ENERGY LEVELS AND gf VALUES OF IONS IN INDEPENDENT PARTICLE MODEL CALCULATIONS

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ENERGY LEVELS AND gf VALUES OF IONS IN INDEPENDENT PARTICLE MODEL

CALCULATIONS

W. Eissner, M. Jones

To cite this version:

W. Eissner, M. Jones. ENERGY LEVELS AND gf VALUES OF IONS IN INDEPENDENT PARTI- CLE MODEL CALCULATIONS. Journal de Physique Colloques, 1970, 31 (C4), pp.C4-149-C4-154.

�10.1051/jphyscol:1970424�. �jpa-00213879�

JOURNAL DE PHYSIQUE

Colloqlrc C4, sirpplc't~letlt al: t1° 1 1-12. Tot~le 3 1. Nor.-Dkc. 1970. page C4-139

ENERGY LEVELS AND ^gf VALUES OF IONS IN INDEPENDENT PARTICLE MODEL CALCULATIONS

by W. EISSNER and M. JONES Department of Physics University College London

Rbum6.

Nous comparons les donnees atomiques, pour les ions, obtenues avec des fonctions d'onde radiales deduites du modele statistique, avec les resultats obtenus par d'autres methodes plus raffinkes et aussi avec les resultats experimentaux. Nous avons calcule les niveaux d'energie avec et sans corrections relativistes, et quelques forces d'oscillateur. Un programme, dkcrit par Eissner et Nussbaumer, a ete modifie et elargi pour ces calculs.

Abstract.

We compare atomic data for ions, obtained with statistical model radial wave functions, with results from more refined methods and with observations. We have calculated statistical model tertn

without and with relativistic corrections, and some oscillator strengths.

For this purpose a computer programme described by Eissner and Nussbaumer has been extended.

A. Introduction.

There is considerable interest in ionic radial functions for calculating atomic data, such as term energies, oscillator strengths, or collision cross sections. Layzer [ I ] showed with his Z dependent theory how important configuration mixing becomes in highly ionized atoms. We know from Bagus and and Moser's multi-configuration Hartree-Fock calcu- lations [2] how much effort is needed to get good term separations, and good energy ratios. The history of He calculations is well known. Seaton [3] recently com- mented on Hartree-Fock theory ; unrestricted Hartree- Fock theory is a special case of configuration-inter- action theory.

The emphasis in this paper is on radial wave func- tions, that can be generated quickly in a self-suffi- cient computer program and that are fairly reliable and adequate for approximate methods. in which they are applied. We would like to illustrate both the advantages and the limitations of radial wave func- tions calculated in a statistical model potential.

B. Term energies.

Eissner and Nussbaumer [4]

described a computer program for studying energy levels along isoelectronic sequences. It assumes eiectrostatic interaction and allows fully for configura- tion mixing. Its salient features are

it uses Slater state expansions to solve the algebraic (angular) problems, and statistical model radial wave functions. Corres- pondingly it consists of two primary branches. The first branch requires a list of configurations C,,jor the N electrons as input. It computes the structure of the resulting Hmatrix elements in terms of Slater integrals, which arise from the interelectronic interaction,

R,(ab, cd)

J d r , J dr2 Pa(rl) Pb(r2) x

0

and of integrals representing the kinetic energy and the potential energy of the N electrons in the field of the electric charge Z of the nucleus,

( ~ 2 ) Following [4] we write

< CzSLMs ML 1 H I C' a' Sf L' Ms, ML, >

2 ~:b,~,.(ab, cd ; 1) R,(ab, cd)

-

\

- C.d E C'

We shall sometimes use the abbreviation

With these data stored one may repeatedly cycle through the second primary branch, which essentially requires Z as input. A scaled statistical model poten- tial V ( Z , N, I

r) of Thomas-Fermi-Dirac type

see G o n ~ b i s [5]

is integrated from a potential equation, subject to the physical conditions

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1970424

C4- 1 50 W.

EISSNER AND M.

Z 2 - N + 1

lim V(r)

, lim V(r)

r-+O

r

r . (B5)

When computing bound state radial functions Pn Ar) = rRnLr)

as solutions of

one may use different scaling factors 1, for func- tions P,, t o different angular quantum numbers I.

But since orthogonal one-particle functions are required when deriving (B3), the same potential V(r) must be used for radial functions with the same 1.

With the integrals (B1) and (B2) computed the SL- matrices are evaluated and diagonalized :

The scaling parameter R, in the potential V may be chosen variationally,

where

g i

( 2 S i + 1) ( 2 L i + ^{1 ) .} (B9) One minimizes over the JL lowest terms ; the para- meter JL is input. This facility allows in (B8) to leave out terms that would be buffered by terms arising from configurations not included.

In Table I we compare results obtained from this program with Bagus and Moser's very satisfactory multiconfiguration Hartree-Fock calculations [2] for the carbon sequence. We have picked out a lowly ionized member, 0 111 (upper entries), and a medium ionized element, Si IX. A clear pattern with not unexpected trend emerges. Bagus and Moser's emphasis is on obtaining good term separations ((( equal error approximation

rather than good correlation ener- gies. Our third combination of configurations, which has no equivalent among their examples, is quite instructive. The configurations C,,

1 s2 2 s2 2 p 3 p and C , ,

1 s2 2 s 2 p2 3 s of the complex (2, 3, l), which has completely been included, couple via a term Ac,,c,,, in (B3) with the Hartree-Fock configura- tion C,. Terms of this type usually dominate off- diagonal elements. However, one finds from [4] that the absolute value of A does not depend upon SL, if CSL is not degenerate (and if it is, then the sum over the squares is independent of SL) ; thus the effect of including C,, and C,, is to shift each term of the ground conj?guration by a similar amount. Bagus and Moser discussed criteria for selecting configurations in previous papers. We compare u~itk their combina- tions (i), (ii), and (iv).

There is surprisingly good agreement for the biggest mixing coefficients, from the Hartree-Fock configuration C, itself and from the quasi-degenerate configuration C2

I s2 2 p4. O L I ~ method overesti- mates the contribution of the ground state configura- tion to its ow11 terms slightly ; so a considerably smaller portion remains for the other configurations.

The limitations of the method described so far with regard to increasing Z are obvious in Eissner and Nussbaumer's f i g ~ ~ r e s in 141. Disagreement grows in a regions where the radial functions should become rather more reliable.

C. Relativistic corrections.

Jones [6] has impro- ved Eissner and Nussbaumer's results [4] considerably by including relativistic corrections of the low-Z Pauli Hamiltonian

His one-particle-operators are the mass correction, the Darwin term, and the classical spin-orbit interaction with the field Z of the nucleus

Among the two particle operators only the spin- other-orbit interaction has been taken into account, because it has an important effect on the accuracy of the intermediate wave functions.

gij(so 4 so')

rij3(vij x pi).& + ²

+

The H matrix elements, in a CuSLJM, representation, are reduced in a Slater state expansion, with states CuSLJM, constructed from CaSLMs M, by using a Clebsch-Gordan formula. Jones evaluates the two- particle interaction g i j in the same approximation as Blume and Watson [7]. However, he allows for more than one open subshell nl ; and he includes contribu- tions to off-diagonal elements, < ^C ... ( H,,, ( C' ... >.

This is a complete extension to a configuration mixing program.

The spin-orbit parameter 5 is the radial integral resulting from the separation and reduction of

and it is defined by

Col~~parison o f t~i~rlticorzfiSuratiotl results for 0 I11 (first entries) and Si IX (lower lines), for the 3 rernls o f ' 1 s' 2 s' 2 p'. Columns headed i. and i. refer to the present calculations, while headings ( I ) , (II), ( I V ) refer to Bagus and Moser's paper [2]. { S L ) sfanris for E(SL), < ^{S L} I CSL > ^{is tlle}

- coeficient < i 1 t > in eqlration (B7) i. m e a ~ u i.,

R,

.... in (B 6), while

i. indicates independent variation of A, and I.,

A,

... in equation (B8) ; the A

3 ground confguration terrns have been nzinimized

N"N,.,

j z % v I c

I

N N N N C2, C 5 = 1 sz 2 s2 3 p2 C1 = I s2 2 s2 2 pr ^' C 1 , C z = I sz 2 p4 ^{N N N N} L n ( n * *

i

Configurations

----

included :

1 ,: 1; '!

I

x 2 e V

I !

I i uu"

i

!-Ip-

¹

1.8 I 2 2 6 1.279 1.230 1.271

1.252

i

H F

,

1

^j

Z both the spin-other-orbit and spin-spin interaction.

(C6) Jones follows them in this.

Defined in this way, i accounts fi)r the interaction of the spin of an opcn shell electron with its orbit in the field of the nucleus only, and omits the effect of both, outer shell-core interactions and outer shell-outer shell interactions. These are two-body interactions resulting from spin-other-orbit and spin-spin interactions.

Blume and Watson [7] have shown that

can be evaluatcd muc11 more accurately if contributions from the spin-other-orbit inter:~ctioris in g i j arc also included.

They neglcct the

rcsidual intcraction

wliicll is the interaction between open shell electrons arising I'ronl

Term cncrgics and tlic coupling matrix are worked out just as in part I3

the matrix elemcnts of the total Hamiltonian must be calculated with radial functions defined by (B8). It would be completely wrong to minimize tile H 111atrix including H,,,, (Cl). Bethe arid Salpeter's [8] argument for using the Brcit inter- tion as a perturbation only, holds mutatis mutandis when applying the variational principle hcre.

In figure I results along the sodium sequence are

cornparetl with prwious calculations ([4] and God-

frcdscn

a n d with observations compiled by

Charlotte kloorc [ l o ]

tlic centre

gravity is usctl.

C4-152

W. EISSNER AND M. JONES

FIG.

- Sodium I Isoelectronic Sequence, 6E versus Z for the transitions 3 p 2P-3 s 2S and 3 d 2D-3

zP. Experi- ment, - - - - Non-relativistic calculation, - Relativistic

Calculation.

Clearly, it would have been quite inadequate to fit the scaling parameters in calculations without relativistic corrections to obserue'd data.

Figure 2 shows the term separations 6E(2P112 - 2S,,2) and 6E(2P312 - 2S,,2) as a function of Z ; the dots refer to observations [lo].

Figure 3 compares observed spin-orbit parameters with the ones calculated in the present wol k, with the ones calculated by Condon and Odabasi [l I], and with those calculated by Froese [12]. Condon and Odabasi define their spin-orbit parameter thus :

1t1lier.e V is the Hartree potential. This is equivalent to calculating the direct part of the Blume and Watson expression. Froese's calculations include the full Blume and Watson treatment.

D. Electric radiation data.

The first primary branch of the structure program [4] has been extended to compute electric multipole transition coefficients

in a configuration representation. Oscillator strengths or gf values

for purely electric radiation of order 2k, E$'. may be expressed in terms of reduced radiation elements,

FIG.

2. - Sodium I Sequence, 6E v. Z for the transitions 3

2pi/2-3 s 2S1/2 and 3

2P3/2-3 s 2S1/2. Experiment,

Relativistic Calculation.

FIG. 3.

Sodium I Sequence,

versus Z. Expe- riment

- - - - Condon and Odabasi

l], - - - - - - Froese

[121, - Present Calculation.

ENERGY LEVELS A N D

VALUES O F IONS

where

and

< ^{CaSL 11 E ' ~ ' 11} C' a' St L' >

= 6,,, 1 ^{< CaSL I( C'k'(nl, n' 1') I1 C' a' SL' > x}

n l s C n'l' E C'

In the case C' # C the sum reduces to not more than one element

see dipole selection rule for pure configuration transitions ; the two configurations differ in the pair

17' 1'). To calculate transitions between fine structure levels CaSLJ one may apply

< CcrSLJ (1 c'~'( ...) 11 C' a' SL' J' >

x W(LJL1 J' ; Sk)

x < CaSL 11 c("( ...) 11 Ca' SL' > _(D5)

(Racah [13]) and use the analogue relation to (D2).

Therefore the algebraic problem reduces to computing the structure and the angular coefficients of (D4).

Derivation and computation is straightforward.

The F!,:! in (D3) are the factors to the one-particle operators

that forms the radiation operator. One expresses (D6)

k ( k )

in spherical tensor operator components, r C, .

Reduced matrix elements of a tensor operator are found by evaluating a single matrix element (for a transition between levels b

CcrSLM, M, and b') and by applying the Wigner-Eckart theorem. Nuss- baumer [14] describes a Slater state method for dipole transitions b

b'. A general formulation for matrix elements to any k follows from Racah's paper [13]. A Slater state method relieves from the cumbersome labour of checking phase conditions.

In the second primary branch of [4] gf'values are evaluated using statistical model radial functions in (D4). Radial functions based on (B6) automatically satisfy the orthogonality requirements, H ' I I ~ I E J Hartree- Fock functions for initial and final states would have 10 be ortliogonalized to each other. This involves yet another configuration expansion.

In Table I 1 some 0 111 results are compared with observations from Wiese, Smith and Glennon's

Wavelengt/~s and absorption oscillator strengths for 0 111, compared wit11 Wiese, Smith and Glennon's tables [15]. 0 TI1 is calculated using configurations C , and C2 of Table I , together with conjiguration C'

1 s 2 2 s 2 p 3

Transition 2 s' 2

SL-

2 s 2 p 3 S L '

--

-

3P 3D0

3P 3PO 3P

ID

'DO

ID

]PO

'S 'PO

f (absorption) present obser-

work ved

[I51

- -

0.148 0.15 0.182 0.18 0.202 0.19 0.391 0.37 0.254 0.25 0.374 0.35

present AIA obser-

work ved

[I51

821 834.50 695 703.36 480 507.93 557 599.598 496 525.795 544 597.818

Fine structure transitions in Si X ,

gf values compared with Garstang's calculations [16]

Transition

present work

-

< 10-5

0.000 01 0.000 02 0.000 01 0.000 10 0.167 0.042 0.31 0.080 0.066 0.34 0.18 0.17 0.76

gf

Garstang

[I61

2.9 x 10-5 1.4 x 10-5 2.0 x 10-6 3.0 x 10-5 1.3 x 10-5 0.18 0.027 0.31 0.087 0.053 .24 .I8 .13 .72

tables [I51 ; they label them as

E

more than 50 % uncertainty. We also calculated neutral helium, where they have compiled extremely precise experi- mental data. This is a most unfavourable case for a statistical model potential, j3et it gives quite tolerable results except in the most sensitive cases, wliicli differ by a factor 2. Most gf's for dipole transitions

1s nl

Is

up to n

4 are wrong by less

than 20 %. Finally Table 111 contains some prelimi- nary results for fine structure transitions, compared with Garstang's results [16].

Acknowledgements.

We wish to thank Prof.

M. J. Seaton for advice and constant encouragement.

W. Eissner has held a fellowship under the exchange

arrangement between the Deutsche Forschungs-

gemeinschaft and the Royal Society, and M. Jones

has held a Science Research Council Award, while

this work was done.

W.

EISSNER AND M. JONES

References [I] LAYZER (D.), Ann. Phys., N. Y., 1959, 8, 271.

[2] BAGUS (P. S.) and MOSER (C. M.), J. Phys. B, 1969, 2, 1214.

[3] SEATON (M. J.), Comments on Atomic and Mole- cular Physics, 1970, 1, 177.

[4] EISSNER (W.) and NUSSBAUMER (H.),

Phys. B, 1969, 2, 1028.

[5] GOMBAS p.), Handb. Phys., 1956, 36, 109 (Berlin

Springer-Verlag).

[6] JONES (M.), to be published in J. Phys. B.

[7] BLUME (M.) and WATSON (R. E.), Proc. Roy. Soc.

(London), 1962, A 270, 127.

[8] BETHE (H. A.) and SALPETER (E. E.), 1957, Quantum mechanics of one- and two-electron atoms, (Berlin

Springer-Verlag).

[9] GODFREDSEN (E.), Ast~.ophys. J., 1966, 145, 308.

[lo] MOORE (C. E.), 1949, Atomic Energy Levels, Natn.

Bur. Stand. Circ. No. 467 (Washington

U. S.

Govt Printing Ofice).

[ l l ] CONDON (E. U.) and ODABASI (H.), 1966, Spin-orbit interactions in self-consistent fields, JILA rept.

No 61, University of Colorado.

[I21 FROESE (C.), Can. J. Phys., 1967,

1501.

[I31 RACAH (G.), Phys. Rev., 1942, 62,438.

[14] NUSSBAUMER (H.), Mon. Not. R , astr. Soc., 1969, 145, 141.

1151 WIESE (W. L.), SMITH (M. W.) and GLENNON (B. M.), 1966, Atomic Transition Probabilities, Vol. I Natn. Bur. Stand. NSRDS-NBS4 (Washington

U. S. Govt Printing Office).

[16] GARSTANG (R. H.), Ann. Astrophys., 1962, 25, 109.