A Simple, General Method for Predicting the Relative Intensities of First-Order, NMR Spin-Spin Coupled Multiplets John Homer and Mansour Sultan-Mohammadi The University of Aston in Birmingham, Gosta Green. Birmingham B4 7ET, England Preliminary undergraduate instruction in the principles of nuclear magnetic resonance spectroscopy normally includes discussion of the analysis of first-order, spin-spin coupled multiplets. I t is well known that the number of multiplet lines is given by 2nI 1 when n magnetically equivalent nuclei of spin I are adjacent to the resonant nucleus and cause the splitting of its resonance.' Similarly well known are the principles for predicting the distribution of line relative intensities in first-order multiplets.' Because these can be tedious to implement per se it proves advantageous to have simple rules that can be used speedily to predict the line relative intensities. While such rules for spin % nuclei are often referred to, there appears to have been little reference to a general rule that permits the rapid evaluation of the relative intensities of the components of first-order multiplets that arise from c o u ~ l i n with e anv number of eauivalent nuclei of
ciated by reference to Fig. l(a) which is the result of assuming that each additional nucleus splits each line in the preceding pattern into two. This is mirrored in Fig. l(b) where each of the elements of any row are obtained by summing the two numbers directly above it and to its left. This is a specific case of a general rule. When first-order multiplets arise from coupling to n nuclei of spin I the relative intensities of the components can be deduced by deriving appropriate Pascal-type "triangles". For this, each element of a particular row is deduced by summing the 21 I numbers above and to the left in theprecedingrow. This is illustrated in Figure 2 for the case of I = 3i2. It can be seen from this that, for example, when n = 3, the 'H resonance of Na+ B3 H8- will occur as a 1:3:6:10:12:12:10:6:3:1 decet due to coupling with the three l1B n u ~ l e i . ~ With the availability of multinuclear-pulsed F-T NMR
Fundamentally, both the number of multiplet lines and their relative intensities depend on the possible combinations, 2 , m,, of the allowed spins of each nucleus, i;for each nucleus the quantum number m can adopt the values I, I-1,I-2 . . . . -I. Consequently, the number of lines arising from coupling to n equivalent nuclei is 2nI 1and the relative intensities of these is given by the number of times each value of 2 , m, occurs. In the case of coupling t o n spin -'iz nuclei the relative intensities can be generated rapidly from the coefficients of the binomial series or, probably more popularly, using Pascal's triangle. The relevance of the latter (Fig. l(b)) can be appre-
acquainted with the principles of analyzing spectra involving a variety of nuclear spins. I t is possible that the triangulation rules outlined above may prove helpful in the analysis of first-order spin-coupled multiplets.
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' Akilt. J. W., "N.M.R. and Chemistry," Chapman and Hall, New York, 1972~
Jesson, J. P., and Muetterties, E. L., "Dynamic Nuclear Magnetic Resonance Spectroscopy," (Editors: Jackman, L. M., and Cotton. F. A,). Academic Press, New York, 1975.
1 Number of I = 3 neighbors (n)
(a)
(b)
Figure 1. (a) The relative intensity diswibution of first-ordw multiplet lines caused by coupling to n nuclei of spin %. (b) Pascal's "triangle."
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Journal of Chemical Education
Figure 2. Construction of Pascal-type "triangle" far nnuclei of spin 312. Inthis example, 21 1 = 4 so that four numbers from the preceding row must be added to obtain an item for the row below, as indicated.
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