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Problem solved: I just have to include the solution of the homogeneous equation into the solution of the differential equation. The solution to the differential equation is $$V_H(r) = -\frac{K}{r} - \frac{r+1}{r} \exp(-2r).$$ By requiring that $\lim_{r \to 0} V_H(r)$ is real we get $K=-1$ and $$V_H(r) = \frac{1}{r} - \frac{r+1}{r} \exp(-2r).$$


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TEBD is an approximation method that relies on an ansatz for the wavefunction in terms of matrix-product states. If you consider the formula for MPS in the link, you see that the different degrees of freedom are coupled in a "one-by-one style". This works good for spin chains as these use short-range interaction potentials, typically nearest ...


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