Chapter 9: Molecular Structure

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Welcome to the deep dive, the shortcut to being well informed.

Today, we're tackling something pretty fundamental,

molecular structure.

Yeah, how atoms actually stick together.

Exactly.

We're diving deep into the quantum mechanical theories behind it all, not just how they stick, but the energy side of things too.

Right.

Our goal is to give you a clear college level overview,

without needing the textbook right in front of you.

And we're basically synthesizing the core ideas from physical chemistry, specifically looking at the two main ways chemists think about bonding.

Good job.

Balance bond theory, or VB, which is more localized, and then the big one, molecular orbital theory, MO theory.

Okay, VB and MO.

Got it.

But before we get into the differences, I understand there's a really crucial simplification they both rely on.

Oh, absolutely.

You can't really do any serious calculations without it.

It's called the Born -Oppenheimer approximation.

Okay, let's unpack this.

Born -Oppenheimer.

What's the core idea?

Well, think about a simple molecule, even hydrogen, H2.

You've got two protons, two electrons, calculating all their interactions precisely.

Well, it's basically impossible analytically.

There are many moving parts.

Exactly.

So Born -Oppenheimer leverages a huge difference in mass.

Nuclei are thousands of times heavier than electrons.

Okay.

So the electrons are zipping around incredibly fast, while the nuclei are, relatively speaking, almost standing still, like hummingbirds around statues.

Okay, I like that image.

So the approximation says,

let's just freeze the nuclei in place for a moment, treat them as fixed points, and then we solve for the electron behavior, the electronic structure for that specific nuclear arrangement.

So you calculate the energy with the nuclei at one distance,

then move them a tiny bit, calculate again.

Precisely.

You repeat that over and over.

And this separation, this trick, lets us map out the energy landscape for the molecule.

And that map is the potential energy curve we always see for diatomic molecules.

That's the one.

Or a potential energy surface, if you have more atoms.

And the minimum point on that curve, that tells you almost everything important.

Like the bond length.

The equilibrium bond length.

Yes.

And also the depth of that minimum tells you the bond dissociation energy, dsubsub, how much energy it takes to break the bond.

So without Born -Oppenheimer,

modern computational chemistry just wouldn't work.

Pretty much no.

It's that fundamental.

Okay.

So assumption made.

Let's look at the first theory.

Valence bond theory.

This feels more intuitive, right?

Shared electron pairs, like we learned in intro chem.

It does.

It connects nicely to Lewis structures.

The core idea in VB is that a bond forms when an electron in an atomic orbital on one atom pairs its spin with an electron in an orbital on another atom.

Spin pairing.

Okay.

So let's take H2 again.

Two hydrogen atoms, A and B.

Each has one electron, one and two.

Right.

Naively you might think, okay, electron one belongs to atom A, electron two belongs to atom B.

So the wave function would be something like a sub sub one times sub sub sub two.

Makes sense.

But quantum mechanics says electrons are indistinguishable.

Once they're close, you can't tell which is which.

Electron one could be on B and two on A.

That's just as valid.

Sub is spam, two A.

So VB theory says the reality is a mix of both.

Exactly.

A superposition.

The VB sum.

A is sub sub sub plus sub sub sub sub sub one.

And adding them together,

that does something important.

It leads to constructive interference.

The wave functions reinforce each other in the region between the two nuclei.

Ah, so you get more electron probability density right there in the middle.

Precisely.

A buildup of negative charge that attracts both positive nuclei and holds the molecule together.

That's your sigma.

It's cylindrically symmetrical around the bond axis.

Okay.

That's the single bond.

What about double or triple bonds?

Like an N2?

For that, VB theory introduces the pi bond.

This happens when pi orbitals overlap side by side, not head on like the sigma bond.

Side by side.

Imagine two p orbitals parallel to each other.

They interact above and below the intranuclear axis.

This creates two lobes of electron density separated by a nodal plane where the axis lies.

So N2 nitrogen gas, which has a triple bond.

In VB terms, that's described as one strong sigma bond from head on overlap and then two pi bonds perpendicular to each other from the side by side overlap of the remaining key orbitals matches the Lewis structure nae nae perfectly.

VB also introduced the idea of resonance, right?

I always think of benzene.

Benzene is the classic example, yes.

But resonance is more fundamental.

It's the idea that if you can draw multiple valid Lewis structures or electron arrangements for the same nuclear framework, the actual structure is a blend, a hybrid of all of

No, definitely not.

It is the hybrid.

Think about a simple polar bond like an HCl.

VB describes this as a resonance hybrid.

A hybrid of what?

A hybrid between a purely covalent form, HCl, sharing electrons, and a purely ionic form.

Hsup plus calsup electron transferred.

The actual wave function is a subcovalent sub plus a subionic sub.

And that lambda?

Lambda is a coefficient.

Its square tells you the probability or the weight of that ionic structure contributing to the overall real picture of the bond.

It quantifies the bond polarity.

So resonance isn't just for fancy rings.

It's how VB handles polarity.

Fundamentally, yes.

And this blending always leads to extra stability.

That's resonance stabilization.

The hybrid is always lower in energy, more stable than any single contributing structure would be on its own.

That's key to

Okay.

But VB runs into trouble with predicting molecular shapes, doesn't it?

Like water.

It does.

If you just use the basic atomic orbital, say, oxygen's p orbitals, they're at 90 degrees to each other.

So VB would initially predict a 90 degree bond angle for water.

But it's actually about 104 .5 degrees.

Big difference.

Huge difference.

And also, how does carbon, with only two unpaired electrons in its ground state, two SN2p payoff, form four bonds in

Right.

VB theory needs a couple of, well, let's call them conceptual fixes.

The first is promotion.

Promoting an electron.

Yeah.

You conceptually invest a bit of energy to bump one of carbon's two electrons up into the empty 2p orbital.

Now you have four unpaired electrons, one, two, three, two p, ready to form four bonds.

Okay, that explains the number of bonds.

But what about the shape, the 104 .5 degrees in water, or the tetrahedral shape of methane?

That's where hybridization comes in.

This is the idea that atomic orbitals on the same atom can mix together, or hybridize, to form new hybrid atomic orbitals that point in specific directions.

Ah, so you mix the s and p orbitals.

Exactly.

For methane, needing four identical bonds pointing tetrahedrally, VB proposes mixing the 1 -2s and the 3 -2p orbitals on carbon.

Like this.

Four equivalence pole hybrid orbitals.

They naturally point towards the corners of a tetrahedron with angles of about 109 .5 degrees.

Perfect for methane.

So hybridization is like a mathematical tool VB uses to make the orbital picture match the observed geometry.

That's a good way to put it.

It's an ad hoc adjustment, but a very useful one.

If you need trigonal planar geometry, like in ethene, C2H4, you mix 1s and 2ps to get spot hybrids 120 degrees apart, leaving 1p orbital unhybridized for the pi bond.

And linear means p.

Ns1p gives two spot hybrids pointing 180 degrees apart, like in acetylene, C2H2.

It allows VB to rationalize pretty much any VSPR shape.

Okay, so VB gives us pairs, resonance,

and uses hybridization to fix geometry.

Now, at the thought of the coin,

molecular orbital theory.

This sounds more modern.

It is.

MO theory is really the foundation for most modern quantitative understanding of bonding.

The big shift is that it treats electrons as belonging to the entire molecule, not just between two atoms.

Electrons are delocalized in molecular orbitals.

How do you build these molecular orbitals?

The simplest approach, especially for diatomics, is called the linear combination of atomic orbitals, or LCAO -MUMMO.

It's pretty much what it sounds like.

You approximate the molecular orbital by just adding or subtracting the atomic orbitals of the atoms involved.

Say, sub -a -sub and sub -sub.

So, sub -a -sub plus sub -sub, or sub -a -sub,

sub -sub.

Ignoring normalization for a sec.

Exactly.

The plus combination, sub plus sub, represents constructive interference.

Like in VB, this builds up electron density between the nuclei.

This is the bonding molecular orbital.

We often call it, if it's symmetrical, around the axis.

Lure energy, more holds things together.

Right.

The minus combination, sub -sub, represents destructive interference.

The wave functions cancel out in the middle, creating a nodal plane zero electron density between the nuclei.

Sounds bad for bonding.

It is.

This is the antibonding molecular orbital, often labeled sigma star.

It's higher in energy than the original atomic orbitals.

Placing electrons here actively destabilizes the molecule.

Now, I remember hearing something specific here, that the antibonding orbital isn't just destabilizing, it's more destabilizing than the bonding orbital is stabilizing.

Why is that?

That's a crucial point, yes.

The antibonding orbital energy goes up more than the bonding orbital energy goes down.

Okay, why?

It's partly due to electron repulsion being greater when density is forced away from the internuclear region, but also kinetic energy.

That node forces the wave function to curve sharply, and higher curvature means higher kinetic energy, according to quantum mechanics.

This extra push pushes the nuclei apart more strongly than the bonding attraction pulls them together.

So if you fill both the SO and sorbitals equally, like if you tried to make helium dimer, here's.

The net effect is repulsive, the destabilization wins, that's why he doesn't form a stable molecule.

You need more electrons in bonding MOs than antibonding MOs.

Makes sense.

And filling these MOs, we just use the same rules as for atoms.

Yep.

The Aufbau principle, fill lowest energy first.

The Pauli exclusion principle, max two electrons per orbital, opposite spins.

And Hund's rule, fill degenerate orbitals singly first with parallel spins.

And for molecules with identical atoms, like NO or ORO, there are those extra symmetry labels, or G and U.

Right.

Giraud and undiraud.

It's about inversion symmetry.

If you imagine inverting the orbital through the center point of the molecule.

Like reflecting everything through the midpoint.

Exactly.

If the sine of the wave function stays the same after inversion, it's girade or G even.

If the sign flips, it's undiraud or U odd.

Sigma bonding orbitals are usually G, sigma star are usually U, pi bonding or U pi star or G.

It helps classify them.

Okay.

So we fill up these MOs, bonding and antibonding.

And that lets us calculate something really useful.

The bond order.

Yes.

The bond order B is a fantastic measure of the net bonding.

It's simply half the difference between the number of electrons in bonding orbitals, N, and the number in antibonding orbitals, N.

So B equals pi half N.

Half the net bonding electrons.

Right.

For N euros, you fill up the MOs, count electrons, and you find B pi half 10, four equals three.

A bond order of three.

Which matches its triple bond perfectly.

Higher bond order means stronger, shorter bond.

Generally, yes.

It correlates very well with bond strength and bond length.

Bond order zero means no stable bond, like in Heria bound.

Now,

the classic test case.

The molecule that really showed MO theory's power over VB theory.

Oxygen.

O -O.

Ah, yes.

The O story is a great one.

VB theory, using Lewis structures, draws a double bond for O or O.

O -O.

Yeah.

All electrons paired up.

So VB predicts O -Rho should be diamagnetic, repelled by a magnet.

Correct.

But experimentally, oxygen is paramagnetic.

It's attracted to a magnetic field.

Quite strongly, actually, VB theory had no simple explanation for this.

So how does MO theory explain it?

When you build the MO diagram for O -Rho, which has 12 valence electrons, you fill up the sigma and pi bonding orbitals.

The last two electrons need to go into the next available orbitals, which are the degenerate pi star antibonding orbitals.

The same energy level.

Right.

So Hunn's rule applies.

The two electrons go into separate orbitals, and crucially, their spins are unpaired electrons.

Exactly.

Two unpaired electrons.

This immediately predicts that O should have a net electron spin, S1, and therefore be paramagnetic.

MO theory got it right.

Where VB failed, it was a major triumph.

That is convincing.

And we can actually experimentally see these molecular orbital energy levels you're talking about.

We can, using a technique called photoelectron spectroscopy, or PES.

PES.

Okay.

How does that work?

It's clever.

You shine high energy photons, usually UV light or x -rays, onto your sample.

The photon energy is high.

Planck's constant times frequency.

Got it.

If that energy is high enough, it can knock an electron completely out of one of the molecule's orbitals.

Ionize the molecule.

Right.

And the ejected electron flies off with some kinetic energy, E sub sub, which we can measure very precisely.

Okay.

The key insight is conservation of energy.

The energy of the photon that went in must equal the energy needed to remove the electron, the ionization energy is subi sub, plus the kinetic energy the electron has left over, E sub sub.

So high is subi sub plus E sub sub.

We know high.

We measure E sub sub so we can find a subi sub.

Exactly.

And you don't just get one ionization energy.

You get a whole spectrum of them because electrons can be knocked out of different molecular orbitals.

Ah, so each peak in the PE spectrum corresponds to ionizing an electron from a specific MO.

Precisely.

And there's a theorem, Koopman's theorem, which says that approximately the ionization energy is subi sub for removing an electron from a specific orbital is equal to the negative of that orbital's energy, subi sub.

Wow.

So PE's directly maps out the energy levels of the molecular orbitals we calculate.

That's pretty direct evidence.

It really is.

It provides experimental validation for the MO energy level diagrams.

Okay.

MO theory works great for identical atoms like N, U, or O.

How does it handle molecules where the atoms are different?

Heteronuclear diatomics like HF or CO.

If we connect this to the bigger picture, the LCAO principle still applies.

S subi sub is subi sub plus C sub sub sub sub.

But now the atomic orbitals subi sub and sub sub don't start at the same energy level.

One atom is typically more electronegative than the other.

Like fluorine is way more electronegative than hydrogen and HF.

Right.

Electronegativity means the atom holds onto its electrons more tightly, so its atomic orbitals are lower in energy.

In HF, the F2P orbitals are much lower energy than the H1's orbital.

So how does that affect the molecular orbitals formed?

The resulting bonding MO is still lower in energy, but it's no longer an equal mix.

The lower energy atomic orbital contributes more to the bonding MO.

Meaning the electrons in the bonding MO spend more time near the fluorine.

Exactly.

The coefficient sub F sub will be larger than C sub sub in the bonding MO.

Sub bond sub, C sub sub sub sub, plus C sub sub sub sub sub.

The electron density is pulled towards the more electronegative atom.

This is how MO theory naturally describes a polar covalent bond.

And the anti -bonding orbital.

It's skewed the other way.

The higher energy atomic orbital, H1's in this case, contributes more to the anti -bonding MO.

So this electronegativity thing is really central.

We mentioned Pauling scale, but also Mulliken.

Yeah.

Electronegativity is just that power to attract electrons in a bond.

Pauling scale is the most famous, derived empirically from bond energy data.

Mulliken defined it more fundamentally based on atomic properties.

The average of the atom's ionization energy and its electron affinity,

a suba sub, Mulliken sub agrees half I plus is to be C sub.

They measure the same underlying tendency.

When we're doing these LCAO calculations, especially with different atoms or many atoms, how do we know we found the best possible combination, best coefficients like C sub s sub, C sub sub?

Ah, that brings us to a cornerstone of quantum chemistry, the variation principle.

Sounds important.

It is.

It provides the mathematical justification for how we optimize our calculations.

The principle states if you calculate the energy of a system using any approximate trial wave function, the energy you calculate will always be greater than or equal to the true ground state energy of the system.

It can never be lower.

Okay.

So the true energy is the absolute minimum possible.

Exactly.

So in the LCAO method, those coefficients, C sub a sub, C sub sub, et cetera, are treated as variable parameters.

We use computational methods to systematically adjust these coefficients, trying to minimize the calculated energy of the resulting molecular orbital.

Keep tweaking the mix until the energy can't go any lower.

Precisely.

When you find the set of coefficients that gives the lowest possible energy, the variation principle guarantees that this is the best approximation to the true molecular orbital that you can achieve with that particular set of atomic orbitals, that basis set.

And does this optimization process tell us anything about which atomic orbitals interact best?

It absolutely does.

A key outcome is that the strongest bonding interactions, the biggest energy lowering,

occur when the atomic orbitals that are combining have similar energies.

Ah, that's why core electrons, like one's electrons, usually don't participate much in bonding.

Their energy is way too low compared to the valence orbitals of the other atom.

Exactly.

And it's why the H1's and F2P orbitals interact strongly in HF.

They're reasonably close in energy while the F1's orbital is basically left alone.

Okay.

This makes sense for diatomics, but what about bigger molecules,

polyatomics?

Things get really complex fast.

They do.

The MOs now spread over all the atoms in the molecule.

Finding the coefficients involves solving much larger mathematical problems, specifically involving something called secular determinants.

It quickly becomes a job for computers.

But there was an early simplified approach for certain types of polyatomics.

Yes.

For planar conjugated systems, think molecules with alternating single and double bonds, like butadiene or benzene.

There's the Huckel approximation.

Huckel.

What does it approximate?

It makes some pretty drastic simplifications to the math.

It completely ignores any overlap between atomic orbitals, sets overlap integrals s to zero, and it assumes electrons in P orbitals only interact with electrons in P orbitals on adjacent atoms, ignoring interactions further away.

It also assigns just two parameters, an F for the energy of a P orbital and the meta -libby for the interaction energy between adjacent P orbitals.

Sounds very simplified.

Is this still useful?

Remarkably so for understanding systems.

It allows you to easily calculate the energy levels of the molecular orbitals.

And critically, it lets you calculate the delocalization energy.

Which is?

It's the extra stability of molecule canes because its electrons are spread out, delocalized over the whole system, compared to if they were stuck in localized double bonds.

For benzene, the Huckel method predicts a large delocalization energy, two actually, which accounts for its unusual stability, its aromaticity.

But those Huckel assumptions are pretty severe.

What do chemists use today for accurate calculations on any molecule?

Today, it's all about computational chemistry.

The general approach is usually iterative, called the self -consistent field SCF method.

You guess the electron distribution, calculate the electric field it creates, use that field to calculate a new electron distribution, and repeat until the distribution stops changing until it's self -consistent.

Okay, SCF.

And within that, there are different levels of rigor.

Exactly.

Broadly, you have,

one, semi -empirical methods.

These still simplify things, estimating some of the complex mathematical integrals using parameters derived from experimental data.

Faster, but less accurate.

Two, ab initio methods.

Latin for from the beginning.

These try to calculate everything from fundamental physics, evaluating all the integrals numerically.

Very computationally expensive, especially for large molecules.

They often use mathematical functions called Gaussian -type orbitals, GTOs, to make the integrals more manageable.

And the third type.

I feel like I hear about this one a lot.

That would be density functional theory, or DFT.

DFT has become incredibly popular, kind of a sweet spot.

Why?

What's different about DFT?

Instead of focusing on the incredibly complicated many -electron wave function, DFT focuses on the much simpler electron density.

It's a function of just three spatial coordinates, not three n -coordinates like the wave function for n electrons.

There's a theorem proving that the ground state energy is uniquely determined by this density.

So calculating based on density is easier.

Yeah.

Faster.

Much, much less computationally demanding than high -level ab initio methods, but often gives results of similar accuracy.

It's become the workhorse for a huge amount of computational chemistry research.

And when these calculations are done, how do we visualize the results?

It's not just numbers.

Right.

We often visualize the MOs themselves as isodensity surfaces, showing where the probability of finding an electron is constant.

Or, perhaps even more useful for predicting reactivity, we generate electrostatic potential ESP surfaces.

ESP maps.

Yeah.

These map the electrostatic potential created by the nuclei and electrons onto an density surface.

They're usually color -coded.

Red for regions of negative potential, electron -rich, attractive to positive charge selectrophiles.

And blue for regions of positive potential, electron -poor, attractive to negative charges nucleophiles.

So you can literally see where a molecule is likely to react.

It gives you a powerful visual guide to chemical reactivity grounded directly in the quantum mechanical calculations.

So let's just quickly wrap up what we've covered in this deep dive.

We started with that essential Born -Oppenheimer approximation, the bedrock that lets us separate nuclear and electronic motion.

Right.

Treating nuclei stationary while electrons zip around.

Then we saw valence bond theory intuitive localized pairs using resonance and hybridization as tools to explain polarity and geometry.

But it needed those fixes.

Exactly.

Then we moved to molecular orbital theory, the more powerful delocalized view.

Using LCAO, we built bonding and antibonding orbitals, filled them using standard rules and calculated bond order.

And MO theory nailed things like the paramagnetism of O -euros and is backed up by experimental techniques like PES.

And finally, we saw how these ideas extend to polar bonds using electronegativity, how the variation principle ensures we find the best computational solution, and how modern methods like DFT allow us to tackle complex polyatomic molecules and visualize reactivity using ESP maps.

So when you boil it all down, understanding why molecules have the shapes, stabilities, and reactivities they do, it really comes down to how those electron waves, those atomic orbitals, interfere with each other across the molecule.

Constructively for bonding,

destructively for antibonding.

That's the heart of it.

The energy landscape is dictated by that quantum mechanical interference.

Now here's something to think about further.

We talked about MOs and their energy levels, like the HOMO, highest occupied, and LUMO, lowest unoccupied.

Consider a conjugated molecule like butadazine which absorbs UV light.

That absorption corresponds to exciting an electron from the HOMO to the LUMO.

Knowing that higher energy orbitals generally have more nodes, how might that single electron excitation changing the occupancy of those orbitals actually change the effective bond order between different carbon atoms and butadiene?

And how might that fleeting change in bonding influence its photochemical reactivity?

Something to mull over.

Interesting how light itself can tweak the bonding picture.

Definitely food for thought.

Thank you for joining us for this deep dive into the fascinating quantum world of molecular structure.