### Relationships

A system of 3 different protons has 12 lines. The same system, made of ²H nuclei, has 27 lines. Both systems can be described with 3 shifts and 3 coupling constants. That's obvious. Add a common line-width: 7 parameters to describe 27 or 12 lines.

There are many other "obvious" relationships that often reduce the degree of freedom of the problem. The group CH₃-CH-CH₃ contains many more hydrogens, but is described by only 2 or 3 parameters (because of the magnetic equivalence). A para- (or ortho-) di-substituted benzene, if the substituent is the same, is symmetric: all parameters are duplicated. Tin is a mixture of isotopes, 3 of them are magnetically active. If you want to simulate the proton spectrum of a tin compound, you have to declare 3 or 4 systems, but their parameters (shifts, couplings), neglecting the isotope effects, are the same. The above 3 examples can be handled by the software. Summarizing:

- magnetic equivalence
- symmetry
- isotopic mixture

with the symmetry-aware equivalent:

(pictures courtesy of www.inmr.net)

While it's important that the software directly implements the listed concepts, it's also useful that it understands additional relations defined directly by the user and specific for his/her problem. The first picture exemplifies the idea. The user renounced to the advantage of symmetry and the table of couplings is large. The number of parameters is still low, because he declared 4 relationships. For example, he declared that the shift of hydrogen C is the same as hydrogen B. This is the simplest form of relation (equality), that is implemented through symbolic parameters.

When are relations really useful? First example: transverse relaxation times. Quite often these simulation are performed assuming that all the nuclei have the same T₂ (same line-width). In a few cases, when one or two nuclei relax faster, you are forced to declare the individual widths. To keep the total number of parameters low, I prefer an intermediate solution: dividing the nuclei in two classes. There are only two parameters for the line-widths. This is possible if the software allows me to declare symbolic values. Excel allows me to fill a cell with numbers, text or expressions, at will, but doesn't understand NMR. I have cited Excel just to make the concept clear. The above picture shows that is possible to use symbols instead of numbers inside an NMR software.

Exploring the same web site you can see that symbolic values have been able to solve all sorts of problems:

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