X = «Αω*})1/2 ' y = « J C Ü2» I / 2 ' * = J(Aw2))1/2 ' where
/ « A \
1 / 2η
/ « B \1 /2/ « C \
1 / 2 / y | 0 / nEquations (4.24) define a unit vector η = (cos a, cos ]8, cos y), whose components may also be expressed in terms of the polar angles φ, θ:
cos α = sin Θ cos φ, cos β = sin θ sin 7?, cos y = cos θ. (4.25) In terms of the reduced chemical shifts, equations (4.14) take the form
χ cos a. + y cos β ζ cos y = 0, (4.26)
*2 + s2 = 1, (4.27)
cos β cos y y ρ3/2. (4.28)
Equation (4.26) requires the reduced shifts to lie in a plane passing through the origin and perpendicular to n. Equation (4.27) is the equation of the unit sphere about the origin, so that (4.26) and (4.27) together
4. M O M E N T ANALYSIS OF H I G H - R E S O L U T I O N SPECTRA 323
( a ) r = I , Δ ωΒ = Δ ω 0 > 0 ( b ) r = I , Δ α % = Δ ω 0 < 0
Δ ωΑ 0 Δ ωΒ, Δ ωΑ Δ ω β , Δ ω ο Ο Δ ωΑ
( c ) Γ = - i , Δ ωΑ = Δ ω0 > Ο ( d ) r = ~ i , Δ ωΑ = Δ ω ς < Ο
Δ ωβ Ο Δ ωΑ Δ ωι 0 Δ ωΑ Δ ωι 0 Ο Δ ωΒ
( e ) r = - 2 , Δ ωΑ = Δ ωΒ > Ο ( f ) r = - 2 , Δ ωΑ = Δ ω Β < Ο
Δ ω0 Ο Δ ωΑ Δ ω „ Δ α ^ , Δ ω β Ο ι Δ ωε
F I G . 7.14. Solutions of t h e first t h r e e m o m e n t e q u a t i o n s for nA = «B = nc , a n d ρ = 21/3.
( α ) r = o o , Δ ω0 > 0 ( b ) r = - o o , Δ ωΑ > 0
Δ ωΑ Δ ωΒ Δ ω ς Δ ω 0 Δ ωΒ Δ ωΑ
( c ) r = - I , Δ ως > 0 ( d ) r = - I , Δ ωΒ > 0
Δ ωΒ Δ ω . Δ ωΒ
( e ) r = Ο , Δ ωΑ> Ο ( f ) r = Ο , Δ ω β > 0
Δωβ Δ ω ς Δ ωΑ Δ ωΑ Δ α ^ Δ ωΒ
F I G . 7.15. S o l u t i o n s of t h e first t h r e e m o m e n t e q u a t i o n s for nA = nB = nc , a n d P = 0.
324 7. PERTURBATION AND M O M E N T C A L C U L A T I O N S
state that χ, y, and ζ lie on the unit circle defined by the intersection of the plane (4.26) with the sphere (4.27). T h e intersection of this circle with the surface (4.28) defines the possible values of the reduced chemical shifts. These geometric considerations suggest the introduction of new variables (w, v, w), defined by the orthogonal transformation:
χ = u cos Θ cos φ — v sin φ + w sin θ cos <p,
y = u cos θ sin φ + ν cos φ + w sin θ sin <p, (4.29)
ζ = —u sin 0 + w cos θ.
Substituting (4.29) in (4.26) through (4.28), one finds that w, v, and w satisfy the equations
«2+ Ü2 = 1, (4.30)
2«3 cot 20 + 3WÜ2 cot θ + = P3 / 2 /4 e 3 1x
sin θ
w = 0. (4.32)
Equations (4.30) through (4.32) are quite useful in the study of particular three-group systems. Their use will be illustrated by a hypothetical example that also illustrates the existence of nontrivial multiple solutions of (4.14).
Consider a three-group system with nA — nB = 1, nc = 2, whose chemical shifts relative to <ω> are ΔωΑ = —6 cps, ΔωΒ = —2 cps, Δω€ = 4 cps. From these data one finds
(Δω2} = 18 (cps)2, <Λω3> = - 2 4 (cps)3, p3/2 = - ? V2.
Consider now the problem of computing ΔωΑ , JcoB , and Δω€ , given the above values for <^1ω2>, <Ζΐω3>, and ρ3/2. Since tan φ = tan 0 = 1 , equations (4.30) and (4.31) yield the cubic equation
3 2 Λ/2 _
whose roots are
^(V2 + V3)
(i(V2-V3).
4. M O M E N T ANALYSIS OF H I G H - R E S O L U T I O N SPECTRA 325 T h e corresponding values of ν are given by ± ( 1 — u2)1/2 (w = 0 for all values of u and v). From these results one finds the solutions given in the accompanying tabulation, where R± = (Λ/6 ± 2 )1 / 2. Only three
(1) (2) (3)
J o >A - 6 2(1 - h V6 2(1 - R+V2) - V6
Jo>B - 2 2(1 + V6 2(1 + R+V2) - V6
J c oc 4 - ( 2 - 1- V6) - ( 2 - V6)
of the six solutions are given in the tabulation; the remaining three solutions are obtained by interchanging ΔωΑ and ΔωΒ in solutions (1), (2), and (3). It should be verified that these solutions satisfy the equations
ΔωΑ + ΔωΒ +
2
Δωο =0,
(ZICOA)2 + (ΔωΒ)2 + 2(Δωα)2 = 72, (Ζ!ωΑ)3 + (ΔωΒγ + 2(Δω0)3 = - 9 6 .
Geometrically speaking, the existence of two imaginary solutions means that the circle (4.30) intersects the plane curve (4.31) at four points. Errors in the experimental moments could introduce an imaginary component in an otherwise acceptable solution. Hence imaginary solutions should not be summarily dismissed without investigating the possibility that such solutions are imaginary only by virtue of experi-mental errors in <Jo>2> and (Δω3}.
(α)
Δ ωΑ Δ CDB
0
Δ ω ς(b)
Δ ωΑ Δ ω ς Δ cog
F I G . 7.16. M u l t i p l e solutions of t h e first t h r e e m o m e n t e q u a t i o n s for nA = «B = 1, nc = 2, <Jo>2> = 18 ( c p s )2 <Jco3> = —24 ( c p s )3. T h e s e data also yield c o m p l e x solu-tions (see t h e text).
326 7. P E R T U R B A T I O N AND M O M E N T C A L C U L A T I O N S
Solution (3) is a real, acceptable solution whose rejection in favor of solution (1) would require additional information. Solutions (1) and (3) are sketched in Fig. 7.16.
E. Approximate Moment Calculations
T h e moment method can also be applied to systems of the type described in Section 2. T h e selection rules for such systems show that one obtains a set of moment equations for each spectral region. Consider, for example, the An ABn ß ··· Χη χΥΛ ··· system. Let / denote the common spin quantum number of the NAB... = nA + nB + "" nuclei forming groups A, B, and let Γ and NXY... denote the corresponding quantities for groups X, Y, ... . T h e moment equations of the irreducible Α/ ΑΒ/ β · · · Χ /χΥ /γ * * · component for the AB · · · region are obtained from (4.2), using (2.5) without the negative sign for Jff, and J±(AB ···) for J±.
T h e moments for the AB ··· region of the complete system may be obtained by a procedure similar to that used for the An ABn ß ··· system.
Similar remarks apply to the calculation of the moments for the X Y ··· region of the spectrum. T h e results for the first three moments of each spectral region are nth moments computed with respect to the mean frequencies of the AB ··· and X Y ··· regions of the spectrum.
4. M O M E N T ANALYSIS OF H I G H - R E S O L U T I O N SPECTRA 327 F. Remarks on the Moment Method
T h e most attractive feature of the moment method is the directness of its approach to the problem of spectral analysis—the parameters of interest appear as unknown quantities in a system of simultaneous algebraic equations. T h e theoretical moments are exactly calculable, so that one can always set up a number of moment equations equal to the number of chemical shifts and coupling constants. In principle, therefore, the moment method provides a direct, general method for the analysis of complex spectra. In practice, the usefulness of the method is rather limited, owing to some inherent difficulties in the method itself, and in the determination of experimental moments.
Experimental moments are calculated by expressions of the form
where A$ and œi denote the integrated intensity and resonance frequency of the tth resonance, respectively. Measurements of resonance frequencies are usually more reliable than intensity measurements, and it is fortunate that repeated multiplications of experimental intensities are not required.
However, these operations are performed on experimental frequencies, and even small errors in frequency measurements can introduce serious errors in the calculation of higher moments.
Additional errors are introduced when the spectrum contains over-lapping signals that require a decomposition into subareas and the assignment of fractional areas to each resonance. Furthermore, un-observed resonances may have large-frequency arms whose contributions to higher moments may be significant. This is an inherent difficulty of the moment method, since the contributions of all resonances are taken into account in the calculation of theoretical moments.
Finally, there are the ambiguities that arise whenever the moment equations possess multiple solutions.
Although the moment method is not suitable for the study of incom-pletely resolved spectra, the lower moments can often be used to obtain partial analyses of well-resolved spectra. For example, the second and third moments of the proton resonances of 2-bromothiophene (Fig. 7.4) at 60 Mcps are
(Δω2} = 61.558 (cps)2, (Αω*> = 253.146 (cps)3.
From these data one finds that ρ = 0.65 and, from Fig. 7.12, that r = - 3 . 5 4 , - 1 . 3 9 , - 0 . 7 2 , 0 . 2 9 , 0.39, 2.53. The value r = - 3 . 5 4 yields
ωΑ β / 2 π = 13.4 cps, ajAC/2n = 18.0 cps. A somewhat better
corre-328 7. P E R T U R B A T I O N AND M O M E N T C A L C U L A T I O N S
spondence with the results of the direct analysis could be obtained by using the properties of the A B C system derived in Section 1 to introduce corrections for the mixed transitions not included in the calculation of <Ja>2> and <Ζΐω3>.
R E F E R E N C E S
1. W . A . A n d e r s o n , Phys. Rev. 102, 151 (1956).
2. L . I. Schiff, " Q u a n t u m M e c h a n i c s , " 2 n d ed., C h a p . V I I . M c G r a w - H i l l , N e w York, 1955.
3. E . O . Bishop a n d R. E . R i c h a r d s , Mol. Phys. 3, 114 (1960).
4. W . Brügel, T . Ankel, a n d F . K r ü c k e b e r g , Ζ. Electrochem. 64, 1121 (1960).
5. S. Castellano a n d J. S. W a u g h , / . Chem. Phys. 34, 2 9 5 (1961); 35, 1900 (1961).
See also J. R. C a v a n a u g h , ibid. 39, 2378 (1963); 40, 248 (1964).
6. J. R. C a v a n a u g h , P . S. L a n d i s , a n d P . L . Corio, S o c o n y M o b i l T e c h n i c a l R e p o r t ( u n p u b l i s h e d ) .
7. G . A. Williams a n d H . S. G u t o w s k y , / . Chem. Phys. 25, 1288 (1956) 8. P . L . Corio, Chem. Rev. 60, 363 (1960); / . Mol. Spectry. 8, 193 (1962).
9. (a) P . T . N a r a s i m h a n a n d M . T . Rogers, / . Chem. Phys. 31, 1430 (1959); 34, 1049 (1961); (b) S. Alexander, / . Chem. Phys. 32, 1700 (1960); (c) J. A . P o p l e a n d T . Schaefer, Mol. Phys. 3, 547 (1960); (d) P . D i e h l a n d J. A . P o p l e , ibid., 557 (1960).
10. (a) H . S. G u t o w s k y , C . H . H o l m , A . Saika, a n d G . A . Williams, / . Am. Chem. Soc.
79, 4596 (1957); (b) F . S. M o r t i m e r , / . Mol. Spectry. 3, 355 (1959); (c) A . D . C o h e n a n d N . S h e p p a r d , Proc. Roy. Soc. {London) A252, 4 8 8 (1959); (d) R. W . F e s s e n d e n a n d J. S. W a u g h , / . Chem. Phys. 30, 944 (1959); (e) C . N . Banwell a n d N . S h e p p a r d , Proc. Roy. Soc. (London) A263, 136 (1961).
11. R. C . H i r s t , D . M . G r a n t , a n d E . G . P a u l , / . Chem. Phys. 44, 4305 (1966).
12. W . A . A n d e r s o n a n d H . M . M c C o n n e l l , / . Chem. Phys. 26, 1946 (1957). See also H . P r i m a s a n d H . G u n t h a r d , Helv. Phys. Acta 31, 4 3 (1955).
13. D . R. W h i t m a n , / . Mol. Spectry. 10, 2 5 0 (1963).
14. C . N . Banwell a n d H . P r i m a s , Mol. Phys. 6, 225 (1963).