Exchange Perturbation Theory for Multiatomic Electron System and Its Application to Spin Arrangement in Manganite Chains

Abstract

A new methodology of binding energy calculation with respect to different spin arrangements for a multiatomic electron system is developed from the first principle in the frame of the exchange perturbation theory (EPT). We developed EPT formalism in the general form of the Rayleigh-Srchödinger expansion with a symmetric Hamiltonian, taking into account an exchange and nonadditive contributions of a superexchange interaction. The expressions of all corrections to the energy and wave function were reduced to the nonsymmetric Hamiltonian form. The EPT method is extended for the case of degeneracy in the total spin of a system. As an example of the application of the developed EPT formalism for the degeneracy case, spin arrangements were considered for the key ⟨Mn⟩–O–⟨Mn⟩ (⟨Mn⟩: Mn³⁺ or Mn⁴⁺) fragments in manganites. In ⟨Mn⟩–O–⟨Mn⟩ for La_1/3Ca_2/3MnO₃ are in good agreement the obtained estimations of Heisenberg parameter and binding energy with the available experimental data.

1. Introduction

Analytical calculation of the interaction potential in multiatomic electron systems with intermediate interatomic distances is essentially a problem in quantum chemistry. Intermediate distances indicate the distances on which interatomic interaction becomes small enough to be considered as a perturbation, with the exchange contribution being exponentially small, but exceeding van der Waals interaction. The quantum chemical analysis of a multiatomic system for the intermediate interatomic distances with respect to electronic spin ordering should definitely take into account intercenter electronic permutations.

The most efficient method for such distances is the perturbation theory method, because of its possibility to present the result in the form of expansion, each term of which has definite physical interpretation [1, 2]. The variation procedures of atom and molecule wave function calculation or density functional approximation (DFT) method [3] in the frame of the mean field theory allow us to calculate pair potential of the particle interaction for the ground state. However, a common procedure in this case is forcing the wave function parameters to fit the experimental form of the potential [1, 2]. Excited state determination needs additional convergence conditions. There is a quantum chemical Hartree-Fock method based on the Slater determinant, where antisymmetric wave function uses definite electron spin 𝑧-projections. The disadvantage of this method is concerned with the impossibility of determining the total spin value of the electronic system involved in interaction because of the noncommutating of the total spin quadrate operator of the system 𝑆² and 𝑠̂_𝑖𝑧 operators of the electronic spin 𝑧-projection for each particle, for example, [𝑆²,𝑠̂_𝑖𝑧]≠0. It is important that interatomic potentials in the analytical form be found from the first principles, and exchange perturbation theory method gives such an opportunity. However, constructing the exchange perturbation theory formalism is connected with the two main following problems [1, 2].

First, the base of multiatomic system wave functions, being antisymmetric in intercenter permutations, is nonorthogonal and hence overcompleted (this is known as the overcompleteness catastrophe).

Second, the perturbation operator and unperturbed part of the Hamiltonian with respect to electron intercenter permutations are asymmetric. The total Hamiltonian 𝐻=𝐻₀+𝑉 of the system includes two terms: that of unperturbed Hamiltonian 𝐻₀ of the noninteracting multiatomic system and the 𝑉 perturbation, describing all electron and nuclei intercenter interactions. The total Hamiltonian of the system is symmetric with respect to intercenter permutations. In other words, [𝐻,𝐴]=0, but [𝐻₀,𝐴]≠0 and [𝑉,𝐴]≠0, where 𝐴 is the antisymmetrization operator.

It is also necessary we take into account nonadditive superexchange contributions to the system energy and include pairwise nonadditive interactions in a symmetric Hamiltonian formalism. The constants of the superexchange interaction are usually estimated by combinations of pair integrals [4, 5], which actually means that the specific non-additive part of multicenter interaction is discarded.

There are many formal EPT variants classified according to the way of overcoming these problems (a detailed description of this classification can be found in Kaplan’s monograph [1, 2]). All EPT variants can be divided into two groups, the first of which consisting of theories with non-symmetric Hamiltonian known as symmetry-adapted perturbation theories, for example [6–9]. It uses the fundamental basis of the zeroth functions, which are non-symmetric in intercenter electron permutations. These functions are eigenfunctions of non-symmetric Hamiltonian of the unperturbed system. Wave function antisymmetrization is done post factum in each interpolation step. Finally, this analysis requires the use of an additional variation procedure. The second group consists of approaches, allowing for the general Rayleigh Schrödinger form use by setting up a special zeroth symmetric Hamiltonian, for which the antisymmetric functions are eigenfunctions. An attempt to modify the Hamiltonian in such a way was made in [10, 11] (known as the Sternheimer procedure), which led to a non-Hermitian form of the total Hamiltonian and actually limited the use of the method only by the two-electron systems. The symmetric Hamiltonian treatment has been abandoned due to several unsolvable problems in this approach. The main problem is that the perturbation corrections have no well-defined meaning in the limit of complete basis set. In contrast, methods with non-symmetric Hamiltonian known as symmetry-adapted perturbation theories became the main-stream method for intermolecular interaction calculations.

The Ritchie EPT [12], where the specific projection operators Λ_𝑝 to the subspace of wave functions, corresponding to the determined permutation 𝑝, were sophistically entered and employed in the ETP formalism in the Wigner form of expansion, can also be assigned to the second group. The explicit forms of these operators were not given. A similar way of the Hamiltonian symmetrized form construction was used in [13], where the perturbation theory in the Rayleigh-Schrödinger form was developed; however, the Λ_𝑝 operators were not obtained analytically either. The symmetric nonperturbed Hamiltonian and symmetric form of the perturbation operator were developed from the first principles in [14–16] for the two-center system, but only the attempts of developing analytical forms of the projection operators were made.

In the present paper we have developed a general EPT form for the multiatomic electron system for the case of the nonorthogonal basis of spatial parts of the wave function antisymmetrized with respect to the intercenter electron permutations. Due to the one-to-one correspondence between the symmetries of the wave function spin and spatial parts we create an algorithm permitting to determine the total spin value for the system of electrons involved in interaction. The analytical form of the Λ-projection operators and the symmetric form of the unperturbed Hamiltonian as well as that of the perturbation operator have been obtained. The resultant expressions of the corrections to the energy and wave functions were analytically reduced to the non-symmetric perturbation form. Such simple form is more useful for the applications; nevertheless, it contains all exchange and superexchange inputs. The EPT formalism is extended for the case of degeneracy in the total spin for multiatomic electronic system. As an example of the EPT developed formalism application for the case of degeneracy we consider the spin arrangement problem in the key manganese ⟨Mn⟩–O–⟨Mn⟩ fragments (⟨Mn⟩: Mn³⁺ or Mn⁴⁺) in manganites. On the basis of the analytical equations obtained, calculation of the binding energy and splitting in the total spin system have been performed for the La_1/3Ca_2/3MnO₃ compound. The estimations obtained are in good agreement with the available experimental data.

2. Exchange Perturbation Theory

Exchange interaction between closely spaced atoms involves participation of out-shell electrons of these atoms. These shells can include not only the outermost valence shell, but also inner shells adjacent to the last shell, providing an appreciable contribution to the overlap energy. In general, a total of 𝑁>2 electrons will participate in the interaction of each pair of atoms. The total system Hamiltonian can be indicated as a Hamiltonian 𝐻₀ of the independent atoms plus perturbation operator 𝑉 taking into account all interatomic interactions:𝐻=𝐻⁰+𝑉. (1) The spatial part of the zero approximation wave functionΦ𝑟₁,…,𝑟_𝑁=_𝛼𝜓_𝛼𝑟_𝑖,…,𝑟_𝑗 (2) is a simple product of atomic wave functions. We may leave the wave function spin part, if the unperturbed Hamiltonian and the perturbation do not directly contain spin operators, such as relativistic inputs of the spin-orbit and the spin-spin interactions. The spatial eigenfunction Φ_𝑛(𝑟₁,…,𝑟_𝑁) of the unperturbed Hamiltonian 𝐻⁰Φ_𝑛𝑟₁,…,𝑟_𝑁=𝐸⁰_𝑛Φ_𝑛𝑟₁,…,𝑟_𝑁, (3) being antisymmetrized in intercenter permutations Ψ⁰_𝑛(𝑟₁,…,𝑟_𝑁)=𝐴Φ_𝑛(𝑟₁,…,𝑟_𝑁), will not be an eigenfunction of the 𝐻⁰ Hamiltonian. Here, 𝐴 is the antisymmetrization operator. The antisymmetric wave function corrections calculated in the non-symmetric perturbation 𝑉 will contain nonphysical contributions [1].

The spatial part of the wave function antisymmetrized with the Young operator 𝜔^[𝜆]_𝑝,𝑡, where [𝜆] is the form of Young tableaux and 𝑡 is the number of the standard tableaux, can be written as [2]Ψ⁰_𝑛𝑟₁,…,𝑟_𝑁=𝐴Φ_𝑛𝑟₁,…,𝑟_𝑁=_𝑡𝜔^[𝜆]_𝑝=0,𝑡Φ_𝑛^𝑝=0=1𝑓^𝑃_𝑛^𝑃_𝑝=0(−1)^𝑔𝑝Φ^𝑝_𝑛𝑟₁,…,𝑟_𝑁, (4) where 𝑝 is the number of the permutation, 𝑔_𝑃 is the permutation parity factor, and Φ^𝑝_𝑛(𝑟₁,…,𝑟_𝑁) is the same simple product as (2), but with the electron permutation of number 𝑝, where 𝑃 is the total number of the possible permutations, 1/𝑓^𝑝_𝑛 is normalization 𝑓^𝑃_𝑛=∑^𝑃_𝑝=0(−1)^𝑔𝑃(Φ⁰_𝑛∣Φ^𝑝_𝑛). We will use the notation for the non-symmetric state vector—|Φ^𝑝_𝑛) and for the antisymmetric state vector—|Ψ⁰_𝑛⟩. Here is the normalization condition (Φ⁰_𝑛∣Ψ⁰_𝑛⟩=1, which differs by a factor √𝑃 from the commonly used condition ⟨Ψ⁰∣Ψ⁰⟩=1.

First, we will construct a symmetric form of the unperturbed Hamiltonian. The function Φ^𝑝_𝑛(𝑟₁,…,𝑟_𝑁) in which electrons have been interchanged is also an eigenfunction of the unperturbed system Hamiltonian 𝐻⁰_𝑃 with the same permutation, that is, 𝐻⁰_𝑃Φ^𝑃_𝑖=𝐸⁰_𝑖Φ^𝑃_𝑖. Then we can write out the following set of equations: 𝐻⁰_𝑝=0Φ_𝑖^𝑝=0=𝐸⁰_𝑖Φ_𝑖^𝑝=0,𝐻⁰_𝑝=1Φ_𝑖^𝑝=1=𝐸⁰_𝑖Φ_𝑖^𝑝=1,⋮𝐻⁰_𝑃Φ^𝑃_𝑖=𝐸⁰_𝑖Φ^𝑃_𝑖. (5) We sum up these equations so that, on the right, we obtain the function corresponding to the coordinate part of the Young tableaux, for which the symmetrization procedure is carried out in (4). The result is ^𝑝=𝑃_𝑝=0(−1)^𝑔𝑝𝐻⁰_𝑝Φ^𝑝_𝑖=𝐸⁰_𝑖^𝑝=𝑃_𝑝=0(−1)^𝑔𝑝Φ_𝑖^𝑝=0, (6) or with respect to (4) we get the following expression:^𝑝=𝑃_𝑝=0(−1)^𝑔𝑝𝐻⁰_𝑝Φ^𝑝_𝑖=𝐸⁰_𝑖𝑓^𝑃_𝑖Ψ⁰_𝑖. (7) We introduce a projector Λ^𝜋_𝑛=|Φ^𝜋_𝑛)(Φ^𝜋_𝑛| on the 𝜋-permutation vector of the state corresponding to energy level 𝐸⁰_𝑛. Applying it to the antisymmetrized vector |Ψ⁰_𝑛⟩ with respect to the normalization condition we getΛ^𝜋_𝑛Ψ⁰_𝑛=(−1)^𝑔𝜋Φ^𝜋_𝑛. (8) Using (8), equation (7) can be presented in the following form: 1𝑓^𝑃_𝑖^{𝑝=𝑃𝑝=0}𝐻⁰_𝑝Λ^𝑝_𝑖Ψ⁰_𝑖=𝐸⁰_𝑖Ψ⁰_𝑖, (9) or𝐻₀Ψ⁰_𝑖=𝐸⁰_𝑖Ψ⁰_𝑖, (10) where the symmetrized unperturbed Hamiltonian has the form𝐻₀=^𝑃_𝑝=0𝐻⁰_𝑝𝑛1𝑓^𝑃_𝑛Λ^𝑝_𝑛. (11) We can represent the perturbation operator in the symmetric form𝑉=^𝑃_𝑝=0𝑉_𝑝𝑛1𝑓^𝑃_𝑛Λ^𝑝_𝑛. (12)

The total Hamiltonian of the interacting system is always invariant under the electron permutations so that the solution of the Schrödinger equation can be antisymmetric under the electron permutations𝐻|Ψ⟩=𝐸|Ψ⟩, (13) where 𝐻=𝐻₀+𝑉 with respect to (11) and (12).

Solving (13) by the approximation method, we search for the perturbative corrections to the energy of the unperturbed system and the wave function (10). The perturbation operator in the form of (12) provides the same symmetry for the corrections as does the initial wave function. If interaction is small, we can represent the wave function and energy in the form of the following series:Ψ_𝑖=Ψ_𝑖⁽⁰⁾+Ψ_𝑖⁽¹⁾𝐸+⋯,_𝑖=𝐸_𝑖⁽⁰⁾+𝐸_𝑖⁽¹⁾+⋯. (14) In the first approximation in the perturbation, the Schrödinger equation takes the form𝐻₀Ψ_𝑖⁽¹⁾+𝑉Ψ_𝑖⁽⁰⁾=𝐸_𝑖⁽¹⁾Ψ_𝑖⁽⁰⁾+𝐸_𝑖⁽⁰⁾Ψ_𝑖⁽¹⁾. (15) The intermediate normalization condition is Φ_𝑖∣Ψ_𝑖=Φ_𝑖∣Ψ⁰_𝑖, (16) that is, (Φ_𝑖|{|Ψ_𝑖⟩−|Ψ⁰_𝑖⟩}=0. Which means that all corrections to the vector |Ψ⁰_𝑖⟩ of the zeroth approximation lie in the subspace orthogonal to this vector.

There is a projection operator 𝑃_𝑖=|Ψ⁰_𝑖⟩(Φ_𝑖| in the subspace of vectors parallel to |Ψ⁰_𝑖⟩.

Since 𝑃_𝑖𝐻₀Ψ_𝑖⁽¹⁾=𝐸⁰_𝑖𝑃_𝑖Ψ_𝑖⁽¹⁾ (17) is valid (see Appendix A), applying the projection operator 𝑃_𝑖 to (15) results in (Φ_𝑖|𝑉|Ψ⁰_𝑖⟩|Ψ⁰_𝑖⟩=𝐸_𝑖⁽¹⁾|Ψ⁰_𝑖⟩. Then the first-order correction to the energy is𝐸_𝑖⁽¹⁾=Φ_𝑖𝑉Ψ_𝑖⁽⁰⁾. (18) Now we introduce the projector O_𝑖=1−𝑃_𝑖 (19) on the subspace of vectors orthogonal to the vector |Ψ⁰_𝑖⟩. The first-order corrections to the antisymmetrized vector |Ψ⁰_𝑖⟩ will have the following form of expansion:Ψ_𝑖⁽¹⁾=_𝑛^′𝐶_𝑛Ψ_𝑛⁽⁰⁾, (20) where 𝑛≠𝑖. Inserting the expansion (20) to (15) and applying the projector (19) to the result, we obtain the following equation:𝑂_𝑖𝑉Ψ_𝑖⁽⁰⁾=_𝑛𝐶_𝑛𝐸⁰_𝑖−𝐸⁰_𝑛Ψ_𝑛⁽⁰⁾. (21) Using the property of completeness ∑_𝑛|Φ^𝑝_𝑛)(Φ^𝑝_𝑛|=𝐼̂ of the orthogonal basis of the non-symmetric zeroth functions, we can get the left-hand side of (21) as follows (the details can be found in Appendix A):𝑂_𝑖𝑉Ψ_𝑖⁽⁰⁾=1𝑃_𝑛^′𝑓^𝑝_𝑛Ψ_𝑛⁽⁰⁾Φ_𝑛⁽⁰⁾𝑂_𝑖𝑉Ψ_𝑖⁽⁰⁾. (22) From (21) and (22) we obtain the first-order corrections to the wave vectorΨ_𝑖⁽¹⁾=1𝑃^𝑛𝑓^𝑝_𝑛Φ_𝑛⁽⁰⁾𝑂_𝑖𝑉Ψ_𝑖⁽⁰⁾𝐸⁰_𝑖−𝐸⁰_𝑛Ψ⁰_𝑛,Φ_𝑖⁽¹⁾=1𝑃^𝑛𝑓^𝑝_𝑛Φ_𝑛⁽⁰⁾𝑂_𝑖𝑉Ψ_𝑖⁽⁰⁾𝐸⁰_𝑖−𝐸⁰_𝑛Φ⁰_𝑛. (23) Since (Φ_𝑖|𝑂_𝑖𝑉|Ψ_𝑖⁽⁰⁾⟩=(Φ_𝑖|{1−|Ψ_𝑖⟩(Φ_𝑖|}𝑉|Ψ_𝑖⁽⁰⁾⟩≡0 and (Φ_𝑛|𝑂_𝑖=(Φ_𝑛|−(Φ_𝑛|Ψ_𝑖⁽⁰⁾⟩(Φ_𝑖|≈(Φ_𝑛| (see Appendix A), we get the final expressions:Ψ_𝑖⁽¹⁾=1𝑃_𝑛^′𝑓^𝑃_𝑛Φ_𝑛⁽⁰⁾𝑉Ψ_𝑖⁽⁰⁾𝐸⁰_𝑖−𝐸⁰_𝑛Ψ_𝑛⁽⁰⁾,Φ_𝑖⁽¹⁾=1𝑃_𝑛^′𝑓^𝑃_𝑛Φ_𝑛⁽⁰⁾𝑉Ψ_𝑖⁽⁰⁾𝐸_𝑖⁽⁰⁾−𝐸_𝑛⁽⁰⁾Φ_𝑛⁽⁰⁾. (24) Here the intermediate normalization condition {(Φ_𝑖⁽⁰⁾|+(Φ_𝑖⁽¹⁾|}{|Ψ_𝑖⁽⁰⁾⟩+|Ψ_𝑖⁽¹⁾⟩}=1 is used.

Using (9), (11), and (12), it can be shown that the integral (Φ_𝑛⁽⁰⁾|𝑉|Ψ_𝑖⁽⁰⁾⟩ is equal to the integral ⟨Ψ_𝑛⁽⁰⁾|𝑉_𝑝=0|Φ_𝑖⁽⁰⁾), where perturbation 𝑉 has the simple non-symmetric form corresponding to the number of permutations 𝑝=0. Then (18) and (24) can be rewritten in the form of non-symmetric perturbation operator:𝐸_𝑖⁽¹⁾=Ψ_𝑖⁽⁰⁾𝑉_𝑝=0Φ_𝑖,Ψ_𝑖⁽¹⁾=1𝑃_𝑛^′𝑓^𝑃_𝑛Ψ_𝑛⁽⁰⁾𝑉_𝑝=0Φ_𝑖⁽⁰⁾𝐸_𝑖⁽⁰⁾−𝐸_𝑛⁽⁰⁾Ψ_𝑛⁽⁰⁾,Φ_𝑖⁽¹⁾=1𝑃_𝑛^′𝑓^𝑃_𝑛Ψ_𝑛⁽⁰⁾𝑉_𝑝=0Φ_𝑖⁽⁰⁾𝐸_𝑖⁽⁰⁾−𝐸_𝑛⁽⁰⁾Φ_𝑛⁽⁰⁾. (25) Higher-order corrections are found in a similar way. So, for the second order we obtain the following equation:𝐻₀Ψ_𝑖⁽²⁾+𝑉Ψ_𝑖⁽¹⁾=𝐸_𝑖⁽⁰⁾Ψ_𝑖⁽²⁾+𝐸_𝑖⁽¹⁾Ψ_𝑖⁽¹⁾+𝐸_𝑖⁽²⁾Ψ_𝑖⁽⁰⁾, (26) from which, after applying the operator 𝑃_𝑖, we find the second-order correction to the energy: 𝐸_𝑖⁽²⁾=_𝑘^′𝑓^𝑃_𝑘𝑃Φ_𝑖⁽⁰⁾𝑉Ψ⁰_𝑘Φ_𝑘⁽⁰⁾𝑉Ψ_𝑖⁽⁰⁾𝐸_𝑖⁽⁰⁾−𝐸_𝑘⁽⁰⁾. (27)

Hence for the simple non-symmetric form it is 𝐸_𝑖⁽²⁾=_𝑘^′𝑓^𝑃_𝑘𝑃Ψ_𝑖⁽⁰⁾𝑉_𝑝=0Φ_𝑘⁽⁰⁾Ψ_𝑘⁽⁰⁾𝑉_𝑝=0Φ_𝑖⁽⁰⁾𝐸_𝑖⁽⁰⁾−𝐸_𝑘⁽⁰⁾. (28)

Similarly, applying the operator 𝑂_𝑖, we find the wave vector correction:Ψ_𝑖⁽²⁾=_𝑛^′_𝑘^′𝑓^𝑃_𝑛𝑓^𝑃_𝑘𝑃²Ψ⁰_𝑛Ψ⁰_𝑛𝑉_𝑝=0Φ⁰_𝑘Ψ⁰_𝑘𝑉_𝑝=0Φ⁰_𝑖𝐸⁰_𝑖−𝐸⁰_𝑘𝐸⁰_𝑖−𝐸⁰_𝑛−_𝑛^′𝑓^𝑃_𝑛𝑃²Ψ⁰_𝑛Ψ⁰_𝑛𝑉_𝑝=0Φ⁰_𝑖Ψ⁰_𝑖𝑉_𝑝=0Φ⁰_𝑖𝐸⁰_𝑖−𝐸⁰_𝑛²−Ψ⁰_𝑖2_𝑛^′𝑓^𝑃_𝑛𝑃²Ψ⁰_𝑖𝑉_𝑝=0Φ⁰_𝑛Ψ⁰_𝑛𝑉_𝑝=0Φ⁰_𝑖𝐸⁰_𝑖−𝐸⁰_𝑛²,Φ_𝑖⁽²⁾=_𝑛^′_𝑘^′𝑓^𝑃_𝑛𝑓^𝑃_𝑘𝑃²Φ⁰_𝑛Ψ⁰_𝑛𝑉_𝑝=0Φ⁰_𝑘Ψ⁰_𝑘𝑉_𝑝=0Φ⁰_𝑖𝐸⁰_𝑖−𝐸⁰_𝑘𝐸⁰_𝑖−𝐸⁰_𝑛−_𝑛^′𝑓^𝑃_𝑛𝑃²Φ⁰_𝑛Ψ⁰_𝑛𝑉_𝑝=0Φ⁰_𝑖Ψ⁰_𝑖𝑉_𝑝=0Φ⁰_𝑖𝐸⁰_𝑖−𝐸⁰_𝑛²−Φ⁰_𝑖2_𝑛^′𝑓^𝑃_𝑛𝑃²Ψ⁰_𝑖𝑉_𝑝=0Φ⁰_𝑛Ψ⁰_𝑛𝑉_𝑝=0Φ⁰_𝑖𝐸⁰_𝑖−𝐸⁰_𝑛². (29) Here we used the intermediate normalization conditionΦ⁰_𝑖+Φ_𝑖⁽¹⁾+Φ_𝑖⁽²⁾Ψ⁰_𝑖+Ψ_𝑖⁽¹⁾+Ψ_𝑖⁽²⁾=1. (30) For the third-order correction to the energy, we get the following equation:𝐸_𝑖⁽³⁾=_𝑛^′_𝑘^′𝑓^𝑃_𝑛𝑓^𝑃_𝑘𝑃²×Ψ⁰_𝑖𝑉_𝑝=0Φ⁰_𝑘Ψ⁰_𝑘𝑉_𝑝=0Φ⁰_𝑛Ψ⁰_𝑛𝑉_𝑝=0Φ⁰_𝑖𝐸⁰_𝑛−𝐸⁰_𝑖𝐸⁰_𝑘−𝐸⁰_𝑖−Φ⁰_𝑖𝑉Ψ⁰_𝑖×_𝑛^′𝑓^𝑃_𝑛𝑓^𝑃_𝑛𝑃²Ψ⁰_𝑖𝑉_𝑝=0Φ⁰_𝑛Ψ⁰_𝑛𝑉_𝑝=0Φ⁰_𝑖𝐸⁰_𝑖−𝐸⁰_𝑛². (31) The set of the antisymmetrized wave function is nonorthogonal, but this set satisfies the property of completeness. Applying the operator ∑_𝑛|Ψ_𝑛⟩(Φ_𝑛| to an arbitrary antisymmetrized vector |Ψ⟩, we get_𝑛Ψ_𝑛Φ_𝑛∣Ψ=_𝑛^𝑃_𝑝=0Φ^𝑝_𝑛Φ_𝑛∣Ψ⋅(−1)^𝑔𝑝1𝑓^𝑃_𝑛. (32) Since this arbitrary vector |Ψ⟩ is antisymmetric, it can be written out in the following form:_𝑛Ψ⁰_𝑛Φ_𝑛|Ψ⟩=_𝑛^𝑃_𝑝=0(−1)^𝑔𝑝𝑓^𝑃_𝑛Φ^𝑝_𝑛Φ_𝑛∣Ψ=_𝑛^𝑃_𝑝=0(−1)^𝑔𝑃𝑓^𝑃_𝑛Φ^𝑝_𝑛Φ^𝑝_𝑛∣Ψ(−1)^𝑔𝑝=^𝑃_𝑝=01𝑓₀|Ψ⟩=𝑃𝑓₀|Ψ⟩, (33) where 𝑓^𝑃_𝑛≈𝑓₀=∑^𝑝=𝑃_𝑝=0(Φ₀∣Φ^𝑝₀)(−1)^𝑔𝑝 and ∑_𝑛|Φ^𝑝_𝑛)(Φ^𝑝_𝑛|=1 are used. Then we get the equation𝑓₀𝑃^𝑛Ψ_𝑛Φ_𝑛=_𝑛Φ_𝑛Φ_𝑛. (34) This property of completeness (34) was proved with the use of two approximations:Φ⁰_𝑖∣Φ^𝑝_𝑛≪Φ⁰_𝑛∣Φ^𝑝_𝑛∼Φ⁰_𝑖∣Φ^𝑝_𝑖,𝑓^𝑃_𝑛≈𝑓₀=^𝑝=𝑃_𝑝=0Φ₀∣Φ^𝑝₀(−1)^𝑔𝑝. (35)

Thus, we have reduced the expressions for the high-order corrections to the energy and wave function for multiatomic system to the form of the exchange perturbation theory of Rayleigh-Schrödinger formalism with nonsymmetric Hamiltonian, so-called symmetry-adapted perturbation theories. In doing so the intercenter overlapping of the wave functions and superexchange effects were taken into account. The antisymmetrization with the Young tableaux makes the set of zero approach wave functions nonorthogonal; nevertheless, this set can meet the property of completeness under the condition of (35). The obtained EPT expansion is two-parametrical. The first small parameter is included in the perturbation, and it has the order of vanishing 𝑎_𝐵/𝑅, where 𝑎_𝐵 is Bohr’s radius, whereas 𝑅 is interatomic distance. The second parameter is the degree of interatomic overlapping, which has the order of vanishing exp(−𝑅/𝑎_𝐵).

It should be underlined that for the intercenter distances 𝑅>15𝑎_𝐵 the expressions for all corrections to the energy and wave function in the frame of EPT naturally become a standard expansion of Rayleigh-Schrödinger perturbation theory. In this case all exchange integrals containing the intercenter permutations (Φ^′|𝑉|Φ)≪(Φ|𝑉|Φ) are much less than the simple Coulomb contributions and the overlapping integrals(Φ∣Φ^′)→0.

2.1. The Case of Degeneracy

The zeroth wave vector for a multicenter system can be antisymmetrized by various Young tableaux, which differ for different values of total system spin. In other words, a multiatomic system of noninteracting atoms is degenerate in total spin, with the degeneracy lifted by allowing for the ordinary intercenter Coulomb interaction. Thus, by the zeroth approximation we haveΨ⁰_𝑛𝛼=𝐴_𝛼Φ⁰_𝑛,𝐻_0𝛼Ψ⁰_𝑛𝛼=𝐸⁰_𝑛Ψ⁰_𝑛𝛼,𝐻_0𝛼=^𝑃_𝑝=0,𝑛1𝑓^𝛼_𝑛𝐻⁰_𝑝Λ^𝑝_𝑛, (36) where {|Ψ⁰_𝑛𝛼⟩} is the set of wave vectors antisymmetrized by different Young tableaux α and corresponding to the same energy level 𝐸⁰_𝑛 of the system.

The wave vector of an interacting multicenter system of electrons is taken in the form Ψ_𝑖=_𝛽𝐶⁰_𝛽Ψ⁰_𝛽𝑖+𝜑, (37) where 𝜑 is a wave function correction. If we substitute (37) in the complete Schrödinger equation (13), we obtain 𝐻_𝛽𝐶⁰_𝛽Ψ⁰_𝛽+𝐻𝜑=𝐸⁰_𝑖+𝜀_𝛽𝐶⁰_𝛽Ψ⁰_𝛽+𝐸⁰_𝑖+𝜀𝜑, (38) where 𝜀 is an energy correction. Since the total Hamiltonian is invariant under all permutations, it can be taken outside the summation sign in accordance with the Young tableau:_𝛽𝐻⁰_𝛽+𝑉_𝛽𝐶⁰_𝛽Ψ⁰_𝛽+𝐻𝜑=𝐸⁰_𝑖+𝜀_𝛽𝐶⁰_𝛽Ψ⁰_𝛽+𝐸⁰_𝑖+𝜀𝜑. (39) Using (36), we can shift all terms containing the required wave function correction 𝜑 to the left-hand side of (39) and all the other terms to the right-hand side. The result is𝐻𝜑−𝐸⁰_𝑖+𝜀𝜑=_𝛽𝜀−𝑉_𝛽𝐶⁰_𝛽Ψ⁰_𝛽. (40) In (40) we drop all the terms whose order is higher than the first one, thus getting𝐻⁰_𝛾−𝐸⁰_𝑖𝜑=_𝛽𝜀−𝑉_𝛽𝐶⁰_𝛽Ψ⁰_𝛽. (41) The solution of the homogeneous analog of (41) for 𝜑 is |𝜑⟩=|Ψ⁰_𝛾⟩. Then, according to the Fredholm alternative [17], the nonhomogeneous equation (41) has a solution only if the bra-vector ⟨Ψ⁰_𝛾| is orthogonal to the entire right-hand side: _𝛽𝜀Ψ⁰_𝛾∣Ψ⁰_𝛽−Ψ⁰_𝛾𝑉_𝛽Ψ⁰_𝛽𝐶⁰_𝛽=0. (42) Thus, we have a system of equations (42) for determining the coefficients 𝐶⁰_𝛽 of the regular zeroth wave function. The system has a solution only if the main determinant of (42) is equal to zero: 𝜀Δ_𝛾𝛽−Ψ⁰_𝛾𝑉_𝛽Ψ⁰_𝛽=0, (43) whereΔ_𝛾𝛽=Ψ⁰_𝛾∣Ψ⁰_𝛽. (44) This is the secular equation for determining the energy corrections. Because of Ψ⁰_𝛾𝑉_𝛽Ψ⁰_𝛽=Δ_𝛾𝛽Φ⁰𝑉_𝛽Ψ⁰_𝛽, (45)the secular equation becomes 𝜀−Φ_𝛽=1⁰𝑉_𝛽Ψ⁰_𝛽⋅detΔ_𝛾𝛽=0. (46) Then for the corrections 𝜀 to the energy we obtain definite expressions:𝜀=Φ⁰𝑉_𝛼Ψ⁰_𝛼, (47) or, in the simple non-symmetric form,𝜀=Ψ⁰_𝛼𝑉_𝑝=0Φ⁰ (48) only if the conditiondetΔ_𝛼𝛽≠0 (49) is satisfied. Then the set of zeroth wave functions antisymmetrized by the Young tableaux is regular.

3. Application of EPT to Manganites

In the last decades many experiments have been devoted to the investigation of perovskites and manganites [18–21]. Scientific interest to such systems has emerged due to their remarkable physical properties. Depending on the composition they show a variety of magnetic and electrical phenomena, including ferromagnetic, antiferromagnetic, charge, and orbital ordering. So, for example, the initial three-component LaMno₃ and AmnO₃ (A: Sr, Ca, or Ba) are antiferromagnetic compounds with the spin moments located on the manganese ions. The four-component stoichiometric La_1−𝑥A_𝑥MnO₃ with 0.2≤𝑥≤0.5 is strongly ferromagnetic and demonstrates the metal-type conductivity below the Curie temperature, whereas the same system with 𝑥>0.5 is insulator with antiferromagnetic behavior below Neel temperature 𝑇_𝑁.

Usually the microscopic description of magnetic ordering in crystalline structures including manganites [19–22] is either a statistical analysis of spin systems on the basis of the Heisenberg Hamiltonian or the analysis of magnetic susceptibility in the single-electron approximations in Stoner’s [23–25], Hubbard’s [4], Anderson’s [23–25] or Hohenberg-Kohn’s type [3] models. The main parameter characterizing the spin system is the parameter of exchange (or superexchange) interaction. Hubbard-type model considers only the systems with electrons strongly localized at the centers. Therefore, the Coulomb exchange interaction in such system is calculated in the Wannier function approach. The use of a “truncated” Wannier-function basis artificially reduces the contribution of the intercenter exchange energy and, therefore, essentially eliminates the intercenter correlation effects. The Stoner model is used in the cases of nearly free electron gas. Anderson [23–25] was the first to point out the need for a more careful quantum chemical description of interatomic (interionic) interaction. Anderson’s model considers the intracenter exchange (Hund’s exchange) as well as intercenter exchange interactions; however, remaining in the frame of Hubbard’s and Stoner’s models, it does not take real superexchange interactions into account. As it was pointed out by Edio Dagotto [19–21], “the Heisenberg parameters 𝐽 usually have a large error and, as a consequence, it is common practice to simply consider them as free parameters to be determined from experiments.”

We will apply the EPT algorithm developed in the present work for the analysis of magnetic ordering and energy splitting for the key fragments of ⟨Mn⟩–O–⟨Mn⟩ (⟨Mn⟩: Mn³⁺ or Mn⁴⁺), which is critical for the interplay of charge, orbital, and spin ordering in manganites. On the basis of the energy splitting analysis we will estimate the main parameters of magnetic ordering (Heisenberg parameter and the resultant magnetic moment) for the La_1/3Ca_2/3MnO₃ manganites.

Magnetic structures of La_1−𝑥Ca_𝑥MnO₃ (𝑥≥0.5) were determined from neutron diffraction measurements carried out by Wollan and Koehler [26]. According to [26], in the most studied La_1/2Ca_1/2MnO₃ and La_1/3Ca_2/3MnO₃ compounds magnetic structures belong to CE type. For instance, in La_1/2Ca_1/2MnO₃ compound below 𝑇_N≈170 K, it consists of two ferromagnetic zigzag (⋯Mn³⁺–Mn⁴⁺–Mn³⁺–Mn⁴⁺⋯) chains coupled antiferromagnetically in the (𝑎−𝑏) basal plane. These planes are stacked along the 𝑐-axis but with the opposite spins.

There are three possible three-center fragments in the key manganese chain, those of Mn³⁺–O²⁻–Mn⁴⁺, Mn⁴⁺–O²⁻–Mn⁴⁺, and Mn³⁺–O²⁻–Mn³⁺. The magnetic form factors of the Mn⁴⁺ and Mn³⁺ ions correspond to the 3𝑑³ and 3𝑑⁴ states, respectively. For the Mn³⁺–O²⁻–Mn⁴⁺ fragment we consider the following microscopic model: owing to electrostatic dipole interaction, we assume that one electron of oxygen is drawn off from O²⁻ ion by Mn⁴⁺ resulting in the hybridization state of 2𝑝⁵ of oxygen and 3𝑑_𝑧2 of manganese; then we get a virtual (or intermediate) (Mn³⁺–O⁻ + 1e ↔ Mn^4+/3+) state. Due to the overlapping of the electron wave functions and their hybridization, the (Mn³⁺–O⁻ + 1e ↔ Mn^4+/3+) fragment has a common electron cloud, making this fragment electrostatically stable with approximately identical two manganese ions with valence 3+. The presence of significant O_2p hole numbers in the manganites is confirmed experimentally [27, 28]. Then in our calculations, because of the noted hybridization, we will take into account only three “active” electrons centred on the three different ions: 3𝑑_𝑧2, 3𝑑_𝑧2(𝑟)⃗+2𝑝⁵(𝑟⃗−𝑅) for the manganese ions, and 2𝑝⁵ for the oxygen ion.

The complete electron wave function of the ion chain (for brevity, we denote this chain by I-II-III) can be antisymmetrized in three different ways corresponding to the following spin configurations: (I↑II↑III↑), (I↑II↑III↓), and (I↑II↓III↑). The corresponding Young tableaux are given in Figure 1.

Figure 1 

At the beginning, the electrons with numbers 1, 2, and 3 belong to the ions I, II, III, respectively. In this case the coordinate parts of the wave function corresponding to the Young tableaux 𝛼, 𝛽, 𝛾 are [1³], [2,1], and [2,1] type, respectively:Ψ⁰_𝛼𝑟₁,𝑟₂,𝑟₃=1−2𝐼²₁+2𝐼₂𝐼²₁−𝐼²₂⁻¹Ψ_I𝑟₁Ψ_I𝑟₂Ψ_I𝑟₃Ψ_II𝑟₁Ψ_II𝑟₂Ψ_II𝑟₃Ψ_III𝑟₁Ψ_III𝑟₂Ψ_III𝑟₃,Ψ_𝛽𝑟₁,𝑟₂,𝑟₃=𝜔^[21]₁₁Φ_Mn𝑟₁Φ_O𝑅2−𝑟₂Φ_Mn𝑅−𝑟₃,Ψ_𝛾𝑟₁,𝑟₂,𝑟₃=𝜔^[21]₁₂Φ_Mn𝑟₁Φ_O𝑅2−𝑟₂Φ_Mn𝑅−𝑟₃, (50) where, for instance, 𝑟₁ is the position vector of the electron 1, Ψ_I(𝑟₁) is a one electron wave function centered at the center I, and 𝑅 is the radius vector from the first manganese ion to the second one. The Young operator 𝜔^[21]_𝑝𝑡 of antisymmetrization for the tableau [2,1] type is presented in Appendix B.

The integrals corresponding to the exchange densities are𝐼₁=Ψ^∗_I(𝑟)Ψ_II(𝑟)𝑑³𝑟=Ψ^∗_II(𝑟)Ψ_III(𝑟)𝑑³𝐼𝑟,₂=Ψ^∗_I(𝑟)Ψ_III(𝑟)𝑑³𝑟. (51) For the initial distribution of the numbered electrons over the centers, the perturbation operator is𝑧𝑉=¹𝑧₂𝑒²𝑅_I−𝑅_II+𝑧²₁𝑒²𝑅_I−𝑅_III+𝑧₁𝑧₂𝑒²𝑅_II−𝑅_III−𝑧₂𝑒²𝑟₁−𝑅_II−𝑧₁𝑒²𝑟₁−𝑅_III−𝑧₁𝑒²𝑟₂−𝑅_I−𝑧₁𝑒²𝑟₂−𝑅_III−𝑧₁𝑒²𝑟₃−𝑅_I−𝑧₂𝑒²𝑟₃−𝑅_II+𝑒²𝑟₁−𝑟₂+𝑒²𝑟₂−𝑟₃+𝑒²𝑟₁−𝑟₃=𝑉_I,II+𝑉_II,III+𝑉_III,I, (52) where, for instance, 𝑉_I,II=𝑧₁𝑧₂𝑒²𝑅_I−𝑅_II−𝑧₂𝑒²𝑟₁−𝑅_II−𝑧₁𝑒²𝑟₂−𝑅_I+𝑒²𝑟₁−𝑟₂, (53) describes the interaction of the ion cores I and II, the interaction of the electrons with the “foreign” nuclei, and the interaction of the electrons with each other, respectively. 𝑅_I, 𝑅_II, and 𝑅_III are position vectors of the interacting ions I, II, and III, respectively; 𝑧₁, 𝑧₂, and 𝑧₃ stand for the ion charges. Direct calculation yields the following values of the scalar product parameters:Δ_𝛼𝛽=Δ_𝛼𝛾=0,Δ_𝛽𝛾=21−𝐼₂𝐼²₁,Δ_𝛾𝛾=41−𝐼₂𝐼²₁,Δ_𝛼𝛼=61−2𝐼²₁1−𝐼₂−𝐼²₂,Δ_𝛽𝛽=41−𝐼²₁+𝐼²₂−𝐼₂𝐼²₁. (54) The determinant condition isΔ_𝛼𝛼000Δ_𝛽𝛽Δ_𝛽𝛾0Δ_𝛾𝛽Δ_𝛾𝛾≠0, (55) Thus the condition to find a solution is met. In this case the energy corrections are𝜀_𝛼=Ψ⁰_𝛼𝑉_𝑝=0Φ⁰,𝜀_𝛽=Ψ⁰_𝛽𝑉_𝑝=0Φ⁰,𝜀_𝛾=Ψ⁰_𝛾𝑉_𝑝=0Φ⁰, (56) whereΦ⁰=Φ_Mn𝑟₁Φ_𝑂𝑅2−𝑟₂Φ_Mn𝑅−𝑟₃. (57) As an example, we consider the La_1/3Ca_2/3MnO₃ manganite with the known magnetic ground state of CE type. For La_1/3Ca_2/3MnO₃ due to Jahn-Teller effect, 𝑅_Mn³⁺_−O=2.19 Å, 𝑅_Mn⁴⁺_−O=1.97 Å [18] in the zigzig chain, and then |𝑅_I−𝑅_II|=2.19 Å, |𝑅_II−𝑅_III|=1.97 Å. Then the energy corrections calculated for all possible spin configurations in Mn³⁺–O²⁻–Mn⁴⁺ fragment are𝜀_𝛼()𝜀↑↑↑=−0.0165a.u,_𝛽()𝜀↑↓↓=−0.0173a.u,_𝛾(↑↓↑)=−0.0197a.u. (58) We note that the numerical values of the three-center superexchange contributions are of the same order as the pair couple exchange. They can yield the exchange energy correction of 10 to 30%, depending on the factor 𝐼₁, 𝐼₂ accounting for the extent of the third electron overlap.

In the manganites, spin orientation according to inside Hund exchange 𝐽_Hund=1.5 eV [18] is ferromagnetic. For the considered fragment the magnetization and the Heisenberg parameter are 𝑀=7𝜇_𝐵 and 𝐽₁≈𝜀_𝛽−𝜀_𝛾=0.0024 a.u. =0.065 eV, correspondingly, with the mixing of the magnetization state 𝑀=9𝜇_𝐵 and the 𝐽^′₁≈𝜀_𝛽−𝜀_𝛼=0.0008 a.u. =−0.022 eV parameter.

The Mn⁴⁺–O²⁻–Mn⁴⁺ fragment has an absolutely symmetric spatial form with respect to the intercenter and intracenter electron permutations as shown in Figure 2.

Figure 2 

Its symmetry corresponding to the spin part is described by the only possible Young diagram (see Figure 3).

Figure 3 

The total spin 𝑆 of such system equals zero and corresponds to the antiferromagnetic state. In the same way, for the Mn³⁺–O2⁻–Mn³⁺ fragment we also obtain antiferromagnetic state with 𝑆=0. Then we estimate the Heisenberg parameter for the fragments under consideration. For 𝑅=3.94 Å [18] between two Mn⁴⁺ ions in the (𝑎-𝑏) basal plane the estimation results in 𝐽=−0.0102 eV corresponding to the antiferromagnetic state. For the Mn³⁺–O2⁻–Mn³⁺ and Mn⁴⁺–O2⁻–Mn⁴⁺ fragments located between two zigzag chains in the basal plane with 𝑅=3.94 Å [18] as well, we obtained 𝐽_Mn³_−Mn³=−0.0134 eV and 𝐽_Mn⁴_−Mn⁴=−0.0102, respectively. For the interplanar interactions in the 𝑐 direction both fragments have the absolutely symmetric spatial form with respect to intercenter and intracenter electron permutations, with the distance in this case being 𝑅≈3.94 Å [18] for both triads under considerations. Our estimations result in the following values for the Heisenberg parameter: 𝐽_Mn³_−Mn³=−0.0134 eV for the Mn³⁺–Mn³⁺ fragment and 𝐽_Mn⁴_−Mn⁴=−0.0102 eV for the Mn⁴⁺–Mn⁴⁺ fragment. The antiferromagnetic character of exchange interaction obtained for these fragments is in agreement with the experimentally detected magnetic structure [26].

The estimation of the average Heisenberg ⟨𝐽⟩ parameter with respect to degeneracy factor from (58) equals ⟨𝐽⟩≈0.04 eV; then the average magnetization per manganat ion is 𝜇≈4.1𝜇_𝐵. From the statistic Heisenberg model [4] the Weiss temperature 𝜃=(2/3)𝐽, then ⟨𝐽⟩≈0.023 eV. The estimated value of the Heisenberg parameter is in good agreement with 𝜃≈175 K obtained in the experiment for the undoped compound [29].

4. Conclusions

The general EPT form for a multicenter electron system has been developed from the first principles, and the degree of approach has been presented for each order of expansion. The analytical form of the Λ-projection operators and the symmetric form of the Hamiltonian unperturbed part and that of the perturbation operator have been obtained. In the frame of the exchange perturbation theory we have reduced the expressions for the high-order corrections to the energy and wave function to the standard form of the Rayleigh-Schrödinger perturbation theory. The EPT formalism is extended for the degeneracy case in the total spin for the multicentered electronic system. This formalism does not need any additional variation procedure; each term of the expansion obtained has a certain physical interpretation. The contributions of superexchange interaction enter in the first-order terms. The main advantage of the presented formalism is the logical procedure of obtaining energy and wave function corrections, where it retains the one-to-one correspondence of the symmetry of spin and spatial parts of the wave function. Each energy correction corresponds to certain spin configuration. The presented ETP formalism is the only method allowing us to find the total spin value and spin configuration for the multicenter electron system.

As an example, using the developed EPT formalism with degeneracy in total spin, we obtained the analytical expressions of energy splitting for the key ⟨Mn⟩–O–⟨Mn⟩ fragments (⟨Mn⟩: Mn³⁺ or Mn⁴⁺) in manganites. On the basis of these expressions the Heisenberg parameter and energy splitting were estimated for the La_1/3Ca_2/3MnO₃ manganite. The Heisenberg parameter values obtained and the average magnetic moment of the system are in agreement with the known magnetic structure [26] and the experimentally obtained value of the Weiss temperature [29, 30] for the compound under consideration. The developed algorithm can be used for the analysis of Mn-site doping effect in manganites.

Appendices

A. Details of the Theoretical Derivations

Proof of (17), 𝑃_𝑖𝐻₀|Ψ_𝑖⁽¹⁾⟩=𝐸⁰_𝑖𝑃_𝑖|Ψ_𝑖⁽¹⁾⟩. One has 𝑃_𝑖𝐻₀Ψ_𝑖⁽¹⁾=Ψ⁰_𝑖𝑓^𝑃_𝑖^𝑃_𝑝=0Φ⁰_𝑖𝐻^𝑝₀Φ^𝑝_𝑖Φ^𝑝_𝑖∣Ψ_𝑖⁽¹⁾=𝐸⁰_𝑖Ψ⁰_𝑖𝑓^𝑃_𝑖^𝑃_𝑝=0Φ⁰_𝑖∣Φ^𝑝_𝑖Φ^𝑝_𝑖∣Ψ_𝑖⁽¹⁾=𝐸⁰_𝑖Ψ⁰_𝑖Φ⁰_𝑖∣Ψ_𝑖⁽¹⁾=𝐸⁰_𝑖𝑃_𝑖Ψ_𝑖⁽¹⁾, (A.1) where the completeness property ∑_𝑛|Φ^𝑝_𝑛)(Φ^𝑝_𝑛|=𝐼̂ is used.

The Properties of Projector on the Vector Subspace, Orthogonal to the Vector |Ψ⁰_i⟩
Since (1) 𝑂_𝑖|Ψ⁰_𝑖⟩≡0, (2) 𝑂_𝑖|Ψ⁰_𝑛⟩≈|Ψ⁰_𝑛⟩, for the non-symmetric bra-vector: Φ_𝑛𝑂_𝑖=Φ_𝑛−Φ_𝑛∣Ψ⁰_𝑖Φ_𝑖≈Φ_𝑛,Φ_𝑖𝑂_𝑖=Φ_𝑖−Φ_𝑖∣Ψ⁰_𝑖Φ_𝑖≡0. (A.2) Since (Φ⁰_𝑖∣Φ⁰_𝑛)=0 and (Φ⁰_𝑖|Φ^𝑝_𝑛)≪(Φ⁰_𝑛|Φ^𝑝_𝑛)≈(Φ⁰_𝑖|Φ^𝑝_𝑖) we have an expression: 𝑃_𝑖Ψ⁰_𝑛=Ψ⁰_𝑖Φ⁰_𝑖∣Ψ⁰_𝑛=Ψ⁰_𝑖1𝑓_𝑃^𝑃𝑝=0Φ⁰_𝑖∣Φ^𝑝_𝑛(−1)^𝑔𝑝≈0. (A.3)

Proof of (22), 𝑂_𝑖𝑉|Ψ⁰_𝑖⟩=1/𝑃∑_𝑛^′𝑓^𝑝_𝑛|Ψ⁰_𝑛⟩(Φ⁰_𝑛|𝑂_𝑖𝑉|Ψ⁰_𝑖⟩. One has 1𝑃^{𝑃𝑝=0𝑛≠𝑖}Φ^𝑝_𝑛Φ^𝑝_𝑛𝑂_𝑖𝑉Ψ⁰_𝑖=1𝑃^{𝑃𝑝=0𝑛≠𝑖}Φ^𝑝_𝑛(−1)^𝑝Φ⁰_𝑛𝑂_𝑖𝑉Ψ⁰_𝑖=1𝑃^𝑛≠𝑖𝑓^𝑝_𝑛Ψ⁰_𝑛Φ⁰_𝑛𝑂_𝑖𝑉Ψ⁰_𝑖 (A.4)

Proof of (44). One has Ψ⁰_𝛾𝑉_𝛽Ψ⁰_𝛽=1𝑓^𝛾₀^𝑃𝑝=0(−1)^𝑔𝑝𝛾Φ^𝑝𝛾𝑉_𝛽Ψ⁰_𝛽=1𝑓^𝛾₀^{𝑃𝑝𝛾=0}(−1)^{𝑔𝑝𝛾+𝑔𝑝𝛽}Φ⁰𝑉_𝛽Ψ⁰_𝛽=Δ_𝛾𝛽Φ⁰𝑉_𝛽Ψ⁰_𝛽,Δ_𝛾𝛽=Ψ⁰_𝛾∣Ψ⁰_𝛽=1𝑓^𝛾₀^𝑃𝑝=0(−1)^𝑔𝑝Φ^𝑝𝛾∣Ψ⁰_𝛽=^𝑃_𝑝=0(−1)^𝑔𝑝𝛾Φ⁰∣Ψ⁰_𝛽=1𝑓^𝛾₀^𝑃𝑝=0(−1)^{𝑔𝑝𝛾+𝑔𝑝𝛽}. (A.5)

B. The Young Operators of Antisymmetrization

Young Operators of Antisymmetrization for the Tableaus [2, 1] Type
Consider 𝜔^[21]₁₁=1√122+2𝑃₁₂−𝑃₂₃−𝑃₁₃−𝑃₁₂₃−𝑃₁₃₂,𝜔^[21]₁₂=12𝑃₂₃−𝑃₁₃+𝑃₁₂₃−𝑃₁₃₂,𝜔^[21]₂₁=12𝑃₂₃−𝑃₁₃−𝑃₁₂₃+𝑃₁₃₂,𝜔^[21]₂₂=1√122−2𝑃₁₂+𝑃₂₃+𝑃₁₃−𝑃₁₂₃−𝑃₁₃₂. (B.1) Here the 𝑃_𝑖𝑗𝑘 and 𝑃_𝑖𝑗 are operators of the numbered 𝑗 and 𝑘 electrons permutation.
The total antisymmetric function is Ψ()=11,2,3𝑓_𝜆^𝑟𝜓_𝑟^[𝜆]𝜒^[𝜆_]̃𝑟=𝜔^[21]₁₁Φ_Mn𝑟₁Φ_𝑂𝑅2−𝑟₂Φ_Mn𝑅−𝑟₃⊗𝜔^[21]₂₂+𝜔𝛼1𝛽2𝛼3_[21]12Φ_Mn𝑟₁Φ_𝑂𝑅2−𝑟₂Φ_Mn𝑅−𝑟₃⊗𝜔^[21]₂₁𝛼1𝛼2𝛽3, (B.2) where 𝛼1 and 𝛽2 are spinors, corresponding to the particles numbered 1 and 2.

Acknowledgments

F. E. Orlenko is grateful to the Science and High School Committee of St. Petersburg Government (Russia) for the Grant-2010 for young scientists. This work was supported by the Ministry of Education and Science of the Russian Federation (Federal program “Scientific and Pedagogical Staff of Innovative Russia for 2009–2013"). The authors are thankful to Professor N. V. Popova of the Foreign Language Department of St. Petersburg State Polytechnic University for the help with language corrections.