Category:Forces - Revision history

Huebsch: /* DFT Forces */

2024-06-30T11:13:07Z

DFT Forces

← Older revision		Revision as of 11:13, 30 June 2024
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	</math>		</math>

	where <math> \nabla_{A} </math> denotes the gradient with respect to ionic position <math>\mathbf{R}_{A}</math>. The DFT forces will depend on the chosen [[:Category:Exchange-correlation functionals\|exchange-correlation functional]] via the electronic ground-state density <math> n~~^{KS-DFT}~~({\bf r}) </math>. Therefore, the choice of the proper exchange-correlation functional for the system of interest is crucial for obtaining proper forces and, hence, the correct material properties.		where <math> \nabla_{A} </math> denotes the gradient with respect to ionic position <math>\mathbf{R}_{A}</math>. The DFT forces will depend on the chosen [[:Category:Exchange-correlation functionals\|exchange-correlation functional]] via the electronic ground-state density <math> n({\bf r}) </math>. Therefore, the choice of the proper exchange-correlation functional for the system of interest is crucial for obtaining proper forces and, hence, the correct material properties.

	===RPA-forces===		===RPA-forces===

Huebsch: /* DFT Forces */

2024-06-30T11:12:18Z

DFT Forces

← Older revision		Revision as of 11:12, 30 June 2024
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	<math>		<math>
	\mathbf{F}_{A}=-\nabla_{A} E_{tot~~}^{KS-DFT~~}=\nabla_{A}\sum_{A}\int\frac{Z_{A}}{\left\vert{\bf r}-{\bf R}_{A}\right\vert}n~~^{KS-DFT}~~({\bf r})d^{3}r -\nabla_{A}\frac{1}{2}\sum_{A\ne B}\frac{Z_{A}Z_{B}}{\left\vert{\bf R}_{A}-{\bf R}_{B}\right\vert},		\mathbf{F}_{A}=-\nabla_{A} E_{tot}=\nabla_{A}\sum_{A}\int\frac{Z_{A}}{\left\vert{\bf r}-{\bf R}_{A}\right\vert}n({\bf r})d^{3}r -\nabla_{A}\frac{1}{2}\sum_{A\ne B}\frac{Z_{A}Z_{B}}{\left\vert{\bf R}_{A}-{\bf R}_{B}\right\vert},
	</math>		</math>

Huebsch: /* DFT Forces */

2024-06-30T11:11:41Z

DFT Forces

← Older revision		Revision as of 11:11, 30 June 2024
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	===DFT Forces===		===DFT Forces===

	One way to compute the potential energy's negative gradient is through DFT. In DFT there is no classical potential energy function <math> V(\{\mathbf{r}_{i}\}) </math> but a Hamiltonian <math> \mathcal{H} </math> depending on the ionic positions <math> \mathbf{R}_{i}</math> and the electronic positions <math> \mathbf{r}_{i}</math>. The ~~exact form of the Energy~~ is given by		One way to compute the potential energy's negative gradient is through DFT. In DFT there is no classical potential energy function <math> V(\{\mathbf{r}_{i}\}) </math> but a Hamiltonian <math> \mathcal{H} </math> depending on the ionic positions <math> \mathbf{R}_{i}</math> and the electronic positions <math> \mathbf{r}_{i}</math>. The total energy is given by

	<math>		<math>
	E_{tot~~}^{KS-DFT~~} = -\frac{1}{2}\sum_{i}\psi_{i}^{*}({\bf r})\nabla^{2}\psi_{i}({\bf r}) - \sum_{A}\frac{Z_{A}}{\left\vert{\bf r}-{\bf R}_{A}\right\vert}n({\bf r}) + \frac{1}{2}\frac{n({\bf r})n({\bf r'})}{\left\vert{\bf r}-{\bf r'}\right\vert} + E_{\rm xc} + \frac{1}{2}\sum_{A\ne B}\frac{Z_{A}Z_{B}}{\left\vert{\bf R}_{A}-{\bf R}_{B}\right\vert},		E_{tot} = -\frac{1}{2}\int \sum_{i}\psi_{i}^{*}({\bf r})\nabla^{2}\psi_{i}({\bf r}) d{\bf r} - \int \sum_{A}\frac{Z_{A}}{\left\vert{\bf r}-{\bf R}_{A}\right\vert}n({\bf r})d{\bf r} + \int \int \frac{1}{2}\frac{n({\bf r})n({\bf r'})}{\left\vert{\bf r}-{\bf r'}\right\vert} d{\bf r'}d{\bf r}+ E_{\rm xc} + \frac{1}{2}\sum_{A\ne B}\frac{Z_{A}Z_{B}}{\left\vert{\bf R}_{A}-{\bf R}_{B}\right\vert},
	</math>		</math>

	where <math> n(\mathbf{r}) </math> denotes the electronic ground-state density and <math> \psi_{i} </math> are the Kohn-Sham orbitals. <math> E_{XC} </math> is the exchange-correlation energy. To obtain the force acting on ion A, the Hellmann-Feynman theorem has to be used.		where <math> n(\mathbf{r}) </math> denotes the electronic ground-state density and <math> \psi_{i} </math> are the Kohn-Sham orbitals. <math> E_{\rm xc} </math> is the exchange-correlation energy. To obtain the force acting on ion A, the Hellmann-Feynman theorem has to be used.

	<math>		<math>

Huebsch: /* DFT Forces */

2024-06-30T11:01:51Z

DFT Forces

← Older revision		Revision as of 11:01, 30 June 2024
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	===DFT Forces===		===DFT Forces===

	One way to compute the potential energy's negative gradient is through DFT. In DFT there is no classical potential energy function <math> V(\{\mathbf{r}_{i}\}) </math> but a Hamiltonian <math> \mathcal{H} </math> depending on the ionic positions <math> \mathbf{R}_{i}</math> and the electronic positions <math> \mathbf{r}_{i}</math>. The exact form of the ~~Hamiltonian~~ is given by		One way to compute the potential energy's negative gradient is through DFT. In DFT there is no classical potential energy function <math> V(\{\mathbf{r}_{i}\}) </math> but a Hamiltonian <math> \mathcal{H} </math> depending on the ionic positions <math> \mathbf{R}_{i}</math> and the electronic positions <math> \mathbf{r}_{i}</math>. The exact form of the Energy is given by

	<math>		<math>
	~~\mathcal~~{H} = -\frac{1}{2}\sum_{i}\psi_{i}^{*}({\bf r})\nabla^{2}\psi_{i}({\bf r}) - \sum_{A}\frac{Z_{A}}{\left\vert{\bf r}-{\bf R}_{A}\right\vert}n({\bf r}) + \frac{1}{2}\frac{n({\bf r})n({\bf r'})}{\left\vert{\bf r}-{\bf r'}\right\vert} + E_{\rm xc} + \frac{1}{2}\sum_{A\ne B}\frac{Z_{A}Z_{B}}{\left\vert{\bf R}_{A}-{\bf R}_{B}\right\vert},		E_{tot}^{KS-DFT} = -\frac{1}{2}\sum_{i}\psi_{i}^{*}({\bf r})\nabla^{2}\psi_{i}({\bf r}) - \sum_{A}\frac{Z_{A}}{\left\vert{\bf r}-{\bf R}_{A}\right\vert}n({\bf r}) + \frac{1}{2}\frac{n({\bf r})n({\bf r'})}{\left\vert{\bf r}-{\bf r'}\right\vert} + E_{\rm xc} + \frac{1}{2}\sum_{A\ne B}\frac{Z_{A}Z_{B}}{\left\vert{\bf R}_{A}-{\bf R}_{B}\right\vert},
	</math>		</math>

Jona: /* RPA-forces */

2024-02-09T18:29:49Z

RPA-forces

← Older revision		Revision as of 18:29, 9 February 2024
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	The operators <math>\rho^{1}</math> and <math>\gamma^{1}</math> are associated with the functional derivatives <math>\delta E/\delta V^{KS}</math> and <math>\delta E/\delta S</math> respectively. S defines the overlap operator between the Kohn-Sham orbitals of the used DFT approximation. The first term of the force equation can be associated with the exchange energy and the second term of the equation can be associated with the correlation part.		The operators <math>\rho^{1}</math> and <math>\gamma^{1}</math> are associated with the functional derivatives <math>\delta E/\delta V^{KS}</math> and <math>\delta E/\delta S</math> respectively. S defines the overlap operator between the Kohn-Sham orbitals of the used DFT approximation. The first term of the force equation can be associated with the exchange energy and the second term of the equation can be associated with the correlation part.
	{{NB\|mind\|It is recommended to use the Perdew-Burke-Ernzerhof (PBE) XC potential}}		{{NB\|mind\|It is recommended to use the Perdew-Burke-Ernzerhof (PBE) XC potential.}}
	{{NB\|mind\|It is recommended to use the [[Available PAW potentials#Recommended potentials for GW/RPA calculations\|GW POTCAR-files]]}}		{{NB\|mind\|It is recommended to use the [[Available PAW potentials#Recommended potentials for GW/RPA calculations\|GW POTCAR-files]].}}

	===Machine-learned forces===		===Machine-learned forces===

Jona: /* RPA-forces */

2024-02-01T14:42:02Z

RPA-forces

← Older revision		Revision as of 14:42, 1 February 2024
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	</math>		</math>

	The operators <math>\rho^{1}</math> and <math>\gamma^{1}</math> are associated with the functional derivatives <math>\delta E/\delta V^{KS}</math> and <math>\delta E/\delta S</math> respectively. S defines the overlap operator between the Kohn-Sham orbitals of the used DFT approximation. ~~The trace operation can be interpreted as a double integral over the entire space, involving <math>d\mathbf{r}</math> and <math>d\mathbf{r}'</math>.~~		The operators <math>\rho^{1}</math> and <math>\gamma^{1}</math> are associated with the functional derivatives <math>\delta E/\delta V^{KS}</math> and <math>\delta E/\delta S</math> respectively. S defines the overlap operator between the Kohn-Sham orbitals of the used DFT approximation. The first term of the force equation can be associated with the exchange energy and the second term of the equation can be associated with the correlation part.
	The first term of the force equation can be associated with the exchange energy and the second term of the equation can be associated with the correlation part.
	{{NB\|mind\|It is recommended to use the Perdew-Burke-Ernzerhof (PBE) XC potential}}		{{NB\|mind\|It is recommended to use the Perdew-Burke-Ernzerhof (PBE) XC potential}}
	{{NB\|mind\|It is recommended to use the [[Available PAW potentials#Recommended potentials for GW/RPA calculations\|GW POTCAR-files]]}}		{{NB\|mind\|It is recommended to use the [[Available PAW potentials#Recommended potentials for GW/RPA calculations\|GW POTCAR-files]]}}

Jona: /* How To */

2024-01-31T08:39:02Z

How To

← Older revision		Revision as of 08:39, 31 January 2024
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	*[[Best practices for machine-learned force fields\|Machine learning best practice]]		*[[Best practices for machine-learned force fields\|Machine learning best practice]]
	*[[Molecular dynamics calculations\|Molecular dynamics simulation]]		*[[Molecular dynamics calculations\|Molecular dynamics simulation]]
			*[[ACFDT/RPA calculations\|ACFDT/RPA calculations]]

	== References ==		== References ==

	[[Category:VASP]][[Category:Ionic minimization]]		[[Category:VASP]][[Category:Ionic minimization]]

Jona: /* RPA-forces */

2024-01-31T08:33:59Z

RPA-forces

← Older revision		Revision as of 08:33, 31 January 2024
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	The operators <math>\rho^{1}</math> and <math>\gamma^{1}</math> are associated with the functional derivatives <math>\delta E/\delta V^{KS}</math> and <math>\delta E/\delta S</math> respectively. S defines the overlap operator between the Kohn-Sham orbitals of the used DFT approximation. The trace operation can be interpreted as a double integral over the entire space, involving <math>d\mathbf{r}</math> and <math>d\mathbf{r}'</math>.		The operators <math>\rho^{1}</math> and <math>\gamma^{1}</math> are associated with the functional derivatives <math>\delta E/\delta V^{KS}</math> and <math>\delta E/\delta S</math> respectively. S defines the overlap operator between the Kohn-Sham orbitals of the used DFT approximation. The trace operation can be interpreted as a double integral over the entire space, involving <math>d\mathbf{r}</math> and <math>d\mathbf{r}'</math>.
	The first term of the force equation can be associated with the exchange energy and the second term of the equation can be associated with the correlation part.		The first term of the force equation can be associated with the exchange energy and the second term of the equation can be associated with the correlation part.
	{{NB\|~~tip~~\|It is recommended to use the Perdew-Burke-Ernzerhof (PBE) ~~pseudopotential~~}}		{{NB\|mind\|It is recommended to use the Perdew-Burke-Ernzerhof (PBE) XC potential}}
			{{NB\|mind\|It is recommended to use the [[Available PAW potentials#Recommended potentials for GW/RPA calculations\|GW POTCAR-files]]}}

	===Machine-learned forces===		===Machine-learned forces===

Jona: /* RPA-forces */

2024-01-29T09:08:02Z

RPA-forces

← Older revision		Revision as of 09:08, 29 January 2024
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	Note that the RPA is a correction to the underlying functional. Therefore, the choice		Note that the RPA is a correction to the underlying functional. Therefore, the choice
	of the proper [[:Category:Exchange-correlation functionals\|exchange-correlation functional]] is still crucial in the RPA approach for obtaining forces.		of the proper [[:Category:Exchange-correlation functionals\|exchange-correlation functional]] is still crucial in the RPA approach for obtaining forces.
			The RPA forces are computed by the following equation

			<math>
			\mathbf{F}_{A}=-Tr[\rho^{(1)}\nabla_{A}V^{KS}-\gamma^{(1)}\nabla_{A}S]
			</math>

			The operators <math>\rho^{1}</math> and <math>\gamma^{1}</math> are associated with the functional derivatives <math>\delta E/\delta V^{KS}</math> and <math>\delta E/\delta S</math> respectively. S defines the overlap operator between the Kohn-Sham orbitals of the used DFT approximation. The trace operation can be interpreted as a double integral over the entire space, involving <math>d\mathbf{r}</math> and <math>d\mathbf{r}'</math>.
			The first term of the force equation can be associated with the exchange energy and the second term of the equation can be associated with the correlation part.
			{{NB\|tip\|It is recommended to use the Perdew-Burke-Ernzerhof (PBE) pseudopotential}}

	===Machine-learned forces===		===Machine-learned forces===

Huebsch at 11:49, 23 October 2023

2023-10-23T11:49:58Z

← Older revision		Revision as of 11:49, 23 October 2023
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	Newton's second law of motion states that the change of motion of an object is proportional to the force acting on the object and oriented in the same direction as the force vector. Therefore, the force is defined as the change of particle momentum with time		Newton's second law of motion states that the change of motion of an object is proportional to the force acting on the object and oriented in the same direction as the force vector. Therefore, the force is defined as the change of particle momentum with time

	<math>		:<math>
	\mathbf{F}(t) = m\frac{d\mathbf{v}(t)}{d t} = m\mathbf{a}(t),		\mathbf{F}(t) = m\frac{d\mathbf{v}(t)}{d t} = m\mathbf{a}(t),
	</math>		</math>
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	where <math> V(\{\mathbf{r}_{i}\})</math> is the potential energy of the system. With Lagrange's equation of the second kind		where <math> V(\{\mathbf{r}_{i}\})</math> is the potential energy of the system. With Lagrange's equation of the second kind

	<math>		:<math>
	\frac{d}{dt}\frac{\partial L}{\partial \mathbf{v}_{i}}=\frac{\partial L}{\partial \mathbf{r}_{i}}		\frac{d}{dt}\frac{\partial L}{\partial \mathbf{v}_{i}}=\frac{\partial L}{\partial \mathbf{r}_{i}}
	</math>		</math>
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	the relation		the relation

	<math>		:<math>
	\mathbf{F}_{i} = -\frac{\partial V(\{\mathbf{r}_{i}\})}{\partial \mathbf{r}_{i}} = -\nabla V(\{\mathbf{r}_{i}\}).		\mathbf{F}_{i} = -\frac{\partial V(\{\mathbf{r}_{i}\})}{\partial \mathbf{r}_{i}} = -\nabla V(\{\mathbf{r}_{i}\}).
	</math>		</math>
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	The machine learning force field decomposes the total DFT energy into local atomic contributions <math> E_{B}(\{\mathbf{R}_{C}\}) </math> depending on all atomic positions in the system. Therefore, the force acting on ion A is computed by		The machine learning force field decomposes the total DFT energy into local atomic contributions <math> E_{B}(\{\mathbf{R}_{C}\}) </math> depending on all atomic positions in the system. Therefore, the force acting on ion A is computed by

	<math>		:<math>
	\mathbf{F}_{A} = -\nabla_{A}\sum_{B=1}^{N}E_{B}(\{\mathbf{R}_{C}\})=-\sum_{B}^{N}w_{B}\frac{dK_{B}(\{\mathbf{R}_{C}\})}{d\mathbf{R}_{A}},		\mathbf{F}_{A} = -\nabla_{A}\sum_{B=1}^{N}E_{B}(\{\mathbf{R}_{C}\})=-\sum_{B}^{N}w_{B}\frac{dK_{B}(\{\mathbf{R}_{C}\})}{d\mathbf{R}_{A}},
	</math>		</math>
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	===Stress and pressure===		===Stress and pressure===

	The ~~[[ISIF\|~~stress tensor]] provides valuable information about how forces are distributed throughout a material, both in magnitude and direction. It includes normal stresses, which act perpendicular to a given plane, and shear stresses, which act parallel to the plane. Together, these components allow ~~engineers and scientists to predict~~ how materials will behave under various conditions, such as tension, compression, or shear. The stress tensor can be computed from a viral theorem including pair forces or with a finite difference approach deforming the simulation box.		The stress tensor (see {{TAG\|ISIF}}) provides valuable information about how forces are distributed throughout a material, both in magnitude and direction. It includes normal stresses, which act perpendicular to a given plane, and shear stresses, which act parallel to the plane. Together, these components allow predicting how materials will behave under various conditions, such as tension, compression, or shear. The stress tensor can be computed from a viral theorem, including pair forces, or with a finite difference approach deforming the simulation box.
	Pressure, often denoted as P, is a scalar component of the stress tensor. It represents the normal force per unit area acting on a surface within the material. In the stress tensor, pressure is related to the diagonal components σxx, σyy, and σzz:

	<math>		Pressure, often denoted as P, is a scalar component of the stress tensor. It represents the normal force per unit area acting on a surface within the material. In the stress tensor, pressure is related to the diagonal components <math>\sigma_{xx}</math>, <math>\sigma_{yy}</math>, and <math>\sigma_{zz}</math>:
	P = \frac{1}{3}(\sigma_{xx} + \sigma_{yy} + \sigma_{zz})
			:<math>
			P = \frac{1}{3}(\sigma_{xx} + \sigma_{yy} + \sigma_{zz}).
	</math>		</math>

	So, the pressure is the average of the normal components of the stress tensor in the three spatial directions. In electronic structure calculations, finite basis sets are used to express		In other words, the pressure is the average of the normal components of the stress tensor in the three spatial directions. In electronic structure calculations, finite basis sets are used to express the electron density. Due to this finiteness of the basis set, errors on the stress tensor and the pressure are introduced. The error in the pressure is referred to as [[Energy vs volume Volume relaxations and Pulay stress\|Pulay stress]] and can be corrected with the tag {{TAG\|PSTRESS}} or by increasing {{TAG\|ENCUT}}.
	the electron density. Due to this finiteness of the basis set errors on the stress tensor and the pressure are introduced. The error in the pressure is referred to as
	[[Energy vs volume Volume relaxations and Pulay stress\|Pulay stress]] and can be corrected ~~for~~ with the tag {{TAG\|PSTRESS}}.

	===Force constants and phonons===		===Force constants and phonons===

	The forces are defined by the negative gradient of the potential energy. The force constant matrix is defined by		The forces are defined as the negative gradient of the potential energy. The force-constant matrix is defined by

	<math>		:<math>
	\Phi_{I\alpha J\beta} (\{\mathbf{R}^0\}) =		\Phi_{I\alpha J\beta} (\{\mathbf{R}^0\}) =
	\left. \frac{\partial E(\{\mathbf{R}\})}{\partial R_{I\alpha} \partial R_{J\beta}} \right\|_{\mathbf{R} =\mathbf{R^0}}		\left. \frac{\partial E(\{\mathbf{R}\})}{\partial R_{I\alpha} \partial R_{J\beta}} \right\|_{\mathbf{R} =\mathbf{R^0}}
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	</math>		</math>

	and is therefore the gradient of the force. The force constant matrix is a fundamental concept in solid-state physics and materials science, especially in the context of understanding the vibrational properties of crystals. It is a mathematical representation of the interatomic forces and their interactions within a crystal lattice. This matrix is used to describe the relationships between atomic displacements and the resulting forces that occur in a crystal. By Fourier transforming the ~~Force~~ constant matrix the [[Phonons: Theory\|dynamical matrix]] is obtained. By computing the eigenvalues of the dynamic matrix on various reciprocal lattice points the phonon dispersion relation can be obtained.		and is, therefore, the gradient of the force. The force-constant matrix is a fundamental concept in solid-state physics and materials science, especially in the context of understanding the vibrational properties of crystals. It is a mathematical representation of the interatomic forces and their interactions within a crystal lattice. This matrix is used to describe the relationships between atomic displacements and the resulting forces that occur in a crystal. By Fourier transforming the force-constant matrix, the [[Phonons: Theory\|dynamical matrix]] is obtained. By computing the eigenvalues of the dynamic matrix on various reciprocal lattice points, the phonon dispersion relation can be obtained.
	Understanding phonons is essential as they influence materials properties such as the electrical conductivity, thermal conductivity, and mechanical properties of materials.		Understanding phonons is essential as they influence materials properties such as the electrical conductivity, thermal conductivity, and mechanical properties of materials.