Quantity definitions

A more thorough discussion of pseudopotentials, from which many of the equations below have been adapted can be found here.

Norm-conserving pseudopotentials

Following the procedure of Vanderbilt 1991, separable norm-conserving pseudopotentials can be directly produced using the following procedure:

solve the all-electron Kohn-Sham equations for the isolated atom, yielding the all-electron potential $V(r)$ and atomic wave-functions $|\phi_{l,n}\rangle$ with energies $\varepsilon_{l,n}$
generate a local potential $V_{\mathrm{loc}}(r)$ such that $V_{\mathrm{loc}}(r) = V(r)$ for $r > r_L$ ($r_L$ is the inner pseudization radius for the local potential); $V_\mathrm{loc}(r)$ for $r < r_L$ can be any smooth regular function
generate pseudo-atomic wavefunctions $|\tilde{\phi}_{l,n}\rangle$ such that $\tilde{\phi}_{l,n}(r) = \phi_{l,n}(r)$ for $r > r_{c,l,n}$ ($r_{c,l,n}$ is the inner cutoff radius for the n-th pseudo-atomic wavefunction at angular momentum $l$); $\tilde{\phi}_{l,n}(r)$ for $r < r_{c,l,n}$ can be any smooth regular function
generate the corresponding functions $|\chi_{l,n}\rangle$ (vanishing for $r > r_{c,l,n}$):

\[|\chi_{l,n}\rangle \equiv (\varepsilon_{l,n} - T - V_{\mathrm{loc}})|\tilde{\phi}_{l,n}\rangle\]

generate the KB projectors $|\beta_{l,m}\rangle$:

\[|\beta_{l,m}\rangle \equiv \sum_{m} (B^{-1})_{l,nm} |\chi_{l,m}\rangle\]

where $B_{l,nm} = \langle \tilde{\phi_{l,n}} | \chi_{l,n} \rangle$ and $|\beta_{l,m}\rangle$ satisfy $\langle \beta_{l,n} | \tilde{\phi}_{l,m} \rangle = \delta_{nm}$

Form of the pseudopotentials

PseudoPotentialIO assumes a the above separable (Kleinman-Bylander) form for all the pseudopotentials. Therefore, the total pseudopotential is defined as:

\[\hat{V}^{\mathrm{PsP}} \rightarrow \hat{V}_{\mathrm{KB}} = \hat{V}^\prime_{\mathrm{loc}} + \hat{V}_{\mathrm{NL}}\]

The non-local part of the potential $\hat{V}_{l,\mathrm{NL}}$ at angular momentum $l$ is defined as

\[\hat{V}_{l,\mathrm{NL}} \equiv \sum_{nm} | \beta_{l,n} \rangle D_{l,nm} \langle \beta_{l,m} |\]

where $\beta_{l,n}$ is the n-th non-local projector at angular momentum $l$ in Kleinman-Bylander (KB) form, and $D_{l,nm}$ are the KB energies or projector coupling coefficients at angular momentum $l$.

Storing the quantities

The local part of the potential $\hat{V}^\prime_{\mathrm{loc}}$ is stored in numerical pseudopotentials as the vector Vloc, without any prefactor, i.e. the stored quantity is

\[\hat{V}_{\mathrm{loc}}(r)\]

The KB projectors $\beta_{l,n}$ are stored in numerical pseudopotentials as β[l][n] with a prefactor of $r^2$, i.e. the stored quantity is

\[r^2 \beta_{l,n}(r)\]

The KB energies / projector coupling coefficients are stored in all pseudopotentials as D[l][n,m].
If available, the $\chi_{l,n}$ functions are stored in numerical pseudopotentials as χ[l][n] with a prefactor of $r^2$, i.e. the stored quantity is

\[r^2 \chi_{l,n}(r)\]

If available, the pseudo-atomic valence charge density $\rho_{\mathrm{val}}(r) = \sum_{l=0}^{l_\mathrm{max}} \sum_{m=-l}^{l} \sum_{n} |\tilde{\phi}_{l,n}(r)|^2$ is stored with a prefactor of $r^2$, i.e. the stored quantity is

\[r^2 \rho_{\mathrm{val}}(r)\]

If available, the core charge density (non-linear core correction) $\rho_{\mathrm{core}}(r)$ is stored with a prefactor of $r^2$, i.e. the stored quantity is

\[r^2 \rho_{\mathrm{core}}(r)\]