Direct Product Groups

Let ^ = {a, b, c, ...} be a finite group of order g, <&' = {a\ b\ c\ ...}

a finite group of order g\ and consider the set of Kronecker products

1 1 N o t e t h a t t h e lexicographical o r d e r i n g of t h e e l e m e n t s of t h e p r o d u c t basis p r e c l u d e s any a m b i g u i t i e s in p r o d u c t s of t h e form | S?y | Sf'y.

392 8. THE ANALYSIS OF SYMMETRICAL SPIN SYSTEMS

(cf. Chapter 4) 9 ® Ψ = {a ® a\ a ® b\ b ® a\ b ® è', ...}. If multiplication of the elements in S? ® ST is defined by

(Χ ® X')(J> ® y') = xy ® (5.12) then ^ ® ^ ' is a group of order gg' called the direct product of ^ and

T h e proof of this statement requires a verification of the group axioms.

(1) Since xy and xy' are elements of <& and respectively, xy ® is, by definition, an element of ^ ® T h u s ^ ® & contains the pro-duct of (χ ® x') and (y ® y'), whenever it contains χ ® χ and j> ®y'.

(2) Multiplication of the elements in ® ST is associative, since, by (5.12),

[(χ ® x')(y ® Y)](# ® z') = (xy ® x'y')(z ® #') = (XY)* ® (x'y')z\

(χ ® *')[(V ®y')(z ® z')] = (χ ® x')(yz ®y'zf) = x(yz) ® x'(y'z').

These expansions are equal, since the elements of ^ and <&' obey the associative law of multiplication:

x(yz) = (xy)z, x'(y'z') = (x'y')z'.

(3) T h e identity of ^ ® <§' is e ® e\ the Kronecker product of the identities of and ST. For if χ ® x' is any element in S? ® Â?', then

(E ® E')(# ® #') = (ex ® E V ) = Χ ® x\

(χ ® #')(£ ® e') = xe ® # Y = Χ ®

(4) If χ ® is any element of ^ ® ST, the inverse element is also in 9 ® &. Indeed, the inverse element is (χ ® χ')~λ = χ~λ ® χ'~λ, since

(χ ® X'^X-1 ® X'_ 1) = xx~x ® x'x''1 = e ® E', ( X- 1 ® X'_ 1)(X ® Χ') = X_ 1X ® X'- 1X' = e (χ) e .

T h e direct product of two groups is often denoted <& X <&\ rather than ^ ® T h e latter notation has been used here, since the multi-plication rule (5.12) is formally identical to the operator product of two Kronecker products, defined in Chapter 2. T h e ordering conventions adopted for Kronecker products will also be used here. Thus, in any product a ® a\ the first factor will be an element of the first group, the second factor an element of the second group.

If & ® &' is the direct product of Ή and W ^y the set of all elements of the form {a ® e'y b ® e\ ...} is a subgroup of order g. T h e one-to-one

5. T H E X A P P R O X I M A T I O N F O R S Y M M E T R I C A L S Y S T E M S 393

correspondence a <-> a ® e\b <-> £ ® é,shows that {a ® e\ b ® e\ ...}

is isomorphic to . Similarly, the set of all elements of the form [e (x) a\ e ® b\ ...} is a subgroup of ^ ® ^ ' isomorphic to If

^ = the set of elements {a ® a, è ® b, ...} provides another sub-group isomorphic to ^ .

The significance of direct product groups in the study of systems conforming to the weak-coupling approximation is that the symmetry group of the zero-order hamiltonian (5.10) is the direct product of the symmetry groups of J ^^A and J ^^x . T o prove this theorem, it must be recalled that the right-hand members of (5.10) are abbreviations for

® 1χ a nd 1 ^A ® ^ x y where 7A and lx are identity operators for the AA' ··· and X X ' ··· subsystems. N o w let ^ and ST denote the symmetry groups of J4?A and J^x , ^ ® <&' their direct product. If Q is in ^ , and R is in then ρ (χ) is in ^ ® ST, so that

ρ ® <°>ρ -ι

^{® fl-1 = Q^AQ-1}

® tfJx*-

¹

+ ρ/ΑΟ"

¹

®

^{R^xR1}

= ^ Α ® Ι Χ Λ - 1 Α ® ^ Χ = ^ \

since

ρ^ΐρ-

¹

=

^$?A

,

^{R^xR'1} = . This proves that

^r<

^{0) is}

invariant under a similarity transformation with any element of & ® <&'.

In other words, every element in ^ ® & is contained in the symmetry group of <°>.

Consider now a generic element in the symmetry group of J f( 0 ). This element, by virtue of the manner in which J ^( 0) was constructed, can be written in the form S ® T, where dim S = dim J^f^A and dim Τ = dim £F^X . Note that S and Τ are not required to be elements of 9 and respectively. By definition,

S ® T j f ^ S - 1 ® T -1 = { 0) = S J ^ S -1 ® 7X + 1 A ® T : ^ x T - \

Subtracting this result from (5.10), one obtains

[ S J TAS- i - ® 7X + 7A ® [ r j rxr - i - J ?x] = 0 ,

so that S ^ S - 1 = Jf^A and T^XT~^X = Jf^x . It follows that 5 is in ^ , and Τ is in so that S ® Τ is in ^ ® <&'. This shows that every element in the symmetry group of ^( 0 ) is in ^ ® <&\ and completes the proof.

T h e preceding theorem shows that the symmetry groups of (5.4), (5.5), and (5.6) are ^2 ® ^2, 3 )2 ® ^2, and ^3 ® <^3 , respectively. It must not be concluded, however, that these groups are the symmetry groups of the complete hamiltonian operators. If ® & is the symmetry

394 8. T H E A N A L Y S I S O F S Y M M E T R I C A L S P I N S Y S T E M S

group of Jf{0\ the symmetry group of 3f consists of those elements R ® S in & ® Ψ for which

Q ® RVWQ-1 ® R1 = V(l\ (5.13) Therefore, the symmetry group of the complete hamiltonian is a

sub-group of 9 ® <&'.

T o illustrate this result, consider the A A ' X X ' system. T h e symmetry group of the zero-order hamiltonian is ^2 ® ^2 = {Ε ® Ε, Ε ® C2 , C2 ® £ , C2 ® C2}, where the first factor in each element of ^2 ® ^2

refers to the AA' nuclei, the second factor to the X X ' nuclei. It is easily verified that, with Va) given by (5.7), the only elements of ^2 ® ^2

satisfying (5.13) are E®E and C2 ® C2 . These elements form a subgroup of ^2 ® ^2 , which is an isomorphic image of ^2 = {E, C2}, the symmetry group of the composite spin system. The isomorphism is given by

E<->E(x)E, C

² <-> C^{2 ( g )} C² . (5.14) In general, the symmetry group of the complete hamiltonian will be

an isomorphic image of the symmetry group of the nuclear configuration, realized as a subgroup of the direct product of the symmetry groups of the component subsystems. Care must be taken, however, to verify that the elements of the subgroup satisfy (5.13), since the requirement that the correct subgroup be an isomorphic image of the symmetry group of Jti? is only a necessary condition. For example, the subgroups {Ε ® Ε, Ε ® C2} and {Ε ® Ey C2 ® E) are isomorphic to ^2 > b ut

neither Ε ® C2 nor C2 ® Ε satisfy (5.13) in the case of an A A ' X X ' system.

From the foregoing remarks, it is not difficult to show that, for the AA'A"A'"XX' and AA'A"XX'X" systems, the correct realizations of

&2 and @s as subgroups of <^2 ® ^2 and &3 ® &3 are

@

²

= {£ ® E,

c 2 ® c 2 , c y ®

E,

c; ® C2},

^ 3 = ® £ , c 3 ® c 3 , c 3 2 ® c 3 , c 2 ® c 2 , c y ® c y , c 2 ® c 2 } .

T h e isomorphisms between the elements of these groups and those of Tables 8.1 and 8.2 are explicitly given by

3²: Ε <-> E (x) E^Y C2^ C2® C2, C2' < ^ C2' ® £ , C2'<-> C£ ® C2, (5.15)

^³: Ε <-+ Ε (χ) E, C³^ C³® C³, C^{3 2} C^{3 2} ® C^{3 2},

C ' t ) ^ ^ C ² ( X ) C ' t j , ^ ^ C * 2 ( X ) C * 2 , Ce} ^ ^ C ' 2 ( g ) C * 2 .

(5.16)

5. T H E X A P P R O X I M A T I O N FOR S Y M M E T R I C A L SYSTEMS 395

(5.19)

12 T h e s y m b o l ( g ) has b e e n i n t r o d u c e d into t h e l e f t - h a n d m e m b e r s of these e q u a t i o n s to indicate t h a t t h e projection o p e r a t o r s are c o n s t r u c t e d from realizations of t h e a p p r o p r i a t e s y m m e t r y g r o u p s as s u b g r o u p s of direct p r o d u c t g r o u p s .

In general, if ^ = <£', the symmetry group of the hamiltonian operator will consist of all elements of the form Ε ® E, R ® R, S ® S, ... . C. Decomposition of the Hamiltonian Matrix

From the theory developed in Section 3, and the isomorphisms (5.14) through (5.16), it follows that the symmetry constants of the motion (projection operators) for the ΑΑ'ΧΧ', AA'A"A"'XX', and AA'A"XX'X" systems are1 2

^ 2 : Pa(®) ⁼ UE ®

Ε + C

2 ® C 2 } ,

(5.17)

Pd®) =i{E®E-C

²

® C

²

};

^ 2 : Pai(®) =

\{E®E + C

²

®C

²

+ C

²

®E + C'

^T

®

^{C 2 } ,}

P^(®)

⁼

i{E ® Ε

⁺ c² ® c² - ^{c y}

® Ε

^{- c ;} ® c²},

(5.18) P*Ji®) = î{E ® Ε - c²® c² + c y ® Ε - c; ® c²} , P*,(®) =

\{E®E-C

²

®C

²

-C

²

®E + C'

²

®

^{C 2 } ;}

P^ai(®) =

UE®E+ C

³

®C

³

+ C

³

*®C

³

*

+ C

²

® C

²

+ C

²

® C

²

'

+ C 2 ® C 2 } ,

* W ® ) = U

^E

®E + C

³

®C

³

+ C

³

>® C*

- c ² ® c ² - c y ® c²' - c²' ® c²} ,

= i { £

®E~ \(C

³

®C

³

+

C 3 2 ® C 3 2 )

+ 1( C2 ® C 2 - 2 C 2 ' ® C 2 ' + C 2 ®

CD),

PA®) =i{E®E- Uc

³

®c

³

+ C

^{3 2}

® C

^{3 2}

)

-

^{i( c2 ® c2} - 2 c y ® c y + c 2 ' ® c y ; } .

Although these operators were constructed with the ΑΑ'ΧΧ', ΑΑ'Α"Α"'ΧΧ', and AA'A"XX'X" systems as models, they apply also to irreducible components such as Α/ ΑΑ ^ΑΧ/ χΧ ^χ .

396 The projection operators for a given system decompose the spin

8. THE ANALYSIS OF SYMMETRICAL SPIN SYSTEMS

space into disjoint subspaces. This decomposition is independent of any assumptions concerning the spin-spin interactions, so that the description of the direct sum decomposition of the hamiltonian matrix given in Section 3 also applies to the present discussion. In the weak-coupling approximation /S( A A ' ···) and /2( X X ' ···), more precisely 72(AA' ···) (χ) l^x and 1A ® /2( X X ' ···), commute with each other and Jf.

Furthermore, the structure of (5.17) through (5.19) shows t h a t /2( A A ' ···), 7S(XX' ···), and Jf also commute with the projection operators derived from the appropriate symmetry groups. This reveals the particular advantage of the preceding formulation of the weak-coupling approxima-tion for symmetrical systems—all nontrivial constants of the moapproxima-tion are explicitly taken into account.

The construction of symmetrized bases for AA' ··· X X ' ··· systems may be carried out according to the general prescription described in Section 3. However, the calculations can often be expedited by the rules for representation multiplication, and will be illustrated for some particular AA' ··· X X ' ··· systems.

1. The A A ' X X ' System. Table 8.5 gives a set of symmetrized vectors that constitute a basis for the space of the AA' subsystem. In the construction of this basis, it was shown that tr Ε = 4, tr C2 = 2.

Since the trace is a matrix invariant, these relations also hold for the matrices of Ε and C2 defined by the vectors of Table 8.5.

T h e vectors of Table 8.5 may also be used as a symmetrized basis for the X X ' nuclei, so that tr Ε = 4, tr C2 = 2, when Ε and C2 refer to the X X ' nuclei. From these results one can calculate the ranks of Pa{®) and P#(®) relative to the product basis {| A>| X>}:

tr Pa(®) = J{4 · 4 + 2 · 2} = 10, tr Ρ „ ( ® ) = ^{4 · 4 - 2 · 2} = 6, since tr R (x) S = tr R tr S. It follows that Pa(®) and Pa(®) ^{w i l}l decompose the 16-dimensional spin space into disjoint subspaces of dimensions 10 and 6, in agreement with the results previously obtained for the AA'BB' system.

T h e only remaining problem is the determination of the symmetry species of the products | A ) | X>. One can formally construct 10 linearly independent vectors with Ol symmetry, and six linearly independent vectors with âd symmetry by applying Pa{®) and P&{0) to all elements of {I A>| X » . In the present case, the following procedure is preferable.

T h e elements of {| A>} are all of the form Ρ^αΦ, Ρ&Φ'where P^a and P%

are given by (3.17) and Φ, Φ' are appropriate product kets. Similarly,

5. T H E X A P P R O X I M A T I O N FOR SYMMETRICAL SYSTEMS 397

= 0, (5.21)

(5.22) the elements of {| X ) } are of the form Ρ^αΨ, Ρ®Ψ'. Thus there are four types of products in the basis {| A ) | X ) } :

Ρ^αΦΡ^αΨ = Ρ^αΦ ® Ρ^αΨ = P^a® Ρα{Φ ® Ψ), ΡαΦΡ,,Ψ' = ΡαΦ ® Ρ^Ψ' =Ρα® ΡΑΦ ® Ψ'), Ρ^Φ'ΡαΨ = Ρ^Φ' ® Ρ^αΨ = Ρ* ® Ρ^α{Φ' ® Ψ), ΡαΦ'Ρύ,Ψ' = Ρ^Φ' ® PJ¥' =P^m® Ρ»(Φ' ® Ψ').

These results, which follow from the theory of Kronecker products developed in Chapter 4, show that the application of Pa(®) or P®{®) to Ρ^ΦΡ^Ψ is equivalent to applying Pa(®)P^ ® P y or Pm(®)Py ®Py to the vector Φ ® Ψ. This suggests an examination of the operator products Pa{ ®)Pa ® Pa , etc. N o w

Pa ® Pa

=

\{E ® Ε + Ε ® C² + C² ® Ε + C² ® C²), Pa®P® = i{E®E -E®C2 + C2®E ~C2® C2},

(5.20) Ρ® ® Pa

=

Î{E ® Ε + Ε ® C2 - C2 ® Ε - C2 ® C2},

P®®P® = \{E®E-E®C²-C²®E + C²® C²), by (3.17). Applying Pa{®) and P@(®) to these equations, one finds, by the rule for multiplying Kronecker products and the ^2 multiplication table,

Pa{®)Pa®Pa = Pa®Pa, Pa{®)Pa ®P® = Pa(®)P® ® Pc Pa{®)P»®Pa=P»®P*;

P®(®)Pa ®Pa = P®(®)P® ® Pi P*(®)Pa ®Pm=Pa®Pa, P®{®)P® ®P^a =

P®®Pa-Equations (5.21) show that all products of the form ΡαΦ ® Pa^^{1 or} Psf& ® P&P ^{a re} eigenvectors of Pa(®) corresponding to the eigenvalue

1, so that such products belong to the symmetry species 01.

Equations (5.21) also show that the application of Pa(®) to products of the form ΡαΦ ® Ρ@Ψ or Ρ&Φ ® ΡαΨ yields the zero vector, so that such products do not belong to the 01 representation. On the other hand, equations (5.22) show that these products are eigenvectors of P®(®) corresponding to the eigenvalue 1.

398 8. T H E ANALYSIS OF S Y M M E T R I C A L S P I N SYSTEMS

T h e basis {| A ) | X ) } is given in Table 8 . 1 1 , from which one may readily infer the factorization of the secular determinant. In general, mixing occurs only between those products of a given symmetry species with the same eigenvalue of I^z and the same eigenvalues of I^Z(AA' ···) and /²( Χ Χ ' ···). In the present case, the only 01 states mixed by V^{1) are 3(Œ)⁰ and 4(Œ)⁰ ; the only 08 states mixed by V⁽¹⁾ are \{ß\ and 2{ß\ . Hence the hamiltonian matrix is the direct sum of twelve L X L subma-trices and two 2 X 2 submasubma-trices (Fig. 8.9). All diagonal elements of 3tf are of the form ß / >^{m A A}, + ß /^{t m x x}, , and may be obtained directly from (3.24).

For example, one obtains ß / ,^{m A A}, by setting Ν = 2, I^A = \ , m = m^AA' ; the value of I is 1 or 0, accordingly as | A> — | Œ} or | âSy.

F I G . 8.9. Factorization of t h e secular d e t e r m i n a n t for t h e A A ' X X7 system.

2. The AA'A"A'"XX' System. A basis for the Α Α Ά ' Ά ' " subsystem is given by the first 14 entries of Table 8.9, and the vectors L(6?)⁰>o >

2(Oi)

^Q

Q

, given in Section 3.C. It was noted in Section 3.A that, relative to the spin space of the Α Α Ά ' Ά ' " system, tr Ε = 16, tr C2 = tr C2 r = tr C2 = 4. For the X X ' subsystem, one uses Table 8.5 and the trace relations tr Ε = 4, tr C² = 2. With these results, equations (5.18) yield

tvP^a{®) = l{\6 · 4 + 4 - 2 + 4 · 4 + 4 · 2 } = 24, tr Pai(®) = i{ 1 6 · 4 + 4 · 2 - 4 · 4 - 4 · 2 } = 1 2 , tr P^²(®) = i{ 1 6 · 4 - 4 · 2 + 4 · 4 - 4 · 2 } = 16, tr P^3(®) = i{ 1 6 · 4 - 4 · 2 - 4 · 4 + 4 · 2 } = 12.

T h e product basis contains eight types of products: | Œ}\ Œ},

I ---ι I ^ 3> l These products may be considered to be

generated by the application of Pa ® Pa , ® Pa , ..., P^ ® P ^ ,

5. T H E X A P P R O X I M A T I O N FOR SYMMETRICAL SYSTEMS 399

which can be obtained from (3.17) and (3.19), to appropriate product kets. It is not difficult to show that

P^(®)=^P^®^P^ + ^P^®P^,

P^{®) = Pm% ®Pa + Pa®P&, ( 5'2 3 )

P*^t(®) =P*ⁱ®Pa + P*¹®P*.

From these results it follows that the symmetry species of the products I &*y\ ^{a re} given by the rules

ay\ay =

| * i > | #

> = 1

a>,

» i > i oty = 1 ^ 3 > l » ' > = 1 * l > ,

* , > i

oty =

^{> = 1} ^ 2 > ,

%>\ oiy =

I * 1 > I 0 ' > = I ^ 3 > ·

(5.24)

T h e proof of these rules makes use of the mutual orthogonality and idempotence of projection operators. For example, to prove that

I Oiy \ (Τ) = I CT>, one multiplies the first of equations (5.23) from the right with P^a ® P^a to obtain

Pa(®)Pa ®Pa = (Pa ® PaWa ® Pa) + {P*^% ® P®)(Pa ® Pa)

= Pa² ® Pa² + P*^tPa ® P^a -Pa® Pa,

since P^a2 = P^a , and P@Pa = P®Pa = 0· Similar calculations establish the symmetry species of other types of products. From these results, it is easy to show that the hamiltonian matrix decomposes according to the scheme given in Table 8.16.

3. The AA'A"XX'X" System. T h e product basis for this system is obtained by first identifying the vectors of Table 8.8 with the AA'A"

nuclei, then the XX'X" nuclei, and then forming the 64 products {I A>| X » . Since ^ = <S' = @³ , equations (2.33) may be used to compute the ranks of the projection operators (5.19):

tr PŒl(®) = £ { 8 · 8 + 2 · 2 + 2 · 2 + 4 · 4 + 4 · 4 + 4 · 4 } = 20, tr Pa2(®) = i{ 8 · 8 + 2 · 2 + 2 · 2 - (4 · 4 + 4 · 4 + 4 · 4)} = 4,

tr P , « g » - H8 * 8 - i( 2 · 2 + 2 - 2) + 1(4 · 4 - 2 · 4 · 4 + 4 - 4)} = 20, tr iV(<8» = i{ 8 · 8 - i( 2 - 2 + 2 · 2) - J(4 · 4 - 2 · 4 · 4 + 4 · 4)} = 20.

In document Analysis of Symmetrical (Pldal 63-72)