Thermodynamic integration: redox potential - Revision history

Csheldon: Move the TPT note under Step 4, where the GGA integration it follows on from is described

2026-09-18T14:12:02Z

Move the TPT note under Step 4, where the GGA integration it follows on from is described

← Older revision		Revision as of 14:12, 18 September 2026
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	Run a molecular dynamics (MD) simulation, then calculate the local potential <math>\phi</math> of the water slab. Using this, the difference between the Oxygen ''1s'' level between the water slab and the water molecules far away from the ions in the redox cells can be calculated. This provides the [[Vacuum reference\|chemical potential relative to the vacuum]] <math>e \Delta \bar{\phi}</math>.		Run a molecular dynamics (MD) simulation, then calculate the local potential <math>\phi</math> of the water slab. Using this, the difference between the Oxygen ''1s'' level between the water slab and the water molecules far away from the ions in the redox cells can be calculated. This provides the [[Vacuum reference\|chemical potential relative to the vacuum]] <math>e \Delta \bar{\phi}</math>.
	{{NB\|important\|This step requires using GGAs on many structures, so will be more time-consuming.}}		{{NB\|important\|This step requires using GGAs on many structures, so will be more time-consuming.}}
	{{NB\|mind\|Thermodynamic perturbation theory (TPT) from RPBE+D3 to PBE0, for both Fe<sup>3+</sup> and Fe<sup>2+</sup>, is not covered here because of its cost. It can be replicated using Δ-learning from RPBE to PBE0, using the same procedure as from MLFF to RPBE.}}

	=== Step 3: TI from oxidized to reduced state ===		=== Step 3: TI from oxidized to reduced state ===
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	Perform a thermodynamic integration from [[Thermodynamic integration between a machine-learned force field and a density functional\|MLFF (λ=0) to RPBE+D3 (λ=1)]]. This should be done separately for Fe<sup>3+</sup> and Fe<sup>2+</sup>.		Perform a thermodynamic integration from [[Thermodynamic integration between a machine-learned force field and a density functional\|MLFF (λ=0) to RPBE+D3 (λ=1)]]. This should be done separately for Fe<sup>3+</sup> and Fe<sup>2+</sup>.
	{{NB\|important\|This step requires using GGAs on many structures so will be more time consuming.}}		{{NB\|important\|This step requires using GGAs on many structures so will be more time consuming.}}
			{{NB\|mind\|Thermodynamic perturbation theory (TPT) from RPBE+D3 to PBE0, for both Fe<sup>3+</sup> and Fe<sup>2+</sup>, is not covered here because of its cost. It can be replicated using Δ-learning from RPBE to PBE0, using the same procedure as from MLFF to RPBE.}}

	=== Step 5: Analysis ===		=== Step 5: Analysis ===

Csheldon: Renumber the analysis step now that the TPT step is a note

2026-09-18T14:11:23Z

Renumber the analysis step now that the TPT step is a note

← Older revision		Revision as of 14:11, 18 September 2026
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	{{NB\|important\|This step requires using GGAs on many structures so will be more time consuming.}}		{{NB\|important\|This step requires using GGAs on many structures so will be more time consuming.}}

	=== Step 6: Analysis ===		=== Step 5: Analysis ===
	Once each step is completed, you will have the following values:		Once each step is completed, you will have the following values:
	* the chemical potential - <math>e \Delta \bar{\phi}</math>		* the chemical potential - <math>e \Delta \bar{\phi}</math>

Csheldon: Use the Step N: heading form; state how the terms combine; fold the TPT step into a note; give real recommendations

2026-09-18T14:11:02Z

Use the Step N: heading form; state how the terms combine; fold the TPT step into a note; give real recommendations

← Older revision		Revision as of 14:11, 18 September 2026
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	== Step-by-step instructions ==		== Step-by-step instructions ==
	=== 1. Preparing the MLFFs ===		=== Step 1: Preparing the MLFFs ===
	First, [[Training machine-learned force fields for redox reactions\|train the MLFFs]] for the water slab (<code>128H2O_slab</code>) and the redox pairs (<math>[\mathrm{Fe}(\mathrm{H}_2\mathrm{O})_n]^{3+}</math> and <math>[\mathrm{Fe}(\mathrm{H}_2\mathrm{O})_n]^{2+}</math>).		First, [[Training machine-learned force fields for redox reactions\|train the MLFFs]] for the water slab (<code>128H2O_slab</code>) and the redox pairs (<math>[\mathrm{Fe}(\mathrm{H}_2\mathrm{O})_n]^{3+}</math> and <math>[\mathrm{Fe}(\mathrm{H}_2\mathrm{O})_n]^{2+}</math>).

	=== 2. Calculating the vacuum potential ===		=== Step 2: Calculating the vacuum potential ===
	Run a molecular dynamics (MD) simulation, then calculate the local potential <math>\phi</math> of the water slab. Using this, the difference between the Oxygen ''1s'' level between the water slab and the water molecules far away from the ions in the redox cells can be calculated. This provides the [[Vacuum reference\|chemical potential relative to the vacuum]] <math>e \Delta \bar{\phi}</math>.		Run a molecular dynamics (MD) simulation, then calculate the local potential <math>\phi</math> of the water slab. Using this, the difference between the Oxygen ''1s'' level between the water slab and the water molecules far away from the ions in the redox cells can be calculated. This provides the [[Vacuum reference\|chemical potential relative to the vacuum]] <math>e \Delta \bar{\phi}</math>.
	{{NB\|important\|This step requires using GGAs on many structures, so will be more time-consuming.}}		{{NB\|important\|This step requires using GGAs on many structures, so will be more time-consuming.}}
			{{NB\|mind\|Thermodynamic perturbation theory (TPT) from RPBE+D3 to PBE0, for both Fe<sup>3+</sup> and Fe<sup>2+</sup>, is not covered here because of its cost. It can be replicated using Δ-learning from RPBE to PBE0, using the same procedure as from MLFF to RPBE.}}

	=== 3. TI from oxidized to reduced state ===		=== Step 3: TI from oxidized to reduced state ===
	Perform a thermodynamic integration from the [[Thermodynamic integration between machine-learned force fields\|oxidized state (λ=0; Fe<sup>3+</sup>) to the reduced state (λ=1; Fe<sup>2+</sup>)]].		Perform a thermodynamic integration from the [[Thermodynamic integration between machine-learned force fields\|oxidized state (λ=0; Fe<sup>3+</sup>) to the reduced state (λ=1; Fe<sup>2+</sup>)]].
	{{NB\|mind\|This is a thermodynamic integration between two MLFFs, so is very quick.}}		{{NB\|mind\|This is a thermodynamic integration between two MLFFs, so is very quick.}}

	=== 4. TI from MLFF to GGA ===		=== Step 4: TI from MLFF to GGA ===
	Perform a thermodynamic integration from [[Thermodynamic integration between a machine-learned force field and a density functional\|MLFF (λ=0) to RPBE+D3 (λ=1)]]. This should be done separately for Fe<sup>3+</sup> and Fe<sup>2+</sup>.		Perform a thermodynamic integration from [[Thermodynamic integration between a machine-learned force field and a density functional\|MLFF (λ=0) to RPBE+D3 (λ=1)]]. This should be done separately for Fe<sup>3+</sup> and Fe<sup>2+</sup>.
	{{NB\|important\|This step requires using GGAs on many structures so will be more time consuming.}}		{{NB\|important\|This step requires using GGAs on many structures so will be more time consuming.}}

	=== ~~5. TPT from GGA to hybrid (not covered here) ===~~		=== Step 6: Analysis ===
	~~Due to the high cost of converting from RPBE to PBE0, thermodynamic perturbation theory (TPT) is not covered here~~:
	* From RPBE+D3 to PBE0 (Fe<sup>3+</sup>)
	* From RPBE+D3 to PBE0 (Fe<sup>2+</sup>)
	~~{{NB\|tip\|TPT can be replicated using Δ-learning from RPBE to PBE0, using the same procedure as from MLFF to RPBE.}}~~

	~~=== 6.~~ Analysis ===
	Once each step is completed, you will have the following values:		Once each step is completed, you will have the following values:
	* the chemical potential - <math>e \Delta \bar{\phi}</math>		* the chemical potential - <math>e \Delta \bar{\phi}</math>
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	:<math>\Delta A^{\mathrm{FP}_\mathrm{GGA}} = \Delta A ^{\mathrm{ML}} + \Delta A^{\mathrm{FP}_\mathrm{GGA}-\mathrm{ML}}_1 - \Delta A^{\mathrm{FP}_\mathrm{GGA}-\mathrm{ML}}_0</math>		:<math>\Delta A^{\mathrm{FP}_\mathrm{GGA}} = \Delta A ^{\mathrm{ML}} + \Delta A^{\mathrm{FP}_\mathrm{GGA}-\mathrm{ML}}_1 - \Delta A^{\mathrm{FP}_\mathrm{GGA}-\mathrm{ML}}_0</math>

	~~Take a summation over these to obtain~~ the free energy difference from Fe<sup>3+</sup> to Fe<sup>2+</sup> on the GGA level <math>\Delta A^{\mathrm{FP}_\mathrm{GGA}}</math>. Switch the sign to get the absolute redox potential <math>U^{\mathrm{FP}_\mathrm{GGA}}_{\mathrm{redox}}</math>. Finally, take the standard hydrogen electrode (SHE) (4.44 V {{Cite\|trasatti:pac:1986}}) away to compare to experiment <math>U^{\mathrm{FP}_\mathrm{GGA}}_{\mathrm{redox,SHE}}</math>. The results below match the literature to within 30 meV.		The vacuum alignment is already contained in <math>\Delta A^{\mathrm{ML}}</math>. Combining the remaining terms as above gives the free energy difference from Fe<sup>3+</sup> to Fe<sup>2+</sup> on the GGA level <math>\Delta A^{\mathrm{FP}_\mathrm{GGA}}</math>. Switch the sign to get the absolute redox potential <math>U^{\mathrm{FP}_\mathrm{GGA}}_{\mathrm{redox}}</math>. Finally, take the standard hydrogen electrode (SHE) (4.44 V {{Cite\|trasatti:pac:1986}}) away to compare to experiment <math>U^{\mathrm{FP}_\mathrm{GGA}}_{\mathrm{redox,SHE}}</math>. The results below match the literature to within 30 meV.

	The results are summarized in the following table:		The results are summarized in the following table:
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	== Recommendations and advice ==		== Recommendations and advice ==
	*~~These~~ are ~~provided~~ in ~~each section independently~~.		*The vacuum-reference step and the MLFF:GGA integrations are by far the most expensive parts of the workflow, since both evaluate a GGA on many structures. Training the MLFFs in the first place can also be time consuming. The MLFF:MLFF integration, by contrast, is very quick.
			<!--*TPT can be performed using Δ-ML. This is not yet available, so it is hidden for now.-->

	== Related tags and articles ==		== Related tags and articles ==

Csheldon: /* 1. Preparing the MLFFs */

2026-09-18T08:33:55Z

1. Preparing the MLFFs

← Older revision		Revision as of 08:33, 18 September 2026
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	== Step-by-step instructions ==		== Step-by-step instructions ==
	=== 1. Preparing the MLFFs ===		=== 1. Preparing the MLFFs ===
	First, [[Training machine-learned force fields for redox reactions~~#Training_of_water_slab~~\|train the MLFFs]] for the water slab (<code>128H2O_slab</code>) and the redox pairs (<math>[\mathrm{Fe}(\mathrm{H}_2\mathrm{O})_n]^{3+}</math> and <math>[\mathrm{Fe}(\mathrm{H}_2\mathrm{O})_n]^{2+}</math>).		First, [[Training machine-learned force fields for redox reactions\|train the MLFFs]] for the water slab (<code>128H2O_slab</code>) and the redox pairs (<math>[\mathrm{Fe}(\mathrm{H}_2\mathrm{O})_n]^{3+}</math> and <math>[\mathrm{Fe}(\mathrm{H}_2\mathrm{O})_n]^{2+}</math>).

	=== 2. Calculating the vacuum potential ===		=== 2. Calculating the vacuum potential ===

Csheldon: Point at the published title; hold the unpublished density-functionals link; make categories live

2026-09-18T08:07:30Z

Point at the published title; hold the unpublished density-functionals link; make categories live

← Older revision		Revision as of 08:07, 18 September 2026
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	\| <math>\Delta A_{\mathrm{ML}}</math>		\| <math>\Delta A_{\mathrm{ML}}</math>
	\|-		\|-
	\| [[~~Construction:~~Thermodynamic integration between a machine-learned force field and a density functional\|MLFF:GGA]]		\| [[Thermodynamic integration between a machine-learned force field and a density functional\|MLFF:GGA]]
	\| MLFF (Fe<sup>3+</sup>)		\| MLFF (Fe<sup>3+</sup>)
	\| RPBE+D3 (Fe<sup>3+</sup>)		\| RPBE+D3 (Fe<sup>3+</sup>)
	\| <math>\Delta A^{\mathrm{FP}_\mathrm{GGA}-\mathrm{ML}}_0</math>		\| <math>\Delta A^{\mathrm{FP}_\mathrm{GGA}-\mathrm{ML}}_0</math>
	\|-		\|-
	\| [[~~Construction:~~Thermodynamic integration between a machine-learned force field and a density functional\|MLFF:GGA]]		\| [[Thermodynamic integration between a machine-learned force field and a density functional\|MLFF:GGA]]
	\| MLFF (Fe<sup>2+</sup>)		\| MLFF (Fe<sup>2+</sup>)
	\| RPBE+D3 (Fe<sup>2+</sup>)		\| RPBE+D3 (Fe<sup>2+</sup>)
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	=== 4. TI from MLFF to GGA ===		=== 4. TI from MLFF to GGA ===
	Perform a thermodynamic integration from [[~~Construction:~~Thermodynamic integration between a machine-learned force field and a density functional\|MLFF (λ=0) to RPBE+D3 (λ=1)]]. This should be done separately for Fe<sup>3+</sup> and Fe<sup>2+</sup>.		Perform a thermodynamic integration from [[Thermodynamic integration between a machine-learned force field and a density functional\|MLFF (λ=0) to RPBE+D3 (λ=1)]]. This should be done separately for Fe<sup>3+</sup> and Fe<sup>2+</sup>.
	{{NB\|important\|This step requires using GGAs on many structures so will be more time consuming.}}		{{NB\|important\|This step requires using GGAs on many structures so will be more time consuming.}}

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	\| -4.95		\| -4.95
	\|-		\|-
	\| [[~~Construction:~~Thermodynamic integration between a machine-learned force field and a density functional\|'''4 - TI (GGA)''']]		\| [[Thermodynamic integration between a machine-learned force field and a density functional\|'''4 - TI (GGA)''']]
	\| <math>\Delta A^{\mathrm{FP}_\mathrm{GGA}-\mathrm{ML}}_1 - \Delta A^{\mathrm{FP}_\mathrm{GGA}-\mathrm{ML}}_0</math>		\| <math>\Delta A^{\mathrm{FP}_\mathrm{GGA}-\mathrm{ML}}_1 - \Delta A^{\mathrm{FP}_\mathrm{GGA}-\mathrm{ML}}_0</math>
	\| -0.055		\| -0.055
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	* [[Vacuum reference]]		* [[Vacuum reference]]
	* [[Thermodynamic integration between machine-learned force fields]]		* [[Thermodynamic integration between machine-learned force fields]]
	* [[~~Construction:~~Thermodynamic integration between a machine-learned force field and a density functional\|Thermodynamic integration between a machine-learned force field and a density functional]]		* [[Thermodynamic integration between a machine-learned force field and a density functional\|Thermodynamic integration between a machine-learned force field and a density functional]]
	;Theory		;Theory
	* [[Thermodynamic integration]]		* [[Thermodynamic integration]]

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

2026-09-18T08:03:52Z

Point the links at the published titles; categories made live

← Older revision		Revision as of 08:03, 18 September 2026
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	\| '''Term'''		\| '''Term'''
	\|-		\|-
	\| [[~~Construction:~~Thermodynamic integration between ~~MLFFs~~\|MLFF:MLFF]]		\| [[Thermodynamic integration between machine-learned force fields\|MLFF:MLFF]]
	\| ''Oxidized'' (Fe<sup>3+</sup>)		\| ''Oxidized'' (Fe<sup>3+</sup>)
	\| ''Reduced'' (Fe<sup>2+</sup>)		\| ''Reduced'' (Fe<sup>2+</sup>)
	\| <math>\Delta A_{\mathrm{ML}}</math>		\| <math>\Delta A_{\mathrm{ML}}</math>
	\|-		\|-
	\| [[Construction:Thermodynamic integration between ~~MLFF~~ and ~~GGAs~~\|MLFF:GGA]]		\| [[Construction:Thermodynamic integration between a machine-learned force field and a density functional\|MLFF:GGA]]
	\| MLFF (Fe<sup>3+</sup>)		\| MLFF (Fe<sup>3+</sup>)
	\| RPBE+D3 (Fe<sup>3+</sup>)		\| RPBE+D3 (Fe<sup>3+</sup>)
	\| <math>\Delta A^{\mathrm{FP}_\mathrm{GGA}-\mathrm{ML}}_0</math>		\| <math>\Delta A^{\mathrm{FP}_\mathrm{GGA}-\mathrm{ML}}_0</math>
	\|-		\|-
	\| [[Construction:Thermodynamic integration between ~~MLFF~~ and ~~GGAs~~\|MLFF:GGA]]		\| [[Construction:Thermodynamic integration between a machine-learned force field and a density functional\|MLFF:GGA]]
	\| MLFF (Fe<sup>2+</sup>)		\| MLFF (Fe<sup>2+</sup>)
	\| RPBE+D3 (Fe<sup>2+</sup>)		\| RPBE+D3 (Fe<sup>2+</sup>)
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	== Step-by-step instructions ==		== Step-by-step instructions ==
	=== 1. Preparing the MLFFs ===		=== 1. Preparing the MLFFs ===
	First, [[~~Construction:~~Training ~~MLFF~~ for redox reactions#Training_of_water_slab\|train the MLFFs]] for the water slab (<code>128H2O_slab</code>) and the redox pairs (<math>[\mathrm{Fe}(\mathrm{H}_2\mathrm{O})_n]^{3+}</math> and <math>[\mathrm{Fe}(\mathrm{H}_2\mathrm{O})_n]^{2+}</math>).		First, [[Training machine-learned force fields for redox reactions#Training_of_water_slab\|train the MLFFs]] for the water slab (<code>128H2O_slab</code>) and the redox pairs (<math>[\mathrm{Fe}(\mathrm{H}_2\mathrm{O})_n]^{3+}</math> and <math>[\mathrm{Fe}(\mathrm{H}_2\mathrm{O})_n]^{2+}</math>).

	=== 2. Calculating the vacuum potential ===		=== 2. Calculating the vacuum potential ===
	Run a molecular dynamics (MD) simulation, then calculate the local potential <math>\phi</math> of the water slab. Using this, the difference between the Oxygen ''1s'' level between the water slab and the water molecules far away from the ions in the redox cells can be calculated. This provides the [[~~Construction:~~Vacuum reference\|chemical potential relative to the vacuum]] <math>e \Delta \bar{\phi}</math>.		Run a molecular dynamics (MD) simulation, then calculate the local potential <math>\phi</math> of the water slab. Using this, the difference between the Oxygen ''1s'' level between the water slab and the water molecules far away from the ions in the redox cells can be calculated. This provides the [[Vacuum reference\|chemical potential relative to the vacuum]] <math>e \Delta \bar{\phi}</math>.
	{{NB\|important\|This step requires using GGAs on many structures, so will be more time-consuming.}}		{{NB\|important\|This step requires using GGAs on many structures, so will be more time-consuming.}}

	=== 3. TI from oxidized to reduced state ===		=== 3. TI from oxidized to reduced state ===
	Perform a thermodynamic integration from the [[~~Construction:~~Thermodynamic integration between ~~MLFFs~~\|oxidized state (λ=0; Fe<sup>3+</sup>) to the reduced state (λ=1; Fe<sup>2+</sup>)]].		Perform a thermodynamic integration from the [[Thermodynamic integration between machine-learned force fields\|oxidized state (λ=0; Fe<sup>3+</sup>) to the reduced state (λ=1; Fe<sup>2+</sup>)]].
	{{NB\|mind\|This is a thermodynamic integration between two MLFFs, so is very quick.}}		{{NB\|mind\|This is a thermodynamic integration between two MLFFs, so is very quick.}}

	=== 4. TI from MLFF to GGA ===		=== 4. TI from MLFF to GGA ===
	Perform a thermodynamic integration from [[Construction:Thermodynamic integration between ~~MLFF~~ and ~~GGAs~~\|MLFF (λ=0) to RPBE+D3 (λ=1)]]. This should be done separately for Fe<sup>3+</sup> and Fe<sup>2+</sup>.		Perform a thermodynamic integration from [[Construction:Thermodynamic integration between a machine-learned force field and a density functional\|MLFF (λ=0) to RPBE+D3 (λ=1)]]. This should be done separately for Fe<sup>3+</sup> and Fe<sup>2+</sup>.
	{{NB\|important\|This step requires using GGAs on many structures so will be more time consuming.}}		{{NB\|important\|This step requires using GGAs on many structures so will be more time consuming.}}

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	\| '''Lit. {{Cite\|jinnouchi:ncm:2024}} (eV)'''		\| '''Lit. {{Cite\|jinnouchi:ncm:2024}} (eV)'''
	\|-		\|-
	\| [[~~Construction:~~Vacuum reference\|'''2 - Vacuum reference''']]		\| [[Vacuum reference\|'''2 - Vacuum reference''']]
	\| <math>e \Delta \bar{\phi}</math>		\| <math>e \Delta \bar{\phi}</math>
	\| -3.71		\| -3.71
	\| -3.74		\| -3.74
	\|-		\|-
	\| [[~~Construction:~~Thermodynamic integration between ~~MLFFs~~\|'''3 - TI (MLFF)''']]		\| [[Thermodynamic integration between machine-learned force fields\|'''3 - TI (MLFF)''']]
	\| <math>\Delta A_{\mathrm{ML}}</math>		\| <math>\Delta A_{\mathrm{ML}}</math>
	\| -4.98		\| -4.98
	\| -4.95		\| -4.95
	\|-		\|-
	\| [[Construction:Thermodynamic integration between ~~MLFF~~ and ~~GGAs~~\|'''4 - TI (GGA)''']]		\| [[Construction:Thermodynamic integration between a machine-learned force field and a density functional\|'''4 - TI (GGA)''']]
	\| <math>\Delta A^{\mathrm{FP}_\mathrm{GGA}-\mathrm{ML}}_1 - \Delta A^{\mathrm{FP}_\mathrm{GGA}-\mathrm{ML}}_0</math>		\| <math>\Delta A^{\mathrm{FP}_\mathrm{GGA}-\mathrm{ML}}_1 - \Delta A^{\mathrm{FP}_\mathrm{GGA}-\mathrm{ML}}_0</math>
	\| -0.055		\| -0.055
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	== Related tags and articles ==		== Related tags and articles ==
	;How-tos		;How-tos
	* [[~~Construction:~~Vacuum reference]]		* [[Vacuum reference]]
	* [[~~Construction:~~Thermodynamic integration between ~~MLFFs~~]]		* [[Thermodynamic integration between machine-learned force fields]]
	* [[Construction:Thermodynamic integration between ~~MLFF~~ and ~~GGAs~~]]		* [[Construction:Thermodynamic integration between a machine-learned force field and a density functional\|Thermodynamic integration between a machine-learned force field and a density functional]]
	;Theory		;Theory
	* [[Thermodynamic integration]]		* [[Thermodynamic integration]]
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	<references/>		<references/>

	~~<!--~~[[Category:Howto]][[Category:Advanced molecular-dynamics sampling]][[Category:Machine-learned force fields]][[Category:Electrochemistry]]-->		[[Category:Howto]][[Category:Advanced molecular-dynamics sampling]][[Category:Machine-learned force fields]]
			<!--[[Category:Electrochemistry]]-->

Csheldon: Csheldon moved page Construction:Thermodynamic integration: redox potential to Thermodynamic integration: redox potential

2026-09-18T07:58:10Z

Csheldon moved page Construction:Thermodynamic integration: redox potential to Thermodynamic integration: redox potential

← Older revision	Revision as of 07:58, 18 September 2026
(No difference)

Csheldon: Repeat the composition equation in the analysis step, where the terms are combined

2026-09-17T16:22:19Z

Repeat the composition equation in the analysis step, where the terms are combined

← Older revision		Revision as of 16:22, 17 September 2026
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	** Reduced: Fe<sup>2+</sup> - <math>\Delta A^{\mathrm{FP}_\mathrm{GGA}-\mathrm{ML}}_1</math>		** Reduced: Fe<sup>2+</sup> - <math>\Delta A^{\mathrm{FP}_\mathrm{GGA}-\mathrm{ML}}_1</math>
	** Oxidized: Fe<sup>3+</sup> - <math>\Delta A^{\mathrm{FP}_\mathrm{GGA}-\mathrm{ML}}_0</math>		** Oxidized: Fe<sup>3+</sup> - <math>\Delta A^{\mathrm{FP}_\mathrm{GGA}-\mathrm{ML}}_0</math>

			:<math>\Delta A^{\mathrm{FP}_\mathrm{GGA}} = \Delta A ^{\mathrm{ML}} + \Delta A^{\mathrm{FP}_\mathrm{GGA}-\mathrm{ML}}_1 - \Delta A^{\mathrm{FP}_\mathrm{GGA}-\mathrm{ML}}_0</math>

	Take a summation over these to obtain the free energy difference from Fe<sup>3+</sup> to Fe<sup>2+</sup> on the GGA level <math>\Delta A^{\mathrm{FP}_\mathrm{GGA}}</math>. Switch the sign to get the absolute redox potential <math>U^{\mathrm{FP}_\mathrm{GGA}}_{\mathrm{redox}}</math>. Finally, take the standard hydrogen electrode (SHE) (4.44 V {{Cite\|trasatti:pac:1986}}) away to compare to experiment <math>U^{\mathrm{FP}_\mathrm{GGA}}_{\mathrm{redox,SHE}}</math>. The results below match the literature to within 30 meV.		Take a summation over these to obtain the free energy difference from Fe<sup>3+</sup> to Fe<sup>2+</sup> on the GGA level <math>\Delta A^{\mathrm{FP}_\mathrm{GGA}}</math>. Switch the sign to get the absolute redox potential <math>U^{\mathrm{FP}_\mathrm{GGA}}_{\mathrm{redox}}</math>. Finally, take the standard hydrogen electrode (SHE) (4.44 V {{Cite\|trasatti:pac:1986}}) away to compare to experiment <math>U^{\mathrm{FP}_\mathrm{GGA}}_{\mathrm{redox,SHE}}</math>. The results below match the literature to within 30 meV.

Csheldon: Upright multi-letter math subscripts; write the Gamma point in prose; tidy Latin abbreviations

2026-09-17T16:03:14Z

Upright multi-letter math subscripts; write the Gamma point in prose; tidy Latin abbreviations

← Older revision		Revision as of 16:03, 17 September 2026
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	\| ''Oxidized'' (Fe<sup>3+</sup>)		\| ''Oxidized'' (Fe<sup>3+</sup>)
	\| ''Reduced'' (Fe<sup>2+</sup>)		\| ''Reduced'' (Fe<sup>2+</sup>)
	\| <math>\Delta A_{ML}</math>		\| <math>\Delta A_{\mathrm{ML}}</math>
	\|-		\|-
	\| [[Construction:Thermodynamic integration between MLFF and GGAs\|MLFF:GGA]]		\| [[Construction:Thermodynamic integration between MLFF and GGAs\|MLFF:GGA]]
	\| MLFF (Fe<sup>3+</sup>)		\| MLFF (Fe<sup>3+</sup>)
	\| RPBE+D3 (Fe<sup>3+</sup>)		\| RPBE+D3 (Fe<sup>3+</sup>)
	\| <math>\Delta A^{~~FP_~~{GGA}-ML}_0</math>		\| <math>\Delta A^{\mathrm{FP}_\mathrm{GGA}-\mathrm{ML}}_0</math>
	\|-		\|-
	\| [[Construction:Thermodynamic integration between MLFF and GGAs\|MLFF:GGA]]		\| [[Construction:Thermodynamic integration between MLFF and GGAs\|MLFF:GGA]]
	\| MLFF (Fe<sup>2+</sup>)		\| MLFF (Fe<sup>2+</sup>)
	\| RPBE+D3 (Fe<sup>2+</sup>)		\| RPBE+D3 (Fe<sup>2+</sup>)
	\| <math>\Delta A^{~~FP_~~{GGA}-ML}_1</math>		\| <math>\Delta A^{\mathrm{FP}_\mathrm{GGA}-\mathrm{ML}}_1</math>
	\|-		\|-
	\|}		\|}

	Finally, the absolute redox potential on a GGA-level <math>\Delta A^{~~FP_~~{GGA}}</math> is calculated:		Finally, the absolute redox potential on a GGA-level <math>\Delta A^{\mathrm{FP}_\mathrm{GGA}}</math> is calculated:

	:<math>\Delta A^{~~FP_~~{GGA}} = \Delta A ^{ML} + \Delta A^{~~FP_~~{GGA}-ML}_1 - \Delta A^{~~FP_~~{GGA}-ML}_0</math>		:<math>\Delta A^{\mathrm{FP}_\mathrm{GGA}} = \Delta A ^{\mathrm{ML}} + \Delta A^{\mathrm{FP}_\mathrm{GGA}-\mathrm{ML}}_1 - \Delta A^{\mathrm{FP}_\mathrm{GGA}-\mathrm{ML}}_0</math>

	== Input files ==		== Input files ==
	=== {{FILE\|POSCAR}}s ===		=== {{FILE\|POSCAR}}s ===
	The chemical potential requires four different systems: bulk H<sub>2</sub>O with 64 molecules (<code>64H2O_bulk</code>), Fe<sup>3+</sup> in 64 H<sub>2</sub>O (<code>Fe3P_64H2O</code>), Fe<sup>2+</sup> in 64 H<sub>2</sub>O (<code>Fe2P_64H2O</code>), H<sub>2</sub>O slab with 128 molecules (<code>128H2O_slab</code>)~~. I.e.~~, a bulk water reference, <math>[\mathrm{Fe}(\mathrm{H}_2\mathrm{O})_n]^{3+}</math>, <math>[\mathrm{Fe}(\mathrm{H}_2\mathrm{O})_n]^{2+}</math>, and the water slab.		The chemical potential requires four different systems: bulk H<sub>2</sub>O with 64 molecules (<code>64H2O_bulk</code>), Fe<sup>3+</sup> in 64 H<sub>2</sub>O (<code>Fe3P_64H2O</code>), Fe<sup>2+</sup> in 64 H<sub>2</sub>O (<code>Fe2P_64H2O</code>), H<sub>2</sub>O slab with 128 molecules (<code>128H2O_slab</code>), that is, a bulk water reference, <math>[\mathrm{Fe}(\mathrm{H}_2\mathrm{O})_n]^{3+}</math>, <math>[\mathrm{Fe}(\mathrm{H}_2\mathrm{O})_n]^{2+}</math>, and the water slab.

	[[File:Redox_structures.png\|800px\|thumb\|center\|'''Figure 1'''. Top left: Fe<sup>3+</sup> in 64 H<sub>2</sub>O (<code>Fe3P_64H2O</code>), top center: Fe<sup>2+</sup> in 64 H<sub>2</sub>O (<code>Fe2P_64H2O</code>), top right: bulk H<sub>2</sub>O with 64 molecules (<code>64H2O_bulk</code>), and bottom: H<sub>2</sub>O slab with 128 molecules (<code>128H2O_slab</code>).]]		[[File:Redox_structures.png\|800px\|thumb\|center\|'''Figure 1'''. Top left: Fe<sup>3+</sup> in 64 H<sub>2</sub>O (<code>Fe3P_64H2O</code>), top center: Fe<sup>2+</sup> in 64 H<sub>2</sub>O (<code>Fe2P_64H2O</code>), top right: bulk H<sub>2</sub>O with 64 molecules (<code>64H2O_bulk</code>), and bottom: H<sub>2</sub>O slab with 128 molecules (<code>128H2O_slab</code>).]]
Line 1,065:		Line 1,065:

	=== {{FILE\|KPOINTS}} ===		=== {{FILE\|KPOINTS}} ===
	The {{FILE\|KPOINTS}} file contains only the Gamma-point:		The {{FILE\|KPOINTS}} file contains only the Γ point:

	Gamma-point only		Gamma-point only
Line 1,107:		Line 1,107:
	* the chemical potential - <math>e \Delta \bar{\phi}</math>		* the chemical potential - <math>e \Delta \bar{\phi}</math>
	* the free energy difference from MLFF to MLFF (with chemical potential reference)		* the free energy difference from MLFF to MLFF (with chemical potential reference)
	** from Fe<sup>3+</sup> to Fe<sup>2+</sup> - <math>\Delta A_{ML}</math>		** from Fe<sup>3+</sup> to Fe<sup>2+</sup> - <math>\Delta A_{\mathrm{ML}}</math>
	* the free energy difference from MLFF to GGA		* the free energy difference from MLFF to GGA
	** Reduced: Fe<sup>2+</sup> - <math>\Delta A^{~~FP_~~{GGA}-ML}_1</math>		** Reduced: Fe<sup>2+</sup> - <math>\Delta A^{\mathrm{FP}_\mathrm{GGA}-\mathrm{ML}}_1</math>
	** Oxidized: Fe<sup>3+</sup> - <math>\Delta A^{~~FP_~~{GGA}-ML}_0</math>		** Oxidized: Fe<sup>3+</sup> - <math>\Delta A^{\mathrm{FP}_\mathrm{GGA}-\mathrm{ML}}_0</math>

	Take a summation over these to obtain the free energy difference from Fe<sup>3+</sup> to Fe<sup>2+</sup> on the GGA level <math>\Delta A^{~~FP_~~{GGA}}</math>. Switch the sign to get the absolute redox potential <math>U^{~~FP_~~{GGA}}_{redox}</math>. Finally, take the standard hydrogen electrode (SHE) (4.44 V {{Cite\|trasatti:pac:1986}}) away to compare to experiment <math>U^{~~FP_~~{GGA}}_{redox, SHE}</math>. The results below match the literature to within 30 meV.		Take a summation over these to obtain the free energy difference from Fe<sup>3+</sup> to Fe<sup>2+</sup> on the GGA level <math>\Delta A^{\mathrm{FP}_\mathrm{GGA}}</math>. Switch the sign to get the absolute redox potential <math>U^{\mathrm{FP}_\mathrm{GGA}}_{\mathrm{redox}}</math>. Finally, take the standard hydrogen electrode (SHE) (4.44 V {{Cite\|trasatti:pac:1986}}) away to compare to experiment <math>U^{\mathrm{FP}_\mathrm{GGA}}_{\mathrm{redox,SHE}}</math>. The results below match the literature to within 30 meV.

	The results are summarized in the following table:		The results are summarized in the following table:
Line 1,127:		Line 1,127:
	\|-		\|-
	\| [[Construction:Thermodynamic integration between MLFFs\|'''3 - TI (MLFF)''']]		\| [[Construction:Thermodynamic integration between MLFFs\|'''3 - TI (MLFF)''']]
	\| <math>\Delta A_{ML}</math>		\| <math>\Delta A_{\mathrm{ML}}</math>
	\| -4.98		\| -4.98
	\| -4.95		\| -4.95
	\|-		\|-
	\| [[Construction:Thermodynamic integration between MLFF and GGAs\|'''4 - TI (GGA)''']]		\| [[Construction:Thermodynamic integration between MLFF and GGAs\|'''4 - TI (GGA)''']]
	\| <math>\Delta A^{~~FP_~~{GGA}-ML}_1 - \Delta A^{~~FP_~~{GGA}-ML}_0</math>		\| <math>\Delta A^{\mathrm{FP}_\mathrm{GGA}-\mathrm{ML}}_1 - \Delta A^{\mathrm{FP}_\mathrm{GGA}-\mathrm{ML}}_0</math>
	\| -0.055		\| -0.055
	\| -0.049		\| -0.049
	\|-		\|-
	\| Absolute redox potential		\| Absolute redox potential
	\| <math>U^{~~FP_~~{GGA}}_{redox} </math>		\| <math>U^{\mathrm{FP}_\mathrm{GGA}}_{\mathrm{redox}} </math>
	\| 5.03		\| 5.03
	\| 5.00		\| 5.00
	\|-		\|-
	\| Redox potential		\| Redox potential
	\| <math>U^{~~FP_~~{GGA}}_{redox, SHE}</math>		\| <math>U^{\mathrm{FP}_\mathrm{GGA}}_{\mathrm{redox,SHE}}</math>
	\| 0.59		\| 0.59
	\| 0.56		\| 0.56

Csheldon: Use the friendly imperative for reader actions and impersonal phrasing for reported results

2026-09-17T15:54:35Z

Use the friendly imperative for reader actions and impersonal phrasing for reported results

← Older revision		Revision as of 15:54, 17 September 2026
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	=== 5. TPT from GGA to hybrid (not covered here) ===		=== 5. TPT from GGA to hybrid (not covered here) ===
	Due to the high cost of converting from RPBE to PBE0, ~~we omit describing~~ thermodynamic perturbation theory (TPT):		Due to the high cost of converting from RPBE to PBE0, thermodynamic perturbation theory (TPT) is not covered here:
	* From RPBE+D3 to PBE0 (Fe<sup>3+</sup>)		* From RPBE+D3 to PBE0 (Fe<sup>3+</sup>)
	* From RPBE+D3 to PBE0 (Fe<sup>2+</sup>)		* From RPBE+D3 to PBE0 (Fe<sup>2+</sup>)
Line 1,112:		Line 1,112:
	** Oxidized: Fe<sup>3+</sup> - <math>\Delta A^{FP_{GGA}-ML}_0</math>		** Oxidized: Fe<sup>3+</sup> - <math>\Delta A^{FP_{GGA}-ML}_0</math>

	Take a summation over these to obtain the free energy difference from Fe<sup>3+</sup> to Fe<sup>2+</sup> on the GGA level <math>\Delta A^{FP_{GGA}}</math>. Switch the sign to get the absolute redox potential <math>U^{FP_{GGA}}_{redox}</math>. Finally, take the standard hydrogen electrode (SHE) (4.44 V {{Cite\|trasatti:pac:1986}}) away to compare to experiment <math>U^{FP_{GGA}}_{redox, SHE}</math>. ~~In our testing, the~~ results ~~matched~~ the literature to within 30 meV.		Take a summation over these to obtain the free energy difference from Fe<sup>3+</sup> to Fe<sup>2+</sup> on the GGA level <math>\Delta A^{FP_{GGA}}</math>. Switch the sign to get the absolute redox potential <math>U^{FP_{GGA}}_{redox}</math>. Finally, take the standard hydrogen electrode (SHE) (4.44 V {{Cite\|trasatti:pac:1986}}) away to compare to experiment <math>U^{FP_{GGA}}_{redox, SHE}</math>. The results below match the literature to within 30 meV.

	~~Our~~ results are summarized in the following table:		The results are summarized in the following table:
	{\|cellpadding="5" cellspacing="0" border="1"		{\|cellpadding="5" cellspacing="0" border="1"
	\| '''Step'''		\| '''Step'''