Peculiarities of a helium interaction with hydrogen and free electrons from the point of view of the exchange perturbation theory solutions

Abstract

We consider an adiabatic potential of a Hydrogen-Helium interaction by using a new approach based on the exchange perturbation theory (EPT). This method allows to present results analytically. We find the solution as an expansion in an antisymmetric basis thanks to the proved completeness property for a nonorthogonal set of antisymmetric wave functions. We use the developed invariant form with respect to interatomic electron permutations for both the Hamiltonian describing an unperturbed system and the perturbation operator. The invariant form for both operators allows one to obtain properly antisymmetric corrections to the antisymmetric wave function of the zero approximation. The probability of electron scattering with electron capture by the helium atom is discussed. We use an EPT formalism developed for the time-dependent perturbations. A formalism of the Time-dependent EPT has a form of an invariant perturbation theory that accounts for intercentre electron permutations among the overlapping states. The expressions for the matrix elements of an S-matrix and a T-matrix account for interatomic electron permutations. We use the obtained differential cross section formulas for the description of the electron capture during the scattering processes. The differential cross section of the electron capture by Helium atom has a complex shape of peak for the scattering energy around 0.7 atomic units, associated with the adiabatic creation of the negative ion He(-1).

Keywords

Intercentre electron exchange nonorthogonal basis of the antisymmetrized wave vectors stationary case case of degeneracy and time dependent case of the exchange perturbation theory Helium-Hydrogen adiabatic potential electron-Helium collision electron capture

1. Introduction

Besides, of hydrogen molecule, helium hydride is the second of the most elementary molecules. In their pioneer work, Micelle and Harris [1] have predicted the existence of the bound exited states of helium hydride by using a molecular orbital calculation. The ground state of this system is repulsive; it is shown by numerous experiments and theoretical calculations [2]. Now, it is not only helium hydride, but also the systems, like, NeH and ArH, have only repulsive ground state, nevertheless they possess a series of bound excited Rydberg states. The dissociation of mentioned systems occurs into a ground-state of Ne, Ar, He and to an excited H atom [2, 3, 4]. The first observation supporting the existence of bound, excited states of HeH was performed in the experiment of Möller et al. [5]. In another work [6] authors reported on the spectroscopic proof of the existence of rather long-lived excited states of HeH. It leads to broad uv bound-free fluorescence continua. In their experiment W. Ketterle, H. Figger, and H. Walther observed the discrete emission spectra [7]. There was the ultimate proof of the existence the bound states of helium hydride. These bound states are discussed also in the monography [10].

Theoretical investigations of noble (rare) gases hydrides have been performed by numerical calculations with using an Atomic Orbital technique (LCAO), or Hartree-Fock methods (restricted and unrestricted) [8, 9] and in the late works of Taku Onishi [11, 12, 13, 14]. The reference spaces consisted of the configurations symmetry-adapted functions. The multi-reference approach uses an antisymmetric wave function basis, constructed half-intuitive due to the absence of a regular method for the probe wave functions choice. Another theoretical treatment uses the atomic basis sets consisted of contracted Gaussian-type functions [2] constructed to avoid their overlapping, hence basis overfilling, to solve the non-orthogonality problem. Anyway, the variation techniques, like as Hartree-Fock approximation (LCAO-SCF-HF), or MRD-CI, have indeterminacy in the initial wave functions basis.

Obviously, a perturbation theory (PT) method gives certain advantages in comparison with variation methods. So, the representation of the interaction energy in PT as a sum of different terms, each having a specific physical meaning, allows to associate this energy with the physical characteristics of the interacting system of atoms [15, 16]. The method of the exchange perturbation theory (EPT) makes it possible to analyse the two- or three-center molecular system analytically and to present the results in a quadrature form. Currently there are two types of multi-electronic systems according to the construction of the antisymmetric wave functions and their inputs to exchange contributions. For example, in the single-centred system, any atomic system has a zeroth approximation of antisymmetrized wave function as an eigenfunction of the multielectronic system Hamiltonian without electron-electron interaction. In other words, the “zeroth” Hamiltonian $\hat{H}_{0}$ describing the non-interactive system is invariant with respect to the antisymmetrization operation $\hat{A}$ , that takes into account all electron permutations: $[\hat{H}_{0},\hat{A}]=0$ . For such a system, a perturbation operator $\hat{V}$ accounting for all electron-electron interactions is also invariant to the antisymmetrization operation: $[\hat{V},\hat{A}]=0$ . Hence, a total Hamiltonian $\hat{H}=\hat{H}_{0}+\hat{V}$ of the above mentioned multielectron single centred system is invariant for the electron permutation operation $[\hat{H},\hat{A}]=0$ . In this case, the zeroth approximation wave functions antisymmetrized with respect to all electron permutations form an orthogonal basis. It is commonly accepted in this case to use the antisymmetrized wave functions in a Slater determinant form for this “single-centered” case. The second type of multielectronic systems is manifold centered, or molecular systems. For this case, both the “zeroth” Hamiltonian $\hat{H}_{0}$ describing a manifold centred multielectron system without interatomic interaction and perturbation operator $\hat{V}$ that describes an interatomic interaction are non invariant to the antisymmetrisation operation, accouning for the electron permutations among the atomic centres: $[\hat{H}_{0},\hat{A}]\neq 0$ , $[\hat{V},\hat{A}]\neq 0$ . At that time, the total Hamiltonian of the system preserves its invariance $[\hat{H},\hat{A}]=0$ [11, 12, 13, 14]. The most pronounced exchange effects for such multi-atomic systems manifest themselves at the so-called intermediate atomic distances when the electrons are redistributing among the atomic centres. At these distances, the exchange effects are already more significant than van der Waals contributions. However, at the same time they are weak compared to intra-atomic interaction. Therefore, we can consider them as a perturbation. Exchange contributions related to the permutation symmetry of the particles system wave function decrease exponentially when the distance between interacting atoms increases. During this time, the van der Waals interaction are still weaker by order of magnitude. This fact allows us to construct a perturbation theory for multi-centered electron system taking into account the electron indistinguishability in any approximation order and preserves an antisymmetric form of total wave function of the system in all its corrections. In this case, corrections to the energy should contain both regular dispersion contributions and overlap integrals that yield appropriate exchange contributions. It would seem that for a system of electrons, Pauli Exclusion Principle requirements can be readily taken into account by applying the usual perturbation theory to the set of un-symmetrized states and then to perform the antisymmetrization of the wave function corrections. In particular, antisymmetrization might be performed post factum in each term of the perturbation theory expansion. However, this procedure is unsatisfactory and may lead to incorrect results. When creating a formalism of an exchange perturbation theory for manifold centred system, one should remember that this formalism should overcome two fundamental difficulties [15, 18]:

(I)
The first of them is the previously mentioned non-invariance, the Hamiltonian ${\hat{H}}_{0}$ describing the non-interacting two and more atomic subsystems and perturbation operator $\widehat{V}$ , representing the interaction of the above subsystems to the antisymmetrization operation with respect to electron permutations among these atomic centers. At the same time, the total Hamiltonian $\hat{H}$ of the whole interacting system remains invariant. This inconsistency means a serious problem concerning the fact that a zeroth approximation wave function antisymmetrized with respect to intercentre electron permutations is not an eigenfunction of the non-invariant $\hat{H}_{0}$ Hamiltonian [15, 16, 18]. Therefore, to avoid such an inconvenience the exchange perturbation theory (EPT) formalism has been formulated in such a way that first, the antisymmetrized wave function of zeroth approximation should to be an eigenfunction of the Hamiltonian which describes the non-interactive subsystems forming the total system. Second, corrections of any order to the wave function accounting for the perturbation should have properly antisymmetric form.
(II)
The second difficulty is related to the fact that a set of wave functions of zeroth approximation antisymmetrized with respect to electron permutations among the centers forms a non-orthogonal basis. It is commonly believed that such a basis is overfilled, and this problem is called “an overfilling catastrophe”. However, this problem was solved in [17, 18, 20, 21, 22, 23, 24, 25, 26] where it was shown that under certain conditions a non-orthogonal basis of antisymmetrized wave functions constitutes a complete set.

There exists quite a large number of EPT formalisms. An overview of EPT formalisms can be found in Ref. [15]. In this book, Kaplan classifies EPT formalisms into two groups: (I) formalisms with non-symmetric unperturbed Hamiltonian $[\hat{H}_{0},\hat{A}]\neq 0$ and non-symmetric perturbation $[\hat{V},\hat{A}]\neq 0$ ; (II) formalisms with symmetric zero-approximation Hamiltonian $[\hat{H}_{0},\hat{A}]=0$ and symmetric perturbation $[\hat{V},\hat{A}]=0$ . The first type of formalisms, also called symmetry-adapted EPT formalisms, have an advantage of being a simple representation of the perturbation operator. Therefore, they are often favoured as bases for software packages. The drawback of these formalisms is that they require an antisymmetrisation procedure, doing post factum. It means that the antisymmetrization is performed after the perturbation operator action on to the non-symmetrised wavefunction of the system. Such algorithm leads to the necessity of applying the additional variation procedure for obtaining the corrections within the perturbation theory, as seen in Mayer’s formalism [25, 26]. This problem has been previously described in detail in [15, 17, 18, 19, 20]. (II) The second type of formalisms allows the standard perturbation theory to be applied by constructing a symmetric zero-approximation Hamiltonian. An attempt to construct a symmetric unperturbed Hamiltonian whose eigenfunctions are antisymmetric functions leads to a non-Hermitian Hamiltonian (see Sternheimer’s work [27]). Thus, the practical applicability of this Hamiltonian is restricted to two-electron systems. The Ritchie formalism [28] in the Brillouin-Wigner form, developed for a two-electron two-center system, and the later Rumyantsev formalism [19] in the Rayleigh-Schr?dinger version, also belong to the class of symmetric zeroth-Hamiltonian and perturbation formalisms. Both of the above formalisms introduced abstract projection $\Lambda$ -operators to reduce the zeroth Hamiltonian $\hat{H}_{0}$ of unperturbed system and perturbation operator $\hat{V}$ to a symmetric form. The $\Lambda$ -operator plays a key role in the successive approximation procedure but has an indefinite form and remains as a symbolic operator in these formalisms. It cast doubt on the Hermitian character of the total Hamiltonian ( $\hat{H}_{0}$ and $\hat{V}$ ) parts mentioned as well as on the validity of the original steady state Schr?dinger equation for a zero approximation antisymmetric function.

In the present work, we use the EPT formalism developed in a general form [16, 17, 18, 20, 21, 22, 23, 24]. The presentation of the energy corrections and the wave function corrections contain a special symmetric form of the perturbation operator. We consider an adiabatic potential of the Hydrogen-Helium interaction by using EPT formalism and represent the result in the analytical form. We are accounting the indistinguishability principle of identical particles during a description of heavy atomic particles scattering processes. In this case the EPT formalism developed for the time dependent perturbations is useful. As shown in [16, 17, 18], the EPT formalism for non-stationary case has a form of an Invariant exchange perturbation theory (IEPT). General expressions obtained for the scattering (S) matrix and the transition (T) matrix for arbitrary types of interaction consistently take into account the intercentre electron permutations between overlapping non-orthogonal states. This formalism was applied for the description of a heavy atomic particles collision process with respect to the electron permutations among the scattered particles accounting for the exchange effects [20, 21, 22, 23, 24]. A scattering process of a double-charged helium ion (the alpha particle) on a Lithium atom, or a proton scattering on a Lithium atom were considered by use of IEPT with accounting a charge exchange with electron capture by heavy ion. Now we dicuss a problem of an electron capture by Helium-atom with a creation of a negative ion He ${}^{1-}$ during the elastic scattering process. We consider this system from two points of view, the first, as an elastic scattering process with the electron capture, for which we alculate the cross section of this process. The second is a stationary problem of determining the adiabatic interaction potential of electron and Helium atom in the analytical form and the computation of a binding energy. The random phase exchange approximation (RPEA) metod was used for the description such a process involving electric-dipol interaction with photon. The describing effects determined by the second-order perturbation corrections. We consistently take into account all exchange effects which put their contributions in the first-order corrections to the scattering amplitude and to the cross section.
2. Exchange perturbation theory (EPT)

2.1 Stationary case

For the description of Hydrogen – Helium system in the adiabatic approximation, we will use EPT for the stationary case [13]. There are some notations of this formalism:

1) In zero-order approximation, the wave function for the system with neglecting the interaction among particles is represented as a simple product of the isolated particles wave functions. The wave function describing each atom $\alpha$ is antisymmetrized with respect to all intracentre (intra-atomic) electron permutations:

$\displaystyle\Phi\left(r_{1}\xi_{1},\ldots,r_{N}\xi_{N}\right)=\prod_{\alpha}% \psi_{\alpha}\left(r_{i}\xi_{i},\ldots,r_{j}\xi_{j}\right)_{\alpha},$ (1)

where $r_{i}$ and ${\xi}_{i}$ a position vector and a spin variable, respectively, of $i$ -th electron belonging to the atom $\alpha$ . This wave function is an eigenfunction of the Hamiltonian $H_{0}$ that describes a non-interactive atomic system. It includes kinetic energy operators of all electrons, potential energy operators of electrons interactions with “own” atomic center, and the mutual interaction among the electrons belonging to the same center.

$\displaystyle H^{0}|\Phi_{n})=E_{n}^{0}|\Phi_{n}),$ (2)

where $\{E_{n}^{0}\}$ is an energy spectrum of the stationary non-interactive system. An eigenvector $|\Phi_{n})$ corresponds to the energy level $E^{0}_{n}$ . The round brackets denote here the non-symmetric wave vectors with respect to the electronic permutations among the $\alpha$ -centers.

2) An antisymmetrized wave function of the system is obtained by using Young diagrams (YD) [26], or by the Slater determinant with accounting only inter-atomic permutations:

$\displaystyle\Psi_{n}^{0}\left(r_{1},\ldots r_{N};\xi_{1}\ldots\xi_{N}\right)=% \hat{A}\Phi_{n}\left(r_{1},\ldots r_{N};\xi_{1}\ldots\xi_{N}\right),$ (3)

where $\hat{A}$ denotes the antisymmetrisation operator. We rewrite this function in the form of antisymmetric vector $|\Psi_{n}^{0}\rangle$ as

$\displaystyle|\Psi_{n}^{0}\rangle=\frac{1}{f_{n}^{P}}\sum_{p}^{P}(-1)^{g_{p}}|% \Phi_{n}^{p})$ (4)

where $g_{p}$ is a parity of the electron permutation p among the centres, and $f_{n}^{P}=(\Phi_{n}^{p}|\sum_{p}^{P}(-1)^{g_{p}}|\Phi_{n}^{p})$ – is a normalization factor of the chosen normalization condition:

$\displaystyle\left(\Phi_{n}^{p=0}\right.\left|\Psi_{n}^{0}\right\rangle=1,$ (5)

3) As has been shown in [9], the completeness property of the unperturbed system antisymmetric functions basis has a form:

$\displaystyle\sum_{n}{\left|\Psi_{n}^{0}\right\rangle}\frac{f_{n}}{P}\left(% \Phi_{n}^{p=0}\left|=\sum_{n}\right|\Phi_{n}^{p=0}\right)\left(\Phi_{n}^{p=0}% \right|=\hat{1}.$ (6)

A symmetric form of the Hamiltonian that describes the system without interatomic interaction, developed from the first principals in [11, 12, 13, 27] is:

$\displaystyle\hat{H}_{0}=\left(\sum_{p=0}^{p=P}H_{p}^{0}\Lambda^{p}\right)=% \sum_{p=0}^{P}H_{p}^{0}\sum_{n}\frac{\left|\left.\Phi_{n}^{p}\right)\left(% \left.\Phi_{n}^{p}\right|\right.\right.}{f_{n}^{P}}$ (7)

where $H_{p}^{0}$ is a zeroth Hamiltonian correspondent to the p-th interatomic electron permutation among the centers. Then a Schrödinger equation for the multiatomic system without interatomic interaction has the form that takes into account identical principle for the electrons that belong to different atomic centres. The eigenvector of this Hamiltonian has the antisymmetric form with accounting the interatomic electron permutations:

$\displaystyle\hat{H}_{0}{\left|\Psi_{i}^{0}\right\rangle}=E_{i}^{0}{\left|\Psi% _{i}^{0}\right\rangle}.$ (8)

A symmetric form for the perturbation operator that takes into account inter-atomic interactions:

$\displaystyle\hat{V}=\left(\sum_{p=0}^{p=P}V_{p}\Lambda^{p}\right)=\sum_{p=0}^% {P}V_{p}\sum_{n}\frac{\left|\left.\Phi_{n}^{p}\right)\left(\left.\Phi_{n}^{p}% \right|\right.\right.}{f_{n}^{P}}.$ (9)

The first corrections to the energy obtained [12, 13, 27] in the form using the symmetric perturbation operator Eq. (9), are

$\displaystyle E_{i}^{1}=\left(\left.\Phi_{i}^{p=0}\right|\right.\hat{V}{\left|% \Psi_{i}^{0}\right\rangle},$ $\displaystyle E_{i}^{2}=\sum_{k}{}^{{}^{\prime}}\frac{f_{k}^{P}}{P}\frac{\left% (\Phi_{i}^{p=0}\right.|\hat{V}{\left|\Psi_{k}^{0}\right\rangle}\left(\Phi_{k}^% {p=0}|\right.\hat{V}{\left|\Psi_{i}^{0}\right\rangle}}{(E_{i}^{0}-E_{k}^{0})}.$

The obtained expressions of the energy corrections and the antisymmetric wave vector corrections analytically exactly reduced to the symmetry-adapted form that involves an ordinary non-symmetrized perturbation operator corresponding to the original zeroth electrons permutation are [12, 13, 14, 15, 16]:

$\displaystyle E_{i}^{1}={\left\langle\Psi_{i}^{0}\right|}V_{p=0}\left|\left.% \Phi_{i}^{p=0}\right)\right.;$ $\displaystyle E_{i}^{2}=\sum_{k}{}^{{}^{\prime}}\frac{{\left\langle\Psi_{i}^{0% }\right|}V_{p=0}\left|\left.\Phi_{k}^{p=0}\right)\right.\frac{f_{i}^{p}}{f_{k}% ^{p}}\frac{f_{k}^{P}}{P}\frac{f_{k}^{p}}{f_{i}^{p}}{\left\langle\Psi_{k}^{0}% \right|}V_{p=0}\left|\left.\Phi_{i}^{p=0}\right)\right.}{(E_{i}^{0}-E_{k}^{0})};$ $\displaystyle E_{i}^{3}=\sum_{n}{}^{{}^{\prime}}\sum_{k}{}^{{}^{\prime}}\left(% \frac{{\left\langle\Psi_{i}^{0}\right|}V_{p=0}\left|\left.\Phi_{k}^{p=0}\right% )\right.\frac{f_{k}^{P}}{P}{\left\langle\Psi_{k}^{0}\right|}V_{p=0}\left|\left% .\Phi_{n}^{p=0}\right)\right.\frac{f_{n}^{P}}{P}{\left\langle\Psi_{n}^{0}% \right|}V_{p=0}\left|\left.\Phi_{i}^{p=0}\right)\right.}{(E_{n}^{0}-E_{i}^{0})% (E_{k}^{0}-E_{i}^{0})}\right.--\left.{\left\langle\Psi_{i}^{0}\right|}V_{p=0}% \left|\left.\Phi_{i}^{p=0}\right)\right.\sum_{n}{}^{{}^{\prime}}\frac{f_{i}^{P% }}{P}\frac{{\left\langle\Psi_{i}^{0}\right|}V_{p=0}\left|\left.\Phi_{n}^{p=0}% \right)\right.\frac{f_{n}^{P}}{P}{\left\langle\Psi_{n}^{0}\right|}V_{p=0}\left% |\left.\Phi_{i}^{p=0}\right)\right.}{(E_{i}^{0}-E_{n}^{0})^{2}}\right).$ $\displaystyle\left\langle\Psi_{i}^{1}\right|=\sum_{n\neq i}{}^{{}^{\prime}}% \frac{f_{n}^{P}}{P}\frac{{\left\langle\Psi_{i}^{0}\right|}V_{p=0}\left|\left.% \Phi_{n}^{p=0}\right)\right.}{\left(E_{i}^{0}-E_{n}^{0}\right)}{\left\langle% \Psi_{n}^{0}\right|},$ $\displaystyle\left|\left.\Phi_{i}^{1}\right)\right.=\sum_{n\neq i}{}^{{}^{% \prime}}\left|\left.\Phi_{n}^{p=0}\right)\right.\frac{f_{n}^{P}}{P}\frac{{% \left\langle\Psi_{n}^{0}\right|}V_{p=0}\left|\left.\Phi_{i}^{p=0}\right)\right% .}{\left(E_{i}^{0}-E_{n}^{0}\right)},$ (10) $\displaystyle\left|\left.\Phi_{i}^{2p=0}\right)\right.=\sum_{n}{}^{{}^{\prime}% }\sum_{k}{}^{{}^{\prime}}\left|\left.\Phi_{n}^{p=0}\right)\right.\frac{\frac{f% _{n}^{P}}{P}{\left\langle\Psi_{n}^{0}\right|}V_{p=0}\left|\left.\Phi_{k}^{p=0}% \right)\right.\frac{f_{k}^{P}}{P}{\left\langle\Psi_{k}^{0}\right|}V_{p=0}\left% |\left.\Phi_{i}^{p=0}\right)\right.}{\left(E_{i}^{0}-E_{k}^{0}\right)\left(E_{% i}^{0}-E_{n}^{0}\right)}--\sum_{n}{}^{{}^{\prime}}\left|\left.\Phi_{n}^{p=0}% \right)\right.\frac{\left(\frac{f_{n}^{P}}{P}\right){\left\langle\Psi_{n}^{0}% \right|}V_{p=0}\left|\left.\Phi_{i}^{p=0}\right)\right.}{\left(E_{i}^{0}-E_{n}% ^{0}\right)^{2}}\left(\frac{f_{i}^{P}}{P}\right){\left\langle\Psi_{i}^{0}% \right|}V_{p=0}\left|\left.\Phi_{i}^{p=0}\right)\right.-\sum_{n}{}^{{}^{\prime% }}\frac{\left|\left.\Phi_{i}^{p=0}\right)\right.}{2}\frac{\left(\frac{f_{i}^{P% }}{P}\right){\left\langle\Psi_{i}^{0}\right|}V_{p=0}\left|\left.\Phi_{n}^{p=0}% \right)\right.\left(\frac{f_{n}^{P}}{P}\right){\left\langle\Psi_{n}^{0}\right|% }V_{p=0}\left|\left.\Phi_{i}^{p=0}\right)\right.}{\left(E_{i}^{0}-E_{n}^{0}% \right)^{2}},$ $\displaystyle{\left\langle\Psi_{i}^{2}\right|}=\sum_{n}{}^{{}^{\prime}}\sum_{k% }{}^{{}^{\prime}}\frac{{\left\langle\Psi_{i}^{0}\right|}V_{p=0}\left|\left.% \Phi_{k}^{p=0}\right)\right.\frac{f_{k}^{P}}{P}{\left\langle\Psi_{k}^{0}\right% |}V_{p=0}\left|\left.\Phi_{n}^{p=0}\right)\right.}{\left(E_{i}^{0}-E_{k}^{0}% \right)\left(E_{i}^{0}-E_{n}^{0}\right)}\frac{f_{n}^{P}}{P}{\left\langle\Psi_{% n}^{0}\right|}--\sum_{n}{}^{{}^{\prime}}\frac{{\left\langle\Psi_{i}^{0}\right|% }V_{p=0}\left|\left.\Phi_{i}^{p=0}\right)\right.\frac{f_{i}^{P}}{P}{\left% \langle\Psi_{i}^{0}\right|}V_{p=0}\left|\left.\Phi_{n}^{p=0}\right)\right.}{% \left(E_{i}^{0}-E_{n}^{0}\right)^{2}}\left(\frac{f_{n}^{P}}{P}\right){\left% \langle\Psi_{n}^{0}\right|}-\frac{1}{2}\sum_{n}{}^{{}^{\prime}}\frac{{\left% \langle\Psi_{i}^{0}\right|}V_{p=0}\left|\left.\Phi_{n}^{p=0}\right)\right.% \frac{f_{n}^{P}}{P}{\left\langle\Psi_{n}^{0}\right|}V_{p=0}\left|\left.\Phi_{i% }^{p=0}\right)\right.}{\left(E_{i}^{0}-E_{n}^{0}\right)^{2}}\left(\frac{f_{i}^% {P}}{P}\right){\left\langle\Psi_{i}^{0}\right|}.$

Such a form is more convenient for practical applications. It is important to underline, that these successively obtained corrections to the energy and to the state vector are simplified post factum, i.e., after the corrections are obtained by a regular method. The Eq. (2.1) obtained without any additional approximations preserves all exchange and super-exchange contributions. The total energy correction, represented in invariant form, using the developed T-operator on the energy surface is

$\displaystyle E_{n}=E_{n}^{0}+{\left\langle\Psi_{n}^{0}\right|}\hat{T}\left.% \left|\Phi_{n}^{p=0}\right.\right).$ (11)

Here, the operator equation, determining the $T$ -operator is

$\displaystyle\hat{T}=V_{p=0}^{{\mathbb{N}}}+V_{p=0}^{{\mathbb{N}}}\left(E_{i}^% {0}-H_{p=0}^{0}+i\eta\right)^{-1}\left(\frac{\hat{f}}{P}\right)^{-1}\hat{T},$ (12)

where a renormalized perturbation operator

$\displaystyle V_{p=0}^{{\mathbb{N}}}=\left(\frac{\hat{f}}{P}\right)V_{p=0}.$

During the expansion of operator $(E_{i}^{0}-H_{p=0}^{0}+i\eta)^{-1}$ we should take into account the unit operator in the form Eq. (6), that corresponds to the form of Resolvents operators $\hat{R}$ and $\hat{R}^{0}$ :

$\displaystyle\hat{R}=\sum_{k\neq i}\frac{{\left|\Psi^{0}{}_{k}\right\rangle}% \left(\frac{f_{k}^{P}}{P}\right)\left(\left.\Phi_{k}^{p=0}\right|\right.}{% \left(E_{i}^{0}-E_{k}^{0}\right)},\hat{R}^{0}=\sum_{k\neq i}\frac{{\left|\Psi^% {0}{}_{i}\right\rangle}\left(\frac{f_{i}^{P}}{P}\right)\left(\left.\Phi_{i}^{p% =0}\right|\right.}{\left(E_{i}^{0}-E_{k}^{0}\right)}.$ (13)

Then the general expression for the energy is

$\displaystyle E=\left(\left.\Phi^{0}_{i}\right|\right.\left(E_{i}^{0}+\hat{V}+% \hat{V}\hat{R}\hat{V}+\hat{V}\hat{R}\hat{V}\hat{R}\hat{V}-\hat{V}\hat{R}^{0}% \hat{V}\hat{R}\hat{V}\right){\left|\Psi_{i}^{0}\right\rangle},$ (14)

and the general expression for the antisymmetric wave vector is

$\displaystyle{\left|\Psi\right\rangle}=\left(\hat{1}+\hat{R}\hat{V}+\hat{R}% \hat{V}\hat{R}\hat{V}-\hat{R}\hat{V}\hat{R}^{0}\hat{V}-\frac{1}{2}\hat{R}^{0}% \hat{V}\hat{R}\hat{V}\right){\left|\Psi_{i}^{0}\right\rangle}.$ (15)

Taking into account a Hermitian properties of Resolvents $R=\hat{R}^{{\dagger}}$ and $R^{0}=\hat{R}^{0{\dagger}}$ the expression for energy and wave vectors in the symmetry-adapted form are:

$\displaystyle E_{i}=E_{i}^{0}+{\left\langle\Psi_{i}^{0}\right|}V_{p=0}+V_{p=0}% \textit{RV}_{p=0}+\left\{V_{p=0}\textit{RV}_{p=0}\textit{RV}_{p=0}-V_{p=0}R^{0% }V_{p=0}\textit{RV}_{p=0}\right\}+...\left.\left|\Phi_{i}^{p=0}\right.\right).% \left|\left.\Phi_{i}\right)\right.=\left\{\hat{1}+\textit{RV}+\textit{RVRV}-% \textit{RVR}^{0}V-\frac{1}{2}R^{0}\textit{VRV}\right\}\left|\left.\Phi_{i}^{p=% 0}\right)\right.;$ (16) $\displaystyle{\left\langle\Psi\right|}={\left\langle\Psi_{i}^{0}\right|}\left(% \hat{1}+V_{p=0}R+V_{p=0}\textit{RV}_{p=0}R-V_{p=0}R^{0}V_{p=0}R-\frac{1}{2}V_{% p=0}\textit{RV}_{p=0}R^{0}\right).$

The developed expressions explicitly contain both direct and exchange integrals. All possible configurations of super exchange integrals contributions automatically presented. The obtained expressions for the energy and for the wave vector in symmetry-adapted form use the ordinary nonsymmetrized perturbation operator; this fact simplifies the computation procedure. The represented EPT series is in the Rayleigh–Schrödinger (RS) form.

2.2 Case of degeneracy

A wave function of zeroth approximation that describes a multicentred electron system, could be antisymmetrized in different ways referred to the different values of a total spin of the system. Technically such an antisymmetrisation is possible by using different Young Diagrams (YD), $\hat{A}_{\alpha}$ numbered by index $\alpha$ . Therefore, instead to Eq. (3) we have for the mentioned antisymmetric eigent vectors of the Hamiltonian Eq. (7)

$\displaystyle{\left|\Psi_{n\alpha}^{0}\right\rangle}=\hat{A}_{\alpha}\left|% \left.\Phi_{n}^{0}\right)\right.,$ (17) $\displaystyle\hat{H}_{0}\left|\Psi_{n\alpha}^{0}\right\rangle=E_{n}^{0}{\left|% \Psi_{n\alpha}^{0}\right\rangle}.$

As commonly used, we search the correct wave vectors in the zeroth approximation as linear combinations of antisymmetric wave vectors:

$\displaystyle{\left|\Psi_{n{\rm A}}^{\textit{0corr}}\right\rangle}=\sum_{% \alpha}C_{\alpha}^{0}{\left|\Psi_{n\alpha}^{0}\right\rangle}.$ (18)

A system of homogeneous linear equations for the quantities $C_{\alpha}^{0}$ and a secular equation which determines the energy corrections $\varepsilon$ , developed in [12, 13, 15, 16] are:

$\displaystyle\left\{\begin{array}[]{l}{\sum\limits_{\alpha=1}^{\alpha=\Sigma}C% _{\alpha}^{0}\left\{\varepsilon\left\langle\Psi_{n1}\right.{\left|\Psi_{n% \alpha}^{0}\right\rangle}-{\left\langle\Psi_{n1}\right|}\hat{V}{\left|\Psi_{n% \alpha}^{0}\right\rangle}\right\}=0}\\ \ldots\\ {\sum\limits_{\alpha=1}^{\alpha=\Sigma}C_{\alpha}^{0}\left\{\varepsilon\left% \langle\Psi^{0}{}_{n\beta}\right.{\left|\Psi_{n\alpha}^{0}\right\rangle}-{% \left\langle\Psi_{n\beta}\right|}\hat{V}{\left|\Psi_{n\alpha}^{0}\right\rangle% }\right\}=0}\\ \ldots\\ {\sum\limits_{\alpha=1}^{\alpha=\Sigma}C_{\alpha}^{0}\left\{\varepsilon\left% \langle\Psi_{n\Sigma}^{0}\right.{\left|\Psi_{n\alpha}^{0}\right\rangle}-{\left% \langle\Psi_{n\Sigma}^{0}\Sigma\right|}\hat{V}{\left|\Psi_{n\alpha}^{0}\right% \rangle}\right\}=0}\end{array}\right.$ (19)

$\displaystyle\det\left|\varepsilon\left\langle\Psi_{n\beta}^{0}\right.{\left|% \Psi_{n\alpha}^{0}\right\rangle}-{\left\langle\Psi_{n\beta}^{0}\right|}\hat{V}% {\left|\Psi_{n\alpha}^{0}\right\rangle}\right|=0,$ (20) $\displaystyle\quad\Delta_{n\beta n\alpha}=\left\langle\Psi^{0}{}_{n\beta}% \right.{\left|\Psi_{n\alpha}^{0}\right\rangle}=\frac{1}{f_{n}^{P\beta}}\sum_{p% =0}^{P}\left(-1\right)^{g_{p\beta}+g_{p\alpha}}\left(\Phi_{n}^{p=0}{\left|\Psi% _{n\alpha}^{0}\right\rangle}\right.=\frac{1}{f_{n}^{P\beta}}\sum_{p=0}^{P}% \left(-1\right)^{g_{p\alpha}+g_{p\beta}},$

or in another form,

$\displaystyle\det\left|\Delta_{n\beta n\alpha}\right|\prod_{\alpha=1}^{\Sigma}% \left(\varepsilon-\left(\Phi_{n}^{p=0}\left|\hat{V}\right.{\left|\Psi_{n\alpha% }^{0}\right\rangle}\right.\right)=0.$

In this case the energy corrections, obtained with accounting of a following condition $\left|\Delta_{n\beta n\alpha}\right|\neq 0$ , are

$\displaystyle\varepsilon_{n\alpha}=\left(\Phi_{n}^{p=0}\left|\hat{V}\right.{% \left|\Psi_{n\alpha}^{0}\right\rangle}\right.,$ (21)

or, in symmetry-adapted form $\varepsilon_{n\alpha}={\left\langle\Psi_{n\alpha}^{0}\right|}\left.V_{p=0}% \right|\left.\Phi_{n}{}^{p=0}\right)$ .

The obtained energy corrections determine the energy splitting of the system depending on the value of the total spin. Substituting Eq. (21) in turn these roots in the system Eq. (19) and solving, we obtain the coefficients $C_{\alpha}^{0}$ , that determine the set of the correct wave vectors in the zeroth approximation.

3. The adiabatic potential of the He-H interaction

We apply EPT formalism for describing and computing a Helium – Hydrogen adiabatic potential of an interaction. As an initial wave function of the system noninteractive He-H we take the form of the simple product of two atomic wave functions, dependent on their spatial and spin parts

$\displaystyle\Phi(\textbf{r}_{1},\textbf{r}_{2},\textbf{r}_{3};\xi_{1},\xi_{2}% ;\xi_{3})=\Phi_{\textit{He}}(\textbf{R}-\textbf{r}_{1},\textbf{R}-\textbf{r}_{% 2};\xi_{1},\xi_{2})\Phi_{H}(\textbf{r}_{3})\chi(;\xi_{3}).$ (22)

Here R is a vector originating at the Hydrogens nucleus and pointing to the Helium nucleus; $\textbf{r}_{i}$ – is the radius vector of the electron, numbered i, originating at the Hydrogen nucleus; $\xi_{i}$ – is the spin variable of the electron i. The atomic function $\Phi_{\textit{He}}(\textbf{R}-\textbf{r}_{1},\textbf{R}-\textbf{r}_{2};\xi_{1}% ,\xi_{2})$ of Helium atom antisymmetrized with respect to the internal electron permutations has the form:

$\displaystyle\Phi_{\textit{He}}(\textbf{R}-\textbf{r}_{1},\textbf{R}-\textbf{r% }_{2};\xi_{1},\xi_{2})$ (23) $\displaystyle\quad=\frac{1}{\sqrt{2}}\left(\psi_{n1}(\textbf{R}-\textbf{r}_{1}% )\psi_{n2}(\textbf{R}-\textbf{r}_{2})\pm\psi_{n2}(\textbf{R}-\textbf{r}_{1})% \psi_{n1}(\textbf{R}-\textbf{r}_{2})\right)\cdot\frac{1}{\sqrt{2}}\left(\alpha 1% \beta 2\mp\beta 1\alpha 2\right),$

where $\alpha$ and $\beta$ are the one-electron spin functions, corresponding to the “up” and “down” spin states, respectively. The Hamiltonian unperturbed system has the form

$\displaystyle H^{0}(1,2,3)=-\frac{\hbar^{2}}{2m}(\nabla_{1}^{2}+\nabla_{2}^{2}% +\nabla_{3}^{2})-\frac{2e^{2}}{r_{He1}}-\frac{2e^{2}}{r_{He2}}+\frac{2e^{2}}{r% _{12}}-\frac{e^{2}}{r_{H3}},$ (24)

where we denote $r_{\textit{He1}}=\left|\textbf{R}-\textbf{r}_{1}\right|$ the distance between electron, numbered 1, and its own Helium nucleus; $r_{12}=\left|\textbf{r}_{1}-\textbf{r}_{2}\right|$ – a distance between electrons, numbered 1 and 2, respectively, that belong to the same centre, the Helium nucleus, with $r_{H3}=\left|\textbf{r}_{3}\right|$ being a distance between electron, numbered 3 and its own centre, originating at the Hydrogen nucleus. The function Eq. (3) is an eigenfunction for the Hamiltonian Eq. (24). For the ground state, we use the eigenfunction in the form Eq. (22) accounting the Eq. (3) in the form of ${}^{1}S_{\textit{0He}}$ -state of Helium atom:

$\displaystyle\Phi_{0}(\textbf{r}_{1},\textbf{r}_{2},\textbf{r}_{3};\xi_{1},\xi% _{2};\xi_{3})=\left(\varphi_{1s}(\textbf{R}-\textbf{r}_{1})\varphi_{1s}(% \textbf{R}-\textbf{r}_{2})\right)\cdot\psi_{H}(\textbf{r}_{3})$ $\displaystyle\quad\cdot\frac{1}{\sqrt{2}}\left(\alpha 1\beta 2-\beta 1\alpha 2% \right)\chi_{H}(\xi_{3}),$ (25) $\displaystyle\Phi_{\textit{He}}(\textbf{R}-\textbf{r}_{1},\textbf{R}-\textbf{r% }_{2};\xi_{1},\xi_{2})=^{1}S_{\textit{0He}}(1,2)=\left(\varphi_{1s}(\textbf{R}% -\textbf{r}_{1})\varphi_{1s}(\textbf{R}-\textbf{r}_{2})\right)$ $\displaystyle\quad\cdot\frac{1}{\sqrt{2}}\left(\alpha 1\beta 2-\beta 1\alpha 2% \right).$

For the three-electron system, He-H, where both electrons occupy the 1 s ${}^{2}$ orbital in a singlet state, only single Young diagram is possible. It is depicted in Fig. 1a. This Young diagram allows two possible combinations shown in Fig. 1b and c.

Figure 1.

Young diagrams for three-electron system. (a) Young diagram. (b) and (c) filled Young tableaux with two possible combinations.

Antisymmetrization of the atomic wavefunfction given by Eq. (22) over interatomic electrons permutations performed using the four independent Young’s operators corresponding to the Young diagram is depicted in Fig. 1: $\omega_{11}^{[21]};\omega_{12}^{[21]};\omega_{21}^{[21]};\omega_{22}^{[21]}$ . Here the superscript [21] describes the ordering of empty boxes in the Young diagram: two boxes in the upper row and one box in the bottom row, and the subscript describes the different arts of the boxes filling [23]. For example,

$\displaystyle\omega_{11}^{[21]}=\frac{1}{\sqrt{12}}(2+2P_{12}-P_{23}-P_{13}-P_% {123}-P_{132})$

and

$\displaystyle\omega_{12}^{[21]}=(P_{23}-P_{13}-P_{123}+P_{132}).$ (26)

$P_{12}$ denotes the permutation of electrons 1 and 2; $P_{123}$ denotes the cyclic permutation of electrons 1, 2 and 3.

$\displaystyle\Psi_{1}(\textbf{r}_{1},\textbf{r}_{2};\textbf{r}_{3})=\omega_{11% }^{[21]}\left(\varphi_{1s}(\textbf{R}-\textbf{r}_{1})\varphi_{1s}(\textbf{R}-% \textbf{r}_{2})\right)\Phi_{H}(r_{3})=\frac{1}{\sqrt{12}}(2+2P_{12}-P_{23}-P_{% 13}-P_{123}-P_{132})\Phi_{\textit{He}}(\textbf{R}-\textbf{r}_{1},\textbf{R}-% \textbf{r}_{2})\Phi_{H}(r_{3});$ (27) $\displaystyle\Psi_{2}(\textbf{r}_{1},\textbf{r}_{2};\textbf{r}_{3})=\omega_{12% }^{[21]}\Phi_{\textit{He}}(\textbf{R}-\textbf{r}_{1},\textbf{R}-\textbf{r}_{2}% )\Phi_{H}(\textbf{r}_{3})=(P_{23}-P_{13}-P_{123}+P_{132})\Phi_{\textit{He}}(% \textbf{R}-\textbf{r}_{1},\textbf{R}-\textbf{r}_{2})\Phi_{H}(\textbf{r}_{3})=0.$

The corresponding spin parts are

$\displaystyle{\rm X}_{\textit{He\_ H1}}=\omega_{22}^{[21]}\chi(123)=\omega_{22% }^{[21]}{\left|\alpha 1\beta 2\alpha 3\right\rangle}=\frac{1}{\sqrt{12}}\left% \{2{\left|\alpha 1\beta 2\alpha 3\right\rangle}+{\left|\alpha 1\alpha 2\beta 3% \right\rangle}+{\left|\alpha 1\beta 2\alpha 3\right\rangle}-2{\left|\beta 1% \alpha 2\alpha 3\right\rangle}\right.-\left.{\left|\alpha 1\alpha 2\beta 3% \right\rangle}-{\left|\beta 1\alpha 2\alpha 3\right\rangle}\right\}=\frac{% \sqrt{3}}{2}\left\{{\left|\alpha 1\beta 2\alpha 3\right\rangle}-{\left|\beta 1% \alpha 2\alpha 3\right\rangle}\right\}$ (28) $\displaystyle{\rm X}_{\textit{He\_ H2}}=\omega_{21}^{[21]}\chi(123)=\omega_{21% }^{[21]}{\left|\alpha 1\beta 2\alpha 3\right\rangle}=\frac{1}{2}\left\{-{\left% |\alpha 1\alpha 2\beta 3\right\rangle}+{\left|\alpha 1\beta 2\alpha 3\right% \rangle}-{\left|\alpha 1\alpha 2\beta 3\right\rangle}+{\left|\beta 1\alpha 2% \alpha 3\right\rangle}\right\}=\frac{1}{2}\left\{-2{\left|\alpha 1\alpha 2% \beta 3\right\rangle}+{\left|\alpha 1\beta 2\alpha 3\right\rangle}+{\left|% \beta 1\alpha 2\alpha 3\right\rangle}\right\}.$

We obtain the antisymmetric wave function by applying the normalized Young operator [23] to the wave function of the Hydrogen–Helium system as follows:

$\displaystyle\Psi=\frac{1}{f_{0}}\left(\Psi_{1}(\textbf{r}_{1},\textbf{r}_{2},% \textbf{r}_{3}){\rm X}_{\textit{HeH1}}(1,2,3)+\Psi_{2}(\textbf{r}_{1},\textbf{% r}_{2},\textbf{r}_{3}){\rm X}_{\textit{HeH2}}(1,2,3)\right).$ (29)

Here is the normalized factor $f_{0}$ , determined by

$\displaystyle f_{0}={\left\langle\left(\Psi_{1}(\textbf{r}_{1},\textbf{r}_{2},% \textbf{r}_{3}){\rm X}_{\textit{HeH1}}(1,2,3)+\Psi_{2}(\textbf{r}_{1},\textbf{% r}_{2},\textbf{r}_{3}){\rm X}_{\textit{HeH2}}(1,2,3)\right)\right|}\left.\Phi_% {\textit{He}}(\textbf{R}-\textbf{r}_{1},\textbf{R}-\textbf{r}_{2};\xi_{1},\xi_% {2})\Phi_{H}(\textbf{r}_{3})\chi_{H}(\xi_{3})\right)$ (30)

Taking into account the orthogonality of the wavefunction spin parts, we reduce the normalisation factor $f_{0}$ to the expression

$\displaystyle f_{0}=\frac{\sqrt{3}}{2}{\left\langle\Psi_{1}(\textbf{r}_{1},% \textbf{r}_{2},\textbf{r}_{3})\right|}\left.\Psi_{{\rm He}}(\textbf{r}_{1},% \textbf{r}_{2},)\psi_{{\rm H}}(\textbf{r}_{3})\right).$ (31)

The same form of antisymmetrized wave function represented by Eq. (29) we obtained by using a Slater determinant:

$\displaystyle\Psi=\frac{1}{\sqrt{3!}}\left|\begin{array}[]{ccc}{\varphi_{1s% \alpha}(\textbf{R}-\textbf{r}_{1})}&{\varphi_{1s\beta}(\textbf{R}-\textbf{r}_{% 1})}&{\psi_{H\alpha}(\textbf{r}_{1})}\\ {\varphi_{1s\alpha}(\textbf{R}-\textbf{r}_{2})}&{\varphi_{1s\beta}(\textbf{R}-% \textbf{r}_{2})}&{\psi_{H\alpha}(\textbf{r}_{2})}\\ {\varphi_{1s\alpha}(\textbf{R}-\textbf{r}_{3})}&{\varphi_{1s\beta}(\textbf{R}-% \textbf{r}_{3})}&{\psi_{H\alpha}(\textbf{r}_{3})}\end{array}\right|=\frac{1}{% \sqrt{3}}\left(\varphi_{1s}(\textbf{R}-\textbf{r}_{1})\varphi_{1s}(\textbf{R}-% \textbf{r}_{2})\cdot\frac{1}{\sqrt{2}}\left(\alpha 1\beta 2-\beta 1\alpha 2% \right)\psi_{H}(\textbf{r}_{3})\alpha 3\right.-\varphi_{1s}(\textbf{R}-\textbf% {r}_{1})\varphi_{1s}(\textbf{R}-\textbf{r}_{3})\cdot\frac{1}{\sqrt{2}}\left(% \alpha 1\beta 3-\beta 1\alpha 3\right)\psi_{H}(\textbf{r}_{2})\alpha 2\left.-% \varphi_{1s}(\textbf{R}-\textbf{r}_{3})\varphi_{1s}(\textbf{R}-\textbf{r}_{2})% \cdot\frac{1}{\sqrt{2}}\left(\alpha 3\beta 2-\beta 3\alpha 2\right)\psi_{H}(% \textbf{r}_{1})\alpha 1\right)=\frac{1}{\sqrt{3}}\left({}^{1}S_{\textit{0He}}(% 1,2)\psi_{H\alpha}(3)-^{1}S_{\textit{0He}}(1,3)\psi_{H\alpha}(2)-^{1}S_{% \textit{0He}}(3,2)\psi_{H\alpha}(1)\right).$ (32)

The energy of He-H interaction in the first approximation equals to the first energy correction in the symmetry-adapted form (see Eq. (16)). An ordinary non-symmetric form of the perturbation operator that corresponds to the original zeroth permutation of electrons is:

$\displaystyle V_{p=0}\equiv V(12,3)=-\frac{2e^{2}}{r_{He3}}-\frac{e^{2}}{r_{H1% }}-\frac{e^{2}}{r_{H2}}+\frac{2e^{2}}{R}+\frac{e^{2}}{r_{23}}+\frac{e^{2}}{r_{% 13}}.$ (33)

Then, we can write the adiabatic potential of the He-H interaction in the first approximation using the Eq. (2.1) in the form

$\displaystyle E(R)=E_{i}^{1}={\left\langle\Psi\right|}V_{p=0}\left|\left.\Phi^% {p=0}\right)\right.,$ (34)

where bra-vector ${\langle\Psi|}$ has the form Eq. (32) and ket-vector $\left|\left.\Phi^{p=0}\right)\right.$ determined by Eq. (3).

Then, the expression Eq. (34) has the form:

$\displaystyle U_{\textit{int}}(R)=\frac{1}{f_{0}}{\int\!\!\!\!\int\!\!\!\!\int% }\Psi_{1}(\textbf{r}_{1},\textbf{r}_{2},\textbf{r}_{3})V(12,3)\left(\varphi_{1% s}(\textbf{R}-\textbf{r}_{1})\varphi_{2s}(\textbf{R}-\textbf{r}_{2})\right)% \cdot\psi_{H}(\textbf{r}_{3})d^{3}\textbf{r}_{1}d^{3}\textbf{r}_{2}d^{3}% \textbf{r}_{3}.$ (35)

We transform this equation into the following expression:

$\displaystyle U_{\textit{int}}(R)=\frac{1}{f_{0}(R)}\left(4K(R)-3A_{23}(R)-A_{% 31}(R)\right),$ (36)

where

$\displaystyle f_{0}(R)=\frac{2(1-S^{2}(R))}{3}.$

We introduce the following two types of the exchange contributions, with the first of them being

$\displaystyle A_{23}(R)=\left(\varphi_{\textit{He}}(1)\varphi_{\textit{He}}(3)% \psi_{H}(2)\right.\!\left|-\frac{2}{r_{He3}}-\frac{1}{r_{H1}}-\frac{1}{r_{H2}}% \left.+\frac{2}{R}+\frac{1}{r_{23}}+\frac{1}{r_{13}}\right|\right.\!\left.% \varphi_{\textit{He}}(1)\varphi_{\textit{He}}(2)\psi_{H}(3)\right)=\textit{4SI% }+S\left(\mathop{\odot\sim\bullet}\limits_{\longleftarrow}{}\right)+\mathop{% \overrightarrow{\bullet\sim\bullet}}\limits_{\longleftarrow}.$ (37)

The second type of the exchange contribution has the form:

$\displaystyle A_{13}=\left(\varphi_{\textit{He}}(2)\varphi_{\textit{He}}(3)% \psi_{H}(1)\right.\left|-\frac{2}{r_{\textit{He3}}}-\frac{1}{r_{H1}}-\frac{1}{% r_{H2}}\left.+\frac{2}{R}+\frac{1}{r_{23}}+\frac{1}{r_{13}}\right|\right.\left% .\varphi_{\textit{He}}(1)\varphi_{\textit{He}}(2)\psi_{H}(3)\right)=\textit{3% SI}-\frac{S^{2}}{R}+S\left(\mathop{\odot\sim\bullet}\limits_{\longleftarrow}{}% \right)+\mathop{\overrightarrow{\bullet\sim\bullet}}\limits_{\longleftarrow}{}.$ (38)

Finally, we have the H-He potential:

$\displaystyle E(R)=\frac{1}{8\left(1-S^{2}\right)}\left\{4K+\textit{15SI}-2% \left(\mathop{\odot\sim\bullet}\limits_{\longleftarrow}{}\right)-4\mathop{% \overrightarrow{\bullet\sim\bullet}}\limits_{\longleftarrow}{}+S^{2}/R\right\}.$ (39)

We denote the overlap integral S as following

$\displaystyle S=\int\psi_{H}(\textbf{r})\varphi_{\textit{He}}(\textbf{R}-% \textbf{r})d^{3}\textbf{r}.$

The “direct” contribution $K$ is the integral

$\displaystyle K=\left(\varphi_{\textit{He}}(1)\varphi_{\textit{He}}(2)\psi_{H}% (3)\right.\!\left|-\frac{2}{r_{\textit{He3}}}-\frac{1}{r_{H1}}-\frac{1}{r_{H2}% }\!\left.+\frac{2}{R}+\frac{1}{r_{23}}+\frac{1}{r_{13}}\right|\right.\left.% \varphi_{\textit{He}}(1)\varphi_{\textit{He}}(2)\psi_{H}(3)\right),$ (40)

The exchange contribution to the interaction of the electron with the H-nucleus is:

$\displaystyle I=\int\psi_{H}(\textbf{r})\frac{1}{\left|\vec{r}_{H}\right|}% \varphi_{\textit{He}}(\textbf{r})d^{3}\textbf{r}.$ (41)

The exchange contribution to the electron-electron interaction is:

$\displaystyle\mathop{\odot\sim\bullet}\limits_{\longleftarrow}{}={\int\!\!\!\!% \int}\varphi_{\textit{He}}(\textbf{r})\varphi_{\textit{He}}(\textbf{r}^{\prime% })\frac{1}{\left|\textbf{r}-\textbf{r}^{\prime}\right|}\varphi_{\textit{He}}(% \textbf{r}^{\prime})\psi_{H}(\textbf{r})d^{3}\textbf{r}d^{3}\textbf{r}^{\prime}.$ (42)

The double-exchange contribution to the electron-electron interaction is the following:

$\displaystyle\overrightarrow{\mathop{\bullet\sim\bullet}\limits_{% \longleftarrow}{}}={\left\langle\varphi_{\textit{He}}(\textbf{r})\psi_{H}(% \textbf{r}^{\prime})\right|}\frac{1}{\left|\textbf{r}-\textbf{r}^{\prime}% \right|}{\left|\psi_{H}(\textbf{r})\varphi_{\textit{He}}(\textbf{r}^{\prime})% \right\rangle}.$ (43)

The initial electron states in the Helium atom are given by [23] where the parameter $\alpha$ is taken from [24, 25]:

$\displaystyle\psi(\textbf{R}-\textbf{r}_{i})=(\alpha^{3}/\pi)^{1/2}\exp(-% \alpha|\textbf{R}-\textbf{r}|),$ (44) $\displaystyle\alpha=27/16.$

The ground state of the electron in the Hydrogen atom is described by the single-electron wave function:

$\displaystyle\psi_{H}(r_{H})=\psi_{nlm}(r,\theta,\varphi)\frac{1}{\sqrt{(2n% \cdot(n-l-1)!\cdot(n+l)!)}}\cdot\left(\frac{2}{na_{0}}\right)^{\frac{2}{3}}% \cdot\exp\left(-\frac{r}{na_{0}}\right)\cdot\left(\frac{2r}{na_{0}}\right)^{l}% L_{n-l-1}^{2l+1}\left(\frac{2r}{na_{0}}\right)\cdot Y_{l,m}(\theta,\varphi)$ (45)

where $a_{0}$ - is a Bohr radius, $L_{n-l-1}^{2l+1}(\frac{2r}{na_{0}})$ – generalized Laguerre polynomial of degree $n-\ell-1$ , $Y_{l,m}(\theta,\varphi)$ is a spherical harmonic function of degree $\ell$ and order $m$ .

To evaluate all above-mentioned integrals we use the elliptical coordinates for each $i$ -th electron (see Ref. [29]

$\displaystyle\mu_{i}=\frac{r_{Hei}+r_{Hi}}{R};$ $\displaystyle\nu_{i}=\frac{r_{Hei}-r_{Hi}}{R};$ (46) $\displaystyle\phi_{i},$

where $\phi_{i}$ is the rotation angle around the axis connecting the two nuclei. The volume element $d\tau$ in these coordinates is

$\displaystyle d\tau=\frac{R^{3}}{8}(\mu^{2}-\nu^{2})d\mu d\nu d\phi,$ (47)

where the integration is between the limits: $1\leqslant\mu\leqslant\infty$ , $-1\leqslant\nu\leqslant 1$ , $0\leqslant\phi\leqslant 2\pi$ .

A Computation of the potential $E(R)$ , given by the Eqs (36) and (39) with respect to expressions Eqs (37) and (38) for the exchange contributions and Eq. (40) for the direct Coulomb contribution gives the following terms. Curves, presented on the Fig. 2a–d do not include the contributions of the atomic energy eigenvalues. $E(R)$ show only potential of the interatomic interaction in the Hartree units $E_{h}$ , whereas an interatomic distance has been taken in Bohr radius $a_{0}$ .

Figure 2.

a. The potential of the interatomic interaction in the Hartree units $E_{h}$ , whereas an interatomic distance is taken in Bohr radius $a_{0}$ for the ground state of He-atom and ground state ( $n=$ 1, $l=$ 0, $m=$ 0), or (100) and excited s-states of Hydrogen atom (200), (300), (400) and $p$ -state (210); b. A comparison of the interatomic interaction potential (in the Hartree units $E_{h}$ , whereas an interatomic distance is taken in Bohr radius $a_{0}$ for the ground state of He-atom and ground state ( $n=$ 1, $l=$ 0, $m=$ 0), or (100) and excited s-states of Hydrogen atom (200), (300), (400); c. The potential of the interatomic for the ground state of He-atom and excited s-states of Hydrogen atom (200), (300), (400) in more detailed resolution; d. A comparison of the interatomic interaction potential for the ground state of He-atom and excited p-states of Hydrogen atom (210), (310), (410) with the s-state (200); e. A comparison of bond potentials He-H for the excited H-states ( $n=$ 3, $l=$ 1, $m=$ 0) (p-state), d-state-(320), p-state (410), d-state-(420); f. Comparison of bond potentials He-H for excited H states ( $n=$ 3, $l=$ 2, $m=$ 0) (d-state), s-state-(400), p-state (410), d-state-(420).

Analysing the interaction potential we may conclude that the most preferable excited states of Hydrogen atom to create the Helium hydride are following states:

2-s-state ( $n=$ 2, $l=$ 0, $m=$ 0) has a bond energy $E_{\textit{bond}}=$ 0.01 Hartree, with a distance R* of the energy minimum R* $=$ 2 (in a Bohr radius)

2-p-state ( $n=$ 2, $l=$ 1, $m=$ 0) has a bond energy $E_{\textit{bond}}=$ 0.007 Hartree with a distance R* of the energy minimum R* $=$ 1.85 (in a Bohr radius)

3-d-state ( $n=$ 3, $l=$ 2, $m=$ 0) has a bond energy $E_{\textit{bond}}=$ 0.009 Hartree with a distance R* of the energy minimum R* $=$ 2.45 (in a Bohr radius)

It is well known, that the ground state of Hydrogen atom is repulsive and does not form the helium hydride state. Our results shown on the Fig. 2a demonstrate such a repulsive potential. It is interesting to note, that the bond state of Helium with a Hydrogen-atom in the excited state (210) (see Fig. 2d) is higher than the same potential well for the (200) Hydrogen state. Such a bond state forms the term ${}^{2}\Pi_{1/2}$ of the Helium hydride and has a high positive energy due to the repulsive interaction of 2 $p_{z}$ -electron belonging to the Hydrogen-atom with s-cloud of Helium-atom electrons. There is very deep 2 $p_{z}$ –1 $s^{2}$ electrons overlapping belonging to the two different centers with respect to spin states that forms an interference picture inside the symmetric electron cloud that that involve both of Hydrogen and Helium centers. Namely, such symmetric electron distribution with respect to its interaction with both Hydrogen and Helium centers produces the potential well, which is higher than in the case of $n$ s–1s ${}^{2}$ interaction. The overlapping of $n$ s–1s ${}^{2}$ states forms the interference electron distribution between the centres whereas the electron overlapping cloud 2 $p_{z}$ –1s ${}^{2}$ envelops the both of nucleus.

Values of the interaction energy $E(R)$ should also include the atomic eigenstates energy:

$\displaystyle E(R)=E_{He}^{0}+E_{H}+U_{\text{int}}(R)=-\left(Z-\frac{5}{16}% \right)^{2}\frac{e^{2}}{a_{0}}-\frac{e^{2}}{2a_{0}n^{2}}+U_{\text{int}}(R)=-% \frac{e^{2}}{a_{0}}\left[\left(Z-\frac{5}{16}\right)^{2}+\frac{1}{2n^{2}}% \right]+U_{\text{int}}(R)$ (48)

Here $Z$ -is a real Helium nuclear charge, $Z=$ 2. So that, the each mentioned term would have the following energy shift:

For the line X ${}^{2}\Sigma^{+}$ ( $n=$ 1, $l=$ 0):

$\displaystyle-\frac{e^{2}}{a_{0}}\left[\left(2-\frac{5}{16}\right)^{2}+\frac{1% }{2}\right]=-3,35\frac{e^{2}}{a_{0}}=-3,35E_{h};$ $\displaystyle A^{2}\Sigma^{+}(n=2,l=0):-\frac{e^{2}}{a_{0}}\left[\left(2-\frac% {5}{16}\right)^{2}+\frac{1}{2\cdot 2^{2}}\right]=-2,97\frac{e^{2}}{a_{0}}=-2,9% 7E_{h};$ $\displaystyle B^{2}\Pi(n=2,l=1):-\frac{e^{2}}{a_{0}}\left[\left(Z-\frac{5}{16}% \right)^{2}+\frac{1}{2n^{2}}\right]=-2,97\frac{e^{2}}{a_{0}}=-2,97E_{h};$ $\displaystyle C^{2}\Sigma^{+}(n=3,l=0):-\frac{e^{2}}{a_{0}}\left[\left(2-\frac% {5}{16}\right)^{2}+\frac{1}{2\cdot 3^{2}}\right]=-2,903\frac{e^{2}}{a_{0}}=-2,% 903E_{h};$ $\displaystyle C^{2}\Pi(n=3,\textit{l}=1):-\frac{e^{2}}{a_{0}}\left[\left(Z-% \frac{5}{16}\right)^{2}+\frac{1}{2n^{2}}\right]=-2,903\frac{e^{2}}{a_{0}}=-2,9% 03E_{h};$ $\displaystyle C^{2}\Delta(n=3,l=2):-\frac{e^{2}}{a_{0}}\left[\left(Z-\frac{5}{% 16}\right)^{2}+\frac{1}{2n^{2}}\right]=-2,903\frac{e^{2}}{a_{0}}=-2,903E_{h}.$

The results are in a good agreement with the experimental dates [7, 8] and with the computations with using Molecular Orbital technics [2, 7, 31, 32].

4. Time-dependent exchange perturbation theory

When a time-dependent perturbation added to a multicentre system, we search a solution of the perturbed equation for antisymmetrized states:

$\displaystyle i\hbar\frac{\partial}{\partial t}{\left|\Psi\right\rangle}=\left% (\hat{H}_{0}+\hat{V}\left(t\right)\right){\left|\Psi\right\rangle},$ (49)

As in the standard perturbation theory, we seek the solution ${\left|\Psi\left(t\right)\right\rangle}$ in form of sum:

$\displaystyle{\left|\Psi\left(t\right)\right\rangle}=\sum_{n}C_{n}\left(t% \right)\exp\left(-\frac{i}{\hbar}E_{n}^{0}t\right){\left|\Psi_{n}^{0}\right% \rangle},$ (50)

where the expansion coefficients $C_{n}\left(t\right)$ are functions of time, $E_{n}^{0}$ – are eigenvalue of energy, determined by Eq. (8). Retaining successively terms of nth-degree order of as smallest and acting by the projector $\hat{O}_{i}=\hat{1}-\hat{P}_{i}$ on both sides of the following equation determined the nth-correction to the wave vector we obtain the expressions for the expansion coefficients se in [18, 22, 23]. Continuing this process, we derive the following expression for an $n$ -th correction to the wave vector [23]:

$\displaystyle{\left|\Psi^{(n)}\right\rangle}=\sum_{n_{n}}{}^{{}^{\prime}}\sum_% {n_{n-1}}{}^{{}^{\prime}}\ldots\sum_{n_{1}}\left\{\exp\left(-\frac{i}{\hbar}E_% {n}^{0}t\right){\left|\Psi_{n}^{0}\right\rangle}\times\right.$ $\displaystyle\quad\times\frac{f_{n_{1}}^{P}\cdot f_{n_{2}}^{P}\ldots f_{n}^{P}% }{\left(i\hbar P\right)^{n}}\cdot\int_{0}^{t}dt_{n}\exp\left(i\omega_{nn_{n-1}% }t_{n}/\hbar\right)\left.\left(\Phi_{n}^{p=0}\right.\right|\hat{V}(t_{n}){% \left|\Psi_{n_{n-1}}^{0}\right\rangle}$ $\displaystyle\quad\int_{0}^{t_{n}}\exp\left(i\omega_{n_{n-1}n_{n-2}}t_{n-1}% \right)\left(\left.\Phi_{n_{n-1}}^{p=0}\right|\hat{V}(t_{n-1}){\left|\Psi_{n_{% n-2}}^{0}\right\rangle}\right.dt_{2}\ldots$ (51) $\displaystyle\left.\int_{0}^{t_{2}}\exp\left(i\omega_{n_{1}i}t_{1}\right)\left% (\left.\Phi_{n_{1}}^{p=0}\right|\hat{V}(t_{1}){\left|\Psi_{i}^{0}\right\rangle% }\right.dt_{1}\right\};$

The principle differences to compare the standard perturbation theory are: first, in the matrix elements, the ket-vectors are the vectors antisymmetrized with respect to the inter-center electron permutations , whereas the bra-vectors are the simple product of atomic wave functions and corresponds to the zeroth permutation $p=$ 0 (see Eqs (1) and (4)). Second, each correction to the initial antisymmetric wave vector has a properly symmetric form. Each matrix elements contributed here contain both direct and exchange terms. The perturbation operator has the symmetric form Eq. (9). We should also indicate the order of time instants in the matrix elements: $0<t_{1}<t_{2}<\ldots t_{n}<t$ . Using the Dyson [33, 34]chronological operator $\hat{\tau}$ , and putting the additional factor $\frac{1}{n!}$ [34], we have for the $n$ -th correction to the antisymmetric ket-vector:

$\displaystyle{\left|\Psi^{n}\right\rangle}=\frac{1}{n!}\sum_{n_{n}}{}^{{}^{% \prime}}\sum_{n_{n-1}}{}^{{}^{\prime}}\ldots\sum_{n_{1}}\left\{\exp\left(-% \frac{i}{\hbar}E_{n}^{0}t\right){\left|\Psi_{n}^{0}\right\rangle}\right.\times% \cdot\hat{\tau}\int_{0}^{t}\ldots\int_{0}^{t}dt_{1}\ldots dt_{n}\left(\exp% \left(i\omega_{nn_{n-1}}t_{n}/\hbar\right)\right.\left(\Phi_{n}^{p=0}|\hat{V}(% t_{n})|\Psi_{n_{n-1}}^{0}\right\rangle\exp\left(i\omega_{n_{n-1}n_{n-2}}t_{n-1% }\right)\left(\Phi_{n_{n-1}}^{p=0}|\hat{V}(t_{n-1})|\Psi_{n_{n-2}}^{0}\right% \rangle\ldots\left.\exp\left(i\omega_{n_{1}i}t_{1}\right)\left(\left.\Phi_{n_{% 1}}^{p=0}\right|\hat{V}(t_{1}){\left|\Psi_{i}^{0}\right\rangle}\right.\right)% \left.\frac{f_{n_{1}}^{P}\cdot f_{n_{2}}^{P}\ldots f_{n}^{P}}{\left(i\hbar P% \right)^{n}}\right\}.$ (52)

Now we reduce the obtained expressions of the antisymmetric wave vector corrections to the symmetry-adapted form that involves an ordinary non-symmetrized perturbation operator corresponding to the original zeroth electrons permutation. Such a form is more simple and convenient for practical use. To do this, we transform all matrix elements in the following way:

$\displaystyle\left(\Phi^{0}_{n}\left|\hat{V}(t)\right|\Psi_{k}^{0}\right% \rangle=\left(\Phi^{0}{}_{n}\left|\sum_{p=0}^{P}V_{p}(t)\Lambda^{p}\right|\Psi% _{k}^{0}\right\rangle=\left(\Phi^{0}_{n}\left|\sum_{p=0}^{P}\frac{\left(-1% \right)^{g_{p}}}{f_{k}^{p}}V_{p}(t)\right|\Phi_{k}^{p}\right)=\sum_{p=0}^{P}% \frac{\left(-1\right)^{g_{p}}}{f_{k}^{p}}\left(\Phi^{p}_{n}|V_{p=0}(t)|\Phi_{k% }^{p=0}\right)=\frac{f_{n}^{p}}{f_{k}^{p}}\left\langle\Psi_{n}^{0}|V_{p=0}(t)|% \Phi_{k}^{p=0}\right).$ (53)

Then, we obtain the $n$ -th correction to non-symmetric ket-vector and for the antisymmetric bra-vector in the symmetry-adapted form

$\displaystyle\left|\Phi^{n,p=0}\right)=\frac{1}{n!}\sum_{n_{n}}{}^{{}^{\prime}% }\sum_{n_{n-1}}^{{}^{\prime}}\ldots\sum_{n_{1}}\left\{\exp\left(-\frac{i}{% \hbar}E_{n}^{0}t\right)\left.\left|\Phi^{p=0}{}_{n}\right.\right)\right.\times% \cdot\hat{\tau}\int_{0}^{t}\ldots\int_{0}^{t}dt_{1}\ldots dt_{n}\left(\exp% \left(i\omega_{nn_{n-1}}t_{n}/\hbar\right)\right.{\left\langle\Psi_{n_{n}}^{0}% \right|}\left.V_{p=0}(t_{n})\right|\left.\Phi^{p=0}{}_{n_{n-1}}\right)\exp% \left(i\omega_{n_{n-1}n_{n-2}}t_{n-1}\right){\left\langle\Psi_{n_{n-1}}^{0}% \right|}\left.V_{p=0}(t_{n-1})\right|\left.\Phi^{p=0}{}_{n_{n-2}}\right)\ldots% \left..\left.\exp\left(i\omega_{n_{1}i}t_{1}\right){\left\langle\Psi_{n_{1}}^{% 0}\right|}\left.V_{p=0}(t_{1})\right|\left.\Phi_{i}^{p=0}\right)\right)\frac{f% _{n_{1}}^{P}\cdot f_{n_{2}}^{P}\ldots f_{n}^{P}}{\left(i\hbar P\right)^{n}}% \right\};$ (54)

$\displaystyle{\left\langle\Psi_{n}^{n}\right|}=\frac{1}{n!}\sum_{n_{n}}{}^{{}^% {\prime}}\sum_{n_{n-1}}{}^{{}^{\prime}}\ldots\sum_{n_{1}}\left\{\frac{f_{n_{1}% }^{P}\cdot f_{n_{2}}^{P}\ldots f_{n}^{P}}{\left(i\hbar P\right)^{n}}\right.% \times\cdot\hat{\tau}^{{\dagger}}\int_{0}^{t}\ldots\int_{0}^{t}dt_{1}\ldots dt% _{n}\left({\left\langle\Psi_{i}^{0}\right|}\left.V_{p=0}(t_{1})\right|\left.% \Phi^{p=0}{}_{n_{n-1}}\right)\exp\left(i(\omega_{in_{n-1}}t_{1}\right)\right.{% \left\langle\Psi_{n_{n-1}}^{0}\right|}\left.V_{p=0}(t_{2})\right|\left.\Phi^{p% =0}{}_{n_{n-2}}\right)\exp\left(i\omega_{n_{n-1}n_{n-2}}t_{2}\right)\ldots% \left.\left.{\left\langle\Psi_{n_{1}}^{0}\right|}\left.V_{p=0}(t_{n})\right|% \left.\Phi_{n_{n}}^{p=0}\right)\exp\left(i\omega_{n_{1}n_{n}}t_{n}\right)\exp% \left(i\omega_{n_{1}i}t_{1}\right)\right){\left\langle\Psi_{n_{n}}^{0}\right|}% \exp\left(\frac{i}{\hbar}E_{n_{n}}t\right)\right\};$

The expressions Eq. (52) and (54) define the corrections to the wave vector of a multi-centred system with accounting for the exchange effects to any order of the perturbation theory. The general expression for the expansion coefficients that determine a perturbation induced transitions from the initial state $i$ to the final states $f$ has an invariant form:

$\displaystyle C_{fi}(t)=\left\langle\Psi_{f}^{0}\left|\hat{\tau}\exp\left(-% \frac{i\hat{f}}{\hbar P}\int_{0}^{t}\hat{W}(t^{\prime})dt^{\prime}\right)% \right|\Phi_{i}^{p=0}\right),$ (55)

where we introduce the perturbation operator $\hat{W}(t)$ in the interaction representation and a normalization operator $\hat{f}$ as

$\displaystyle\hat{W}(t)=e^{\frac{i}{\hbar}H_{0}^{p=0}t}V^{p=0}(t)e^{-\frac{i}{% \hbar}H_{0}^{p=0}t}\ \text{and}\ \hat{f}{\left|\Psi_{n}^{0}\right\rangle}=f_{n% }^{P}{\left|\Psi_{n}^{0}\right\rangle}.$ (56)

An expression for the probability of the transition from the initial state $i$ into a final state $f$ during the acting period of perturbation has a form:

$\displaystyle w_{\textit{fi}}(t)=\left|C_{\textit{fi}}(t)\right|^{2}=\left|% \left\langle\Psi_{f}^{0}\left|\hat{\tau}\exp\left(-\frac{i\hat{f}}{\hbar P}% \int_{0}^{t}\hat{W}(t^{\prime})dt^{\prime}\right)\right|\Phi_{i}^{p=0}\right)% \right|^{2}$ (57)

It is assumed that both states belong to the discrete spectrum.

If the final state belongs to the continuous spectrum of energy, this probability has a following form:

$\displaystyle\textit{dw}_{\textit{fi}}(t)=\left|C_{\textit{fi}}(t)\right|^{2}d% \nu_{f}=\left|\left\langle\Psi_{f}^{0}\left|\hat{\tau}\exp\left(-\frac{i\hat{f% }}{\hbar P}\int_{0}^{t}\hat{W}(t^{\prime})dt^{\prime}\right)\right|\Phi_{i}^{p% =0}\right)\right|^{2}d\rho(E_{f}),$ (58)

where $d\rho(E_{f})$ is the density of states in continuous spectrum. We denoted the number of final states per unit interval of energy by $\rho(E_{f})$ . It is important to underline, that unlike common used theory [33, 34], in Eqs (55)–(58) we use bra-vector in the antisymmetric form regards to electron permutations between the incident heavy particles, that gives rise for accounting all exchange contributions in above mentioned expressions.

In the case where we choose the initial – final time interval in the expression Eq. (55) is expanded to ] – $\infty$ , $+\infty$ [, the matrix elements $C_{fi}(\infty)$ denoted by ${\left\langle\Psi_{f}^{0}\right|}\hat{S}\left|\left.\Phi_{i}^{p=0}\right)\right.$ are called the matrix elements of the S matrix [34]:

$\displaystyle C_{fi}(\infty)=\left\langle\Psi_{f}^{0}|S|\Phi_{i}^{0(0)}\right)% =\left\langle\Psi_{f}^{0}\left|\hat{\tau}\exp\left(-\frac{i\hat{f}}{\hbar P}% \int_{-\infty}^{\infty}\hat{W}(t)dt\right)\right|\Phi_{i}^{p=0}\right)\equiv% \equiv{\left\langle\Psi_{f}^{0}\right|}\left\{\!1+\frac{\hat{f}}{i\hbar P}\!% \int_{-\infty}^{\infty}\hat{W}(t)dt)+\!\left(\frac{\hat{f}}{i\hbar P}\right)^{% 2}\int_{-\infty}^{\infty}dt_{1}\int_{-\infty}^{\infty}dt_{2}\hat{W}(t_{1})\hat% {W}(t_{2}))+\ldots\right\}\left|\Phi_{i}^{p=0}\right)=\sum_{\alpha}^{\infty}% \left\langle\Psi_{f}^{0}\right|\hat{S}^{(\alpha)}\left|\Phi_{i}^{p=0}\right).$ (59)

Here is as above bra-vector ${\left\langle\Psi_{f}^{0}\right|}$ has antisymmetric form and ket-vector $\left|\Phi_{i}^{p=0}\right)$ corresponds to the 0-th permutation of the initial state. Due to the interaction between two atomic centers is changing slower in time than the interaction between electrons that belong to each centers, adiabatic approximation is appropriate for description of such a system. In this case, the interaction between atoms is a time-independent perturbation. The matrix elements of the S-matrix are written in a common way:

$\displaystyle{\left\langle\Psi_{f}^{0}\right|}\hat{S}\left|\left.\Phi_{i}^{p=0% }\right)\right.=-2\pi i\delta(E_{f}-E_{i}){\left\langle\Psi_{f}^{0}\right|}% \hat{T}\left|\left.\Phi_{i}^{p=0}\right)\right..$ (60)

Here the introduced operator $\hat{T}$ is the operator of transition on the energy surface and its matrix element ${\left\langle\Psi_{f}^{0}\right|}\hat{T}\left|\left.\Phi_{i}^{p=0}\right)\right.$ the transition matrix element on the energy surface. To compare to the commonly used matrix elements ${\left\langle\psi_{f}\right|}\hat{T}{\left|\psi_{i}\right\rangle}$ , developed for the orthogonal basis, the matrix elements obtained here, account for all possible electron permutations between two subsystems. The operator equation developed in Refs [18, 22, 23] for the general case of non-orthogonal basis of antisymmetric wave vectors has the form:

$\displaystyle\hat{T}=V_{0}^{{\mathbb{N}}}+V_{0}^{{\mathbb{N}}}\left(E_{i}-H_{p% =0}^{0}+i\eta\right)^{-1}\left(\frac{\hat{f}}{P}\right)^{-1}\hat{T}$ (61)

Using the method of successive approximations, the solution got from this equation is

$\displaystyle\hat{T}=V_{p=0}^{{\mathbb{N}}}+V_{p=0}^{{\mathbb{N}}}\left(E_{i}-% H_{p=0}^{0}+i\eta\right)^{-1}\left(\frac{\hat{f}}{P}\right)^{-1}V_{p=0}^{{% \mathbb{N}}}+V_{p=0}^{{\mathbb{N}}}\left(E_{i}-H_{p=0}^{0}+i\eta\right)^{-1}% \left(\frac{\hat{f}}{P}\right)^{-1}V_{p=0}^{{\mathbb{N}}}\left(E_{i}-H_{p=0}^{% 0}+i\eta\right)^{-1}\left(\frac{\hat{f}}{P}\right)^{-1}V_{p=0}^{{\mathbb{N}}}+\ldots$ (62)

Here $V_{p=0}^{{\mathbb{N}}}=\left(\frac{\hat{f}}{P}\right)V_{p=0}$ is a renormalized perturbation operator. Operator $\hat{T}$ satisfied Eq. (61) may be rearranged to (see for details Ref. [18, 23]):

$\displaystyle\hat{T}=V_{p=0}^{{\mathbb{N}}}+V_{p=0}^{{\mathbb{N}}}\left(E_{i}-% H+i\eta\right)^{-1}\left(\frac{\hat{f}}{P}\right)^{-1}V_{p=0}^{{\mathbb{N}}}.$ (63)

Here, the total Hamiltonian $H=(\hat{H}^{0}+\hat{V}\left(t\right))=H_{p=0}^{0}+V_{p=0}=H_{p}^{0}+V_{p}$ is always invariant respectively to the antisymmetrisation operator $[H,\hat{A}]=0$ .

A transition probability per unit time has a form:

$\displaystyle w_{\textit{fi}}=\frac{\left|{\left\langle\Psi_{f}^{0}\right|}% \hat{S}\left|\left.\Phi_{i}^{p=0}\right)\right.\right|^{2}}{\lim_{\tau\to% \infty}\int_{-\tau}^{\tau}dt}=\frac{2\pi}{\hbar}\delta(E_{f}-E_{i})\left|{% \left\langle\Psi_{f}^{0}\right|}\hat{T}\left|\left.\Phi_{i}^{p=0}\right)\right% .\right|^{2}.$ (64)

By dividing the result of Eq. (64) by the flux density of incident particles ( $j_{i}=\hbar k_{i}/\mu_{i}$ , where $k_{i}$ is the wavevector of the relative movement of the incident particles and $\mu_{i}$ is their reduced mass), we obtain the expression of the cross section of scattering events and reactions, accounted the intercentre permutations

$\displaystyle\sigma_{fi}=\frac{2\pi\mu_{i}}{\hbar^{2}k_{i}}\delta(E_{f}-E_{i})% \left|{\left\langle\Psi_{f}^{0}\right|}\hat{T}\left|\left.\Phi_{i}^{p=0}\right% )\right.\right|^{2}.$ (65)

If the final state is within the continuous spectrum, by common way we multiply the expression for the cross section given by Eq. (65) by the number of final states in the volume per unit energy interval $\rho(E_{f})$ . Then for the differential cross section, we have following expression:

$\displaystyle\frac{{\rm d}\sigma_{fi}}{{\rm d}\Omega}=j_{i}^{-1}\frac{{\rm d}w% _{fi}}{{\rm d}\Omega}=\frac{\mu_{i}\mu_{f}k_{f}}{\left(2\pi\hbar^{2}\right)^{2% }k_{i}}\left|{\left\langle\Psi_{f}^{0}\right|}\hat{T}\left|\left.\Phi_{i}^{p=0% }\right)\right.\right|^{2}.$ (66)

Such form is convenient for practical applications. It is important to underline, that these successively obtained corrections to the energy and to the state vector are simplified post factum, i.e., after the corrections are obtained by a regular method (see in details Ref. [18]). The Eqs (64)–(66) preserve all multicentre exchange and superexchange contributions. The developed expressions explicitly contain both direct and exchange integrals. All possible configurations of super exchange integrals contributions automatically presented. The obtained expressions for the energy and for the wave vector in symmetry-adapted form use the ordinary non-symmetric perturbation operator; this fact simplifies the computation procedure.

5. The electron-Helium collision with electron capture

We apply IEPT formalism for describing a process of non-radiative electron capture by excited Helium atom $(1s2s)^{3}S$ with a negative Helium ion formation in the state $(1s2s2p)^{4}P$ :

$\displaystyle e+\textit{He}\left({}^{3}S\right)\to\textit{He}^{-}\left({}^{4}P% \right).$ (67)

We consider this problem as a case of complex particle collision in such a way that the composition of particles changes in the collision [34]. We shall call such a collision reaction. The life-time of the state is enough long to be observed in the experiments and to use it in the tandem accelerators [28, 29]. For our calculations we use the first approximation of IEPT formalism. All computations are performing in the Hartree atomic units, where $e=m_{e}=\hbar=1$ .

An initial state correspondent to the free electron motion, considered in the center-mass system and a Helium atom. The center-mass system is directly binding with the position of Helium nucleus. Then we take as the Hamiltonian unperturbed system in the form

$\displaystyle H^{0}(1,2,3)=-\frac{\hbar^{2}}{2m}(\nabla_{1}^{2}+\nabla_{2}^{2}% +\nabla_{3}^{2})-\frac{2e^{2}}{r_{1}}-\frac{2e^{2}}{r_{2}}+\frac{e^{2}}{r_{12}},$ (68)

where we denote $r_{i}=\left|R-r_{i}\right|$ the distance between electron, numbered $i$ ( $i=$ 1, or 2) and its own Helium nucleus; $r_{12}=|r_{1}-r_{2}|$ – a distance between electrons, numbered 1 and 2, respectively, that belong to the same centre. The eigenvalue $E_{i}$ , of the Hamiltonian Eq. (3) is a total energy of initial system:

$\displaystyle E_{i}=-\frac{\hbar^{2}}{2m}k_{i}^{2}+\varepsilon_{n_{i}},$ (69)

Where the first term corresponds to the kinetic energy of incident electron, and $\varepsilon_{n_{i}}$ – is the internal energy of the excited Helium atom in the state ${}^{3}S_{\textit{0He}}$ (here is $n_{i}={}^{3}S_{\textit{0He}}$ ). The eigenvector for this Hamiltonian is the initial excited state of Helium atom, taken in the form of ${}^{3}S_{\textit{0He}}$ -state $(1s2s)$ . The wave function of the noninteractive system He–e we take the form of the simple product of the atomic wave function $\Psi_{\textit{He}}(\textbf{R}-\textbf{r}_{1},\textbf{R}-\textbf{r}_{2};\xi_{1}% ,\xi_{2})$ and of the free electron wave function $\psi_{e}(\textbf{r}_{3})\chi(;\xi_{3})$ dependent on their spatial and spin parts

$\displaystyle\Phi(\textbf{r}_{1},\textbf{r}_{2},\textbf{r}_{3};\xi_{1},\xi_{2}% ;\xi_{3})=\Psi_{\textit{He}}(\textbf{R}-\textbf{r}_{1},\textbf{R}-\textbf{r}_{% 2};\xi_{1},\xi_{2})\psi_{e}(\textbf{r}_{3})\chi_{e}(;\xi_{3}).$ (70)

Here R is a vector originating at the arbitrary center and pointing to the Helium nucleus; $\textbf{r}_{i}$ – is the radius vector of the electron, numbered $i$ , originating at the Hydrogen nucleus; $\xi_{i}$ – is the spin variable of the electron i. The atomic function $\Psi_{\textit{He}}(\textbf{R}-\textbf{r}_{1},\textbf{R}-\textbf{r}_{2};\xi_{1}% ,\xi_{2})$ of Helium atom antisymmetrized with respect to the internal electron permutations has the form:

$\displaystyle\Psi_{\textit{He}}(\textbf{R}-\textbf{r}_{1},\textbf{R}-\textbf{r% }_{2};\xi_{1},\xi_{2})$ (71) $\displaystyle\quad=\frac{1}{\sqrt{2}}\left(\psi_{n1}(\textbf{R}-\textbf{r}_{1}% )\psi_{n2}(\textbf{R}-\textbf{r}_{2})\pm\psi_{n2}(\textbf{R}-\textbf{r}_{1})% \psi_{n1}(\textbf{R}-\textbf{r}_{2})\right)\cdot\frac{1}{\sqrt{2}}\left(\alpha 1% \beta 2\mp\beta 1\alpha 2\right),$

where $\alpha$ and $\beta$ are the one-electron spin functions, corresponding to the “up” and “down” spin states, respectively. Because of chosen center-mass system, the originating point to the Helium nucleus, in this coordinate system $\textbf{R}=0$ . We consider as initial the excited state of Helium atom $(1s2s)^{3}S$ , then the spin part of the He wave function ${\rm X}_{\textit{He}}(\xi_{1}\xi_{2})$ may be chosen in the form ${\rm X}_{\textit{He}}(\xi_{1}\xi_{2})=\alpha_{1}\alpha_{2}$ , correspondent to the total spin $S=$ 1. In this case, the spatial part of the total wave function $\Psi_{\textit{He}}(\textbf{R}-\textbf{r}_{1},\textbf{R}-\textbf{r}_{2};\xi_{1}% ,\xi_{2})\equiv\Psi_{\textit{He}}(\textbf{r}_{1},\textbf{r}_{2};\xi_{1},\xi_{2})$ (with accounting $n1=$ 1 $s$ and $n2=$ 2 $s$ ) will have a form:

$\displaystyle\Phi_{\textit{He}}(\textbf{r}_{1},\textbf{r}_{2})=\frac{1}{\sqrt{% 2!}}\left|\begin{array}[]{cc}{\varphi_{1s}\left(\textbf{r}_{1}\right)}&{% \varphi_{1s}\left(\textbf{r}_{2}\right)}\\ {\varphi_{2s}\left(\textbf{r}_{1}\right)}&{\varphi_{2s}\left(\textbf{r}_{2}% \right)}\end{array}\right|=\frac{1}{\sqrt{2}}\left[\varphi_{1s}\left(\textbf{r% }_{1}\right)\varphi_{2s}\left(\textbf{r}_{2}\right)-\varphi_{1s}\left(\textbf{% r}_{2}\right)\varphi_{2s}\left(\textbf{r}_{1}\right)\right]$ (72)

We take a spin function of the free electron as $\chi_{e}(\xi_{3})=\alpha_{3}$ . Then the vector of the initial state, corresponding to the permutation $p=$ 0, but antisymmetrized only with accounting the internal permutations in Helium- atom has a form:

$\displaystyle\left|\left.\Phi_{i}^{p=0}\right)=\Phi_{\textit{He}}(\textbf{r}_{% 1},\textbf{r}_{2}){\rm X}_{\textit{He}}(\xi_{1}\xi_{2})\psi_{e}\left(\textbf{r% }_{3}\right)\chi_{e}\left(\xi_{3}\right)=\right.\left|\left.\Psi_{\textit{He}}% (1,2),\psi_{e}\left(3\right)\right)\right.,$ (73)

Taking into account the operator Eq. (63), instead of solution in the form Eq. (5) we choose the final state wave function as an eigenfunction of the total Hamiltonian $\hat{H}=H^{0}(1,2,3)+V_{p=0}-\frac{\hbar^{2}}{2\mu_{\textit{He}}}\Delta_{R}$ in the adiabatic case with accounting the first correction to the wave function. The perturbation operator in the form correspondent to the permutation $p=$ 0 is

$\displaystyle V_{p=0}=\frac{e^{2}}{r_{13}}+\frac{e^{2}}{r_{23}}-\frac{2e^{2}}{% r_{3}},$ (74)

where, as above, we denote $r_{3}=\left|\textbf{r}_{3}\right|$ the distance between electron, numbered 3, and a Helium nucleus. $r_{13}=\left|\textbf{r}_{1}-\textbf{r}_{3}\right|$ – is a distance between electrons, numbered 1 and 3, respectively, and by the same way $r_{23}=|\textbf{r}_{2}-\textbf{r}_{3}|$ . The term $-\frac{\hbar^{2}}{2\mu_{\textit{He}^{1-}}}\Delta_{R}$ means the operator of kinetic energy of the negative He- ion , where $\mu_{\textit{He}^{1-}}$ is its mass. A basic vector of final state correspondent to the permutation $p=$ 0 has a form:

$\displaystyle\left|\left.\Phi_{f}^{p=0}\right)=\right.\left|\left.\Psi_{% \textit{He}}(1,2),\phi_{e}\left(3\right)e^{iK\cdot R}\right)\right.=\Phi_{% \textit{He}}(\textbf{r}_{1},\textbf{r}_{2})\phi_{2p}\left(\textbf{r}_{3}\right% )e^{iK\cdot R}\alpha_{1}\alpha_{2}\alpha_{3},$ (75)

where $\phi_{e}\left(3\right)$ is a function of the third electron, correspondent to the binding state and $e^{iK\cdot R}$ – is a plane wave describing the $\textit{He}^{1-}$ motion.

The antisymmetric form of the final state vector is:

$\displaystyle{\left|\Psi_{f}^{0}\right\rangle}=\hat{A}\left|\left.\Psi_{He}(1,% 2),\phi_{e}\left(3\right)\right)\right.e^{iK\cdot R}=\frac{1}{f_{0}}\Psi_{0}(% \textbf{r}_{1},\textbf{r}_{2},\textbf{r}_{3})e^{iK\cdot R}{\rm X}_{0}(\xi_{1}% \xi_{2}\xi_{3}).$ (76)

Here spin part ${\rm X}_{0}(\xi_{1}\xi_{2}\xi_{3})=\alpha_{1}\alpha_{2}\alpha_{3}$ is symmetric and corresponds to the total spin S=3/2. Then, the spatial part should be antisymmetric. Then, we have the expression for the spatial part $\Psi_{0}(r_{1},r_{2},r_{3})$ :

$\displaystyle\Psi_{0}=\frac{1}{\sqrt{2}}\left\{\frac{1}{\sqrt{3!}}\left|\begin% {array}[]{ccc}{\varphi_{1s}(\textbf{r}_{1})}&{\varphi_{1s}(\textbf{r}_{2})}&{% \varphi_{1s}(\textbf{r}_{3})}\\ {\varphi_{2s}(\textbf{r}_{1})}&{\varphi_{2s}(\textbf{r}_{2})}&{\varphi_{2s}(% \textbf{r}_{3})}\\ {\phi_{2p}(\textbf{r}_{1})}&{\phi_{2p}(\textbf{r}_{2})}&{\phi_{2p}(\textbf{r}_% {3})}\end{array}\right|\right.--\left.\frac{1}{\sqrt{3!}}\left|\begin{array}[]% {ccc}{\varphi_{2s}(\textbf{r}_{1})}&{\varphi_{2s}(\textbf{r}_{2})}&{\varphi_{2% s}(\textbf{r}_{3})}\\ {\varphi_{1s}(\textbf{r}_{1})}&{\varphi_{1s}(\textbf{r}_{2})}&{\varphi_{1s}(% \textbf{r}_{3})}\\ {\phi_{2p}(\textbf{r}_{1})}&{\phi_{2p}(\textbf{r}_{2})}&{\phi_{2p}(\textbf{r}_% {3})}\end{array}\right|\right\}=\frac{\sqrt{2}}{\sqrt{3!}}\left|\begin{array}[% ]{ccc}{\varphi_{1s}(\textbf{r}_{1})}&{\varphi_{1s}(\textbf{r}_{2})}&{\varphi_{% 1s}(\textbf{r}_{3})}\\ {\varphi_{2s}(\textbf{r}_{1})}&{\varphi_{2s}(\textbf{r}_{2})}&{\varphi_{2s}(% \textbf{r}_{3})}\\ {\phi_{2p}(\textbf{r}_{1})}&{\phi_{2p}(\textbf{r}_{2})}&{\phi_{2p}(\textbf{r}_% {3})}\end{array}\right|=\frac{1}{\sqrt{3}}[\varphi_{1s}\left(\textbf{r}_{1}% \right)\varphi_{2s}\left(\textbf{r}_{2}\right)\phi_{2p}\left(\textbf{r}_{3}% \right)+\varphi_{1s}\left(\textbf{r}_{3}\right)\varphi_{2s}\left(\textbf{r}_{1% }\right)\phi_{2p}\left(\textbf{r}_{2}\right)++\varphi_{1s}\left(\textbf{r}_{2}% \right)\varphi_{2s}\left(\textbf{r}_{3}\right)\phi_{2p}\left(\textbf{r}_{1}% \right)-\varphi_{1s}\left(\textbf{r}_{3}\right)\varphi_{2s}\left(\textbf{r}_{2% }\right)\phi_{2p}\left(\textbf{r}_{1}\right)-\varphi_{1s}\left(\textbf{r}_{1}% \right)\varphi_{2s}\left(\textbf{r}_{3}\right)\phi_{2p}\left(\textbf{r}_{2}% \right)--\varphi_{1s}\left(\textbf{r}_{2}\right)\varphi_{2s}\left(\textbf{r}_{% 1}\right)\phi_{2p}\left(\textbf{r}_{3}\right)]$ (77)

We determine normalization factor $f_{0}$ by following condition:

$\displaystyle\left\langle\Psi_{f}^{0}(1,2,3)|\left.\Phi_{f}^{p=0}(1,2,3\right)% \right.=1.$ (78)

The eigenenergy of the total Hamiltonian $\hat{H}=H^{0}(1,2,3)+V_{p=0}-\frac{\hbar^{2}}{2\mu_{He}}\Delta_{R}$ , where the antisymmetric vector Eq. (31) is eigenvector of the total Hamiltonian $\hat{H}{\left|\Psi_{f}^{0}\right\rangle}=E_{f}{\left|\Psi_{f}^{0}\right\rangle}$ , have following expressions:

$\displaystyle E_{f}=\frac{\hbar^{2}}{2\mu_{f}}K_{f}^{2}+\varepsilon_{n_{f}},$ (79)

Because of the energy conservation low, ( $E_{i}=E_{f}$ ), the energy of the relative motion determined by

$\displaystyle\frac{\hbar^{2}}{2\mu_{f}}K_{f}^{2}=\frac{\hbar^{2}}{2\mu_{i}}k_{% i}^{2}+\varepsilon_{\textit{He}}-\varepsilon_{\textit{He}^{1-}}.$

$\displaystyle K_{f}^{2}=\frac{2M_{\textit{He}^{1-}}}{\hbar^{2}}\left(\frac{% \hbar^{2}}{2m_{e}}k_{i}^{2}+\varepsilon_{\textit{He}}-\varepsilon_{\textit{He}% ^{1-}}\right)>0$ (80)

For the interaction of electron scattering by Helium atom, we have the perturbation operator $V_{p=0}$ in the form Eq. (74). Experimentally, one observes the flux of one of the reaction products corresponding to the transition into one bond state. The flux of negatives ions into unit solid angle in the direction ${\bm{n}}$ can be e[pressed in terms of the scattering amplitude and is equal to $\left(\frac{\hbar K}{M_{\textit{He}^{1-}}}\right)\left|{\left\langle\Psi_{f}^{% 0}\right|}\frac{f_{0}^{2}}{3!}V_{p=0}\left|\left.\Phi_{i}^{p=0}\right)\right.% \right|^{2}$ . Dividing this flux by the flux density of incident particles $\left(\frac{\hbar k_{i}}{m_{e}}\right)$ , we get a cross-section for this reaction, using the general form Eqs (65) and (66) (see [34]):

$\displaystyle\frac{d\sigma_{\textit{if}}}{d\Omega}=\frac{0,4m_{e}\mu_{\textit{% He}^{1-}}K_{f}}{\left(2\pi\hbar^{2}\right)^{2}k_{i}}\left|\left\langle\Psi_{f}% ^{0}\left|\frac{f_{0}^{2}}{3!}V_{p=0}\right|\Phi_{i}^{p=0}\right)\right|^{2}.$ (81)

By integrating the expression Eq. (81) in the solid angle, we obtain the total cross section of the open channel scattering process as follows:

$\displaystyle\sigma_{\textit{if}}=2\pi\int_{0}^{\pi}\frac{d\sigma_{\textit{if}% }}{d\Omega}\sin\theta d\theta.$ (82)

During our computations, we use the program Wolfram Mathematica 10.0.

For constructing zeroth approximation wave functions, we use one-electron Slater atomic orbitals [33, 37, 39]:

$\displaystyle\varphi_{1s}\left(\textbf{r}\right)=\frac{2^{1,5}}{\sqrt{\pi}}e^{% -2r},$ $\displaystyle\varphi_{2s}\left(\textbf{r}\right)=\frac{1,15^{2,5}}{4\sqrt{2\pi% }}\textit{re}^{-0,575r},$ (83) $\displaystyle\phi_{2p}\left(\textbf{r}\right)=\frac{0,8^{2,5}}{4\sqrt{2\pi}}% \textit{re}^{-0,4r}\cos\theta$

For a free motion electron state, we use the function $\psi\left(\textbf{r}\right)=\frac{1}{\left(2\pi\right)^{{\textstyle\frac{3}{2}% }}}e^{ikr\cos\theta}$ .

Now we show the way that determines the hydrogen-like wave function $\phi_{2p}\left(\textbf{r}_{3}\right)=\gamma_{1}r_{3}e^{-\gamma r_{3}}\cos(\theta)$ for the third electron, centered on the Helium-atom in the form with determined parameters $\phi_{2p}\left(\textbf{r}\right)=\frac{0,8^{2,5}}{4\sqrt{2\pi}}re^{-0,4r}\cos\theta$ . For that, we solve the stationary Schrödinger equation using the EPT formalism to determine the first energy correction and the binding energy of the third electron deposited on a Helium atom to form a negative helium ion. Doing it, we use Eqs (14) and (16) to determine corrections to the energy and to the wave function, respectively. We consider the total Hamiltonian, describing the stationary three-electron system interacting with the alpha-particle as

$\displaystyle H_{\textit{stat}}(1,2,3)=\left[-\frac{\hbar^{2}}{2m}(\nabla_{1}^% {2}+\nabla_{2}^{2}+\nabla_{3}^{2})-\frac{2e^{2}}{r_{1}}-\frac{2e^{2}}{r_{2}}-% \frac{2e^{2}}{r_{3}}\right]+\frac{e^{2}}{r_{12}}+\frac{e^{2}}{r_{13}}+\frac{e^% {2}}{r_{23}},$

where the distances $r_{i}$ ( $i=$ 1, 2, 3), of electrons are accounted from the nucleus (alpha-particle). Then we divide the total Hamiltonian on two parts, a Hamiltonian of the non-perturbed system in the form, accounting an effective screened charge of Helium nucleus attracting the third electron is $\gamma e$ :

$\displaystyle H_{\textit{stat}}^{0}(1,2,3)=\left[-\frac{\hbar^{2}}{2m}(\nabla_% {1}^{2}+\nabla_{2}^{2})-\frac{2e^{2}}{r_{1}}-\frac{2e^{2}}{r_{2}}+\frac{e^{2}}% {r_{12}}\right]-\frac{\hbar^{2}}{2m}\nabla_{3}^{2}-\frac{\gamma e^{2}}{r_{3}}$ (84)

and a perturbation operator, accounting the repulsive potential interaction of the third electron with two electrons belonged to the Helium atom, and also an interaction with the nucleus

$\displaystyle V_{\textit{(stat)}p=0}=\frac{e^{2}}{r_{13}}+\frac{e^{2}}{r_{23}}% -\frac{(2-\gamma)e^{2}}{r_{3}}$ (85)

Then, using Eqs (2.1) and (16), the first energy correction will be

$\displaystyle E_{f}^{1}={\left\langle\Psi_{f}^{0}\right|}V_{\textit{(stat)}p=0% }\left|\left.\Phi_{f}^{p=0}\right)\right.={\left\langle\frac{1}{f_{0}}\Psi_{0}% (1,2,3){\rm X}_{0}(1,2,3)\right|}V_{\textit{(stat)}p=0}\left|\left.\Psi_{% \textit{He}}(1,2),\phi_{e}\left(3\right)\right)\right.={\int\!\!\!\!\int\!\!\!% \!\int}d\textbf{r}_{1}d\textbf{r}_{2}d\textbf{r}_{3}\left(\frac{1}{f_{0}}\Psi^% {*}{}_{0}(\textbf{r}_{1},\textbf{r}_{2},\textbf{r}_{3})V_{\textit{(stat)}p=0}% \Phi_{\textit{He}}(\textbf{r}_{1},\textbf{r}_{2})\phi_{2p}\left(\textbf{r}_{3}% \right)\right)\cdot\left\langle\alpha_{3}^{{\dagger}}\alpha_{2}^{{\dagger}}% \alpha_{1}^{{\dagger}}|\alpha_{1}\alpha_{2}\alpha_{3}\right\rangle$ (86)

Using the perturbation operator in the form Eq. (85) to determine the analytical form of the self-consistent potential curve for the third electron we calculate the following mean value:

$\displaystyle U(r_{3})={\int\!\!\!\!\int}d\textbf{r}_{1}d\textbf{r}_{2}\left(% \frac{1}{f_{0}}\Psi^{*}_{0}(\textbf{r}_{1},\textbf{r}_{2},\textbf{r}_{3})V_{% \textit{(stat)}p=0}\Phi_{\textit{He}}(\textbf{r}_{1},\textbf{r}_{2})\psi\left(% \textbf{r}_{3}\right)\right)\cdot\left\langle\alpha_{3}^{{\dagger}}\alpha_{2}^% {{\dagger}}\alpha_{1}^{{\dagger}}|\alpha_{1}\alpha_{2}\alpha_{3}\right\rangle,$ (87)

where we take a probe wave function $\phi_{2p}\left(r_{3}\right)$ in the Slater form (Hydrogen-like form):

$\displaystyle\phi_{2p}\left(\textbf{r}_{3}\right)=\gamma_{1}r_{3}e^{-\gamma r_% {3}}\cos(\theta),$ (88)

with a normalization factor $\gamma_{1}=\frac{\gamma^{{\textstyle\frac{5}{2}}}}{\sqrt{\pi}}$ . We consider the parameter $\gamma$ as a variable. Using the hydrogen-like wave function Eq. (88) with the mentioned parameter $\gamma$ , it is possible to obtain its optimal value from analysing the following potential curve Eq. (89). The analytic expression of $U(r_{3},\gamma)$ as function of the distance $r_{3}$ and the parameter $\gamma$ is

$\displaystyle U(r_{3},\gamma)=\frac{r_{3}}{9}\sqrt{\frac{\gamma^{5}}{\pi}}e^{-% 2\gamma r_{3}}\left(-1+\sqrt{2\gamma^{5}}\pi e^{-2r_{3}}\left\{\left(-1+e^{4r_% {3}}-4r_{3}(1+2r_{3})\right)e^{-2r_{3}}\right.\right.\left.\left.-1+e^{2r_{3}}% -2r_{3}(1+r_{3}+r_{3}^{3})\right\}\right)$ (89)

The curve is shown on the Fig. 3a. It is easy to see that this mean-field potential accounting the exchange effects can capture the electron and settle it in the most preferable bond state $\phi_{2p}\left(r\right)=\frac{0,8^{2,5}}{4\sqrt{2\pi}}re^{-0,4r}\cos\theta$ Eq. (5) with the parameter $\gamma\approx 0.4$ . The binding energy as a function of a parameter $\gamma$ , calculated by the Eq. (86), has the expression:

$\displaystyle E^{\left(1\right)}=E_{\textit{bond}}(\gamma)=\frac{1}{18}\left(-% 3\sqrt{2\pi}+8\sqrt{2}{\pi}^{{5}/{2}}\left(\frac{\gamma\left(24+6\gamma\left(1% 2+5\gamma\left(3+2\gamma\right)\right)\right)}{4{\pi}^{2}{\left(2+\gamma\right% )}^{6}}\right.\right.\left.\left.+\frac{3\gamma 2(1+2\gamma(4+\gamma(14+\gamma% (28+5\gamma(-5+2\gamma)))))}{32{\pi}^{2}{(1+\gamma)}^{8}}\right)\right).$ (90)

The binding energy dependence of the parameter $\gamma$ is shown on the Fig. 3b.

Figure 3.

a) The potential curve for the free electron moving in the mean-field of Helium atom, where the axes r3 corresponds to the distances of electron from the Helium nucleus and the second axes corresponds to the variable parameter $\gamma$ . b) Binding energy (in Hartree atomic units) as a function of the parameter $\gamma$ .

From Eq. (44) we find the value of binding energy, taken into account $\gamma\approx 0.4$ , that equals to $E_{\textit{bond}}=-0.23\;a.u.$

The main characteristic that determines the cross section is the absolute value of the scattering amplitude, that is, the absolute value of the matrix element in Eq. (81). According to Eq. (55), the cross-section $\sigma_{cap}$ of electron capture is proportional to the square on the absolute value of the matrix element $\sigma_{\textit{cap}}\propto|{\langle\Psi_{f}^{0}|}\frac{f_{0}^{2}}{3!}V_{p=0}% |\Phi_{i}^{p=0})|^{2}$ . We show in the Fig. 4 the dependence of this square on the absolute value of the matrix element as a function of the angle $\theta$ and the wave vector k of the incident electron, to compare it to the well-known experiment dates [35], referred in the monography [36]. It is easy to see, that singularity in the k-dependence on the cross section takes place in the interval [ $k\sim$ 0.9, $k\sim$ 1.5] and decrease dependently on the scattering angle $\theta$ $\pi$ /2, as shown in [35].

Figure 4.

a) A quadrat of the matrix element absolute value $|{\langle\Psi_{f}^{0}|}\frac{f_{0}^{2}}{3!}V_{p=0}|\Phi_{i}^{p=0})|^{2}$ in a.u. as function of the angle $\theta$ and the incident electron wave vector $k$ [1.19, 4.5] in a.u. b) Variation of the scattering cross section dependent on the scattering angle. Experimental results taken from [36] E $=$ 19,4 eV ( $k=$ 1.19 a.u.)

Using Eq. (81) we calculated analytically the dependence of the cross section of the electron capture on the attack angle

$\displaystyle\frac{d\sigma_{if}}{d\Omega}=0.228\cdot\frac{\sqrt{k^{2}-1.42}}{k% }\times\left\{\textit{Abs}\left[-\frac{0.228\cos\theta}{\left(0.4-ik\cos\theta% \right)^{3}}+8\sqrt{2}\pi^{5/2}\right.\right.\times\left.\left.\right]\right\}% ^{2}$ (91)

This dependence is shown on the Fig. 5. There is a peak observed in experiments [35, 36]. That demonstrate a good agreement of our calculation with the real observations. Analysing the shape of peaks shown on the Fig. 5a and b, the extremal value of reaction probability corresponds to the electron wave vector $k*=$ 1.19 (1/ $a_{B}$ ), which corresponds to energy 19.4 eV and a scattering (attack) angle $\theta*=$ 1.26 rad ( $\sim$ 72.2 ${}^{\circ}$ ). It is easy to see that the shape of the peak is more complex than the usual expected for slow scattering events dependence $\sim\frac{1}{k^{2}}$ [34]. The curve shape [40] for the flux intensity of electron beam elastic scattered under the angle 72 ${}^{\circ}$ shows sharp peak for the electron beam energy 19.4 eV.

Table 1

Total cross section

$k_{i}$ (a.u.) [1/ $a_{0}$ ]	$\sigma_{if}$ (a.u.) [ $a_{0}^{2}$ ]
2 $\pi$	0.000230568
$\pi$	0.00170531
2 $\pi$ /3	0.00500652
2 $\pi$ /4	0.00838681
1.5	0.008686
1.48	0.00873762
1.45	0.00877739
1.43	0.00877343
1.4	0.00871029
1.35	0.00839925
1.3	0.00769388
1.25	0.00626647
1.23	0.00530712
1.2	0.00271013
1.195	0.00182438
1.193	0.00129405
1.192	0.0009151
1.19102	0.0001009

Figure 5.

a) An effective cross-section $\frac{d\sigma_{\textit{if}}}{d\Omega}$ in a.u. $[\frac{a_{0}^{2}}{\textit{steradian}}]$ as a function of the initial wave vector $k$ [1, 5] (a.u.) and a scattering angle $\theta$ [0, $\pi$ ] . b) An effective cross-section $\frac{d\sigma_{\textit{if}}}{d\Omega}$ as a function of the initial wave vector $k$ [1, 10] (a.u.) and a scattering angle $\theta$ [0, $\pi$ ] c) Scattering amplitude of the elastic process, scattering angle vary [0, $\pi$ /2].

Figure 6.

a) Shape of peak around a critical values of wave vector $k$ and attack-angle $\theta$ . b) Experiment dates [40]. The results of the first observation of resonance in He (1s2s ${}^{2}$ ) ${}^{2}$ S obtained by passing an electron beam through a gas.

Table 1 gives the calculated values of the total cross section of the reaction (a capture-detached process) as dependent of the wave vector value. It can be seen that, just like the effective cross section, the total cross section has a peak for wave vectors of about 1.4 [1/ $a_{0}$ ]. This feature can be associated with the processes of electron capture and detachment by the resonance level of helium atom.

6. Results and conclusion

A number of theoretical and experimental investigations (see reviews [41, 42, 43]) have shown that the behavior of the photodetachment cross section in the vicinity of ionization thresholds for inner shells of negative ions differs significantly from that in neutral atoms. Analogous effects were considered and taken into account earlier in calculation of the cross section of photodetachment of inner electrons from the Li 1 [42, 43, 44] ions. For the description such a process involving electric-dipol interaction with photon the random phase exchange approximation (RPEA) metod was used. The describing effects determined by the second-order perturbation corrections. Effects of atomic core relaxation are also taken into account. In our consideration we don’t take into account photo-processes giving the second-order corrections to the cross section. Multielectron correlations play a decisive role in the description of processes negative ions creation. We consistently take into account all exchange effects by using EPT-formalism that allows to develop characteristics of scattering events in general form, using a scattering matrix obtained with respect to all electron permutations among the scattered heavy atomic particles. Namely multi-central electron correlations put their contributions in the first-order corrections to the scattering amplitude and to the cross-section. EPT-formalism gives a clear algorithm for accounting a muli-centered electron correlations during the scattering process. The results are in a good agreement with the experimental dates [7, 8] and with the computations with using Molecular Orbital technics [2, 7, 35, 36, 40].

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