# Computation of Troullier-Martins pseudowavefunctions

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).

• 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? – davidhigh Dec 27 '19 at 18:57
• 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... – Alone Programmer Dec 28 '19 at 2:53
• 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. – Tommi Höynälänmaa Dec 28 '19 at 10:55