Vacuum reference theory - Revision history

Csheldon at 12:37, 18 September 2026

2026-09-18T12:37:16Z

← Older revision		Revision as of 12:37, 18 September 2026
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	Finding a reference to compare between experiments or calculations can be challenging. Experimentally, there is usually a standard reference taken, e.g., the standard hydrogen electrode (SHE) under standard conditions. Computationally, it can be difficult to even compare the energies of cells if a reference is not carefully taken. The local potential cannot be used for a cell because it is relative to the individual cell. Instead, it is more useful to find a common reference, e.g., the vacuum.		Finding a reference to compare between experiments or calculations can be challenging. Experimentally, there is usually a standard reference taken, e.g., the standard hydrogen electrode (SHE) under standard conditions. Computationally, it can be difficult to even compare the energies of cells if a reference is not carefully taken. The local potential cannot be used for a cell because it is relative to the individual cell. Instead, it is more useful to find a common reference, e.g., the vacuum.

	This page derives the vacuum alignment <math>e\Delta\bar{\phi}</math> that enters the free-energy difference of an electrochemical half-cell, and states the approximations it rests on. For the procedure that computes it, see [[Vacuum reference]].		This page derives the vacuum alignment <math>e\Delta\bar{\phi}</math> that enters the free-energy difference of an electrochemical half-cell, and states the approximations it rests on {{Cite\|jinnouchi:ncm:2024}}. For the procedure that computes it, see [[Vacuum reference]].

	The derivation proceeds in four steps. The free energy of the half-cell reaction is written with an explicit electron-reservoir term, that term is expressed relative to the energy zero of the periodic calculation, the result is averaged along the integration path, and the offset itself is estimated from the oxygen core levels.		The derivation proceeds in four steps. The free energy of the half-cell reaction is written with an explicit electron-reservoir term, that term is expressed relative to the energy zero of the periodic calculation, the result is averaged along the integration path, and the offset itself is estimated from the oxygen core levels.

Csheldon: /* Endpoint approximation for the half-cell reaction */

2026-09-18T12:35:54Z

Endpoint approximation for the half-cell reaction

← Older revision		Revision as of 12:35, 18 September 2026
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	[[File:Vacuum_ref.png\|600px\|thumb\|center\|'''Figure 1'''. Alignment of the periodic bulk-water calculation with the vacuum reference. The average O <sub>1s</sub> level of water molecules far from the Fe ion, <span style="color:#A82C35"><math>\left\langle\epsilon_{1s,\mathrm{bulk}}\right\rangle_{\lambda}</math></span>, is used as the bulk reference. Water molecules within half a unit cell of the ion are excluded. The O <sub>1s</sub> level in the bulk-like region of the water slab, <span style="color:#3E70EA"><math>\left\langle\epsilon_{1s,\mathrm{slab}}\right\rangle</math></span>, connects this reference to the vacuum level estimated from the slab calculation. Their difference gives the approximate potential offset <span style="color:#8342A4"><math>e\Delta\bar{\phi}_{\lambda}</math></span> and hence the chemical potential of the vacuum electron expressed in the bulk energy reference. Reproducing Fig. 1 of Ref. {{Cite\|jinnouchi:ncm:2024}}.]]		[[File:Vacuum_ref.png\|600px\|thumb\|center\|'''Figure 1'''. Alignment of the periodic bulk-water calculation with the vacuum reference. The average O <sub>1s</sub> level of water molecules far from the Fe ion, <span style="color:#A82C35"><math>\left\langle\epsilon_{1s,\mathrm{bulk}}\right\rangle_{\lambda}</math></span>, is used as the bulk reference. Water molecules within half a unit cell of the ion are excluded. The O <sub>1s</sub> level in the bulk-like region of the water slab, <span style="color:#3E70EA"><math>\left\langle\epsilon_{1s,\mathrm{slab}}\right\rangle</math></span>, connects this reference to the vacuum level estimated from the slab calculation. Their difference gives the approximate potential offset <span style="color:#8342A4"><math>e\Delta\bar{\phi}_{\lambda}</math></span> and hence the chemical potential of the vacuum electron expressed in the bulk energy reference. Reproducing Fig. 1 of Ref. {{Cite\|jinnouchi:ncm:2024}}.]]

	~~== Endpoint approximation for the half-cell reaction ==~~
	For the Fe<sup>2+</sup>/Fe<sup>3+</sup> reaction, the path-averaged O <math>1s</math> level is approximated by the mean of the endpoint values,		For the Fe<sup>2+</sup>/Fe<sup>3+</sup> reaction, the path-averaged O <math>1s</math> level is approximated by the mean of the endpoint values,

Csheldon: /* Averaging along the integration path */

2026-09-18T12:35:04Z

Averaging along the integration path

← Older revision		Revision as of 12:35, 18 September 2026
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	where <math>\Delta\bar{\phi}_{\lambda}</math> is the vacuum-to-bulk potential offset at coupling parameter <math>\lambda</math>.		where <math>\Delta\bar{\phi}_{\lambda}</math> is the vacuum-to-bulk potential offset at coupling parameter <math>\lambda</math>.
	~~== Averaging along the integration path ==~~
	Since the alignment depends on <math>\lambda</math>, the consistently aligned free-energy difference may be written as		Since the alignment depends on <math>\lambda</math>, the consistently aligned free-energy difference may be written as

Csheldon: Split the derivation into named sections, add an overview paragraph, drop the tip box that repeated the text above it

2026-09-18T12:33:12Z

Split the derivation into named sections, add an overview paragraph, drop the tip box that repeated the text above it

← Older revision		Revision as of 12:33, 18 September 2026
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	This page derives the vacuum alignment <math>e\Delta\bar{\phi}</math> that enters the free-energy difference of an electrochemical half-cell, and states the approximations it rests on. For the procedure that computes it, see [[Vacuum reference]].		This page derives the vacuum alignment <math>e\Delta\bar{\phi}</math> that enters the free-energy difference of an electrochemical half-cell, and states the approximations it rests on. For the procedure that computes it, see [[Vacuum reference]].

	== ~~Theory~~ ==		The derivation proceeds in four steps. The free energy of the half-cell reaction is written with an explicit electron-reservoir term, that term is expressed relative to the energy zero of the periodic calculation, the result is averaged along the integration path, and the offset itself is estimated from the oxygen core levels.

			== Free energy with an electron reservoir ==
	During the Fe<sup>2+</sup>/Fe<sup>3+</sup> half-cell reaction, one electron is exchanged with an electron reservoir. The Helmholtz free-energy difference is		During the Fe<sup>2+</sup>/Fe<sup>3+</sup> half-cell reaction, one electron is exchanged with an electron reservoir. The Helmholtz free-energy difference is

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	where <math>\left\langle U_1-U_0\right\rangle_{\lambda}</math> is the ensemble-averaged potential-energy difference between an interacting system (<math>\lambda=1</math>) and non-interacting system (<math>\lambda=0</math>) taken from thermodynamic integration with coupling parameter <math>\lambda</math>, <math>\mu</math> is the electron chemical potential expressed relative to the same energy reference as the calculated energies, and <math>n</math> is the signed number of electrons exchanged with the reservoir.		where <math>\left\langle U_1-U_0\right\rangle_{\lambda}</math> is the ensemble-averaged potential-energy difference between an interacting system (<math>\lambda=1</math>) and non-interacting system (<math>\lambda=0</math>) taken from thermodynamic integration with coupling parameter <math>\lambda</math>, <math>\mu</math> is the electron chemical potential expressed relative to the same energy reference as the calculated energies, and <math>n</math> is the signed number of electrons exchanged with the reservoir.

			== Aligning the bulk calculation to the vacuum ==
	The desired absolute reference is an electron at rest in vacuum, <math>\mu_{\mathrm{vac}}\equiv0~\mathrm{eV}.</math>		The desired absolute reference is an electron at rest in vacuum, <math>\mu_{\mathrm{vac}}\equiv0~\mathrm{eV}.</math>
	This does not mean that the chemical potential appearing in the periodic bulk calculation can be set to zero. The bulk calculation contains no vacuum region, and its electrostatic potential has an arbitrary additive constant. Consequently, <math>\mu_{\mathrm{vac}}</math> must first be expressed relative to the energy zero of each bulk calculation.		This does not mean that the chemical potential appearing in the periodic bulk calculation can be set to zero. The bulk calculation contains no vacuum region, and its electrostatic potential has an arbitrary additive constant. Consequently, <math>\mu_{\mathrm{vac}}</math> must first be expressed relative to the energy zero of each bulk calculation.
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	where <math>\Delta\bar{\phi}_{\lambda}</math> is the vacuum-to-bulk potential offset at coupling parameter <math>\lambda</math>.		where <math>\Delta\bar{\phi}_{\lambda}</math> is the vacuum-to-bulk potential offset at coupling parameter <math>\lambda</math>.
			== Averaging along the integration path ==
	Since the alignment depends on <math>\lambda</math>, the consistently aligned free-energy difference may be written as		Since the alignment depends on <math>\lambda</math>, the consistently aligned free-energy difference may be written as

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	</math>		</math>

			== Estimating the offset from the O 1s levels ==
	For bulk-like water molecules, the O <math>1s</math> energy may be written schematically as		For bulk-like water molecules, the O <math>1s</math> energy may be written schematically as

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	[[File:Vacuum_ref.png\|600px\|thumb\|center\|'''Figure 1'''. Alignment of the periodic bulk-water calculation with the vacuum reference. The average O <sub>1s</sub> level of water molecules far from the Fe ion, <span style="color:#A82C35"><math>\left\langle\epsilon_{1s,\mathrm{bulk}}\right\rangle_{\lambda}</math></span>, is used as the bulk reference. Water molecules within half a unit cell of the ion are excluded. The O <sub>1s</sub> level in the bulk-like region of the water slab, <span style="color:#3E70EA"><math>\left\langle\epsilon_{1s,\mathrm{slab}}\right\rangle</math></span>, connects this reference to the vacuum level estimated from the slab calculation. Their difference gives the approximate potential offset <span style="color:#8342A4"><math>e\Delta\bar{\phi}_{\lambda}</math></span> and hence the chemical potential of the vacuum electron expressed in the bulk energy reference. Reproducing Fig. 1 of Ref. {{Cite\|jinnouchi:ncm:2024}}.]]		[[File:Vacuum_ref.png\|600px\|thumb\|center\|'''Figure 1'''. Alignment of the periodic bulk-water calculation with the vacuum reference. The average O <sub>1s</sub> level of water molecules far from the Fe ion, <span style="color:#A82C35"><math>\left\langle\epsilon_{1s,\mathrm{bulk}}\right\rangle_{\lambda}</math></span>, is used as the bulk reference. Water molecules within half a unit cell of the ion are excluded. The O <sub>1s</sub> level in the bulk-like region of the water slab, <span style="color:#3E70EA"><math>\left\langle\epsilon_{1s,\mathrm{slab}}\right\rangle</math></span>, connects this reference to the vacuum level estimated from the slab calculation. Their difference gives the approximate potential offset <span style="color:#8342A4"><math>e\Delta\bar{\phi}_{\lambda}</math></span> and hence the chemical potential of the vacuum electron expressed in the bulk energy reference. Reproducing Fig. 1 of Ref. {{Cite\|jinnouchi:ncm:2024}}.]]

	~~{{NB\|tip\|Although~~ the ~~vacuum electron has <math>\mu_{\mathrm{vac}}~~=~~0</math> on the vacuum scale, its chemical potential expressed relative to the arbitrary energy zero of the periodic bulk calculation is <math>\mu_{\mathrm{bulk},\lambda}~~=~~e\Delta\bar{\phi}_{\lambda}</math> and is generally nonzero.}}~~		== Endpoint approximation for the half-cell reaction ==

	For the Fe<sup>2+</sup>/Fe<sup>3+</sup> reaction, the path-averaged O <math>1s</math> level is approximated by the mean of the endpoint values,		For the Fe<sup>2+</sup>/Fe<sup>3+</sup> reaction, the path-averaged O <math>1s</math> level is approximated by the mean of the endpoint values,

Csheldon: Cite the 2025 core-level paper; drop the self-piped links left over from the namespace move

2026-09-18T12:30:34Z

Cite the 2025 core-level paper; drop the self-piped links left over from the namespace move

← Older revision		Revision as of 12:30, 18 September 2026
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	Finding a reference to compare between experiments or calculations can be challenging. Experimentally, there is usually a standard reference taken, e.g., the standard hydrogen electrode (SHE) under standard conditions. Computationally, it can be difficult to even compare the energies of cells if a reference is not carefully taken. The local potential cannot be used for a cell because it is relative to the individual cell. Instead, it is more useful to find a common reference, e.g., the vacuum.		Finding a reference to compare between experiments or calculations can be challenging. Experimentally, there is usually a standard reference taken, e.g., the standard hydrogen electrode (SHE) under standard conditions. Computationally, it can be difficult to even compare the energies of cells if a reference is not carefully taken. The local potential cannot be used for a cell because it is relative to the individual cell. Instead, it is more useful to find a common reference, e.g., the vacuum.

	This page derives the vacuum alignment <math>e\Delta\bar{\phi}</math> that enters the free-energy difference of an electrochemical half-cell, and states the approximations it rests on. For the procedure that computes it, see [[~~Vacuum reference\|~~Vacuum reference]].		This page derives the vacuum alignment <math>e\Delta\bar{\phi}</math> that enters the free-energy difference of an electrochemical half-cell, and states the approximations it rests on. For the procedure that computes it, see [[Vacuum reference]].

	== Theory ==		== Theory ==
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	This does not mean that the chemical potential appearing in the periodic bulk calculation can be set to zero. The bulk calculation contains no vacuum region, and its electrostatic potential has an arbitrary additive constant. Consequently, <math>\mu_{\mathrm{vac}}</math> must first be expressed relative to the energy zero of each bulk calculation.		This does not mean that the chemical potential appearing in the periodic bulk calculation can be set to zero. The bulk calculation contains no vacuum region, and its electrostatic potential has an arbitrary additive constant. Consequently, <math>\mu_{\mathrm{vac}}</math> must first be expressed relative to the energy zero of each bulk calculation.

	A water-slab calculation is used to establish this alignment. The electrostatic potential in its nominal vacuum region provides an estimate of the vacuum level. Because the slab and vacuum regions are finite, this estimate may be affected by interfacial fluctuations and interactions between periodic images and is therefore an approximation to the macroscopic vacuum level. The O <math>1s</math> level is used to transfer the reference from the slab to the bulk calculation. It is a deep, strongly localized state whose energy follows shifts of the local electrostatic potential. Its dependence on the local chemical environment is reduced by averaging over many water molecules and configurations. In the bulk calculation, water molecules within half a unit cell of the Fe ion are excluded to avoid ion-specific core-level shifts.		A water-slab calculation is used to establish this alignment. The electrostatic potential in its nominal vacuum region provides an estimate of the vacuum level. Because the slab and vacuum regions are finite, this estimate may be affected by interfacial fluctuations and interactions between periodic images and is therefore an approximation to the macroscopic vacuum level. The O <math>1s</math> level is used to transfer the reference from the slab to the bulk calculation. It is a deep, strongly localized state whose energy follows shifts of the local electrostatic potential. Its dependence on the local chemical environment is reduced by averaging over many water molecules and configurations. In the bulk calculation, water molecules within half a unit cell of the Fe ion are excluded to avoid ion-specific core-level shifts. A more detailed account of core-level calculations is given in Ref. {{Cite\|jinnouchi:chemsci:2025}}.

	The chemical potential of the vacuum electron expressed relative to the bulk energy zero at coupling parameter <math>\lambda</math> is approximated by		The chemical potential of the vacuum electron expressed relative to the bulk energy zero at coupling parameter <math>\lambda</math> is approximated by
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	== Related tags and articles ==		== Related tags and articles ==
	;How-tos		;How-tos
	* [[~~Vacuum reference\|~~Vacuum reference]]		* [[Vacuum reference]]
	* [[~~Thermodynamic integration: redox potential\|~~Thermodynamic integration: redox potential]]		* [[Thermodynamic integration: redox potential]]
	;Theory		;Theory
	* [[Thermodynamic integration]]		* [[Thermodynamic integration]]

Csheldon: Csheldon moved page Vacuum reference: theory to Vacuum reference theory

2026-09-18T12:27:11Z

Csheldon moved page Vacuum reference: theory to Vacuum reference theory

← Older revision	Revision as of 12:27, 18 September 2026
(No difference)

Csheldon: /* Theory */

2026-09-18T12:26:36Z

Theory

← Older revision		Revision as of 12:26, 18 September 2026
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	The desired absolute reference is an electron at rest in vacuum, <math>\mu_{\mathrm{vac}}\equiv0~\mathrm{eV}.</math>		The desired absolute reference is an electron at rest in vacuum, <math>\mu_{\mathrm{vac}}\equiv0~\mathrm{eV}.</math>
	This does not mean that the chemical potential appearing in the periodic bulk calculation can be set to zero. The bulk calculation contains no vacuum region, and its electrostatic potential has an arbitrary additive constant. Consequently, <math>\mu_{\mathrm{vac}}</math> must first be expressed relative to the energy zero of each bulk calculation.		This does not mean that the chemical potential appearing in the periodic bulk calculation can be set to zero. The bulk calculation contains no vacuum region, and its electrostatic potential has an arbitrary additive constant. Consequently, <math>\mu_{\mathrm{vac}}</math> must first be expressed relative to the energy zero of each bulk calculation.
	A water-slab calculation is used to establish this alignment. The electrostatic potential in its nominal vacuum region provides an estimate of the vacuum level. Because the slab and vacuum regions are finite, this estimate may be affected by interfacial fluctuations and interactions between periodic images and is therefore an approximation to the macroscopic vacuum level.
	The O <math>1s</math> level is used to transfer the reference from the slab to the bulk calculation. It is a deep, strongly localized state whose energy follows shifts of the local electrostatic potential. Its dependence on the local chemical environment is reduced by averaging over many water molecules and configurations. In the bulk calculation, water molecules within half a unit cell of the Fe ion are excluded to avoid ion-specific core-level shifts.		A water-slab calculation is used to establish this alignment. The electrostatic potential in its nominal vacuum region provides an estimate of the vacuum level. Because the slab and vacuum regions are finite, this estimate may be affected by interfacial fluctuations and interactions between periodic images and is therefore an approximation to the macroscopic vacuum level. The O <math>1s</math> level is used to transfer the reference from the slab to the bulk calculation. It is a deep, strongly localized state whose energy follows shifts of the local electrostatic potential. Its dependence on the local chemical environment is reduced by averaging over many water molecules and configurations. In the bulk calculation, water molecules within half a unit cell of the Fe ion are excluded to avoid ion-specific core-level shifts.

	The chemical potential of the vacuum electron expressed relative to the bulk energy zero at coupling parameter <math>\lambda</math> is approximated by		The chemical potential of the vacuum electron expressed relative to the bulk energy zero at coupling parameter <math>\lambda</math> is approximated by

Csheldon: Point the links at the published titles; categories made live

2026-09-18T08:03:54Z

Point the links at the published titles; categories made live

← Older revision		Revision as of 08:03, 18 September 2026
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	Finding a reference to compare between experiments or calculations can be challenging. Experimentally, there is usually a standard reference taken, e.g., the standard hydrogen electrode (SHE) under standard conditions. Computationally, it can be difficult to even compare the energies of cells if a reference is not carefully taken. The local potential cannot be used for a cell because it is relative to the individual cell. Instead, it is more useful to find a common reference, e.g., the vacuum.		Finding a reference to compare between experiments or calculations can be challenging. Experimentally, there is usually a standard reference taken, e.g., the standard hydrogen electrode (SHE) under standard conditions. Computationally, it can be difficult to even compare the energies of cells if a reference is not carefully taken. The local potential cannot be used for a cell because it is relative to the individual cell. Instead, it is more useful to find a common reference, e.g., the vacuum.

	This page derives the vacuum alignment <math>e\Delta\bar{\phi}</math> that enters the free-energy difference of an electrochemical half-cell, and states the approximations it rests on. For the procedure that computes it, see [[~~Construction:~~Vacuum reference\|Vacuum reference]].		This page derives the vacuum alignment <math>e\Delta\bar{\phi}</math> that enters the free-energy difference of an electrochemical half-cell, and states the approximations it rests on. For the procedure that computes it, see [[Vacuum reference\|Vacuum reference]].

	== Theory ==		== Theory ==
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	== Related tags and articles ==		== Related tags and articles ==
	;How-tos		;How-tos
	* [[~~Construction:~~Vacuum reference\|Vacuum reference]]		* [[Vacuum reference\|Vacuum reference]]
	* [[~~Construction:~~Thermodynamic integration: redox potential\|Thermodynamic integration: redox potential]]		* [[Thermodynamic integration: redox potential\|Thermodynamic integration: redox potential]]
	;Theory		;Theory
	* [[Thermodynamic integration]]		* [[Thermodynamic integration]]
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	<references/>		<references/>

	~~<!--~~[[Category:Theory]][[Category:Advanced molecular-dynamics sampling]][[Category:Electrostatics]][[Category:Potential]][[Category:Electrochemistry]]-->		[[Category:Theory]][[Category:Advanced molecular-dynamics sampling]][[Category:Electrostatics]][[Category:Potential]]
			<!--[[Category:Electrochemistry]]-->

Csheldon: Csheldon moved page Construction:Vacuum reference: theory to Vacuum reference: theory

2026-09-18T08:00:33Z

Csheldon moved page Construction:Vacuum reference: theory to Vacuum reference: theory

← Older revision	Revision as of 08:00, 18 September 2026
(No difference)

Csheldon: Drop a banned intensifier; the INCAR contains tags, not parameters

2026-09-17T16:10:20Z

Drop a banned intensifier; the INCAR contains tags, not parameters

← Older revision		Revision as of 16:10, 17 September 2026
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	The desired absolute reference is an electron at rest in vacuum, <math>\mu_{\mathrm{vac}}\equiv0~\mathrm{eV}.</math>		The desired absolute reference is an electron at rest in vacuum, <math>\mu_{\mathrm{vac}}\equiv0~\mathrm{eV}.</math>
	This does not mean that the chemical potential appearing in the periodic bulk calculation can ~~simply~~ be set to zero. The bulk calculation contains no vacuum region, and its electrostatic potential has an arbitrary additive constant. Consequently, <math>\mu_{\mathrm{vac}}</math> must first be expressed relative to the energy zero of each bulk calculation.		This does not mean that the chemical potential appearing in the periodic bulk calculation can be set to zero. The bulk calculation contains no vacuum region, and its electrostatic potential has an arbitrary additive constant. Consequently, <math>\mu_{\mathrm{vac}}</math> must first be expressed relative to the energy zero of each bulk calculation.
	A water-slab calculation is used to establish this alignment. The electrostatic potential in its nominal vacuum region provides an estimate of the vacuum level. Because the slab and vacuum regions are finite, this estimate may be affected by interfacial fluctuations and interactions between periodic images and is therefore an approximation to the macroscopic vacuum level.		A water-slab calculation is used to establish this alignment. The electrostatic potential in its nominal vacuum region provides an estimate of the vacuum level. Because the slab and vacuum regions are finite, this estimate may be affected by interfacial fluctuations and interactions between periodic images and is therefore an approximation to the macroscopic vacuum level.
	The O <math>1s</math> level is used to transfer the reference from the slab to the bulk calculation. It is a deep, strongly localized state whose energy follows shifts of the local electrostatic potential. Its dependence on the local chemical environment is reduced by averaging over many water molecules and configurations. In the bulk calculation, water molecules within half a unit cell of the Fe ion are excluded to avoid ion-specific core-level shifts.		The O <math>1s</math> level is used to transfer the reference from the slab to the bulk calculation. It is a deep, strongly localized state whose energy follows shifts of the local electrostatic potential. Its dependence on the local chemical environment is reduced by averaging over many water molecules and configurations. In the bulk calculation, water molecules within half a unit cell of the Fe ion are excluded to avoid ion-specific core-level shifts.