The modified Bardeen integral

In this subsection, we show that using Schrbdinger s equations, the tunneling matrix element can be converted to a surface integral similar to Bardeen s. Using Schrbdinger s equation for the tip states, Eq. (2.25), the matrix element is converted into 

The delta function factor in Eq. (2.30), that is, the condition of elastic transition, requires that = E.. Using Schrbdinger s equation for the sample states, Eq. (2.23), and noticing that in the tip body, fir, the potential of the sample Us is zero, the transition matrix element is converted into 

By writing down the kinetic energy term T = — (ftV2m)V explicitly, and using Green s theorem, the transition matrix element is finally converted into a surface integral similar to Bardeen s, in terms of modified wavefunctions  

The matrix element has the dimension of energy. In Chapter 7, we will show that the physical meaning of Bardeen s matrix element is the energy lowering 

By integrating over all the states in the tip and the sample, taking into account the occupation probabilities, the tunneling current is 

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In this subsection, we show that by evaluating the modified Bardeen integral, Eq. (7.14), with the distortion of the hydrogen wavefunction from another proton considered, as shown by Holstein (1955), an accurate analytic expression for the exact potential of the hydrogen molecular ion is obtained. 

In Chapter 2, we showed that the tunneling current can be determined with a perturbation approach. The central problem is to calculate the matrix elements. Those are determined by the modified Bardeen surface integral, evaluated from the wavefunctions of the tip and the sample (with proper corrections) on a separation surface between them, as shown in Fig. 3.1 ... 

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