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The computation of Troullier-Martins pseudowavefunctions has been described in [1]. The pseudowavefunction $R^{\textrm{PP}}_l$ is defined by $$ R^{\textrm{PP}}_l(r) = \left\{ \begin{array}{ll} R^{\textrm{AE}}_l(r) ; & r \geq r_{\textrm{cl}} \\ f_l(r) ; & r \leq r_{\textrm{cl}} \end{array} \right. $$ Here $$ f_l(r) = r^l \exp(p(r)) $$ and $$ p(r) = c_0 + c_2 r^2 + c_3 r^3 + c_4 r^4 . $$ where $c_i$ are unknown coefficients. Note that the $c_1$ coefficient is not present. Functions $R^{\textrm{AE}}_l$ are known. The coefficients $c_i$ are defined by equations $$\int_0^{r_{\textrm{cl}}} \vert f_l(r) \vert^2 r^2 dr = \int_0^{r_{\textrm{cl}}} \vert R^{\textrm{AE}}_l(r) \vert^2 r^2 dr $$ $$ f_l(r_{\textrm{cl}}) = R^{\textrm{AE}}_l(r_{\textrm{cl}}) $$ $$ f'_l(r_{\textrm{cl}}) = (R^{\textrm{AE}})'_l(r_{\textrm{cl}}) $$ and $$ f''_l(r_{\textrm{cl}}) = (R^{\textrm{AE}})''_l(r_{\textrm{cl}}) $$ How are these equations solved? What algorithm should be used?

[1] Troullier and Martins. Efficient pseudopotentials for plane-wave calculations. Phys. Rev. B vol. 43, no. 3, pp. 1993-2006 (1991).

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    $\begingroup$ I don't know this method, but at first glance I would rather express the polynomial in powers of $(r-r_{cl})$ instead of $r$, as this seems to simplify the matching equations. For which application do you need pseudopotentials, and of what? $\endgroup$ – davidhigh Dec 27 '19 at 18:57
  • $\begingroup$ It's a system of nonlinear equations that can't be solved by analytical solutions mostly because of that first equation in terms of integral. You need to solve it by using Newton-Raphson probably... $\endgroup$ – Alone Programmer Dec 28 '19 at 2:53
  • $\begingroup$ I use the pseudopotentials to compute atoms and small molecules in a three-dimensional basis set. The HGH potential for oxygen atom handles 2s (and 2p) electrons as valence electrons. As the 2s wavefunctions behave irregularly near the origin I thought that I should use the Troullier-Martins pseudopotentials instead of HGH. $\endgroup$ – Tommi Höynälänmaa Dec 28 '19 at 10:55

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