Functionals.md

@@ -83,19 +83,19 @@ The term in ASLDA functional that depends on the currents $`\bm{j}_{\sigma}`$ in
 
 *Note 1*: SLDA functional exhibits much better convergence properties than ASLDA, i.e `st-wslda` codes typically converge in a significantly smaller number of iterations.  
 
-*Note 2*: SLDA functional provides by factor about 2x better performance in case of `td-wslda` codes than ASLDA variant.  
+*Note 2*: SLDA functional provides by a factor about 2x better performance in case of `td-wslda` codes than ASLDA variant.  
 
-Below we provide plot showing relative energy change (in %) between simulations of SLDA and ASLDA that compute the energy of N fermions (both even and odd particle numbers) in the spin-symmetric unitary Fermi gas trapped in an isotropic harmonic oscillator.  For more info see [supplemental material of this paper](https://arxiv.org/abs/1306.4266).  
+Below we provide a plot showing relative energy change (in %) between simulations of SLDA and ASLDA that compute the energy of N fermions (both even and odd particle numbers) in the spin-symmetric unitary Fermi gas trapped in an isotropic harmonic oscillator.  For more info see [supplemental material of this paper](https://arxiv.org/abs/1306.4266).  
 ![slda_vs_aslda](uploads/073efe244d1ab67bed1314af5cf0fc21/slda_vs_aslda.png)
 
 # BdG - Bogoliubov de-Gennes functional
 The BdG functional is equivalent to  Bogoliubov de-Gennes mean-field approximation. This approximation is valid in BCS regime, under assumption $`-ak_F<1`$. The functional has form:
 ```math
-\mathcal{E}_{\textrm{edf}}(n,\nu,\ldots)  =  \frac{\tau_{\uparrow}}{2} + 
-    \frac{\tau_{\downarrow}}{2} 
+\mathcal{E}_{\textrm{edf}}(n,\nu,\ldots)  =  \alpha_\uparrow\frac{\tau_{\uparrow}}{2} + 
+    \alpha_\downarrow\frac{\tau_{\downarrow}}{2} 
     + g\nu^{\dagger}\nu 
 ```  
-The (bare) coupling constant $`g`$ is related to scattering length by the formula $`g=4\pi\hbar^2a/m_r`$, where $`m_r`$ is reduced mass.  
+The (bare) coupling constant $`g`$ is related to scattering length by the formula $`g=4\pi\hbar^2a/m_r`$, where $`m_r`$ is twice of reduced mass, i.e.: $`\frac{1}{m_r}=\frac{1}{2}(\alpha_\uparrow+\alpha_\downarrow)`$.
 
 Below we provide predictions of BdG functional for total energy and paring gap as a function of coupling constant $`-ak_F`$ computed for uniform and spin-symmetric system. Results are compared with predictions of BCS theory, and as expected the agreement is observed for $`-ak_F<1`$.  
 ![BdG-E](uploads/a3e8e87c80efd2c4a31d2ea50dec6191/BdG-E.png)