Chapter 8: Atomic Structure and Spectra

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Welcome back to the Deep Dive.

Today we're tackling something huge, really getting into the heart of atomic physics.

We are.

We're looking at the quantum mechanical view of atomic structure and spectra.

Yeah, this is like the instruction manual for the periodic table, isn't it?

It explains why everything is the way it is, chemically speaking.

That's a good way to put it.

Our source material really digs into electronic structure, basically, how electrons arrange themselves around a nucleus.

And our mission today is to walk you through that, step by step.

We'll start simple and build up.

Exactly.

We'll start with the one case where the math works out perfectly, the hydrogenic atom, and then layer on the complexity needed for, well, pretty much everything else.

And how we actually see this stuff using light, the spectra part.

Crucial.

It's foundational stuff.

You really can't grasp chemistry or molecules without getting this.

Okay, let's unpack this, starting with the one electron world.

Right.

Hydrogenic atoms.

So hydrogenic doesn't just mean hydrogen, right?

It's broader than that.

No, not at all.

It's any nucleus could be helium with one electron knocked off, he plus jute or lithium, le2 plus derr, even uranium losing 91 electrons.

Crazy, I know.

As long as it's got just that one electron left.

Precisely.

And the beauty is with only one electron, there's no electron repulsion messing things up.

So the Schrodinger equation, the master equation, you can actually solve it exactly.

You can, analytically.

And the solution, the wave function, it splits neatly into two parts because of the atom symmetry.

An angular part and a radial part.

The angular part gives us those shapes that we sort of recognize the spherical harmonics.

These define the basic form of the atomic orbital.

Okay.

And these orbitals are pinned down by those three quantum numbers.

Everyone remembers those, hopefully.

First is nuller.

The principal quantum number, positive integers.

One, two, three, and so on.

It tells you the shell, K, L, M shells, and most importantly, the energy level.

Exactly.

And that energy formula is really insightful.

It's negative, meaning the electron's trapped, it's bound.

And the energy depends on zitterwiller, the nuclear charge, squared and a squared.

That's right.

Numbered on is proportional to minus EG2 over $200.

So bigger nucleus, tighter binding, higher light, looser binding.

Makes sense.

Then number two,

the orbital angular momentum quantum number.

This one dictates the shape of the orbital and the magnitude of its angular momentum.

Light legs is an orbital, all leg is orbital, less is $2, and so on.

And the rule is, go goes from zero up to better one, so no 1p orbitals, for instance.

Correct.

Can't have less than, equal to, or greater than one.

And the third one is $1, the magnetic quantum number.

This tells you how that angular momentum is oriented in space, specifically its component along, say, the z -axis.

It runs from a PoB through dollars up to plus about.

So for a PoB orbital, where $1, you get an equal one to the zero plus the zero point three possible p orbitals.

Exactly.

Px, py, pt, againtually.

Now, the amazing thing is, this mathematical framework born from quantum theory, it perfectly predicts the old experimental results.

The Rydberg formula, which described the spectral lines of hydrogen linemen Balmer series.

The specific colors of light hydrogen emits or absorbs.

The formula just falls right out of the quantum mechanical energy levels.

It was stunning proof.

Total vindication for the theory.

Yeah.

Okay, visualizing these things without pictures.

Corbitals, orbitals, the Shtiers, right?

Symmetrical.

Perfectly spherical.

And here's a tricky point.

The wave function itself, CRI, is not zero at the nucleus for solar orbitals.

It actually has its maximum value there.

So the electron can be right at the nucleus.

Probability density is highest there.

The density, yes.

But think about the total probability of finding the electron at a certain distance.

That's the radial distribution function, PR.

Okay, how does that differ?

PR is the probability density.

So even if the density is highest at three dollars, the surface area there is zero.

Exactly.

So PR starts at zero, rises to a maximum at some distance.

Which is the Bohr radius, T dollars, for the hydrogen one dollar orbital.

That's the one.

And then it falls off again.

So the most probable distance isn't zero, even though the density is highest there.

It's a balance.

Volume versus density.

Got it.

And that's crucial, because P and orbitals are different.

Fundamentally different in this respect.

For P orbitals, the wave function is proportional to two.

For two or dollars, it's proportional to two orbitals.

Meaning they are zero at the nucleus.

Shu dollars makes them zero.

Correct.

They have zero probability density at the nucleus.

They also have angular nodes planes where the wave function is zero.

This difference in behavior near the nucleus is key.

Okay, here's where it gets really interesting then.

What happens when we add more electrons?

Like helium.

Chaos.

Well, mathematical chaos anyway.

That exact solution we loved, gone.

Why?

What breaks it?

Electron repulsion.

That one dollar twelve term in the Schrödinger equation, the repulsion between electron one and electron two, it couples their motions.

You can't separate the variables anymore.

So we can't solve it perfectly.

What do we do?

Give up?

Huh.

No, we approximate.

We use the orbital approximation.

We pretend, essentially, that each electron lives in its own orbital, described by its own wave function, even though they're interacting.

So the total wave function is just a product of these individual ones.

Seems like a big simplification.

It is, but it's the only way forward analytically, and it forms the basis for almost all our thinking about atomic structure.

How do we make it better then?

Computationally.

Methods like Hartree -Fock, self -consistent field, or HFSEF.

You basically guess the orbitals, calculate the average field each electron feels from the others, solve for better orbitals in that field, recalculate the field, solve again.

You iterate back and forth until the orbitals in the field they generate are consistent with each other.

Self -consistent.

Okay, clever.

So using this approximation, we start building up atoms, but there are rules, new rules.

Two absolutely critical non -classical rules.

First,

electron spin.

That intrinsic angular momentum, sort of like the electron is spinning, though not literally.

Right.

Discovered via the Stern -Gerlach experiment, electrons have spin quantum numbers six -ton pulled into it.

They are fermions.

And second rule.

The big one.

The Pauli Principle.

In its deep form, it says the total wave function for identical fermions must change sign if you swap any two of them.

It has to be anti -symmetric.

Okay, deep.

But the practical consequence?

The Pauli Exclusion Principle.

No two electrons in an atom can have the same set of all four quantum numbers.

One -octers and the spin projection multiple -octers.

Meaning, if two electrons are in the same orbital,

same dolla now and well, their spins must be opposite.

One spin up, one spin down.

Exactly.

Max two electrons per orbital, and they must be paired.

This principle is arguably why matter occupies space.

It prevents everything collapsing.

And it immediately messes with the energy levels, right?

In hydrogen, two like a dollar, two dollars have the same energy.

Not anymore.

Not anymore.

That degeneracy is lifted in many electron atoms.

And the reason comes back to those orbital shapes and the nucleus.

It's penetration and shielding.

Shielding makes sense.

The inner electrons block some of the positive charge of the nucleus from the outer electrons.

Right.

So an outer electron doesn't feel the full nuclear charge.

It feels a reduced effective nuclear charge.

Zess is a minus some shielding constant.

Sigma.

But penetration?

That's about how close the electron gets.

Yes.

Remember how sp orbitals are non -zero at the nucleus, but feet and T out orbitals are zero there.

That means an alike electron, even if it's mostly far out, has a small but significant chance of being found inside the inner electron shells very close to the nucleus.

It penetrates the shield.

And when it penetrates, it feels a much stronger pull, less shielding.

Exactly.

It experiences a higher ZF momentarily.

So on average, an electron feels a stronger attraction than a P electron in the same shell.

Which makes it more tightly bound, lower energy.

Precisely.

That's why for a given one, like 1, 3, 3, the energy order is $3, $3, 3.

So it's all about penetration and shielding.

And this order gives us the building up principle, or Aufbau principle.

Fill the lowest energy orbitals first.

$1, then two billers, then two piers, then three piers, then three piers is the $4 before three piers.

That's the sequence dictated by these energy effects.

But then you hit, say, the PAR orbitals, three of them, all with the same energy, degenerate.

How do you fill those?

Hunn's maximum multiplicity rule.

The rule of the bus seat.

Electrons prefer to occupy separate degenerate orbitals with parallel spins before they start pairing up upper para wood.

Why parallel spins?

Seems counterintuitive, like magnets repelling.

It's quantum mechanics again, specifically spin correlation.

The Pauli principle, in its deeper form, forces electrons with parallel spins to stay spatially further apart.

They have an anti -symmetric spatial wave function.

Yes.

Staying farther apart minimizes their electrostatic repulsion.

So the state with the most parallel spins, maximum multiplicity, has the lowest energy.

Wow.

Okay, so these rules,

Pauli, Hund, penetration shielding, they build the entire periodic table and explain its trends.

Absolutely, like atomic radii.

Across a period, zeta increases because you're adding protons, but electrons are going into the same shell, shielding each other imperfectly.

So electrons get pulled in tighter, atoms get smaller.

Until you start a new shell, then the radius jumps up.

Right.

And ionization energy, the energy to remove an electron,

generally increases across a period because of that increasing z -ray.

But there are those little dips, aren't there?

Like from beryllium to boron or nitrogen to oxygen.

Yep.

Explained perfectly by the structure.

B to B, you're removing the first electron from a 2 -key A to orbital boron, which is higher energy and better shielded than the 2 -A CLA beryllium, so it's easier to remove.

And out.

Nitrogen has three unpaired electrons in 2PO upro -upro.

Oxygen has four upro -upro.

That first paired electron in oxygen experiences extra repulsion from its orbital partner, making it slightly easier to remove than expected.

It all fits.

What about things like the lanthanide contraction?

A great example of orbital effects.

The $4 orbitals, filled across the lanthanide series, are really diffuse shapes.

They are terrible at shielding the outer electrons from the increasing nuclear charge.

So as you add 14 protons across the lanthanides.

The Zeph experienced by the outer 6 -dezils electrons increases much more than you'd expect.

This pulls the atom together, making the elements after the lanthanides, like hafnium, tantalum, unexpectedly small and dense.

Okay, that makes sense.

Now how do we actually measure these energy levels?

That brings us to spectra.

Atomic spectra.

The lines you see, emission or absorption, correspond to photons carrying energy delta A equal to the difference between two allowed energy states, or terms.

Terms, okay.

And not just any transition is allowed.

No, you have selection rules.

These come from conservation laws.

Mainly, conservation of angular momentum.

The photon itself carries one unit of angular momentum.

So the atom's angular momentum has to change to compensate when it absorbs or emits one.

Exactly.

For hydrogenic atoms, the main rule we learn is delta L eq P m l o n 1.

The orbital angular momentum quantum number must change by one.

So an electron can jump from socal to P to his dollars, or P to a dollar, but not soap to a dollar, or T to a - Correct.

Those are forbidden transitions, at least in the simplest approximation.

Now, for many electron atoms, it gets more complex.

We need to talk about the total state.

Right.

We need term symbols.

They look complicated, like $2 plus one l j doll, but they efficiently pack in all the crucial angular momentum info for the entire atom's electronic state.

Okay, let's break that down.

Socal is the total spin angular momentum number.

Yes.

You combine all the individual electron spins to get a total dollars.

The superscript, $2 plus couple one, is called the multiplicity.

$2 plus one n is a singlet state.

$2 plus one plus four is a doublet.

Three is a triplet.

Tells you how many unpaired spins, essentially.

Then dollars.

Dollars is the total orbital angular momentum number.

You combine the individual dollar values vectorially, $1 is an S term, careful capital S, $1 is a P term, one two is D, $1 three is F, and so on.

And G dollar subscript.

Dep life is the total angular momentum number, combining both total orbital and total spin angular momenta.

This vector coupling is usually done via the Russell -Saunders coupling scheme, especially for lighter atoms.

So you can figure out dollars,

figure out segerballers, then combine them to get possible values of devils.

Exactly.

And the different deagle values corresponding to the same dollar and dollars often have slightly different energies.

This splitting is called fine structure.

And what causes that fine structure splitting?

Spin orbit coupling.

It's fundamentally a magnetic effect.

An electron orbiting the nucleus creates a magnetic field.

The electron's own intrinsic spin is also magnetic.

Like two tiny magnets interacting.

Pretty much.

The energy of that interaction depends on the relative orientation of the spin orbital magnetic moments, which is captured by the total angular momentum dollar song.

So a term like, say, a P term, all more in dollar with one unpaired electron, LH2 splits into levels with different gel values.

Right.

Diopola can range from LH2 to LH plus Sokol.

So for L dollars, S12, $2 can be $12, $2 or $32.

You get two levels, like 2P12 and 2P32.

That's the famous splitting of the sodium D lines.

And this spin orbit coupling, it gets stronger for heavier atoms.

Dramatically stronger.

It scales roughly as Z400 dollars.

For heavy elements, spin orbit coupling isn't just fine structure.

It's a huge effect that dominates their electronic properties in chemistry.

Russell Saunders' coupling starts to break down.

That's a really key point.

A seemingly small effect becomes dominant just because the nucleus is heavier.

Absolutely.

Now, Hunn's rules come back again here, but this time to tell us the relative energies of the different terms arising from a configuration.

Okay, rule one.

Term with the maximum multiplicity, largest dollar, lies lowest in energy.

Again, spin correlation minimizes repulsion.

Rule two.

For terms with that same maximum multiplicity, the one with the maximum dollar lies lowest.

Hand -wavingly, higher dollars means electrons are orbiting in a more correlated way, keeping farther apart.

And rule three helps with the dollar levels from skin orbit coupling.

Yes.

For shells less than half -filled, the level with the lowest dollars is lowest in energy.

For shells more than half -filled, the highest dollar is the lowest.

Got it.

So, term symbols really are the language of atomic spectra.

They tell you the state in which transitions between states are likely, following selection rules like delta S equals a Kero, and delta L equals zero PMO, and delta J equals a zero EA1.

Exactly.

They are incredibly concise summaries of the quantum state.

Okay, let's recap this deep dive then.

We started with the hydrogenic atom, the solvable ideal case that gave us the language of orbitals, dollar, L, MP, and perfectly matched early spectral data.

Then we faced the complexity of many electron atoms.

The math got impossible exactly, forcing approximations like the orbital approximation and computational methods like Hartree -Fock.

We brought in the non -classical rules, electron spin and the crucial Pauli exclusion principle, which enforces structure.

And saw how interactions led to shielding and penetration, splitting the Dillard's PD SDS energy levels and dictating the Aufbau principle in periodic trends.

Hunn's rules helped fill degenerate orbitals.

Finally, we connected this structure to observable spectra.

Transitions between energy terms give spectral lines governed by selection rules.

And we introduced term symbols, two SPS plus one LJD to describe these multi -electron states, explaining fine structure via spin -orbit coupling, which becomes critical for heavy elements.

Hunn's rules helped order the terms too.

It's a lot, but it pieces together logically.

It does.

And it's worth remembering that while the orbital approximation is central, modern computational chemistry goes beyond simple Hartree -Fock to get incredibly accurate results, refining our picture constantly.

So what does this all mean, the takeaway?

It means that these abstract quantum rules, how electrons arrange themselves, their spin, their angular momentum, how they shield each other, these rules dictate literally everything about chemistry.

The size of atoms, how they bond, how they react, how they interact with light.

It all traces back to this quantum foundation.

Precisely.

The quantum mechanics of the electron is the bedrock of chemistry and materials science.

Amazing stuff.

Thank you for joining us for this deep dive into atomic structure and spectra.