LMAXFOCK: Difference between revisions
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Description: {{TAG|LMAXFOCK}} sets the maximum angular momentum quantum number ''l'' for the augmentation of charge densities in Hartree-Fock type routines. | Description: {{TAG|LMAXFOCK}} sets the maximum angular momentum quantum number ''l'' for the augmentation of charge densities in Hartree-Fock type routines. | ||
---- | ---- | ||
In the PAW method, the difference between the charge density of the all-electron partial waves and | In the PAW method, the difference between the charge density of the all-electron partial waves <math>\phi_\beta</math> and | ||
the pseudo partial waves | the pseudo partial waves <math>\tilde \phi_\beta</math> | ||
<math> | <math> | ||
Q_{\alpha\beta}(r)= \phi^*_\alpha(r)\phi_\beta(r) - \tilde \phi^*_\alpha(r)\tilde \phi_\beta(r) | Q_{\alpha\beta}(r)= \phi^*_\alpha(r)\phi_\beta(r) - \tilde \phi^*_\alpha(r)\tilde \phi_\beta(r) | ||
</math> | </math> | ||
is usually restored on spherical grids centered at each atom | is usually restored on spherical grids centered at each atom | ||
(one-center terms inside the PAW spheres). To describe long range electrostatic effect, the ''moments'' of the differences of the all-electron and pseudo charge density | (one-center terms inside the PAW spheres). To describe long range electrostatic effect, the ''moments'' of the differences of the all-electron and pseudo charge density | ||
also need to added on the plane wave grid. This is done up to a certain ''l'' quantum number. | also need to added on the plane wave grid. This is done up to a certain ''l'' quantum number. | ||
These augmentation charges restore the moments of the all-electron density on the plane wave | These augmentation charges exactly restore the moments of the all-electron density on the plane wave | ||
grid. For the charge | grid. For the charge densities used in the Hartree and DFT term, | ||
''l'' quantum number required | this augmentation is done exactly up to the maximum ''l'' quantum number required by the POTCAR files, | ||
the augmentation is controlled by {{TAG|LMAXFOCK}}. | whereas for the Fock exchange, the augmentation on the plane wave grid is controlled by {{TAG|LMAXFOCK}}. | ||
Specifically, the exchange | Specifically, when the exchange energy is evaluated | ||
<span> | |||
<math> | <span id="ExFock"> | ||
:<math> | |||
= -\frac{e^2}{2}\sum_{\mathbf{q}m}f_{\mathbf{q}m} | E_{\mathrm{x}}= -\frac{e^2}{2}\sum_{\mathbf{k}n,\mathbf{q}m} | ||
f_{\mathbf{k}n} f_{\mathbf{q}m} \times | |||
\frac{ | \int\int d^3\mathbf{r} d^3\mathbf{r}' | ||
\frac{\psi_{\mathbf{k}n}^{*}(\mathbf{r})\psi_{\mathbf{q}m}^{*}(\mathbf{r}') | |||
\psi_{\mathbf{k}n}(\mathbf{r}')\psi_{\mathbf{q}m}(\mathbf{r})} | |||
{\vert \mathbf{r}-\mathbf{r}' \vert} | {\vert \mathbf{r}-\mathbf{r}' \vert} | ||
</math> | </math> | ||
</span> | </span> | ||
the overlap density <math> \psi_{\mathbf{k}n}^{*}(\mathbf{r})\psi_{\mathbf{q}m}(\mathbf{r}) </math> | |||
the | between two Bloch orbitals needs to be calculated on the plane wave grid. The tag {{TAG|LMAXFOCK}} | ||
To | controls up to which ''l'' quantum number, augmentation charges are added on the plane wave grid. | ||
To accelerate convergence with respect to {{TAG|LMAXFOCK}}, VASP uses a small trick: the contributions from the Hartree-Fock one-center terms are evaluated for the pseudo orbitals also only up to ''l''={{TAG|LMAXFOCK}}, whereas the one-center terms for the exact all-electron orbitals are evaluated up to the maximum required ''l'' (twice the angular quantum number of the partial wave with the highest ''l''). The default is {{TAG|LMAXFOCK}}=4, and it might be necessary to increase this parameter, if the system contains f-electrons. Since this increases the computational load considerably (factor 2), it is recommended to perform tests, whether the results are already reasonably converged using the default {{TAG|LMAXFOCK}}=4. | |||
To be compatible w.r.t. old releases, VASP also reads the flag {{TAG|HFLMAX}} to the same effect as {{TAG|LMAXFOCK}}. | To be compatible w.r.t. old releases, VASP also reads the flag {{TAG|HFLMAX}} to the same effect as {{TAG|LMAXFOCK}}. | ||
Revision as of 16:18, 14 March 2017
LMAXFOCK = [integer]
Default: LMAXFOCK = 4
Description: LMAXFOCK sets the maximum angular momentum quantum number l for the augmentation of charge densities in Hartree-Fock type routines.
In the PAW method, the difference between the charge density of the all-electron partial waves [math]\displaystyle{ \phi_\beta }[/math] and the pseudo partial waves [math]\displaystyle{ \tilde \phi_\beta }[/math] [math]\displaystyle{ Q_{\alpha\beta}(r)= \phi^*_\alpha(r)\phi_\beta(r) - \tilde \phi^*_\alpha(r)\tilde \phi_\beta(r) }[/math] is usually restored on spherical grids centered at each atom (one-center terms inside the PAW spheres). To describe long range electrostatic effect, the moments of the differences of the all-electron and pseudo charge density also need to added on the plane wave grid. This is done up to a certain l quantum number. These augmentation charges exactly restore the moments of the all-electron density on the plane wave grid. For the charge densities used in the Hartree and DFT term, this augmentation is done exactly up to the maximum l quantum number required by the POTCAR files, whereas for the Fock exchange, the augmentation on the plane wave grid is controlled by LMAXFOCK.
Specifically, when the exchange energy is evaluated
- [math]\displaystyle{ E_{\mathrm{x}}= -\frac{e^2}{2}\sum_{\mathbf{k}n,\mathbf{q}m} f_{\mathbf{k}n} f_{\mathbf{q}m} \times \int\int d^3\mathbf{r} d^3\mathbf{r}' \frac{\psi_{\mathbf{k}n}^{*}(\mathbf{r})\psi_{\mathbf{q}m}^{*}(\mathbf{r}') \psi_{\mathbf{k}n}(\mathbf{r}')\psi_{\mathbf{q}m}(\mathbf{r})} {\vert \mathbf{r}-\mathbf{r}' \vert} }[/math]
the overlap density [math]\displaystyle{ \psi_{\mathbf{k}n}^{*}(\mathbf{r})\psi_{\mathbf{q}m}(\mathbf{r}) }[/math] between two Bloch orbitals needs to be calculated on the plane wave grid. The tag LMAXFOCK controls up to which l quantum number, augmentation charges are added on the plane wave grid. To accelerate convergence with respect to LMAXFOCK, VASP uses a small trick: the contributions from the Hartree-Fock one-center terms are evaluated for the pseudo orbitals also only up to l=LMAXFOCK, whereas the one-center terms for the exact all-electron orbitals are evaluated up to the maximum required l (twice the angular quantum number of the partial wave with the highest l). The default is LMAXFOCK=4, and it might be necessary to increase this parameter, if the system contains f-electrons. Since this increases the computational load considerably (factor 2), it is recommended to perform tests, whether the results are already reasonably converged using the default LMAXFOCK=4.
To be compatible w.r.t. old releases, VASP also reads the flag HFLMAX to the same effect as LMAXFOCK.