Localizing Charge - VASP
Contents
Introduction
Localization of charge (polarons) often fails with DFT because of self-interaction errors inherent in typical exchange-correlation functionals. For instance, an oxygen vacancy in TiO2 may lead to reduction of two Ti4+ cations to Ti3+ (O2- + 2Ti4+ --> 1/2 O2 + 2Ti3+). Each Ti3+ atom has one unpaired electron. DFT calculations however lead to the two unpaired electrons being spread across several Ti sites (delocalized solution), which is not correct.
We have previously modeled unpaired electrons in TiO2 where we can control where Ti3+ electrons form. See for instance these papers.
“Electron Transport via Polaron Hopping in Bulk TiO2: A Density Functional Theory Characterization”, Deskins NA, Dupuis M, Physical Review B, 75, 195212, 2007 doi:10.1103/PhysRevB.75.195212
“Intrinsic Hole Migration Rates in TiO2 from Density Functional Theory”, Deskins NA, Dupuis M, Journal of Physical Chemistry C, 113, 346-458, 2009 doi:10.1021/jp802903c
“Localized Electronic States from Surface Hydroxyls and Polarons in TiO2(110)”, Deskins NA, Rousseau R, Dupuis M, Journal of Physical Chemistry C, 113, 14583-14586, 2010 doi:10.1021/jp9037655
Below is described a procedure to control where localization of unpaired electrons occurs. Note that we also have a website describing an even more efficient method of modeling localized charge (polarons): modelingpolarons.org
Factors Affecting Localization
Standard DFT often fails in modeling localized electrons. If one applies the DFT+U method (or hybrid functionals) it is possible to get correct wavefunctions with localized electrons. Even still, getting electrons localized at specific sites is not guaranteed with DFT+U (see for instance B. Meredig, A. Thompson, H.A. Hansen, C. Wolverton, A. van de Walle, "Method for locating low-energy solutions within DFT + U", Phys. Rev. B. 82 (2010) 2–6.).
Primarily the (1) initial geometry and (2) initial wavefunction control where electrons localize (or do not localize!). Electron polarons (localized electrons in semiconductors) tend to have elongated bond lengths near the polaron site. An initial geometry with elongated bond lengths (such as longer Ti-O bond lengths in TiO2) may help localize an electron. An artificial wavefunction with an electron localized at specific sites may also be used to start a calculation to localize electrons.
Description of Procedure
In our work we localized electrons at specific Ti sites in TiO2. Ti3+ sites formed either due to defects (O vacancies for instance) or excess number of electrons (mimicking presence of an unpaired electron).
We created initial geometries by stretching Ti-O bond lengths around the polaron sites. This is straight-forward and can be performed with molecular editing software or by editing input files directly.
We also created wavefunctions that had localized electrons using vanadium pseudopotentials. Vanadium has one more proton in its nucleus than titanium, so has stronger positive attraction to any electrons. Thus, in a system of Ti and V, electrons are more strongly attracted to V. By running a calculation with a V atom, we can generate a wavefunction where the electron is localized at the V site. This wavefunction can then be used for a calculation without any V atoms (the real system).
Please note, that even using these procedures may not always give localized solutions that you may want. Final solutions may not give electrons localized at the desired sites! Sometimes many simulation attempts are needed to get the electron localized at the desired site. This may mean trying different simulation parameters (such as U value, stretched bond length, initial geometry, etc.) until you get a solution that is acceptable.
Input Files
Below are sample input files for VASP. The cell size, k-point mesh, input parameters, etc. have not been tested for convergence. These files are only for illustration purposes and not to be used for publication-quality results.
These files are for a bulk (2x2x2) rutile system. We want to model one excess electron near a HO site. The H atom induces formation of Ti3+ (H + O2- + Ti4+ --> HO- + Ti3+ ).
We use the following KPOINTS file for all calculations. POTCAR files have Ti, O, H, and V atoms (I used PAW_PBE pseudopotentials with pv type for Ti and V).
KPOINTS Automatic mesh 0 MonkhorstPack 1 1 2 0. 0. 0.
Generate localized wavefunction.
The first calculation generates a WAVECAR file that has an electron localized at desired site. We use the V pseudopotential for the site to form Ti3+ (POTCAR has Ti, V, O, and H atoms). The total number of electrons for the (2x2x2) system with an H atom is 353; we want the same number of electrons in this system. In this example a very large U value is applied to U.
INCAR ISMEAR = 0 SIGMA = 0.05 ENCUT = 400 NCORE = 4 NELMIN = 2 NELM = 60 EDIFF = 1E-5 EDIFFG = -1E-2 NSW = 200 IBRION = 1 ISPIN = 2 NELECT = 353 LDAU = .TRUE. LDAUTYPE = 1 LDAUL = 2 2 1 1 LDAUU = 4.0 15 0 0 LDAUJ = 0.0 0 0 0
POSCAR
V substituted file. First atom type is Ti, second is V, third is O, fourth is H
1.00000000000000
9.3399999999999999 0.0000000000000000 0.0000000000000000
0.0000000000000000 9.3399999999999999 0.0000000000000000
0.0000000000000000 0.0000000000000000 5.9199999999999999
15 1 32 1
Selective dynamics
Direct
-0.0004699846303929 -0.0032521031703214 -0.0150057787074976 T T T
0.2252812914556076 0.2503486410659392 0.2496480853510218 T T T
0.5040401977766118 -0.0175910099222484 -0.0098935819159592 T T T
0.7612607986623962 0.2510027065308989 0.2374553844619291 T T T
0.0065393869454851 0.5008877313757687 -0.0115290896271588 T T T
0.2417791839728632 0.7531822118864184 0.2434893407022365 T T T
0.4953976959534083 0.5030287771335586 -0.0102910427339003 T T T
0.7574359142294389 0.7445929140285682 0.2383072940781929 T T T
-0.0045374335092064 0.0000389405546199 0.4943091115189039 T T T
0.2623142078546065 0.2525168738801275 0.7495659041841092 T T T
0.7380280576189752 0.2484201880419621 0.7463050094637427 T T T
-0.0097154391835515 0.4954972004858848 0.4987088388185281 T T T
0.2511566467570405 0.7483755898646669 0.7331026805363690 T T T
0.7502316340570636 0.7425302314109283 0.7371534580822683 T T T
0.4987259393896031 0.4973109568307729 0.4829847838375106 T T T
0.5043313906435358 -0.0012963277606170 0.5007210317731002 T T T
0.1532788030320936 0.1495024002598312 0.0042077149116071 T T T
0.3445867030912693 0.3470370873735698 0.0027209542887362 T T T
0.0946437970256201 0.4039121503976735 0.2537819595960750 T T T
0.4153352401715846 0.1188960891157516 0.2508008613363913 T T T
0.6558043558691304 0.1507403161527646 0.0055610269945205 T T T
0.8487168461882529 0.3495109827369624 0.0042176090245850 T T T
0.5922926687957819 0.3942791487099890 0.2551711265220347 T T T
0.9042408397779853 0.0990053110576561 0.2557481030603395 T T T
0.1530373247066900 0.6519043644238529 0.0035899592464614 T T T
0.3484796287716045 0.8529261055674779 0.0102733584154136 T T T
0.0961011711252115 0.9001632139967597 0.2549315153597910 T T T
0.4032134243585848 0.5972560498413075 0.2530173787613980 T T T
0.6527857332925765 0.6506885114855878 0.0057599279362317 T T T
0.8460104069033896 0.8481890384215569 0.0061649765416283 T T T
0.5938802652015220 0.9031857612142125 0.2601903694574049 T T T
0.9028888813661187 0.5987401917084600 0.2546767904919052 T T T
0.1548198723819734 0.1496910261185387 0.5024952234481855 T T T
0.3446506819485423 0.3488816861736828 0.5056429107633921 T T T
0.0963905613494719 0.4022229754500734 0.7524796237292541 T T T
0.4083992374630906 0.0946763823797750 0.7487168039838024 T T T
0.6546268734460808 0.1503920301163800 0.5051843708212360 T T T
0.8470441404243731 0.3494151356870118 0.5049234116112693 T T T
0.5962133850530550 0.4002788127941702 0.7558903703808348 T T T
0.9045197407782197 0.0970934659418627 0.7529299311809232 T T T
0.1539701400868104 0.6528636518509016 0.5039158829421505 T T T
0.3463228730230190 0.8504234235871974 0.5023663603305436 T T T
0.0969836679261209 0.9002396360526088 0.7550468540123395 T T T
0.4018223770689703 0.5995778255918651 0.7555346430473812 T T T
0.6527451186439509 0.6489838682628037 0.5053310082365458 T T T
0.8462297516413920 0.8468098103155415 0.5054389873393840 T T T
0.5989517353018746 0.8974174536362854 0.7540901084780324 T T T
0.9040101525090046 0.5981797115586129 0.7538115766246487 T T T
0.4812854837328195 0.2020937377266750 0.2403568713021547 T T T
Utilize localized wavefunction.
In the first step we created a WAVECAR file with the localized electron. We use that WAVECAR to start a new calculation in this step, but without V (POTCAR has Ti, Ti, O, and H atoms). The POSCAR file could be taken from previous calculation, or a "stretched" POSCAR file could be used (as below). Note that the number of electrons is still 353. I use slightly smaller U value for the Ti3+ site. You may need to experiment with different U values.
INCAR
ISMEAR = 0
SIGMA = 0.05
ENCUT = 400
NCORE = 4
NELMIN = 2
NELM = 60
EDIFF = 1E-5
EDIFFG = -1E-2
NSW = 200
IBRION = 1
ISPIN = 2
NELECT = 353
LDAU = .TRUE.
LDAUTYPE = 1
LDAUL = 2 2 1 1
LDAUU = 3.0 10 0 0
LDAUJ = 0.0 0 0 0
POSCAR
Non-V calculation. First atom type is Ti, second is also Ti, third is O, fourth is H
1.00000000000000
9.3399999999999999 0.0000000000000000 0.0000000000000000
0.0000000000000000 9.3399999999999999 0.0000000000000000
0.0000000000000000 0.0000000000000000 5.9199999999999999
15 1 32 1
Selective dynamics
Cartesian
-0.004389656447869686 -0.030374643610801878 -0.08883420994838577 T T T
2.1041272621953753 2.338256307555872 1.4779166652780495 T T T
4.707735447233553 -0.16430003267380008 -0.05857000494247847 T T T
7.11017585950678 2.3443652789985956 1.4057358760146208 T T T
0.06107787407083085 4.6782914110496785 -0.06825221059278008 T T T
2.258217578306543 7.034721859019148 1.4414568969572406 T T T
4.627014480204834 4.698288778427438 -0.06092297298468979 T T T
7.07445143890296 6.954497817026828 1.4107791809429022 T T T
-0.04237962897598778 3.637047801498661E-4 2.926309940191911 T T T
2.4500147013620257 2.3585076020403912 4.437430152769927 T T T
6.893182058161231 2.3202445563119265 4.4181256560253575 T T T
-0.09074220197437097 4.627943852538164 2.952356325805686 T T T
2.3458030807107586 6.989828009335988 4.339967868775304 T T T
7.007163462092974 6.935232361378071 4.363948471847028 T T T
4.658100273898894 4.644884336799419 2.859269920318064 T T T
4.7104551886106245 -0.012107701284162782 2.964268508096754 T T T
1.4316240203197548 1.3963524184268237 0.024909672276714027 T T T
3.218439806872456 3.241326396069142 0.016108049389318306 T T T
0.8839730642192918 3.772539484714271 1.5023892008087638 T T T
3.8792311432026 1.1104894723411198 1.3347410991114367 T T T
6.125212683817679 1.407914552866822 0.032921279807561356 T T T
7.927015343398282 3.264432578763229 0.024968245425543198 T T T
5.532013526552602 3.682567248951297 1.5106130690104451 T T T
8.445609443526383 0.924709605278508 1.5140287701172095 T T T
1.429368612760485 6.088786763718788 0.021252558739051488 T T T
3.2547997327267866 7.966329826000244 0.06081828181924852 T T T
0.8975849383094756 8.407524418729734 1.5091945709299628 T T T
3.766013383509182 5.5783715055178105 1.497862882267476 T T T
6.097018748952667 6.077430697275392 0.03409877338249166 T T T
7.901737200477659 7.922085618857342 0.03649666112643953 T T T
5.546841676982215 8.435755009740744 1.3803269871878371 T T T
8.432982151959548 5.592233390557015 1.507686599712079 T T T
1.4460176080476317 1.3981141839471514 2.9747717228132595 T T T
3.219037369399386 3.258554948862198 2.9934060317192817 T T T
0.9002878430040675 3.756762590703686 4.454679372477185 T T T
3.814448877905266 0.8842774114270984 4.582403479584111 T T T
6.264214997986397 1.5346615612869897 2.9906914752617157 T T T
7.911392271563646 3.2635373673166903 2.9891465967387143 T T T
5.568633016395531 3.7386041114975503 4.474870992654543 T T T
8.44821437886857 0.9068529718969977 4.457345192591066 T T T
1.4380811084108094 6.09774650828742 2.983182027017531 T T T
3.1046556340349983 7.8029547763044255 2.9740088531568167 T T T
0.9058274584299693 8.408238200731367 4.46987737575305 T T T
3.7530210018241834 5.600056891028019 4.472765086840497 T T T
6.096639408134504 6.061509329574587 2.991559568760351 T T T
7.903785880330602 7.909203628347161 2.9921988050491533 T T T
5.5942092077195085 8.381879016962905 4.604213442189953 T T T
8.443454824434102 5.586998505957445 4.462564533617921 T T T
4.495206418064535 1.8875555103671446 1.4229126781087558 T T T
Run Final Solution
The final step would be to use the WAVECAR and CONTCAR from the previous step (with localized electron at desired site), but edit INCAR so the U values are reasonable.
Check Localization
At each step we need to verify that we have the solution we want. Two methods to check electron localization are Bader analysis or looking at spin density.
Bader Analysis
Bader analysis can be performed with the Bader code of the Henkelman group (see here ). Note that I was using VASP 5.3 which added some extra lines to CHGCAR for each atom type (a line containing "Ti V O H"). The old VASP version doesn't include these lines, so I had to manually delete them from CHGCAR in order for the Bader code to work.
Visualizing Spin Density.
Programs like VMD and others can visualize the spin density. The CHGCAR file has the spin density information at the end of the file. You should be able to see nice localized electrons from the spin density data. If you don't see nice localized charge (you should see d orbital in example above), then you need to retry the calculation - it didn't work.
Localizing Holes
The above description pertains to electron localization. When holes form (such as O2- forming O-), similar procedures can be used (see paper below), except the U correction can be applied to the oxygen p orbitals.
“Intrinsic Hole Migration Rates in TiO2 from Density Functional Theory”, Deskins NA, Dupuis M, Journal of Physical Chemistry C, 113, 346-458, 2009 doi:10.1021/jp802903c
