Integrating over all orbitals - Revision history

Schlipf at 13:00, 28 February 2025

2025-02-28T13:00:30Z

Schlipf: Schlipf moved page K-point integration to Integrating over all orbitals: Preparation for refactoring

2025-02-28T12:59:56Z

Schlipf moved page K-point integration to Integrating over all orbitals: Preparation for refactoring

← Older revision	Revision as of 12:59, 28 February 2025
(No difference)

Schlipf: /* Improved functional form by Methfessel and Paxton */

2025-02-24T09:39:36Z

Improved functional form by Methfessel and Paxton

← Older revision		Revision as of 09:39, 24 February 2025
Line 109:		Line 109:
	The method described previously has two shortcomings:		The method described previously has two shortcomings:
	*The forces calculated by VASP are a derivative of the free electronic energy F (see {{TAG\|Forces}}). Therefore the forces can not be used to obtain the equilibrium groundstate, which corresponds to an energy-minimum of <math>E(\sigma \to 0)</math>. Nonetheless the error in the forces is generally small and acceptable.		*The forces calculated by VASP are a derivative of the free electronic energy F (see {{TAG\|Forces}}). Therefore the forces can not be used to obtain the equilibrium groundstate, which corresponds to an energy-minimum of <math>E(\sigma \to 0)</math>. Nonetheless the error in the forces is generally small and acceptable.
	*The parameter <math>\sigma</math> must be chosen with great care. If <math>\sigma</math> is too large the energy <math>E(\sigma \to 0)</math> will converge to the wrong value even for an infinite k-point mesh. If <math>\sigma</math> is too small the convergence speed with the number of k-points will deteriorate. An optimal choice for <math>\sigma</math> for several cases is given in table 1. Aluminium possesses an extremely simple DOS, Lithium and Tellurium are also simple nearly free electron metals, therefore <math>\sigma</math> might be large. For Copper <math>\sigma</math> is restricted by the fact that the d-band lies approximately 0.5 eV beneath the Fermi-level. Rhodium and Vanadium posses a fairly complex structure in the DOS at the Fermi-level, <math>\sigma</math> must be small. The only way to get a good math>\sigma</math> is by performing several calculations with different k-point meshes and different parameters for <math>\sigma</math>.		*The parameter <math>\sigma</math> must be chosen with great care. If <math>\sigma</math> is too large the energy <math>E(\sigma \to 0)</math> will converge to the wrong value even for an infinite k-point mesh. If <math>\sigma</math> is too small the convergence speed with the number of k-points will deteriorate. An optimal choice for <math>\sigma</math> for several cases is given in table 1. Aluminium possesses an extremely simple DOS, Lithium and Tellurium are also simple nearly free electron metals, therefore <math>\sigma</math> might be large. For Copper <math>\sigma</math> is restricted by the fact that the d-band lies approximately 0.5 eV beneath the Fermi-level. Rhodium and Vanadium posses a fairly complex structure in the DOS at the Fermi-level, <math>\sigma</math> must be small. The only way to get a good <math>\sigma</math> is by performing several calculations with different k-point meshes and different parameters for <math>\sigma</math>.

	{\| class="wikitable"		{\| class="wikitable"

Huebsch at 11:33, 21 October 2024

2024-10-21T11:33:53Z

← Older revision		Revision as of 11:33, 21 October 2024
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	<references/>		<references/>
	----		----
	[[Category:Electronic minimization]][[Category:~~Electronic ground-state properties~~]][[Category:Theory]]		[[Category:Electronic minimization]][[Category:Crystal momentum]][[Category:Theory]]

Schlipf at 15:30, 10 June 2024

2024-06-10T15:30:13Z

← Older revision		Revision as of 15:30, 10 June 2024
Line 165:		Line 165:
	<references/>		<references/>
	----		----
	[[Category:Electronic minimization]][[Category:~~Density of states~~]][[Category:Theory]]		[[Category:Electronic minimization]][[Category:Electronic ground-state properties]][[Category:Theory]]

Huebsch at 13:05, 6 April 2022

2022-04-06T13:05:22Z

← Older revision		Revision as of 13:05, 6 April 2022
Line 165:		Line 165:
	<references/>		<references/>
	----		----
	[[Category:Electronic ~~Minimization]][[Category:Electronic Minimization Methods~~]][[Category:Density of ~~States~~]][[Category:Theory]]		[[Category:Electronic minimization]][[Category:Density of states]][[Category:Theory]]

Ces at 13:15, 2 June 2021

2021-06-02T13:15:10Z

← Older revision		Revision as of 13:15, 2 June 2021
Line 11:		Line 11:
	<math>		<math>
	\sum_n \frac{1}{\Omega_{\mathrm{BZ}}} \int_{\Omega_{\mathrm{BZ}}}		\sum_n \frac{1}{\Omega_{\mathrm{BZ}}} \int_{\Omega_{\mathrm{BZ}}}
	\epsilon_{n\bold{k}} \, \Theta(\epsilon_{n\bold{k}}-\mu) \, d \bold{k}, ~~test~~		\epsilon_{n\bold{k}} \, \Theta(\epsilon_{n\bold{k}}-\mu) \, d \bold{k},
	</math>		</math>

Ces at 13:15, 2 June 2021

2021-06-02T13:15:00Z

← Older revision		Revision as of 13:15, 2 June 2021
Line 11:		Line 11:
	<math>		<math>
	\sum_n \frac{1}{\Omega_{\mathrm{BZ}}} \int_{\Omega_{\mathrm{BZ}}}		\sum_n \frac{1}{\Omega_{\mathrm{BZ}}} \int_{\Omega_{\mathrm{BZ}}}
	\epsilon_{n\bold{k}} \, \Theta(\epsilon_{n\bold{k}}-\mu) \, d \bold{k},		\epsilon_{n\bold{k}} \, \Theta(\epsilon_{n\bold{k}}-\mu) \, d \bold{k}, test
	</math>		</math>

Vaspmaster: /* Improved functional form by Methfessel and Paxton */

2019-09-23T17:56:55Z

Improved functional form by Methfessel and Paxton

← Older revision		Revision as of 17:56, 23 September 2019
Line 108:		Line 108:

	The method described previously has two shortcomings:		The method described previously has two shortcomings:
	*The forces calculated by VASP are a derivative of the free electronic energy F (see {{TAG\|Forces}}). Therefore the forces can not be used to		*The forces calculated by VASP are a derivative of the free electronic energy F (see {{TAG\|Forces}}). Therefore the forces can not be used to obtain the equilibrium groundstate, which corresponds to an energy-minimum of <math>E(\sigma \to 0)</math>. Nonetheless the error in the forces is generally small and acceptable.
	obtain the equilibrium groundstate, which corresponds to an energy-minimum of <math>E(\sigma \to 0)</math>. Nonetheless the error in the forces is generally small and acceptable.
	*The parameter <math>\sigma</math> must be chosen with great care. If <math>\sigma</math> is too large the energy <math>E(\sigma \to 0)</math> will converge to the wrong value even for an infinite k-point mesh. If <math>\sigma</math> is too small the convergence speed with the number of k-points will deteriorate. An optimal choice for <math>\sigma</math> for several cases is given in table 1. Aluminium possesses an extremely simple DOS, Lithium and Tellurium are also simple nearly free electron metals, therefore <math>\sigma</math> might be large. For Copper <math>\sigma</math> is restricted by the fact that the d-band lies approximately 0.5 eV beneath the Fermi-level. Rhodium and Vanadium posses a fairly complex structure in the DOS at the Fermi-level, <math>\sigma</math> must be small. The only way to get a good math>\sigma</math> is by performing several calculations with different k-point meshes and different parameters for <math>\sigma</math>.		*The parameter <math>\sigma</math> must be chosen with great care. If <math>\sigma</math> is too large the energy <math>E(\sigma \to 0)</math> will converge to the wrong value even for an infinite k-point mesh. If <math>\sigma</math> is too small the convergence speed with the number of k-points will deteriorate. An optimal choice for <math>\sigma</math> for several cases is given in table 1. Aluminium possesses an extremely simple DOS, Lithium and Tellurium are also simple nearly free electron metals, therefore <math>\sigma</math> might be large. For Copper <math>\sigma</math> is restricted by the fact that the d-band lies approximately 0.5 eV beneath the Fermi-level. Rhodium and Vanadium posses a fairly complex structure in the DOS at the Fermi-level, <math>\sigma</math> must be small. The only way to get a good math>\sigma</math> is by performing several calculations with different k-point meshes and different parameters for <math>\sigma</math>.

Karsai: /* Finite temperature approaches - smearing methods */

2019-05-06T17:50:26Z

Finite temperature approaches - smearing methods

← Older revision		Revision as of 17:50, 6 May 2019
Line 82:		Line 82:
	F =E - \sum_{n\bold{k}} w_{\bold{k}} \sigma S(f_{n\bold{k}}).		F =E - \sum_{n\bold{k}} w_{\bold{k}} \sigma S(f_{n\bold{k}}).
	</math>		</math>

	The calculated forces are now the derivatives of this free energy <math>F</math>		The calculated forces are now the derivatives of this free energy <math>F</math>
	(see {{TAG\|Forces}}).		(see {{TAG\|Forces}}).