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Schrödinger equation for two-electron atomic states with conserved angular and I.K. Dmitrieva, G.I. Plindov

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I.K. Dmitrieva, G.I. Plindov. Schrödinger equation for two-electron atomic states with con- served 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 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 , 220728 Minsk, U.S.S.. (**) 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 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 function expansion in eigenfunctions of a symmetric top is used to reduce the six-dimensio- nal Schrödinger equation for a two-electron 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 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 . 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 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 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 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 » 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’( 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 S. Equations similar to (11) have been previously obtained in the ri, r2 and ri2 coordinates [9]. The use of one- coordinates has resulted in very tedious equations which are difficult to study. Recently [10], an equation for a two-electron system similar to (9) has been obtained using a vector-para- meter technique for the rotation group. As the coordinate system chosen in [10] is not related to the principal axes of inertia, the wave functions are not separated with respect to the state parity. Equation (11) reduces the initial six-dimensional Schr6dinger equation to system of three-dimensional equations. Solution of such a multi-component system in a general case remains a complex problem. For small J, the problem is somewhat simplified If J = 0, as is well known, there exists only one equation : where Ho is represented by (7). For J = 1, the system (11) contains one equation for even P* function :

and two equations for the components of odd Po function :

In the general case, the system (11) for a certain value of J is divided into (J + 1) equations for the states with n = ( - I)i and J equations for the states with n = ( - I y + ’ (for J = 2, 3, see Appendix). Two states with momenta differing by unity ( j and j +1) and having parity n = ( - I Y are described by equations of the same structure. From (14), it follows that such pairs are formed by the different multiplicity states : IS*_3p*; 3Se-lpe; IpO_3 DO; 3pO_ ’Do; IDe_3Fe, etc.

3. Fock expansion

Let us now construct an exact wave function of the two-electron atom of arbitrary J, s and n. As the potential energy, V(R, Q), (6), is an uniform function of R, it would seem natural to look for a solution of (1) in the form of a R series expansion with coefficients depending on 0. However, as Fock [2] showed, the wave function of the Ise may be given only as a double series expansion in R and In R This result was formally gene- ralized in [11] for an arbitrary atomic state. In [12], the expansion of Demkov and Ermolaev [11] was corrected 1497 to account for the spin of electrons. We write down the exact wave function in the form of a generalized Fock series, without using the expansion in hyperspherical functions Y,,,(Q) :

Here [n/2] is the integer part of the number n/2. When the expansion (18) is substituted into (1) and the coeflicients of R"(In R)k are taken equal to zero, we obtain the following recurrence relation that determines the exponent a and the functions Pn,k(Q) :

Here, the operators A’(0) and U(Q) are determined by (5) and (6). The governing equations of system (19) are of the form :

The functions P:’k(Q) are the hyperspherical functions [4, 5]. As is shown in [4, 5], the eigenvalues of (20a) are even for even states and odd for odd ones. The exponent a also depends on j. For the lower state

Moreover, for S- and P*-states, ai. depend on the multiplicity and is equal, respectively, to :

The functions cp: k(Q) are determined from (20a) within a global coefficient. In order to find out this coefficient, one must impose an orthogonality condition on CP:’k(Q) with respect to RHS of(20c), i.e.

Equation (21) is sufficient to completely determine P:tk(Q) if the condition

is satisfied. If for some state Ii = 0, then the coefficient of R’ In R is equal to zero, and the leading logarithmic term in (18) is of relative order R4 ln R. The functions Oj" 2k I 1,k(0) are particular solutions to (20b) with the correct beha- viour at singularities, because the homogeneous solution to (20b) does not possess the required symmetry. The functions Oj,’,k - 1 (g2), Oj’s + l,k -etc. can be found as a linear combination of the solutions of the appropriate homogeneous equations. Let us consider even S- and P*-states. It is possible only for these states, to completely separate the depen- dence on the external angles in the wave function :

A ’(0) in (19)-(20) is replaced by the operator Âj(ç, (p) : 1498

When solving, for the lower states, the first two equations of the system (20) for k = 0, 1, we obtain within the normalization factor :

Equations (23) through (26) give the first expansion terms of the lower I,lSe - and ’,’Pe-states wavefunction near the point of triple collision of electrons with the nucleus. The first-order function for a tse-state was found in [2], for a ’S’-state, in [13] and for t,3pe-states is given for the first time in the present study. The coefficients of the first logarithmic term for ’,’S"-states were first found in [14]. Pluvinage [15] has recently obtained them from differential equations. For the 3Sl-state, his value differs from the one given in [14]. We have calculated this coefficient using orthogonality condition (21) on cP:l(ç, T) and found that Pluvinage’s value [15] is exact. Com- parison of (23) with (26) and of (24) with (25) shows the similarity of pairs iSe-3Pe and 3S°-lpe, which was pointed out in the previous section. Note that all these wave functions have non-analytic terms of relative order R Z In R, whose coefficient differs only by a factor for different S, pe_States. The next term of the Fock expansion is proportional to cPz,o(ç, cp). The first attempt to obtain 02,0(0) for the ground state was made in [14]. It was found that cP2,O(Q) can be represented as an expansion in hyperspherical functions Y m,(fJ)

and the first coefficients dml were estimated The exact values of the first 20 coefficients in the expansion (27) were obtained in [16]. Unfortunately, the authors of [14,16] did not examine the convergence of the obtained expan- sions. The study [15] is more advanced, because the functions 02,0 for ’,’S-states are given explicitly in the form of an expansion in Legendre polynomials, and the absolute convergence of the obtained series is proved Com- parison of (15) and (16) shows that the procedure of [15] is applicable to 1,3pe-states if the series expansion in the associated Legendre polynomials Pli (cos 0) is used It is of interest to relate expansions (23), (26) to the Z perturbation theory. As given in [14], when the S-states are examined, all terms but the zeroth orders one of the Z -1 perturbation theory have non-analytic logarithmic-type dependence. One can show that the value of coefficients of relative order R 2 In R calculated to first order of Z-1 perturbation theory coincides with the one given in (23) and (24). These results can be extended completely to P’-states of a two-electron atom. Using the scaling x = ZR in (23)-(26), one easily shows that the effect of logarithmic terms is most substantial for wave functions of a negative and decreases rapidly with increasing Z.

4. Approximate solutions. In the previous sections, the similarity of wave function pairs lse_3pe and 3S,_Ip, was shown. With this law, the methods for studying S-states may be extended to the more complex P’-states. In particular, the well-known 1499

method of solution of the S-state Schrodinger equation by expansion in Legendre polynomials may be easily extended to P’-states. To show this, let us re-write (15) and (16) in the rl, r2 cos 0 coordinates :

The part of the kinetic energy operator in (28) and (29) that depends on the angle 0 may be presented in a form : general I

Eigenfunctions of t 8 are the associated Legendre polynomials Pi (cos) 0 with j I; j = 0 for S-states ; j = 1 for P-states :

It is natural to find the eigenfunction of equation (29) in the form of an expansion in P/(cos 0) :

Substitution of (30) into (291’simplifies the three-dimensional equation into an infinite system of two-dimensional equations for the functions 01(rl, r2) :

When system (31) is truncated at :nax = M, it may be solved by iteration. The simplest approximation, the analog of the S-limit for equation (28), contains only one equation :

Here r, = max (rl, r2), r, = min (r. i, r2). Equations (31) may be used to calculate both 1pe and 3Pe-states. From conditions (14) it follows that the function of a ’P"-state must be symmetric with respect to ri +-- r2 permutation, and the function of a 1 pe-state, antisymmetric. Let us now show that (25)-(26) may be useful for constructing variational wave functions of pe -states. Let us rewrite these equations in the Hylleraas coordinates : S = ri + r2, u = r,21 t = ri - r2 convenient in variatio- nal calculations :

The simplest basis that takes into account the specific features of a wave function near a nucleus and at R -+ 00 is

. _ ., 1500

Compare (35) and (36) at small s, u and t with (33) and (34). No limitations to the first coefficients Gn,l,m for 3Pe-state follow from (34); from (33) it follows that in (35) Booo = Bolo = B020 = 0. Further improvement of the basis may be made by introducing logarithmic terms in (35) and (36) :

where the first non-vanishing terms are proportional to ai S 2, fil t2, y, U2(3pe_States) and

5. Discussion and conclusion, logarithmic term is of relative order R’ In R It should be noted that the coefficients in (25) and (26) do not The choice of coordinates related to the principal axes depend on the energy and on the main quantum of inertia for a system makes it possible to three-particle number; hence, these expressions specify the proper- separate the part of the kinetic energy operator that ties of a wave function for P"-states in the vicinity of on the total momentum. this depends angular Using the triple collision of electrons with the nucleus and of a rotor dependence eigen-functions symmetric = (R -+ 0) for all states in a channel with a 2(’P") /!, y) and /!, y), we have simplified the Dk,m(a, Dl:;"( 0152, and a = 4(’Pe). For other P" channels, the solution six-dimensional Schrodinger equation for a two- of (20a) through (20c) with the appropriate value of a electron atom a finite of three-dimen- system coupled allows determination of the first expansion terms. The sional equations (11). This system splits naturally into exact wave function can be constructed for P°, Do, D’ two each states with conserved subsystems, describing and other states, expanding each of the components values of the of the momen- operators squared angular :1 (R, ç, (p) in (11) in Fock series. tum J 2, its component and parity A. The multipli- j. The (25)-(30) may be useful for precise of states is chosen When the structure expansions city using (14). variational calculations of even P-states wave functions of for J is a chosen equations given known, specially and . As has been already shown for S-states basis be used for each state, in may thereby resulting [17,18], the introduction of logarithmic terms provides substantial improvement of the variational procedure high accuracy with a small number of variational convergence. For equations with small J, numerical parameters. The exact behaviour of the wave function methods may be employed alongside with the varia- near the nucleus may be adopted also to calculate tional ones. The series expansion in the associated some expectation values such as averages of ( p 1 ), is a convenient numerical Legendre polynomials of the Is splitting, etc. method that can be adopted to solve the equation for P’-state. We have extended the Fock series expansion Acknowledgments. to states of arbitrary J. Upon solving (20a)420c), we have found the first terms of the Fock expansion for The authors are indebted to the referees who mention- the lower 1,3pe-states and revealed that the leading ed references [9] and [15].

Appendix.

Let us write down the explicit form of the equations for D- and F-states. For D’-states, the system (11) consists of two components :

Even D*-states are described by three equations : 1501

Fe-states are also described by three equations :

Fl-states are described by four equations :

References

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