Most magnetic resonance spectroscopists will invariably have seen Pascal’s triangle (i.e. the triangle of binomial coefficients) in introductory undergraduate courses or school curricula:
Pascal’s triangle is widely used as a pedagogical tool to explain first-order multiplet patterns.
However, magnetic resonance is full of other, less well-known combinatorial structures. One of the most useful is closely related: a Catalan triangle* (so named due to the leftmost columns giving the Catalan numbers):
In basic terms, this Catalan triangle (which adds up just like Pascal’s triangle) provides the distribution of eigenstates in a symmetric N-spin-1/2 system, immensely simplifying the treatment of multispin problems. For example, an A2 spin system such as H2 can be treated as a sum of 1 spin-0 (singlet) and 1 spin-1 (triplet) particle. An A4 spin system such as CH4 can be treated as a sum of 2 spin-0, 3 spin-1, and 1 spin-2 particles:
Deep in the annals of NMR theory, the coefficients of this Catalan triangle are also known as the “Clebsch-Gordan multiplicities“.
The old NMR texts referred to the above decomposition as the composite particle method showing that this was a much easier way to treat multispin systems than going Rambo and invoking symmetry groups, character tables, molecular rotation, and what have you. The simple triangle itself appears in some form in early references such as Grimley’s paper (1963) and Corio’s famous book (1967):
It is rather unfortunate that in today’s literature, there is an apparent insistence on obfuscating such basic combinatorial results by invoking unnecessarily complicated, apocryphal chains of group theoretical arguments – behavior that early quantum physicists would have called gruppenpest.
A beautiful recent example of the appearance of the coefficients of this Catalan triangle in an unexpected context is this excellent paper, which rigorously explores weighted multiplets; for example, the intensity ratios of the simple z1 multiplets (generated by the INEPT sequence) are given by a “mirrored” Catalan triangle:
Historically, this Catalan triangle can be traced back to Wigner’s timeless book (1959) (where it appears as an expression) and the classic combinatorics paper of Forder (1961), although there are surely older, less relevant appearances in the literature.
*Note: unfortunately, due to the ubiquity of Catalan numbers in mathematics, “Catalan triangle” and “Catalan’s triangle” have been used in the literature to refer to several different number triangles, generating immense confusion. Since there are many Catalan triangles, care should be taken with terms such as “Catalan’s triangle” or “the Catalan triangle” without an appropriate reference, and one should use terms such as “a Catalan triangle” or simply “triangle of Clebsch-Gordan multiplicites”. I personally like “Catalan-type triangle”.
†There are some pretty results, such as the number of spin-0 (singlet) states in a (2N)-spin-1/2 system being given by the N-th Catalan number. This is also true for the number of spin-1/2 states in a (2N-1)-spin-1/2 system.
2 thoughts on “Catalan Triangle and Clebsch-Gordan Multiplicities”