 All right, welcome to this lecture on electronic spectroscopy, where the topic will be to talk about the spectroscopy or the types of light that are absorbed by molecules when you change their electronic states rather than their rotational or vibrational states. So we know that molecules can change their position in several different ways. They can translate, they can rotate, they can vibrate. But each of those motions involves changing the positions of the atoms of the molecule, the nuclei themselves, changing the locations of the individual atoms. So since we're moving the nuclei around, those are called nuclear states of the molecule. The energy levels that are produced when we solve the Schrodinger's equation for those types of motion are nuclear energy levels. Nuclear in this case doesn't have anything to do with radioactivity. The word nuclear here just means having to do with the nuclei. So we know that the spacing between those energy levels, the translational energy levels are based very closely together in most cases. Rotational energy levels are somewhat larger, but still less than kT in most cases, and the vibrational energy levels are spaced with an even larger spacing. But molecules, in addition to having nuclei, also have electrons. If we want to fully describe the state of the molecule, we need to specify not just where the nuclei are, but where the electrons are in that molecule. So if the electrons have their own potential energy, they have their own energy levels, and we can calculate the electronic energy levels of the molecule. So the question that's going to be relevant is what is the spacing between those electronic energy levels? Because as we've seen, if the spacing between the levels is very small, then there's many of them occupied at room temperature. If the spacing is large, then only the lowest levels, or perhaps even only the ground state level are substantially occupied at room temperature. So the main question is going to be how far apart are those electronic energy levels? So what we could do, of course, is what we did for translations and rotations and vibrations. We can write down the Schrodinger equation if we're able to solve, write down the potential energy of the equation. We can solve Schrodinger's equation, but we're not going to do that for the electronic states. Instead, I'll just point out that every time we solve that equation so far, the energy has ended up looking like some h's in a numerator, a mass in the denominator, and then some details that depend on specifically what the problem was. And the h's in the m's show up because there's an h squared over an m in the original Schrodinger equation in the kinetic energy term. So in general, what that means is the lighter an object is, the larger its energy levels will be. And that is one of the most important things that determines the energies of electronic energy levels. When we're talking about nuclei, the smallest nuclei we can have is a hydrogen atom, or hydrogen nucleus. Its mass is about one gram per mole, or in SI units, that's about 10 to the minus 27th kilograms. Mass of an electron, of course, is much lighter. Electrons are about 2,000 times lighter than a hydrogen atom. So the mass of an electron is much, much smaller than one gram per mole. It's only about 10 to the minus 30th kilograms. So that's about three orders of magnitude smaller than the mass of a hydrogen atom. So what that means is the energy levels for electronic states where we're moving light electrons around are going to be several orders of magnitude larger than the energy levels associated with moving nuclei around. So the spacings are going to be quite large. In terms of actual quantitative numbers, so we know rotation energy spacings are smaller than vibrational energy spacings, and we've just convinced ourselves that they're probably going to be even smaller than electronic energy spacings. When they're expressed in terms of wave numbers, we've already seen that rotational excitations usually take a few wave numbers, somewhere in the microwave portion of the spectrum. Vibrational energy excitations might take a few hundred or a few thousand wave numbers depending on what type of bond we're talking about, and that's in the infrared portion of the spectrum. Whereas electronic energy levels, as we'll see in just a second, typically require maybe 10,000 or 100,000 wave numbers in order to be excited. And that falls typically in the ultraviolet or the visible portions of the spectrum. So again, several orders of magnitude larger than the energy required to excite nuclear emotions. All right, so what are these electronic states? What do they mean exactly when I say an electronic state of a molecule? Let's start with what we know about vibrational and rotational states. If a molecule looks like a harmonic oscillator, it's got equally spaced vibrational levels and then stacked on top of those are lots of rotational states with smaller spacing. Each vibrational level has its own family of rotational levels. If we zoom out a little bit, we can see that those vibrational levels live within this well, which is not quite harmonic. It becomes anharmonic if we move to longer bond lengths. But that one potential energy well, that one bond, is one of only several different bonds that can cause the molecule to experience. So there's this family and this figure, the green, blue, purple, yellow and red curves are all different electronic states of the molecule. I show you another picture. These are all pictures based on real molecules. The picture on the left is for an O2 molecule, the energy levels on the right are for a nitrogen molecule and two. So you can see that there's several different curves. Each of those corresponds to a different electronic state. Each of those electronic states has its own family of vibrational levels. And the scale is such that we can't see the very finely spaced rotational levels stacked on top of the vibrational levels. So now, in terms of actual units, the spacing in this nitrogen example, the spacing between the ground vibrational level in the lowest electronic state and the ground vibrational level in the next electronic state, works out to be about 50,000 wave numbers. So that's just an illustration of the fact that the electronic states are spaced by tens of thousands of wave numbers typically. So what are these electronic states? What do they mean? If I ask you to describe the electronic state of an atom, let's start with atoms because they're a little simpler than diatomic molecules, if I were to ask, let's say, a general chemistry student, tell me what you can about the electrons that live in a nitrogen atom. That general chemistry student might give me an electron configuration. Nitrogen has nine electrons, so the electron configuration is 1s2, 2s2, 2p5, for example. So that's a way of specifying the state of the electrons in the nitrogen atom. Some of them live in S orbital, some of them live in P orbitals. There's another way of describing the symmetry, the properties of those electrons a little more compactly. We give it something called a term symbol. I won't explain exactly what this means, but we can say that the term symbol that describes those electrons in a nitrogen atom is this superscript 4 followed by an S. We would pronounce that quadruplet S. It's in a quadruplet S state. And again, I won't specify exactly what that means. Think of that as just a label we use to describe the electrons in this nitrogen atom. There's more than one possible electronic state for the electrons in that atom. So for example, if the molecule was not in its ground state, if I didn't put the electrons in the lowest possible energy levels, I could excite one of them. For example, let me take one of these electrons. Actually, there's a typo here. Take one of these electrons from the 2p5 configuration and excite it up to a 3s1 configuration. And that's a different electronic state of the molecule. I've taken one of the electrons out of the 2p orbital and I've promoted it to a 3s orbital. That would give it a different term symbol. So the point is we can put electrons in different orbitals, give them different energies. Those all correspond to different states of the molecule. That's really all we mean by electronic states. When we move on to molecules rather than atoms, things get a little bit more complicated. Those of you who remember MO theory or the very small minority of you who enjoyed MO theory might be able to remember that the electron configuration for an N2 molecule is this long string sigma 1s2, sigma 1s anti-bonding 2, et cetera. So that's just a way of describing where are the electrons in the nitrogen molecule? Are they in sigma orbitals or pi orbitals? Are they in bonding orbitals or anti-bonding orbitals? And just like the nitrogen atom, we can give that electron configuration a term symbol to describe the symmetry of the electrons in various ways. So I'll give a few details of what this term symbol means in just a second. But again, the ground electronic state of the molecule where all the electrons are in the lowest possible energy levels is not the only one. We can excite that some of those electrons out of the lower lying levels up to higher lying levels and that would correspond to some different electronic state. So rather than writing out electron configurations, I'll just say that the term symbols for the ground state or the excited states, we might write them as singlet sigma or doublet sigma or triplet sigma and so on. So each of those term symbols looks something like this. There might be superscripts before or after a capital Greek letter and there might be a subscript that is a G or a U. And those all have different meanings. In this class, we won't get into the details of how to understand the term symbols in detail. I'll just mention what they are, but you won't be responsible for them. The capital Greek letter, sigma or pi or delta, that's a lot like the S or P or D that would describe orbitals of an electron and a hydrogen atom. S, sigma being the Greek version of an S, pi being the Greek version of a P and delta being the Greek version of a D. Essentially what's that's describing is the total angular momentum of the electrons in this molecule. There's a subscript that describes whether the electrons are symmetric or anti-symmetric with respect to reflection of a particular type. There's another superscript which corresponds to a different type of symmetry called parity. Those, we're not gonna think about too much. The one we are gonna be concerned with is this superscript number in front of the Greek letter which describes the spin of the molecule in particular the total spin of the molecule in a way that describes its multiplicity. So for example, we know that electrons can have spin of plus one half or minus one half, spin up or spin down. If the total spin in the molecule is zero, if the spin quantum number S is zero, then this quantity two S plus one would be one and we call that molecule a singlet. It's in a singlet state. There's only one way for that spinless molecule to exist. On the other hand, if the spin is one half and the spin could either be up or down, two S plus one, one S is equal to half, that turns out to be two and that's the doublet molecule which refers to the fact that there's two configurations the spin can exist in. So all we need to know for right now is this superscript number describes the spin of the molecule. Okay, so what else can we say about the electronic states of the molecule? Look at the curves on the right side of the screen here. If I put the electrons in a different configuration, if I promote some of them to an antibonding orbital, if I move them from a sigma orbital to a pi orbital and so on, that's gonna change where the electrons are in the molecule. It's gonna change, for example, what the bond order of the molecule is. If I promote, if I take let's say a nitrogen atom and I promote some of the atoms from bonding orbitals to antibonding orbitals, I can convert that triple bonded nitrogen atom in its ground state to a double bonded or to a single bonded molecule. So I can change the bond order of the molecule by moving the electrons into different orbitals. By changing the bond order, I can certainly weaken or sometimes strengthen the bond. So on this diagram, the way I recognize a weaker or a stronger bond is by how much energy it takes to dissociate the bond. So in this case, the ground state of the molecule, nitrogen is a triple bonded molecule in its ground state. It takes a lot of energy, this much energy to dissociate that molecule but in one of the different electron configurations, one of the different electronic states, it takes less energy to dissociate the molecule. So each one of these curves has a different dissociation energy corresponding to how strong the bond is. I can also, by changing the electrons, that also has an effect on the bond length of the molecule and that's also visible in this figure. The ground state of the molecule, the bottom of the well is located at this particular bond length, whereas some of the excited states have the bond lengths that are a little bit different. And that makes sense if I weaken the bond, if I take nitrogen with a triple bond, compared to nitrogen with a double bond or a single bond, the weaker bonds tend to have longer bond lengths. And you can see that in this figure, that the weaker bonds tend to have longer bond lengths. We can also see that the different curves on this diagram all have different curvatures, which we think of as the spring constant. Remember when we solve for the vibrational energies, the curvature of this well, when we treat it as a harmonic oscillator, determine the frequency of the harmonic oscillator, the frequency with which the molecules bond vibrates. So some of these molecules have relatively soft spring constants, some have relatively stiff spring constants, and that affects the spacing of the vibrational energy levels and the frequency of the molecules. And the last difference between all these different curves is, of course, what happens when you stretch the bond to the point when the bond dissociates, if I have N2 molecules and I stretch the bond, what I end up with after dissociation is two nitrogen atoms. And those nitrogen atoms themselves have electrons. You can see on the diagram that sometimes when I dissociate the bonds, the products have one energy, and sometimes when I dissociate a bond, the products have a different energy. And that's accounted for the fact that the nitrogen atoms themselves dissociated into different electron configurations, either dissociated into nitrogen atoms in the ground state or I dissociated into nitrogen atoms that themselves may be in an excited electronic configuration. So some of those details we can get just by looking at this diagram. All right, so what we have learned about electronic states so far is that by putting electrons into different orbitals, different positions within the molecule, different wave functions, that can certainly change the properties of the molecule, I can change its energy, I can change its bond length, I can change its vibrational frequency. Every one of those electronic states is not just one state, it is a description of the molecule that allows it to vibrate with different frequencies and therefore have a completely different vibrational ladder of vibrational states that live within that electronic state. So this electronic state has its own vibrational ladder, this electronic state has its own vibrational ladder and so on. The spacing between those electronic states is quite large compared to the spacing between vibrational energy states. It's on the order of tens or hundreds of thousands of wave numbers so that requires UV or visible light to excite it. And then when we want to describe these electronic states, usually what we do is we give them this cryptic label like singlet sigma plus G and the important part of that label for us in the near future will be the spin term, the singlet doublet triplet portion of that term. So what we're gonna do next is combine all that information and use it to describe the types of light that molecules will be able to absorb but that will be in the next lecture.