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Nov 23, 2015 - Yubing Sun , Songhua Lu , Xiangxue Wang , Chao Xu , Jiaxing Li .... Jing Zhou , Jiong Li , Yu Wang , Shitong Yang , Shuao Wang , Jingye...
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EXAFS and DFT Studies on the Complexation Mechanism of Amidoximate Ligand to Uranyl Carbonate Linjuan Zhang, Jing Su, Shitong Yang, Xiaojing Guo, Yunpeng Jia, Ning Chen, Jing Zhou, Shuo Zhang, Shuao Wang, Jiong Li, Jingye Li, Guozhong Wu, and Jian-Qiang Wang Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.5b03217 • Publication Date (Web): 23 Nov 2015 Downloaded from http://pubs.acs.org on November 30, 2015

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EXAFS and DFT Studies on the Complexation Mechanism of Amidoximate Ligand to Uranyl Carbonate Linjuan Zhang1, Jing Su1, Shitong Yang2, Xiaojing Guo1, Yunpeng Jia1, Ning Chen3, Jing Zhou1, Shuo Zhang1, Shuao Wang2, Jiong Li1, Jingye Li1, Guozhong Wu1*, Jian-Qiang Wang1* 1

Shanghai Institute of Applied Physics, Chinese Academy of Sciences, Shanghai 201800, P.R.

China. 2

School for Radiological and Interdisciplinary Sciences (RAD-X), Soochow University, Suzhou

215123, P.R. China 3

Canadian Light Source Inc., Saskatoon, Saskatchewan, S7N 2V3, Canada.

KEYWORDS. Uranyl speciation, EXAFS, Local coordination structure, Amidoximate ligand

ABSTRACT. To shed some light on the uranium extraction mechanism of amidoximate (AO) ligands from uranyl carbonate solution, we present an experimental data taken using extended X-ray absorption fine structure (EXAFS) at U L3 edge and theoretical calculation results. The EXAFS data was well simulated and confessedly shows that AO ligands directly substitutes CO32- group in the equatorial plane to form a stable [UO2(CO3)3-x(AO)x](4-x)- complex. Density functional theory calculation indicates that although they have a slightly weaker electrostatic

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attraction than CO32- ligands, AO ligands display stronger binding capability to uranyl because of the remarkable orbital hybridization between U5f/6d and (N,O)2p in the uranyl–AO complex. This finding provides strong evidence supporting the substitutional mechanism, which implies that cationic adsorbents is preferred in further designing higher efficient adsorbents, and furthermore highlights the crucial role of covalent effect in extraction process.

1. INTRODUCTION To satisfy the increasing demand for nuclear fuel caused by the rapid development of advanced nuclear energy systems, extracting uranium from seawater is a promising approach due to the fact that uranium content is estimated to be 4.5 billion tons1. In the condition of seawater, uranium element mainly exists as a form of uranyl tricarbonate [UO2(CO3)3]4-, which has a high stability constant due to strong bond between uranyl and carbonate ligand.1,2 Considering the fact that the concentration of uranium is as low as 3.3 µg L-1, the cost-effective and operable extracting method was realized to be using solid adsorbent materials. Among more than 200 synthesized functionalized polymers, materials with amidoximate (AO) ligands, i.e., RC(NH2)(NOH), exhibit high selective and efficiency to uranium and thus have attracted intensive attention2-9. Although extensive theoretical and experimental investigations have been carried out, the sequestering mechanism remains in dispute. One view is that uranyl ions will initially dissociate from the uranyl carbonate complex and then combine with AO ligands.7,10 The other argues a picture of direct sorption of [UO2(CO3)3]4- with functional ligands because that the dissociation enthalpy of [UO2(CO3)3]4- is up to 39.2 kJ/mol.11 These two controversial points lead to opposite directions in designing adsorbents with improved efficiencies, that is, the former tends to cationic adsorbents whereas the latter recommends anionic adsorbents. To

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resolve this dispute, acquirements of the final coordination properties of the uranyl ions and bonding formation is mandatory. To this end, extensive theoretical and experimental investigations have been performed and the picture of the coordination mode between AO and uranyl ions is increasingly clear. In non-aqueous media, single crystal samples also display η2 binding motif was revealed by refining of X-ray diffraction pattern.12-14 For non-crystal system, theoretical calculations based on the density functional theory (DFT) agree that the η2 binding motif is the preferred form both in gas phase and in aqueous solution.15-18 From the view of experiment, we recently also confirmed this coordination mode in an aqueous UO22+-AO complex by means of X-ray absorption spectroscopy supplemented with first-principles calculations. Nevertheless this information is valuable but it is insufficient to settle the dispute whether AO can substitute the CO32- group from uranyl tri-carbonate under the condition of seawater. The key to solve this problem is to directly determine the local structure of uranyl tri-carbonate in the solution with the presence of AO ligands. Unfortunately, to date few results regarding it were published due to the local structure in solution is difficult obtained.11 Element-selective local probes and accurate analysis are then required. X-ray absorption spectroscopy is a suitable tool, successfully applied in the investigation of complex in the solution environment. If we indeed prove that AO ligands can bind to uranyl ions, another significant issue we have to face, that is, why AO can destroy the strong bond between uranyl and carbonate ligand, and thus a detail electronic structure is also essential. Here, we report a comprehensive investigation regarding on the local structure and electronic structure of uranyl carbonate complex solution under the condition with or without AO ligands through extended X-ray absorption fine structure (EXAFS) combined with density functional

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theory (DFT) calculations. Theoretical EXAFS spectra have been obtained based on multi-scattering calculations and relation between features in the spectrum and corresponding scattering atoms was figured out. Then through the comparison with experimental spectra, the most suitable structural model was confirmed. It was found that AO ligands directly substitutes CO32- group in the equatorial plane to form a stable [UO2(CO3)3-x(AO)x](4-x)- complex. DFT calculation indicates that although slightly weaker electrostatic attraction than carbonate ligands, AO ligands still exhibits stronger binding ability to the uranyl due to remarkable orbital hybridization between U5f/6d and (N,O)2p in the uranyl-AO complex. Our results suggest that cationic adsorbents with the ligands that can bind to uranyl via strong covalent bond are the promising systems used for extracting uranium from seawater. 2. EXPERIMENTLA AND CALCULATION 2.1. Experimental Materials. The raw materials included UO2(NO3)2·6H2O, (NH4)2CO3, CH3C(NH2)(NOH) (referred to as AO), and MilliQ water with a conductivity of 18.2 MΩ cm-1. First, we prepared a solid (NH4)4UO2(CO3)3 sample. Then, UO2(NO3)2·6H2O (0.007 mol) was directly dissolved in a specific amount of deionized water. Subsequently, (NH4)2CO3 aqueous solutions (0.12 mol) were slowly added into the uranyl nitrate solution. The yellow precipitate was isolated by centrifugation and collected after drying. To prepare a uranyl carbonate solution, (NH4)4UO2(CO3)3 powder was directly dissolved in deionized water with pH 8.16. To prepare uranyl–AO/CO32- solutions, a desired concentration of an aqueous AO solution was added into uranyl carbonate solutions with a concentration of 200 mM AO. In each solution sample, UO22+ concentration was maintained at 40 mM. 2.2. EXAFS Methods. X-ray absorption spectroscopic were collected at the beamline 14 W1 of the Shanghai Synchrotron Radiation Facility with a Si(111) double crystal monochromator in

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transmission mode for the uranium L3-edge spectra. The electron beam energy of the storage ring was 3.5 GeV, and the maximum stored current was approximately 210 mA. Energy calibration was performed using a zirconium foil (~17,998 eV). Each sample was measured thrice, and the spectra were averaged. The uranium L3-edge EXAFS data were analyzed using the standard procedures in Demeter19. The double-electron excitations affect the EXAFS signal and can influence the result of data analysis20-23. Thus, in uranium L3-edge EXAFS experimental spectra the double-electron excitations were subtracted as a reflection of the data translated to the position in energy of the excitation using the standard procedures in Demeter. Figure S1 shows the uranium L3-edge EXAFS data before and after subtracting the double-electron excitation in k and R space, in which the feature at very low distances (R~1Å) improved obviously. Actually, the coordination numbers is expected to be influenced, while the bond length is little influenced by double-electron excitations. Theoretical EXAFS data were calculated using FEFF 9.024. Fitting procedure was performed on the k3-weighted FT-EXAFS from 2.4 to 18.4 Å-1. An R window of 1-4 Å was used for the fitting. The amplitude reduction factor S02 was fixed at 0.9 in EXAFS fits, and the shifts in the threshold energy ∆E0 were constrained to be the same value for all fitted shells. 2.3. DFT Methods. All density-functional calculations were implemented in Gaussian 09 program25 using the exchange–correlation hybrid functional B3LYP, 26,27 which has been widely utilized and proven to be sufficiently accurate for extensive systems. The cc-PVDZ basis sets were adopted for hydrogen, carbon, nitrogen, and oxygen atoms. Geometry optimizations were performed without symmetry restrictions. Frequency calculations were conducted to verify the structures at the energy minima. Solvation effects were considered by using the conductor-like polarizable continuum model (CPCM).28-31 Natural population analyses and Wiberg Bond

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Indices were calculated to understand the bonding of uranyl-AO/CO32- complexes based on the B3LYP results using the natural bond orbital method32, implemented in the NBO 3.1 program.33 Charge decomposition analysis (CDA) developed by Franking et al.34,35 was performed as implemented in the Multiwfn 3.3 software.36 3. RESULTS AND DISCUSSION 3.1 Local structure of uranyl ions in the uranyl carbonate solution

Figure 1. (a) Aqueous U(VI) speciation distribution as a function of pH at [U(VI)] = 40 mmol/L, [CO32-] = 120 mmol/L, and 25°C. (b) Experimental Fourier Transform at the uranium L3-edge EXAFS data of the uranyl carbonate solutions in the absence and presence of AO molecule and their corresponding fits in R space: -, experimental data; ●, theoretical fit. The pH-dependent uranium aqueous speciation distributions of the U(VI) –CO32- complexes in aqueous solutions were calculated by Visual MINTEQ ver. 3.037 with the stability constants from the literature38 and shown in Figure 1a. The number of carbonate ligands that bind to uranyl ions increases with the pH value. When pH < 5.5, UO2CO3 is dominant in the uranyl–carbonate

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complexes. Within the pH range of 5.5–6.5, the [UO2(CO3)2]2- species is main formation. When pH exceeds 7, the most of uranyl complexes exist in the form of [UO2(CO3)3]4-, which is the same as the existing form in seawater. To confirm this conclusion, EXAFS measurement was performed. Figure 1b plots the experimental Fourier transforms (FTs) at the uranium L3-edge of EXAFS data for uranyl carbonate solution in the absence and presence of AO molecule and their corresponding fits in R space. According to previous reports39-42, tricarbonate coordinated uranyl cations in the bidentate-coordinated motif in the [UO2(CO3)3]4− species, which can be verified by the EXAFS fits and the fitting results are given in Table 1. Table 1. EXAFS structural parameters of uranyl carbonate solutions in the absence and presence of AO molecule. Sample

Uranyl carbonate solution

Uranyl AO/CO32solution

a

Oax, Oeq

Bond Typea

CNb

R (Å)c

σ2 (Å2)d

Oax (SS)

2.0 ± 0.3

1.80

0.0017

Oeq (SS)

6.0 ± 0.4

2.45

0.0073

C(SS)

3.0 ± 0.5

2.92

0.0046

Odist (MS)

3.0 ± 0.5

4.18

0.0026

Oax (SS)

2.0 ± 0.2

1.80

0.0019

Oeq (SS)

6.0 ± 0.5

2.43

0.0080

C(SS)

2.2 ± 0.4

2.90

0.0031

Odist (MS)

2.2 ± 0.4

4.17

0.0024

R factor

0.01

0.01

and Odist refer to coordinated oxygen atoms on the axial plane, equational plane and

distant oxygen atoms from the carbonate ligands, respectively. SS: single-scattering path. MS: multiple scattering paths corresponding to the linear U-C-Odist arrangement. b CN: coordination number. cError: R ≤ ±0.02Å. dDebye-Waller factors. Error: σ2 ≤ ±0.0008 Å2.

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3.2 Uranium L3-edge EXAFS features in the uranyl carbonate solution with and without AO ligands. To simulate the effect of [UO2(CO3)3]4- speciation by AO functionalized adsorbed materials in seawater, we added the excess AO ligands into the [UO2(CO3)3]4- solution. Considering the extremely low U concentration in seawater, when AO is grafted on materials its effective concentration is relatively higher and thus present research is carried in highly concentration AO system. After adding the AO molecule into the uranyl carbonate solution, significant relative intensity improvement can be seen in the experimental EXAFS data in R space shown in Figure 1b. First, the intensity of the Oeq shell decreased, which implied larger disorder. Second, the intensity of the Odist shell decreased, which corresponded to decreasing coordination number. All these results were confirmed by quantitative fits shown in Table 1, which suggests coordinated carbonate molecule with well-symmetry were substituted by new ligands, i.e. AO. Figure 2 shows a comparison of the experimental EXAFS oscillation function for U L3-edge in the absence and presence of AO molecule solutions. Firstly, the main EXAFS oscillations of uranium L3-edge are in rough agreement in k space in the absence and presence of AO aqueous solutions. Secondly, distinct characteristics within the k range of 6-10 Å-1 in the EXAFS spectra of the uranyl ions in the CO32- and AO/CO32- solutions can be seen. As marked by the lines, we can observe that the increasing valley between peaks A and B and the decreasing intensity of peak C with the addition of AO ligands. The EXAFS oscillation is well-known to be the superimposition of individual scattering paths from ligand shell atoms; thus, pattern changes in EXAFS oscillations directly reflect local coordination structure in a uranyl system. Such changes may indicate that the new coordination complexes that are different from the pure tricarbonate complexes are formed. Table 2.

Local structure environment of (NH4)4UO2(CO3)3.

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Path

path ID

R

Rank

CN

P1

U-Oax

1.7913

100

2

P2

U-Oeq(CO3)

2.4415

P3

U-Oeq(CO3)

2.4644

P4

U-Oeq(CO3)

2.4671

P5

U-C(CO3)

2.8926

2 100

2 2 2

27.01 P6

U-C(CO3)

2.9172

1

Figure 2. Comparison of experimental uranium L3-edge k3-weighted EXAFS oscillation data after subtracting the double-electron excitation in the absence and presence of AO molecule solutions. Beneath the experimental data are the theoretical contributions of each individual path of standard tricarbonate uranyl species (ICSD-16093). 3.3 Scattering contribution of different paths to k-weighted EXAFS spectra. To understand the origin of feature changes in the k3-weighted EXAFS functions affected by AO ligands, we extracted the scattering contribution of different paths near the absorber atom based on the ideal tricarbonate uranyl model. As shown in Table 2, six paths were considered,

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including U-Oax. U-Oeq, and U-C, which significantly influence the patterns of experimental spectra. In Figure 2, we compared the k3-weighted experimental EXAFS function and the theoretical contribution of each individual path of standard tricarbonate uranyl species. According to the path analysis in k space, we ascribed peaks A and C to the contributions of U-6*Oeq (P2, P3, and P4) and U-3*C (P5 and P6), respectively, whereas peak B only originates from the contribution of U-2*Oax (P1). The results implied that peaks A and C will change if AO molecules substitute the carbonate ligands in the equatorial plane of uranyl compounds because of the different bond lengths of U-Oeq for carbonate and AO ligands and the missing U-C. And thus it is reasonable to believe that AO ligands replace the carbonate ligands that bind to uranyl ions during uranium extraction from seawater. 3.4 Determining the competitive effect of AO ligands in the uranyl carbonate solution. To verify the aforementioned speculation, we compared the structural changes under different modes when AO ligands gradually substitute carbonate ligands and calculated their corresponding EXAFS spectra. Due to that there is no standard structural model can be used for spectral simulation, DFT calcualtion was employed to obtain the optimized structure. In the Figure S2, we consider all possible binding motifs for AO ligands and uranyl cations, including (a) monodentate binding to oxygen atom of the oxime ligand, (b) bidentate chelation involving the oxime oxygen atom and amide nitrogen atom, and (c) η2 binding with the N-O bond. The results support that η2 mode is the most preferred binding motif in the uranyl-AO complex, which is consistent with previous reference11,12,13,15,17. Obtained stable structural models of the possible [UO2(CO3)3-x(AO)x](4-x)- complex are shown in Figure 3 and local structural details are listed in Table 3. According to the DFT/B3LYP calculations, the slightly decrease of the bond distance between uranium and axial oxygen (U-Oax) with growing of AO coordination number

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can be found, implying that AO ligands have comparable binding capacities with carbonate groups. In addition, several remarkable changes can be presented. (1) The bond length of U-O (CO32-) decreases by 0.02-0.03 Å, and the bond length of U-C (CO32-) bonds decreases with a reduced coordination number. (2) The bond length of U-N(AO) is obviously longer than the U-O(AO) on the equatorial plane; this result may be attributed to the different binding strengths of the oxime oxygen and oxime nitrogen atoms. (3) The average bond length of uranium-ligand (U-Leq) on the equatorial plane also decreases by 0.01-0.03 Å, which may be caused by the charge distribution around the uranyl center induced by the decreasing coordination number of U-C. Figure 3b shows the corresponding EXAFS oscillation spectra based on the optimized structural models. With the carbonate ligands replaced by the AO ligands, several significant changes occur: (1) the position of peak A shifts to a higher k. (2) The valley between peaks A and B significantly increases, which corresponds to the stronger intensity of peak B. (3) The intensity of peak C decreases. Such trends are consistent with the experimental results shown in Figure 2. Therefore we believe that the CO32- ions can be replaced by AO ligands from the [UO2(CO3)3]4- complex during uranium extraction from seawater. In addition, a comparison between the estimated distances from DFT and EXAFS helps to identify the intrinsic structure, which shows that η2 mode with one AO ligand substitute the carbonate molecule is the main existence form in our uranyl AO/CO32- solution. Table 3. Optimized distances (in Å) between uranium and ligand atoms, R(U-Oax) (axial oxygen), R(U-Leq) (equatorial ligands), and R(U-C) from CPCM DFT/B3LYP calculations of the [UO2(CO3)3-x(AO)x](4-x)- complexes.

Complexes

R(U-Oax)

R(U-Leq)

Average R(U-Leq)

(U-C)

CN

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U-Oc: 2.440, 2.440, [UO2(CO3)3]4-

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2.905

1.827 2.441, 2.441, 2.441,

2.440

2.905

6

1.828 2.442

2.906

U-Oc: 2.415,2.422,2.415, [UO2(CO3)2(AO)]3-

1.826

2.417

2.877 2.414

1.827

U-OAO: 2.372

6 2.877

U-N: 2.444 U-Oc: 2.394,2.394 [UO2CO3(AO)2]2-

1.825 U-OAO: 2.355,2.354

2.398

2.850

6

1.823 U-N: 2.448,2.441 [UO2(AO)3]a

Oax, Oc

1.819,

U-OAO: 2.330, 2.332, 2.332

1.820

U-N: 2.447 ,2.447,2.449

2.390

6

and OAO refer to coordinated oxygen atoms in uranyl, carbonate ligands and AO ligands,

respectively.

Figure 3. (a) Proposed structural models of the [UO2(CO3)3-x(AO)x](4-x)- complexes when AO ligands gradually substitute carbonate ligands that bind to uranyl ions; (b) Calculated k3-weighted EXAFS oscillations of the [UO2(CO3)3-x(AO)x](4-x)- species at uranium L3-edge, as well as the experimental pattern of uranyl carbonate solution for comparison.

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3.5 Metal-ligand bonding nature in the [UO2(CO3)3-x(AO)x](4-x)- complexes. To understand the nature of metal-ligand bonding in the [UO2(CO3)3-x(AO)x](4-x)- complexes, we first performed charge decomposition analysis at the B3LYP level of theory. Some relevant uranyl-AO/CO32bonding molecular orbitals (MOs) are illustrated in Figure 4. The shape of MOs involving the metal-ligand bonding change with the coordinated ligands. The MO plots provide a pictorial description of the U−O and U−N σ bond, which mainly originate from the interaction of U 5f/6d orbitals and N or O 2p orbitals. The compositions of the MOs that involve metal-ligand bonding undergo obvious changes with the replacement of carbonate ligands by AO ligands. As shown in Figure 4, introductioning AO ligands can increase the contribution of U 5f/6d orbitals to U-Leq bonding, which may be responsible for the excellent binding capability of AO ligands to uranyl even in the stable [UO2(CO3)3]4- solution.

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Figure 4. Main uranyl-ligand bonding molecular orbitals and their atomic orbital compositions in the [UO2(CO3)3-x(AO)x](4-x)- complexes.

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Table 4. Wiberg bond indices (WBIs) of U-N and U-O bonds and natural charge analysis based on the DFT/B3LYP methoda. ∆Q(CO32-)

Compounds

U-Oax

U-OC

[UO3(CO3)3]4species

2.09

0.60

Mono-AO

2.11

0.61

0.68

0.49

+0.97

+0.70

+0.77

bi-AO species

2.13

0.62

0.69

0.49

+0.95

+0.71

+0.80

tri-AO species

2.15

0.71

0.49

+0.95

+0.72

a

U-OAO

U-N

Q(U)

∆Q(AO)

+1.01

+0.75

Note: Oxygen atoms in uranyl, CO32-, and AO ligands are labeled as Oax, OC, and OAO,

respectively. ∆Q(AO/CO32-) indicates the natural charge change from a free AO/CO32- ligand to a coordinated AO/CO32- ligand. The bonding nature of these [UO2(CO3)3-x(AO)x](4-x)- complexes, including bond orders and atomic charges, have been investigated through a natural bond orbital

analysis at the

DFT/B3LYP level of theory, and the results are summarized in Table 4. The WBIs of the U−Oax bond order slightly increase as the coordinated AO ligands increase; this result is consistent with the slightly decreasing U−Oax bond lengths in Table 3. Clearly, the WBIs of U-OAO bonds are approximately 0.7, those of U-OC bonds are approximately 0.6, and those of U-N are approximately 0.5. These comparisons show the following bond order trend: U-OAO > U-OC > U-N, which indicates that AO ligand is comparable with carbonate molecule coordination. Furthermore, according to the natural population analysis, natural charges in the uranium atoms of these four complexes remain nearly unchanged because carbonate molecules are being replaced by AO ligands, which indicate a similar electron donation from AO ligands to the uranium as that from carbonate molecules. This result is also supported by the slight difference between ∆Q(AO) and ∆Q(CO32-), whose values show AO and CO32- coordination transfer

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∼0.7-0.8 e− to uranium. Although they have a slightly weaker electrostatic attraction than carbonate ligands, AO ligands demonstrate stronger binding capability to uranyl, in which a strong orbital interaction between U 5f/6d and (N,O)2p should be the key reason, as shown in Figure 4. 4. CONCLUSIONS In summary, the complexation mechanism of AO ligands to uranyl carbonate was systematically investigated through EXAFS experimental and theoretical analyses combined with DFT calculations. When the AO ligands was added into the uranyl carbonate solution, significantly different characteristics became evident within the k range of 6-10 Å-1 of the EXAFS spectra, which strongly correlates to the local structure around uranyl ions. With regard to the replacement of carbonate ligands by AO ligands, the trend of the theoretical spectra is exactly consistent with that of the experimental results, which supports the suggestion that carbonate ions can be substituted by AO ligands from the [UO2(CO3)3]4- complexes. The electronic mechanism was further studied by DFT calculation. According to the charge decomposition analysis, the contributions of U 5f/6d orbitals to the uranyl-AO bonding MOs increase with the introduction of AO ligands in the uranyl carbonate solutions. This strong covalent interaction should be responsible for the observation that AO ligands have stronger binding ability to uranyl than CO32- group. ASSOCIATED CONTENT Supporting Information. The Supporting Information is available free of charge on the ACS Publications website. Comparison of experimental uranium L3-edge k3-weighted EXAFS oscillation data and Fourier Transform data before and after subtracting the double-electron

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excitation. Optimized geometries and total energies after zero-point energy correction (in hartrees) [UO2(CO3)3-x(AO)x](4-x)- complexes. AUTHOR INFORMATION Corresponding Author *Email: [email protected]; [email protected] Notes The authors declare no competing financial interest. ACKNOWLEDGMENTS This work was partly supported by the Program of International S&T Cooperation, (2014DFG60230, ANSTO-SINAP), National Natural Science Foundation of China (Grants No. 11405254, 11575280, 21571185, 21306220, 91326105), Joint Funds of the National Natural Science Foundation of China (Grant U1232117). Strategic Priority Research Program of the Chinese Academy of Sciences (Grant No. XDA02040104), Knowledge Innovation Program of Chinese Academy of Sciences, and Youth Innovation Promotion Association (2014237), Chinese Academy of Sciences. We thank Christoph Hennig and Ling-Yun Jang for hints and fruitful discussions about the double-electron excitations.

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