Download A straightforward way to determine relative intensities of spin

A Straightforward Way To Determine Relative Intensities
of Spin-Spin Splitting Lines
of Equivalent Nuclei in NMR Spectra
Ronald H. Orcutt
East Tennessee State University, Johnson City, TN 37614
+
The well-known n
1 rule predicts the degree of spinspin splitting of proton NMR lines by equivalent protons;
the relative intensities of the members of the resulting multiplets are given by the binomial coefficients C(n,r), as defined by
where n is the number of equivalent nuclei causing the multiplet splitting, and r, ranging from 0 to n, is the index
number of the particular line within the multiplet. A simple
way of obtaining a set of relative intensities is by use of
Pascal's triangle (see Table 1) rather than calculation of
binomial coefficients.
Starting with a row of two 1's (corresponding to one nucleus giving rise to the splitting), successive rows can be generated by noting that each entry is the sum of the two entries
above and to the left of the one desired. If an element is
nonexistent, its value is taken to be zero.
Because of the increasingly widespread availability of FTNMR spectrometers with multinuclear capability, it is desirable to have a similar, simple rule for predicting relative
intensities to be expected in spin systems other than (-'12,
+%I.
A quick look at standard NMR reference^',^ does not
reveal such a rule for other spin systems.
The degree of splitting is given by the formula
Table 1.
1
2
3
1
1
1
1
1
1
4
5
6
(S1+ S2
I
3
6
10
15
Pascal's Trlangle
1
4
10
20
1
5
15
1
6
1
+ S3P. Expansion yields the result:
+ +
S13+ SZ3+ S33 + 3S12(S2 S3) 3S2'(51
3S3'(Sl+ S2) 6S1S2S3
+
+
+ 53)
The term S13 represents the situation when all three equivalent deuterons are in the -1 state; since only one such term is
present, the degeneracy of the -3 (-1, -1, -1) spin state is
1. Similarly, the three terms S12S3 represent a spin state of
-1 (-1, -1, +I), as do the terms SZ2S1(0,0,-1); thus, the
degeneracy of the -1 state is 6. Continuing this process of
where I is the nuclear spin of t h e n equivalent nuclei causing
the splitting. The possible spin states in a particular case can
he generated by taking all possible combinations of the 21
1nuclear spins for each of t h e n equivalent atoms and evaluating relative intensities by counting the number of comhinations that vield the same net suin. One wav- of eeneratine
all such possible spin states is t o k i t e the set of nuclear spins
as a ~olvnomialand then raise the ~olvnomialto a Dower
equai tothe number of equivalent nuclei: Such an expansion
can he carried out either by algebraic manipulation or by use
of the multinomial expansion. This process generates all the
terms; evaluation of the total spin for each term followed by
counting the number of terms with the same total spin gives
the degeneracy for each total spin state and, hence, a determination of the relative intensity to he expected.
Consider, for example, the splitting of the C-13 resonance
due to equivalent deuterium atoms (I= 1) in the CD3 moieties in molecules such as CD30H or (CD3)2CO. In these
cases seven line multiplets Are expected.
If one labels the three equivalent spins S1, S2, and S3
(corresponding to the set of values 1-1,0, +I)), then all the
possible spin combinations will be generated as the terms of
+
'
Becker. E. D. HighResolutionNMR TheoryandChernical Applica
tion: Academic: Orlando. FL. 1969.
~ S o h a r ,P. Nuclear Magnetic Resonance Spectroscopy; CRC:
Boca Raton, FL, 1983.
'%NMR spectra of acetone. d6, and rnethandl.64.
Volume 64
Number 9
September 1987
763
Table 2.
Relative Intensities of Multiplets for Nuclear Spins Ranging from 1 to 512
collecting like total spins gives the degeneracies as follows:
Total Spin State
Degeneracy
-3
-2
-1
0
1
2
3
1
3
6
7
6
3
1
Thus heptuplets with intensities in the ratios 1:3:6:7:6:3:1
are predicted. This is consistent with the observed spectra as
seen in the figure.
The above procedure was carried out for systems of three,
four, five, and six spins for up to six equivalent nuclei; the
results are shown in Table 2.
T o the author's astonishment, a rule like that for Pascal's
764
Journal of Chemical Education
triangle for generating the resulting relative intensities
works for all the cases examined. One can generate Table 2
by first writing down a row of l's, 21 + 1in number; additional rows are then constructed hv s i m ~ l taking
v
the sum of the
21 + 1entries in the row abo;e and to the Gft of the entry
desired (nonexistent entries being assigned values of zero as
in Pascal's triangle), until the row number is equal to the
number of equivalent nuclei in the moiety under consideration. These results can also he applied to the analysis of the
splitting of ESR lines by n equivalent nuclei.
The author has not given a general proof of this rule, hut
offers it as a convenient way of summarizing the results
obtained in all the cases tried.
Acknowledgment
The author wishes to express his gratitude to his late
colleague Boris Franzus, who suggested the problem and
enthusiastically greeted this solution.