Lecture 2.1: Atomic and Molecular Orbitals
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[Slide 1] Hello and welcome back to nanoHUB-U's course on Organic Electronic Devices. Over the first five lectures in this course, we've talked a lot about organic semiconductors. And in particular, we've talked about the molecular characteristics of these materials, how to synthesize these materials and how to characterize the molecular, thermal and structural properties of organic semiconductors. We now have a kind of a toolkit by which we can talk about all the different kinds of materials that we could make, desire or want. Now, we're going to move on and think about the electronic orbital theory of these materials. And by talking about electronic theory, we'll be able to think about how the design organic semiconductors to achieve the performance we'd like to for certain device applications. And once we understand the electronic structure, we'll be able to talk about electronic transport and once we're able to talk about electronic transport, we're able to talk about the devices you could make from organic and Polymer based semiconductors. So, let's get started with today's lecture where we will cover atomic and molecular orbital theory. 

[Slide 2] And really in this lecture, what we'll talk about first is the atomic orbital levels, and the classic example of this is that regarding the hydrogen atom. It's a very simple example but it's a great model for thinking about how we talk about the structure of orbitals within molecules. One, we have that under our belt we'll talk about molecular orbital theory and extend it from a simple hydrogen atom how to bond the carbon-carbon bonds and you'll see we'll start to build up what looks like the beginning of small molecular organic semiconductors or even polymer semiconductors. Then at the very end of the lecture today, we'll talk about the origin of transport bands and organic semiconductors and hopefully, we'll see how those can arise from a simple molecular orbital kind of picture. And at the end of the lecture today, I hope we've accomplished the following learning objectives, OK? And the first of those is we need to be able to define what we mean by an atomic or molecular orbital. We need to be able to talk about the corresponding energies associated with these molecular orbitals. We also like to be able to talk about the interplay between electron density and molecules and comment on what I mean when I say a bonding or an antibonding orbital. Finally, we should be able to describe what I mean when I say the HOMO or the LUMO level of a material, and how that will in turn interplay into this molecular orbital theory. So, with those concepts in mind let's get started and start walking down this path towards how electrons can move and how electrons look when they're next to a molecular semiconductor. 

[Slide 3] And one thing that we need to keep in mind during all of this is the idea that electrons can exhibit either particle like behavior or wave like behavior. So, this wave particle duality will play a key role in how we think about these materials. And during this course, sometimes, it will be more helpful if we think about it in terms of ways and sometimes it'll be more helpful if we think about the electrons in terms of particles. And really depend on what application we're looking at, OK? So, when we think about electrons as particles sometimes that's easier to do, sometimes it's easier to think about them as particles when we're talking about how we have a bonding structure. So, in just a second, we'll talk about feeling up different orbital shells with particles and we'll draw them as arrows and it looks very easy if we say that there's one electron there. This whole time we need to keep in mind that we're also going to be talking about waves as well. We need to think about that when we have electrons we can have two electrons that have identical quantum numbers but with different spins, so, well, you have either a spin up or a spin down. And this is in conjunction with the Pauli Exclusion Principle it says if electron has four quantum numbers no other electron can have the same four quantum numbers. So, we'll have three but then the last one will be the spin up or the spin down state. We'll talk a little bit about Hund's rule and in future lectures when we get to the device section of the course we'll talk about electrons as particles as they move through the materials. And for a lot of this second segment of lectures, we'll be talking about way up electrons as ways. And in the next lecture, we'll introduce Schrodinger's equation and that will really introduce this wave kind of behavior of the materials and from this, we'll be able to derive a lot of mathematical expressions to describe the probability of finding electron in a given space. And by solving Schrodinger's equation, not only we'll be able to think about the probability, we'll also be able to think about the energy of those electrons. So, really the moral of this slide is that when we're going through this course, please remember to think about electrons both as particles and as waves and to be able to tailor your thinking about them as appropriate for the model we're using. Now, in reality, these electrons are going to be in both state simultaneously but sometimes one model looks at a particle behavior and sometimes it looks as a wavelike behavior. 

[Slide 4] So, you guys all have seen this periodic table before, right? And we have a variety of elements and some of our favorite elements for organic materials are these carbons, right? And we also have hydrogen and in this hydrogen here we know that we know that we have hydrogen has a one here because it has one electron, right? Carbon has six electrons, right, if we move down to sulfur for thinking about thiophenes, we know that it has 16 electrons. So, where these guys are placed on a periodic table is corresponding to the number of electrons it has. And maybe if you remember back to chemistry class you'll remember that these electrons fill in a very certain way. And there's a very nice easy table to remember how these electrons fill. And it goes like this and we start here and for hydrogen it was one electron and it fills this 1s shell first, right? And the exact number and letter associated with these won't matter so much in this course. But the different numbers and letters and there will be more numbers coming up have to do with the quantum numbers of the materials. But we just want to talk about the general shape and the general probability distribution of different types of electrons. So, hydrogen will have a 1s electron and it turns out that 1s orbital can occupy two electrons, one will spin up and one will spin down. You can move to a 2s and then on to a 2p, a 3s through 3p and on to 4s. And organic semiconductors generally we don't have that many electrons that we move beyond the state although there are some metals that will be incorporated into our materials that might. But this is a general feeling strategy for our electrons, so for instance if I wanted to fill six electrons which would be carbon, I could put two in the 1s, I could put another two in the 2s and I could put another two in the 2p and that's how I fill up my six electrons. And we'll go over some examples of how to do that in just a second. But this is just a general feel of how we would fill up the orbitals in our materials. 

[Slide 5] So, what are these different S, Ps and things of that nature look like. Well, what I've shown here, is basically the probability distribution associated with each one of those kinds of orbitals. So, here we see that we have S type, we have P type, we have D type and we have F type. And what you see is the probability distribution of finding electron where the origin of those materials is the nuclei of that atom. So, in this case, for instance the nuclei of the hydrogen atom would be setting right here. And around it, that one electron has a high probability of being found somewhere within a sphere there. And as that red ball that you see right there. If we move to some of the head P type orbitals, we see that the nuclei is still right here in the center and then we have a lobe on either side. So, sometimes these are called dumbbell shaped lobes, and those are the P orbitals and you can see that they can either be aligned on the X axis, the Z axis or the Y axis. So, we'll talk about Px, Pz and Py type orbitals. You'll note the S group that we'll talk about a lot as well is symmetric all the way around, so they'll just be an S type orbital. Now, you may be wondering how these different probability distributions arise and why they're different shapes for different letters, S, P, D and F. Well, we'll talk about that in just a couple of lectures but it has to do with solving the Schrodinger equation. But just for right now let's back up and think that we can basically have one kind of S orbital, three kinds of P orbitals, five kinds of D orbitals and seven kinds of F orbitals. And the relative shapes of the probabilities are shown there. 

[Slide 6] So, let's go back to our friend hydrogen, we know that it has one electron and let's take just a hydrogen atom, it's not bonded together yet, it's not H2, it's just H. And if I had that, I know that I have a nuclei which I have drawn as a black dot there and around it is some electron distribution. And that electron is in the 1s orbital and I've drawn it as a spin up arrow here. So, this is spin up and as a single electron so it's a single headed arrow. If I wanted to look at another hydrogen atom I could draw that as well. So, now, there's some metric like that, we have two hydrogen atoms separated from each other but if I start to bring them into contact you know that these hydrogen atoms who wants to try to bond. When they bond they're going to form H2. And when that happens we'll create a molecular--, we'll create a molecular orbital here, for the H2 molecule. And this is made of an atomic orbital of 1s and 1s and they come together to form a sigma molecular orbital. And you'll note that the number of molecular orbitals is going to be equal to the number of atomic orbitals from which they are made. So, I have two atomic orbitals one from each one of the hydrogen atoms but I've only displayed one molecular orbital, that's called the bonding molecular orbital. You'll note that in this bonding diagram here that we've obeyed Pauli exclusion principle by now having one be spin up and one be spin down. So, they're not the exact same electron. But what we'll see here is that when they come together they'll bond. And you'll see that they'll now share the electron distribution with the nuclei spread out evenly and then the electron distribution throughout. We can also have an alternate strategy, we can have what we call the antibonding orbital or sigma*. So, now, that's where our second molecular orbital arises and you can see that there's a gap and energy between the bonding and the antibonding. And you can see maybe it's apparent that the reason why is called antibonding is because it's a higher in energy than the two composite atomic orbitals and the bonding lowers the energy and this is more favorable to happen spontaneously. If we did have the hydrogen atom in an antibonding state it would have a different electron distribution. In fact the two atoms that remain relatively in the same spot, the two nuclei, but instead we would have a distribution of the electrons in a different manner. So, now, they're not sharing it in the center so much as in the case of the bonding orbital but now they're distributed out further away and we have what we call a nodal surface or a nodal plane here in between the two nuclei that there's a zero probability of finding the electron. So, that's how we move from atomic orbitals to molecular orbitals. In the case of the hydrogen atom it's fairly straight forward because we only have to deal with one electron from each system. 

[Slide 7] Now, we can move on to more complicated systems. And now we're going to start talking about carbon. And we'll talk about Frontier Orbitals a lot because that's a lot of a time where the bonding action happens. And when I say frontier orbital what I mean is the orbital that's the highest occupied one. So, in the last case we had hydrogen the highest form was 1s but in carbon, remember, we said we'd go from 1s to 2s to 2p so our frontier orbitals are the two as in 2p kind of level. So, if we look at this we've talk about this already, we filled up the 2s orbital and we filled up one, two of the 2ps, right? So, that's how we have four electrons there, the other two to make six are in the 1s orbital, we're only talk about that frontier orbital. Now, what can happen is that we can have hybridization. And this increases the number of unpaired electrons. So, now, we've gone from two paired electrons and two unpaired electrons to four unpaired electrons by this hybridization. And this is part of Hund's rule that says that unpaired electrons are preferred in terms of energy. So, what we've done here is that we've hybridized an S orbital at two of the P orbitals to form three what we call sp2 orbitals. So, there's 1s and 2ps in it so it's sp2 and we've left one Pz orbital all by itself. So, now we have sp2 hybridization. And it turns out that this kind of hybridization will be very powerful when we talk about the bonding structure of conjugated macromolecules or conjugated small molecules. So, here's this again we've taken a 2s and at Px and a Py to form this bonding structure and when unchanged Pz orbital. And what does this allows to do, well, now we can start thinking about it, right? So, one of those bonds is going to go on to a sigma bond which going to connect two of the carbons together. We have to still remember that we have some CH bonds here and here. Another CH bond here and here but now we have this pi bonds, right? If you remember from all our chemical structures we drew in the first five lectures those are going to be those double bonds that we see a lot. So, this sp2 hybridization naturally occurs in conjugated materials. 

[Slide 8] Let's look at an example of 1,3-Butadiene, this will be our model material for quite sometime here but here we see that we have a-- a double bond, single bond, double bond, and we have one, two, three, four carbons and they all have sp2 hybridization. So, let's look at the molecular orbital diagram. We're going to say that we have that 1Pz and 3 sp2s, and there's going to four of those, right? So, if we look at that, these molecules here, these orbitals are not going in to anything besides the sigma bond and the CH bonds. Well, we're really interested is this Pz bonds because they're going to tell us about our pi bonding. So, let's just look at the 4 Pz bonds, remember, this is for each one of the molecules we'll have a Pz-- I'm sorry, for each one of the carbon atoms who have a Pz and we have one, two, three, four carbons. So, if we have four carbons, we have 4Pzs and they'll all have ones spin up right now, all right? And they can either have bonding orbitals or antibonding orbitals. And I'll call those pi1, pi2 or pi3*, and pi4* where the star indicates that it is some antibonding orbital. So, how is this going to fill? Well, it's going to first pair down below because those are lower in energy. And we happen to fill up all for electrons in the two orbitals that are bonding orbitals right there. So, now, we have a nice structure where you have a difference between our boding and antibonding. And in this case we need to talk about terminology. So, we have a level here that's occupied by electrons and a low that's slightly higher that's occupied by electrons, right? Because energy increases this way. So, if this is our highest occupied molecular orbital that's our HOMO level. We then have a gap in energy and then we have the next lowest energy state and that's our lowest on occupied molecular orbital or our LUMO level. So, that's what we talk about when we talk about HOMO and a LUMO energy level is the difference between where the electrons are occupied and where they're unoccupied. And that gap between the highest occupied and the lowest unoccupied molecular orbital levels will play critical role in some of our transport properties later on in this course. 

[Slide 9] So, now, we've just talked about an isolated molecule, right? Just a single butadiene chain, all right? And it's in the isolated environment, right? But when we talk about organic electronic devices everything I've shown you have been very solid state. So, we're going to have one gram per centimeter cubed, right, we're going to have Avogadro's number of molecules all aligned together and the likelihood that they all have the exact same confirmation is very unlikely. So, if we had a single molecule of butadiene it would have a discrete HOMO level and a discrete LUMO level. But maybe the guy right next to it maybe it's just slightly changed, maybe it has a different energetics because the chains came to different way slight rotation of the bond. So, it has a slightly raised value for its HOMO and LUMO. And there maybe the guy next to it has a slightly altered HOMO-LUMO level as well. And if I have 10 to the 23rd different molecules looking like this becomes too hard to distinguish any individual molecule, right? And what we give is bands. And we'll call this the solid state HOMO band and the LUMO band, where we have all this very narrowly confined molecular orbitals right next to one another. And in terms out there in very close proximity remember that there's roughly 26 milli electron volts of energy available to us at room temperature. So, if these things are in the 10s of milli electron volts away from one another more or less at room temperature they're just as discussible to anything as the other one. So, we need to think about these bands as being very discrete and very separated though. So, we'll have an average HOMO level and it will be the top of it will be here so this is the HOMO and the bottom of it will be LUMO but they'll be discrete very closely space bands very near it. We can see how solid state transport starting to occur. And if you're familiar with the solid state physics you'll see that is clearly lends itself to start calling this the band gap, right? The different between the HOMO energy level and a LUMO energy level. So, that's your band gap energy that you might recall from classic semiconductor physics. 

[Slide 10] Now, I would like to draw a couple analogies between the two cases right here, that is organic materials and inorganic materials. The bonding in these materials is different, the transport in this materials is going to be different. But sometimes we like to draw analogies between the organic world and the inorganic or classical semiconductor world. And a few of this include the fact that we have a HOMO energy and that's a keen to something like a valence energy or a valence band energy that you may have heard about in inorganic semiconductors. And a similar manner we'll talk about the LUMO level and that's a keen to a conduction band in inorganic semiconductors. And the band gap energy again is what we call the difference the HOMO and LUMO that would be the difference in energy between the conduction and valence bands in inorganic systems. So, there are analogies that we try to have between the more-- more established inorganic world and up in coming organic semiconductor kind of literature. So, you should be aware of those and be able to incorporate them as we go throughout this course. 

[Slide 11] With that I hope what you seen today is the fact that we've been able to talk about atomic orbitals and how they an correspond back to molecular orbitals, what they're general probability distribution shapes look like in real space. And hybridization plays an important role and conjugate the polymers. We talked about the HOMO level and the LUMO level of individual molecules and how that can transport or transfer into the solid state and we've talk about the bands of HOMO and band of LUMO as well. With that we're going to well set off for our next time I see you when we talk about using quantum mechanics to think about solving the Schrodinger's equation and setting up the probability distribution of electrons. Thank you for your attention and I look forward seeing you again on the next installment of organic electronic devices.