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Schrödinger Equation for Two-Electron Atomic States with Conserved Angular Momentum and Parity I.K

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Schrödinger Equation for Two-Electron Atomic States with Conserved Angular Momentum and Parity I.K Schrödinger equation for two-electron atomic states with conserved angular momentum and parity I.K. Dmitrieva, G.I. Plindov To cite this version: I.K. Dmitrieva, G.I. Plindov. Schrödinger equation for two-electron atomic states with con- served angular momentum and parity. Journal de Physique, 1986, 47 (9), pp.1493-1501. 10.1051/jphys:019860047090149300. jpa-00210345 HAL Id: jpa-00210345 https://hal.archives-ouvertes.fr/jpa-00210345 Submitted on 1 Jan 1986 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. J, Physique 47 (1986) 1493-1501 SEPTEMBRE 1986, 1493 Classification Physics Abstracts 31.10 Schrödinger equation for two-electron atomic states with conserved angular momentum and parity I. K. Dmitrieva (*) and G. I. Plindov (**) (*) Heat and Mass Transfer Institute, BSSR Academy of Sciences, 220728 Minsk, U.S.S.R. (**) Power Engineering Institute, BSSR Academy of Sciences, Minsk, U.S.S.R. (Reçu le 24 septembre 1985, révisé le 16 avril 1986, accepté Ie 28 mai 1986) Résumé. 2014 En développant la fonction d’onde d’un atome à deux électrons sur la base des fonctions propres d’une toupie symétrique, on réduit l’équation de Schrödinger à 6 dimensions en un système fini d’équations à 3 dimensions pour les états correspondant à des valeurs définies du carré du moment angulaire total, de sa composante le long de z et de la parité. Le développement en série de Fock est étendu aux états de moment angulaire arbitraire. Les pre- miers termes de la série de Fock pour les états pairs Pe sont obtenus jusqu’au terme logarithmique. La méthode proposée peut être utilisée pour résoudre les équations décrivant les états Pe grâce à un développement en poly- n6mes de Legendre associés P1l(cos 03B8). Abstract. 2014 The wave function expansion in eigenfunctions of a symmetric top is used to reduce the six-dimensio- nal Schrödinger equation for a two-electron atom to a finite system of three-dimensional equations for eigenstates of the squared angular momentum, of its z component and of parity. The Fock series expansion for 1Se wave func- tion is extended to the states of arbitrary J. Its first terms are found for the lower Pe states, including the leading logarithmic term. The proposed method can be used to solve equations for Pe states with series expansion in the associated Legendre polynomial Pl1(cos 0). 1. Introduction In order to avoid this difficulty, it is necessary to obtain an explicit form of the Schrodinger equation For a time the two-electron was of long problem for two-electron states with a given value of the total interest to This is due to the fact that the physicists. angular momentum J and parity 7c, which is the first two-electron atom is the in which the simplest system, goal of the present work. We will also extend the Fock electron interaction is of from importance. Starting series expansion to states of arbitrary J and give an the works [I], many attempts were early of Hylleraas explicit form of exact S- and P"-wave functions near the to an solution of the made find exact Schrodinger triple collision point (R - 0) of the electrons with the equation. Fock [2] contributed to the study of a wave nucleus. function with zero angular momentum by introducing a hyperspherical radius R and obtained an exact solu- 2. Equations. tion to the problem as a double series expansion in R and In R During the last two decades special interest The Schrodinger equation for a two-electron atom with an nucleus is written atomic was centred on electron correlations in doubly excited infinitely heavy (in in the non-relativistic as : states. The experiments have put in evidence that a units) approximation description of such states in terms of one single-par- ticle configuration is impossible [3]. Two-electron wave functions, with conserved total angular momen- where tum and parity, must be constructed to interpret the experimental data. Usually this problem is solved by expanding two-electron wave functions in one- electron ones with the help of a vector coupling pro- In (1) and (2), ri, r2, V2 and V2 are the radius-vector cedure, which results in infinite series summations. and the Laplace operator of the first and second elec- Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:019860047090149300 1494 trons, r12 = (ri + r2 - 2 ri r2 cos 0)1/2 is the distance angles ç and T. The first angle is determined by prin- between them and Z is the nuclear charge. cipal moments of inertia The Hamiltonian H, the squared total angular momentum j2, its component Jz, the parity operator di and the electron exchange operator P12 form a set of commuting operators. Hence, the wave function At fixed values of Rand ç, the electrons move simulta- is a simultaneous of f/1(r l’ r2) eigenfunction these neously over an ellipse. The electron position on the operators : ellipse is determined by the second internal angle w (Fig. 1). As follows from (2), the potential energy of the system depends on three coordinates only while the kinetic one is function of six coordinates. According to Fock [2], it is convenient to use a hyperspherical radius : A set of anglers 92 may be chosen for the remaining five coordinates. Then, the kinetic energy T is expressed as : Fig. 1. - Internal coordinates of the two-electron atom. The electron positions relative to the nucleus are where A 2(Q) is the grand angular momentum : described by : Here Yuy(S) are hyperspherical functions [4, 5]. In this case, there exists a different choice of hyperspherical angles. We use the set of 0 proposed by Smith [6] to exploit the rotational invariance. For this purpose, it is convenient to adopt three Euler’s angles (a, fl, y) for the where #,,2 = ± 3 a/2. The Cartesian components atom orientation in a space-fixed frame and internal refer to the principal axes of inertia. A detailed study of the operator A2 properties [4-6] allows A2to be expressed as a function of the compo- nents of the angular momentum J [6] : The components of the total angular momentum J refer to the body-fixed coordinate system, which coincides with the principal axes of inertia. The potential energy in the R, Q-coordinates is written as : Combining (3), (5) and (6) we get the following expression for the Hamiltonian H in the R, Q-coordinates : 1495 where From (7) it follows that consideration of the two-electron problem gives a generalized centrifugal barrier, with a non zero value even for S-states. It is obvious that for R - 0, the kinetic energy T is the dominant part of the operator H. Therefore, at small R, eigenvalues of the operator A 2, (4), are « good » quantum numbers. This fact will be used to study the behaviour of the wave function near the nucleus. Equation (7) reveals that even the use of the Hamiltonian with a selected orbital momentum does not allow us to completely separate the external coordinates cx, p and y. Diagonalization of Hamiltonian (7) must then be made of the momentum as a finite using eigenfunctions angular operator Dl,m(’ ex, p, y) [7]. Expressing ql(R, 0) sum : substituting (8) into (1) and considering the action of the operators J2 on Dl,m(a., f3, y) [7], we obtain the following system of coupled three-dimensional differential equations for t/1m(R, ç, cp) : Two-dimensional equations similar to (9) appear in studies of the hyperspherical functions properties [4, 8]. Some shortcoming of (9) is. the existence of an imaginary part in the effective Hamiltonian. It is more convenient to expand the wave function #(l§ 0) in eigenfunctions of a symmetric top [7]. Let us write : where into and and the effect of the on We obtain a Substituting (10) (1) (7) considering operators J 2 Di,m(a,’ (3, y). system of coupled differential equations for the functions 0’(X ç, (p) : The functions #(j, R, lp) or ql£ (R, ç, lp) with even or odd indices are separately coupled in (9) or (11). Therefore, linear combinations (8) and ( 1 0) are divided into two independent parts : One contains only the functions with even m and the other with odd m Considering that the action of the parity operator f does not change the internal angles ç and T and two Euler angles a and P and transforms y into y + n and using the properties of the D-functions [7] it is easy to show that .J e(R, Q) describes the even states of an atom while #o(l§ Q) describes the odd states. 1496 It should be noted that according to the Wigner-Eckart theorem, equations (9) and (11) do not depend on k, which results in a (2 k + l)-fbld generacy of the wave function t/1(R, Sl). Let us consider the action of the electron exchange operator P 12 on wave function (10) : The action of the on is to two successive transformations -+ y + 7r and operator P12 Dj.(oc,’ p, y) equivalent y p -+ p + ’It Y -+ - y, thus From (12) and (13), we obtain the following transformation property for the functions q5.’ (R, ç, T) belonging to a given multiplicity : Thus, equations (14) and (11) make it possible to solve the problem of constructing the Schrodinger equation wavefunction for a two-electron atom with conserved total angular momentum J, parity n and spin S.
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