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The moment equations for three-group systems
JOURNAL
OF MAGNETIC
RESONANCE
43,40-48
(1981)
The Moment Equations for Three-Group Systems P. L. CORIO AND M. L. TROVER Department
of Chemistry,
University
of Kentucky,
Lexington,
Kentucky
40506
Received August 29, 1980 The moment equations for determining the chemical shifts of three-group systems are examined qualitatively and quantitatively. The nature of the solutions depends upon the parameter p = (Ao3)/(Ao * ) 312.There are three critical values of p that separate real and imaginary solutions, and formulas are given that permit their calculation for any three-group system. At a critical value, the moment equations have double roots. Satisfactory values of the chemical shifts can be obtained provided p is not one of the critical values, but large errors arise as p approaches a critical value. INTRODUCTION
The moment method (1) has been sparingly used in the analysis of high-resolution spectra, but recent work (2) has again demonstrated that it can be quite useful. It remains unlikely, however, that the method will’ever be used to provide complete analyses for systems with three or more spins. Consider, for example, the case of N identical nuclei with spin quantum number I in three groups A, B, C containing nA, na, and nc nuclei. The spectrum is determined by six parameters: three chemical shifts AmA, Aog, Awe, measured from the mean resonance frequency, and three coupling constants JAB, J.&c, JBc. The six moment equations required for determining these parameters are equivalent to an algebraic equation of degree 720, whereas it is necessary to solve an algebraic equation of only the 6th degree for a partial analysis based on the first three moment equations, namely, nAfhA
+ n&‘hB + ncAwc = 0,
nA(AwA)2 + nB(AWB)’ + nc(Awc)* = N(Aw*),
nA(hJA)3 -I- na(A~a)~ f nc(Awc)3 = N(Aw3),
[II
in which (A&) and (Ao3), respectively, denote the second and third moments, computed with respect to the mean resonance frequency. But even when the spectra are well resolved and the computations feasible, there is an inherent peculiarity that should be considered before consigning the problem to a computer. Specifically, the assumption that small errors in the moments result in small errors in the calculated parameters is not always tenable. This situation, which occurs when the equivalent algebraic equation has multiple roots, is analogous to the problem encountered in second-order perturbation theory when the eigenvalues are degenerate. This point will be illustrated for systems with three groups of nuclei using a procedure (3) that is more suitable for dealing with the question 40 0022-2364/81/040040-09$02.00/O
Copyright 8 1981 by Academic Press. Inc. All rights of reproduction in any form reserved.
MOMENT
EQUATIONS
FOR
HIGH-RESOLUTION
SYSTEMS
41
than the largely graphical methods previously used (1; 3, pp. 322-326; 4), and more suitable in numerical computations. The analysis also provides a qualitative description of the solutions of system [l]. The problem, however, is of a general nature; the discussion is limited to three-group systems for simplicity and because the moment method is more likely to be applied to such systems. The procedure used to determine multiple solutions and the conditions for their existence may be extended to any number of moment equations that relate an equal number of unknown parameters. In the subsequent discussion, “moment equations” will always mean system [l]. To avoid trivial cases, it will be assumed that none of the integers nA, nB, nc is zero and that the second moment does not vanish. The latter assumption excludes the case AwA = Aoa = Aw c = 0, and permits use of the (positive) square root of the. second moment as the unit of frequency. The third moment can be supposed nonnegative; for if (AwA,AoB,Aw,-) is a solution of the moment equations with (80~) positive, then (-Am*, -AoB, -Am,-) is a solution of these equations with the third moment replaced by - ( Ao3). It will also be assumed that lineshape corrections (5), if necessary, have been introduced into the experimental moments. Finally, as the first three moments do not contain the spin quantum number or the coupling constants, it need not be assumed that Z = l/2 or that the groups are magnetically equivalent. TRANSFORMATION
OF THE
MOMENT
Geometrical considerations (3, pp. 322-326) variables U, V, w through the equations
EQUATIONS
suggest the introduction
of new
AwA = (Aw~)~‘~(u cot 6 - v tan + cosec 0 + w), AWN = (Ao~)~‘~(u cot 8 + ZJcot C#J cosec 8 + w), Awe = (Aw~)~‘~(-u tan 0 + w), tan 4 = (ng/nA)l12,
PI
cos 0 = (nclAy2,
131
where 8, C$ are the polar angles of a fixed unit vector with rectangular components (nA/N)1’2, (nB/N)“2, (nc/N)“2, so that both 8 and $ are between zero and 7~/2. The nonsingular linear transformation [2] reduces the first moment equation to the identity w = 0, and transforms the second and third moment equations into u2 + u2 = 1, ]41 au3 + 3cuv2 + dv3 = p,
[51
where a=
nc - nA - nB [&(~A
d=
+ N(nA
[NnAnB(nA
,
c = (nA;nB)“2,
161
nB)11’2 -
b) +
(Aw3) nB)]“2
’
’
=
+2)3/2
’
171
42
CORIO AND TROVER
As a consequence of our assumptions, the dimensionless, experimentally determinable parameter p is nonnegative.’ The solution of the moment equations is thus equivalent to the determination of the points of intersection of the cubic curve [5] with the unit circle [4]. The left-hand member of [5] is a binary cubic form whose discriminant (6) is A = a2dz + 4aC3 Or, in terms Of nA, nB, and nc, A = -(nAt~BnJ-l(nA
+ nB - nC)(nA - 12B+ nC)(-nA
+ nB + nc).
The qualitative properties of the cubic curve [5] are determined which iS pOSitiVe when nA, 128, and nc SatiSfy -,tA
-
ttB
+
n,
>
0,
HA
-
nB
+
ttc
>
0,
-nA
+
@I
by the sign of A, nB
+
n,
>
0,
[91
or either of two other sets of inequalities obtainable from Eq. [9] by cyclical perInUtatiOnS Of A, B, C; it iS zero if nA, )zB, and nc Satkfy nA
+
-
nB
nc
=
1101
0,
or either of two other equalities obtainable from Eq. [lo] by cyclical permutations of A, B, C; finally, the discriminant if negative if and only if nA + ng - nc > 0,
nA - ng + nc > 0,
-nA + nB + nc > 0.
[111
Since the labeling of the groups is arbitrary, the ambiguities in the conditions for A > 0, and A = 0 will be removed by two conventions: (1) when A > 0, the groups are to be labeled so that the inequalities [9] are satisfied; (2) when A = 0, the groups are to be labeled so that Eq. [lo] is satisfied. The advantage of these conventions is that the parameter a, defined by the first part of Eq. [6], is positive, negative, or zero with A. It is only necessary, therefore, to study Eqs. [4] and [5] in three cases: (1) A > 0, a > 0; (2) A = a = 0; (3) A < 0, a < 0. Actually, we shall not study [4] and [5], but an equivalent system consisting of the unit circle and the cubic curve L(=
p - dv3
[=I
a + (3c - u)v” ’
which is easily derived from [4] and [5]. MULTIPLE
SOLUTIONS
An equation for v may be obtained by using Eq. [12] to eliminate Eq. [5]. For brevity, let p, 4, r, s be defined by (3c - u)(a - c)
d
* = d2 + (3c - a)’ r=
’
u(u - 2c) d2 + (3c - a)’
1 In Refs. (3) and (4), p denotes (Ao3)z’3/(Aoz); fractional exponent in Eq. [5].
u from
’
’ = d2 + (3c - a)2 ’ .Y=
p2 - a2 d2 + (3c - u)~ *
the definition
[I31
used here avoids an inconvenient
MOMENT
EQUATIONS
FOR HIGH-RESOLUTION
SYSTEMS
43
The equation for u may then be written 19 + 3pv4
- 2qv3
+ 3rv2
+ s = 0.
[I41
Unless the coefficients satisfy special conditions, this equation must be solved numerically. The corresponding values of u are obtained2 from [12], and the chemical shifts from [2] with w = 0. The internal chemical shifts tiGG, = wG - wc, = hoc - Aoc, can then be computed from the equations iV(nA + ng)(Aw2)
1’2u,
1 --f-l. = Nyy:“]“‘[(&)1’2u
WAB = -
[151
lZAnB
@AC
[161
As an illustration, take the case of the protons in 2-bromothiophene (3, pp. 279, 327) whenn, = nB = n, = 1. At60MHz, (AWN) = 61.558Hz2, (Ao3) = 253.146Hz3, from which p = 0.5241. The solutions of Eq. [14] are 20.2428, t0.7187, kO.9615. A quick check of these roots is provided by [ 141, which shows that the sum of the roots is zero, and their product s = -0.02816. The root u = -0.7187 gives u = 0.6953, and Eqs. [15] and [16] yield WAB= 13.81 Hz, WAC= 18.48 Hz, in excellent agreement with the values oAB = 13.92 Hz, oAC = 18.57 Hz, obtained by direct analysis (3, pp. 279, 327). It is evident from Eqs. [2] and [ 121 that the chemical shifts are real or imaginary accordingly as v is real or imaginary. The nature of the roots of Eq. [14] is determined by its double roots, or, equivalently, by those values of p for which [14] has double roots. If u is a double root, then v satisfies Eq. [14] and v5 + 2pv3
- qv2 + rv = 0,
[I71
which, except for a numerical factor, is the derivative of [ 141. The theory of algebraic elimination (7) may be applied to [ 141 and [ 171 to determine the double roots and the corresponding values of p; but the same results can be obtained with considerably less effort on noting that, by virtue of the linear relations [2], Eq. [ 141 has multiple solutions if and only if system [l] has multiple solutions. Now system [l] obviously has multiple solutions when conditions of equality subsist among the three quantities AwA, Ao~, Ao,. All three chemical shifts cannot coincide since the first moment equation would require Aw, = AwB = Aw, = 0, contradicting the assumption (Ao2) # 0. It can be concluded, therefore, that [14] does not have a triple root. Double roots occur when any two of Au*, AWN, Aw, are equal. From Eqs. [2], [4], and [5], we obtain the following results. nc PAB
=
sgn
uAB
= +-1,
l4
-
nA
[nc(nA
vAB
-
+
=
0;
nB
nB)]l/2
’
[I91
2 It is preferable in numerical computations to calculate u with Eq. [4], using [12] to determine the algebraic sign.
44
CORIO
AND
TROVER
TABLE DOUBLE nA:nB:nC 1:2:6 1:1:3 1:2:3 1:1:2 2:3:4 1:l:l
PAB
UAB
0.7071 0.4082 0 0 0.2236 0.7071
1 1 1 1
n The discriminant fifth and sixth entries.
PAC
=
sgn
uAC
=
t
=
sgn
VAB 0 0 0 0 0 0
-1 -1 is positive
v
ROOTS
for
1
OF SOMETHREE-GROUP
SYSTEMS"
PAC
UAC
VAC
PBC
UBC
431
1.3363 1.5000 0.7071 0.9428 0.7071 0.7071
0.7559 0.6124 0.7071 0.5774 0.6324 0.5000
0.6547 0.7906 0.7071 0.8165 0.7746 0.8660
2.4749 1.5000 1.7889 0.9428 1.3363 0.7071
0.5000 0.6124 0.4472 0.5774 0.4781 0.5000
-8.8660 -0.7906 -0.8944 -0.8165 -0.8783 -0.8660
the
first
two
entries,
zero
for
the
next
pair,
negative
for
the
N’n,(n,
vAC
=+
PII
Ao, = Ao,: N2n,(n, PBC
uBC
v
- nB) - 3NnAnBnc - (nc - nA - nB>ncni [NnB
ET
+ nAnc]3'2[nA(nA+ nB)]"* uBC
3 WI WI
=+
In Eq. [ 181 sgn u denotes the sign of u which is to be chosen in [ 191 so that the numerator pAB is positive. A similar remark applies to sgn v in the remaining equations. Note that when nA = ng, pAc = PBS-,UAC = uBc, and VAc = -vBc. Table 1 gives the double roots and the values of p AB, pAc, pBc for several particular cases. Only the ratios of nA, nB, and nc need be specified since the values of p and the corresponding values of u and v are homogeneous functions of degree zero in n A, na, and nc. Thus these quantities have the same values under varying experimental conditions and for all systems with nAr = hn,, nBf = An,, ncr = An,, where h is a positive integer. DESCRIPTION
OF THE
SOLUTIONS
A qualitative description of the solutions of Eq. [14] and, therefore, those of the system [I], is easily carried through with the results of the preceding section. Case 1: A > 0
Since a > 0, the denominator of [ 121 is never zero, so u describes a continuous cubic curve, asymptotic to the line u = -[dl(3c - a)]v, with a relative maximum p/a at v = 0 when p > 0, a point of inflection when p = 0. These properties are
MOMENT
EQUATIONS
FOR HIGH-RESOLUTION
SYSTEMS
45
FIG. 1. Graphs of the cubic curve, Eq. [12], for several values of p with nA:nB:nc = 1:2:6. In this figure and those that follow, the center and radius of the circle fix the origin and the unit of length: the u axis is vertical, the u axis horizontal.
illustrated for the case nA:nB:nC = 1:2:6 in Fig. 1, which includes the unit circle [4]. When p = 0, the cubic curve intersects the unit circle in two points, so there are two real solutions and four imaginary solutions (in complex conjugate pairs, since 1141 has real coefficients). For 0 I p < pAB, there are two real solutions and four imaginary solutions. As p increases, the maximum of the cubic curve approaches the unit circle and is tangent to it at p = pAB, when there are four real solutions (including the double root u = 0), and two complex solutions. For oAB < p < pAC, the cubic curve intersects in four distinct points, so there are two imaginary solutions and four distinct real solutions. The vertical axis separates the real roots into two pairs, and the members of each pair approach one another as p increases further, one pair coalescing to a new double root for p = pAC2, when there are again four real roots and two complex roots. For pAC < p < pBC, that portion of the cubic curve with u > 0 does not touch the unit circle, so [14] has two distinct real solutions and four imaginary solutions. As p increases further, the remaining pair of real roots approach each other and merge to a double root when p = pBc. For p > ~a,-, all solutions are imaginary. In the special case nA = n,, the vertical axis is an axis of symmetry for the cubic curve [12] and the double roots. Furthermore, as the solutions for p < 0 may be obtained from those for p > 0 by replacing u and u with --u and -u, Fig. 1 may be converted to the case p < 0 by a 180” rotation about the center of the unit circle. These remarks are also applicable in the following cases. Case 2: A = 0
In this case a = 0 and c = 1. The denominator so the curve has two branches with asymptotes u = are shown in Fig. 2 for several values of p and p -+ 0, the branches tend to the asymptotes, giving so there are four real solutions and two imaginary
of [ 121 vanishes when v = 0, -(d/3)v and v = 0. The curves nA:nB:~c = 1:2:3. In the limit a double root at u = 1, v = 0, solutions. When 0 < p < pAC,
46
CORIO
AND
TROVER
~AxB:~c = 1:2:3. FIG. 2. Graphs GIraphs of the cubic curve, Eq. [12], for several values of p with ~AXB:~C
there are four distinct real solutions and two complex solutions. When p = pAC, there are four real solutions, one a double root, and two complex solutions. For PAc < p < pBc, there are four complex solutions and two distinct real solutions which merge to a double root when p = pBC. All solution are imaginary when P ’
PBC-
Case 3: A < 0 The parameter a is negative in this case, so that [12] has three branches and three asymptotes: u = -[dl(3c - a)@, v = 4 la 1/(3c - a). It is only in this case
FIG. 3. Graphs of the cubic curve, Eq. [12], for several values of p with nA:ngxC = 1:l:l. only in this case that all three branches can be simultaneously tangent to the unit circle.
It is
MOMENT
EQUATIONS
FOR
HIGH-RESOLUTION
SYSTEMS
47
that the moment equations can have six real and distinct solutions, and this occurs when p is less than the smallest of p AB, pAC, pBc; the solutions are all imaginary when p is greater than the largest of these numbers. In the important case illustrated in Fig. 3, pA B = PAC = PBC = 2-l’*, so there are six real and distinct solutions for p < 2-1’2, six imaginary solutions for p > 2-l’*, and three distinct double roots when p = 2-l’*. ESTIMATION
OF ERRORS
An error Ap in the experimental value of p introduces an error Av into the calculation of v that can be computed by solvingf(v + Au, p + Ap) = 0, where f(v,p) denotes the left-hand member of [14]. The error can be estimated by expandingf(v + Au, p + Ap) about the point (v,p), wheref(v,p) = 0. Supposing, for the moment, that terms of the second and higher order in Au and Ap can be neglected, and that v is not a double root, the first-order approximation to f(u + Av, p + Ap) yields Av =
dv3 - p 3v[d* + (3~ - a)*](v’
+ 2pv* - qv + r) I
AP.
]241
The quantity within the curly brackets is of the order of unity, so Av is of the order of Ap and the omitted terms are negligible when Ap is small compared with p. Thus, a small error in p introduces a small error into u, provided that v is not a double root. When u is a double root, the denominator of [24] vanishes. In this case it is necessary to include higher-order terms in the expansion. Neglecting terms of the second and higher order in Ap and those of the third and higher order in Au, we obtain 3[d” + (3c - a)*](W
+ 6pv* - 2qv + r)(Av)*
- 6dv*ApAv
+ 2Ap(p - dv3) = 0.
[25]
The roots of this quadratic equation, which may be real or imaginary, depending upon the sign and magnitude of Ap, give the correction for each component of the double root. In the special case v = 0, Eq. [25] yields, since p = Ia 1, Av = 5
P61
For the system nA:nB:nc = 1:2:6, a = +2-112, 2c - a = 1.5(2)“*, and [26] gives Au = %0561(Ap)“*, so the corrections are real for Ap > 0, imaginary for Ap < 0. Taking Ap = +O.Ol, the errors are kO.056, almost six times that in p. (The values obtained by solving [14] with p + Ap = 0.7171 are +0.056, -0.055.) The corresponding errors in u may be calculated with [ 121, noting (Table 1) that u = 1 when v = 0. For Av = +0.056, Eq. [12] gives AU = -0.016, which is comparable with p. There is, however, no inherent asymmetry in u and u. The transformation [2] includes a rotation with the Euler angle $ = 0; by choosing $ = 7~/2, the roles of the u and u are reversed. Equation [25] also simplifies in the special case nA = nB, when q = d = 0,
48
CORIO AND TROVER
so that Au = ?
-~PAP 3(3c - n)*(5v4 + 6pu* + r) I
If the double root v # 0, this equation of Eq. [17] to Au = +
can be simplified
112 .
further
j271
with the help
112
-PAP 6(3c - u)~v*(v* + 1)
1
[281
’
The calculated errors Au and Au may be used to determine the errors in the Aw, by means of Eqs. [2], or the errors in the uGGPby means of Eqs. [15] and [16]. To simplify the discussion, we consider only the error 80~~ in wAB. If cr denotes the fractional error in the second moment and terms of the second and higher order in o can be neglected, then 60 AB
=
-
N(nA
+ nBNAo*) nAnB
1’2~0.5av
+
AvI.
~291
I
When v = 0, the error in SWABis determined by the error in p. Taking the case nA:nB:nc = 1:2:6, Ap = +O.Ol, Av = +0.056, and (Ati*)“* = 8.0 Hz, Eq. [29] gives I&JAB 1 = 1.7 Hz. To bring the error down to 0.1 Hz, Ap must be 4 x 10w5. For the zero double root of a system with 11A:ng:nc = 1: 1: 1, there is some improvement in these results because the factor N(nA + nB)/nAnB is smaller. In this case, taking 1Ap 1 = 0.01, (Aw * ) l’* = 8.0 HZ, the error 1&JAB I is 1.1 HZ; to bring the magnitude down to 0.1 Hz, ) Ap) should be 1 x 10P4. Now the fractional error in p is Aplp = - 1.5~ + 7, where T is the fractional error in the third moment. Since p is of the order of unity at double roots, Ap = - 1.5~ + 7. Unless there is a fortitous cancellation, the errors in the second and third moments would have to be less than 0.01%. On the other hand, if v is not a double root, errors of the order of 1% in o and 7 can produce errors in oAB of the order of tenths of a hertz. In summary, the moment method can be useful for partial analyses of wellresolved three-group systems provided p is not close to one of the values PAB, pAc, pBc. When p iS ClOSe to one of the critical values, substantial errors in the chemical shifts can occur, and their avoidance requires extremely accurate measurements of the second and third moments. Indeed, the errors may be large enough at double roots to render the method quite unsatisfactory for preliminary data in iterative schemes. Since the critical values of p can be calculated in advance, the difficulties can be anticipated and errors estimated. REFERENCES 1. W. A. ANDERSON AND H. M. MCCONNELL, J. Chem. Phys. 26, 1946 (1957). 2. H. D~LHAINE, M. ENGELHARDT, G. HKCELE, AND W. K~CKELHAUS, J. Magn. Reson. 41, l(l980). .?. P. L. CORIO, “Structure of High-Resolution NMR Spectra,” Academic Press, New York, 1%6. 4. P. L. CORIO, Chem. Rev. 60, 363 (1960). 5. H. PRIMAS AND Hs. H. G~NTHARD, Helv. Phys. Acta 31, 413 (1958). 6. L. E. DICKSON, “Algebraic Theories,” pp. 7-10, Dover, New York, 1959. 7. M. B&HER, “Introduction to Higher Algebra,” Chap. XV, Macmillan, New York, 1907.
OF MAGNETIC
RESONANCE
43,40-48
(1981)
The Moment Equations for Three-Group Systems P. L. CORIO AND M. L. TROVER Department
of Chemistry,
University
of Kentucky,
Lexington,
Kentucky
40506
Received August 29, 1980 The moment equations for determining the chemical shifts of three-group systems are examined qualitatively and quantitatively. The nature of the solutions depends upon the parameter p = (Ao3)/(Ao * ) 312.There are three critical values of p that separate real and imaginary solutions, and formulas are given that permit their calculation for any three-group system. At a critical value, the moment equations have double roots. Satisfactory values of the chemical shifts can be obtained provided p is not one of the critical values, but large errors arise as p approaches a critical value. INTRODUCTION
The moment method (1) has been sparingly used in the analysis of high-resolution spectra, but recent work (2) has again demonstrated that it can be quite useful. It remains unlikely, however, that the method will’ever be used to provide complete analyses for systems with three or more spins. Consider, for example, the case of N identical nuclei with spin quantum number I in three groups A, B, C containing nA, na, and nc nuclei. The spectrum is determined by six parameters: three chemical shifts AmA, Aog, Awe, measured from the mean resonance frequency, and three coupling constants JAB, J.&c, JBc. The six moment equations required for determining these parameters are equivalent to an algebraic equation of degree 720, whereas it is necessary to solve an algebraic equation of only the 6th degree for a partial analysis based on the first three moment equations, namely, nAfhA
+ n&‘hB + ncAwc = 0,
nA(AwA)2 + nB(AWB)’ + nc(Awc)* = N(Aw*),
nA(hJA)3 -I- na(A~a)~ f nc(Awc)3 = N(Aw3),
[II
in which (A&) and (Ao3), respectively, denote the second and third moments, computed with respect to the mean resonance frequency. But even when the spectra are well resolved and the computations feasible, there is an inherent peculiarity that should be considered before consigning the problem to a computer. Specifically, the assumption that small errors in the moments result in small errors in the calculated parameters is not always tenable. This situation, which occurs when the equivalent algebraic equation has multiple roots, is analogous to the problem encountered in second-order perturbation theory when the eigenvalues are degenerate. This point will be illustrated for systems with three groups of nuclei using a procedure (3) that is more suitable for dealing with the question 40 0022-2364/81/040040-09$02.00/O
Copyright 8 1981 by Academic Press. Inc. All rights of reproduction in any form reserved.
MOMENT
EQUATIONS
FOR
HIGH-RESOLUTION
SYSTEMS
41
than the largely graphical methods previously used (1; 3, pp. 322-326; 4), and more suitable in numerical computations. The analysis also provides a qualitative description of the solutions of system [l]. The problem, however, is of a general nature; the discussion is limited to three-group systems for simplicity and because the moment method is more likely to be applied to such systems. The procedure used to determine multiple solutions and the conditions for their existence may be extended to any number of moment equations that relate an equal number of unknown parameters. In the subsequent discussion, “moment equations” will always mean system [l]. To avoid trivial cases, it will be assumed that none of the integers nA, nB, nc is zero and that the second moment does not vanish. The latter assumption excludes the case AwA = Aoa = Aw c = 0, and permits use of the (positive) square root of the. second moment as the unit of frequency. The third moment can be supposed nonnegative; for if (AwA,AoB,Aw,-) is a solution of the moment equations with (80~) positive, then (-Am*, -AoB, -Am,-) is a solution of these equations with the third moment replaced by - ( Ao3). It will also be assumed that lineshape corrections (5), if necessary, have been introduced into the experimental moments. Finally, as the first three moments do not contain the spin quantum number or the coupling constants, it need not be assumed that Z = l/2 or that the groups are magnetically equivalent. TRANSFORMATION
OF THE
MOMENT
Geometrical considerations (3, pp. 322-326) variables U, V, w through the equations
EQUATIONS
suggest the introduction
of new
AwA = (Aw~)~‘~(u cot 6 - v tan + cosec 0 + w), AWN = (Ao~)~‘~(u cot 8 + ZJcot C#J cosec 8 + w), Awe = (Aw~)~‘~(-u tan 0 + w), tan 4 = (ng/nA)l12,
PI
cos 0 = (nclAy2,
131
where 8, C$ are the polar angles of a fixed unit vector with rectangular components (nA/N)1’2, (nB/N)“2, (nc/N)“2, so that both 8 and $ are between zero and 7~/2. The nonsingular linear transformation [2] reduces the first moment equation to the identity w = 0, and transforms the second and third moment equations into u2 + u2 = 1, ]41 au3 + 3cuv2 + dv3 = p,
[51
where a=
nc - nA - nB [&(~A
d=
+ N(nA
[NnAnB(nA
,
c = (nA;nB)“2,
161
nB)11’2 -
b) +
(Aw3) nB)]“2
’
’
=
+2)3/2
’
171
42
CORIO AND TROVER
As a consequence of our assumptions, the dimensionless, experimentally determinable parameter p is nonnegative.’ The solution of the moment equations is thus equivalent to the determination of the points of intersection of the cubic curve [5] with the unit circle [4]. The left-hand member of [5] is a binary cubic form whose discriminant (6) is A = a2dz + 4aC3 Or, in terms Of nA, nB, and nc, A = -(nAt~BnJ-l(nA
+ nB - nC)(nA - 12B+ nC)(-nA
+ nB + nc).
The qualitative properties of the cubic curve [5] are determined which iS pOSitiVe when nA, 128, and nc SatiSfy -,tA
-
ttB
+
n,
>
0,
HA
-
nB
+
ttc
>
0,
-nA
+
@I
by the sign of A, nB
+
n,
>
0,
[91
or either of two other sets of inequalities obtainable from Eq. [9] by cyclical perInUtatiOnS Of A, B, C; it iS zero if nA, )zB, and nc Satkfy nA
+
-
nB
nc
=
1101
0,
or either of two other equalities obtainable from Eq. [lo] by cyclical permutations of A, B, C; finally, the discriminant if negative if and only if nA + ng - nc > 0,
nA - ng + nc > 0,
-nA + nB + nc > 0.
[111
Since the labeling of the groups is arbitrary, the ambiguities in the conditions for A > 0, and A = 0 will be removed by two conventions: (1) when A > 0, the groups are to be labeled so that the inequalities [9] are satisfied; (2) when A = 0, the groups are to be labeled so that Eq. [lo] is satisfied. The advantage of these conventions is that the parameter a, defined by the first part of Eq. [6], is positive, negative, or zero with A. It is only necessary, therefore, to study Eqs. [4] and [5] in three cases: (1) A > 0, a > 0; (2) A = a = 0; (3) A < 0, a < 0. Actually, we shall not study [4] and [5], but an equivalent system consisting of the unit circle and the cubic curve L(=
p - dv3
[=I
a + (3c - u)v” ’
which is easily derived from [4] and [5]. MULTIPLE
SOLUTIONS
An equation for v may be obtained by using Eq. [12] to eliminate Eq. [5]. For brevity, let p, 4, r, s be defined by (3c - u)(a - c)
d
* = d2 + (3c - a)’ r=
’
u(u - 2c) d2 + (3c - a)’
1 In Refs. (3) and (4), p denotes (Ao3)z’3/(Aoz); fractional exponent in Eq. [5].
u from
’
’ = d2 + (3c - a)2 ’ .Y=
p2 - a2 d2 + (3c - u)~ *
the definition
[I31
used here avoids an inconvenient
MOMENT
EQUATIONS
FOR HIGH-RESOLUTION
SYSTEMS
43
The equation for u may then be written 19 + 3pv4
- 2qv3
+ 3rv2
+ s = 0.
[I41
Unless the coefficients satisfy special conditions, this equation must be solved numerically. The corresponding values of u are obtained2 from [12], and the chemical shifts from [2] with w = 0. The internal chemical shifts tiGG, = wG - wc, = hoc - Aoc, can then be computed from the equations iV(nA + ng)(Aw2)
1’2u,
1 --f-l. = Nyy:“]“‘[(&)1’2u
WAB = -
[151
lZAnB
@AC
[161
As an illustration, take the case of the protons in 2-bromothiophene (3, pp. 279, 327) whenn, = nB = n, = 1. At60MHz, (AWN) = 61.558Hz2, (Ao3) = 253.146Hz3, from which p = 0.5241. The solutions of Eq. [14] are 20.2428, t0.7187, kO.9615. A quick check of these roots is provided by [ 141, which shows that the sum of the roots is zero, and their product s = -0.02816. The root u = -0.7187 gives u = 0.6953, and Eqs. [15] and [16] yield WAB= 13.81 Hz, WAC= 18.48 Hz, in excellent agreement with the values oAB = 13.92 Hz, oAC = 18.57 Hz, obtained by direct analysis (3, pp. 279, 327). It is evident from Eqs. [2] and [ 121 that the chemical shifts are real or imaginary accordingly as v is real or imaginary. The nature of the roots of Eq. [14] is determined by its double roots, or, equivalently, by those values of p for which [14] has double roots. If u is a double root, then v satisfies Eq. [14] and v5 + 2pv3
- qv2 + rv = 0,
[I71
which, except for a numerical factor, is the derivative of [ 141. The theory of algebraic elimination (7) may be applied to [ 141 and [ 171 to determine the double roots and the corresponding values of p; but the same results can be obtained with considerably less effort on noting that, by virtue of the linear relations [2], Eq. [ 141 has multiple solutions if and only if system [l] has multiple solutions. Now system [l] obviously has multiple solutions when conditions of equality subsist among the three quantities AwA, Ao~, Ao,. All three chemical shifts cannot coincide since the first moment equation would require Aw, = AwB = Aw, = 0, contradicting the assumption (Ao2) # 0. It can be concluded, therefore, that [14] does not have a triple root. Double roots occur when any two of Au*, AWN, Aw, are equal. From Eqs. [2], [4], and [5], we obtain the following results. nc PAB
=
sgn
uAB
= +-1,
l4
-
nA
[nc(nA
vAB
-
+
=
0;
nB
nB)]l/2
’
[I91
2 It is preferable in numerical computations to calculate u with Eq. [4], using [12] to determine the algebraic sign.
44
CORIO
AND
TROVER
TABLE DOUBLE nA:nB:nC 1:2:6 1:1:3 1:2:3 1:1:2 2:3:4 1:l:l
PAB
UAB
0.7071 0.4082 0 0 0.2236 0.7071
1 1 1 1
n The discriminant fifth and sixth entries.
PAC
=
sgn
uAC
=
t
=
sgn
VAB 0 0 0 0 0 0
-1 -1 is positive
v
ROOTS
for
1
OF SOMETHREE-GROUP
SYSTEMS"
PAC
UAC
VAC
PBC
UBC
431
1.3363 1.5000 0.7071 0.9428 0.7071 0.7071
0.7559 0.6124 0.7071 0.5774 0.6324 0.5000
0.6547 0.7906 0.7071 0.8165 0.7746 0.8660
2.4749 1.5000 1.7889 0.9428 1.3363 0.7071
0.5000 0.6124 0.4472 0.5774 0.4781 0.5000
-8.8660 -0.7906 -0.8944 -0.8165 -0.8783 -0.8660
the
first
two
entries,
zero
for
the
next
pair,
negative
for
the
N’n,(n,
vAC
=+
PII
Ao, = Ao,: N2n,(n, PBC
uBC
v
- nB) - 3NnAnBnc - (nc - nA - nB>ncni [NnB
ET
+ nAnc]3'2[nA(nA+ nB)]"* uBC
3 WI WI
=+
In Eq. [ 181 sgn u denotes the sign of u which is to be chosen in [ 191 so that the numerator pAB is positive. A similar remark applies to sgn v in the remaining equations. Note that when nA = ng, pAc = PBS-,UAC = uBc, and VAc = -vBc. Table 1 gives the double roots and the values of p AB, pAc, pBc for several particular cases. Only the ratios of nA, nB, and nc need be specified since the values of p and the corresponding values of u and v are homogeneous functions of degree zero in n A, na, and nc. Thus these quantities have the same values under varying experimental conditions and for all systems with nAr = hn,, nBf = An,, ncr = An,, where h is a positive integer. DESCRIPTION
OF THE
SOLUTIONS
A qualitative description of the solutions of Eq. [14] and, therefore, those of the system [I], is easily carried through with the results of the preceding section. Case 1: A > 0
Since a > 0, the denominator of [ 121 is never zero, so u describes a continuous cubic curve, asymptotic to the line u = -[dl(3c - a)]v, with a relative maximum p/a at v = 0 when p > 0, a point of inflection when p = 0. These properties are
MOMENT
EQUATIONS
FOR HIGH-RESOLUTION
SYSTEMS
45
FIG. 1. Graphs of the cubic curve, Eq. [12], for several values of p with nA:nB:nc = 1:2:6. In this figure and those that follow, the center and radius of the circle fix the origin and the unit of length: the u axis is vertical, the u axis horizontal.
illustrated for the case nA:nB:nC = 1:2:6 in Fig. 1, which includes the unit circle [4]. When p = 0, the cubic curve intersects the unit circle in two points, so there are two real solutions and four imaginary solutions (in complex conjugate pairs, since 1141 has real coefficients). For 0 I p < pAB, there are two real solutions and four imaginary solutions. As p increases, the maximum of the cubic curve approaches the unit circle and is tangent to it at p = pAB, when there are four real solutions (including the double root u = 0), and two complex solutions. For oAB < p < pAC, the cubic curve intersects in four distinct points, so there are two imaginary solutions and four distinct real solutions. The vertical axis separates the real roots into two pairs, and the members of each pair approach one another as p increases further, one pair coalescing to a new double root for p = pAC2, when there are again four real roots and two complex roots. For pAC < p < pBC, that portion of the cubic curve with u > 0 does not touch the unit circle, so [14] has two distinct real solutions and four imaginary solutions. As p increases further, the remaining pair of real roots approach each other and merge to a double root when p = pBc. For p > ~a,-, all solutions are imaginary. In the special case nA = n,, the vertical axis is an axis of symmetry for the cubic curve [12] and the double roots. Furthermore, as the solutions for p < 0 may be obtained from those for p > 0 by replacing u and u with --u and -u, Fig. 1 may be converted to the case p < 0 by a 180” rotation about the center of the unit circle. These remarks are also applicable in the following cases. Case 2: A = 0
In this case a = 0 and c = 1. The denominator so the curve has two branches with asymptotes u = are shown in Fig. 2 for several values of p and p -+ 0, the branches tend to the asymptotes, giving so there are four real solutions and two imaginary
of [ 121 vanishes when v = 0, -(d/3)v and v = 0. The curves nA:nB:~c = 1:2:3. In the limit a double root at u = 1, v = 0, solutions. When 0 < p < pAC,
46
CORIO
AND
TROVER
~AxB:~c = 1:2:3. FIG. 2. Graphs GIraphs of the cubic curve, Eq. [12], for several values of p with ~AXB:~C
there are four distinct real solutions and two complex solutions. When p = pAC, there are four real solutions, one a double root, and two complex solutions. For PAc < p < pBc, there are four complex solutions and two distinct real solutions which merge to a double root when p = pBC. All solution are imaginary when P ’
PBC-
Case 3: A < 0 The parameter a is negative in this case, so that [12] has three branches and three asymptotes: u = -[dl(3c - a)@, v = 4 la 1/(3c - a). It is only in this case
FIG. 3. Graphs of the cubic curve, Eq. [12], for several values of p with nA:ngxC = 1:l:l. only in this case that all three branches can be simultaneously tangent to the unit circle.
It is
MOMENT
EQUATIONS
FOR
HIGH-RESOLUTION
SYSTEMS
47
that the moment equations can have six real and distinct solutions, and this occurs when p is less than the smallest of p AB, pAC, pBc; the solutions are all imaginary when p is greater than the largest of these numbers. In the important case illustrated in Fig. 3, pA B = PAC = PBC = 2-l’*, so there are six real and distinct solutions for p < 2-1’2, six imaginary solutions for p > 2-l’*, and three distinct double roots when p = 2-l’*. ESTIMATION
OF ERRORS
An error Ap in the experimental value of p introduces an error Av into the calculation of v that can be computed by solvingf(v + Au, p + Ap) = 0, where f(v,p) denotes the left-hand member of [14]. The error can be estimated by expandingf(v + Au, p + Ap) about the point (v,p), wheref(v,p) = 0. Supposing, for the moment, that terms of the second and higher order in Au and Ap can be neglected, and that v is not a double root, the first-order approximation to f(u + Av, p + Ap) yields Av =
dv3 - p 3v[d* + (3~ - a)*](v’
+ 2pv* - qv + r) I
AP.
]241
The quantity within the curly brackets is of the order of unity, so Av is of the order of Ap and the omitted terms are negligible when Ap is small compared with p. Thus, a small error in p introduces a small error into u, provided that v is not a double root. When u is a double root, the denominator of [24] vanishes. In this case it is necessary to include higher-order terms in the expansion. Neglecting terms of the second and higher order in Ap and those of the third and higher order in Au, we obtain 3[d” + (3c - a)*](W
+ 6pv* - 2qv + r)(Av)*
- 6dv*ApAv
+ 2Ap(p - dv3) = 0.
[25]
The roots of this quadratic equation, which may be real or imaginary, depending upon the sign and magnitude of Ap, give the correction for each component of the double root. In the special case v = 0, Eq. [25] yields, since p = Ia 1, Av = 5
P61
For the system nA:nB:nc = 1:2:6, a = +2-112, 2c - a = 1.5(2)“*, and [26] gives Au = %0561(Ap)“*, so the corrections are real for Ap > 0, imaginary for Ap < 0. Taking Ap = +O.Ol, the errors are kO.056, almost six times that in p. (The values obtained by solving [14] with p + Ap = 0.7171 are +0.056, -0.055.) The corresponding errors in u may be calculated with [ 121, noting (Table 1) that u = 1 when v = 0. For Av = +0.056, Eq. [12] gives AU = -0.016, which is comparable with p. There is, however, no inherent asymmetry in u and u. The transformation [2] includes a rotation with the Euler angle $ = 0; by choosing $ = 7~/2, the roles of the u and u are reversed. Equation [25] also simplifies in the special case nA = nB, when q = d = 0,
48
CORIO AND TROVER
so that Au = ?
-~PAP 3(3c - n)*(5v4 + 6pu* + r) I
If the double root v # 0, this equation of Eq. [17] to Au = +
can be simplified
112 .
further
j271
with the help
112
-PAP 6(3c - u)~v*(v* + 1)
1
[281
’
The calculated errors Au and Au may be used to determine the errors in the Aw, by means of Eqs. [2], or the errors in the uGGPby means of Eqs. [15] and [16]. To simplify the discussion, we consider only the error 80~~ in wAB. If cr denotes the fractional error in the second moment and terms of the second and higher order in o can be neglected, then 60 AB
=
-
N(nA
+ nBNAo*) nAnB
1’2~0.5av
+
AvI.
~291
I
When v = 0, the error in SWABis determined by the error in p. Taking the case nA:nB:nc = 1:2:6, Ap = +O.Ol, Av = +0.056, and (Ati*)“* = 8.0 Hz, Eq. [29] gives I&JAB 1 = 1.7 Hz. To bring the error down to 0.1 Hz, Ap must be 4 x 10w5. For the zero double root of a system with 11A:ng:nc = 1: 1: 1, there is some improvement in these results because the factor N(nA + nB)/nAnB is smaller. In this case, taking 1Ap 1 = 0.01, (Aw * ) l’* = 8.0 HZ, the error 1&JAB I is 1.1 HZ; to bring the magnitude down to 0.1 Hz, ) Ap) should be 1 x 10P4. Now the fractional error in p is Aplp = - 1.5~ + 7, where T is the fractional error in the third moment. Since p is of the order of unity at double roots, Ap = - 1.5~ + 7. Unless there is a fortitous cancellation, the errors in the second and third moments would have to be less than 0.01%. On the other hand, if v is not a double root, errors of the order of 1% in o and 7 can produce errors in oAB of the order of tenths of a hertz. In summary, the moment method can be useful for partial analyses of wellresolved three-group systems provided p is not close to one of the values PAB, pAc, pBc. When p iS ClOSe to one of the critical values, substantial errors in the chemical shifts can occur, and their avoidance requires extremely accurate measurements of the second and third moments. Indeed, the errors may be large enough at double roots to render the method quite unsatisfactory for preliminary data in iterative schemes. Since the critical values of p can be calculated in advance, the difficulties can be anticipated and errors estimated. REFERENCES 1. W. A. ANDERSON AND H. M. MCCONNELL, J. Chem. Phys. 26, 1946 (1957). 2. H. D~LHAINE, M. ENGELHARDT, G. HKCELE, AND W. K~CKELHAUS, J. Magn. Reson. 41, l(l980). .?. P. L. CORIO, “Structure of High-Resolution NMR Spectra,” Academic Press, New York, 1%6. 4. P. L. CORIO, Chem. Rev. 60, 363 (1960). 5. H. PRIMAS AND Hs. H. G~NTHARD, Helv. Phys. Acta 31, 413 (1958). 6. L. E. DICKSON, “Algebraic Theories,” pp. 7-10, Dover, New York, 1959. 7. M. B&HER, “Introduction to Higher Algebra,” Chap. XV, Macmillan, New York, 1907.