Physical Chemistry Electronic Transitions

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

In the last four lessons, we have been discussing vibration spectroscopy, rotational spectroscopy, vibration rotation spectroscopy.0004

Today, we will talk about electronic transitions.0011

Let us get started.0016

Let us go ahead and do blue today.0020

Diatomic molecules absorbing radiation in the visible ultraviolet range.0032

They experience transitions to excited electronic states.0053

We call these electronic transactions.0072

We send the electronic up to a higher level of energy.0081

Electronic transmissions, remember when we did vibrational transition,0087

we had just rotational transitions that is the microwave range.0094

The infrared range, we have vibrational transitions.0098

But with the vibrational, you got the rotational also.0101

With electronic transitions, you get the vibrational and rotational also.0104

Electronic transitions are accompanied by both vibrational and rotational transitions.0111

In general, the rotational transitions we are not going to worry about because there are reasonably insignificant.0136

It is the vibration transitions that we are going to be concerned with.0141

Each electronic state has its own potential energy curve.0146

That is what you see here.0163

This is the ground state, this is the first excited state, or they say E1 could be any of the excited states.0164

Anything above the ground state has a potential energy curve that has is a set of vibrational energy states.0171

And the excited electronic state, this one right here on top, it has its own set of vibrational energy levels.0180

Each electronic state has its own potential energy curve.0188

They do not necessarily need to look alike.0194

One is not a copy of the other.0201

It might look like it here but they are not.0204

Let me see, should I do it on this page?0213

The total energy than of the molecule, the diatomic molecule leaving off the transitional energy,0223

leaving off the energy of motion.0237

We are just going to be concerned with the electronic energy, the vibrational energy, the rotational energy.0240

The molecule is a total, it = the electronic energy + the vibrational energy + the rotational energy.0246

That simple.0267

E total = electronic + ν sub E × R + ½ - X sub E ν sub E × R + ½².0274

This is the vibrational energy under the anharmonic oscillator + B × J × J + 1 - D × J² × J + 1².0295

This is the rotational energy in non rigid rotator that accounts for the centrifugal distortion.0320

This one right here, E electronic is the energy at the minimum of the potential energy curve of the potential energy curve.0331

In other words, in the ground vibrational state is that energy right there.0362

Whatever that happens to be.0367

In the first excited state, it is that right there.0368

It is the energy at the minimum of the potential energy curve.0372

That is what the electronic energy is.0376

Transitions between the vibrational states during electronic transitions,0388

in other words during the transition from one electronic state to another, are called vibronic transitions.0405

In general, we will ignore the rotational term.0429

It makes our equations a little bit easier to deal with.0442

Ignore the rotational terms in the above equation.0449

The reason is because on the scale of electronic energies, when we are talking about 10⁻¹⁸ J,0467

the rotational energies are insignificant.0486

The rotational energies 10⁻²⁴ J are very small.0494

For the most part, we can just ignore them.0504

We have E total = electronic E × R + ½ ν sub E × R + ½².0532

This is the total energy of molecule.0557

Our selection rule for the vibronic transitions, δ R = + or -1, + or -2, + or -3, and so on.0562

You can jump from 0 to 1, 0 to 2, 0 to 3.0586

You can jump from 1 to 2, 1 to 3, 1 to 4, 1 to 5.0595

You have to go just from one level at the time.0601

As we said before, at normal temperatures most molecules are in the R = 0 vibrational state.0608

Most of the vibronic transitions would happen from the ground state, the R = 0.0639

It happen from the 0 vibrational state.0648

Most of the vibronic transitions originate there.0654

In other words, R = 0.0670

You are going to have the 0 to 1 transition.0673

You are going to have 0 to 2.0676

You are going to have 0 to 3.0678

You are going to have 0 to 1, 0 to 2, 0 to 3, 0 to 4.0682

Those are the transitions that are going to take place.0690

As we said before, each electronic state has its own potential energy curve.0705

That is the potential energy curve for the upper state.0710

That is the potential energy curve for the lower state.0714

In general, the ground state.0717

It is not necessary to write this down.0733

I will just go ahead and tell you.0735

The upper states are usually designated with a single prime.0738

You are going to see it right there.0762

The lower energy states, they are usually designated with a double prime.0768

Personally, I prefer the designations U for upper and L for lower.0786

The primes tend to confuse me especially when you start taking the difference between energy levels.0791

All of the sudden you have got single primes and double primes.0796

We have already seen symbolism heavy.0799

Primes and double primes, it is just a personal thing.0802

In general, in your book more than likely you are going to see single primes and double primes.0805

Single for upper and double for lower.0809

For my notation, I will just use U and L.0811

Sometimes I will go ahead and use 0 for the ground state.0815

We are going to investigate the vibronic transitions from the lower R value of 0 to the upper R value 1, 2, 3, and so on.0821

We are going to go from the ground vibrational state up.0858

In the upper electronic state, we would be hitting the 1, 2, 3, 4, 5 vibrational states.0861

That is what we are going to actually look at.0867

Each transition 0 to 1, 0 to 2, 0 to 3, 0 to 4, represents a line in the spectrum.0869

Each transition is a line in the spectrum.0878

Actually in the case of electronic spectrum, you are not going to see necessarily a line.0889

What you are going to see is a peak.0893

In electronic spectrum, under visible UV spectroscopy, you are going to see just a bunch of peaks.0895

Those represent the lines, the absorption, the transmission, things like that.0905

This set of transitions 0 to 1, 0 to 2, 0 to 3, 0 to 4, 0 to 5, it is called a progression.0912

This collection of transitions 0 to 1, 0 to 2, 0 to 3, there a lot of them by the way, not just 1, 2, 3, 4, 5.0930

We are talking 60 or 70 sometimes.0943

This collection of transitions is called a progression.0947

0 to 1, 0 to 2, 0 to 3, 0 to 4, 0 to 5, that is a question that we see on the actual spectrum that we take.0961

Each one of those peaks represents a vibronic transition.0972

When you look in your spectrum, that is what you are going to be seeing.0976

We have the energy total is equal to the electronic energy + that × R + ½ - X sub E ν sub E R + ½², that gives us the energy.0981

The frequencies that we observe in the spectrum, the peaks that we see,1006

the frequencies we see in the electronic spectrum of a particular diatomic molecule is going to be,1012

We will call it ν observed.1024

Ν observed, I will specify that we are talking about vibronic, although we should know that because that is the lesson we are in.1029

Again, the observed frequency that we see in the spectrum is going to be the difference between one energy level and another.1037

The upper - the lower.1043

It is going to be the total energy of the upper level - the total energy of the lower level.1045

In this particular case, it is the ground state 0.1055

Sometimes, I will use 0 instead of L but again we know what it is going to be upper – lower.1059

The energy of the upper, that is that one.1067

The energy of the lower, that is that one.1069

Let us go ahead and put these values in.1072

We are moving the lower level, it is the ground state.1074

It is going to be R = 0 vibrational state.1078

In this particular case, R is going to equal 0.1086

A lot of symbolism here, I apologize.1090

The electronic energy, the upper state + ν sub E upper × R upper + ½ -1099

X sub E upper ν sub E upper × R upper + ½² - the lower energy.1120

This is the upper energy - the lower energy.1137

The lower energy R value is equal to 0.1140

That is going to be the energy electronic 0 state + ½ ν 0 – X sub E 0 ν sub E.1143

I will stick with upper and lower so I’m not going to use the 0.1170

This is going to be the energy of the electronic of the lower state + ν sub E lower state × ½ - R 0 ½² is ¼.1173

Let us go ahead and do a little bit of algebra here.1215

I’m going to separate out the terms, multiply these out, put some terms together.1219

It is going to be, our ν observed for vibronic is going to be the electronic energy of1227

the upper state - the electronic energy of the lower state.1235

That takes care of the electronic energies.1240

+ ν sub E upper × R in the upper + ½ ν sub E of the upper.1244

It is not necessary for me to actually go through this algebra.1257

I could just write down the equation, but I think it is nice to go through them.1264

It is part of your scientific and your mathematical literacy here with physical chemistry1268

in quantum mechanics spectroscopy, whatever it is.1277

I just think it is nice to go through the mathematics, it makes it a lot more clear1280

instead of dropping down like a stone.1284

Some equation being dropped in your lab.1286

All we are doing is we are taking the upper energy - lower energy.1288

The rest is just very very careful algebra with this insane symbolism.1291

I apologize for that.1295

- X sub E upper ν sub E upper × R upper² – X sub E upper ν sub E upper.1300

That is crazy, I have no idea how they kept all of it straight all those years.1320

In some of the problems, you might be asked to derive these equations, - ½ ν sub E lower + ¼.1331

The negative × negative is positive.1351

¼ X sub E upper ν sub E upper.1355

Our final equation comes down to this.1366

When I put some things together here, I'm going to get ν observed equal1372

to the upper electronic state - the lower electronic state +,1381

I'm going to combine different terms, ½ ν sub E upper - 1/4 X sub E upper ν sub E upper - ½ ν sub E lower1395

- ¼ X sub E lower ν sub E lower + ν sub E upper × R - X sub E upper ν sub E upper.1421

This X sub E ν sub E, this is a single parameter.1445

We just write it together.1447

× R × R + 1.1450

This equation right here, this gives us the frequency of the line that we see.1454

Let us break this down even further.1461

The frequency of the transition 0 to 1, 0 to 2, things like that.1463

1, 2, 3, 4, 5, 6, there are 6 terms in this equation.1475

The difference of the first two terms, in other words the E upper - E lower electronic is sometimes called T sub E.1490

It is the difference of the minima of the two potential energy curves.1540

In other words, it is going to be this energy - this energy.1565

That is TE.1571

If you want to put a little line there, a little line here, go like this.1575

The difference between been the minima.1582

The third and fourth terms of the equation that we just had,1585

they are just the anharmonic 0 point vibrational energies of the two states.1598

In other words, the third term in that equation, that represents the energy of that level.1623

The fourth term represents the energy of that level.1629

The 0 point energy of the two states.1635

The ground state, in other words.1638

For the first 4 terms taken together, the difference between the energy minima and1657

the difference between these two ground state energies,1664

Let me go ahead and write this down.1669

The first 4 terms taken together represents the frequency of the 0 to 0 vibronic transition.1671

The transition that goes from this level 0 in the lower electronic state to the R = 0 of the upper electronic state.1715

That, the frequency of that transition, that is what those 4 terms taken together represent.1727

The E upper, the E lower, and that third term and that 4th term.1735

We often symbolize this as ν 00 or sometimes ν 0 to 0, with a little arrow.1741

Some variation, thereof.1754

You put a comma, you do not put a comma, it is up to you.1756

Again, it represents the vibronic transition from the R = 0 state, ground state to the R = 0 state of the upper state.1761

The ground state of the upper electronic state, that is what that represents.1770

If we use the symbolism, either that one or this one, if we use the symbolism1776

to make our 6 term equation more compact, we get the following.1796

We get that the observed frequency of transition is equal to this ν 00 + ν sub E upper × R – X sub E upper ν sub E upper.1810

I think I should just put that is as 1, that is okay.1831

× R × R + 1.1834

Here, R is equal to 1, 2, 3, and so on.1838

Here, R is the vibration quantum number of the upper state, the one in that electronic state.1845

If we set R = 0 in this equation, we get the observed frequency of the 0 to 0 transition.1871

That is what we actually get.1902

The 0 to 0 vibronic transition, when we set R = 0 in this equation.1907

Let us see what we have got here.1925

Let us go ahead and go to blue.1933

Let me write it over here.1941

This is very very important, please make sure you understand that each electronic state, each potential energy curve,1944

it has its own set of spectroscopic parameters.1966

In other words, it has its own ν sub E, X sub E, ν sub E.1993

It has its own B sub E and so on.2003

It is very important.2006

Those are the parameters that we are actually going to be solving for many of the problems.2007

Let us talk about this thing called the Franck-Condon principle, which actually is what this image really represents.2013

We see a lower electronic state, we see an upper electronic state.2019

We should have the vibration levels but all the vibration levels, we also have the actual wave functions.2022

These wave functions right here.2031

This image shows the wave functions.2034

If we were to take the square of the wave function, ψ², what we would get is the probability density.2037

The only difference between the wave function of a probability density is the same exact picture.2051

All of these curves, they would all be above the axes.2055

In other words, like this one right here, it would be curved up.2059

Everything would be above the axis because you square something, you end up getting something positive.2066

Now, let us talk about the Franck Condon principle.2073

Let me go ahead and do this in red.2085

We see that each electronic state has its own potential energy curve.2090

The minima of each state, the minima of the various states, in this particular case2098

I have 2 electronic states but I will say various because it is more than one electronic state, many of them.2114

The minima of the various electronic states do not necessarily lie on top of each other.2120

In other words, you notice this minima is right here.2162

This minima is right here.2166

There is a difference between them.2169

That difference is very important and you will see in a minute.2172

Do not necessarily lie on top of each other.2174

In other words, the R sub E for this state is different than the R sub E for this state.2180

Again, we know that already, they have different parameters.2189

The Franck Condon principle says that the electronic transitions happens very fast2193

because the electronic transitions happen in time frames that are instantaneous,2216

compared to the motion of the nuclei of the atoms involved.2241

Let us go ahead and say that.2269

The much more massive nuclei, in other words the electrons can move a lot faster and2274

move to other states a lot faster than the nuclei can actually adjust to the new state.2280

That is what is happening, the electron is so much smaller than a nucleus.2286

When it moves to a higher electronic state, it is there in a minute.2291

It is going to take a lot longer, relatively speaking, for the nuclei to adjust to that new electronic state2295

compared to the motion of much more massive nuclei.2302

Because of that, we can represent vibronic transitions as vertical lines.2309

This is one electronic state, this is another electronic state.2314

It is already been adjusted.2317

One electron actually move from one state to the other.2321

It is just going to jump straight up.2325

This electronic state, these wave functions represent the different vibrational levels of that state.2329

Here, the wave functions represent the different vibration levels of that state.2338

When electron makes a jump to a higher electronic state, it is just going to jump time wise because it happened quickly.2342

Relative to the motion of the nuclei, we can represent them as just a vertical leap.2348

Graphically, we represent it as just a straight vertical line from the ground state .2353

Therefore, on a diagram like the one above, like the ones that you see in your book,2367

the transition from the lower state to the upper electronic state is represented vertically.2383

When we represent a vibronic transition, we are representing it vertically.2397

There is a state, there is another state, it is going to go this way.2401

Where it lands have a relative, based on the wave function is the extent to which we are actually going to see that line of the spectrum.2405

Therefore, on the diagram, the transition is represented vertically.2415

Let me write up here.2426

Each curve in these diagrams shows the wave function for each value of R, the probability density ψ² look the same.2433

Except all the shadings are above the X axis.2476

Nothing that we do not know from our previous work in quantum mechanics.2486

What the Franck Condon principle does is, it gives us the relative intensities of the vibronic transitions.2495

Let us say the vibronic transition lines.2535

The lines that we see, some of them are going to be very strong lines.2537

Some of them are going to be very weak lines.2541

The strength and the weakness of those lines depends on the probability density of the electron2543

is going to be in that particular state there.2551

Here is what is going on.2563

We would be going from, let us say to 0 to 1 transition.2565

You look over here, the 0 transition from the ground state R = 0 up to level 1.2569

First of all, notice that this particular transition.2582

Because this in this particular image, for this state E1 and E0, the R value of E1 is if a significantly larger than the RE value of the E sub 0.2587

They are not on top of each other.2606

The transition that takes place, the vibronic transition we said it was vertically,2607

it actually ends up passing the one level and go straight to the two level.2611

At the two level, notice where it hits.2617

It actually hits where the density is rather high.2620

In this particular case, we might not even see a line for the 0 to 1 transition.2623

Because it does not even touch the potential energy curve from here, the place of maximum density2629

and the 0 vibration state for the lower electronic state to a place of maximum density.2636

For the upper actually ends up hitting for the level 2, that is where it hits.2642

That particular line is going to be very intense.2649

Maximum intensity, maximum intensity.2652

Here, you may or may not see a line for the 0 to 1 transition.2656

You are definitely not going to see one for the 0 to 0 transition.2660

You may or may not.2662

Again, a little bit of density out here but it is outside of the potential energy curve so you might not see anything at all.2665

What about the 0 to 3 transition?2672

The 0 to 3 transition, if we go straight up, we hit right about there.2675

It is a place of minimum density.2682

We will still probably see one but it may not be very strong.2684

It may not be very intense.2689

How about the 0 to 4 transition?2691

Let us see, where 0 to 4?2693

Over here, 0 to 4 transition we would go straight up.2695

Sorry about that, the 0 to 4 transition straight up.2699

It is probably going to be a little bit more intense than the 0 to 3 but not quite as intense as the 0 to 2.2703

And that is what the Franck Condon principal says.2709

When you have one electronic state, you have another electronic state,2712

the transitions are going to take place vertically on these diagrams.2715

The intensity of the transition 0 to 1, 0 to 2, 0 to 3, depends on where you are actually going to hit2719

maximum probability density, minimum probability density, or somewhere in between.2727

You are just going to get a series of lines that have different intensities.2733

That is all the Franck Condon principle.2739

Let us see here.2745

Let me remind you.2752

It can happen that the upper states R sub E is significantly larger than the R sub E for the lower state.2757

In other words, the upper state can lie much further, not over but in a shifted away from the lower state,2791

such that the 00 transition may not even appear.2804

Sometimes the 0 to 1, 0 to 2 transitions do not even appear.2823

Sometimes the first transition that you see in the spectrum is maybe 0 to 3, 0 to 4, 0 to 5, and so on.2826

The relative intensities of each of those lines is going to depend on the probability density at that particular position.2833

Let us go ahead and do example and see if we can make sense.2844

The following data table lists the observed frequencies of the first 3 vibronic transitions of hydrogen gas2850

to a certain excited electronic state.2857

We see a line at 121 to 76 inverse cm for the 0 to 0 transition.2863

We see a line at 123 to 70 for the 0 to 1 transition.2869

And we see 124 to 438 for the 0 to 2 transition.2874

In this case, we do see 3 lines.2878

We do not know what the relative intensities are.2879

At this point, that is a separate problem, I do not know.2882

We see the 0 to 0, 0 to 1, 0 to 5 vibronic transitions.2885

These are the frequencies that we see, that we observe on the lines on the spectrum.2889

Use this data to find the values of the spectroscopic parameters, ν sub E upper and X sub E ν for the upper excited state.2894

We are going from 0 to 0, 0 to 1, 0 to 2.2907

We want you to use this information, these 3 lines on this electronic spectrum to actually2910

find spectroscopic parameters for the upper electronic state, and this is how we do it.2917

Let me see our equation.2926

Let me go ahead and do this in red, I think.2928

I’m getting really tired of writing here.2935

I apologize if my writing is sloppier.2937

Our equation for the observed frequencies, the vibronic transitions is ν observed is equal to ν 00 +2940

ν sub E upper × R – X sub E upper ν sub E upper × R × R + 1.2955

Let us go ahead and take R = 0, 1, 2.2968

When R is equal to 0, it represents the 0 to 0 transition.2971

In other words, just the ν sub 00.2975

Let us go ahead and do the 0 to 0 transition.2979

The 0 to 0 transition, let me actually write down each one so we have everything.2981

When R is equal to 0 that represents the 0 to 0 vibronic transition.2989

Our ν observed is going to equal ν sub 00 +, if R is 0 this term is 0.2996

If R is 0, this term is 0. 0 + 0, we end up with ν observed = ν sub 00 that is equal to 120,176.3005

This one of the equations that we want.3025

We will call it equation 1.3028

Let us go ahead and deal with the R = 1 case.3032

This represents the transition from 0 to 1.3035

In this particular case, ν observed is equal to ν 00 +,3038

We are putting 1 now, R into the equation.3046

+ ν sub E × 1 - X sub E upper ν sub E upper × 1 × 1 + 1.3051

We end up with ν observed = ν 00 + ν sub E upper -2 X sub E upper ν sub E upper.3066

This one was equal to 122,370 inverse cm.3084

This is our second equation that we have.3090

Let us go ahead and find the R = 2.3096

This represents the transition from the 0 to 2 vibronic line.3100

Here we have ν observed = ν 00.3107

Ν sub E upper × 2 – X sub E upper ν sub E upper × 2 × 2 + 1.3116

The equation that we get is ν 00 + 2 × ν sub E upper -6 × X sub E ν sub E.3133

Upper upper ~ ~, and that one, the table said is 124,438 inverse cm.3147

In order to solve this, I have 3 equations and a couple unknowns.3158

I'm going to go ahead and take equation number 2.3162

This is this one right here, it is going to be equation number 3.3168

I’m going to take the equation 2 - equation 1.3173

When I take equation 2 - equation 1, I end up with the following.3180

I end up with ν sub E upper -2 X sub E upper ν sub E upper is equal to 2194 inverse cm.3190

We will call this one equation A.3207

When I take equation 3 - equation 1, I end up with 2 ν sub E upper -6 X sub E ν sub E both upper, both ~, and I end up with 4262.3210

This one is going to be my equation B.3242

I’m going to solve equation A and equation B simultaneously.3247

I'm not going to keep writing out these X sub E and ν sub E stuff.3252

I’m just going to call it S and T.3258

I’m going to let S equal to this ν sub E upper and I’m going to let T = this X sub E upper ν sub E upper.3261

Remember that is a single a parameter taken together.3273

What I end up with is the following equation.3278

I get S -2 T = 2194 and I get 2S - 6T = 4262.3280

2S - 4T multiply the top by 2, I get 4388.3295

I will not do this for you but what the hell.3303

-6 T = 4262.3306

I subtract and I end up with 2T = 126.3312

T = 63.3318

I get S - 2 × 63 is equal to 2194.3325

I want to make sure my numbers are right here.3343

I get S is equal to 2320.3345

There we go, we said that S was equal to ν sub E, that is equal to 2320 inverse cm.3351

This is ν sub E upper.3360

That is what we are doing. We are finding the parameters for the upper state and T is equal to X sub E upper ~ ν sub E upper ~.3363

It is a single parameter, that is equal to 263 inverse cm.3374

There you go.3382

We finished this problem, I thought you guys might like to see what a particular table of parameters of states actually looks like,3385

if you happen to be interested.3395

If not, not a big deal.3397

If so, this is what it looks like.3398

This is from the NIST website, the National Institute of Standards and Technology.3403

They have a bunch of databases, a bunch of spectroscopic databases, all kinds of things.3409

You should check it out.3416

If you want to see for yourself, basically what you are going to do is you are going to go to,3418

web book.NIST.gov/chemistry.3428

Under general search, click formula or however you want to search.3444

I generally just click formula.3456

Enter molecule in the box on line 1.3462

Check off the box that says constants of diatomic molecules and click search.3478

After that, you are going to scroll down to give you some information and they will give you this very long table.3502

I have only taken a section of this table and they will go all the way down.3510

Scroll down until you see the table and we are interested in the ground state, it is at the bottom of the table.3515

This is the bottom of the table photograph that I actually took.3538

The ground state is going to be represented by something like this.3548

You are going to see an X, you are going to see this singlet sigma +,3551

Do not worry about the term symbol for the electronic state,3558

I will be explaining what those mean in subsequent lessons but you want to look for this X and3561

you want to see this 0 here for the P sub E.3566

It is the ground state electronic energy, we set that equal to 0.3572

Notice, the first column, this is for the NIST website, this table is ω E.3579

For our purposes, this is our ν sub E.3585

Ν sub E, ω E, in this particular table you will also see it with an ω.3589

That is 29946.3594

This O sub E X sub E, this is the X sub E ν sub E.3598

That is that for the ground state.3605

Do not worry about that, here the B sub E that is the rotational constant.3609

There is the α sub E, that was the constant that had to do with the vibration rotation interaction.3614

Here is the dissociate energy.3623

Do not worry about that, do not worry about that.3626

Here is the R sub E, the equilibrium bond length.3630

This is the transition as represented.3633

And here is the ν sub 00.3636

Very important, that was the difference between energies of the ground vibrational state in lower electronic state3640

and the ground vibrational state of the upper electronic state.3646

That is what those columns mean.3650

For the states are concerned, here is the ground state, you might jump up to let us say that excited electronic state.3653

You have a whole different set of parameters for each electronic state, for each potential energy curve.3665

I hope that helps.3678

This is not something that you need, most of the information is going to be provided for you in your problems.3680

You have tables in your books but I figured if you want to see, you can go ahead and see for yourself.3684

Thank you so much for joining us here at www.educator.com.3689

We will see you next time, bye.3692