Benchmarking the DFT+U Method for Thermochemical Calculations of

Article pubs.acs.org/JPCA

Benchmarking the DFT+U Method for Thermochemical Calculations of Uranium Molecular Compounds and Solids George Beridze†,‡ and Piotr M. Kowalski*,†,‡ †

Institute of Energy and Climate Research, Nuclear Waste Management and Reactor Safety, Forschungszentrum Jülich, Wilhelm-Johnen-Straße, 52428 Jülich, Germany ‡ JARA High-Performance Computing, Schinkelstraße 2, 52062 Aachen, Germany ABSTRACT: Ability to perform a feasible and reliable computation of thermochemical properties of chemically complex actinide-bearing materials would be of great importance for nuclear engineering. Unfortunately, density functional theory (DFT), which on many instances is the only affordable ab initio method, often fails for actinides. Among various shortcomings, it leads to the wrong estimate of enthalpies of reactions between actinide-bearing compounds, putting the applicability of the DFT approach to the modeling of thermochemical properties of actinide-bearing materials into question. Here we test the performance of DFT+U methoda computationally affordable extension of DFT that explicitly accounts for the correlations between felectrons - for prediction of the thermochemical properties of simple uraniumbearing molecular compounds and solids. We demonstrate that the DFT+U approach significantly improves the description of reaction enthalpies for the uranium-bearing gas-phase molecular compounds and solids and the deviations from the experimental values are comparable to those obtained with much more computationally demanding methods. Good results are obtained with the Hubbard U parameter values derived using the linear response method of Cococcioni and de Gironcoli. We found that the value of Coulomb on-site repulsion, represented by the Hubbard U parameter, strongly depends on the oxidation state of uranium atom. Last, but not least, we demonstrate that the thermochemistry data can be successfully used to estimate the value of the Hubbard U parameter needed for DFT+U calculations.

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INTRODUCTION Understanding the properties of actinide-bearing materials and their interaction with the environment is a challenge for nuclear waste storage strategies.1−4 Lack of in-depth understanding of the atomic-scale processes that determine the properties of materials upon incorporation of radionuclides highly limits the availability of materials that could be used as host phases for nuclear waste storage. On the one hand, the relevant experimental studies are usually very difficult to perform and often time-consuming. On the other hand, with the increasing power of computing resources, it is now possible to simulate the atomic-scale behavior of actinide-bearing materials using ab initio methods of computational quantum chemistry.1,5 When reliable and time-feasible, such computer-aided studies could to great extent complement the experimental techniques used in characterization of nuclear waste forms and substantially enhance our knowledge on the atomic-scale processes that influence performance of these materials and their safety within the repositories.6 Unfortunately, having strongly correlated 5f electrons, actinide-bearing materials are very difficult to handle by the standard density functional theory (DFT) methods. It is then proposed that more computationally demanding methods (at least hybrid functionals) should be used in calculations of such systems.7 However, even with the most powerful computing resources, DFT is often the only affordable © 2014 American Chemical Society

quantum chemistry method that allows for the ab initio modeling of chemically complex materials. Therefore, an assessment of the performance of available DFT-based methods and a careful choice of the computational strategy are required to perform meaningful ab initio simulations of complex, actinide-bearing materials. Although there exist various studies of actinide-bearing compounds and solids by DFT,8−11 its applicability for computation of such materials is questionable.7,12,13 DFT often fails not only on the quantitative but also on the qualitative level. Wen et al.13 have shown that the hybrid functional, such as HSE, has to be used to correctly compute the semiconducting state of actinide oxide solids, which are often described as metals by DFT. On the quantitative level, when used for the prediction of physical and chemical parameters, DFT can lead to significant errors. For instance, the difference between predicted by DFT and the measured enthalpies of reactions for various uranium-bearing molecular compounds can be as large as 100 kJ/mol.7,14,15 By testing the different computational methods, including various DFT generalized gradient approximations (GGA) and hybrid Received: October 7, 2014 Revised: November 19, 2014 Published: November 20, 2014 11797

dx.doi.org/10.1021/jp5101126 | J. Phys. Chem. A 2014, 118, 11797−11810

The Journal of Physical Chemistry A

Article

We are especially interested in the performance of the DFT+U method with the Hubbard U parameter (or the effective Hubbard U parameter Ueff = U − J) derived for each compound and solid by the linear response procedure proposed by Cococcioni and de Gironcoli.36 To our knowledge there exist no systematic studies of the performance of the DFT+U method for prediction of the thermochemical parameters of actinides. In particular, the DFT+U calculation with the Hubbard U parameter derived by the linear response method reported here is the first such study conducted for the actinidebearing materials.

functionals as well as post Hartree−Fock methods such as MP2 and CCSD(T), Shamov et al.7 and Schreckenbach and Shamov12 have shown that, to reach the experimental accuracy (20 kJ/mol), at least hybrid functionals such as PBE0 have to be used in computation of the enthalpies of reactions involving uranium-bearing molecular compounds. In line with these findings, most of the meaningful DFT studies of actinidebearing materials utilize the hybrid functionals such as B3LYP, PBE0, or HSE.16−20 Moreover, studies of Shamov et al.,7 Schreckenbach and Shamov,12 and Odoh and Schreckenbach21 indicate that the result of such hybrid calculations can also depend on the number of core electrons modeled by pseudopotentials and that at least 32 electrons of the uranium atom (5s25p65d106s26p65f36d17s2) should be treated explicitly, which further increases the computational cost. We note, however, that this result is still in dispute. For instance, IcheTarrat and Marsden15 have shown that the explicit treatment of 32 electrons of the uranium atom only marginally improves the performance of either DFT or hybrid-DFT functionals over the case when only 14 valence electrons of uranium (6s26p65f36d17s2) are treated explicitly. We will show that our results implicitly support the later conclusion. The usage of hybrid functionals or any post-Hartree−Fock method requires substantial computational resources. This limits the applicability of these methods to the simplest molecular compounds and solids. Materials that are interesting for nuclear engineering have usually complex chemical compositions and structures with supercells containing often more than hundred atoms. This is especially true when solid solutions with diluted concentration of elements are of interest.2,6 Thus, DFT and its simple modifications, such as DFT+U, remain the only choice if one wants to compute complex materials and simulate their dynamical behavior by methods such as ab initio molecular dynamics22−24 on nowadays supercomputing resources. Wen et al.25 have shown that the DFT+U method with a reasonable choice of the Hubbard U parameter can reproduce the band gaps of actinide oxides and correctly predict insulating state for these solids. This method has been successfully used in the description of UO2 and its metastable states,26,27 in the calculation of U(VI) aqua complexes on titania particles,14 and for the investigation of incorporation of uranium in the ferric garnet matrices.28 It also predicts correctly the magnetic state of actinide compounds29 and the full phonon dispersion of strongly correlated materials.30 These successes have been achieved sometimes at a cost of worse description of lattice parameters25 or even anomalous change in volume.31 In all these studies the same Hubbard U parameter value for uranium of 4.5 eV (or Ueff = U − J = 4.5 eV − 0.5 eV = 4 eV) has been used, which has been derived from the spectroscopic measurements of UO2.32,33 We note, however, that there is no guarantee that the value derived for one system should be easily transferable to another, especially that other estimates for neutral uranium atoms provide somehow lower values of U = 2.3 eV34 and U = 1.9 eV.35 In this contribution we systematically test the performance of the DFT+U method for the prediction of the thermochemical parameters, namely the reaction enthalpies, of simple uraniumbearing molecular complexes and solids, most of which were considered in previous studies. We do that to compare our results with already published predictions of much more demanding computational methods such as hybrid functionals, MP2 or CCSD(T)7 and with the available experimental data.

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COMPUTATIONAL METHOD DFT Setup. All calculations were performed using the planewave Quantum ESPRESSO code.37 We used the PBE,38 PBEsol,39 and BPBE38,40 exchange−correlation functionals. The core electrons of the computed atoms were replaced by the ultrasoft pseudopotentials41 and the 6s26p65f36d17s2 electrons of the uranium atom were treated explicitly (such a pseudopotential is often referred as “large core pseudopotential”). We performed the calculations with a large core pseudopotential because the main focus of our studies is the benchmarking of the feasible computational method that could be used in further modeling of chemically complex actinidebearing materials important, for instance, in nuclear waste management. We therefore do not focus here on benchmarking the small core size pseudopotentials or the all-electron methods that are currently computationally unfeasible for computation of such materials. All calculations were spin polarized and the spin configurations were carefully checked to achieve the ground state electronic configurations of the considered systems. We applied the energy cutoff of 50 Ryd, which gave energy differences within 0.5 kJ/mol, which is sufficient for our studies. The atomic configurations were relaxed to the equilibrium positions, so the maximum component of the residual forces on the ions was less than 0.005 eV/Å. The DFT equilibrium geometries of the molecular compounds were confirmed by supplementary calculations of the vibrational frequencies. The crystalline solids were treated as continuum in all three spatial dimensions by applying the periodic boundary conditions. For the calculations of solids, the Methfessel− Paxton42 k-points grids were used and their sizes were chosen to give the energy differences within 0.5 kJ/mol. The calculations of solids were performed by simultaneous relaxation of the lattice parameters, and the ionic positions so the resulting pressure was 0 GPa with a tolerance of 0.01 GPa. Because the differences in the energies between the antiferromagnetic and ferromagnetic states of the considered solids were usually smaller than 1 kJ/mol, we consider the ferromagnetic spin arrangement for them. The exceptions are α-U and UO2 solids for which the antiferromagnetic states were considered as the ground state. Following Shamov et al.7 we also neglect the spin−orbit interaction in all the calculations. This is because these effects should not affect the computed reaction enthalpies by more than 20 kJ/mol, which is at the level of experimental accuracy and much smaller than the error in predicted reaction enthalpies obtained by using the standard DFT approximation (∼100 kJ/mol7,15). Modified PBE Functional. To test the tendency of different DFT functionals in predicting the reaction enthalpies, we performed a series of calculations with different choices of μ and κ parameters of the PBE functional exchange enhancement 11798



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Table 1. List of the 17 Considered Reactions between the Gas-Phase Uranium-Containing Moleculesa UF6 → UF5 + F

(1)

UO3 → UO2 + O

UF5 → UF4 + F

(2)

UOF4 + UO3 → 2UO2 F2

UF4 → UF3 + F

(3)

2UOF4 → UF6 + UO2 F2

UCl 6 → UCl5 + Cl

(11) (12)

(4)

UO3 + H 2O → UO2 (OH)2

UCl5 → UCl4 + Cl

(5)

UF6 + H 2O → UOF4 + 2HF

UCl4 → UCl3 + Cl

(6)

UF6 + 2H 2O → UO2 F2 + 4HF

UOF4 → UF4 + O

UF6 + 3H 2O → UO3 + 6HF

(7)

UO2 F2 → UO2 + 2F UF6 + 2UO3 → 3UO2 F2 a

(10)

(8)

(13) (14) (15) (16)

UO2 F2 + 2H 2O → UO2 (OH)2 + 2HF

(17)

(9)

The reactions 1 and 2, 7−12, and 14−16 were computed by Shamov et al.7

function.38 Here we do not limit the choice of these parameters by any physical behaviors, such as the slowly varying density limit or the Lieb−Oxford bound,38 but simply treat μ and κ as free parameters. Such an exercise of modifying the exchange part of the PBE functional, when one should rather expect correction of the correlation part of an exchange−correlation functional, was inspired by the fact that the hybrid PBE0 functional, which admixes the Hartree−Fock exchange with the PBE exchange, keeping the PBE correlations,43,44 performs exceptionally well for uranium.7 By the modified PBE functional in further discussion we mean the case with μ = 0.219 51 (PBE value) and κ = 6.0. Further discussion of the modified functional approach is provided in the Results and Discussion. DFT+U Method. In the current studies we utilize two kinds of the DFT+U approaches. First, we performed calculations with the fixed values of the Hubbard U parameter of 0.5, 1.5, 3, 4.5, and 6 eV. We also performed calculations with the Hubbard U parameter derived by the linear response approach proposed by Cococcioni and de Gironcoli.36 In this procedure the molecular geometries were fixed to the DFT structures. We used the atomic f-orbitals produced by uspp-736 package41 as projectors and the elements of response matrices were derived by the finite differences, setting α parameter to ±0.01, as described in the work of Cococcioni and de Gironcoli.36 The U values have been derived for uranium in all uranium-bearing molecular compounds and solids considered in this paper. Because the uranium atom d-shell could also be partially filled, we checked the effect of applying the +U correction to the d-orbitals of uranium. Similarly to the case of f-orbitals, we computed the Hubbard U values for the d-shell by the linear response approach of Cococcioni and de Gironcoli.36 However, because the current version of the Quantum ESPRESSO code does not allow for simultaneous application of the DFT+U method to two or more orbitals of the same species, in the subsequent calculations we computed only the correction to the primary DFT+U calculations (i.e., the calculations with the Hubbard U applied to f-electrons only). We did this by computing the Hubbard energy resulting from the application of DFT+U method to the d-orbitals and by fixing the molecular structures to the atomic configurations optimized with the primary DFT+U method. We will show that this approach is justified by the smallness of the correction. Last but not least, because it is known that the DFT+U calculations of uranium solids, namely UO2, can lead to a menagerie of metastable states depending on the initially chosen occupations of f-orbitals,45 in our studies we took

special care to ensure that the computed states are the real ground states. We did this following the procedure outlined in Dorado et al.45 by (1) performing set of calculations with the different initial occupations of f-orbitals, (2) breaking the cubic symmetry of electrons, and (3) checking the correctness of the final occupations. We also verified that all the considered solids show large band gaps. However, in spite of the applied precautions, because of the complexity of the problem itself, there is no full guarantee that a ground state has been reached in all the cases. Nevertheless, results of Dorado et al.45 for UO2 show that in the case when the ground state is not reached, the resulted metastable state should be higher in energy by no more than a fraction of kJ/mol. Such an error is negligibly small in comparison with the experimental accuracy of the considered reaction enthalpies (20 kJ/mol).

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CONSIDERED SYSTEMS The set of uranium-bearing molecular compounds, including U(VI), U(V), U(IV), and U(III) halogenide, oxide, and oxyhalide molecules computed here contains all the compounds considered by Shamov et al.,7 Iche-Tarrat and Marsden,15 and Batista et al.46 In addition, we computed the series of uranium chlorides for which good experimental data exist. All the considered reactions are given in Table 1. We selected the already computed reactions to make a broad comparison of our results with the results obtained by using different computational methods, including hybrid functionals and higher level post Hartree−Fock methods such as MP2 and CCSD(T).7 The thermochemistry data used as a reference were derived from the formation enthalpies taken from Morss et al.47 and Guillaumont et al.48 The reaction enthalpies were computed by taking the differences in the total energies of the reactants. The initial structural parameters of computed solids were taken from different experimental studies, and the adequate references are indicated in the relevant text and tables. Molecular Compounds. The stable geometries of the molecules found in our calculations are in agreement with previous studies of the same systems.7,12,15,46 Uranium hexahalogenides (UX6, where X = F, Cl) have Oh symmetry. The symmetry of uranium pentahalogenides (UX5) is C4v and that of UX4 molecules is C2v. UX3 and UO2F2 molecules have C3v and C2v symmetries, respectively. UO3 molecule has Tshaped planar geometry and UO2 has linear geometry. The two possible structures of UOF4 are shown in Figure 1. Our DFTPBE calculation indicates that the square pyramidal structure is more stable than the trigonal bipyramide form by 2.4 kJ/mol, which is in agreement with the studies of Shamov et al.7 11799



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bond distance and the Hartree−Fock calculation also shows surprisingly good agreement with the experiment. The allelectron PBE0 calculation7 with 1.997 Å gives the best match to the experimental value. Because the PBEsol functional recovers correctly the slowly varying density limit of the exact exchange energy functional, it usually improves description of the structural parameters over PBE.39,63 This is also visible in our calculations. PBEsol results in 2.009 Å for the U−F bond length of UF6, which is in better agreement with the experiment than the PBE value. On the one hand, the U−F bond length obtained with the modified PBE functional is in the worst agreement with the experimental value. On the other hand, this functional predicts much better the enthalpies of reactions than the PBE or PBEsol functionals, including the dissociation enthalpy of UF6, which will be discussed later. The fact that, by switching to a different GGA functional, one can improve the prediction of the reaction enthalpies at a cost of worsening the structural parameters and vice versa, is a known shortcoming of current GGA-DFT functionals. We investigated further such behavior by computing the U−F bond length in UF6 and the dissociation enthalpies of UF6 and UF5 molecules with a modification of PBE functional. The modification was made by treating κ (PBE value of 0.804) and μ (PBE value of 0.219 51) parameters of the exchange energy functional enhancement function Fx(s),38

Figure 1. Two possible structures of UOF4 molecule: the trigonal bipyramide (left) and the square pyramidal configuration (right).

Crystalline Solids. We consider 12 solids: α-U,49 UF6,50 αUF5,51 β-UF5,52 UF4,53 UF3,54 UCl6,55 UCl4,56 UCl3,56 UO2,57 α-UO3,58 and U3O8.59 The initial structures were taken from the indicated references.

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RESULTS AND DISCUSSION Molecular Compounds: Structural Parameters. Three of the considered molecules, UF6, UCl6, and UCl3, have measured structural parameters (the U−F,Cl bond lengths and the Cl−U−Cl angle in the case of UCl3). We therefore start our analysis by performing an assessment of performance of different methods in prediction of the bond lengths in these compounds. First, we compare the U−F bond length computed for the gas-phase UF6 molecule to the experimental value and to the previous theoretical estimates.60−62 All the results are given in Table 2. The measured U−F bond length is 1.99960 and 1.997 Å.61 The PBE functional overestimates the bond length by slightly more than 1%, which is a known feature of this functional.63 The BLYP functional62 gives an even worse prediction, which is also expected.64 The hybrid B3LYP functional used in the same studies improved the value of the

Fx(s) = 1 + κ −

κ 1 − μs 2 /κ

(1a)

where s ∝ ∥∇n∥/n is the reduced electron density (n) gradient,38 as free parameters. The result is given in Figure 2. On the one hand, it is clearly seen that the smaller μ results in a better bond length. In fact, our result with μ being half the value of that of the PBE functional, which closely resembles the

Table 2. Bond distances in UF6, UCl6, and UCl3 Molecules between Uranium and Halogen Atoms and U−Cl−U Bond Angle in UCl3 Molecule Obtained Using Different Computational Methodsa

a

functionals and methods

U−F

PBE PBEsol BPBE modified PBE PBE+U (U = 0.5 eV) PBE+U (U = 1.5 eV) PBE+U (U = 3.0 eV) PBE+U (U = 4.5 eV) PBE+U (U = 6.0 eV) PBE+ULR PBEsol+U (U = 0.5 eV) PBEsol+U (U = 1.5 eV) PBEsol+U (U = 3.0 eV) PBEsol+U (U = 4.5 eV) PBEsol+U (U = 6.0 eV) PBEsol+ULR BLYPb PBE0b HFc,d B3LYPb MP2b expe,f,g,h

2.024 2.009 2.025 2.048 2.025 2.027 2.031 2.034 2.039 2.030 2.010 2.012 2.015 2.024 2.015 2.015 2.043 1.997 1.984 2.014 2.005 1.999 (1.997)

U−Cl(UCl6)

U−Cl(UCl3)

Cl−U−Cl

2.469 2.445 2.506 2.471 2.470 2.472 2.477 2.486 2.500 2.474 2.446 2.449 2.453 2.462 2.474 2.451

2.509 2.488 2.519 2.559 2.525 2.550 2.557 2.570 2.581 2.535 2.495 2.504 2.538 2.548 2.563 2.509

106.4 105.0 106.4 111.8 106.4 106.7 106.3 106.7 106.4 106.3 105.4 105.1 105.8 106.3 106.2 105.1

2.565

109.5

2.592 2.521 2.549 ± 0.008

114.5 106.9 95 ± 3

2.48

2.42 (2.461)

The bond lengths are given in Å and angles in degrees. bReference 62. cReference 65. dReference 66. eReference 60. fReference 67. gReference 68. Reference 61.

h

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approach in general results in a larger U−F bond length for UF6 than the one predicted by the applied GGA functional. However, we will show that, in line with the above conclusions, it also results in much better prediction of the gas-phase reaction enthalpies. The same trends as for UF6 are observed in the case of UCl6 that has the same geometry and the oxidation state of uranium. Also in this case the PBEsol functional results in a better match to the measured U−Cl bond length. The experimental U−Cl bond length for uranium trichloride, UCl3, is 2.549 ± 0.008 Å.69 All the GGA functionals underestimate this bond length by ∼0.05 Å. In that case the DFT+U method improves the bond length and the modified PBE functional also results in a good match to the experiment. However, all the methods overestimate the Cl−U−Cl bond angle. Interestingly, for that parameter the hybrid functionals, PBE0 and B3LYP, give worse prediction than the PBE and PBEsol methods. Molecular Compounds: Reaction Enthalpies. Several reactions involving gas-phase molecules were selected to asses the performance of different methods in the prediction of enthalpies of reactions involving uranium-bearing molecules. As already mentioned, the experimental data exist for all of them and the enthalpies of most of the considered reactions were already computed using different methods of computational quantum chemistry. All the reactions are listed and labeled in Table 1. This labeling we will use subsequently in all further discussion in the text. DFT Calculations. All the reaction enthalpies computed with the standard GGA functionals are given in Table 3. First, despite the different pseudopotentials used in our and Shamov et al.7 studies (we used the large core ultrasoft pseudopotentials, whereas Shamov et al.7 used the all electrons or the small core pseudopotential calculations), the PBE results of the two studies are well consistent. The difference, measured by the mean absolute error (MAE), between the two calculations is only 3.2 kJ/mol.a This result indirectly validates our computational setup. The MAE obtained with the PBE functional for all

Figure 2. Left panel: enthalpies of reactions 1 (red, upper set of curves) and 2 (black, lower set of curves) computed with the PBE-like functionals and different values of κ (horizontal axis) and μ = 0.109755 (dotted lines), 0.21951 (solid lines, PBE), and 0.43209 (dashed lines). μ and κ determine the enhancement function of the exchange functional (eq 1a). Right panel: U−F bond length of UF6. The different results are marked using the convention applied in the left panel. In both panels, the horizontal lines indicate the experimental values.

case of PBEsol functional (where μ = 10/81 = 0.123 456 839), gives better U−F bond length than PBE. On the other hand, the computed reaction enthalpy become worse and it cannot be easily improved by just modifying κ. For a fixed value of μ, the increase of κ always leads to a better description of the reaction enthalpies, but at a cost of worsening the bond length. The larger the μ, the worse the U−F bond length that results in good enthalpy is. In the subsequent sections we will show the performance of the modified PBE functional, where, keeping the PBE value of μ = 0.219 51, we set κ to 6. We chose such a set of parameters because it results in a relatively good prediction of the dissociation enthalpies of UF6 and UF5 molecules, which is indicated in Figure 2. The DFT+U

Table 3. Reaction Enthalpies Computed with Different DFT Methods for the Reactions Given in Table 1 and the Respective Experimental Values47,48,a 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 mean error MAE a

PBE

PBEsol

BPBE

PBEmodif

PBE-BPBE corr

exp

422 514 619 278 312 414 554 1153 −207 678 −113 19 −141 178 337 608 131 78 95

458 549 641 317 347 447 613 1204 −206 718 −114 23 −157 190 356 638 125 101 119

403 497 607 254 302 395 529 1131 −208 658 −105 1 −133 173 344 620 143 70 88

324 413 541 176 217 339 441 964 −214 568 −114 14 −63 112 210 421 149 4 51

328 429 558 155 261 320 430 1045 −213 577 −72 −69 −105 152 373 666 189 35 74

313 ± 17 385 ± 22 618 ± 27 205 ± 20 204 ± 20 412 ± 25 405 ± 20 1033 ± 20 −311 ± 22 576 ± 20 −170 ± 40 −25 ± 31 −182 ± 11 65 ± 12 189 ± 12 437 ± 12 272 ± 11 20

Term PBEmodif indicates the modified PBE functional. The enthalpies are given in kJ mol−1. 11801



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Table 4. Reaction Enthalpies Computed with the PBE+U Method for the Reactions Given in Table 1 and the Respective Experimental Values47,48,a reaction

0.5 eV

1.5 eV

3 eV

4.5 eV

6 eV

ULR

ULR, d

exp

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 mean error MAE

410 496 609 266 294 399 528 1176 −232 693 −127 23 −147 175 328 607 133 71 90

386 460 627 241 254 411 474 1107 −283 608 −158 32 −158 170 308 603 138 47 63

349 405 594 200 193 330 391 1120 −363 598 −204 45 −173 160 275 593 146 14 55

311 349 559 157 136 300 311 1079 −443 536 −247 52 −186 145 238 579 154 −23 70

271 310 511 100 99 261 248 1048 −497 497 −278 59 −187 130 201 550 162 −55 90

273 402 622 222 198 357 346 1048 −260 579 −121 −19 −149 127 272 538 117 8 45

270 410 621 223 198 361 352 1015 −225 560 −98 −28 −146 125 278 530 106 8 50

313 ± 17 385 ± 22 618 ± 27 205 ± 20 204 ± 20 412 ± 25 405 ± 20 1033 ± 20 −311 ± 22 576 ± 20 −170 ± 40 −25 ± 31 −182 ± 11 65 ± 11 189 ± 12 437 ± 12 272 ± 11 20

a

The ULR,d column represents the PBE+ULR results corrected with the PBE+U calculations applied to the d-orbitals (see text for details). The enthalpies are given in kJ mol−1.

Table 5. Reaction Enthalpies Computed with the PBEsol+U Method for the Reactions Given in Table 1 and the Respective Experimental Values47,48,a

a

reaction

0.5 eV

1.5 eV

3 eV

4.5 eV

6 eV

ULR

exp

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 mean error MAE

447 532 634 305 329 429 586 1188 −228 696 −127 27 −162 187 348 635 126 90 107

423 496 617 280 289 438 533 1157 −281 648 −159 37 −172 182 326 630 132 68 84

387 441 615 240 228 397 452 1113 −358 583 −204 51 −185 171 292 618 140 33 60

349 385 584 198 169 332 373 1075 −436 524 −247 57 −197 156 255 601 149 −6 63

309 328 553 148 123 300 292 1096 −513 526 −288 64 −208 140 217 582 157 −35 83

322 432 653 261 231 428 410 1030 −244 558 −109 −25 −161 139 302 574 111 28 50

313 ± 17 385 ± 22 618 ± 27 205 ± 20 204 ± 20 412 ± 25 405 ± 20 1033 ± 20 −311 ± 22 576 ± 20 −170 ± 40 −25 ± 32 −182 ± 11 110 ± 11 189 ± 12 437 ± 12 272 ± 11 20

Columns represent results obtained with the fixed values of U and with U = ULR. The enthalpies are given in kJ mol−1.

better match to the measured values for all the considered reactions. We found that this behavior is not restricted to the reactions between uranium-bearing compounds but has been also observed for the bond energies of metal−ligand complexes.70 Because the different GGA functionals result in different systematic offsets from the measured values, we decided to test the possibility of using a combination of two functionals to get better estimation of the reaction enthalpies. Especially instructive is the mixture of the BPBE and PBE results, because both functionals differ only by the exchange functional (they have the same PBE correlation energy functional38). In Table 3 in column labeled “PBE-BPBE corr”

the considered reactions is 95 kJ/mol. Because the mean error (ME) is of similar value, this indicates a systematic offset from the measured values. Such a significant systematic error is unacceptably large, having the largest experimental error of 40 kJ/mol. The PBEsol functional results in even larger error, which as we already discussed, is a known feature of this functional. In their studies, Shamov et al.7 used the PBE, BPBE, and OLYP standard GGA-DFT functionals. These calculations resulted in a systematic offset for all the considered reactions, with BPBE performing slightly better than PBE, and OLYP resulting in reduction of the error by ∼50%. Our calculations show a similar trend. The BPBE functional gives a slightly 11802



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we give the reaction enthalpies obtained by mixing the results of PBE and BPBE functionals as EPBE + 5(EBPBE − EPBE), where E indicates enthalpy. As shown in Table 3, such a mixing results in significant reduction of the mean error, although the MAE is reduced by only 21 kJ/mol. Interestingly, the modified PBE functional, with a ME of only 4.1 kJ/mol, significantly improves the prediction of reaction enthalpies. It gives results that are better than predictions of the PBE or PBEsol functionals, not only for the reactions 1 and 2 used in the construction of the functional but also for most of the considered reactions. This is a promising result. However, we will show that such a simple modification of the PBE functional does not give similarly good results for solids. Shamov et al.,7 Odoh and Schreckenbach,21 and Schreckenbach and Shamov12 concluded that calculations with the large core pseudopotential (14 valence electrons for uranium, which is also our setup) can result in significant underperformance of the PBE0 functional in prediction of the reaction enthalpies and suggest to use a small core or allelectron calculations to get converged energies. Because with the PBE and BPBE functionals we obtained results that are consistent with the all-electron calculations of Shamov et al.7 and Schreckenbach and Shamov,12 we suspect that the size of the core is not that important, when a GGA functional is used. Interestingly, Iche-Tarrat and Marsden15 have shown that the 32 electrons core pseudopotential only marginally improves, or even worsens in some cases, the performance over the 14 electrons core pseudopotential in the case of GGA and hybrid functionals. One exception is PBE0, which is also shown in results of Iche-Tarrat and Marsden.15 Therefore, we can only suspect that the number of valence electrons can impact the result, when that functional is used in calculations. DFT+U Studies. In most of the studies that utilize the DFT +U method, the Hubbard U parameter, and sometimes J (representing strength of the on-site electron exchange), is chosen in a way that particular, known properties of the investigated systems, such as the lattice parameters or the band gaps, could be reproduced by calculations. Some studies use ab initio based methods for the estimation of Hubbard U value,24 such as the linear response method,36 which we use in our studies, the constrained random-phase approximation (cRPA)71 or simple approaches such as the Slater transition state.72 Hubbard U values can be also derived from the spectroscopic measurements.32−35,73,74 In our studies we first computed the reaction enthalpies for different Hubbard U parameter values of 0.5, 1.5, 3, 4.5, and 6 eV that are kept the same for all the reactants and products. All the results are given in Tables 4 and 5. It is clearly visible that the DFT+U method improves the prediction of the enthalpies of reactions over the PBE and PBEsol functionals. In the next step, we determined the values of the Hubbard U parameter in a way that it minimizes the deviation from the measured reaction enthalpies (i.e., minimizes the MAE). These values were obtained by fit of the MAE obtained for different values of U, and reported in Tables 4 and 5, by parabola. The U value is given by the minimum of the fitted curve and the fitting results are illustrated in Figure 3. For the PBE+U and PBEsol+U methods, we obtained U = 3.0 ± 1.0 eV and 3.8 ± 1.0 eV respectively. The uncertainties on these values were estimated from Figure 3 taking into account the mean experimental error of (20 kJ/ mol)/√17 ∼ 5 kJ/mol.b In addition to these studies, we derived the Hubbard U parameters for each considered molecular compound using the linear response method

Figure 3. Mean absolute error (MAE) of the computed enthalpies of reactions given in Table 1 as a function of the Hubbard U parameter. Filled and open circles represent the results of PBE+U and PBEsol+U calculations, respectively. The curves represent fits by a parabola that were used to estimate the optimal U values (see discussion in the text).

proposed by Cococcioni and de Gironcoli.36 We therefore define so derived Hubbard U parameters as ULR (after linear response), and subsequently the DFT+U method as DFT+ULR. The obtained values of ULR are reported in Table 6. Table 6. Hubbard U Parameter Values for the UraniumBearing Molecules Calculated Using the Linear Response Method of Cococcioni and de Gironcoli36,a Hubbard ULR molecule

PBE (f-electrons)

PBEsol (f-electrons)

PBE (d-electrons)

UF6 UF5 UF4 UF3 UCl6 UCl5 UCl4 UCl3 UOF4 UO2F2 UO2(OH)2 UO3 UO2

3.1 2.3 1.8 1.9 2.2 2.2 1.7 1.4 2.8 3.1 2.8 2.6 2.0

3.0 2.3 1.7 1.9 2.2 2.2 1.7 1.5 2.7 3.1 2.8 2.6 1.9

0.6 0.6 0.7 0.8 0.5 0.5 0.5 0.6 0.6 0.7 0.6 0.6 0.6

a

The values are given in electronvolts. The last column represents the values computed for the d-orbitals.

Interestingly, on average, these values are slightly smaller than the ones that minimize the MAE. However, we notice that the derived Hubbard U parameters decrease with the decreasing oxidation state of uranium. When we consider only the values computed for U(VI) complexes, the obtained U value of 2.8 eV is in good agreement with the values obtained from minimization of the MAE. This is not surprising as most of the considered reactions involve U(VI) complexes. However, our result indicates that the thermochemistry data can be used to determine the Hubbard U parameter value independently from other experimental methods such as spectroscopy.32−35,73,74 Our studies also show that the linear response approach of Cococcioni and de Gironcoli36 gives U values that minimize the error of DFT+U reaction enthalpies. 11803



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In Table 6 we also provide the Hubbard U values computed for the d-orbitals of uranium. These are substantially smaller than the ones derived for the f-orbitals. As a consequence, the related Hubbard energy correction to the reaction enthalpies is usually not larger than 10 kJ/mol per reacting uranium atom (Table 4), which is small in comparison to an average 70 kJ/ mol correction to DFT resulting from the application of DFT +U method to f-electrons. The ME and the MAE are also almost identical to the ones obtained without applying DFT+U to d-electrons. This justifies the ommission of the DFT+U treatment of d-electrons of uranium in the previous studies (e.g., refs 25−29 and 31). We also neglect this effect in further analysis. In most of the up to date studies of the uranium-bearing materials the same Hubbard U parameter value for uranium of 4.5 eV (or Ueff = U − J = 4 eV) has been applied. This value was derived from the spectroscopic measurements of UO2.32,33 Our ULR values reported in Table 6 are somehow lower, but consistent with other estimates for neutral uranium atoms of 2.3 eV34 and 1.9 eV.35 We will elaborate more on this problem when discussing results for solids. As we have shown, the DFT+ULR method substantially reduces the error of the predicted reaction enthalpies for both PBE and PBEsol functionals. In Figure 4 we compare the MAE

currently limited to no more than a few dozens of atoms, and such calculations are usually orders of magnitude more resource consuming than the DFT+U method, which in that aspect performs similarly to the standard DFT. Our results thus strongly suggest that the DFT+U method, with carefully derived Hubbard U parameters, such as ULR, is a good tool for efficient calculations of complex actinide-bearing materials, including chemically complex solids. Therefore, in the next step, we tested the performance of DFT+ULR method for the prediction of structural parameters and reaction enthalpies of simple uranium-bearing solids. Uranium Solids. We extended the calculations and analysis performed for the molecular complexes into computation of simple uranium-bearing solids. We wanted to check if by using the mixture of GGA functionals, the modified PBE functional and the DFT+U method one can also improve the description of structures and the prediction of reaction enthalpies in the case of periodic systems. As for molecular compounds, in these calculations the Hubbard U parameter value has been derived for each solid using the linear response approach36 and the results for the values of ULR are given in Table 7. The ULR values derived for the solids are consistent with those derived for the molecular complexes (Table 6). Here we also observe that ULR is larger for the U(VI)-bearing solids (UF6 and UO3) and is the smallest for the U(III) carrying materials. Structures. In Tables 8, 9, and 10 we provide the lattice parameters of all the considered solids, computed with different methods and measured experimentally. All the methods usually overestimate the lattice lengths and volumes of the uranium fluorides solids (see the cases of UF6, α-UF5, and β-UF5, Table 8). This is consistent with the results obtained for the gas-phase UF6. Similarly to this case, the PBEsol functional gives the best match to the experiment. The DFT+U method, in general, worsens the agreement, in comparison with the prediction of respective GGA functionals, and the modified functional gives the worst prediction. Interestingly, all the methods overestimate the lattice parameters of UCl6 solid (Table 9), which is different from the result for the gas-phase UCl6 (Table 2). However, the PBEsol+ULR method results in good prediction of the lattice parameters of UCl4 and UCl3. α-U is the only pure-uranium solid considered here. Both the GGA functionals result in relatively good predictions of its lattice constants. This result is consistent with other computational studies.8 However, it is not well described by the DFT+U method, including the DFT+ULR cases, which significantly overestimates its volume. Surprisingly, a very good description of the α-U structure is obtained with the modified PBE functional, which badly overestimates the volumes of many other solids (Tables 8−10). Actinide oxides are major constituents of most actinidecarrying minerals, including the ones of interest for nuclear waste management.2 Therefore, we are especially interested in the performance of the utilized methods in the case of uranium oxides. Here we analyze three such crystalline solids: UO2, αUO3, and U3O8. The results are given in Table 10. We notice that for the oxides, the PBE and PBEsol functionals result in much better prediction of the lattice parameters than in case of uranium fluorides and chlorides. For UO2, the DFT+U method improves the agreement with experiment, and DFT+ULR results in a good match to the experimental lattice parameters. A similar trend is also observed for U3O8. α-UO3 is also well described by PBEsol+ULR, although with PBE+ULR its volume is significantly overestimated, which is similar to the case of αU. This may be somehow related to usage of the PBE

Figure 4. Mean absolute error for the 11 reactions considered by Shamov et al.7 obtained using different methods. The PBE0, CCSD(T), and MP2 results are those of Shamov et al.7

errors of the selected methods obtained for the reactions reported by Shamov et al.7 We immediately see that the DFT +ULR method reduces the error by a half and in fact it gives even slightly better results than the MP2 method and it is only slightly worse than the PBE0 hybrid functional, which is claimed to give good results for actinides.7,13 Figure 5 illustrates the deviations of the results obtained with all the considered methods from the experimental data and PBE0, MP2, and CCSD(T) results of Shamov et al.7 Interestingly, the absolute deviation from PBE0 prediction of the DFT+ULR method is only 27 kJ/mol, with the mean deviation being even 0 kJ/mol in the case of PBE+ULR, which suggests that DFT+ULR behaves in a similar way and carries similar errors as the PBE0 hybrid functional. We notice that the best match to the experimental values is obtained with the CCSD(T) method. However, because of the bad scaling, applicability of such advanced post Hartree−Fock methods is limited to a few atoms containing molecular compounds.5 Applicability of hybrid functionals is also 11804



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Figure 5. Mean absolute error and mean error (in kJ mol−1) of the selected 11 gas-phase reactions considered by Shamov et al.7 (Table 1) obtained with different methods indicates on the horizontal axes as (1) PBE, (2) PBEsol, (3) BPBE, (4) modified PBE, (5) PBE-BPBEcorr, (6) PBE+U(U = 0.5 eV), (7) PBE+U(U = 1.5 eV), (8) PBE+U(U = 3 eV), (9) PBE+U(U = 4.5 eV), (10) PBE+U(U = 6 eV), (11) PBE+ULR, (12) PBEsol+U(U = 0.5 eV), (13) PBEsol+U(U = 1.5 eV), (14) PBEsol+U(U = 3 eV), (15) PBEsol+U(U = 4.5 eV), (16) PBEsol+U(U = 6 eV), (17) PBEsol+ULR, (18) PBE0, (19) MP2, and (20) CCSD(T). Here the experimental values47,48 and the calculations of Shamov et al.7 using PBE0, MP2, and CCSD(T) methods were chosen as reference and the references are indicated in the left-upper corners.

Table 7. Hubbard U Parameter Values for the UraniumBearing Solids Calculated Using the Linear Response Method of Cococcioni and de Gironcoli36,a

gaps of the considered materials is discussed (e.g., see ref 25). This is because standard DFT usually underestimates the band gaps by as much as ∼40%.5,76,77 In Table 11 we provide the values of the band gap obtained in all our DFT+U calculations of solids. The experimental band gap of UO2 is 2.1 ± 0.1 eV.78 In the previous DFT+U studies that utilize the Hubbard U value of 4.5 eV and J = 0.51 eV, Dorado et al.26 got the band gap of 2.4 eV, which is only slightly larger than the experimental value. Our calculations with U = 4.5 eV give band gaps of 2.0 and 1.9 eV for PBE+U LR and PBEsol+U LR methods, respectively, which are very consistent with the experimental value. However, due to smaller values of the Hubbard U parameter derived by the linear response method (Table 7), both the DFT+ULR calculations resulted in a much smaller band gap of 0.7 eV (PBE+ULR) and 0.4 eV (PBEsol+ULR), which is inconsistent with the aforementioned measurements. We notice, however, that in principle the DFT is a ground state theory and the Kohn−Sham eigenvalues have no strict physical meaning.5,76,77 This would imply that the interpretation of the Kohn−Sham band gap as the fundamental band gap, although performed on the regular basis, is also not well justified. In fact, on the one hand, Perdew and Levy76 and Perdew77 have shown that due to the discontinuity in the derivative of the exchange− correlation energy, even having the exact ground state density and the exact Kohn−Sham potential, the Kohn−Sham band gap is still expected to be underestimated. From this point of view, the underestimation of the band gap by a DFT-based method cannot be seen as a flaw. On the other hand, from the construction of the Hubbard energy functional it is expected that the DFT+U method with the correct Hubbard U parameter should result in the discontinuity in the derivative

Hubbard ULR

a

solids

PBE

PBEsol

UF6 α-UF5 β-UF5 UF4 UF3 UCl6 UCl4 UCl3 U3O8 α-UO3 UO2 α-U

2.9 2.3 2.3 1.4 0.8 2.7 1.6 0.8 2.4 2.7 1.8 1.9

2.8 2.3 2.3 1.3 0.8 2.7 1.6 0.9 2.3 2.7 1.7 1.9

The values are given in electronvolts.

functional in the DFT+U calculations. The overall result clearly indicates that the DFT+ULR method could be successfully used for the prediction of structural parameters of uranium oxidecontaining solids. Interestingly, this result is in line with our recent studies of the performance of DFT+ULR method for Ln2O3 and monazite-type orthophosphates (LnPO4). In those studies we found that the PBEsol+ULR method significantly outperforms the PBE and PBEsol functionals in predicting the structural parameters of Ln-bearing materials. Band Gaps. When solids are investigated by the DFT+U method, often the ability of this method to reproduce the band 11805



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Table 8. Lattice Parameters of Uranium Fluoride Solids and α-Uranium Calculated Using Different Methods and Measured47,48,a α-UF5

UF6 a PBE PBEsol BPBE modified PBE PBE+U(3 eV) PBE+U(4.5 eV) PBE+U(6 eV) PBE+ULR PBEsol+U(3 eV) PBEsol+U(4.5 eV) PBEsol+U(6 eV) PBEsol+ULR exp

PBE PBEsol BPBE modified PBE PBE+U(3 eV) PBE+U(4.5 eV) PBE+U(6 eV) PBE+ULR PBEsol+U(3 eV) PBEsol+U(4.5 eV) PBEsol+U(6 eV) PBEsol+ULR exp a

b

β-UF5

c

V

a=b

c

V

a=b

c

V

5.58 5.46 6.04 6.41 5.67 5.70 5.70 5.66 5.57 5.58 5.58 5.53 5.21

565 530 720 852 590 598 602 588 561 563 565 551 462

6.93 6.75 7.37 8.37 6.95 6.95 6.96 7.12 6.85 6.78 6.79 6.77 6.52

4.53 4.48 4.54 4.67 4.57 4.59 4.61 4.56 4.52 4.53 4.56 4.51 4.47

217.8 204.1 246.7 326.9 220.5 221.8 223.0 231.6 212.1 208.6 210.0 206.5 189.9

11.56 11.41 11.57 18.36 11.64 11.67 11.72 11.62 11.51 11.55 11.61 11.49 11.46

5.40 5.28 5.49 6.70 5.43 5.45 5.45 5.43 5.31 5.33 5.33 5.30 5.20

721 687 735 2260 736 743 749 733 703 711 718 700 682

10.62 10.42 11.50 12.16 10.74 10.82 10.82 10.74 10.59 10.60 10.62 10.55 9.90

9.53 9.32 10.36 10.92 9.69 9.69 9.75 9.68 9.51 9.52 9.53 9.45 8.96 UF4

a

b

c

V

a=b

c

V

a

b

c

V

12.88 12.72 12.90 14.90 13.02 13.08 13.13 12.96 12.87 12.93 12.98 12.80 12.73

10.85 10.74 10.86 11.53 10.95 11.00 11.06 10.89 10.83 10.89 10.95 10.78 10.75

8.43 8.32 8.43 11.19 8.51 8.54 8.58 8.47 8.41 8.45 8.49 8.38 8.43

950 916 953 1554 979 993 1008 965 946 960 974 930 929

7.18 7.05 7.20 7.49 7.27 7.33 7.38 7.22 7.16 7.21 7.24 7.09 7.18

7.42 7.30 7.39 7.58 7.48 7.50 7.55 7.41 7.37 7.40 7.43 7.31 7.35

331 314 332 369 343 349 356 335 327 333 338 318 328

2.73 2.68 2.72 2.76 3.76 3.82 3.86 3.61 3.65 3.74 3.77 3.47 2.84

5.82 5.69 5.82 5.89 6.22 6.37 6.47 5.95 5.96 6.19 6.34 5.52 5.87

4.92 4.86 4.93 4.97 6.05 6.18 6.28 5.88 5.93 6.06 6.17 5.77 4.94

78.1 74.3 78.2 80.8 141.3 150.6 156.9 126.4 128.8 140.2 147.2 110.5 82.1

α-U

UF3

All length values are given in Å, and volumes are given in Å3.

Table 9. Lattice Parameters of Uranium Chloride Solids Calculated Using Different Methods and Measured47,48,a UCl6 PBE PBEsol BPBE modified PBE PBE+U(3 eV) PBE+U(4.5 eV) PBE+U(6 eV) PBE+ULR PBEsol+U(3 eV) PBEsol+U(4.5 eV) PBEsol+U(6 eV) PBEsol+ULR exp a

UCl4

UCl3

a=b

c

V

a=b

c

V

a=b

c

V

11.76 11.25 13.89 15.33 11.76 11.78 11.78 11.76 11.25 11.24 11.24 11.28 10.95

6.46 6.13 7.81 8.50 6.46 6.45 6.45 6.46 6.15 6.15 6.19 6.14 6.02

774 672 1305 1729 775 774 775 774 673 674 677 676 625

8.36 8.23 8.36 11.21 8.44 8.49 8.54 8.40 8.33 8.37 8.42 8.28 8.30

7.76 7.47 7.87 15.28 7.82 7.90 7.88 7.87 7.56 7.57 7.58 7.54 7.49

543 506 551 1918 558 569 574 556 524 531 538 517 515

7.55 7.37 7.57 11.76 7.61 7.66 7.68 4.56 7.42 7.47 7.52 7.40 7.44

4.28 4.14 4.27 3.91 4.39 4.42 4.44 4.32 4.35 4.35 4.37 4.23 4.32

210.6 194.8 212.1 445.9 220.1 224.3 227.4 213.6 207.0 211.1 214.3 199.9 207.5


of the exchange−correlation energy and reproduce the fundamental band gap.24 Thus, we cannot exclude the possibility that the linear response method underestimates the U values and therefore the band gaps of the considered materials. We notice that this is similar to the case of lanthanide oxides, for which the DFT+ULR underestimates the band gaps by ∼1.7 eV.75 At the same time Blanca-Romero et al.75 found that the PBEsol+ULR method results in very good prediction of

the structural parameters of these materials. Similarly here, the DFT+ULR method results in good prediction of the reaction enthalpies, also for solids, which will be discussed in the next section. The problem of potential underestimation of the Hubbard U parameters by the linear response method of Cococcioni and de Gironcoli36 could be further addressed by computation of the Hubbard U parameter values with other method, such as cRPA.71 We note, however, that such an 11806



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Table 10. Lattice Parameters of Uranium Oxide Solids Calculated Using Different Methods and Measured47,48,a UO2 V

a

b

c

V

a

b

c

V

5.37 5.30 5.41 5.49 5.54 5.58 5.62 5.52 5.48 5.52 5.55 5.45 5.47

154.8 149.0 158.6 165.4 170.0 173.1 176.4 167.7 164.3 167.5 170.7 161.6 163.5

7.02 6.77 7.02 7.28 7.23 7.28 7.34 7.20 7.14 7.19 7.25 7.11 6.70

11.58 11.69 11.58 11.62 11.53 11.55 11.57 11.52 11.43 11.45 11.48 11.43 11.95

4.16 4.12 4.16 4.20 4.17 4.18 4.19 4.17 4.14 4.15 4.15 4.14 4.14

338 326 338 355 347 351 355 346 338 341 345 336 332

3.74 3.73 3.74 3.44 3.41 3.42 3.42 3.41 3.71 3.70 3.74 3.80 3.96

7.15 7.00 7.15 10.77 9.96 9.97 10.34 9.96 7.14 7.27 7.29 6.94 6.86

4.12 4.09 4.12 4.18 4.17 4.18 4.19 4.17 4.10 4.12 4.14 4.11 4.17

110.2 106.7 110.2 174.7 141.6 142.2 148.4 141.5 108.7 111.0 113.0 108.5 113.2

PBE PBEsol BPBE modified PBE PBE+U(3 eV) PBE+U(4.5 eV) PBE+U(6 eV) PBE+ULR PBEsol+U(3 eV) PBEsol+U(4.5 eV) PBEsol+U(6 eV) PBEsol+ULR exp a

α-UO3

U3O8

a=b=c


consider the energies of atoms, not the relevant molecules (i.e., for instance, we use the energy of a F atom, not 1/2F2). The results are provided in Tables 13 and 14. On the one hand, the PBE and PBEsol functionals result in significant, systematic errors of 168 and 184 kJ/mol, respectively. Similarly to the case of the enthalpies of reaction between the molecular compounds, the mixture of PBE and BPBE functionals and the modified PBE functional result in improvements over the PBE and PBEsol functionals, although the improvement is not that pronounced as for the molecules. On the other hand, the DFT+U method, similarly to the case of reactions between the molecular compounds, significantly improves the reaction enthalpies and the best results are obtained with the DFT +ULR method. The MAE is significantly reduced to 31 and 47 kJ/mol for PBE+ULR and PBEsol+ULR, respectively, which is comparable to the error obtained for the enthalpies of reactions between the uranium-bearing molecules. In general, even the DFT+U method with fixed U value ranging from 3 to 6 eV substantially improves the prediction of DFT, and such calculations with U = 4.5 eV (the value that reproduces the band gap of UO2) give the best results. The results obtained for solids are thus in line with the conclusions reached from the studies of molecular compounds. Both benchmark calculations indicate that the DFT+U method could allow for meaningful, computer-aided simulations of actinide-bearing materials, including those relevant for nuclear waste management, and that very good results for the thermochemistry can be obtained, when the Hubbard U parameter is derived ab initio for each considered molecular or solid structure.

Table 11. Values of Band Gaps (eV) for the UraniumBearing Solids Calculated with PBE+U and PBEsol+U Methods for Different U Values PBE+U

PBEsol+U

solids

3 eV

4.5 eV

6 eV

ULR

3 eV

4.5 eV

6 eV

ULR

UF6 α-UF5 β-UF5 UF4 UF3 UCl6 UCl4 UCl3 U3O8 α-UO3 UO2 α-U

3.5 3.3 1.8 2.0 2.6 2.0 1.4 2.3 2.1 2.2 1.2 1.4

3.7 4.4 2.5 2.9 3.5 1.9 1.5 3.3 2.4 2.5 2.0 2.0

3.9 4.4 2.3 3.6 4.1 1.8 1.5 4.0 2.4 2.9 2.6 2.1

3.5 2.8 1.5 1.0 1.0 2.0 1.1 1.0 1.8 2.1 0.7 0.4

3.6 3.1 1.7 1.9 2.4 2.1 1.5 2.2 1.8 1.1 1.1 1.2

3.8 4.3 2.4 2.8 3.1 2.0 1.6 2.9 2.5 1.5 1.9 2.0

4.0 4.7 2.8 3.6 3.8 1.8 1.6 3.6 2.5 1.4 2.5 2.2

3.6 2.6 1.3 1.0 0.9 2.0 0.9 0.5 1.2 1.2 0.4 0.3

investigation as well as the in-depth investigation of the electronic structure of the investigated materials is well beyond the scope of this paper, which aims into benchmarking of the DFT+U method, and DFT+ULR approach in particular, for the thermochemical calculations. Such an analysis is also complicated due to lack of experimental information on the band gaps of the most of the investigated materials. We also notice that the performance of DFT+U for the prediction of the band gaps and the electronic densities of states was already a subject of many previous studies (e.g., refs 25 and 26). Reaction Enthalpies. We computed the enthalpies of nine reactions between the considered solids. All the reactions are provided in Table 12. We note that in the calculation of reaction enthalpies for the gas-phase species, F, O, and Cl, we

■

CONCLUSIONS In this paper we have shown that the DFT+U method can be successfully used in calculation of actinide-bearing materials. We confirmed previous finding of Shamov et al.7 that the standard DFT systematically overestimates the enthalpies of reactions between various uranium-bearing compounds by ∼100 kJ/mol. We were especially interested in testing the performance of the DFT+U method with the Hubbard U parameter derived using the linear response method of Cococcioni and de Gironcoli.36 We found that it significantly reduces the error of the enthalpies of reactions to ∼50 kJ/mol, which is at the level of the MP2 method,7 and in general gives results that are consistent with the all-electrons hybrid functional (PBE0) calculations proposed as the best method

Table 12. List of the Nine Considered Reactions between the Uranium-Bearing Solids α‐UF5 + F → UF6

(18)

UO2 + O → α ‐UO3

β‐UF5 + F → UF6

(19)

3UO2 + 2O → U3O8

(24)

UF4 + F → α ‐UF5

(20)

UCl4 + 2Cl → UCl 6

(25)

(21)

UCl3 + Cl → UCl4

UF4 + F → β ‐UF5 UF3 + F → UF4

(23)

(26)

(22) 11807



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Table 13. Reaction Enthalpies Computed with Different DFT Methods for the Reactions Given in Table 12 and the Respective Experimental Values47,48,a

a

reaction

PBE

PBEsol

BPBE

PBEmodif

PBE-BPBE corr

18 19 20 21 22 23 24 25 26 mean error MAE

−324 −363 −407 −368 −554 −490 −1208 −490 −310 −168 168

−352 −369 −414 −397 −578 −510 −1191 −515 −335 −184 184

−327 −378 −403 −352 −541 −439 −1086 −492 −278 −143 143

−280 −277 −443 −446 −522 −487 −1037 −389 −298 −131 131

−341 −439 −385 −288 −488 −236 −598 −499 −148 −46 128

exp −200 −192 −240 −248 −491 −250 −818 −290 −276

± ± ± ± ± ± ± ± ±

8 6 10 8 9 3 4 6 5

7

Term PBEmodif indicates the modified PBE functional. The enthalpies are given in kJ mol−1.

Table 14. Reaction Enthalpies Computed with the DFT+U Method for the Reactions Given in Table 12 and the Respective Experimental Values47,48,a PBE

a

PBEsol

reaction

3 eV

4.5 eV

6 eV

ULR

3 eV

4.5 eV

6 eV

ULR

18 19 20 21 22 23 24 25 26 mean error MAE

−256 −295 −302 −263 −471 −369 −898 −317 −223 −43 59

−219 −257 −246 −208 −432 −282 −702 −229 −178 28 55

−186 −217 −184 −153 −399 −207 −503 −171 −159 92 97

−201 −241 −275 −235 −491 −333 −839 −266 −229 −12 31

−284 −305 −319 −298 −491 −392 −964 −351 −281 −76 76

−248 −268 −263 −242 −455 −306 −773 −264 −214 −3 41

−211 −229 −206 −188 −423 −221 −578 −194 −164 65 77

−241 −262 −281 −260 −518 −362 −921 −297 −268 −45 47

exp −200 −192 −240 −248 −491 −250 −818 −290 −276

± ± ± ± ± ± ± ± ±

8 6 10 8 9 3 4 6 5

7

Columns represent results obtained with the fixed values of U and with U = ULR. The enthalpies are given in kJ mol−1.

for computation of actnide-bearing materials.7 Comparing with DFT, the DFT+ULR method gives a better prediction for the enthalpies of all the considered reactions. It also improves the description of structures of uranium oxides. The derived Hubbard U parameters are consistent with the values obtained experimentally34,35 but are smaller (by ∼2 eV) than the value of 4.5 eV used in most of the previous calculations and obtained from the spectroscopic measurements of UO2.32,33 We also found that the value of Hubbard U parameter decreases with decrease of the oxidation state of uranium. Our results thus suggest that, because of serious limitation in applicability of the computationally expensive post Hartree− Fock methods such as CCSD(T) or the hybrid DFT functionals for investigation of chemically complex, actinidebearing materials, the DFT+U method with the Hubbard U parameter derived by ab initio methods could be a good choice as a feasible method for computation of such materials.

■

■

ACKNOWLEDGMENTS

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ADDITIONAL NOTES

Funded by the Excellence Initiative of the German federal and state governments and the Jülich Aachen Research Alliance High-Performance Computing. We thank the JARA-HPC awarding body for time on the RWTH computing cluster awarded through JARA-HPC Partition.

a b

Considering only the reactions computed by Shamov et al.7 Computed as the standard error of the arithmetic mean.

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REFERENCES

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AUTHOR INFORMATION

Corresponding Author

*P. M. Kowalski. E-mail: [email protected]. Phone: +49 2461619356. Notes

The authors declare no competing financial interest. 11808



Article

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