Fallout from my learning about Molecular Orbital theory and Hartree-Fock.
I’ve said repeatedly that Organic Chemistry is along the spectrum of pursuits that uses Quantum Mechanics. Organic Chemists learn a brutal regimen of details for constructing ball-and-stick models of complicated molecules. I’ve also recently discovered that chemistry –to this day– is teaching a fundamental lie to undergraduates about quantum mechanics… not because they don’t actually know the truth, but because it’s easier and more systematic to teach.
As a basic example, let’s use the model of methane (CH4) for a small demonstration.
This image is taken pretty much at random from The New World Encyclopedia via a Google image search. The article on that link is titled “covalent bond” and they actually do touch briefly on the lie.
A covalent bond is a structure that is formed between two atoms where each atom donates one electron to form a paired structure. You have probably heard of sigma- and pi- bonds.
This image of Ethylene (a fairly close relative of methane) is taken from Brilliant and shows details of the two most major types of covalent bonds. Along this path, you might even remember my playing around in the first post I made in this series, where I directly plotted sigma- and pi- bonds from linear combinations of hydrogenic orbitals.
These bond structure ideas seem to emerge predominantly based on papers by Linus Pauling in the 1930s. The notion is that the molecule is fabricated out of overlapping atomic orbitals to make a structure sort of resembling a balloon animal, as seen in the figure above containing ethylene. Organic chemistry is largely about drawing sticks and balls.
With methane, you have four sticks joining the balls together. We understand the carbon to be in Sp3 hybridization, which is directly a construct offered by Linus Pauling in 1931, describing a four orbital system, with four sigma bonds, involving carbon with tetrahedral symmetry which is three parts p and one part s. The orbitals are formed specifically from hydrogenic s- and p- types. If you count, you’ll see that there are 8 electrons involved in the bonding in this model.
I used to think this was the story.
The molecular orbital calculations tell me something different. First I will recall for you the calculated density for methane achieved by closed-shell Hartree-Fock.
This density sort of looks like the thing above, I will admit. To see the lie, you have to glance a little closer.
This is a collection of the molecular orbitals calculated by STO-3G and the energy axis is not to perfect scale. The reported energies are high given the incompleteness of the basis. The arrows show the distribution of the electrons in the ground state with one spin up and spin down electron in each orbital. The -11.03 Hartree orbital is the deep 1s electrons of the carbon and these are so tightly held that the density is not very visible at this resolution. The -0.93 orbital is the next out and the density is mainly like a 2s orbital, though when you threshold to see the diffuse part of the wave function, it has a sort of tetrahedral shape. Note, this shape only emerges if you threshold so that it becomes visible. The next three orbitals at -0.53 are degenerate in energy and have these weird blob-like shapes that actually don’t really look like anything; one of them sort of looks like a Linus Pauling Sp-hybrid, but we’re stumped by the pesky fact that there are three rather than four. The next four orbitals above zero are virtual orbitals and are unpopulated in the ground state of the molecule –these could be called anti-bonding states.
Focusing on the populated degenerate orbitals:
These three seem to throw a wrench at everything that you might ever think from Linus Pauling. They do not look like the stick-like bonds that you would expect from your freshman chemistry balloon animal intuition. Fact is that these three are selected in the Hartree-Fock calculation as a composite rather than as individual orbitals. They occur at the same energy, meaning that they are fundamentally entangled with each other and the filter placed on finding them finds all three together in a mixture. This has to be the case because these orbitals examined in isolation do not preserve the symmetry of the molecule.
With methane, we must expect the eigenstates to have tetrahedral symmetry: the symmetry transformations for tetrahedral symmetry (120 degree rotations around each of the points) would leave the Hamiltonian unaltered (it transforms back into itself), so that the Hamiltonian and the symmetry operators commute. If these operators commute, the eigenstates of the molecule’s Hamiltonian must be simultaneous eigenstates of tetrahedral symmetry. This is basic quantum mechanics.
You can see by eye that these orbitals are not.
Now, with this in mind, you can look at the superposition of these which was found during the Hartree-Fock calculation:
This is the probability distribution for the superposition of the three degenerate eigenstates above. Now we have a thing that’s tetrahedral. Note, there is no thresholding here, this is the real intensity distribution for this orbital collection. This manifold structure contains 6 electrons in three up-down spin pairs where they are in superpositions of three unknown (unknowable) degenerate states.
The next lower energy set has two electrons in up-down and looks like this:
This is the -0.93 orbital without thresholding so that you can see where the orbital is mostly distributed as a 2s-like orbital close to the Carbon atom in the center. It does have a diffuse fringe that reaches the hydrogens, but it’s mainly held to the carbon.
I have to conclude that the tetrahedral superposed orbital thing is what holds the hydrogens onto the molecule.
Where are my stick-like bonds? If you stop and think about the Linus Pauling Sp-hybrids, you realize that those orbitals in isolation also don’t preserve symmetry! Further, we’ve got a counting conundrum: the orbitals holding the molecule together have six electrons, while the ball-and-stick covalent sigma-bonded model has eight. In the molecular orbital version, two of the electrons have been drawn in close to the carbon, leaving the hydrogen atoms sitting out in a six-electron tetrahedral shell state.
This vividly shows the effect of electronegativity: carbon is withdrawing two of the electrons to itself while only six remain to hold the four hydrogen nuclei. There is not even one spin-up-down two electron sigma-bond in sight!
And so we hit the lie: there is no such thing as sigma- and pi- bonds!
…there is no spoon…
The ideas of the sigma- and pi-bonds come from a model not that different from the Bohr atom. They have power to describe the multiplicity that comes from angular momentum closure, having originated as a description in the 1930s explaining bonding effects noticed in the 1910 to 1920 range, but they are not a complete description. The techniques to produce the molecular orbitals originated later: ’50s, ’60s, ’70s and ’80s. These newer ideas are crazily different from the older ones and require a good dose of pure quantum mechanics to understand. I have a Physical Chemistry book for Chemists from the early 2000s that does not contain a good treatment of molecular orbital theory, stopping only with basically the variational methods Pauling and the workers in the 1930s were using. I asked one of my coworkers, who is versed in organic chemistry models, how many electrons she thought were in the methane bonding system and she said “8,” exactly as I would have prior to this little undertaking.
There’s a conspiracy! We’re living in a lie man!
I spent some time looking at Ethylene, which is the molecule featuring the example of the balloon animal Pi-bond in the image above. I found a structure that resembles a Pi-bond at the highest energy occupied orbital of the molecule.
Density of Ethylene:
Density of Ethylene:
I’ve added two images of the density so that you can see the three dimensional structure.
Ethylene -0.32 hartrees molecular orbital, looks like a pi-bond:
The -0.53 hartrees orbital looks sort of sigma-like between the carbons:
The rest of the orbitals look nothing like conventional sigma- or pi- bonds. The hydrogens are again attached by a manifold of probability density which probably allows the entire system to be entangled and invertible based on symmetry.
Admittedly, ethylene has only one pi-bond and the first image above probably qualifies as the pi-bond. I would point out, however, that in the case of ethylene, the stereotypical sigma- and pi- configurations between the carbons matches the symmetry of the molecule, which has a reflection symmetry plane between the carbons and a 180 degree rotation axis along the long axis of the molecule. The sigma- and pi- bond configurations can be symmetry preserving here, but for the carbons only.
One other interesting observation is that the deep electrons in the 1s orbitals of the carbons are degenerate in energy, leading these orbitals to be entangled:
This also matches the reflection symmetry of the molecule (and would in fact be required by it). There are four electrons in this orbital and you can’t tell which are which, so the probability distribution allows them to be in both places at once… either on the one carbon or on the other. Note, this does not mean that they are actually in both places; it means that you could find them in one place or the other and that you cannot know where they are unless you look –I think this distinction is important and frequently overlooked.
An interesting accessory question here is what happens if you twist ethylene? Molecules like ethylene are not friendly to rotation along the long axis of the double bond because that supposedly breaks the pi-bonding. So, I did that. The total energy of the molecule increases from -77.07 to -76.86; that isn’t a huge amount, but it would constitute a barrier to rotation around the double bond.
Twisted Ethylene density, rotated about the bond by 90 degrees:
In this case, you do get what appear –sort of– to look like four-fold degenerate sigma-bonds attaching the hydrogens:
But, the multiplicity is about two-fold degenerate, suggesting only four electrons in the orbital instead of eight, which badly breaks the sigma-bond idea (of two electrons to a sigma-bond). This again suggests strong electron withdrawing by carbon, and stronger with ethylene than methane.
The highest energy occupied state has an energy increased from -0.32 in the planar state to -0.177 in the twisted state… and it looks like a broken pi-bond:
I think that the conventional idea about why ethylene is rigid is probably fairly accurate. The pictures here might be regarded as a transition state between the two planar cases where the molecule has a barrier to twisting, but is permitted to do so at some slow rate.
In the twisted case, the deep 1s electrons on the carbons are broken from reflection symmetry and they become distinctly localized to one carbon or the other.
Overall, I can see why you would teach the ideas of the sigma- and pi- bonds, even though they are probably best regarded as special cases. If you’re not completely aware that they are special cases, and that pictures like the one on Brilliant.org are broken, then we have a problem.
This exercise has been a very helpful one for me, I think. I’ve heard a huge amount about symmetries and about different organic chemistry conventions. Performing this series of calculations really helps to bridge the gap. Seeing actual examples is eye-opening. Why aren’t there more out there?
As I’ve continued to learn more about electronic bonds, I’ve learned that the structural details have been continuously argued for a long time. It becomes clear pretty quickly that the molecular orbital structures tend to exclude those notions you encounter early in schooling. Still, molecular orbitals have broken-physics problems themselves when you try to pull them apart by splitting a molecule in half. You end up having to be molecular orbital-like when the molecule is intact, but atomic orbital-like when the molecule is pulled apart into its separate atoms.
I found a paper from 1973 By Goddard and company which rescues some of the valance bond ideas as Generalized Valence Bonds (GVB). Within this framework, the molecular orbitals are again treated as linear combinations of atomic parts and answers the protestations of symmetry by saying simply that if you can make a combination of atomic orbitals that globally preserve the symmetry in a molecule, then that combination is an acceptable answer. GVB adds to the older ideas by putting in the notion that bonds can push and deform each other, which certainly fits with the things you start to see when you examine the molecular orbitals.
You can have sigma and pi bonds if you make adjustments. I’m not sure yet how the GVB version of methane would be constructed, but the direct treatment of carbon in the paper slays the idea of Sp-hybridization, as I understand it, while still producing the expected geometry of molecules.
Still thinking about this.
I’ve been strongly aware that my little Python program is simply not going to cut it in the long haul if I desire to be able to make some calculations that are actually useful to a modern level of research. I decided to learn how to use GAMESS.
For a poor academic with some desire to do quantum mechanics/ molecular mechanics type calculations, GAMESS is a godsend.
More than that actually. GAMESS is like stumbling over an Aston Martin Vanquish sitting in an alley way, unlocked, with the keys in the ignition, where the vanity plate says “wtng4U.” It isn’t actually shareware, but it could be called licensed freeware. GAMESS is an academic project whose roots existed clear back in the 1970s, roughly parallel to Gaussian, which still exists today and is accessible to people whom the curators deem reasonable. My academic email address probably helped with the vetting and I can’t say I know exactly how far they are willing to distribute their admittedly precious program.
To give you an idea of the performance gap between my little go-cart and this porche: the methane calculations I made above took 17 seconds for my Python program… GAMESS did it in 0.1 seconds. Roughly 170-fold! This would bring benzene down from two hours for my program to maybe a few minutes with GAMESS.
This image, produced by a GAMESS satellite program called wxMacMolPlt, is a methane coordinate model with a GAMESS calculated electron density depicted as a mesh to demonstrate a probability isosurface. What GAMESS adds to where I was in my own efforts is a sophistication including direct calculations of orbital electron occupancy. Under these calculations, it’s clear that electrons are withdrawn from the hydrogens, but maybe not quite as extremely as my crude estimations above would suggest: the orbitals associated with the hydrogens have 93% to 96% electron occupancy… withdrawn, but not so withdrawn as to be empty (I estimated 6 for 8 electrons above, or more like 75% occupancy, which was relatively naive). This presumably comes from the fringes of the 2s orbital centered on the carbon. Again, the analysis is very different from the simple notations of sigma- and pi-bonding, where the electrons are clearly set in clouds defined by the whole molecule rather than as distinct localizations.
I’ve really just learned how to make GAMESS work, so my ability to do this is very much limited. And, admittedly, since I have no access to real computer infrastructure (just a quadcore CPU) it will never reach its full profound ability. In my hands, GAMESS is still an atomic bomb used as a fly swatter. We’ll see if I can improve upon that.
Hit a few bumps learning how to make GAMESS dance, but it seems I’ve managed to turn it against the basic pieces I was able to attack on my own.
Here is Ethylene, both a model and a thresholded form of the total electron density.
I also went and found those orbitals in the carbon-carbon bond.
The first is the sigma-like bond at -0.54 and the second is the pi-like bond at -0.33. The numbers here are slightly off from what I quote above because the geometry is optimized and STO-3G ends up optimizing slightly shorter than X-ray observed bond lengths. These are somewhat easier to see than the clouds I was able to produce with my own program (though I think my work might be a little prettier). I’ve also noticed that you can’t plot density of orbital super-positions with the available GAMESS associated programs, as I did with methane above. I can probably get tricky by processing molecular orbitals on my own to create the superpositions and then plot them –GAMESS handily supplies all the eigenvectors and basis functions in its log files.
In the build of GAMESS that I acquired, I’ve stumbled over an apparent bug. The program can’t distinguish tetrahedral symmetry in a normal manner… it’s converting the Td point group of methane into what appears to be a D2h point group, apparently. I was able to work around this by calling symmetry C1. Considering that I started out with no idea how to enter anything at all, I take this as a victory. As open freeware, they work with a smaller budget and team, so I think the goof is probably understandable –though it sure felt malicious when I realized that the problem was with GAMESS itself. I’m not savvy enough with programming to dig in and fix this one myself, I think, though the pseudo-open source nature of GAMESS would certainly allow that.
Given how huge an effort my own python SCF program ended up requiring, I’m not too surprised that GAMESS has small problems floating around. As an academic product, they have funding limits. At the very least, I’m impressed that it cranks out in seconds what took my program minutes… that speed extends my range a lot. I was able to experiment with true geometry optimization in GAMESS where my program stopped with me scrounging atomic coordinates out of the literature.
This is an image of pyrophosphate, calculated with the 6-311G basis set in GAMESS by restricted Hartree-Fock. This includes geometry optimization and is in a polarized continuum model for representation of solvation in water. The wireframe itself represents an equi-probability surface in the electron density profile while the coloration of the wireframe represents the electrostatic potential at that surface (blue for negative, red for positive).
This was an attempted saddle point search in GAMESS trying to find if a transition state exists in the transfer of a proton from one water molecule to another (for formation of hydronium and hydroxide)
This is the weirdest thing I’ve seen using GAMESS yet. This is not exactly a time simulation, it’s an attempted geometry minimization showing the computer trying different geometries in an attempt to locate a saddle point in the potential energy surface. I’m befuddled a bit by this search type because I’m trying to study a reaction pathway in my own work. Unfortunately, this sort of geometry search is operating in a counter-intuitive fashion and I’m not certain whether or not it’s broken in the program. However, when you see two oxygens fighting over a proton… well… that’s just cool. If the waters are set close enough so that they enjoy a hydrogen bond, the energy surface appears to have no extrema except where the protons are located as two waters. If you back the waters off from one another so that they are out of hydrogen bonding distance and pull the hydrogen out so that it is hydrogen bonding with both oxygens, you get this weird behavior where the proton bounces around until the nuclei are close enough to go back to the hydrogen bonded water configuration. I need to pin the oxygens away from each other, which won’t happen in reality.
Not sure what I think.