Relations between the central-like and tensor-like pseudopotentials

In the following we present the recoupling formulae which connect the two alternative forms of the pseudopotential of the Eqs. (1) and (22). We have,

$$\hat{V}_{\tilde{n} \tilde{L},v_{12} S}^{\tilde{n}' \tilde{L}'} = \frac{1}{2} i^{v_{12}} \left(1-{\textstyle{\frac{1}{2}}}\delta_{v_1,v_2}\right) \sqrt{2S+1}$$

$$\times \left( \sum_{J=\vert\tilde{L}'-v_1\vert}^{\tilde{L}'+v_1} (-1)^{J... ... \left[K_{\tilde{n}\tilde{L}} \sigma^{(2)}_{v_2} \right]_{J}\right]_{0} \right.$$

$$+ \sum_{J=\vert\tilde{L}'-v_1\vert}^{\tilde{L}'+v_1} (-1)^{J+v_2+... ...ht]_{J} \left[K_{\tilde{n}\tilde{L}} \sigma^{(1)}_{v_2} \right]_{J}\right]_{0}$$

$$+ \sum_{J=\vert\tilde{L}-v_1\vert}^{\tilde{L}+v_1}(-1)^{J+v_2+\ti... ...]_{J} \left[K_{\tilde{n}'\tilde{L}'} \sigma^{(2)}_{v_2} \right]_{J}\right]_{0}$$

$$+ \left. \sum_{J=\vert\tilde{L}-v_1\vert}^{\tilde{L}+v_1}(-1)^{J+... ...left[K_{\tilde{n}'\tilde{L}'} \sigma^{(1)}_{v_2} \right]_{J}\right]_{0} \right)$$

$$\times \left(1-\hat{P}^{M}\hat{P}^{\sigma}\hat{P}^{\tau}\right) \hat{\delta}_{12}(\bm{r}'_1\bm{r}'_2;\bm{r}_1\bm{r}_2) .$$ (92)

Analogously, the recoupling formula which allows to express the tensor-like pseudopotential through the central-like one reads,

$$\hat{\tilde{V}}_{\tilde{n} \tilde{L}, v_{12}J}^{\tilde{n}' \tilde{L}'} = \frac{1}{2} i^{v_{12}} \left(1-{\textstyle{\frac{1}{2}}}\delta_{v... ...J+1} \sum_{S=\vert\tilde{L}'-\tilde{L}\vert}^{\tilde{L}'+\tilde{L}} \sqrt{2S+1}$$

$$\times \left( (-1)^{S+J+v_1+\tilde{L}} \left(\begin{array}{ccc} \tilde{L... ...right]_{S} \left[ \sigma^{(1)}_{v_1} \sigma^{(2)}_{v_2} \right]_{S}\right]_{0}$$

$$+(-1)^{J+v_2+\tilde{L}} \left(\begin{array}{ccc} \tilde{L}' & \ti... ...right]_{S} \left[ \sigma^{(1)}_{v_2} \sigma^{(2)}_{v_1} \right]_{S}\right]_{0}$$

$$+ (-1)^{S+J+v_1+\tilde{L}'} \left(\begin{array}{ccc} \tilde{L} & ... ...right]_{S} \left[ \sigma^{(1)}_{v_1} \sigma^{(2)}_{v_2} \right]_{S}\right]_{0}$$

$$\left. + (-1)^{J+v_2+\tilde{L}'} \left(\begin{array}{ccc} \tilde{... ...S} \left[ \sigma^{(1)}_{v_2} \sigma^{(2)}_{v_1} \right]_{S}\right]_{0} \right)$$

$$\times \left(1-\hat{P}^{M}\hat{P}^{\sigma}\hat{P}^{\tau}\right) \hat{\delta}_{12}(\bm{r}'_1\bm{r}'_2;\bm{r}_1\bm{r}_2) .$$ (93)

According to the recoupling of the Eq. (92), we give the list of the relations between the parameters of the two forms of the pseudopotential. For the second order terms we have,

$$C_{00,00}^{20} = \tilde{C}_{00,00}^{20} ,$$ (94)

$$C_{00,20}^{20} = \tilde{C}_{00,21}^{20},$$ (95)

$$C_{00,22}^{22} = \tilde{C}_{00,21}^{22},$$ (96)

$$C_{11,00}^{11} = \tilde{C}_{11,01}^{11},$$ (97)

$$C_{11,20}^{11} = \frac{1}{3}\tilde{C}_{11,20}^{11}+\frac{1}{\sqrt{3}}\tilde{C}_{11,21}^{11}+\frac{\sqrt{5}}{3}\tilde{C}_{11,22}^{11} ,$$ (98)

$$C_{11,11}^{11} = -\tilde{C}_{11,11}^{11} ,$$ (99)

$$C_{11,22}^{11} = \frac{\sqrt{5}}{3}\tilde{C}_{11,20}^{11}-\frac{\sqrt{5}}{2\sqrt{3}}\tilde{C}_{11,21}^{11}+\frac{1}{6}\tilde{C}_{11,22}^{11};$$ (100)

at the fourth order,

$$C_{00,00}^{40}=\tilde{C}_{00,00}^{40} , \\$$ (101)

$$C_{00,20}^{40}= \tilde{C}_{00,21}^{40} , \\$$ (102)

$$C_{00,22}^{42}= \tilde{C}_{00,21}^{42} , \\$$ (103)

$$C_{11,00}^{31}= \tilde{C}_{11,01}^{31} , \\$$ (104)

$$C_{11,20}^{31}= \frac{1}{3}\tilde{C}_{11,20}^{31} + \frac{1... ...{11,21}^{31} + \frac{\sqrt{5}}{3}\tilde{C}_{11,22}^{31} , \\$$ (105)

$$C_{11,11}^{31}= -\tilde{C}_{11,11}^{31} , \\$$ (106)

$$C_{11,22}^{31}= \frac{\sqrt{5}}{3}\tilde{C}_{11,20}^{31} -\... ...lde{C}_{11,21}^{31} + \frac{1}{6}\tilde{C}_{11,22}^{31} , \\$$ (107)

$$C_{11,22}^{33}= \tilde{C}_{11,22}^{33} , \\$$ (108)

$$C_{20,00}^{20}= \tilde{C}_{20,00}^{20} , \\$$ (109)

$$C_{20,20}^{20}= \tilde{C}_{20,21}^{20} , \\$$ (110)

$$C_{20,22}^{22}= \tilde{C}_{20,21}^{22} , \\$$ (111)

$$C_{22,00}^{22}= \tilde{C}_{22,02}^{22} , \\$$ (112)

$$C_{22,20}^{22}= \frac{1}{\sqrt{5}}\tilde{C}_{22,21}^{22}+\f... ...22,22}^{22} + \sqrt{\frac{7}{15}}\tilde{C}_{22,23}^{22} , \\$$ (113)

$$C_{22,11}^{22}= -\tilde{C}_{22,12}^{22} , \\$$ (114)

$$C_{22,22}^{22}= \frac{\sqrt{35}}{10}\tilde{C}_{22,21}^{22}-... ...{C}_{22,22}^{22} + \frac{1}{\sqrt{15}}\tilde{C}_{22,23}^{22};$$ (115)

at the sixth order,

$$C_{00,00}^{60}=\tilde{C}_{00,00}^{60} , \\$$ (116)

$$C_{00,20}^{60}=\tilde{C}_{00,21}^{60} , \\$$ (117)

$$C_{00,22}^{62}=\tilde{C}_{00,21}^{62} , \\$$ (118)

$$C_{11,00}^{51}=\tilde{C}_{11,01}^{51} , \\$$ (119)

$$C_{11,20}^{51}= \frac{1}{3}\tilde{C}_{11,20}^{51} + \frac{1... ...}_{11,21}^{51} +\frac{\sqrt{5}}{3}\tilde{C}_{11,22}^{51}, \\$$ (120)

$$C_{11,11}^{51}= -\tilde{C}_{11,11}^{51} , \\$$ (121)

$$C_{11,22}^{51}= \frac{\sqrt{5}}{3}\tilde{C}_{11,20}^{51} -\... ...lde{C}_{11,21}^{51} + \frac{1}{6}\tilde{C}_{11,22}^{51} , \\$$ (122)

$$C_{11,22}^{53}=\tilde{C}_{11,22}^{53} , \\$$ (123)

$$C_{20,00}^{40}=\tilde{C}_{20,00}^{40} , \\$$ (124)

$$C_{20,20}^{40}=\tilde{C}_{20,21}^{40} , \\$$ (125)

$$C_{20,22}^{42}=\tilde{C}_{20,21}^{42} , \\$$ (126)

$$C_{22,22}^{40}=\tilde{C}_{22,21}^{40} , \\$$ (127)

$$C_{22,00}^{42}=\tilde{C}_{22,02}^{42} , \\$$ (128)

$$C_{22,20}^{42}= \frac{1}{\sqrt{5}}\tilde{C}_{22,21}^{42}+\f... ...22,22}^{42} + \sqrt{\frac{7}{15}}\tilde{C}_{22,23}^{42} , \\$$ (129)

$$C_{22,11}^{42}=-\tilde{C}_{22,12}^{42} , \\$$ (130)

$$C_{22,22}^{42}= \frac{\sqrt{35}}{10}\tilde{C}_{22,21}^{42}-... ...{22,22}^{42} + \frac{1}{\sqrt{15}}\tilde{C}_{22,23}^{42}, \\$$ (131)

$$C_{22,22}^{44}=\tilde{C}_{22,23}^{44} , \\$$ (132)

$$C_{31,00}^{31}=\tilde{C}_{31,01}^{31} , \\$$ (133)

$$C_{31,20}^{31}= \frac{1}{3}\tilde{C}_{31,20}^{31} + \frac{1... ..._{31,21}^{31} +\frac{\sqrt{5}}{3}\tilde{C}_{31,22}^{31} , \\$$ (134)

$$C_{31,11}^{31}=-\tilde{C}_{31,11}^{31} , \\$$ (135)

$$C_{31,22}^{31}=\frac{\sqrt{5}}{3}\tilde{C}_{31,20}^{31} -\f... ...lde{C}_{31,21}^{31} + \frac{1}{6}\tilde{C}_{31,22}^{31} , \\$$ (136)

$$C_{31,22}^{33}=\tilde{C}_{31,22}^{33} , \\$$ (137)

$$C_{33,00}^{33}=\tilde{C}_{33,03}^{33} , \\$$ (138)

$$C_{33,20}^{33}=\sqrt{\frac{5}{21}}\tilde{C}_{33,22}^{33}+\f... ...C}_{33,23}^{33}+\sqrt{\frac{3}{7}}\tilde{C}_{33,24}^{33}, \\$$ (139)

$$C_{33,11}^{33}=-\tilde{C}_{33,13}^{33} , \\$$ (140)

$$C_{33,22}^{33}=\sqrt{\frac{2}{7}}\tilde{C}_{33,22}^{33}-\fr... ...3,23}^{33}+\frac{\sqrt{5}}{2\sqrt{14}}\tilde{C}_{33,24}^{33}.$$ (141)