Density Functional Theory Calculations of the Redox Potentials of

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Density Functional Theory Calculations of the Reduction Potentials of Actinide (VI) / Actinide (V) Couple in Water Helen Mary Steele, Dominique Guillaumont, and Philippe Moisy J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/jp401875f • Publication Date (Web): 19 Apr 2013 Downloaded from http://pubs.acs.org on April 23, 2013

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The Journal of Physical Chemistry

Density Functional Theory Calculations of the Reduction Potentials of Actinide (VI) / Actinide (V) Couple in Water Helen M. Steele*, Dominique Guillaumont and Philippe Moisy CEA, Nuclear Energy Division, RadioChemistry & Processes Department, F-30207 Bagnols sur Cèze, France DFT, Actinyl, Redox Potentials.

ABSTRACT The measured reduction potential of an actinide at an electrode surface involves the transfer of a single electron from the electrode surface on to the actinide centre. Before electron transfer takes place the complexing ligands and molecules of solvation need to become structurally arranged such that the electron transfer is at its most favourable. Following the electron transfer there is further rearrangement to obtain the minimum energy structure for the reduced state. As such there are three parts to the total energy cycle required to take the complex from its ground state oxidized form, to its ground state reduced form. The first part of the energy comes from the structural rearrangement and solvation energies of the actinide species before the electron transfer or charge transfer process, the second part, the energy of the electron transfer

*

Corresponding author, E-mail : [email protected]

ACS Paragon Plus Environment

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and the third part, the energy required to reorganise the ligands and molecules of solvation around the reduced species. The time resolution of electrochemical techniques such as cyclic voltammetry is inadequate to determine to what extent bond and solvation rearrangement occurs before or after electron transfer, only for a couple to be classed as reversible it is fast in terms of the experimental time. Consequently, the partitioning of the energy theoretically is of importance to obtain good experimental agreement. Here we investigate the magnitude of the instantaneous charge transfer through calculating the fast one electron reduction energies of AnO2(H2O)n2+, where An = U, Np and Pu, for n= 4,5,6 in solution without inclusion of the structural optimization energy of the reduced form. These calculations have been performed using a number of DFT functionals, including the recently developed functionals of Zhao and Truhlar. The results obtained for calculated electron affinities in the aqueous phase for the AnO2(H2O)52+/+ couples are within 0.04V of accepted experimental redox potentials, nearly an order of magnitude improvement on previous calculated standard potentials E0 values, obtained using both DFT and high level multireference approaches. 1. Introduction Understanding electrochemical processes is essential if we are to be able to predict most aspects of fundamental actinide chemistry1 and importantly those relating to control within chemical separations and the environment. For actinide systems the accurate theoretical calculation of redox potentials remains one of classical chemistry’s most important technical challenges. This is due in part to the complexity of the electrochemical process, often the size of the chemical system and the necessity for thermodynamic accuracy. These challenges are exacerbated due to the large number of electrons, characteristic of actinide systems for which correlation and the effects of relativity must be included.


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The electrochemistry of the actinyl moiety AnO2(H2O)52+/+, where An = U, Np and Pu was first investigated over a decade ago by Hay et al.,2 with improvements in the modelling of solvation, DFT functionals and the inclusion of strong correlations being responsible for significant improvements in the performance of single reference calculations,3,4,5 with more accurate results typically being obtained using a multireference approach for the electron affinity.6 Theoretical calculations of redox potentials generally utilise a free energy type cycle (Figure 1) and for organic systems such an approach gives good experimental agreement, within 10’s of mV’s7. Results for transition metals8,9 and the actinyl moiety typically have not shown the same close experimental agreement. Recently published calculations of the actinyl couple have demonstrated improvements for actinyl systems with an average experimental agreement of ~0.2V for both DFT4 and CASPT26 calculations. .

Ox(g) +

e

∆Gs(ox)

Ox(aq) +

∆G0EA(gas)

-

∆Gs(red)

=0

e-

Red(g)

∆G0(solv)

Red(aq)

Figure 1: Schematic free energy cycle used to calculate redox potentials E0.


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Within this free energy cycle, ∆Gs(ox) and ∆Gs(red) are the solvation energies of the structurally and electronically optimized, oxidized and reduced species. The electron affinity (EA) is calculated as the change in energy in forming the reduced species at the same geometry as the oxidized species, as such is typically accompanied by vibrational excitation, but such vertical EA’s being the most probable type of electron transfer.10 Reduction energies calculated using such a free energy cycle have similar contributions in magnitude from, the EA of the actinyl centre and the solvation energies of both the oxidized and reduced complexes.

Under a non adiabatic regime electron or charge transfer process measured at an electrode surface may have been proceeded by, and/or followed by optimization of the complex and any associated water molecules of solvation. At the point of electron transfer, the geometry of the oxidized species and associated solvent configuration will contain aspects of both the molecular configurations of the oxidized and reduced forms11, the extent to which it resembles either being so far undetermined. To obtain such a solute and solvent configuration, composed of both the oxidized and reduced forms would be technically challenging. Therefore, as a preliminary starting structure for the electron or charge transfer we have used the optimized oxidized structure, due to the results obtained further starting structures were not investigated.

Aqueous actinyl geometry optimization is a function of both fast and slow processes, including optimization of the strongly bound actinyl oxygens and of the more weakly bound equatorial ligands, plus reorganization of the bulk water molecules of solvation. Such a range in bond strengths would be expected to have different time frames and combined with the effects of mass transfer and diffusion at the electrode the time dependence is difficult to ascertain12, potentially


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leading to discrepancies when being compared to values calculated using the specific break down of energies shown in Figure1.

In this paper we present results for non adiabatic electron affinities calculated in the aqueous phase without structural optimization of the reduced form, using the optimized actinyl (VI) structure for the starting point of the electron/charge transfer. The calculated energy is obtained using a very similar approach to that shown in Figure 1, but the solvation energy of the reduced species is calculated at the geometry of the oxidized complex, leading to EA’s calculated in the aqueous phase, the results obtained are compared to standard potentials (extrapolated at zero ionic strength) for the AnO2(H2O)n2+/+ where An=U, Np and Pu, for n=5 in the equatorial plane, the values obtained for n=5 are subsequently compared with calculated values for n=4 and 6.

2. Theoretical Calculations

All calculations were performed using Gaussian 0913 code, using a number of well established density functionals including the hybrid B3LYP and the PBE functionals14 which have previously been shown to perform well in redox calculations3. The recently developed functionals, M06L which contains no HF exchange and the meta hybrid M06 functional, recommended for transition metals of Zhao and Truhlar15 were both assessed.

For U, Np and Pu the ECP60MWB8SEG basis set16 was implemented along with the small core MWB60 effective core potentials pseudopotential from the Stuttgart group.17 All light elements were represented with a relatively small 6-31g* basis sets. Calculations using further polarisation and diffuse functions for the light O and H elements did not show any improvements


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in test calculations. Due to the computational efficiency of the DFT formalism symmetry constraints on the orbitals were not required, but all structures were given a starting Dnh symmetry, where n=4,5 and 6. Applying symmetry constraints on the orbitals was shown to have a small but non negligible effect (~0.05 eV) on the calculated redox couples. A significant number of f-orbital configurations for the occupied orbitals were assessed in order to obtain the minimum energy structures of both the oxidized and reduced forms through a number of orbital rotations prior to optimization. All potential spin projector quantum numbers, Ms, were assessed to find the lowest energy spin state with the initial assumption of the highest spin projector quantum number being valid in all cases. There were no contributions from other potential electron configurations as such spin contamination (Ms(Ms+1)) was not an issue for any of the calculations.

Primary coordination around the linear actinyl unit is satisfied through the inclusion of 4-6 explicit equatorial water molecules, as such this part of the solvation is not included in the reported solvation energies and considered part of the solute, this complex is placed within an SCRF PCM18 continuum model with no further explicit water molecules being included. All oxidized AnO2(H2O)n2+ geometries were optimized in the gas phase and then re-optimized in the presence of the PCM solvation model. UFF radii were used to obtain the solute cavity with a scaled Van der Waals surface and the standard water dielectric was used. Vibrational analysis of the oxidized species was preformed to ensure they were true minima and to asses the affect on the actinyl bond due to changes in the number of equatorial ligands. Changes to the thermal contributions and zero point energies are very small compared with the EA and solvation energies, being between 2 to 3 orders of magnitude smaller and therefore were not included.


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Previously, the ad hoc addition of both multiplet and spin orbit corrections of Hay et al.2 have been seen as crucial in reproducing good agreement with experimental results and trends. However for the calculation of EA’s in the aqueous phase we have only included the spin orbit correction. It has been highlighted previously3 that calculations based on the fixed geometry bare actinyl ions may not be representative of the fully hydrated actinyl complex and that the inclusion of the multiplet correction is only possible and justified if HF based SO-CI results can be transferred directly to DFT calculations, therefore assuming the complete neglect of the multiplet part in the correlation energy, as such this correction has not been included. To enable direct comparison with experimentally measured redox potentials the EA’s calculated in the aqueous phase presented are referenced to the SHE electrode and accordingly a value of 4.44 V 19 is used to adjust the absolute values.

3. Results and Discussions

Structures of AnO2(H2O)52+ were fully optimized using four DFT functionals (B3LYP, PBE, M06L and M06) in both the gaseous phase and with a PCM solvation model for the oxidized An(VI) complexes without applying any symmetry constraints. A broken symmetry pseudo D5h structure was used as an initial starting structure. The minimum energy structures obtained for Np had 4 vertical and 1 planar water molecule in the equatorial plane, to a lesser extent the Pu structure also had a single non vertical water molecule but it was not fully rotated into the horizontal plane whereas the local minimum structures for the U structure had a more pronounced pseudo D5h structure with all equatorial water molecules in the vertical plane. When compared with experimentally measured bond lengths,20 all gas phase calculated bond lengths are systematically shorter by between 0.01 and 0.05Å, with the largest


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difference found for the Pu actinyl bonds calculated using the PBE and M06 functionals. When calculated in conjunction with the PCM solvation model experimentally measured bond lengths are reproduced well.3,20 The error in the calculated actinyl bond An=Oyl is between 0.00 and 0.04 Å and for the calculated An-H2O bond lengths a maximum overestimation error of 0.05 Å (These structures are available in the supporting literature).

Calculated and Experimental Redox Potentials. EA’s in the aqueous phase have been calculated for the AnO2(H2O)52+/+ (An= U, Np and Pu) couples, these are given in Table 1. These values are compared with reported values calculated thermodynamically following the approach described in Hay et al.2 and experimentally measured redox potentials, E0’s.

It is difficult to determine accurately electrochemical data in dilute solutions, therefore formal potential measurements are normally made at relatively high ionic strength and corrected (for which there are a number of accepted techniques) to obtain the correct activity coefficient and often further temperature correction may be required to enable a reference to standard state conditions. The OECD (Organisation for Economic Co-operation and Development) recommended values for the U, Np and Pu standard potentials E0 (extrapolated at zero ionic strength), are 0.0878 ±0.0013 V/SHE, 1.159 ±0.004 V/SHE and 0.936 ±0.0005 V/SHE respectively21(c). The apparent (formal) standard potential E’0 (HClO4 1M) used by OECD for the zero ionic strength value for U are extrapolated from formal potentials measured between 0.10M – 3.05M NaClO4, which it reports range between 0.067 – 0.074 V/SHE. Within the OECD review for U in 1M NaClO4 the reported formal potentials vary between 0.06 and 0.081 V/SHE. For Np formal potentials measured under standard conditions in 1M HClO4 range between 1.1361 – 1.140 V/SHE with the formal potential used by the OECD review being 1.137 V/SHE.


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For Pu reported formal potentials under standard conditions range from 0.912 – 0.941 V/SHE with a value of 0.913 V/SHE being used. Such a range in formal potentials may in part be attributed to differences in experimental conditions and techniques used (polarography, voltammetry and flow coulometry). Another set of apparent standard potentials are reported in J. Phys. Chem. Ref. Data22 and are given as 0.160, 1.236 and 0.966V/SHE and there is a further comprehensive IUPAC review by Kihara23 for the U, Np and Pu (VI/V) couples for which standard potentials E0 in acidic media are reported as 0.05-0.163, 1.15-1.236 and 0.931.013V/SHE for U, Np and Pu respectively. These reviews show a variation in experimental actinyl(VI/V) in water apparent standard potentials and further show the scarcity of data particularly with respect to U, Np and Pu (VI/V) couples under non acidic conditions. Likewise the value of the absolute potential of the SHE remains under scrutiny.24 The results presented within this paper are within experimental accuracy of the accepted experimental values and accordingly actual trends are as important as absolute values.

Table 1: Electron Affinities calculated in the aqueous phase for AnO2(H2O)52+/+ (An= U, Np and Pu) including the spin orbit corrections of Hay et al2. Comparison has been made with respect to experimental measured values21c and thermodynamically calculated values using CASPT2, M06L and PBE-Prioda approaches. MUSE (Mean Unsigned Error) values are calculated with respect to standard potential (V). B3LYP

PBE

M06

M06L

CASPT26 M06L4

Prioda3

Std. potential E0 21c


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U

-0.024

0.052

0.136

0.136

0.00

-0.54

-0.51

0.088

Np

1.177

1.056

0.989

1.227

1.53

1.33

0.87

1.159

Pu

1.232

1.379

1.374

0.946

0.73

0.49

0.43

0.936

MUSE

0.13

0.19

0.22

0.04

0.22

0.42

0.46

Using this approach, which does not include any energetic contributions from the reorganisation of the reduced species in solution gives EA’s calculated in the aqueous phase values for all functionals in close agreement with standard potential values for the three actinyl couples.

The M06L functional has a very good 0.04 V MUSE, accuracy comparable to experimental error for all three actinyl couples. This is a smaller MUSE value than that reported by Austin et al4 (0.42 V) using the same DFT functional, pseudopotentials and basis sets and compares well with the 0.22 V MUSE value reported using the multireference CASPT2 approach6. For the PBE functionals, EA’s calculated in the aqueous phase have a MUSE of 0.19 V compared to 0.46 V thermodynamically calculated using the four component Prioda-PBE code3. For the M06 functionals a MUSE of 0.22 V is obtained similar to the 0.23 V calculated by Austin et al4. These authors undertook an extensive study of DFT functionals and explicit water molecule configurations and found that the M06 functional gave the closest experimental agreement. The B3LYP results presented in this paper show a slight improvement with respect to the results obtained for both the PBE and M06 functionals with a MUSE of 0.13 V, but the Np and Pu couples being incorrect in their relative values. These B3LYP results also compare favourably with a previous reported value of 0.46 V by Austin et al4.


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The energetic contributions to the EA’s calculated in the aqueous phase can be broken down into three, these contributions are given in the tables below and subsequently discussed: the EA calculated in the gaseous phase (Table 2); the solvation energy of the optimized oxidized complexes (Table 3(a)) and the solvation energy of the reduced complexes at the geometry of the oxidized structure (Table 3(b)).

Table 2: Electron affinities (EA’s) calculated in the gas phase including spin orbit corrections,2 for the B3LYP, PBE, M06 and M06L functionals for AnO2(H2O)52+/+. Comparison of EA’s calculated in the gaseous phase for the M06L functional and the standard potential E0’s are given, both calculated and experimental values are normalised with respect to the Np(VI/V) couple (V).

EAgas

Normalised with respect to Np

B3LYP PBE

M06

M06L

M06L

Std potential E0 21c

U

-9.55

-9.36

-10.07

-9.75

-1.16

-1.07

Np

-10.82

-10.72

-10.68

-10.91

0.00

0.00

Pu

-11.25

-11.12

-11.31

-10.66

-0.25

-0.22


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Table 2 shows EA’s calculated in the gaseous phase for the DFT functionals studied and the normalised values relative to Np for the M06L functional and standard potential E0’s. Through normalisation with respect to Np of both the standard experimental potential E0’s and M06L calculated gaseous phase EA’s, a very good correlation is demonstrated for this functional. However, there is significant variation in the calculated gaseous phase EA’s, varying by over 0.5V for both U and Pu. For the B3LYP, PBE and M06 functionals the relative ordering of the Pu and Np is incorrect. This disparity in ordering results from the relatively large Pu EA’s calculated in the gaseous phase using those functionals (Table 2), but for all functionals the EA calculated in the gaseous phase for U is correctly calculated as the smallest.

With the notable exception of the M06L functional, reproduction of the experimental Pu redox couple does not closely agree with experimental values (Table 1). This difference appears to originate in the Pu EA’s calculated in the gaseous phase using those functionals (Table 2) which are more negative than the corresponding M06L values. The error in the Pu values may have arisen from using a single reference DFT. The correct trend (U

Density Functional Theory Calculations of the Redox Potentials of

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