ANGEWANDTE CHEMIE VOLUME 5 . NUMBER 7 JULY 1966 P A G E S 623-688 Classification and Analysis of NMR Spectra BY DR. B. DISCHLER I NSTITUT FUR ELEKTROWERKSTOFFE, F R A U N H O F E R - G E S E L L S C H T , FREIBURG (GERMANY) 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 (1) 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 field. [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 1959. [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, (I-ai) (2) 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: x, n I95 1 : H = vi Iz(i) i= 1 + k-1 i=l n 2 Jik I(i) I(k) (3) k=z 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). 623 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 VJAB ; “8““ (b) a letter from the middle of the alphabet (M, . . .) if JAM is approximately one order of magnitude smaller than V ~ ~ A[lo]; M O H (c) a letter from the end of the alphabet (X, . . .) if J A x is two or more orders of magnitude smaller than Vo8AX @ H-CF3@ @ H-ClCH,I,@ F Fig. 1. H 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 difference. [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]. + 624 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 1 2 Iz(i)‘l‘= miY’ (74 I(i) I(k) yr= (2Pik-l)Y (7b) a n k-I n 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. ( a +ppet) ((Hpq)) (9) 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 functions. (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 : n 2 (1--0i) I+(i) (10) In Equation (10) I+ is a “shift operator” with the effect that I+ct=O (lla) andI+p=a (llb) 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)]: --f ~ [ 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, . . . I 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 ~~ v(@-P.) Spectrum b + (Ep.y’q) H(X)= ~ and 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). pp. (c) The Eigenvalue Problem Parameter (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 2 Ami=+ 1 i= 1 A m i = o or (12) + I. 625 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 numbers. 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. G H Y = HGY 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. 626 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 (15) 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. Iqtr 9 2 Itp+ 2 FAY.”,) (16) P 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 (1966) No. 7 ito u ' V n AX 6 > > fJ. l tt 'A I/% ' 23 \\ I I '\ (b) The AB2 System ii7AB Yo& 111 2.3 141 1- -lrrr-rrr$ I Ya8=0 A, tt Fig. 3. Calculated spectra for the two-spin system AB, with energylevel diagrams and spectra for the limiting cases AX and A2 1201. ~ C l ~ J 50 Hz 0 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. 6L7 t.580 I 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). (8 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 / 5.6 7,8 I i II 1 3 I \ A3 Fig. 5 . Spin coupling in the ABz system (schematic). ~ 1 Val. 5 lbl ICI 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. )V~ Angew. Chem. internat. Edit. la I 3 (1966) 1 No. 7 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]. s23 627 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. - 0 lOHz Figure 7 shows the AB2 spectrum of the protons in 1,2,3-trichlorobenzene. (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 = rn % + 1/2): v A = v A + ~1J A X ; v B = v B ~21 J B x -4 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 (17) 1/2): = - In the analysis of the ABX spectrum, all the parameters can be found from the two AB subspectra. The defining equations are: , ? I , ? I , , , . , , - f 2- f 3-f4 IJABI=fl-f2=f3-f4=fl 1 V o 8 =~ ~(D’+ D”) JAx = fo - ,, fo JBx = , fo - f, r 1 + 1 I, - 2 (D’ - D”) (D’ - D”) The symbols used have the following meanings : D ABX , , = V A - , -vB= (f;-fk)(f;-f;) ~- ,, D =vA-vB=+ I , - I , ‘ 1 fo = 2 (VA+ VB) I 1 , (fl = I + f4) 1 = , I 2 ( f 2 + f3) The resonance lines of the X nucleus permit the calculation of two additional quantities: fs (27) flO) (f7 - f8) (28) J A X + J B X = f6 JAX - JBX rmjjl 71 78 1- 3 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. 628 = 1 (fS ~V,~AB ~ ~ 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. Ll 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 t=26L 1” 2‘ 3’ h = 8 2 H Z , ‘Vo6~e=-11LH~ 67 1‘ 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 (1961). 629 (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 Hpp= c Upq1Eq 7f ' ~ (E,) L ' f q p , JqPI Spectrum 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 A2Bz 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). 630 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. 7 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 ,i-diclilorobenzene Rcf “0 Concn. (MHz) ( Z) -- 30 I00 I.o 8.3 1oc 100 100 40 56.4 60 60 60 < 5 lbl 95 [cl Average (lines 3 to 6) [a1 < 8,l - 7.7 7.44 7.5 7.5 8.3 7.9 8.17 7.9 x. I 7.54 8.02 la1 ~ la1 1.7 1.6 1.61 1.7 1.5 0 0.5 0.36 0.5 0.3 I5.?.0 1 ,J:3 I/ + .If4= 8.1 H z 1271; [b] Mole- %, i n cyclohexane: T a b l e 2. 7.5 0.5 10.1 14.60 35.23 16.16 - 0.26 0.252 0.259 0.254 0.286 0.263 ~ [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 Iteration steps [411 (Hz) 0 - 0 0.09 I Af __ __ h l o m e n t m e t h o d 1271 Direct analysis 1421 Algebraic I391 0 0.099 A I t e r a t i o n [44] 9 9 Fitting [45a] < 8.4[-] - 7.44 0.20 7.44 0.05 0.097 7.55 0.076 ;t 0.10 7.53 0.04 + 8.12 : 0.20 8.17 z 0.05 8.1C f 0.07 8.09 2 0.10 [a 1 y .- ~ 1.57 0.20 1.61 x 0.05 1.51 f 0.06 0.26 ’ 0.20 0.35 : : 0.05 0.37 +_ 0.10 1/J;3 ~ 3:4 - - 0.20 ~ 7.552 .- 7.553 & 0.037 -- 0.037 7.628 - 7.606 I 0.015 + 0.002 15.23 0.05 15.12 ~: 0.07 15.24 0.02 15.18 - - 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. + JAB 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 spectra. 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 15.15 - Average (lines 2 t o 5 ) [a] 15.3 ~ J [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 mectrum Difficulty grading Method of analysis b) direct c) algebraic d) iteration e) fitting *) simplification by division into subspectra (cf. Sect. 1Vc) I Multidets 11 AB AB2 I1 A2X2 I11 111 111 AzB2 AzBzX ABU,X IV IV IV ABC AB C X V V ABCD AB2Cz VI VI ABX AB2X [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). 632 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 line. 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 (1960). [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 (1962). [62] R. Freeman and W. A . Anderson, J. chern. Physics 37, 2053 (1962). 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 (1964). Ex amp1 e A AB Nuclei 1 2 ABC I IABCDl - 3 4 - 81, 1 1 I I 1 I 2 [(;)I' Matrix elements [b] Energy levels 2 6 12 ( 4 1 20 70 12870 i=O - --1 8 (16 15 156 5 10 - - ~ _ _ - Allowed transitions [cl max. miii I I 1 14 2 1 1 2 5 6 1 2 " 1 1~(~)~~1) 11440 186 i=O 2.-(:); n Total intensity normalized [d] experimental 1 4 1 12 12 3 32 4 1024 8 n n 2n-I [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 parameters. [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 future. 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 BY PROF. W. KEIDERLING AND DR. H. P. WETZEL MEDIZINISCHE UNIVERSIT.&TSKLINIK,FREIBURG (GERMANY) 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 (1966) No. 7 n-1 -----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 633

1/--страниц