HAL Id: jpa-00210345
https://hal.archives-ouvertes.fr/jpa-00210345
Submitted on 1 Jan 1986
HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
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�
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.
2014En 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.
2014The 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).
Classification Physics Abstracts
31.10
1. Introduction
For a long time the two-electron problem was of
interest to physicists. This is due to the fact that the two-electron atom is the simplest system, in which the electron interaction is of importance. Starting from
the early works of Hylleraas [I], many attempts were made to find an exact solution of the Schrodinger equation. Fock [2] contributed to the study of a wave
function with zero angular momentum by introducing
a hyperspherical radius R and obtained an exact solu- tion to the problem as a double series expansion in R
and In R During the last two decades special interest
was centred on electron correlations in doubly excited
states. The experiments have put in evidence that a
description of such states in terms of one single-par-
ticle configuration is impossible [3]. Two-electron
wave functions, with conserved total angular momen-
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-
cedure, which results in infinite series summations.
In order to avoid this difficulty, it is necessary to obtain an explicit form of the Schrodinger equation
for two-electron states with a given value of the total
angular momentum J and parity 7c, which is the first
goal of the present work. We will also extend the Fock series expansion to states of arbitrary J and give an explicit form of exact S- and P"-wave functions near the
triple collision point (R - 0) of the electrons with the nucleus.
2. Equations.
The Schrodinger equation for a two-electron atom with an infinitely heavy nucleus is written (in atomic units) in the non-relativistic approximation as :
where
In (1) and (2), ri, r2, V2 and V2 are the radius-vector 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
between them and Z is the nuclear charge.
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 f/1(r l’ r2) is a simultaneous eigenfunction of these operators :
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 :
where A 2(Q) is the grand angular momentum :
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
atom orientation in a space-fixed frame and internal
angles ç and T. The first angle is determined by prin- cipal moments of inertia
At fixed values of Rand ç, the electrons move simulta-
neously over an ellipse. The electron position on the ellipse is determined by the second internal angle w
(Fig. 1).
Fig. 1.
-Internal coordinates of the two-electron atom.
The electron positions relative to the nucleus are
described by :
where #,,2 = ± 3 a/2. The Cartesian components 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 :
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 using eigenfunctions of the angular momentum operator Dl,m( ex, p, y) [7]. Expressing ql(R, 0) as a finite
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
Substituting (10) into (1) and (7) and considering the effect of the operators J 2 on Di,m(a, (3, y). We obtain a
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 operator P12 on Dj.(oc, p, y) is equivalent to two successive transformations y -+ y + 7r and
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.
Equations similar to (11) have been previously obtained in the ri, r2 and ri2 coordinates [9]. The use of one- particle 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
ground state 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
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 hydrogen ion 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
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
general form : 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
. _ .,
