Symmetry classification of two-particle operators in atomic spectroscopy

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Symmetry classification of two-particle operators in atomic spectroscopy

B.G. Wybourne

To cite this version:

B.G. Wybourne. Symmetry classification of two-particle operators in atomic spectroscopy. Journal

de Physique, 1969, 30 (1), pp.39-46. �10.1051/jphys:0196900300103900�. �jpa-00206762�

SYMMETRY

CLASSIFICATION OF

TWO-PARTICLE

OPERATORS IN ATOMIC SPECTROSCOPY

(1)

By

^B. G. WYBOURNE

(2),

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

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

(Reçu

^le 1 er avril

1968.)

Résumé. 2014 On utilise les méthodes de la théorie des groupes ^et

l’algèbre

^des

pléthysmes

pour obtenir ^une classification

complète,

^{selon les}

propriétés

^de

symétrie,

^des

opérateurs

^scalaires

à deux

particules qui

^ont des éléments de matrice non nuls dans les

configurations

Pn, dn, fn

et

(d

+ s)n. ^On étudie l’identification d’interactions réelles et effectives ayant ^les

propriétés

de

symétrie.

Abstract. ²⁰¹⁴ The methods of group

theory

^{and the}

algebra

^of

plethysm

^{are used to}

give

a

complete

^symmetry classification of the scalar

two-particle operators

^{that have}

non-vanishing

matrix elements in the

configurations

pn, dn, fn ^and

(d

+

s)n.

The identification of both real and effective interactions

having

^these symmetry

properties

^{is discussed.}

1. Introduction. ^- Atomic interactions _may

always

be

expanded

^{as a linear} combination of a suitable set

of

products

^{of tensor}

operators [1].

^{In the case}

of LS

coupled

^{states the basic}

one-particle

operators

are the double tensors :

In these operators ^{x is the rank of the}

spin part

^of the operator and k the orbital rank. These basic

one-particle

^tensor operators ^{form the}

building

^blocks

for

describing

^an

N-particle

interaction.

Any

^two-

particle

interaction _may be

expanded

^{as a linear}

combination of two-fold

products

^{of the Wl,,k) tensors.}

A

typical two-particle

^tensor operator may be written

as :

In the case of scalar

two-particle

^operators

K == Q == 0

and x ⁼ k. For the sake of

simplicity,

^{we shall limit}

our attention to scalar

operators

but there is no

difficulty

whatsoever in

extending

^the

theory

^{to include}

more

general

operators.

(1)

^Research

sponsored

ⁱⁿ

part by

the Air Force Office of Scientific Research, ^{Office of}

Aerospace

Research, 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 construction of

N-particle operators

^{that have} well-defined transformation

properties

^{under the} sym- metry

operations

^{of the} groups used to

classify

^the

atomic

eigenfunctions

^{can lead to a} substantial sim-

plifications

ⁱⁿ the calculation of the matrix elements of atomic interactions. If the symmetry

properties

of the basic

one-particle

^tensor operators ^are

known,

there is no

difficulty

ⁱⁿ

obtaining

^a

symmetry

^des-

cription

^of any

N-particle

operator constructed from them

[2].

The number of

independent symmetrized N-particle

operators ^that may be constructed from the basic operators ^{and which}

have,

^at

least,

^some

non-vanishing

matrix elements when

acting

^on

symmetrized

^atomic

states, ^is

clearly

^{finite due to} symmetry restrictions.

The

symmetrized N-particle

operators may be cons-

tructed from

particular

linear combinations of N-fold

products

of the basic

one-particle

^tensor operators.

When the

complete

^{set of}

symmetrized N-particle

operators ^is

constructed,

any

N-particle

^{atomic inter-}

action _{may in} turn be

represented

^{as a} linear combi- nation of the

symmetrized N-particle

operators ^with each

operator having

well-defined symmetry pro-

perties. Thus,

^{if the matrix elements of the} symme- trized

N-particle

^{operators are}

known,

^{then the matrix}

elements of _any

N-particle

^atomic interaction _may be

expressed

^{as a linear} combination of these.

Our

object,

^{in the} present paper, is ^to establish a

complete

^{list of the} symmetry classifications available for

describing

^the symmetry

properties

^{of scalar two-}

particle

operators. ^{We shall} not,

however,

attempt their

explicit

construction at this time. The _sym-

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0196900300103900

40

TABLE I

metry

description

of the basic

one-particle

operators will first be considered and then the symmetry ^clas- sification of the scalar

two-particle

operators

given

for the

particular

^{cases of}

pn, dn, f n

^and

(d

^-t-

S) n

^confi-

gurations.

^{We shall then} show that the number of

independent symmetrized

^scalar

two-particle

opera- tors is not more than the number of

independent

matrix elements

appearing

in the energy matrices of the two-electron

configuration. Finally,

^{we shall re-}

mark _upon the identification of both real and effective interactions with the

symmetrized

^scalar

two-particle

operators.

2.

Symmetry description

^of

one-particle

^{tensor ope-} rators. ^-

By allowing

^{the tensor}

component W (,,k)

^to

operate ^{on the}

symmetric

^and

antisymmetric

^states

of the 12

configuration

^{it is not difficult to show that}

the

correspondence :

holds.

Furthermore,

^{the tensor ranks x and k are}

restricted to the values x ⁼

0, 1

^{and k =}

0, 1,

^{..., 21.}

Values of

x + k

^{even are} associated with the anti-

symmetric

^{states of} 12 and hence the operators ^with

must transform

collectively

^as

the ( 12 ) representation

of

U4l+2.

Likewise the

operators

^{W(xk) with x}

+ k

odd transform in the same way ^as the

symmetric

^states

of l2 and hence the

operators :

transform as

the ( 2 ) representation

^of

UI,12,

The transformation

properties

of the double tensors

W(xkl _may be further

investigated by reducing

^the

symmetry

^{of the} group

U4112

^{in the same} way ^as is

used to

classify

^{the states of 12.}

Using

the chain of groups :

yields

^the symmetry classification of the double tensors

listed in table I.

In the

particular

^{case of}

f-electrons

^{where the}

exceptional

group

G2

^{is used to}

classify

^{the states of}

the

f n configurations

^a further classification of the W(xk) operators ^is

possible. Proceeding

^as

above,

^we

readily

^{find that the four sets of tensors}

[1] :

for a

given

component 7T ^{form the} components ^of

tensors

transforming

^{as the}

(20), (11), (10)

^and

(00) representations

^of

G2, respectively.

The classification of

one-particle

^{double tensor}

operators ^{for mixed}

configurations

has been discussed

by

^Feneuille

[3, 4].

We shall limit our discussion

to the

(d

⁺

S) n configurations.

^The

complete

^set

of double tensors in this case involves the twelve operators

Wlxk) (dd),

^{the two} operators

w(00)(ss)

^and

W(10)(ss)

and the four operators

w+(x")(ds)

^and

w-(Xk)(ds),

^where

following

^Feneuille

[4]

^we

put :

The

W+(,,k) (ds) operators

^transform

according

^{to the}

antisymmetric

^{states of}

(d + S)2

^{while the}

W-(,,k) (ds)

TABLE II

operators ^transform

according

^{to the}

symmetric

^states

of

(d

⁺

s)2.

^The

complete

symmetry classification of the double tensors

appropriate

^to

(d

⁺

s) n configu-

rations is

given

^{in table II.}

3.

Symmetry

classification of scalar

two-particle operators

for ln

configurations.

^{- The}

symmetry description

^of

N-particle

operators may be ^most

readily approached using

Littlewood’s

[5-7] algebra

of

plethysm [2].

^{The basic}

one-particle

^{double ten-}

sors Wlxk) _span

the {12} and ( 2 ) representations

^of

the

unitary

^group

U1112. Two-particle

operators formed

by products

^of

degree

^{two in} these basic forms will transform under

U4l+2 according

^{to the represen-}

tations contained in the

separation

^{of the}

plethysm :

The first term on the

right-hand-side

of eq.

(9) gives

^the symmetry types ^{that arise in} the formation of

products

^of

degree

^{two in} the double ^tensors Wl,,kl with x

+ k

even, the second term the symmetry types that arise in the formation of

products

^of

degree

^two

in W(,,k) with ^x

+ k odd,

^{while the last term}

gives

^the

symmetry

types

that arise in the formation of bilinear

products

^{of double tensors}

transforming as {12}

^with

those

transforming (2)

^of

U41+2.

^{This last term}

effectively

describes the transformation

properties

^of

non-hermitian

two-particle

operators ^{and as discussion} is limited to the usual hermitian

operators

^{we shall}

not consider them further.

The plethysm 1121 & 121

^and

121 & 121

^may

be

readily

^{evaluated to}

give [2] :

The

eigenfunctions

^{of In} type

configurations

^{all trans-}

form under

U41+2 as { 1 n}

^and

since I lnl I lnl 4) 14}

we may

delete (4)

^{as a}

possible

symmetry type ^for

describing two-particle operators

^{that have at least}

some

non-vanishing

^matrix elements. Thus under

U41+2 only

^the symmetry

types {14} and {22}

^need

be considered further. A

complete symmetry

^classi- fication could be obtained

by simply decomposing

each of these

representations

^of

U41+2 according

^to

the chain of _groups of eq.

(7).

We shall

primarily

^be concerned with operators that have well-defined

symplectic symmetry

^under

SP41+2

^{rather than}

unitary symmetry

^under

U41+21

and it is

advantageous

^{to first}

decompose

^the

unitary characters ( 12) and (2)

^into

symplectic

^characters

prior

^to

evaluating

^the

plethysms.

^{Thus :}

Evaluating

^{each term}

gives :

Operators transforming

^under

Sp,,,, as 4 )

^will

have null matrix elements in In

configurations

^and

hence

only

^the

symplectic symmetries 0 ), (

¹²

X ( 14 ) and 22 )

^are available for the symmetry ^des-

cription

^of

two-particle

operators. ^The symmetry

description

^of

possible two-particle

operators may be extended

by decomposing

^each

symplectic

^character

into those of

SU2 X R21,1

and thence to

SU2

^X

R3.

In the case of

fn configurations

^the group

G2

^{is also}

42

TABLE III

TABLE IV

TABLE V

at our

disposal. Restricting

^{ourselves to} consideration

only

^{of scalar}

two-particle operators

^{limits us to} ope- rators

transforming

^under

SU2

^X

R3

^like

1S,

^{3P and}

5D terms.

Only

operators ^{of even}

parity

will be hermitian and it is convenient to associate a

symbol

^{with each}

symmetry

type which shows how the

one-particle operators

^are

compounded

^to form the final even

parity operator.

^{To do}

this,

^{we shall} associate the

letter g

^with

one-particle operators transforming as ( 1 2 )

under

U4,,,

^and the letter ^u with those

transforming as (2). ^Thus,

^the symmetry types

arising

^from

12 1 & f 2 1

will be labelled _gg and those

arising

^from

2 1 (x) { 2 }

will be labelled uu... It is also useful to dis-

play

^the

plethysm involving

^the

symplectic

^characters

to

give

^further

insight

^{into the structure} of the final

operators. Upon making

^the

appropriate

^character

decompositions,

^we obtain the final results

displayed

in tables III to V for

pn, dn

^and

fn configurations

respectively.

44

TABLE VI

4.

Symmetry

classification of scalar

two-particle operators

^for

(d + S) nconfigurations.

^- The sym- metry classification of scalar

two-particle operators

for

(d

^-)-

s)n configurations proceeds

^{in the same}

manner as for In

configurations except

^{that the chain} of _groups used is :

The determination of the relevant

branching rules,

ⁱⁿ

terms of

plethysm,

has been discussed

by Wybourne

and Butler

[8].

^The

complete

^{list of}

symmetry

types available for the

description

of scalar

two-particle

operators ⁱⁿ

(d

⁺

s) n is given

^{in table VI.}

5.

Quasi-spin

classification of the

operators.

^{- It}

is useful in

making practical

calculations to construct

operators ^{that have not}

only

well-defined

symplectic

symmetry ^under

SP2,

but also well-defined

quasi-spin

so as to

enjoy

^{the full}

advantages

^{of the}

seniority

scheme

[9, 10]. Quasi-spin

^{ranks x} may be associated with the various

symplectic symmetries by studying decomposition

^{of the}

representations

^of

R4v

^{into those}

of

SU2

^X

SP2V.

^In

assigning quasi-spin

^{ranks to the}

symmetry types associated with

N-particle operators

we may restrict our attention to those

representations of R4.

^{of the}

type [11-10 - 0] having

^{an even number}

of

symbols

^{1 and not more than 2N}

[11].

^The

relevant

decompositions

appear in table VII with the

quasi-spin multiplicity (2K

⁺

1) being given

^{as a}

left

superscript.

Inspection

of table VII shows that operators ^trans-

forming as 22 )

^under

SP2V

^have pure

quasi-spin

K - 0 while those

transforming as 14 )

^have

quasi- spin

^{K = 2. The}

symplectic symmetries 0 )

^and

12 )

^are both associated with

quasi-spin

^{ranks of}

K =

0,

^{1 and 2.}

6. The number of

independent operators.

^{- Our}

list of symmetry

types

available for the

description

^of

scalar

two-particle operators

includes twelve entries for

pn, twenty-three

^for

dn, thirty-seven

^for

fn

^and

forty-nine

^for

(d + s) n configurations.

^{Sets of} sym- metrized

operators

could be constructed from linear combinations of the

two-particle operators

of eq.

(3) transforming according

^{to each} symmetry type. ^If the matrix elements of the

symmetrized two-particle

operators ^{are known in the} two-electron

configura- tion,

^{then those of} the n-electron

configuration

may

always

^be constructed from these

by

^an

appropriate

chain calculation

using

coefficients of fractional

parentage [1]. However,

^{in each case considered} the number of

symmetrized

operators ^that may be constructed exceeds the number

of independent

^matrix

elements in the

respective

^{two electron}

configurations and, hence,

the matrix elements of many of the

symmetrized ^operators

^{must be}

expressible simply

^as

a linear combination of the matrix elements of ^a subset of the

complete

^set. The determination of the maximum number of

independent symmetrized

^ope-

rators of a

given symmetry

type may be made

by application

^{of the}

Wigner-Eckart

^{theorem to the two}

electron matrix elements

[2, 12].

As an

example,

consider the case of the

thirty-seven

entries in table V of the symmetry

types

^{for two-}

particle

^scalar

operators

ⁱⁿ

f n configurations.

^There

are five entries

S,, S2, S3, S8 and 89 corresponding

to the

symmetry

types

[000] (00)

^under

R7

^and

G2.

The matrix elements of operators

transforming

^as

[000] (00)

^{will be}

independent

^{of L} and since there

are

only

^three

representations

^of

R7 [000], [200]

and

[110J

^{to label the states of}

f 2 only

^three

indepen-

dent operators

transforming

^as

[000] (00)

^{can be}

constructed. The symmetry

types S1, S2, S8

may be

represented by

^a

single symmetrized

operator ^trans-

forming as ( 0 ) [000] (00)’S

^and

S3

^and

S9

^taken

as

independent operators. In f 2 C([X] [À’J [111])

^{= 0}

except ^for

[;k]

⁼

[X’]

⁼

[110]

^{where :}

and, hence,

^the

symmetry type ( 22 ) [111] (00)lS

may be

represented by

^a

single symmetrized

operator since the matrix elements

of S,

^and

Slo

^will

simply

^be

proportional

^{to one another. For}

exactly

^{the same}

reason ( 22 ) [400] (40)lS

may be

represented by

^a

single symmetrized

operator

and 14 ) [220] (22)’S and 22 ) [220] (22)lS by

^{one each.}

Thus,

^there

can

only

^{be seven}

independent symmetrized operators

in

f n transforming

^{like 1S states under}

SU2

^X

R3.

Proceeding

^{in this manner we find} that nine inde-

pendent symmetrized

operators ^{transform as 3P under}

SU2

^X

R3

^{and five as 5D. Hence in}

f n

^{these are}

just

twenty-one

independent symmetrized

^{scalar two-}

particle

operators

corresponding

^to the number of

independent

^{matrix elements in}

f2. Similarly,

ⁱⁿ

pn

there are seven

independent symmetrized

^{scalar two-}

particle

operators, ⁱⁿ dn fourteen and in

(d

⁺

S) n thirty.

TABLE VII

46

7. Real and effective interactions. ^{- It is a common}

practice

^{in atomic}

spectroscopy

^to allow for the effects of

configuration

interaction on the _energy level struc- ture of In type

configurations by treating

^the

(I

⁺

1)

Slater radial

integrals

associated with the Coulomb interaction as

adjustable parameters

^{and to introduce} effective

N-particle

operators with further associated parameters. If attention is limited to effective scalar

two-particle

operators

transforming

^under

SU2

^X

R3

as IS we have in addition to the

(7+1) symmetrized

Coulomb operators ^made up of scalar

products

^{of even}

rank orbital ^tensor

operators,

^l

symmetrized

^scalar

operators ^{which are} constructed from scalar

products

of odd rank orbital tensor operators. ^These I symme- trized

operators

^{transform in}

precisely

^{the same}

manner as the

symmetrized

forms of the orbit-orbit interaction and

part

^{of the contact}

spin-spin

^inter-

action.

Thus,

^the

parameters usually

associated with the I

symmetrized

effective scalar

two-particle

opera- tors absorb not

only

the effects of

perturbations

^pro-

duced

by

Coulomb interaction between

configurations

but also a number of intra_ln interactions that are

normally

excluded in

making specific

calculations.

Furthermore,

^{since the}

general symmetrized operators

may be constructed to include operators Wllkl ^we

must conclude that part ^{of the effects} of electro-

statically

correlated

spin-dependent

interactions will be absorbed

by

^the parameters.

Generally,

^{the use} of effective

operators

^{tends to} accommodate more effects than the

original

^{reasons that}

inspired

their introduction. This constitutes both the

strength

and weakness of the method of effective interactions. While the method is

well-adapted

^to

correlating

^a considerable mass of energy level data in terms of a few

parameters,

^the

physical significance

and

composition

^{of the} parameters ^{are obscured.}

The future work will

undoubtedly

be directed more

towards ab initio calculations

using

^{atomic wave func-}

tions ^to calculate all radial

integrals. Klapisch’s

approach [14]

^of

expressing

the central field

potential

in terms of a few

adjustable

parameters ^{and then}

calculating explicitly

^{both the} radial and

angular

parts

using

^an iterative method to

modify

the central field would seem to be

capable

of considerable deve-

lopment

while still

retaining

^{all of the}

advantages brought

^about

by

^the

separation

of the radial and

angular

^variables.

An extreme form of the method of effective inter- actions would be ^{to treat} the

independent

^matrix

elements of l2 as

independent parameters

^{and add to} these the

three-particle

^scalar operators

[11, 15]

^that

transform as 1S as further parameters

(in

^dn

configura-

tions this would result in a total of sixteen

parameters).

This would have the effect of

giving

^a

complete

^des-

cription

^{of all}

possible two-particle

scalar interactions

together

^{with a}

complete description

of the second- order Coulomb

configuration

interaction. Such an

approach

^would

undoubtedly give

^{a very}

good

^des-

cription

^of the energy

levels, including

^{their term}

separations

^and

multiplet separations,

^for

systems having

^{three or more}

equivalent

^d

or f

^electrons.

This

description

could be useful for the

prediction

^of

unknown levels and their associated

properties though,

of _course, much of the information

concerning

^the

details of the relevant interactions would be lost in the parameters.

8. Conclusion. ^- A

simple

^method

has been

^pre-

sented for

determining

^the

symmetry

classification of

N-particle operators

^{and in}

particular

^{scalar two-}

particle

operators. ^The

problem

^{of the}

explicit

construction of these

operators

^remains.

Clearly

^once

this has been

completed

^it will then be

possible

^to

calculate the matrix elements of the ^set of

independent symmetrized

operators ^{and then}

obtain,

^at any

time,

the matrix elements of any scalar

two-particle

^inter-

action as a linear combination of these basic matrix elements.

REFERENCES [1]

JUDD (B. R.), Operator Techniques

^{in Atomic}

Spectroscopy,

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

[2] ^SMITH

(P. R.)

^{and WYBOURNE}

(B. G.), J.

^Math.

Phys., ^{1968, 9, 1040.}

[3] FENEUILLE

(S.), J. Physique,

^{1967, 28, 61.}

[4]

FENEUILLE

(S.), J. Physique,

^{1967, 28, 315.}

[5]

LITTLEWOOD

(D. E.), Theory

^of

Group

Characters, 2nd ed., 1950, Oxford

University

Press, London.

[6] LITTLEWOOD

(D. E.),

^{Phil. Trans.} Roy. Soc., London, 1944, ^A 239, 305.

[7]

LITTLEWOOD

(D. E.),

Phil. Trans. Roy. Soc., London, 1944, ^A 239, 387.

[8] ^WYBOURNE

(B. G.)

^{and BUTLER}

(P. H.), J. Physique,

1969, 30, to be

published.

[9]

^LAWSON

(R. D.)

^and MACFARLANE

(M. H.),

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[10] JUDD

(B.

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