JOURNAL OF THE OPTICAL SOCIETY OF AMERICA

Configuration Mixing in the N=4 Electron Isoelectronic Sequence*HALis ODABASI

Joint Institute for Laboratory Astrophysics, University of Colorado, Boulder, Colorado 80302(Received 6 December 1968)

The theoretical central-field-approximation predictions for the 3P and 1P terms of the Is2 2s 3p configura-tion in the N= 4 electron isoelectronic sequence are in violent disagreement with observation. Correlationeffects in the form of configuration interaction have been included by considering the mixing among nineodd configurations. Even though the terms are displaced considerably, they remain quite pure, and afterconfiguration mixing the agreement between theory and observation is good. In order to show the use of suchtheoretical calculation, weighted oscillator strengths for the 2s2 1 Si-2s3p 'Pi, 'Pi transitions are alsocomputed.

INDEX HEADINGS: Spectra; Sources.

For the last decade, correlation effects in atoms havebeen heavily emphasized in atomic and molecularphysics. As is well known, the central-field approxima-tion contains only a small portion of the interelectronrepulsion, that is, the sum of ri;' terms in the N-electron hamiltonian. The remaining part is included inthe form of perturbation. Sinanoklu's1 many-electrontheory (MET) and the Z-dependent theory introducedby Hylleraas2 and developed by Layzer3 and others4

are among the most important achievements on thesubject. The configuration-interaction approach5 is oneof the oldest and simplest of the existing theories.The main objection to the method of configurationmixing has been that it requires the inclusion of a largenumber of configurations and, as a result, quite a lotof labor. Second, it does not give physical insight intothe problem. Indeed, to account for the configurationmixing of the terms, adjustable parameters have often

* This research was supported by the Advanced ResearchProjects Agency of the Department of Defense and was monitoredby the Army Research Office-Durham under Contract No. DA-31-124-ARO-D-139.

l 0. Sinanoglu, J. Chem. Phys. 33, 1212 (1960); Proc. Roy.Soc. (London) A260, 379 (1961); Phys. Rev. 122, 491, 493 (1961);Proc. Natl. Acad. Sci. (U. S.) 47, 1217 (1961); J. Chem. Phys. 36,706, 3198 (1962); Advances in Clhemtical Physics, I. Prigogine, Ed.(Interscience Publ., John Wiley & Sons, Inc., New York, 1964),Vol. VI, p. 315; 0. Sinanoglu and D. F. Tuan, J. Chem. Phys.38, 1740 (1963); 41, 2677 (1964); V. McKoy and 0. Sinanoilu,J. Chem. Phys. 41, 2689 (1964); 0. Sinanoilu and I. Oksuz, Phys.Rev. Letters 21, 507 (1968).

2 E. A. Hylleraas, Z. Physik 65, 209 (1930); E. A. Hylleraas andJ. Midtdal, Phys. Rev. 103, 829 (1956).

a D. Layzer, Monthly Notices Roy. Astron. Soc. 114, 692 (1954);Ann. Phys. (N. Y.) 8, 27 (1959); Phys. Rev. 132, 735 (1963);D. Layzer and J. Bahcall, Ann. Phys. (N. Y.) 17, 177 (1962).

4 Ch. W. Scherr, J. Chem. Phys. 317, 33 (1960); Z. Hordk,Proc. Phys. Soc. (London) 77, 980 (1961); J. Linderberg, Phys.Rev. 121, 816 (1961); C. S. Sharma and C. A. Coulson, Proc.Phys. Soc. (London) 80, 81 (1962); W. B. Sommerville and A. L.Stewart, Proc. Phys. Soc. (London) 80, 97 (1962); C. S. Sharma,Proc. Phys. Soc. (London) 80, 839 (1962); R. Perrin and A. L.Stewart, Proc. Phys. Soc. (London) 81, 28 (1963); R. J. S.Crossley and C. A. Coulson, Proc. Phys. Soc. (London) 81, 211(1963).

6 E. A. Hylleraas, Z. Physik 48, 469 (1928); 54, 347 (1929);E. U. Condon, Phys. Rev. 36, 1121 (1930); E. U. Condon andG. H. Shortley, The Theory of Atomtic Spectra (Cambridge Univ.Press, New York, 1935), Ch. XV.

been introduced6 ; though the inclusion of more andmore parameters improves the agreement betweencalculated and observed term energies, there is noassurance that the chosen values of the parameters arerealistically related to the ones given by theory.Therefore, a purely theoretical approach7 is moreinstructive. Zare's5 recent work, employing hfs basissets, shows that in the age of computers, configuration-mixing calculations are not more time consuming thanany other existing methods and it is possible to get asmuch physical information as most of the otherapproaches yield.

During a survey' of the N= 4 electron isoelectronicsequence we found that the observed term energiesof the iS2 2s3p configuration are in violent disagreementwith theoretical expectations. The theory, withoutconfiguration mixing, predicts that the 3P term alwayslies lower than the '1, but the observations show thatit is so for only the first two elements, Be i and B II,and that the 1P term lies lower than the 3P for the rest.This presents one of the best-known examples of con-figuration mixing.

The primary purpose of this work was to explaintheoretically the observed behavior of the 3P and 'PRussell-Saunders terms in the 2s3p excited configura-tion of the N= 4 sequence. After that was done, werealized that it is possible to calculate the weightedoscillator strengths for the transitions between the

R. F. Bacher and S. Goudsmit, Phys. Rev. 46, 948 (1934);R. E. Trees, Phys. Rev. 83, 756 (1951); 84, 1089 (1951); 85,381 (1952); R. E. Trees and C. K. Jorgensen, Phys. Rev. 123,278 (1961); 129, 1220 (1963); R. E. Trees, J. Opt. Soc. Am. 54,651 (1964); G. Racah, Phys. Rev. 85, 381 (1952); G. Racah andY. Shadmi, Phys. Rev. 119,156 (1960); S. Meshkov, Phys. Rev.91, 871 (1953); 93, 270 (1954); S. Meshkov and C. W. Ufford,Phys. Rev. 94, 75 (1954); N. Sack, Phys. Rev. 102, 1302 (1956);K. Rajnak and B. G. Wybourne, Phys. Rev. 132, 280 (1963); 134,A596 (1964); J. Chem. Phys. 41, 565 (1964); B. G. Wybourne,Phys. Rev. 137, A364 (1965); K. Rajnak, J. Opt. Soc. Am. 55, 126(1965).

7 R. K. Nesbet and R. E. Watson, Phys. Rev. 110, 1073 (1958);R. E. Watson, Phys. Rev. 119, 170 (1960); A. W. Weiss, Phys.Rev. 122, 1826 (1961); H.4P. Kelly, Phys. Rev. 131, 684 (1963);136, B896 (1964).

8 R. N. Zare, J. Chem. Phys. 45, 1966 (1966); 47, 3561 (1967).9E. U. Condon and H. Odabasi, JILA Report No. 95 (1968).

583

VOLUME 59, NUMBER 5 MAY 1969

HALIS ODABASI

TABLE I. Matrix elements of configuration interaction amongthe 2snp, 2pus, and 2pnd, odd configurations for the 3P and 1PRussell-Saunders terms.

2snp: 2snp

2puts: 2puts

2pnd: 2pnd

2snp: 2sinp

E(up)Ep(us)

E (nd)

E(up,nip)

2snp: npins E(2s,mns)

2snp: it'pins E" (m1s,1up)

2snp: 2pind E" (up,vid)

2puts: 2pins E(Lus,111s)

2pus: 2pizd E" (its,ind)

2pnd: 2pind E(id,ind)

E(2d) = E+E'(Wt) +E" (il)

E=2I(ls)+F0 (ls,ls)

E'(its) = I (2p) +1 (ts) +2F0 (1s,2p) -:3G' (1s,2p)+2F0 (ls,ns) -GO (ls,us)

E'(up) =1I(2s) +I(up) +2F (1s,2s)-G'(1s,2s)+2F°O(ls,np) - 13GI (1s,tP)

E' (Ed) =I(2p) +I(Ed) +2 P(1s,2p)-L Gl (Is,2p)+2F0 (ls,upd)--lG2(ls,nd)

E" (us) = F° (2p,its) L -3 GI (2p,us) I

E" (up) = F0 (2s,utp) -4-i3G' (2s,np)

E" (nd) = F1 (2p,nd) ±lF2 (2p,nd) i (1/15)GI (2p,nd)-l ) (9/35)G3(2pnd)

E' (us,ins) =I (uts,ms) -+2R0 (its,ls; ns,ls) -RO (ns,ls; ls,vts)

E(utp,inp) =I(Iup,vp) +2R 0 (utp,ls; vtp,ls)--3R'(atp,ls; ls,inp)E'(ud,mzd) =I (nd,ind) +2R 0 (nd,ls; md,ls) - R 2 (nd,Is; ls,1md)

E" (ns,ins) t=RO(2p,ns; 2 p,ms) +4RI (2p,us; mus,2p)

E" (1p,mup) = RO (2s,utp; 2s,nup) 4- R' (2s,utp; iup, 2s)

E" (nd,md) = RO (2p,nd; 2p,ind) +' ;R 2(2p,nd; 2p,md)(l1/l5)Rl (2 p,izd; m1d ,2 p) -4 (9135)R3 (2 p,nd; inzd,2 p)

E(2s,ms) = -4[1(2s,mns) +2R'(2s,ls; its,ls)-R0(2s,ls; ls,ius)+RO(2p,2s; 2p,ms)-4&R'(2p,2s; ms,2p)]

E" (ts,mp) = 4RO(2p,us; utp,2s)+xR1 (2p,uts; 2s,vtp)

E" (uts,ind) = - (V2/5)R 2(2p,us; 2p,mnd)OF (V2/3)R' (2p,ns; ind,2p)

E" (np,mnd) =- (v2/5)R2 (2s,np; ind,2p): (\Z/3)Rl(2s,itp; 2p,mid)

The upper signs are for the 1P term.

ground and the 2s3p excited configurations with theassumption that the 2s2 ground configuration does notmix with the other even configurations such as 2sns andnW2 (there are theoretical calculations' showing that theground state remains quite pure for Be). The predic-tions of the more dependable oscillator strengths con-stitute one of the very useful applications of this typeof theoretical calculation.

METHOD OF COMPUTATION

The theory of configuration interaction for the caseof two electrons outside a closed core is given by Zare.8

Therefore, we do not repeat it here. For a given term,the matrix elements between interacting configurationsof the same parity are radial integrals with fixedcoefficients that come from the angular part of thebasis sets (angular correlation).

The choice of a limited number of the most effectiveconfigurations to include in the calculation presents amajor problem. Although perturbation theory indicatesthat the most mixed states are those lying closest toeach other, there is another factor to consider, which isthe angular correlation. Probably a dependable ap-proach is to include the configurations one at a timeand, according to their effects, retain or discard them.Doing so, we decided to include nine odd configurationsfor the problem at hand. These are: (IS 2) 2s2p, 2s3p,2p3s, 2s4p, 2p4s, 2s5p, 2p5s, 2p3d, and 2p4d. Thematrix elements among these nine configurations arepresented in Table I. The radial integrals involved are(in atomic units)

I Enadf 1(+)f R2Zr2 by Ldrl r2 r

X [U(r)+ (Z/r)lRnbl(r)dr,

1(a)=I(a,a),W x r<k-

Rlk (asb; c,d) = / ;-R,, (r,)R, (ri)Joar>k~l

XRb(r 2)Rd(r 2)drldr2,

Fk(a,b) = Rk(a,b; a,b) and Gk(a,b) = Rk(a,b; b,a),

(1)

(2)

(3)(4)

where Ra (r) is the radial part of the basis set defined as

Ra (r) = rPa (r); (5)

e., is the one electron energy and U(r) is the centralfield potential.

These radial integrals were evaluated for the twelveelements in the isoelectronic sequence by using hfs10 1

wave functions and a modified program first writtenby Zare. Then the matrices were diagonalized. Thediagonal elements are the term energies with configura-tion mixing, and rows of the unitary matrices whichare used in the diagonalization process are the eigen-vectors. The wave functions after configuration mixingare linear combinations of the initial basis sets,

Zi= aiibj,iq

(6)

where aij is an element of the unitary matrix A thatdiagonalizes the energy matrix HI; that is, ASIA is

"J. C. Slater, Phys. Rev. 81, 385 (1951).11 F. Herman and S. Skillman, Atomuic Structure Calculations

(Prentice-Hall, Englewood Cliffs, N. J., 1963).

584 Vol. 59

M 1 CONFIGURATION MIXING IN N=4 SEQUENCE

diagonal (j is transpose of A). We call aj "mixingcoefficients." Because of the properties of unitarymatrices, the eigenvectors are orthonormal.

The line strength in the Russell-Saunders couplingscheme for electric dipole transition is defined as' 2

SL(J,J')= I (0SLJI!P(1) jj0'S'L'J')1 (7)

(We distinguish between multiplet and line strength byputting a subscript M or L to S.) Then for the specifictype of transition that we are considering

SL1= SL1 (nl n'l' SLJ; ni n"l" SL'J')

= (- 1)ML+LI+S+J+1"+1[(2L+ 1) (2L'+ 1)

L IIf/ I

()L+LI+S+J+l E(2J+ 1) (2J'+ 1) 12S+ 1

[L J S]X ' SM2 (8)

J' L' Iwhere

SM1 = (-1) +l>[l> (41>2-1) (2S+1) (2L+ 1) (2L'+ 1)]1

Vl L I]X} (9)

in which 1> stands for the greater of 1' and 1". [For atransition of the type 12 -4 11', it is necessary to multiplythe right-hand side of Eq. (9) by \r and change 1' toI and 1" to 1'.]

When the dipole length definition is chosen,

(n .02CDWm0c>_

W C

w -.02z

w - .04

-.06[

4 6 8 10 12z

FIG. 1. The theoretical term energies before and after configura-tion mixing, relative to the old configuration average (configura-tion average before mixing) for the Is' 2s3p excited configurationof the N = 4 sequence. E Theoretical, before configuration mixing.

Theoretical, after configuration mixing.

L-S couplings respectively, and the matrices A and Bare the unitary matrices which diagonalize initial- andfinal-state hamiltonians (as before AHI is diagonal).

The oscillator strengths in terms of dipole integrals,or multiplet strengths, are defined' 4 as

D

[41>2- 1]ID= f RnrRn'1 'dr. (10)

Inclusion" of the magnetic effect in the hamiltonianof the system presents off-diagonal elements, that is,the hamiltonian is no longer diagonal with respect tothe quantum labels L and S. On the other hand, thetotal quantum number J and parity are still good labels.The diagonalization of the energy matrix introducesanother mixing among the states with the same J butdifferent L and S. The new wave functions are the linearcombinations of the pure LSJM ones. This is calledintermediate coupling; now the line strengths are

(SL)IC= B (SL6)LSA, (11)

where subscripts IC and LS stand for intermediate and

12 B. W. Shore and D. H. Menzel, Principles of Atomic Spectra(John Wiley & Sons, Inc., New York, 1968), pp. 449-464.

13 E. U. Condon and G. H. Shortley, TDe Theory of AtomicSpectra (Cambridge Univ. Press, New York, 1935), pp. 264-271.

a(1',lt)= 3 (E SM,/X-E,)[0E/(21'+1)]D2

= 0.03038 Sm/XE, (12)

where E.,,, and En,,, are the energies of lower andupper states respectively and X is the wavelength of thetransition, between the same states, in microns.

A more useful quantity is the weighted oscillatorstrength" which is symmetrical between emission andabsorption. Furthermore, it is additive for lines andmultiplets

gtf=E fg,J

wheregt = (2L+ 1) (2S+ 1)

andg= (2J+1)

(13)

(14a)

(14b)

are statistical weights.

'4 S. Chandrasekhar, Astrophys. J. 102, 223 (1945).1" C. W. Allen, Astrophysical Quantities (The Athlone Press,

London, 1963), p. 58.

.06

.04

585May 1969

L J S VX (2J+ 1) (2J'+ 1) ji J' Ll L Ll

X

TABLE II. Coefficients of configuration mixing for the 3P and Ip terms in the is2 2s3p excited configuration of the N=4 sequence.(First entry is for the 3P and second is for the 1P term.)

a ai X22 a2 3 a 2 4 a 2 5 a 2 6 a27 a28 a29

Z=4,Be i -0.06191 0.98163 -0.02755 -0.14555 0.01804 -0.06731 0.01041 -0.07351 -0.015890.33660 0.86791 -0.16049 -0.30050 -0.02834 -0.12301 -0.01765 -0.06889 -0.03303

5,B II -0.02848 0.98863 -0.05926 -0.08222 0.00884 -0.03943 0.00589 -0.09879 -0.010270.07359 0.95692 -0.25712 -0.07439 -0.04503 -0.03640 -0.02331 0.05744 -0.00759

6,C iII -0.01586 0.98938 -0.07243 -0.05709 0.00472 -0.02780 0.00369 -0.10737 -0.005250.03008 0.95429 -0.27878 -0.04790 -0.04061 -0.02428 -0.02004 0-.07581 -0.00457

7,N iv -0.00995 0.98941 -0.07882 -0.04370 0.00281 -0.02146 0.00258 -0.11118 -0.002150.01593 0.95133 -0.29036 -0.03675 -0.03585 -0.01887 -0.01720 0U08435 -0.00410

8,0 v -0.00675 0.98935 -0.08230 -0.03536 0.00181 -0.01745 0.00195 -0.11312 -0.000280.00960 0.94913 -0.29781 -0.03018 -0.03180 -0.01559 -0.01497 0.08920 -0.00403

9,F vi -0.00484 0.98930 -0.08436 -0.02968 0U00124 -0.01470 0.00155 -0.11423 0.000870.00625 0.94749 -0.30311 -0.02574 -0.02846 -0.01333 -0.01321 0.09229 -0.00401

10,Ne vII -0.00361 0.98926 -0.08565 -0.02555 0.00089 -0.01270 0.00129 -0.11489 0.001600.00428 0.94622 -0.30706 -0.02249 -0.02570 -0.01167 -0.01181 0.09440 -0.00397

11,Na viii -0.00277 0.98923 -0.08649 -0.02243 0.00066 -0.01117 0.00110 -0.11531 0.002050.00304 0.94520 -0.31015 -0.02000 -0.02341 -0.01038 -0.01067 0.09593 -0.00390

12,Mg ix -0.00218 0.98922 -0.08707 -0.01998 0.00050 -0.00997 0.00095 -0.11558 0.002330.00221 0.94438 -0.31264 -0.01802 -0.02148 -0.00936 -0.00973 0.09707 -0.00380

13,Al x -0.00175 0.98921 -0.08748 -0.01801 0.00039 -0.00900 0.00084 -0.11577 0.002510.00163 0.94368 -0.31472 -0.01639 -0.01983 -0.00852 -0.00893 0.09797 -0.00370

14,Si xi -0.00143 0.98920 -0.08776 -0.01640 0.00031 -0.00820 0.00076 -0.11589 0.002610.00121 0.94309 -031644 -0.01505 -0.01841 -0.00782 -0.00826 0.09867 -0.00359

15,P xisi -0.00118 0.98920 -0.08798 -0.01504 0.00025 -0.00753 0.00068 -0.11598 0.002660.00090 0.94258 -0.31794 -0.01391 -0.01718 -0.00723 -0.00768 0.09925 -0.00348

' (2A) = a2nb (2s2pA) +a224(2s3PA) +a2 3b (2p3sA) +a24b(2s4PA) +a25D(2p4sA)+a26'(2s5pA) +a 24(2p5sA) +a,23 (2p3dA) +a20 (2p4dA)

RESULTS AND DISCUSSION

The theoretical term energies relative to the oldconfiguration average before and after mixing, for theis 2 2s3p excited configuration, are graphed in Fig. 1.

I I I I I I I I I I I

1 , I , I , I ,I I )

in]m

o0

w

z

Lir-.

4 6 8z

10 12 14

FIG. 2. The observed and theoretical term energies, relative totheir configuration averages (theoretical configuration average isnow the one found after mixing) for the is2 2sp3p excited configura-tion of the N=4 sequence. A Observed. Theoretical.

Table II presents the mixing coefficients for the termsof the same configuration. Though the terms are con-siderably displaced because of the mixing among thenine configurations considered, they remain quite pure.Furthermore, the lP term is mixed more than the 3P

with terms of the other configurations. This is respon-sible for the observed phenomena mentioned at thebeginning of the paper. The term energies with con-figuration mixing were found relative to the new con-figuration average and are plotted in Fig. 2 with theobserved ones.'6 "17

The energy levels are computed by taking the theoret-ical term energies and adding to them the magneticinteraction. The levels with J= 1, that is 3P, and 'Pi,are mixed because of this interaction. This is taken intoaccount by diagonalizing two-by-two matrices givenby Obi and Yanagawal8 and corrected by Garstang.19

Such an approach ignores the matrix elements ofmagnetic interaction between different configurations,but when the contributions to these matrix elementsfrom the spin-orbit part of the magnetic interactionwere included the result did not change. The best wayto compare the effect of the magnetic interaction is toconsider the intervals. That is done in Fig. 3 for thetheoretical as well as observed levels.

The agreement between theoretical term energies

16 C. Moore, Atomic Energy Levels, Vol. 1, Nati. Bur. Std.(U. S.) Circ. 467 (U. S. Gov't. Printing Off., Washington, D. C.,1949).

'1 The newly determined values of x's in Moore's tables for theN=4 sequence are kindly provided to us by Professor B. Edlen.

18 S. Obi and S. Yanagawa, Pub]. Astron. Soc. Japan, 7, 125(1955).

'9 R. H. Garstang, Proc. Cambridge Phil. Soc. 57, 115 (1961).

586 HALIS OIDABASI Vol. 59

.C

_ (

May 1969 CONFIGURATION MIXING IN N=4 SEQUENCE 587

TABLE III. The theoretical weighted oscillator strengths, gif and gf, in the N=4'sequence. (First entry is gif value for the Is2 2s' IS- ls2 2s3p 1P transition and second is taken from Allen." Third and fourth entries are gf values for the is2 2s' 'SO- is2 2s3p 'Pi and theIs' 2s'So - Is' 2s3p 3Pi transition, respectively. The values in parentheses are the energy differences, E (3Pi) -E ('So), in Rydbergunits, used in finding the latter gf value.)

4,Be i 5,B II 6,C i1 7,N iv 8,0 v 9.F vi lONe vii 11,Na vii 12,Mg ix 13,AI x 14,Si xt 15,P XKT

0.299 0.045 0.180 0.279 0.350 0.403 0.446 0.479 0.505 0.526 0.546 0.561*-- ... 0.2 0.3 0.3 ... ... ... ... ... ..

0.299 0.045 0.180 0.279 0.350 0.402 0.443 0.471 0.489 0.496 0.494 0.4794.68XI0-9 6.95X10-6 3.10X10- 1.37X10-4 4.73X10-4 1.36X10-

3 3.40X10- 7.67X10-3 1.57XI0-2 2.97X10' 5.17X10' 8.19X10'

(0.53575) (1.31213) (2.36665) (3.69958) (5.31121) (7.20197) (9.3708) (11.8172) (14.5417) (17.5440) (20.8234) (24.3824)

Reference 15, pp. 68-69.

after configuration mixing and the observed ones in ' I I I I

general is good. But there is a disturbing fact, that is,theoretical and observed curves are not parallel, indicat-ing that the theory does not give the trend correctly. .8

The inclusion of the magnetic effect helps, but is notenough. Indeed the observations show an increasing n6amount of mixing along the isoelectronic sequence. Wethink that is due to the diving 2p3s and 2p3d con-figurations (these two configurations get lower and .4 -lower along the isoelectronic sequence). Because of theapproximations involved8 the theory does not predictit correctly. Furthermore, there might be some other .2

diving odd configurations which do not contributesignificantly at the beginning of the sequence but whoseimportance depends strongly on the nuclear charge.

Theory predicts the magnetic interaction energiesconsistently high. However, the trend is correctly -.2

I I I , I I I I I I I -.I

- - 4 I

4 6 8 10 12 14z

FIG. 4. The behavior of [ggf]* in the isoelectronic sequence.

2 P 3P)a given. This is evident from the fact that the theoretical

W 103- curves are parallel to the observed ones in Fig. 3.in -Because of the assumption that the ground con-

P -3p) 3 ~figuration remains pure the expression for the multiplet>, I 0 strength with configuration mixing is'0

SM(2s' IS; 21P)= fa2,SM'(2s2 15; 2s2p 'P)

t /X +a225M'(2s2 '5; 2s3p 'P)64 /-+a24SM1( 2 s2 IS; 2s4p lP)

±a26SM(2s2 IS; 2s5p 'PI1. (15)

IXl In order to find the desired multiplet strengths, the

X - proper dipole-length integrals were evaluated byemploying hfs basis sets and substituted in Eq. (9).The needed mixing coefficients were taken from Table

4 6 8 10 12 14 II. Then the weighted oscillator strengths, as given byz Eqs. (12)-(14), were computed; they are given in

FIG. 3. The observed and theoretical intervals of the 3P2, D Table III. Examination of these values reveals aRussell-Saunders level for the is2 2s3p excited configuration ofthe N=4 sequence. X Observed. Fj Theoretical. 20 R. N. Zare, J. Chem. Phys. 47, 3561 (1967).

HALIS ODABASI

peculiar behavior: for Be i the gif is larger than the onefor B ii. This is because, at the beginning of the se-quence, the contribution from the 2s' 'S-2s2p 'P transi-tion is very strong. When the square roots of theseweighted oscillator strengths are plotted against thenuclear charge they fall on a smooth curve (Fig. 4), asis to be expected. Fisher2 ' found the same behaviorin the NV= 13 sequence.

The magnetic mixing creates the possibility oftransitions between singlet and triplet systems. Theline strengths are computed as given in Eqs. (8) and(11). To be consistent with the previous calculations,in finding the weighted oscillator strength we need theobserved energy differences between 2s2 'S, and 2s3p 3P

21 C. Froese Fisher, J. Quant. Spectry. Radiative Transfer, 8,755 (1968).

levels, which are known only up to F vi. We extrap-olated these energy differences, up to P xii, by plottingthe difference between theoretical and observed energydifferences, that is,

[E(2s3p 3 P,)-E(2s2 'Sl)T.bS

-[E(2s3p 3PO)-E(2s2 iS1)]theor,

against the nuclear charge, which turns out to be linear.In Table III, the weighted oscillator strengths as wellas the energy differences between the 2s2 'S and 2s3p 'P1levels are presented.

ACKNOWLEDGMENT

The author wishes to express his gratitude to Dr.E. U. Condon and to Dr. R. H. Garstang for theirkind encouragement and valuable help.

JOURNAL OF THE OPTICAL SOCIETY OF AMERICA

588 Vol. 59

VOLUME 59, NUMBER 5 MAY 1969