Benchmarking the DFT+ U Method for Thermochemical Calculations of Uranium Molecular Compounds and Solids

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Article

Benchmarking the DFT+U Method for ThermochemicalCalculations of Uranium Molecular Compounds and Solids.

George Beridze, and Piotr M. KowalskiJ. Phys. Chem. A, Just Accepted Manuscript • Publication Date (Web): 20 Nov 2014

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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 Julich, Wilhelm-Johnen-Straße, 52428 Julich, Germany, and JARA

High-Performance Computing, Schinkelstraße 2, 52062 Aachen, Germany

E-mail: [email protected]

Phone: +49 2461619356.

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Abstract3

Ability to perform a feasible and reliable computation of thermochemical proper-4

ties of chemically complex actinide-bearing materials would be of great importance for5

nuclear engineering. Unfortunately, density functional theory (DFT), which on many6

instances is the only affordable ab initio method, often fails for actinides. Among7

various shortcomings, it leads to wrong estimate of enthalpies of reactions between8

actinide-bearing compounds, putting the applicability of DFT approach to the model-9

ing of thermochemical properties of actinide-bearing materials into question. Here we10

test the performance of DFT+U method - a computationally affordable extension of11

∗To whom correspondence should be addressed†Institute of Energy and Climate Research, Nuclear Waste Management and Reactor Safety,

Forschungszentrum Julich, Wilhelm-Johnen-Straße, 52428 Julich, Germany‡JARA High-Performance Computing, Schinkelstraße 2, 52062 Aachen, Germany

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DFT that explicitly accounts for the correlations between f electrons - for prediction12

of the thermochemical properties of simple uranium-bearing molecular compounds and13

solids. We demonstrate that the DFT+U approach significantly improves the descrip-14

tion of reaction enthalpies for the uranium-bearing gas-phase molecular compounds15

and solids and the deviation from the experimental values are comparable to those16

obtained with much more computationally demanding methods. Good results are ob-17

tained with the Hubbard U parameter values derived using the linear response method18

of Cococcioni and de Gironcoli. We found that the value of Coulomb on-site repulsion,19

represented by the Hubbard U parameter, strongly depends on the oxidation state of20

uranium atom. Last but not least, we demonstrate that the thermochemistry data can21

be successfully used to estimate the value of Hubbard U parameter needed for DFT+U22

calculations.23

Keywords: Ab initio calculations, Density functional calculations, Gas-phase reactions,24

Uranium, Actinides25

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INTRODUCTION26

Understanding the properties of actinide-bearing materials and their interaction with the27

environment is a challenge for nuclear waste storage strategies1–4. Lack of in-depth un-28

derstanding of the atomic-scale processes that determine the properties of materials upon29

incorporation of radionuclides highly limits the availability of materials that could be used30

as host phases for nuclear waste storage. The relevant experimental studies are usually very31

difficult to perform and often time consuming. On the other hand, with the increasing power32

of computing resources it is now possible to simulate the atomic scale behavior of actinide-33

bearing materials using ab initio methods of computational quantum chemistry1,5. When34

reliable and time-feasible, such computer-aided studies could to great extent complement the35

experimental techniques used in characterization of nuclear waste forms and substantially36

enhance our knowledge on the atomic-scale processes that influence performance of these37

materials and their safety within the repositories6. Unfortunately, having strongly corre-38

lated 5f electrons, actinide-bearing materials are very difficult to handle by the standard39

density functional theory (DFT) methods. It is then proposed that more computationally40

demanding methods (at least hybrid functionals) should be used in calculations of such sys-41

tems7. On the other hand, even when utilizing the most powerful computing resources, DFT42

is often the only affordable quantum chemistry method that allows for the ab initio modeling43

of chemically complex materials. Therefore, an assessment of the performance of available44

DFT-based methods and a careful choice of the computational strategy are required in order45

to perform meaningful ab initio simulations of complex, actinide-bearing materials.46

Although there exist various studies of actinide-bearing compounds and solids by DFT8–11,47

its applicability for computation of such materials is questionable7,12,13. DFT often fails not48

only on the quantitative, but also on the qualitative level. Wen et al. 13 have shown that49

the hybrid functional, such as HSE, has to be used to correctly compute the semiconducting50

state of actinide-oxide solids, which are often described as metals by DFT. On the quan-51

titative level, when used for the prediction of physical and chemical parameters, DFT can52

lead to significant errors. For instance, the difference between predicted by DFT and the53

measured enthalpies of reactions for various uranium-bearing molecular compounds can be54

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as large as 100 kJ/mol7,14,15. By testing the different computational methods, including55

various DFT generalized gradient approximations (GGA) and hybrid functionals as well as56

post Hartree-Fock methods such as MP2 and CCSD(T), Shamov et al. 7 and Schreckenbach57

and Shamov 12 have shown that in order to reach the experimental accuracy (20 kJ/mol) at58

least hybrid functionals such as PBE0 have to be used in computation of the enthalpies of59

reactions involving uranium-bearing molecular compounds. In line with these findings, most60

of the meaningful DFT studies of actinide-bearing materials utilize the hybrid functionals61

such as B3LYP, PBE0 or HSE16–20. Moreover, studies of Shamov et al. 7 , Schreckenbach and62

Shamov 12 and Odoh and Schreckenbach 21 indicate that the result of such hybrid calculations63

can also depend on the number of core-electrons modeled by pseudopotentials and that at64

least 32 electrons of uranium atom (5s25p65d106s26p65f 36d17s2) should be treated explicitly,65

which further increases the computational cost. We note however, that this result is still66

in dispute. For instance, Iche-Tarrat and Marsden 15 have shown that the explicit treat-67

ment of 32 electrons of uranium atom only marginally improves the performance of either68

DFT or hybrid-DFT functionals over the case when only 14 valence electrons of uranium69

(6s26p65f 36d17s2) are treated explicitly. We will show that our results implicitly support70

the later conclusion.71

The usage of hybrid functionals or any post-Hartree-Fock method requires substantial72

computational resources. This limits the applicability of these methods to the simplest73

molecular compounds and solids. Materials that are interesting for nuclear engineering have74

usually complex chemical compositions and structures with supercells containing often more75

than hundred atoms. This is especially true when solid solutions with diluted concentration of76

elements are of interest2,6. Thus, DFT and its simple modifications, such as DFT+U , remain77

the only choice if one wants to compute complex materials and simulate their dynamical78

behavior by methods such as ab initio molecular dynamics22–24 on nowadays supercomputing79

resources. Wen et al. 25 have shown that the DFT+U method with a reasonable choice of80

the Hubbard U parameter can reproduce the band gaps of actinide oxides and correctly81

predict insulating state for these solids. This method has been successfully used in the82

description of UO2 and its metastable states26,27, in the calculation of U(VI) aqua complexes83

on titania particles14 and for the investigation of incorporation of uranium in the ferric garnet84

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matrices28. It also predicts correctly the magnetic state of actinide compounds29 and the full85

phonon dispersion of strongly correlated materials30. These successes have been achieved86

sometimes at a cost of worse description of lattice parameters25 or even anomalous change87

in volume31. In all these studies the same Hubbard U parameter value for uranium of 4.5 eV88

(or Ueff = U − J = 4.5 eV − 0.5 eV = 4 eV) has been used, which has been derived from the89

spectroscopic measurements of UO232,33. We note however, that there is no guarantee that90

the value derived for one system should be easily transferable to another, especially that91

other estimates for neutral uranium atoms provide somehow lower values of U = 2.3 eV3492

and U = 1.9 eV35.93

In this contribution we systematically test the performance of DFT+U method for the94

prediction of the thermochemical parameters, namely the reaction enthalpies, of simple95

uranium-bearing molecular complexes and solids, most of which were considered in pre-96

vious studies. We do that in order to compare our results with already published predic-97

tions of much more demanding computational methods such as hybrid functionals, MP2 or98

CCSD(T)7 and with the available experimental data. We are especially interested in the99

performance of DFT+U method with the Hubbard U parameter (or the effective Hubbard100

U parameter Ueff = U − J) derived for each compound and solid by the linear response101

procedure proposed by Cococcioni and de Gironcoli 36 . Up to our knowledge there exist102

no systematic studies of the performance of DFT+U method for prediction of the thermo-103

chemical parameters of actinides. In particular, the DFT+U calculation with the Hubbard104

U parameter derived by the linear response method reported here is the first such study105

conducted for the actinide-bearing materials.106

COMPUTATIONAL METHOD107

DFT Setup108

All calculations were performed using the plane-wave Quantum ESPRESSO code37. We109

used the PBE38, PBEsol39 and BPBE38,40 exchange-correlation functionals. The core elec-110

trons of the computed atoms were replaced by the ultrasoft pseudopotentials41 and the111

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6s26p65f 36d17s2 electrons of uranium atom were treated explicitly (such a pseudopotential112

is often referred as ”large core pseudopotential”). We performed the calculations with a113

large core pseudopotential because the main focus of our studies is the benchmarking of114

the feasible computational method that could be used in further modeling of chemically115

complex actinide-bearing materials important, for instance, in nuclear waste management.116

We therefore do not focus here on benchmarking the small core size pseudopotentials or117

the all-electron methods that are currently computationally unfeasible for computation of118

such materials. All calculations were spin polarized and the spin configurations were care-119

fully checked in order to achieve the ground state electronic configurations of the considered120

systems. We applied the energy cutoff of 50 Ryd which gave energy differences within121

0.5 kJ/mol, which is sufficient for our studies. The atomic configurations were relaxed to the122

equilibrium positions, so the maximum component of the residual forces on the ions was less123

than 0.005 eV/A. The DFT equilibrium geometries of the molecular compounds were con-124

firmed by supplementary calculations of the vibrational frequencies. The crystalline solids125

were treated as continuum in all three spatial dimensions by applying the periodic boundary126

conditions. For the calculations of solids, the Methfessel-Paxton42 k-points grids were used127

and their sizes were chosen to give the energy differences within 0.5 kJ/mol. The calcula-128

tions of solids were performed by simultaneous relaxation of the lattice parameters and the129

ionic positions so the resulting pressure was 0 GPa with a tolerance of 0.01GPa. Because130

the differences in the energies between the antiferromagnetic and ferromagnetic states of the131

considered solids were usually smaller than 1 kJ/mol we consider the ferromagnetic spin ar-132

rangement for them. The exceptions are α-U and UO2 solids for which the antiferromagnetic133

states were considered as the ground state.134

Following Shamov et al. 7 we also neglect the spin-orbit interaction in all the calculations.135

This is because these effects should not affect the computed reaction enthalpies by more136

than 20 kJ/mol, which is at the level of experimental accuracy and much smaller than the137

error in predicted reaction enthalpies obtained by using the standard DFT approximation138

(∼ 100 kJ/mol7,15).139

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Modified PBE Functional140

In order to test the tendency of different DFT functionals in predicting the reaction en-141

thalpies we performed series of calculations with different choice of µ and κ parameters142

of PBE functional exchange enhancement function38. Here we do not limit the choice of143

these parameters by any physical behaviors, such as the slowly varying density limit or the144

Lieb-Oxford bound38, but simply treat µ and κ as free parameters. Such an exercise of145

modifying the exchange part of PBE functional, when one should rather expect correction146

of the correlation part of an exchange-correlation functional, was inspired by the fact that147

hybrid PBE0 functional, which admixes the Hartree-Fock exchange with the PBE exchange,148

keeping the PBE correlations43,44, performs exceptionally good for uranium7. By the mod-149

ified PBE functional in further discussion we mean the case with µ = 0.21951 (PBE value)150

and κ = 6.0. Further discussion of the modified functional approach is provided in Results151

and Discussion section.152

DFT+U Method153

In the current studies we utilize two kinds of the DFT+U approaches. First, we performed154

calculations with the fixed values of the Hubbard U parameter of 0.5 eV, 1.5 eV, 3 eV, 4.5 eV155

and 6 eV. We also performed calculations with the Hubbard U parameter derived by the156

linear response approach proposed by Cococcioni and de Gironcoli 36 . In this procedure157

the molecular geometries were fixed to the DFT structures. We used the atomic f -orbitals158

produced by uspp-736 package41 as projectors and the elements of response matrices were159

derived by the finite differences, setting α parameter to ±0.01, as described in the work of160

Cococcioni and de Gironcoli 36 . The U values have been derived for uranium in all uranium-161

bearing molecular compounds and solids considered in this paper.162

Because uranium atom d-shell could also be partially filled, we checked the effect of163

applying +U correction to the d-orbitals of uranium. Similarly to the case of f -orbitals, we164

computed the Hubbard U values for the d-shell by the linear response approach of Cococcioni165

and de Gironcoli 36 . However, because the current version of Quantum ESPRESSO code does166

not allow for simultaneous application of the DFT+U method to two or more orbitals of the167

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Table 1: List of the considered 17 reactions between the gas-phase uranium-containingmolecules. The reactions 1-2, 7-12 and 14-16 were computed by Shamov et al. 7 .

(1) UF6 → UF5 + F (10) UO3 → UO2 +O(2) UF5 → UF4 + F (11) UOF4 +UO3 → 2UO2F2

(3) UF4 → UF3 + F (12) 2UOF4 → UF6 +UO2F2

(4) UCl6 → UCl5 + Cl (13) UO3 +H2O → UO2(OH)2(5) UCl5 → UCl4 + Cl (14) UF6 +H2O → UOF4 + 2HF(6) UCl4 → UCl3 + Cl (15) UF6 + 2H2O → UO2F2 + 4HF(7) UOF4 → UF4 +O (16) UF6 + 3H2O → UO3 + 6HF(8) UO2F2 → UO2 + 2F (17) UO2F2 + 2H2O → UO2(OH)2 + 2HF(9) UF6 + 2UO3 → 3UO2F2

same species, in the subsequent calculations we computed only the correction to the primary168

DFT+U calculations (i.e. the calculations with the Hubbard U applied to f -electrons only).169

We did this by computing the Hubbard energy resulting from the application of DFT+U170

method to the d-orbitals and by fixing the molecular structures to the atomic configurations171

optimized with the primary DFT+U method. We will show that this approach is justified172

by the smallness of the correction.173

Last but not least, because it is known that the DFT+U calculations of uranium solids,174

namely UO2, can lead to menagerie of metastable states depending on the initially chosen175

occupations of f orbitals45, in our studies we took special care to assure that the computed176

states are the real ground states. We did this following the procedure outlined in Dorado177

et al. 45 by: (1) performing set of calculations with the different initial occupations of f178

orbitals, (2) breaking the cubic symmetry of electrons and (3) checking the correctness of179

the final occupations. We also verified that all the considered solids show large band gaps.180

However, in spite of the applied precautions, because of the complexity of the problem itself181

there is no full guaranty that a ground state have been reached in all the cases. Nevertheless,182

results of Dorado et al. 45 for UO2 show that in case when the ground state is not reached, the183

resulted metastable state should be higher in energy by no more than a fraction of kJ/mol.184

Such an error is negligibly small comparing with the experimental accuracy of the considered185

reaction enthalpies (20 kJ/mol).186

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CONSIDERED SYSTEMS187

The set of uranium-bearing molecular compounds, including U(VI), U(V), U(IV) and U(III)188

halogenide, oxide and oxyhalide molecules computed here contains all the compounds con-189

sidered by Shamov et al. 7 , Iche-Tarrat and Marsden 15 and Batista et al. 46 . In addition we190

computed the series of uranium-chlorides for which good experimental data exist. All the191

considered reactions are given in Table 1. We selected to consider the already computed192

reactions in order to make a broad comparison of our results with the results obtained by193

using different computational methods, including hybrid functionals and higher level post194

Hartree-Fock methods such as MP2 and CCSD(T)7. The thermochemistry data used as a195

reference were derived from the formation enthalpies taken from Morss et al. 47 and Guillau-196

mont et al. 48 . The reaction enthalpies were computed by taking the differences in the total197

energies of the reactants. The initial structural parameters of computed solids were taken198

from different experimental studies and the adequate references are indicated in the relevant199

text and tables.200

Molecular Compounds201

The stable geometries of the molecules found in our calculations are in agreement with202

previous studies of the same systems7,12,15,46. Uranium hexahalogenides (UX 6, where X =203

F, Cl) have Oh symmetry. The symmetry of uranium pentahalogenides (UX5) is C4v and204

that of UX4 molecules is C2v. UX3 and UO2F2 molecules have C3v and C2v symmetries205

respectively. UO3 molecule has T-shaped planar geometry and UO2 has linear geometry.206

The two possible structures of UOF4 are shown in Figure 1. Our DFT-PBE calculation207

indicates that the square pyramidal structure is more stable than the trigonal bipyramide208

form by 2.4 kJ/mol, which is in agreement with the studies of Shamov et al. 7 .209

Crystalline Solids210

We consider twelve solids: α-U49, UF650, α-UF5

51, β-UF552, UF4

53, UF354, UCl6

55, UCl456,211

UCl356, UO2

57, α-UO358 and U3O8

59. The initial structures were taken from the indicated212

references.213

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

RESULTS AND DISCUSSION214

Molecular compounds: structural parameters215

Three of the considered molecules, UF6, UCl6 and UCl3, have measured structural parame-216

ters (the U−F,Cl bond lengths and the Cl−U−Cl angle in the case of UCl3). We therefore217

start our analysis by performing an assessment of performance of different methods in pre-218

diction of the bond lengths in these compounds. First, we compare the U-F bond length219

computed for the gas-phase UF6 molecule to the experimental value and to the previous220

theoretical estimates60–62. All the results are given in Table 2. The measured U-F bond221

length is 1.999 A60 and 1.997 A61. The PBE functional overestimates the bond length by222

slightly more than 1%, which is a known feature of this functional63. The BLYP functional62223

gives even worse prediction, which is also expected64. The hybrid B3LYP functional used in224

the same studies improved the value of the bond distance and the Hartree-Fock calculation225

also shows surprisingly good agreement with the experiment. The all-electron PBE0 calcu-226

lation7 with 1.997 A gives the best match to the experimental value. Because the PBEsol227

functional recovers correctly the slowly varying density limit of the exact exchange energy228

functional, it usually improves description of the structural parameters over PBE39,63. This229

is also visible in our calculations. PBEsol results in 2.009 A for the U-F bond length of UF6230

that is in better agreement with the experiment than the PBE value. The U-F bond length231

obtained with the modified PBE functional is in the worst agreement to the experimental232

value. On the other hand, this functional predicts much better the enthalpies of reactions233

than the PBE or PBEsol functionals, including the dissociation enthalpy of UF6, which will234

be discussed later. The fact that by switching to a different GGA functional one can improve235

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0 1 2 3 4 5 6κ

200

250

300

350

400

450

500

550

600

Rea

ctio

n E

ntha

lpy

(kJ/

mol

e)

0 1 2 3 4 5 6κ

1.98

1.99

2

2.01

2.02

2.03

2.04

2.05

2.06

2.07

2.08

2.09

2.1

2.11

2.12

2.13

2.14

2.15

U-F

bon

d le

ngth

(Å

)

Figure 2: Left panel: The 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 ofκ (horizontal axis) and µ =0.109755 (dashed lines), 0.21951 (solid lines, PBE) and 0.43209(dotted lines). µ and κ determine the enhancement function of the exchange functional (seeEq. 1). Right panel: the U − F bond length of UF6 . The different results are markedusing the convention applied in right panel. On both panels, the horizontal lines indicatethe experimental values.

the prediction of the reaction enthalpies at a cost of worsening the structural parameters236

and vice versa, is a known shortcoming of current GGA-DFT functionals. We investigated237

further such behavior by computing the U-F bond length in UF6 and the dissociation en-238

thalpies of UF6 and UF5 molecules with a modification of PBE functional. The modification239

was made by treating κ (PBE value of 0.804) and µ (PBE value of 0.21951) parameters of240

the exchange energy functional enhancement function Fx(s),38

241

Fx(s) = 1 + κ− κ

1− µs2/κ, (1)

where s ∝ ‖∇n‖/n is the reduced electron density (n) gradient38, as free parameters. The242

result is given in Figure 2. It is clearly seen that the smaller µ results in a better bond length.243

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In fact, our result with µ being half the value of that of the PBE functional, which closely244

resembles the case of PBEsol functional (where µ = 10/81 = 0.123456839), gives better U-F245

bond length than PBE. On the other hand, the computed reaction enthalpy become worse246

and it can not be easily improved by just modifying κ. For a fixed value of µ, increase of κ247

always leads to better description of the reaction enthalpies, but at a cost of worsening the248

bond length. The larger the µ, the worse the U-F bond length that results in good enthalpy,249

is. In the subsequent sections we will show the performance of the modified PBE functional,250

where keeping the PBE value of µ = 0.21951 we set κ to 6. We chose such a set of parameters251

because it results in relatively good prediction of the dissociation enthalpies of UF6 and UF5252

molecules, which is indicated in Figure 2. The DFT+U approach in general results in larger253

U-F bond length for UF6 than the one predicted by the applied GGA functional. However,254

we will show that, in line with the above conclusions, it also results in much better prediction255

of the gas-phase reaction enthalpies.256

The same trends as for UF6 are observed in the case of UCl6 that has the same geometry257

and the oxidation state of uranium. Also in this case the PBEsol functional results in258

better match to the measured U− Cl bond length. The experimental U-Cl bond length for259

uranium trichloride, UCl3 is 2.549±0.008 A.69 All the GGA functionals underestimate this260

bond length by ∼ 0.05 A. In that case the DFT+U method improves the bond length and261

the modified PBE functional also results in a good match to the experiment. However, all the262

methods overestimate the Cl-U-Cl bond angle. Interestingly, for that parameter the hybrid263

functionals, PBE0 and B3LYP, give worse prediction than the PBE and PBEsol methods.264

Molecular Compounds: Reaction Enthalpies265

Several reactions involving gas-phase molecules were selected to asses the performance of266

different methods in the prediction of enthalpies of reactions involving uranium-bearing267

molecules. As already mentioned, the experimental data exist for all of them and the en-268

thalpies of most of the considered reactions were already computed using different methods269

of computational quantum chemistry. All the reactions are listed and labeled in Table 1.270

This labeling we will use subsequently in all further discussion in the text.271

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Table 2: The bond distances in UF6, UCl6 and UCl3 molecules between uranium and halogenatoms and U−Cl−U bond angle in UCl3 molecule obtained using different computationalmethods. The bond lengths are given in A and angles in degrees.

Functionals and methods U− F U− Cl(UCl6) U− Cl(UCl3) Cl− U− ClPBE 2.024 2.469 2.509 106.4PBEsol 2.009 2.445 2.488 105.0BPBE 2.025 2.506 2.519 106.4modified PBE 2.048 2.471 2.559 111.8PBE+U (U=0.5 eV) 2.025 2.470 2.525 106.4PBE+U (U=1.5 eV) 2.027 2.472 2.550 106.7PBE+U (U=3.0 eV) 2.031 2.477 2.557 106.3PBE+U (U=4.5 eV) 2.034 2.486 2.570 106.7PBE+U (U=6.0 eV) 2.039 2.500 2.581 106.4PBE+ULR 2.030 2.474 2.535 106.3PBEsol+U (U=0.5 eV) 2.010 2.446 2.495 105.4PBEsol+U (U=1.5 eV) 2.012 2.449 2.504 105.1PBEsol+U (U=3.0 eV) 2.015 2.453 2.538 105.8PBEsol+U (U=4.5 eV) 2.024 2.462 2.548 106.3PBEsol+U (U=6.0 eV) 2.015 2.474 2.563 106.2PBEsol+ULR 2.015 2.451 2.509 105.1BLYPa 2.043PBE0a 1.997 2.565 109.5HFb,c 1.984 2.48B3LYPa 2.014 2.592 114.5MP2a 2.005 2.521 106.9exp.d,e,f,g 1.999 (1.996) 2.42 (2.461) 2.549±0.008 95±3

[a]Ref.62, [b] Ref.,65[c] Ref.,66 [d] Ref.,60 [e] Ref.,67[f] Ref.,68 [g] Ref.61

DFT Calculations272

All the reaction enthalpies computed with the standard GGA functionals are given in Table273

3. First of all, despite the different pseudopotentials used in our and Shamov et al. 7 studies274

(we used the large core ultrasoft pseudopotentials, while Shamov et al. 7 used the all electrons275

or the small core pseudopotential calculations) the PBE results of the two studies are well276

consistent. The difference, measured by the mean absolute error (M.A.E), between the277

two calculations is only 3.2 kJ/mol1. This result indirectly validates our computational278

setup. The M.A.E. obtained with the PBE functional for all the considered reactions is279

95 kJ/mol. Because the mean error (M.E.) is of similar value, this indicates a systematic280

1Considering only the reactions computed by Shamov et al. 7

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Table 3: The reaction enthalpies computed with different DFT methods for the reactionsgiven in Table 1 and the respective experimental values47,48. Term PBEmodif indicates themodified PBE functional. The enthalpies are given in kJ mol−1.

PBE PBEsol BPBE PBEmodif PBE-BPBE corr. exp.(1) 422 458 403 324 328 313±17(2) 514 549 497 413 429 385±22(3) 619 641 607 541 558 618±27(4) 278 317 254 176 155 205±20(5) 312 347 302 217 261 204±20(6) 414 447 395 339 320 412±25(7) 554 613 529 441 430 405±20(8) 1153 1204 1131 964 1045 1033±20(9) -207 -206 -208 -214 -213 -311±22(10) 678 718 658 568 577 576±20(11) -113 -114 -105 -114 -72 -170±40(12) 19 23 1 14 -69 -25±31(13) -141 -157 -133 -63 -105 -182±11(14) 178 190 173 112 152 65±12(15) 337 356 344 210 373 189±12(16) 608 638 620 421 666 437±12(17) 131 125 143 149 189 272±11

mean error 78 101 70 4 35M.A.E. 95 119 88 51 74 20

offset from the measured values. Such a significant systematic error is unacceptably large,281

having the largest experimental error of 40 kJ/mol. The PBEsol functional results in even282

larger error, which as we already discussed, is a known feature of this functional. In their283

studies Shamov et al. 7 used the PBE, BPBE and OLYP standard GGA-DFT functionals.284

These calculations resulted in a systematic offset for all the considered reactions, with BPBE285

performing slightly better than PBE, and OLYP resulting in reduction of the error by ∼ 50%.286

Our calculations show similar trend. The BPBE functional gives slightly better match to the287

measured values for all the considered reactions. We found that this behavior is not restricted288

to the reactions between uranium-bearing compounds but has been also observed for the289

bond energies of metal-ligand complexes70. Because the different GGA functionals result290

in different systematic offsets from the measured values, we decided to test the possibility291

of using a combination of two functionals in order to get better estimation of the reaction292

enthalpies. Especially instructive is the mixture of the BPBE and PBE results, because293

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both functionals differ only by the exchange functional (they have the same PBE correlation294

energy functional38). In Table 3 in column labeled by PBE-BPBE corr. we give the reaction295

enthalpies obtained by mixing the results of PBE and BPBE functionals as EPBE+5(EBPBE−296

EPBE), where E indicates enthalpy. As shown in Table 3, such a mixing results in significant297

reduction of the mean error, although M.A.E. is reduced by only 21 kJ/mol. Interestingly, the298

modified PBE functional, with M.E of only 4.1 kJ/mol, significantly improves the prediction299

of reaction enthalpies. It gives results that are better than predictions of the PBE or PBEsol300

functionals, not only for the reactions (1) and (2) used in the construction of the functional,301

but for most of the considered reactions. This is a promising result. However, we will show302

that such a simple modification of the PBE functional does not give similarly good results303

for solids.304

Shamov et al. 7 , Odoh and Schreckenbach 21 and Schreckenbach and Shamov 12 concluded305

that calculations with the large core pseudopotential (14 valence electrons for uranium, which306

is also our setup) can result in significant under-performance of the PBE0 functional in pre-307

diction of the reaction enthalpies and suggest to use a small core or all-electron calculations308

in order to get converged energies. Because with the PBE and BPBE functionals we ob-309

tained results that are consistent with the all-electron calculations of Shamov et al. 7 and310

Schreckenbach and Shamov 12 , we suspect that the size of the core is not that important,311

when a GGA functional is used. Interestingly, Iche-Tarrat and Marsden 15 have shown that312

the 32 electrons core pseudopotential only marginally improves, or even worsens in some313

cases, the performance over the 14 electrons core pseudopotential in the case of GGA and314

hybrid functionals. One exception is PBE0, which is also shown in results of Iche-Tarrat and315

Marsden 15 . Therefore we can only suspect that the number of valence electrons can impact316

the result, when that functional is used in calculations.317

DFT+U Studies318

In most of the studies that utilize the DFT+U method the Hubbard U parameter, and319

sometimes J (representing strength of the on-site electron exchange), is chosen in a way320

that particular, known properties of the investigated systems, such as the lattice parameters321

or the band gaps, could be reproduced by calculations. Some studies use ab initio based322

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0 1 2 3 4 5 6U (eV)

40

50

60

70

80

90

100

110

120

M.A

.E. (

kJ/m

ol)

Figure 3: The mean absolute error (M.A.E) of the computed enthalpies of reactions givenin Table 1 as a function of the Hubbard U parameter. Filled and open circles represent theresults of PBE+U and PBEsol+U calculations, respectively. The curves represent fits by aparabola that were used to estimate the optimal U values (see discussion in the text).

methods for the estimation of Hubbard U value24, such as the linear response method36,323

which we use in our studies, the constrained random phase approximation (cRPA)71 or324

simple approaches such as the Slater transition state72. Hubbard U values can be also derived325

from the spectroscopic measurements32–35,73,74. In our studies we first computed the reaction326

enthalpies for different Hubbard U parameter values of 0.5 eV, 1.5 eV, 3 eV, 4.5 eV and 6 eV327

that are kept the same for all the reactants and products. All the results are given in Tables 4328

and 5. It is clearly visible that the DFT+U method improves the prediction of the enthalpies329

of reactions over the PBE and PBEsol functionals. In the next step, we determined the values330

of Hubbard U parameter in a way that it minimizes the deviation from the measured reaction331

enthalpies (i.e. minimizes the M.A.E.). These values were obtained by fit of the M.A.E.332

obtained for different values of U , and reported in Tables 4 and 5, by parabola. The U value333

is given by the minimum of the fitted curve and the fitting results are illustrated in Figure334

3. For the PBE+U and PBEsol+U methods we obtained U = 3.0± 1.0 eV and 3.8± 1.0 eV335

respectively. The uncertainties on these values were estimated from Figure 3 taking into336

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account the mean experimental error of 20 kJ/mol/√17 ∼ 5 kJ/mol.2 In addition to these337

studies, we derived the Hubbard U parameters for each considered molecular compound338

using the linear response method proposed by Cococcioni and de Gironcoli 36 . We therefore339

define so derived Hubbard U parameters as ULR (after linear response), and subsequently340

the DFT+U method as DFT+ULR. The obtained values of ULR are reported in Table 6.341

Interestingly, on average these values are slightly smaller than the ones that minimize the342

M.A.E. However, we notice that the derived Hubbard U parameters decrease with decreasing343

the oxidation state of uranium. When we consider only the values computed for U(VI)344

complexes, the obtained U value of 2.8 eV is in good agreement with the values obtained345

from minimization of the M.A.E. This is not surprising as most of the considered reactions346

involve U(VI) complexes. On the other hand, our result indicates that the thermochemistry347

data can be used to determine the Hubbard U parameter value independently from other348

experimental methods such as spectroscopy32–35,73,74. Our studies also show that the linear349

response approach of Cococcioni and de Gironcoli 36 gives U values that minimize the error350

of DFT+U reaction enthalpies.351

In Table 6 we also provide the Hubbard U values computed for the d-orbitals of uranium.352

These are substantially smaller than the ones derived for the f -orbitals. As a consequence,353

the related Hubbard energy correction to the reaction enthalpies is usually not larger than354

10 kJ/mol per reacting uranium atom (see Table 4), which is small comparing to an aver-355

age 70 kJ/mol correction to DFT resulting from the application of DFT+U method to f356

electrons. The M.E. and the M.A.E. are also almost identical to the ones obtained without357

applying DFT+U to d electrons. This justifies the ommission of the DFT+U treatment of358

d-electrons of uranium in the previous studies (eg.25–29,31). We also neglect this effect in359

further analysis.360

In most of the up to date studies of the uranium-bearing materials the same Hubbard U361

parameter value for uranium of 4.5 eV (or Ueff = U −J = 4 eV) has been applied. This value362

was derived from the spectroscopic measurements of UO232,33. Our ULR values reported in363

Table 6 are somehow lower, but consistent with other estimates for neutral uranium atoms364

of 2.3 eV34 and 1.9 eV35. We will elaborate more on this problem when discussing results for365

2Computed as the standard error of the arithmetic mean.

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Table 4: The reaction enthalpies computed with the PBE+U method for the reactions givenin Table 1 and the respective experimental values47,48. The ULR, d column represents thePBE+ULR results corrected with the PBE+U calculations applied to the d-orbitals (see textfor details). The enthalpies are given in kJ mol−1.

0.5 eV 1.5 eV 3 eV 4.5 eV 6 eV ULR ULR, d exp.(1) 410 386 349 311 271 273 270 313±17(2) 496 460 405 349 310 402 410 385±22(3) 609 627 594 559 511 622 621 618±27(4) 266 241 200 157 100 222 223 205±20(5) 294 254 193 136 99 198 198 204±20(6) 399 411 330 300 261 357 361 412±25(7) 528 474 391 311 248 346 352 405±20(8) 1176 1107 1120 1079 1048 1048 1015 1033±20(9) -232 -283 -363 -443 -497 -260 -225 -311±22(10) 693 608 598 536 497 579 560 576±20(11) -127 -158 -204 -247 -278 -121 -98 -170±40(12) 23 32 45 52 59 -19 -28 -25±31(13) -147 -158 -173 -186 -187 -149 -146 -182±11(14) 175 170 160 145 130 127 125 65±11(15) 328 308 275 238 201 272 278 189±12(16) 607 603 593 579 550 538 530 437±12(17) 133 138 146 154 162 117 106 272±11

mean error 71 47 14 -23 -55 8 8M.A.E. 90 63 55 70 90 45 50 20

solids.366

As we have shown, the DFT+ULR method substantially reduces the error of the predicted367

reaction enthalpies for both PBE and PBEsol functionals. In Figure 4 we compare the368

M.A.E errors of the selected methods obtained for the reactions reported by Shamov et al. 7 .369

We immediately see that the DFT+ULR method reduces the error by a half and in fact370

it gives even slightly better results than MP2 method and it is only slightly worse than371

PBE0 hybrid functional, which is claimed to give good results for actinides7,13. Figure 5372

illustrates the deviations of the results obtained with all the considered methods from the373

experimental data and PBE0, MP2 and CCSD(T) results of Shamov et al. 7 . Interestingly,374

the absolute deviation from PBE0 prediction of the DFT+ULR method is only 27 kJ/mol,375

with mean deviation being even 0 kJ/mol in the case of PBE+ULR, which suggests that376

DFT+ULR behaves in similar way and carries similar errors as the PBE0 hybrid functional.377

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Table 5: The reaction enthalpies computed with the PBEsol+U method for the reactionsgiven in Table 1 and the respective experimental values47,48. Columns represent resultsobtained with the fixed values of U and with U = ULR. The enthalpies are given in kJmol−1.

reactions 0.5 eV 1.5 eV 3 eV 4.5 eV 6 eV ULR exp.(1) 447 423 387 349 309 322 313±17(2) 532 496 441 385 328 432 385±22(3) 634 617 615 584 553 653 618±27(4) 305 280 240 198 148 261 205±20(5) 329 289 228 169 123 231 204±20(6) 429 438 397 332 300 428 412±25(7) 586 533 452 373 292 410 405±20(8) 1188 1157 1113 1075 1096 1030 1033±20(9) -228 -281 -358 -436 -513 -244 -311±22(10) 696 648 583 524 526 558 576±20(11) -127 -159 -204 -247 -288 -109 -170±40(12) 27 37 51 57 64 -25 -25±32(13) -162 -172 -185 -197 -208 -161 -182±11(14) 187 182 171 156 140 139 110±11(15) 348 326 292 255 217 302 189±12(16) 635 630 618 601 582 574 437±12(17) 126 132 140 149 157 111 272±11

mean error 90 68 33 -6 -35 28M.A.E. 107 84 60 63 83 50 20

We notice that the best match to the experimental values is obtained with the CCSD(T)378

method. However, because of the bad scaling, applicability of such advanced post Hartree-379

Fock methods is limited to a few atoms containing molecular compounds5. Applicability of380

hybrid functionals is also currently limited to no more than a few dozens of atoms and such381

calculations are usually orders of magnitudes more resources consuming than the DFT+U382

method, which in that aspect performs similarly to the standard DFT. Our results thus383

strongly suggest that the DFT+U method, with carefully derived Hubbard U parameter,384

such as ULR, is a good tool for efficient calculations of complex actinide-bearing materials,385

including chemically complex solids. Therefore, in the next step, we tested the performance386

of DFT+ULR method for the prediction of structural parameters and reaction enthalpies of387

simple uranium-bearing solids.388

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Table 6: The Hubbard U parameter values for the uranium-bearing molecules calculatedusing the linear response method of Cococcioni and de Gironcoli 36 . The values are given ineV. The last column represents the values computed for the d-orbitals.

Molecule Hubbard ULR

PBE (f electrons) PBEsol (f electrons) PBE (d electrons)UF6 3.1 3.0 0.6UF5 2.3 2.3 0.6UF4 1.8 1.7 0.7UF3 1.9 1.9 0.8UCl6 2.2 2.2 0.5UCl5 2.2 2.2 0.5UCl4 1.7 1.7 0.5UCl3 1.4 1.5 0.6UOF4 2.8 2.7 0.6UO2F2 3.1 3.1 0.7UO2(OH)2 2.8 2.8 0.6UO3 2.6 2.6 0.6UO2 2.0 1.9 0.6

Uranium Solids389

We extended the calculations and analysis performed for the molecular complexes into com-390

putation of simple uranium-bearing solids. We wanted to check if by using the mixture of391

GGA functionals, the modified PBE functional and the DFT+U method one can also im-392

prove the description of structures and the prediction of reaction enthalpies in the case of393

periodic systems. As for molecular compounds, in these calculations the Hubbard U pa-394

rameter value has been derived for each solid using the linear response approach36 and the395

results for the values of ULR are given in Table 7. The ULR values derived for the solids are396

consistent with those derived for the molecular complexes (Table 6). Here we also observe397

that ULR is larger for the U(VI)-bearing solids (UF6 and UO3) and is the smallest for the398

U(III) carrying materials.399

Structures400

In Tables 8, 9 and 10 we provide the lattice parameters of all the considered solids, computed401

with different methods and measured experimentally. All the methods usually overestimate402

the lattice lengths and volumes of the uranium fluorides solids (see the cases of UF6, α-UF5403

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0102030405060

7080

90100110120130140150160

170180

Mea

n ab

solu

te e

rror

(kJ

/mol

)

PB

E

BP

BE

PB

Eso

l

mod

ified

PB

E PB

E-B

PB

E c

orr.

PB

Eso

l+U LR

PB

E+

U LR

PB

E0

CC

SD

(T)

MP

2

Figure 4: The mean absolute error for the eleven reactions considered by Shamov et al. 7 ob-tained using different methods. The PBE0, CCSD(T) and MP2 results are those of Shamovet al. 7 .

and β-UF5, Table 8). This is consistent with the results obtained for the gas-phase UF6.404

Similarly to this case, the PBEsol functional gives the best match to the experiment. The405

DFT+U method, in general, worsens the agreement, when comparing with the prediction406

of respective GGA functionals, and modified functional gives the worst prediction. Interest-407

ingly, all the methods overestimate the lattice parameters of UCl6 solid (Table 9), which is408

different from the result for the gas-phase UCl6 (see Table 2). However, the PBEsol+ULR409

method results in good prediction of the lattice parameters of UCl4 and UCl3. α-U is the410

only pure-uranium solid considered here. Both the GGA functionals result in relatively good411

predictions of its lattice constants. This result is consistent with other computational stud-412

ies8. However, it is not well described by the DFT+U method, including the DFT+ULR413

cases, which significantly overestimates its volume. Surprisingly, very good description of414

the α-U structure is obtained with the modified PBE functional, which badly overestimates415

the volumes of many other solids (see Tables 8-10).416

Actinide oxides are major constituents of most actinide-carrying minerals, including the417

ones of interest for nuclear waste management2. Therefore, we are especially interested in418

performance of the utilized methods in the case of uranium-oxides. Here we analyze three419

such crystalline solids: UO2, α-UO3 and U3O8. The results are given in Table 10. We notice420

that for the oxides, the PBE and PBEsol functionals result in much better prediction of the421

lattice parameters than in case of uranium fluorides and chlorides. For UO2 the DFT+U422

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1 2 3 4 5 6 7 8 9 1011121314151617181920-50

050

100150200

The mean error (M.E.)

1 2 3 4 5 6 7 8 9 1011121314151617181920

050

100150200

The mean absolute error (M.A.E.)

1 2 3 4 5 6 7 8 9 1011121314151617181920-50

050

100150200

1 2 3 4 5 6 7 8 9 1011121314151617181920

0050

100150200

1 2 3 4 5 6 7 8 9 1011121314151617181920-50

050

100150200

1 2 3 4 5 6 7 8 9 1011121314151617181920

050

100150200

1 2 3 4 5 6 7 8 9 1011121314151617181920-50

050

100150200

1 2 3 4 5 6 7 8 9 1011121314151617181920

050

100150200

experimental experimental

PBE0 PBE0

MP2 MP2

CCSD(T) CCSD(T)

Figure 5: The mean absolute error and mean error (in kJmol−1) of the selected 11 gas-phase reactions considered by Shamov et al. 7 (see Table 1) obtained with different methodsindicates 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 = 3eV), (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 usingPBE0, MP2 and CCSD(T) methods were chosen as reference and the references are indicatedin the left-upper corners.

method improves the agreement with experiment, and DFT+ULR results in a good match423

to the experimental lattice parameters. Similar trend is also observed for U3O8. α-UO3424

is also well described by PBEsol+ULR, although with PBE+ULR its volume is significantly425

overestimated, which is similar to the case of α-U. This may be somehow related to usage426

of the PBE functional in the DFT+U calculations. The overall result clearly indicates that427

the DFT+ULR method could be successfully used for the prediction of structural parameters428

of uranium-oxides-containing solids. Interestingly, this result is in line with our recent stud-429

ies of the performance of DFT+ULR method for Ln2O3 and monazite-type orthophosphates430

(LnPO4)75. In those studies we found that the PBEsol+ULR method significantly outper-431

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Table 7: The Hubbard U parameter values for the uranium-bearing solids calculated usingthe linear response method of Cococcioni and de Gironcoli 36 . The values are given in eV.

Solids Hubbard ULR

PBE PBEsolUF6 2.9 2.8α-UF5 2.3 2.3β-UF5 2.3 2.3UF4 1.4 1.3UF3 0.8 0.8UCl6 2.7 2.7UCl4 1.6 1.6UCl3 0.8 0.9U3O8 2.4 2.3α-UO3 2.7 2.7UO2 1.8 1.7α-U 1.9 1.9

forms the PBE and PBEsol functionals in predicting the structural parameters of Ln-bearing432

materials.433

Band Gaps434

When investigating solids by the DFT+U method often the ability of this method to repro-435

duce the band gaps of the considered materials is discussed (eg.25). This is because standard436

DFT usually underestimates the band gaps by as much as ∼ 40%5,76,77. In Table 11 we437

provide the values of the band gap obtained in all our DFT+U calculations of solids. The438

experimental band gap of UO2 is 2.1±0.1 eV78. In the previous DFT+U studies that utilize439

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,440

which is only slightly larger than the experimental value. Our calculations with U = 4.5 eV441

give band gaps of 2.0 eV and 1.9 eV for PBE+ULR and PBEsol+ULR methods respectively,442

which are very consistent with the experimental value. However, due to smaller values of443

the Hubbard U parameter derived by the linear response method (see Table 7), both the444

DFT+ULR calculations resulted in much smaller band gap of 0.7 eV (PBE+ULR) and 0.4 eV445

(PBEsol+ULR), which is inconsistent with the aforementioned measurements. We notice446

however, that in principle the DFT is a ground state theory and the Kohn-Sham eigen-447

values have no strict physical meaning5,76,77. This would imply that the interpretation of448

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Table 8: The lattice parameters of uranium fluoride solids and α-uranium calculated usingdifferent methods and measured47,48. All length values are given in A and volumes are givenin A3.

UF6 α-UF5 β-UF5

a b c V a=b c V a=b c VPBE 10.62 9.53 5.58 565 6.93 4.53 217.8 11.56 5.40 721PBEsol 10.42 9.32 5.46 530 6.75 4.48 204.1 11.41 5.28 687BPBE 11.50 10.36 6.04 720 7.37 4.54 246.7 11.57 5.49 735modified PBE 12.16 10.92 6.41 852 8.37 4.67 326.9 18.36 6.70 2260PBE+U(3 eV) 10.74 9.69 5.67 590 6.95 4.57 220.5 11.64 5.43 736PBE+U(4.5 eV) 10.82 9.69 5.70 598 6.95 4.59 221.8 11.67 5.45 743PBE+U(6 eV) 10.82 9.75 5.70 602 6.96 4.61 223.0 11.72 5.45 749PBE+ULR 10.74 9.68 5.66 588 7.12 4.56 231.6 11.62 5.43 733PBEsol+U(3 eV) 10.59 9.51 5.57 561 6.85 4.52 212.1 11.51 5.31 703PBEsol+U(4.5 eV) 10.60 9.52 5.58 563 6.78 4.53 208.6 11.55 5.33 711PBEsol+U(6 eV) 10.62 9.53 5.58 565 6.79 4.56 210.0 11.61 5.33 718PBEsol+ULR 10.55 9.45 5.53 551 6.77 4.51 206.5 11.49 5.30 700exp. 9.90 8.96 5.21 462 6.52 4.47 189.9 11.46 5.20 682

UF4 UF3 α-Ua b c V a=b c V a b c V

PBE 12.88 10.85 8.43 950 7.18 7.42 331 2.73 5.82 4.92 78.1PBEsol 12.72 10.74 8.32 916 7.05 7.30 314 2.68 5.69 4.86 74.3BPBE 12.90 10.86 8.43 953 7.20 7.39 332 2.72 5.82 4.93 78.2modified PBE 14.90 11.53 11.19 1554 7.49 7.58 369 2.76 5.89 4.97 80.8PBE+U(3 eV) 13.02 10.95 8.51 979 7.27 7.48 343 3.76 6.22 6.05 141.3PBE+U(4.5 eV) 13.08 11.00 8.54 993 7.33 7.50 349 3.82 6.37 6.18 150.6PBE+U(6 eV) 13.13 11.06 8.58 1008 7.38 7.55 356 3.86 6.47 6.28 156.9PBE+ULR 12.96 10.89 8.47 965 7.22 7.41 335 3.61 5.95 5.88 126.4PBEsol+U(3 eV) 12.87 10.83 8.41 946 7.16 7.37 327 3.65 5.96 5.93 128.8PBEsol+U(4.5 eV) 12.93 10.89 8.45 960 7.21 7.40 333 3.74 6.19 6.06 140.2PBEsol+U(6 eV) 12.98 10.95 8.49 974 7.24 7.43 338 3.77 6.34 6.17 147.2PBEsol+ULR 12.80 10.78 8.38 930 7.09 7.31 318 3.47 5.52 5.77 110.5exp. 12.73 10.75 8.43 929 7.18 7.35 328 2.84 5.87 4.94 82.1

the Kohn-Sham band gap as the fundamental band gap, although performed on the regular449

basis, is also not well justified. In fact, Perdew and Levy 76 and Perdew 77 have shown that450

due to the discontinuity in the derivative of the exchange-correlation energy, even having451

the exact ground state density and the exact Kohn-Sham potential, the Kohn-Sham band452

gap is still expected to be underestimated. From this point of view the underestimation of453

the band gap by a DFT-based method can not be seen as a flaw. On the other hand, from454

the construction of the Hubbard energy functional it is expected that the DFT+U method455

with the correct Hubbard U parameter should result in the discontinuity in the derivative of456

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Table 9: The lattice parameters of uranium chloride solids calculated using different methodsand measured47,48. All length values are given in A and volumes are given in A3.

UCl6 UCl4 UCl3a=b c V a=b c V a=b c V

PBE 11.76 6.46 774 8.36 7.76 543 7.55 4.28 210.6PBEsol 11.25 6.13 672 8.23 7.47 506 7.37 4.14 194.8BPBE 13.89 7.81 1305 8.36 7.87 551 7.57 4.27 212.1modified PBE 15.33 8.50 1729 11.21 15.28 1918 11.76 3.91 445.9PBE+U(3 eV) 11.76 6.46 775 8.44 7.82 558 7.61 4.39 220.1PBE+U(4.5 eV) 11.78 6.45 774 8.49 7.90 569 7.66 4.42 224.3PBE+U(6 eV) 11.78 6.45 775 8.54 7.88 574 7.68 4.44 227.4PBE+ULR 11.76 6.46 774 8.40 7.87 556 4.56 4.32 213.6PBEsol+U(3 eV) 11.25 6.15 673 8.33 7.56 524 7.42 4.35 207.0PBEsol+U(4.5 eV) 11.24 6.15 674 8.37 7.57 531 7.47 4.35 211.1PBEsol+U(6 eV) 11.24 6.19 677 8.42 7.58 538 7.52 4.37 214.3PBEsol+ULR 11.28 6.14 676 8.28 7.54 517 7.40 4.23 199.9exp. 10.95 6.02 625 8.30 7.49 515 7.44 4.32 207.5

the exchange-correlation energy and reproduce the fundamental band gap24. Thus we can457

not exclude the possibility that the linear response method underestimates the U values and458

therefore the band gaps of the considered materials. We notice that this is similar to the case459

of lanthanide-oxides, for which the DFT+ULR underestimates the band gaps by ∼ 1.7 eV75.460

At the same time Blanca-Romero et al. 75 found that the PBEsol+ULR method results in461

very good prediction of the structural parameters of these materials. Similarly here, the462

DFT+ULR method results in good prediction of the reaction enthalpies, also for solids which463

will be discussed in the next section. The problem of potential underestimation of the Hub-464

bard U parameters by the linear response method of Cococcioni and de Gironcoli 36 could be465

further addressed by computation of the Hubbard U parameter values with other method,466

such as cRPA71. We note however that such an investigation as well as the in-depth inves-467

tigation of the electronic structure of the investigated materials is well beyond the scope of468

this paper, which aims into benchmarking of the DFT+U method, and DFT+ULR approach469

in particular, for the thermochemical calculations. Such an analysis is also complicated due470

to lack of the experimental information on the band gaps of the most of the investigated471

materials. We also notice that the performance of DFT+U for the prediction of the band472

gaps and the electronic densities of states was already a subject of many previous studies473

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Table 10: The lattice parameters of uranium oxide solids calculated using different methodsand measured47,48. All length values are given in A and volumes are given in A3.

UO2 U3O8 α-UO3

a=b=c V a b c V a b c VPBE 5.37 154.8 7.02 11.58 4.16 338 3.74 7.15 4.12 110.2PBEsol 5.30 149.0 6.77 11.69 4.12 326 3.73 7.00 4.09 106.7BPBE 5.41 158.6 7.02 11.58 4.16 338 3.74 7.15 4.12 110.2modified PBE 5.49 165.4 7.28 11.62 4.20 355 3.44 10.77 4.18 174.7PBE+U(3 eV) 5.54 170.0 7.23 11.53 4.17 347 3.41 9.96 4.17 141.6PBE+U(4.5 eV) 5.58 173.1 7.28 11.55 4.18 351 3.42 9.97 4.18 142.2PBE+U(6 eV) 5.62 176.4 7.34 11.57 4.19 355 3.42 10.34 4.19 148.4PBE+ULR 5.52 167.7 7.20 11.52 4.17 346 3.41 9.96 4.17 141.5PBEsol+U(3 eV) 5.48 164.3 7.14 11.43 4.14 338 3.71 7.14 4.10 108.7PBEsol+U(4.5 eV) 5.52 167.5 7.19 11.45 4.15 341 3.70 7.27 4.12 111.0PBEsol+U(6 eV) 5.55 170.7 7.25 11.48 4.15 345 3.74 7.29 4.14 113.0PBEsol+ULR 5.45 161.6 7.11 11.43 4.14 336 3.80 6.94 4.11 108.5exp. 5.47 163.5 6.70 11.95 4.14 332 3.96 6.86 4.17 113.2

(eg.25,26).474

Reaction Enthalpies475

We computed the enthalpies of nine reaction between the considered solids. All the reactions476

are provided in Table 12. We note that in the calculation of reaction enthalpies for the gas-477

phase species, F, O and Cl, we consider the energies of atoms, not the relevant molecules478

(i.e. for instance we use the energy of F atom, not 1/2 F2). The results are provided479

in Tables 13 and 14. The PBE and PBEsol functionals result in significant, systematic480

errors of 168 kJ/mol and 184 kJ/mol respectively. Similarly to the case of the enthalpies of481

reactions between the molecular compounds, the mixture of PBE and BPBE functionals and482

the modified PBE functional result in improvements over the PBE and PBEsol functionals,483

although the improvement is not that pronounced as for the molecules. On the other hand484

the DFT+U method, similarly to the case of reactions between the molecular compounds,485

significantly improves the reaction enthalpies and the best results are obtained with the486

DFT+ULR method. The M.A.E. is significantly reduced to 31 kJ/mol and 47 kJ/mol for487

PBE+ULR and PBEsol+ULR respectively, which is comparable to the error obtained for the488

enthalpies of reactions between the uranium-bearing molecules. In general, even the DFT+U489

26

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Table 11: The values of band gaps (in eV) for the uranium-bearing solids calculated withPBE+U and PBEsol+U method for different U values.

PBE+U PBEsol+USolids 3 eV 4.5 eV 6 eV ULR 3 eV 4.5 eV 6 eV ULR

UF6 3.5 3.7 3.9 3.5 3.6 3.8 4.0 3.6α-UF5 3.3 4.4 4.4 2.8 3.1 4.3 4.7 2.6β-UF5 1.8 2.5 2.3 1.5 1.7 2.4 2.8 1.3UF4 2.0 2.9 3.6 1.0 1.9 2.8 3.6 1.0UF3 2.6 3.5 4.1 1.0 2.4 3.1 3.8 0.9UCl6 2.0 1.9 1.8 2.0 2.1 2.0 1.8 2.0UCl4 1.4 1.5 1.5 1.1 1.5 1.6 1.6 0.9UCl3 2.3 3.3 4.0 1.0 2.2 2.9 3.6 0.5U3O8 2.1 2.4 2.4 1.8 1.8 2.5 2.5 1.2α-UO3 2.2 2.5 2.9 2.1 1.1 1.5 1.4 1.2UO2 1.2 2.0 2.6 0.7 1.1 1.9 2.5 0.4α-U 1.4 2.0 2.1 0.4 1.2 2.0 2.2 0.3

Table 12: List of the 9 considered reactions between the uranium-bearing solids.

(18) α-UF5 + F → UF6

(19) β-UF5 + F → UF6

(20) UF4 + F → α-UF5

(21) UF4 + F → β-UF5

(22) UF3 + F → UF4

(23) UO2 +O → α-UO3

(24) 3UO2 + 2O → U3O8

(25) UCl4 + 2Cl → UCl6(26) UCl3 + Cl → UCl4

method with fixed U value ranging from 3 eV to 6 eV substantially improves the prediction of490

DFT, and such calculations with U = 4.5 eV (the value that reproduces the band gap of UO2)491

give the best results. The results obtained for solids are thus in line with the conclusions492

reached from the studies of molecular compounds. The both benchmark calculations indicate493

that the DFT+U method could allow for meaningful, computer-aided simulations of actinide-494

bearing materials, including those relevant for nuclear waste management, and that very good495

results for the thermochemistry can be obtained, when Hubbard U parameter is derived ab496

initio for each considered molecular or solid structure.497

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Table 13: The reaction enthalpies computed with different DFT methods for the reactionsgiven in Table 12 and the respective experimental values47,48. Term PBEmodif indicates themodified PBE functional. The enthalpies are given in kJ mol−1.

PBE PBEsol BPBE PBEmodif PBE-BPBE corr. exp.(18) -324 -352 -327 -280 -341 -200±8(19) -363 -369 -378 -277 -439 -192±6(20) -407 -414 -403 -443 -385 -240±10(21) -368 -397 -352 -446 -288 -248±8(22) -554 -578 -541 -522 -488 -491±9(23) -490 -510 -439 -487 -236 -250±3(24) -1208 -1191 -1086 -1037 -598 -818±4(25) -490 -515 -492 -389 -499 -290±6(26) -310 -335 -278 -298 -148 -276±5

mean error -168 -184 -143 -131 -46M.A.E. 168 184 143 131 128 7

CONCLUSIONS498

In this paper we have shown that the DFT+U method can be successfully used in cal-499

culation of actinide-bearing materials. We confirmed previous finding of Shamov et al. 7500

that the standard DFT systematically overestimates the enthalpies of reactions between501

various uranium-bearing compounds by ∼100 kJ/mol. We were especially interested in test-502

ing the performance of DFT+U method with the Hubbard U parameter derived using the503

linear response method of Cococcioni and de Gironcoli 36 . We found that it significantly504

reduces the error of the enthalpies of reactions to ∼50 kJ/mol, which is at the level of MP2505

method7, and in general gives results that are consistent with the all-electrons hybrid func-506

tional (PBE0) calculations proposed as the best method for computation of actnide-bearing507

materials7. Comparing with DFT, the DFT+ULR method gives better prediction for the508

enthalpies of all the considered reactions. It also improves the description of structures of509

uranium-oxides. The derived Hubbard U parameters are consistent with the values obtained510

experimentally34,35, but are smaller (by ∼ 2 eV) than the value of 4.5 eV used in most of the511

previous calculations and obtained from the spectroscopic measurements of UO232,33. We512

also found that the value of Hubbard U parameter decreases with decrease of the oxidation513

state of uranium.514

Our results thus suggest that, because of serious limitation in applicability of the computationally-515

28

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Table 14: The reaction enthalpies computed with the DFT+U method for the reactionsgiven in Table 12 and the respective experimental values47,48. Columns represent resultsobtained with the fixed values of U and with U = ULR. The enthalpies are given in kJmol−1.

PBE PBEsol exp.reaction 3 eV 4.5 eV 6 eV ULR 3 eV 4.5 eV 6 eV ULR

(18) -256 -219 -186 -201 -284 -248 -211 -241 -200±8(19) -295 -257 -217 -241 -305 -268 -229 -262 -192±6(20) -302 -246 -184 -275 -319 -263 -206 -281 -240±10(21) -263 -208 -153 -235 -298 -242 -188 -260 -248±8(22) -471 -432 -399 -491 -491 -455 -423 -518 -491±9(23) -369 -282 -207 -333 -392 -306 -221 -362 -250±3(24) -898 -702 -503 -839 -964 -773 -578 -921 -818±4(25) -317 -229 -171 -266 -351 -264 -194 -297 -290±6(26) -223 -178 -159 -229 -281 -214 -164 -268 -276±5

mean error -43 28 92 -12 -76 -3 65 -45M.A.E. 59 55 97 31 76 41 77 47 7

expensive post Hartree-Fock methods such as CCSD(T) or the hybrid DFT functionals for516

investigation of chemically complex, actinide-bearing materials, the DFT+U method with517

the Hubbard U parameter derived by ab initio methods could be a good choice as a feasible518

method for computation of such materials.519

ACKNOWLEDGMENT520

Funded by the Excellence Initiative of the German federal and state governments and the521

Julich Aachen Research Alliance - High-Performance Computing. We thank the JARA-522

HPC awarding body for time on the RWTH computing cluster awarded through JARA-HPC523

Partition.524

References525

(1) Chroneos, A.; Rushton, M.; Jiang, C.; Tsoukalas, L. Nuclear Wasteform Materials:526

Atomistic Simulation Case Studies. J. Nucl. Mater. 2013, 441, 29–39.527

29

Page 29 of 39



123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

(2) Schlenz, H.; Heuser, J.; Neumann, A.; Schmitz, S.; Bosbach, D. Monazite as a Suitable528

Actinide Waste Form. Z. Kristallogr. 2013, 228, 113–123.529

(3) Ewing, R.; Weber, W.; Lian, J. Nuclear Waste Disposal-pyrochlore (A(2)B(2)O(7)):530

Nuclear Waste Form for the Immobilization of Plutonium and “Minor” Actinides. J.531

Appl. Phys. 2004, 95, 5949–5971.532

(4) Ewing, R.; Wang, L. Phosphates as Nuclear Waste Forms. Rev. Mineral. Geochem.533

2002, 48, 673–699.534

(5) Jahn, S.; Kowalski, P. M. Theoretical Approaches to Structure and Spectroscopy of535

Earth Materials. Rev. Mineral. Geochem. 2014, 78, 691–743.536

(6) Li, Y.; Kowalski, P. M.; Blanca-Romero, A.; Vinograd, V.; Bosbach, D. Ab initio537

Calculation of Excess Properties of Solid Solutions. J. Solid State Chem. 2014, 220,538

137–141.539

(7) Shamov, G. A.; Schreckenbach, G.; Vo, T. N. A Comparative Relativistic DFT and Ab540

initio Study on the Structure and Thermodynamics of the Oxofluorides of Uranium(IV),541

(V) and (VI). Chem. Eur. J. 2007, 13, 4932–4947.542

(8) Li, J.; Ren, Q.; Lu, C.; Lu, L.; Dai, Y.; Liu, B. Structure, Formation Energies and543

Elastic Constants of Uranium Metal Investigated by First Principles Calculations. J.544

Alloys Compd. 2012, 516, 139–143.545

(9) Buzko, V.; Chuiko, G.; Kushkhov, K. DFT Study of the Structure and Stability of546

Pu(III) and Pu(IV) Chloro Complexes. Russ. J. Inorg. Chem. 2012, 57, 62–67.547

(10) Perez-Villa, A.; David, J.; Fuentealba, P.; Restrepo, A. Octahedral Complexes of the548

Series of Actinides Hexafluorides AnF6. Chem. Phys. Lett. 2011, 507, 57–62.549

(11) Pantazis, D. A.; Neese, F. All-Electron Scalar Relativistic Basis Sets for the Actinides.550

J. Chem. Theory Comput. 2011, 7, 677–684.551

(12) Schreckenbach, G.; Shamov, G. A. Theoretical Actinide Molecular Science. Acc. Chem.552

Res. 2010, 43, 19–29.553

30

Page 30 of 39



123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

(13) Wen, X.-D.; Martin, R. L.; Henderson, T. M.; Scuseria, G. E. Density Functional554

Theory Studies of the Electronic Structure of Solid State Actinide Oxides. Chem. Rev.555

2013, 113, 1063–1096.556

(14) Odoh, S. O.; Pan, Q.-J.; Shamov, G. A.; Wang, F.; Fayek, M.; Schreckenbach, G. The-557

oretical Study of the Reduction of Uranium(VI) Aquo Complexes on Titania Particles558

and by Alcohols. Chem. Eur. J. 2012, 18, 7117–7127.559

(15) Iche-Tarrat, N.; Marsden, C. J. Examining the Performance of DFT Methods in Ura-560

nium Chemistry: Does Core Size Matter for a Pseudopotential? J. Phys. Chem. A561

2008, 112, 7632–7642.562

(16) Wen, X.-D.; Martin, R. L.; Scuseria, G. E.; Rudin, S. P.; Batista, E. R.; Burrell, A. K.563

Screened Hybrid and DFT+U Studies of the Structural, Electronic, and Optical Prop-564

erties of U3O8. J. Phys.: Condens. Matter 2013, 25, 025501.565

(17) Bhattacharyya, A.; Mohapatra, P.; Manchanda, V. Role of Ligand Softness and Diluent566

on the Separation Behaviour of Am(III) and Eu(III). J. Radioanal. Nucl. Chem. 2011,567

288, 709–716.568

(18) Castro, L.; Lam, O. P.; Bart, S. C.; Meyer, K.; Maron, L. Carbonate Formation from569

CO2 via Oxo versus Oxalate Pathway: Theoretical Investigations into the Mechanism570

of Uranium-Mediated Carbonate Formation. Organometallics 2010, 29, 5504–5510.571

(19) Hennig, C.; Ikeda-Ohno, A.; Tsushima, S.; Scheinost, A. C. The Sulfate Coordination of572

Np(IV), Np(V), and Np(VI) in Aqueous Solution. Inorg. Chem. 2009, 48, 5350–5360.573

(20) Majumdar, D.; Balasubramanian, K. Theoretical Studies on the Nature of Uranylsili-574

cate, Uranylphosphate and Uranylarsenate Interactions in the Model H2UO2SiO4·3H2O,575

HUO2PO4 ·3H2O, and HUO2AsO4 ·3H2O molecules. Chem. Phys. Lett. 2004, 397, 26–576

33.577

(21) Odoh, S. O.; Schreckenbach, G. Performance of Relativistic Effective Core Potentials in578

DFT Calculations on Actinide Compounds. J. Phys. Chem. A 2010, 114, 1957–1963.579

31

Page 31 of 39



123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

(22) Marx, D.; Hutter, J. Ab initio molecular dynamics: Theory and implementation. Mod-580

ern Methods and Algorithms of Quantum Chemistry, Forschungszentrum Julich, NIC581

Series. 2000; p 301.582

(23) Kowalski, P. M.; Camellone, M. F.; Nair, N. N.; Meyer, B.; Marx, D. Charge Local-583

ization Dynamics Induced by Oxygen Vacancies on the TiO2(110) Surface. Phys. Rev.584

Lett. 2010, 105, 146405.585

(24) Himmetoglu, B.; Floris, A.; de Gironcoli, S.; Cococcioni, M. Hubbard-corrected DFT586

energy functionals: The LDA+U description of correlated systems. Int. J. Quantum587

Chem. 2014, 114, 14–49.588

(25) Wen, X.-D.; Martin, R. L.; Roy, L. E.; Scuseria, G. E.; Rudin, S. P.; Batista, E. R.;589

McCleskey, T. M.; Scott, B. L.; Bauer, E.; Joyce, J. J. et al. Effect of Spin-orbit Coupling590

on the Actinide Dioxides AnO2 (An=Th, Pa, U, Np, Pu, and Am): A Screened Hybrid591

Density Functional Study. J. Chem. Phys. 2012, 137, 154707.592

(26) Dorado, B.; Amadon, B.; Freyss, M.; Bertolus, M. DFT+U Calculations of the Ground593

State and Metastable States of Uranium Dioxide. Phys. Rev. B 2009, 79, 235125.594

(27) Dudarev, S. L.; Manh, D. N.; Sutton, A. P. Effect of Mott-Hubbard Correlations on595

the Electronic Structure and Structural Stability of Uranium Dioxide. Philos. Mag. B596

1997, 75, 613–628.597

(28) Rak, Z.; Ewing, R. C.; Becker, U. Role of Iron in the Incorporation of Uranium in Ferric598

Garnet Matrices. Phys. Rev. B 2011, 84, 155128.599

(29) Elgazzar, S.; Rusz, J.; Oppeneer, P. M.; Colineau, E.; Griveau, J.-C.; Magnani, N.;600

Rebizant, J.; Caciuffo, R. Ab initio Computational and Experimental Investigation of601

the Electronic Structure of Actinide 218 Materials. Phys. Rev. B 2010, 81, 235117.602

(30) Floris, A.; de Gironcoli, S.; Gross, E. K. U.; Cococcioni, M. Vibrational properties of603

MnO and NiO from DFT + U -based density functional perturbation theory. Phys. Rev.604

B 2011, 84, 161102.605

32

Page 32 of 39



123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

(31) Bajaj, S.; Sevik, C.; Cagin, T.; Garay, A.; Turchi, P.; Arryave, R. On the Limitations606

of the DFT+U Approach to Energetics of Actinides. Comput. Mater. Sci. 2012, 59,607

48–56.608

(32) Kotani, A.; Yamazaki, T. Systematic Analysis of Core Photoemission Spectra for Ac-609

tinide Di-Oxides and Rare-Earth Sesqui-Oxides. Prog. Theor. Phys. Supp. 1992, 108,610

117–131.611

(33) Baer, Y.; Schoenes, J. Electronic Structure and Coulomb Correlation Energy in UO2612

Single Crystal. Solid State Commun. 1980, 33, 885–888.613

(34) Johansson, B. Nature of The 5f Electrons in the Actinide Series. Phys. Rev. B 1975,614

11, 2740–2743.615

(35) Herbst, J. F.; Watson, R. E.; Lindgren, I. Coulomb Term U and 5f Electron Excitation616

Energies for the Metals Actinium to Berkelium. Phys. Rev. B 1976, 14, 3265–3272.617

(36) Cococcioni, M.; de Gironcoli, S. Linear Response Approach to the Calculation of the Ef-618

fective Interaction Parameters in the LDA+U Method. Phys. Rev. B 2005, 71, 035105.619

(37) Giannozzi, P.; Baroni, S.; Bonini, N.; Calandra, M.; Car, R.; Cavazzoni, C.; Ceresoli, D.;620

Chiarotti, G. L.; Cococcioni, M.; Dabo, I. et al. QUANTUM ESPRESSO: a Modular621

and Open-source Software Project for Quantum Simulations of Materials. J. Phys.:622

Condens. Matter 2009, 21, 395502.623

(38) Perdew, J. P.; Burke, K.; Ernzerhof, M. Generalized Gradient Approximation Made624

Simple. Phys. Rev. Lett. 1996, 77, 3865–3868.625

(39) Perdew, J. P.; Ruzsinszky, A.; Csonka, G. I.; Vydrov, O. A.; Scuseria, G. E.; Con-626

stantin, L. A.; Zhou, X.; Burke, K. Restoring the Density-Gradient Expansion for627

Exchange in Solids and Surfaces. Phys. Rev. Lett. 2008, 100, 136406.628

(40) Becke, A. D. Density-functional Exchange-energy Approximation with Correct Asymp-629

totic Behavior. Phys. Rev. A 1988, 38, 3098–3100.630

33

Page 33 of 39



123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

(41) Vanderbilt, D. Soft Self-consistent Pseudopotentials in a Generalized Eigenvalue For-631

malism. Phys. Rev. B 1990, 41, 7892–7895.632

(42) Methfessel, M.; Paxton, A. T. High-precision Sampling for Brillouin-zone Integration633

in Metals. Phys. Rev. B 1989, 40, 3616–3621.634

(43) Adamo, C.; Barone, V. Toward Reliable Density Functional Methods Without Ad-635

justable Parameters: The PBE0 Model. J. Chem. Phys. 1999, 110, 6158–6170.636

(44) Perdew, J. P.; Ernzerhof, M.; Burke, K. Rationale for Mixing Exact Exchange With637

Density Functional Approximations. J. Chem. Phys. 1996, 105, 9982–9985.638

(45) Dorado, B.; Amadon, B.; Freyss, M.; Bertolus, M. DFT+U Calculations of the Ground639

State and Metastable States of Uranium Dioxide. Phys. Rev. B 2009, 79, 235125.640

(46) Batista, E. R.; Martin, R. L.; Hay, P. J.; Peralta, J. E.; Scuseria, G. E. Density Func-641

tional Investigations of the Properties and Thermochemistry of UF6 and UF5 Using642

Valence-electron and All-electron Approaches. J. Chem. Phys. 2004, 121, 2144–2150.643

(47) Morss, L.; Edelstein, N.; Fuger, J.; Katz, J. The Chemistry of the Actinide and Trans-644

actinide Elements (Set Vol.1-6); Springer, 2011.645

(48) Guillaumont, R.; Agency, O. N. E.; Mompean, F. Update on the Chemical Thermo-646

dynamics of Uranium, Neptunium, Plutonium, Americium and Technetium; Chemical647

Thermodynamics Series; Elsevier Science & Technology Books, 2003.648

(49) Lander, G. H.; Mueller, M. H. Neutron Diffraction Study of α-uranium at Low Tem-649

peratures. Acta Crystallogr. Sect. B 1970, 26, 129–136.650

(50) Taylor, J.; Wilson, P. The Structures of Fluorides X. Neutron Powder Diffraction Profile651

Studies of UF6 at 193K and 293K. J. Solid State Chem. 1975, 14, 378–382.652

(51) Eller, P. G.; Larson, A.; Peterson, J.; Ensor, D.; Young, J. Crystal Structures of -UF5653

and U2F9 and Spectral Characterization of U2F9. Inorg. Chim. Acta 1979, 37, 129–133.654

34

Page 34 of 39



123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

(52) Ryan, R. R.; Penneman, R. A.; Asprey, L. B.; Paine, R. T. Single-crystal X-ray Study655

of β-uranium Pentafluoride. The Eight Coordination of UV . Acta Crystallogr. Sect. B656

1976, 32, 3311–3313.657

(53) Zachariasen, W. H. Crystal Chemical Studies of the 5f-series of Elements. XII. New658

Compounds Representing Known Structure Types. Acta Crystallogr. 1949, 2, 388–390.659

(54) Laveissiere, J. Application de la Diffraction de Neutrons a l’etude de la Structure660

Cristalline du Trifluorure d’Uranium UF3. Bull. Soc. Franc. Mineral. Crist. 1967, 90,661

308–310.662

(55) Taylor, J. C.; Wilson, P. W. Neutron and X-ray Powder Diffraction Studies of the663

Structure of Uranium Hexachloride. Acta Crystallogr. Sect. B 1974, 30, 1481–1484.664

(56) Schleid, T.; Meyer, G.; Morss, L. R. Facile Synthesis of UCl4 and ThCl4, Metallothermic665

Reductions of UCl4 with Alkali Metals and Crystal Structure Refinements of UCl3, UCl4666

and Cs2UCl6. J. Less-Common MET. 1987, 132, 69–77.667

(57) Barrett, S. A.; Jacobson, A. J.; Tofield, B. C.; Fender, B. E. F. The Preparation and668

Structure of Barium Uranium Oxide BaUO3+x. Acta Crystallogr. Sect. B 1982, 38,669

2775–2781.670

(58) Loopstra, B. O.; Cordfunke, E. H. P. On the Structure of alpha-UO3. Recl. Trav. Chim.671

Pays-Bas 1966, 85, 135–142.672

(59) Loopstra, B. O. Neutron Diffraction Investigation of U3O8. Acta Crystallogr. 1964, 17,673

651–654.674

(60) Weinstock, B.; Crist, R. H. The Vapor Pressure of Uranium Hexafluoride. J. Chem.675

Phys. 1948, 16, 436–441.676

(61) Seip, H. Studies on the Failure of the First Born Approximation in Electron Diffraction.677

I. Uranium Hexafluoride. Acta Chem. Scand. 1965, 19, 1955–1968.678

35

Page 35 of 39



123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

(62) Hay, P. J.; Martin, R. L. Theoretical Studies of the Structures and Vibrational Frequen-679

cies of Actinide Compounds Using Relativistic Effective Core Potentials with Hartree–680

Fock and Density Functional Methods: UF6, NpF6, and PuF6. J. Chem. Phys. 1998,681

109, 3875–3881.682

(63) Csonka, G. I.; Perdew, J. P.; Ruzsinszky, A.; Philipsen, P. H. T.; Lebegue, S.; Paier, J.;683

Vydrov, O. A.; Angyan, J. G. Assessing the Performance of Recent Density Functionals684

for Bulk Solids. Phys. Rev. B 2009, 79, 155107.685

(64) Mattsson, A. E.; Armiento, R.; Paier, J.; Kresse, G.; Wills, J. M.; Mattsson, T. R. The686

AM05 Density Functional Applied to Solids. J. Chem. Phys. 2008, 128, 084714.687

(65) Privalov, T.; Schimmelpfennig, B.; Wahlgren, U.; Grenthe, I. Structure and Thermody-688

namics of Uranium(VI) Complexes in the Gas Phase: A Comparison of Experimental689

and ab Initio Data. J. Phys. Chem. A 2002, 106, 11277–11284.690

(66) Malli, G. L. Ab initio All-electron Fully Relativistic DiracFock Self-consistent Field691

Calculations for UCl6. Mol. Phys. 2003, 101, 287–294.692

(67) Ezhov, Y.; Komarov, S.; Sevastyanov, V.; Bazhanov, V. Electron Diffraction Determi-693

nation of Molecular Constants of Uranium Hexachloride. J. Struct. Chem. 1993, 34,694

473–474.695

(68) Thornton, G.; Edelstein, N.; Rosch, N.; Egdell, R. G.; Woodwark, D. R. The Electronic696

Structure of UCl6: Photoelectron Spectra and Scattered Wave X Calculations. J. Chem.697

Phys 1979, 70, 5218–5221.698

(69) Bazhanov, V.; Ezhov, Y.; Komarov, S. Electron Diffraction Study of Uranium Trichlo-699

ride. J. Struct. Chem. 1990, 31, 986–989.700

(70) Schultz, N. E.; Zhao, Y.; Truhlar, D. G. Density Functionals for Inorganometallic and701

Organometallic Chemistry. J. Phys. Chem. A 2005, 109, 11127–11143.702

(71) Sasıoglu, E.; Friedrich, C.; Blugel, S. Effective Coulomb Interaction in Transition Metals703

from Constrained Random-phase Approximation. Phys. Rev. B 2011, 83, 121101.704

36

Page 36 of 39



123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

(72) Anisimov, V. I.; Gunnarsson, O. Density-functional Calculation of Effective Coulomb705

Interactions in Metals. Phys. Rev. B 1991, 43, 7570–7574.706

(73) Yamazaki, T.; Kotani, A. Systematic Analysis of 4f Core Photoemission Spectra in707

Actinide Oxides. J. Phys. Soc. Jpn. 1991, 60, 49–51.708

(74) Hufner, S.; Wertheim, G. K. Estimates of the Coulomb Correlation Energy from X-Ray709

Photoemission Data. Phys. Rev. B 1973, 7, 5086–5090.710

(75) Blanca-Romero, A.; Kowalski, P. M.; Beridze, G.; Schlenz, H.; Bosbach, D. Performance711

of DFT+U Method for Prediction of Structural and Thermodynamic Parameters of712

Monazite-type Ceramics. J. Comput Chem. 2014, 35, 1339–1346.713

(76) Perdew, J. P.; Levy, M. Physical Content of the Exact Kohn-Sham Orbital Energies:714

Band Gaps and Derivative Discontinuities. Phys. Rev. Lett. 1983, 51, 1884–1887.715

(77) Perdew, J. P. Density functional theory and the band gap problem. Int. J. Quantum716

Chem. 1985, 28, 497–523.717

(78) Schoenes, J. Optical properties and electronic structure of UO2. J. Appl. Phys. 1978,718

49, 1463–1465.719

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Figure 6: Table of Contents Image.

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Table of Contents Image 50x50mm (300 x 300 DPI)

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Benchmarking the DFT+ U Method for Thermochemical Calculations of Uranium Molecular Compounds and Solids