deletions | additions       diff --git a/Delta_sigma_i__HB_is__.tex b/Delta_sigma_i__HB_is__.tex  index 1037253..8697251 100644  --- a/Delta_sigma_i__HB_is__.tex  +++ b/Delta_sigma_i__HB_is__.tex 
...
$\Delta\sigma^i_{HB}$ is the effect of hydrogen bonding to the amide H ($\Delta\sigma^i_{1^\circ HB}$) and O ($\Delta\sigma^i_{2^\circ HB}$) atoms of residue $i$ on the chemical shielding of the backbone atoms (i.e. this term is zero for C$\beta$)   $$\Delta \sigma^i_{HB}=\Delta\sigma^i_{1^\circ HB}(r_{\mathrm{HO}},\theta,\rho)+\Delta\sigma^i_{2^\circ HB}(r_{\mathrm{OH}},\theta_{\mathrm{O}},\rho_{\mathrm{O}})$$  $\Delta\sigma^i_{1^\circ HB}$ is computed using the structural models shown in Figure \ref{fig:HB} as the change in chemical shielding relative to free N-methyl acetamide computed at the OPBE/6-31G(d,p)//PM6 level of theory for a variety of orientations (see subsection \ref{subsec:HBscan} for more information) of the free monomer geometries. Note that thethe  carbonyl carbon formally belongs to the residue $i-1$. $\Delta\sigma^i_{1^\circ HB}$. $\Delta\sigma^i_{2^\circ  HB}$ is included only when another amide group is hydrogen bonded to the amide oxygen and is computed as the change in the chemical shielding of the top amide group in Figure \ref{subsec:HBscan}a. Note in this case that the amide nitrogen and hydrogen formally belong to residue $i+1$ and that $r_\mathrm{HO}$, $\theta$, and $\rho$ are defined relative to the carbonyl oxygen of residue $i$ rather than the amide proton as for $\Delta\sigma^i_{1^\circ HB}$. $r_\mathrm{HO}$, $\theta$, and $\rho$ are therefore labeled $r_\mathrm{HO}$, $\theta_{\mathrm{O}}$, and $\rho_{\mathrm{O}}$ in Eq xx.  Here $r_{\mathrm{HO}}$ refers to the O-H HB distance involving the amide H and $\theta_{\mathrm{H}}$ and $\rho_{\mathrm{H}}$ are the corresponding angles (Figure \ref{fig:HB}). Similarly, $r_{\mathrm{OH}}$ refers to the OH HB distance involving the amide O and $\theta_{\mathrm{O}}$ and $\rho_{\mathrm{O}}$ are the corresponding angles. For H^{N}, N, C$\alpha$, and H$\alpha$ is computed using the amide proton of residue $i$, i.e. $\Delta\sigma^i_{1^\circ HB}(r_{\mathrm{H(i)O}},\theta_{\mathrm{H}i},\rho_{\mathrm{H}i})$, while for C’, which is bonded to the N atom of the subsequent residue, $\Delta\sigma^i_{1^\circ HB}$ is computed using the amide proton of residue $i+1$. is computed for the amide C, H$\alpha$ and C$\alpha$ using the amide O of residue i, while for the amide H and N, it is computed using the amide O atom of the preceding $(i-1)$ residue.    
...