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 | The PAW formalism is a generalization of ideas of both Vanderbilt-type<ref name="vander:90"/> | ||
ultrasoft-pseudopotentials | ultrasoft-pseudopotentials<ref name="kresshaf:94"/> (USPP) and the | ||
linearized augmented-plane-wave | linearized augmented-plane-wave<ref name="andersen:75"/> (LAPW) method. The method was first | ||
proposed and implemented by Blöchl | 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 | Joubert<ref name="kressjoub:99"/>, and the generalization of the PAW method to non collinear | ||
magnetism has been | magnetism has been | ||
discussed by Hobbs, Kresse and Hafner | 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"/>). | ||
In the PAW method the one electron wavefunctions <math>\psi_{ | In the PAW method the one electron wavefunctions <math>\psi_{n\mathbf{k}}</math>, in the following simply | ||
called orbitals, are | called orbitals, are | ||
derived from the pseudo orbitals <math>\widetilde{\psi}_{ | derived from the pseudo orbitals <math>\widetilde{\psi}_{n\mathbf{k}}</math> by means of a | ||
linear transformation: | linear transformation: | ||
<math> | ::<math> | ||
|\psi_{ | |\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}_{ | \langle \widetilde{p}_{i} |\widetilde{\psi}_{n\mathbf{k}} \rangle. | ||
</math> | </math> | ||
The pseudo orbitals | The pseudo orbitals | ||
<math>\widetilde{\psi}_{ | <math>\widetilde{\psi}_{n\mathbf{k}}</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, | and expanded in plane waves (see below). In the interstitial region between the PAW spheres, | ||
the orbitals <math>\widetilde{\psi}_{ | 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 | ||
norm of the all-electron wave function is not reproduced. | norm of the all-electron wave function is not reproduced. | ||
The last equation is required to map the auxiliary quantities <math>\widetilde{\psi}_{ | The last equation is required to map the auxiliary quantities <math>\widetilde{\psi}_{n\mathbf{k}}</math> | ||
onto the corresponding exact orbitals. | onto the corresponding exact orbitals. | ||
The PAW method implemented in VASP exploits the frozen core (FC) approximation, | 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 | 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}_{ | \langle \mathbf{r} | \widetilde{\psi}_{n\mathbf{k}} \rangle = | ||
\frac{1}{\Omega^{1/2}} \sum_{\mathbf{G}} C_{ | \frac{1}{\Omega^{1/2}} \sum_{\mathbf{G}} C_{n\mathbf{kG}} | ||
e^{i(\mathbf{G}+\mathbf{k})\cdot \mathbf{r}}, | e^{i(\mathbf{G}+\mathbf{k})\cdot \mathbf{r}}, | ||
</math> | </math> | ||
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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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Starting from the completeness relations it is possible to show that, in the PAW | 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 <math>\psi_{ | 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{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. | 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}_{ | \widetilde{n}(\mathbf{r}) = \langle \widetilde{\psi}_{n\mathbf{k}}| \mathbf{r}\rangle\langle \mathbf{r} | ||
| \widetilde{\psi}_{ | | \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}) | ||
\widetilde{\phi}_\beta (\mathbf{r}) | \widetilde{\phi}_\beta (\mathbf{r}) | ||
\langle\widetilde{\psi}_{ | \langle\widetilde{\psi}_{n\mathbf{k}}|\widetilde{p}_\alpha\rangle | ||
\langle\widetilde{p}_\beta| \widetilde{\psi}_{ | \langle\widetilde{p}_\beta| \widetilde{\psi}_{m\mathbf{k}}\rangle | ||
</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}) | ||
\phi_\beta (\mathbf{r}) | \phi_\beta (\mathbf{r}) | ||
\langle\widetilde{\psi}_{ | \langle\widetilde{\psi}_{n\mathbf{k}}|\widetilde{p}_\alpha\rangle | ||
\langle\widetilde{p}_\beta| \widetilde{\psi}_{ | \langle\widetilde{p}_\beta| \widetilde{\psi}_{m\mathbf{k}}\rangle. | ||
</math> | </math> | ||
The quantities with a superscript 1 are one- | 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 | 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_{ | \rho_{\alpha\beta} = \sum_{n\mathbf{k}} | ||
f_{ | f_{n\mathbf{k}} \langle \widetilde{\psi}_{n\mathbf{k}} | \widetilde{p}_{\alpha} \rangle | ||
\langle \widetilde{p}_{\beta} | \widetilde{\psi}_{ | \langle \widetilde{p}_{\beta} | \widetilde{\psi}_{n\mathbf{k}} \rangle, | ||
</math> | </math> | ||
where the occupancy <math>f_{ | where the occupancy <math>f_{n\mathbf{k}}</math> is one for occupied orbitals | ||
and zero for unoccupied one electron orbitals. | and zero for unoccupied one electron orbitals. | ||
== The compensation density == | == The compensation or augmentation density == | ||
The PAW method would yield exact overlap densities on the plane wave grid | The PAW method would yield exact overlap densities on the plane wave grid | ||
if the density were calculated as | if the density were calculated as | ||
<math> | ::<math> | ||
n(\mathbf{r})= \langle \widetilde{\psi}_{ | n(\mathbf{r})= \langle \widetilde{\psi}_{n\mathbf{k}}| \mathbf{r}\rangle\langle \mathbf{r} | ||
| \widetilde{\psi}_{ | | \widetilde{\psi}_{m\mathbf{k}} \rangle + \sum_{\alpha, \beta} | ||
( | ( | ||
\phi^\ast_\alpha(\mathbf{r}) | \phi^\ast_\alpha(\mathbf{r}) | ||
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\widetilde{\phi}_\beta (\mathbf{r}) | \widetilde{\phi}_\beta (\mathbf{r}) | ||
) | ) | ||
\langle\widetilde{\psi}_{ | \langle\widetilde{\psi}_{n\mathbf{k}}|\widetilde{p}_\alpha\rangle | ||
\langle\widetilde{p}_\beta| \widetilde{\psi}_{ | \langle\widetilde{p}_\beta| \widetilde{\psi}_{m\mathbf{k}}\rangle | ||
</math> | </math> | ||
In practice, the second term changes far | In practice, the second term changes far too rapidly in real space | ||
to be represented on a plane wave grid. Since even the norm | 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 | 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. | 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. | ||
Its purpose is to approximate | |||
<math> | ::<math> | ||
Q_{\alpha,\beta}({\mathbf r}) = \phi^\ast_\alpha(\mathbf{r}) | |||
\phi_\beta (\mathbf{r}) | \phi_\beta (\mathbf{r}) | ||
- | - | ||
\widetilde{\phi}^\ast_\alpha(\mathbf{r}) | \widetilde{\phi}^\ast_\alpha(\mathbf{r}) | ||
\widetilde{\phi}_\beta (\mathbf{r}) | \widetilde{\phi}_\beta (\mathbf{r}). | ||
</math> | </math> | ||
This compensation density is chosen such that the sum of the pseudo charge density | 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 | and the compensation density | ||
<math>\widetilde{n}^{1} + \widehat{n}</math> has exactly the same moments as the exact | <math>\widetilde{n}^{1} + \widehat{n}</math> has exactly the same moments as the exact | ||
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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} | ||
Y_{LM}^{\ast}(\widehat{\mathbf{r}-\mathbf{R}_\alpha})\,d\mathbf{r} = 0 | Y_{LM}^{\ast}(\widehat{\mathbf{r}-\mathbf{R}_\alpha})\,d\mathbf{r} = 0 | ||
\forall \mathbf{R}_\alpha, L, M. | \quad \forall \quad \mathbf{R}_\alpha, L, M. | ||
</math> | </math> | ||
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the augmentation sphere. | the augmentation sphere. | ||
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 | have been published elsewhere<ref name="kressjoub:99"/>. The compensation charge density is written in the form | ||
of a one- | 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}_{ | \langle \widetilde{\psi}_{n\mathbf{k}} | \widetilde{p}_{\alpha} \rangle | ||
\langle \widetilde{p}_{\beta} | \widetilde{\psi}_{ | \langle \widetilde{p}_{\beta} | \widetilde{\psi}_{m\mathbf{k}} \rangle, | ||
</math> | </math> | ||
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}). | ||
</math> | </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> | 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. | is defined in Eq. (25) of Ref. <ref name="kressjoub:99"/>. | ||
== References == | == References == | ||
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<ref name="hobbs:00">[https://doi.org/10.1103/PhysRevB.62.11556 D. Hobbs, G. Kresse, and J. Hafner, Phys. Rev. B. 62 (2000). ]</ref> | <ref name="hobbs:00">[https://doi.org/10.1103/PhysRevB.62.11556 D. Hobbs, G. Kresse, and J. Hafner, Phys. Rev. B. 62 (2000). ]</ref> | ||
</references> | </references> | ||
---- | |||
[[Category:Electronic minimization]][[Category:Projector-augmented-wave method]][[Category:Theory]] |
Revision as of 08:32, 19 October 2023
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 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 , 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
- ↑ D. Vanderbilt, Phys. Rev. B 41, 7892 (1990).
- ↑ G. Kresse, and J. Hafner, J. Phys.: Condens. Matter 6, 8245 (1994).
- ↑ O.K. Andersen, Phys. Rev. B 12, 3060 (1975).
- ↑ a b c P.E. Blöchl, Phys. Rev. B 50, 17953 (1994).
- ↑ a b c d G. Kresse, and D. Joubert, Phys. Rev. B 59, 1758 (1999).
- ↑ D. Hobbs, G. Kresse, and J. Hafner, Phys. Rev. B. 62 (2000).