Main definitions

Here we provide some basic definitions for a number of quantities which are calculated by the code:

• A charge in a sphere $P_{R}$ of radius $R$ is defined as
  
  $$Q=\int_{\mathbf{r}\in P_{R}}\rho(\mathbf{r})\textrm{d}\mathbf{r}$$
  
  
  where the sphere is centred at point $\mathbf{P}$.
• A charge in a volume $V$ made by several overlapping spheres,
  
  $$V=\bigcup_{R}P_{R}\,,$$
  
  
  is defined similarly, with $P_{R}$ replaced by $V$.
• Partial electronic density for the state (``band'') $n$ is defined as:
  

  $$\rho_{n}(\mathbf{r})=\frac{1}{N}\sum_{\mathbf{k}\in BZ}\left... ...hbf{k}}\left\vert\psi_{n\mathbf{k}}(\mathbf{r})\right\vert^{2}$$ (3.1)

  
  
  where SP means a set of (special) k-points (i.e. the actual k-point sampling), $\psi_{n\mathbf{k}}(\mathbf{r})$ is the wave function and $\omega_{\mathbf{k}}$ is the weighting factor.
• Partial electronic density for an island of bands $n\in[n_{1},n_{2}]$ is
  

  $$\rho_{[n_{1},n_{2}]}(\mathbf{r})=\sum_{n=n_{1}}^{n_{2}}\rho_{n}(\mathbf{r})$$ (3.2)

  
  

• Partial $z-$densities, integrated over the surface plane ($x$ and $y$), useful for slab (surface) calculations:
  

  $$\rho_{[n_{1},n_{2}]}(z)=\int_{\mathbf{r}\in(layer)}\rho(\mathbf{r})\textrm{d}x\textrm{d}y$$ (3.3)

  
  

• Total electronic Density of States (DOS):
  

  $$N(\epsilon)=\frac{1}{N}\sum_{n}\sum_{\mathbf{k}\in BZ}\delta(\epsilon-\epsilon_{n\mathbf{k}})$$ (3.4)

  
  
  where the sum over $n$ is run over all states (both occupied and unoccupied) and $\epsilon_{n\mathbf{k}}$ are Kohn-Sham eigenvalues.
• Local DOS (projected on a sphere):
  

  $$N_{\mathbf{P}_{R}}(\epsilon)=\frac{1}{N}\sum_{n}\sum_{\mathbf{k}\in BZ}A_{n\mathbf{k}}\delta(\epsilon-\epsilon_{n\mathbf{k}})$$ (3.5)

  
  
  where the weighting factors are defined as:
  

  $$A_{n\mathbf{k}}=\int_{\mathbf{r}\in P_{R}}\vert\Psi_{n\mathbf{k}}(\mathbf{r})\vert^{2}\textrm{d}\mathbf{r}$$ (3.6)

  
  
  with the integration performed over the sphere $P_{R}$ of radius $R$ centered at point $\mathbf{P}$.
• Angular-momenta ($s,p,d$) projected DOS is defined as above, but with the following factors:
  

  $$A_{n\mathbf{k}}=\int_{\mathbf{r}\in P_{R}}\,\sum_{m=-l}^{l}\... ...{\mathbf{r}})\right\rangle \right\vert^{2}\textrm{d}\mathbf{r}$$ (3.7)

  
  
  where $S_{lm}(\widehat{\mathbf{r}})$ is a real spherical function for the momenta $l=0,1,2$.
• A DOS projected on a layer (for the slab calculations). The factors in this case are:
  

  $$A_{n\mathbf{k}}=\int_{\mathbf{r}\in(layer)}\vert\Psi_{n\mathbf{k}}(\mathbf{r})\vert^{2}\textrm{d}\mathbf{r}$$ (3.8)

  
  
  where the layer is thought to be perpendicular to the $z$-axis.