Second-order functional

At NLO, one obtains the EDF corresponding to the $i=1$ term of the regularized pseudopotential (43) by applying on all possible bilinear densities the relative-momentum operator:

$$\bm{k}'^*{}^2+ \bm{k}^2 = -{\textstyle{\frac{1}{4}}}\left(\bm{\nabla}'_1{}^2-2\bm{\nabla}'_... ... +\bm{\nabla} _1 ^2-2\bm{\nabla} _1\cdot\bm{\nabla} _2+\bm{\nabla} _2^2\right).$$ (77)

Using once again for densities the same notations as in Ref. [23], we get the quasilocal term,

$$\hspace*{-2cm} \langle V_1^L \rangle = -{\textstyle{\frac{1}{2}}}\int\! \mathrm d\mathbf r_1\,\mathrm d\mathbf r_2\,g_a(\mathbf r_1-\mathbf r_2)$$

$$\times \biggl\{ A_1^{\rho_0} \Bigl[ \left[{\textstyle{\frac{1}{2}}}\,\De... ...rho_0(\mathbf r_2) +\mathbf j_0(\mathbf r_1)\cdot\mathbf j_0(\mathbf r_2)\Bigr]$$

$$+ A_1^{\rho_1} \Bigl[ \left[{\textstyle{\frac{1}{2}}}\,\Delta\rho... ...rho_1(\mathbf r_2) +\mathbf j_1(\mathbf r_1)\cdot\mathbf j_1(\mathbf r_2)\Bigr]$$

$$+ A_1^{\mathbf s_0} \Bigl[ \left[{\textstyle{\frac{1}{2}}}\,\Delt... ...f s_0(\mathbf r_1)-\mathbf T_0(\mathbf r_1)\right]\cdot\mathbf s_0(\mathbf r_2)$$

$$~~~~~~~~ -{\textstyle{\frac{1}{4}}}\,\boldsymbol\nabla \otimes\ma... ... s_0(\mathbf r_2) +\mathsf J_0(\mathbf r_1)\cdot\mathsf J_0(\mathbf r_2) \Bigr]$$

$$+ A_1^{\mathbf s_1} \Bigl[ \left[{\textstyle{\frac{1}{2}}}\,\Delt... ...f s_1(\mathbf r_1)-\mathbf T_1(\mathbf r_1)\right]\cdot\mathbf s_1(\mathbf r_2)$$

$$~~~~~~~~ -{\textstyle{\frac{1}{4}}}\,\boldsymbol\nabla \otimes\ma... ...hbf r_2) +\mathsf J_1(\mathbf r_1)\cdot\mathsf J_1(\mathbf r_2) \Bigr] \biggr\}$$ (78)

and the nonlocal term,

$$\hspace*{-2cm} \langle V_1^{N}\rangle = -{\textstyle{\frac{1}{4}}}\int\! \mathrm d\mathbf r_1\,\mathrm d\mathbf r_2\,g_a(\mathbf r_1-\mathbf r_2)$$

$$\times \biggl\{ B_1^{\rho_0} \left[ \rho_0(\mathbf r_2,\mathbf r_1)\Delt... ...f r_2) -2\rho_0(\mathbf r_2,\mathbf r_1)\tau_0(\mathbf r_1,\mathbf r_2) \right.$$

$$\left. ~~~~~~-{\textstyle{\frac{1}{2}}}\boldsymbol\nabla \rho_0(\... ... r_2,\mathbf r_1)\cdot\boldsymbol\nabla \rho_0(\mathbf r_1,\mathbf r_2) \right]$$

$$+ B_1^{\rho_1} \left[ \rho_1(\mathbf r_2,\mathbf r_1)\Delta\rho_1... ...f r_2) -2\rho_1(\mathbf r_2,\mathbf r_1)\tau_1(\mathbf r_1,\mathbf r_2) \right.$$

$$\left. ~~~~~~-{\textstyle{\frac{1}{2}}}\boldsymbol\nabla \rho_1(\... ... r_2,\mathbf r_1)\cdot\boldsymbol\nabla \rho_1(\mathbf r_1,\mathbf r_2) \right]$$

$$+ B_1^{\mathbf s_0} \left[ \mathbf s_0(\mathbf r_2,\mathbf r_1)\c... ... s_0(\mathbf r_2,\mathbf r_1)\cdot \mathbf T_0(\mathbf r_1,\mathbf r_2) \right.$$

$$\left.~~~~~~ -{\textstyle{\frac{1}{2}}}\boldsymbol\nabla \otimes\... ... J_0(\mathbf r_2,\mathbf r_1)\cdot \mathsf J_0(\mathbf r_1,\mathbf r_2) \right]$$

$$+ B_1^{\mathbf s_1} \left[ \mathbf s_1(\mathbf r_2,\mathbf r_1)\c... ... s_1(\mathbf r_2,\mathbf r_1)\cdot \mathbf T_1(\mathbf r_1,\mathbf r_2) \right.$$

$$\left.~~~~~~ -{\textstyle{\frac{1}{2}}}\boldsymbol\nabla \otimes\... ...thbf r_2,\mathbf r_1)\cdot\mathsf J_1(\mathbf r_1,\mathbf r_2) \right]\biggr\}.$$ (79)

In Eqs. (79) and (80), similarly as in Sec. 2, depending on the context, the nabla operator $\boldsymbol\nabla$ acts on the local densities as $\boldsymbol\nabla _1$ or $\boldsymbol\nabla _2$ and on the nonlocal densities as $\boldsymbol\nabla _1+\boldsymbol\nabla _2$.

By taking the local limit of functionals (79) and (80), we again recover the quasilocal Cartesian EDF, which reads

$$\langle V_1\rangle= {\textstyle{\frac{1}{4}}}\,t_1\int\! \mathrm d\mathbf r\, {\textstyle{\frac{3}{4}}}\left(1+z_1\right) \left[\tau_0\rho_0-\mathbf j_0^2-{\textstyle{\frac{3}{4}}}\rho_0\Delta\rho_0\right]$$

$$-\left[{\textstyle{\frac{1}{4}}}\left(1+z_1\right)+{\textstyle{\frac... ...[ \tau_1\rho_1-\mathbf j_1^2-{\textstyle{\frac{3}{4}}}\rho_1\Delta\rho_1\right]$$

$$-\left[{\textstyle{\frac{1}{4}}}\left(1+z_1\right)-{\textstyle{\frac... ...thsf J_0^2-{\textstyle{\frac{3}{4}}}\,\mathbf s_0\cdot\Delta\mathbf s_0 \right]$$

$$-{\textstyle{\frac{1}{4}}}\left(1+z_1\right) \left[\mathbf T_1\cdot\... ...hsf J_1^2-{\textstyle{\frac{3}{4}}}\,\mathbf s_1\cdot\Delta\mathbf s_1 \right].$$ (80)

Finally, one obtains the EDF corresponding to the $i=2$ term of the regularized pseudopotential (44) using the relative-momentum operator:

$$\bm{k}'^*\cdot\bm{k} = {\textstyle{\frac{1}{4}}}\left(\bm{\nabla}'_1\cdot\bm{\nabla}_1-\... ...a}_1 +\bm{\nabla}'_2\cdot\bm{\nabla}_2-\bm{\nabla}'_1\cdot\bm{\nabla}_2\right),$$ (81)

with the result

$$\hspace*{-2cm} \langle V_2^L \rangle = {\textstyle{\frac{1}{2}}}\int\!\mathrm d\mathbf r_1\,\mathrm d\mathbf r_2\, g_a(\mathbf r_1-\mathbf r_2)$$

$$\times \biggl\{ A_2^{\rho_0} \left[ \rho_0(\mathbf r_1)\tau_0(\mathbf r_... ...o_0(\mathbf r_2) -\mathbf j_0(\mathbf r_1)\cdot\mathbf j_0(\mathbf r_2) \right]$$

$$+ A_2^{\rho_1} \left[ \rho_1(\mathbf r_1)\tau_1(\mathbf r_2) -{\t... ...o_1(\mathbf r_2) -\mathbf j_1(\mathbf r_1)\cdot\mathbf j_1(\mathbf r_2) \right]$$

$$+ A_2^{\mathbf s_0} \left[ \mathbf s_0(\mathbf r_1)\cdot\mathbf T... ...s_0(\mathbf r_2) -\mathsf J_0(\mathbf r_1)\cdot\mathsf J_0(\mathbf r_2) \right]$$

$$+ A_2^{\mathbf s_1} \left[ \mathbf s_1(\mathbf r_1)\cdot\mathbf T... ...bf r_2) -\mathsf J_1(\mathbf r_1)\cdot\mathsf J_1(\mathbf r_2) \right] \biggr\}$$ (82)

and

$$\hspace*{-2cm} \langle V_2^{N}\rangle = {\textstyle{\frac{1}{2}}}\int\!\mathrm d\mathbf r_1\,\mathrm d\mathbf r_2\, g_a(\mathbf r_1-\mathbf r_2)$$

$$\times \biggl\{ B_2^{\rho_0} \left[{\textstyle{\frac{1}{4}}} \boldsymbol... ...bf r_2) -\rho_0(\mathbf r_2,\mathbf r_1)\tau_0(\mathbf r_1,\mathbf r_2) \right]$$

$$+ B_2^{\rho_1} \left[{\textstyle{\frac{1}{4}}} \boldsymbol\nabla ... ...bf r_2) -\rho_1(\mathbf r_2,\mathbf r_1)\tau_1(\mathbf r_1,\mathbf r_2) \right]$$

$$+ B_2^{\mathbf s_0} \left[ {\textstyle{\frac{1}{4}}}\boldsymbol\n... ... J_0(\mathbf r_2,\mathbf r_1)\cdot \mathsf J_0(\mathbf r_1,\mathbf r_2) \right.$$

$$\left.~~~~~~ - \mathbf s_0(\mathbf r_2,\mathbf r_1)\cdot \mathbf T_0(\mathbf r_1,\mathbf r_2)\right]$$

$$+ B_2^{\mathbf s_1} \left[ {\textstyle{\frac{1}{4}}}\boldsymbol\n... ... J_1(\mathbf r_2,\mathbf r_1)\cdot \mathsf J_1(\mathbf r_1,\mathbf r_2) \right.$$

$$\left.~~~~~~ - \mathbf s_1(\mathbf r_2,\mathbf r_1)\cdot \mathbf T_1(\mathbf r_1,\mathbf r_2) \right] \biggr\},$$ (83)

whereas the corresponding quasilocal Cartesian EDF reads,

$$\langle V_2\rangle={\textstyle{\frac{1}{4}}}\,t_2 \int\!\mathrm d\mathbf r \biggl\{ \left[{\textstyle{\frac{5}{4}}}(1-z_2)+x_2-y_2\right] \left[ \rho_0\tau_0+{\textstyle{\frac{1}{4}}}\,\rho_0\Delta\rho_0-\mathbf j_0^2 \right]$$

$$+ \left[{\textstyle{\frac{1}{4}}}(1-z_2)+{\textstyle{\frac{1}{2}}}(x... ...rho_1\tau_1+{\textstyle{\frac{1}{4}}}\,\rho_1\Delta\rho_1-\mathbf j_1^2 \right]$$

$$+ \left[{\textstyle{\frac{1}{4}}}(1-z_2)+{\textstyle{\frac{1}{2}}}(x... ...extstyle{\frac{1}{4}}}\,\mathbf s_0\cdot\Delta\mathbf s_0-\mathbf J_0^2 \right]$$

$$+ {\textstyle{\frac{1}{4}}}(1-z_2) \left[ \mathbf s_1\cdot\mathbf T_... ...frac{1}{4}}}\,\mathbf s_1\cdot\Delta\mathbf s_1-\mathbf J_1^2 \right] \biggr\}.$$ (84)

The second-order Cartesian EDF (79-80) and (83-84) is exactly equivalent to that in the spherical-tensor representation. The corresponding relations of conversions between the parameters of the second-order regularized pseudopotential (3) and those of its Cartesian form (38) read

$$C_{00,00}^{20,0} = \sqrt{3}t_1+ \frac{\sqrt{3}}{2}t_1x_1,$$ (85)

$$C_{00,00}^{20,1} = -\sqrt{3}t_1y_1- \frac{\sqrt{3}}{2}t_1z_1 ,$$ (86)

$$C_{00,20}^{20,0} = -\frac{3}{2} t_1x_1 ,$$ (87)

$$C_{00,20}^{20,1} = \frac{3}{2} t_1z_1,$$ (88)

$$C_{11,00}^{11,0} = \sqrt{3}t_2+ \frac{\sqrt{3}}{2}t_2x_2 ,$$ (89)

$$C_{11,00}^{11,1} = -\sqrt{3}t_2y_2- \frac{\sqrt{3}}{2}t_2z_2,$$ (90)

$$C_{11,20}^{11,0} = -\frac{3}{2} t_2x_2 ,$$ (91)

$$C_{11,20}^{11,1} = \frac{3}{2} t_2z_2.$$ (92)