Abstract
Understanding magnetically induced currents (MICs) in aromatic or metallic nanostructures is crucial for interpreting local magnetic shielding and NMR data. Direct measurements of the induced currents have been successful only in a few planar molecules but their indirect effects are seen in NMR shifts of probe nuclei. Here, we have implemented a numerically efficient method to calculate gaugeincluding MICs in the formalism of auxiliary density functional theory. We analyze the currents in two experimentally synthesized goldbased, hydrogencontaining ligandstabilized nanoclusters [HAu_{9}(PPh_{3})_{8}]^{2+} and [PtHAu_{8}(PPh_{3})_{8}]^{+}. Both clusters have a similar octet configuration of Au(6s)derived delocalized “superatomic” electrons. Surprisingly, Ptdoping in gold increases the diatropic response of the superatomic electrons to an external magnetic field and enhances the aromaticity of [PtHAu_{8}(PPh_{3})_{8}]^{+}. This is manifested by a stronger shielding of the hydrogen proton in the metal core of the cluster as compared to [HAu_{9}(PPh_{3})_{8}]^{2+}, causing a significant upfield shift in agreement with experimental proton NMR data measured for these two clusters. Our method allows the determination of local magnetic shielding properties for any component in large 3D nanostructures, opening the door for detailed interpretation of complex NMR spectra.
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Introduction
Magnetically induced currents (MICs) originate from the response of electrons of an object placed in an external magnetic field B^{1,2}. A diamagnetic object (atom, molecule, cluster, nanoparticle, nanostructure) generates an internal ring electron current in a plane normal to B in such a way that it induces a magnetic field opposite to the external one. This classical ring current is called a diatropic current, increasing the shielding of the part of the object inside the current ring against the effect of the external field. Systems with strong diatropic currents are called aromatic^{2,3}. Antiaromatic systems have a strong nonclassical current component (paratropic current) that tends to create an internal magnetic field parallel to the external one. Nonaromatic systems have a vanishing net current due to similar diatropic and paratropic contributions. Although recent efforts have been made to directly measure ring currents in atoms and planar molecules^{4}, it is still a considerable challenge to detect them in more complex systems such as 3D clusters or nanoparticles. Their effects can be studied indirectly since they affect the shielding/deshielding properties visible in NMR shifts of probe nuclei, which can be calculated routinely from various approaches using the density functional theory (DFT)^{5,6} even for fairly large nanostructures^{7,8,9}. However, after the pioneering work from Jusélius and colleagues^{10}, efficient selfconsistent methods to calculate, analyse and visualize local MICs inside complex nanostructures have been lacking, preventing detailed analyses of the local electronic structure and magnetic shielding.
Ligandstabilized metal clusters^{11,12} have emerged as tunable nanomaterials with a range of potential applications in catalysis^{13,14,15,16}, nanomedicine^{17}, biological sensors^{18}, and CO_{2} reduction^{19,20} among others. In most cases, the metal core of these clusters is composed of noble metals, possibly doped with transition metals. The core is protected chemically by organic ligand molecules which modify its electronic structure. The electronic structure of ligandstabilized metal clusters is commonly interpreted in terms of the superatom concept, introduced in 2008^{11}. Similar to electrons residing in quantized energy shells in the ordinary atoms, metal valence electrons in superatoms can respond to an external electromagnetic field and generate MICs in these rather complicated 3D nanostructures.
Here, we have implemented a numerically efficient method to calculate gaugeincluding MICs in the formalism of auxiliary density functional theory. We validate the method by studying two hydrogencontaining goldbased, ligand stabilized nanoclusters [HAu_{9}(PPh_{3})_{8}]^{2+} (1) and [PtHAu_{8}(PPh_{3})_{8}]^{+} (2). We are able to explain the measured^{21,22,23} anomalous difference in the proton NMR shift of the hydrogen in the metal core of these superatoms (15.4 ppm in 1 vs. 5.4 ppm in 2) by analyzing the local diatropic and paratropic MICs around the hydrogen atom. Surprisingly, Ptdoping of the gold core in (2) makes the system more aromatic, also increasing the magnetic shielding around the hydrogen. Our work presents a methodological advance for detailed analyses of the local electronic structure of the metal core and the protecting layer of superatoms, offering new possibilities to understand their growth, structure, and functions.
Results
Computation of the gaugeincluding magnetically induced current
Historically, computations of magnetic properties led to a problem that arises from the choice of the gauge origin. In a homogeneous medium, a magnetic field B can be defined through a vector potential A(r) (B = ∇ × A(r)) but the opposite does not hold due to the arbitrary scalar function found in B = ∇ × (A(r) + ∇ Φ(r))^{10}. In standard DFT computations the use of finite basis sets introduces a gauge dependence in the magnetic calculations that can be overcome only in the limit of a complete basis set. To avoid this problem, the idea to use magnetic fielddependent basis functions goes back to the calculation of ring currents by the seminal work of London^{24}. In this way the gauge dependence on the origin can be eliminated. These functions are known as gauge—including atomic orbitals (GIAO) defined in the Coulomb gauge (∇ ⋅ A(r) = 0). In largescale computations small basis sets can be used to achieve rapid convergence^{5,10}.
The shielding tensor σ_{C,λη} for the nucleus C is defined as the second derivative of the energy with respect to the external magnetic field B and the nuclear magnetic moment μ_{C} in the limit of zero magnetic field and zero nuclear magnetic moment (λ, η = x, y, z)^{25}.
An alternative expression to compute the nuclear magnetic shielding tensor via the Biot–Savart law^{1,26,27,28} is
where ε_{ηατ}(η, α, τ = x, y, z) is the Levi–Civita tensor, C is the position of the nucleus C and
are the firstorder elements of the tensor for the MIC density. Equating Eqs. (1) and (2) one can obtain the MIC that only includes direct dependence on basis functions, first derivatives of basis functions, density matrix, and magnetically perturbed density matrix. In this work, we take advantage of the high efficiency to calculate the nuclear shielding tensors^{5} employing GIAOs to compute gaugeincluding magnetically induced currents (GIMIC). The implementation was done in the deMon2k^{29} code (see short summary in Methods and the detailed discussion of relevant algebra in Supplementary Methods).
Atomic and electronic structure of the studied superatoms
We studied two previously reported phosphineprotected goldbased clusters, [HAu_{9}(PPh_{3})_{8}]^{2+} (1, ref. ^{21,30}) and [HPtAu_{8}(PPh_{3})_{8}]^{+} (2, ref. ^{22,23}). The position of the hydrogen atom in the metal core has been previously deduced by NMR analysis in clusters 1^{21} and 2^{22,23}. Their DFToptimized structures are shown in Fig. 1. Cluster 1 shows a C_{4v} symmetry in the core (formally (HAu_{9})^{2+}) and a C_{4} symmetry including the ligand layer. Cluster 2 shows a lower C_{2} symmetry both in the metal core (formally (HPtAu_{8})^{+}) and in the ligand layer. The two “crownlike” metal cores have an atom (Au/Pt) at the center and can be characterized by four distances d1–d4 and their respective distortions d1_{a,b}–d3_{a,b} (Fig. 1).
For the cluster 1 the distance d3 (2.77 Å) is longer than d4 (2.70 Å) both preserving a perfect square on top and bottom from the central Au atom (square edge being 3.57 and 3.40 Å respectively). In cluster 2 there are three characteristic distances from the central Pt atom, a shorter one to the top square (2.66 Å) and two longer (2.68 and 2.71 Å) to the bottom parallelogram. Both parallelograms are distorted from the perfect square showing longer distances on top (edges 3.65 and 3.58 Å) than on bottom (edges 3.10 and 3.15 Å). This indicates the effect of the central atom (Au/Pt) on the geometry. For cluster 1 the distance between H and the central Au is 1.72 Å and for cluster 2 the corresponding distance to central Pt atom is shorter, 1.63 Å.
The electronic structure of 1 and 2 was analyzed by projecting the frontier orbitals to clustercentered spherical harmonics Y_{lm}. The projected density of states (Supplementary Note 1 and Supplementary Fig. 1) shows that both clusters are 8electron superatoms with a closedshell configuration and significant HOMOLUMO energy gaps (1.88 eV for 1 and 1.93 eV for 2) as expected from their chemical formula and overall charge^{11}. The Bader atomic charge analysis^{31} showed for both hydrogens nearly similar, vanishing charges, −0.09 e in 1 and −0.06 e in 2. This, combined with analysis of the superatomic orbitals (Supplementary Fig. 1) indicates that both hydrogens can be thought of as neutral atoms, contributing their selectron to the superatomic electron system. Frontier orbital analysis of 1 done in ref. ^{21} led to similar conclusions.
NMR chemical shifts
First, we calculated proton shifts for the ortho, meta, and parapositions in PPh_{3} ligands for clusters 1 and 2 and compared with experimental data (Table 1 and Supplementary Table 1). This served also as an internal test for the DFT approximations (xc functional and basis set) in the calculations.
In cluster 1 the measured signals^{21} fragmented into three groups. A broad signal at 7.25 ppm was assigned to the oproton. This smeared out triplet peak was explained by the spin–spin coupling with the neighboring phosphorous and the mproton. The 6.62 ppm triplet signal was assigned to the mproton and the 7.05 ppm triplet was assigned to the pproton. The DFT results showed a 7.26–7.44 ppm chemical shift for the oproton depending on the level of theory employed. This proton showed the highest standard deviation (0.78–0.89 ppm) independently of the level of theory employed within the three o, m, and pproton analyzed, in good agreement with the chemical shifts observed. Computed values for the mproton were in the range of 7.16–7.27 ppm with standard deviation of 0.38–0.39 ppm, and for the pproton in the range of 7.44–7.53 ppm with standard deviation of 0.22–0.26 ppm.
Since the ligand layer of cluster 2 is similar to that of cluster 1, the measured o, m, and pproton shifts showed almost identical values to the ones measured for 1. The computed values were within 0.4 ppm from the experimental values for o and mprotons and within 0.2 ppm for pprotons (Table 1 and Supplementary Table 1). We note that all our computed values for the proton shifts are based on a single structure of the cluster and a full exploration of the distribution of the proton shifts due to ligand dynamics and potential solvent effects is not possible at the moment.
Next, we turn to the discussion of the NMR shift of the hydrogen in the metal cores of 1 and 2. The hydrogen peak of 1 was observed at 15.1 ppm^{21} in comparison with the previously reported 5.4 ppm for 2^{22,23} (Table 2 and Supplementary Table 2). This anomalously large difference (Δ = 9.7 ppm) is rather surprising in view of similar atomic geometries of 1 and 2. Our computed values are notably shifted upfield (Table 2 and Supplementary Table 2) but show consistently a qualitatively similar difference in the range of Δ = 4.3–4.5 ppm, independent of the level of DFT approximations. The systematic upfield shift and smaller Δ obtained in the computations could be explained by the limitations in the description of the electron density and the scalarrelativistic approximation used in the effective core potential^{32,33,34,35,36}. These approximations preclude the calculations of the shielding tensor on the true allelectron density around the Pt and Au atoms including also the full spinorbit effects.
The anomalous difference of the hydrogen shifts in 1 and 2 prompted the authors of ref. ^{21} to speculate the reasons arising either from different hydrogen charge or different magnetic shielding around the hydrogen in these superatoms. However, our charge analysis indicated essentially identical hydrogen charge in both systems as discussed above. Hence, we next seek the explanation by examining the computed magnetically induced currents in 1 and 2. We chose to use the PBE xc functional and the SDD pseudopotential in these computations.
Magnetically induced currents
The ninecomponent current density tensor \({{{{\mathcal{J}}}}}_{\tau }^{(\lambda )}\) (Eq. (3)) is a complex object and difficult to visualize. But, given a particular direction of the external magnetic field, one can analyze the current density as a vector function J(r). We chose an external magnetic field B = 1 T in a direction pointing from the central Au/Pt atom of clusters 1 and 2 to the hydrogen site and analysed J(r) in a cube containing 20 × 20 × 20 grid points centered at the central atom.
3D visualizations of J(r) (Supplementary Fig. 2 and Supplementary Movies 1 and 2) reveal a surprising qualitative indirect effect of the central Pt atom in cluster 2 on MICs around the corehydrogen site. It is seen that Ptdoping of the gold changes significantly the diatropic current around the corehydrogen as compared to cluster 1. To understand this interesting effect, a more quantitative analysis of this phenomenon is presented in Figures 2 and 3 and in Table 3. Figure 2 shows the projected vectorial field of J(r) as well as the diatropic and paratropic contributions in the xy planes containing either the corehydrogen or the central Au/Pt atoms in 1 and 2. Ptdoping in 2 has a tiny paratropic current basically at the metal site but induces a large area of diatropic current in the metal core of 2. In cluster 1, the corresponding gold site induces a deep maximum in the diatropic current immediately around the metal site but the paratropic contribution comes into play in the vicinity, neutralizing the diatropic effect. This interplay is seen quantitatively in Fig. 3 that shows circularly integrated currents in the xy planes containing the central metal site. Radial integrals of the contributions up to 4 Å radius show diatropic/paratropic components of (−2.619/0.715) Å^{2}nA/T in 2 as compared to (−1.626/1.184) Å^{2}nA/T in 1 (table 3). This causes the total integrated current to be −1.905 Å^{2}nA/T in 2 vs. −0.442 Å^{2}nA/T in 1 indicating an enhanced aromaticity in that plane in cluster 2.
The enhanced aromaticity in the Ptdoped cluster 2 has an indirect effect also in the plane containing the corehydrogen, revealing a key mechanism to understand the previously reported proton NMR shifts.^{21,22,23} Analogously to the central xy plane discussed above, the Ptdoped cluster has also a larger diatropic environment around the corehydrogen as compared to the allgold cluster. Radially integrated diatropic/paratropic currents up to 4 Å in the plane containing the corehydrogen are (−1.297/0.667) Å^{2}nA/T for 2 and (−1.034/1.253) Å^{2}nA/T for 1. As a result, the total integrated current around the corehydrogen in 1 is paratropic (0.218 Å^{2}nA/T) while in 2 it is diatropic (−0.63 Å^{2}nA/T). This qualitative and quantitative difference in the magnetic shielding between 1 and 2 provides the explanation for the reported^{21} downfield shift of the corehydrogen in 1.
Discussion
We have implemented and validated a numerically efficient method to calculate, analyse and visualize magnetically induced currents in nanostructures. The method is based on using gaugeincluding atomic orbitals in the framework of the auxiliary density functional theory. To validate the method, we studied diatropic and paratropic currents inside a metal core of two experimentally synthesized and characterized goldbased, hydrogencontaining ligandprotected clusters [HAu_{9}(PPh_{3})_{8}]^{2+} and [HPtAu_{8}(PPh_{3})_{8}]^{+}. Our computed proton NMR shifts for the hydrogens reproduce the anomalous trend observed in the experiment, where the hydrogen in [HAu_{9}(PPh_{3})_{8}]^{2+} was significantly downfieldshifted. Analysis of the MICs in the metal cores showed that Pt doping of the gold core in [HPtAu_{8}(PPh_{3})_{8}]^{+}, surprisingly, increases the overall contribution of diatropic currents in the core, making the cluster more aromatic. This results in a better magnetic shielding of the hydrogen in [HPtAu_{8}(PPh_{3})_{8}]^{2+} causing an upfield proton chemical shift while the net effect of the shielding currents is paratropic in a plane of the hydrogen in [HAu_{9}(PPh_{3})_{8}]^{2+} inducing deshielding and downfield proton chemical shift.
We expect that the methodological advancement reported in this work will allow for detailed studies of local electronic structure in ligandstabilized metal clusters which will yield an improved understanding of their physicochemical properties and tunability for diverse applications.
Methods
Ground state DFT calculations
The structures of clusters 1 and 2 were optimized using the realspace grid based GPAW^{37,38} program with a uniform grid spacing of 0.2 Å. Pt, Au, C, and P atoms were described by valence of 16, 11, 4, and 5, respectively, and the frozen core approximation was used for inner electrons. The PAW setups for Pt and Au include scalarrelativistic effects. The PBE^{39} functional was employed for the exchangecorrelation interaction. Structure optimization was continued until the force on each atom was less than 0.05 eV/Å. To identify the superatom symmetries of the frontier orbitals, the Y_{lm} analysis^{11} was performed for all the clusters with a cutoff radius of 4.0 Å. A Bader charge analysis^{31} was performed for the studied systems using the density obtained in a single point computation in GPAW.
NMR chemical shifts and magnetic currents
The magnetic shielding tensors and gaugeincluding magnetically induced currents were computed in deMon2k^{29} code employing the Auxiliary Density Functional Theory (ADFT) approach. For the ^{1}H NMR chemical shift and the MIC computations the geometries optimized in GPAW were used with PBE^{39} and BP86^{40,41} level for exchangecorrelation effects. The Stuttgart–Dresden (SDD) pseudopotential^{42} (Au and Pt are described by a valence of 19 and 18 electrons respectively) along with the DZVP^{43} basis set (P, C, and H) and the effective core potentials LANL2DZ^{44} along with D95^{45} basis set (P, C, and H) were employed. All the computations were performed in combination with the GENA2* auxiliary function set. Each ^{1}H chemical shift was referenced to the ^{1}H TMS chemical shift computed in the respective level of theory.
Code availability
Both DFT codes (GPAW and deMon2k) used in this work are free to download for academic use at the respective sites. GPAW can be downloaded at https://wiki.fysik.dtu.dk/gpaw/. The gaugeincluding MIC calculations are implemented in the version 6.1.8 of the deMon2k code (http://www.demonsoftware.com/public_html/index.html).
Change history
20 January 2022
A Correction to this paper has been published: https://doi.org/10.1038/s4146702228053w
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Acknowledgements
This work was supported by the Academy of Finland (grants 294217, 319208, 315549), and through H.H.’s Academy Professorship. E.S. acknowledges the The Finnish Cultural Foundation for a PhD study grant. The computations were made at the Nanoscience Center of the University of Jyväskylä by utilizing the FCCI  Finnish Computing Competence Infrastructure (persistent indentifier urn:nbn:fi:researchinfras2016072533). B.Z.G. acknowledges the funding from CONACyT project CB2015258647.
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O.L.E. conceived the concept, performed all DFT calculations, analyzed the results together with E.S. and S.M. and wrote the first manuscript draft. B.Z.G. implemented the GIMIC method in deMon2k. H.H. supervised the work. All authors commented on the manuscript draft that was finalized by H.H.
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LópezEstrada, O., ZunigaGutierrez, B., Selenius, E. et al. Magnetically induced currents and aromaticity in ligandstabilized Au and AuPt superatoms. Nat Commun 12, 2477 (2021). https://doi.org/10.1038/s4146702122715x
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DOI: https://doi.org/10.1038/s4146702122715x
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