Charge sloshing - Revision history

Huebsch at 09:30, 17 October 2024

2024-10-17T09:30:42Z

← Older revision		Revision as of 09:30, 17 October 2024
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	To prevent charge sloshing {{VASP}} uses [[:Category:Density_mixing \| density mixing]] in the [[self-consistency cycle]].		To prevent charge sloshing {{VASP}} uses [[:Category:Density_mixing \| density mixing]] in the [[self-consistency cycle]].

			== Related tags and articles ==
			{{TAG\|ICHARG}}, {{TAG\|AMIX}}, {{TAG\|BMIX}}, {{TAG\|AMIX_MAG}}, {{TAG\|BMIX_MAG}}, [[Troubleshooting electronic convergence]]

	[[Category:Theory]][[Category:Electronic minimization]][[Category:Density mixing]]		[[Category:Theory]][[Category:Electronic minimization]][[Category:Density mixing]]

Vaspmaster at 07:08, 16 July 2024

2024-07-16T07:08:01Z

← Older revision		Revision as of 07:08, 16 July 2024
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	To prevent charge sloshing {{VASP}} uses [[:Category:Density_mixing \| density mixing]] in the [[self-consistency cycle]].		To prevent charge sloshing {{VASP}} uses [[:Category:Density_mixing \| density mixing]] in the [[self-consistency cycle]].

	[[Category:Theory]]		[[Category:Theory]][[Category:Electronic minimization]][[Category:Density mixing]]

Huebsch at 14:50, 15 July 2024

2024-07-15T14:50:14Z

← Older revision		Revision as of 14:50, 15 July 2024
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	To prevent charge sloshing {{VASP}} uses [[:Category:Density_mixing \| density mixing]] in the [[self-consistency cycle]].		To prevent charge sloshing {{VASP}} uses [[:Category:Density_mixing \| density mixing]] in the [[self-consistency cycle]].

			[[Category:Theory]]

Vaspmaster at 14:00, 15 July 2024

2024-07-15T14:00:28Z

← Older revision		Revision as of 14:00, 15 July 2024
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	To prevent charge sloshing {{VASP}} uses [[density mixing]] in the [[self-consistency cycle]].		To prevent charge sloshing {{VASP}} uses [[:Category:Density_mixing \| density mixing]] in the [[self-consistency cycle]].

Vaspmaster at 13:43, 15 July 2024

2024-07-15T13:43:35Z

← Older revision		Revision as of 13:43, 15 July 2024
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	* The aformentioned change in the orbitals leads to a ''long-wavelength'' change in the charge density:		* The aformentioned change in the orbitals leads to a ''long-wavelength'' change in the charge density:

	::<math>\rho({\bf r})=2 \Delta {\rm Re}\, e^{i 2 \delta{\bf k}\cdot{\bf r}}</math>		::<math>\delta\rho({\bf r})=2 \Delta {\rm Re}\, e^{i 2 \delta{\bf k}\cdot{\bf r}}</math>

	:and the consequent change in the Hartree potential due to this long-wavelength change in the density is given by:		:and the consequent change in the Hartree potential due to this long-wavelength change in the density is given by:

	::<math>V_{\rm H}({\bf r})=2 \Delta \frac{4 \pi e^2}{ \| 2\delta {\bf k}\|^2}{\rm Re} \,e^{i 2 \delta{\bf k}\cdot{\bf r}}.</math>		::<math>\delta V_{\rm H}({\bf r})=2 \Delta \frac{4 \pi e^2}{ \| 2\delta {\bf k}\|^2}{\rm Re} \,e^{i 2 \delta{\bf k}\cdot{\bf r}}.</math>

	:The factor <math>1/\|\delta {\bf k}\|^2</math> in the above is in fact the principal origin of charge sloshing: ''a long-wavelength change in the charge density leads to a strongly amplified change in the electrostatic potential.''		:The factor <math>1/\|\delta {\bf k}\|^2</math> in the above is in fact the principal origin of charge sloshing: ''a long-wavelength change in the charge density leads to a strongly amplified change in the electrostatic potential.''

Vaspmaster at 09:58, 15 July 2024

2024-07-15T09:58:11Z

← Older revision		Revision as of 09:58, 15 July 2024
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	:and thus the response in the potential increases with ''L''<sup>2</sup>, the square of the maximum extent of the supercell.		:and thus the response in the potential increases with ''L''<sup>2</sup>, the square of the maximum extent of the supercell.

	:Consequently, the maximum stable step size in a direct optimization algorithm decreases inversely proportional to the square of the maximum extent of the supercell (<math>\Delta = 1/L^2</math>).		:Consequently, the maximum stable step size in a [[Direct_optimization_of_the_orbitals \| direct optimization algorithm]] decreases inversely proportional to the square of the maximum extent of the supercell (<math>\Delta = 1/L^2</math>).

Vaspmaster at 09:56, 15 July 2024

2024-07-15T09:56:12Z

← Older revision		Revision as of 09:56, 15 July 2024
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	To prevent charge sloshing {{VASP}} uses [[density mixing]] in the self-consistency cycle.		To prevent charge sloshing {{VASP}} uses [[density mixing]] in the [[self-consistency cycle]].

Vaspmaster at 09:55, 15 July 2024

2024-07-15T09:55:17Z

← Older revision		Revision as of 09:55, 15 July 2024
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	To prevent charge sloshing {{VASP}} uses [[~~Category:Density_mixing~~ density mixing]] in the self-consistency cycle.		To prevent charge sloshing {{VASP}} uses [[density mixing]] in the self-consistency cycle.

Vaspmaster at 09:54, 15 July 2024

2024-07-15T09:54:45Z

← Older revision		Revision as of 09:54, 15 July 2024
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	To prevent charge sloshing {{VASP}} uses density mixing in the self-consistency cycle.		To prevent charge sloshing {{VASP}} uses [[Category:Density_mixing density mixing]] in the self-consistency cycle.

Vaspmaster: Created page with "Fig. 1: A handwaving illustration of charge sloshing. A very handwaving illustration of the phenomenon of charge sloshing during the electronic optimization is shown in Fig. 1: * Suppose we have a system with two chemically identical sites (Site 1 and 2), and suppose that in step ''N'' of the electronic optimization the situation is as follows: : At the beginning of iteration ''N'', Site 1 is occupied by 2 electrons, Site 2 is empty...."

2024-07-15T09:52:01Z

Created page with "600px|thumb|Fig. 1: A handwaving illustration of charge sloshing. A very handwaving illustration of the phenomenon of charge sloshing during the electronic optimization is shown in Fig. 1: * Suppose we have a system with two chemically identical sites (Site 1 and 2), and suppose that in step ''N'' of the electronic optimization the situation is as follows: : At the beginning of iteration ''N'', Site 1 is occupied by 2 electrons, Site 2 is empty...."

New page

[[File:sloshing1.png|600px|thumb|Fig. 1: A handwaving illustration of charge sloshing.]]

A very handwaving illustration of the phenomenon of charge sloshing during the electronic optimization is shown in Fig. 1:

* Suppose we have a system with two chemically identical sites (Site 1 and 2), and suppose that in step ''N'' of the electronic optimization the situation is as follows:
: At the beginning of iteration ''N'', Site 1 is occupied by 2 electrons, Site 2 is empty. After the refinement of the one-electron orbitals, however, the lowest eigenstate associated with Site 2 lies below the occupied state at Site 1.

* In step ''N+1'' the situation is reversed: the two electrons that were on Site 1 now are on Site 2, and after the refinement of the one-electron orbitals the lowest lying state at Site 1 will lie below the occupied state at Site 2.

* In subsequent steps the situation will bounce back-and-forth between the two aforementioned states, without ever reaching the "groundstate" depicted on the right in Fig. 1, where both Site 1 as well as Site 2 are occupied by a single electron.

A bit more mathematical explanation for the occurrence of charge sloshing goes as follows:

* Consider a metal, with "free electron"-like states at the Fermi-level, and consider a very large supercell, in which a single '''k'''-point suffices to sample the Brillouin zone, and hence all states are folded back to this point in the Brillouin zone.
:The state just below the Fermi-level is approximately given by <math>\psi_n = e^{i ({\bf k}_F-\delta {\bf k}){\bf r}}</math>, and the one above the Fermi-level by <math>\psi_m = e^{i ({\bf k}_F+\delta {\bf k}) {\bf r}}</math>. The former is occupied and the latter is not.

* During the electronic optimization, orbitals <math>\psi_n</math> and <math>\psi_m</math> will hybridize. This happens because the gradient of the total energy with respect to the orbitals has a so-called ''subspace rotational'' component:

::<math>
| s_n \rangle = \sum^N_{m=1} \frac{1}{2} {\bf H}_{nm} (f_n - f_m) \hat{S} \vert \psi_m \rangle
</math>

:where <math>\{ f_i | i=1,..,N \}</math> are the partial occupancies, and

::<math>
{\bf H}_{nm}=\langle \psi_m \vert \hat{H} \vert \psi_n \rangle
</math>

:is the Hamiltonian expressed within the subspace spanned by the current orbitals <math>\{ \psi_i | i=1,..,N \}</math>.
:A suitable search direction along the subspace rotational part of the gradient is given by a small unitary rotation between orbitals <math>\psi_n</math> and <math>\psi_m</math>:

::<math> |\psi'_n \rangle = | \psi_n \rangle + \Delta | \psi_m \rangle \qquad |\psi'_m \rangle = | \psi_m \rangle - \Delta | \psi_n \rangle </math>

:where <math>\Delta</math> denotes the stepsize.

* The aformentioned change in the orbitals leads to a ''long-wavelength'' change in the charge density:

::<math>\rho({\bf r})=2 \Delta {\rm Re}\, e^{i 2 \delta{\bf k}\cdot{\bf r}}</math>

:and the consequent change in the Hartree potential due to this long-wavelength change in the density is given by:

::<math>V_{\rm H}({\bf r})=2 \Delta \frac{4 \pi e^2}{ | 2\delta {\bf k}|^2}{\rm Re} \,e^{i 2 \delta{\bf k}\cdot{\bf r}}.</math>

:The factor <math>1/|\delta {\bf k}|^2</math> in the above is in fact the principal origin of charge sloshing: ''a long-wavelength change in the charge density leads to a strongly amplified change in the electrostatic potential.''

* In our example, the smallest possible <math>|\delta {\bf k}|</math> is proportional to ''L'', the maximum extent of the supercell:

::<math>|\delta {\bf k}| \propto 2 \pi / L </math>,

:and thus the response in the potential increases with ''L''<sup>2</sup>, the square of the maximum extent of the supercell.

:Consequently, the maximum stable step size in a direct optimization algorithm decreases inversely proportional to the square of the maximum extent of the supercell (<math>\Delta = 1/L^2</math>).

The reasoning above elucidates two important aspects of charge sloshing:

* Metals and systems with a small gap are more prone to charge sloshing than wide-gap insulators.

* Problems due to charge sloshing increase with increasing (super)cell size.

To prevent charge sloshing {{VASP}} uses density mixing in the self-consistency cycle.