Molecular dynamics and free energy calculations of dicyclohexano-18-crown-6 diastereoisomers with Sm²⁺, Eu²⁺, Dy²⁺, Yb²⁺, Cf²⁺ and three halide salts in THF and acetonitrile using the AMOEBA force field

Abstract

With the continual development of lanthanides (Ln) in current technological devices, an efficient separation process is needed that can recover greater amounts of these rare elements. Dicyclohexano-18-crown-6 (DCH18C6) is a crown ether that may be a promising candidate for Ln separation, but additional research is required. As such, molecular dynamics (MD) simulations have been performed on four divalent lanthanide halide salts (Sm²⁺, Eu²⁺, Dy²⁺ and Yb²⁺) and one divalent actinide halide salt (Cf²⁺) bound to three diastereoisomers of DCH18C6. Dy²⁺, Yb²⁺, Cf²⁺, DCH18C6 and tetrahydrofuran (THF) solvent were parameterized for the AMOEBA polarizable force field for the first time, whereas existing parameters for Sm²⁺ and Eu²⁺ were utilized from our previous efforts. A coordination number (CN) of six for Ln²⁺/An²⁺-O solvated in THF indicated that the cations interacted almost entirely with the oxygens of the polyether ring. A CN of one for Ln²⁺/An²⁺-N solvated in acetonitrile for systems containing iodide suggested that the N atom of acetonitrile was competitive with I⁻ for cation interactions. Fluctuation between five and six CNs for Dy²⁺ and Yb²⁺ suggested that although the cations remained in the polyether ring, the size of the ring may not be an ideal fit as these cations possess comparatively smaller ionic radii. Gibbs binding free energies of Sm²⁺ in all DCH18C6 diastereoisomers solvated in THF were calculated. The binding free energy of the cis-syn-cis diastereoisomer was the most favorable, followed by cis-anti-cis, and then trans-anti-trans. Finally, two major types of conformation were observed for each diastereoisomer that were related to the electrostatic interactions and charge density of the cations.

Graphical Abstract

1. Introduction

The macrocyclic polyether compounds, known as crown ethers (CE), have been an interesting topic in chemistry for decades. CEs can bind both positively and negatively charged ions or compounds making them very attractive for multiple applications. CEs can encapsulate cations as a consequence of electrostatic interactions between the cation and the oxygens on the polyether ring allowing for the solvation of the cation into non-polar media and, interestingly, drawing the salts into the solvent as well.^1–4 This ability is very important in the extraction or separation of metal ions, and, of particular interest here, divalent lanthanide (Ln²⁺) and actinide (An²⁺) cations. Examples of current Ln and An separation methodologies include: Ln separation, Plutonium Uranium Reduction Extraction (PUREX) and Spent Nuclear Fuel (SNF).^5–7,8 However, new and more efficient separation methods are urgently needed as lanthanides play a central role in the development of new electric vehicles comprising almost 15 kg of lanthanides and new electronic devices like smart phones having nine lanthanide elements. This, indeed, impose a warning for depleting lanthanides sources globally.⁹ Beside separation problem, crown ethers can benefit significant improvement in luminescence properties of both divalent and trivalent lanthanides that is of great interest to technological devices.^10–12

CEs have shown very strong binding to Ln and An ions and can provide the stabilization necessary to study the behavior of these elements.^{7, 13, 14 15, 16} Among mid-size macrocyclic polyethers, dicyclohexano-18-crown-6 (DCH18C6) has shown promising interactions with metal ions. DCH18C6 is a byproduct of dibenzo-18-crown-6 and has five common diastereoisomers (Figure 1) including cis-syn-cis, cis-anti-cis, trans-syn-trans, trans-anti-trans, and cis-trans, (ordered based on the most favor isomer) of which the first two are well studied (but not with Ln²⁺ or An²⁺ systems) and obtain better complexation products. The difference among these diastereoisomers lies within the spatial orientation of the hydrogen pairs on the cyclohexane rings, where for cis-syn-cis, both pairs have the cis form and stay in the plane (or out of the plane), for cis-anti-cis, obtaining the cis form, one pair stays in the plane and the other out of the plane or vice versa, in trans-syn-trans and trans-anti-trans, they both have the trans form, and for trans-syn-trans they are symmetrically oriented while for trans-anti-trans they are anti-symmetrically oriented, and finally for cis-trans one pair has the cis form and one has the trans form.^17–22 Given the potential of DCH18C6 for the separation of lanthanides and tuning luminescence properties of Ln²⁺ and An²⁺, theoretical work using molecular dynamics (MD) simulations has been employed here to study the conformations of cis-syn-cis, cis-anti-cis and trans-anti-trans with Sm²⁺, Eu²⁺, Dy²⁺, Yb²⁺, Cf²⁺ in THF and acetonitrile with halogen based salts, i.e., Cl, Br and I. Interestingly, previous studies^{23, 24} have shown that Sm²⁺ has a redox potential of −1.55 V and is counted as an electrochemical analog for Cf²⁺ with a redox potential of −1.6 V. Correspondingly, the present simulations can help broaden the understanding of the Cf²⁺ species. While the cis-syn-cis and cis-anti-cis diastereoisomers are prevalent in the literature, given the ease of formation from the synthesis of dibenzo-18-crown-6, the trans-anti-trans diastereoisomer is less-studied. Interestingly, trans-anti-trans formed a bowl shape structure with the cations not seen in the two more common diastereoisomers.¹⁹ Given the highly charged system, the polarizable AMOEBA^25–31 force field was employed and required the development of new parameters.

Figure 1.

2D schemes of DCH18C6 isomers

2. Computational Details

2.1. Parameterization process

To perform MD simulations on the Ln²⁺/An²⁺-DCH18C6 systems, the Dy²⁺, Yb²⁺, Cf²⁺ cations needed to be parameterized according to the standard AMOEBA parameterization protocol as conducted in our previous study.³² The MM parameterization portion was revisited to achieve better fitting in which, during the optimization process, the bound of the vdW potential well was tested with 0.6 – 0.9, 0.7 – 0.9 and 0.8 – 0.9 kcal/mol values. Although marginally, the 0.6 – 0.9 kcal/mol bound obtained the best minimized value. Moreover, the weight on this constraint/parameter was slightly increased led to obtain a better fitting. This procedure was applied to Sm²⁺ and Eu²⁺, too.

Parameters of acetonitrile were taken from AMOEBA’s 2009 parameters set, “amoeba09.prm”. THF and DCH18C6 were parameterized using the Poltype package²⁹ that automates the parameterization of small molecules. All quantum mechanical (QM) calculations were carried out at the MP2 level of theory using Gaussian 09.³³ The basis sets used were: 6–31G(d) for optimization, 6–311G(d) for single point calculations of electron density, 6–311++G (2d,2p) for the electrostatic potential, 6–31G(d) for torsion optimization and 6–311++G(d,p) for torsion single point calculations (for fitting step). Distributed Multipole Analysis³⁴ (DMA) was carried out using the GDMA package^{35, 36} to derive the initial atomic multipoles. A Wiberg bond order³⁷ tolerance of 0.01 was considered for fragmentation of the molecule and six folded Fourier terms have been considered. All molecular mechanical (MM) steps have been run using the Tinker package³⁸. For THF, a MM dipole moment of 1.878 D was obtained from final parameters that showed good agreement with QM dipole moment of 1.901 D, with a relative error of 1.2%. A structural RMSD of 0.08 was obtained from the parameters when compared to the structure from QM calculations. For DCH18C6, a MM dipole moment of 1.865 D was obtained from the final parameters and agreed well with the QM-computed dipole moment of 1.828 D, with a relative error of 2.0%. A structural RMSD of 0.27 was acquired from the parameters in comparison with the QM-derived structure.

2.2. Molecular Dynamics

2.2.1. MD simulations

The Tinker package³⁸ was used to carry out the MD simulations at a constant temperature of 298.15 K using the Bussi thermostat³⁹, a vdW cutoff of 12 Å, and a 2 femtosecond (fs) MD timestep with the trajectory saved at every 1 ps for a total simulation time of 60 ns. The r-RESPA integration method⁴⁰ was used for producing the trajectories. Two types of simulations were performed, one with a DCH18C6 isomer plus a cation and two halides (Cl⁻, Br⁻, or I⁻) in a box of approximately 500 THF molecules, the other one in a box of approximately 790 acetonitrile molecules, with both boxes having x, y, z, dimensions of ~41.0 Å using periodic boundary conditions (PBC). MD simulations of trans-anti-trans isomer and acetonitrile solvent were run only for Sm²⁺ and Eu²⁺. The Dy²⁺, Yb²⁺, Cf²⁺, simulations have been carried out once with cis-syn-cis and once with cis-anti-cis plus a cation and two halides in only THF. Particle-Mesh Ewald (PME) summation⁴¹ was used to model long-range electrostatics with a cutoff of 7 Å in real space. For AMOEBA’s induced dipole polarization, a convergence criterion of 1e⁻⁵ was considered. First, the systems were heated up to room temperature (298 K) with the NVT ensemble for 1 ns then equilibrated to a pressure of 1 bar using the Monte Carlo barostat⁴² with the NPT ensemble for 1 ns, and, finally, the MD production steps were done with a Monte Carlo barostat⁴² with the NPT ensemble for 60 ns. In addition, the VMD package⁴³ was used for visualization of the systems, and all the plots in this work have been generated with the python package, Matplotlib.⁴⁴

2.2.2. Free energy calculations

Further MD simulations were performed in triplicate to compute the mean and standard deviation of the absolute Gibb’s binding free energy (ΔG_bind) of Sm²⁺ and the cis-syn-cis/cis-anti-cis/trans-anti-trans stereoisomers. Alchemical free energy calculation applies an unphysical path to connect two physically meaningful end states, here the Sm²⁺ fully coupled state and the Sm²⁺ fully decoupled state. In this study, to evaluate the absolute binding free energies of Sm²⁺, a double-decoupling approach was employed. Based on a linear scaling function, U_elec(λ) = U_elec λ, eighteen independent alchemical simulations were performed to gradually turn off electrostatic interactions with the environment; the λ values of the corresponding alchemical states were respectively set as 1.0, 0.97, 0.93, 0.9, 0.87, 0.83, 0.8, 0.77, 0.73, 0.68, 0.65, 0.62, 0.6, 0.55, 0.5, 0.4, 0.0, where the fully coupled state was represented by λ=1.0 and the fully decoupled state by λ=0.0. Then based on a softcore scaling function, ${\text{U}}_{\text{vdw}}({\lambda }^{\prime })={{\lambda }^{\prime }}^5{\epsilon }_{ij}\frac{{1.07}^7}{0.7{\left(1-{\lambda }^{\prime }\right)}^2+{\left(r_{ij}+0.07\right)}^7}\left(\frac{1.12}{0.7{\left(1-{\lambda }^{\prime }\right)}^2+r_{ij}^7+0.12}-2\right)$, in which ε_ij stands for the well depth between atoms i and j; and r_ij denotes the current distance, additional nineteen independent alchemical simulations were carried out to gradually turn off van der Waals forces from the environment; the λ’ values of the corresponding alchemical states were respectively set as 1.0, 0.95, 0.9, 0.85, 0.8, 0.75, 0.7, 0.65, 0.6, 0.55, 0.5, 0.45, 0.4, 0.35, 0.3, 0.25, 0.2, 0.15, 0.1, 0.0, where the fully coupled state was represented by λ’=1.0 and the fully decoupled state by λ’=0.0. These simulations were performed both in the solvent environment to calculate solvation free energy, ΔG_solv, and inside the crown ether to calculate complexation free energy, ΔG_comp. For solvation free energy calculation, Sm²⁺ and two Br⁻ were solvated in a cubic THF periodic box with a length of 30.0 Å. All these individual simulations were performed via first heating to the room temperature through 4 ns NVT ensemble MD, then equilibrating through 5 ns NPT ensemble MD, and finally the NPT ensemble production step for 1 ns. Each frame was saved every 0.1 ps. For complexation free energy calculations, crown ether, Sm²⁺ and two Br⁻ were solvated in a cubic THF periodic box with a length of 41.0 Å and a similar procedure to the above was followed except that near the fully coupled state (both λ=1.0 and λ’=1.0), the lengths of production simulations were set as 1.5 ns. Moreover, in complexation simulations, a flat-bottomed restraint with a force constant of 5 kcal/mol and a distance of 2.7 Å was applied to keep Sm²⁺ around the crown ether center. In this decoupled state, this flat bottom restraint leads to a standard state binding free energy correction (ΔG_correction) of 1.58 kcal/mol. Absolute binding free energy, ΔG_bind, was computed according to the following,

$$\Delta {\text{G}}_{\mathit{\text{bind}}}=\Delta {\text{G}}_{\mathit{\text{solv}}}-\Delta {\text{G}}_{\mathit{\text{comp}}}+\Delta {\text{G}}_{\mathit{\text{correction}}}.$$ (1)

Free energy estimation between each pair of neighboring states was performed using the Bennet Acceptance Ratio (BAR)⁴⁵ method as implemented in the bar program of the Tinker package⁴⁶. Compared with the two common free energy calculations approaches, thermodynamic integration (TI)⁴⁷ and thermodynamic perturbation (TP)⁴⁸, BAR has higher precision⁴⁹. For a pair of adjacent states (i and j), where numbers of samples are n_i and n_j, free energy difference can be computed as follows:

$$\Delta {\text{G}}_{ij}=k_{B}T\left(ln\frac{{\sum }_{j=0}^{n_j}f(U_i-U_j+C)}{{\sum }_{i=0}^{n_i}f(U_j-U_i-C)}-ln\frac{n_j}{n_i}\right)+C$$ (2)

where k_B is the Boltzmann constant; T is the temperature; U_i and U_j are the potential energies respectively calculated based on the functions of state i and state j; f is the so-called Fermi function, $f\left(x\right)=\frac{1}{1+\text{exp}\left(\beta x\right)}$; the constant C, is the quantity of interest in Bennett’s approach. A few steps before obtaining the equation (2), we have

$$\frac{Q_i}{Q_j}=\frac{{\langle f(U_i-U_j+C)\rangle }_j}{{\langle f(U_j-U_i-C)\rangle }_i}\cdot ex{p}^{(\beta C)}$$ (3)

where, $Q={\sum }_i{e}^{-\beta U_i}$, is the partition function, the brackets denote the ensemble average (over state i or j) and $\beta =\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{${\text{K}}_{\mathrm{B}}\text{T}$}\right.$. As such, although equation (3) is valid for any value of C, the optimal choice is $\text{C}\text{=}ln\frac{Q_i n_j}{Q_j n_i}$. However, this appears to be problematic since it requires us to have the knowledge of $\frac{Q_i}{Q_j}$ that we wish to compute. Practically, C can be obtained self-consistently through an iterative process. To do so, after choosing an initial guess, the optimal value of C is searched through such that the argument of the natural logarithm in equation (2) becomes equal one, satisfying the following condition:

$${\sum }_{j}f\left(U_i-U_j+C\right)={\sum }_{i}f\left(U_j-U_i-C\right)$$ (7)

then the free energy difference can be calculated through the equation (8):^49–56

$$\Delta {\text{G}}_{ij}=-k_{B}T\left(ln\frac{n_j}{n_i}\right)+C$$ (8)

3. Results and discussions

3.1. Parameterization

For the validation procedure of the parameterization process, the reader is referred to references^{26, 32}. The resulting curves of parameterization of Dy²⁺, Yb²⁺, Cf²⁺ are shown in Figure S.1 and the modified parameters are provided in Table 1.

Table 1.

AMOEBA force field parameters for Ln²⁺ and An²⁺ polarization and vdW.

	α	a	R0 (Å)	ε (kcal/mol)
Sm2+	2.5	0.25	3.78	0.6
Eu2+	2.5	0.22	3.74	0.6
Dy2+	2.5	0.25	3.67	0.6
Yb2+	2.5	0.24	3.51	0.6
Cf2+	2.5	0.28	3.74	0.6

	Pairwise vdW
		R0 (Å)	ε (kcal/mol)
Acetamide O	Sm2+	2.95	0.60
Acetamide O	Eu2+	2.88	0.67
Acetamide O	Dy2+	2.88	0.60
Acetamide O	Yb2+	2.81	0.62
Acetamide O	Cf2+	2.89	0.66

AMOEBA considers four parameters for ions, α is the polarizability, a is the Thole’s damping function⁵⁷, R₀ is the vdW minimum energy distance, and ε is the vdW potential well.

In addition to the parameter validation of DCH18C6 described in section 2.1, the average interaction distance of Ln²⁺-O (oxygens of the crown ether) from both QM and MM calculations in the gas phase have been compared in Table 2.

Table 2.

Average gas phase interaction distances (in Å) and intermolecular energy (in kcal/mol) for Ln²⁺/An²⁺-O (O of DCH18C6).

Ln2+/An2+-O	QM dist.	AMOEBA dist.	% Relative error	QM energy	AMOEBA energy	% Relative error
cis-syn-cis
Sm2+ - O	2.71	2.72	% 0.48	−216.99	−215.74	%0.58
Eu2+ - O	2.70	2.70	% 0.001	−218.11	−204.70	%6.14
Dy2+ - O	2.66	2.67	% 0.02	−228.89	−216.72	%5.32
Yb2+ - O	2.63	2.65	% 0.83	−235.64	−223.39	%5.12
Cf2+ - O	2.68	2.71	% 1.12	−232.80	−209.90	%9.84
cis-anti-cis
Sm2+ - O	2.73	2.76	% 1.13	−216.82	−201.08	%7.26
Eu2+ - O	2.73	2.75	% 0.8	−218.80	−201.60	%7.86
Dy2+ - O	2.71	2.72	% 0.37	−228.23	−211.70	%7.24
Yb2+ - O	2.68	2.70	% 1.0	−235.04	−217.65	%7.40
Cf2+ - O	2.71	2.76	% 1.54	−233.51	−206.18	%11.70
trans-anti-trans
Sm2+ - O	2.72	2.75	% 1.1	−221.78	−201.42	%9.18
Eu2+ - O	2.71	2.74	% 0.09	−222.97	−202.29	%9.28

3.2. Sm²⁺ and Eu²⁺

3.2.1. Radial distribution function

Normalized Radial distribution functions (RDFs) have been computed for multiple combinations of Sm²⁺ and Eu²⁺ cations, the three DCH18C6 diastereoisomers (cis-syn-cis, cis-anti-cis and trans-anti-trans), and three halide anions (Cl⁻, Br⁻ and I⁻). In terms of solvent, the cis-syn-cis and trans-anti-trans systems were studied in both THF and acetonitrile, whereas the cis-anti-cis system was performed solely in THF to reduce the number of simulations since cis-syn-cis and cis-anti-cis to some extend behave similarly. Acetonitrile was included to observe the effect of a more polar solvent. Figure 2 provides (1) the normalized RDF plots of Sm²⁺-O and Eu²⁺-O for cis-syn-cis in THF/acetonitrile (left y-axis), (2) the coordination number (CN) curves (right y-axis). Figure 3 shows the CN curve of Sm²⁺-N and Eu²⁺-N. As observed from the RDF plots, the choice of anion (Cl⁻/Br⁻/I⁻) did not impact the coordination between Sm²⁺/Eu²⁺ and the oxygens of the polyether ring. In addition, a stable CN of 6 for all simulations has been computed for cation-O interactions. Since acetonitrile does not contain oxygen, the CN curves of Sm²⁺/Eu²⁺-O find a plateau in plots. In addition, the CN curves of Ln²⁺-N show that for systems containing I⁻ a CN of one for nitrogen of acetonitrile was observed. This can be rationalized through the hard/soft acid base rules, in which soft acids prefer soft bases and hard acids prefer hard bases. Eu²⁺ and Sm²⁺ as hard acids do not tend to interact with I⁻, a soft base, thus, they tend to interact more with the nitrogen atoms of acetonitrile. Notably, although MD simulations were performed in triplicate based on the LnI₂ hypothesis, for all three simulations, at some point in the trajectory, one iodide anion dissociates and a stable LnI-DCH18C6 forms. It is likely that due to the repulsion of the second I- ion, the binding of the first I- ion to the complex cannot complete with the acetonitrile solvation of this anion.

Figure 2.

Normalized radial distribution functions and CNs of Sm²⁺-O and cis-syn-cis DCH18C6 in THF (top left) and in acetonitrile (top right). Normalized RDF of Eu²⁺-O and cis-syn-cis DCH18C6 in THF (bottom left) and in acetonitrile (bottom right).

Figure 3.

CN of Sm²⁺-N-acetonitrile (left), Eu²⁺-N-acetonitrile (right) for the cis-syn-cis system.

Again, for simulations with THF we observe that the CN curves of Sm²⁺/Eu²⁺-O increase more than six only after ~6 Å, showing that cations had almost no interaction with the O atoms of THF. To further investigate, the percentage of available O atoms within range of 3.5 Å of Sm²⁺/Eu²⁺ was analyzed and shown in the Table 3. It was found that the cations were in sole contact with the oxygens of the polyether ring the entire time, and if any contact was made with the O atom of THF, they could not draw the cation out of the ring of crown ether.

Table 3.

The percentage of Sm²⁺/Eu²⁺-O coordination from MD simulations.

system	5 coordinated O %	6 coordinated O %	7 coordinated O %
Sm 2+
SmCl2 cis-syn-cis (THF)	0.01%	99.99%	0.01%
SmBr2 cis-syn-cis (THF)	0.0%	99.86%	0.14%
SmI2 cis-syn-cis (THF)	0.0%	100%	0.0%
SmCl2 cis-syn-cis (MeCN)	0.01%	99.99%	0.0%
SmBr2 cis-syn-cis (MeCN)	0.00%	100%	0.0%
SmI2 cis-syn-cis (MeCN)	0.01%	99.99%	0.0%
SmCl2 cis-anti-cis (THF)	0.03%	99.97%	0.0%
SmBr2 cis-anti-cis (THF)	0.02%	99.98%	0.01%
SmI2 cis-anti-cis (THF)	0.01%	99.99%	0.0%
SmCl2 trans-anti-trans (THF)	0.08%	99.39%	0.53%
SmBr2 trans-anti-trans (THF)	0.0%	99.99%	0.01%
SmI2 trans-anti-trans (THF)	0.01%	99.92%	0.07%
SmCl2 trans-anti-trans (MeCN)	0.08%	99.92%	0.0%
SmBr2 trans-anti-trans (MeCN)	0.0%	100%	0.0%
SmI2 trans-anti-trans (MeCN)	0.08%	99.92%	0.0%
Eu 2+
EuCl2 cis-syn-cis (THF)	0.0%	100%	0.0%
EuBr2 cis-syn-cis (THF)	0.01%	99.99%	0.0%
EuI2 cis-syn-cis (THF)	0.0%	100%	0.0%
EuCl2 cis-syn-cis (MeCN)	0.01%	99.99%	0.0%
EuBr2 cis-syn-cis (MeCN)	0.0%	100%	0.0%
EuI2 cis-syn-cis (MeCN)	0.02%	99.98%	0.0%
EuCl2 cis-anti-cis (THF)	0.03%	99.96%	0.0%
EuBr2 cis-anti-cis (THF)	0.1%	99.84%	0.06%
EuI2 cis-anti-cis (THF)	0.02%	99.98%	0.0%
EuCl2 trans-anti-trans (THF)	0.07%	99.45%	0.48%
EuBr2 trans-anti-trans (THF)	0.11%	99.72%	0.17%
EuI2 trans-anti-trans (THF)	0.13%	99.74%	0.13%
EuCl2 trans-anti-trans (MeCN)	0.08%	99.92%	0.0%
EuBr2 trans-anti-trans (MeCN)	0.13%	99.87%	0.0%
EuI2 trans-anti-trans (MeCN)	0.09%	99.91%	0.0%

3.2.2. Interactions of Eu²⁺/Sm²⁺-X and Eu²⁺/Sm²⁺-O

The interaction distances between Ln²⁺-O in solution for the DCH18C6 and halide systems are reported in Table 4. The error bars of all Sm²⁺/Eu²⁺ systems are shown in Figure 4 and have been computed with the three-sigma (three times standard deviation) rules. As observed from the error bars, the distances between the cations and oxygens remained around 2.7 Å. As expected, the distance of Sm²⁺/Eu²⁺-Cl⁻ < Sm²⁺/Eu²⁺-Br⁻ < Sm²⁺/Eu²⁺-I⁻. Moreover, the systems with iodide in acetonitrile showed different behavior in comparison to the systems solvated in THF. This may be attributed to the polarity of acetonitrile (dielectric constant of 37.0),⁵⁸ in which the ions tend to be more dispersed in the solvent.

Table 4.

The interaction distance of Ln²⁺-O and Ln²⁺-X (X = I⁻/Br⁻/Cl⁻)

MD trajectory	Ln2+-O (Å)	Ln2+-I− (Å)	Ln2+-Br− (Å)	Ln2+-Cl− (Å)
Sm 2+
SmCl2 cis-syn-cis (THF)	2.76			2.84
SmBr2 cis-syn-cis (THF)	2.75		3.00
SmI2 cis-syn-cis (THF)	2.75	3.21
SmCl2 cis-syn-cis (MeCN)	2.74			2.87
SmBr2 cis-syn-cis (MeCN)	2.75		3.00
SmI2 cis-syn-cis (MeCN)	2.71	3.25
SmCl2 cis-anti-cis (THF)	2.74			2.87
SmBr2 cis-anti-cis (THF)	2.72		3.00
SmI2 cis-anti-cis (THF)	2.74	3.25
SmCl2 trans-anti-trans (THF)	2.73			2.86
SmBr2 trans-anti-trans (THF)	2.73		3.00
SmI2 trans-anti-trans (THF)	2.72	3.26
SmCl2 trans-anti-trans (MeCN)	2.72			2.89
SmBr2 trans-anti-trans (MeCN)	2.72		3.00
SmI2 trans-anti-trans (MeCN)	2.71	3.36
Eu 2+
EuCl2 cis-syn-cis (THF)	2.76			2.87
EuBr2 cis-syn-cis (THF)	2.75		3.00
EuI2 cis-syn-cis (THF)	2.75	3.21
EuCl2 cis-syn-cis (MeCN)	2.74			2.86
EuBr2 cis-syn-cis (MeCN)	2.73		3.03
EuI2 cis-syn-cis (MeCN)	2.72	3.28
EuCl2 cis-anti-cis (THF)	2.74			2.87
EuBr2 cis-anti-cis (THF)	2.72		3.04
EuI2 cis-anti-cis (THF)	2.73	3.25
EuCl2 trans-anti-trans (THF)	2.73			2.87
EuBr2 trans-anti-trans (THF)	2.73		3.05
EuI2 trans-anti-trans (THF)	2.72	3.25
EuCl2 trans-anti-trans (MeCN)	2.72			2.88
EuBr2 trans-anti-trans (MeCN)	2.70		3.04
EuI2 trans-anti-trans (MeCN)	2.69	3.33

Figure 4.

Error bars for the interaction distance of (top) Sm²⁺-O and Sm²⁺-X⁻ and (bottom) Eu²⁺-O and Eu²⁺-X⁻.

3.2.3. Conformational analysis within the MD trajectories

To investigate if different conformations of DCH18C6 appeared within the MD trajectories, two different approaches have been employed. First the cpptraj⁵⁹ program of Amber Tools was used to analyze the ten most frequent conformations for each trajectory. In this case, solvent molecules were removed to only consider differences in the crown ether molecules, and the k-means method was used to cluster the conformations. The second approach monitored the potential energy components including vdW, electrostatic interactions and torsion, over the course of the trajectory. As shown in Figure 5, all simulations in THF gave three nearly straight lines, which indicated that no significant fluctuations were present in those energies. For systems with I⁻ in acetonitrile, where one iodide stays away from the crown ether, fluctuations were observed in the electrostatic interactions due to the separation of the iodide and the cation and not because of a conformational change as observed in the other approach.

Figure 5.

Potential energy components of SmI₂ - cis-syn-cis in THF (left) and in MeCN (right).

For both Sm²⁺ and Eu²⁺ systems, among ten conformations of cis-syn-cis isomer (obtained from cpptraj program), there are two major types as shown in Figures 6a and 6b. The rest are the conformations with changes in the orientation of the cyclohexyl rings under those two categories. The conformation where the rings are aligned (Figure 6a) largely occurred for systems with Cl⁻> Br⁻ > I⁻, indicating that the orientation of the rings can be affected greatly with the anions having higher charge density, i.e., larger electrostatic interactions, that again is associated with hard/soft acid base rules. The other conformation type (Figure 6b) where the rings are not aligned mostly occurred for systems including iodide. This indicates that for the systems having less electrostatic interactions Figure 6b is more favorable. It should be noted that although this was done through an iterative process of k-means clustering, both types of conformations were observed for all systems. For systems with stronger electrostatic interactions Figure 6a were dominant where for systems having less electrostatic interactions Figure 6b and 6c were dominant.

Figure 6.

a) SmCl₂ - cis-syn-cis, b) SmBr₂ - cis-syn-cis, c) SmI₂ - cis-anti-cis, d) SmBr₂ - cis-anti-cis, e) SmCl₂ – trans-anti-trans, f) SmI₂ – trans-anti-trans.

For cis-anti-cis (Figure 6c and 6d) the trend observed for cis-syn-cis was not obtained here. The structure in Figure 6d was the most prevalent conformation where the other conformation (Figure 6c) was obtained less frequently.

For trans-anti-trans (Figures 6e and 6f), as the name suggests, on the rings one hydrogen is located in the plane and the other is out of plane in an antisymmetric fashion. This avoids having large changes in the orientation of the rings. However, the most significant change in the conformations is the bending of the entire molecule. For I⁻ systems, where the interaction distance of Ln²⁺-halides is the largest, a bowl shape of crown ether encapsulating the cation occurs. For Cl⁻ where the Ln²⁺-halides distance is the least, an even shape of the molecule was observed as chlorides try to interact with the cation strongly. Consequently, because of steric hinderance crown ether cannot obtain the bowl shape. For Br⁻, the bent shape of the structure falls between the latter systems, not as bent as I⁻ systems, nor as flat as Cl⁻ systems.

Since both Sm²⁺ and Eu²⁺ have followed the same trend, only the conformations of Sm²⁺ systems are shown here in Figure 6 and the rest are provided in Figure S2.

3.2.4. Free energy computations

To the best of the authors’ knowledge, no binding free energies for the Ln²⁺ and DCH18C6 systems have been previously reported, thus the ΔG_bind computed here are without direct comparison to experimental data.

A harmonic restraint correction was also applied to all ΔG_comp. Agreement between the forward and backward FEP for every two intermediate states was < 1 kcal/mol. The mean ΔG_bind and standard deviations of Sm²⁺ and DCH18C6 isomers are provided in Table 5. As expected, cis-syn-cis, being the most stable isomer of DCH18C6 had the most favorable ΔG_bind in comparison to the other two isomers. Again, as expected and studied, the next most stable isomer was found to be cis-anti-cis, and finally, trans-anti-trans with a slight difference of ~ 0.6 kcal/mol with latter isomer.^17–21

Table 5.

Gibbs binding free energies (ΔG_bind), complexation free energy (ΔG_comp), solvation free energy (ΔG_solv) of Sm²⁺ - DCH18C6 (in kcal/mol). Harmonic restraint correction is only applied to ΔG_bind here.

DCH18C6 isomer	ΔGbind	ΔGbind STD Dev	ΔGcomp	(ΔGsolv)
Sm2+ - cis-syn-cis	−18.62	±0.43	−383.75	−365.13
Sm2+ - cis-anti-cis	−16.48	±0.50	−381.61	−365.13
Sm2+ - trans-anti-trans	−15.87	±0.83	−380.77	−365.13

3.3. Dy²⁺, Yb²⁺ and Cf²⁺

3.3.1. Radial distribution function

The current MD simulations have consistently shown that the choice of halide has little effect on the coordination of oxygens (Figure 7). In addition, the CN curves, which reveal the number of oxygen atoms at each spherical layer, show a consistent coordination number, six, in the layers with the radius from ~3.5 Å to ~6 Å. This suggests that the cations only interact with the oxygens of crown ether. Similarly, the number of available oxygen atoms within range of 3.5 Å of Dy²⁺/Yb²⁺/Cf²⁺ have been analyzed and are shown in the Table 6. By going toward the late lanthanides in which the radii decrease, starting from Dy²⁺ some fluctuations between five- and six-coordinated frames were observed. This was more pronounced in Yb²⁺ as it is located at the end of the lanthanide series. The calculations suggest that DCH18C6 may not be a good fit for late lanthanides, although they remained inside the ring of the crown ether. However, Cf²⁺ was dominantly observed as a six-coordinated oxygen and had almost negligible five-coordinated frames.

Figure 7.

Normalized RDF and CN curves of Dy²⁺-O (top left), Yb²⁺-O (top right) and Cf²⁺-O (bottom left) of cis-syn-cis in THF.

Table 6.

Computed Dy²⁺/Yb²⁺/Cf²⁺-O coordination number percentage.

MD trajectory	5 coordinated O %	6 coordinated O %	7 coordinated O %
Dy2+
DyCl2 cis-syn-cis (THF)	2.85%	97.15%	0.0%
DyBr2 cis-syn-cis (THF)	0.15%	99.78%	0.07%
DyI2 cis-syn-cis (THF)	1.0%	99.0%	0.0%
DyCl2 cis-anti-cis (THF)	0.16%	99.84%	0.0%
DyBr2 cis-anti-cis (THF)	0.15%	99.85%	0.0%
DyI2 cis-anti-cis (THF)	0.28%	99.69%	0.03%
Yb2+
YbCl2 cis-syn-cis (THF)	17.97%	82.03%	0.0%
YbBr2 cis-syn-cis (THF)	20.85%	79.15%	0.0%
YbI2 cis-syn-cis (THF)	0.14%	99.86%	0.0%
YbCl2 cis-anti-cis (THF)	7.93%	92.07%	0.0%
YbBr2 cis-anti-cis (THF)	1.51%	98.49%	0.0%
YbI2 cis-anti-cis (THF)	1.45%	98.55%	0.0%
Cf2+
CfCl2 cis-syn-cis (THF)	0.03%	99.97%	0.0%
CfBr2 cis-syn-cis (THF)	0.21%	99.78%	0.0%
CfI2 cis-syn-cis (THF)	0.02%	99.98%	0.0%
CfCl2 cis-anti-cis (THF)	0.04%	99.96%	0.0%
CfBr2 cis-anti-cis (THF)	0.09%	99.91%	0.0%
CfI2 cis-anti-cis (THF)	0.06%	99.94%	0.0%

3.3.2. Interactions of Dy²⁺/Yb²⁺/ Cf²⁺-X and Dy²⁺/Yb²⁺/ Cf²⁺-O

As shown in Table 7, the interaction distances of Dy²⁺/Yb²⁺-O were very close; however, the Yb²⁺-O distance is smaller than Dy²⁺-O as ytterbium has slightly smaller radius from being located near the end of the lanthanide series. Despite Cf²⁺ and Dy²⁺ both having a nf¹⁰ ground state, the Cf²⁺-O interactions were close to that of Sm²⁺ and Eu²⁺ (Table 4). This may be rationalized through the redox potential, but is more likely due to the similar size found between Cf²⁺ and Sm²⁺, where Sm²⁺ is often considered both a good electrochemical and similar sized lanthanide analog for Cf²⁺ chemistry.¹¹ However, not only are the Cf²⁺-I interactions shorter than Sm²⁺, they are also shorter than Dy²⁺ and Yb²⁺ even though Dy²⁺ and Yb²⁺ are smaller ions. This corresponds to the actinide elements being softer in nature than the lanthanides.⁶⁰ This observation is displayed in Figure 8 where if a trendline is built off of the Dy²⁺-X and Yb²⁺-X, and the Cf²⁺-X interaction is placed in the middle, the Cf²⁺-X interactions with a harder lewis acid Cl⁻ fall on the trendline while the softer lewis acid I⁻ falls below it. Similar trends have been observed where actinide-ligand bond distances are lanthanide-like with hard donor atoms,^{61, 62} and not with soft donor atoms.^{63, 64} Again, following Ln trends for soft bases⁶⁴, where actinides stay below the trendline, the same behavior observed here (Figure 8) for both diastereoisomers which is an indicative of a good agreement between our proposed AMOEBA potentials for lanthanides and experimental data.

Table 7.

The interaction distance of Ln²⁺-O and Ln²⁺-X (X = I⁻/Br⁻/Cl⁻)

MD trajectory	Ln2+-O (Å)	Ln2+-I− (Å)	Ln2+-Br− (Å)	Ln2+-Cl− (Å)
Dy2+
DyCl2 cis-syn-cis (THF)	2.66			2.84
DyBr2 cis-syn-cis (THF)	2.69		3.03
DyI2 cis-syn-cis (THF)	2.66	3.26
DyCl2 cis-anti-cis (THF)	2.66			2.86
DyBr2 cis-anti-cis (THF)	2.67		3.03
DyI2 cis-anti-cis (THF)	2.64	3.28
Cf2+
CfCl2 cis-syn-cis (THF)	2.71			2.83
CfBr2 cis-syn-cis (THF)	2.73		2.98
CfI2 cis-syn-cis (THF)	2.71	3.20
CfCl2 cis-anti-cis (THF)	2.71			2.84
CfBr2 cis-anti-cis (THF)	2.73		2.99
CfI2 cis-anti-cis (THF)	2.72	3.20
Yb2+
YbCl2 cis-syn-cis (THF)	2.63			2.80
YbBr2 cis-syn-cis (THF)	2.61		2.99
YbI2 cis-syn-cis (THF)	2.63	3.23
YbCl2 cis-anti-cis (THF)	2.64			2.80
YbBr2 cis-anti-cis (THF)	2.64		2.98
YbI2 cis-anti-cis (THF)	2.62	3.23

Figure 8.

Error bars for the interaction distance of Dy²⁺/Yb²⁺/Cf²⁺-O (top left) Dy²⁺/Yb²⁺/Cf²⁺-X⁻ in cis-syn-cis (top right) and in cis-anti-cis (bottom).

3.3.3. Conformational analysis within the MD trajectories

For Dy²⁺/Yb²⁺/Cf²⁺ the same trend of conformations obtained for Sm²⁺/Eu²⁺ for cis-syn-cis was observed, i.e., systems with stronger electrostatic interactions (Cl⁻ as counter anion) preferred the conformation with aligned rings (Figure 6a), whereas systems with lower electrostatic interactions (I⁻ as counter anion) preferred the conformation with non-aligned rings (Figure 6b). As such, no trend was observed for cis-anti-cis (in accordance with Sm²⁺/Eu²⁺ systems) and the structure in Figure 6c was the dominant conformation. The conformations of Dy²⁺/Yb²⁺/Cf²⁺ and cis-syn-cis/cis-anti-cis systems are available in Figure S3.

4. Conclusion

In this study, polarizable parameters were developed for the AMOEBA force field for (1) the divalent lanthanides Dy²⁺ and Yb²⁺, (2) the divalent actinide Cf²⁺, (3) the crown ether DCH18C6 and (4) the THF solvent for the first time. QM level calculations were used to validate the parameters with good agreement reported. MD simulations were carried out for the mentioned cations plus Sm²⁺ and Eu²⁺ with DCH18C6 diastereoisomers solvated in THF. The systems were also simulated in acetonitrile, but only for Sm²⁺ and Eu²⁺ with the cis-syn-cis and trans-anti-trans diastereoisomers. RDF plots and coordination numbers (CNs) were analyzed and reported. In almost all systems, a CN of 6 was an indicative of the cation solely coordinating with the oxygen of the crown ethers. However, for acetonitrile-solvated systems containing iodides, a CN <1 was found for the interaction between the cations and nitrogen atoms. This illustrated that the nitrogen atom of acetonitrile can compete with iodide for interactions with the cation, as opposed to systems that included chlorides and bromides where a CN of zero computed for Ln²⁺-N suggested a lack of interaction. Distances of Ln²⁺/An²⁺-O (oxygens of crown ether) and Ln²⁺/An²⁺-X (halide) were reported here for the first time in the liquid phase. Conformational analyses were performed for all systems to identify the most dominant DCH18C6 diastereoisomer plus any other possible conformations. For systems with higher electrostatic interactions, where the ions have higher charge densities, different behavior was observed. For cis-syn-cis and cis-anti-cis different orientations of the attached hexane rings were observed. For trans-anti-trans systems including I⁻ and Br⁻, the crown ether formed a bowl shape. However, Cl⁻ systems did not find the bowl shape, which may be due to steric hinderance arising from the closer interacting distance between Cl⁻ and the cation. Finally, Gibbs binding free energy computations were carried out for SmBr₂ systems with all diastereoisomers for the first time. Free energy values of cis-syn-cis were more favorable than the other two diastereoisomers. In addition, the binding free energy of the cis-anti-cis isomer was slightly more negative than trans-anti-trans, reproducing the DCH18C6 experimental trend.

Data Availability Statement

AMOEBA ion parameterization codes can be found in the github page of the author (H.A): https://github.com/hesam-a

Orca binary files can be downloaded from: https://orcaforum.kofo.mpg.de/app.php/portal

Tinker package can be obtained from their github page at: https://github.com/TinkerTools/tinker or from Dr. Ponder’s website at: https://dasher.wustl.edu/tinker/.

First “fftw” package, then tinker (from source directory) should be compiled.

The parameter file “amoeba09.prm” can be obtained from “params” directory available in the package.

VMD package can be obtained from: https://www.ks.uiuc.edu/Development/Download/download.cgi?PackageName=VMD or can be installed with Anaconda package with “conda install -c conda-forge vmd” command.

All additional data are available from the corresponding author upon reasonable request.