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Classification and Analysis of NMR Spectra.

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JULY 1966
P A G E S 623-688
Classification and Analysis of NMR Spectra
The use of nuclear magnetic resonance as the basis of a spectroscopic method in organic
analysis is steadily growing[l]. The object of the present article is to present a general
picture of the classification and interpretation of the NMR spectra obtained at high resolutions. In simple cases complete interpretation is possible with the aid of only a few
equations; otherwise it is often necessary to use a laborious mathematical procedure,
which is discussed here. For a more profound study, the reader is referred to more detailed
publications 12-61.
I. Introduction
Nuclear magnetic resonance arises as a result of transitions between the energy levels of a nuclear magnetic
dipole in a static magnetic field. The existence of discrete
states can be understood in terms of the quantum
theory. The observation of nuclear magnetic resonance
requires the presence of an external alternating magnetic
field of suitable frequency, by which detectable transitions are induced.
When nuclear magnetic resonance signals were first
observed, it was found that this “nuclear Zeeman
effect” could be explained by the simple Equation (1):
1945: h v o = r H
where h is Planck’s constant; vo the resonance or Larmor frequency, p the magnetic moment of the nucleus, I the
nuclear spin, and H the strength of the external magnetic
[l a ] K. H . Hauser, Angew. Chem. 68, 729 (1956).
[lb] J . D . Roberts, Angew. Chem. 75, 20 (1963); Angew. Chem.
internat. Edit. 2, 53 (1963).
[2] J . A . Pople, W. G. Sclrneider, and H . J. Bernstein: High-resolution Nuclear Magnetic Resonance. McGraw Hill, New York
[3] L. M . Jackmann: Applications o f Nuclear Magnetic Resonance Spectroscopiy in Organic Chemistry. Pergamon Press, London 1959.
[4] P. L. Corio, Chem. Reviews 60, 363 (1960).
[5] J. D . Roberts: An Introduction to the Analysis of Spin-Spin
Splitting in High-Resolution Nuclear Magnetic Resonance
Spectra. Benjamin, New York 1961.
[6a] J. Ranff, Fortschr. Physik 9, 149 (1961).
[6b] H . Strehlow: Magnetische Kernresonanz und chemische
Struktur. Steinkopff, Darmstadt 1962.
Angew. Chem. internnt. Edit.
1 Vol. 5 (1966) 1 No. 7
With better resolution it became possible to recognize
differences in the shielding of the external magnetic
field by the electrons of atoms of identical species in
different chemical environments. This “chemical shift”
was taken into account by Equation (2).
1949: hvi
’I H,
where i is a subscript for numbering the nuclei in a molecule, H,
the field strength of the external magnetic field at resonance, and a a shielding parameter ( I 0-4-10-5 for protons).
Later the indirect nuclear spin interaction caused by
spin polarization of the electron cloud was discovered.
A complete description of the spectrum therefore requires the introduction of a spin Hamiltonian operator,
which is given by:
I95 1 : H =
vi Iz(i)
i= 1
2 Jik I(i) I(k)
where I(i) is the angular momentum operator for the nucleus i,
Iz(i) the z component of I(i) [the negative z-direction is
given by the direction of the static magnetic field], and
Jik is the parameter for spin-spin coupling between the
nuclei i and k.
This energy operator will be discussed further in Section 111.
11. Types of NMR Spectra
It is usual and convenient to divide nuclear resonance
spectra into types. A short formula is used to characterize both the spectrum and the arrangement (the symmetry) of the nuclei in the molecule (see Figs. 1 and 2).
The characterization requires that all the nuclei in the
molecule should be reduced to the “spin system”. This
spin system consists of the nuclei having the spin 1/2
(e.g. 1H, 19F, 31P) that interact with one another by
indirect spin coupling 171.
The first nucleus, i.e. that with the highest spin coupling (and any other equivalent nucleus), is always
denoted by A and the other nuclei by
(a) a subsequent letter from the beginning of the alphabet (B, . . .) if JAB is comparable in magnitude with
(b) a letter from the middle of the alphabet (M, . . .)
if JAM is approximately one order of magnitude
smaller than V ~ ~ A[lo];
(c) a letter from the end of the alphabet (X, . . .) if J A x
is two or more orders of magnitude smaller than
@ H-CF3@
@ H-ClCH,I,@
Fig. 1.
Designation of N M R spin systems.
If the approximate values of the Larmor differences
Vo8ik[81 and the coupling parameters Jik are known,
the type to which a spectrum belongs can be given by
two rules :
1. Chemically non-equivalent nuclei in a molecule are
assigned different letters [91.
2. The choice of the letter is determined by the magnitude of the spin-spin coupling relative to the Larmor
[7] The spin system does not include, for example, carbon and
oxygen nuclei (spin = 0), except when the isotopes I3C and 1 7 0
are involved. Moreover, nuclei with spin P 1/2 (chlorine, bromine, etc.) are not normally included, since they have a quadrupole moment (Q $- 0) and do not exhibit spi:i coupling because
of quadrupole relaxation.
[8] The quantity known as the “Larmor difference” is defined by
Vi-Vk = Vo8&.
(vi, vk = Larmor frequencies at constant magnetic field Ho in the
absence of any spin coupling; vo = the frequency of the radiofrequency transmitter; 8ik = a dimensionless parameter related
to the 7 Scale [22] by 8ik = Ti-Tk.)
[9] As a result of symmetries in the molecular structure, the
nuclei in the spin system may have:
(a) equal Larmor frequencies, i.e. vi=vk,
(b) equal spin coupling parameters, i.e. Jij = Jkj, for all i, k, and
j with vj
v i = vk.
If condition (a) is satisfied, the nuclei i and k are described as
“chemically equivalent” or “isochronous”. If condition (b) is also
satisfied, the nuclei i and k are described as “magnetically equivalent”. The special significance of this latter property will be
seen in Section IIIe.
In the examples of the AB2 type in Figures 1 and 7, the isochronous B protons are magnetically equivalent (JAB = JAB), whereas
in the examples of the A2B2 type in Figures 1 and 10, the A and
B protons are chemically, but not magnetically equivalent
(JAB 9 J ~ B ) ;see also [12].
Fig. 2. Types of N M R spectra for three chemically non-equivalent
nuclei, schematically represented by envelope curves.
Case (a) corresponds to the so-called complex spectra,
in which the Larmor differences are important as well
as the coupling parameters. In Case (b), the signals of
the non-equivalent nuclei occur in separate groups, in
which the line splitting and the line intensities depend
only slightly on the Larmor differences. In Case (c), the
resonance signrtls of the non-equivalent nuclei are so far
apart that they are usually recorded as separate groups
of lines; within these sub-spectra, the splittings and
intensities of the lines are independent of the Larmor
differences, i.e. they are determined only by the spin
coupling. Thus the use of Rule 2 takes into account
both the external appearance of the spectra and the
various effects of the Larmor differences on the line
splittings and intensities.
Rules 1 a n d 2 ar e sometimes n o t sufficient to give an unambigous designation for mo r e complex spin systemsr121; a n extension of the rules is therefore desirable [131.
[lo] Rule 2b is a subsequent refinement, and has not yet been
generally accepted. Some authors use the letter K instead of M.
[ l l ] The condition for 2c is automatically satisfied when the
nuclei present are different (e.g. 1H and 19F).
[I21 Regarding nomenclature in A2B2 spin systems, it should be
noted that notations such as Az’B’2, AABB‘, or AA*BB* are
occasionally used in order to indicate that the nuclei are not
magnetically equivalent [9]. However, the usual notations AzB2
and A2X2 are preferable. There is no danger of confusion, since
there are very few A2B2 systems with magnetically equivalent
nuclei. Two rare exceptions are cyclopropene and difluoromethane, which may be denoted by A*2B*2 or A*zX*2 [14bI.
[ 131 The existing rules may be supplemented as follows. The omission of one or two letters in the alphabetic sequence should
correspond to cases (b) and (c) of Rule 2. This suggestion is
followed in the last three examples in Figures 1 and 2.
Angew. Chern. internat.
Edit. Vol. 5 (I966) No. 7
III. Basic Principles for the Mathematical
Treatment of NMR Spectra
In the quantum-mechanical picture, the lines of a
nuclear magnetic resonance spectrum corresoond to
transitions between the stationary states of a spin
system. The three fundamental equations are:
and pp. The first factor in the product spin function is
then the spin function of the first nucleus, the second
that of the second nucleus, and so on. Thus the permutation operator P12 [see Equation (8)] has the following
effect :
Ptzctct = cta; P&ct = ctp;
1 2
Iz(i)‘l‘= miY’
I(i) I(k) yr= (2Pik-l)Y
mi is the magnetic quantumnumber;Pik the permutation operator,
which permutes the subscripts of the nuclei i a n d . k , and 1 the
identity operator (without effect).
The first term in Equation (8) gives the “nuclear Zeeman
energy” which is proportional to the resonance frequency vi [or the magnetic field strength H*, see Equation (l)]. The second term gives the energy due to
indirect spin coupling, which is proportional to the
coupling parameter Jik.
T h e values of t h e parameters vi a n d Jik a r e inserted in t he
Hamiltonian operator [Equation (S)]. T h e desired eigenvalues
E, a n d eigenfunctions Ypt h a t satisfy Equation (4) must then
b e calculated i n t h e usual man n er by means of secular equations, i.e. t h e energy matrix ((H,,)) is calculated with a set of
starting functions and brought i n t o t h e diagonal f o r m by a
unitary transformation. T h e diagonal elements a r e t h en t he
desired eigenvalues, and t h e transformation matrix consists of
t h e eigenvectors t h at link t h e eigenfunctions with t h e starting
(d) Transition Probability and Selection Rules
The calculation of the line intensities by means of Equation (6) involves an operator H(’), which is defined as
follows for the resonance absorption of radio-frequency
energy :
2 (1--0i) I+(i)
In Equation (10) I+ is a “shift operator” with the effect
Equation ( l l b ) states that the nuclear spin is “flipped”
from the p into the a orientation 1151. No absGrption can
occur in the state CI [Equation (lla)]. The transition
p cc leads to a change in the magnetic quantum
number m from -112 to +1/2, i.e. Am = +l. Equations
(10) and (11) give the selection rule [Equation (12)]:
[ 14a] J . A. Pople and T. Schaefer, Molecular Physics3,547 (1960).
[14b] P. Diehl and J . A. Pople, Molecular Physics 3, 557 (1960).
[14c] P. Diehl, Helv. chim. Acta 48, 567 (1965).
[I51 It was assumed in Equations (3) and (8) that the vector of
the static magnetic field points in the negative z direction; this is
usually the case. In the state ct (in = + I / * ) , the spin is then antiparallel to the magnetic field.
Angew. Chem. internal. Edit.
c -
+---+ Ifqp, IqpI
i= 1
The functions on which the Hamiltonian operator H is
allowed to act are the product spin functions.
The spin functions corresponding to the two orientations of a single nucleus (I = 112) are denoted by the
symbols a and p [151. For a spin system consisting of two
nuclei there are four product spin functions: aa,ap, pa,
Here the step a denotes the calculation of the complete energy
matrix, Hp, being equal to ‘ppHlpq,
b denotes reduction to the diagonal form by solution of the eigenvalue problem, and
c denotes calculation of the frequencies and intensities of the
allowed transitions.
(b) The Spin Functions
+ (Ep.y’q)
The entire calculations of nuclear magnetic resonance
spectra have been programmed for electronic computers,
so that now the theoretical spectrum corresponding to
a given set of parameters can be obtained without effort
(see Section V d, e). The course of the calculation is
shown in Scheme (9).
I Vi, Jik I +a+
H in Equation (4) is replaced by the spin Hamiltonian
operator from Equation (3). For nuclei with spin 1/2,
the complicated angular momentum operator is given
by Equations (7), which together with Equation (3)
give Equation (8).
(c) The Eigenvalue Problem
(a) The Spin Hamiltonian Operator
In general the stationary states are described by linear
combinations of product functions. For example, the
following functions occur in any A2 spin system (cf.
Fig. 3).
The general equation of state [Equation (4)]is used to
calculate the energy eigenvalues E, and the eigenfunctions ‘I?
of,the stationary states for a given Hamiltonian
operator H. The line frequency f,, for a transition from
the state ‘r,of energy E, to the state \r, with energy
Ed is given by the Bohr frequency condition [Equation
( 3 1 . The corresponding intensity I,, is propxtional to
the matrix element with the operator of the transition
probability H(’) [Equation (6)].
P~zctp= pa; P1$p
Vol. 5 (1966)
No. 7
Ami=+ 1
i= 1
A m i = o or
+ I.
Equation (12) is applicable if the transition is between
only t w o energy levels described by simple product
spin functions. However, the eigenfunctions are mostly
linear combinations, for which the less accurate selection rule of Equation (13) is used for the sake of simplicity [161. A further condition for allowed transitions is
given later [Equation (15)].
F, is the z-component of the total spin.
(e) Some Important Rules
The signal due to isochronous nuclei I91 is usually split
because of spin coupling. There are compounds, however, in which no splitting occurs in spite of spin
coupling. For example, the “spectra” of unsubstituted
benzene and methane consist of a single line each (see
Rule 2). Moreover, it is a well-known fact that the spin
coupling between the protons of a methyl group that
can rotate is not detectable, but is by no means zero
(see Rule 1).
The nuclei in these structures, are magnetically equivalentI91, and the following two rules are generally
valid [I71 :
Rule 1. The spin coupling between magnetically equivalent nuclei does not affect the spectrum.
Rule 2. If a spin system contains only one type of chemically equivalent (isochronous) nuclei, these are always
magnetically equivalent [18al,and the“spectrum” consists
of a single signal, irrespective of the couplings present.
An explanation for the peculiarities described in the
two rules is found in the occurrence of “good” quantum
If two operators are commutable [Equation (14a)],
their eigenvalues g are known as “good” quantum
numbers [Equation (14b)l. The reason for their importance in quantum mechanics is the fact that they cause
certain matrix elements to disappear [Equation (14c)l.
GYP = gpYp; G Y q= gqYq
‘FpH’IPq=0 when g, $. gq
If H in Equation (I&) is taken as the operator of the
transition probability H(x) from Equation (lo), it
follows from Equation (6) that all transitions in which
at least one “g00d” quantum number changes are
forbidden. For magnetically equivalent nuclei, the parameters for the mutual spin coupling occur only in the
form of good quantum numbers. Thus owing to the
substraction involved in the calculation of the allowed
frequencies pquation (5)], these coupling parameters
cannot influence the spectrum.
[16] As an example of this, transition 9 in the AX2 scheme in
Figure 6 is forbidden according to Equation (12), though it
appears to be allowed according to Equation (13).
1171 H . S. Gutowsky, D. W. McCall, and C. P. Slichter, J. chem.
Physics 21, 279 (1953).
[18a] In An, the spin coupling with all other (imaginary) nuclei
is zero.
The Hamiltonian opxator can also commute with some
other operators, e.g. the F, operator of the total spin
[see Equation (13)]. The energy matrix ((Hpq)) therefore breaks down into independent sub-matrices for the
various values of F,.
The molecule under examination often possesses elements of symmetry, and the operators H and Hex) can
then commute with certain symmetry operators S. Symmetry eigenvalues s corresponding to these, e.g. with
the values +1 and -1 for symmetric and antisymmetric
functions, respectively, can then be defined as good
quantum numbers.
Another good quantum number is the “particle spin”
I*, which is obtained when a group of magnetically
equivalent [91 nuclei are considered together as a “spin
particle” [18bl. In a freely rotating methyl group, for
example, there are only states with I* = 312 or I* = ‘/z.
Thus in the magnetic field there are two molecular
species, which cannot be interconverted by nuclear
resonance. The situation is similar to that of ortho- and
para-hydrogen in optical spectroscopy.
Owing to the ability of S and I* to commute with H(’),
another selection rule for allowed transitions in addition
to Equations (12) and (13) results:
41*=A s = O
Since S and I* may also commute with the Hamiltonian
operator, the matrix elements H,, disappear whenever
the condition of Equation (15) is not satisfied by the
functions involved.
There is a simple, but very useful, summation rule
[Equation (16)] for the line intensities. This rule is based
on the energy-level diagram and requires that the total
intensity for n nuclei of spin = 1/2 be normalized to
n2n-1 [191.
Rule 3. The sum of the intensities of all transitions that
terminate in the state Yt differs from the sum of the
intensities of all transitions that start at \ r t by exactly
twice the value of the z component of the total spin in
the state yt.
2 FAY.”,)
The states p and q over which the sums are to be taken
are given by the energy-level diagram; q passes through
at most ),(:
values and p through (,TI) values,
where r = 2F,(Yt).
IV. Examples of Simple Spin Systems
In the equipment used to record the following spectra,
the magnetic field strength was varied and the radio
frequency v, was kept constant.
(a) The AB System
Figure 3 shows a number of calculated spectra for
two coupled non-equivalent nuclei. The energy-level
[ISb] D. R . Whitman, L. Oiisager, M . Snunders, and H . E. Dubb,
J . chem. Physics 32, 67 (1960).
[I91 G. Giomousisand J. D. Swalen, J. chem. Physics36,2077 (1962).
Angew. Chem. internat. Edit.
1 Vol. 5
No. 7
AX 6
fJ. l
(b) The AB2 System
111 2.3
Fig. 3. Calculated spectra for the two-spin system AB, with energylevel diagrams and spectra for the limiting cases AX and A2 1201.
50 Hz
In the three-spin AB2 system (Fig. 51, JAB = J'AB
because of symmetry. Consequently, the two B nuclei
are magnetically equivalent [91 and form a spin particle
with the possibilities I*B = 1 and 1"s = 0. The BB
coupling is not observed (cf. Section IIle, Rule l), and
the energy-level diagram breaks down vertically into
six and two states (cf. Fig. 6).
In the limiting case of an A3 system ( V , ~ A B = 0) there
is a further vertical division. The good quantum
numbers are then:
I*A = 3/2, ~ 2 =
3 +1 [Fig. 6, (a)];
I*A = 1/2, ~ 2 =
3 $1 [Fig. 6, (b)];
I*A = 112, ~ 2 =
3 -1 [Fig. 6, (c)] 1231.
On this basis it is possible to deduce the following
properties of the AB2 spectrum:
1) Line 3 must remain invariant owing to the fact that
it is isolated; its frequency is always VA and its intensity
is 1;
2) as the limiting case A3 is approached, Lines 1, 2,
and 8 become weaker and weaker and are ultimately
forbidden ;
3) the "combination line" 9 is particularly weak, since
it is forbidden in both limiting cases AX2 and A3.
diagram is given for the limiting cases AXand A2. The
forms of the spectra depend on the ratio J A B : V ~ ~ A B .
In the proton magnetic resonance spectrum of 1,2-dibenzo[d,,f ] dithiacyclooctadiene [211, the methylene
protons in the eight-membered ring give rise to an AB
spectrum (Fig. 4).
Fig. 4. Resonance of the methylene protons of I ,2-dibenzo[d, L]dithiacyclooctadiene [211 in CS2 a t v, = 6 0 M H z and 24 "C
0.67 ppm.; J = 13.7 Hz).
I n the rigid eight-membered ring, the
protons are situated i n sterically different
positions, with a Larmor difference of
40.1 and a spin coupling of 13.7 Hz.
These values were used for the calculation
of the control spectrum (Fig. 4, bottom).
The resonance of the protons of the ophenylene residue is situated at a lower
field strength, and is not shown !221
3 I
Fig. 5 . Spin coupling in the ABz system
1 Val. 5
I 1 I = + lf,-l'+ f6-18 I ;
[20] The arrows by the energy levels show the spin orientation,
the static magnetic field being directed upward. The number of
"beads" shown on the transitions corresponds to the intensity.
[211 A. Liittiinghaus et al., Chem. Ber., in press.
i221 The Larmor frequencies in the spectrum are given relative
to the tetramethylsilane signal on the dimensionless T scale,
where q = 10
~ O ~ ( V ~ - V T M S ppm.
Angew. Chem. internat. Edit.
la I
1 No.
Fig. 6.
Calculated spectra for the three-spin syste n ABz with the
enerpy-level diagrams and spectra f o r the limiting cases AX2 and A3 1201.
1231 The symmetry operation corresponding to the eigenvalue
is p23.
[24] Corio [4] gives Tables for AB, AB2, AB3, and A2B3,
most of which are reproduced by Strehlow [6b].
Apart from the normalization factor, the AB2 spectrum
depends only on the ratio JAB :VJAB. Some calculated
AB2 spectra are to be found in the literature as a function
of this ratio[24*251. They show that there is no crossing
of lines, i.e. the order of the lines always permits an unambiguous assignment.
Figure 7 shows the AB2 spectrum of the protons in
(c) The ABX System
New features of the ABX system in comparison with
the above examples are first the dependence of the
spectrum on the relative signs of the spin coupling parameters J, second the resulting problem of the assignment
of the lines, and third the fact that the problem can be
simplified [I43 if the ABX system is broken down into
two AB systems. The energy-level diagram can thus be
subdivided into two parts with the magnetic quantum
numbers mx = +1/2 and mx = -1/2 (see Fig. 8). The two
subschemes are linked only bq the transitions of the
nucleus X; otherwise they represent two AB systems
similar to that discussed in Section IVa. Here, however,
the spectrum contains “effective” Larmor frequencies
(v‘A, V’B, V‘ A, v“B), which depend on the spin coupling
with the X nucleus in the following manner:
Subsystem I (mx =
+ 1/2):
v A = v A + ~1J A X ; v B = v B ~21 J B x
Fig. 7. Magnetic resonance of the protons of 1,2,3-trichlorobenzene in
CC14 at v,, = 60 MHz and 24 “C as an example of an AB2 spectrum
(8 = 0.24 ppm.; J = 8.2 Hz).
Subsystem 11 (mx
= -
In the analysis of the ABX spectrum, all the parameters
can be found from the two AB subspectra. The defining
equations are:
- f 2- f 3-f4
V o 8 =~ ~(D’+ D”)
The symbols used have the following meanings :
= V A
D =vA-vB=+
I ,
I ,
fo = 2 (VA+ VB)
+ f4)
2 ( f 2 + f3)
The resonance lines of the X nucleus permit the calculation of two additional quantities:
flO) (f7 - f8)
J A X + J B X = f6
Fig. 8. Term diagram and two calculated spectra for the three-spin
system ABX.
The influence of the signs of JAX and JBX can be recognized in the two
calculated spectra.
Whereas the spectrum is affected only by the magnitude
of the coupling parameter JAB, the relative signs of JAX
[25] K . B. Wiberg and B. J. Nist: The Interpretation of N M R
Spectra. Benjamin, New York, 1961. - This book contains tables
and spectra for the AB, AB2, AB3, AB4, and A2B3 system types
and a collection of spectra whose form depends on more than one
parameter (ABX, ABC, and A2B2 systems).
Angew. Chem. internat. Edit. Vol. 5 (1966) 1 No. 7
and JBX affect the frequencies and intensities throughout the spectrum (see Fig. 8).
The proton magnetic resonance spectrum of 1,2,4-trichlorobenzene (Fig. 9) appears to be an ABC spectrum,
but is in fact of the ABM type ( V , ~ A M = 8.4 JAM;
V , ~ B M = 14.2 JBM; cf. Section 11). In an ABM analysis, there are no simple defining equations similar to
those that apply to ABX systems [Equations (19)-(28)].
However, it has been found that these equations can be
used to good approximation, since the parameters obtained are practically the same as those obtained by
precise ABM analysis [261.
can be calculated by summation of simple products 1271.
It has been shown by theory [281 that these moments can
be equated to expressions containing the Larmor frequencies and coupling parameters. The advantage of
this method is the fact that assignment of the lines is not
necessary. It has the disadvantage that a complete analysis is impossible except for the simplest typesof spectra,
since it gives only the parameters that are explicitly contained in the spectral moments. However, the method is
extremely useful, for example, when only the chemical
shifts in A2B2 spectra are required (see Section VI,
Table 2) or when the overlapping of the lines is too
serious to permit the use of other methods, e.g. in broadline NMR spectra of solids.
(b) Direct Analyses
2 88
2 73
h = 8 2 H Z , ‘Vo6~e=-11LH~
II“ i+’
A very simple case is encountered with “multiplet
spectra” (first-order spectra), e.g. of the type A,Xt with
magnetically equivalent [91 nuclei. The coupling parameters can be read off directly from the multiplet splittings, and the Larmor frequencies from the multiplet
centers. In other direct analyses, the parameters are obtained by substitution of the line frequencies in defining
equations; this possibility exists for spectra of the types
AB, AB2, and ABX [see Figs. 3 and 6, and Eqs. (19) to
(28)1, or of the types A2X2, AzBz, A2B2X, and AB2X
(see Table 4).
(c) The Algebraic Method
V. Methods for the Analysis of NMR Spectra
The information about a molecule that can be obtained
using nuclear magnetic resonance spectroscopy are usually based on the screening parameter o and the spin
coupling constant J. Thus, analysis of an NMR spectrum, i.e. the determination of the numerical values of
these quantities from the observed positions of the lines,
is the reverse of the problem of calculating the spectrum
corresponding to a given set of parameters (cf. Section
IIIc). Scheme (9) should now be followed from right
to left. This is possible for steps a and c, but there is no
simple solution for the inversion of step b. Various
approaches to the solution of this principal difficulty
in the analysis of complex spectra have been suggested.
The most important methods will de bescribed briefly
in this section.
In the algebraic method [29-311, the parameters are calculated from the solution of a system of equations. Such a
system of equations is assigned to each eigenvalue
probleni or to its secular equation (Vikta’s square root
law). Other equations follow from the invariant rules
for unitary transformations. The system of equations is
solved numerically after insertion of the experimental
energy eigenvalues. The observed line frequencies must
first be converted into the eigenvalues with the aid of
the energy-level diagram. Although explicit expressions
have been derived even for complex systems such as
AB2C2[291, the practical use of this method is mainly
confined to the type ABC [30,311, since the experimental
errors mount up too rapidly in the long equations obtained for larger systems.
1271 The spectral moment of the n-th order with respect to a
frequency fo is defined by
where Ik = is the intensity. For example, for A2B2 spectra with fo
in the center of symmetry[28]:
(a) The Method of Moments
[28] W . A . Anderson and H . M . McConneN, J. chem. Physics 26,
For any spectrum in which the line intensities and the
distances of the frequencies from an arbitrary reference
point fo have been determined, the “spectral moments”
[26] G . Englert and W. Briigel, unpublished results.
Angew. Chem. internat. Edit.
Vol. 5 (1966) 1 No. 7
1496 (1957).
[29] D. R . Whirmann, J. molecular Spectroscopy 10, 250 (1963).
[30] W. Briigel, Tli. A n k d , and F. Kruckeberg, Z. Elektrochem.,
Ber. Bunsenges. physik. Chem. 64, 1121 (1960).
[31] S. Castellano and J. S . Wangli, J. chern. Physics 34, 295
(d) The Iteration Method
The principle of the iteration method is shown in the
following scheme W 3 3 1 . An approximate set of parameters @:,):J
is established, e.g. by comparison
with similar substances, from theoretical data, or by a
preliminary analysis. These parameters are then used to
calculate an approximate spectrum (Steps a, b, and c),
and the assignment of the lines is transferred to the experimental spectrum (Step d). It is then possible to
determine the eigenvalues Ep from the experimental
frequencies fqp (Step e). The crucial step now is the
transformation of these eigenvalues back into the matrix
elements H,, (Step f). This is carried out in approximation by means of the unitary transformation matrix U
with which the eigenvalue problem was previously solved
for the approximate parameters, i.e. U is given by the
matrix of the approximate eigenvectors.
V i . J,k
c Upq1Eq 7f '
(E,) L ' f q p , JqPI
In another method [361, the correct set of parameters is
calculated by the method of least squares. A set of
approximate parameters is used as in Section Vd to
calculate a theoretical spectrum, and partial derivaiives
of these approximate frequencies with respect to the
individual parameters are determined numerically. The
deviations of the calculated line frequencies from the
experimental values and the partial derivatives give the
corrections for the approximate parameter (minimum
condition). As in Newton's approximation, the resulis
converge towards the correct values if the starting values
are reasonably good.
VI. Comparison of Various Analytical Methods
Using the
Spectrum of o-Dichlorobenzene
The proton magnetic resonance spectrum of o-dichlorobenzene (cf. Fig. 10) has been analysed several times by
a number of different processes. It can be seen from
Table 1 that the spin coupling parameters were obtained
with an accuracy of only 0.1-0.2 Hz. The methods
described in [34,35,3*, 393 are exact analyses and
would give accurate parameters for accurate line fre-
A set of parameters (vi, Jik) is then calculated from the
diagonal elements H,, (Step g). This set is better than
the approximate set (vp, J:,)
since the experimental
eigenvalues E, are more accurate than the approximation Eg. The eigenvalue problem is now solved with
the new parameters (Steps h and i) and transformed
back with the aid of the improved eigenvectors Up,
(Step j). The cycle g-h-i-j-g is repeated until no further
improvement of the parameters is obtained (Step k).
(e) The Fitting Method
An attempt may be made to find a set of parameters to
fit any spectrum by simple trial and error. The better the
agree nent between the frequencies and intensities calculated from the tentative parameters and the experimental spectrum, the closer is the set of parameters used
to the correct values (trial and error fitting). This method
can be regarded as an analytical procedure only if it is
carried out systematically, e.g. by calculation of a succession of theoretically possible spectra and determination of the correct parameters by interpolation. It works
well if only a few independent parameters are involved [41 or if a series of similar spectra are to be analysed [34,351.
[32] J . D . Swalen and C . A. Reilly, J. chem. Physics 37, 21 (1962).
[33a] R . A . Hoffmanand S . Gronowith, Arkiv Kemi 15,45 (1959).
[33b] R . A. Hoffman, J. chem. Physics 33, 1256 (1960).
[34] J . Martin and B. P . Dailey, J. chem. Physics 37, 2594 (1962).
[35] M . Grant, R. C. Hirst, and H . S. Gutowsky, J. chem. Physics
38, 470 (1963).
f i g . 10. Masnetic resonance of the protons o f o-dichlorobenzene in
CCld at vo = 60 MHz and 24 'C [45b]. For the analysis of this A2Rz
spectrum, see Part VI.
quencies. Thus the deviations in the coupling constants
are due only to the limited accuracy of the practical
measurements. Since the measurements always involve
some inaccuracy, the influence of these errors on the
accuracy of the parameters must be taken into account
in a comparison of the analytical methods.
It can be seen from Table 2 that the various methods
differ in this respect. The values in Table 2 all refer to
[36] S. Castellano and A. A . Botlmer-By, J. chem. Physics 41,
3863 (1964).
[37] J. A. Pople, W. G. Sclmeider, and H. J. Bernstein, Canad. J.
Chem. 35, 1060 (1957).
[38] B. Dischler and G . Englert, 2. Naturforsch. 16a, I180 (1961).
[39] D . R . Whitman, J. chem. Physics 36, 2085 (1962).
Angew. Chem. internat. Edit.
1 Vol. 5 (1966) 1 No.
The analysis by the iteration process NMRITL441 was carried
out following the scheme in Section Vd. Divergence began
after two iterations, though the approximate parameters
(from [371) had been distinctly improved. The analysis was
repeated with these improved values, and the final set of
parameters was obtained after seven iterations. The error
the Same spectrum 1401, and the Same approximate parameters were used where necessary (in accordance
with [371). The quality of the analysis was assessed on the
between the
basis of the frequency deviation nf[413
calculated spectrum and the experimental spectrum.
T a b l e I . Results of t h e analysis by various mctbods of independently recorded spectra of
< 5 lbl
95 [cl
Average (lines 3 to 6)
< 8,l
x. I
I/ + .If4= 8.1 H z 1271; [b] Mole- %, i n cyclohexane:
T a b l e 2.
[c] Vol- %, in cyclohexane.
Results of spectral analysis b y v a r i o u s m e t h o d s f o r t h e s a m e experimental soectrum of o-dichlorobenzene
__ Method
0.09 I
__ __
h l o m e n t m e t h o d 1271
Direct analysis 1421
Algebraic I391
I t e r a t i o n [44]
Fitting [45a]
< 8.4[-]
;t 0.10
: 0.20
z 0.05
f 0.07
2 0.10
[a 1
x 0.05
f 0.06
’ 0.20
: 0.05
+_ 0.10
- 0.20
& 0.037
-- 0.037
- 7.606
+ 0.002
- - 8.4 Hz.
The following points should be mentioned in connection with
the values given in Table 2:
The second moment of the spectrum gives a very reliable
value for the Larmor difference V ~ S A B , and with the aid of
the fourth moment[*71 it is found that J i B J& = 70.6.
It follows that JAB= 513 = 8.0-8.3 cps if a value between
1.5 and 2.5 Hz is assumed for
= 514. The moment
method does not yield any information about the values of
JAA = Ji2 and JBB = J34.
The direct analysis 1381 was carried out using defining equations 1421 but without the use of approximate parameters. The
limits of error correspond to the maximum deviation between the frequencies of the experimental and calculated
In the algebraic analysis carried out by Whitman[391,summation rules [431 for frequencies and intensities were used in the
assignment of the lines. The estimated accuracy value of
f 0.05 Hz 1391 is very optimistic.
[40] The spectrum published by Whit~na~z
[39] c3ntains uncxtainties ol‘ u p to 0.2 Hz in the frequencies; this is desirable for
the test of Table 2 . Better accuracy can rrow be obtained if the
spectrum is recorded several times and the resultj avcraged.
[41] The root mean square error for t lines is:
L k=l
Average (lines 2 t o 5 )
[42] B. Dischler and W. Maier, Z. Naturforsch. I6a, 318 (1961);
B. Dischler, ibid. ZOa, 888 (1965).
[43] Cf. Rule 3 in Part Ille. The following trivial rulc applies to
frequencies: Th:: distance between two ertergy levels is independent of the transitions chosen. I n Figure 3, for example, f l + f4 =
f i + f3.
Aiigew. Cheni. ititernat. Edit.
Vol. 5 (1966)
1 No. 7
limits were calculated from the program, with the assumption
that the experimental eigenvalues are accurate to +. 0.1 Hz.
In the fitting method LAOCOON I1 “‘51, a comparison spectrum was first calculated from the approximate parameters.
The correctly assigned experimental line positions were then
introduced, but the expected convergence was not observed.
In a second attempt, the four innermost lines, which showed
particularly marked deviation from the comparison spectrum,
were omitted. This gave a useful set of parameters in five steps,
and these parameters were then used in four further steps
using all the lines, in order to obtain the final values. The
limits of error given by the program is the change in the parameters at which the mean frequency deviation r f [ 4 1 1 would
be exactly doubled. Thes- “probable” errors are only indirectly connected with the inaccuracy of the parameters, which
is, for example, about 10 times greater at v1.
The fitting method gives the best results, both with regard to
Af[411 and in relation to the average values in Tables 1 and 2,
i.e. in this example at least, the errors in the measurements
are best averaged out by LAOCOON 11. This is as expected,
since in this method the genuine information from the line
positions is used exactly in accordance with their significance
for the individual parameters. Though line positions are also
used directly in the direct analysis method, the distribution of
the weigths is rigid and unequal, as in any defining equation
[44] Thanks are due to Dr. W . Briigel, Hauptlaboratorium
BASF, Ludwigshafen, for the analysis by NMRIT [32].
[45a] The analysis with LAOCOON 11 (Least-squares Adjustment
Of Calculated On Observed N MR spectra). [33]was kindlycarried
out by Dr. K . Giitither, Institut f u r Organischz Chemie, Universi-
tat Koln.
[45b] Dr. H . G i n f l i e r also supplied the depicted spectrum of o-
dichlorobenzene, which was recorded with a varian A60 spectrometer.
63 1
of generally validity. W i t h t h e analytical a n d iteration methods
(b) Experimental Aids
some o f t h e original accuracy is lost in t h e conversion of
frequencies i n t o energies. In view of t h e uncertain convergence behavior encountered in t h e methods of [441 a n d [45al,
it seems advisable t o use t w o steps: for example, a set of parameters m a y b e calculated by t h e simple direct method “Q],
a n d these m a y then b e further improved, if necessary, by t h e
fitting method [45al.
In order t o m a k e a general assessment of t h e efficiencies of
t h e various analytical methods, it would naturally be necessary t o work out a large n u m b e r of examples.
Vn. Suggestions for the Analysis
(a) Arrangement in Order of Difficulty
The very different mathematical requirements involved
in the analysis of the individual spectra have here been
arranged in order of difficulty. Table 3 shows the criteria
used in the assessment, and Table 4 shows a selection of
analyses that have been carried out.
Table 3. Classification of spectral analyses according t o their difficulty.
Difficulty grading
Without calculation
Directly analysable
Assignment required
Central spectrum required
Difficult calculation
Good approximate parameters required
Table 4. Examples o f investigated spectra, arranged to illustrate the
various degrees of difficulty in accordance with Table 3.
Type of
Method of analysis
b) direct
c) algebraic
d) iteration
e) fitting
*) simplification by division into subspectra
(cf. Sect. 1Vc)
[46] W. A . Anderson, Physic. Rev. 102, 151 (1956).
[47] E. L. Muetterties and W. D . Phillips, J. Arner. chern. SOC.79,
322 (1957).
[48] H . S. Gutowsky, C. H . Holm, A . Saika, and G . A . Williams,
J. Arner. chern. SOC.79, 4596 (1957).
1491 R. J. Abraham, E. 0. Bishop, and R. E. Richards, Molecular
Physics 3, 485 (1960).
[50] H . M. McConnell, A . D . McLean, and C. A . Reilly, J. chern.
Physics 23, 1152 (1955).
[51a] G. W. Flynn and J . D . Baldeschwieler, J. chern. Physics 38,
226 (1963).
[51b] G . W . Flynn, M . Matsushirna, J . D . Baldeschwieler, and
N . C . Craig, J. chern. Physics 38, 2295 ( I 963).
Substitution by isotopes is useful in NMR spectroscopy.
For instance proton magnetic resonance spectra can be
simplifiedby partial deuteration of the substances, since
the lines due to the hydrogen atoms that have been
replaced by deuterium disappear, and the H-D spin
coupling constants are smaller by a factor of 1/0.154
than the corresponding H-H coupling constants.
If the measuring frequency is increased from 60 to 100
MHz, the Larmor differences are almost doubled,
whereas the spin couplings remain unchanged. Thus the
spectrum approaches the multiplet type, which is easier
to interpret and to analyse. Conversely, a change in
frequency from 60 to 15 MHz leads to a more complex
spectrum, owing to the smaller Larmor differences.
Comparison of the spectra obtained with all three frequencies often makes it possible to decide between
several combinations of signs in the spin couplings 1491.
The Larmor differences can also be varied within certain
limits by solvent effects [5*l; this permits the unambiguous determination of the signs of the coupling parameters
in systematic studies [593.
An important aid is nuclear magnetic double resonance
(NMDR). In normal operation the spectrum is recorded
in a radio-frequency field HI, the strength of which is
kept so small that the energy-level diagram remains
stationary, so that the Dirac perturbation theory, on
which the calculation of Equations (S), (0,
(lo), and
(12) is based, is valid. In double resonance, a second
radio-frequency field H 2 is applied parallel to Hi ; the
much greater field strength of H 2 leads to a change in
the energy-level diagram whenever its frequency v2
coincides exactly or approximately with a resonance
In “spin decoupling” experiments [60,611, the multiplet
splitting caused by one or more X nuclei is caused to
disappear by the application of the Larmor frequency
vx with saturation field strength. In addition to simplifying the spectrum, spin decoupling yields information
on the interrelationship between certain lines.
This interrelationship between lines is specially studied
in “perturbation field experiments” [621, where the frequency of a single resonance line is applied permanently.
1521 B. D. Nageswara-Rao and P . Venkateswarlu, Proc. Indian
Acad. Sci., Sect. A 54, 1 (1961).
[53] R. E. Richards and T. Schaefer, Proc. Roy. SOC.(London),
Ser. A 246, 429 (1958).
[54] C. A . Reilly and J. D . Swalen, J. chern. Physics 32, 1378
[55] C . S. Johnson Jr., M . A . Weiner, J. S . Waugh, and D . Seyferth, J. Arner. chern. SOC.83, 1306 (1961).
1561 W. Briigel, Z. Elektrochern., Ber. Bunsenges. physik. Chern.
66, 159 (1962).
[57] C. A . Reillyand J. D . Swalen, J. chern. Physics34,980(1961).
1581 S. Clough, Molecular Physics 2, 349 (1959).
[59] H . Dreeskamp and E. Sackmann, Z. physik. Chem. N.F. 34,
261 (1962).
[60] A . L. Bloom and J . N . Shoolery, Physic. Rev. 97, 1261 (1955).
[61] W. A. Anderson and R. Freeman, J. chern. Physics 37, 85
[62] R. Freeman and W. A . Anderson, J. chern. Physics 37, 2053
Angew. Chern. internat. Edit.
Vol. 5 (1966) 1 No. 7
The field strength must be large enough to perturb the
energy levels involved in the resonance in question, the
perturbation being either (a) energetic [621 or (b) in the
occupation numbers [61,631, depending on the relaxation
time. It is then found that for (a), the lines due to all
transitions that begin or terminate at these two levels
are split into two equally strong componenLs[621with
a perturbation parameter that depends almost linearly
on the field strength H2, while for (b) these lines undergo
characteristic intensity changes 1643. Since the lines in
which the same levels are involved are known in perturbation field experiments, this method assists in obtaining
a definite assignment of the lines.
The observation of multiple quantum transitions
(Am = 2, 3 . . .) which occur near the saturation intensity of H1[65,661 yields additional information about the
energy-level diagram, which may allow a particular
choice in the signs of spin coupling parameters.
The proton magnetic resonance spectrum of gaseous
1,l-difluoroethylene (at 10-26 atm) contains both sharp
and broad lines [51al; this is in agreement with theoretical
calculations which assume an interaction between the
spin of the fluorine nuclei and the molecular rotation as
the relaxation mechanism. The different line widths are
of great assistance in the assignment of the lines, i.e. in
the determination of the relative signs of coupling parameters [511.
Vm. Outlook
For n nuclei of spin 1/2, the number of energy levels
increases witb 2 n , and the number of lines roughly with
n3 (see Table 5). As the size of the spin system becomes
larger, the number of lines increases so rapidly that they
[63] R . A. Hoffman, B. Gestblom, and S. Forsen, J. chem. Physics
40,3734 (1964).
[64] R . Kaiser, J. chem. Physics 39, 2435 (1963).
[65] K. A. McLauchlan and D . H. Whiffen, Proc. chem. SOC.
(London) 1962, 144.
[66] A . D. Cohen and D . H . Whiffen, Molecular Physics 7 , 449
Ex amp1 e
1 1 I I 1 I 2 [(;)I'
Matrix elements [b]
Energy levels
( 4
i=O -
--1 8
- - ~ _ _ -
Allowed transitions [cl
1 2 5 6 1 2 "
1 1~(~)~~1)
Total intensity
normalized [d]
[a] For n > I , the number of icdependent parameters is always two less, since the
form of the spectrum depends only o n the ratio of the Larmor differences to coupling
[bl These values take into account the breakdown into submatrices Fz
[cl The nuinher of these is variable, owing to the selection rule Equstion (15). The
minimum number is found, for example, in A n systems, and the ma%iTumnumber
in the above examples.
[dl The normalization is based o n the summation rule (cf. Section IIle, Rule 3)
can no longer be resolved. Moreover, the average
intensity of the lines rapidly decreases. Analysis thus
becomes very difficult or even impossible. It is therefore
to be expected that the simpler spin systems studied so
far (see Table 4) will still occupy the foreground in the
The author is grateful to the Director of this Institute,
ProJ R. Mecke, for his support of this work. The experimental examples are taken from investigations carried
out in the Institute. Thanks are due also to Dr. H. Friebolin
and Dr. G. Englert for their valuable co-operation and for
supplying the spectra (recorded with a Varian DP 60
spectrometer, Type V 4302), and to Dr. W. Briigel, Ludwigshafen, and Dr. H. Gunther, Cologne, for carrying
out the test calculations on the spectrum of o-dichlorobenzene.
Received: July 9th, 1965
[A 521 IE]
German version: Anpew. Chem. 78, 653 (1966)
Translated by Express Translation Service, London
Problems of Iron Metabolism
with Special Reference to Biochemical, Physiological, and Clinical Aspects
A brief account of the functions and chemical structures of the principal iron compounds
found in the human body is followed by a discussion of the physiology of iron metabolism,
starting with its resorption, continuing with its transport and assimilation by the body
cells, and concluding with its elimination. Special attention is paid f 6 the problems of
intracellular and extracellular transport of iron. A number of diseases with characteristic
disturbances of the iron metabolism are also briefly described.
Wherever respiration occurs, iron is present in some
form. Iron is essential to the elementary metabolic
processes in the cell. In the catalytic processes of the
Angew. Chem. internat. Edit.
1 Vol. 5
No. 7
-----g = 2 Or -2
respiratory chain it acts as an electron carrier, and it is
responsible for the transport of molecular oxygen in
higher organisms. These important functions depend
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