Projector-augmented-wave formalism: Difference between revisions

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== Basics of the PAW formalism  ==
 
The PAW formalism is a generalization of ideas of both Vanderbilt-type~<ref name="vander:90"/>
ultrasoft-pseudopotentials (USPP)~<ref name="kresshaf:94"/> and the
linearized augmented-plane-wave (LAPW) method~<ref name="andersen:75"/>. The method was first
proposed and implemented by Blöchl~<ref name="bloechl:94"/>. The formal relationship between Vanderbilt-type
ultrasoft pseudopotentials and the PAW method has been derived by Kresse and
Joubert~<ref name="kressjoub:99"/>, and the generalization of the PAW method to non collinear
magnetism has been
discussed by Hobbs, Kresse and Hafner~<ref name="hobbs:00"/>.
We briefly summarize the basics of the PAW method below (following Refs. <ref name="bloechl:94"/>
and <ref name="kressjoub:99"/>).
 
In the PAW method the one electron wavefunctions <math>\psi_{nk}</math>, in the following simply
called orbitals, are
derived from the pseudo orbitals <math>\widetilde{\psi}_{nk}</math> by means of a
linear transformation:
 
<math>
|\psi_{nk} \rangle = |\widetilde{\psi}_{nk} \rangle +
      \sum_{i}(|\phi_{i} \rangle - |\widetilde{\phi}_{i} \rangle)
                  \langle \widetilde{p}_{i} |\widetilde{\psi}_{nk} \rangle.
</math>
 
 
The pseudo orbitals
<math>\widetilde{\psi}_{nk}</math>, where <math>nk</math> 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 <math>\widetilde{\psi}_{nk}</math>  are identical to the exact orbitals <math>{\psi}_{nk}</math>.
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 <math>\widetilde{\psi}_{nk}</math>
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 <math>\alpha</math> is a shorthand for the atomic site <math>\mathbf{R}_\alpha</math>, the angular momentum
quantum numbers <math>l_\alpha,m_\alpha</math> and an additional index <math>\varepsilon_\alpha</math> referring to
the reference
energy. The pseudo orbitals are expanded in the reciprocal space using plane waves
 
<math>
\langle \mathbf{r} | \widetilde{\psi}_{nk} \rangle =
    \frac{1}{\Omega^{1/2}} \sum_{\mathbf{G}} C_{a\mathbf{G}}(\mathbf{r})
                                                  e^{i(\mathbf{G}+\mathbf{k})\cdot \mathbf{r}},
</math>
 
where <math>\Omega</math> is the volume of the Wigner-Seitz cell. The all-electron (AE)
partial waves
<math>\phi_{\alpha}</math> are solutions of the radial Schrödinger equation for a
non-spinpolarized reference atom
at a specific energy  <math>\varepsilon_\alpha</math> and for a specific angular momentum <math>l_\alpha</math>:
 
<math>
\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}).
</math>
 
The notation <math>\widehat{\mathbf{r}-\mathbf{R}_\alpha}</math> is used to clarify that
the spherical harmonics <math>Y</math> depends on the orientation but not on
the length of the vector <math>\mathbf{r}-\mathbf{R}_\alpha</math>.
Note that the radial component of the partial wave <math>u_{\alpha}</math> is independent of
<math>m_\alpha</math>, 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 <math>\widetilde{\phi}_{\alpha}</math> are equivalent to the AE partial waves outside a core
radius <math>r_{c}</math> and match continuously onto <math>\phi_{\alpha}</math> inside the core radius:
 
<math>
\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}).
</math>
 
The core radius <math>r_{c}</math> is usually chosen approximately around half the nearest
neighbor distance. The projector functions <math>\widetilde{p}_{\alpha}</math> are dual to the partial
waves:
 
<math>
\langle \widetilde{p}_{i} | \widetilde{\phi}_{j} \rangle = \delta_{ij}.
</math>
 
 
== 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 overlapdensity) related to two orbitals <math>nk</math> and <math>mk</math>
 
<math>
n(\mathbf{r}) = \psi^{\ast}_{nk}(\mathbf{r})\,\psi_{mk}(\mathbf{r})
</math>
 
can be rewritten as  (for details we refer to Ref.~<ref name="bloechl:94"/>):
 
<math>
n(\mathbf{r}) =  \widetilde{n}  (\mathbf{r}) -
\widetilde{n}^{1}(\mathbf{r})+
n^{1}(\mathbf{r}).
</math>
 
Here, the constituent charge densities are defined as:
 
<math>
\widetilde{n}(\mathbf{r}) =  \langle \widetilde{\psi}_{nk}| \mathbf{r}\rangle\langle  \mathbf{r}
| \widetilde{\psi}_{mk} \rangle
</math>
 
<math>
\widetilde{n}^{1}(\mathbf{r}) =  \sum_{\alpha, \beta}
                                      \widetilde{\phi}^\ast_\alpha(\mathbf{r})
                                      \widetilde{\phi}_\beta (\mathbf{r})
                                      \langle\widetilde{\psi}_{nk}|\widetilde{p}_\alpha\rangle
                                      \langle\widetilde{p}_\beta| \widetilde{\psi}_{mk}\rangle
</math>
 
<math>
n^{1}(\mathbf{r}) =  \sum_{\alpha, \beta}
                                      \phi^\ast_\alpha(\mathbf{r})
                                      \phi_\beta (\mathbf{r})
                                      \langle\widetilde{\psi}_{nk}|\widetilde{p}_\alpha\rangle
                                      \langle\widetilde{p}_\beta| \widetilde{\psi}_{mk}\rangle.
</math>
 
The quantities with a superscript 1 are one-centre 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 values.
The indices <math>\alpha</math> and <math>\beta</math> are restricted to those pairs that correspond to one atom
<math>\mathbf{R}_\alpha=\mathbf{R}_\beta</math>.
For a complete set of projectors the one-centre pseudo
charge density <math>\widetilde{n}^{1}</math> is exactly
identical to <math>\widetilde{n}</math> within the augmentation spheres.
Furthermore, it is often necessary to define <math>\rho_{\alpha\beta}</math>, the occupancies of each
augmentation channel <math>(\alpha,\beta)</math> inside each PAW sphere. These  are calculated from the pseudo orbitals
applying the projector functions and summing over all bands
 
<math>
\rho_{\alpha\beta} = \sum_{nk}
              f_{nk} \langle \widetilde{\psi}_{nk} | \widetilde{p}_{\alpha} \rangle
                  \langle \widetilde{p}_{\beta} | \widetilde{\psi}_{nk} \rangle,
</math>
 
where the occupancy <math>f_{nk}</math> is one for occupied orbitals
and zero for unoccupied one electron orbitals.
 
== The compensation density ==
 
The PAW method would yield exact overlap densities on the plane wave grid
if the density were calculated as
 
<math>
n(\mathbf{r})= \langle \widetilde{\psi}_{nk}| \mathbf{r}\rangle\langle  \mathbf{r}
| \widetilde{\psi}_{mk} \rangle + \sum_{\alpha, \beta}
(
                                      \phi^\ast_\alpha(\mathbf{r})
                                      \phi_\beta (\mathbf{r})
-
                                      \widetilde{\phi}^\ast_\alpha(\mathbf{r})
                                      \widetilde{\phi}_\beta (\mathbf{r})
)
                                      \langle\widetilde{\psi}_{nk}|\widetilde{p}_\alpha\rangle
                                      \langle\widetilde{p}_\beta| \widetilde{\psi}_{mk}\rangle
</math>
 
In practice, the second term changes far to rapidly in real space
to be represented on a plane wave grid. Since even to norm
of the pseudo-orbitals is not right, it does not suffice to calculate
electrostatic 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 <math>\widehat{n}</math>, is introduced.
It's purpose is to approximate
 
<math>
                                      \phi^\ast_\alpha(\mathbf{r})
                                      \phi_\beta (\mathbf{r})
-
                                      \widetilde{\phi}^\ast_\alpha(\mathbf{r})
                                      \widetilde{\phi}_\beta (\mathbf{r})
</math>
 
This compensation density is chosen such that the sum of the pseudo charge density
and the compensation density
<math>\widetilde{n}^{1} + \widehat{n}</math> has exactly the same moments as the exact
density <math>n^{1}</math> within each augmentation sphere centered at the position
<math>\mathbf{R}_\alpha</math>. This requires that
 
<math>
\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
 
\forall \mathbf{R}_\alpha, L, M.
</math>
 
This implies that the electrostatic potential originating from <math>n^{1}</math>
is identical to that of <math>\widetilde{n}^{1}+\widehat{n}</math> outside
the augmentation sphere.
Details on the construction of the compensation charge density in the VASP program
have been published elsewhere~<ref name="kressjoub:99"/>. The compensation charge density is written in the form
of a one-centre multipole expansion
 
<math>
  \widehat{n}(\mathbf{r}) = \sum_{\alpha,\beta,LM} \widehat{Q}_{\alpha,\beta}^{LM}(\mathbf{r})\,
              \langle \widetilde{\psi}_{nk} | \widetilde{p}_{\alpha} \rangle
              \langle \widetilde{p}_{\beta} | \widetilde{\psi}_{mk} \rangle,
</math>
 
where the functions <math>\widehat{Q}_{\alpha\beta}^{LM}(\mathbf{r})</math> are given by
<math>
\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}).
</math>
The moment <math>L</math> of the function  <math>g_L(r)</math> is equal to 1. The quantity <math>q_{\alpha\beta}^{LM}</math>
is defined in Eq. (25) of Ref.~<ref name="kressjoub:99"/>.
 
 
== References ==
<references>
<ref name="vander:90">[http://journals.aps.org/prb/abstract/10.1103/PhysRevB.78.121201 D. Vanderbilt, Phys. Rev. B 41, 7892 (1990).]</ref>
<ref name="kresshaf:94">[http://journals.aps.org/prb/abstract/10.1103/PhysRevB.81.115126 G. Kresse, and J. Hafner, J. Phys.: Condens. Matter 6, 8245 (1994). ]</ref>
<ref name="andersen:75">[http://journals.aps.org/prb/abstract/10.1103/PhysRevB.77.045136 O.K. Andersen, Phys. Rev. B 12, 3060 (1975). ]</ref>
<ref name="bloechl:94">[http://journals.aps.org/prb/abstract/10.1103/PhysRevB.90.075125 P.E. Blöchl, Phys. Rev. B 50, 17953 (1994). ]</ref>
<ref name="kressjoub:99">[http://pubs.acs.org/doi/abs/10.1021/ct5001268 G. Kresse, and D. Joubert, Phys. Rev. B  59, 1758 (1999). ]</ref>
<ref name="hobbs:00">[http://pubs.acs.org/doi/abs/10.1021/ct5001268 D. Hobbs, G. Kresse, and J. Hafner, Phys. Rev. B. 62 (2000). ]</ref>
</references>

Revision as of 17:16, 15 March 2017

Basics of the PAW formalism

The PAW formalism is a generalization of ideas of both Vanderbilt-type~[1] ultrasoft-pseudopotentials (USPP)~[2] and the linearized augmented-plane-wave (LAPW) method~[3]. The method was first proposed and implemented by Blöchl~[4]. The formal relationship between Vanderbilt-type ultrasoft pseudopotentials and the PAW method has been derived by Kresse and Joubert~[5], and the generalization of the PAW method to non collinear magnetism has been discussed by Hobbs, Kresse and Hafner~[6]. We briefly summarize the basics of the PAW method below (following Refs. [4] and [5]).

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 _{{nk}} , 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 }_{{nk}} 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 _{{nk}}\rangle =|\widetilde {\psi }_{{nk}}\rangle +\sum _{{i}}(|\phi _{{i}}\rangle -|\widetilde {\phi }_{{i}}\rangle )\langle \widetilde {p}_{{i}}|\widetilde {\psi }_{{nk}}\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 }_{{nk}} , 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 }_{{nk}} 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 }_{{nk}} . 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 }_{{nk}} 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 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 } , the angular momentum quantum numbers 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/":): l_{\alpha },m_{\alpha } and an additional 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/":): \varepsilon _{\alpha } referring to the reference energy. The pseudo orbitals are expanded in the reciprocal space using plane 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 {\mathbf {r}}|\widetilde {\psi }_{{nk}}\rangle ={\frac {1}{\Omega ^{{1/2}}}}\sum _{{{\mathbf {G}}}}C_{{a{\mathbf {G}}}}({\mathbf {r}})e^{{i({\mathbf {G}}+{\mathbf {k}})\cdot {\mathbf {r}}}},

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

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}}|\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 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 {{\mathbf {r}}-{\mathbf {R}}_{\alpha }} is used to clarify that the spherical harmonics 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/":): Y depends on the orientation but not on the length of the vector 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}}-{\mathbf {R}}_{\alpha } . Note that the radial component of the partial wave 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/":): u_{{\alpha }} is independent of 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/":): 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 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 {\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 overlapdensity) 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/":): nk 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/":): mk

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 _{{nk}}^{{\ast }}({\mathbf {r}})\,\psi _{{mk}}({\mathbf {r}})

can be rewritten as (for details we refer to Ref.~[4]):

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 }_{{nk}}|{\mathbf {r}}\rangle \langle {\mathbf {r}}|\widetilde {\psi }_{{mk}}\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 }_{{nk}}|\widetilde {p}_{\alpha }\rangle \langle \widetilde {p}_{\beta }|\widetilde {\psi }_{{mk}}\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 }_{{nk}}|\widetilde {p}_{\alpha }\rangle \langle \widetilde {p}_{\beta }|\widetilde {\psi }_{{mk}}\rangle .

The quantities with a superscript 1 are one-centre 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 values. 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 _{{nk}}f_{{nk}}\langle \widetilde {\psi }_{{nk}}|\widetilde {p}_{{\alpha }}\rangle \langle \widetilde {p}_{{\beta }}|\widetilde {\psi }_{{nk}}\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_{{nk}} is one for occupied orbitals and zero for unoccupied one electron orbitals.

The compensation 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 }_{{nk}}|{\mathbf {r}}\rangle \langle {\mathbf {r}}|\widetilde {\psi }_{{mk}}\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 }_{{nk}}|\widetilde {p}_{\alpha }\rangle \langle \widetilde {p}_{\beta }|\widetilde {\psi }_{{mk}}\rangle

In practice, the second term changes far to rapidly in real space to be represented on a plane wave grid. Since even to norm of the pseudo-orbitals is not right, it does not suffice to calculate electrostatic 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. It's 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/":): \phi _{\alpha }^{\ast }({\mathbf {r}})\phi _{\beta }({\mathbf {r}})-\widetilde {\phi }_{\alpha }^{\ast }({\mathbf {r}})\widetilde {\phi }_{\beta }({\mathbf {r}})

This compensation 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 within each augmentation sphere centered at the position . This requires that

This implies that the electrostatic potential originating from is identical to that of outside the augmentation sphere. Details on the construction of the compensation charge density in the VASP program have been published elsewhere~[5]. The compensation charge density is written in the form of a one-centre multipole expansion

where the functions are given by The moment of the function is equal to 1. The quantity is defined in Eq. (25) of Ref.~[5].


References

  1. ↑ D. Vanderbilt, Phys. Rev. B 41, 7892 (1990).
  2. ↑ G. Kresse, and J. Hafner, J. Phys.: Condens. Matter 6, 8245 (1994).
  3. ↑ O.K. Andersen, Phys. Rev. B 12, 3060 (1975).
  4. ↑ a b c P.E. Blöchl, Phys. Rev. B 50, 17953 (1994).
  5. ↑ a b c d G. Kresse, and D. Joubert, Phys. Rev. B 59, 1758 (1999).
  6. ↑ D. Hobbs, G. Kresse, and J. Hafner, Phys. Rev. B. 62 (2000).
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