## Math, Physics, and Chemistry — RVW #3

June 14th, 2007*In his third post, Retread looks at some of the physics and math needed and used by chemists.*

Now that the answer book to Jones has arrived (23 May) I’m going to start over from the beginning, this time doing all the problems. Mathematicians are fond of saying that it isn’t a spectator sport, and that problems must be done to really grasp the material. I’m not so sure that is true for chemistry, but I’ll take excimer’s advice.

I’d gone through the first 300 or so pages of Jones and it’s apparent the two textbooks are wildly different in the way they approach the beginner.

English & Cassidy start out with hydrocarbon structures. Molecular orbitals make their appearance on p. 63 (remember the book only has 442 pages of text). They are introduced as follows: “A somewhat more easily visualized physical picture of a double bond is given by a relatively recent theoretical development known as the molecular orbital theory.” Recall that there are only six three dimensional drawings in the first 100 pages of E&C. Four of them involve orbitals.

Jones drenches the reader in atomic and molecular orbitals using all the graphics at his disposal. One needs to step back in time to realize the mathematical armor the high school graduate possessed in ‘56 entering college. Outside a few of the great academic high schools in the USA (Central High in Philly, Bronx High School of Science) calculus was simply not taught in US public high schools. The thinking was that is was so hard that it would destroy the brain of anyone under 18. There was a giant dose of anxiety on taking it for the first time. Just about all the undergraduates in the math courses I audited had a year or two of calculus in high school and some were exposed to it in the 8th grade.

So junior chem majors back then didn’t have the apparatus to tackle the Schrodinger equation etc. Formal quantum mechanics wasn’t taught to us back then. Even though the department had Walter Kauzmann, who wrote the influential “Quantum Chemistry” in ‘57, and who taught physical chemistry (which we all took), quantum mechanics wasn’t really discussed in the course.

The department did introduce us to quantum mechanical thought. As juniors we were required to read a book “The Logic of Modern Physics” by P. W. Bridgman, written in 1927 in the early heyday of quantum mechanics. I found it extremely irritating. It argued that all we could know was numbers on a dial reporting the results of a measurement. The notion of particle trajectory was to be abandoned, etc. etc. Jones skirts the issue on p. 10 where he talks about the node in the 2S orbital, a place where an electron is NEVER found. An electron following a trajectory as we know it macroscopically could never pass through the node. If you like thinking about such things, I recommend just about anything a physicist named Mark P. Silverman writes. In particular, Ch. 5 “And Yet It Moves: Exotic Atoms and the Invariance of Charge” in his book “A Universe of Atoms, An Atom in the Universe” deals with the issue of the ‘motion’ of an electron in an atom.

I have a friend, a retired philosophy prof from Columbia, who dismisses all biology and chemistry as ‘anecdotal’. The only thing he regards as solid is Godel’s proof. I told him he better hope he’s wrong if he ever gets sick. In a similar vein, Bridgman is as wrong as he can be when it comes to chemistry. Why? Because the theory behind chemistry may have its origin in numbers on a dial, but it gives rise to gazillions of successful predictions about reactions, structure and spectra. Theory is immaterial, but it guides chemists (it’s the old Cartesian dichotomy between flesh and spirit). Following chemistry in a peripheral fashion during my years in medicine by reading what appeared in Nature and Science (on a superficial level), it seemed that the work in gas phase kinetics was just confirming (always good) what we ‘knew’ back from ‘58 – ‘62. The one surprise was the reversal (p. 245) of the acidity of primary, secondary and tertiary alcohols in the gas phase.

**Previous Comments**

- Uncle Al Says:

June 14th, 2007 at 11:51 am Who needs more than LCAO until the Woodward-Hoffmann rules? Will green-as-grass undergrads sweat cross-correlated NOE-determined protein structure problems?Education targets no market. A university is chop suey. Dump it all in, stir it around with some heat, and hope the result is edible to somebody.

End synthesis labs! They say everything post-industrial, socially engineered, limp-wristed, Enviro-whiner America has evolved beyond. Objective qualification is discrimination. Diversity is all about dialectical purity.

- Darksyde Says:

June 14th, 2007 at 1:19 pm I went on a date with a kindergarten teacher, and on this date we ran into one of her 5-year old students, he was reading chapter books and solving for x. He’ll probably be doing calculus by 7. A lot of math genii come from repressive environments where minds are tempered into submission for the greater good of the motherland; it was refreshing to see that this was a sufficient (but not necessary) condition. Alas, there was no followup date.As an undergraduate in general chemistry one of our test questions was to use a sense of intuition for symmetry arguments to derive the molecular orbitals (and energy diagrams) for the H3+ molecular ion. Later in inorganic chemistry we really did learn some symmetry (although the real math behind character tables and group theory was glossed over — I remember asking a question about the product of symmetry operations and getting “I’m just a poor southern boy” as an answer) with a modicum of rigour.

I think perhaps something that chemistry has in common with mathematics is the need to parlay intuition into rigorous derivation, as exemplified by mechanistic evaluations. On the other hand my love for mathematics is because it’s completely useless (I hate PDEs and linear algebra and love topology) and I like chemistry because I need to use my hands, although one could argue the stuff I do is equally useless.

- Ψ*Ψ Says:

June 15th, 2007 at 3:13 am Darksyde: I have nothing but love for linear algebra and other logic-maths. It’s calculus I hate. - Darksyde Says:

June 15th, 2007 at 12:47 pm I’m not sure what that means. Linear Algebra, to me, is pretty applied (as is calculus). “Algebra” is group theory, ring theory, Galois theory, and other topics like vector spaces algebraic topology. “Analysis” “covers” calculus, but from a truly theoretical level, like advanced integration/measure theory (Lebesgue, Stieltjes, Daniell, and Gauge integrals), and generalizations of real number vector spaces like Banach and Hilbert Spaces, properties of manifolds, Lp spaces and fourier transforms, etc.In the end, I never really understood the “process of proof” until about the third quarter of analysis where I had to prove that a particular pathological function had a (Reimann) integral of zero. So in my experience, “calculus” was a very “logic-intensive” process. I think I can still prove the chain rule from first principles and I created a pretty clever (nonstandard) proof of polynomial derivative rules.

Of course it all doesn’t matter, modern mathematics is so convoluted that in the end you need to have a full understanding of algebra to understand analysis, analysis to understand number theory, and number theory to understand algebra.

Which is why I quit. I’m just not good enough.

- Retread Says:

June 17th, 2007 at 2:22 pm I’m not sure that a chemistry site is the place for discussions about math (but I find comments like psi*psi’s and Darksyde’s quite interesting). Paul can tell us to quit if he wishes. I’ll respond to psi*psi first (and obliquely to Darksyde) and wait for Paul.Linear algebra is quite discrete (at least in the finite dimensional case) just like chemistry. I’m sure every chemist at one time or other wonders how continuous math can apply to chemistry. Sure it works when you have Avogadro’s number of things to play with. But Cauchy sequences get as close as you wish to the real number they approximate (actually define), at levels of accuracy way below the Planck length. Probably that’s why the discreteness of quantum mechanics does so well with atoms, molecules and spectra etc. etc.

One of the difficulties with calculus and analysis is that mathematicians grant themselves the ability to see infinity whole — and all at once — not just by approximating it closely. Consider the Bolzano Weierstrass theorem — it says that any infinite sequence confined to a bounded interval of real numbers must contain at least one subsequence which converges to a real number. This implies that you can lop the interval into two parts, look at each part and decide which part contains an infinite # of points, before proceeding further (they both might contain an infinite # of points however, but that doesn’t matter).

It took me long time to realize this. I asked a mathematician and fellow chamber musician about this. She replied that students always had trouble with non-constructive proofs, and that what I really needed to do was “take the Lord into your heart” (exact words) and believe that it could be done. For an excellent discussion of this and the difficulties it gives rise to see pp. 111 – 114 of “Vector Calculus, Linear Algebra and DIfferential Forms” 2nd Ed by Hubbard and Hubbard. This may be why psi*psi hates calculus so.

But Darksyde is quite correct, that all mathematics is one endless set of mirrors, each branch reflecting and enlarging the others. Consider the fundamental theorem of Algebra — every finite polynomial of degree n has n roots (no more no less — but allowing repetitions) in the complex numbers. Although purely algebraic, it can’t be proved without using analysis and seeing infinity whole.

On the other hand, the closest thing I ever had to a religious experience was sitting in the quantum mechanics course (for graduate chemists) and watching quantum numbers magically appear in the recursion relations in the Legendre and Laguerre polynomial solutions to the Schrodinger equation. The relations are required so that solutions of the equations go to zero at infinity. And of course the equations are differential equations with all the calculus, infinite divisibility and analysis they imply. There’s no escape — the discrete emerges — like Venus rising from the seashell — from the continuous

- Paul Says:

June 18th, 2007 at 5:28 am Math is fine by me, though I must admit that I haven’t used pencil calculus in quite some time. Thank goodness for Origin and Mathematica.