Figure 1

(Color online) Upper panel: Slater parameters $F^{\left(0\right)}$ (full black line), $F^{\left(2\right)}$ (red line with squares), $F^{\left(4\right)}$ (green line with diamonds), $F^{\left(6\right)}$ (blue line with circles), and parameter $J$ 10 times (dashed black line) as function of screening length $\lambda$ of the Yukawa potential in US. Lower panel: comparison of Slater parameters ratios $A_1=F^{\left(4\right)}/F^{\left(2\right)}$ (red lines) and $A_2=F^{\left(6\right)}/F^{\left(2\right)}$ (black lines) calculated by a screened Yukawa potential (full line with squares for $A_1$ and full line with circles for $A_2$) with the fixed ones in Refs. 6, 18 (full line for $A_1$ and dashed line for $A_2$) and in Refs. 16, 17 (dashed-dotted line for $A_1$ and dotted line for $A_2$). The radial wave function used for the calculation of Slater integrals in Eq. (5) corresponds to the energy at the center of $5f$ band and the MT radius of $\text{U}$ is $R_{\text{MT}}^{\text{U}}=2.79\text{a}\text{.u}\text{.}$Reuse & Permissions

Figure 2

(Color online) Spin (upper panel) and orbital magnetic moment (lower panel) of US, $a=10.36\text{a}\text{.u}\text{.}$, calculated with $\text{LDA}+\text{SOC}+U$ approach and INT DC (see DC corrections in the method section). We compare the results obtained by calculating the $F^{\left(k\right)}$ with a screened Yukawa potential (full black line and screening length in the upper axis) with ones obtained by fixing the ratios $A_1$ and $A_2$ to ones of Refs. 6, 18 (dashed red line) and to ones of Refs. 16, 17 (dashed-dotted blue line). In the fixed ratios calculations we fixed the parameter $J=0.46\text{eV}$ to one in Ref. 16. The MT radii of $\text{U}$ and $\text{S}$ are set, respectively, to $R_{\text{MT}}^{\text{U}}=2.79\text{a}\text{.u}\text{.}$ and $R_{\text{MT}}^{\text{S}}=2.13\text{a}\text{.u}\text{.}$. The parameter $R_{\text{MT}}^{\text{U}}{\left|\vec{G}+\vec{k}\right|}_{\text{max}}$, governing the number of plane waves in the $\text{APW}+\text{lo}$ method, is chosen to be 9.76. The Brillouin zone (BZ) is sampled with 1728 $k$ points uniformly spaced.Reuse & Permissions

Figure 3

(Color online) Spin $M_{\text{sp}}$ (red lines) and orbital $M_{\text{orb}}$ (blue lines) magnetic moments of paramagnetic $\text{PuS}$ (full line with squares for $M_{\text{sp}}$, full line with circles for $M_{\text{orb}}$), $a=10.46\text{a}\text{.u}\text{.}$, and ferromagnetic $\text{PuP}$ (dashed line for $M_{\text{sp}}$, dashed-dotted line for $M_{\text{orb}}$), $a=10.49\text{a}\text{.u}\text{.}$, as function of $U$ for different types of DC. The Slater parameters are calculated SC by using a Yukawa potential with screening length $\lambda$ reported in the upper axis. For AMF DC the moments vanish in both compounds at $U\approx 2.0\text{eV}$, for FLL DC both compounds stay magnetic for all values of $U$. Only for the INT DC there is a range of $U$ $\left(U\gtrsim 3.5\text{eV}\right)$ for which we obtain the experimental magnetic structure of both compounds (Ref. 1); $\text{PuS}$ becomes nonmagnetic while $\text{PuP}$ stays magnetic. The green lines refer to the $5f$ charge of $\text{PuS}$ (full line with triangles) and $\text{PuP}$ (dotted line). The black lines refer to INT DC factor $\alpha$ of $\text{PuS}$ (full line) and $\text{PuP}$ (dashed-two-dots line). The MT radii of Pu and $\text{S}\left(\text{P}\right)$ are set, respectively, to $R_{\text{MT}}^{\text{Pu}}=2.82\text{a.u.}$ and $R_{\text{MT}}^{\text{S}\left(\text{P}\right)}=2.15\text{a.u.}$; the basis set cut off $R_{\text{MT}}^{\text{Pu}}\mid {\vec{G}+\vec{k}\mid }_{\text{max}}$ is set to be 9.1. The integration in the BZ is performed with 1728 with $k$ points uniformly spaced.Reuse & Permissions

Figure 4

(Color online) Spin (upper panel) and orbital magnetic moment (lower panel) of US, $a=10.36\text{a}\text{.u}\text{.}$, calculated with $\text{LDA}+\text{SOC}+U$ approach. We compare the results obtained by calculating the $F^{\left(k\right)}$ with a screened Yukawa potential (full black line and screening lenght in the upper axis) with ones obtained by fixing the ratios $A_1=F^{\left(4\right)}/F^{\left(2\right)}$ and $A_2=F^{\left(6\right)}/F^{\left(2\right)}$ to ones of Refs. 6, 18 (dashed red line) and to ones of Refs. 16, 17 (dashed-dotted blue line). In the fixed ratios calculations we varied $J$ to the one determined by Yukawa potential at every different screening length, $J\left(\lambda \right)$. Other calculation details are the same as the ones reported in Fig. 2.Reuse & Permissions

Figure 5

(Color online) Relevant exchange energy channels in Eq. (29), $E_X^{k_{1}pr}$, of US calculated with $\text{LDA}+\text{SOC}+U$ method and INT DC. The Slater parameters are calculated SC by using a screened Yukawa potential with screening length $\lambda$ reported in the upper axis. Other calculation details are reported in Fig. 2.Reuse & Permissions

Figure 6

(Color online) Upper panel: spin (red line with squares) and orbital (blue line with circles) moment of ferromagnetic $\text{NpN}$, $a=9.25\text{a}\text{.u}\text{.}$. Lower panel: relevant exchange energy channels in Eq. (29), $E_X^{k_{1}pr}$, of $\text{NpN}$ calculated with $\text{LDA}+\text{SOC}+U$ approach and INT DC. The Slater parameters are calculated SC by using a screened Yukawa potential with screening length $\lambda$ reported in the upper axis. The MT radii of Np and $\text{N}$ are set, respectively, to $R_{\text{MT}}^{\text{Np}}=2.60\text{a}\text{.u}\text{.}$ and $R_{\text{MT}}^{\text{N}}=1.80\text{a}\text{.u}\text{.}$; the parameter $R_{\text{MT}}^{\text{Np}}{\left|\vec{G}+\vec{k}\right|}_{\text{max}}$ is set to 9.4. The integration in the BZ is performed with 1728 $k$ points uniformly spaced.Reuse & Permissions

Figure 7

(Color online) Upper panel: spin (red line with squares) and orbital (blue line with circles) moment of NpSb in AFM-3k structure, $a=11.82\text{a}\text{.u}\text{.}$. Lower panel: relevant exchange energy channels in Eq. (29), $E_X^{k_{1}pr}$, of NpSb calculated with $\text{LDA}+\text{SOC}+U$ and INT DC. The Slater parameters are calculated SC by using a screened Yukawa potential with screening length $\lambda$ reported in the upper axis. The MT radii of Np and Sb are set, respectively, to $R_{\text{MT}}^{\text{Np}}=2.95\text{a}\text{.u}\text{.}$ and $R_{\text{MT}}^{\text{Sb}}=2.65\text{a}\text{.u}\text{.}$; the parameter $R_{\text{MT}}^{\text{Np}}{\left|\vec{G}+\vec{k}\right|}_{\text{max}}$ is set to be 9.2. The integration in the BZ is performed with 1728 $k$ points uniformly spaced.Reuse & Permissions

Figure 8

(Color online) Relevant exchange energy channels in Eq. (29), $E_X^{k_{1}pr}$, for paramagnetic $\text{PuS}$, $a=10.46\text{a}\text{.u}\text{.}$, and ferromagnetic $\text{PuP}$, $a=10.49\text{a}\text{.u}\text{.}$, as function of $U$ for INT DC. The Slater parameters are calculated SC by using a Yukawa potential with screening lenght $\lambda$ reported in the upper axis. Other calculation details are reported in Fig. 3.Reuse & Permissions

Figure 9

(Color online) Upper panel: spin (dashed red line) and orbital (dashed-dotted blue line) magnetic moments of high-$T_c$ superconductor ${\text{PuCoGa}}_5$, $a=7.84\text{a}\text{.u}$ and $c=12.57\text{a}\text{.u}\text{.}$, calculated with $\text{LDA}+\text{SOC}+U$ and INT DC, as function of parameter $U$. The interpolation factor $\alpha$ is plotted with a dotted black line. The Slater parameters are calculated SC by using a Yukawa potential with screening lenght $\lambda$ reported in the upper axis. We used an AFM-1k structure with $\mathbf{q}=\left(0,0,1/2\right)$. Lower panel: relevant exchange energy channels in Eq. (29), $E_X^{k_{1}pr}$, of ${\text{PuCoGa}}_5$. The MT radii of Pu, Co, and Ga are set, respectively, to $R_{\text{MT}}^{\text{Pu}}=2.7\text{a}\text{.u}\text{.}$ and $R_{\text{MT}}^{\text{Co},\text{Ga}}=2.2\text{a}\text{.u}\text{.}$. The parameter $R_{\text{MT}}^{\text{Pu}}{\left|\vec{G}+\vec{k}\right|}_{\text{max}}$ is set to 9.5. The integration in the BZ is performed with 576 $k$ points uniformly spaced.Reuse & Permissions