Projector-augmented-wave formalism: Difference between revisions

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== Basics of the PAW formalism  ==
== Basics of the PAW formalism  ==


The PAW formalism is a generalization of ideas of both Vanderbilt-type~<ref name="vander:90"/>
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  
ultrasoft-pseudopotentials<ref name="kresshaf:94"/> (USPP) and the  
linearized augmented-plane-wave (LAPW) method~<ref name="andersen:75"/>. The method was first  
linearized augmented-plane-wave<ref name="andersen:75"/> (LAPW) method. The method was first  
proposed and implemented by Blöchl~<ref name="bloechl:94"/>. The formal relationship between Vanderbilt-type
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  
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
Joubert<ref name="kressjoub:99"/>, and the generalization of the PAW method to noncollinear
magnetism has been
magnetism has been
discussed by Hobbs, Kresse and Hafner~<ref name="hobbs:00"/>.  
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"/>  
We briefly summarize the basics of the PAW method below (following Refs. <ref name="bloechl:94"/>  
and <ref name="kressjoub:99"/>).  
and <ref name="kressjoub:99"/>).  
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linear transformation:
linear transformation:


<math>
::<math>
|\psi_{n\mathbf{k}} \rangle = |\widetilde{\psi}_{n\mathbf{k}} \rangle +
|\psi_{n\mathbf{k}} \rangle = |\widetilde{\psi}_{n\mathbf{k}} \rangle +
       \sum_{i}(|\phi_{i} \rangle - |\widetilde{\phi}_{i} \rangle)
       \sum_{i}(|\phi_{i} \rangle - |\widetilde{\phi}_{i} \rangle)
                   \langle \widetilde{p}_{i} |\widetilde{\psi}_{n\mathbf{k}} \rangle.
                   \langle \widetilde{p}_{i} |\widetilde{\psi}_{n\mathbf{k}} \rangle.
</math>
</math>


The pseudo orbitals  
The pseudo orbitals  
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and expanded in plane waves (see below). In the interstitial region between the PAW spheres,
and expanded in plane waves (see below). In the interstitial region between the PAW spheres,
the orbitals <math>\widetilde{\psi}_{n\mathbf{k}}</math>  are identical to the exact orbitals <math>{\psi}_{n\mathbf{k}}</math>.
the orbitals <math>\widetilde{\psi}_{n\mathbf{k}}</math>  are identical to the exact orbitals <math>{\psi}_{n\mathbf{k}}</math>.
Inside the spheres the pseudo orbitals are however only a computational
Inside the spheres, the pseudo-orbitals are however only a computational
tool and an inaccurate
tool and an inaccurate
approximation to the true orbitals, since even the
approximation to the true orbitals, since even the
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which is not an inherent characteristic of the PAW method, but has been made in all  
which is not an inherent characteristic of the PAW method, but has been made in all  
implementations so far.
implementations so far.
In the present case the core electrons are also kept frozen in the configuration for  
In the present case, the core electrons are also kept frozen in the configuration for  
which the PAW dataset was generated.
which the PAW dataset was generated.


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energy. The pseudo orbitals are expanded in the reciprocal space using plane waves
energy. The pseudo orbitals are expanded in the reciprocal space using plane waves


<math>
::<math>
\langle \mathbf{r} | \widetilde{\psi}_{n\mathbf{k}} \rangle =
\langle \mathbf{r} | \widetilde{\psi}_{n\mathbf{k}} \rangle =
     \frac{1}{\Omega^{1/2}} \sum_{\mathbf{G}} C_{n\mathbf{kG}}
     \frac{1}{\Omega^{1/2}} \sum_{\mathbf{G}} C_{n\mathbf{kG}}
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at a specific energy  <math>\varepsilon_\alpha</math> and for a specific angular momentum <math>l_\alpha</math>:
at a specific energy  <math>\varepsilon_\alpha</math> and for a specific angular momentum <math>l_\alpha</math>:


<math>
::<math>
  \langle \mathbf{r}|\phi_{\alpha}\rangle  = \frac{1}{|\mathbf{r}-\mathbf{R}_\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})
                   u_{\alpha}(|\mathbf{r}-\mathbf{R}_\alpha|)Y_{\alpha}(\widehat{\mathbf{r}-\mathbf{R}_\alpha})
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radius <math>r_{c}</math> and match continuously onto <math>\phi_{\alpha}</math> inside the core radius:
radius <math>r_{c}</math> and match continuously onto <math>\phi_{\alpha}</math> inside the core radius:


<math>
::<math>
  \langle \mathbf{r}|\widetilde{\phi}_{\alpha}\rangle  = \frac{1}{|\mathbf{r}-\mathbf{R}_\alpha|}
  \langle \mathbf{r}|\widetilde{\phi}_{\alpha}\rangle  = \frac{1}{|\mathbf{r}-\mathbf{R}_\alpha|}
                   \widetilde{u}_{\alpha}(|\mathbf{r}-\mathbf{R}_\alpha|)
                   \widetilde{u}_{\alpha}(|\mathbf{r}-\mathbf{R}_\alpha|)
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waves:
waves:


<math>
::<math>
\langle \widetilde{p}_{i} | \widetilde{\phi}_{j} \rangle = \delta_{ij}.
\langle \widetilde{p}_{i} | \widetilde{\phi}_{j} \rangle = \delta_{ij}.
</math>
</math>
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method, the total charge density (or more precisely the overlap density) related to two orbitals <math>\psi_{n\mathbf{k}}</math> and <math>\psi_{m\mathbf{k}}</math>  
method, the total charge density (or more precisely the overlap density) related to two orbitals <math>\psi_{n\mathbf{k}}</math> and <math>\psi_{m\mathbf{k}}</math>  


<math>
::<math>
n(\mathbf{r}) = \psi^{\ast}_{n\mathbf{k}}(\mathbf{r})\,\psi_{m\mathbf{k}}(\mathbf{r})
n(\mathbf{r}) = \psi^{\ast}_{n\mathbf{k}}(\mathbf{r})\,\psi_{m\mathbf{k}}(\mathbf{r})
</math>
</math>


can be rewritten as  (for details we refer to Ref.<ref name="bloechl:94"/>):
can be rewritten as  (for details we refer to Ref. <ref name="bloechl:94"/>):


<math>
::<math>
n(\mathbf{r}) =  \widetilde{n}  (\mathbf{r}) -
n(\mathbf{r}) =  \widetilde{n}  (\mathbf{r}) -
  \widetilde{n}^{1}(\mathbf{r})+
  \widetilde{n}^{1}(\mathbf{r})+
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Here, the constituent charge densities are defined as:
Here, the constituent charge densities are defined as:


<math>
::<math>
\widetilde{n}(\mathbf{r}) =  \langle \widetilde{\psi}_{n\mathbf{k}}| \mathbf{r}\rangle\langle  \mathbf{r}
\widetilde{n}(\mathbf{r}) =  \langle \widetilde{\psi}_{n\mathbf{k}}| \mathbf{r}\rangle\langle  \mathbf{r}
| \widetilde{\psi}_{m\mathbf{k}} \rangle
| \widetilde{\psi}_{m\mathbf{k}} \rangle
</math>
</math>


<math>
::<math>
\widetilde{n}^{1}(\mathbf{r}) =  \sum_{\alpha, \beta}
\widetilde{n}^{1}(\mathbf{r}) =  \sum_{\alpha, \beta}
                                       \widetilde{\phi}^\ast_\alpha(\mathbf{r})
                                       \widetilde{\phi}^\ast_\alpha(\mathbf{r})
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</math>
</math>


<math>
::<math>
n^{1}(\mathbf{r}) =  \sum_{\alpha, \beta}
n^{1}(\mathbf{r}) =  \sum_{\alpha, \beta}
                                       \phi^\ast_\alpha(\mathbf{r})
                                       \phi^\ast_\alpha(\mathbf{r})
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</math>
</math>


The quantities with a superscript 1 are one-centre quantities and
The quantities with a superscript 1 are one-center quantities and
are usually only evaluated on radial grids. Furthermore, one can usually
are usually only evaluated on radial grids. Furthermore, one can usually
drop the complex conjugation for the partial waves, since they are real valued.  
drop the complex conjugation for the partial waves, since they are real-valued.  
The indices <math>\alpha</math> and <math>\beta</math> are restricted to those pairs that correspond to one atom  
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>.
<math>\mathbf{R}_\alpha=\mathbf{R}_\beta</math>.
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applying the projector functions and summing over all bands
applying the projector functions and summing over all bands


<math>
::<math>
\rho_{\alpha\beta} = \sum_{n\mathbf{k}}
\rho_{\alpha\beta} = \sum_{n\mathbf{k}}
               f_{n\mathbf{k}} \langle \widetilde{\psi}_{n\mathbf{k}} | \widetilde{p}_{\alpha} \rangle
               f_{n\mathbf{k}} \langle \widetilde{\psi}_{n\mathbf{k}} | \widetilde{p}_{\alpha} \rangle
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if the density were calculated as
if the density were calculated as


<math>
::<math>
n(\mathbf{r})= \langle \widetilde{\psi}_{n\mathbf{k}}| \mathbf{r}\rangle\langle  \mathbf{r}
n(\mathbf{r})= \langle \widetilde{\psi}_{n\mathbf{k}}| \mathbf{r}\rangle\langle  \mathbf{r}
| \widetilde{\psi}_{m\mathbf{k}} \rangle + \sum_{\alpha, \beta}  
| \widetilde{\psi}_{m\mathbf{k}} \rangle + \sum_{\alpha, \beta}  
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Hartree or exchange energies from the pseudo densities only.
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  
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.  
an additional quantity, the  compensation density <math>\widehat{n}</math>, is introduced.  
It's purpose is to approximate  
Its purpose is to approximate  


<math>
::<math>
Q_{\alpha,\beta}({\mathbf r})    =      \phi^\ast_\alpha(\mathbf{r})
Q_{\alpha,\beta}({\mathbf r})    =      \phi^\ast_\alpha(\mathbf{r})
                                       \phi_\beta (\mathbf{r})
                                       \phi_\beta (\mathbf{r})
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<math>\mathbf{R}_\alpha</math>. This requires that  
<math>\mathbf{R}_\alpha</math>. This requires that  


<math>
::<math>
\int_{\Omega_{r}}[n^{1}(\mathbf{r}) -\widetilde{n}^{1}(\mathbf{r}) -
\int_{\Omega_{r}}[n^{1}(\mathbf{r}) -\widetilde{n}^{1}(\mathbf{r}) -
                       \widehat{n}(\mathbf{r})]|\mathbf{r}-\mathbf{R}_\alpha|^{L}
                       \widehat{n}(\mathbf{r})]|\mathbf{r}-\mathbf{R}_\alpha|^{L}
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Details on the construction of the compensation charge density in the VASP program  
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
have been published elsewhere<ref name="kressjoub:99"/>. The compensation charge density is written in the form
of a one-centre multipole expansion
of a one-center multipole expansion


<math>
::<math>
   \widehat{n}(\mathbf{r}) = \sum_{\alpha,\beta,LM} \widehat{Q}_{\alpha,\beta}^{LM}(\mathbf{r})\,
   \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{\psi}_{n\mathbf{k}} | \widetilde{p}_{\alpha} \rangle
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where the functions <math>\widehat{Q}_{\alpha\beta}^{LM}(\mathbf{r})</math> are given by
where the functions <math>\widehat{Q}_{\alpha\beta}^{LM}(\mathbf{r})</math> are given by
<math>
::<math>
\widehat{Q}_{\alpha \beta}^{LM}(\mathbf{r}) = q_{\alpha \beta}^{LM}\,g_{L}(|\mathbf{r}-\mathbf{R}_i|)
\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}).
Y_{LM}(\widehat{\mathbf{r}-\mathbf{R}_\alpha}).
Line 233: Line 232:
</references>
</references>
----
----
[[The_VASP_Manual|Contents]]


[[Category:Electronic Minimization]][[Category:PAW]][[Category:Theory]]
[[Category:Electronic minimization]][[Category:Projector-augmented-wave method]][[Category:Theory]]

Latest revision as of 07:00, 6 September 2024

Basics of the PAW formalism

The PAW formalism is a generalization of ideas of both Vanderbilt-type[1] ultrasoft-pseudopotentials[2] (USPP) and the linearized augmented-plane-wave[3] (LAPW) method. 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 noncollinear 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 , in the following simply called orbitals, are derived from the pseudo orbitals by means of a linear transformation:

The pseudo orbitals , where 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 are identical to the exact orbitals . 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 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 is a shorthand for the atomic site , the angular momentum quantum numbers and an additional index referring to the reference energy. The pseudo orbitals are expanded in the reciprocal space using plane waves

where is the volume of the Wigner-Seitz cell. The all-electron (AE) partial waves are solutions of the radial Schrödinger equation for a non-spinpolarized reference atom at a specific energy and for a specific angular momentum :

The notation is used to clarify that the spherical harmonics depends on the orientation but not on the length of the vector . Note that the radial component of the partial wave is independent of , 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 are equivalent to the AE partial waves outside a core radius and match continuously onto inside the core radius:

The core radius is usually chosen approximately around half the nearest neighbor distance. The projector functions are dual to the partial waves:

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 and

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

Here, the constituent charge densities are defined as:

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 and are restricted to those pairs that correspond to one atom . For a complete set of projectors the one-centre pseudo charge density is exactly identical to within the augmentation spheres. Furthermore, it is often necessary to define , the occupancies of each augmentation channel inside each PAW sphere. These are calculated from the pseudo orbitals applying the projector functions and summing over all bands

where the occupancy 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

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 , is introduced. Its purpose is to approximate

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 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-center 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