400 8. THE ANALYSIS OF SYMMETRICAL SPIN SYSTEMS

T A B L E 8.16

D E C O M P O S I T I O N OF THE H A M I L T O N I A N M A T R I X FOR THE A A ' A ' A ^ X X ' SYSTEM

3 1 x 1

2 1x1,1x1 1 χ 1

1 1 χ 1, 2 χ 2, 3 x 3 1 χ 1, 2 χ 2

0 1 χ 1, 1 x 1,4 x 4 1 χ 1, 1 χ 1, 2 χ 2 -1 1 χ 1, 2 χ 2, 3 χ 3 1 χ 1, 2 χ 2

-2 1x1,1x1 1 χ 1

-3 1 χ 1

m ^ 2

3

2 1x1,1x1 1 χ 1

1 1 χ 1, 2 χ 2 1 χ 1, 2 χ 2

0 1 χ 1, 1 χ 1,4 χ 4 1 χ 1, 1 χ 1, 2 χ 2

-1 1 χ 1, 2 χ 2 1 χ 1, 2 χ 2

-2 1x1,1x1 1 χ 1

-3

Hence a symmetrized basis for the AA'A"XX'X" system contains 20 vectors with Ol^x symmetry, four vectors with Ol² symmetry, and 40 vectors with ê symmetry.

The analysis of the symmetry species of the products | «$01 «$0 will be carried out for product bases that include products of the form

I

^A2}\

«$0, I «SOI

^ŒÙ- These products do not occur in the AA'A"XX'X"

system, since Table 8.8 does not contain any vectors with 0L² symmetry.

In a general basis, there will be 16 types of products | «$01 ') ⁼ Ρ^Φ ® Ρ^Ψ = P<r ® Ρ#»(Φ ® Ψ), where Φ and Ψ are product kets.

By expanding the Kronecker products Ρ#> ® P#>, , one can show that PA,(®) + ^PA®) = PA, ® PA, + Ρ^Α% ® Ρ a, + Ρ* ® ^PA^X + Ρ^ΛΧ ® P*

+ PA² ® P* + PS ® PA.² + P$® Ρ* + Ρ/ ® P*\

(5.25)

5. T H E X A P P R O X I M A T I O N FOR SYMMETRICAL SYSTEMS 401

Pa,(®) + PA®) = Pa, ® Pa, + Pa, ® Pa, + Pa, ®Pg+P$® Pa,

+ Pa, ® P/ + Ps ® Pa,

+.iV ®

P/ + P« ® Ps • (5.26) T h e interpretation of these equations is simple. Equation (5.25) states that all elements of the product basis that are eigenvectors of the projection operators on the right side contribute to the 0ίλ and ê' species of a symmetrized basis. This result can be sharpened for particular elements of the product basis. If S is any element in <®3 , then SP^Œi = P^Œi , so that

Pampoti ®Pa1 = iX(S® S)PŒl ® Ρ a, s

= i X SP a, ® SPai = P^ai ® P^{a i}.

S

Therefore, any product of the form | 0l^)\ 01^} belongs to the symmetry species 0lx . Furthermore, ΕΡα^ = CJP^a<i = C³²P^a2 = P^a^, whereas

^2Pa2 = C2'P^2 = C2Pa2 — ~pa2 > s o that

ρα^λ(®)^Ρα² ®^pa² = ^ ( S ® S)P^Œ2 ® P^a.² = Ρ a, ® Ρ a, . s

Hence any basis element of the form | Œ2}\ Ol²y belongs to the symmetry species | Οί^. Products such as | ST>\ <f >, \ S?y\ g'\ \ <T>| ST\ \ Ol where £f = 01 ^ or 012, belong to $'. This may be proved by showing that such products do not belong to Ot^x . For example, the application of PaH®) to P/ ®P^Œl gives

P^ai(®)PS ®^pa¹ = i l ^{/ SP}<?' ® ^{S P}^ = i X ^{S P}* ® ^P^

s s

= έ t (Σ s) ^Ρ* \ ® ^Ροι, = ^Pot^pS ® ^ρα^λ = 0, since Ρα and PS are mutually orthogonal. Hence, by (5.25), any product of the form Ρ / Φ ® Pa^ belongs to S". One has, therefore, the following rules:

I

A XY \ A , Y = \ A 2 Y \ O I 2 Y = \ A , Y , (5.27)

I

A , Y \ Ê F Y = \ Ê ' Y \ A X Y = \ A 2 Y \ SY

= 1

S Y \ A 2 Y = \

<r>.

^(5.28)

Products of the form | êy\ êy and | S'y\ S'y contain the symmetry species Œ1 and ê \ so that all products of this form must be

resym-402 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

Basis vector ( ^ ) m

+ ( O l / . ( O l / . } ( # l ) l

- ^ { ( ^ ) l /( ^ ) _ l2 2 + ( O i / 2 ( 0 - i / J ( ^ i ) o /

^ y { W - l / . W l / . + ( O - l / l t O l / l ) ( # l ) o

y j i W - i / . W - i / i + ( 0 - i / 2 ( 0 - i / J ( ^ i ) - i

^ y { ( * ) l / . ( O l / . - ( O l / l W l / l )

^ { ( f l i / î t O - i / i - ( ^ ) ι / « - ι / 2 } W o

- K ) - l / 2 ( O l / J ( ^ 2 ) θ

^ j i W - l / . i O - l / . - ( O - l / . W - l / i } ( « . ) - !

^ ( W l / i K ' l l / i + ( O l A / J ( * ) l

^ { ( * ) l / . ( 0 - l / . + W o

^5 « O- I/ 2 W 1 / 2 + ( « 0 - 1 / 2 ( ^ ) 1 / 2 } Κ ) θ

^ { W - l / 2 ( 0 - l / 2 + ( O- I/ 2 W- I/ 2 } K ) - !

^5

^{- ( O l}

^ ( W l / i W - l / . - ( O l / 2 ( 0 - l / J ( O o

^ { ( ^ ) - l / 2 ( ^ ) l/ 2 - ( 0 - ΐ / 2 ( Ο ΐ / 2 > ( O o

- ( 0 - 1 / 2 ( 0 - 1 / 2 ) ( O - l T A B L E 8 . 1 7

RESYMMETRIZED BASIS VECTORS FOR THE A A ' A ' X X ' X " S Y S T E M

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 403

metrized. For this purpose, one applies P ^ ( ® ) and P / ( ® ) to these products, and obtains basis vectors of Œ¹ and $ ' symmetry in the usual manner.

A similar analysis of the projection operators on the right side of (5.26) shows that

I

« i> l

OLÙ = I ^

²

>l

^{« i >}

= I

( 5 . 2 9 )

I

^{« i> l < 0}

= I *>l

^{# i >}

= I

^2>l

Ο = I <OI ^

²

> = I

< 0 - ( 5 . 3 0 )

Products of the form | «f>| é") and | i ? ' > | * 0 contain the symmetry species 6 ?² and <?, so that all such products must be resymmetrized with Pa^t(®) and P , ( ® ) .

T h e application of the preceding results to the AA'A"XX'X" system shows that it is necessary to resymmetrize | # ) | < ^ ) , | < f > | | ê' > | < f > , I < f '> | <?'>, to obtain a symmetrized basis for the composite system. T h e resymmetrized vectors are given in Table 8.17. For notational simplicity

( < ^ ) i/ 2, ± i/ 2 d ( < Π ιa n / 2, ± ι/ 2 h a e v b ene written ( « ? )± 1/2 , ( < H ± i/ 2 · TE H

symmetry species of all other products are given by (5.27), (5.28), and (5.30). From these results one may deduce the decomposition of the hamiltonian matrix given in Table 8.18.

T A B L E 8.18

DECOMPOSITION OF THE HA M I L T O N I A N MA T R I X FOR THE A A ' A " X X ' X " SYSTEM

3 l x l

2 1 χ 1, 1 χ 1 1 χ 1, 1 χ 1 1 l x l , l x l , 2 x 2 l x l l x l , l x l , 3 x 3 0 1 x 1 , 1 x 1 , 2 x 2 , 2 x 2 1 x 1 , 1 x 1 3 x 3 , 3 x 3

- 1 1 x 1 , 1 x 1 , 2 x 2 l x l 1 x 1 , 1 x 1 , 3 x 3

- 2 1 x 1 , 1 x 1 1 x 1 , 1 x 1 - 3 l x l

D. Remarks on the Theory of Symmetrical Systems

T h e discussion of this chapter has been principally concerned with the theory of finite groups, their representations, and the application of these concepts to the study of symmetrical spin systems. However, some further remarks concerning the three-dimensional rotation group Θ³ are appropriate at this point. It has already been mentioned that &³ is of fundamental importance in the quantum mechanical theory of

404 8. T H E ANALYSIS OF S Y M M E T R I C A L SPIN SYSTEMS

angular momentum. In fact, one can say that the theory of Θ³ and its irreducible representations is the theory of angular momentum.

&s has a denumerably infinite number of irreducible representations.

There is exactly one irreducible representation for every dimension from 1 to infinity. T o translate these results into more familiar terms, consider a particle of spin / , and the associated basis {| 7, m}}, whose elements are eigenvectors of I2 and Iz . In group-theoretical language, the vectors {| 7, m}} constitute a basis for an irreducible representation of &3 of dimension 2 7 + 1 . T h e spin quantum number labels the irreducible representation, the dimension of the spin space specifies the dimension of the irreducible representation, and the eigenvalues of Iz

label the basis vectors.

T h e addition of two angular momenta \^x and I² , to obtain their resultant I, is equivalent to reducing the product basis {| Ιλ , m^)\ I2 , # * 2 / } >

which generates a (2IX + l ) ( 2 /² + l)-dimensional representation of &³ , into its nonequivalent irreducible components. The latter are labeled by the total spin quantum numbers / = Ix + I2 , Ιλ -f- I2 — 1, [ I1 — 7² |, each occurring exactly once in the decomposition. T h e coefficients expressing the eigenvectors of I² and Iz in terms of the product bases are called the Clebsch-Gordan coefficients. In general, the spin multiplicity gj gives the number of times the irreducible representation labeled by the quantum number 7 occurs in the reduction. Of course, all gj = 1 for the addition of two angular momenta. These remarks indicate the motivation behind the term "irreducible component'' introduced in Chapter 5 to describe systems of the form A^{/ A}B^{/ ß}C^{/ c} ···. The reduction of a given spin system into its irreducible components is nothing more than a recognition, in physical terms, of the irreducible representations Finally, two contingencies of the weak-coupling approximation will be noted. Occasionally, certain subblocks of the hamiltonian matrix reveal additional decompositions that are not predicted by the theory given above. These decompositions depend upon the particular system under study, and cannot be anticipated. The reason is that the additional decompositions do not occur throughout the complete hamiltonian matrix, so that they cannot be attributed to an unrecognized constant of the motion.

A useful occurrence in the spectra of weakly coupled symmetrical systems is the appearance of certain sets of lines that can be made to correspond with the spectra of other systems (79, 27). For example, some of the resonances observed in the spectrum of 1,3, 5-trifluoro-benzene can be interpreted as an A B C spectrum (19). The recognition of such patterns can materially simplify the analysis. However, to

5. T H E X A P P R O X I M A T I O N FOR SYMMETRICAL SYSTEMS 405

establish such correlations it is necessary to compare the matrix elements in certain subblocks of the hamiltonian matrix with those appearing in the hamiltonian matrix of the correlated system. This detailed examination of matrix elements precludes the existence of general rules for determining such correlations—each spin system must be separately examined for any existing correlations. This situation should be con-trasted with the theory of the irreducible components of multispin systems. One can determine, in advance, which irreducible components will appear in the spectrum of a given system, and their relative weights—

without calculating a single matrix element.

In document Analysis of Symmetrical (Pldal 72-77)