Spin System: AB2, General Information

Appearance of an AB₂ NMR Spectrum

From refs. [2,3], "BPS" expressions:

• This spin system may baffle people that expect their spectrum to show the simple appearance of an AX₂ spin system, i.e. a triplet and a doublet, but observe a multiline spectrum instead.
• For reasons of symmetry, the AB₂ spin system has J_AB = J'_AB; the coupling J_BB is not observable.
• The spectrum of an AB₂ spin system consists of 8 (9) peaks. The ninth peak, a combination line, is usually too weak to be observed; however, it can reach about 1.5 % of line 3 for Δν/J around 0.9.
• The analysis gives J_AB and the chemical shifts of A and B. The appearance of the spectrum does not depend on the sign of J_AB, although the detailed line assignment to specific transitions will be different.
• The analysis starts with labeling the peaks from 1 to 8 (9), with 1 starting in the A part. If A is to higher frequencies from B, the figure and order of labeling are reversed.
• The assignment can be verified by checking the following relations:
   
  
  • if the relative intensity of transition 3 corresponds to 1, the total sum of intensities should be 12 (intensity sum rule)
  • the intensities (integral) of B should be twice that of A
  • the separation (ν₁ - ν₃) can be found three times in the spectrum (property of repeated spacings):
    (ν₁ - ν₃) = (ν₂ - ν₄) = (ν₅ - ν₈) = J_AB + M'₊ (in the Garbisch [1] nomenclature)
    (broken green lines in the picture above; so far, using the BPS expressions I managed to show that the 2nd and 3rd splitting are equal, but numerically it appears that all three are identical, indeed, cf. the simulation report)
  • the separation (ν₁ - ν₂) can also be found three times in the spectrum:
    (ν₁ - ν₂) = (ν₃ - ν₄) = (ν₆ - ν₇) = J_AB - M'_- (in the Garbisch [1] nomenclature)
    (broken red lines in the picture above; so far, using the BPS expressions I managed to show that the 1st and 3rd splitting are equal, but numerically it appears that all three are identical, indeed, cf. the simulation report)
• Once the lines are assigned correctly, the three fundamental parameters can be evaluated:
   
  
  ν_B = (ν₅ - ν₇) / 2
  ν_A = ν₃
  J_AB = (|ν₁ - ν₄| + |ν₆ - ν₈|) / 3
• Bovey [3] states that J_AB is not given directly by any of the spectral spacings. It is also true that J_AB is not manifested by repeated spacings (as in the AX₂ case). However, one can show that the sum of the two spacings |ν₁ - ν₄| and |ν₆ - ν₈| corresponds to 3 J_AB.
• The above expressions for the transition energies and intensities are used on the page on
  AB₂ spectrum simulation.
• The analysis of an AB₂ NMR spectrum can be performed interactively on the page on
  AB₂ spectrum analysis.

Additional Information on the AB₂ NMR Spectrum

• A series of nice examples is featured on Hans Reich's website.
  Also, his software WINDNMR is a nice teaching tool for several spin systems (and runs well under Wine on Linux). A successor is in the making: UW-DNMR.

References

1. Garbisch, E. W., Jr., J. Chem. Educ. 1968, 45, 402-416, DOI: 10.1021/ed045p402. The equations for the transition frequencies and intensities look more attractive for an implementation to calculate spectra, but they contain errors. The calculations here use the expressions from [2] and [3].
2. Bernstein, H. J.; Pople, J. A.; Schneider, W. G., Can. J. Chem. 1957, 35, 65-81, DOI: 10.1139/v57-011
3. Bovey, F. A.; Jelinski, L.; Mirau, P. A., Nuclear Magnetic Resonance Spectroscopy, 2nd ed., Academic Press, San Diego, 1988, DOI: 10.1016/C2009-0-21711-9