Coefficients of fractional parentage and LL-coupling

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Coefficients of fractional parentage and LL-coupling

B.G. Wybourne

To cite this version:

B.G. Wybourne. Coefficients of fractional parentage and LL-coupling. Journal de Physique, 1969, 30

(1), pp.35-38. �10.1051/jphys:0196900300103500�. �jpa-00206761�

COEFFICIENTS

OF FRACTIONAL PARENTAGE AND LL-COUPLING

(1)

Par B. G. WYBOURNE

(2),

Laboratoire Aimé-Cotton, C.N.R.S. and Faculté des Sciences, Orsay, ^Essonne

and Physics Department, University ^of Canterbury, Christchurch, ^{New Zealand.}

(Reçu

le 1 er avril

1968.)

Résumé. 2014 Les coefficients de

parenté

fractionnelle nécessaires aux calculs en

couplage

^LL

des

configurations

d’électrons

équivalents d

^{et f sont obtenus} directement à

partir

^{des tables}

disponibles

^de coefficients de

parenté

fractionnelle en

couplage

^LS. Les résultats

complets

sont tabulés pour les ^deux types ^de

configurations.

Les détails

pratiques

^{des calculs en}

couplage

^{LL ont} été considérés.

Abstract. ²⁰¹⁴ The coefficients of fractional parentage

required

^for

making computations

in the

LL-coupling

^{scheme for}

equivalent

^electron

configurations

^{of d and f electrons are}

shown to be

directly

^{derivable from}

existing

tabulations of coefficients of fractional

parentage

in the

LS-coupling

^scheme.

Complete

^{results are} tabulated for both

configurations.

^Practical

details of

making

calculations in the

LL-coupling

^{scheme are} considered.

LE J OURN AL PHYSIQUE 30, JANVIER 1969,

1. Introduction. ^-

Judd [1]

^has

recently

^shown

that atomic calculations can sometimes be substan-

tially simplified by considering

^two spaces, ^one

(the

A

space)

ⁱⁿ which all the

spins

^are

"up"

^{and the}

other

(the

^B

space)

^{in which}

they

^{all are} "down".

Mathematically,

^this amounts to

representing

^the pro- duct

[ 1 /2] [1]

^as

([J] A

+

[J]B)

^where

[1] A corresponds

to the orbital functions in

"spin up space"

^and

[1] B

^to

the set in

"spin

^down

space".

^{The wave} functions thus span the

representation ([IJA

⁺

[1] B)

^{of the}

imprimi-

tive rotation group in two sets of variables

[2].

The

plethysm appropriate

^{to the}

n-particle configu-

ration _{may be}

expanded

^to

yield [3] :

The S-functions

[4] {I n}, {1 rx}

^and

{ ln-,, I

may be taken as

labelling

^the

antisymmetric representa-

tions of the

unitary

groups

L+ U2 +1

^and

UB respectively

^{where :}

(1)

^Research

sponsored

ⁱⁿ

part by

the Air Force Office of Scientific Research, Office of

Aerospace

^Re- search, United States Air Force, ^{under A.F.O.S.R.}

Grant No 1275-67.

(2)

^{This work was done in}

part

while the author was a

visiting

^{Professor with} Laboratoire Aim6-Cotton, ^under

the

sponsorship

^{of D.G.R.S.T.}

The

description

^{of the}

n-particle

^{states is extended}

by making

^the

decompositions :

The states of the In

configurations (I 3)

may then be

uniquely

^labelled

by

^the sequence :

where

[1 X]

^and

[1 Y]

label the irreducible representa- tions of

R’,l

^and

RB+1 respectively

^{and if oc}

1,

directly

from Newell’s

[5]

modification rules for ortho-

gonal

groups.

The

representations {1} and f ln-o, I

^{remain irre-}

ducible _upon restriction of the

unitary

group ^to the

orthogonal

group

and, hence,

^the

symbols [1"]

^and

[1 Y]

ⁱⁿ eq.

(4) give

^no additional classification of the states.

They

^are,

however,

^{retained to make the}

correspondence

with other classification schemes more

transparent.

The

opposition

^{of the}

spins

^{in the two} spaces, A and

B,

^{means that we need}

only

^{ensure that the}

Pauli exclusion

principe

is satisfied in each space

separately and, then, couple

^the

antisymmetrized

orbital states of these two spaces

by

^{the usual}

angular

momentum

coupling techniques

^to be assured that the resultant

eigenfunctions

^are

totally antisymmetric

with respect ^to both the orbital and

spin

^functions.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0196900300103500

36

In this respect, ^the

problem

^of

constructing antisym-

metric

eigenfunctions

ⁱⁿ

LL-coupling

^is

entirely

^ana-

logous

^{to that of}

constructing antisymmetric eigen-

functions in

J J-coupling [6].

^{As such we} expect ^the

problem

^of

constructing

coefficients of fractional parentage

(c.f.p.)

^for

LL-coupled

^{states to} be substan-

tially simpler

^{than that}

required

^for

LS-coupled

states

[7, 8].

In this _paper, we shall first show how the

c.f.p.

may be

simply

constructed and

give

the results for dn and

f n configurations.

^We shall then

briefly

^discuss

their

application

^to the calculation of the

off-diagonal

matrix elements of the

spin-orbit

interaction.

2. Construction of coefficients of fractional _paren-

tage.

^{- In}

LL-coupling

^{we need}

only

^construct

c.f.p.

for the states of the

subconfigurations :

The same set of

c.f.p.

may be used for calculations in the A and _{B spaces} and hence we shall limit our

discussion to the _{A space}

c.f.p.

^In the formulas that follow the A

subscripts

will be omitted.

A

typical state I {I rx} [111] LML > may be

repre- sented as a linear combination of states formed

by coupling

^the

eigenfunction

of the a-th

particle

^to

those of the

(a 2013 1) LL-coupled particles by writing :

where the

symbols i

^{and "t’ have been} introduced to serve as

auxiliary

labels for

distinguishing

^states

having

^{the same set of}

question

^numbers. The coeffi- cients of this linear combination _may be factored

using

^a theorem due to Racah

[7]

^to

give :

The first factor on the

right

in the usual Clebsch- Gordan coefficient which the second factor is the

c.f.p.

The

c.f.p.

may be further factored to

give :

Since the

c.f.p.

^{must form an} orthonomal ^set we are at

liberty

^to choose for the first factor the value

x-l ^ex-2

(- 1)

^{2 for rJv odd and}

(- 1) s

^{for x even. This}

somewhat bizarre choice is made to retain a close

relationship

^{with the}

corresponding phase

^choice

made

by

^{Racah for}

LS-coupling c.f.p.

Thus,

ⁱⁿ

LL-coupling

^the

c.f.p.

may be taken ^as

just :

where ^we have

suppressed

^the

symbols ( I ) [1]

^asso-

However,

^the

right-hand-side

^of eq.

(9)

^{is identical}

to the

corresponding

^{factor that arises in the}

c.f.p.

^for

states of maximum

multiplicity

ⁱⁿ

LS-coupling [7, 8]

and hence the

c.f.p

^for

LL-coupled

^{states of dn and}

f n

may be deduced

directly

^{from known results.}

3.

LL-coupling c.f.p.

for dn

configurations.

^-

Applying

eq.

(9)

^{to the case of}

equivalent

d-electrons leads

immediately

^to the results :

Using

^{the results}

given by

^Racah

[7]

^to obtain the

right-hand-side

of the above

equations gives directly

the results of table I.

TABLE I

4.

LL-coupling c.f.p.

for

f n configurations.

^-

Applying

eq.

(9)

^{to the case of}

equivalent f-electrons

leads to the results :

The

right-hand-side

^{of these}

equations

may be

readily

evaluated from the known tables

of ( W U

+

f ) I wu >

and (

^UL -)-

f I UL >

^as

given,

^for

example, by Judd [8].

^{The results are}

presented

^{in table II.}

5. Calculations in the

LL-coupling

scheme. ^-

Judd [1]

^{has discussed the} calculation of the

diagonal

matrix elements for ^a number of interactions. In these cases, it is

possible

^{to make the} calculations

entirely

^{in terms} of the known

LS-coupling

^matrix

elements for states of maximum

multiplicity

^without

using

^the

LL-coupling c.f.p. However,

calculations of the

off-diagonal

matrix elements of interactions that do not commute with the

angular

^momentum

operators

L2, LZ,

^or

SZ usually require LL-coupling c.f.p.

We shall restrict our discussion to the calcula- tion of the

off-diagonal spin-orbit

interaction matrix elements.

The

diagonal

matrix elements of the

spin-orbit

interaction involve

just

^the

components Es,il,,i

^of

i

the scalar

product £ (si li)

while the

off-diagonal

2

matrix elements involve the components :

where

Wil!i

^and

Wlli

^are

components

of the double

tensor operator ^{W(1I) as defined}

by Judd [8]. Equa-

TABLE II

38

tion

(10),

^{as it}

stands,

^{is not in a} suitable form for

calculating

matrix elements of the

spin-orbit

^interac-

tion with

LL-coupled

^{states. The} calculation becomes very

simple

^{if the}

components

^{of the}

operator

^W(Il,

are

expressed

^{in terms of}

coupled products

^{of the}

creation and annihilation

operators a+

^and

ba

^defined

in Judd’s

paper

[1], leading

^to the final result :

Alternatively,

the above result _may be derived

using

conventional

recoupling techniques.

As a

simple example

^{of the}

application

^{of the above}

formula to find an

off-diagonal spin-orbit

^matrix

element in the

LL-coupling

^{scheme we have :}

Equation (11)

^{has the} appearance of

being

^substan-

tially simpler

^{than the}

corresponding expression

^for

the

spin-orbit

interaction matrix element with

LS-coupled

^states..

Notably,

^{there is no summation}

over

c.f.p., only

^a

simple product.

^The

advantages

of

LL-coupling

^for

practical

calculations are however

largely illusory

since while the calculation of individual matrix elements is

substantially simpler

ⁱⁿ

LL-coupling

the number of matrix elements

required

^is

vastly

greater since while in

LS-coupling

^the

spin-orbit

interaction commutes with J2 and

I z, allowing

^{us to}

label the matrices with the

quantum

^numbers

JMJ

in

LL-coupling

^{we are}

only

left with

MJ

^{to label}

the matrices.

One noticeable

computational advantage

^{of the}

LL-coupling

^{scheme is the} compactness of the table of

c.f.p.

^For

example,

Nielson and Koster

[10]

list 211

c.f.p.

^{for the}

LS-coupled

^states of dn whereas in

LL-coupling only

⁹

c.£ p.

^are

required.

^{Even more}

dramatic reductions occur in

fn and configurations.

However,

^{the same} reductions in size of the

c.f.p.

tables can be obtained in the

JJ-coupling

^scheme

which has the

advantage of requiring

the construction of matrices that can be characterized

by J

^{and of}

being

the natural scheme for the

study

of relativistic effects.

REFERENCES AND FOOTNOTES

[1] JUDD (B. R.),

Phys. Rev., 1967, 162, 28.

[2]

LITTLEWOOD

(D. E.),

Proc. Lond. Math. Soc.

(3),

1956, 6, 251.

[3]

^SMITH

(P. R.)

^{and WYBOURNE}

(B. G.), J.

^Math.

Phys.,

1968, 9, 1040.

[4]

LITTLEWOOD

(D. E.),

^The

Theory

^of

Group

^Charac- ters, ^2nd Edition, 1950, ^Oxford

University

Press, London.

[5]

^NEWELL

(M. J.),

^Proc. Roy. ^Irish Acad., 1951, ^A 54, 153.

[6]

^{DE SHALIT}

(A.)

^{and TALMI}

(I.),

Nuclear Shell

Theory,

1963, ^Academic Press, Inc., New York.

[7] ^RACAH

(G.), Group Theory

^and

Spectroscopy,

^1965,

Springer-Verlag,

^Berlin.

[8]

JUDD

(B. R.), Operator Techniques

^{in Atomic}

Spectroscopy,

1963, McGraw-Hill Book Co., Inc., New York.

[9] JUDD (B. R.),

^Second

Quantization

^{and Atomic}

Spectroscopy,

^{1967, The}

Johns Hopkins

Press, Baltimore.

[10]

^NIELSON

(C. W.)

and KOSTER

(G. F.), Spectroscopic

Coefficients for the pn, ^{dn and} fn

Configurations,

1963, The M.I.T. Press, Massachusetts Institute of

Technology, Cambridge.