Uses of Fourier / Laplace transforms in chemistry (apart from spectroscopy)

I'm just now learning about the Fourier transform, which seems like a pretty useful tool, and I know it has uses in spectroscopy (e.g. FTIR) since that is what Google shows when searching for applications of the Fourier transform in chemistry. Interestingly, I found out that the Laplace transform can be used in chemical kinetics.

I was wondering if there were any other cool uses for the Fourier/Laplace transform in chemistry? And if so, any links to resources to learn more about that particular use would be appreciated.

• (+1) Fourier transform techniques are used extensively in signal processing (check out the signal processing stack exchange for examples) and chemists use various signal processing techniques for things like resolution enhancement of chromatograms. I predict that you will get answers or comments that are informative in this regard.
– Ed V
Apr 2 at 21:12
• I don't have the knowledge to go much deeper, but in molecular dynamics simulations Fourier transfroms are used to calculate the electrostatic interaction in periodic systems. For more details see en.wikipedia.org/wiki/Ewald_summation Apr 2 at 23:24
• Generously used in crystallography and for discussing the electronic properties of crystals or other semicrystalline nature, e.g. solids with repeating or symmetrical subunits. Apr 3 at 7:48
• It is essential to NMR, EPR, and xray diffraction from crystals, as well as IR /Raman spectroscopy. It is not used in kinetics but various maths transforms may be used in solving these equations and also in solving quantum equations. The Hadamard transform can be used in pump-probe chemical kinetics. Apr 3 at 9:18

For the principle «what is small, gets large; what is large, gets small» when applying Fourier transformation and inverse Fourier transformation, it is one important tool in crystallography to relate between the regular arrangement of molecules in crystals at Angstrom scale, and the intensity recordings on film (old school) and electronic detector (very often area CCD) where signals are much wider apart from each other. (This is often is referred to the relationship between direct and reciprocal space, too.) Thus, scrutinizing the diffraction pattern's symmetry from special directions (figure b) below) allows the trained eye to relate to possible symmetries of the crystalline sample (a) , and the ratio of vector lengths describing the smallest unit of a crystal structure, the unit cell. Today, assessing position and intensities of these data is typically assisted by a computer.

(credit, open access)

As shown by Tsutaoka et al. (Eur. J. Phys. 35 (2014) 055021, doi 10.1088/0143-0807/35/5/055021), it is possible to simulate such patterns with a laser pointer passing two film gratings slightly rotated to each other; the 2D crystal is then build by the points where the two gratings S1 and S2 intersect each other (note the vector ratio $$\vec{a} / \vec{b}$$ vs. $$\vec{a^*} / \vec{b^*}$$, and angles enclosed), e.g.

ImageMagick is one program allowing you to experiment with FT on 2D images but recommend to consult it altogether with the above mentioned open access publication by Aubert and Lecomte (J. Appl. Cryst. (2007) 40, 1153-1165, doi 10.1107/S0021889807043622) for additional examples and background.

• I still think it is absolutely magic to remember the TEM transmitted beam visual, (starmap-like) set the aperture to just one particular "star" then flip the magic switch to get a heat map of the structure where just that one phase lights up and the rest are dim. Absolute witchcraft. Apr 5 at 15:51

The Fourier Transform is important in many applications beyond spectroscopy, for instance in computational and theoretical chemistry. One example encountered in learning chemistry is in quantum theory. The FT is intimately related to the Heisenberg Uncertainty principle (for instance for position-momentum). You might want to look at a libretext on the subject.

As implied in other answers, a practical utility of the FT lies in revealing or exploiting regularity in data.

When evaluating integrals in some computational QM problems, application of the FT can provide a time-saving device. See for instance Ref 1, a thesis detailing this.

A very important example of its application in computational chemistry is in computing the Ewald summation:

Ewald summation, named after Paul Peter Ewald, is a method for computing long-range interactions (e.g. electrostatic interactions) in periodic systems. It was first developed as the method for calculating electrostatic energies of ionic crystals, and is now commonly used for calculating long-range interactions in computational chemistry. Ewald summation is a special case of the Poisson summation formula, replacing the summation of interaction energies in real space with an equivalent summation in Fourier space. In this method, the long-range interaction is divided into two parts: a short-range contribution, and a long-range contribution which does not have a singularity. The short-range contribution is calculated in real space, whereas the long-range contribution is calculated using a Fourier transform.

If you want to evaluate a Madelung constant you might also employ the FT.

References

1. Jaime Axel Rosal Sandberg. New efficient integral algorithms for quantum chemistry. Doctoral Thesis in Theoretical Chemistry and Biology, School of Biotechnology, Royal Institute of Technology, Stockholm, Sweden 2014 (open access repository link).
• @BuchThorn If this is the reference you intended, keep the link; if not revoke the edit. For future reference: For various reasons, PhD theses may be more difficult to access, than a publication in a scientific journal. But if there is a public repository of them, please add one permanent link. Apr 3 at 12:39
1. Fourier transform (FT) is simply a bridge which lets us view the data in reciprocal domains. For example, if you collected data with respect to time, FT of this data will allow you to see the same information in the (1/time) domain. In FTIR, you collect an interferogram with respect to distance (length units), FT of this data yields information in (1/length). That is why your FTIR x-axis is in wavenumbers (cm$$^{-1}$$).

2. The second advantage of FT is that simplifies many calculations into simple multiplication or division.

You should explore a little bit more about discrete Fourier transform (DFT). That is what is used in spectroscopy and analytical chemistry. FT is for continuous function, DFT is for sampled data. The most important thing to note is that there is no loss or gain of information before or after (D)FT of any data.

FT finds extensive use in denoising analytical chemistry data. All instrumental signals have noise to some extent. In most cases, the signal is present at low frequencies and noise has higher frequencies. You can cut-off high frequency noise and do an inverse transform to get a signal which has higher signal to noise ratio.

FT techniques can resolve overlapping peaks if you know what broadened the signal. Visit Prof. Tom O'Haver website or see his book (free pdf). Window 1 (top) is the broadened signal which could be a chromatogram or a spectrum, Window 2 is a function which caused broadening of the peaks. Window 1 (bottom) is "recovered" signal.

https://terpconnect.umd.edu/~toh/spectrum/Deconvolution.html

FT is also required in determining X-ray crystal structure.

• "You can cut-off high frequency noise and do an inverse transform to get a signal which has higher signal to noise ratio." I haven't heard anyone using this technique in practice. I played with this approach a bit and it wasn't satisfactory: either not much smoothing or the signal starts to look wavy. "there is no loss or gain of information before or after (D)FT of any data" - well, except for the noise added due to rounding by computers :) The more calculations the higher the error. Though it's probably nothing relative to the chemical and electronic noise. Apr 4 at 18:14
• Hi Stanislav, I am afraid you are not doing this correctly. I routinely use DFT to denoise analytical data and it is very common in signal processing. It is called "windowing" in the literature. If you want post a separate question with details, and I will try to address it. The loss in info due to computers is negligible for all practical purpose until and unless someone is doing theoretical calculations. Apr 4 at 20:15
• I've found articles on windowing with FFT, thanks. Will dig deeper into the subject someday when I get back to it. Out of curiosity: why do you use FT instead of Savitzky-Golay or Moving Average? They seem to be the default choices for smoothing in analytical chemistry. At least I hear about them most. Apr 4 at 20:27
• People run away from DFT because it is not as simple as moving average or Savitsky-Golay. I have written a long review on DFT applications for chromatography/analytical chemistry with detailed steps. Hopefully it should be out in 2-3 months, if no major problems occur. Apr 4 at 20:30
• Cool. Would you be able to post a link here once it's available? Apr 4 at 20:31

Laplace transforms are used all the time in Chemical Process Control (I'm a chemical engineer, not a chemist). Most (open-loop) chemical processes can modeled as a first order system, possibly with delay. A tank being heated using a heat-exchanger on its input will show the same dynamics as a you see in reaction kinetics.

There are various ways to measure the dynamics of a process. Once you do that, you can characterize them as first-order + delay and use that Laplace process description to get a dynamic model of the process.

Non open-loop processes (whether because they are under control or because they have a natural loop in the process) can also be modeled using Laplace equations.

Forgive me if I get the terminology wrong - I've been doing mostly software and not process control for the last several decades