J Mol Model (2014) 20:2398 DOI 10.1007/s00894-014-2398-y

ORIGINAL PAPER

Developing polarizable potential for molecular dynamics of Cm(III)-carbonate complexes in liquid water Riccardo Spezia & Yannick Jeanvoine & Rodolphe Vuilleumier

Received: 27 February 2014 / Accepted: 23 July 2014 / Published online: 3 August 2014 # Springer-Verlag Berlin Heidelberg 2014

Abstract In this work we have developed a polarizable potential to study Cm(III) forming complexes with carbonate anions in liquid water. The potential was developed by employing an extension of the procedure that we used to study the hydration of lanthanoids(III) and actinoids(III). Force field performances were benchmarked against DFT results obtained by both geometry optimization and Car-Parrinello molecular dynamics. With this polarizable potential, we run extended molecular dynamics simulations in liquid water from which we were able to identify structural and dynamical properties of such systems. In particular, water exchange dynamics were analyzed in detail. We obtained an average of three water molecules in the first shell of Cm(III) that show a relatively fast exchange dynamic (faster than for bare ions). Summarizing these results, we were able to draw an analogy to the results from the lanthanoid(III) series. In particular, it seems that Cm(III) behaves more like Nd(III) than Gd(III), as one would expect based on the recent hydration results and on f orbital occupancy.

This paper belongs to Topical Collection QUITEL 2013 R. Spezia : Y. Jeanvoine CNRS, UMR 8587, Evry Cedex, France R. Spezia (*) : Y. Jeanvoine Laboratoire Analyse et Modélisation pour la Biologie et l’Environnement, Université d’Evry Val d’Essonne, Bd F.Mitterrand, 91025 Evry Cedex, France e-mail: [email protected] R. Vuilleumier Départment de Chimie, Ecole Normale Supérieure, 24, rue Lhomond, Paris, France R. Vuilleumier UMR 8640 CNRS-ENS-UPMC, UPMC Univ Paris 06, 4 Place Jussieu, 75005 Paris, France

Keywords Actinoids . Carbonate complexes . Hydration structure and dynamics . Polarizable molecular dynamics

Introduction The current renewed interest in nuclear power technology and nuclear waste management is accompanied by the need for a fundamental understanding of the elements involved in the process and their behavior. Understanding the aqueous chemistry of actinoids (An) and lanthanoids (Ln) plays a key role in radioactive waste management [1]. Most of the heavier actinoids (from Am) are stable at the 3+ oxidation state in aqueous solutions [2]. However, since experimental studies of heavy actinoids are difficult due to their radioactivity and low abundance, lanthanoids are often used as their analogues [3–6]. This corresponds to the usual picture of f-elements behaving as hard cations. While hydration properties of lanthanoids were largely studied by both experimental and theoretical approaches [7–21], the understanding of how their actinoid analogues behave is a much more recent endeavor [22–32]. Recently, by combining EXAFS and polarizable molecular dynamics simulations, it was possible to point out a strong analogy between the two series, which explains why the separation process between different actinoids and lanthanoids is such a difficult procedure [33, 34]. These results confirm the simple picture for which the coordination of bare ions in water is essentially driven by their charges and ionic radii, corresponding to a hard-hard interaction. Based on this physical picture, we were able to build polarizable potentials for interaction of Ln(III) and An(III) in water based on a few quantum chemical calculations and linear extrapolation of Buckingham parameters [35, 36]. The same approach was shown to be applicable to study the hydration of Ln(III) carbonate complexes [37], providing results in agreement with DFT calculations and

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available experimental data [38–40]. Carbonate complexes are interesting due to their ubiquitous presence in geological media and because they have been proposed to be involved in a separation procedure for Ln(III) and An(III) ions in solution [41]. Classical (polarizable) molecular dynamics simulation is a technique that is well-fitted to study structural, dynamical, and thermodynamical properties of systems in liquid phase by correctly taking into account temperature effects and by performing (if simulation time is sufficient) a correct statistical sampling. For this one needs a well parametrized force field between An(III) and carbonates that is not present in the literature. We have thus investigated the possibility of using our procedure to generate a force field between Cm(III) and the carbonate anion in liquid water following the same approach used in our previous studies. Here DFT calculations (both static and dynamic, employing the Car-Parrinello method [42]) were used to parametrize the force field, using as a starting point our previous knowledge on such parameters and the relationship between them across the two series. Polarizable molecular dynamics simulations were then performed to study statistical properties of Cm(III)-carbonate complexes in liquid water; we focused our attention on water exchange dynamics that are related to time-resolved laser-induced fluorescence spectroscopy (TRLFS) experiments [5, 43, 44]. We have used Cm(III) since its f valence orbitals are half filled, 5f 7, and thus single determinant DFT can provide reliable results that we can use to benchmark our force field development. In this paper, we want to show how a polarizable force field can be successfully obtained by minimizing the number of quantum chemical calculations, thus opening the possibility to extend this procedure to the full series — for which ab initio calculations would be even more difficult and subtle due to the multi-reference nature of these systems.

Methods DFT calculations We have first performed geometry optimization (called static DFT calculations hereafter) of the complex formed by Cm(III) binding three carbonates in a bidentate fashion (see Fig. 1) using the B3LYP functional [45, 46]. For the Cm atom, we used the energy-consistent pseudopotentials (ECP) of the Stuttgart/Cologne group, which are semi-local pseudopotentials adjusted to reproduce atomic valenceenergy spectra. Amongst the available pseudopotentials, we chose the ECP60MWB [47–49] small core with 60 core electrons, also with multi electron fit (M) and quasi relativistic reference data (WB); we have used the ECP60MWB_SEG basis set for Cm. For C and O atoms we used the 6-31+G(d) basis set.

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Fig. 1 Chemical structure of the Cm(III)-carbonate complex

High spin state was considered corresponding to seven unpaired 5f electrons of Cm(III) — its electronic configuration is [Rn]5f 76d07s0. Initial structures were obtained from our previous study of Ln(III) carbonate complexes and then minimized, checking that the minimization result was a real minimum by frequency analysis (no imaginary frequencies). These calculations were performed with Gaussian 09 package [50]. Then, we performed DFT-based dynamics in liquid water by using the Car-Parrinello molecular dynamics (CPMD) method [42]. We immersed the Cm(III) tri-carbonate cluster (bidentate) in a box composed of 120 water molecules (using the initial box size equilibrated in our previous study [37]) corresponding to a box length of 15.52 Å (cubic box). The BLYP functional [46, 51] was employed with plane waves basis set and a cut-off of 110 Ry. For C, O, and H we have used standard Troullier-Martins semi-core pseudo-potentials [52] (PP) used previously in similar CPMD simulations [37, 53–55], that are standard pseudo-potentials available in CPMD libraries [56]. For Cm(III) we have used the semicore Troullier-Martins pseudo potential developed recently by us [32]. As detailed in the original work, we used 5f 76s 26p66d 0 as a reference configuration for Cm(III) outer orbitals. The 6s, 6p, 6d, and 5f orbitals were included in the PP with cut-offs of 1.33, 1.59, 2.36, and 1.20 a.u., respectively. We have considered the system in the high spin state (eigthfold multiplicity) within the local spin density. When using pseudo-potentials with plane waves, the semilocal KleinmanBylander form was used with the p channel as the local channel [57]. Molecular dynamics simulations were then performed using a fictious mass of 150 a.u. and a time-step for equation of motion numerical integration of 2 a.u. (=0.0484 fs). Nosé-Hoover thermostat chain [58, 59] was used with a target temperature of 300 K to simulate NVT ensemble. After an equilibration of 0.5 ps, a trajectory of 2 ps was accumulated to obtain average quantities. All CPMD simulations were performed by means of the CPMD code [60].

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Polarizable molecular dynamics simulations

The induced dipoles are obtained through the resolution of the self-consistent equations

Polarizable classical molecular dynamics (CLMD) simulations in liquid water were done as follows. A cluster composed of one Cm(III) and three carbonates, (CO3)2−, was immersed in a box composed of 216 water molecules. Box sizes have been obtained from our previous study on Ln(III) carbonate complexes, i.e., after an equilibration in the NpT ensemble to obtain box edges (see ref. [37] for further details). Thus, the same box edge of 19.5 Å was used in the present simulations. Note that we have shown that such a box is sufficient to obtain structural and dynamical properties of carbonate complexes [37], as well as for ions in liquid water, studied extensively by our group in the last years [35, 36, 61, 62]. We have used this box size to determine the best set of parameters that enable us to study Cm(III)-carbonate complexes in liquid water. Periodic boundary conditions (PBC) were applied to each simulation box in order to mimic bulk conditions. Long-range interactions were calculated by using the smooth particle mesh Ewald method [63]. The SHAKE algorithm was used to constrain bonds and angles of water molecules [64]. Once the system equilibrated at 300 K using velocity rescaling, we switch to the NVE ensemble. Trajectories were thus 3 ns long for each system. The total energy of our system is modeled as a sum of potential energy terms: V tot ¼ V elec þ V LJ þ V Buck ;

ð1Þ

where V elec is the electrostatic energy term composed of the solvent-solvent and solvent-solute interactions,

V

elec

" #  1 X qi q j 1 ¼ þ 3 −qi p j þ q j pi ⋅ rij þ pi ⋅ Tij ⋅ p j 2 ij rij rij þ

1X p ⋅ ðαi Þ−1 ⋅ pi : 2 i i

pi ¼ αi ⋅ Ei þ

! Tij ⋅ p j :

ð3Þ

i≠ j

The resolution of this self-consistent problem becomes rapidly extremely time consuming as the system grows. In order to reduce the computing time, we have used an alternative way of resolving such a problem for each time step of the dynamics. In particular, we have used a Car-Parrinello type of dynamics (extended Lagrangian) of additional degrees of freedom associated with induced dipoles [67]. Thus, the Hamiltonian of the system is now: H ¼ V tot þ

1X 1X mi v2i þ mpi v2pi ; 2 i 2 i

ð4Þ

where vpi is the velocity of the induced dipole pi treated as an additional degree of freedom in the dynamics, and mpi is its associated fictitious mass, identical for each atom. The dynamic of the induced dipole degrees of freedom is fictitious, such that it only serves the purpose of keeping the dipoles as close as possible to their values that would be obtained through the exact resolution of self-consistent equations. Simulations were performed using the velocity-Verlet algorithm to numerically solve the equations of motion using a 1 fs time step. More details are reported in our previous studies [37, 61]. V LJ is the standard Lennard-Jones non-bonding potential and it is used for water-water, water-carbonate, and carbonatecarbonate interaction. For water we used TIP3P parameters [68] and for carbonate those present in the literature [69]. V Buck is the Buckingham potential, used for Cm-water and Cm-carbonate interactions:

ð2Þ Following Thole’s induced dipole model [65], each atomic site i carries one permanent charge qi and one induced dipole pi associated with the isotropic atomic polarizability tensor, αi. The polarization catastrophe is avoided using a screening function for the dipole-dipole interactions at short distances. Isotropic polarizabilities are assigned at each atomic site. Here we used atomic polarizabilities determined by van Duijnen and Swart [66] for water (O: 0.85 Å3 and H: 0.41 Å3), and what we have previously obtained for Cm(III) [36], (1.238 Å3) and carbonates [37] (O: 0.71 Å3 and C: 0.52 Å3). For charges we used modified TIP3P charges for water [61], +3 for Cm(III) and our recently developed charges for carbonate [37].

X

V Buck ¼

N water  X i

AW e−bw ri þ

 X  N carb  CW CC −bC r j A e þ þ ; C r9i r9i i ð5Þ

where i and j run over water and carbonate molecules respectively. For Cm-water we used the same parameters recently developed [36], while for Cm-carbonates we have developed parameters of Cm-O interaction similarly to that done for carbonates. Three sets of parameters were used (the rationale of them is explained in the Force field development and validation section). All parameters used in this work are summarized in Table 1.

2398, Page 4 of 9 Table 1 Force fields parameters used. POL1, POL2, and POL3 are the three different Buckingham parameter sets for Cm-carbonate interaction. Polarizability, α, is in Å3, εLJ in kcal mol−1, σLJ in Å, bW and bC in Å−1, and CW and CC in kcal mol−1 Å6

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q

α

εLJ

σLJ

bW

CW

bC

CC

Cm POL1 POL2 POL3 OW

+3.000 – – – −0.658

1.238 – – – 0.852

– – – – 0.155

– – – – 3.165

3.551 – – – –

7980.60 – – – –

– 3.579 3.615 3.551 –

– 8192.54 7576.60 8192.54 –

HW CC OC

0.329 0.781 −0.927

0.414 0.520 0.710

– 0.120 0.154

– 3.296 3.165

– – –

– – –

– – –

– – –

All CLMD simulations were performed with MDVRY package [70].

X W ðCmÞ =X W ðGd Þ ¼ γ X rðCmÞ =rðGd Þ

ð8Þ

X C ðCmÞ =X C ðGd Þ ¼ γ X rðCmÞ =rðGdÞ ;

ð9Þ

Results Force field development and validation In our previous works, we have shown that using polarizable potential combined with Buckingham potential allows us to derive interaction parameters for elements belonging to the lanthanide and actinide series; we use the simple physical consideration that ionic radii decrease across the series. Using ionic radii in developing parameters for Buckingham potential was already done by Madden and co-workers in the case of molten salts [71–73]. We have shown that for b and C parameters of Eq. 5, the following simple linear relationships hold: bnew ¼ bref þ k b Δr

ð6Þ

C new ¼ C ref þ k C Δr;

ð7Þ

where bref and Cref are two reference values obtained, for example, from quantum chemical calculations, kb and kC are two proportionality constants and Δr is the difference in ionic radii. These simple relationships previously allowed us to obtain new parameters for atoms in the series where the ionic radii were known, the Ln(III) and An(III) series in water [35, 36]. They were also extended to study Ln(III)-carbonate complexes in water [37]. Having experimentally determined ionic radii for both the Ln(III) and An(III) series, we can now build three different force fields that we will test against DFT results to assess their validity. A first, labeled POL1, is based on the assumption that 5f 7 and 4f 7 ions behave similarly, such that the ratio between Gd and Cm parameters is a function of the ration between their ionic radii:

where XW(Cm) and XW(Gd) are Buckingham parameters (b or C) of ion-water interaction, for Cm and Gd respectively, XC(Cm) and XC(Gd) are the interaction parameters for carbonates, r(Cm) and r(Gd) are ionic radii and γX is a proportionality factor. The second potential, POL2, assumes that Cm(III) behaves like Gd(III) in the interaction with carbonates (i.e., same Buckingham parameters) and that the only difference in interaction with carbonates is in the difference in ionic polarizability. The third potential, POL3, uses bC value of Cm(III)-water potential, while CC uses the same as POL1. This choice corresponds to a short range repulsion equal to Cm-water, while the long range attractive contribution behaves differently and these parameters can be extrapolated by our previous knowledge of ionic radii (i.e., the same argument that we used to build the whole POL1 force field). Results obtained are summarized in Table 2, where we used the Cm-OC(n) distances (OC(n) are the oxygen atoms of carbonate that point toward Cm) to assess the quality of the model. This was also done for Ln(III)-carbonates [37]. We note that both POL1 and POL2 simulations underestimate Cm-OC(n) distance with respect to DFT results (geometry minimization and DFT dynamics provide the same CmOC(n) distance). POL3 is in good agreement with DFT results, Table 2 Structural results for [Cm(CO3)3]3− complexes. Distances are in Å, angles in degrees

B3LYP CPMD POL1 POL2 POL3

Cm-OC(n)

Cm-C

Cm-OC(f)

OC(n)-C-OC(n)

Cm-C-OC(f)

2.409 2.409 2.365 2.348 2.411

2.860 2.866 2.637 2.619 2.692

4.122 4.115 3.895 3.874 3.952

114.0 115.5 127.0 126.8 127.4

179.9 167.1 172.0 172.2 171.8

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using the Cm-OC(n) distance as the key parameter to asses the quality of the force field. The other distances, Cm-C and CmOC(f) (OC(f) are the oxygen atoms of carbonates far from Cm) are also improved, but still provide shorter values. This is due, as for Ln(III)-carbonate complexes, to the opening of carbonate as shown by the OC(n)-C-OC(n) angles — this is largely discussed in our previous work ref. [37]. Here our aim was to point out that it is possible to extend our Ln(III)-carbonate force fields to Cm(III) in order to have information on how the cluster behaves in water. The POL3 force field, that is in better agreement with respect to DFT results, will be used in the following section to study properties of Cm(III)-carbonate complexes. Fig. 3 Time evolution of C-Cm-C angles obtained by six independent polarizable molecular dynamics simulations with POL3 force field

Complex fluctuations and water exchange dynamics Structural results in liquid water are determined by the radial distribution functions (RDFs) and (eventually) angular distributions (or evolution of some angle as a function of time). Here, we first report in Fig. 2 Cm(III)-carbonate ions RDFs obtained by the POL3 force field, showing that even in the case of strong interactions, simulations in liquid phase at room temperature provides not negligible, even if relatively small, distance fluctuations. We obtained, from RDF peaks, full width at half maximum (FWHM) values of 0.14, 0.125, and 0.165 Å for Cm-OC(n), Cm-C, and Cm-OC(f) distances respectively. While for Cm-OC and Cm-C distances we obtained such small fluctuations, the overall complex showed an extended dynamic that we can quantify by plotting the C-Cm-C angles, shown in Fig. 3. Angles can vary about 100°, corresponding to a breathing motion of the complex (when one angle changes from ∼80° to ∼180° another one changes from ∼180° to ∼80°). Cm-OW distance (OW are water molecule oxygen atoms) fluctuates more (FWHM=0.23 Å), as shown in Fig. 4, since

Fig. 2 Radial distribution function (RDF) between Cm(III) and carbonate atoms, as obtained from POL3 polarizable molecular dynamics simulation

interaction with water molecules is weaker and there are several exchanges between water molecules in the first Cm(III) shell and the bulk. This is reflected in Cm-OW RDF by the fact that it does not go exactly to zero. Furthermore, integrated RDF provides the coordination number as a function of the distance, and shows a slightly increasing plateau between 2.93 and 3.84 Å corresponding to three water molecules. Note that a very strong structuration will correspond to a plateau almost parallel to the x axis. Based on Cm-OW RDF, we could define a first hydration shell and then count the number of water molecules, nH2O, in the first shell of Cm(III) as a function of time. In order to have more statistics on exchange dynamics, we have performed six independent simulations of 3 ns each, and the results for nH2O as a function of time are shown in Fig. 5. The most probable nH2O is 3, reflecting the integrated RDF result, but we obtain many exchanges: one water molecule can leave and one water molecule can enter, where the latter is more probable than the former. In Table 3 we summarized Cm-OW distances

Fig. 4 Cm-OW radial distribution function (RDF) obtained from POL3 polarizable molecular dynamics simulations. In dashed we also show the coordination number

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Fig. 5 Time evolution of the number of water molecules in Cm(III) first shell as obtained by six polarizable molecular dynamics simulations with POL3 force field

Fig. 6 Snapshot from polarizable molecular dynamics simulation of Cm(III) with three carbonates in liquid water. Water molecules in first shell are also highlighted

(maximum of RDF) and the average number of water molecules obtained for each simulation. By averaging all nH2O obtained from different POL3 simulations we obtain =3.14. In Fig. 6 we show a snapshot for the POL3 simulation that is representative of what looks like the Cm(III)-carbonate complex in liquid water. Based on these six trajectories, we calculated the mean residence time (MRT) by using the Impey method [74]. At this end, we calculated the average nion(t) as follows:

This function is then fitted with a mono-exponential function,

t X 1 X p ðt n ; t; t *Þ; Nt n¼1 j j

N

nion ðt Þ ¼

ð10Þ

where Pj(tn,t;t*) is a property of the water molecule j. Its value is 1 if the molecule j is in the first coordination shell of the ion at both time steps tn and t+tn and if it stays there for any continuous period longer than t* (here we used t*=2ps as in the original paper of Impey — used in our previous work of Ln(III)-carbonates [37]). Its value is 0 otherwise; Nt is the number of steps.

nion ðt Þ ∼ nion e−t=τ ;

ð11Þ

where nion =, obtaining residence time τ. Results on MRT are summarized in Fig. 7 and corresponding τ are reported in Table 3, where we also show results for POL1 and POL2 force fields. We should note that, even with some differences between the six trajectories, POL3 provides exchange times that are clearly slower than those of POL1 and POL2. Averaging all POL3 results, we obtained a time of 176.421 ps; excluding the two extreme values (POL3-2 and POL3-4) the resulting value is very similar, 167.339 ps. Note that the average exponential, using nion =3.14 and τ=176.421, shown in the same Fig. 7, fits well with all curves but POL3-2.

Table 3 Structural and dynamical results for water molecules in first Cm(III) hydration shell of for [Cm(CO3)3]3− complexes. Distances are in Å, mean residence time, τ, in ps

POL1 POL2 POL3-1 POL3-2 POL3-3 POL3-4 POL3-5 POL3-6

Cm-Ow



τ

2.67 2.68 2.65 2.66 2.65 2.66 2.65 2.65

3.00 2.88 3.19 3.15 3.17 3.22 3.17 2.94

90.467 72.982 157.416 232.179 180.101 156.989 166.124 165.716

Fig. 7 Time evolution of nion(t) as defined in Eq. 10 obtained from six polarizable molecular dynamics simulations using POL3 force field (black lines). In red line we show the exponential fit (Eq. 11) done for each individual curve and in green dashed line we show the average decay function (Eq. 11 using < nion > =3.14 and τ=176.421 ps)

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Inspecting the POL3-2 behavior in more detail, we notice that the fast kinetics (up to about 150 ps) is well represented by the average exchange rate, while the slow part (tail of the curve) behaves differently. Results can be compared with what was obtained for Ln(III)-carbonate systems. Cm-OW distance is similar to what was obtained for Sm and Eu, but other parameters (nH2O and τ) are similar to Nd results [37]. Furthermore the Cm-OC(n) distance is very similar to what was obtained for Nd(III)carbonate complex.

Conclusions In this paper, we have developed a polarizable force field to study Cm(III) with carbonates in liquid water. Based on simple physical considerations, we have tested some force field sets against DFT results, both statical (geometry optimization) and dynamical (Car-Parrinello simulations). One force field was in particular in agreement with DFT results and thus was used to further study structural and dynamical properties of the Cm(III) tricarbonate complex in liquid water. We found, similarly to Ln(III)-carbonate complexes, that water molecules can enter in the first hydration shell — three in the case of Cm(III) on average — with a relatively fast exchange rate. The whole structural and dynamical properties of the Cm(III)-carbonate complex allow us to compare these results with the Ln(III) series complexes; they seem to be similar to Nd(III) ones. This suggests that in the case of complexes with carbonate we cannot use Ln(III) atoms as analogues to their An(III) vertical counterparts. Gd(III), 4f 7 ion, does not have the same properties of Cm(III), 5f 7 ion, when forming complexes with carbonates; Cm(III) shows similar properties to Nd(III), 4f 4 ion. This is probably due to the greater softness of the carbonates with respect to water, since Cm(III) hydration properties are more similar to Gd(III) [8]. The possibility of developing a polarizable force field based on simple physical considerations (i.e., the behavior of ionic radii along the series), can open the way of studying the whole series by molecular dynamics in liquid water, and thus establish a stronger relationship with Ln(III) ions. These results could then be compared with wave-function properties of Ln(III) and An(III) carbonate complexes to have the full picture; this is currently under investigation by our group. Acknowledgments We thank ANR 2010 JCJC 080701 ACLASOLV (Actinoids and Lanthanoids Solvation) for grant support and Grand Equipement National de Calcul Intensif (GENCI) (grant x2013071870) for generous allocation of computing time. We thank Mrs. Elizabeth A. Kish for careful reading of the manuscript.

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