Category: Combinatorial Aspects of Magnetic Resonance

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The Quantum Pascal Pyramid

Posted on by Mohamed Sabba

In the previous blog post, I discussed a close relative of Pascal’s triangle – a Catalan triangle – and briefly alluded to z1 multiplets at the end.

In this post I will discuss another less well-known (but very useful!) combinatorial structure – the Pascal (difference) pyramid, which has a beautiful correspondence to the multispin (single-quantum) coherences observed in NMR experiments.

Suppose one has an INS (or AXN) spin system. Begin by considering the following cumulative tensor product:

The expansion may be organized by the spin product rank q, yielding a convenient operator basis of I-spin longitudinal (z-) spin-operators which Sorensen denoted ZNq. Some examples for the cases N=3 and N=4 are:


In my mind, I like to call these “the binomial Z-operators” since the number of operators forms binomial patterns much like NMR spectra (1:3:3:1, 1:4:6:4:1, etc.) for reasons I hope are obvious. (If it isn’t obvious, consider the basic combinatorial problem: How many unique words with length q can be formed by combining N different letters?)

Now, suppose you wanted to observe single-quantum coherences involving a product of any of these ZNq operators with, say, Sx. What would the S-spin spectrum look like? Some visual examples are given here:



Intuitively, one can reason that each ZNq.Sx operator has to be a sum of the “pure” single-transition operators corresponding to each m=J×{-N/2,…,+N/2} line component making up the S-spin spectrum. One might notice that each ZNq operator changes sign exactly q times. But it’s not so clear to see how.

It turns out there is an incredible direct map between the operators describing pure populations of states with azimuthal quantum number m (which I will denote PNm) and ZNq . The relationship is given by:

The map can be represented by an (N+1)×(N+1) matrix. Some matrices illustrating this relationship are shown below:

Naturally, this type of relationship has appeared in other fields of quantum mechanics (see equation 11). But to my knowledge, the only magnetic resonance paper which describes something resembling a “Pascal’s pyramid” is the excellent paper on relaxation in the AX4 spin system of 15NH4+ by Nicolas Werbeck and D. Flemming Hansen, where the combinatorial structure is referred to as a “modified Pascal triangle”.

EDIT (February 23, 2024, 15:56 UK time): after writing this article, I realized something neat. Recall the de Moivre-Laplace theorem, which is the famous mathematical result that the binomial distribution (corresponding to the multiplets associated with ZN0) converges to a normal (Gaussian) distribution as N→∞. I have observed a generalization of the de Moivre-Laplace theorem: that the columns of Pascal’s pyramid, i.e the multiplets associated with ZNq, converge to the q-th derivatives of a Gaussian distribution. The q-th derivative of a generic Gaussian function has a general form that can be written in terms of a regularized hypergeometric function:




That is to say, the ZNq operator may be approximated by a q-th order Gaussian derivative:

We know from basic mathematics that a derivative is a measure of a rate of change. We can appreciate the corresponding physical picture in more than one way:

1. each ZNq operator transforms under, say, an I-spin x-rotation of a flip angle β, with an increasing “responsiveness” to rotation that depends on the spin product rank q. The self-evolution of each ZNq operator would be given by:

2. each ZNq operator can be converted into I-spin multiple quantum coherence with a (maximum) coherence order q. This well-known property is commonly exploited in NMR experiments. Of course, by definition, an MQC of coherence order q (which I’ll dub qQC for q-quantum coherence) has the following property under an I-spin z-rotation with a flip-angle β:

EDIT (May 16, 2024, 14:20 UK time): I have written this blog post as a small paper on arXiV, which goes into a bit more detail.

Catalan Triangle and Clebsch-Gordan Multiplicities

Posted on by Mohamed Sabba

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.