Exploring High-Symmetry Lanthanide-Functionalized Polyoxopalladates as Building Blocks for Quantum Computing

Abstract

The structural, electronic, and magnetochemical properties of the star-shaped polyoxopalladate [Pd₁₅O₁₀(SeO₃)₁₀]¹⁰⁻ (POPd) and its lanthanide-functionalized derivatives have been investigated on the basis of density functional theory, followed by a ligand field analysis using the Radial Effective Charge (REC) model. Our study predicts that heteroPOPd is a robust cryptand that enforces D_5h symmetry around the encapsulated Ln³⁺ centers. This rigid coordination environment favors an interesting potential magnetic behavior in the Er and Ho derivatives, and the presence of a cavity in the structure suggests an effective insulation of the electronic system from the lattice phonons, which may be of interest for molecular spintronics and quantum computing applications.

1. Introduction

Quantum computing is an emerging area of research based on the explicit use of quantum-mechanical phenomena to implement logic operations for the purposes of highly efficient information and communication technologies. In contrast to conventional computers which store information in one of the two definite states ‘0’ or ‘1’, the basic units of a quantum computer are quantum bits (qubits), which can be a superposition of the basis states. Within this new paradigm, magnetic molecules are quantum two-level systems that have been proposed as promising candidates to build up quantum computers [1,2,3,4,5,6]. In this regard, magnetic molecules exhibit remarkable structural and chemical versatility, tunable physicochemical properties, and inherent scalability, considering the enormous opportunities of (self-)organization [7]. Although the current prospects for their technological implementation remain elusive due to challenges such as loss of quantum coherence [8], a series of recent breakthroughs in the field have been encouraging [9,10,11,12,13,14].

In this context, polyoxometalate (POM) chemistry has shown some advantageous chemical, structural, and electronic features such as robustness, highly symmetrical encapsulating environments, and absence of nuclear spins around the magnetic ion [15]. Indeed, lanthanide-based polyoxometalates (POMs) are among the first examples of single-ion magnets (SIMs) [16] and have also contributed significantly to the field of molecular spin qubits [17,18]. Some relevant examples in molecular magnetism include two families of lanthanide-containing polyoxomolybdates [19] and the cube-shaped polyoxopalladates (POPds) [20].

In this work, we focus on a different series of lanthanide-containing POPds as model systems with the aim to rationally design new molecular nanomagnets. The theoretical investigation we present herein is based on the polynuclear star-shaped heteoropoly-15-palladate(II) polyanion [Pd₁₅Se₁₀O₄₀]¹⁰⁻ (1) (Figure 1a) [21,22]. This negatively charged nanocluster is the selenite analogue of the star-shaped polyanion of general formula [Pd₁₅O₁₀(PO₄)₁₀]²⁰⁻ reported in 2009 [23] and belongs to the family of non-classical late transition metal-based POMs (Ni, Cu, Pd, Pt, and Au) [24,25,26,27,28]. The chemical structure of 1 presents an internal cavity defined by oxo ligands positioned at the vertices of a virtual pentagonal prism that is suitable for encapsulation of metal cations and water molecules (Figure 1b) [21,22,23,29,30]. Among the other available phosphate- or phenyl arsenate-capped polyanions [23,29,30], we decided to study 1 because it contains the lowest number of atoms and a relatively low negative charge, while retaining the high overall D_5h symmetry. Using density functional theory (DFT), we initially explore the diamagnetic [NaPd₁₅Se₁₀O₄₀]⁹⁻ (1-Na) POPd system and we find a suitable level of theory that can accurately estimate the geometry of the inner cavity. As a second step, we calculate and discuss the structural and electronic properties of three lanthanide-encapsulating model systems with general formula [LnPd₁₅Se₁₀O₄₀]⁷⁻, where Ln = La(1-La), Gd(1-Gd) and Lu(1-Lu). Finally, we use the optimized molecular geometry of the Gd³⁺ derivative to explore the spin energy levels and ground doublet composition of the Tb³⁺(1-Tb), Dy³⁺(1-Dy), Ho³⁺(1-Ho), and Er³⁺(1-Er) derivatives. Our approach provides a general protocol based on the former two steps, i.e., DFT calculations followed by a ligand field analysis, for the inexpensive estimation of the potential of coordination complexes as molecular nanomagnets and spin qubits prior to beginning the experimental work.

2. Results and Discussion

2.1. Structural and Electronic Properties

The structural and electronic properties of 1, 1-Na, 1-La, 1-Gd, and 1-Lu were investigated by means of relativistic density functional theory (DFT) as implemented in the ADF code (ADF 2017, SCM, Theoretical Chemistry, Vrije Univesity, Amsterdam, The Netherlands) [31]. The geometry of 1-Na in water was followed at different theoretical levels (Table S1) that have been successfully used previously for an accurate estimation of the structure of noble metal-based POMs [32,33,34,35]. Considering that for the ligand field analysis it is more essential to have geometries that primarily provide an accurate estimation of positions of the oxo ligands that coordinate to the encapsulated lanthanide centers, we have found that geometry optimization calculations of the model system 1-Na at BP86/TZ2P/ZSC/COSMO-water level provide the most accurate approximation of the coordination environment around the Na⁺ center, while the Pd–O and Se–O bonds are also well approximated (see Table S1). Subsequently, polyanions 1, 1-La, 1-Gd, and 1-Lu were optimized at this level as well. To confirm energy minima, vibrational analysis was performed at the same theoretical level (Figure S4). All insights on the electronic structure and bonding energies are discussed based on single-point calculation performed at (U)B3LYP/TZ2P/ZSC/COSMO-water level (see Supplementary Materials).

Our calculations show that polyanions 1, 1-Na, 1-La, 1-Gd, and 1-Lu exhibit very narrow Se–O, Pd–O(Pd), Pd–O(Se) bond lengths in the ranges of 1.731–1.733 Å, 1.994–2.008 Å, and 2.058–2.092 Å, respectively (Table S2). The interatomic distances d₁(O∙∙∙O) and d₂(O∙∙∙O) that define the internal pentagonal cavity (Figure 1b) are found in the range of 2.646–2.701 Å and 2574–2672 Å, respectively. The changes induced by encapsulation of the different cations are in the range of <0.1 Å, which implies that the internal cavities are generally unaffected by the encapsulation. The calculated Ln–O bonds/distances are in the range of 2.559–2.644 Å, which are characteristic for lanthanide-containing POM systems [16,17,18,19,20].

The optimized systems 1-La, 1-Gd, and 1-Lu exhibit a gap energy of ca 3.5–3.6 eV between the highest occupied and the lowest unoccupied molecular orbital (i.e., HOMO and LUMO). The high gap energy is comparable to that of the reported polyanions 1 (3.4 eV) and 1-Na (3.5 eV), which substantiates that the proposed lanthanide-encapsulating systems exhibit significant intrinsic stability making them promising candidates for future preparation (see Table S4). The HOMO in systems 1-La, 1-Gd, and 1-Lu is doubly degenerate and is composed of Pd-centered $d_{xy}$- and $d_{yz}$-type orbitals and O-centered p-type orbitals (Figure 2a). The LUMO in 1-La, 1-Gd, and 1-Lu is also doubly degenerate and populates mainly O and Pd centers but also minorly Se⁴⁺ and Ln³⁺ centers (Figure 2a). The antibonding LUMO exhibits nodes between the Pd-centered $d_{x^2-y^2}$-type orbitals and O-centered p-type orbitals, suggesting that population of this molecular orbital (e.g., via electrochemical reduction) can destabilize these bonds. The transition between the HOMO to LUMO corresponds predominantly to d–d transitions accompanied by metal-to-ligand charge transfer (Pd → O) (Table S5).

In our view, polyanions 3–9 can be potentially prepared in a two-step approach, where polyanions 1 or 2 serve as precursors and are reacted with the lanthanide trications to yield [LnPd₁₅Se₁₀O₄₀]⁷⁻, either by direct encapsulation or by Na⁺ cation exchange. Some aspects of the DFT calculations substantiate this opportunity. First, polyanions 1 and 1-Na exhibit pentagonal entrance defined by five oxo ligands that is comparable in size to that of the Preyssler-type polyoxoanions, (Figure S1) which is well-known to permeate lanthanide cations [16]. The inscribed virtual radius of the pentagonal entrance in 1 is ca. 1.79 Å, which is significantly larger than the effective ionic radius of the bare lanthanide trications [36]. Second, Mulliken charge analysis (Table S3) and molecular electrostatic potential (Figure 2b) of 1 show that the inner oxygen atoms defining the inner cavity are highly negatively charged and thus very nucleophilic. The high nucleophilicity of the inner cavity is the driving force for encapsulation of Na⁺, showing favorable binding energies of −201.2 kcal/mol, which is in line with the experimental evidence [21]. In contrast, the lanthanide trications show more negative binding energies which increase along the lanthanide series (La³⁺ = −983.9 kcal/mol, Gd³⁺ = −1031.1 kcal/mol, and Lu³⁺ = −1060.6 kcal/mol).

2.2. Ligand Field Analysis

Starting from the DFT-relaxed structure of the 1-Gd, we have applied the Radial Effective Charge (REC) model [37] using the SIMPRE computational package [38] to the 1-Tb, 1-Dy, 1-Ho, and 1-Er derivatives. This parametric model introduces covalency effects by placing an effective charge at an effective distance determined by a radial displacement D_r (see Section Five of Supplementary Materials for details). One of its main advantages is that, for similar families of homoleptic coordination complexes, such parameters are reusable, allowing an inexpensive determination of the spectroscopic and magnetic properties [39]. Thus, we have taken advantage of the semi-empirical information of the recently reported [LnPd₁₂(AsPh)₈O₃₂]⁵⁻ series [20] to provide an estimation of the ligand field splitting of the ground-J multiple as well as the ground state composition of each derivative. As the coordination number in these two families of POPds is different, 8 in the case of [LnPd₁₂(AsPh)₈O₃₂]⁵⁻ and 10 in [LnPd₁₅Se₁₀O₄₀]⁷⁻ and we do not have access to the X-ray structures, we have explored different possibilities for the REC parameters, always keeping constant the total f = D_r·Z_i product [40]. The scanning of the REC parameters at different levels of covalency allows us to evaluate the robustness of the composition of the ground state with the aim of identifying which candidates are more suitable to exhibit single-molecule magnet (SMM) behavior. The resulting energy level pattern as a function of the value of the radial displacement for 1-Tb, 1-Dy, 1-Ho, and 1-Er is reported in Figure 3.

A general trend that one can observe in the evolution of the energy levels versus D_r is that the larger the covalent correction, the larger the total ligand field splitting. This is related to the 1/(R_i^K+1) dependence of the k = 2, 4, 6 ligand field parameters in the model, where R_i is the effective distance between the point charge and the Ln³⁺ ion (see Section Five of Supplementary Materials, Equations (2a)–(2c)). Because of the almost perfect D_5h geometry around the lanthanide, the extra-diagonal ligand field B_kq parameters, which are responsible of the mixing between M_J microstates, are practically negligible. This results in almost pure wave functions (~100% |M_J>) (Table S7). Regarding the ground state compositions, one can notice that for the non-Kramers ions (1-Tb and 1-Ho), the same M_J is the most stable along our exploration. In the case of 1-Tb, the equatorial geometry of {Pd₁₅Se₁₀O₄₀} yields to a diamagnetic state M_J = 0, whereas for 1-Ho, the ground state is composed by two ±4 microstates. This indicates that we can practically discard the 1-Tb as a molecular nanomagnet, whereas the 1-Ho derivative is a potential candidate as a quantum two-level system, as long as small distortions in the real structure of the complex create a sufficiently large tunneling splitting between these two states [17]. On the other hand, the ground states of the 1-Dy and 1-Er derivatives present two different possibilities based on our magnetostructural investigation. Thus, we cannot state accurately the main M_J contribution to the ground doublet of these systems, as it will be very sensitive to the real coordinates of the compounds. However, whereas the 1-Dy analogue can be either ±1/2 or ±11/2, the 1-Er analogue is clearly the most promising of the series, as the model consistently predicts a ground state determined by a pure high M_J microstate (either ±15/2 or ±13/2). This preference of the polyoxoanion for stabilizing larger spin microstates in 1-Er is a direct consequence of the equatorial distribution of the electrostatic charge around the magnetic center [5,41]. Thus, according to our calculations, we can conclude that compound 1-Ho presents an interesting composition of the ground state to design a molecular spin qubit with possible operating points, whereas 1-Er has a very favorable coordination environment to exhibit slow relaxation of the magnetization. In comparison with the [LnPd₁₂(AsPh)₈O₃₂]⁵⁻ family [20], where the lack of anisotropy prevented SMM behavior in all the analogues of the series, the [LnPd₁₅Se₁₀O₄₀]⁷⁻ derivatives offer a promising pathway to develop POPd-based nanomagnets. Finally, the internal rigid cavity of the POM is expected to reduce the loss of quantum information by shielding the electronic structure from lattice vibrations, which means larger coherence times.

3. Conclusions

In summary, we have theoretically investigated the structural, electronic, and magnetic properties of a high-symmetry molecular structure based on POPds, providing a general protocol for the inexpensive screening of potential molecular nanomagnets and quantum two-level systems. Our calculations based on density functional theory and a ligand field approach indicate that the rigid near-perfect D_5h coordination environment practically cancels all the extra-diagonal contributions to the ligand field Hamiltonian, leading to almost pure spin states in all the systems of the series. The equatorially expanded charge distribution around the lanthanide favors the stabilization of a high-spin ground doublet in the 1-Er derivative, which can be considered as a potential single-ion magnet. On the other hand, the ground state of the 1-Ho derivative offers an interesting playground to break ±4 degeneracy through ligand field-induced tunneling splitting which can be of interest for quantum computing applications. Our efforts to shed more experimental light on these systems are currently underway.