T h e second method is preferable in the weak-coupling limit. For then, the several factors in a product spin function can be labeled with their ζ components of angular momenta, and these labels determine the additional decompositions of the hamiltonian matrix in the weak-coupling approximation. T h e disadvantage of the method is that it may be necessary to carry out a resymmetrization of the product func-tions to determine their symmetry species (cf. Section 5).
An especially simple case occurs with spin systems containing invariant nuclei. If a symmetrized set of basis vectors for the non-invariant nuclei is given, a symmetrized basis for the complete system is obtained by forming the products of the given set with the kets for the invariant nuclei. T h e symmetry species of any such product is given by the symmetry species of the noninvariant factor. For example, in the case of a monosubstituted benzene, the protons ortho and meta to the substituent, which is presumed to have no influence on the magnetic resonance spectrum, may be initially considered as an AA'BB' system, and the para proton an invariant nucleus. A symmetrized basis for the entire system is given by the 64 products of | + > and | — >, with the vectors of Table 8.10 (or Table 8.11). T h e symmetry species of these products are given by the rules (Œ)m\ ± > = (Œ)m±1/2 , (ß)J ± > = ( ^ )m± i / 2 · E. The AA'BB' System
T h e reduction of the determinantal equation for a symmetrical spin system will be explicitly illustrated for the AA'BB' system (16-18). The stationary hamiltonian operator is
= — {ωΑ(1Αζ + IA'Z) + ωΒ(ΙΒζ + IB>Z) + JAA'IA ' IA' + /BB'IB * IB'
+ / A B (IA ' IB + IA' ' IB') + / A B ' (IA ' IB' + IA' · IB)}, (3.35) and is defined with respect to a 16-dimensional spin space. According to the discussion of Section 3.B, (3.35) admits Iz , Pa = \(E -\- C2), and Pm = \(E — C2) as constants of the motion. These operators decompose the spin space into disjoint subspaces, and the hamiltonian matrix into a direct sum of lower dimensional submatrices.
T h e direct sum decomposition of the hamiltonian matrix is auto-matically accomplished by choosing the vectors of the initial basis to be eigenvectors of Iz , Pa , and P% . For this purpose, one may use the vectors of Table 8.10 or Table 8.11 as a basis. T o facilitate the subsequent discussion of the A A ' X X ' system, the vectors of Table 8.11 will be used as an initial basis. Relative to this basis, the hamiltonian matrix is the direct sum of a 10 X 10 matrix, generated by basis vectors with Ol
sym-3. S Y M M E T R I Z A T I O N OF BASIS VECTORS 3 7 7
metry, and a 6 X 6 matrix, generated by basis vectors with symmetry (Fig. 8 . 5 ) . Each of these submatrices decomposes into smaller sub-matrices whose dimensions are determined by the degeneracies of the Iz eigenvalues. Thus the 10 X 10 matrix is the direct sum of five
sub-matrices of dimensions 1, 2 , 4 , 2 , 1, and the 6 x 6 matrix is the direct sum of three two-dimensional matrices (Fig. 8 . 5 ) . Therefore, the analysis of the eigenvalue problem requires the solution of algebraic equations of the first, second, and fourth degrees.
2
ο
F I G . 8.5. Factorization of the secular d e t e r m i n a n t for t h e A A ' B B ' system.
T h e 4 x 4 matrix generated by the basis vectors n(Œ)0 (η = 1, 2 , 3, 4 ) ,
IS
Jf(0) =
where
0 2^+
0 δ — \ K+ + } L + l r H
-2 + IT
- \ L _ τκ+
( 3 . 3 6 )
K± = JAA' ± JBB' » L± — JAB ± JAB> .
T h e eigenvalues of ( 3 . 3 6 ) will be denoted Ωί, i = 1, 2 , 3, 4 , where Ω1 -> —δ — \Κ+ + > Φ
β . *
-(3.37)
δ — \K+ + j£+ ,
4 ^
as the off-diagonal elements of ( 3 . 3 6 ) tend to zero. T h e matrix that diagonalizes ( 3 . 3 6 ) will be denoted (a{j). Since ( 3 . 3 6 ) is a real, symmetric matrix, there is no essential loss of generality in assuming that (aid) is orthogonal.
378 8. T H E A N A L Y S I S OF S Y M M E T R I C A L S P I N SYSTEMS
T h e quartic equation det[Jf (0) — XI] = 0 can be solved by radicals (cf. Reference 1, Chapter 5), but the expressions are quite cumbersome, and of limited practical value. In practice, the eigenvalues and eigen-vectors of J f ( 0 ) are obtained numerically, using a set of trial parameters.
There are, however, three properties of the eigenvalues and eigenvectors of (3.36) that can be deduced by inspection.
(1) T h e eigenvalues and eigenvectors of (3.36) are independent of K_.
(2) T h e replacement of δ with —δ is equivalent to interchanging Ω1
and ß2 , and the corresponding eigenvectors. This property follows from an interchange of the first and second rows of det[JfJ(0) — XI], and an interchange of the first and second columns in the resulting deter-minant.
(3) T h e eigenvalues and eigenvectors of (3.36) are unaltered if L_ is replaced by — L_ . For a change in the sign ofL_ is equivalent to removing a factor of —1 from the fourth row and the fourth column of d e t [ ^ ( 0 ) — XI]. T h e removal of these factors simply multiplies the determinant by (—1)(—1) = + 1 .
It should be noted that the eigenvalues and eigenvectors of (3.36) are altered if K+ is replaced by — K+ , or if L+ is replaced by — L+ (unless K+ = 0, or L+ = 0).
T h e solutions of the linear and quadratic equations for the remaining eigenvalues and eigenvectors are given in Table 8.12, where
Rx = [(δ + K_f +LJfl\ R2 = [KJ +LJfl\
R3 = [(δ - K_f +LJ]V\ RA = [δ2 +L+*\V\
δ + K_ + R1 ' ^2 K_+R,
Η ±= Η
-δ - K_ + Rs ' ^4 δ + P4 ' δ = ω A — COB ·
With this table, and the selection rules derived in Section 3.B, one obtains the resonance frequencies and relative intensities given in Table 8.13. It turns out that the spectrum is symmetrical with respect to the mean resonance frequency, so that only half of the 28 allowed transitions are tabulated.
Detailed descriptions of the types of AA'BB' spectra predicted by Table 8.13 for various values of the internal chemical shifts and
spin-3. S Y M M E T R I Z A T I O N OF BASIS VECTORS 379
TABLE 8.12
EIGENVALUES AND EIGENVECTORS OF THE AA'BB' SYSTEM
Eigenvector E i g e n v a l u e
spin coupling constants have been given in the literature (17-19) and will not be repeated here. T h e following discussion will note some general properties of such spectra and some special cases of interest.
From Table 8.13, and the properties of the (Œ\ states mentioned above, it is easy to show that the appearance of an AA'BB' spectrum is unchanged under any of the following transformations: (1) δ —• — δ;
380 8. THE ANALYSIS OF SYMMETRICAL SPIN SYSTEMS
3. S Y M M E T R I Z A T I O N O F B A S I S V E C T O R S 381 (2) L_ -> —L_\ (3) K_ - > — i£_ . Therefore, a given experimental spectrum will, at best, yield | δ |, | K± |, \L± |, and the sign of K+
relative to L+ . From this information, one can deduce the magnitudes of the coupling constants and their relative signs. It is not possible, however, to distinguish JAA' from /Β Β' , or /A B' from /A'B' . T h e assign-ment of these parameters to particular pairs of nuclei in the molecule, and the determination of the sign of δ, requires supplementary information.
T h e frequencies of the transitions recorded in Table 8.13 have the property that, as all —> 0, transitions 4 and 7 approach 3S/2, while the remaining frequencies approach δ/2. Thus the frequencies of all transitions in the table are greater than or less than \(ωΑ + ωΒ), accord-ingly, as ωΑ > ωΒ or ωΑ < ωΒ . It should not be concluded, however, that this property persists for nonvanishing values of the coupling constants. In fact, it is not uncommon for "crossovers" to occur.
T h e assignment of transitions in an observed AA'BB' spectrum normally commences with the attempt to assign those transitions whose frequencies and intensities are given explicitly in terms of the spectral parameters. T h e assignment of transitions 1 and 2, together with their images Γ and 2', is particularly important. These transitions form an AB type of quartet, except that / is replaced by / + J\ so that their assignment yields | δ | and | / + / ' |. T h e assignment may not be obvious, since there will usually be several sets of four resonance frequencies that resemble an AB spectrum. A simple check on the assignment is provided by the relation
Int 1 + Int 2 = 4.
Since the frequency separation of transitions 1 and 2 is independent of the applied field, the assignment may also be checked by experiments at different polarizing field strengths.
T h e next step in the analysis is the assignment of the 38 transitions, from which one obtains \K_ \ and T h e assignment of these transitions may be checked by the relations
Int 11 + Int 12 = 2, Int 13 + Int 14 = 2.
T h e determination of | K+ | and the sign of K+ relative to L+ requires an investigation of the transitions involving the (Œ)0 states. This investi-gation is usually carried out by "trial-and-error" techniques, using numerical diagonalizations of (3.36). T h e assignment of these transitions may be checked with the help of the relations
Int 3 + Int 4 + Int 5 + Int 6 = 4(1 + £+/2i?4),
Int 7 + Int 8 + Int 9 + Int 10 = 4(1 - LJ2R
A).
382 8. T H E ANALYSIS OF SYMMETRICAL S P I N SYSTEMS
which follow from Table 8.13 by expanding the squares of transitions 3 to 10, and using the orthogonality relations satisfied by the atj .
T h e protons in o-dichlorobenzene (77) provide a good illustration of a well-resolved AA'BB' spectrum (Fig. 8.6). Only four transitions are unresolved: 4, 4', 7,7'. T h e sign of δ and the assignment of coupling constants were inferred from observed results of related systems (77).
F I G . 8 . 6 . E x p e r i m e n t a l a n d theoretical p r o t o n m a g n e t i c r e s o n a n c e spectra for p u r e o-dichlorobenzene at 6 0 M c p s .
T h e discussion of the AA'BB' system will be concluded with brief descriptions of five special cases for which the eigenvalues and eigen-vectors of Jf(0) can be readily calculated.
(1) δ = 0. T h e simplest limiting case occurs when δ = 0. T h e system is then equivalent to an A4 system, so that the spectrum consists of a single line at ωΑ = ωΒ . T h e verification that Table 8.13 also yields this result may be easily carried out, upon noting that the secular equa-tion det[J^(0) — A7] = 0 is equivalent to a pair of linear equaequa-tions and a quadratic equation. On solving these equations, one need only solve linear equations to obtain the . With these results one may apply perturbation theory to calculate the spectrum of the AA'BB' system in the strong coupling limit (| /A A- |, | /B B- |, | /A B |, | /A-B | > | δ |).
(2) JAB = JAB' — 0. In this case the eigenvalues of Jf(0) are given by the diagonal elements of (3.36), so that (a^) = T h e spectrum consists of two lines of equal intensity, one at ωΑ , the other at ωΒ .
(3) JAA' JBB' — JA'B = 0. In this case, the eigenvalues of (3.36) are ± \JAB , \JAB ± (δ2 + / 1Β )1 / 2· With these eigenvalues one can determine the eigenvectors of J f ( 0 ) , and obtain an explicit reduction of Table 8.13. It is much simpler, however, to note that the hamiltonian operator is the sum of the hamiltonian operators for two independent,
3. S Y M M E T R I Z A T I O N O F B A S I S V E C T O R S 383 but otherwise identical, AB systems. T h e spectrum will thus have the appearance of an AB quartet.
(4) / A B = / A B < ^ 0 . When /A B = /A B- , L_ = Q1 = Q2 = Q3 = 0.
Furthermore, ß4 = f K+ , so that au = au = δΐ 4 , i = 1, 2, 3, 4. From these results it can be shown that the intensities of transitions 6, 10, 12, and 14 vanish, and that Table 8.13 correctly reduces to the A2B2 system.
Note that transitions 4 and 7 are mixed transitions, and that transitions 11 and 13 have the common resonance frequency \(ωΑ -f- ωΒ -f- δ). It will be recalled that the spectrum of an A2B2 system depends only upon the absolute magnitudes of /A B and δ.
(5) I δ I —> oo. An important special case of the AA'BB' system occurs when | δ | —> oo, but all coupling constants remain finite (13).
In this limit,1 0 AA'BB' - > AA'XX', and the proper reduction of Table 8.13 is obtained by replacing B, B' with X, X', and allowing I δ I —>* oo. T h e only (Œ)0 states mixed by the spin-spin interactions are 3(O)0 and 4(Œ) ο , so that
Ωλ = —ωΑ + ωχ — \ΚΛ -f \1^Λ ,
where
Rb = (K+*+LJfl\
T h e corresponding eigenvectors are
1
Wo > (ΓΤ£)7)νΛ
3^ο] +
Ö5[4Wo]}, 2(ίΤ)ο - jï^jyTÀQdmol - [4(0%]}, whereUsing these results, one obtains the frequencies and intensities given in Table 8.14. T h e table shows that the AA' region of the spectrum generally consists of a symmetrical 10-line spectrum, with ωΑ at the center of symmetry. If all 10 lines are resolved, the frequency separation
1 0 A m o r e systematic p r o c e d u r e for arriving at t h e w e a k - c o u p l i n g limit in symmetrical systems is d e s c r i b e d in Section 5.