Does Current Quantum Mechanics Help Your Chemistry?

The usual approach to the chemical bond is to "solve the Schrödinger equation", and this is done by attempting to follow the dynamics of the electrons. As we all know, that is impossible; the equation as usually presented requires you to know the potential field in which every particle moves, and since each electron is in motion, the problem becomes insoluble. Even classical gravity has no analytical solution for the three-body problem. We all know the answer – there are various assumptions and approximations made, and as Pople noted in his Nobel lecture, validation of very similar molecules allows you to assign values to the various difficult terms and you can get quite accurate answers for similar molecules.

However, you can only be sure of that if there are suitable examples from which to validate. So, quite accurate answers are obtained,*but* the question remains, is the output of any value in increasing the *understanding* of what is going on for chemists? In other words, can they say *why* A behaves differently to a seemingly similar B?

There is a second issue. Because validation and the requirement to obtain results equivalent to those observed, can we be sure they are obtained the right way? As an example, in 2006 some American chemists decided to test some programs that were considered tolerable advanced and available to general chemists on some quite basic compounds. The results were quite disappointing, even to the extent of showing that benzene was non-planar. (Moran, D. and five others. 2006.*J. Amer. Chem. Soc. 128*: 9342-9343.)

There is a third issue, and this seems to have passed without comment amongst chemists. In the state vector formalism of quantum mechanics, it is often stated that you cannot factorise the overall wave function. That is the basis of the Schrödinger cat paradox. The whole cat is in the superposition of states that differ on whether or not the nucleus has decayed. If you can factorise the state, the paradox disappears. You may still have to open the box to see what has happened to the cat, but the cat, being a macroscopic being, has behaved classically and was either dead or alive before you opened it. This, of course, is an interpretive issue. The possible classical states are "cat alive" (that has amplitude A) and "cat dead" (which has amplitude B). According to the state vector formalism, the actual state has amplitude (A B), hence thinking that the cat is in a superposition of states. The interesting thing about this is it is impossible to prove this wrong, because any attempt to observe the state collapses it to either A or B, and the "or" is the exclusive form. Is that science or another example of the mysticism that we accuse the ancients of believing, and we laugh at them for it? Why won't the future laugh at us? In my opinion, the argument that this procedure aids calculation is also misleading; classically you would calculate the probability that the nucleus had decayed, and the probability the rest of the device worked, and you could lay bets on whether the cat was alive or dead.

Accordingly, I am happy with factorizing the wave function. Indeed, every time you talk about a p orbital interacting with . . . you have factorized the atomic state, and in my opinion chemistry would be incomprehensible unless we do this sort of thing. However, I believe we can go further. Let us take the hydrogen atom, and accept that a given state has the action equal to*nh* associated with any state. We can factorise that (Schiller, R. 1962. *Phys Rev 125* : 1100 – 1108 ) such that

*nh* = [(*n*_{r} + ½) + (* l* + ½)*h*

Here, while the quantum numbers count the action, they also count the number of radial and angular nodes respectively. What is interesting is the half quanta; why are they there? In my opinion, they have separate functions from the other quanta. For example, consider the ground state of hydrogen. We can rewrite (1) as

*h* = [( ½ ) + ( ½)]*h* (2)

What does (2) actually say? First there are no nodes. The second is the state actually complies with the Uncertainty Principle. Suppose instead, we put the RHS of (2) simply equal to 1. If we assign that to angular motion solely, we have the Bohr theory, and we know that is wrong. If we assign it to radial motion solely, we have the motion of the electron as lying on a line through the nucleus, which is actually a classical possibility. While that turns up in most text books, again I consider that to be wrong because it has zero angular uncertainty. You know the angular momentum (zero) and you know (or could know if you determined it) the orientation of the line. (The same reasoning shows why Bohr was wrong, although of course at the time he had no idea of the Uncertainty Principle.)

There is another good point about (2): it asserts the period involves two "cycles". That is a requirement for a wave, which must have a crest and a trough. If you have no nodes separating them, you need two cycles. Now, I wonder how many people reading this (if any??) can see what happens next?

Which gets me to a final question, at least for this post: how many chemists are actually happy with what theory offers them? Comments would be appreciated.

However, you can only be sure of that if there are suitable examples from which to validate. So, quite accurate answers are obtained,

There is a second issue. Because validation and the requirement to obtain results equivalent to those observed, can we be sure they are obtained the right way? As an example, in 2006 some American chemists decided to test some programs that were considered tolerable advanced and available to general chemists on some quite basic compounds. The results were quite disappointing, even to the extent of showing that benzene was non-planar. (Moran, D. and five others. 2006.

There is a third issue, and this seems to have passed without comment amongst chemists. In the state vector formalism of quantum mechanics, it is often stated that you cannot factorise the overall wave function. That is the basis of the Schrödinger cat paradox. The whole cat is in the superposition of states that differ on whether or not the nucleus has decayed. If you can factorise the state, the paradox disappears. You may still have to open the box to see what has happened to the cat, but the cat, being a macroscopic being, has behaved classically and was either dead or alive before you opened it. This, of course, is an interpretive issue. The possible classical states are "cat alive" (that has amplitude A) and "cat dead" (which has amplitude B). According to the state vector formalism, the actual state has amplitude (A B), hence thinking that the cat is in a superposition of states. The interesting thing about this is it is impossible to prove this wrong, because any attempt to observe the state collapses it to either A or B, and the "or" is the exclusive form. Is that science or another example of the mysticism that we accuse the ancients of believing, and we laugh at them for it? Why won't the future laugh at us? In my opinion, the argument that this procedure aids calculation is also misleading; classically you would calculate the probability that the nucleus had decayed, and the probability the rest of the device worked, and you could lay bets on whether the cat was alive or dead.

Accordingly, I am happy with factorizing the wave function. Indeed, every time you talk about a p orbital interacting with . . . you have factorized the atomic state, and in my opinion chemistry would be incomprehensible unless we do this sort of thing. However, I believe we can go further. Let us take the hydrogen atom, and accept that a given state has the action equal to

Here, while the quantum numbers count the action, they also count the number of radial and angular nodes respectively. What is interesting is the half quanta; why are they there? In my opinion, they have separate functions from the other quanta. For example, consider the ground state of hydrogen. We can rewrite (1) as

What does (2) actually say? First there are no nodes. The second is the state actually complies with the Uncertainty Principle. Suppose instead, we put the RHS of (2) simply equal to 1. If we assign that to angular motion solely, we have the Bohr theory, and we know that is wrong. If we assign it to radial motion solely, we have the motion of the electron as lying on a line through the nucleus, which is actually a classical possibility. While that turns up in most text books, again I consider that to be wrong because it has zero angular uncertainty. You know the angular momentum (zero) and you know (or could know if you determined it) the orientation of the line. (The same reasoning shows why Bohr was wrong, although of course at the time he had no idea of the Uncertainty Principle.)

There is another good point about (2): it asserts the period involves two "cycles". That is a requirement for a wave, which must have a crest and a trough. If you have no nodes separating them, you need two cycles. Now, I wonder how many people reading this (if any??) can see what happens next?

Which gets me to a final question, at least for this post: how many chemists are actually happy with what theory offers them? Comments would be appreciated.

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