In 1911 Kamerlingh Onnes  discovered the superconductivity (SC) of the Hg (at critical Tc = 4.19 K), while he was doing his experiments on the resistivity of gold and mercury wires at low temperature. At that time, Kamerlingh was the only one who could reach very low temperatures because he was the first who obtained the liquid helium. At that moment, a new state of the matter was discovered, the SC state. For explaining the mechanism of SC, it was required the creation of quantum mechanics (1925), see     , the formulation of the Pauli Exclusion Principle (1925)  , the creation of quantum field theory (QFT) , and many other developments in quantum mechanics, before in 1957 Bardeen-Cooper-Schrieffer (BCS)   formulated their famous microscopic theory of superconductivity.
The BCS theory was based on creation of Cooper pairs (pair of electrons that attract each other, instead of repelling, through the interaction with the lattice vibrations). Later on, Gor’kov  generated the microscopic formulation of the phenomenological macroscopic Ginzburg-Landau (G-L) theory  . After, Eliashberg  created a new approach to the conventional superconductors, broadening the range of application of the BCS theory to systems with strong electron-phonon interaction.
For many years the critical temperature Tc was low, the maximum critical temperature was obtained for Nb3Sn, Tc = 18.5 K. In 1986 Bednorz and Müller  discovered the high Tc (~30 K) SC in the cuprates family. For YBa2Cu3O7 the lowest Tc = 65 K  was obtained for zero pressure. As was shown in    for different materials, the increase of pressure leads to an increase of Tc.
One of the long-standing challenges was the observation of room-temperature SC. For many years numerous laboratories failed to increase Tc. The progress arises after Drozdov et al.  using high pressure obtained for sulfur hydride system a Tc = 203 K. At last, in 2020 Snider et al.  obtained the really room-temperature SC with the Tc = 287.7 K (15˚C) for a photochemically transformed carbonaceous sulfur hydride. They used the diamond anvil cell with a palladium thin film that assisted the synthesis by protecting the sputtered yttrium from oxidation and promoting subsequent hydrogenation. These types of materials are characterized by high frequencies vibration that increases the electron-phonon coupling, which is needed for high Tc phonon mediated SC, that is, for conventional SC.
The discovery in 2008 by Hosono and coworkers   of the superconductivity in the La[O(1-x) Fx]FeAs with (x = 0.05 − 0.12) represented the rise of a new era with the family of high-Tc Fe-based superconductors (Fe-SC), which are also named as iron-based superconductors (IBSC). This family is composed by six groups of IBSC compounds . Among them, the Ba-based crystals, BaFe2As2, are widely used  - . They have a high-quality single crystal and are easily growing. It is very important that for this crystal it is easy to produce SC materials with a variety of chemical doping. It is for this material that the SC phase was first observed by Co substitution on the Fe site . The parent compound is a paramagnetic semimetal, it turns into superconductor upon electron doping by d-electrons atoms (substitution of Fe atoms by Co, Ni, Rh or Pd) or upon hole doping in the plane of the Ba atoms (e.g., substitution of Ba atoms by K).
IBSC materials where intensively studied by theorists, see  -  and references therein. It was shown that the IBSC material has a quite complicated band structure and several disconnected Fermi surfaces (FSs)   . According to these studies all five 3d orbitals of the Fe are involved in the formation of the FSs. IBSC belong to the broad category of strongly correlated superconductors such as heavy fermions and cuprates high-Tc SC, although the latter has rather different mechanism of SC. We recommend the readers the popular and comprehensive reviews by Norman  , Mazin , Wang and Lee , Chubukov , Kordyuk , Baquero  and Prosorov et al. .
From the first year of the discovery of the IBSC, it has been accepted that the superconductivity in these materials is non-conventional, presenting an anti-ferromagnetic (AFM) order. As was proposed by Mazin et al.  , these new superconducting materials tend to form AFM order, and the magnetism existing in the parent crystal at zero doping is suppressed by the AFM spin fluctuations, similar results were obtained also by Singh and Du . The AFM spin fluctuations can induce s-wave pairing with sign change of the order parameter between the electron like FSs and hole like FSs, denoted as s±. At the same time, Kuroki et al.  applied multiorbital random-phase approximation  to the model of five d-orbitals and obtained similar results as in  , but they also accepted the d-wave symmetry. Other types of symmetry have been proposed by Onari and Kontani , being d-wave symmetry, and also opposing to the s±-wave symmetry .
The parent compound in IBSC can be considered as some kind of Mott insulator      and the physics of a Mott insulator may play an important role in the IBSC mechanism. From this model also follows the anti-ferromagnetism and s± pairing     . As was discussed in the review by Lee et al. , the Anderson resonating valence bond (RVB) theory, that was first proposed for cuprates, can be applied to the Mott insulating model naturally.
The RVB theory  for high Tc superconductors was proposed after the discovery of the cuprates. In this theory the antiferromagnetic lattice is melted into a spin-liquid phase composed by singlet pairs. When doping is applied, the singlets become charged giving rise to the superconducting state. This theory takes into account the separation between spin and charge, then the electronic excitation spectra can be presented as two different branches: charged spinless holons and chargeless spinons  .
In our previous publications devoted to IBSC   , we performed the comparative studies of the electronic structure of the pure Ba4Fe5As8 cluster and doped with substitutions of Fe atom by two pairs of dopants Co, Ni and Rh, Pd. The Embedded Cluster Method at the Möller-Plesset second order electron correlation level (ECM-MP2)    was used and the detailed charge and spin distribution at the Natural Bond Orbital (NBO) analysis    was obtained. In these calculations, spinless electron on the 3d orbitals was obtained, pointing out on the Anderson RVB model as the possible mechanism for superconductivity for this new type of material.
In this article we study the electronic structure of the isolated Ba4Fe5As8 cluster doped by Rh and Pd using unrestricted Möller-Plesset second order (MP2) method. The presented results obtained by the GAUSSIAN 2016 A.03  suite of programs and will be compared with our previous results performed also by GAUSIAN 2016 A.03  but for the embedded cluster. We will analyze the energy difference between the isolated cluster and the embedded cluster and the NBO orbital population as well.
The embedded cluster method at the Möller-Plesset second order electron correlation level (ECM-MP2) was used. The ECM-MP2 methodology includes two stages. At the first stage, the cluster representing the crystal is selected and the quantum-mechanical MP2 calculations are performed with the unrestricted Hartree-Fock (UHF) method, as the zero-order approximation. A detailed description of MP2 is given in Appendix 3 of book .
The complete structural information is taken from . The selected cluster composed by 17 atoms is depicted on Figure 1. This selection must maintain the symmetry of the crystal. Since we study the influence of local effects in the electronic structure, we placed one of the Fe atoms, which will be substituted by dopants in the centre of the cluster.
At the second stage, the cluster is embedded in a background charges that reproduce the Madelung potential for the infinite crystal. Two conditions must be fulfilled: 1) the symmetry of the crystal must be preserved; 2) the cluster with the background charges must be neutral. The background charges are taken from our previous studies  . Then the cluster with the background charges is calculated at the MP2 level. The charges are modified and the whole system is recalculated, repeating this process until self-consistency is achieved, see  .
The calculations are performed with the Gaussian 2016 A.03 suite of programs . The triply split valence basis set is used (6-311G(d)) for Fe    and As    and all electrons are taken into account for both atoms. For heavier atoms, the relativistic Wood-Boring pseudopotential   for the core electrons on Ba , Rh  and Pd  was used, and its associated basis sets were used for the valence electrons. The UHF calculations and then the MP2 calculations are performed using unrestricted HF results as initial guess. The electron and spin distribution are studied using the NBO analysis   .
Figure 1. The cluster Ba4Fe5As8 from different directions and the unit cell of studied crystal BaFe2As2 (right).
3. Results and Discussion
3.1. The Isolated and Embedded Cluster Energy and Its Dependence on the Multiplicity of the State
In Table 1 we present the energy and multiplicity for ground state of the pure and doped isolated cluster Ba4Fe5As8 calculated by GAUSSIAN 2016 A.03 . The multiplicity is defined as M = 2S + 1 where S is the total spin of the state. The eigenvalues of the S2 operator, S2 = S(S + 1), which is used for checking the spin contamination of the state and the corrected spin contamination values are given in parenthesis. In our non-relativistic quantum-mechanic calculations, the operator S2 commutes with the Hamiltonian that does not depend on the spin, therefore the spin S is a good quantum number.
The presented new results correspond to the isolated cluster. According to Table 1, for the pure cluster the energy diminishes till the multiplicity M = 8 and this was the reason for the calculation until M = 10, where the energy begins to increase. Thus, as follows from Table 1 the ground state in this case corresponds to multiplicity M = 8 (S = 7/2). The ground state for the cluster doped by Rh is the singlet state, which is a non-magnetic state. However, we are interested in magnetic states. Thus, we should analyse states beginning from triplet state, S = 1. It follows that the ground state has M = 5 (S = 2). In the case of Pd doping, a large spin contamination is observed for two multiplicities M = 2 and M = 4, therefore these results should not be trusted. Nevertheless, the most probable that the ground state for the Pd doping has M = 6 (S = 5/2).
In Table 1, it is also presented our old results from , where also unrestricted MP2 calculations were used, but for the embedded cluster. In this case, for the pure cluster the ground state has M = 6 (S = 5/2). When the cluster is doped by Rh, the ground state is a non-magnetic, S = 0. Thus, the lowest energy for a magnetic state for the cluster doped by Rh corresponds to M = 3 (S = 1). In the case of Pd, the ground state has M = 4 (S = 3/2). The values of the operator S2 after correction on the spin contamination practically agree with the correct value S(S + 1), except when S = 1⁄2. This indicates that all calculated energies with only one mentioned exception can be accepted as correct.
3.2. Natural Bond Orbital Analysis
In Table 2 and Table 3, the atomic charge and the valence orbital population at the NBO level for the central and nearest neighbors (n.n.) atoms are presented. The outer atoms of As and Ba are not presented because they are on the boundary of the cluster. Although the excited Rydberg orbitals are not presented, they are taken into account for calculating the atomic charges.
According to Table 2, the central atom of the pure cluster is almost neutral for the embedded cluster, whereas it is almost one electron for the isolated cluster. After doping by Rh and Pd, a large negative charge appears on both dopant atoms, for the isolated and embedded cluster. For As(n.n.), a decrease in the negative charge is observed, it is associated with the charge transfer from As(n.n.) to the dopant atom. Whereas, in all Fe(n.n.) a small change in their charge is observed. Thus, there is a charge transfer from As(n.n.) atoms to dopants. This situation is the same for isolated and embedded cluster.
Table 1. Energy of the states calculated at the MP2 level using GAUSSIAN 2016 A.03 according to different multiplicities for the embedded and isolated cluster, pure and doped.
Table 2. NBO charge distribution at the ground state of the embedded and isolated cluster, pure and doped, at the MP2 level.
Table 3. NBO detailed valence orbital population at the ground state of the embedded and isolated cluster, pure and doped, at the MP2 level.
It is instructive to compare the obtained valence orbital population for the embedded and isolated pure cluster with the valence orbital population of free atoms: Fe: [Ar] 3d64s2 and As: [Ar] 4s24p3. According to Table 2, for the embedded pure cluster, the Fe atoms in the pure cluster show a decrease in its 4s orbital population of 1.6e for the central atom, 1.53e for the Fe(n.n.)a, and 0.97e for the Fe(n.n.)b. The population of the 3d orbitals is increased by 1.45e on the central atom, 0.64e on the Fe(n.n.)a, and 0.36e on the Fe(n.n.)b. On As(n.n.) a decrease is observed in the 4s orbital population of 0.51e and an increase of 1.53e on the orbital 4p. On the other hand, for the isolated pure cluster, there is also a decrease on the 4s orbital by 1.55e for the central Fe, by 1.5e for the Fe(n.n.)a, and by 1.22e for the Fe(n.n.)b; a and b denote the crystallographic directions. The 3d orbital population increased by 0.65e for the central atom, by 0.57e for the Fe(n.n.)a, and by 0.39e for the Fe(n.n.)b. On As(n.n.) a decrease by 0.19e on the orbital 4s and an increase by 1.51e on the orbital 4p are observed.
Let us return to Table 2 and Table 3. In comparison with the population of free atoms (Rh: [Kr] 4d85s1 and Pd: [Kr] 4d10, as follows from Table 2, for the isolated and embedded cluster for both doping, it is observed the charge transfers from As(n.n.) atoms to the dopant atoms, that can be due to the screening effect. As follows from Table 3 for both doping, the 3d orbital population on Fe(n.n.) depends on direction of the orbitals.
In Table 4 and Table 5, the spin orbital population at the NBO level for the ground state of embedded and isolated cluster, pure and doped, are presented. As follows from Table 4 for the embedded pure cluster, the spin on the central Fe is equal to 0.32 ħ, whereas for the isolated cluster, the spin on the central Fe atom is almost cero. For the embedded cluster the spin is practically absent on both dopant atoms, whereas for the isolated cluster the Rh dopant has S = −0.51 ħ. In the case of the isolated cluster, all atoms are practically spinless except As(n.n.) for Rh doping and Fe(n.n.)a for Pd doping. For embedded doped clusters, the distribution of the spin orbital population does not change comparing with the pure clusters. In the case of the isolated cluster for Rh doping, the β-electrons transfer to the Rh atom, whereas α-electron transfers to As(n.n.). For Pd doping, the β-electrons transfer to Fe(n.n.)a.
As follows from Table 5 for the detailed spin valence orbital population, for the Rh doping there is a β-spin density population on dxy and dyz, whereas for Pd doping there is the zero-spin density on all orbitals. We would like to mention that for Pd doping, the spin density population does not depend on direction of Fe(n.n.). Also, for Rh doping there is spin density population for the Fe(n.n.)a on the dxy and dxz where α and β-spin density populations are observed.
The spin distribution obtained in Table 5 is in agreement with the charge distribution in Table 3. We would like to mention that for the embedded cluster in the case of Rh doping, the orbitals dxz and dyz of Fe(n.n.)a, and dyz and of Fe(n.n.)b are practically occupied by one electron with zero spin population. For the Pd doping, on the orbital dyz of Fe(n.n.)a there is also one electron with zero spin population. The spinless electron resembles the spinless holons proposed by Anderson in his RVB model of high Tc-SC  .
Table 4. NBO spin distribution at the ground state of the embedded and isolated cluster, pure and doped, at the MP2 level.
Table 5. NBO detailed spin valence orbital population at the ground state of the embedded and isolated cluster, pure and doped, at the MP2 level.
As follows from the discussion of our calculations by unrestricted open shell ECM-MP2, the ground state for the isolated cluster is characterized by a different multiplicity than the ground state of the embedded cluster. The background charges modify the energy of the cluster and the valence orbital population. It is also revealed that the calculation by unrestricted open shell MP2 method leads in some cases to high spin contamination of the state.
For the isolated cluster doped by Rh and Pd, we obtained a decrease in population of some valence orbitals. The orbital population for Fe(n.n.) depends on direction, this is in agreement with experiments. For the doped isolated cluster, a charge transfer from the As(n.n.) atoms to the central atom was observed, as in the case of the embedded cluster. Thus, for the embedded and isolated clusters for Rh and Pd doping, the charge transfers from nearest neighbor atoms to the dopants, whereas only for the isolated cluster doped by Rh we obtained spin transfer.
It is important to mention that for both dopants, the spin disappears on the dopants and the charge and spin transfer are completely independent. Thus, obtained in our calculations charge and spin orbital distributions, are in agreement with the spinless electrons proposed by Anderson (Anderson’s holon). This indicates the possibility of the superconductivity mechanism in this material proposed by Anderson in his RVB theory.
The authors thank the DGTIC computer staff for providing access to the MITZLI cluster of Universidad Nacional Autónoma de México. This work was partly supported by grants from DGAPA PAPIT IN111519. We also gratitude Lic. Alejandro Pompa-García and Tec. Cain González for their technical support.
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