A combined stochastic search and density functional theory study on the neutral and charged silicon-based clusters MSi₆ (M = La, Ce, Yb and Lu)

Abstract

The neutral and charged silicon clusters Si₆^q (q = 0, −1) are doped systematically with an external atom of different lanthanides: La, Ce, Yb and Lu. The structural, electronic, and magnetic properties of the doped clusters, MSi₆^q (M = La, Ce, Yb and Lu; q = 0, −1) are investigated using the Saunders “Kick” global stochastic search technique combined with density functional theory (DFT) calculations. The accuracy and consistency of structure optimization of MSi₆^q clusters are tested with three exchange correlation functionals (PBEPBE, BPW91 and B3LYP). DFT calculations show that the pentagonal bipyramid structures of MSi₆⁻ (M = Yb, Lu) and Si_6/7⁻ with high symmetry are ground state structures, which give simulated photoelectron spectra in good agreement with available experiments. The detailed comparison between previously published experimental photoelectron spectra, and the present theoretical simulation helps to identify the ground state structures. The neutral species shows the impurity as a four-coordinate atom in the equatorial plane of pentagonal bipyramid. Furthermore, the current magnetic property analysis indicates that the magnetic moments of the doped rare earth atoms remain largely localized, and the atomic-like magnetism is maintained in the doped clusters.

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1. Introduction

Since silicon is the keystone of modern semiconductor technology, it attracts great attention as building blocks for silicon-based nanomaterials. Silicon clusters have been the subject of extensive experimental^1–6 and theoretical^7–12 studies in the past two decades. It was observed that these clusters tend to form compact, diamond-like structures rather than the fullerene-like cages that are characteristic of their congener, carbon.¹³ These results are ascribed to the fact that silicon atoms generally prefer sp³ hybridization to sp². By the doping of a suitable metal atom into a pure Si cluster, a highly stabilized cluster will open up a new avenue to cluster-assembled materials made from Si atoms. More significantly, however, the lion's share of that interest stems from the realization that the addition of “impurity” atoms can significantly alter the properties of silicon. For example, the notion of altering the electronic properties of silicon by doping group IIIA (Ga, Al) and group VA (P, As) elements into it has been in use for over half a century and has led to the revolutionary rise of modern computers. As is well known, the first hints that silicon cages may be stabilized by the introduction of a doping atom emerged from the early, pioneering experimental work of Beck who observed enhanced ion intensities of metal–silicon clusters, MSi_n⁺ (M = Mo, Cr, W) with around n = 16, when a transition metal atom was added to the cluster.¹⁴ Since then, many stable MSi_n species have been reported, in which various factors accounting for their formation have been examined experimentally^15–23 and theoretically.^24–34 Recently, infrared multiple photon dissociation (IR-MPD) and mass spectrometry have been used to identify the structures of small neutral vanadium- and manganese-doped silicon clusters Si_nX (n = 6–9, X = V and Mn).¹⁵ Recent theoretical work on doping silicon anion clusters with yttrium suggests that the following two growth behaviors play an important role: (1) the metal atom acting as a linker between two subclusters and (2) the metal atom being endohedrally encapsulated in a silicon cage.²⁵

The rare earths (RE) are special transition metals, possessing many of the same properties (such as optical property, magnetism) as the transition elements. RE compounds are usually used as catalysts and synthetic products in the production of petroleum. Up to now, only a few reports exist on RE-doped silicon clusters.^{17,25,29,35–37} However, interest in their potential applications has spurred considerable activity over the past couple of years. We have recently published the results of comparing experimental photoelectron spectroscopy and theoretical density functional theory (DFT) studies of MSi₆^−/0 (M = Pr, Gd, Ho) clusters.³⁵ This article provides an interesting example for evaluating the accuracies of various DFT methods, in general, used to verify the proposed structural types for these species. Experimentally, the validity of the concept can be proven by using photoelectron spectroscopy, which is shown to be a powerful experimental technique for probing electronic structures of size-selected clusters.^38–45 Especially, Bowen and co-workers have investigated the lanthanide–silicon cluster anions LnSi_n⁻ (3 ≤ n ≤ 13; Ln = Ho, Gd, Pr, Sm, Eu, Yb)⁴⁶ and EuSi_n⁻ (3 ≤ n ≤ 17)¹⁷ by using photoelectron spectroscopy. They proposed that the building blocks of silicon-based clusters may exist as magnetic materials. They also proposed that the Eu atom encapsulated into the Si_n clusters at the size n = 12, which has been confirmed by theoretical calculation.²⁹ In addition, Nakajima and co-workers have previously reported the electronic properties of silicon clusters, containing a transition or lanthanide metal atom from group 3, 4, or 5, by using anion photoelectron spectroscopy at 213 nm.⁴⁷

Here, we report the generation of a series of anionic and neutral lanthanide metal-doped silicon clusters MSi₆ (M = La, Ce, Yb and Lu) using Saunders “Kick” global stochastic search technique combined with DFT geometric optimization. The aims of the present work are as follows: (1) to obtain the various global minimum structures of MSi₆^q clusters; and (2) to compare the results of our extensive computations with previously experimental findings.^46,47 Good agreement between previously measured photoelectron spectroscopy and theoretical simulations helps to identify the ground states of the anionic and neutral RE-doped MSi₆ clusters. The rest of this article is organized as follows. We describe our computational details and theoretical backgrounds in Section 2. In Section 3, we introduce and compare various geometric structures, simulated photoelectron spectra (PES), as well as electronic and magnetic properties. Finally, the conclusions are given in Section 4.

2. Computational methods

The calculations in this work began with unbiased searches for the low-lying structures of a series of anionic and neutral RE-doped silicon clusters MSi₆ (M = La, Ce, Yb and Lu) by using the Saunders “Kick” (SK) global search technique⁴⁸ combined with DFT geometric optimization (SK-DFT) within the generalized gradient approximation in the Perdue–Burke–Ernzerhof (PBE)⁴⁹ functional form using the GAUSSIAN09 program.⁵⁰ This SK stochastic method generates structures randomly and facilitates the thorough exploration of unknown isomers much more easily than manual methods. Specifically, all the atoms are placed at the same point initially, and they are then “kicked” randomly within a sphere of radius R (R is the maximum kick distance), e.g., a hollow sphere of inside radius 10 Å. The method developed further in this work generates up to 500 unbiased starting geometries, and then submits them automatically for optimization with GAUSSIAN09. We have recently successfully employed the SK-DFT method for global minimum searches on a series of bimetallic cluster and vanadium oxides.^51–57 The scheme of this research consists of the following four steps. Firstly, the initial SK global minimum search was performed using the DFT method known in the literature as “PBEPBE” functional⁴⁹ with a scalar relativistic effective core potential and a LANL2DZ basis set for the RE atom and a 6–31G basis set for Si atom (the PBEPBE/RE/LANL2DZ/Si/6–31G level), which is implemented in the GAUSSIAN09 package. Many structural isomers were generated for each cluster. The kick method runs at the PBEPBE/RE/LANL2DZ/Si/6–31G level some 500 times until no new minima appeared. If this procedure is repeated enough number of times, eventually all isomeric structures for the cluster will be found. Secondly, the distinct isomers were ranked according to their relative energies at the PBEPBE/RE/LANL2DZ/Si/6–31G level. Moreover, the effect of spin multiplicity is also taken into account in the present study. Thirdly, we selected the top-eight lowest-energy isomers for each species, and those with their energy value within ∼1.50 eV from the lowest-energy isomer are to be further optimized using the PBEPBE exchange–correlation functional with the larger RE/SDD/Si/6–311+G(d) basis sets (the PBEPBE/RE/SDD/Si/6–311+G(d) level). Here, the SDD basis set⁵⁸ along with the quasi-relativistic MWB (28 core electrons) pseudopotential^58,59 was chosen for RE atoms. For transition metals compounds, lanthanides and actinides, the SDD basis set is standard practice, it is slightly superior and usually gives very nice structures.⁶⁰ For Si, we adopted the 6–311+G(d) basis set. Such diffuse functions improve the predicted properties of species with extended electronic densities such as anions. Finally, the top-several isomers were ranked again by their relative energies at high PBEPBE/RE/SDD/Si/6–311+G(d) level, and those with their energy value within 0.40 eV from the lowest-energy isomer were all regarded as potential candidate lowest-energy structures. All the optimization procedure was carried out without any symmetry constraints. This was followed by frequency calculations to ensure that all found structures are at local minima of the potential energy surface and have real frequencies.

The reliability of the present computational method (at PBEPBE/RE/SDD/Si/6–311+G(d) level) was validated by performing calculations on the features of simulated PES spectra and the first adiabatic/vertical detachment energy (ADE/VDE) for which experimental results are available (see Fig. 2, 5, 6 and Table 3).^4,6,46,47 Furthermore, geometry optimization of MSi₆ (M = La, Ce, Yb and Lu) clusters were tested with several exchange–correlation functionals (PBEPBE,⁴⁹ BPW91^61,62 and B3LYP^63–65) for accuracy and consistency. The calculated results using these three different DFT levels are presented in Table 2. These results indicate that the geometries of all the clusters obtained by three functionals are similar, although the order of isomers is reversed in some cases. Here, only PBEPBE results will be discussed in the text. In addition, due to the possible multireference structure of wavefunction, the DFT functional we employed here may be insufficient in giving a very accurate description. The multi-reference configuration interaction (MRCI) method, which is based on multiconfiguration self-consistent field (MC-SCF) wavefunction, would be more optimal for the present clusters. Unfortunately, it is currently not computationally feasible to study the RE-doped clusters using this advanced theoretical method. Geometries are regarded as optimized when the maximum force, the root-mean-square (RMS) force, the maximum displacement of atoms, and the RMS displacement of atoms have magnitudes less than 0.00045, 0.0003, 0.0018, and 0.0012 a.u., respectively. Because the computed zero-point energy (ZPE) corrections of the isomers of a specific cluster size are small and almost the same, they are not expected to affect the relative energy ordering. Hence, the relative energies of isomers are obtained from the total electronic energies (excluding ZPE).

We used three criteria in comparing the theoretical results with the experimental data to select our most likely candidate structures: (1) the relative energies; (2) the first VDE and ADE; (3) the number of distinct peaks of simulated PES in the low binding energy range of ≤266 nm (4.661 eV) and their relative positions. The first criterion addresses typical errors in DFT calculations, which is about several tenths of an electron volt in relative energies for MSi₆^q clusters. In the present study, we used a cut-off energy of 0.40 eV at the PBEPBE/RE/SDD/Si/6–311+ G(d) level to obtain low-lying isomers. Next, we calculated the first VDE (which is defined as the energy difference between the neutral clusters at optimized anion geometry clusters and optimized anion clusters: VDE = E_{neutral at optimized anion geometry} − E_{optimized anion}) and ADE (which is the energy difference between the optimized anion geometry and the optimized neutral geometry: ADE = E_{optimized neutral} − E_{optimized anion}). Finally, the simulated spectra were then compared to the measured PES spectra to distinguish from top candidates for the low-lying isomers. VDEs were calculated using the generalized Koopmans' theorem by adding a correction term to the eigenvalues of the anion.⁶⁶ The correction term was calculated as δE = E₁ − E₂ − ε_HOMO, where E₁ and E₂ are the total energies of the anion and neutral, respectively, in their ground states at the anion equilibrium geometry and ε_HOMO corresponds to the eigenvalue of the highest occupied molecular orbital (HOMO) of the anion. The simulated PES spectra were fitted with a full width at half-maximum (FWHM) of 0.20 eV.⁵³ These methods have been used successfully in a number of previous studies, and they have been shown to yield VDEs in good agreement with PES data.^38–45 We paid particular attention to the number of peaks below 4.661 eV. Only those isomers that meet all three criteria are considered to be the superlative candidates for MSi₆⁻ clusters.

3. Results and discussions

The optimized anion and neutral ground-state and low-lying structures at the PBEPBE/RE/SDD/Si/6–311+G(d) level are presented in Fig. 1 for both pure Si_6/7^q and RE-doped MSi₆^q (M = La, Ce, Yb and Lu; q = 0, −1) clusters. Table 1 gives various structural and energetic characteristics of parent Si_6/7^q and doped MSi₆^q clusters. The calculated relative energies for the top-eight lowest-energy isomers of MSi₆ (M = La, Ce, Yb and Lu) with three density functional methods are presented in Table 2. The calculated first ADE and VDE of the anionic top-several isomers, as well as some experiment results, are given in Table 3. Natural charges populations of the lowest-energy RE-doped MSi₆^q (M = La, Ce, Yb and Lu; q = 0, −1) clusters are listed in Table 4. Table 5 gives the magnetic moment (μ_B) of 4f, 5d, 6s, and 6p states for RE atom, total magnetic moment (μ_B) of the RE atom, and total magnetic moment of the lowest-energy RE-doped MSi₆^q (M = La, Ce, Yb and Lu; q = 0, −1) clusters. Fig. 2–6 show the simulated PES spectra of the top-several isomers (within ∼0.40 eV), along with experimental PES spectra for comparison. The calculated ADEs are obtained by optimizing the geometry of the neutrals with the ground state geometries of the anions as starting points. The HOMO (SOMO for open-shell species) and the Mülliken spin density pictures of RE-doped MSi₆^q (M = La, Ce, Yb and Lu; q = 0, −1) clusters are presented in Fig. 7–10.

Fig. 1 Equilibrium geometries of the lowest-energy isomers for pure Si₆^q (Si₇^q) (q = 0, −1) clusters and RE-doped MSi₆^q (M = La, Ce, Yb and Lu; q = 0, −1) clusters at the PBEPBE/RE/SDD/Si/6–311+G(d) level of theory. The dark cyan spheres stand for silicon atoms and the black ones for RE atoms.

Table 1 Various structures with the symmetry type (Sym), the spin multiplicity (SM), the binding energy (BE) per atom, the doping energy (DE) per atom, HOMO–LUMO energy gap E_gap, and energy separation of the isomeric structures from the lowest-energy structures (ΔE) of both pure Si_6/7^q (q = 0, −1) clusters and MSi₆^q (M = La, Ce, Yb and Lu; q = 0, −1) clusters

Cluster	The lowest-energy isomer						Low lying isomer
	Isomer	SM	Sym	BE (eV)	DE (eV)	Egap (eV)	Isomer	ΔE (eV)	Isomer	ΔE (eV)
Si6	c	1	C2v	3.42		2.15	d	0.62	e	1.45
Si6^−	a	2	C2v	3.55		0.55	b	0.19	d	0.85
Si7	f	1	D5h	3.56	4.37	2.12	g	0.96	h	0.97
Si7^−	f	2	D5h	3.62	4.04	0.59	g	0.79	h	0.87
LaSi6	i	2	C1	3.59	4.59	0.32	m	0.29	k	0.30
LaSi6^−	j	1	C5v	3.82	4.45	1.04	k	0.29	l	0.34
CeSi6	i	3	Cs	5.35	16.93	0.17	l	0.39	k	0.42
CeSi6^−	j	2	Cs	5.63	16.73	0.49	k	0.34	l	0.37
YbSi6	i	1	C2v	3.16	1.58	0.62	o	0.71	p	0.84
YbSi6^−	i	2	C2v	3.67	1.17	0.42	j	0.34	p	0.56
LuSi6	i	2	C2v	3.35	2.90	0.68	m	0.47	j	0.51
LuSi6^−	j	1	C5v	3.81	3.01	1.36	i	0.35	l	0.53

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Cluster	Low lying isomer
	Isomer	ΔE (eV)	Isomer	ΔE (eV)	Isomer	ΔE (eV)	Isomer	ΔE (eV)	Isomer	ΔE (eV)
LaSi6	l	0.34	o	0.61	p	0.62	j	0.63	n	0.75
LaSi6^−	m	0.36	n	0.65	o	0.72	i	0.82	p	1.44
CeSi6	m	0.48	j	0.62	o	0.75	n	0.86	p	0.91
CeSi6^−	m	0.38	i	0.62	n	0.71	o	0.86	p	1.35
YbSi6	j	0.90	l	0.92	n	0.95	m	0.96	k	1.19
YbSi6^−	o	0.63	l	0.75	m	0.76	k	0.83	n	1.02
LuSi6	l	0.52	k	0.60	m	0.64	n	0.93	o	0.95
LuSi6^−	m	0.53	k	0.63	o	0.79	p	0.95	n	1.02

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Table 2 Relative energies in eV for selected low-lying isomers of MSi₆ (M = La, Ce, Yb and Lu) with different density functional methods

Cluster	Isomer	Sym	PBE	B3LYP	BPW91	Cluster	Isomer	Sym	PBE	B3LYP	BPW91
LaSi6	i	C1	0.00	0.00	0.00	YbSi6	i	C2v	0.00	0.00	0.00
	j	Cs	0.63	0.63	0.62		j	Cs	0.90	0.77	0.88
	k	C1	0.30	0.28	0.28		k	C1	1.19	1.15	1.18
	l	C1	0.34	0.32	0.34		l	C1	0.92	0.55	0.92
	m	Cs	0.29	0.24	0.27		m	Cs	0.96	0.74	0.93
	n	Cs	0.75	0.65	0.72		n	Cs	0.95	0.74	0.93
	o	C3v	0.61	0.51	0.56		o	C3	0.71	0.48	0.65
	p	Cs	0.62	0.79	0.81		p	C2v	0.84	0.66	0.82
CeSi6	i	Cs	0.00	0.00	0.00	LuSi6	i	C2v	0.00	0.00	0.00
	j	C1	0.62	0.41	0.69		j	Cs	0.51	0.50	0.51
	k	C1	0.42	0.09	0.42		k	C1	0.60	0.61	0.60
	l	Cs	0.39	0.22	0.40		l	Cs	0.52	0.47	0.51
	m	Cs	0.48	0.15	0.39		m	Cs	0.47	0.48	0.47
	n	Cs	0.86	0.45	0.85		n	Cs	0.92	0.84	0.90
	o	C1	0.75	0.34	0.70		o	Cs	0.95	0.88	1.00
	p	Cs	0.91	0.70	0.87		p	Cs	0.64	0.52	0.83

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Table 3 Experimentally measured adiabatic and vertical detachment energies (ADEs and VDEs) from the photoelectron spectra compared to the calculated ground state or low-lying anion state ADEs and VDEs for pure Si_6/7⁻ and RE-doped MSi₆ (M = La, Ce, Yb and Lu) clusters. All energies are in eV

Cluster	Isomer	RE	ADE^a^,^b	VDE^a
a Numbers in parentheses represent the experimental uncertainties in the last digit.b Electron affinity of the neutral species.c From ref. 4.d From ref. 6.e From ref. 46.f From ref. 47.
Si6^−	a	0.00	2.24	2.45 (2.39 ± 0.01)^c
	b	0.19	2.05	2.77
Si7^−	f	0.00	1.91 (1.85 ± 0.02)^d	2.30 (2.39 ± 0.04)^c
LaSi6^−	j	0.00	2.73	2.81
	k	0.29	2.11	2.18
	l	0.34	2.10	2.19
	m	0.36	2.03	2.08
CeSi6^−	j	0.00	2.66	2.83
	k	0.34	2.12	2.19
	l	0.37	2.06	2.16
	m	0.38	2.09	2.13
YbSi6^−	i	0.00	1.83 (1.80 ± 0.05)^e	1.95 (1.98)^e
	j	0.34	2.39	2.42
LuSi6^−	j	0.00	2.86 (2.10)^f	3.07 (3.20)^f
	i	0.35	2.00	2.16

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Table 4 Population of the natural charges of the lowest-energy RE-doped MSi₆^q (M = La, Ce, Yb and Lu; q = 0, −1) clusters

Cluster	Isomer	RE	Si-1	Si-2	Si-3	Si-4	Si-5	Si-6
a Atoms bonding to RE atom.
LaSi6	i	0.47	−0.10^a	−0.21^a	−0.10^a	−0.21^a	0.07	0.08
LaSi6^−	j	0.16	−0.17^a	−0.17^a	−0.17^a	−0.17^a	−0.17^a	−0.29
CeSi6	i	0.61	−0.12^a	−0.28^a	−0.12^a	−0.26^a	0.09	0.09
CeSi6^−	j	0.10	−0.17^a	−0.17^a	−0.17^a	−0.16^a	−0.16^a	−0.28
YbSi6	i	0.91	−0.21^a	−0.34^a	−0.21^a	−0.34^a	0.09	0.09
YbSi6^−	i	0.33	−0.26^a	−0.36^a	−0.26^a	−0.36^a	−0.05	−0.05
LuSi6	i	0.56	−0.10^a	−0.30^a	−0.10^a	−0.30^a	0.13	0.13
LuSi6^−	j	0.30	−0.19^a	−0.19^a	−0.19^a	−0.19^a	−0.19^a	−0.33

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Table 5 Magnetic moment (μ_B) of 4f, 5d, 6s, and 6p states for RE atom, total magnetic moment (μ_B) of the RE atom, and total magnetic moment of the lowest-energy RE-doped MSi₆^q (M = La, Ce, Yb and Lu; q = 0, −1) clusters

Cluster	Isomer	RE Moment (μB)					Molecule (μB)
		4f	5d	6s	6p	Total
LaSi6	i	0.02	0.51	0.09	0.01	0.63	1
LaSi6^−	j	0.00	0.00	0.00	0.00	0.00	0
CeSi6	i	0.87	0.31	0.52	0.07	1.77	2
CeSi6^−	j	1.02	0.04	0.00	0.00	1.06	1
YbSi6	i	0.00	0.00	0.00	0.00	0.00	0
YbSi6^−	i	0.01	0.01	0.67	0.17	0.86	1
LuSi6	i	0.00	0.10	0.59	0.16	0.85	1
LuSi6^−	j	0.00	0.00	0.00	0.00	0.00	0

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Fig. 2 Comparison of the experimental photoelectron spectra of Si₆⁻ and Si₇⁻ clusters with their simulated spectra. Each VDE was fitted with a full width at half-maximum (FWHM) of 0.20 eV to obtain the simulated PES spectra. The experimental PES spectra are cited from ref. 4.

Fig. 3 Structures, relative energies, and simulated photoelectron spectra for the four low-lying isomers of LaSi₆⁻. Each VDE was fitted with a full width at half-maximum (FWHM) of 0.20 eV to obtain the simulated PES spectra.

Fig. 4 Structures, relative energies, and simulated photoelectron spectra for the four low-lying isomers of CeSi₆⁻. Each VDE was fitted with a full width at half-maximum (FWHM) of 0.20 eV to obtain the simulated PES spectra.

Fig. 5 Structures, relative energies, and simulated photoelectron spectra for the two low-energy isomers of YbSi₆⁻. Each VDE was fitted with a full width at half-maximum (FWHM) of 0.20 eV to obtain the simulated PES spectra. The black curve is for the experimental PES for YbSi₆⁻. The experimental PES spectra are cited from ref. 46.

Fig. 6 Structures, relative energies, and simulated photoelectron spectra for the two low-energy isomers of LuSi₆⁻. Each VDE was fitted with a full width at half-maximum (FWHM) of 0.20 eV to obtain the simulated PES spectra. The black curve is for the experimental PES for LuSi₆⁻. The experimental PES spectra are cited from ref. 47.

Fig. 7 (a) The HOMO picture of LaSi₆⁻ (C_5v, ¹A′). (b) The SOMO picture of LaSi₆ (C₁, ²A). (c) The Mülliken spin density of LaSi₆ (C₁, ²A).

Fig. 8 (a) The SOMO picture of CeSi₆⁻ (C_s, ²A′′). (b) The SOMO and (c) SOMO-1 pictures of CeSi₆ (C_s, ³A′′). (d) The Mülliken spin density of CeSi₆⁻ (C_s, ²A′′). (e) The Mülliken spin density of CeSi₆ (C_s, ³A′′).

Fig. 9 (a) The SOMO picture of YbSi₆⁻ (C_2v, ²A₁). (b) The HOMO picture of LaSi₆ (C_2v, ¹A₁). (c) The Mülliken spin density of YbSi₆⁻ (C_2v, ²A₁).

Fig. 10 (a) The HOMO picture of LuSi₆⁻ (C_5v, ¹A′). (b) The SOMO picture of LuSi₆ (C_2v, ²A₁). (c) The Mülliken spin density of LuSi₆ (C_2v, ²A₁).

3.1. Geometric structure and stability

In this section, we discuss the structural details of a few low-lying isomers of MSi₆^q clusters, which were obtained by Saunders “Kick” (SK) global search technique combined with DFT geometric optimization (SK-DFT). We have obtained many low-lying isomers and determined the lowest-energy structures for RE-doped MSi₆^q (M = La, Ce, Yb and Lu; q = 0, −1) using the computation scheme SK-DFT described in Section 2 (computational methods). Here, in order to discuss the effects of doped impurity atom on pure silicon clusters, we carried out an additional SK-DFT calculation on pure Si_6/7^−,0 clusters. Many isomers are found and the top three isomers of Si_6/7^−,0 clusters are shown in Fig. 1. For Si₆⁻ cluster, the lowest-energy structure is isomer a, which can be viewed as a Si₂-capping of a bent rhombus. The point group symmetry is C_2v. The second lowest-energy isomer b is a regular octahedron with D_4h symmetry, and it is only 0.19 eV above the ground state. Another structure d obtained for Si₆⁻ can be formed from capping of a trigonal bipyramid. It is less stable than isomer a by 0.85 eV. For neutral Si₆ cluster, the lowest-energy isomer c is a relaxed octahedron. In fact, initial structures of isomers a and b are also considered in the present calculation. When six Si atoms were placed in the shape of isomer a or b, they move away from their original position during the geometry optimization process. The other two isomers, i.e., Si-capped triangular bipyramid-shaped isomer d and planar isomer e, are 0.62 and 1.45 eV higher than ground state structure c, respectively. The lowest-energy structures of Si₆⁻ and Si₆ are in agreement with previously reported results by using different theoretical techniques.^5,8,67 As for pure neutral and anionic Si₇ clusters, three low-lying isomers (f–h) of neutral structures are similar to the anionic structures, even the order of isomers is the same. The lowest-energy structure is a regular pentagonal bipyramid with D_5h symmetry. Our calculated structure on neutral Si₇ is in line with the early theoretical results.^3,8,12 Isomers g and h are about 0.96, 0.97 eV (for neutral Si₇) and 0.79, 0.87 eV (for anionic Si₇), which is less stable compared to ground state structure f.

The ground-state structures and some low-lying metastable isomers of RE-doped MSi₆^q (M = La, Ce, Yb and Lu; q = 0, −1) species are shown in Fig. 1, in which the dark cyan spheres stand for silicon atoms and the black ones for RE atoms. Remarkably, our calculations show that all the low-lying isomers (top-eight) for MSi₆^q are three-dimensional (3D) structures. The spin multiplicity of ground state is kept constant for all other higher energy isomers (Table 1 and 2). It must be pointed out that the RE atom cannot be encapsulated inside the bare Si₆^q cluster and still remains on the surface of the lowest-energy MSi₆^q cluster using three different density functional methods. The most stable structure for RE-doped MSi₆⁻ (M = La, Ce, and Lu) is a 3D structure with the lanthanide metal atom sitting on top (or bottom) of the regular pentagonal bipyramid (see Fig. 1, isomer j). These most stable structures are formed by replacing a silicon atom on the top (bottom) of the inerratic pentagonal bipyramid shaped pure Si₇ cluster. The other seven low-lying metastable 3D isomers are all much higher in energy than the perfect pentagonal bipyramid structure for all three clusters (0.29–1.44 eV for LaSi₆⁻, 0.34–1.35 eV for CeSi₆⁻ and 0.34–1.02 eV for LuSi₆⁻). The M–Si distance in the lowest-energy structures for RE-doped MSi₆⁻ (M = La, Ce, and Lu) has the following relationship: R_La–Si (2.96 Å) > R_Ce–Si (2.89 Å) > R_Lu–Si (2.82 Å), which is mainly due to the difference in atomic radius of lanthanide-metal R_La = 2.43 Å > R_Ce = 2.42 Å > R_Lu = 2.24 Å. It should be noted that all the ground state structures of MSi₆⁻ (M = La, Ce, and Lu) were retained after replacing a silicon atom on the top for the pure Si₇ cluster, but electronic features had been significantly altered, which presented the reason for a far cry from photoelectron spectra of pure silicon clusters. On the other hand, results for the most stable structures of the RE-doped silicon clusters YbSi₆⁻ and MSi₆ (M = La, Ce, Yb, and Lu) are also obtained. All the ground states of the neutral MSi₆ (M = La, Ce, Yb, and Lu) and YbSi₆⁻ clusters also have a 3D structure, and no 2D isomers were obtained. The most stable structure for neutral MSi₆ (M = La, Ce, Yb, and Lu) and YbSi₆⁻ is a 3D structure with the lanthanide metal atom sitting on side of the distorted pentagonal bipyramid (See Fig. 1, isomer i). For the neutral MSi₆, the high symmetry 3D pentagonal bipyramid arrangement is not a stable structure, and is much higher in energy than the most stable distorted pentagonal bipyramid by 0.63, 0.62 0.90 and 0.51 eV for M = La, Ce, Yb, and Lu, respectively. As for YbSi₆⁻, the structure j is higher than i by 0.34 eV. According to the above descriptions of geometries, we find that all the lowest energy structures of neutral MSi₆ (M = La, Ce, Yb, and Lu) and YbSi₆⁻ clusters were obtained by substituting one Si in the equatorial plane by an M atom from the ground state structure (Fig. 1, isomer f) of the Si₇^−,0 cluster; however, the anionic ground state of MSi₆ (M = La, Ce, and Lu) clusters were obtained by substituting one Si by one M atom from the bottom (or top) of the pure Si₇^−,0 cluster.

It is interesting to know if the structure and stability of the clusters will change when an atom with a full 4f shell is doped into Si₆ cluster compared with the case of by doping open 4f RE atoms. Here, we will compare the results of YbSi₆ and LuSi₆ with open 4f RE-doped MSi₆ (M = Ce, Pr, Eu, Gd, and Ho) clusters.^29,35 The point group symmetry of the lowest-energy structure of YbSi₆ and LuSi₆ is C_2v symmetry. It should be mentioned that the early RE-doped MSi₆ clusters have relatively low symmetry compared to the full 4f shell RE-doped clusters, e.g. CeSi₆ (C_s) PrSi₆ (C_s), GdSi₆ (C_s), and HoSi₆ (C_s). It seems that the more f-electrons, the higher the point group symmetry is. This may be attributed to Jahn–Teller distortion. For all the RE-doped MSi₆ clusters, the additional RE atom always takes one of the equatorial positions of the pentagon, forming a distorted pentagonal bipyramid-shaped structure.

Table 1 gives various structural and energetic characteristics of both pure Si_6/7^q (q = 0, −1) and MSi₆^q (M = La, Ce, Yb and Lu; q = 0, −1) clusters. In order to analyze the stability of Re-doped clusters, we calculated the binding energies (BE) per atom and the doping energy (DE), which are defined by the following formulas: BE = [5E(Si) + E(Si^q) + E(M) − E(MSi₆^q)]/7, DE = E(Si₆^q) + E(M) − E(M Si₆^q) (M = La, Ce, Yb and Lu; q = 0, −1), where E(Si), E(Si^q), E(M), E(MSi₆^q) and E(Si₆^q) represent the total energies of the ground-state of Si, Si^q, RE atom M, MSi₆^q and Si₆^q clusters, respectively. The BE per atom value of the lowest-energy MSi₆^q clusters is displayed in Table 1. The notable feature is that the BE per atom in the RE-doped MSi₆^q clusters is larger than that in the pure Si_6/7^q clusters, except for the case of YbSi₆ and LuSi₆ clusters. This suggests that the Si-M bond is stronger than the Si–Si bond, and the doped RE atom with open f shells can stabilize the pure pentagonal bipyramid structure. In fact, the high stability of these clusters has been observed in EuSi_6/7/8.²⁹ To further study the stabilities of MSi₆^q clusters, we will discuss the DE of the ground state structure. From Table 1, we can see that both neutral and anionic CeSi₆ clusters have the highest DE of 16.73 and 16.93 eV among the clusters considered in the present study. Thus, CeSi₆^0,− should be the most stable one among these clusters. The highest occupied molecular orbital-lowest unoccupied molecular orbital (HOMO–LUMO) energy gap is another useful quantity for examining the kinetic stability of clusters. A large energy gap corresponds to a high strength required to perturb the electronic structure. To calculate the HOMO–LUMO gap we consider HOMO and LUMO of both the spins (up and down). Comparing the difference between the energy gap of pure and doped clusters in Table 1 one can see that the E_gap of the LuSi₆⁻ with 1.36 eV is the largest one among the doped clusters but still smaller than the pure Si₆ and Si₇ clusters. This suggests that LuSi₆⁻ is relatively more chemically stable than the neighboring clusters. Overall, after doping an RE atom into the silicon cluster, the chemical stability of RE-doped cluster is less reactive than pure cluster.

3.2. Comparison of the PES spectrum between experiment and theory

The well-resolved PES spectra for pure Si_6/7⁻ and doped MSi₆⁻ (M = Yb and Lu) in Fig. 2, 5 and 6 at 355 nm (3.496 eV), 266 nm (4.661 eV) and 213 nm (5.83 eV) photon energy are presented by the Castleman,⁴ Bowen⁴⁶ and Nakajima⁴⁷ groups, respectively, and they could serve as electronic fingerprints for the MSi₆⁻ clusters, and thus allow comparisons with theoretical calculations to verify the identified global minimum structures. The photo-electron spectra of present species are simulated by adding the occupied orbital energies relative to the HOMO (SOMO for open-shell systems) to the VDE and fitting them with a FWHM of 0.20 eV.⁵³ The simulated photoelectron spectra for the two low-lying energy isomers of MSi₆⁻ (M = Yb and Lu) are presented in Fig. 5 and 6 for comparison. The low-lying energy isomers whose simulated spectra agree best with the experiment are assigned as the primary structure for most of the clusters. It must be pointed out that for MSi₆⁻ (M = La and Ce) clusters, to date no experimental PES has been reported; thus, to facilitate the future comparison with further experiments, we have drawn the simulated PES spectra of the four lowest-lying isomers within 0.40 eV at the PBEPBE/RE/SDD/Si/6–311+G(d) level in Fig. 3 and 4. Furthermore, experimentally measured ADE/VDE from the anion photoelectron spectroscopy are compared with the calculated values at the PBEPBE/RE/SDD/Si/6–311+G(d) level of theory in Table 3. The calculated first ADE/VDE energies of the host clusters Si₆⁻ and Si₇⁻ are also presented in Table 3 for comparison. The VDEs of each cluster anion corresponds to the first peak maximum of each spectrum in Fig. 2–6. As shown in Table 3, the ADE/VDE are 2.73/2.81, 2.66/2.83, 1.83/1.95, and 2.86/3.07 eV, respectively, for M = La, Ce, Yb and Lu, which are bigger than those of Si_6/7⁻ clusters (theoretical: 2.24/2.45 eV for Si₆⁻, 1.91/2.30 eV for Si₇⁻, experimental: the VDE is 2.39 eV for Si₆⁻, 1.85 ± 0.02/2.39 eV for Si₇⁻), except for the case of the YbSi₆⁻ cluster. The calculated ADE/VDE for pure Si_6/7⁻ agree with experimentally measured results.

3.2.1. Si₆⁻ and Si₇⁻. For pure Si_6/7⁻, the simulated PES spectra of the lowest-energy isomers a for Si₆⁻ and f for Si₇⁻ agree with the experimental spectra,⁴ which reproduce two prominent PES bands below 3.5 eV. The calculated first VDEs values are 2.45 and 2.30 eV, which are in agreement with the experimental values (2.39 ± 0.01 eV for Si₆⁻ and 2.39 ± 0.04 eV for Si₇⁻).⁴ The calculated ADE for pure Si₇⁻ is 1.91 eV, which is also in line with experimentally measured results (1.85 ± 0.02 eV).⁶ For Si₆⁻, the second lowest-energy isomer b is 0.19 eV higher in energy than a, and considering the inherent accuracy of the DFT, we could not exclude the probability of isomer b. However, the calculated first VDE (2.77 eV) is higher than the experimental value (2.39 ± 0.01 eV).⁴ According to the second criterion, we could exclude the possibility of the b isomer in the Si₆⁻ cluster beam. Thus, we can conclude that structures a and f should be the reasonable isomers for pure Si_6/7⁻. All other higher energy structures should have been negligible contribution to the PES spectrum. The very large X–A energy gap (1.10 eV for Si₆⁻, 1.50 eV for Si₇⁻) between the first and second VDEs indicates that the neutral Si_6/7 clusters are closed-shell singlets with strongest stability.

3.2.2. LaSi₆⁻. The ground state of LaSi₆⁻ is closed shell (C_5v, ¹A′) with a valence configuration of (1a′)²(1a′′)²(2a′)²(3a′)²(2a′′)²(4a′)²(5a′)²(6a′)²(3a′′)²(4a′′)²(7a′)²(5a′′)²(8a′)²(9a′)². The first VDE corresponds to electron detachment from the 9a′ HOMO, which is an antibonding σ orbital (Fig. 7(a)). The SOMO (14a) of the neutral cluster upon removing an electron from the anion HOMO is shown in Fig. 7(b) and its spin density (Fig. 7(c)) is mainly on the La atom (0.68). As shown in Fig. 3, the simulated PES patterns on the basis of the four isomers appear to be rather different, the one isomer j being a bit loose and consisting of four electronic states within 0–4.661 eV, while, the others isomers (k, l, and m) presented more congested situations and consisting of six or seven electronic states. VDE calculations for the four low-lying isomers of LaSi₆⁻ were performed using the PBEPBE/RE/SDD/Si/6–311+G(d) level of theory. The lowest-energy structure j gives the first ADE/VDE values (2.73/2.81 eV), which are bigger than the pure silicon clusters (2.24/2.45 eV for Si₆⁻, and 1.91/2.30 eV for Si₇⁻).

3.2.3. CeSi₆⁻. For CeSi₆⁻, to date, no experimental PES spectrum has been reported; thus, we only present our theoretical detachment spectrum from the lowest-energy and three others isomers of the CeSi₆⁻ cluster at the PBEPBE/RE/SDD/Si/6–311+G(d) level in Fig. 4. The simulated PES spectra and calculated ADE/VDE of the four low-lying isomers are presented in Fig. 4 and Table 3. The ground state of CeSi₆⁻ is open shell (²A′′) with a valence configuration of (1a′)²(1a′′)²(2a′)²(3a′)²(2a′′)²(4a′)²(5a′)²(3a′′)²(6a′)²(4a′′)²(7a′)²(5a′′)²(8a′)²(9a′)²(6a′′)¹. The 6a′′ SOMO of CeSi₆⁻ (Fig. 8(a)) is primarily a nonbonding σ orbital. The Mülliken spin density (Fig. 8(d) and (e)) shows that the unpaired spin is primarily occupied on the Ce atom. After removal of an electron, the ground state structure of the neutral CeSi₆ is slightly different from CeSi₆⁻, although it retains the C_s symmetry. The other low-lying isomers (k, l, and m) are found to be nearly degenerate in energy (ΔE = 0.34, 0.37 and 0.38 eV), and their simulated spectra are a lot different from the ground state structure. They reproduce much more prominent PES bands than the global minimum structure j. The first calculated VDE (2.83 eV), corresponding to the electron detachment from the 6a′′ SOMO is larger than three others isomers (2.18 eV for k, 2.19 eV for l and 2.08 eV for m, Table 3). Detachment from the fully occupied MOs below the 6a′′ SOMO can lead to both singlet (S) and triplet (T) final states. At the PBEPBE level, the calculated VDEs from the (9a′)² (T, S), (8a′)² (T, S), and (5a′′)² (T, S) orbitals correspond to the intense and broad A band (ranging from 3.14 to 3.25 eV).

3.2.4. YbSi₆⁻. The C_2v, (²A₁) structure (Fig. 5 i) is clearly the ground state for YbSi₆⁻ with alternative structures being at least 0.34 eV higher in energy. The ground state of YbSi₆⁻ is open shell (²A₁) with a valence configuration of (1a₁)²(1b₂)²(2a₁)²(1b₁)²(3a₁)²(2b₂)²(4a₁)²(1a₂)²(5a₁)²(2b₁)²(3b₂)²(6a₁)²(4b₂)²(7a₁)²(3b₁)²(5b₂)²(8a₁)²(4b₁)²(2a₂)²(9a₁)²(6b₂)¹. Experimental photoelectron spectra of YbSi₆⁻ generally differ from those of Ho-, Gd-, and Pr-doped LnSi₆⁻ members with their most characteristic distinction being filled comparably low EBE peaks.⁴⁶ The X band in the experimental spectra is a low electron binding energy peak, which is broad and congested, suggesting that it may contain multiple detachment channels. Indeed, the calculated VDEs for the first seven detachment channels, from the SOMO 6b₂ (1.95 eV), 9a₁ (1.97 eV and 1.99 eV), 2a₂ (2.02 eV and 2.04 eV) and the 4b₁ (2.05 eV and 2.06 eV) orbital (T and S) are very close to each other, consistent with the broad X band. The 6b₂ SOMO is mainly a nonbonding σ orbital, as shown in Fig. 9(a) and (c) for the spin density. On removal of an electron, the neutral YbSi₆ retains the C_2v structure with a shorter Yb–Si bond by about 0.10–0.13 Å. The calculated VDEs from the 8a₁ orbital, 2.62 eV(S) and 2.64 eV(T), corresponding to the removal of the spin up and spin down electrons, are in good agreement with the observed VDE of the A band (2.60 ± 0.05 eV).⁴⁶ The simulated PES spectrum of the second-lowest energy isomer j is different from the experimental features, suggesting structure j should not be a reasonable isomer. Furthermore, the theoretical first ADE/VDE (2.39/2.42 eV) values of the second-lowest energy isomer j agree very poorly with the experimental data (1.80 ± 0.05/1.98 eV).⁴⁶ Overall, the simulated PES spectrum and calculated ADE/VDE from the global minimum of YbSi₆⁻ are in very good agreement with the experimental observations (Table 3 and Fig. 5), lending considerable credence to the global minimum identified.

3.2.5. LuSi₆⁻. The comparisons between experiment and theory for the ground state of LuSi₆⁻ are presented in Table 3 and Fig. 6. The ground state of LuSi₆⁻ (structure j) is closed shell (¹A′) with a valence configuration of (1a′)²(1a′′)²(2a′)²(3a′)²(2a′′)²(4a′)²(5a′)²(6a′)²(3a′′)²(7a′)²(4a′′)²(8a′)²(5a′′)²(9a′)². The first calculated VDE corresponds to electron detachment from the 9a′ HOMO, which is mainly a bonding σ orbital (Fig. 10(a)). The SOMO (7a₁) of the neutral cluster on removing an electron from the anion HOMO is shown in Fig. 10(b), and its spin density (Fig. 10(c)) is seen to concentrate on the Lu atom. The simulated PES patterns on the basis of the two isomers (Fig. 6) appear to be rather different, i.e., the PES of the one isomer j is extremely loose and consists of only two bands within 0–4.661 eV, while, the other isomer i presented extremely congested situations and consisting of five bands. For the ground state structure of LuSi₆⁻, the calculated VDE of 3.07 eV is in good agreement with the experimental value of 3.20 eV (Table 3);⁴⁷ however, the calculated ADE of 2.86 eV is larger than the experimental value of 2.10 eV. As seen from Table 3, this situation is ambiguous when considering the first ADE. We note that the second isomer i is 0.35 eV above the global minimum structure, but the computed ADE (2.00 eV) agrees very well with the experimental results (2.10 eV).⁴⁷ Usually, it is rare to have such a high-energy isomer coexisting in the cluster beam experiments. Considering their very close ADE, it is possible that isomer i is a minor isomer. These results lead us to conclude that the global minimum structure j and minor isomer i might be coexisting in the LuSi₆⁻ cluster beam.

3.3. Electronic and magnetic properties

We performed natural population analysis (NPA) for the lowest-energy structures of MSi₆^q clusters, and the summarized natural charges and populations are given in Table 4. As seen from Table 4, charge always transfers from the RE atom to the Si atom for RE-doped MSi₆^q clusters. The transferred charges are from 0.47 e to 0.91 e for neutral MSi₆ and by 0.10 e–0.33 e for anionic species, which indicates that the RE atom acts as an electron donor. Note that the Yb atom loses the largest charge (0.91 e) from its neighbouring Si atoms. The NPA of all the Si atoms bonding to RE atom is negative. The charge distribution is dependent on the symmetry of the cluster. The magnetic properties are one of the most interesting in cluster physics. The physicochemical property of size-selected cluster is quite different from that of single molecular or bulk material. Unfortunately, no Stern–Gerlach experiment has been carried out for present species up to now; however, the reliability of the present DFT (PBEPBE) was validated by performing calculations on the simulated PES spectra, the first ADE/VDE on pure Si_6/7⁻ and many other RE-doped MSi₆⁻ (M = Yb, Lu, Pr, Gd, and Ho) clusters. Magnetic moment (μ_B) of 4f, 5d, 6s, and 6p states for the RE atom, total magnetic moment (μ_B) of the RE atom, and total magnetic moment of the lowest-energy RE-doped MSi₆^q (M = La, Ce, Yb and Lu; q = 0, −1) clusters are presented in Table 5. As shown in Table 5, it can be seen that the total magnetic moment of the CeSi₆ cluster has the largest value (2 μ_B) among the MSi₆^q species. It can also be seen that the total magnetic moments of the MSi₆^q clusters and the magnetic moments on the RE do not quench when the RE is encapsulated in a Si₆ outer frame structure, which is consistent with the results of the EuSi_n.²⁹ Interestingly, the overwhelming majority of total magnetic moments are contributed by RE atoms. These results seem to indicate that the majority of f electrons do not form chemical bonds between the RE atoms and their silicon cluster environment. To further understand the magnetic properties of MSi₆^q clusters, we performed a detailed analysis of the local magnetic moment of RE atom in MSi₆^q clusters. For free Yb and Lu atoms, the configurations of valence electrons are [Yb] 4f¹⁴6s² and [Lu] 4f¹⁴5d¹6s², which have closed 4f shells and primarily donate their 6s electrons to Si₆/Si₆⁻, forming charge-transfer YbSi₆⁻/LuSi₆ clusters with little distortions. However, for the open 4f shell of Ce dopant, the free Ce atom with the configuration of valence electrons 4f¹5d¹6s², the 4f orbitals obtain electrons by 0.13 e for CeSi₆ and the electron transfer is very small for CeSi₆⁻ from the Si₆/Si₆⁻ clusters.

Certain trends of the magnetic moments of MSi₆^q in this work can be described as follows: (1) Table 5 shows that LaSi₆, CeSi₆^0,−, YbSi₆⁻ and LuSi₆ are magnetic clusters, and the magnetic moments are 1, 2, 1, 1 and 1 μ_B, respectively. LaSi₆⁻, YbSi₆ and LuSi₆⁻ species present a non-magnetic ground state. (2) For all the RE-doped MSi₆^q clusters, RE atoms carry most of the magnetic moments. The magnetism of the clusters considered in this study is mainly located on the RE atoms, and only very small magnetic moments are found on the parent clusters. This is quite consistent with the result of Eu-doped Si clusters.²⁹ For CeSi₆⁻ clusters, Si₆⁻ contributes a small negative magnetic moment to the total moment, while for other RE-doped clusters, they carry a small positive magnetic moment. (3) The dopant RE atoms remain largely localized, and the atomic-like magnetism is maintained in the doped clusters. The pure Si₆ cluster is usually non-magnetic, but when being doped with different RE atoms, it sometimes becomes a magnetic cluster.

4. Conclusions

In conclusion, we have investigated the structural, electronic, and magnetic properties of a series of RE-doped neutral and charged silicon clusters MSi₆^q (M = La, Ce, Yb and Lu; q = 0, −), using previous experimental anion photoelectron spectroscopy and density-functional calculations at the PBEPBE level. Stochastic search procedure by Saunders “kick” method combined with extensive DFT calculations are used for global minimum searches (SK-DFT). Optimized geometries of the ground-state and low-lying energy states of both the anion and neutral species are reported using SK-DFT and the results are compared with experimental photoelectron spectroscopy results from the literature. The global minimum search revealed that the MSi₆⁻ (M = La, Ce, and Lu) is a three-dimensional structure with the lanthanide-metal atom sitting on top (or bottom) of the regular pentagonal bipyramid, which is obtained by substituting one Si by M atom from the bottom (or top) of the pure Si₇^−,0 cluster. Instead, the neutral species and YbSi₆⁻ show the impurity as a four-coordinate atom in the equatorial plane of a pentagonal bipyramid, which is obtained by substituting one Si in the equatorial plane by an M atom from the ground state structure of the Si₇ cluster. The CeSi₆^−,0 clusters have the largest binding energy and doping energy, indicating that the neutral and anionic CeSi₆ clusters should be the most stable among MSi₆^q species. The study of magnetic properties indicates that the magnetic moment of the doped RE atoms remains largely localized and the atomic-like magnetism is maintained in the doped clusters. The pure Si₆ cluster is usually non-magnetic, but when being doped with different RE atoms, it sometimes becomes a magnetic cluster. The current Si₆ doping with different RE atoms suggest that a new class of pentagonal bipyramid clusters with varying magnetic properties may exist. The systematic methods, including structural characteristics and magnetic properties, presented in this study are useful for the analysis of the theoretical and experimental data and provide insight into more complicated cluster systems.

Acknowledgements

This work was supported by the Natural Science Foundation of Fujian Province of China (no. 2012J05005 and 2014J05006), the Programs for New Century Excellent Talents and Excellent Youth Talents in Fujian Province University (NCETFJ and EYTFJ, no. JA13009), and the Fundamental Research Funds for the Central Universities (JB-ZR1201).

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