p. 808 — “These wavefunctions contain al the observable information about the system.” — A huge assumption, and in fact a postulate of quantum mechanics. OK, actually, since QM has never made an incorrect prediction.

p. 809 — “In classical mechanics, the forces on a system create two kinds of energy — kinetic and potential”. Hmm. How does force ‘create’ energy? It does so by doing work. Work is force * distance, and if you do a dimensional analysis, you find that force * distance has the dimensions of kinetic energy (mass * velocity^2) — It’s worth working through this yourself, if you’ve been away from physics for a while. Recall that potential energy is the general name for energy which has to do with location relative to something else.

After reading Lawrence Krause’s biography of Feynmann (which goes much more into the actual physics than other biographies including Gleick’s), I cracked open the 3 volumes of the Feynmann lectures on physics and have begun reading. It’s amazing how uncannily accurate his speculations were. particularly about things which weren’t known in the 60’s but which are known now.

He says that we really don’t know what energy is (even though we know 9 forms in which it appears) just that it’s conserved. Even so, the conservation law allows all sorts of physics problems to be solved. To really get into why energy is conserved, you have to read about Noether’s theorem — which I’m about to do, using a book called “Emmy Noether’s Wonderful Theorem” by Dwight E. Neuenschwander.

Later (Lecture 4 page 4) Feynmann defines potential energy as the general name of energy which has to do with location relative to something else.

p. 809 — The QM course I audited 2 years ago, noted that the Schrodinger equation really can’t be derived, but is used because it works. However, the prof then proceeded to give us a nice pseudo-derivation based on the standard equation for a wave propagating in space and time, Einsteins E = h * nu, and De Broglie’s p = h/lambda, and differenating the wave equation twice with respect to position, then twice with respect to time and equating what he got.

However, to get the usual Hamiltonian, he had to arbitrarily throw in a term for potential energy (because it works).

p. 810 “The energy E is simply a number” — should have said “The energy E is simply a **real** number” which is exactly why the complex conjugate must be used. If you *really* want to know what’s going on see — the 10 articles in the category == Linear Algebra Survival Guide for QM.

One point for the unitiated (into the mysteries of quantum mechanics) to consider. “The more nodes an orbital has, the higher is its energy. Recall from Chapter 1 that nodes are points of zero electron density, where the wavefunction changes sign.” Well, a point of zero electron density, or a point at which the wavefunction equals zero, means the electron is never (bold) found there. So why is the probability of finding an electron on both sides of the node not zero. You need to abandon the notion that an electron has a trajectory within an atom. Having done so, what does angular momentum mean in quantum mechanics?

p. 814 — “This is the fundamental reason that a bond forms; the kinetic energy of the electrons in the bonding region is lower than the kinetic energy of the electrons in isolated atomic orbits” — this is because the wave function amplitude changes less between the nuclei. However, since we’ve had to abandon the notion of a trajectory — what does kinetic energy actually mean in the quantum mechanical situation (see note on pp. 811 – 812).

p. 814 — “The greater the overlap between to orbitals the lower the kinetic energy” — to really see this you have to look at figure 14.6 on p. 813 — The greater the overlap, the shallower the depression of the wave function amplitude between the nuclei, which implies less change in amplitude with distance, whch implies a smaller Laplacian (a second derivative) and less kinetic energy for the electrons here. So this is why overlapping atomic orbitals result in lower kinetic electron energy at sites of overlap — e. g. why bonds form (bold). Great stuff ! ! ! !

Continuing on, the next paragraph explains where Morse potentials (p. 422) come from, and why populating antisymmetric orbitals causes repulsion (the change in orbital sign increases the Lagrangian greatly along with it the electron kinetic energy, despite the fact the the potential energy of the antisymmetric orbitals is favorable for keeping the atoms close — e.g. bonding).

p. 815 — What does the Born Oppenheimer approximation (which keeps internuclear distances fixed) do to the calculation of vibrational energies — which depend on nuclear motion? The way the energy of the solutions of the Schrodinger Equation using the BO approximation is gradually approached (moving the nuclei around and calculating energy) clearly won’t work for CnH2n+2 with n > 2. There will be more than a single minimum. What about a small protein? Clearly these situations the Born Oppenheimer approximation is hopeless. Because of the difficulty in understanding A&Ds discussion of the secular equation (see comments on p. 828), I’ve taken to reading other books (which have the advantage of devoting hundreds of pages to A&Ds 60 to computational chemistry), notably Cramer’s “Essentials of Computational Chemistry” — He notes that lacking the Born Oppenheimer approximation, the concept of the potential energy surface vanishes.

p. 816 – 817 — Antisymmetric wave functions and Slater determinants are interesting ways to look at the Pauli exclusion principle. The Slater determinant is basically a linear combination of orbitals — why is this allowed? — because the orbitals are the solution of a differential equation, and the differential of a sum of functions is the same as the sum of differentials of the functions and orbitals are the solution to a differential equation.

p. 823 — What’s a diffuse function? Also polarization orbitals strike me as a gigantic kludge. I suppose the proof of the pudding is in the prediction of energy levels, but there appear to be an awful lot of adjustible parameters lurking about.

p. 824 — “the motions of the electrons are pairwise correlated to keep the electrons apart” — but electrons don’t really have trajectories — see note on pp. 811 – 812. I got all this stuff from Mark P. Silverman “And Yet It Moves” Chapter 3

p. 825 — Nice to see why electron correlation is required, if you want to study Van der Waals forces between molecules, The correlation energy could be considered a intramolecular Van der Waals force.

What is a single point energy? — I couldn’t find it in the index.

p. 826 — The descriptions of the successive kludges required for the ab initio approach to orbitals are rather depressing. However, there’s no way around it. You are basically trying to solve a many body problem when you solve the Schrodinger equation. It’s time to remember what a former editor of Nature (John Gribbin) said “*It’s important to appreciate, though, that the lack of solutions to the three-body problem is not caused by our human deficiencies as mathematicians; it is built into the laws of mathematics* “

p. 828 — I was beginning not to take all this stuff seriously, until I found that the Hartree Fock approach produces results agreeing with experiment. However, given the zillions of adjustable parameters involved in getting to any one energy, it better produce good results for n^2 molecules, where n is the number of adjustable parameters. Fortunately, organic chemistry can provide well over n^2 molecules with n carbons and 2n+2 hydrogens.

Along these lines, how did the secular equation get its name. Is there a religious equation?

What can you do with an approximate wavefunction produced by any of these methods. The discussion in A&D so far is all about energy levels. However, unlike wavefunctions, operators on wavefunctions are completely known, so you can use them to calculate other properties (Cramer doesn’t give an example).

p. 830 – 831 — Even so, given the solutions of the secular equation for a very simple case, you see why the energy of a bonding orbital is less than two separate atomic orbital — call the amount B (for bonding). More importantly the energy of the antibonding oribtal his greater than the two separate atomic orbitals by a greater amount than B — explaining why filling a both a bonding and the corresponding antibonding orbital results in repulsion. This is rule #8 the rules of Qualitative Molecular Orbital theory (p. 28).

It’s amazing that Huckel theory works at all, ignoring as it does electron electron repulsion.

p. 836 — If everything in Density Functional Theory (DFT) depends on the electron density — how do you ever find it? Isn’t this what the wavefunctions which are the solutions to the Schrodinger equation actually are? I’m missing something here and will have to dig into Cramer again.

p. 838 — Most energy diagrams of molecular orbitals made from two identical atomic orbitals have the bonding and antibonding orbitals have them symmetrically disposed lower and higher than the atomic orbitals. This is from use of the Huckel approximation, which simply ignores overlap integrals. The truth is shown in the diagram on p. 831.

p. 838 — Another statement worth the price of the book — the sigma amd pi orbitals are of opposite symmetry (different symmetry) and so the sigma and pi orbitals don’t mix. The sigma electrons provide part of the potential field experienced by the pi electrons.

p. 839 — Spectacular to see how well Huckel Molecular Orbital Theory works for fulvene — even if the bonding and antibonding orbitals* are* symmetrically disposed.

p. 843 — With all these nice energy level diagrams, presumably spectroscopy has been able to determine the difference between them, and see how well the Huckel theory fits the pattern of energy levels (if not the absolute values).

p. 846 — Table 1.1 should be table 1.4 (I think)

p. 849 — How in the world was the bridged [10] annulene made?

p. 853 — Why is planar tetracoordinate carbon a goal of physical organic chemistry? The power of the molecular orbital approach is evident — put the C’s and H’s where you want in space, and watch what happens — sort of like making a chemical unicorn. Why not methane with 3 hydrogens in an equilateral triangle, the carbon in the center of the triangular plane, and the fourth hydrogen perpendicular to the central carbon?

p. 854 — What are the observations supporting the A < S ordering of molecular orbitals in 1, 4 dihydrobenzene? The arguments to rationalize the unexpected strike me as talmudic, not that talmudic reasoning is bad, just that no one calls it scientific.

p. 856 — Good to see that one can calculate NMR chemical shifts using ab initio calculations (hopefully without tweaking parameters for each molecule). Bad to see that it is too complicated to go into here. More reading to do (next year) after Anslyn (probably Cramer), with a little help from two computational chemist friends.

p. 858 — How in the world did anyone ever make Coates’ ion?

p. 861 — Have the cyclobutanediyl and cyclpentanediyl radicals ever been made?

p. 863 — “Recall that the 3d orbitals are in the same row of the period(ic) table as the 4s and 4p orbitals’ — does anyone have an idea why this is so? Given the periodic table, the 4s orbitals fill before the 3d, which fill before the 4p (lowest energy orbital fill first presumably). The higher energy of the 4p than 3d may explain by d2sp3 orbitals are higher in energy than the leftover 3d orbitals — d(xy), d(yz) and d(zx). Is this correct? However the diagram in part B. of Figure 14.33 on p. 864 still shows that (n+1)s is of higher energy than nd, even though the periodic table would imply the opposite.

It’s not clear why the t(2g) combinations of d(xy), d(yz) and d(zx) orbitals don’t interact with the 6 ligand orbitals, since they are closer to them in energy. Presumably the geometry is wrong? Presumably d(z2) and d(x2 – y2) are used to hybridize with the p orbitals because they are oriented the same way, and d(xy), d(yz) and d(zx) are offset from the p orbitals by 45 degrees. Is this correct?

This is the downside of self-study. I’m sure a practicing transitional metal organic chemist could answer these questions quickly, but without these answers it’s back to the med school drill — that’s the way it is, and you’d best memorize it.

p. 865 — Frontier orbitals have only been defined for the Huckel approximation at this point — the index has them discussed in the next chapter. On p. 888 they are defined as HOMO and LUMO which have been well defined previously.

p. 866 — The isolobal work is fascinating, primarily because it allows you to predict (or at least rationalize) things.

This section does elaborate a bit on organometallic bonding details, lacking in chapter 12. However, no reactions are discussed, and electrons are not pushed. Perhaps later in the book, but I doubt it.

It’s clear from reading chapter 12 that organometallics have revolutionized synthesis in the past 50 years. I’ll need to read further next year, in order to reach the goal of enjoying new and clever syntheses as they come out.

Does anyone out there have any thoughts about Cramer’s book? Any recommendations for other computational chemistry books? I’ve clearly got to go farther.

## Comments

Single point calculation means that energy was computed only for one point at the at the potential energy surface without any geometry optimization done. I. e. energy for one defined geometry.

With t(2g) orbitals and their interactions… You’re basically right. They don’t posses the right symmetry to overlap with ligands’ orbitals capable of sigma bonding. However t(2g) orbitals can interact with pi-type ligands’ orbitals and this effect is called very prominent with ligands like CO or CN, where backdonation happens. That means the t(2g) electrons are donated to antibonding and/or p orbitals of a ligand. Thus high field splitting.

Anyway it’s remarkable that you found a courage and pleasure to go through all of these books. Cramer may be too heavy stuff, at least from what I hear from my friends who do computational chemistry. QMM by Atkins and Friedman contains a lot of basic stuff. But if you understand Cramer and have the time, stick with it. There isn’t probably other such comprehensive book on this topic.

Re: p. 808 & 809 – I’ve always felt that since classical mechanics and electrodynamics are actually the classical limits of quantum theory, one should be able to derive them from quantum mechanics, not the other way around. ;) Having said that – the classical Hamiltonian is also H = T + V, so that potential energy term is present if one starts out with classical physics. It does not just get arbitrarily tossed in there, in my judgment. It’s what works, of course, but one can also ask the same question of how the classical Hamiltonian is constructed.

Re: “what things mean in quantum mechanics” – The more I think about quantum mechanics, I find myself both more baffled and less confused. That our intuitive notions of waves and particles are conflated at such small scales don’t surprise me (we didn’t evolve to pick up on such things at such small scales – so “how does the electron get through the node” just tells me someone hasn’t come to grips with that dichotomy), but that – for example – suddenly quantities have become differential operators bothers me. I could measure the mass of a toy car, measure its velocity along a straight track, and calculate the momentum. Can’t do that in QM….

For what it’s worth, though, if one looks at the relativistic orbitals, a lot of those nodes disappear, and there’s a very small but finite density at those points. Not in all cases, though, if my memory serves.

People have actually done computer simulations of vibrational frequencies of amide I bands in proteins, and found they correspond with experiment. At least, it was close enough that such research still goes on and didn’t make the field crawl into a hole and give up from what I remember of those talks a few years back.

There is a somewhat famous case study of computational quantum chemistry being used to demonstrate that experimental results were incorrect (the structure of triplet methylene as determined experimentally was challenged and eventually the experimentalists saw that the the prediction held). Admittedly, this was back in the 1970s, and it is only methylene, but the point holds, I’d say.

Re: density functional theory – historically, the electron density was modeled by plugging in an electron gas model, but has since been replaced by various approximations (oh, there’s that word!), some of the latest which rely upon Hartree-Fock calculations. DFT has its origins in solid state physics, so most of the early developments are rooted there, and probably reek as too idealized to most chemists.

Re: secular equation – there’s no “religious” equation, for one. Heh. I remember it has something to do with a secondary definition of “secular” but it’s a carry-over from that.

MJ — Thanks for taking the trouble to respond. Sorry for being so late getting back. The idea that that trajectories need to be abandoned in QM isn’t my idea but that of a physicist, Mark P. Silverman. His books are worth a look, if you’re into getting migraines from considering what quantum mechanics really means about the world. He’s written 4 according to Amazon.

Cramer is pretty good as I have told you before. You may also want to take a look at the “source”, Pople and Beveridge’s “Approximate Molecular Orbital Theory” which gives you a very good sense of the actual recipes to calculate various molecular properties.

I don’t know where the word “secular” came from but a similar equation was first used in the calculation of planetary orbits and celestial mechanics.

-The idea that trajectories need to be abandoned in QM isn’t my idea but that of a physicist, Mark P. Silverman

I don’t think Mark P. Silverman came up with that thought. It was one of the basic motivations for QM postulated by the founding fathers (Bohr, Heisenberg etc.)

Quite true, but Silverman goes into a lot of detail and background, and is worth reading if you have the time. I do think Silverman did one of the earliest experiments where widely separated in time single electrons were sent through the two slit apparatus, showing that each electron in fact interferes with itself. He’s written several books, and I’m not sure exactly where these discussions occur.