Functionals.md

@@ -120,7 +120,7 @@ SLDAE is extension of SLDA-type functional to arbitrary value of $`\lambda=|ak_F
 	\mathcal{E}  = A_\lambda \frac{\tau}{2}
 	        + \frac{3}{5}B_\lambda n \varepsilon_F
 	        + \frac{C_\lambda }{n^{1/3}} |\nu|^2
-                + (1 - A_\lambda) \frac{\vb{j}^2}{2n}
+                + (1 - A_\lambda) \frac{j^2}{2n}
 ``` 
 where functional parameters $`A_\lambda`$, $`B_\lambda`$ and $`C_\lambda`$ are constructed in such a way to reproduce properly quasiparticle properties in the whole regime of $`\lambda`$, and also satisfy constraints in limits $`\lambda\rightarrow 0`$ and $`\lambda\rightarrow \infty`$. Precisely, the SLDAE is constructed in such a way to reproduce data for the:
 * the ground-state energy per unit volume $`E = 3n\xi_\lambda \varepsilon_F/5`$, and thus corresponding chemical potential $`\mu/\varepsilon_F = \zeta_\lambda`$ which must verify  the thermodynamic relationship $`\zeta_\lambda = \xi_\lambda + (\lambda/5) \xi_\lambda^\prime`$,
@@ -128,7 +128,8 @@ where functional parameters $`A_\lambda`$, $`B_\lambda`$ and $`C_\lambda`$ are c
 * the effective mass of particle $`\alpha_\lambda=m/m^\star`$.
 The plot below shows $`\lambda=|ak_F|`$ dependence of these quantities in the SLDAE functional. 
 ![sldae_qp_properties](uploads/bc7018319141055f9f032d7526a2dc9c/sldae_qp_properties.png)
-For more details see https://arxiv.org/abs/2201.07626.
+For more details see https://arxiv.org/abs/2201.07626.  
+**NOTE**: the functional has been constructed for spin-symmetric systems, $`N_a=N_b`$.
 
 ## SLDAE with $`m/m^\star=1`$
 The last term of SLDAE functional (depending on currents $`j`$) introduces a significant cost to the computation process. In many cases, it is sufficient to proceed with the functional that neglects corrections related to the effective mass (actually the effective mass is not known with high precision). The W-SLDA Toolkit provides a variant of the SLDAE functional where the constraint $`\alpha_\lambda = m/m^\star=1`$ is imposed. In order to activate it you need to use flag (in `predefines.h`):