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Looking online I seem to have found 2 different definitions for chemical shift $\delta$ (in the context of NMR). The first definition uses the chemical shift of the reference in both the numerator and denominator: $$\delta = \frac{\nu - \nu_{ref}}{\nu_{ref}}$$ This can be found on the wikipedia page here: https://en.wikipedia.org/wiki/Chemical_shift

The second uses the frequency of the spectrometer as the denominator, i.e. $$ \delta = \frac{\nu - \nu_{ref}}{\nu_{spec}}$$ This can be found, for example, here: https://chem.libretexts.org/Courses/Athabasca_University/Chemistry_350%3A_Organic_Chemistry_I/13%3A_Structure_Determination-_Nuclear_Magnetic_Resonance_Spectroscopy/13.03%3A_Chemical_Shifts

Now, I am aware that the "frequency" of the spectrometer is really just an approximate frequency at which protons tend to resonate in that field, and so these numbers would be roughly similar. My question is, is there an actual difference between the two (which is largely ignored because they give similar values) or are they somehow identical?

Anyways, thanks in advance.

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First, let's be careful about what the quantities refer to. The reference frequency, of course, refers to the frequency of a reference compound. In the case of $\ce{^1H}$ or $\ce{^{13}C}$, the reference frequency is given by the methyl groups of tetramethylsilane.

The spectrometer frequency, however, could be considered slightly ambiguous. It may either refer to the carrier frequency, which is placed in the middle of the spectral window. If your spectrum runs from $\pu{0 ppm}$ to $\pu{10 ppm}$, then the carrier frequency is $\pu{5 ppm}$, or whatever that corresponds to in terms of $\pu{MHz}$. On the other hand, the spectrometer frequency itself usually refers to the frequency that corresponds to $\pu{0 ppm}$. (In TopSpin, these two parameters are respectively called SFO1 and BF1; this link contains more explanation about these parameters.)

Given that the reference frequency is defined to be at $\pu{0 ppm}$, it follows that the reference frequency is actually the same thing as the spectrometer frequency, which—in a strict sense—makes this question trivial to answer: it can be either one, since they're both the same thing.*

Officially, the definition is with respect to the reference. As per the IUPAC Gold Book:

chemical shift, $\delta$ (in NMR)

The fractional variation of the resonance frequency of a nucleus in nuclear magnetic resonance (NMR) spectroscopy in consequence of its magnetic environment. The chemical shift of a nucleus, $\delta$, is expressed as a ratio involving its frequency, $\nu_\text{cpd}$, relative to that of a standard, $\nu_\text{ref}$, and defined as:

$$\delta = \frac{\nu_\text{cpd} - \nu_\text{ref}}{\nu_\text{ref}}$$

The above equation assumes that the numerator is measured in $\pu{Hz}$, the denominator in $\pu{MHz}$, and the chemical shift in $\pu{ppm}$. (This implicit convention is a bit annoying, but when actually doing NMR spectroscopy it comes quite naturally.)

Why not the carrier frequency, though? The carrier frequency is also proportional to the field strength, so it has no bearing on the argument that the chemical shift should be independent of field strength. I'd argue that it doesn't make sense to use that because that's a parameter that you can change, by varying the centre of the spectral window. If you acquire two different spectra with different spectral windows, it doesn't make sense that the chemical shifts should differ between the two of them.

On top of that, we can motivate this choice by comparing the chemical shift formula against the expression for the relative error of a measurement: the relative error $\eta$ is given by

$$\eta = \left| \frac{x - x^*}{x^*} \right|$$

where $x$ is the measured value and $x^*$ is the true value. Note that the denominator contains the quantity that we are trying to compare it against. In the case of the chemical shift, it is not an error that we are trying to measure, but rather a "relative deviation" from the reference frequency. It is logically consistent, therefore, that the denominator contains the quantity that we are trying to compare against.


Footnote

* Philosophically, however, they're different things. For a given magnetic field strength, the frequency of TMS is a physical constant. That constant is the reference frequency. However, when calibrating a spectrometer, you'll need to determine what that reference frequency is, and then set the spectrometer frequency to that number. If you don't determine it accurately, then the spectrometer frequency will be off (and the resulting spectra will be shifted by a little bit).

    Using an analogy: let's return to 2018, before the redefinition of the kilogram. At that time, the reference kilogram was defined by a block of metal; so you could say that the reference mass was the mass of that block. This is a quantity that is a fixed constant, and can't be changed. On the other hand, if I had to calibrate a new scale, the scale mass might be said to be the mass that makes the scale show $\pu{1 kg}$. It follows that the scale mass should be the same as the reference mass, for a well-calibrated scale. However, for a poorly-calibrated scale, this may differ.

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  • $\begingroup$ So if you have a signal at 1.000000 ppm, and you choose the wrong denominator, it would come out as 0.999995 ppm ? $\endgroup$
    – Karsten
    Oct 7, 2021 at 15:46
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    $\begingroup$ @orthocresol The 2001 recommendations in PAC recall that the 1972 rules included a factor $\times 10^6$ for proton NMR (eq. 4), then define $\delta$ as in the first equation by the OP (eq. 5) as more general (and open to ppb «for some isotopic effects») and show one to calculate $\delta$ in ppm in [Hz] (numerator) and [MHz] denominator. IUPAC 2008 then only shows the first equation of the OP (eq. 1) in lines of «"ppm" is exchangeable with $\pu{\times 10^{-6}}$ [...] as is "%" with $\times 0.01$». $\endgroup$
    – Buttonwood
    Oct 7, 2021 at 15:57
  • $\begingroup$ @Buttonwood Thanks for that. Just to be clear, do you just mean to clarify the missing factor of $10^6$ (which I admit, I didn't even think about!), or there something more than that? $\endgroup$ Oct 7, 2021 at 16:01
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    $\begingroup$ @orthocresol Preparing an answer, I encountered this Hz/MHz division in one of the equations. Maybe this is a reason why we find the computation of $\delta$[ppm] with division over spectrometer frequency (in MHz) $\times{} 10^6$. But it is not about an error in your answer I'm fine with. $\endgroup$
    – Buttonwood
    Oct 7, 2021 at 16:08
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    $\begingroup$ @KarstenTheis Hmm, thinking about it a bit more, I suspect the numerator plays a larger role. I reason about it this way: the denominator is a large, positive number anyway, so hardly matters whether it's 699.90 or 699.95 or 700.00. The numerator, however, refers to a frequency difference (i.e. difference of two large numbers) which can range from anywhere between, let's say, 0 Hz to 20000 Hz. Shifting the reference frequency (or recalibrating) will affect the denominator by a tiny fraction, but the numerator may shift by quite a bit. $\endgroup$ Oct 7, 2021 at 16:17

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