Projector-augmented-wave formalism

Basics of the PAW formalism

The PAW formalism is a generalization of ideas of both Vanderbilt-type ultrasoft-pseudopotentials (USPP) and the linearized augmented-plane-wave (LAPW) method. The method was first proposed and implemented by Blöchl. The formal relationship between Vanderbilt-type ultrasoft pseudopotentials and the PAW method has been derived by Kresse and Joubert, and the generalization of the PAW method to non collinear magnetism has been discussed by Hobbs, Kresse and Hafner. We briefly summarize the basics of the PAW method below (following Refs. and ).

In the PAW method the one electron wavefunctions Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): \psi _{{n{\mathbf {k}}}} , in the following simply called orbitals, are derived from the pseudo orbitals Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): \widetilde {\psi }_{{n{\mathbf {k}}}} by means of a linear transformation:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): |\psi _{{n{\mathbf {k}}}}\rangle =|\widetilde {\psi }_{{n{\mathbf {k}}}}\rangle +\sum _{{i}}(|\phi _{{i}}\rangle -|\widetilde {\phi }_{{i}}\rangle )\langle \widetilde {p}_{{i}}|\widetilde {\psi }_{{n{\mathbf {k}}}}\rangle .

The pseudo orbitals Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): \widetilde {\psi }_{{n{\mathbf {k}}}} , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): nk is the band index and k-point index, are the variational quantities and expanded in plane waves (see below). In the interstitial region between the PAW spheres, the orbitals Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): \widetilde {\psi }_{{n{\mathbf {k}}}} are identical to the exact orbitals Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\psi }_{{n{\mathbf {k}}}} . Inside the spheres, the pseudo-orbitals are however only a computational tool and an inaccurate approximation to the true orbitals, since even the norm of the all-electron wave function is not reproduced. The last equation is required to map the auxiliary quantities Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): \widetilde {\psi }_{{n{\mathbf {k}}}} onto the corresponding exact orbitals. The PAW method implemented in VASP exploits the frozen core (FC) approximation, which is not an inherent characteristic of the PAW method, but has been made in all implementations so far. In the present case, the core electrons are also kept frozen in the configuration for which the PAW dataset was generated.

The index Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): \alpha is a shorthand for the atomic site $\mathbf {R} _{\alpha }$, the angular momentum quantum numbers $l_{\alpha },m_{\alpha }$ and an additional index $\varepsilon _{\alpha }$ referring to the reference energy. The pseudo orbitals are expanded in the reciprocal space using plane waves

$\langle \mathbf {r} |{\widetilde {\psi }}_{n\mathbf {k} }\rangle ={\frac {1}{\Omega ^{1/2}}}\sum _{\mathbf {G} }C_{n\mathbf {kG} }e^{i(\mathbf {G} +\mathbf {k} )\cdot \mathbf {r} },$

where $\Omega$ is the volume of the Wigner-Seitz cell. The all-electron (AE) partial waves $\phi _{\alpha }$ are solutions of the radial Schrödinger equation for a non-spinpolarized reference atom at a specific energy $\varepsilon _{\alpha }$ and for a specific angular momentum $l_{\alpha }$:

$\langle \mathbf {r} |\phi _{\alpha }\rangle ={\frac {1}{|\mathbf {r} -\mathbf {R} _{\alpha }|}}u_{\alpha }(|\mathbf {r} -\mathbf {R} _{\alpha }|)Y_{\alpha }({\widehat {\mathbf {r} -\mathbf {R} _{\alpha }}})={\frac {1}{|\mathbf {r} -\mathbf {R} _{\alpha }|}}u_{l_{\alpha }\varepsilon _{\alpha }}(|\mathbf {r} -\mathbf {R} _{\alpha }|)\,Y_{l_{\alpha }m_{\alpha }}({\widehat {\mathbf {r} -\mathbf {R} _{\alpha }}}).$

The notation ${\widehat {\mathbf {r} -\mathbf {R} _{\alpha }}}$ is used to clarify that the spherical harmonics $Y$ depends on the orientation but not on the length of the vector $\mathbf {r} -\mathbf {R} _{\alpha }$. Note that the radial component of the partial wave $u_{\alpha }$ is independent of $m_{\alpha }$, since the partial waves are calculated for a spherical atom. Furthermore, the spherical harmonics depend on the angular quantum numbers only and not on the reference energy. The pseudo partial waves ${\widetilde {\phi }}_{\alpha }$ are equivalent to the AE partial waves outside a core radius Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): r_{{c}} and match continuously onto Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): \phi _{{\alpha }} inside the core radius:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): \langle {\mathbf {r}}|\widetilde {\phi }_{{\alpha }}\rangle ={\frac {1}{|{\mathbf {r}}-{\mathbf {R}}_{\alpha }|}}\widetilde {u}_{{\alpha }}(|{\mathbf {r}}-{\mathbf {R}}_{\alpha }|)Y_{{\alpha }}(\widehat {{\mathbf {r}}-{\mathbf {R}}_{\alpha }})={\frac {1}{|{\mathbf {r}}-{\mathbf {R}}_{\alpha }|}}\widetilde {u}_{{l_{\alpha }\varepsilon _{\alpha }}}(|{\mathbf {r}}-{\mathbf {R}}_{\alpha }|)\,Y_{{l_{\alpha }m_{\alpha }}}(\widehat {{\mathbf {r}}-{\mathbf {R}}_{\alpha }}).

The core radius Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): r_{{c}} is usually chosen approximately around half the nearest neighbor distance. The projector functions Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): \widetilde {p}_{{\alpha }} are dual to the partial waves:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): \langle \widetilde {p}_{{i}}|\widetilde {\phi }_{{j}}\rangle =\delta _{{ij}}.

Charge and overlap densities

Starting from the completeness relations it is possible to show that, in the PAW method, the total charge density (or more precisely the overlap density) related to two orbitals Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): \psi _{{n{\mathbf {k}}}} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): \psi _{{m{\mathbf {k}}}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): n({\mathbf {r}})=\psi _{{n{\mathbf {k}}}}^{{\ast }}({\mathbf {r}})\,\psi _{{m{\mathbf {k}}}}({\mathbf {r}})

can be rewritten as (for details we refer to Ref. ):

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): n({\mathbf {r}})=\widetilde {n}({\mathbf {r}})-\widetilde {n}^{{1}}({\mathbf {r}})+n^{{1}}({\mathbf {r}}).

Here, the constituent charge densities are defined as:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): \widetilde {n}({\mathbf {r}})=\langle \widetilde {\psi }_{{n{\mathbf {k}}}}|{\mathbf {r}}\rangle \langle {\mathbf {r}}|\widetilde {\psi }_{{m{\mathbf {k}}}}\rangle

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): \widetilde {n}^{{1}}({\mathbf {r}})=\sum _{{\alpha ,\beta }}\widetilde {\phi }_{\alpha }^{\ast }({\mathbf {r}})\widetilde {\phi }_{\beta }({\mathbf {r}})\langle \widetilde {\psi }_{{n{\mathbf {k}}}}|\widetilde {p}_{\alpha }\rangle \langle \widetilde {p}_{\beta }|\widetilde {\psi }_{{m{\mathbf {k}}}}\rangle

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): n^{{1}}({\mathbf {r}})=\sum _{{\alpha ,\beta }}\phi _{\alpha }^{\ast }({\mathbf {r}})\phi _{\beta }({\mathbf {r}})\langle \widetilde {\psi }_{{n{\mathbf {k}}}}|\widetilde {p}_{\alpha }\rangle \langle \widetilde {p}_{\beta }|\widetilde {\psi }_{{m{\mathbf {k}}}}\rangle .

The quantities with a superscript 1 are one-center quantities and are usually only evaluated on radial grids. Furthermore, one can usually drop the complex conjugation for the partial waves, since they are real-valued. The indices Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): \alpha and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): \beta are restricted to those pairs that correspond to one atom Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\mathbf {R}}_{\alpha }={\mathbf {R}}_{\beta } . For a complete set of projectors the one-centre pseudo charge density Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): \widetilde {n}^{{1}} is exactly identical to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): \widetilde {n} within the augmentation spheres. Furthermore, it is often necessary to define Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): \rho _{{\alpha \beta }} , the occupancies of each augmentation channel Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): (\alpha ,\beta ) inside each PAW sphere. These are calculated from the pseudo orbitals applying the projector functions and summing over all bands

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): \rho _{{\alpha \beta }}=\sum _{{n{\mathbf {k}}}}f_{{n{\mathbf {k}}}}\langle \widetilde {\psi }_{{n{\mathbf {k}}}}|\widetilde {p}_{{\alpha }}\rangle \langle \widetilde {p}_{{\beta }}|\widetilde {\psi }_{{n{\mathbf {k}}}}\rangle ,

where the occupancy Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): f_{{n{\mathbf {k}}}} is one for occupied orbitals and zero for unoccupied one electron orbitals.

The compensation or augmentation density

The PAW method would yield exact overlap densities on the plane wave grid if the density were calculated as

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): n({\mathbf {r}})=\langle \widetilde {\psi }_{{n{\mathbf {k}}}}|{\mathbf {r}}\rangle \langle {\mathbf {r}}|\widetilde {\psi }_{{m{\mathbf {k}}}}\rangle +\sum _{{\alpha ,\beta }}(\phi _{\alpha }^{\ast }({\mathbf {r}})\phi _{\beta }({\mathbf {r}})-\widetilde {\phi }_{\alpha }^{\ast }({\mathbf {r}})\widetilde {\phi }_{\beta }({\mathbf {r}}))\langle \widetilde {\psi }_{{n{\mathbf {k}}}}|\widetilde {p}_{\alpha }\rangle \langle \widetilde {p}_{\beta }|\widetilde {\psi }_{{m{\mathbf {k}}}}\rangle

In practice, the second term changes far too rapidly in real space to be represented on a plane wave grid. Since even the norm of the pseudo-orbitals does not agree with the norm of the all-electron orbitals, it does not suffice to calculate Hartree or exchange energies from the pseudo densities only.

Hence, in order to treat the long-range electrostatic interactions in the Hartree and exchange term an additional quantity, the compensation density Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): \widehat {n} , is introduced. Its purpose is to approximate

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): Q_{{\alpha ,\beta }}({{\mathbf r}})=\phi _{\alpha }^{\ast }({\mathbf {r}})\phi _{\beta }({\mathbf {r}})-\widetilde {\phi }_{\alpha }^{\ast }({\mathbf {r}})\widetilde {\phi }_{\beta }({\mathbf {r}}).

This compensation density (sometimes also referred to as augmentation density) is chosen such that the sum of the pseudo charge density and the compensation density Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): \widetilde {n}^{{1}}+\widehat {n} has exactly the same moments as the exact density $n^{1}$ within each augmentation sphere centered at the position $\mathbf {R} _{\alpha }$. This requires that

$\int _{\Omega _{r}}[n^{1}(\mathbf {r} )-{\widetilde {n}}^{1}(\mathbf {r} )-{\widehat {n}}(\mathbf {r} )]|\mathbf {r} -\mathbf {R} _{\alpha }|^{L}Y_{LM}^{\ast }({\widehat {\mathbf {r} -\mathbf {R} _{\alpha }}})\,d\mathbf {r} =0\quad \forall \quad \mathbf {R} _{\alpha },L,M.$

This implies that the electrostatic potential originating from $n^{1}$ is identical to that of ${\widetilde {n}}^{1}+{\widehat {n}}$ outside the augmentation sphere. Details on the construction of the compensation charge density in the VASP program have been published elsewhere. The compensation charge density is written in the form of a one-center multipole expansion

${\widehat {n}}(\mathbf {r} )=\sum _{\alpha ,\beta ,LM}{\widehat {Q}}_{\alpha ,\beta }^{LM}(\mathbf {r} )\,\langle {\widetilde {\psi }}_{n\mathbf {k} }|{\widetilde {p}}_{\alpha }\rangle \langle {\widetilde {p}}_{\beta }|{\widetilde {\psi }}_{m\mathbf {k} }\rangle ,$

where the functions ${\widehat {Q}}_{\alpha \beta }^{LM}(\mathbf {r} )$ are given by

${\widehat {Q}}_{\alpha \beta }^{LM}(\mathbf {r} )=q_{\alpha \beta }^{LM}\,g_{L}(|\mathbf {r} -\mathbf {R} _{i}|)Y_{LM}({\widehat {\mathbf {r} -\mathbf {R} _{\alpha }}}).$

The moment $L$ of the function $g_{L}(r)$ is equal to 1. The quantity $q_{\alpha \beta }^{LM}$ is defined in Eq. (25) of Ref. .

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