Pulay stress - Revision history

Huebsch at 09:04, 24 October 2025

2025-10-24T09:04:10Z

← Older revision		Revision as of 09:04, 24 October 2025
Line 21:		Line 21:

	[[File:Shape_pulay_grids.png\|700px\|thumb\|centre\|Figure 3. Cell volume and lattice positions are kept constant, while the shape is free to change (ISIF = 5). The shape changes from cubic to hexagonal. The blue spherical basis changes to the red ellipsoid basis, along the direction of the sheer. On restarting, a spherical basis returns.]]		[[File:Shape_pulay_grids.png\|700px\|thumb\|centre\|Figure 3. Cell volume and lattice positions are kept constant, while the shape is free to change (ISIF = 5). The shape changes from cubic to hexagonal. The blue spherical basis changes to the red ellipsoid basis, along the direction of the sheer. On restarting, a spherical basis returns.]]

			==References==
			<references>
			</references>

			==Related tags and articles==

			{{TAG\|PSTRESS}}

			[[volume relaxation]], [[energy cutoff and FFT meshes]]

	[[Category:Ionic minimization]][[Category:Howto]]		[[Category:Ionic minimization]][[Category:Howto]]

	~~==References==~~

Csheldon: /* Introduction */

2025-08-06T13:28:47Z

Introduction

← Older revision		Revision as of 13:28, 6 August 2025
Line 2:		Line 2:

	= Introduction =		= Introduction =
	The energy for a periodic system, e.g. band structures, is calculated using a finite number of plane waves and a finite number of k-points. A fixed number of plane waves or plane wave energy cutoff may be used to set a constant basis.{{cite\|gomesdacosta:nielsen:kunc:1986}} In VASP, a constant energy cutoff is used, cf. {{TAG\|ENCUT}}. The number of plane waves <b><i><span>N</span></b><sub>PW</sub></i> (Note: the number of plane waves in VASP can be found using by searching for <code>NPLWV</code> in the {{TAG\|OUTCAR}} file) is related to the energy cutoff <b><i><span>E</span></b><sub>cutoff</sub></i> and the size of the cell <span>'''Ω'''</span><sub>0</sub>:		The energy for a periodic system, e.g., band structures, is calculated using a finite number of plane waves and a finite number of k-points. A fixed number of plane waves or plane-wave energy cutoff may be used to set a constant basis {{cite\|gomesdacosta:nielsen:kunc:1986}}. In VASP, a constant energy cutoff is used, cf. {{TAG\|ENCUT}}. The number of plane waves <b><i><span>N</span></b><sub>PW</sub></i> (Note: the number of plane waves in VASP can be found using by searching for <code>NPLWV</code> in the {{TAG\|OUTCAR}} file) is related to the energy cutoff <b><i><span>E</span></b><sub>cutoff</sub></i> and the size of the cell <span>'''Ω'''</span><sub>0</sub>:
	::<math> N_{PW} \propto\ \Omega_0\ E_{cutoff}^{3/2} </math>		::<math> N_{PW} \propto\ \Omega_0\ E_{cutoff}^{3/2} </math>

	<b><i><span>N</span></b><sub>PW</sub></i> is constant in a relaxation calculation, which means that <b><i><span>E</span></b><sub>cutoff</sub></i> must change to compensate for changes in <span>'''Ω'''</span><sub>0</sub>. All the initial G-vectors within a sphere are included in the basis. However, when comparing cells of different sizes, i.e. during a relaxation, the cell shape is relaxed, so the direct and reciprocal lattice vectors change. The number of reciprocal G-vectors in the basis is kept fixed but the length of the G-vectors changes, indirectly changing the energy cutoff. In other words, the shape of the cutoff region changes from a sphere to an ellipsoid. This can be solved by using an infinite number of k-points and plane waves. In practice, a large enough plane wave energy cutoff and number of k-points leads to converged energies.{{cite\|payne:francis:1990}} All energy changes are strictly consistent with the stress tensor; however, when the basis set is too small, i.e. prematurely truncated, this results in discontinuities in the total energy between cells of varying volumes. These discontinuities between energy and volume create stress that decreases the equilibrium volume (cf. Fig. 1 (top)), due to the diagonal components of the stress tensor being incorrect. This is called the ''Pulay stress''.		<b><i><span>N</span></b><sub>PW</sub></i> is constant in a relaxation calculation, which means that <b><i><span>E</span></b><sub>cutoff</sub></i> must change to compensate for changes in <span>'''Ω'''</span><sub>0</sub>. All the initial G-vectors within a sphere are included in the basis. However, when comparing cells of different sizes, i.e., during a relaxation, the cell shape is relaxed, so the direct and reciprocal lattice vectors change. The number of reciprocal G-vectors in the basis is kept fixed, but the length of the G-vectors changes, indirectly changing the energy cutoff. In other words, the shape of the cutoff region changes from a sphere to an ellipsoid. This can be solved by using an infinite number of k-points and plane waves. In practice, a large enough plane wave energy cutoff and number of k-points leads to converged energies {{cite\|payne:francis:1990}}. All energy changes are strictly consistent with the stress tensor; however, when the basis set is too small, i.e., prematurely truncated, this results in discontinuities in the total energy between cells of varying volumes. These discontinuities between energy and volume create stress that decreases the equilibrium volume (cf. Fig. 1 (top)), due to the diagonal components of the stress tensor being incorrect. This is called the ''Pulay stress''.

	The pressure of the cell, being proportional to the trace of the stress tensor, can be used to visualize this. When the cell volume is below the equilibrium volume, the pressure is positive; ~~contrastingly~~, it is negative when above the equilibrium volume, so at equilibrium, this is zero. Plotting the magnitude of the pressure vs. volume curve and the total energy allows comparison between these two minima. In Figure 1 it is clear that the the absolute pressure-volume and energy-volume minima coincide for a converged basis, while the pressure equilibrium is much lower than the energy equilibrium for the unconverged basis. This is the effect of the Pulay stress.		The pressure of the cell, being proportional to the trace of the stress tensor, can be used to visualize this. When the cell volume is below the equilibrium volume, the pressure is positive; conversely, it is negative when above the equilibrium volume, so at equilibrium, this is zero. Plotting the magnitude of the pressure vs. volume curve and the total energy allows comparison between these two minima. In Figure 1 it is clear that the the absolute pressure-volume and energy-volume minima coincide for a converged basis, while the pressure equilibrium is much lower than the energy equilibrium for the unconverged basis. This is the effect of the Pulay stress.

	= Further explanation =		= Further explanation =

Csheldon: /* Introduction */

2025-04-15T10:17:53Z

Introduction

← Older revision		Revision as of 10:17, 15 April 2025
Line 2:		Line 2:

	= Introduction =		= Introduction =
	The energy for a periodic system, e.g. band structures, is calculated using a finite number of plane waves and a finite number of k-points. A fixed number of plane waves or plane wave energy cutoff may be used to set a constant basis.{{cite\|gomesdacosta:nielsen:kunc:1986}} In VASP, a constant energy cutoff is used, cf. {{TAG\|ENCUT}}. The number of plane waves <b><i><span>N</span></b><sub>PW</sub></i> (Note: the number of plane waves in VASP can be found using by searching for ''NPLWV'' in the {{TAG\|OUTCAR}} file) is related to the energy cutoff <b><i><span>E</span></b><sub>cutoff</sub></i> and the size of the cell <span>'''Ω'''</span><sub>0</sub>:		The energy for a periodic system, e.g. band structures, is calculated using a finite number of plane waves and a finite number of k-points. A fixed number of plane waves or plane wave energy cutoff may be used to set a constant basis.{{cite\|gomesdacosta:nielsen:kunc:1986}} In VASP, a constant energy cutoff is used, cf. {{TAG\|ENCUT}}. The number of plane waves <b><i><span>N</span></b><sub>PW</sub></i> (Note: the number of plane waves in VASP can be found using by searching for <code>NPLWV</code> in the {{TAG\|OUTCAR}} file) is related to the energy cutoff <b><i><span>E</span></b><sub>cutoff</sub></i> and the size of the cell <span>'''Ω'''</span><sub>0</sub>:
	::<math> N_{PW} \propto\ \Omega_0\ E_{cutoff}^{3/2} </math>		::<math> N_{PW} \propto\ \Omega_0\ E_{cutoff}^{3/2} </math>

Csheldon: /* Introduction */

2024-08-30T09:11:30Z

Introduction

← Older revision		Revision as of 09:11, 30 August 2024
Line 2:		Line 2:

	= Introduction =		= Introduction =
	The energy for a periodic system, e.g. band structures, is calculated using a finite number of plane waves and a finite number of k-points. A fixed number of plane waves or plane wave energy cutoff may be used to set a constant basis.{{cite\|gomesdacosta:nielsen:kunc:1986}} In VASP, a constant energy cutoff is used, cf. {{TAG\|ENCUT}}. The number of plane waves <b><i><span>N</span></b><sub>PW</sub></i> (~~N.B.~~ the number of plane waves in VASP can be found using by searching for ''NPLWV'' in the {{TAG\|OUTCAR}} file) is related to the energy cutoff <b><i><span>E</span></b><sub>cutoff</sub></i> and the size of the cell <span>'''Ω'''</span><sub>0</sub>:		The energy for a periodic system, e.g. band structures, is calculated using a finite number of plane waves and a finite number of k-points. A fixed number of plane waves or plane wave energy cutoff may be used to set a constant basis.{{cite\|gomesdacosta:nielsen:kunc:1986}} In VASP, a constant energy cutoff is used, cf. {{TAG\|ENCUT}}. The number of plane waves <b><i><span>N</span></b><sub>PW</sub></i> (Note: the number of plane waves in VASP can be found using by searching for ''NPLWV'' in the {{TAG\|OUTCAR}} file) is related to the energy cutoff <b><i><span>E</span></b><sub>cutoff</sub></i> and the size of the cell <span>'''Ω'''</span><sub>0</sub>:
	::<math> N_{PW} \propto\ \Omega_0\ E_{cutoff}^{3/2} </math>		::<math> N_{PW} \propto\ \Omega_0\ E_{cutoff}^{3/2} </math>

Csheldon: /* Further explanation */

2024-08-19T14:36:34Z

Further explanation

← Older revision		Revision as of 14:36, 19 August 2024
Line 21:		Line 21:

	[[File:Shape_pulay_grids.png\|700px\|thumb\|centre\|Figure 3. Cell volume and lattice positions are kept constant, while the shape is free to change (ISIF = 5). The shape changes from cubic to hexagonal. The blue spherical basis changes to the red ellipsoid basis, along the direction of the sheer. On restarting, a spherical basis returns.]]		[[File:Shape_pulay_grids.png\|700px\|thumb\|centre\|Figure 3. Cell volume and lattice positions are kept constant, while the shape is free to change (ISIF = 5). The shape changes from cubic to hexagonal. The blue spherical basis changes to the red ellipsoid basis, along the direction of the sheer. On restarting, a spherical basis returns.]]



	[[Category:Ionic minimization]][[Category:Howto]]		[[Category:Ionic minimization]][[Category:Howto]]

	==References==		==References==

Csheldon: /* Further explanation */

2024-08-19T14:29:50Z

Further explanation

← Older revision		Revision as of 14:29, 19 August 2024
Line 18:		Line 18:
	[[File:Pulay_stress_grids.png\|700px\|thumb\|centre\|Figure 2. Cell shape and lattice positions are kept constant, while the volume <b><span>V</span></b> is free to change (ISIF = 7). The initial volume <b><span>V</span></b><sub>1</sub> changes to the final volume <b><span>V</span></b><sub>2</sub>. Two cases are given, one for volume increasing on relaxation (top) and one for it decreasing (bottom). The change in real space is given on the left, while the change in reciprocal space and the subsequent effect on the G-vectors is given on the right. Blue is the initial basis, while red is the new, restarted basis. The relation between <b><i><span>E</span></b><sub>cutoff</sub></i>, <b><i><span>N</span></b><sub>PW</sub></i>, and G-vectors is given for the initial and final volumes.]]		[[File:Pulay_stress_grids.png\|700px\|thumb\|centre\|Figure 2. Cell shape and lattice positions are kept constant, while the volume <b><span>V</span></b> is free to change (ISIF = 7). The initial volume <b><span>V</span></b><sub>1</sub> changes to the final volume <b><span>V</span></b><sub>2</sub>. Two cases are given, one for volume increasing on relaxation (top) and one for it decreasing (bottom). The change in real space is given on the left, while the change in reciprocal space and the subsequent effect on the G-vectors is given on the right. Blue is the initial basis, while red is the new, restarted basis. The relation between <b><i><span>E</span></b><sub>cutoff</sub></i>, <b><i><span>N</span></b><sub>PW</sub></i>, and G-vectors is given for the initial and final volumes.]]

	Alternatively, the shape of the cell could change. As the shape changes, the G-vectors continue to be directed along the lattice coordinates, meaning that some shorten while others lengthen, see Fig. 3. This results in a shift from a spherical basis, where all G-vectors are of equal length, to one where some are stretched and others compressed, i.e. an ellipsoid. This changes the effective <b><i><span>E</span></b><sub>cutoff</sub></i> along each lattice parameter. On resetting the calculation, the cutoff is once again spherical. This draws an analogy to the symmetry breaking of the Bravais lattice seen for gradient-corrected functionals {{TAG\|GGA_COMPAT}}.		Alternatively, the shape of the cell could change. As the shape changes, the G-vectors continue to be directed along the lattice coordinates, meaning that some shorten while others lengthen, see Fig. 3. This results in a shift from a spherical basis, where all G-vectors are of equal length, to one where some are stretched and others compressed, i.e. an ellipsoid. This changes the effective <b><i><span>E</span></b><sub>cutoff</sub></i> along each lattice parameter. On resetting the calculation, the cutoff is once again spherical. This draws an analogy to the symmetry breaking of the Bravais lattice seen for gradient-corrected functionals (cf. {{TAG\|GGA_COMPAT}}), where the spherical symmetry of the G-vectors is broken for non-cubic cells.

	[[File:Shape_pulay_grids.png\|700px\|thumb\|centre\|Figure 3. Cell volume and lattice positions are kept constant, while the shape is free to change (ISIF = 5). The shape changes from cubic to hexagonal. The blue spherical basis changes to the red ellipsoid basis, along the direction of the sheer. On restarting, a spherical basis returns.]]		[[File:Shape_pulay_grids.png\|700px\|thumb\|centre\|Figure 3. Cell volume and lattice positions are kept constant, while the shape is free to change (ISIF = 5). The shape changes from cubic to hexagonal. The blue spherical basis changes to the red ellipsoid basis, along the direction of the sheer. On restarting, a spherical basis returns.]]

Csheldon at 14:28, 19 August 2024

2024-08-19T14:28:29Z

← Older revision		Revision as of 14:28, 19 August 2024
Line 18:		Line 18:
	[[File:Pulay_stress_grids.png\|700px\|thumb\|centre\|Figure 2. Cell shape and lattice positions are kept constant, while the volume <b><span>V</span></b> is free to change (ISIF = 7). The initial volume <b><span>V</span></b><sub>1</sub> changes to the final volume <b><span>V</span></b><sub>2</sub>. Two cases are given, one for volume increasing on relaxation (top) and one for it decreasing (bottom). The change in real space is given on the left, while the change in reciprocal space and the subsequent effect on the G-vectors is given on the right. Blue is the initial basis, while red is the new, restarted basis. The relation between <b><i><span>E</span></b><sub>cutoff</sub></i>, <b><i><span>N</span></b><sub>PW</sub></i>, and G-vectors is given for the initial and final volumes.]]		[[File:Pulay_stress_grids.png\|700px\|thumb\|centre\|Figure 2. Cell shape and lattice positions are kept constant, while the volume <b><span>V</span></b> is free to change (ISIF = 7). The initial volume <b><span>V</span></b><sub>1</sub> changes to the final volume <b><span>V</span></b><sub>2</sub>. Two cases are given, one for volume increasing on relaxation (top) and one for it decreasing (bottom). The change in real space is given on the left, while the change in reciprocal space and the subsequent effect on the G-vectors is given on the right. Blue is the initial basis, while red is the new, restarted basis. The relation between <b><i><span>E</span></b><sub>cutoff</sub></i>, <b><i><span>N</span></b><sub>PW</sub></i>, and G-vectors is given for the initial and final volumes.]]

	Alternatively, the shape of the cell could change. As the shape changes, the G-vectors continue to be directed along the lattice coordinates, meaning that some shorten while others lengthen, see Fig. 3. This results in a shift from a spherical basis, where all G-vectors are of equal length, to one where some are stretched and others compressed, i.e. an ellipsoid. This changes the effective <b><i><span>E</span></b><sub>cutoff</sub></i> along each lattice parameter. On resetting the calculation, the cutoff is once again spherical.		Alternatively, the shape of the cell could change. As the shape changes, the G-vectors continue to be directed along the lattice coordinates, meaning that some shorten while others lengthen, see Fig. 3. This results in a shift from a spherical basis, where all G-vectors are of equal length, to one where some are stretched and others compressed, i.e. an ellipsoid. This changes the effective <b><i><span>E</span></b><sub>cutoff</sub></i> along each lattice parameter. On resetting the calculation, the cutoff is once again spherical. This draws an analogy to the symmetry breaking of the Bravais lattice seen for gradient-corrected functionals {{TAG\|GGA_COMPAT}}.

	[[File:Shape_pulay_grids.png\|700px\|thumb\|centre\|Figure 3. Cell volume and lattice positions are kept constant, while the shape is free to change (ISIF = 5). The shape changes from cubic to hexagonal. The blue spherical basis changes to the red ellipsoid basis, along the direction of the sheer. On restarting, a spherical basis returns.]]		[[File:Shape_pulay_grids.png\|700px\|thumb\|centre\|Figure 3. Cell volume and lattice positions are kept constant, while the shape is free to change (ISIF = 5). The shape changes from cubic to hexagonal. The blue spherical basis changes to the red ellipsoid basis, along the direction of the sheer. On restarting, a spherical basis returns.]]

Csheldon: /* Introduction */

2024-08-13T15:56:54Z

Introduction

← Older revision		Revision as of 15:56, 13 August 2024
Line 2:		Line 2:

	= Introduction =		= Introduction =
	The energy for a periodic system, e.g. band structures, is calculated using a finite number of plane waves and a finite number of k-points. A fixed number of plane waves or plane wave energy cutoff may be used to set a constant basis.{{cite\|gomesdacosta:nielsen:kunc:1986}} In VASP, a constant energy cutoff is used, cf. {{TAG\|ENCUT}}. The number of plane waves <b><i><span>N</span></b><sub>PW</sub></i> is related to the energy cutoff <b><i><span>E</span></b><sub>cutoff</sub></i> and the size of the cell <span>'''Ω'''</span><sub>0</sub>:		The energy for a periodic system, e.g. band structures, is calculated using a finite number of plane waves and a finite number of k-points. A fixed number of plane waves or plane wave energy cutoff may be used to set a constant basis.{{cite\|gomesdacosta:nielsen:kunc:1986}} In VASP, a constant energy cutoff is used, cf. {{TAG\|ENCUT}}. The number of plane waves <b><i><span>N</span></b><sub>PW</sub></i> (N.B. the number of plane waves in VASP can be found using by searching for ''NPLWV'' in the {{TAG\|OUTCAR}} file) is related to the energy cutoff <b><i><span>E</span></b><sub>cutoff</sub></i> and the size of the cell <span>'''Ω'''</span><sub>0</sub>:
	::<math> N_{PW} \propto\ \Omega_0\ E_{cutoff}^{3/2} </math>		::<math> N_{PW} \propto\ \Omega_0\ E_{cutoff}^{3/2} </math>

	<b><i><span>N</span></b><sub>PW</sub></i> is constant in a relaxation calculation ~~(N.B. the number of plane waves in VASP can be found using by searching for ''NPLWV'' in the {{TAG\|OUTCAR}} file)~~, which means that <b><i><span>E</span></b><sub>cutoff</sub></i> must change to compensate for changes in <span>'''Ω'''</span><sub>0</sub>. All the initial G-vectors within a sphere are included in the basis. However, when comparing cells of different sizes, i.e. during a relaxation, the cell shape is relaxed, so the direct and reciprocal lattice vectors change. The number of reciprocal G-vectors in the basis is kept fixed but the length of the G-vectors changes, indirectly changing the energy cutoff. In other words, the shape of the cutoff region changes from a sphere to an ellipsoid. This can be solved by using an infinite number of k-points and plane waves. In practice, a large enough plane wave energy cutoff and number of k-points leads to converged energies.{{cite\|payne:francis:1990}} All energy changes are strictly consistent with the stress tensor; however, when the basis set is too small, i.e. prematurely truncated, this results in discontinuities in the total energy between cells of varying volumes. These discontinuities between energy and volume create stress that decreases the equilibrium volume (cf. Fig. 1 (top)), due to the diagonal components of the stress tensor being incorrect. This is called the ''Pulay stress''.		<b><i><span>N</span></b><sub>PW</sub></i> is constant in a relaxation calculation, which means that <b><i><span>E</span></b><sub>cutoff</sub></i> must change to compensate for changes in <span>'''Ω'''</span><sub>0</sub>. All the initial G-vectors within a sphere are included in the basis. However, when comparing cells of different sizes, i.e. during a relaxation, the cell shape is relaxed, so the direct and reciprocal lattice vectors change. The number of reciprocal G-vectors in the basis is kept fixed but the length of the G-vectors changes, indirectly changing the energy cutoff. In other words, the shape of the cutoff region changes from a sphere to an ellipsoid. This can be solved by using an infinite number of k-points and plane waves. In practice, a large enough plane wave energy cutoff and number of k-points leads to converged energies.{{cite\|payne:francis:1990}} All energy changes are strictly consistent with the stress tensor; however, when the basis set is too small, i.e. prematurely truncated, this results in discontinuities in the total energy between cells of varying volumes. These discontinuities between energy and volume create stress that decreases the equilibrium volume (cf. Fig. 1 (top)), due to the diagonal components of the stress tensor being incorrect. This is called the ''Pulay stress''.

	The pressure of the cell, being proportional to the trace of the stress tensor, can be used to visualize this. When the cell volume is below the equilibrium volume, the pressure is positive; contrastingly, it is negative when above the equilibrium volume, so at equilibrium, this is zero. Plotting the magnitude of the pressure vs. volume curve and the total energy allows comparison between these two minima. In Figure 1 it is clear that the the absolute pressure-volume and energy-volume minima coincide for a converged basis, while the pressure equilibrium is much lower than the energy equilibrium for the unconverged basis. This is the effect of the Pulay stress.		The pressure of the cell, being proportional to the trace of the stress tensor, can be used to visualize this. When the cell volume is below the equilibrium volume, the pressure is positive; contrastingly, it is negative when above the equilibrium volume, so at equilibrium, this is zero. Plotting the magnitude of the pressure vs. volume curve and the total energy allows comparison between these two minima. In Figure 1 it is clear that the the absolute pressure-volume and energy-volume minima coincide for a converged basis, while the pressure equilibrium is much lower than the energy equilibrium for the unconverged basis. This is the effect of the Pulay stress.

Csheldon: /* Introduction */

2024-08-13T15:56:24Z

Introduction

← Older revision		Revision as of 15:56, 13 August 2024
Line 5:		Line 5:
	::<math> N_{PW} \propto\ \Omega_0\ E_{cutoff}^{3/2} </math>		::<math> N_{PW} \propto\ \Omega_0\ E_{cutoff}^{3/2} </math>

	<b><i><span>N</span></b><sub>PW</sub></i> is constant in a relaxation calculation (N.B. the number of plane waves in VASP can be found using by searching for '''NPLWV''' in the {{TAG\|OUTCAR}} file), which means that <b><i><span>E</span></b><sub>cutoff</sub></i> must change to compensate for changes in <span>'''Ω'''</span><sub>0</sub>. All the initial G-vectors within a sphere are included in the basis. However, when comparing cells of different sizes, i.e. during a relaxation, the cell shape is relaxed, so the direct and reciprocal lattice vectors change. The number of reciprocal G-vectors in the basis is kept fixed but the length of the G-vectors changes, indirectly changing the energy cutoff. In other words, the shape of the cutoff region changes from a sphere to an ellipsoid. This can be solved by using an infinite number of k-points and plane waves. In practice, a large enough plane wave energy cutoff and number of k-points leads to converged energies.{{cite\|payne:francis:1990}} All energy changes are strictly consistent with the stress tensor; however, when the basis set is too small, i.e. prematurely truncated, this results in discontinuities in the total energy between cells of varying volumes. These discontinuities between energy and volume create stress that decreases the equilibrium volume (cf. Fig. 1 (top)), due to the diagonal components of the stress tensor being incorrect. This is called the ''Pulay stress''.		<b><i><span>N</span></b><sub>PW</sub></i> is constant in a relaxation calculation (N.B. the number of plane waves in VASP can be found using by searching for ''NPLWV'' in the {{TAG\|OUTCAR}} file), which means that <b><i><span>E</span></b><sub>cutoff</sub></i> must change to compensate for changes in <span>'''Ω'''</span><sub>0</sub>. All the initial G-vectors within a sphere are included in the basis. However, when comparing cells of different sizes, i.e. during a relaxation, the cell shape is relaxed, so the direct and reciprocal lattice vectors change. The number of reciprocal G-vectors in the basis is kept fixed but the length of the G-vectors changes, indirectly changing the energy cutoff. In other words, the shape of the cutoff region changes from a sphere to an ellipsoid. This can be solved by using an infinite number of k-points and plane waves. In practice, a large enough plane wave energy cutoff and number of k-points leads to converged energies.{{cite\|payne:francis:1990}} All energy changes are strictly consistent with the stress tensor; however, when the basis set is too small, i.e. prematurely truncated, this results in discontinuities in the total energy between cells of varying volumes. These discontinuities between energy and volume create stress that decreases the equilibrium volume (cf. Fig. 1 (top)), due to the diagonal components of the stress tensor being incorrect. This is called the ''Pulay stress''.

	The pressure of the cell, being proportional to the trace of the stress tensor, can be used to visualize this. When the cell volume is below the equilibrium volume, the pressure is positive; contrastingly, it is negative when above the equilibrium volume, so at equilibrium, this is zero. Plotting the magnitude of the pressure vs. volume curve and the total energy allows comparison between these two minima. In Figure 1 it is clear that the the absolute pressure-volume and energy-volume minima coincide for a converged basis, while the pressure equilibrium is much lower than the energy equilibrium for the unconverged basis. This is the effect of the Pulay stress.		The pressure of the cell, being proportional to the trace of the stress tensor, can be used to visualize this. When the cell volume is below the equilibrium volume, the pressure is positive; contrastingly, it is negative when above the equilibrium volume, so at equilibrium, this is zero. Plotting the magnitude of the pressure vs. volume curve and the total energy allows comparison between these two minima. In Figure 1 it is clear that the the absolute pressure-volume and energy-volume minima coincide for a converged basis, while the pressure equilibrium is much lower than the energy equilibrium for the unconverged basis. This is the effect of the Pulay stress.

Csheldon: /* Introduction */

2024-08-13T15:56:18Z

Introduction

← Older revision		Revision as of 15:56, 13 August 2024
Line 5:		Line 5:
	::<math> N_{PW} \propto\ \Omega_0\ E_{cutoff}^{3/2} </math>		::<math> N_{PW} \propto\ \Omega_0\ E_{cutoff}^{3/2} </math>

	<b><i><span>N</span></b><sub>PW</sub></i> is constant in a relaxation calculation (N.B. the number of plane waves in VASP can be found using by searching for NPLWV in the {{TAG\|OUTCAR}} file), which means that <b><i><span>E</span></b><sub>cutoff</sub></i> must change to compensate for changes in <span>'''Ω'''</span><sub>0</sub>. All the initial G-vectors within a sphere are included in the basis. However, when comparing cells of different sizes, i.e. during a relaxation, the cell shape is relaxed, so the direct and reciprocal lattice vectors change. The number of reciprocal G-vectors in the basis is kept fixed but the length of the G-vectors changes, indirectly changing the energy cutoff. In other words, the shape of the cutoff region changes from a sphere to an ellipsoid. This can be solved by using an infinite number of k-points and plane waves. In practice, a large enough plane wave energy cutoff and number of k-points leads to converged energies.{{cite\|payne:francis:1990}} All energy changes are strictly consistent with the stress tensor; however, when the basis set is too small, i.e. prematurely truncated, this results in discontinuities in the total energy between cells of varying volumes. These discontinuities between energy and volume create stress that decreases the equilibrium volume (cf. Fig. 1 (top)), due to the diagonal components of the stress tensor being incorrect. This is called the ''Pulay stress''.		<b><i><span>N</span></b><sub>PW</sub></i> is constant in a relaxation calculation (N.B. the number of plane waves in VASP can be found using by searching for '''NPLWV''' in the {{TAG\|OUTCAR}} file), which means that <b><i><span>E</span></b><sub>cutoff</sub></i> must change to compensate for changes in <span>'''Ω'''</span><sub>0</sub>. All the initial G-vectors within a sphere are included in the basis. However, when comparing cells of different sizes, i.e. during a relaxation, the cell shape is relaxed, so the direct and reciprocal lattice vectors change. The number of reciprocal G-vectors in the basis is kept fixed but the length of the G-vectors changes, indirectly changing the energy cutoff. In other words, the shape of the cutoff region changes from a sphere to an ellipsoid. This can be solved by using an infinite number of k-points and plane waves. In practice, a large enough plane wave energy cutoff and number of k-points leads to converged energies.{{cite\|payne:francis:1990}} All energy changes are strictly consistent with the stress tensor; however, when the basis set is too small, i.e. prematurely truncated, this results in discontinuities in the total energy between cells of varying volumes. These discontinuities between energy and volume create stress that decreases the equilibrium volume (cf. Fig. 1 (top)), due to the diagonal components of the stress tensor being incorrect. This is called the ''Pulay stress''.

	The pressure of the cell, being proportional to the trace of the stress tensor, can be used to visualize this. When the cell volume is below the equilibrium volume, the pressure is positive; contrastingly, it is negative when above the equilibrium volume, so at equilibrium, this is zero. Plotting the magnitude of the pressure vs. volume curve and the total energy allows comparison between these two minima. In Figure 1 it is clear that the the absolute pressure-volume and energy-volume minima coincide for a converged basis, while the pressure equilibrium is much lower than the energy equilibrium for the unconverged basis. This is the effect of the Pulay stress.		The pressure of the cell, being proportional to the trace of the stress tensor, can be used to visualize this. When the cell volume is below the equilibrium volume, the pressure is positive; contrastingly, it is negative when above the equilibrium volume, so at equilibrium, this is zero. Plotting the magnitude of the pressure vs. volume curve and the total energy allows comparison between these two minima. In Figure 1 it is clear that the the absolute pressure-volume and energy-volume minima coincide for a converged basis, while the pressure equilibrium is much lower than the energy equilibrium for the unconverged basis. This is the effect of the Pulay stress.

← Older revision		Revision as of 14:29, 19 August 2024
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	[[File:Pulay_stress_grids.png\|700px\|thumb\|centre\|Figure 2. Cell shape and lattice positions are kept constant, while the volume <b><span>V</span></b> is free to change (ISIF = 7). The initial volume <b><span>V</span></b><sub>1</sub> changes to the final volume <b><span>V</span></b><sub>2</sub>. Two cases are given, one for volume increasing on relaxation (top) and one for it decreasing (bottom). The change in real space is given on the left, while the change in reciprocal space and the subsequent effect on the G-vectors is given on the right. Blue is the initial basis, while red is the new, restarted basis. The relation between <b><i><span>E</span></b><sub>cutoff</sub></i>, <b><i><span>N</span></b><sub>PW</sub></i>, and G-vectors is given for the initial and final volumes.]]		[[File:Pulay_stress_grids.png\|700px\|thumb\|centre\|Figure 2. Cell shape and lattice positions are kept constant, while the volume <b><span>V</span></b> is free to change (ISIF = 7). The initial volume <b><span>V</span></b><sub>1</sub> changes to the final volume <b><span>V</span></b><sub>2</sub>. Two cases are given, one for volume increasing on relaxation (top) and one for it decreasing (bottom). The change in real space is given on the left, while the change in reciprocal space and the subsequent effect on the G-vectors is given on the right. Blue is the initial basis, while red is the new, restarted basis. The relation between <b><i><span>E</span></b><sub>cutoff</sub></i>, <b><i><span>N</span></b><sub>PW</sub></i>, and G-vectors is given for the initial and final volumes.]]

	Alternatively, the shape of the cell could change. As the shape changes, the G-vectors continue to be directed along the lattice coordinates, meaning that some shorten while others lengthen, see Fig. 3. This results in a shift from a spherical basis, where all G-vectors are of equal length, to one where some are stretched and others compressed, i.e. an ellipsoid. This changes the effective <b><i><span>E</span></b><sub>cutoff</sub></i> along each lattice parameter. On resetting the calculation, the cutoff is once again spherical. This draws an analogy to the symmetry breaking of the Bravais lattice seen for gradient-corrected functionals {{TAG\|GGA_COMPAT}}.		Alternatively, the shape of the cell could change. As the shape changes, the G-vectors continue to be directed along the lattice coordinates, meaning that some shorten while others lengthen, see Fig. 3. This results in a shift from a spherical basis, where all G-vectors are of equal length, to one where some are stretched and others compressed, i.e. an ellipsoid. This changes the effective <b><i><span>E</span></b><sub>cutoff</sub></i> along each lattice parameter. On resetting the calculation, the cutoff is once again spherical. This draws an analogy to the symmetry breaking of the Bravais lattice seen for gradient-corrected functionals (cf. {{TAG\|GGA_COMPAT}}), where the spherical symmetry of the G-vectors is broken for non-cubic cells.

	[[File:Shape_pulay_grids.png\|700px\|thumb\|centre\|Figure 3. Cell volume and lattice positions are kept constant, while the shape is free to change (ISIF = 5). The shape changes from cubic to hexagonal. The blue spherical basis changes to the red ellipsoid basis, along the direction of the sheer. On restarting, a spherical basis returns.]]		[[File:Shape_pulay_grids.png\|700px\|thumb\|centre\|Figure 3. Cell volume and lattice positions are kept constant, while the shape is free to change (ISIF = 5). The shape changes from cubic to hexagonal. The blue spherical basis changes to the red ellipsoid basis, along the direction of the sheer. On restarting, a spherical basis returns.]]