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Background: I am a mathematician with very little knowledge in chemistry (I do know some physics). I will probably teach a course "mathematics for (1st year) chemistry students" in the future (1st semester 3h + 1h exercises per week, 2nd semester 2+1h). This will be the only (mandatory) mathematics course in the chemistry Bachelor curriculum (and even the subsequent Master curriculum, as far as I know). The students should have high school mathematics background, such as onedimensional derivates and integration (at least of polynomials), but probably no complex numbers. I will have a lot of freedom in the selection and presentation of topics.

Questions:

  1. Which topics are commonly covered in a "mathematics for chemistry students" courses?

    Closely related: Which textbook are most commonly used?

In more detail:

  • Can I safely assume that mathematical proofs and even rigorous definitions can/should mostly be skipped?

  • How black-boxy is Quantum physics used in the typical chemistry curriculum?

    Even the most basic Quantum physics will need, as minimum, some linear algebra (Hermitian, normal and unitary operators, exponention of operators, etc.) as well as Fourier transformation and a little bit of differential equations. Should these topics be coverd, or is it enough to say "one can show that in an H atom, orbitals look like this", and just talk a bit about complex valued functions in R^3?

    Do you ever do calculations with spins? (Mostly spin 1/2, I would assume? That might be a motivation to show some calculations with complex 2x2 Matrices?)

  • Which computer mathematics software are most commonly used in a chemistry curriculum? (Mathematica? Maple?) I assume it would make sense to do some calculations and exercises using such software?

Update answering some comments:

  • I am in Austria (sorry, do not know how to add an according tag), but I would be interested in how these things are handled in other countries as well.
  • This course has been held before at our department (not by me), but I would like to get "unbiased" input from people from chemistry, and therefore did not list the traditional topics / lecture plan.
  • Of course I will speak to people in the Chemistry department, but I would be happy to get some ideas here first (I do not know the Chemistry department people, and it could be that a randomly selected person who happens to be willing to indulge me has some unorthodox views).
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    $\begingroup$ There is a plethora of textbooks ready to use: Francis' Mathematics for chemists (1984); Barrante's Applied mathematics for physical chemistry (1998); Póta's Mathematical problems for chemistry students (2006); Cockett&Doggett's Maths for chemists (2012); Mortimer's Mathematics for physical chemistry (2013); Kerber's Mathematical Chemistry and Chemoinformatics (2013) and many more. No need to reinvent the wheel, really. As for the software, I strongly advice against using proprietary products such as MathLab or Wolfram Mathematica in introductory courses. Teach them R/Python. $\endgroup$
    – andselisk ♦
    May 28, 2021 at 12:05
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    $\begingroup$ I am sorry, but this is off-topic for Chemistry.SE. Yes, your target audience is chemists, and this place has the most chemists; but Chemistry.SE is for questions about chemistry, not how to teach chemistry, or how to teach maths to chemists. Have you considered Mathematics Educators Stack Exchange? I think I can migrate the question over there; and I think it would be the best to have it migrated, and to put a link at the top of your post pointing people there so that they can contribute if interested. You are also welcome to post in Chemistry Chat. $\endgroup$
    – orthocresol
    May 28, 2021 at 12:36
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    $\begingroup$ @orthocresol I disagree. This question is really about what level of maths is required to study chemistry, so this isn't off-topic for chemistry. Maybe it's more suited for other sites, but it should not be off-topic here $\endgroup$
    – S R Maiti
    May 28, 2021 at 12:44
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    $\begingroup$ Despite providing an answer, I have to agree that the people whose input is most important is the chem dept professors. What do they want students to know for their classes so they don't have to waste time teaching math instead of chemistry? That you don't know them is troubling, as this class should be integrated closely with their department curriculum. $\endgroup$
    – Andrew
    May 28, 2021 at 14:15
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    $\begingroup$ @MaxW Even just a few years ago, I had a first year university chemistry class student, not a chemistry major, who could not multiply 6 times 7. They multiplied 3 times 7 to get 21, then added 21 to that to get 42. They did this in front of the whole large class, whereupon I dumped the use of a projector-linked tablet modality (for “guide on the side” teaching) and reverted to giving lectures. This eliminated inadvertently embarrassing a student. $\endgroup$
    – Ed V
    May 28, 2021 at 19:34

3 Answers 3

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At least in the US, most undergraduate chemistry curricula do not go deep enough into the math of QM that you need to spend a great deal of time on it. Students that do take more advanced QM classes can learn the math at that time. I also am not sure it's safe to assume that all incoming students at that level have a good understanding of 1D calculus, though most will have had some exposure.

With that in mind, here are topics that I'd include (not in any specific order), structured around the relevant applications:

  1. basic calculus review of integration and differentiation in 1D - main intro level application is chemical kinetics. Also applies to intro quantum.

  2. setting up and solving systems of equations, especially including iterative solving methods - intro level application is solving systems of multiple equilibria by mass and charge balance

  3. a module on statistics with a focus on dealing with experimental error and error propagation. Include methods of curve fitting and associated pitfalls. - applies to lab component of curriculum. Probably one of the more commonly overlooked aspects of math in chemistry

  4. Fourier transform and other techniques involved in spectroscopy

  5. Basics of symmetry operations and group theory

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    $\begingroup$ This shows that on topic or not at least OP should have told what is that curriculum in chemistry. I am in Academia since decades and met scientists from all over the world but I have no pale idea of what truly a bachelor is. $\endgroup$
    – Alchimista
    May 28, 2021 at 13:22
  • $\begingroup$ I would add eigenvectors/eigenvalues to the list. We also got to learn some solving of differential equations. $\endgroup$
    – TAR86
    May 28, 2021 at 16:50
  • $\begingroup$ Somewhat surprised multivariate calculus is not there. Year 1, Day 1, lecture 1 for me (UK, mumble, mumble years ago) was classical thermodynamics, and we were getting into exact differentials within next to no time $\endgroup$
    – Ian Bush
    May 28, 2021 at 21:57
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    $\begingroup$ @IanBush - Here in the US, thermo is typically presented in first year undergraduate general chem classes with very little calculus and what calculus there is is univariate. For example, see the Principles of Modern Chemistry textbook by Oxtoby and Nachtrieb. That may be because so many students enter college without having taken calculus at secondary level. They may be required to take it during the first year, but that doesn't help them for first-year chem classes. $\endgroup$
    – Andrew
    May 29, 2021 at 11:20
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    $\begingroup$ @IanBush - To clarify, this is thermo as presented in first-year general chem classes not a more rigorous physical chem specific class which they might take later. It was taught this way when I took general chem more than 25 years ago, so it's not new. Remember that US secondary education is one year shorter than UK and without specialization (nothing similar to A level) and not known for being particularly high quality. Only the more advanced students get decent calculus classes before entering university. $\endgroup$
    – Andrew
    May 29, 2021 at 12:06
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The list of topics will be infinite (if that is allowed in mathematics) if you keep on thinking what chemistry students must know. This should not be the goal of this course to superficially glance over all the possible advanced and superadvanced topics which a chemist may encounter in his MSc/PhD.

We should keep in mind that these are first year students and many students would take separate mathematics / physics courses, and of course, many will not even study chemistry after their bachelors. So, the best thing is to avoid repeating the same dry topics which they will encounter again and again. It is good that you have a lot of flexibility it would be if you make this course highly application oriented. Complementing Andrew's answer...

The most important thing to teach to chemists is "mathematical common sense" in the beginning, which means developing an intuition when they get wrong answers.

(a) Rigorous concept of dimensional analysis and it's requirements. This paper would be a great introduction for first two three classes "Can One Take the Logarithm or the Sine of a Dimensioned Quantity or a Unit? Dimensional Analysis Involving Transcendental Functions", Journal of Chemical Education, 2011, 88, 1, 67–70 (link).

(b) Concepts of linear and non-linear curve fitting, types of graphing (ideally manual on a graph paper and then on computer). Reading the axes on log-log, semi-log, linear axes. Concept of extrapolation and interpolation, different coordinate systems (spherical, polar, Cartesian etc.)

(c) Differential equations relevent to second and third year quantum chemistry (simple harmonic oscillator, H- atom) and rate laws.

(d) Matrix methods to solve a system of linear equations in 3 or more variables.

(e) Obtaining approximate solutions to higher order equations (quadratic, cubic, quartic). They will need them in solving chemical equilibrium. Iterative approximations

(f) Continuous and discrete Fourier transform relevant to chemistry. This would be handy in their third or fourth year spectroscopy.

(g) Numerical differentiation and integration on discrete data sets

(h) Excel based exercises to remove noise from the data which could include moving averages, weighted moving averages, Savitsky-Golay, windowing operations with discrete Fourier transforms.

Given that you are in Austria worth checking the following German classic books.

  1. "Rechnen in der Chemie" von Dr. techn. Ing. Walter Wittenberger: First and Second Parts. It is a nice two volume set. (link)

  2. "Physikalisch-chemisches Rechnen mit einer Einführung in die höhere Mathematik" von Walter Wittenberger und Werner Fritz (link)

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    $\begingroup$ For b) and f) I would add O'Haver's project for that he shows the background and application in a true programming language, as well as (where plausible) in a spreadsheet program. But work in the the later often is more difficult to replicate for errors easy to commit (ex 1, ex 2) and sometimes not due to the user (e.g., McCullough's papers about Excel vs. gnumeric ex 3). $\endgroup$
    – Buttonwood
    May 28, 2021 at 21:18
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    $\begingroup$ Yes. O'Haver is a good friend. He has written a book on Signal Processing which he has kindly made available for free as a pdf. Amazon is selling that book for > $600 which is a rip off. It is not him. $\endgroup$
    – AChem
    May 28, 2021 at 21:23
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    $\begingroup$ @Buttonwood Ah, M. Farooq is too modest: he does not mention that he has co-authored several papers with Tom O’Haver! ;-) $\endgroup$
    – Ed V
    May 28, 2021 at 21:32
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Good luck! Give no formal proofs. Teach only with chemistry examples, say chemical kinetics and simple quantum for differential equations, spend 1 hour teaching and 2 hours student working in groups solving problems. (I have done this and it is the only way that works). Remember that chemists are not mathematicians and only want to know how to solve problems. As these students do lots of practical work they need to know how to use software to do least squares fitting and error analysis. They also need to understand fourier transforms as these are the basis of x-ray diffraction and nuclear magnetic resonance methods, both fundamental knowledge for any chemist. These can be taught largely but not exclusively pictorially. As I said Good Luck!

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