Thermodynamic integration: redox potential
In this article, the procedure of Jinnouchi et al. [1] to calculate the redox potential of an electrochemical half-cell, following the example of the Fe3+/Fe2+ half-cell reaction (oxidized state/reduced state):
[math]\displaystyle{ \mathrm{Fe^{3+} + e^- \;\rightleftharpoons\; Fe^{2+}} }[/math]
The following two models are used:
- [math]\displaystyle{ [\mathrm{Fe}(\mathrm{H}_2\mathrm{O})_n]^{3+} }[/math] - Fe ion in 64 H2O bulk cell with 3 valence electrons put in the background charge
- [math]\displaystyle{ [\mathrm{Fe}(\mathrm{H}_2\mathrm{O})_n]^{2+} }[/math] - Fe ion in 64 H2O bulk cell with 2 valence electrons put in the background charge
Thermodynamic integration (TI) is used to calculate the free energy difference ΔA using the following basic equation:
[math]\displaystyle{ \Delta A = \int_0^1 \langle U_1 - U_0 \rangle_{\lambda} d\lambda - \mu n }[/math]
where U0 and U1 are the potential energies of the non-interacting and interacting states, respectively, λ is a coupling parameter, μ is the chemical potential (relative to vacuum), and n is the number of electrons in the half-cell reaction.
Procedure
The first step is to describe preparation of machine-learned force fields (MLFFs) for the [math]\displaystyle{ [\mathrm{Fe}(\mathrm{H}_2\mathrm{O})_n]^{3+} }[/math]/ [math]\displaystyle{ [\mathrm{Fe}(\mathrm{H}_2\mathrm{O})_n]^{2+} }[/math] pair and water slab. The chemical potential relative to the vacuum [math]\displaystyle{ \mu n }[/math] is then calculated via the difference in local potential [math]\displaystyle{ e \Delta \bar{\phi} }[/math].
Three thermodynamic integrations are described from non-interacting (λ=0) to interacting (λ=1) systems:
| TI | λ=0 | λ=1 | Term |
| MLFF:MLFF | Oxidized (Fe3+) | Reduced (Fe2+) | [math]\displaystyle{ \Delta A_{ML} }[/math] |
| MLFF:GGA | MLFF (Fe3+) | RPBE+D3 (Fe3+) | [math]\displaystyle{ \Delta A^{FP_{GGA}-ML}_0 }[/math] |
| MLFF:GGA | MLFF (Fe2+) | RPBE+D3 (Fe2+) | [math]\displaystyle{ \Delta A^{FP_{GGA}-ML}_1 }[/math] |
Though described in the paper, due to the high cost of converting from RPBE to PBE0, we omit describing thermodynamic perturbation theory (TPT):
- From RPBE+D3 to PBE0 (Fe3+)
- From RPBE+D3 to PBE0 (Fe2+)
| Tip: TPT can be replicated using Δ-learning from RPBE to PBE0, using the same procedure as from MLFF to RPBE. |
Finally, the absolute redox potential on a GGA-level [math]\displaystyle{ \Delta A^{FP_{GGA}} }[/math] is calculated:
- [math]\displaystyle{ \Delta A^{FP_{GGA}} = \Delta A ^{ML} + \Delta A^{FP_{GGA}-ML}_1 - \Delta A^{FP_{GGA}-ML}_0 }[/math]
This absolute redox potential, the redox potential relative to the standard hydrogen electrode (SHE) [math]\displaystyle{ U^{FP_{GGA}}_{redox} }[/math], and the constituent terms are compared with the literature.
Input files
POSCARs
The chemical potential requires four different systems: bulk H2O with 64 molecules (64H2O_bulk), Fe3+ in 64 H2O (Fe3P_64H2O), Fe2+ in 64 H2O (Fe2P_64H2O), H2O slab with 128 molecules (128H2O_slab). I.e., a bulk water reference, [math]\displaystyle{ [\mathrm{Fe}(\mathrm{H}_2\mathrm{O})_n]^{3+} }[/math], [math]\displaystyle{ [\mathrm{Fe}(\mathrm{H}_2\mathrm{O})_n]^{2+} }[/math], and the water slab.
Fe3P_64H2O), top center: Fe2+ in 64 H2O (Fe2P_64H2O), top right: bulk H2O with 64 molecules (64H2O_bulk), and bottom: H2O slab with 128 molecules (128H2O_slab).Click to reveal the
64H2O_bulk POSCARWater
1
12.42128700 0.00000000 0.00000000
0.00000000 12.42128700 0.00000000
0.00000000 0.00000000 12.42128700
H O
128 64
Direct
0.36257633 0.37953597 0.67785342
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Click to reveal the
Fe3P_64H2O POSCARFe_64H2O
1
12.42282200 0.00000000 0.00000000
0.00000000 12.42282200 0.00000000
0.00000000 0.00000000 12.42282200
H O Fe
128 64 1
Direct
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0.00592307 0.09406140 0.79582533
1.10947213 0.28743919 0.33114811
0.13898678 0.54360995 0.97023057
1.15168897 0.42574337 0.16437693
0.96099089 0.62982868 0.29120309
0.66514203 1.00695060 0.04564722
0.15635030 0.24896214 0.68388955
0.26078411 1.11001890 0.32632346
0.75255117 0.09346553 1.23417762
0.93550375 0.42521425 0.27361930
1.12555194 0.49730673 0.58995753
0.97951694 0.64650296 0.50479031
1.02535902 0.32276265 0.18627127
Click to reveal the
Fe2P_64H2O POSCARFe_64H2O
1
12.42282200 0.00000000 0.00000000
0.00000000 12.42282200 0.00000000
0.00000000 0.00000000 12.42282200
H O Fe
128 64 1
Direct
0.12743711 0.48760461 0.68402834
0.07224030 0.59099573 0.69720079
0.39623895 0.90251919 -0.68562329
0.52396846 0.89222892 -0.69202626
0.42964083 0.63656491 0.92024856
0.40471536 0.51747508 0.92244566
0.50949529 0.22553243 0.82975356
0.79695561 -0.29210416 -0.14766098
0.00805766 0.01676094 0.93033834
-0.01969191 0.14077136 0.89733509
0.01773692 1.49221026 -0.45662142
-0.09754205 1.45083180 -0.48707747
0.28568895 0.07977712 0.14809775
0.40460027 0.11889958 0.12632577
-0.44823614 -0.23392047 0.91065326
-0.44655395 -0.27382665 1.03236410
0.55758309 0.63486700 0.51461921
0.54015662 1.52364880 0.45450703
0.67045916 0.85136844 0.58289035
0.71164261 0.89435637 0.47626762
1.16684705 0.13193713 0.89590371
0.33871812 0.20748811 0.69296855
0.38041592 0.31864036 0.75378916
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0.75237735 1.02309221 1.12180723
0.04826055 0.15234086 0.05862687
-0.09230147 -0.06114088 1.02402855
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0.68560087 0.63322667 1.36320530
0.79400335 0.69800156 2.34722324
1.01111679 0.16477327 0.53663862
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1.30154976 0.64025841 0.11667748
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0.77907316 0.40440365 0.75246685
0.73437154 0.47097142 0.84530449
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0.79089209 -0.20809083 -0.24704170
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1.18017291 1.28895470 0.63748305
0.44815316 0.76165600 -0.42603898
0.46425067 0.68605942 -0.32598380
0.26710057 0.21583209 0.85066344
1.08895608 0.13400129 0.17359061
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0.31584241 -0.05457389 -0.25072038
0.46701413 0.39534529 0.33741153
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1.15656687 0.39089427 0.22252886
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1.07353906 0.35801775 -0.07352972
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1.08703359 0.18844511 0.11444108
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1.14696413 1.32734585 0.69656507
0.65811040 0.77819298 1.11522722
0.04903142 0.62925399 0.32627031
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0.66244443 0.93542605 0.32313623
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1.13069333 0.39166539 -0.03166013
1.52690177 0.40275423 0.64871882
0.59258213 0.16054156 0.38717393
1.22868038 0.27296875 0.05742047
Click to reveal the
128H2O_slab POSCARSYSTEM
1
12.50000000 0.00000000 0.00000000
0.00000000 12.50000000 0.00000000
0.00000000 0.00000000 50.00000000
H O
256 128
Direct
-0.15047607 0.45334683 0.32730556
-0.07219808 -0.49010458 0.34403411
0.63805046 0.03123465 0.46999929
0.50838374 0.00754431 0.46372876
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0.21957350 0.63504600 0.31129793
0.49252565 0.97213243 0.52739381
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0.46833659 0.30083696 0.47570094
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0.41854353 0.84406884 0.44927074
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0.89266849 0.40696421 0.52478091
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0.55977900 -0.37296647 0.61748251
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-0.19021572 0.03340561 0.48590483
1.07785299 0.53236279 0.30851784
1.17527108 0.48273981 0.32369409
-0.05242302 0.21986278 0.47826685
0.04602452 0.20858933 0.45856538
0.40878055 0.29190606 0.42949757
0.35385287 0.26786429 0.40109729
0.91361281 1.37763527 0.70485166
0.92601382 1.37974445 0.73573816
0.49180218 0.49176486 0.55630311
0.38888954 0.54455977 0.56839724
0.74382564 0.28498135 0.31337726
0.81303180 0.29929615 0.28888582
0.23876821 1.20570242 0.44319221
0.19992227 1.11701411 0.46333591
0.59407806 1.36278641 0.60873199
0.54524237 1.46654437 0.59522322
2.02346307 1.41217476 0.66399254
1.91147497 1.37915850 0.65989115
0.09845777 0.03688460 0.69616599
0.09766268 -0.08531166 0.69167377
1.25808011 0.44420386 0.36149889
1.31081579 0.38358222 0.33934553
0.07353811 0.81613159 0.44176424
-0.02019300 0.78679404 0.46293839
0.38984424 0.83369934 0.56338610
0.37381688 0.87922811 0.59234427
0.51453372 -0.09541985 0.35009748
0.39484305 -0.06645979 0.35017125
0.55593924 0.38753163 0.51869390
0.66004756 0.46315941 0.51627573
0.64848801 0.61180544 0.37008072
0.57832658 0.64084106 0.34599612
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0.73564205 -0.27427126 0.33331691
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0.56504329 0.06286545 0.46968741
0.24293994 0.69761578 0.32091286
0.48108748 0.89535134 0.53166188
0.49757936 0.32553516 0.45789152
0.76744020 -0.18340073 0.62487691
0.41144876 0.90246316 0.46247378
0.31794376 -0.48573743 0.38681989
0.12877684 0.58271134 0.51624414
0.95851509 0.39985886 0.53493477
0.52243754 -0.30282985 0.61791897
0.76895759 0.37346632 0.56560940
0.66848522 -0.18560993 0.43043112
0.09074693 0.30162769 0.28441615
-0.20154125 1.61789646 0.64720226
0.17181260 -0.12468412 0.29662573
0.15700557 0.00871911 0.61363921
0.42233976 -0.32994901 0.35514866
0.53286785 0.40322511 0.39184579
1.07727341 0.25676911 0.57497686
0.13716955 0.87720865 0.55959520
0.68216863 0.99627546 0.65137040
0.45119708 0.16614064 0.28412456
0.10973328 0.48171438 0.46781222
0.84388934 -0.20664059 0.39174052
0.58303852 -0.54048463 0.31286474
0.32808472 1.59763404 0.62452336
-0.19165309 0.33704380 0.43457786
0.41823919 0.20183034 0.37245093
0.78587144 0.61673307 0.55124293
0.56011355 0.11309492 0.40567704
1.51701295 1.39737040 0.70673631
0.54695764 1.10711476 0.52102138
0.72131688 0.27947218 0.47879616
0.18444949 -0.21210987 0.42113103
0.36668171 0.45488107 0.66482426
0.49064798 -0.46503727 0.44561439
0.82820218 0.69414688 0.75657380
0.65847198 0.61537373 0.40712496
0.76748450 0.96128069 0.73583267
0.81730697 0.91059549 0.51275491
1.27220645 1.14795719 0.57873009
0.64741928 -0.05620926 0.34353694
0.48391157 0.12047094 0.57316210
0.14376608 0.06922992 0.53329165
1.27850839 0.03499489 0.35317314
-0.46354066 0.99485951 0.74685190
0.12508520 0.09293059 0.39933758
0.43296574 0.12732681 0.64290755
0.09700574 1.67696831 0.67957377
-0.02662689 -0.02348385 0.31412699
0.44363826 -0.24009345 0.42143376
0.25025628 1.24855748 0.64562073
0.79823150 1.08209822 0.60445310
0.24357641 1.08550799 0.29686686
0.84487510 0.54791453 0.42274978
1.20665739 1.51259035 0.71454551
-0.14852600 0.27439958 0.38483199
0.79875463 1.36570466 0.63635127
0.20013661 -0.05770220 0.48695782
-0.08803369 1.01243312 0.64786941
0.60681036 0.21343304 0.60836199
0.96760512 0.18267756 0.51744002
1.01869709 -0.36164031 0.72861865
0.89279549 0.03025280 0.42811891
-0.12190623 0.58173239 0.47257350
-0.28364610 0.74658147 0.48027041
0.35461305 0.69479199 0.69387913
0.17636099 0.75539076 0.63132326
0.64567772 0.08234767 0.30555919
0.74693252 0.15345949 0.54261621
-0.02750156 0.20761939 0.32607759
1.27987775 -0.43232854 0.43958099
0.00284833 -0.07336365 0.37560644
0.13403374 -0.30077764 0.37038742
0.98462893 0.81761037 0.60965656
0.51924082 0.63814697 0.49321682
0.28461449 0.88126714 0.66475622
-0.20914894 0.44302019 0.50652580
0.58914466 0.74769132 0.56398020
0.41471483 1.19061656 0.72530883
-0.12231266 0.52176874 0.60092233
0.26021422 1.34539425 0.53435665
1.56662855 1.77066279 0.67736079
1.10461724 0.26953014 0.36735832
0.72369042 0.35764108 0.34663600
0.43147580 -0.68083095 0.32528778
0.81926513 0.85574301 0.56746752
1.25134221 -0.39325035 0.56126973
0.87947711 1.14636757 0.70571444
0.62353030 0.97359658 0.59258366
0.91315731 0.79159814 0.67046472
1.00887481 -0.51613717 0.38007575
0.71990656 0.52980508 0.69520722
0.42715157 0.93815001 0.61949356
1.22065693 1.25957506 0.70894987
0.68302181 0.25679220 0.67873356
1.04556333 1.22974551 0.74538666
0.05950376 0.17484591 0.65818294
-0.25231321 1.76973735 0.70816158
0.45404841 0.29350534 0.51621152
0.47580393 -0.01885257 0.68583286
0.31607039 1.37827042 0.60641490
0.13921945 1.47529964 0.65395905
-0.24949740 1.04980313 0.38476573
1.02198254 0.77527198 0.51809412
-0.13269955 -0.63172943 0.26362895
-0.01665243 1.05122555 0.57819040
0.34830926 0.86183449 0.73481007
0.08547124 0.49199314 0.59517301
-0.22931968 0.07879046 0.47296684
1.15374394 0.51400513 0.30712860
-0.02575549 0.24426930 0.46122795
0.33910288 0.28875076 0.42031444
0.89222545 1.42242644 0.72114703
0.45112644 0.49752204 0.57217511
0.78732813 0.24777689 0.30076703
0.17546728 1.17137686 0.45167807
0.60114018 1.43939441 0.60809185
1.96388141 1.37112395 0.67251971
0.13553192 -0.02618932 0.70064801
1.23640886 0.39518318 0.34779401
0.01704024 0.84626979 0.45444055
0.34009588 0.83637329 0.57759905
0.44929940 -0.11284502 0.35947285
0.58008850 0.46021347 0.52387715
0.65101061 0.61092524 0.34954978
0.79285789 -0.22059881 0.33571374
INCAR
The INCAR files are provided in each section separately.
KPOINTS
The KPOINTS file contains only the Gamma-point:
Gamma-point only 0 Monkhorst Pack 1 1 1 0 0 0
POTCAR
Standard POTCAR files are used throughout:
PAW_PBE H 15Jun2001PAW_PBE O 08Apr2002PAW_PBE Fe_sv 23Jul2007
This is different to in Ref. [1], where GW POTCARs are used.
Step-by-step instructions
1. Preparing the MLFFs
First, train the MLFFs for the water slab (128H2O_slab) and the redox pairs ([math]\displaystyle{ [\mathrm{Fe}(\mathrm{H}_2\mathrm{O})_n]^{3+} }[/math] and [math]\displaystyle{ [\mathrm{Fe}(\mathrm{H}_2\mathrm{O})_n]^{2+} }[/math]).
2. Calculating the vacuum potential
Run a molecular dynamics (MD) simulation, then calculate the local potential [math]\displaystyle{ \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 chemical potential relative to the vacuum [math]\displaystyle{ e \Delta \bar{\phi} }[/math].
| Important: This step requires using GGAs on many structures, so will be more time-consuming. |
3. TI from oxidized to reduced state
Perform a thermodynamic integration from the oxidized state (λ=0; Fe3+) to the reduced state (λ=1; Fe2+).
| Mind: This is a thermodynamic integration between two MLFFs, so is very quick. |
4. TI from MLFF to GGA
Perform a thermodynamic integration from MLFF (λ=0) to RPBE+D3 (λ=0). This should be done separately for Fe3+ and Fe2+.
| Important: This step requires using GGAs on many structures so will be more time consuming. |
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):
- From RPBE+D3 to PBE0 (Fe3+)
- From RPBE+D3 to PBE0 (Fe2+)
| 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:
- the chemical potential - [math]\displaystyle{ e \Delta \bar{\phi} }[/math]
- the free energy difference from MLFF to MLFF (with chemical potential reference)
- from Fe3+ to Fe2+ - [math]\displaystyle{ \Delta A_{ML} }[/math]
- the free energy difference from MLFF to GGA
- Reduced: Fe2+ - [math]\displaystyle{ \Delta A^{FP_{GGA}-ML}_1 }[/math]
- Oxidized: Fe3+ - [math]\displaystyle{ \Delta A^{FP_{GGA}-ML}_0 }[/math]
Take a summation over these to obtain the free energy difference from Fe3+ to Fe2+ on the GGA level [math]\displaystyle{ \Delta A^{FP_{GGA}} }[/math]. Switch the sign to get the absolute redox potential [math]\displaystyle{ U^{FP_{GGA}}_{redox} }[/math]. Finally, take the standard hydrogen electrode (SHE) (4.44 eV [2]) away to compare to experiment [math]\displaystyle{ U^{FP_{GGA}}_{redox, SHE} }[/math]. In our testing, the results matched the literature to within 30 meV.
Our results are summarized in the following table:
| Step | Term | Ours (eV) | Lit. [1] (eV) |
| 2 - Vacuum reference | [math]\displaystyle{ e \Delta \bar{\phi} }[/math] | -3.71 | -3.74 |
| 3 - TI (MLFF) | [math]\displaystyle{ \Delta A_{ML} }[/math] | -4.98 | -4.95 |
| 4 - TI (GGA) | [math]\displaystyle{ \Delta A^{FP_{GGA}-ML}_1 - \Delta A^{FP_{GGA}-ML}_0 }[/math] | -0.055 | -0.049 |
| Absolute redox potential | [math]\displaystyle{ U^{FP_{GGA}}_{redox} }[/math] | 5.03 | 5.00 |
| Redox potential | [math]\displaystyle{ U^{FP_{GGA}}_{redox, SHE} }[/math] | 0.59 | 0.56 |
Recommendations and advice
Related tags and articles
- How-tos
- Construction:Vacuum reference
- Construction:Thermodynamic integration between MLFFs
- Construction:Thermodynamic integration between MLFF and GGAs
- Theory
- Tags
- VCAIMAGES, NCORE_IN_IMAGE1, MDALGO, ML_LMLFF, NELECT, MDALGO, ICORELEVEL, WRT_POTENTIAL, LVHAR, NELECT
References