Problems of theoretical interpretation of the spectra of highly ionized atoms

Nuclear Instruments and Methods 202 (1982) 289-297 North-Holland Publishing Company

289

PROBLEMS OF THEORETICAL INTERPRETATION OF THE SPECTRA OF HIGHLY IONIZED ATOMS Z.B. R U D Z I K A S Institute of Physics of the Academy of Sciences of the Lithuanian SSR, K. Po$elos 54, Vilnius, 232600, USSR

The methods of theoretical and semiempirical investigation of energy spectra and electronic transitions in atoms and ions (non-relativistic approximation, taking into consideration relativistic effects as corrections in the framework of the Breit approximation as well as starting with relativistic wavefunctions) are surveyed. The problem of the identification and classification of the experimentally obtained energy spectra with the help of the optimal coupling scheme is considered. General relativistic and non-relativistic expressions for the operators of electric and magnetic multipole transitions, having unspecified values of the gauge condition of the electromagneticfield potential, are presented. The dependence of the oscillator strengths on this gauge condition is studied. Pecularities of the spectra and the structure of highly ionized atoms are discussed.

I. Introduction

Spectra continue to be a fundamental characteristic of atoms and ions, the main source of the information of their structure and properties. Diagnostics of both laboratory and astrophysical plasma is carried out as a rule on the bases of the spectra. Nowadays the possibilities of theoretical spectroscopy are much enlarged thanks to the wide use of powerful computers. This enables one to investigate fairly complicated mathematical models of the system under consideration and to obtain in this way results which are in quite good agreement with the experimental measurements. Interest in spectroscopy has increased particularly in connection with the rise of non-atmospheric astrophysics and laser physics. A number of new important applied problems have arisen: diagnostics of thermonuclear plasma, creation of lasers generating in the X-ray region, identification of solar and stellar spectra etc. The use of laser-produced plasmas, powerful thermonuclear equipment (Tokamaks), low inductance vacuum sparks, exploding wires, b e a m - f o i l spectroscopy discovered a new extremely interesting and original world of very highly ionized atoms, their radiation being as a rule in far the ultraviolet and even the X-ray wavelengths region. The spectra due to the transitions between highly excited (Rydberg) levels are studied intensely. Investigations of complex atoms and ions, having two, three and even more open shells, are needed. Identification of the spectra of all these abovementioned systems is practically impossible without corresponding theoretical investigations. In this paper, we shall study pecularities of the contemporary theory of the energy spectra and electronic transitions in atoms and ions as well as its role in plasma spectroscopy. Special attention will be paid to recently developed methods of theoretical investigation of highly ionized atoms in the framework of which relativistic effects are accounted for. This is of special interest when dealing with the problems of the non-atmospheric astrophysics and the laboratory h i g h t e m p e r a t u r e plasma, in particular, of controlled thermonuclear fusion. We shall discuss accounting for the relativistic effects both as corrections and in a relativistic approach, as well as the use of other coupling schemes, differing from the traditional L S one, for classification of the energy levels of the systems under consideration. Theoretical calculations are the only way of finding these quantum numbers. In the next section we shall describe the method of theoretical investigation of the spectra of many-electron atoms and ions taking into account relativistic effects both as corrections and in a relativistic approach, starting with the relativistic Hamiltonian and relativistic wave functions. The problem of identification and classification of the energy levels with the help of quantum numbers of various coupling schemes, including intermediate ones, as well as the usage of semiempirical methods for the analyses of the 0167-5087/82/0000-0000/$02.75

© 1982 North-Holland

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energy spectra are studied. Section 3 deals with finding the general relativistic and non-relativistic expressions for the operators of electric and magnetic multipole transitions, having unspecified values of the gauge condition of the electromagnetic field potential. In Section 4 we shall briefly sketch the methods of accounting for correlation effects as well as pecularities of the spectra and structure of highly ionized atoms.

2. Calculation of the energy spectra in non-relativistic and relativistic approximations The Schr~dinger equation cannot be solved exactly for the case of a many-electron atom, therefore one has to use some simplifications. Usually the problem is reduced to the one-particle one, to the description of the motion of each electron in the nuclear charge field and in the screening field of the remaining electrons (the central field approximation). Most effectively it may be done with the help of various modifications of the self-consistent H a r t r e e - F o c k field method. The further refinement of the one-particle approximation requires accounting for correlation and relativistic effects. In atoms and ions, for which relativistic effects are small, they may be taken into consideration as corrections with respect to non-relativistic wave functions in the framework of the Breit operator given below. However, in the cases when they are comparable in order of magnitude with electrostatic interactions, one has to use both a relativistic Hamiltonian and relativistic wave functions. The Hamiltonian of a many-electron atom in the framework of the Breit operator may be written as follows [ 1]:

w.

(1)

Here H 0 denotes the zero-order Hamiltonian, while W represents the relativistic corrections of order c~2 (a is the fine structure constant). They may have the forms H 0 = T + P + Q,

(2)

W = H, + H2 + H3 + H4 + Hs,

(3)

T = -- ~ p ~ / 2 m

(4)

where i

is the kinetic energy of the electrons with respect to the nucleus,

P = - Z ZeZ/ri

(5)

i

is the potential energy of the electrons, moving in the field of the nuclear charge Z e / r ,

Q = ~ eZ/rij i>j

(6)

represents the electrostatic interactions between electrons, H, --

1 8m3c 2

~/p4

(7)

.

stands for the relativistic correction due to the dependence of the electron mass on the velocity,

H2--

2m2c 2 e2

1{ i~j ~ij (PiPj) "-~-(rij(rij'Pi)'Pj)}ri~

(8)

is the orbit-orbit interaction operator,

H a = H~ + H;'

_ ~reh:___._ZZES(r~) era:c: 7

•

~reh: m2c: Z 8(rij)

i>j

(9)

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describes the contact interactions,

H4--2m2c~

ri

i>jri3j[rij×Pi] + i>j 2 7[~,×Pj] "si rij

(lO)

stands for the spin-orbit interaction,

Hs=H~+H~'-- 8¢re2h2~ (s,.sj)8(r~j)+

3m2c2 i>j

e2h 2 ~

1 [ ~ "-~ i>j rij ($i" Sj) --

3(si'r,j)(sj'r~j) ] rij2

(11)

represents the spin-contact (Hi) and spin-spin (Hi') interactions. In these formulas e stands for the absolute value of the electron charge, m is its mass, p denotes its momentum, c is the velocity of light, h is Planck's constant, divided by 2~r, ,} is the distance of the ith electron from the nucleus, s stands for the spin angular momentum of the electron, Z represents the nuclear charge, r~j is the interelectronic distance, 6(r) is the Dirac &function of vectorial argument r. The sum is from ] to N, where N denotes the number of electrons. One-particle operators H 1 and H 3 cause relativistic corrections to the total energy. Two-particle operators H 2, H 3 and H 5 define more precisely the energy of each term, whereas H 4 and H 5 describe their fine structure. If the relativistic effects are sufficiently large and therefore cannot be taken into account as corrections, then as a rule one has to use relativistic wave functions and the relativistic Breit operator. In the case of the N-electron atom the corresponding Hamiltonian may be written as follows (in a.u.): N

N

H~- ~, ( H i l ~ H i 2 ~ - H i 3 ) ~ - ~ (H4.qt-Hi5~t-Hi6j). i= 1 i>j= 1

(12)

Here

Hi' =c(all).p~')),

H~.= 1/rij,

Hi2=jSic2,

HiS_

Hi3~- V(r,),

H 6.-

l (or(l) . = ( i ) ) '

2rij~ i

~b

(13)

2-~i3(a~').rij(1) )(a)(1) .r/51)).

In these formulas the Dirac matrices a (x) and 13 are of the form

o.,_(o

o (')

0

1'

B=

(, 0_) 0

J'

(,4,

where 0 (I) and ] are Pauli and unit matrices of the second order, respectively. H 1, H 2 and H 3 are the one-electron operators. H I describes the electron kinetic energy and one-electron part of the spin-orbit interaction, H 2 denotes the mass effect, whereas H 3 stands for the potential energy, in the Coulomb approximation being of the form -Z/r~. Two-electron operator H 4 corresponds to the electrostatic interaction, H 5 is the operator of the magnetic and a part of the retardation interactions, whereas H 6 stands for the remaining retardation interactions. The sum H s + H 6 is often called the relativistic Breit operator. The expressions for matrix elements of the Hamiltonian (1) in the case of complex electronic configurations may be found in [2], whereas the corresponding formulas for the Hamiltonian (l l) are presented in [3] and in papers quoted therein. Matrix elements of these operators, calculated for some chosen electronic configuration, makeup the energy matrix, which has to be diagonalized. In the case of non-relativistic wave functions it is usually formed starting with LS-coupling, whereas for the relativistic ones the jj coupling scheme has to be used, because in the relativistic approach each shell of equivalent electrons l N splits into two subshells according IV. A T O M I C S T R U C T U R E C A L C U L A T I O N S

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Table 1 W a v e l e n g t h s of the t r a n s i t i o n l s 2 2 s 2 2 p 5 3 d - l s 2 2 s 2 2 p 6 (in A) for F e + 16 a n d M o +32 Ion

L S J - 1So

PT

HFP

DHF

E x p [6]

F e + 16

3 Pi - I S o 3D t i Px -

15.484 15.274 15.023

15.457 15.263 14.996

15.460 15.268 15.004

15.453 15.261 15.012

M o + 32

3 Pi - l So 3D l 1Pt -

4.869 4.812 4.643

4.860 4.802 4.637

4.854 4.803 4.630

4.847 4.804 4.630

to Jl = 1 - 1 / 2 and j 2 = / + 1/2, i.e., instead of l~VaLS we have ljU'alJljU2a2J2J. The radial integrals needed may be calculated while using either numerical or analytical radial orbitals. The usage of numerical radial functions and accounting for relativistic effects as corrections in the framework of operator (3) defines the so-called H a r t r e e - F o c k - P a u l i (HFP) approximation. The general automatized computer program for calculation of the energy spectra in the abovementioned approximation is described in ref. 4. Hamiltonian (12) and the corresponding relativistic numerical radial orbitals form the D i r a c - H a r t r e e - F o c k ( D H F ) approximation. As an illustration of the use of these approximations, in table 1 the wavelengths (in A) of the electric dipole transitions ls22sZ2p53d LSJ-lsZ2s22p 6 1S0, calculated with the help of perturbation theory (PT) [5], H F P and D H F approximations, as well as experimentally measured values [6], are presented. Table 1 shows that all three theoretical methods (particularly the last two) allow one to obtain results fairly close to the experimental data [6]. However, already for M o + 3 2 the relativistic approximation is preferable. It has to be used for the more highly ionized atoms, for which the relativistic effects are sufficiently large and cannot be accounted for as corrections. As a result of diagonalization of the energy matrices we obtain both the numerical values of the energy levels (eigenvalues of the energy operator) and a set of the weights of the wave functions of the initial pure (e.g., LS) coupling scheme (eigenfunctions). In fact, we pass to some intermediate coupling. This considerably complicates the identification and classification of the energy levels. When the intermediate coupling is valid (it is the case for a majority of atoms and ions), only theoretical calculations allow us to determine the structure of the eigenfunction (the weights of the wave functions in a given coupling scheme). The wave functions in intermediate coupling, obtained after diagonalization of the energy matrices, are expressed in terms of the sum of the initial functions of pure coupling, i.e. the true states are the mixture of the states in the initial coupling scheme. Coefficients of this linear combination are the weights of the wave function of the initial coupling scheme. So, eigenfunctions in the intermediate coupling scheme q~(flJ), obtained starting with the wave functions of LS coupling ~(o~iLiSiJ), will be of the form ~,(j~J)---- ~

a(a,L, Si) ~(a,L,S,J ),

(15)

a~LiS,

where a(aiLiSi) denote the weights of the wave functions of the initial coupling scheme. Only J remains as an exact quantum number. The approximate character of the quantum numbers used is caused, on the whole, by the following reasons: the absence of completely pure coupling schemes in the case of many-electron atoms and ions, interaction between angular momenta, particularly characteristic for intermediate quantum numbers, as well as the presence of the quantum numbers, with respect to which some parts of matrix elements of the energy operator are not diagonal. When the numerical value of the weight of one of the functions considerably exceeds the others, then the corresponding level may be with confidence characterized by the quantum numbers of this wave function. When a few weights are of the same order, then for the reliable classification of the levels one has to analyse fine structure of the multiplet as well as the regularities of the changes in fine structure and in weights of the wave functions along isoelectronic sequences. However, a more effective w a y . i s the

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Table 2 The weights of the wave functions of the level 2s2p ( J = 1) {Be} in LS a n d j j coupling Approx.

Level

B+

Fe + 22

Kr + 32

W + 70

U + ss

HFP

I PI 3P 1

1.000 0.000

0.988 -0.153

0.948 -0.317

0.841 --0.541

0.830 -0.558

DHF

s+ p_ 1 s+ p+ l

0.817 --0.577

0.896 -0.440

0.957 -0.285

0.999 -0.039

1.000 0.000

calculation of the energy spectra, starting with several different coupling schemes, and the choice of the one in which, after diagonalization of the energy matrices, there would be one wave function, having a weight, considerably exceeding the others. This may be achieved more easily by using the method of transformation of the wave functions from one coupling scheme to another [7], suitable for the case of complex electronic configurations as well. Table 2 illustrates the change of the coupling scheme (transition from LS to jj coupling) in the Be isoelectronic sequence. If a part of the spectral quantities needed (e.g., energy levels) is already known, then they may be used for so called semiempirical evaluation of the wave functions, and the other features of the system under consideration. Usually for this purpose one utilizes the least squares fitting or the method of effective (model) potentials. In this approach, the numerical values of the radial integrals, present in the energy matrices, are found using the experimental data of the energy levels or by the extrapolation or interpolation of their values along isoelectronic sequences. Further on the energy matrices are calculated, after they are diagonalized. This leads to the whole energy spectrum of the configuration considered. It should be noted that the use of two orders of the perturbation theory in the form of 1/Z expansion makes it possible to find fairly exact values of energy levels, wavelengths and characteristics of electronic transitions [5,8]. However, because of computational difficulties and worsening of the convergence of the expansion this method was successfully applied only to systems having up to ten electrons; besides, it leads to poor results in the case of neutral and not highly ionized atoms. Energy spectra may be classified with the help of various coupling schemes (usually LS, LK, JK and J J). According to the relative magnitude of the non-spherical parts of the electrostatic and magnetic interactions. This is the case even when using the intermediate coupling scheme, because the weights of the wave functions of the two different coupling schemes ~k and aj~ are connected in the following manner:

Cjk = ~ajr(19k[~,).

(16)

r

The use of the algebraic or numerical expressions for the transformation matrices (v~k [~p~) [7] enables one easily to pass from one coupling scheme to the other and to choose in this way the optimal one.

3. General relativistic and non-relativistic expressions for electron transition probabilities Starting with quantum-electrodynamical perturbation theory, in the central field approximation, it is possible to find the following general relativistic expressions [9] for the electric multipole (Ek) transition probabilities (in a.u.):

wIE~k2-- k ( 2 k - ~ 1)C I(t~2

-f-

K

w1~2 k(2k+l)cl(~21vOL~+g

k

O) v - q - r Q - q J

( v O (--*q)- a~ r Q(--~

'

I~l)l 2

(17)

(18)

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Z.B. Rudzikas / Problems of theoretical interpretation

Here

rr.~-qO(k)=- - l {(2k + l) ~,k~ ~" 0 ~k)= - i k ~'q

k+1

t~-._q - -

i(2k + -'I)Vkf2k~-3~ 2

[C(k+, ) X aO)]~)qgk+l(Z )} ,

gk+,(z)[C(k+')×a°)]~)q+~/k(2k - 1)gk_,(z)[C{k-')×a(')]~

(19)

"

(20)

In these formulas ¢0 is the frequency of the photon emitted, K stands for the gauge condition of the electromagnetic field potential, C(qk) represents the spherical function, whereas g~(z) is the spherical Bessel function depending on the argument z = ¢or/c. The relativistic transition probability of the magnetic multipole (Mk) radiation is of the form [10] wM k _ 2 ( 2 k +C 1)~ l(4',lmO(_~14',)l 2 1~2

(21)

mO~)q = gk(z)[C (k) X ot(')]~q) .

(22)

where

In the case of the non-relativistic wave functions we have [11] instead of the formulas (17) and (18) -2

t-2

I 4' 1

14'21

14', I

Q'~k-)q +K

[Q'~)q-coQ(_~[4', [2,

(23)

(24)

where

Q~_~-= --rkC(_k)q

(25)

! 1) [C {k) X L'Ol(k)q} Q'~)q = - r k - I{ kC(k)~.~._ - - q Or + r Ik( k + - _.

(26)

and

In the case of the exact wave functions, the expressions (17), (18) and (23), (24) will lead to the same values of transition probabilities, whereas the terms with the multiplier K will be equal to zero for arbitrary K. For certain K values, the formulas (23) and (24) in the case of k = 1 turn into the usual expressions of the electric dipole transition probabilities in the "length" and "velocity" forms. The probabilities of the non-relativistic magnetic multipole ( M k ) transitions are described by the formula: 1

W ' ~ = 2(k k[(2k + 1)(2k + +1)!!] 21) (c¢°) 2k+

I(%1mQ(-~q~l4")12

(27)

Here the non-relativistic operator of the Mk transitions is as follows:

mQ~_k)q=__ikrk-'?k(2k__l) ( ~ l c

L(1)](k)q_[c(k_l)X _

S (') ](k)

}

,

(28)

where L (° and S (') stand for the orbital and spin angular momenta, correspondingly. The expressions for the leading relativistic corrections to the non-relativistic operators (25) and (26) are presented in ref. 11. The formulas for the matrix elements of both relativistic and non-relativistic operators of electronic transitions between the complex electronic configurations may be taken from refs. 12 and 13. Figs. 1 and 2 illustrate the dependence of the oscillator strength of the E1 transitions lsE2s 2 'S 0 ,-- l se2s2p 'P~ in Be isoelectronic sequence on gauge K. Experimental values of the oscillator strength are plotted in fig. 1 as well (C +2 [14], Ne +6 [15], S +'z [16]). The comparison shows that good agreement

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295

1

C'2

//

0.5

2i

t

2

3

~

5"

6

7-

g

9

tO

ft

I~'

t3

-q

-7-

7-

'1

~

~'

Io It

t~

11, "~9 20

Fig. 1. Comparison of theoretical (DHF) and experimental (asterisks) oscillator strength values for El transition I s22s2p t P I - I s 22s 2 ISo. Fig. 2. Dependence of the oscillator strengths o f the E 1 transition 1s 22s2p I P 1-1S 22s2 IS0 in the Be isoelectronic sequence on gauge condition K ( D H F approximation).

between theoretical and experimental values of oscillator strengths is achieved if K = 1-2. Fig. 2 shows that with the increase of the degree of ionization the dependence of the oscillator strengths on the gauge diminishes; the corresponding curve for W ÷7° and U +Ss approaches a line parallel to the abscissa. Hence, this shows once more that the D H F approximation describes fairly well the structure and the properties of very highly ionized atoms. By the way, it would be interesting to obtain the analogous general expressions for cross-sections of some elementary processes in plasmas (excitation etc.).

4. Conclusion

A collection of general automated algorithms and programs in FORTRAN has been worked out at the Institute of Physics of the Academy of Sciences of the Lithuanian SSR, use of which enables one to calculate energy spectra, oscillator strengths and transition probabilities of the electric and magnetic multipole radiation in the abovementioned approximations for practically any atom and ion of the periodical table. The radial integrals needed may be calculated using numerical (non-relativistic and relativistic versions of the Hartree-Fock equations) or analytical radial wave functions. Below we shall discuss briefly some peculiarities and results of their practical use. The problem of the identification and classification of the energy levels was already considered in section 2. It should be noted that when calculating the electronic transitions in the intermediate coupling scheme one has to use the wave functions (15) in the formulas of the type (23) and (24). Then, for example, the E1 transition probability may be written as follows (E is in a.u., W E1 is in s--l): ~,17EI --

2142 X 107 . -3 2J+i( A E ) I(flJIIQO)llfl'J')l

2-

2142×2j+ 107AEI(flJIIQ'°)IIB'J')I21 ,

(29)

where the reduced matrix element, e.g., of the operator Q(k), is of the form (/3J 11Q(k)II8'J')

=~

aiaj(

~i( J )11Q(k)[I I~j(J') ).

(30)

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Z.B. Rudzikas / Problems of theoretical interpretation

In the intermediate coupling scheme only the selection rules for J and parity remain exact, the rest are valid approximately. For instance, while making such calculations starting with LS coupling, the intercombination lines become allowed, for which AS v~ 0, the range of possible AL values widens significantly. The calculations show [11] that relativistic corrections for the E1 transition operator are as a rule very small, they are most essential for the weak intercombination lines of the light neutral atoms. The relativistic effects influence mostly the transition energies and for the very highly ionized atoms one has to calculate in the relativistic approximation both the energies and electronic transitions. Interesting regularities may be observed when both electric and magnetic transitions are allowed at the same time (e.g., E2 and M1). For the majority of transitions in neutral and not highly ionized atoms, according to the magnitude of transition probability, one type of radiation predominates. However, there are a few cases, when the contributions of both operators are of the same order. For not highly ionized atoms E2 transitions predominate as a rule. However, with the increase of the ionization degree, M1 transitions begin to prevail. Besides, the wavelengths of the usual E1 transitions move to the far ultraviolet or even X-ray spectral region. The lines of the E2 and M1 transitions occure in the visible wavelengths region, therefore they become very useful for diagnostics of high temperature plasmas. The use of the dependence of the electronic transitions on the gauge condition gives one the possibility of finding its optimal value, leading for certain kinds of transitions and types of wave functions to the best coincidence of theoretical and experimental characteristics of the radiation, as well as of treating it as a semiempirical parameter, particularly when dealing with isoelectronic sequences. Electronic configuration r is not always a good quantum number. Giving the single-configuration approximation up is one of the principal methods of accounting for the so-called correlation effects. Then an atom is described by the mixture of different configurations

~p(tcaJ) = ~ a(ffaJ)~p(~,aJ).

(31)

The contribution of each configuration is defined by the quantity l a(ffaJ)l 2 Usually the coefficient with = x is close to unity, the remaining ones being small. This means that the admixture of the other configurations to the one considered is insignificant. However, in a number of cases there is no prevalence of one coefficient over the others, then the single-configuration approximation is not suitable. First of all this concerns the so called quasi-degenerate configurations, having the same number of electrons for each value of the principal quantum number n (e.g., ls22s22p 2 and 1s22p4). For some Z values in isoelectronic sequences one may observe strong interaction of the electrons in the configurations of the type nol4t°+2nJff~n212, (no~ n o ) , even crossing the energy levels of these configurations. In the vicinity of these crossings the role of the superposition of the configurations of the same parity increases drastically, and the single-configuration approximation has no

......

~+p-0

Fig. 3. Qualitative change of the structure of the spectrum of the configuration ls22s2p in the Be isoelectronicsequence.

Z.B. Rudzikas / Problems of theoretical interpretation

297

sense. Let us discuss briefly the structure of multiply charged ions. The calculations show that the mean values of the distance of the electrons from the nucleus decrease very rapidly with the increase of the degree of ionization, the electronic shells are "compressed" under the action of the growing field of the nuclear charge [17]. For the very highly ionized atoms the character of the electronic configuration changes: it is possible to notice the splitting of the electronic shells into subshells. This testifies to the essential changes in the relative role of separate electronic interactions. In fact, as may be seen from fig. 3, on which the energy of the configuration ls22s2p is plotted schematically, taking for each ion the same value for the whole width of the energy spectra, at the very beginning we have practically pure LS-coupling, whereas for U + 88 purejj-coupling already takes place, the levels are grouped in two subconfigurations s+ p_ and s+ p ÷ . This conclusion is maintained by the data of table 2 as well. In conclusion, the present state of the theory and computational methods allows one to calculate the spectral characteristics needed for individual atoms and ions, their isolelectronic sequences, to study the regularities in the changes of the quantities considered along these squences. All this enriches our knowledge about the nature of the objects and phenomena under investigation. The accuracy of the calculations achieved enables one to determine with high precision the wavelength region in which the spectral lines of the ion of interest may be observed, as well as to identify their spectra successfully and to use the theoretically found spectral characteristics for diagnostics of both the laboratory and astrophysical plasmas, including thermonuclear ones.

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [ 11] [12] [13] [14] [15] [16] [17]

H.A. Bethe and E.C. Salpeter, Quantum Theory of One- and Two-Electron Atoms (Academic Press, New York, 1978). A.P. Jucys and A.J. Savukynas, Mathematical Foundations of Atomic Theory (Mintis Publ., Vilnius, 1973). Z.B. Rudzikas and J.M. Kaniauskas, Int. J. Quant. Chem. 10 (1976) 837. S.D. Sad~iuvien~ and P.O. Bogdanovich, Program of Calculations of Complex Energy Spectra of Many-Electron Atoms and Ions (Vilnius, 1980). U.I. Safronova and Z.B. Rudzikas, J. Phys. B10 (1977) 7. V.A. Boiko, S.A. Pikuz and A. Ya. Faenov, Preprint FIAN No. 20 (Moscow, 1976). I.S. Kichkin, A.A. Slepcov, V.I. Sivcev and Z.B. Rudzikas, Lietuvos fizikos rinkinys (Soviet Physics Collection) 16 (1976) 217. U.I. Safronova and Z.B. Rudzikas, J. Phys. B9 (1976) 1989. Z.B. Rudzikas and E.G. Savi~ius, Book of Abstracts of Europ. Conf. on Atomic Physics, vol. 5A, part 1, Heidelberg, (1981) p. 166. J.M. Kaniauskas, I.S. Kichkin and Z.B. Rudzikas, Lietuvos fizikos rinkinys (Soviet Physics Collection) 14 (1974) 463. J.M. Kaniauskas, G.V. Merkelis and Z.B. Rudzikas, Lietuvos fizikos rinkinys (Soviet Physics Collection) 19 (1979) 475. J.M. Kaniauskas and Z.B. Rudzikas, Lietuvos fizikos rinkinys (Soviet Physics Collection) 16 (1976) 491. G.V. Merkelis, E.G. Savi~ius, J.M. Kaniauskas and Z.B. Rudzikas, Spectroscopy of Multiply Charged Ions (Moscow, 1980) p. 66 (in Russian). L. Heroux, Phys. Rev. 180 (1961) 1. G. Beauchemin, J.A. Kernahan, E.J. Knystautas, D.J.G. Irwin and R. Drouin, Phys. Lett. 40A (1972) 194. D.S. Pegg, J.P. Forester, C.R. Vane, S.B. Elston, P.M. Griffin, K.-O. Groeneveld, R.S. Peterson, R.S. Thoe and I.A. Sellin, Phys. Rev. AI5 (1977) 1958. Z.B. Rudzikas, Proc. of the 6th Int. Conf. on Atomic Physics (Zinatne, Riga, 1979) p. 92.

IV. ATOMIC STRUCTURE CALCULATIONS