule general case

In this section, we present an alternative proof of the derivative rule, which provides an expression for the transmission matrix element from an arbitrary tip state expanded in terms of spherical harmonics. In the previous sections, we have expanded the tip wavefunction on the separation surface in terms of spherical harmonics. In general, the expansion is 

The coefficients (3 , are determined by fitting the tip wavefunction on and beyond the separation surface. Inside the tip body, the actual tip wavefunction does not satisfy Eq. (3.2), and the expansion (Eq. (3.30)) does 

On the other hand, the sample wavefunction in the entire volume of the tip body should satisfy the Schrodinger equation, Eq. (3.2). Especially, it should be regular at the origin. Therefore, in the tip body, the sample wavefunction must have the form 

To proceed with the proof, we first state a property of the Bardeen integral If both functions involved, i i and x satisfy the same Schrodinger equation in a region fi, then the Bardeen integral J on the surface enclosing a closed volume w within fl vanishes. Actually, using Green s theorem, Eq. (3.2) becomes 

Because and x satisfy the same Schrddinger equation with the same energy eigenvalue, Eq. (3.32) becomes 

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Fig. 3.2. Derivation of the derivative rule general case. In the shaded region, the tip wavefunction does not satisfy the Schrodinger equation in the vacuum. However, the expansion in Eq. (3.30) satisfies the Schrodinger equation in the vacuum except at the nucleus of the apex atom. Thus the surface on which the Bardeen integral is evaluated can be deformed to be any surface that encloses the nucleus of the apex atom.

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