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Spin Quantum Number: Term Symbols II

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

  • Intro 0:00
  • Microstates 0:25
    • We Started With 21 Possible Microstates
    • ³P State
    • Microstates in ³P Level
    • ¹D State
    • ³P State
    • ²P₂ State
    • ³P₁ State
    • ³P₀ State
    • 9 Microstates in ³P are Subdivided
    • ¹S State
    • Quicker Way to Find the Different Values of J for a Given Basic Term Symbol
    • Ground State
  • Hund's Empirical Rules for Specifying the Term Symbol for the Ground Electronic State 27:29
    • Hund's Empirical Rules: 1
    • Hund's Empirical Rules: 2
    • Hund's Empirical Rules: 3 - Part A
    • Hund's Empirical Rules: 3 - Part B
    • Example: 1s²2s²2p²

Transcription: Spin Quantum Number: Term Symbols II

Hello, and welcome to www.educator.com, welcome back to Physical Chemistry.0000

As I stated in the last lesson, this particular lesson is a continuation of the example that was started in the last lesson which was, 0003

we are developing term symbols for the electron configuration for carbon 1S2, 2S2, 2P2.0013

The P2 electron configuration, we want to find the term symbols for that.0019

Let us go ahead and continue that example.0024

We started with 21 possible microstates.0028

Let me do this in light blue.0033

We started with 21 possible microstates.0038

And by microstates, we just mean the arrangements of electron in a particular orbital.0047

We started with 21 microstates and 6 of these, 6 violated the exclusion principle.0056

That parallel spins in the same suborbital, it is not going to happen.0071

6 violated the exclusion principle.0075

We found that 5 of them were in the D1 level, which is what we wanted.0077

We want to find the term symbol representing that particular electron configuration.0088

One of the term symbols is D1.0092

We have 10 configurations left, we have 10 microstates left, and we need to group those.0095

Let us do the same thing.0107

I’m going to ask you actually to look back over that particular page of microstates.0108

It is a good idea to copy that page, make a screenshot of it so that you can keep referring back and forth to it.0119

Looking back at our page of microstates of the 10 remaining, when you actually chose it, when you actually grouped a certain number of them, 0126

like when we took the 5 and we found that D1, let us go ahead and cross them out, so you do not confuse them with any others.0155

Looking back at our page of microstates.0160

Of the 10 remaining stases, the next largest value of M sub L is equal to 1.0162

And for this value of M sub L, the largest M sub S is also equal to 1.0189

What we have is the following.0213

The fact that M sub L is equal to 1 it implies that L is equal to 1.0215

The possible values of M sub L that are going to be represented are 1, 0, -1.0218

M sub S equaling 1 it implies that S is equal to 1.0233

The possible values, we will find a better way, M sub S is also 1, 0, and -1.0242

L1, L equal to 1, you remember the correlation, when L is equal to 0 that is the S term.0254

When L is equal to 1 that is the P term.0267

S equal to 1 means that 2S + 1 = 2 × 1 + 1 that is equal to 3.0276

Together, we have the P3 term.0288

We found a term of the next set of microstates that fall into a given level, it is the P3 level.0301

The number of the M sub L × the number of the M sub S is equal to 3 × 3 = 9.0313

All those 10 microstates, 9 of those microstates fall into the P3 level.0326

We actually do not say 3P, we say triplet P.0332

We will talk a little bit more about what we actually call these.0334

For example, the 1 D it is actually called singlet D.0337

If you were let us say 2P, they would be doublet P.0342

In this case, it is 3P or it is triplet P state.0346

There are 9 microstates in the triplet P level.0350

All those 9 states, they have the same energy.0366

When I look back, this is what I find.0371

I'm finding all combinations of M sub L and M sub S.0376

The M sub L values are 1, 0, -1.0381

The M sub S values are 1, 0, -1.0386

I’m going to look back and in my microstate, I’m going to find the M sub L, M sub S combination 1 1, 1 0, 1- 1.0390

I’m going to find 0 1, 0 0, 0 -1.0398

I’m going to find -1 1, -1 0, -1 -1.0401

These 9 microstates belong to the triplet P level.0405

They are as follows, we have 1, 2, 3, 4, 5, 6, 7, 8, 9.0411

And we have 123, 123, 123.0418

And we have 123, 123, 123.0424

We have up spin up spin, here M sub L = 1 and M sub S = 1.0431

That is the 1 1 combination, down spin and up spin.0440

M sub L = 1 M sub S =0, that is the 1 -0 combination, down spin down spin.0451

M sub L = 1 M sub S = -1, 1 1, 1 0, 1 -1, 1 1, 1 0, 1 -1 that is a microstate, a microstate.0461

We have this and this, let me just go ahead and draw them all out.0474

That and that, that and that, we have up and up, we have down and up, we have down and down here.0481

M sub L = 0 M sub S = 1, M sub L = 0 M sub S = 0, M sub L = 0 and M sub = -1.0499

This is the 0 1, 0 0, 0 -1, combination.0514

And here we have, M sub L =-1 M sub S = 1, we have M sub L = -1 and M sub S = 0.0518

And we have M sub L =-1 and M sub S is equal to -1.0535

These 9 belong to the triplet P level, these 9 have the same energy.0547

I apologize, it is an old habit of mine to always say 3P, 1D.0564

I really should say triplet P, singlet D, doublet, things like that, quadruplet, quintuplet, whatever. 0565

Just old habits or just for odd reason had not been able to shake things up.0579

We have got 9 microstates of the 10 remaining.0584

This accounts for 9 of them, that leaves 1 microstate.0587

On our page of microstates, there is only one left, its ML value is equal to 0 and it is MS value is equal to 0.0595

M sub L = 0 that implies that L is equal to 0, which means that all the possible M sub L values are just 0.0619

M sub S = 0 implies that S is equal to 0.0644

Those are the ones that we are going to use for term symbol, that implies that all the values for M sub S are also just plain old 0.0649

L equaling 0 means the S term.0659

S = 0 means 2 × 0 + 1 is equal to 1.0665

We have a singlet S level.0673

This microstate is this only one left.0681

For every value, it is the microstate with ML = 0 MS = 0.0687

One value of ML, one value of MS, 1 × 1 is 1.0694

This microstate has M sub L = 0 and M sub S = 0 and it is the microstate with the electrons in the Z orbital paired spin.0698

We have a singlet D state, we have a triplet P state, and we have a singlet S state.0732

15 viable microstates, 15 viable arrangements of 2 electrons in 6 spin orbital arrange themselves in 3 different energies.0747

5 of them are here in the singlet D, 9 of them are going to be the triplet P and 1 of them is going to be in the singlet S.0759

This is how we derive the term symbols.0770

Let us go ahead and finish these.0772

Let us finish the term symbols by finding J, the right subscript, the total angular momentum.0779

Let us go ahead and do the singlet D state first.0798

In this particular case, D means that L is equal to 2.0801

The one here, 2S + 1 = 1, 2S = 0, that means that S =0.0808

This primary term symbol can actually give you the value of L and S.0815

We have used to say configuration to find the term symbol.0819

But if you are just given the term symbol, you can get these numbers.0822

If L is equal to 2, M sub L is equal to 2, 1, 0, -1, and 2, and M sub S is equal to 0.0826

We are going to calculate the M sub J values.0840

We do not need to do the M sub J on that page where we are given the microstates, but we did it anyway, it is not a problem.0842

M sub J, you are going to take all the values of M sub L.0848

Remember, we said M sub J was M sub L + M sub S.0852

We are going to add 2 + 0, 1 + 0, 0 + 0, -1 + 0, -2 + 0.0858

When we do that, we are going to get the 5 values.0866

We add each value of ML with each value of MS to get 2, 1, 0, -1, -2.0871

This sequence 2, 1, 0, -1, -2 for the M sub J means this sequence, that the number J itself is equal to 2.0882

The singlet D state has a total angular momentum equal to 2.0904

The term symbol, the complete term symbol is singlet D2.0909

Notice the correlation that we had.0915

With the hydrogen atom we had L and we had M sub L which takes on the values from -L to + L passing through 0.0917

L, these values of M sub L take on all the values from +L to -L passing through 0.0926

S, all the values of M and S run from -S to + S.0937

We have J, all the values of M sub J run from -J to +J passing through 0.0943

It is that same correlation that goes through all of this.0953

Our complete term symbol is singlet D2, it has 5 microstates in that energy level.0956

Let us go ahead and take care of the P3 or the triplet P state.0963

In this particular case, the P means that the L is equal to 1, the 3 here means 2S + 1 = 3, 2S = 2, S = 1.0978

If L is equal to 1 that means the M sub L values are 1, 0, -1.0991

S is equal to 1 means that the M sub S values are equal to 1, 0, and -1.0997

The M sub J is equal to M sub L + M sub S, I'm going to add each one of these to each one of these.1005

1 + 1, 1 + 0, 1 + -1, I get 2, 1, 0.1020

Next one, 0 + 1, 0 + 0, 0 + -1, I get 1, 0, -1.1027

-1 + 1, -1 + 0, -1 + -1, I get 0, -1, and 2.1035

All of these matter, I look in here and I see a sequence of 2, 1, 0, -1, 2.1045

There is a sequence 2, 1, 0, -1, 2, for the M sub J.1064

That means that J is equal to 2, for one of those.1081

We have a triplet P2 state.1089

However, that just takes care of that.1096

There is also a sequence of 1, 0, -1.1106

There is also a sequence of the M sub J =1, 0, -1.1123

This implies that J is equal to 1.1135

We also have a triplet P1 state.1141

Notice the triplet P states actually have more sub states.1148

What is left over, there is a N sub J = 0 leftover, that implies that J is equal to 0.1153

We also have a triplet P0 state.1168

The 9 microstates in the triplet P level are subdivided as follows.1185

5 of the microstates are in triplet P2.1208

3 microstates in triplet P1 and 1 microstate in triplet P0.1227

If we want to know where this 5, 3, and 1 came from, they are actually degeneracy of each individual fully complete term.1246

These come from 2J + 1.1253

For a complete term symbol, one that actually includes the J value, the degeneracy is 2J + 1.1259

In this case, J is equal to 0, 2 × 0 + 1, 1 microstate falling in that term symbol energy level.1278

Here J is equal to 1, 2 × 1 + 1 is equal to 3, there are 3 microstates in that particular term symbol energy level.1286

Here J is equal to 2, 2 × 2 + 1 is equal to 5.1295

There are 5 microstates in the triplet P2 level.1299

Our last one, we have our singlet S.1307

In this case, L is equal to 0 and S is equal to 0, that means the M sub L is equal to 0, that means M sub S is equal to 0.1311

M sub J is equal to M sub L + M sub S = 0 + 0 is equal to 0, that implies that J is equal to 0.1321

We have singlet S0, that is the complete term symbol.1335

There is actually a quicker way, once you have the initial grouping, the singlet D, the triplet P, singlet S.1345

There is a quicker way to find the different values of J for a given basic term symbol.1360

The basic term symbol is just the left subscript and it is the 2S + 1 and is the L value, without the J. 1388

There is a quicker way to find the different values of J for a given basic term symbol.1397

The allowed values of J are, you start by adding L + S and then you do L + S -1, L + S -2, L + S -3, and you keep going.1408

The lowest number is going to be the absolute value of L – S.1426

Let us do the J values.1430

For the singlet D state, we have L is equal to 2 and we have S is equal to 0.1433

L + S is equal to 2 + 0 it is equal to 2.1440

The absolute value of L - S is the absolute value of 2 -0, which is 2.1445

2 down to 2, only 2.1455

Singlet D2 state, that is how we get that one.1459

Let us do the triplet P state, L is equal to 1, S is equal to 1.1469

L + S = 1 + 1 is equal to 2.1476

The absolute value of L - S = absolute value of 1 -1 which is 0.1482

We have 2, 1, 0, that gives us the triplet P2, triplet P1, triplet P0.1487

Nice and quick.1501

Singlet S state, L = 0 S = 0 L + S 0 + 0 is equal to 0.1503

The absolute value of L - S is 0, that means that 0 is the only one.1512

We have a singlet S0.1521

Let us see, where are we?1530

We have the triplet P2, we have the triplet P1, we have the triplet P0, and we have the singlet S0.1548

All of those 15 viable microstates are distributed among these.1560

And these are the terms symbols that represent the energy of those.1565

It is giving us more information than just a normal electron configuration, 1S2 2S2 2P2.1570

This is telling us something about the spin angular momentum, the orbital angular momentum, and the total angular momentum.1578

All of these have different energies.1585

The question is which one was the ground state?1588

Which do we call the ground state, the state of lowest energy.1597

The one with the lowest energy is the ground state.1609

Which level has the lowest energy, that is the question we are asking.1621

Which level, which term level has the lowest energy?1624

The rules that we use to find that are Hund’s rules.1638

They are the empirical rules that decide which is the lowest energy state for a particular term.1642

Hund’s empirical rules for specifying the term symbol for the ground electronic state.1667

The first rule says this, the state with the largest value of S is the lowest in energy.1706

In other words, the most stable.1737

And as S decreases, stability decreases.1745

Energy goes up, stability decreases.1757

For states that have the same value of S like we just saw second ago.1764

The triplet P has 3 states, triplet P2, triplet P1, triplet P0.1768

How do we decide?1774

For states having the same value of S, the state with the largest value of L is lowest in energy.1779

In other words, the most stable.1819

For states having the same values of S and L, break it down a little bit further.1829

We have part A, if the shell is less than half filled show smallest J, smallest J is the lowest energy.1847

And B, for more than half filled shell show it is the largest J, it is lowest in energy.1882

For our 1S2, 2S2, 2P configuration, we have a singlet D2, triplet P2, a triplet P1, a triplet P0, and we have a singlet S0.1919

Hund’s first rule, largest S means the triplet P state.1942

The second rule did not tell us anything because the L value is the same for each PPP, L = 1.1955

The second one, L was the same for all 3 triplet P states.1963

Let me make this 3 a little bit clearer here.1978

Our third one, 2P2 it is a less than half filled shell that means less than half filled.1982

Therefore, we pick the smallest J between the 2, 1, 0, we choose the 0.1998

Less than half filled, the triplet P0 state is the ground state.2004

The triplet P0 state, we said that there were 5 in this one, 3 in this one, 1 in this one.2023

One particular configuration, that is going to be the ground state.2028

That takes care of term symbol but we are not exactly finished discussing it.2036

But for this particular example, given a configuration in particular the P2 state, 2044

we were able to elucidate 5 term symbols, singlet D2, triplet P2, triplet P1, triplet P0, and a singlet S0.2050

And using Hund’s rules, we were able to actually find out that the ground state is actually the triplet P0.2061

These term symbols give us more information than just the basic electron configuration.2068

They tell us about the spin angular momentum.2073

They tell us about the orbital angular momentum.2078

And it tells us about the total angular momentum.2081

That is all, thank you so much for joining us here at www.educator.com.2086

We will see you next time for a further discussion of term symbols.2090

Take care.2094