The Use of Symmetry in Nuclear Magnetic Resonance

Conference paper
Part of the NMR Basic Principles and Progress / Grundlagen und Fortschritte book series (NMR, volume 1)


It has been the constant quest of the chemist to simplify the calculation of molecular parameters and to understand the theory underlying such calculations. The recognition of symmetry in its broadest sense and its relationship with group theory bring to hand information essential to the understanding and execution of fundamental calculations at a personal level. This approach also leads to a fuller comprehension of the results derived from the computer facility now available to all.


Nuclear Magnetic Resonance Symmetry Group Irreducible Representation Permutation Group Symmetry Operation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Glossary of Symbols

[1], [20]

References, italicised.

[1], [57]

Abbreviated notation for spin-wavefunctions which locates the ß’s.

2 [1] – [2] – [3]

Linear symmetrised combination of spin-wavefunctions, not normalised.

(123), (57) (68)

Permutations (1 → 2 → 3 → 1), (5 ↔ 7) (6 ↔ 8).

A, B, X, ...

Symmetry elements or operations.


Identity operation.


Permutation, (12)


Inversion of all particles through a centre of mass.


Permutation inversion (12)(45)*.

A, B, E

Symmetry species or irreducible representations, italicised.


Representation of the group.

Cnv, Dnh, G

Symmetry Point Groups, italicised.


Permutation Point Group, italicised.


Plane of symmetry.


Rotation axis of symmetry, (bold sub-script).


Centre of symmetry.


Rotation-reflection axis of symmetry, (bold sub-script).


Order of mathematical group.


Dimension of the ith representation = order of the matrices which constitute it.


Class of mathematical group.


Number of spin-wavefunctions of a given set which remain unchanged under operations in class C.


Number of operations in class C.

\( \left. {\begin{array}{*{20}{c}} {\chi _c^{(\gamma )}} \\ {\chi _p^{(\gamma )}} \\ {\chi {{(P)}^{{\Gamma _i}}}} \end{array}} \right\} \)

Symmetry group characters, corresponding to the class C or permutation P and the symmetry species γ or the representation Г i.

vA, va

Larmor frequency and effective Larmor frequency.

JAB, Jab

Scalar coupling constant and effective coupling constant.


Direct coupling constant.

I2, IX, IY, IZ, Ii, Ij

Spin operators.



A3B2, AA′XnXn

NMR spin system notation, primes denote magnetic nonequivalence.


Square brackets denote magnetic non equivalence.

ab, aa′bb′

NMR sub spectral notation (primes as above).

φ, ψ

Spin wavefunctions.


Magnetic quantum number, eigenvalue of I z .


Total magnetic quantum number, sum of m taken over a complete system.

m(XX′), mX

Sum of m taken over a group of nuclei.


Total spin quantum number, associated with I2.


Energy level.


h/2 π Planck’s constant over two pi unit of angular momentum.

\(\vec r, \vec v\)


\(\vec i, \vec j, \vec k\)

Unit vectors.


Cartesian co-ordinates.

M1, M2

Matrices, italicised.


Elements of a matrix in the ith row and jth column.


Unit matrix.


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Copyright information

© Springer-Verlag Berlin Heidelberg 1969

Authors and Affiliations

  1. 1.Chemistry DepartmentUniversity of EssexColchesterEngland

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