where $E_{\textrm{V}}^{\textrm{LAO}}$ and $E_{\textrm{V}}^{\textrm{STO}}$ are the valence band edges of LaAlO$_3$ and SrTiO$_3$, respectively. The simulation cell is a symmetric nonstoichiometric superlattice with 12.5 layers of SrTiO$_3$ and 4.5 layers of LaAlO$_3$. The first method to determine the VBO is to analyze the local density of states, referred to as the LDOS approach [@Balereschi-JPD-1998; @Albina-PRB-2007]. The other method is to use the macroscopic average potential, denoted further as the MA approach [@Baldereschi-PRL-1988; @Baroni-PRB-1991; @Albina-PRB-2007]. We compare our results with other theoretical calculations and the recent experiment in Table \[tab:vbo\].

                               $n$-type (TiO$_2$/LaO)    $p$-type (SrO/AlO$_2$)
  --------------------------- ------------------------- ------------------------
         present study         -0.47 (MA) -0.39 (LDOS)   -0.02 (MA) 0.11 (LDOS)
    Ref. [@Albina-PRB-2007]     0.51 (MA) 0.39 (LDOS)    0.19 (MA) 0.10 (LDOS)
    Ref. [@Cen-NatMat-2008]              1.1                       NA
   Ref. [@Pickett-PRB-2008]         -0.15 (LDOS)                   NA
   Ref. [@Satpathy-PRL-2008]         -0.9 (LDOS)                   NA
     Ref. [@Lee-MRS-2007]             -0.1 (MA)                    NA
    Expt. [@Segal-PRB-2009]        -0.35$\pm$0.18                  NA

  : \[tab:vbo\] The valence band offset (VBO) of LaAlO$_3$/SrTiO$_3$ $n$-type and $p$-type interfaces in eV. Each value is followed by the method used to determine it in parentheses.

We can see from Table \[tab:vbo\] that for the $n$-type interface not only the magnitude of VBO differs, but the sign is not unanimous. One possible origin for the range of theoretical values 