Thermodynamic integration: redox potential
Redox potentials of an electrochemical half-cell can be computed by thermodynamic integration, following the procedure of Jinnouchi, Karsai, and Kresse [1]. The Fe3+/Fe2+ half-cell serves as the example throughout (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.
Three thermodynamic integrations with machine-learned force fields (MLFFs) 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_{\mathrm{ML}} }[/math] |
| MLFF:GGA | MLFF (Fe3+) | RPBE+D3 (Fe3+) | [math]\displaystyle{ \Delta A^{\mathrm{FP}_\mathrm{GGA}-\mathrm{ML}}_0 }[/math] |
| MLFF:GGA | MLFF (Fe2+) | RPBE+D3 (Fe2+) | [math]\displaystyle{ \Delta A^{\mathrm{FP}_\mathrm{GGA}-\mathrm{ML}}_1 }[/math] |
Finally, the absolute redox potential on a GGA-level [math]\displaystyle{ \Delta A^{\mathrm{FP}_\mathrm{GGA}} }[/math] is calculated:
- [math]\displaystyle{ \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
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), that is, 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
0.42947178 0.43286249 0.59010358
0.66870393 1.13989162 0.58842641
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0.44580288 0.01834909 0.16004703
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0.44857483 0.64047308 0.17235572
1.14522157 0.76633728 -0.07286828
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0.56790416 1.14251886 0.92143084
0.60754105 1.14691841 0.80551737
0.75633099 0.25670520 0.32073462
0.75786987 0.15087192 0.26471573
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1.10944135 0.32634895 1.48853144
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1.49704038 0.52822279 0.24728978
0.19113993 0.69295724 0.10392182
Click to reveal the [math]\displaystyle{ [\mathrm{Fe}(\mathrm{H}_2\mathrm{O})_n]^{3+} }[/math] POSCAR
Fe_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
-0.38272023 0.47236734 0.69895078
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1.12555194 0.49730673 0.58995753
0.97951694 0.64650296 0.50479031
1.02535902 0.32276265 0.18627127
Click to reveal the [math]\displaystyle{ [\mathrm{Fe}(\mathrm{H}_2\mathrm{O})_n]^{2+} }[/math] POSCAR
Fe_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
-0.26840230 -0.60381695 0.57433173
0.25751301 1.09723808 -0.39131073
0.28561906 1.17135043 -0.49010853
-0.15559770 0.77405954 1.51947552
-0.03350167 0.77676575 1.52339975
0.75237735 1.02309221 1.12180723
0.04826055 0.15234086 0.05862687
-0.09230147 -0.06114088 1.02402855
-0.07779336 -0.14013315 0.92524744
0.68560087 0.63322667 1.36320530
0.79400335 0.69800156 2.34722324
1.01111679 0.16477327 0.53663862
0.98618018 0.28565957 0.49809698
0.20452861 0.81136804 0.83127978
-0.29824646 -0.48415938 0.54811270
0.37502040 1.26549068 0.41257460
1.30154976 0.64025841 0.11667748
1.75290818 0.59403116 0.98316689
1.65052923 0.63412803 0.91547295
0.75375838 0.00138371 -0.37473678
0.76130867 1.05807834 -0.25911110
-0.05696647 -0.29493528 0.76468746
0.02811791 -0.33267590 0.85098776
0.61065413 0.29830310 0.85951557
0.11763966 0.86822584 0.90729729
0.18919114 0.55109134 0.38993918
0.21099928 0.52226945 0.51012484
0.03269714 0.36292005 0.33650725
0.02620788 0.48115860 0.28181401
1.05541232 0.90157404 0.64524915
1.07849658 0.77961507 0.68820380
-0.62352298 1.63205141 0.30877689
0.43788457 1.73907737 0.26620496
0.64515164 -0.84752695 1.12986454
0.64730542 0.18633377 1.26064334
0.19923207 -0.08426170 0.07502044
0.22493320 -0.20749942 0.08281964
0.78950677 1.18291196 0.57758905
0.83466268 1.23338961 0.48038196
1.10490483 1.28683406 0.74494774
0.29928808 0.66864147 -0.17842015
0.69120317 0.84829619 1.11035412
0.67223805 0.75611954 1.18775106
-0.00106366 0.61277354 0.38525072
0.01326608 0.68609575 0.28998127
-0.07137268 0.05723237 0.64512696
0.03262162 0.06902330 0.71661499
0.77907316 0.40440365 0.75246685
0.73437154 0.47097142 0.84530449
0.22165260 0.78128841 0.57222522
0.24873347 0.68762843 0.48624453
0.21948415 0.65300265 -0.26571256
0.79089209 -0.20809083 -0.24704170
1.11107834 0.01381845 0.32957265
1.03239487 0.10538648 0.35890625
-0.22306466 1.43601128 1.11757892
-0.24593929 1.53904148 0.17695298
0.53330947 -0.13832394 0.74775872
0.48380538 -0.08752260 0.84569979
0.33958593 1.05428999 0.90703938
0.29513801 0.93047632 0.90464110
-0.10080201 -0.17071890 0.20726349
0.00812468 -0.20258160 0.13847835
0.08102303 0.64583254 0.03494449
-0.04079584 0.61193214 0.04209357
0.35121443 1.17006312 0.32377652
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
0.65188184 1.12136051 0.88104293
0.65005020 0.99400031 0.86041374
0.74276310 0.93357167 0.31617568
0.65193906 1.01050888 0.33337941
0.22155564 -0.06334224 -0.33974479
0.31584241 -0.05457389 -0.25072038
0.46701413 0.39534529 0.33741153
0.37353168 0.43618379 0.40490478
0.88534124 0.31114574 0.85608235
0.90593210 0.27739909 0.97341481
0.39857237 -0.04387750 0.56182123
0.47095275 1.02320312 0.49993905
1.25892990 0.45837652 0.17875477
1.15656687 0.39089427 0.22252886
0.51644837 0.17070272 0.40580457
1.27940834 0.60940694 0.99753791
0.42616197 0.40350440 0.06681017
1.44539498 0.30393771 -0.01376590
0.46562325 1.01534518 0.03284939
0.50319974 0.99064681 0.15470387
0.81037140 0.25082530 0.14657752
0.90914722 0.31376690 1.17885883
0.90887977 0.86774057 1.39942038
0.94187889 0.97203833 1.33441155
0.22890408 0.88765248 0.26506994
0.22943934 0.87502350 1.39740388
0.49252993 0.56873322 1.17531223
0.58322974 0.47973985 1.18073167
1.07353906 0.35801775 -0.07352972
1.10345201 0.46291908 -0.00569576
0.56233028 0.47301291 0.65701182
1.50111441 0.39245196 0.57504212
0.64733050 0.17609466 0.44047892
0.76155406 -0.00064089 0.99805907
0.11696365 0.55392898 0.64536596
0.45665944 0.88787932 -0.73431062
0.37333616 0.58173353 0.89358570
0.83287880 -0.23084103 -0.18836803
0.03157586 0.08326143 0.89500066
-0.01967168 1.44066290 -0.50120574
0.32649692 0.14675159 0.14354441
-0.47815402 -0.28476352 0.96183234
0.58825114 1.58190086 0.46693607
0.73530757 0.86486122 0.54696653
0.36850393 0.23869519 0.76098331
-0.24064441 -0.53169325 0.57977533
0.24867886 1.17176032 -0.42128557
-0.09114394 0.73086170 1.49415350
0.77082046 0.96821649 1.06941593
-0.03834292 -0.10477073 0.98185130
0.72904928 0.67702858 2.31155822
0.98643994 0.21068855 0.47648425
0.18404121 0.88355581 0.86864094
1.08703359 0.18844511 0.11444108
1.71734102 0.59616846 0.90840384
0.79123200 0.05674886 -0.33394462
0.02043659 -0.31156425 0.77570162
0.56069314 0.24338431 0.88537227
0.24569014 0.53057214 0.44182948
0.03448211 0.40131521 0.26357804
1.09819988 0.83512130 0.63612042
-0.60356192 0.67216851 0.24349363
0.68641610 -0.81762609 1.18817124
0.21611881 -0.14120573 0.12450673
0.76919341 1.22245249 0.51393968
1.14696413 1.32734585 0.69656507
0.65811040 0.77819298 1.11522722
0.04903142 0.62925399 0.32627031
0.01260756 0.05983530 0.64228866
0.74653920 0.39814097 0.82582336
0.26593357 0.76215637 0.50729285
0.25423481 0.71079817 -0.23075499
1.04656732 0.05170956 0.30685598
-0.27090311 1.50034802 1.11590884
0.55144479 -0.10997314 0.82138721
0.36595179 0.97842307 0.90355956
-0.04879022 -0.22282940 0.18797022
0.02303649 0.60374665 -0.00064438
0.36248716 1.18806896 0.40233898
0.50553342 0.72818132 -0.38174573
0.23339102 0.17685431 0.91622715
0.69519318 1.05764833 0.86986144
0.66244443 0.93542605 0.32313623
0.29157406 -0.03443908 -0.32197353
0.43576289 0.39304384 0.41288487
0.93290635 0.26807660 0.90192508
0.46947458 -0.04604768 0.53317357
1.23797107 0.38686102 0.19701795
1.24279488 0.62426866 1.06753929
1.39453671 0.36177382 0.00797544
0.49872289 1.04632858 0.09816681
0.86030136 0.31356352 0.11502915
0.87609952 0.92053833 1.35250286
0.23398309 0.92340184 1.33460122
0.50301930 0.49047944 1.17320678
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
0.20742016 -0.23715390 0.31145651
0.21957350 0.63504600 0.31129793
0.49252565 0.97213243 0.52739381
0.46254946 0.86699495 0.51338367
0.46833659 0.30083696 0.47570094
0.56435815 0.28642228 0.46051829
0.73586863 -0.12693724 0.63650708
0.75181963 -0.25406949 0.63203814
0.41854353 0.84406884 0.44927074
0.33515626 0.90054023 0.46728072
0.34479509 0.57962028 0.37911763
0.29762780 -0.47903310 0.40660055
0.15115248 0.54108257 0.49957979
0.08260965 0.53074503 0.52635427
0.89266849 0.40696421 0.52478091
-0.06057907 0.40486920 0.55352216
0.55977900 -0.37296647 0.61748251
0.53858714 -0.26767121 0.59987163
0.71275400 0.42250550 0.56832405
0.78839559 0.34546846 0.58417890
0.67385329 -0.19610180 0.44987464
0.59318198 -0.16624091 0.42828513
0.12312508 0.37123899 0.28937208
0.05023666 0.27675971 0.29997236
-0.16681739 1.58406768 0.63195176
-0.21800579 1.57002107 0.66191007
0.20675683 -0.06108281 0.29221359
0.10312049 -0.09294129 0.30396903
0.20277637 0.06067826 0.60416144
0.19887396 -0.00744650 0.62890026
0.42062864 -0.24439336 0.35497596
0.36814211 -0.33879224 0.34193308
-0.46114333 0.32802380 0.39548517
0.45713828 0.41873416 0.39179412
1.05751346 0.29067408 0.55923161
1.04348507 0.18703047 0.57744764
0.21179010 0.87249263 0.56793233
0.09036024 0.85931851 0.57432053
0.60751605 1.01609789 0.65259722
0.72696147 1.05338974 0.65871807
0.44683747 0.21956247 0.29940199
0.48216169 0.19981749 0.26927194
0.10930269 0.40445023 0.47082959
0.17010928 0.50009382 0.45569496
0.78757796 -0.20083679 0.40459572
0.81784835 -0.20993279 0.37400609
0.63044955 -0.50216422 0.32507910
0.61823261 -0.56783464 0.29754461
0.38298254 1.64224442 0.61713447
0.27391580 1.65127025 0.62769264
-0.18349257 0.40823838 0.42743175
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0.37713689 0.15324147 0.36153587
0.47188832 0.15561964 0.38076476
0.84313290 0.66869553 0.54540293
0.78036268 0.55967659 0.53781156
0.62834565 0.07729786 0.39976222
0.58336683 0.13133194 0.42359675
1.57461032 1.43827952 0.71246905
1.47093630 1.44260924 0.69550961
0.56330564 1.09120276 0.50193580
0.61593811 1.12895503 0.52786251
0.73348421 0.31660428 0.46194097
0.75096480 0.20702765 0.47568711
0.18544389 -0.23514793 0.40166768
0.22931162 -0.14674533 0.42153910
0.36292308 0.50876490 0.64987458
0.33531338 0.38470088 0.66057777
1.49747974 -0.54677440 0.44932443
0.51766749 -0.42535318 0.46152212
1.85759374 0.75185320 0.76879203
0.88557936 0.66337125 0.74776712
0.66151301 0.68965700 0.41313588
0.60698800 0.57649179 0.41787791
-0.21783861 0.90704975 0.72268493
-0.18062377 1.02010056 0.73108294
0.79398077 0.91221618 0.53177815
0.89172292 0.88245064 0.51170593
1.21360535 1.19956259 0.58314768
1.23188220 0.10388812 0.56520461
0.69217913 -0.02288592 0.35644459
0.65013288 -0.02349422 0.32542268
0.49314736 0.09990072 0.55428120
0.40698133 0.13735440 0.57630192
0.06940342 0.10352406 0.53021711
0.12918295 0.01423295 0.54673459
1.23761207 0.03882752 0.37085803
0.23384130 0.05388881 0.33858930
-0.50295222 0.92300366 0.74813821
-0.38930037 0.97841032 0.74266531
0.13655853 0.16157403 0.39040653
0.15373516 0.09867462 0.41870070
0.35793276 0.15462364 0.64561652
0.44313351 0.08542197 0.66008140
0.12490452 1.60323574 0.67914603
0.10103912 0.67758518 0.66052594
-0.02910081 -0.04303719 0.33290797
-0.08397074 -0.06715582 0.30625602
0.47100963 -0.31135459 0.42347049
0.44678381 -0.21727665 0.40309666
0.17380547 1.23522872 0.64351497
0.28087835 1.29207879 0.63186950
0.72243965 1.04679669 0.60136565
0.77633776 1.15885707 0.60968516
0.20332595 1.15182338 0.29172154
0.31769260 1.09822536 0.28860721
0.90406410 0.53849761 0.41088426
0.78463589 0.58505950 0.41470974
1.12942702 1.53845314 0.71988868
1.20054880 1.43235266 0.71382259
-0.17158886 0.30160650 0.40269399
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0.74322462 1.32613449 0.64589263
0.77475704 1.41411198 0.62227898
0.14620565 -1.10616729 0.48008239
0.18160223 -0.03968360 0.50606301
-0.11186987 0.94149530 0.65599853
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0.56159298 0.19900069 0.59311202
0.55920469 0.18285975 0.62176711
0.90336505 0.15667383 0.52548512
0.98450787 0.25110265 0.52511525
0.97847305 -0.42779836 0.72340892
1.02887897 0.67051183 0.71108441
0.93053153 -0.03352705 0.43016463
0.94358325 0.08994529 0.42307038
-0.12458051 0.56502080 0.45311411
-0.04730932 0.59270705 0.47590220
-0.26829199 0.80830572 0.49363371
-0.21717255 0.70457633 0.48040899
0.43110180 0.68638080 0.68890088
0.32510025 0.62119216 0.70042335
0.12431384 0.78282478 0.61843684
0.20700484 0.81000752 0.64341301
0.70384786 0.11901625 0.29636532
0.58157041 0.12693352 0.30130944
0.75363478 0.23168055 0.54459190
0.75644483 0.12259613 0.56105372
-0.03231861 0.12955620 0.32499376
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1.35833840 -0.44225252 0.44591484
1.25776038 -0.35606860 0.43991425
0.02974229 -0.01448169 0.38634402
-0.04925278 -0.12297578 0.38637941
0.07972722 -0.25141011 0.36219086
0.18306699 -0.31067445 0.35551374
0.92747522 0.76466556 0.60868816
0.95937680 0.87571382 0.62123825
0.52932163 0.57770897 0.50403614
0.58437494 0.68163282 0.49342216
0.32115740 0.82996586 0.67667456
0.25755504 0.94154338 0.67730506
-0.22810590 0.37614398 0.49430267
-0.18576922 0.49671288 0.49149878
0.65189321 0.70221781 0.55887514
0.58469375 0.80139057 0.54950481
0.45751382 1.14320451 0.73814919
0.45757048 1.25211787 0.71984711
-0.16073265 0.53318711 0.58413419
-0.04460020 0.52255466 0.59636621
1.23763588 0.27151415 0.53650362
0.26869350 1.37267158 0.55194387
1.56531541 1.75888578 0.65789015
1.64170785 1.76839833 0.68268336
1.17071819 0.30450968 0.36193986
1.07205789 0.24647217 0.34988931
0.66268682 0.39143410 0.35576888
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0.45172427 -0.71444999 0.34206563
0.89274414 0.87693241 0.57141470
0.77760242 0.86753703 0.58449204
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0.86559314 0.75071650 0.65769890
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1.05009801 -0.58200279 0.37590291
0.76172957 0.49270651 0.70861525
0.72438777 0.60310359 0.69947487
0.43493202 1.01642204 0.62370834
0.41157215 0.90717107 0.63919892
1.28709531 1.22977646 0.71568664
1.22240102 1.24760104 0.69023856
-0.36929002 0.30112790 0.68799301
0.73835598 1.24904006 0.69307962
1.08438837 1.16887216 0.75281234
1.10337181 1.25880709 0.73309939
0.00881530 0.23511529 0.66153556
0.01992074 0.10863332 0.65331960
-0.18762204 1.76544824 0.69589571
-0.22099427 0.73159748 0.72364889
0.47402075 0.21823941 0.52156939
0.39449064 0.32549396 0.52603852
0.49229702 0.00642229 0.70437200
0.49041094 -0.09636789 0.68559144
0.31519106 1.45346432 0.61090090
0.38603816 1.37161975 0.59824648
0.15729732 1.50568246 0.63546369
0.21559092 1.47629782 0.66066682
-0.19454501 1.01393222 0.39591120
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1.06810038 0.81254123 0.53206673
1.06352248 0.70747723 0.51307936
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-0.09042409 1.07196610 0.58294900
0.02325129 1.03828409 0.59636367
0.32063486 0.92608540 0.72659258
0.35574911 0.80830229 0.72123409
0.10412373 0.41841079 0.59896831
0.14274925 0.51651990 0.58178437
-0.19591148 0.07044928 0.45554844
-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
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0.39484305 -0.06645979 0.35017125
0.55593924 0.38753163 0.51869390
0.66004756 0.46315941 0.51627573
0.64848801 0.61180544 0.37008072
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0.73564205 -0.27427126 0.33331691
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0.95851509 0.39985886 0.53493477
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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
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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 file differs from step to step. Each is given on the how-to page that the corresponding step links to.
KPOINTS
The KPOINTS file contains only the Γ 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
Step 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]).
Step 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. |
Step 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. |
Step 4: TI from MLFF to GGA
Perform a thermodynamic integration from MLFF (λ=0) to RPBE+D3 (λ=1). This should be done separately for Fe3+ and Fe2+.
| Important: This step requires using GGAs on many structures so will be more time consuming. |
| Mind: Thermodynamic perturbation theory (TPT) from RPBE+D3 to PBE0, for both Fe3+ and Fe2+, 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
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_{\mathrm{ML}} }[/math]
- the free energy difference from MLFF to GGA
- Reduced: Fe2+ - [math]\displaystyle{ \Delta A^{\mathrm{FP}_\mathrm{GGA}-\mathrm{ML}}_1 }[/math]
- Oxidized: Fe3+ - [math]\displaystyle{ \Delta A^{\mathrm{FP}_\mathrm{GGA}-\mathrm{ML}}_0 }[/math]
- [math]\displaystyle{ \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]
The vacuum alignment is already contained in [math]\displaystyle{ \Delta A^{\mathrm{ML}} }[/math]. Combining the remaining terms as above gives the free energy difference from Fe3+ to Fe2+ on the GGA level [math]\displaystyle{ \Delta A^{\mathrm{FP}_\mathrm{GGA}} }[/math]. Switch the sign to get the absolute redox potential [math]\displaystyle{ U^{\mathrm{FP}_\mathrm{GGA}}_{\mathrm{redox}} }[/math]. Finally, take the standard hydrogen electrode (SHE) (4.44 V [2]) away to compare to experiment [math]\displaystyle{ 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:
| 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_{\mathrm{ML}} }[/math] | -4.98 | -4.95 |
| 4 - TI (GGA) | [math]\displaystyle{ \Delta A^{\mathrm{FP}_\mathrm{GGA}-\mathrm{ML}}_1 - \Delta A^{\mathrm{FP}_\mathrm{GGA}-\mathrm{ML}}_0 }[/math] | -0.055 | -0.049 |
| Absolute redox potential | [math]\displaystyle{ U^{\mathrm{FP}_\mathrm{GGA}}_{\mathrm{redox}} }[/math] | 5.03 | 5.00 |
| Redox potential | [math]\displaystyle{ U^{\mathrm{FP}_\mathrm{GGA}}_{\mathrm{redox,SHE}} }[/math] | 0.59 | 0.56 |
Recommendations and advice
- 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.
Related tags and articles
- How-tos
- Vacuum reference
- Thermodynamic integration between machine-learned force fields
- Thermodynamic integration between a machine-learned force field and a density functional
- Theory
- Tags