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Structures, Electronic, and Magnetic Properties of CoK_{n} (n = 2–12) Clusters: A Particle Swarm Optimization Prediction Jointed with First-Principles Investigation

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## Abstract

**:**

_{n}(n = 2–12) clusters have been screened out using particle swarm optimization and first principles relaxation. The results show that except for CoK

_{2}the other CoK

_{n}(n = 3–12) clusters are all three-dimensional structures, and CoK

_{7}is the transition structure from which the lowest energy structures are cobalt atom-centered cage-like structures. The stability, the electronic structures, and the magnetic properties of CoK

_{n}clusters (n = 2–12) clusters are further investigated using the first principles method. The results show that the medium-sized clusters whose geometries are cage-like structures are more stable than smaller-sized clusters. The electronic configuration of CoK

_{n}clusters could be described as 1S1P1D according to the spherical jellium model. The main components of petal-shaped D molecular orbitals are Co-d and K-s states or Co-d and Co-s states, and the main components of sphere-like S molecular orbitals or spindle-like P molecular orbitals are K-s states or Co-s states. Co atoms give the main contribution to the total magnetic moments, and K atoms can either enhance or attenuate the total magnetic moments. CoK

_{n}(n = 5–8) clusters have relatively large magnetic moments, which has a relation to the strong Co-K bond and the large amount of charge transfer. CoK

_{4}could be a magnetic superatom with a large magnetic moment of 5 μ

_{B.}

## 1. Introduction

_{13}@Au

_{20}]

^{−}clusters using density functional theory implemented in the DMol

^{3}package, and they pointed out that the transition-metal atom could enhance or attenuate the total magnetic moments [26]. Zhao et al. investigated the geometries and electronic structures of Y

_{n}Al (n = 1–14) clusters and found that doping with an Al atom could attenuate the magnetic moments but enhance the stabilities of the yttrium framework [27]. Our previous work predicted the geometries of the lowest energy and their isomers of ScK

_{n}(n = 2–12) clusters using particle swarm optimization and first principles geometry optimization, and we have also investigated the electronic structures and magnetic properties using the first principles calculation. The results showed that the doping of Sc atoms can improve the magnetic properties and stability of the K

_{n}clusters, and the ScK

_{12}cluster may be a magnetic superatom with a high magnetic moment of 5 μ

_{B}[28]. The VIIIB atom (Fe, Co, and Ni)-doped clusters including VIIIB transition-metal oxides and hydroxides nanoclusters have also attracted great attention due to their interesting electronic structures, physical and chemical properties, and their wide application. For example, Milan Babu Poudel et al. reported that the hierarchical heterostructure comprising ternary metal sulfides covered by nickel–cobalt layered double hydroxide could be used as binder-free cathode material for supercapacitor application [29], and they also reported a superior bifunctional electrocatalyst for oxygen evolution reaction (OER) and hydrogen evolution reaction (HER), which contains both cobalt and nickel atoms [30]. As for the VIIIB atom (Fe, Co, and Ni)-doped clusters, our group has investigated the geometries, electronic structures, and magnetic properties of Ge

_{n}Co (n = 2–12) clusters using the first principles method, and the results show that the total magnetic moments of Ge

_{n}Co (n = 2–12) clusters does not quench, and the doping of the Co atom is beneficial to enhance the stability of host Ge

_{n}clusters [31]. The electronic structures and magnetic moments of the core-shell clusters Co

_{13}@TM

_{20}(TM = Mn, Fe, Co, and Ni) have also been investigated using the first-principles method, and the results show that these clusters have huge magnetic moments, especially for the Co

_{13}@Mn

_{20}cluster, whose magnetic moment is as large as 113 μ

_{B}[32]. After investigating the geometries and electronic structures using the first-principles method implemented in the Vienna Ab Initio Simulation Package (VASP) and Orca code, Hao et al. pointed out that doping of the Co atom in B

_{n}clusters significantly changes their structures, and the Co

_{2}B and Co

_{2}B

_{7}clusters have a large magnetic moment of 3 μ

_{B}[33]. A systematic theoretical investigation of the structure, stability, and electronic properties of Li

_{n}Co clusters showed that the doping with the cobalt atom could enhance the stabilities of host clusters, and greater electron transfer from Li-2s to Co-3d can help to strengthen the bond length of Li-Co [34].

_{n}(n = 2–12) clusters predicted by the particle swarm optimization (PSO) algorithm along with first principles relaxation. The results show that the CoK

_{n}(n = 3–12) clusters are all three-dimensional structures, and CoK

_{7}is the transition structure from which the lowest energy structures are cobalt atom-centered cage-like structures. The stabilities, the magnetic moments, and the electronic structures are also investigated using the first principles calculation. The results show that the cobalt atom-centered cage-like clusters are more stable than smaller-sized clusters. In addition, the electronic configuration of CoK

_{n}clusters could be described as 1S1P1D according to the spherical jellium model. According to the results of the projected density of states, the main components of petal-shaped D molecular orbitals are Co-d and K-s states or Co-d and Co-s states. The main components of sphere-like S molecular orbitals or spindle-like P molecular orbitals are K-s states or Co-s states. Co atoms give the main contribution to the total magnetic moments, and K atoms can either enhance or attenuate the total magnetic moments. CoK

_{n}(n = 5–8) clusters have relatively large magnetic moments, which has a relation with the strong Co-K bond and a large amount of charge transfer. CoK

_{4}could be a magnetic superatom with a large magnetic moment of 5 μ

_{B.}

## 2. Computational Details

_{n}(n = 2–12) clusters. The CALYPSO code has been widely used to predict the ground state structure of clusters and crystals [39,40,41]. Using this powerful structure search method combined with first-principles calculations, Tang et al. have identified a planar CoB

_{6}monolayer as a stable two-dimensional ferromagnet [42], and our group has also investigated the geometries of transition-metal-doped clusters [43].

_{n}(n < 9) and 30 for CoK

_{n}(n ≥ 9) clusters. The maximum number of generations is set as more than 20 (for n < 9) and 30 (for n ≥ 9), that is to say, more than 400 (for n < 9) and 900 (for n ≥ 9) candidate isomers are generated and optimized to get the lowest-energy clusters. The geometry optimization is performed using the VASP code [35,36]. During the calculation, the projected augmented wave method and the Perdew–Burke–Ernzerhof (PBE) exchange-correlation functional [44] under generalized gradient approximation (GGA) are used. The plane-wave cutoff energy is set as 300 eV. The global break condition for the electronic self-consistency (SC) loop is set as 1 × 10

^{−5}eV. The maximum number of electronic SC steps is set as 500, and the maximum number of ionic steps is set as 200. A conjugate gradient algorithm is used to relax the ions into their instantaneous ground state. To avoid the interaction of atoms located in neighbor cells, the CoK

_{n}clusters are placed in a cubic super lattice of 12 × 12 × 12 angstroms (for n < 9) and 15 × 15 × 15 angstroms (for n ≥ 9).

_{n}(n = 2–12) clusters with different spin are carried out to ensure that the obtained structures are the lowest-energy structures. The Gaussian09 calculation is performed using B3LYP [46] functional at a 6-31G** level for the Co atom and a 6-311G** [47] level for K atoms. The authors have carefully checked out the harmonic vibrational frequencies of each lowest- and lower-energy clusters to ensure the obtained clusters are stable clusters without imaginary frequencies. The population analysis of CoK

_{n}clusters is studied using the Multiwfn3.8 program [48,49].

## 3. Results and Discussion

#### 3.1. The Geometries of Lowest-Energy Clusters

_{n}(n = 2–12) clusters along with the bond length of Co-K bonds. As shown in Figure 1, except for CoK

_{2}clusters whose geometries of lowest-energy and their isomer are planar triangular structures, the geometries of the other CoK

_{n}(n = 3–12) clusters are all three-dimensional structures. The geometries of the ground state and meta stable state of CoK

_{3}cluster are a slightly distorted quadrangle with C

_{2V}symmetry. The bipyramid structures were found in the lowest energy structures of CoK

_{n}(n = 4–6) clusters. The lowest energy structure of CoK

_{4}is a triangular bipyramid structure with a C

_{3V}point group. The lowest energy structure of cluster CoK

_{5}is a tetragonal bipyramid structure with a C

_{4V}point group. The CoK

_{6}is a twisted pentagonal bipyramid structure with a C

_{1}point group. The ground state structure of the CoK

_{7}cluster is a Co-atom-centered K-atom-capped tetragonal bipyramid structure with a C

_{3V}point group. Noting that from the CoK

_{7}cluster, the lowest-energy structure becomes a Co-atom-centered polyhedron. The lowest-energy structure of the CoK

_{8}cluster is a Co-atom-centered K-capped pentagonal bipyramid with a C

_{s}point group. At first glance, the lowest-energy structure of the CoK

_{9}cluster looks like a Co-atom-centered K-atom-capped square antiprism, but actually, it is a Co-atom-centered three-K-atom-capped trigonal prism because its point group is D

_{3h}. The lowest-energy structure of the CoK

_{10}cluster is a Co-atom-centered two-K-atom-capped square antiprism. It can also be described as a Co-atom-centered 1-4-4-1-layered structure. The lowest-energy structure of the CoK

_{11}cluster is a distorted Co-atom-centered cage-like structure. The lowest energy structure of the CoK

_{12}cluster is a distorted Co-atom-centered icosahedral structure with a C

_{1}point group.

_{n}Co (n = 1–13) clusters, the geometries of the most stable Ge

_{n}Co (n = 1–3) clusters are planar geometries. The geometries of most stable medium-sized Ge

_{n}Co (n = 4–13) clusters are three-dimensional configurations, and from the Ge

_{9}Co cluster, the dopant cobalt atom encapsulated geometry becomes the lowest-energy structure. As for Li

_{n}Co (n = 1–12) clusters, the smallest Li

_{n}Co (n = 1–3) clusters also adopt planar geometries, and the medium-sized Li

_{n}Co (n = 4–12) clusters adopt three-dimensional configurations. Like the results shown in this paper, from the Li

_{7}Co cluster, the most stable Li

_{n}Co clusters adopt cobalt atom-centered cage-like structures.

#### 3.2. Relative Stability

_{n}(n = 2–12) clusters is further evaluated by the binding energy per atom (E

_{b}), the second-order difference of energies (Δ

_{2}E), and fragmentation energies (E

_{f}), which are defined as follows:

_{9}cluster has the largest binding energy per atom, and then size dependence becomes smooth for n = 9~12. The results show that the large-sized clusters are more stable than small clusters.

_{2}E) and fragmentation energies (E

_{f}), they own similar curves, as shown in Figure 2. The local peaks of Δ

_{2}E localized at n = 4, 6, 9, and the local peaks of E

_{f}are found at n = 4, 6, 8, 11, implying these clusters are more stable than their neighbors.

_{4}, CoK

_{6}, CoK

_{8}, and CoK

_{9}clusters are more stable than their neighbors. The relatively strong stability may have a relation with the relatively smaller bond length and relatively strong atomic interactions. For example, the bond lengths of Co-K in CoK

_{4}are 2.97, 2.97, 2.97, and 3.73 angstrom, respectively, which are smaller than the bond length of Co-K in CoK

_{5}(about 3.20, 3.20, 3.34, 3.34, and 4.60 angstrom). The bond lengths of Co-K in CoK

_{6}are 2.94, 2.97, 3.00, 4.25, and 5.41 angstrom, which are also smaller than their neighbor clusters. Noting that although the cage-like clusters have relatively large bond lengths, the dense packing cage-like structures make these clusters have a relatively large binding energy per atom. Similar conclusions can also be found in other VIIIB atom-doped clusters.

#### 3.3. Magnetic Properties

_{n}(n = 2–12) clusters are obtained using the Gaussian09 calculation at the B3LYP/6-31G**(Co) and B3LYP/6-311G**(K) levels. The obtained total magnetic moments of CoK

_{n}(n = 2–12) clusters are shown in Figure 3 and Table 1. As shown in Figure 3 and Table 1, the magnetic moment of CoK

_{n}(n = 2–4) rapidly increases from 1 μ

_{B}to 5 μ

_{B}, and then gradually decreases to 1 μ

_{B}, except for CoK

_{7}and CoK

_{11}, whose magnetic moments are 4 μ

_{B}and 2 μ

_{B}, respectively. The magnetic moments of CoK

_{n}(n = 4–12) are 5 μ

_{B}(CoK

_{4}), 4 μ

_{B}(CoK

_{5}), 3 μ

_{B}(CoK

_{6}), 4 μ

_{B}(CoK

_{7}), 3 μ

_{B}(CoK

_{8}), 2 μ

_{B}(CoK

_{9}), 1 μ

_{B}(CoK

_{10}), 2 μ

_{B}(CoK

_{11}), and 1 μ

_{B}(CoK

_{12}), respectively.

_{n}(n = 2–12) clusters, the atomic magnetic moment of the Co atom is in the range of 2.1937 μ

_{B}to 2.9430 μ

_{B}, indicating that the Co atom plays an important role in determining the total magnetic moment. As for K atoms, they can enhance (as in the case of CoK

_{4}, CoK

_{5}, CoK

_{6}, CoK

_{7}, and CoK

_{8}) or attenuate (as in the case of CoK

_{2}, CoK

_{3}, CoK

_{9}, CoK

_{10}, CoK

_{11}, and CoK

_{12}) the total magnetic moments of CoK

_{n}clusters.

_{4}cluster exhibits the largest magnetic moment among all these clusters, hence the authors would investigate the molecular orbitals to dig out the origination of the largest magnetic moment. The obtained molecular orbitals are shown in Figure 4. For comparison, the molecular orbitals of CoK

_{5}are also investigated. As shown in Figure 4, the molecular orbitals look like sphere-like s orbitals, spindle-like p orbitals, and petal-shaped d orbitals. For those petal-shaped orbitals, the electrons localize around the Co atom, and the sphere-like and spindle-like electrons delocalize around all atoms. According to the data shown in Figure 4, the authors believe their electrons can be described as 1S

^{2}1P

^{3}1D

^{8}(for CoK

_{4}) and 1S

^{2}1P

^{4}1D

^{8}(for CoK

_{5}) according to the spherical jellium model [45,46].

_{n}clusters. Figure 5 gives the spin-polarized projected density of states of CoK

_{4}and CoK

_{5}clusters. Take CoK

_{4}, for example. As shown in Figure 4, the molecular orbitals of CoK

_{4}are found in the energy range of −2.0~−2.8 eV (spindle-like spin-up molecular orbitals), −3.4~−4.6 eV (petal-shaped spin-up molecular orbitals), −4.8 eV (sphere-like spin up molecular orbital), −2.0~−2.4 eV (petal-shaped spin-down molecular orbitals), and −4.0 eV (sphere-like spin-down molecular orbitals). As shown in Figure 5, in the energy range of −2.0~−2.8 eV, there are K-s, Co-s, and Co-p states (for spin-up states), and K-s, Co-d states (for spin-down states). That is to say, the hybrid molecular orbitals coming from the Co-K interaction are found in this energy range, and the K-s states play an important role in forming the spindle-like spin-up molecular orbitals. The Co-d and K-s atomic states give the main contribution to forming the spin-down petal-shaped molecular orbitals. In the energy range of −3.0~−4.0 eV, there are mainly the Co-d and K-s states in the spin-up states, indicating the spin-up petal-shaped molecular orbitals are mainly coming from the Co-d and K-s atomic states. While there are only Co-s states found nearby −4 eV in spin-down molecular orbitals, indicating the Co-s states give the main contribution to determining the spin-down molecular orbitals. There are Co-d and Co-s states found in the energy range of −4.0~−4.6 eV in the spin-up molecular orbitals, and there are mainly Co-s states found in the spin-up molecular orbitals nearby −4.8 eV. So, the Co-d and Co-s states give the main contribution in forming the spin-up petal-shaped molecular orbitals in the energy range of −4.0~−4.6 eV, and Co-s states determine the spin-up spherical molecular orbitals nearby −4.8 eV. In a word, as for the CoK

_{4}cluster, the main components of petal-shaped D molecular orbitals come from Co-d and K-s states (such as D molecular orbitals in the energy range of −2.0~−2.4 eV) or Co-d and Co-s states (such as D molecular orbitals in the energy range of −4.0~−4.6 eV). In addition, the main components of sphere-like S molecular orbitals or spindle-like P molecular orbitals are K-s states (such as P molecular orbitals in the energy range of −2.0~−2.8 eV) or Co-s states (such as S molecular orbitals nearby −4.8 eV). Similar conclusions can also be found in the CoK

_{5}cluster. The main components of spindle-like P molecular orbitals in the energy range of −2.0~−2.8 eV are K-s states. The main components of spherical S molecular orbitals nearby −4.3 eV and −4.7 eV are Co-s states. The main components of petal-shaped D molecular orbitals in the energy range of −3.6~−5.0 eV are Co-d and K-s states (for D orbitals in the energy range of −3.6~−4.2 eV) and Co-d and Co-s states (for D molecular orbitals in the energy range of −4.6~−5.0 eV).

_{4}cluster has the shortest average bond length of the Co-K bond and a relatively large atomic charge of the Co atom (about −0.8717 e). The electronic configuration of the isolated Co atom is [Ar]3d

^{7}4s

^{2}. In the CoK

_{n}clusters, the spd hybridization was found (as discussed above) and the hybridization makes the electron transfer from the Co-4s states and K-2s states to the Co-3d states and Co-4p states (shown in the NEC results listed in Table 1, and the spin-polarized projected density of states shown in Figure 5). The strong Co-K interaction and a large amount of charge transfer result in the enhanced magnetic moment of the CoK

_{4}cluster. Similar to the CoK

_{4}cluster, enhanced magnetic moments are also found in CoK

_{n}(n = 5–8) clusters. The relatively weak interaction of Co-K coming from the relatively large bond length of Co-K bond in other CoK

_{n}clusters (especially the medium-sized cage-like clusters) makes these clusters have relatively smaller magnetic moments.

## 4. Conclusions

_{n}(n = 2–12) clusters are predicted using the PSO method joined with the first-principles geometries optimization, and then the electronic structures and magnetic moments are further investigated using the DFT calculation. The results show that the lowest-energy structures transform from planar geometry to dense packing structures, and the Co atom moves from the apex position (n < 7) to the central position of cage-like structures (n ≥ 7). Medium-sized clusters with cage-like geometries are more stable than small clusters, and CoK

_{4}, CoK

_{6}, CoK

_{8}, and CoK

_{9}clusters are more stable than their neighbors. The electronic configuration of CoK

_{n}clusters can be described as 1S1P1D according to the spherical jellium model. The main components of petal-shaped D molecular orbitals are Co-d and K-s states or Co-d and Co-s states. The main components of sphere-like S molecular orbitals or spindle-like P molecular orbitals are K-s states or Co-s states. The Co atom plays an important role in determining the total magnetic moments, and K atoms can either enhance or attenuate the total magnetic moments. CoK

_{n}(n = 5–8) clusters have relatively large magnetic moments which originate from the strong interaction of Co-K and a large amount of charge transferring from K to Co atoms. CoK

_{4}could be a magnetic superatom with a large magnetic moment of 5 μ

_{B.}

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**The ground state structure of CoK

_{n}(n = 2–12) clusters (marked as n-a, such as 12-a) and their meta-stable isomers (marked as n-b, such as 12-b). The text below the image also gives the point group, magnetic moment (in μ

_{B}), and the relative energy (in eV) compared with the ground state.

**Figure 2.**The binding energy per atom (E

_{b}), the second-order difference of energies (Δ

_{2}E), and fragmentation energies (E

_{f}) of CoK

_{n}(n = 2–12) clusters.

**Figure 3.**The total magnetic moments of CoK

_{n}(n = 2–12) clusters (black) and local magnetic moments of Co atom in CoK

_{n}(n = 2–12) clusters (red).

**Table 1.**The natural electron configuration of Co atom (NEC(Co)), the magnetic moments of CoK

_{n}clusters (m(CoK

_{n})), Co atom (m(Co)), K atoms (m(K)), the Mulliken atomic charges of Co (q(Co)), and the average bond length of Co-K bond (R(Co-K)).

Cluster | NEC(Co) | m(CoK_{n}) | m(Co) | m(K) | q(Co) | R(Co-K) |
---|---|---|---|---|---|---|

CoK_{2} | 4s^{1.74}3d^{7.45}4p^{0.31} | 1 | 2.1937 | −1.1937 | −0.4264 | 2.98 |

CoK_{3} | 4s^{1.83}3d^{7.43}4p^{0.18} | 2 | 2.4643 | −0.5186 | −0.2175 | 4.08 |

CoK_{4} | 4s^{1.68}3d^{7.51}4p^{1.16} | 5 | 2.8923 | 2.1077 | −0.8717 | 2.91 |

CoK_{5} | 4s^{1.8}3d^{7.4}4p^{1.06} | 4 | 2.8704 | 1.1418 | −0.2659 | 3.54 |

CoK_{6} | 4s^{1.73}3d^{7.56}4p^{1.2} | 3 | 2.7535 | 0.2465 | −0.8875 | 3.69 |

CoK_{7} | 4s^{1.71}3d^{7.48}4p^{2.02} | 4 | 2.9430 | 1.0570 | −0.9731 | 3.63 |

CoK_{8} | 4s^{1.67}3d^{7.43}4p^{2.64} | 3 | 2.2473 | 0.7501 | −0.7206 | 3.53 |

CoK_{9} | 4s^{1.91}3d^{7.39}4p^{2.68} | 2 | 2.3750 | −0.3750 | −0.4794 | 3.65 |

CoK_{10} | 4s^{1.84}3d^{7.4}4p^{2.53} | 1 | 2.4785 | −1.4785 | −0.4925 | 3.89 |

CoK_{11} | 4s^{1.79}3d^{7.42}4p^{2.44} | 2 | 2.3572 | −0.3572 | −0.4844 | 4.16 |

CoK_{12} | 4s^{1.78}3d^{7.41}4p^{2.46} | 1 | 2.4580 | −1.4673 | −0.5966 | 4.26 |

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## Share and Cite

**MDPI and ACS Style**

Jiang, Y.; Aireti, M.; Leng, X.; Ji, X.; Liu, J.; Cui, X.; Duan, H.; Jing, Q.; Cao, H.
Structures, Electronic, and Magnetic Properties of CoK* _{n}* (

*n*= 2–12) Clusters: A Particle Swarm Optimization Prediction Jointed with First-Principles Investigation.

*Nanomaterials*

**2023**,

*13*, 2155. https://doi.org/10.3390/nano13152155

**AMA Style**

Jiang Y, Aireti M, Leng X, Ji X, Liu J, Cui X, Duan H, Jing Q, Cao H.
Structures, Electronic, and Magnetic Properties of CoK* _{n}* (

*n*= 2–12) Clusters: A Particle Swarm Optimization Prediction Jointed with First-Principles Investigation.

*Nanomaterials*. 2023; 13(15):2155. https://doi.org/10.3390/nano13152155

**Chicago/Turabian Style**

Jiang, Yi, Maidina Aireti, Xudong Leng, Xu Ji, Jing Liu, Xiuhua Cui, Haiming Duan, Qun Jing, and Haibin Cao.
2023. "Structures, Electronic, and Magnetic Properties of CoK* _{n}* (

*n*= 2–12) Clusters: A Particle Swarm Optimization Prediction Jointed with First-Principles Investigation"

*Nanomaterials*13, no. 15: 2155. https://doi.org/10.3390/nano13152155