, 1 , — 1 1 1 1 1—ι 1
-io -5 ω
χ+5 +10
F I G . 7.9. E x p e r i m e n t a l a n d theoretical spectra for t h e fluorine n u c l e u s of 2-fluoro-4 , 6 - d i c h l o r o p h e n o l in acetone at 56.2-fluoro-4 M c p s .
The frequencies of the remaining X and M transitions are consistent with (1) and (2), but only (1) predicts the correct relative intensities.
Hence | 8/2π | = 2.55 cps and | /A X — /B X \/2π = 12.21 cps. From (3.10) it follows that JAX and JBX are of opposite sign, and that the magnitudes of these coupling constants are 10.18 and 2.03 cps. Studies of related molecules indicate that the larger coupling constant is asso-ciated with the interaction of the fluorine nucleus and the proton in the " 3 " position. Denoting the latter A, and the other ring proton B, T h e fluorine spectrum is shown in Fig. 7.9. All six predicted resonances in the X region are observed, and since (see Table 7.9)
[ g ( * . £ ) g ( i . - I . -I) + i ]2 + [Q(h - h
i )
- g ( i .- 4 . - i ) ]
2_ , [l + e * ( i , - i , i ) ] [ i + 0
2( £ .
the two lines of greatest intensity correspond to the theoretically predicted X transitions of unit intensity. The frequency separation of these lines is 8.15 cps, in agreement with (3.10).
3. T H E Αη ΑΒ Χη χ S Y S T E M 305
The A B X2 System
T h e theoretical spectrum for the A B X2 system is given in Table 7.11.
The only point worthy of special comment for this system is the fact that Ix = 0 is a possible total spin quantum number for group X. If the AB quartet corresponding to Ix = 0 can be identified in the experi-mental spectrum, | ωΑΒ | and | /A B | may be determined exactly.
The Α2Β Χη χ System
The theoretical spectrum of the Α2Β Χη χ system is given in Table 7.12.
T h e R's and Qys for this system are
R(\,l,mx) = δ — I/ A B + ( / A X — 7B X ) ^ X »
0, mx) = {[δ - y A B + (/AX - JBX)mx]2 + 2 /A B}1 / 2, R(\, - 1 , « χ ) = {[δ + y A B + (/AX - hx)mxf + 2 /A B}1 / 2, R(l, - 2 , m x ) = δ 4 lJAB 4 ( /A X - ]BX)mx ,
Ä( 0 , 0, n i x ) = δ - | /A B + ( /A X - JBX)mx , 0(1, l , mx) - 0 ( 1 , - 2 , mx) - O(0, 0, mx) = 0 ,
the spectrum is described by any one of the following four sets of parameters:
~ = ±2.55 cps, = ±10.18 cps,
ATT ATT
^ = ±2.03 cps, 4^- = +2.49 cps;
ATT ATT
= ±2.55 cps, = ±10.18 cps,
ATT ATT
I** = ±2.03 cps, - ^ S . = _2.49 cps.
ATT ATT
T h e same numerical values for the spectral parameters are obtained if the preceding analysis is carried through on the assumption that the b quartet corresponds to mx = — -|. In this case, however,
R( i
> — έ > — έ ) <
R( i> ~ έ > έ)>
O S TT 8H A AD / A X -N JBX have thesame sign.
TABLE 7.11 RESONANCE FREQUENCIES AND RELATIVE INTENSITIES FOR THE ABX2 SYSTEM
Ο
Transition0 Relative intensity Resonance frequency A Transitions , \\ f Τ I; 1, 1> -> 1 ± ___ZA1___ I {ωΑ -f ωΒ r JAB + (/AX + Λχ) « I4 -1- 1 —1 -1
_1\ 2» ± 2' 2' 2' J> J/CO > δ
i J; 1, Τ f
1, 0> - 1 ± ^ i{ω
Α+
ωΒΤ / AB -f - i,
0)}Ii, - = F i; l ,0> ^ *'
0)i. A : i =F J;
1, —1> 1 ±^ 1 {ω
Α -rω
β-F /AB -
(J ΑΧ+ /ΒΧ ) Ä (i - i
-1) V2> 2'i,±i;J,i ;0,0>- . 1 ±
/abi {
ΩΑ+ O,
B± /AB - R& - h <>)} li±i;i,-i ;0,0 > A( Ï.-Ï .O )
>
U ;iTl;o,o> - ι ± ^ i W A + ω
Βτ J AB + mi , -ο » §
i - Τ i;0,0> K(2' 2,U) Sm
Β Transitions 2Ο > r Ο Α
i±fi f 1,0> - 1
±__^ _ i{«
A+ «B±/AB-Ä (i-iO » >
i ± i i i
1, 1> -> 1±
Α\ i {ω
Α+ ω
Β± /AB + (/ΑΧ + /BX ) üüi-i ^I )
("
Τ,1)-Ä(I-II » :
»,ι 0Λi W + ω
Β±
JAB -R (h
- i ™ *li±ü -l;l,0 > ^ Ο i ± f, i i
1, -1> -> 1± ^ i {ω
Α+ ω
Β± /A B - (/ΑΧ
+ /ßx) ^2» ~ 2'TABLE 7.11 (Continued) Transition0 Relative intensity Resonance frequency X Transitions
i τ ϋ
± b 1.1>i
=Fii ±l;l,0> LI, Τ ± |; 1, -
-i>
2 [Q( i - i, i)£?(i , -i.o> + H
2[i 0
2(i -
bö
2(i -
b 0)]2 [Q( j - j
0)Q(b - b-D + i]
2 [1 +0
2(i
- i, 0)][1I{2«X ± Ä(J, - i 0) Τ #(i - il)}
1
{2α,χ ± - i, -1)Τ
- £, 0)} Hs
W Ii ±ϋ ±I;l,I>- li ±ii ±|;1,0>1
4-1· 1 2> ± 2> 2' ± | ; i,o > i ± i i ± i ι , — 1>
M TransitionsH
{2"X ± /ΑΧ ± 7BX} H ί2ωχ ± 7AX ± JBX}ce
X s I, ± I; I, Τ I;1, 1>
—Ii ±i ;l,0 >
±4;I,TI;l,0>- Ilχ
1 . 1I 2> + 5» 5>
± b
i. -i>
2 [0( i
[1 +0
2(i - i
1)][1 +0
2(i - i
0)]2 [Q(i-è ,0 )-Q(j-i -l)]
2 [1+0
2(i
- I,0)][l+Q
2(i - i
-1)]I {2«.χ ±
Ä (i - i
0) ±R (L - i
1)}I
{2ωΧ ± R(\, - i, -1) ± -1 ,
0)} a Transition in the limit as all /,·,· -* 0. OTABLE 7.12 RESONANCE FREQUENCIES AND RELATIVE INTENSITIES FOR THE A2BX„ SYSTEM Transition" Relative intensity Resonance frequency ! 1,
l ;I - J; /χ,
mx> -|L ,0 ;i,
-I;/X,MX>->- I 1, -1;1, - I;/x,mx>II, 1 ; J , F ,/X,M
X>-> ! 1,0 ; J,I;/
X, m
x>
!L,0; J,I;/X,MX>->; 1 , -L ;II ;/X,M
X> L,L ;I,I ;/X,M
X> 1,0 ;
I|;/x ,mx> —I 1,0 ; J, -4; /χ , m
x>
A Transitions gix{V2 + Q(L, 0, MX)[L -F \/2Q(l, -1, mx]}2 [1 -F 02(l,O,mx)][l + 02(1, -l,mx)]
5 /
X{V 2 4 -0 (1 ,
-l,mx)}2 1 -F 02(1, -L,MX)g /
X{V 2 -0 (1,0 , m
x)}
2 1 -j-02 (l,O,mx)S /
X{0 (1, -L,MX )[V 2
0(L,O,MX) - 1] 4-V2 ?
[1 -r 02(l,O,mx)][l4 -0
2(l ,
-L,*ix)] Β Transitions gIx{l -r V2"Q(l,0,mx)}2 1 -f-02(L,O,MX)£/
X{V20 (l,
0, mx) -[1
-f-V2
0(L,-1 ,
mx)]}2 [1 +02 (l,O,mx)][l -F 02 (1, -l,mx)]Ω A + JAX™X -R
J {Ä (1,
0, mX) -Ä(L, -l,mx)} ^{ΩΑ -R ΩΒ — F J AB ~R( J A X
+ Λχ)^χ + Ä(L, -l,mx)} I{COA -F ΩΒ -F F JAB -R (/ΑΧ JBX)™X+ *(1,0 , mx) }
Ω A + jAXmx -R £{#(!, — 1, mx)-Ä (1,0 , mx) }
4{ΩΑ + ΩΒ + F /AB + (/ΑΧ "F /ßx)mX -#(1,0, mx)} WA + jAxmx — \{R{\, 0, mx) + Ä(L, -l,mx)}TABLE 7.12 (Continued) Transition0 Relative intensity Resonance frequency
i
1, -1;i, 4;/
x , mx> ->I 1, -4; /χ , m
x>
ΙΟ, 0; i4;/x,mx>->|0,0 ; i -i ;/x ,m
x> I
1, 1;I
1.0;i I
1,0;i
1, -1; ! 1, - ! 0, 0; i,I
0,0;11 ,1;
— i; Ιχ , mx> £, -l;/x, wx - 1> — J; Ιχ > wx) -» J, - \\Ιχ y rnx 1>
Β Transitions GIX{L - V2Q(L, -l,mx)}2 1 +Q2(1, -l,mx) X Transitions gix(Ix + mx)(/x -mx + 1)[1 + Q(l, 0, rox)Q(l, 0, mx -I)]2 [1 + 02(1, 0, mx)][l + Q2(l, 0, mx - 1)] gix(Ix + «x)(/x - mx + 1)[1 + 0(1,-1, mx)Q(l, -1, mx - l)]2 4, - è*>7 x » m x>
l ;i, -i ;/x ,
mx-l> — |; Ιχ, ηΐχ) h - i'ylxy mx — 1>i;
^X , mxy -4,
\\ Ιχ , «ιχ[1 + Q2(l, -1, «χ)][1 + Q2(l, -1, mx £/χ(/χ + wx)(/x - mx + 1) £/χ(/χ + mx)(Ix — mx + 1) £/χ(/χ + mx)(Ix — mx + 1)
1)]
1{ωΑ + ωΒ - f/AB + (/ΑΧ + /BX)^X -#(1, -l,mx)} ωχ + 4{/AX + #(1,0, mx) -#(1,0, mx - 1)} ωχ + £{-/AX + W, -hmx) -#(1, -l,mx - 1)} ωχ — 4(2/AX + /BX)
ωχ — 4/ B X ωχ + £(2 /
Αχ + /BX )
1>TABLE 7.12 ( Continued) RESONANCE FREQUENCIES AND RELATIVE INTENSITIES FOR THE Α2ΒΧ„Χ SYSTEM Transition0 Relative intensity Resonance frequency X Transitions Λ Λ glx(Ix + mxXJx- mx +
1)[1 +0 (1,0 , mx)0 (l,O ,m
x-
l)]2! 1 , 0 ; i
J;7χ
, mx> -> ωχ + i{J
AX+ 0 ,
mx -1) ι ι
π ι ι τi s
d +02 d,O,mx)][l+0
2(1,O ,
mx -1) ]
I 1. 0; Ιχ,τηχ- 1> -#(1, 0, mx)} , , £/χ(/χ + "*χ)(/χ-™χ+ 1)[1 +0(1, -l,wx)0(l, -l,mx- l)]2 I 1, -1; i h 'x > ™x> - n , n2n : W1 M n2n : tt; ωχ + J{-JAx + -1, «χ- 1) ι ι ι ι ι r ι\ [1 + 02(1, -1, wx)][l + 02(1, -1, wx - 1)]I
1, -1;YJ X
, mx - 1>-#(1 , -1 ,
mx)} i 0, 0; \y \; Ιχ , mx> £/xdX + mx)(Ix - mx + 1) ωχ + |/Βχ|0,0 ;i,i;/
x,m x - 1 >
M Transitionsgi
X{[V2
+ Od,0,
mx)]0(l,-1 ,
mx) - V20(l,0,
mx)}2 I 1, 1 ; 1, - 7χ , mx> - ωΑ + JAXmx + 0, mx)Μ ι
ι ι τ\ [ 1 +0
2(1,0 , m
x)][l +0
2(1, -1,
wx)]11,
-i; t» è;Jx » wx> + #(i, -i, ™x)} £/χ(/χ + mx)(Ix - mx +1 )[0 (1, 0,
τηχ - 1) - Od,0,
mx)]2! 1 , 1 ; 1 , - 1;
JX , mx>ω
χ+
\{]Αχ+
R(l,0 ,
mx - 1)ι ι
n. ι ι r
1V[ 1 +0
2(1,0 , m
x- 1)][ 1 +0
2(1,0 , m
x)] I 1, 0; £, J
x , mx -1> + 0 ,
mx)} */χ(/χ + mx)(/x - mx + 1)[0(1, -1, mx - 1) - 0(1, -1, mx)]2 I 1, 0; 1, - 1; /χ , mx) - ωχ + £{-/AX + #d, -1, «X - DII
1.11. r wi \ [ 1 +0
2(1, -1,
wx - 1)][1+0
2(1,
-l,mx)]1 1,
— i; i;jx
, mx — i> + — l, mx)} , , Ζΐχϋχ + mx)(Ix - mx +1)[0 (1 , -1,
mx -1 ) - 0 (1, -1 ,
mx)]2i
1, -1;1, 1;/χ
, mx> -ω χ - |{/
AX+
R(l,-1 ,
mx -1) μ
π ι ι ri s [ 1 + 0
2(1, -1 ,
mx -1)][ 1
+ô
2(l, -1 ,
mx)]I 1, 0;
f, - J;/x , mx - 1>- f
R(\, -1, τηχ)} , , 5Τ/χ(/χ + rax)(/x - mx +1 )[0 (1, 0,
mx - 1) -0 (1, 0,
mx)]2I 1, 0; 1,1 ; 7
X , mx> -ω χ + £{/
AX- 0 ,
mx -1) it ι l lr I \ [ 1 +0
2(1,O , m
x- 1)][ 1 +0
2(1,0 , m
x)] I 1, - i;/
x, m
x- 1> -#(1,0 , m
x)}
a In the limit as all —• 0.
3. T H E AnB Xn v S Y S T E M 311
0 ( 1 , 0 , « χ ) J A ^
ρ ( ΐ, - ι, ^ χ )
δ - * / A B + ( / A X - / B X ) " * X + R(h 0, mx) ' / A B ^ 2
δ + I/ A B + (JAX - / B X ) % + R(l, - 1 , mx) ' With the help of these expressions, one may easily obtain the resonance frequencies and relative intensities for any value of nx .
An example of the case nx = 1 is provided by the proton and fluorine resonances of 2,6-dichlorofluorobenzene (77). T h e experimental and theoretical proton spectra are shown in Fig. 7.10; the theoretical spectrum shows the subspectra associated with the two pseudo A2B systems. T h e experimental and theoretical fluorine spectra are shown in Fig. 7.11.
T h e asymmetry of the spectrum with respect to a>x makes it possible to relate the sign of /A B , arbitrarily assumed to be positive, to those of
JAX and /B X .
T h e preceding examples illustrate the practical utility of the X approximation and the concept of effective chemical shifts. Even when
i_ 2
F I G . 7.10. E x p e r i m e n t a l a n d theoretical p r o t o n spectra of 2,6-dichlorofluorobenzene in hexafluorobenzene at 60 M c p s .
312 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 F
1 '-Γ-' V-1
- 5 0 5 cps
F I G . 7. 1 1 . E x p e r i m e n t a l a n d theoretical spectra of t h e fluorine n u c l e u s of 2 , 6 -dichlorofluorobenzene in hexafluorobenzene at 5 6. 4 M c p s .
deviations from the X approximation are observed, the method can be used to obtain approximate, initial estimates of the spectral parameters.
Small deviations from the X approximation can often be satisfactorily accounted for by extending the perturbation calculation to second order.
4. Moment Analysis of High-Resolution Spectra A. Definition of Spectral Moments
T h e perturbation method, although often useful, is of limited value when the absolute values of the ratios / G G ' / ^ G G ' a re comparable to unity.
For this reason the analysis of such systems is usually based upon exact numerical diagonalizations of the hamiltonian matrix. However, the analysis of a spectrum that is a function of several chemical shifts and
4. M O M E N T ANALYSIS OF H I G H - R E S O L U T I O N SPECTRA 313 coupling constants can be a problem of some difficulty, even with the aid of electronic computers. It is not surprising, therefore, that attempts have been made to devise methods of analysis that either entirely obviate the diagonalization of the hamiltonian matrix, or provide relations between the chemical shifts and coupling constants that can be used to simplify the analysis (5, 12-14). That such procedures may be possible is already evident from previous discussions of systems containing two or more nontrivial groups of magnetically equivalent nuclei. In this section an alternative method of spectral analysis—the so-called moment method—will be described. This method (12) does not require the solution of the eigenvalue problem and, in principle, is applicable to all spin systems.
T h e essential idea of the moment method is to compute theoretical expressions for the frequency moments of a high-resolution spectrum which are then equated to the corresponding frequency moments calculated from the observed spectrum. This procedure yields a set of simultaneous algebraic equations for the chemical shifts and coupling constants. T h e basic problem is the computation of the theoretical moments, and it is indeed remarkable that this calculation can be carried out with comparative ease. Unfortunately, the inherent assumptions of the method and the difficulties encountered in the determination of experimental moments considerably limit its applicability. In favorable cases, the moment method can provide significant information.
T h e nth moment of an arbitrary spectrum is defined by the equation
where Qk — Qj is the frequency associated with the transition | j) —• | k), and I 2 = 1Ik j |2 the corresponding relative intensity. T h e denomi-nator of (4.1) is the total intensity, so that
may be interpreted as the probability of observing the frequency (Qk — Qj). Thus <ωη> is the mathematical expectation of the nth power of the frequency, <ω> is the mean frequency, <ω2> the mean square frequency, and so on.
T h e denominator of (4.1) is equivalent to
<"n> Σ . · Σ , ( ^ - ^ · ) Ί / ί 1 2
314 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
T h e numerators of the various moments can also be expressed as traces of matrix products upon noting that ^ k = Qj 8jk . For example, when η = 1,
X iPk — Ikj\2 = 2) {'^kJkj — Ikj^jÙIJk
j,k j,k
= X[^J+W7k = tr[^nr.
j,k
Similar calculations lead to the following expressions for the first four moments:
t r [ j r , / + ] / - 3N t r [ ^ , [JT, /+]][/-, Jt>]
(4.2)
\ωy = — ^ r i x 1 \tr/+7- ' χ 7ωy ~
tr/+7-These equations reveal a pattern of formation that can be used to write down the equations for all higher moments.
T h e distinguishing feature of equations (4.2) is that they are all expressed in terms of trace operations and, since the trace is a matrix invariant, it is not necessary to solve a complicated eigenvalue problem to evaluate the moments—any matrix representation of the operators will suffice.9
B. Calculation of Spectral Moments
T h e derivation of explicit formulas for the theoretical moments requires the specification of the hamiltonian operator and the evaluation of the necessary commutators and traces. T h e hamiltonian operator will be taken to be
# = X "GIGz + / G G 'IG * Ιο- , (4-3) G G < G '
the minus signs usually prefixed to the right-hand members being omitted for convenience. It will be assumed that the total spin quantum numbers for each group G arise by addition of nG angular momenta with spin / , and that all nuclei are identical.
9 See A p p e n d i x V.
4. M O M E N T ANALYSIS OF H I G H - R E S O L U T I O N SPECTRA 315