Evaluation of the modified Bardeen integral

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. 

The effect of distortion as well as the evaluation of Eq. (7.14) has been discussed by Holstein (1955) regarding the charge-exchange interaction between ions and parent atoms, as shown in Fig. 7.5. The exact time-independent Schrodinger equation for the electron, in atomic units, is 

In the absence of the second proton, Eq. (7.19) is the Schrodinger equation for the free hydrogen atom. The ground-state wavefunction is  

The presence of the second proton induces a perturbation to the wavefunction and the energy eigenvalue of the electron. For the perturbed wavefunction, we make the Ansatz 

To make an approximate solution, we neglect the two terms on the right-hand side of the equation, because it is much smaller than the terms on the left hand side (see the following). Thus, we obtain the approximate equation for the correction function g  

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