Spin and orbital magnetism of coinage metal trimers (Cu₃, Ag₃, Au₃): A relativistic density functional theory study

The coinage metal trimers have been studied both experimentally and theoretically for more than three decades.^1–7 These objects are initial blocks to study the larger noble metal clusters and especially for study of gold nanoparticles. However, there was a big challenge to investigate ground state of the coinage metal trimers and substantial efforts have been devoted to the spin magnetism of these trimers. It is worth mentioning that the spin magnetism plays an essential role in transition metals whereas it is generally believed that the orbital magnetism is very small in the bulk phase of transition metal compounds due to the strong hybridization and ligand fields. In a molecule there are more degrees of freedom for valence electrons and therefore large orbital moments may form for individual atoms of transition metal molecules. In particular, the gold trimer with strong spin-orbit interaction in its individual atoms^8,9 is a potential candidate for significant orbital magnetism.

In an earlier study, Xiao and co-workers using projector augmented wave (PAW) method in the framework of scalar relativistic regime have shown that the global minimum for Au₃ is the geometry with an apex angle around 137 degree.¹⁰ They have argued that the spin-orbit coupling does not change the relative stability of Au₃ trimer. These authors have found a triplet magnetic state with 0.33μ_B for individual gold atoms in Au₃ trimer. This finding is problematic since a typical trimer X₃ especially with an obtuse apex angle would have two different atoms in which one apical X(1) atom and two terminal X(2) atoms. Therefore one can expect different electronic configurations for these two type of atoms and thereby different magnetic properties. Arratia-Perez and co-workers in their Dirac spin-restricted calculations have shown that the total spin magnetic moment of Au₃ trimer is 0.811μ_B in which the spin magnetic moment of terminal gold atoms is 0.357μ_B per atom and the spin moment of apical atom is 0.097μ_B.¹¹ These results have been obtained in scalar relativistic regime where the spin-orbit coupling has been dropped in the Dirac spin-restricted calculations. From experimental point of view, the most important results for the spin magnetism of copper, silver, and gold trimers are reported by Howard and co-workers using electron spin resonance (ESR) spectroscopy technique which they have been prepared in hydrocarbon environments.¹² More details of these experimental data are discussed in section III. Based on the mentioned theoretical and experimental studies the neutral coinage metal trimers are magnetic molecules whereas to the best of our knowledge there is no study on the orbital magnetism of coinage metal trimers in the literature. In the present work we study the electronic structures, spin and orbital magnetism of noble metal trimers (Cu₃, Ag₃, and Au₃) using the relativistic density functional theory (RDFT). The paper is organized as follows: Section II contains details about the method of calculations and the numerics. Section III presents the results and discussion (structural properties of the coinage metal trimers, and their spin and orbital magnetic moments). Finally, the paper is summarized in Section IV.

The density functional theory calculations were carried out using the relativistic version of the full potential local orbital method (FPLO-9).¹³ We have used the non-relativistic generalized gradient approximation (GGA) with the Perdew-Burke-Ernzerhof 96 version¹⁴ for the exchange-correlation potential which has been implemented in the FPLO package. In the FPLO scheme the four component Kohn-Sham-Dirac (KSD) equation in the framework of RDFT is solved self-consistently up to all orders.^15,16 The exact form of spin-orbit (SO) coupling which is implicitly contained in the kinetic term of KSD equation, was taken into account in our full relativistic (FR) calculations. In a scalar relativistic (SR) method of K. Koepernik¹⁷ implemented in the FPLO package, mass-velocity and Darwin terms have been taken into account in the KSD equation and the SO coupling has been dropped in the SR approach. The following basis set was adopted as the valance states including (3s, 3p, 3d, 4s, 4p, 4d, 5s), (4s, 4p, 4d, 5s, 5p, 5d, 6s) and (5s, 5p, 5d, 6s, 6p, 6d, 7s) for the copper, silver and gold atoms, respectively. We have performed an exhaustive minima search energy, to predict all possible low-energy structures of Cu, Ag, and Au trimers in the framework of RDFT. In order to optimize the structure of these trimers, the position of all atoms were fully relaxed without any symmetry constraint. In the optimization process, total energies and all forces were converged to 0.001 meV and 0.001 eV/Å, respectively. The atomization energy (AE) per atom for each trimer is calculated as AE = −(E_t-3E_s)/3 where E_t is the total energy of the trimer and E_s is the energy of single atoms.

As starting point of discussion, we have calculated the equilibrium bond lengths of coinage metal dimers. In a non-relativistic approach (without any relativistic correction to the kinetic term of KSD equation) we have found equilibrium bond lengths of 2.258Å, 2.670Å, and 2.500Å for the Cu₂, Ag₂ and Au₂ dimers, respectively. In our scalar relativistic method, the equilibrium bond lengths of Cu₂, Ag₂ and Au₂ dimers were calculated as 2.226Å, 2.586Å, and 2.540Å, respectively. In the full relativistic scheme we have found that the spin-orbit coupling does not change significantly the equilibrium bond lengths of the dimers compared to the results obtained in the SR method. Our findings are in good agreement with the experimental bond lengths measured by Ho and co-workers.¹⁸ These authors using negative ion photoelectron spectroscopy technique have reported 2.219Å for Cu₂, 2.480Å for Ag₂ and 2.472Å for Au₂ neutral dimers. Based on DFT-GGA calculations, Häkkinen and Landman¹⁹ have found 2.57Å for gold dimer and later on Majumder and Kulshreshtha²⁰ have also obtained a bond length of 2.53Å for this dimer. These theoretical values can be compared to our SR-GGA calculated bond length of gold dimer (2.540Å). This comparison is indicated that our relativistic method is reliable.

In Table I the global and local equilibrium apex angles (θ_A), the equilibrium side lengths (D_e), the energy differences between equilibrium triangular and linear trimers (ΔE), the atomization energy per atom (AE), the individual magnetic moments for both apical and terminal atoms and also total magnetic moments of the trimers in SR and FR approaches are presented.

For Cu₃ within SR approach, we have found a global minimum at 67.2° acute apex (AA) angle and a local minimum at 120.1° obtuse apex (OA) angle. The energy difference between these two configurations is 117.16 meV. Jug and co-workers in their non-relativistic DFT calculations have shown that the Cu₃ trimer has a ²B₁ electronic ground state with 67.16° apex angle²¹ and also Ho and co-workers have also shown that the bent structure of Cu₃ trimer with acute apex angle is more stable than its obtuse structure.¹⁸ The local and global equilibrium angles and energy differences (ΔE’s) of Cu₃ trimer are affected marginally in the full relativistic approach. As it can be seen in Table I the most contribution to the total magnetic moment of Cu₃ trimer is localized at the terminal Cu(2) atoms. This feature has been already confirmed after experiment of Howard and co-workers where they have analyzed the Cu₃ in C₁₀H₁₆ hydrocarbon host matrix. They have found spin magnetic moment of 0.026μ_B for the apical Cu(1) atom and spin magnetic moment of 0.290μ_B for the terminal Cu(2) atoms.¹² The calculated orbital moments in FR approach are almost quenched in the Cu₃ trimer. Therefore one can conclude that all 3d states in the Cu₃ are almost full and occupied. It has to be pointed out that in the SR-GGA method the orbital moments are completely quenched since the spin-orbit coupling is dropped form the KSD equation.

In the case of Ag₃ trimer using SR (FR) approach we have found a global minimum at 138.3° (137.0°) and local minimum at 71.6° (71.2°), respectively. The linear Ag₃ trimer is more stable than that of Ag₃ trimer formed at the local minimum in both SR and FR approaches (see Table I). Fournier using Vosko-Wilk-Nusair exchange-correlation functional in the framework of scalar relativistic calculations has shown that the most stable apex angle of Ag₃ trimer is 69°.²² This result can be compared to our finding for local minimum structure. It has to be clarified that the spin-orbit coupling does not change significantly the diagram of minima search energy for Ag₃ trimer. As it is indicated in Table I, the local magnetic moment for apical silver Ag(1) is very small and most contributions to the total magnetic moment of Ag₃ trimer is due to the magnetic moment of its terminal Ag(2) atoms. The calculated magnetic moments of pure Ag₃ can be compared to experimental data (0.060μ_B for apical Ag(1) and 0.440μ_B for terminal Ag(2)) which have been measured by Howard and co-workers for Ag₃ in C₁₀H₁₆ host matrix.¹² In presence of spin-orbit coupling the orbital magnetic moments of Ag₃ trimer for both local and global minimum structures are almost quenched (see also Table I).

Within SR-GGA approach we have found a local minimum at 66.2° AA angle and a global minimum at 137.4° OA angle for Au₃ trimer. This finding can be compared to the obtuse apex angle of 137° found by Xiao and co-workers in the framework of scalar relativistic PAW method. Using scalar relativistic calculations with plane-wave expansion method a similar apex angle has been found by Majumder and Kulshreshtha.²⁰ In the FR-GGA approach we have obtained a global minimum at 147.5° obtuse apex angle for Au₃ trimer. Rusakov and co-workers using small-core shape-consistent relativistic pseudopotentials method and with highly economical Becke-Perdew GGA functional have found a global minimum structure at 144.9° apex angle and also using the Tao-Perdew-Staroverov-Scuseria meta-GGA exchange-correlation functional they have found a local minimum structure with 141.9° apex angle for Au₃ trimer.²³ We have shown that gold trimer with equilibrium obtuse apex angle is more stable than the structure with acute symmetry. As it is depicted in Table I, the energy differences between these two structures are 80 meV and 48 meV in SR-GGA and FR-GGA approaches, respectively. A local minimum at 62° acute apex angle is found within FR-GGA method. This angle is smaller than the acute apex angle obtained in our scalar relativistic calculations (66.2°) by about 6.5%. Our FR-GGA results show that the exact form of spin-orbit coupling do not remove completely the Jahn-Teller distortion in acute angled geometry of Au₃ trimer. Guo and co-workers have argued that the spin-orbit coupling makes an ideal D_3h geometry for Au₃ trimer which is in contrast to our finding.²⁴ Wesendrup and co-workers using relativistic coupled cluster method at the level of scalar relativistic calculations have also shown that the neutral Au₃ with acute symmetry is more stable than the obtuse geometry by about 7 meV which is again in contrast to our results. On the other hand these authors have neglected the spin-orbit coupling because they have proposed that the Ham effect would quenched the spin-orbit coupling in a symmetric object.²⁵ Based on our full relativistic calculations we have shown that the Au₃ trimer is not a equilateral triangle with acute symmetry. In the following, we have shown that the spin-orbit coupling causes spin and orbital magnetism in the asymmetric gold trimer.

In SR-GGA approach the most contribution to the spin magnetism of the Au₃ trimer with obtuse angled geometry is localized at terminal Au(2) atoms with 0.41μ_B while a smaller spin magnetic moment (0.178μ_B) was obtained for apical Au(1) atom. In the presence of spin-orbit coupling, 0.260μ_B (0.150μ_B) and 0.365μ_B (0.351μ_B) were found for spin magnetic moment of apical Au(1) and terminal Au(2) atoms in acute (obtuse) angled structure, respectively. Howard and co-workers using ESR technique have found 0.060μ_B and 0.420μ_B for apical and terminal gold atoms of Au₃ trimer resolved in C₆H₆ host matrix.¹² As it can be seen in Table I the small orbital magnetic moments of individual gold atoms are aligned in opposite direction to their spin magnetic moments in the local minimum structure with 62° apex angle. This finding are obviously in contrast to the third Hund's rule. In our FR-GGA method, a significant orbital magnetic moment (0.13μ_B) was found for apical Au(1) atom of the global minimum structure with 147.5° apex angle. The orbital moment for terminal Au(2) atoms in this structure was found 0.04μ_B per atom.

It can be argued that the magnetic behavior of gold trimer can be understood based on its highest occupied molecular orbital where it is also known as a single occupied molecular orbital (SOMO).¹¹ The model of electronic SOMO states together with SO interaction can give forth an explanation for the large GGA calculated orbital moments in Au₃ trimer. In Fig. 1, we have shown the spin partial density of states (PDOS) for the SOMO of Au₃ trimer with OA angle at its global minimum structure in both SR-GGA and FR-GGA approaches. The PDOS’s at SOMO level were calculated by broadening of the molecular orbitals using a line-shape function Lorentzian with a typical width of 0.1 eV and based on Mulliken analysis.²⁶ As it can be seen in Fig. 1 the most contribution to the spin PDOS of SOMO in the Au₃ trimer is due to the 6s states of terminal Au(2) atoms. Therefore a large spin magnetic moment can be estimated for the terminal gold atoms compared to the small magnetic moment of apical Au(1) atom (see also Table I). The SOMO of Au₃ trimer calculated in the FR-GGA approach was formed with majority and minority spins (Fig. 1(b)) whereas all components of SOMO for Au₃ trimer in the SR-GGA approach are at the majority spins (Fig. 1(a)). Therefore one can conclude that the calculated total spin magnetic moment of Au₃ trimer using SR method is larger than the values obtained in FR approach. In Figs. 1(a) and 1(b), it can be seen that the 5d characters of apical gold atom have more contribution to the PDOS’s of SOMO than the 5d character of terminal gold atoms.

In order to explain the origin of orbital magnetism in the gold trimer we present a very simple model using a tight binding approach proposed by Ebert and co-workers.²⁷ Based on this model, the size of orbital moments can be approximated to first order by the difference of the spin-split 5d-resolved partial densities of states at the SOMO level. Since the PDOS’s at the SOMO levels are Lorentzian, a simple expression was obtained for the orbital magnetic moment of each atom in Au₃ trimer:

where the spin-orbit coupling parameter ξ_SO for the atomic 5d-shell of gold atom is about 1.5 eV. The n(5d,↑) is the occupation number of 5d characters in the majority of spins obtained within the SR-GGA approach. Although this expression for orbital magnetic moments is oversimplified due to the assumption of tight binding in the SOMO level, it still explains significant orbital magnetism in the gold trimer. Based on the Fig. 1 a, we have calculated that the 5d character of terminal Au(2) atom has a small occupation number 0.03 electrons at the SOMO level while the occupation number of 5d character in the apical Au(1) atom is about 0.1 electrons per SOMO level. According to the above equation the orbital magnetic moments for apical Au(1) and terminal Au(2) atoms were found 0.15μ_B and 0.045 μ_B, respectively. These values are in qualitative agreement with the calculated orbital magnetic moments obtained in the FR-GGA approach.

Relativistic density functional theory calculations were performed on the basis of full potential local orbital scheme to investigate the equilibrium structure and magnetic properties of the coinage metal trimers (Cu₃, Ag₃ and Au₃). We have calculated the spin and orbital magnetic moments of individual atoms in these trimers. It was shown that the spin-orbit coupling does not provide significant changes for the structure and magnetic properties of copper and silver trimers while the orbital magnetism of gold trimer is more affected. Therefore one can conclude that the spin-orbit coupling has to be taken into account in particular for the ever-expanding field of magnetism in nanoparticles formed by heavy elements. Further experiments such as x-ray magnetic circular dichroism measurements together with electron spin resonance spectroscopy methods in a gaseous or vacuum environment to check our predictions for spin and orbital magnetism of coinage metal trimers are also desirable.

The computing facilities of high performance computational laboratory at Department of Physics, Iran University of Science and Technology (IUST) were used. Financial support by research division at IUST is gratefully acknowledged.