{"id":10689,"date":"2017-05-05T16:19:48","date_gmt":"2017-05-05T20:19:48","guid":{"rendered":"https:\/\/www.masterorganicchemistry.com\/?p=10689"},"modified":"2022-10-16T05:45:13","modified_gmt":"2022-10-16T10:45:13","slug":"the-pi-molecular-orbitals-of-benzene","status":"publish","type":"post","link":"https:\/\/www.masterorganicchemistry.com\/2017\/05\/05\/the-pi-molecular-orbitals-of-benzene\/","title":{"rendered":"The Pi Molecular Orbitals of Benzene"},"content":{"rendered":"
The Pi Molecular Orbitals of Benzene<\/strong><\/p>\n Today, let’s go through how to draw out the molecular orbitals of benzene. We’ll compare them with the molecular orbitals for (linear) hexatriene. The\u00a0big takeaway is that from contrasting the molecular orbitals of these two 6-electron pi systems, we will unlock the deep, mysterious\u00a0riddle\u00a0of why\u00a0benzene is so unusually stable.<\/p>\n Quick spoiler. Here’s of what the molecular orbitals of benzene look like.<\/p>\n The rest of this post will describe how we came up with this drawing and what it means.<\/p>\n Table of Contents<\/strong><\/p>\n Previously we’ve looked at the molecular orbitals of the allyl system, and of butadiene<\/a>. We learned some key lessons for drawing out the molecular orbitals of (linear) pi systems<\/a> that I will quickly rehash here.<\/p>\n Think of drawing out the pi orbitals as a bit like constructing an apartment building, albeit with some strange municipal building codes and quirky tenant behaviour.<\/p>\n The Ground Floor and the Penthouse are the easiest levels to draw, because they follow from simple rules: align all phases, or alternate all phases.<\/p>\n The tricky part is drawing the orbitals in the intermediate energy levels, due to the quantum-mechanical municipal bylaw that I call the Balanced Node Rule. \u00a0<\/strong>Because (math<\/a>), nodes can’t just be placed anywhere; they’re always arranged\u00a0symmetrically with the respect to the centre\u00a0of the orbital.<\/p>\n The final task\u00a0in drawing the molecular orbitals is to fill up our building with tenants (electrons) using the familiar Aufbau principle<\/a>: starting with the ground floor, each unit (orbital) fills up one electron at a time, to a maximum occupancy of two. [Note 1<\/a>]<\/p>\n Now let’s apply this framework to hexatriene and benzene.<\/p>\n Hopefully straightforward! \u00a0Six p orbitals in the pi systems of benzene and hexatriene will produce six pi molecular orbitals.<\/p>\n The levels in hexatriene stack like a six-story building. Benzene has a different arrangement, for reasons we’ll quickly see below.<\/p>\n Note that the bottom three orbitals are all bonding orbitals and the top three orbitals are antibonding. Only the bottom three “floors” are occupied in neutral hexatriene. \u00a0\u00a0(If every floor in this building were completely filled with tenants (electrons) the building would quickly self-destruct. Talk about a messed-up building) \u00a0<\/em><\/span><\/p>\n Following the pattern described above for butadiene, we draw the “ground floor” first, with the phases of all p-orbitals aligned the same way. It doesn’t matter if you draw the “shaded” or “white” lobes up or down, so long as they are all drawn the same way.<\/p>\n Here’s what they look like for hexatriene and benzene:<\/p>\n The Highest Energy molecular orbitals \u00a0have p orbitals with completely alternating phases.<\/strong><\/p>\n Like the ground floor, the highest-energy molecular orbital (the “penthouse”) \u00a0of a pi system is also straightforward to draw.<\/p>\n Draw all the p orbitals with alternating phases. No two adjacent p orbitals should have lobes with the same phase.<\/p>\n For linear systems, we’ve seen that this gives the highest energy level \u00a0(n\u20131) nodes. For hexatriene (n=6) that means that the highest energy level will have 5 nodes.<\/p>\n What about benzene? This is where things get interesting.<\/strong><\/p>\n In the case of cyclic systems, the (n\u20131) rule fails. Drawing a molecular orbital of benzene with 5 nodes is like solving 5 faces of a Rubik’s cube: impossible. A meelion dollars if you can prove me wrong:<\/em>\u00a0try it! [Note 2<\/a>]<\/em><\/span><\/p>\n We can, however, easily draw an orbital where all the phases alternate. In this case,\u00a0though,\u00a0we can count six places where the phases change. Instead of looking at these as individual nodes, it’s perhaps more helpful to think of these as three\u00a0nodal planes<\/strong>, which cut through the molecule at various points.<\/p>\n This orbital has\u00a0zero overlap between adjacent p orbitals and therefore electrons in this orbital have the minimum possible delocalization. They are therefore the highest energy.<\/p>\n As we said above, the tricky thing in building pi molecular orbitals is knowing where to put\u00a0the nodes in the intermediate levels.<\/p>\n For hexatriene, the second floor (one node) is fairly straightfoward: we put the node in the centre, like this:<\/p>\n It’s impossible to draw a cyclic pi system with one node, but we can draw a system with one nodal plane<\/strong>. Here, for instance, we’ve drawn a nodal plane that cuts through two of the single bonds:<\/p>\n But wait! There’s actually a second way to do it. We\u00a0can also draw a nodal plane through the atoms,<\/strong> like this.<\/p>\n These two molecular orbitals (\u03c02\u00a0<\/sub>and\u00a0\u03c03\u00a0<\/sub>) have the same number of nodal planes, and therefore have the same energy. The way we usually describe this in chemistry is by saying the orbitals are\u00a0degenerate<\/em>.<\/p>\n This is really the key difference in the molecular orbital picture of a cyclic system versus an acyclic system: two units can co-exist\u00a0on the same floor. For benzene, that results in a lowering of energy.<\/strong><\/p>\n Here’s the third, fourth, and fifth “floors” for the hexatriene pi system, which have two, three, and four nodes, respectively.<\/p>\n For benzene, the next level up has two nodal planes. Again, there’s two ways to do it: cut through the bonds, or cut through the atoms. Again, these are of the same energy.<\/p>\n The final step is to arrange all the pi orbitals together (by “floors”) and then to fill each level with electrons (“tenants”).<\/p>\n Here’s the complete molecular orbital picture for hexatriene:<\/p>\n The complete molecular orbital picture for benzene was drawn at the top of the post, but here it is again for your non-scrolling pleasure:<\/p>\n See the key difference? Describing it as “a more efficient stacking of energy levels” isn’t far off.<\/p>\n A cyclic pi system allows for two ways to place a single nodal plane: through-bond, or through-atom. In benzene, this means that the second and third “floors” each have two units with the same energy – they are “degenerate”.<\/p>\n To follow our somewhat crude analogy, both buildings have six “tenants”, but in benzene, those tenants aren’t as high off the ground – and thus have less potential energy.<\/p>\n In chemistry terms,\u00a0\u00a0the highest\u00a0occupied\u00a0molecular orbitals (HOMO) of benzene are lower in energy than the highest occupied molecular orbital (HOMO) of hexatriene. And for our purposes, that lower energy of the pi-electrons translates into lower reactivity. [Note 3<\/a>]<\/p>\n It’s an imperfect analogy, but for our purposes, it will do.<\/p>\n In the next post, let’s see if we can examine the molecular orbitals for\u00a0anti-aromatic<\/b> cyclobutadiene and similarly gain insights into its unusual instability<\/strong>.<\/p>\n Thanks to Tom Struble for his\u00a0contributions to this article.\u00a0<\/strong><\/p>\n<\/p>\n
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\n<\/a>1. Quickie Review: How To Draw Pi Molecular Orbitals For A Given Pi System<\/h2>\n
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\nThe highest energy level (the “penthouse” of our building) has (n\u20131) nodes. We saw that for butadiene (n=4) the highest energy level has three nodes between the orbitals (marked here with red lines).
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<\/a>2. Building The Pi Molecular Orbital Diagram For Benzene: Hexatriene and Benzene Each Have Six Pi Molecular Orbitals<\/strong><\/h2>\n
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<\/a>3. The Lowest-Energy Molecular Orbitals Of Hexatriene And Benzene Have Zero Nodes\u00a0<\/strong><\/h2>\n
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<\/a>4.\u00a0 The “Penthouse” Of The M.O. Diagram\u00a0 (Highest Energy Level) Has The Maximum Number Of Nodes\u00a0<\/strong><\/h2>\n
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<\/a>5. Benzene Has Nodal Planes<\/em>. The Maximum Energy Level Has 3 Nodal Planes<\/h2>\n
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<\/a>6. Where Do We Place The Nodes In The Intermediate Energy Levels Of Benzene?<\/strong><\/h2>\n
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<\/a>7. Filling Out The Rest of The Picture For Benzene<\/strong><\/h2>\n
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<\/a>8. The Benzene Molecular Orbital Diagram: Putting It All Together<\/strong><\/h2>\n
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<\/a>9. Why Is Benzene More Stable Than Hexatriene?<\/h2>\n
\n<\/a>Notes<\/h2>\n