Density Functional Theory Studies of the Electronic Structure and Muon Hyperfine Interaction in [Au25(SR)18]0 and [Au25(SeR)18]0 Nanoclusters.

Density functional theory computational investigation was performed to study the electronic structures, muon sites, and the associated hyperfine interactions in [Au25(SR)18]0 and [Au25(SeR)18]0 where R is phenylethane. The calculated electronic structures show inhomogeneous spin density distribution and are also affected by different ligands. The two most stable muon sites near Au atoms in the thiolated system are MAu11 and MAu6. When the thiolate ligands were replaced by selenolate ligands, the lowest energy positions of muons moved to MAu6 and MAu5. Muons prefer to stop inside the Au12 icosahedral shell, away from the central Au and the staple motifs region. Muonium states at phenyl ring and S/Se atoms in the ligand were found to be stable and the Fermi contact fields are much larger as compared to the field experienced by muons near Au atoms.

Introduction

Magnetic properties of gold nanoparticles and nanoclusters have vast potential in various applications. However, the magnetism of these materials is still unclear, as discussed by Agrachev et al.[1] Contradictory results of various experimental and theoretical investigations could be due to different ligands that cap the gold nanoclusters.[2] Thiolate-passivated gold nanoclusters with icosahedral geometry Au25(SR)18 have been extensively studied because of their high stability, intriguing properties, and ease of synthesis.[3−9] There are three possible charge states for the stable redox species of Au25(SR)18, that is, q = −1, 0, and +1.[10−15] The superatomic electron configurations of the three charge state nanoclusters are 1s21p6, 1s21p5, and 1s21p4, which are diamagnetic, paramagnetic, and diamagnetic, respectively.

Previous studies have reported that when the thiolate ligands are substituted with selenolate ligands, the stability in reactions involving dissociation of the gold–ligand bond increases, but the stability of the nanoclusters in reactions involving intramolecular dissociation of the ligand is reduced.[16−18] This will produce gold nanoclusters with selenolate ligands that are more stable against degradation in solution than the thiolate-protected gold nanoclusters. Among chalcogens, selenolates and tellurides are considered as substitutes for thiolates in the gold nanocluster systems. However, because of the limited stability of tellurium-based monolayers, selenolate appears to be a more suitable candidate.[19]

Goikolea et al. were the first to report the magnetic properties of thiolated-gold nanoparticles probed by muon spin relaxation (μSR) measurement, using 2.1 nm of dodecanethiol-capped Au nanoparticles.[20] They have estimated the internal field at the muon stopping site to be around 0.4 T from the observation of the muon-spin precession. However, this result is unreliable because they used a pulsed muon which has a 70 ns pulse width that gives the detection limit of the internal field at the muon site to be 1 kOe at most.[21] Recently, more explicit evidence for magnetism was found using μSR measurement in butanethiol-capped gold nanoparticles with 2.2 nm in diameter.[2] Their results also indicated the presence of a broad range of internal magnetic fields at the muon stopping site, which is consistent with a spatially inhomogeneous distribution of magnetic moments.

Computed electronic structures of muonated systems can provide an insight into the distribution of spin as well as the Fermi contact and dipolar contributions to the hyperfine field experienced by the implanted muon. In this regard, we have previously reported our computational work on muon sites in [Au25(SR)18]0 nanocluster, where R is a hexyl group.[22] In the current work, we present the results of our density functional theory (DFT) computational investigations on the electronic structure, muon sites, and hyperfine interactions in two nanocluster systems, [Au25(SR)18]0 and [Au25(SeR)18]0, where R is a phenylethane group. As to gauge the effects of different ligands on the electronic structure, muon sites, and muon hyperfine interactions, two chalcogen atoms, sulfur and selenium, were used. The detailed structures used and the computational approach are given in the next section.

Results and Discussion

Spin Density

We now present the spin density distribution in [Au25(SR)18]0 and [Au25(SeR)18]0 nanoclusters. A comprehensive understanding of the spin density distribution in the host systems could provide a general insight into the muon hyperfine interaction in the systems. Figure shows the spin density distribution, viewed in the Au8S6/Au8Se6 plane. To demonstrate the difference in the spin density distribution of [Au25(SR)18]0 and [Au25(SeR)18]0 nanoclusters, the orientations of the Au8S6 and Au8Se6 plane in Figure a,b are kept the same.

Figure 1

Isosurface plots showing spin density distribution: (a) [Au25(SR)18]0 and (b) [Au25(SeR)18]0. The images are viewed in the Au8S6/Au8Se6 plane at isovalue of 0.004 e–/au.[3] The carbon atoms are displayed as a wireframe, and the hydrogen atoms in the coordinating ligands are omitted for clarity. The blue and green surface plots represent the positive and negative spin densities. Spin density at individual atoms are given in Table S1 in the Supporting Information section.

As can be seen in Figure , the distributions of spin density in both systems are rather inhomogeneous and span almost vertically across the Au8S6/Au8Se6 plane but slightly inclined in different directions. A closer look at the spin distribution in the two systems reveals that the selenolate system (Figure b) has a more localized spin density distribution, centered in the vicinity of the central core (Au1). These are expected as the optimized geometrical structures of the two systems are not the same due to different coordinating ligands.[23] The anisotropic effects of these Au nanoclusters were previously reported by Fortunelli et al.,[24] whose work focused on optical and photoluminescence properties. According to Tofanelli et al.,[15] the inhomogeneity of the spin densities for both systems is due to the molecular symmetry distortions, corresponding to the Jahn–Teller symmetry breaking, which is prominent in [Au25(SR)18]0 nanoclusters. The deviation from the idealized polyhedral shape increases as the charge state of Au25(SR)18 systems changes from −1 to +1. Previous DFT[25,26] studies and vibrational analysis[27] reported that the use of certain ligands could also destroy the inversion symmetry and causes significant distortions to the AuS framework. Tlahuice-Flores et al.,[25] however, revealed that low-polarity R groups such as phenylethane ligand in the [Au25(SR)18]− nanocluster have no significant effect on the structure.

The result of a recent μSR experiment on nanogold particles also indicates inhomogeneity in the spin distribution.[2] Dehn et al. have observed a significant μSR in butanethiol-capped gold nanoparticles, indicating the presence of a broad range of internal magnetic fields at the muon stopping site, which is consistent with a spatially inhomogeneous distribution of magnetic moments.[2] They also suggested that the magnetic properties of the nanoparticles are dependent on the number of electrons or density of states at the Fermi level, which in turn depends on the type and number of ligands.

The spin densities of the positively and negatively charged (staple motifs and Au13 core) components for the [Au25(SR)18]0 and [Au25(SeR)18]0 molecular systems are summarized in Table . The spin density distributions in the staple motifs, [−(SR)–Au–(SR)–Au–(SR)−]6/[−(SeR)–Au–(SeR)–Au–(SeR)−]6 and phenylethane component are given separately in Table . In general, spin density distribution in the thiolated system was found to be more delocalized compared to the system with Se ligands. For both systems, most of the spin densities are distributed inside the icosahedral Au13 region, 62, and 75% for the thiolated and selenolated systems, respectively. Furthermore, the spin densities are mainly concentrated around the central gold atom Au1. In both systems, −0.17 of spin density is distributed over the Au12 icosahedral shell area, consisting of 12 Au atoms, Au2–Au13. Spin densities of about 0.02 and −0.11 for each thiolate– and selenolate–gold nanoclusters are spread over a large area throughout the phenylethane components.

Table 1

Spin and Charge Density Distribution, Showing How the Singly Unpaired Spin, As Well As Partial Charges, Are Distributed Accordingly in the Particular Components of the Molecule for Both Nanocluster Systems

	spin density		charge density
structural components	[Au25(SR)18]0	[Au25(SeR)18]0	[Au25(SR)18]0	[Au25(SeR)18]0
Au13 (icosahedral core)	0.62	0.75	–0.76	–1.47
Au12 (icosahedral shell, Au2–Au13)	–0.17	–0.17	+0.68	0.00
[−(SR)–Au–(SR)–Au–(SR)−]6/[−(SeR)–Au–(SeR)–Au–(SeR)−]6 (staple motifs)	0.38	0.25	+0.77	+1.47
(CH2CH2Ph)18	0.02	–0.11	+0.63	–0.23

Charge Density

Figure a,b shows the isosurface plots of charge densities for thiolate– and selenolate–gold nanoclusters, indicating the distribution of partial charge densities throughout the systems. The charge densities were calculated using the natural bond orbital (NBO) method, and the values for essential components of the molecular systems are tabulated in Table . The similar proportions of total charge densities of the negative (Au13) and positive [Au12(SR)18 or Au12(SeR)18] species indicate that there is a strong intermolecular force in the form of electrostatic interaction between the positive species and the negative species in both systems. This result is similar to the findings of previous DFT studies,[28,29] that a significant difference exists in the charge distribution at Au13 core atoms and the outer Au12 atoms in the staple motifs. In particular, the superatomic Au13 core is more electron rich. On the other hand, because of electron transfer to the sulfur of the thiolate ligand, the exterior Au12 atoms in the staple motifs are electron deficient.[28] The charges on Au1 atom are −1.44 and −1.46 for the thiolated and selenolated systems, respectively. The other Au atoms in the icosahedron are relatively neutral. Figure shows the contour plots of electron density distribution on the Au8S6/Au8Se6 plane. Significant differences in the distribution of electrons could be noticed between the Au12(SR)18 and Au12(SeR)18 regions.

Figure 2

Charge density distribution of the (a) [Au25(SR)18]0 and (b) [Au25(SeR)18]0 nanoclusters using NBO population analysis. The color range was set to +1.00 to −1.00. Charge density values at individual atoms are given in Table S1 in the Supporting Information section.

Figure 3

Contour plots of electron density distribution of (a) [Au25(SR)18]0 and (b) [Au25(SeR)18]0 nanoclusters on the Au8S6/Au8Se6 plane.

The calculated average Au1–AuX bond distances, where X = 2–13 represents the Au atoms in the icosahedral shell, are 2.869 and 2.872 Å for the thiolate– and selenolate–gold systems, respectively, while the corresponding average AuX–AuY bond distances among the nearest neighbor gold atoms in the Au12 icosahedral shell are 3.018 and 3.021 Å.

The calculated average bond distances between the gold atoms in the Au12 icosahedral shell and the chalcogenide ligands, Au–S and Au–Se, are 2.456 and 2.539 Å. In comparison, the Au–S and Au–Se average bond distances for Au in the staple motifs are slightly shorter, 2.384 and 2.489 Å, respectively. The average charge on the Au atoms in the Au12 icosahedral shell is +0.26e, while for the sulfur atoms in the ligands is −0.17e, leading to a relatively significant electronegativity difference. Previous experimental and computational data reported that the Au–S bonds in the coordinating ligands exhibit weak charge polarization (Auδ+–Sδ−), implying that Au(I)–thiolate bonding character is present[30] with the possibility of partial ionic character.[31] In contrast, the difference of electronegativity between the Au and Se ligand in the selenolated gold nanocluster is not that significant.

Molecular electrostatic potential (MEP) map can visually analyze the negatively and positively charged region of a molecule. The varying intensities of the electrostatic potential (ESP) mapped surface plots demonstrate the ESP energy values; the red area represents the region where the potential is negative due to high electron density, while the blue area indicates the opposite. The green and yellow regions represent intermediary potentials. In this work, MEP was utilized as an initial prediction tool to determine possible muon stopping sites in both nanocluster systems. A muon that carries one positive charge can be considered an electrophile and is therefore attracted to electron-rich regions in the host systems.

Figure a,b shows the MEP surface plots for the [Au25(SR)18]0 and [Au25(SeR)18]0 nanoclusters. As shown in the figures for both systems, the Au13 inner core regions are filled with red, an indication of favorable sites for the electrophilic attack. Additionally, there are also electron-rich spots in the coordinating ligands around the phenyl groups. Hence, for further muon site investigation, a total of 16 possible muon stopping sites comprising 14 sites near the Au atoms were considered. We have also investigated possible muonium (Mu) states at five sites in the ligand region. They are ortho-, meta-, and para-positions at the phenyl ring and two sites near sulfur (selenium) atoms.

Figure 4

MEP map generated from the calculated total electron density of (a) [Au25(SR)18]0 and (b) [Au25(SeR)18]0 nanoclusters. Both maps are displayed as viewed in the Au8S6/Au8Se6 planes.

Muon Site Estimation and Hyperfine Interaction

Table summarizes the main findings of our computational investigations on muon stopping sites and their associated hyperfine interactions in [Au25(SR)18]0 and [Au25(SeR)18]0 nanoclusters. The “initial site” column specifies the initial muon site before geometry optimization. The initial μ–Au was set to 1.640 Å. The second column, “converged muon site” displays the stable muon site after the geometry optimization procedure. The site with the lowest total energy for each particular system was made as the reference energy for all muon sites in a particular nanocluster. Thus, its relative energy was taken as zero. We will discuss the two most stable muon sites in each system in detail, where the relative energy is less than 0.01 and 0.02 eV, respectively, for the muonated [Au25(SR)18]0 and [Au25(SeR)18]0 nanoclusters.

Table 2

Relative Energy, Spin Densities, and the Distance between μ with Its Nearest Host Atom of the Sixteen Muon Sites for Thiolate– and Selenolate–Gold Nanoclustersa

[Au25(SR)18]0								[Au25(SeR)18]0
initial site	converged muon site	relative energy (eV)	nearest neighbors	μ–host atom (Å)	Aiso (MHz)	Baniso (MHz)	hfcc (MHz)	initial site	converged muon site	relative energy (eV)	nearest neighbors	μ–host atom (Å)	Aiso (MHz)	Baniso (MHz)	hfcc (MHz)
MAu11	MAu11	0.00	Au7, Au1	1.757	–12.10	6.44	–5.66	MAu6	MAu6	0.00	Au1, Au12	1.772	2.85	7.12	9.97
MAu6	MAu6	0.01	Au1, Au12	1.755	–12.67	7.12	–5.55	MAu1	MAu5	0.02	Au1, Au2	1.770	–2.42	7.29	4.87
MAu12	MAu6’	0.01	Au1, Au12	1.755	–12.95	7.46	–5.49	MAu2	MAu5’	0.02	Au1, Au2	1.769	–2.28	7.29	5.01
MAu7	MAu11’	0.08	Au7, Au1	1.752	–5.41	7.12	1.71	MAu5	MAu5″	0.02	Au1, Au2	1.770	–3.13	7.29	4.16
MAu5	MAu5	0.10	Au1, Au2	1.762	21.35	8.14	29.49	MAu13	MAu13	0.04	Au1, Au8	1.766	–4.84	7.97	3.13
MAu9	MAu13’	0.10	Au1, Au8	1.753	1.42	8.14	9.56	MAu8	MAu13’	0.05	Au1, Au8	1.765	–1.71	7.80	6.09
MAu2	MAu5’	0.11	Au1, Au2	1.771	25.05	8.48	33.53	MAu11	MAu11	0.06	Au1, Au7	1.773	18.36	6.78	25.14
MAu3	MAu5″	0.11	Au1, Au2	1.767	22.63	8.99	31.62	MAu14	MAu11’	0.06	Au1, Au7	1.874	–17.36	6.44	–10.92
MAu13	MAu13	0.12	Au1, Au8	1.761	12.95	9.49	22.45	MAu12	MAu10	0.09	Au9, Au12	1.959	20.35	9.49	29.84
MAu1	MAu9	0.13	Au1, Au10	1.850	–17.36	7.29	–10.07	MAu9	MAu10’	0.11	Au9, Au12	1.898	37.01	10.34	47.35
MAu8	MS3	0.25	Au8	1.386	–3.42	5.26	1.84	MAu10	MAu10″	0.11	Au9, Au12	1.792	36.58	10.34	46.92
MAu4	MAu4	0.30	Au1, Au8	1.839	87.82	4.58	92.40	MAu7	MAu4	0.14	Au3, Au7	1.795	37.58	10.51	48.09
MAu10	MAu10	0.39	Au1, Au5	1.799	97.50	2.54	100.04	MAu3	MAu4’	0.15	Au3, Au7	1.791	42.13	10.51	52.64
MAu14	MAu11″	0.50	Au7	1.743	57.36	3.05	60.41	MAu4	MAu4″	0.15	Au3, Au7	1.795	42.84	10.68	53.52

The corresponding hyperfine coupling constant (hfcc) with isotropic Fermi contact (Aiso) and dipolar coupling constants (Baniso) for all muon sites are also given in the table.

Out of the 16 possible muon sites investigated in this work, MAu11 is the site that has the lowest energy in the [Au25(SR)18]0 nanocluster. A muon, which was initially placed at a distance of 1.640 Å from Au11, moved farther away with μ–Au11 distance being 1.757 Å after the geometry optimization procedure.

The site is positioned 0.560 Å above the plane formed by its nearest Au neighbors, Au1, Au7, and Au11, as depicted in Figure a. Figure b shows the interatomic distances between the muon and each of its nearest neighbors and distances among the atoms after the muon’s perturbation. In the muoniated system, these interatomic distances are elongated by about 4–6% to accommodate the presence of the muon and to stabilize the site. As tabulated in Table , the calculated muon isotropic Fermi contact coupling constant Aiso at the MAu11 is −12.10 MHz, an indication of spin polarization effect, while the anisotropic contribution Baniso is 6.44 MHz.

Figure 5

(a) Lowest energy muon site in [Au25(SR)18]0, MAu11. (b) Magnified picture of the corresponding site showing the bond distance between μ and Au11 as well as between μ and other neighboring atoms, Au1 and Au7. (c) Au1, Au11, and Au7 interatomic distances before the introduction of a muon into the host system.

There are two sites labeled MAu6 and MAu6’ with equal relative energy of 0.01 eV. The initial muon positions for MAu6 and MAu6’ are near Au6 and Au12, respectively. After geometry optimization, the muon near Au12 has moved toward Au6. These two sites, MAu6 and MAu6′, are virtually the same with very similar atomic surroundings, as shown in Figures and 7. For the MAu6 site, the μ–Au6 distance is 1.822 Å, just slightly longer than in the case of MAu11. The atomic surrounding of MAu6 is similar to those of MAu11, as can be seen in Figure . Aside from Au6, the other nearest Au neighbors for MAu6 are Au1 and Au12, and the muon is located 0.551 Å above the Au1–Au6–Au12 plane. The increase in the interatomic distances of Au1, Au6, and Au12 is similar to the case of the MAu11 site, that is, about 4–6%. The values of Aiso and Baniso are −12.67 and 7.12 MHz, respectively.

Figure 6

(a) Muon at MAu6 site in the thiolated system. A muon was initially located near Au6, (b) Magnified picture of the corresponding site, showing the bond distance between μ and Au6 as well as between μ and other neighboring atoms, Au1 and Au12. (c) Au1, Au6, and Au12 interatomic distances before the introduction of a muon into the host system.

Figure 7

(a) Muon site MAu6’ in the thiolated system. A muon was initially located near Au12 but moved toward Au6 after geometry optimization. (b) Magnified picture of the corresponding site, showing the bond distance between μ and Au6 as well as between μ and other neighboring atoms, Au1 and Au12.

MAu14 is the only site near an Au atom in the staple motif, and it is not a stable site. A muon that was initially positioned near Au14 moved toward Au11 and resulted in the MAu11″ site. At this MAu11″ position, the muon is 2.151 Å away from Au14. The muon at MAu11″ site is located in the staple motif, unlike the one at MAu11, which is inside the icosahedron. A similar situation also occurred in the selenolated nanocluster. The hyperfine coupling constant (hfcc) values at the various muon sites differ greatly due to a large variation in the contribution of the Fermi contact component. The sign of the Fermi contact coupling constant is negative at some sites where the polarization effect prevails, such as at the three lowest energy sites in the thiolated system. For most sites near Au atoms, the variation in the dipolar field is not so pronounced.

The ordering of muon sites’ stability based on the relative energy for the [Au25(SeR)18]0 nanocluster is different from that for the [Au25(SR)18]0. Moreover, all similar sites such as MAu5, MAu5′, and MAu5″ are grouped together in the ordering list. The lowest energy muon site in the [Au25(SeR)18]0 nanocluster is MAu6, where μ–Au6 distance is 1.772 Å, as shown in Figure . This site has the second-lowest relative energy in the [Au25(SR)18]0 nanocluster. Comparing the data in Figure with Figures and 7, one can notice that there are only small differences in the Au–Au interatomic distances, and the expansion of Au–Au distances to accommodate the presence of a muon are very similar to the ones for the [Au25(SR)18]0 nanocluster. The Fermi contact coupling constant at the MAu6 site is only 2.85 MHz, and the sign, which is positive, is opposite to the case in the [Au25(SR)18]0 nanocluster. The anisotropic component Baniso is 7.12 MHz, that is exactly the same as the one in the [Au25(SR)18]0 system.

Figure 8

(a) Lowest energy muon site in [Au25(SeR)18]0, MAu6. (b) Magnified picture of the corresponding site, showing the bond distance between μ and Au6 as well as between μ and other neighboring atoms, Au1 and Au12. (c) Au1, Au6, and Au12 interatomic distances before the introduction of a muon into the host system.

The muon site with the second-lowest energy in [Au25(SeR)18]0 is MAu5 in which the relative energy is 0.02 eV. A muon that was initially placed near Au1 and Au2 also moved closer toward Au5, indicated as MAu5′ and MAu5″ in Table . At this MAu5 site, the μ–Au5 distance is 1.770 Å, and the distance from Au1, Au2, and Au5 planes is 0.613 Å. Figure shows the surrounding environment of MAu5. Generally, the optimized μ–Au distance [Au25(SeR)18]0 is slightly longer as compared to the μ–Au distance in [Au25(SR)18]0. The Fermi contact and dipolar coupling constants at MAu5 are −3.13 MHz and 7.29 MHz. Note that the isotropic component sign is negative, in contrast to that found at [Au25(SR)18]0 nanocluster, where the sign is positive and with a much larger magnitude of 21.35 MHz. In the [Au25(SR)18]0 nanocluster, MAu5 is the fourth-lowest energy muon site.

Figure 9

(a) Second most energetically favorable muon stopping site in the selenolated system, which is located at the MAu5 site. (b) Magnified picture of the corresponding site, showing the bond distance between μ and Au5 as well as between μ and other neighboring atoms, Au1 and Au2. (c) Au1, Au2, and Au5 interatomic distances before the introduction of a muon into the host system.

The asymmetry and longitudinal field (LF) μSR spectra measured by Dehn et al. show loss of initial amplitude at low temperature which they attributed to a small fraction of muons experiencing very large internal fields.[2] Their spectra at 1.8 K in which the LF ranges from 2 G to 2 kG indicates that the initial symmetry that was suppressed at low field recovered as the field was increased. Goikolea et al. describe spectra, on the other hand, that show missing initial asymmetry, and they suggested that this could be due to the formation of muonium (Mu) states in the ligand.[20] Thus, we have investigated the hyperfine interaction of a Mu at possible sites in the ligand region. Three Mu sites at the phenyl ring, ortho-, meta-, and para-positions, were considered where the double bonds in the ring provide a high electron density region for a Mu to attack. In addition, Mu sites at two sulfur (selenium) atoms in the ligand were also examined. For each site, the calculations were performed in two spin states, doublet (S = 1/2) and triplet (S = 1). In triplet calculations, a muon captures an electron to form a paramagnetic Mu and subsequently is trapped and covalently bonded to a host atom. Table summarizes the results of these calculations on Mu sites in the ligand region.

Table 3

Muonium hfcc at Possible Sites in the Ligand

doublet state						triplet state
Mu site	bond order	Mu–host atom (Å)	Aiso (MHz)	Baniso (MHz)	hfcc (MHz)	Mu site	bond order	Mu–host atom (Å)	Aiso (MHz)	Baniso (MHz)	hfcc (MHz)
[Au25(SR)18]0
Mortho	0.779	1.118	254.35	6.95	261.31	MOrtho	0.799	1.116	485.08	14.41	499.49
Mmeta	0.811	1.111	279.41	9.94	288.90	MMeta	0.827	1.112	402.24	14.58	416.82
Mpara	0.783	1.111	222.90	8.31	231.21	MPara	0.812	1.108	382.17	15.77	397.94
MS1	0.895	1.362	–0.85	0.85	–0.01	MS1	0.882	1.362	135.36	4.75	140.11
MS2	0.847	1.362	38.15	1.86	40.01	MS2	0.847	1.362	52.24	5.76	58.00
[Au25(SeR)18]0
Mortho	0.787	1.117	264.32	7.29	271.61	MOrtho	0.814	1.110	431.85	16.62	448.46
Mmeta	0.810	1.118	387.15	11.02	398.17	MMeta	0.821	1.118	491.06	14.41	505.47
Mpara	0.800	1.110	269.73	10.17	279.90	MPara	0.815	1.109	383.74	14.92	398.66
MSe1	0.823	1.488	–1.28	0.85	–0.43	MSe1	0.851	1.488	302.04	5.26	307.29
MSe2	0.884	1.499	6.69	1.02	7.71	MSe2	0.800	1.499	362.67	2.88	365.55

In doublet state, the calculated Mu hfccs at ortho-, meta-, and para-positions are 261.31, 288.90, and 231.21 MHz. The hfccs increase significantly in the triplet state where the values are 499.49, 416.82, and 397.94 MHz for the ortho-, meta-, and para-positions, respectively. As a comparison, the measured Mu hfcc in some substituted benzenes is in the range of 460–505 MHz.[32,33] Thus, the calculated hfcc values at ortho-, meta-, and para-positions for S = 1 in the thiolated gold nanocluster are similar to the ones in those substituted benzenes. For both S = 1/2 and S = 1 states, the average bond order is about 0.802, signifying the formation of a single covalent bond between Mu and C.

The calculated Mu hfcc at S1 and S2 sites are −0.01 MHz and 40.01 MHz for the doublet state. In the triplet state, the values increase to 140.11 and 58.00 MHz, respectively. The average Mu–S bond order for the two sulfur sites is 0.867, higher than that for the Mu–C at the ortho-, meta-, and para-positions. Thus, Mu also forms a single covalent bond with sulfur atoms S1 and S2.

In the selenolated system, the calculated Mu hfcc values in the doublet state are 271.61, 398.17,, and 279.90 MHz for the ortho-, meta-, and para-sites, which are higher than the values in the thiolated system. The corresponding Mu hfcc values in the triplet state are 448.46, 505.47, and 398.66 MHz. The percentage increase in hfcc in going from doublet to triplet state is substantial but less than that for the case of the thiolated system. The average Mu–C bond order for both spin states is 0.808 which is similar to the thiolated system. Mu hfcc values at Se1 and Se2 sites in the triplet state are 307.29 and 365.55 MHz, respectively, which are higher than the values in the thiolated system. As in the case of thiolated system, Mu also forms a single covalent bond with a Se atom with an average bond order of 0.840. In both thiolated and selenolated systems, the increase in the hfcc values is mainly due to the large Fermi contact contribution, which is typical for Mu hypefine interaction in organic systems.

Conclusions

The results of our present work show that ligand effects are quite significant in affecting the position of the muon in Au nanocluster. In [Au25(SR)18]0, the most stable muon Au site is MAu11, while in [Au25(SeR)18]0, MAu5 is the one with the lowest energy. In our previous work using the hexanethiol ligands, S(CH2)5CH3, the most stable muon site was found to be MAu10. There are of course possibilities that muons stop at multiple Au sites due to the small energy differences among the various Au sites. In terms of hyperfine interaction, the anisotropic component does not change much at different muon Au sites. On the other hand, the isotropic component was found to be quite sensitive to different ligands, varying in both magnitudes and signs. It was found that the hfcc for the most stable Au sites in the [Au25(SR)18]0 and [Au25(SeR)18]0 are −5.66 MHz and 9.57 MHz, respectively.

Our investigation on Mu state in the ligand region indicates that the ortho-, meta-, and para-positions at the phenyl rings, as well as Mu trapping sites at sulfur and selenium atoms are energetically stable. The hfcc values at these Mu sites in the triplet state are high due to a large Fermi contact field. It should be noted that both Goikolea et al. and Dehn et al. systems do not contain any phenyl ring in the ligand, but there are sulfur atoms that can provide multiple Mu trapping sites.[2,20] Thus, the large Mu Fermi contact field at sulfur sites is a factor that could explain the loss of initial asymmetry observed in the μSR spectra.

Computational Details

The initial geometry of the [Au25(SR)18]0 nanocluster used in this investigation was taken from the crystal structure data where R is phenylethane.[15] The crystal structure of the system is triclinic with the P1̅ space group. The starting geometry for the [Au25(SeR)18]0 nanocluster was modified from the thiolated system, where all sulfur atoms were replaced with selenium atoms. As shown in Figure a, the system possesses a core–shell structure of 13 gold atoms, comprising one central Au encapsulated by the Au12 icosahedral shell. The basic framework of the system contains three interlocked Au8S6/Au8Se6 rings indicated using different color codes, as shown in Figure b. All rings have two orthogonal staple motifs, each of which consists of a (−RS–Au–RS–Au–SR−) dimer. For labeling purposes, the rings are divided into two semi-rings labeled as D1, D2, D3, D4, D5, and D6. There is a total of nine crystallographically independent phenylethane ligands, and each is bonded to a sulfur atom or a selenium atom in the staple motifs. A full geometry optimization without symmetry constraints was performed for each nanocluster system to obtain its minimum total energy. This optimized geometry was subsequently utilized as the host system to study possible muon stopping sites.

Figure 10

(a) Structure of [Au25(SeR)18]0 nanocluster modeled in the current study. The labels are used to facilitate the discussion (legend: red = Au; yellow = S). (b) Structure of three interlocked Au8S6 rings.

DFT in the B3LYP framework was employed for all calculations in this work. LanL2DZ basis set was used for Au atoms with the inclusion of scalar relativistic effects. For other atoms, the calculations were carried out using 6-31G++(d,p) basis sets. Because of significant dispersion forces in gold atoms, Grimme’s dispersion with Becke–Johnson damping model[34] was included in all calculations to take into account the empirical dispersion corrections to the DFT hybrid functional. All calculations were performed using G16 Revision B0.1 software package[35] installed at the RIKEN Hokusai GreatWave Supercomputing facility. The setting up of the input file, as well as output analysis and graphical illustration, were accomplished using GaussView 6[36] and Mercury[37] software. The spin densities were examined using Mulliken population analysis,[38] whereas the charge distribution was obtained using natural population analysis.[39]

There is a total of 25 inequivalent Au atoms and 18 inequivalent S/Se atoms in the nanocluster, which allow a muon to be trapped near these sites. Based on our previous work and because of the computational cost, we have visually analyzed the MEP map to identify potential muon stopping sites. The MEP was generated using the host systems without the presence of a muon. This is similar to the minimum ESP search adopted by other works where the map is generated using an unperturbed environment.[40−42] As a result, there are 13 sites near Au atoms in the icosahedral core and one site near Au in the staple motif that were considered in the search for stable muon sites. For identification purposes, the muon sites inside the icosahedral core are labeled as MAu1 to MAu13, and the one in the staple motif is labeled as MAu14. Muon sites near two sulfur/selenium atoms in the staple motifs are designated as MS1/MSe1 and MS2/MSe2. In Figure a, the S1/Se1 atom is indicated as S central, while the S2/Se2 is shown as S terminal. As is customary in muon site calculation, hydrogen with the mass of muonium was used to mimic muonium.[22,43−46] After a positive muon was introduced into the host systems, we performed another geometry optimization that allowed the muon and several host atoms in its vicinity to move. This condition is to allow those atoms to relax into new positions and to stabilize the muon stopping sites.