# Mathematics in inorganic and quantum chemistry

Is mathematics important to learn concepts in inorganic chemistry, such as point groups, point symmetry, because doing these topics requires one to determine the symmetry of the elements and to visualize 3D (such as determining the symmetry of a water molecule?)

As these concepts are derived from mathematics, I wonder whether taking maths courses would be helpful for me to understand these concepts better and if so which maths course would be most useful? Such as linear algebra, abstract algebra?

• Linear algebra is, in retrospect, the most stupid thing to NOT learn the basics in. Of course one should have gotten this in highschool already. – Karl Dec 7 '20 at 22:00

## 5 Answers

For better understanding of molecular symmetry you indeed need some mathematics, namely, some basics of group theory, but aside from that I doubt that you need some mathematics apart form arithmetic and calculus for general, inorganic, organic, and even physical chemistry.

For quantum mechanics you absolutely need to familiarize yourself with the concept of complex numbers and you have to learn some basic approaches for solving differential equations. For an advanced insight into quantum chemistry you need some linear algebra, by which I mean the theory of vector spaces both finite- and infinite-dimensional, not just finite-dimensional part of it, which, in a sense, reduces to matrix algebra.

P.S. Abstract algebra is nothing but a common name for different areas of mathematics which studies different so-called algebraic structures. Say, group theory studies groups, linear algebra studies vector spaces, etc.

• Then how about complex analysis, will it be useful for quantum chemistry? – wei jit Sep 24 '14 at 14:30
• @weijit, sure! But it is very unlikely that you will learn linear algebra without knowing complex numbers, so I forgot to mention that. – Wildcat Sep 24 '14 at 14:34
• What i meant here are those complex analysis that are taught to mathematics majors, which can take one whole semester to learn. Not those simple kinds of complex numbers that are taught in high school – wei jit Sep 24 '14 at 14:41
• Complex analysis isn't needed for chemistry at all (neither is any analysis), but it helps in understanding calculus formally. For the benefit of others this is the mathematical topic of analysis; it doesn't just mean 'analysing things'. – user7232 Sep 24 '14 at 14:43
• @Wildcat - I'd add in differential equations for quantum, but otherwise, I agree 100%. For inorganic, a bit of group theory and linear algebra are more than sufficient. – Geoff Hutchison Sep 24 '14 at 16:02

My personal advice would be to do as many maths courses as you possibly can, they never hurt and even if they are not directly relevant to your chemistry career then they sharpen the mind. There's no such thing as too much maths, although my personal opinion aside there are two school of thought on this—that of the chemist, and that of the mathematician.

In perhaps crude terms, chemists often use very simplified versions of mathematics which would probably make a professional mathematician cry. Linear algebra is very important for quantum mechanics. This topic doesn't really differ too much (what I mean by that is a mathematician inverts a matrix just the same as a chemist does). Linear algebra is therefore useful regardless of where you learn it from.

A professional mathematician would tell you to learn group theory from a mathematician's point of view, but this can often be highly inaccessible to beginners (new notation to learn, much stricter logic which unintuitively seems less obvious to the untrained). The second point is that it's also not necessary for your chemistry career to learn all of group theory if all you need to do is follow the literature.

That being said, the more maths you learn, the more open your options are. Taking abstract algebra for example means that you are exposed to ideas which may be useful to your chemistry career, but such ideas haven't been realised in the chemistry community yet. Abstract algebra lays the foundations for topological chemistry and other higher study, of which the benefits to chemistry are only starting to be realised.

• In ideal world (where humans do not die) there is indeed no such thing as too much math. In reality life is limited, so one can not learn everything and have to prioritize things. – Wildcat Sep 24 '14 at 14:32
• Although i know mathematics is useful for chemistry, im currently focusing on those aspects of mathematics that have a more immediate applications to chemistry, especially those that would help me to understand advanced chemistry concepts better. Im currently considering complex analysis, will these be helpful for quantum chemistry? – wei jit Sep 24 '14 at 14:33
• Clearly my approach isn't shared, so @Wei jit, if you're interested in QM, take a look at Freidman and Atkin's book on Molecular Quantum Mechanics. You won't understand all of it, but take maths course on the bits you don't until you do! – user7232 Sep 24 '14 at 14:42
• Which are the mathematics course most helpful. (Btw Im currently at Multivariable calculus and i still have 1 year to take mathematics courses before i start taking QM. So i just wonder which maths courses could i still take after i finsish multvariable in this 1 year.) – wei jit Sep 24 '14 at 14:45
• You'll be pleasantly surprised at how far linear algebra alone will take you if I'm honest, at an beginner's level. Multivariable calculus is good. Try and become ultra comfortable with it though so it becomes very natural. Stick with the calculus too. Your course structure should bring it all together, so don't panic. I think you're a way off from doing Calculus of Variations yet but it is something to think about. – user7232 Sep 24 '14 at 14:47

for inorganic chemistry you need arithmetic and stereometry. Group theory may be fancy, but more or less ignorable as long as you don't draw molecular orbital diagrams in complicated hi-symmetry cases (understanding ones that are already drawn for you does not require this)

for physical chemistry (including understanding of equilibrium) you also need differential and integral calculus and linear differential equations are required. In case of statistical physics approach familiarity with series is of use. complex numbers may be of use in differential equations.

for quantum chemistry you also need partial differential equations, linear algebra and function approximation by series (generalized Fourier series). It is highly recommended to be familiar with complex numbers, but technically speaking it is possible to get basic understanding of quantum chemistry with very basic understanding of complex numbers. For advanced applications calculus of variations and basic understanding of numerical methods, including ones for function approximations, integration, and iterative solution of eigenvalue problems are of use. Group theory is optional, it is touched in simplification of cases with symmetry, but other than that is of no use.

I also recommend to consider memory excercises. You'll have to remember a huge amount of assorted, badly systematized facts and names.

• Undoubtedly +1 for memory training! Although, I would say that in chemistry we have to remember a relatively huge amount of assorted, badly systematized data. When one compares this ammount to, say, what they have in biology, it seems quite small in fact. :D That is actually why many years ago I eventually decided to enter chemical university rather then biological one: I hate to memorize a lot of things. – Wildcat Sep 27 '14 at 8:29
• Outside of (retro-)synthetic organic chemistry with its terrible amount of named reactions, chemistry is basically void of badly systemised facts and names. I never even thought about going into biology. :D – Karl Dec 7 '20 at 21:51

For inorganic chemistry and group theory, learn linear algebra and matrix mathematics.

For quantum chemistry, learn how to solve partial differential equations. Depending here on how far and what direction you want to go, you may need either a science approach or a numerical methods approach to the higher-level math methods. At an undergraduate level, a solid course on differential equations should be a good start.

• Oh how about probability and statistics? – wei jit Sep 29 '14 at 17:16
• @wei jit: Not really. Those are good prerequisites for courses that involve data analysis. – Jeffrey Weimer Sep 30 '14 at 0:17

You can follow the key aspects of group theory and symmetry for chemistry without ever taking formal abstract algebra courses.

The key aspects are presented in survey and specialized chemistry texts. Probably all the exercises on different molecules (and IR modes and the like) will actually give you MORE of a feeling for the complexity of physical symmetry than a math first attitude would.

This is not to say it hurts, but you totally don't need math based group theory.

Consider, for instance, Arfken has a remark (in horror) in his advanced math book for physics at the 200+ space groups for crystals. But a crystallographer or solid state chemist finds them interesting. [Note that this is not even an aspect of chemistry being very detailed...millions of compounds. It's straight up from the math itself...could be lattices of twigs for all we care. But the mathematician just hates this kind of complexity. A chemist is used to it since he has to organize families of organic compounds and the like.]

Personally I think if you want to dive deeper, would start with specialized books on group theory for chemistry (will have a little more than the touch in your survey inorganic text). Here there are sort of two schools:

A. Molecular, with emphasis on point groups and vibrational modes (IR). Think basic coordination compounds of metals.

B. Solid state, with emphasis on space groups and crystallography. Think metal oxides or similar.

Of course you can and will have crystalized groups of molecules as well, so both are relevant.

Only after digging deep into some of the chemical based stuff would I maybe consider the need to go back and get into an abstract algebra based development of group theory (probably starting at set theory and involving things like rings and fields that don't even apply to your need).

And by the way, if you get seriously into crystallography, you can spend a lot of time on that. And a group theory mathematician won't have a feeling for how to spot flaws in crystal structures (either those that don't make sense chemically OR those that are little logical flaws of the math, e.g. equivalent structures).

If anything learning more vector calc and tensors (and from an applied, simple perspective...not total math theory killer) would help you more. Even that is basically just needed if you become a crystallographer. An experimental inorganic chemist working on molecules really doesn't need it--all his xtal structure determinations are outsourced. Solid state guys tend to do more of this themselves but even here, a lot of it is playing with programs on the computer and the theory of the x-ray geometry is really not needed. More a careful attitude to look for reasonable chemistry and where solved structures have more uncertainty ("thermal parameters").