The derivative rule individual cases

As shown in Eq. (3.1), the transmission matrix elements for different tip states are determined by the Bardeen integral on a surface separating the sample and the tip with one of the tip states. For an s -wave tip state, using Eq. (3.11), 

Using Green s theorem, it can be converted into a volume integral over fir, the tip side from the separation surface. Noticing that the sample wavefunction ip satisfies Schrodinger s equation, Eq. (3.2), in fir, and that the Green s function satisfies Eq. (3.8), we obtain immediately 

Taking the derivative with respect to zo on both sides of Eq. (3.25), and noticing that zo is a parameter in the integral (which does not involve the process of evaluating the integration), and the expression of the tip wavefunction, Eq. (3.14), we find 

Therefore, the tunneling matrix element for a p, tip state is proportional to the j derivative of the sample wavefunction at the center of the apex atom. 

The tunneling matrix elements from the rest of the nine tip wavefunctions can be derived using the relation between the tip and Green s functions established in the previous section. For example, for the d), tip state. 

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