Quantum defect method applied to oscillator strengths

J Quanr

Specrrosc

Radrar

Transfer

Vol 38

No

4. pp

281-293

1987

0022-4073<87

Pnnted tn Great Bntam

53 00 + 0 00

PergamonJournalsLtd

QUANTUM DEFECT METHOD APPLIED TO OSCILLATOR STRENGTHS R E H

CLARK

Los Alamos Nattonal Laborator\ (Recerred

I5

and 4 L

MERTS

Los Alamos December

NM 87545 U S A

1986)

Abstract-We

use Ideas from quantum defect theory to obtam approrlmate energy levels and osctllator strengths The results are easily obtamed reqwre httle computer capdbility and typlcally agree wth detailed calculations to better than 590 The method can be applied to cases ahere hqdropemc and scaling techmques are mappropnate

I

IN-I-RODUCTION

For many appltcatlons, a large number of energy levels and oscillator strengths are required This cn-cumstance frequently forces the user to choose between scaled hydrogemc or other approximate methods or else use rather expensive computer programs for calculatmg a database These databases may become large and cumbersome to mampulate Therefore, one frequently chooses to accept less accurate but efficient computational procedures to obtain estimates of energy levels, oscillator strengths, and cross sections In the present work, we apply quantum defect theory to the calculation of oscillator strengths. and energy levels The six-parameter fit provides matnx elements for a wide range of n. n’ transitions We present results for I4 different Rydberg senes 2

THEORY

One of the earher approximate methods fol obtaining oscillator strengths was published by Bates and Damgaard ’ These authors reahzed that frequently the most Important contnbutlon to the transItIon moment comes from the spatial region outslde of the core region, I e from the region where the solution of the Schrodmger equatton IS known to be the Coulomb functions However, for bound states. the core has the effect of introducing a quantum defect, which can be determmed by knowing the energy levels from experimental observations or from detailed atomic structure calculations In the Bates-Damgaard approxlmatlon, we assume that If we have the energy of the mltlal and the final states of the atom (or Ion). we can then construct approximate solutions from the Coulomb functions m the spatial region r > r,,, where r0 IS sufficiently large to contain practically all of the core charge This procedure leaves the orbltals undefined for the region r < r, However, as we have already mentloned, the region r c r,, frequently does not contnbute greatly to the matnx elements for the transltlon moment Therefore, any reasonably smooth extension of the radial orbital to small r which allows us to normalize the orbltals will be useful One method of progressing IS to look for radial functions that have the property that they are normalized and are asymptotic soluttons of the Schrodlnger equation We achteved this by looking at the forms of the solutions of the Schrodmger equation for the Coulomb potential,? which shows that R,,, IS normahzable If n - I - I IS a positike integer or zero Hence, the radial function remains normahzed If we replace n by n* and I by I* where n * = n - 6,,, and I* = I - 6,, and S,, IS the quantum defect We consider this approach to be an emplncal device to construct functions which have the property that they (I) are bounded and normahzable, (2) are solutions to the Schrodmger equation for large r with energy E,, = Z’,‘(n - d,,)‘. and (3) are hydrogemc solutions for 6,, = 0 In this application, 6,,, will be considered as a parameter to define a wavefunction and It IS adjusted for each level of each ion to best interpolate/extrapolate dipole matnx element data, as described below After the numerical work was well underway, It came to our attention that a more general form of the functions had been obtained by Kostelecky and Nleto3 from studies on supersymmetry Then287

288

R E H CLARK and 4

L MERTS

mtroductlon of a model potential, which has a solution of the specified form. faclhtates study of the propertIes of the functions We shall not concern ourselves alth physlcal interpretation of these functions or their possible connectlons with supersymmetry and atomic physics but rather wll continue to use these functions as an emplncal device The solution to the hydrogen equation can be represented In the form

where x IS the scaled radius, I the Integer angular momentum quantum number. L(P. M, u) the Laguerre polynonual with Integer Index, and r the gamma function It IS not unreasonable to expect that useful results may be obtained from functions of this form. which take mto account some aspects of the quantum defect ansmg from the core This does not Imply that the tnal radial function IS a good solution In the core region (see for example Bates and Damgaard) We shall use Eq (1) by replacing n by n* = n - &, and I by I* = I - 6,, where a,, IS the quantum defect The Laguerre polynomial L(21* + I, n* - I* - I, u) then must be generalized We note from Abramowltz and Stegun4 that this can be done since n * - I* - I IS an Integer q Speclfically. the Laguerre polynormal becomes L(a,q,r)=

i

(-r)&f-(q

+x + I)/[f(k + I)T(k +a + l)r(q -I\ + I,].

kc0

on rewntmg Eq (I), -1*)/2f(n*+I*+l)]‘?x”exp(-u/2q*)L(2/*+l,n*-I*-

R n.,* = n *-Z[r(n*

I.\)

(2)

Using the notation G(r,/)=2n,*n:/(n,*+n:).

N,+=n,*+l,*.

the radial part of the m-pole matrix element T,,=


x (II: “,-

x

- I

v-

G(r,f

N;

=nT

-I,*

L+ =I,*+/:.

(3)

IS

> = I/?(n,*

n: J-2 (n,*)-‘:

)c+I;+“‘+‘[f (A’; )l-(N; ) T(N,+

+ l)f (Iv; + I)]’ 2

- ,

kTo C I[-G(I,S)/n,*lk[-G(I.I

)/~~:l~)

,=O

xf(L’+s+k+m+3) x [k’s’T(N,-

- k)f (N; - s)T(2/,* + k + 2)f (21: + s + 2)]

(4) This form of the matnx element has practical advantages over the approximate methods which use regular and u-regular Coulomb functions In these approxlmatlons, m addltlon to obtaining the two Coulomb functions, we still must carry out numerical integrations to obtain the desired matnx elements The transition energy IS {[(Z - N + 1)/n:]’ - [(Z - N + 1)/n:]‘) In Rydbergs, where N IS the number of bound electrons and Z IS the atomic number We use a different value of the quantum defect, a,,. for the energies from that used for the matnx elements (see Set 3) Also. the quantum defects for the mltlal level n,l, differ from the ones for the final level n,l, 3

FITTING

PROCEDURE

At high values of n, the quantum defect becomes Independent of n, however, devlatlons from a constant value occur at small values of n We have thus chosen to fit the quantum defects for both the energy and matnx element problems to the form b“, = Co + C, in’ + C,,‘d

The lomzatlon energies can thus be expressed as E=

Z-N+1 n -((Co+C,/n2+C2jh4)

2 1

(5)

(Z-N+l)’

=

n*’

Rydbergs

(6)

Quantum

Table

I

defect method

apphed

Energy fit parameters

ION

P

c 11’

s

1 52366-01

c

1’1

P

3 5‘LOe

c

IV

d

1 3938e-03

c

III

s

3 496-e

c III

P

c

to owllaror

strengths

to be used wth

Eq

(6)

Cl

cO

2 >-cl% 02

_-

02

532:e-03

? QQl?e

03

111

2 0>?6e

01

1 603~e

01

0 3.~36.~ 03

d

.a 3318s

112

c II

s

0 14e*e

Cl1

1 u?o3s-I”~1

c II

P

3 ‘228-e

01

8

c

II

d

5 939be

0~

1 ?58’--01 c

Ar VII

s

6 29h>e-01

3 -lL’6’i-O1

Ar VII

., ?:66e

ul

1 017’e-01

Ar VII

1 5151e

01

Ar VII

3 0:41e-02

III

289

-I

o>::e-02

-1

1fJai

II?

2B?Oe-01

1 08E’e-01

and an equation hnear m the fit parameters IS obtalned. VIZ C,, + C, /n’+ C’, n’ = n -(Z

- N + I) I E

(7)

We used Cowan’s’ HF (Hartree-Fock) computer program to calculate values of E over a Rydberg series We then apphed a standard least-squares fittmg procedure to Eq (7) The results for C,,, C,, and C, are gven m Table I for I4 different Rydberg senes In all cases the fit reproduces the energies to better than I% Recently, Magee has apphed this procedure to approx 5000 LS term energies m all eight stages of lomzatlon of oxygen For neutral oxygen. the fit reproduces the energies to better than 5%, for all other stages of lomzatlon the discrepancy IS < 1% We note that this fit IS intended to summanze data for a Rydberg series The user IS cautloned not to use the results for An = 0 transltlons from the ground configuratlon In the apphcatlons here we consider the dipole transition matnx elements, therefore m = I In Eq_ (4) To determine fit coefficients for the matnx element T,{, Eq. (5) can be used m Eq (4) The result IS an equation which IS highly nonlinear m the fit coefficients We use a procedure Identical to that used by Clark er al’ for the nonlmear fit Fu-st. we perform a Taylor series expansion of the function with respect to the fit coefficients and find Tr= r,,O+c(?T,,‘?C,)AC,.

(8)

where the sum IS over SIX coefficients (three for the mmal state and three for the final state) The value of T,,. IS determined from an mitral guess at values of all SIX fit coefficients Equation (8) then IS hnear m the AC,s Having calculated a set of T,, from Cowan’s program, we apply a standard least-squares routme to evaluate the AC,s The values of C, are then updated. and the procedure repeated until every value of AC,/C, IS smaller than some test value. 0 0001 m our work The resulting values of the fit parameters are given m Table 2 The parameters C,,, C,. C? are to be used for the mltlal state. while Cop, C,, and C?, are to be used for the final state We note that the double sum m Eq (4) contains an alternating sign and a gamma function which becomes large when n + n’ IS large When n + n’ 2 20 we found numerical dlfficultles arose m the evaluation of Eq (4) and thus restncted our work to n + n’ G 20 QSRT384-D

290

R E H CLARK and A L MERTS 4

RESULTS

Havmg fitted the dipole matnx elements given by Eq (4) with m = I and the energies given by Eq (6). we calculate the oscillator strength from

Table ic

?

Matrix

element

lit parameters

to be used ulth

Eq (4)

II

1: I’ ”

I”

1- i

1

‘-

III

_

Ill

i

Ii1

c II c II II L 111 -L L

II II

-1

‘iI

Al

Ii

Table Inll

3 Summary I’

h

5

P

of fit compared

to calculations

P I-

I

Cl 15

13 I”

p-5

0 11

t- I”

Pd

11 3n

‘_ III

s P

_ 1

8’3 III

P =

15

S_ III

F d

1 -

c

II

5 P

1 ‘1

i

II

P s

2 2

c

II

P-d

2 8

n i’ I I

5-p

18

A “II

P s

3 0

n ‘L’I I

pd

10

A !I1

d-P

0 68

A i’ I I

d E

1 7

Quantum

defect method

apphed

CIII

to owllator

291

strengths

p-d

IO

IO

Ftg

I

Companson

ol quantum defect owllator strengths tsohd Imes) alth those calculated with CoNan HF program (open circles) for C 111 pd trpe transitions

s

where g, IS the statIstIca weight of the mmal state and ANGFAC IS an angular factor determmed by the specific nature of the transition In the present work we have restncted ourselves to single configurations and configuration averaged quantities Thus ANGFAC reduces to 21,. where I, IS the maximum value of I, or /, We note that the quantum defect approach IS not restncted to single configuratlon cases, for example see Lee* and Seaton ‘The factor [2(Z - N + I)]’ IS needed because Eq (4) gves results for the matrix element for hydrogen In atomic units. while AE IS rn Rydbergs Table 3 gives a summary of the accuracy of our fits to the calculated oscillator strengths The entry labeled “average error” IS an arithmetic average of the absolute values of the errors Figures l-4 show compansons of our fits to thef-values In each graph the circles represent J-values calculated with Cowan’s HF computer program while the lines represent values generated CII

Fig

2

Companson

of quantum

defect program

oscdlator for C /I

s-p

strengths

with

those calculated

s-p type tranwons

wth

Co&an

s HF

292

R E H CLARK and A L MERTS

CITI p - d i

i

i

l

i

i

I

I

I

I

I

I

j

lO

IO

I0

3" I0 0

I I01

F~g 3 Comparison of oscillator strengths calculated solel) from the energy quantum defems V~lth those calculated from Cowan's HF program for C III p-d type transmons

by the fit procedure Data for many n-n" transmons are presented Each hne represents transitions from a fixed value of n going to successively higher values of n' (labeled n, on the graphs) Each hne st~,",'~- ,r~ d ~ o v'X,'~w ¢¢,'vo-~ ,r_, - - - , r -~ } dr,'a~ o'aJ"s ,re ¢ i ~ v'X,'~,~x" ¢¢ivo'~ n 2 -9 ,r = ~-'~," ~-l'g~'c¢- ? ~l"l"o~,sa comparison of our fitting procedure results to results from Cowan's program for C HI p - d transttLons ranging from is'-2p'--Is:2p3d to Is'-2pgp-ls"2p l i d The maxtmum dtscrepancy is 5% Ftgure 2 shows a stmdar comparison for C II s - p transmons Note that Cooper mtmma occur m this case and that results from our fit accurately follow the rather large vanauons in the f-values over the entire range of n-n" values shown Figures 3 and 4 show what happens tf we stmply use the energy level fits to gtve the quantum defects for both energies and matrix elements, t e using the parameters from Table I m both Eqs C n s-p

i0-2

i

i

~q

o

i

i

i

1

10-3

00-4'

10-5

tO-6

10- 7 7 . 10-8 &- io-O

i0 I n 2

Ftg 4 Comparison of oscdlator strengths calculated solely from the energy quantum defects w~th those calculated from Cowan's HF program for C !! s-p type transmons

Quantum

defect method

applied

to osctllator

strengths

293

(6) and (4) Figure 3 shows a comparison of those results (sohd hne) with those of Coaan (open cu-cles) for C III p-d transItIons Quahtatlvely, the results are slmllar to those of Fig I with a maximum error of 8% Figure 4 shows a similar comparison for C II s-p transltlons We observe that although they are about an order of magnitude off. the energy quantum defect results do show the Cooper minima We note that the pure hydrogemc results (6,,= 0) cannot produce this quahtative feature From these comparisons. we conclude that It IS worth the effort to calculate quantum defects speclfically for the dipole matnx elements Oscillator strengths calculated from this approach provide a reasonable approximation to the original values for a wide range of II-11’ values If we use quantum defects calculated only from energies. It appears that quahtatlke features of the oscillator strength behavior are obtained However. quantitative results may be quite far off If Cooper mimma are present REFERENCES I 2 3 4 5 6 7 8 9

D R Bates and 4 Damgaard Phrl 7iians R SOL Lund AZ42, 101 ( 1949) P M Morse and H Feshbxh Werhoo!s of Theorem-al Phlsm McGrawHtII Ne\r York (1953) 1’ A Kostelccky and M M Nteto, Phbs Rev A 32, 3243 (1985) M Abramou~tz and I Stegun Handbook of Marhemarrcal Funcrrons 4ppl Murh Ser 55, Nattonal Bureau of Standards Washmgton DC 20025 (1964) R D Cowan The Theor o/ A~omrc Swucrure und Specrro Unwersny of Cahfomta Press Berkelev Cahf (1981) N H Magee Jr Pnvate conunumcatton (1985) R E H Clark N H Magee, Jr, J B Mann and A L Merts, Ap J 254, 412 (1982) C M Lee Phbs Rev 4 IO, 584 (1974) M J Seaton, Rep Prog Phts 46 (1983)