Quantum Mechanical Calculations of High-T c Fe-Superconductors

In introduction we presented a short historical survey of the discovery of superconductivity (SC) up to the Fe-based materials that are not superconducting in a pure state. For this type of material, the transition to SC state occurs in presence of different dopants. Recently in the Fe-based materials at high pressures, the SC was obtained at room critical temperature. In this paper, we present the results of calculations of the isolated cluster representing infini-tum crystal with Rh and Pd as dopants. All calculations are performed with the suite of programs Gaussian 16. The obtained results are compared with our previous results obtained for embedded cluster using Gaussian 09. In the case of embedded cluster our methodology of the Embedded Cluster Method at the MP2 electron correlation level was applied. In the NBO population analysis two main features are revealed: the independence of charge density transfer from the spin density transfer and, the presence of orbitals with electron density but without spin density. This is similar to the Anderson’s spinless holon and confirms our conclusions in previous publications that the possible mechanism for superconductivity can be the RVB mechanism proposed by Anderson for high T c superconductivity in cuprates.


Introduction
In 1911 Kamerlingh Onnes [1] discovered the superconductivity (SC) of the Hg (at critical T c = 4.19 K), while he was doing his experiments on the resistivity of 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 [12] generated the microscopic formulation of the phenomenological macroscopic Ginzburg-Landau (G-L) theory [13] [14]. After, Eliashberg [15] 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 T c was low, the maximum critical temperature was obtained for Nb3Sn, T c = 18.5 K. In 1986 Bednorz and Müller [16] discovered the high T c (~30 K) SC in the cuprates family. For YBa 2 Cu 3 O 7 the lowest T c = 65 K [17] was obtained for zero pressure. As was shown in [18] [19] [20] for different materials, the increase of pressure leads to an increase of T c .
One of the long-standing challenges was the observation of room-temperature SC. For many years numerous laboratories failed to increase T c . The progress arises after Drozdov et al. [21] using high pressure obtained for sulfur hydride system a T c = 203 K. At last, in 2020 Snider et al. [22] obtained the really room-temperature SC with the T c = 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 T c phonon mediated SC, that is, for conventional SC.
The discovery in 2008 by Hosono and coworkers [23] [24] of the superconductivity in the La[O (1-x) F x ]FeAs with (x = 0.05 − 0.12) represented the rise of a new era with the family of high-T c Fe-based superconductors (Fe-SC), which are also named as iron-based superconductors (IBSC). This family is composed by six groups of IBSC compounds [25]. Among them, the Ba-based crystals, BaFe 2 As 2 , are widely used [26]- [32]. 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 [27]. The parent compound is a paramagnetic semimetal, it turns into superconductor upon electron 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-T c SC, although the latter has rather different mechanism of SC. We recommend the readers the popular and comprehensive reviews by Norman [36] [37], Mazin [38], Wang and Lee [39], Chubukov [40], Kordyuk [41], Baquero [42] and Prosorov et al. [43].
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. [33] [34], 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 [35]. 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. [44] applied multiorbital random-phase approximation [45] to the model of five d-orbitals and obtained similar results as in [33] [35], but they also accepted the d-wave symmetry. Other types of symmetry have been proposed by Onari and Kontani [46], being d-wave symmetry, and also opposing to the s ± -wave symmetry [47].  [66] suite of programs and will be compared with our previous results performed also by GAUSIAN 2016 A.03 [62] 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.

Methodology
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 [67].
The complete structural information is taken from [26]. 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 [57] [58]. 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 [60] [61].
For heavier atoms, the relativistic Wood-Boring pseudopotential [74] [75] for the core electrons on Ba [76], Rh [77] and Pd [77] 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 [63] [64] Journal of Quantum Information Science

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 Ba 4 Fe 5 As 8 calculated by GAUSSIAN 2016 A.03 [66].
The multiplicity is defined as M = 2S + 1 where S is the total spin of the state. The eigenvalues of the S 2 operator, S 2 = 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 S 2 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 [59], 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 S 2 after correction on the spin contamination practically agree with the correct

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 clus-     Let us return to Table 2 and  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  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

Conclusions
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.