Born effective charges - Revision history

Csheldon: /* Related tags and articles */

2025-05-05T07:40:05Z

Csheldon: /* Introduction */

2025-05-05T07:35:57Z

Introduction

← Older revision		Revision as of 07:35, 5 May 2025
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	= Introduction =		= Introduction =
	The dynamical charge is defined as the cell volume <span>'''Ω'''</span~~><sub>0</sub~~>, multiplied by the partial derivative of the macroscopic polarization '''P''' in the direction ''i'' with respect to a rigid displacement of the sublattice of atoms ''κ'' in the direction ''j''.		The dynamical charge is defined as the cell volume <span>'''Ω'''</span>, multiplied by the partial derivative of the macroscopic polarization '''P''' in the direction ''i'' with respect to a rigid displacement of the sublattice of atoms ''κ'' in the direction ''j''.

	However, the polarization is not uniquely defined in periodic systems and depends on the macroscopic electric field <math>\mathcal{E}_i</math> fixed by the periodic boundary conditions. The Born effective charge '''Z''' is the partial derivative of the polarization with respect to position ''u'' for zero macroscopic electric field.{{cite\|gonze:prb:1997}} As polarization is the first derivative of the total energy with respect to the macroscopic electric field, '''Z''' may be rearranged in terms of the partial derivative of the force '''F''' in direction ''j'' on atom ''κ'' with respect to <math>\mathcal{E}_i</math>:		However, the polarization is not uniquely defined in periodic systems and depends on the macroscopic electric field <math>\mathcal{E}_i</math> fixed by the periodic boundary conditions. The Born effective charge '''Z''' is the partial derivative of the polarization with respect to position ''u'' for zero macroscopic electric field.{{cite\|gonze:prb:1997}} As polarization is the first derivative of the total energy with respect to the macroscopic electric field, '''Z''' may be rearranged in terms of the partial derivative of the force '''F''' in direction ''j'' on atom ''κ'' with respect to <math>\mathcal{E}_i</math>:
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	::<math>		::<math>
	Z_{\kappa,ij}^*		Z_{\kappa,ij}^*
	=\frac{\~~Omega_0~~}{e} \frac{\partial \mathcal{P}_i} {\partial u_{\kappa,j}(\textbf{q=0})}		=\frac{\Omega}{e} \frac{\partial \mathcal{P}_i} {\partial u_{\kappa,j}(\textbf{q=0})}
	=\frac{1}{e} \frac{\partial F_{\kappa,j}}{\partial \mathcal{E}_i},		=\frac{1}{e} \frac{\partial F_{\kappa,j}}{\partial \mathcal{E}_i},
	\qquad		\qquad

Csheldon: /* Introduction */

2025-05-05T07:35:32Z

Introduction

← Older revision		Revision as of 07:35, 5 May 2025
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	Z_{\kappa,ij}^*		Z_{\kappa,ij}^*
	=\frac{\Omega_0}{e} \frac{\partial \mathcal{P}_i} {\partial u_{\kappa,j}(\textbf{q=0})}		=\frac{\Omega_0}{e} \frac{\partial \mathcal{P}_i} {\partial u_{\kappa,j}(\textbf{q=0})}
	=\frac{1}{e} \frac{\partial F_{\kappa,j}}{\partial \mathcal{E}_i}		=\frac{1}{e} \frac{\partial F_{\kappa,j}}{\partial \mathcal{E}_i},
	\qquad		\qquad
	{i,j=x,y,z}		{i,j=x,y,z}

Csheldon at 12:54, 13 August 2024

2024-08-13T12:54:11Z

← Older revision	Revision as of 12:54, 13 August 2024
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		The change in polarization from the displacement of an atom is not uniquely defined in periodic systems, where atoms are repeated in different cells and the charge can be generalized.{{cite\|ghosez:michenaud:gonze:1998}} Born effective charges are one way of defining this dynamical charge.

		= Introduction =
		The dynamical charge is defined as the cell volume <span>'''Ω'''</span><sub>0</sub>, multiplied by the partial derivative of the macroscopic polarization '''P''' in the direction ''i'' with respect to a rigid displacement of the sublattice of atoms ''κ'' in the direction ''j''.

		However, the polarization is not uniquely defined in periodic systems and depends on the macroscopic electric field <math>\mathcal{E}_i</math> fixed by the periodic boundary conditions. The Born effective charge '''Z''' is the partial derivative of the polarization with respect to position ''u'' for zero macroscopic electric field.{{cite\|gonze:prb:1997}} As polarization is the first derivative of the total energy with respect to the macroscopic electric field, '''Z''' may be rearranged in terms of the partial derivative of the force '''F''' in direction ''j'' on atom ''κ'' with respect to <math>\mathcal{E}_i</math>:

		::<math>
		Z_{\kappa,ij}^*
		=\frac{\Omega_0}{e} \frac{\partial \mathcal{P}_i} {\partial u_{\kappa,j}(\textbf{q=0})}
		=\frac{1}{e} \frac{\partial F_{\kappa,j}}{\partial \mathcal{E}_i}
		\qquad
		{i,j=x,y,z}
		</math>

		{{NB\|mind\|
		The '''''' does not denote a complex conjugate, '''Z*''' is always a real quantity.
		'''Z''' is given in units of <math>\vert e \vert</math> in VASP.
		*VASP outputs <span><b>Z</b></span><sub><i>ij</i></sub><sup><span>⋆</span></sup> with ''i'' for the macroscopic electric field, and ''j'' for the direction of the force. In literature, <span><b>Z</b></span><sub><i>ji</i></sub><sup><span>⋆</span></sup> is commonly seen, with the force direction ''j'' followed by the electric field direction ''i''. Note, py4vasp follows the latter notation <span><b>Z</b></span><sub><i>ji</i></sub><sup><span>⋆</span></sup> for historic reasons.
		}}

		= How to calculate =
		There are two ways of computing Born effective charges in VASP. The first is done using {{TAG\|LCALCEPS}}, where a finite electric field is applied along the three cartesian directions and the resultant forces on the atoms are calculated:

		{{TAG\|LCALCEPS}} = .TRUE.

		The other approach is done using {{TAG\|LEPSILON}}, where the derivative of the wavefunction with respect to an electric field is calculated using density functional perturbation theory (DFPT):

		{{TAG\|LEPSILON}} = .TRUE.

		These may be used in combination with {{TAG\|IBRION}} to obtain additional dielectric properties:

		{{TAG\|IBRION}} = 5 or 6 ! Calculated using finite differences.
		{{TAG\|IBRION}} = 7 or 8 ! Calculated using DFPT

		For more details, see the pages for each tag. The Born effective charges including local field effects will be given in the {{TAG\|OUTCAR}} file:

		BORN EFFECTIVE CHARGES (including local field effects) (in \|e\|, cummulative output)

		= Excluding local field effects =

		Previously, the local field effects have been included, that is changes in the orbitals due to the electric field induce changes in the Hartree- and exchange-correlation potentials. This may be limited to changes in the Hartree potential, by specifying:

		{{TAG\|LRPA}} = .TRUE.
		{{TAG\|LCALCEPS}} = .TRUE. ! N.B. {{TAG\|LEPSILON}} does not output the final Born effective charges.

		This prints out the Born effective charges excluding local field effects:

		BORN EFFECTIVE CHARGES (excluding local field effects) (in \|e\|, cummulative output)

		These are calculated normally but remain hidden unless explicitly specified.

		== Related tags and articles ==

		{{TAG\|LEPSILON}},
		{{TAG\|LCALCEPS}},
		{{TAG\|IBRION}},
		{{TAG\|LRPA}}

		==References==

		[[Category:Dielectric properties]][[Category:Howto]]

Csheldon: Blanked the page

2024-07-30T15:20:15Z

Blanked the page

← Older revision		Revision as of 15:20, 30 July 2024
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	~~Born effective charges~~

	~~# https://journals.aps.org/prb/abstract/10.1103/PhysRevB.58.6224~~

	~~Equation:~~

	~~::<math>~~
	Z_{\kappa,\beta\alpha}^*
	~~=\Omega_0\frac{\partial \mathcal{P}_{\mathrm{mac},\beta}}{\partial \tau_{\kappa,\alpha}(\textbf{q=0})}~~
	~~=\Omega_0\frac{\partial F_{\kappa,\alpha}}{\partial \mathcal{E}_\beta}~~
	~~\qquad~~
	~~{\alpha,\beta=x,y,z,}.~~
	~~</math>~~

	~~# https://journals.aps.org/prb/abstract/10.1103/PhysRevB.55.10355~~


	~~= Introduction =~~
	~~Insert something from paper/ thesis then edit~~


	~~MIND THAT VASP outputs it oddly?~~

	~~= Calculating =~~
	~~Taken from Dielctric properties:~~
	There are two approaches to compute Born effective charges implemented in VASP: one is done by applying finite electric fields along the three cartesian directions and computing the forces on the atoms which are activated using LCALCEPS or by computing the derivating of the wavefunction with respect to an electric field using density functional perturbation theory (DFPT) using LEPSILON.

	~~Note: This is different in py4vasp.~~



	~~<!--[[Category:Dielectric properties]][[LEPSILON]][[LCALCEPS]][[Category:Howto]]-->~~

	~~==References==~~

Csheldon at 15:15, 30 July 2024

2024-07-30T15:15:38Z

← Older revision		Revision as of 15:15, 30 July 2024
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	::<math>		::<math>
	Z_{\kappa,\beta\alpha}^*		Z_{\kappa,\beta\alpha}^*
	=\Omega_0\frac{\partial \mathcal{P}_{\mathrm{mac},\beta}}{\partial \tau_{\kappa\alpha}(\textbf{q=0})}		=\Omega_0\frac{\partial \mathcal{P}_{\mathrm{mac},\beta}}{\partial \tau_{\kappa,\alpha}(\textbf{q=0})}
	=\Omega_0\frac{\partial F_{\kappa,\alpha}}{\partial \mathcal{E}_\beta}		=\Omega_0\frac{\partial F_{\kappa,\alpha}}{\partial \mathcal{E}_\beta}
	\qquad		\qquad

Csheldon at 15:15, 30 July 2024

2024-07-30T15:15:28Z

← Older revision		Revision as of 15:15, 30 July 2024
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	Z_{\kappa,\beta\alpha}^*		Z_{\kappa,\beta\alpha}^*
	=\Omega_0\frac{\partial \mathcal{P}_{\mathrm{mac},\beta}}{\partial \tau_{\kappa\alpha}(\textbf{q=0})}		=\Omega_0\frac{\partial \mathcal{P}_{\mathrm{mac},\beta}}{\partial \tau_{\kappa\alpha}(\textbf{q=0})}
	=\Omega_0\frac{\partial F_{\kappa\alpha}}{\partial \mathcal{E}_\beta}		=\Omega_0\frac{\partial F_{\kappa,\alpha}}{\partial \mathcal{E}_\beta}
	\qquad		\qquad
	{\alpha,\beta=x,y,z,}.		{\alpha,\beta=x,y,z,}.

Csheldon at 15:15, 30 July 2024

2024-07-30T15:15:11Z

← Older revision		Revision as of 15:15, 30 July 2024
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	=\Omega_0\frac{\partial F_{\kappa\alpha}}{\partial \mathcal{E}_\beta}		=\Omega_0\frac{\partial F_{\kappa\alpha}}{\partial \mathcal{E}_\beta}
	\qquad		\qquad
	{\alpha\beta=x,y,z,}.		{\alpha,\beta=x,y,z,}.
	</math>		</math>

Csheldon at 15:14, 30 July 2024

2024-07-30T15:14:34Z

← Older revision		Revision as of 15:14, 30 July 2024
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	=\Omega_0\frac{\partial F_{\kappa\alpha}}{\partial \mathcal{E}_\beta}		=\Omega_0\frac{\partial F_{\kappa\alpha}}{\partial \mathcal{E}_\beta}
	\qquad		\qquad
	{~~i,j~~=x,y,z}.		{\alpha\beta=x,y,z,}.
	</math>		</math>

Csheldon at 15:13, 30 July 2024

2024-07-30T15:13:59Z

← Older revision		Revision as of 15:13, 30 July 2024
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	::<math>		::<math>
	Z_{\kappa,\beta\alpha}		Z_{\kappa,\beta\alpha}^*
	=\Omega_0\frac{\partial \mathcal{P}_{\mathrm{mac},\beta}}{\partial \tau_{\kappa\alpha}(\textbf{q=0})}		=\Omega_0\frac{\partial \mathcal{P}_{\mathrm{mac},\beta}}{\partial \tau_{\kappa\alpha}(\textbf{q=0})}
	=\Omega_0\frac{\partial F_{\kappa\alpha}}{\partial \mathcal{E}_\beta}		=\Omega_0\frac{\partial F_{\kappa\alpha}}{\partial \mathcal{E}_\beta}

← Older revision		Revision as of 07:40, 5 May 2025
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	{{TAG\|LCALCEPS}},		{{TAG\|LCALCEPS}},
	{{TAG\|IBRION}},		{{TAG\|IBRION}},
	{{TAG\|LRPA}}		{{TAG\|LRPA}},
			{{TAG\|Berry phases and finite electric fields}}

	==References==		==References==

	[[Category:Dielectric properties]][[Category:Howto]]		[[Category:Dielectric properties]][[Category:Howto]]