 Hello! Welcome to day number three. So today we're gonna have a day of quantum chemistry by our senior research scientist, Nathan Fritz Backdrick. He's coming from the UK. He's a continuum. And just to let you know, so pretty much today's morning schedule like our lectures with 10 minute breaks. Then we're gonna have lunch here at 1.30. We have to take the shuttle or walk down to the lab again at the Adratico guest house. So at 2 p.m. we're gonna have quantum chemistry labs today. So make sure everything is charged, the charging points down in the lab. But without any further delay, let's start our day on quantum chemistry. Thank you. So I'm going to take you back quantum computational chemistry. I hope you're looking forward to four hours of me lecturing today. I am looking forward to it. So the idea is essentially to start from the ground, from ground zero basically. So I imagine some of you from computer science background haven't even done quantum chemistry before. And then hopefully even for the physicists, this might be some new stuff. And hopefully try to give you some of the intuition behind quantum chemistry calculation that I think is quite mysterious to a lot of physicists. So why study chemistry? So at the basic level is why do atoms stick together? Which is known as the study of chemical reactions. So the study of chemical reactions essentially is the analysis of bond breaking and bond forming reactions. So you can kind of, you may have seen these curly arrow drawings around, maybe giving you nightmares from your undergraduate part. So these are extremely qualitative and it's kind of amazing that these work describe chemical reactions so well. But you'll see some of just the simple analysis working from atomic orbitals gives you this picture. And it leads to a huge amount of chemical and biological complexity. So particularly if you think, just think about the atomic orbitals, how they interact and the electron density, you can actually build up quite a good picture of chemical bonding. But if you want to go a bit deeper, why study chemistry on computers? So we have essentially, is energy a function of nuclear geometry? Okay. And we have this thing called the potential energy surface. Where each point on this surface is a quantum chemistry calculation. It can be semi empirical. The level of theory is up to you, it's your choice. Obviously you'll get better results if you use a more advanced level of theory. And you can see at the minimum to these point A and C, these are what we call equilibrium geometries. This is where the molecules are stable. It likes to sit in these things. And these are like the, these are what, these are the kind of the relaxed geometries. But obviously to do it to form a bond, we've got to go from A to C, means to go by B. B is known as a transition state. That's where the bond breaking happens. So you can see, if we go along this reaction coordinate, where the reaction coordinate is this hydrogen, sorry, is this hydrogen, this white thing passed between the two oxygens, you can see it doesn't like being in the middle. So there's a high energy here. But obviously to get from A to C by B, we need to increase the energy of the system. Okay. So this is particularly important for biological processes. And studying these accurately because if you look here, we have, for example, this is the oxidization of glucose. This is important for us to be alive. And without an enzyme, it requires a lot more energy to kind of go from this equilibrium geometry of glucose to become dark sodium water. So ideally we want to understand why enzymes can reduce the energy barrier. But obviously the problem is that enzymes these biological systems are huge, loads of degrees of freedom. Therefore, we can't really study all the quantum interaction. So this is one of the motivations of quantum computing is it allows us to break the scaling problem because we can go two to the n number of degrees of freedom. So typically the best results at the moment are kind of quite inaccurate for these strongly correlated systems. And the really famous one is the FAMOCO in the nitrogenase. So the FAMOCO is the reactive center in the nitrogenase enzyme which is found in the soil. It uses sunlight to convert nitrogen, hydrogen, and ammonia, which is important for fertilizers. But we have to deal with the hard process which uses 2% of the most energy on a day-to-day basis. So if we could solve this and if we can figure out why the reaction barrier is so low for FAMOCO, we can save a huge amount of energy. And this is kind of the post a child, this is FAMOCO versus hard processes, why we need quantum computing. Okay, so briefly I just talk about, so the potential area surface is this kind of when you go into high dimensions. So I showed you previously a two-dimensional one. Here we have kind of, this is a three-dimensional one and it actually generalizes to three n minus six because for every atom you have three dimensions with x, y, z, but then obviously you have translational and global translational and rotational which removes six to three to freedom from the problem. So obviously the minimum are these reactions and products and these saddle points, the transition states, and you can do this in numerically formally with kind of Hessian analysis, where the Hessian is the two-dimensional matrix of the second-world derivatives. And then interestingly the molecular properties come from the slopes and the curvature of potential areas. For example, if you've got a very steep well, it's going to want to fall down to that equilibrium geometry very quickly. So there's a very, the reaction, the force required for that reaction to happen is more. But for the purpose of this talk, we're not going to worry about potential area surfaces. So moving on potential area surfaces all causes chemistry, but in order to get a good potential area surface, we need to calculate a single energy point for a given geometry. And then for the rest of the talk, we're just going to be focusing on one awkward point because if you use kind of the cheap methods like DFT or semi-empirical, you'll get potential area surface, but it might not be the right one. So we need to get accurate potential area surfaces. And to do that, we need to use very high-order methods like quantum computing or exact diagonalization if you're familiar with that. Okay. Okay, so I'm going to start kind of what you may have seen before. So just kind of want to introduce the quantum chemistry Hamiltonian. So the quantum chemistry Hamiltonian is the fully interacting Fermion problem, where we have this, essentially we have these one body interactions and these two body interactions. And then we have these fermionic creationation operators, which I'll introduce later. But essentially if you kind of squint, you can kind of see that it's like the Fermion Hubard model, where rather than having nearest neighbor interactions, we have all possible global interactions. Okay. And then rather than having these parameterized weightings for these interactions, these G and H, these are actually these spatially dependent objects, which contain a huge amount of information. And they're actually responsible. There's many, many years of development just, just find these parameters. So these parameters contain all the information of, so this shape here, this orbital, this is just one of these indexes here. So to get all the information for PJ, PQRS, each individual indices, it contains, contains like the shape and the geometry of these molecular, these molecular orbitals. So I'm going to be speaking quite a lot, who for the first lecture and a half about basically just obtaining these molecular orbitals. Because if you start with a, you can just start here in quantum computing. If you want to be naive, you can just say, you may have used pi SCF or some other driver. But what's that? What's that? That gives you your quantum computing Hamiltonian, which you then give to a VQ or phase estimation or whatever. But it's like what? To actually get these weightings is a huge amount of work. And to actually understand the chemistry, to understand how you get the weightings of the interactions. Okay. So we're going to speak about all the basis sets, integral evaluation, and most importantly, Hartree-Fox theory. Okay. So I'm going to start from the total basics. There may be people who haven't studied quantum mechanics at a high level here. Like obviously there's lots of people who do computer science in this field. So essentially, we're only going to be worried about the time independent Schrodinger equation. Because the quantum chemistry Hamiltonian is just dependent on, there's no time dependent term. So we can just use this time independent Schrodinger equation. And essentially, the time independent Schrodinger equation will always have this formula. We'll have a kinetic energy term, and you'll have a potential term. And in three dimensions, the potential energy term is related to, is del squared. Sorry, kinetic energy term is del squared. And this will always be the same. But the behavior of the system is encoded in the potential of the Schrodinger equation. So depending on what application you have, you'll have a different v. And that will determine the overall wave function. Because this is essentially, this is a differential equation. So you can use, for a lot of the toy systems, you can use differential equation methods to solve for psi. And if you have different boundary conditions of terms in your Hamiltonian, you will end up with a different solution to differential equation. But the more of the story is, the Schrodinger equation really depends on the potential and the boundary conditions. So you may have seen it like this. This is what we call Hamilton eigenvalue equation, where we group the operator terms. So I should say this is, this is Planck's constant h bar. And this is the mass of your particle. So we're just talking about a single electron here, as I apologize. Single particle. And then we can group all these terms. We can put them as this h operator here. And then that will generate this kind of, or generate the same wave function, psi, and then this energy term, this energy waiting here. So this is all, this is what we're trying to solve to get the wave function given a potential and some boundary conditions. So Schrodinger when he came up with his wave equation, which I think was in 1927, I think, he then released, I think, six different papers solving time penetrating equation. Using lots of very powerful differential equation methods, which would come from kind of the plus and people like that, looking at planetary motion and things like that. So even though these equations are extremely scary, there's a lot of kind of, the theory was quite well known by this point. So the ubiquitous one that you will probably all solved in your first year or second year physics course is the particle in the box. And it's the simplest example of how potential, one d particle in the box, simple as, simple example how potential and boundary conditions can govern the wave function. And there's some slightly more difficult ones such as like the free particles and free dimensions. This introduces separation of variables, the idea of product wave functions. And then there's also the stationary hydrogen. So this is the one that we're really going to focus on today. So I don't have time to really, so some of the exercises will be to solve these if you haven't before. But what we're really going to focus on today is the solutions of the hydrogen atom. That's where all chemistry can be built up from. So the first, the first kind of solution of the Schroding equation is quite simple model, where you potentially have this one d dimension. And then you have these infinite potentials. And then you have zero potential between zero and L. Okay, so you can see the kinetic energy is kind of represented by this cool drawing that I made. So the Schroding equation looks at that. So it's kind of, it's a bit confusing, because there's no potential in this form of the equation. So we have, and this M here is the weight of the electron or the particle that you choose. But the particle is included in the boundary conditions of the solution. So one of the exercises will be to solve this. But essentially, the moral of the story is, when you solve this equation, you end up with these quantized energy solutions, which are given by this parameter N. So the rest of these are constants and L obviously as well. But given a fixed length of box and a fixed mass, N is the anything that can change. So what this represents essentially is you get standing waves between these two potentials. And each standing wave has the harmonics. These have different energies. So this is really the first indication that Schrodinger showed that you get quantized energy from this Schrodinger equation. So the stationary hydrogen atom is really, is kind of where quantum chemistry really started in my opinion. So here we have a single electron represented by M1. And you have this kinetic energy operator, del squared. And you have this V. So this V is this spherical, this is this spherical potential. Okay, so it's, it's just a totally symmetric spherical potential. Okay, so you can see this now. So it's just one over, so we have one over R squared. And this is the mass of the proton and the mass of electron up here. Okay. So essentially, yeah, so the potential felt by an electron in the sphere will be dependent on a one over R term. Now you can try and solve this, this problem in, in Cartesian coordinates, but it's not very natural because you have spherical potentials, right? So you can switch the polar coordinates, which looks terrifying. But as I said before, this is all quite well known theory from a planetary motion and things like that. So essentially, what you have here is rather than that x, y, z, you have what are called, you have the radial dependence, R, you have the theta, which is the equatorial angle. And then you have the azimuthal. I think that's how you say it. So you basically have rotation around the equation, rotations around the vertical axis. And the equation, then the kinetic energy operator looks like this. And V is just equivalent to one over R. So what this basically allows you to do, if you notice, you have three different terms in your kinetic energy operator. Again, don't try and solve this. This is all the textbook stuff, but I'm just trying to give you the main ideas. You notice you have the R terms here, you have the theta terms here, and you have the phi terms here. Now, what this means is when you have a Hamiltonian object like this, or any differential equation we have non-interacting terms in the linear equation, it basically means that you can solve each one of these, you can solve each variable on its own using a technique called separation of variables. And this is actually really important for Hartree-Fark and a single electron wave function later on. But basically what it means is you can solve an independent system of equations, and then you can basically form the total system by having a product of that, of those independent systems, which is very powerful. So you can kind of see that here. So if we assume that we can have this product wave function, which is dependent, which you have a term for R, a term for theta, and a term for phi, we just substitute this into the previous equation. It looks a bit nasty because you've got this extra potential term, but you can do some rearranging and you get this nice set. You can divide by the wave function, the product wave function, and you get this nice equation. And this can be solved by separation of variables. Now, this takes five chapters in most textbooks. So if you are interested in looking into this, I highly recommend the book by Linus Powling, Introduction to Quantum Mechanics. It was written in 1937, but in my opinion is the best description of how to solve the hydrogen atom. And it really gives you kind of the idea of how these problems are being solved back then before computers. A lot of the modern quantum chemistry textbooks are you just jump into kind of the numerical solutions, which don't give you as much insight into the problem. Okay, so here's the three equations that you get from separation of variables. You've got your phi equation, your theta equation, and you've got your radial equation. Now, these are all, you can solve these using some quite intense differential equations methods. But what this happened, so I've jumped a lot of steps here, but more summary, you get this is the solution of that product equation. Okay. And as with the wave functions in the path to the box solution, which was quantized by some quantum numbers, depending on the harmonics, here we have a similar idea where we have quantum numbers dependent on the radius. It's called a principle quantum number. We have the magnetic quantum number, which is around the equator, L, and then we have, sorry, so we have the angular quantum number, which is around the equator, you have the magnetic quantum, which is the azimuthal degree of freedom. So yeah, it's quite scary, but as I said, like a lot of these polynomials was loads of what happened here in the 1800s. Yeah. So and the main tool they use is the spherical homonites because obviously we have spherical symmetry and the potential of this makes, makes everything easier. Okay, so now you may notice that the energy when you solve this because we have no electrons, extra electrons, you the there's the energy is dependent on the, even though the what the wave function contains an L and that an M is quantum numbers because, because we have no, we have no interacting single electron picture, the solutions of the Schrodinger equation for the hydrogen atom or this spherically symmetrical, particle and spherical particle and spherical symmetric potential, they only contain the principle quantum numbers N. Okay, so I mean, this is, yeah, so it's, if we want to go to larger atoms, larger, like lithium, helium, helium, et cetera, this theory does break down a little bit, because obviously we have interact, you need to take into account the interactions from, from the inner electrons, which called screening, if you've heard of that, nature screening. So, but it's quantitatively, qualitatively actually, actually, qualitatively, it still works quite well. So, so now you can kind of start to do some analysis with these wave functions, looking at the density, where the density is essentially is psi squared. So the solution, I jumped the head slowly. So if you, if you take that equation, you saw that Turks really scary equation actually. And then you put in, say this side here, we have NLM, that's these indexes, say a principle quantum, we have a principle quantum number one, there's only one, then we have a principle quantum number two, we have, we have two, yes, well, we have the S orbital here, and then we have, so that's two zero zero, and then we have the so anything with principle quantum number one, so N zero zero is known as an S orbital. Okay, so that's a totally symmetric solution. So you can see here, we have psi one zero zero. This is just parameterized by this. You can see here, this is just a radial, radial exponential here. It's exponential parameterized by the radian, by the radius. So this is a spherically symmetric 3D thing, or a sphere, basically. And you can see the same applies with the 2s. But now you have kind of this, you have this product, product here, which gives us nodal behavior. So if you see the 2s, rather than being a sphere, is actually this kind of sphere, but with an inner sphere inside it as well. And the inner sphere comes from this product, before. And then you start, when you start to have two one zero and two one one, anything with L equals one, that's a P orbital and that's when you get these dumbbell like shapes here. Okay, but what I'm trying to show is that the shapes of the orbitals, or the hydrogen orbitals all come from the solutions of the spherically symmetric single particle wave function. Okay, so you can do some analysis on these with these things called the radial distribution functions. Now that the, what the radial distribution function is essentially is use Bourne's rule, which essentially is you take the density, you take the density of the wave function solutions, which is just essentially you take one, you basically square the product of them. So here we're looking at the 1s, which is this one zero zero. Remember, this is the simplest one, which is just a spherically symmetric ball. It's just exponentially decaying along along the radius symmetrically. You can see that we have, you can see that the, and there's obviously from when it starts decaying when you get away from the nucleus as well. So you can see here that if we, if we take the, if we analyze this along along the radius, you can see that the probability of finding electron has this distribution function. And then this is the 2s, which is equally symmetric as this function. But you have this nodal character here. So the principal quantum numbers all have these, I just go up in principal quantum and you get an extra node in your wave function, which if in the S orbital picture is a ball, then a ball, then a ball. That makes sense. Okay. But the main point to take is that the hydrogen atom gives you the kind of the shape of where the electrons like to sit. The solution to the hydrogen atom give you the, give you the rough that were the best approximate of how the electrons like to behave in large molecular systems. So this is really to start three quantum chemistry. So having a good understanding of these molecular orbital, these atomic orbital shapes really like gives you some intuition and chemistry. This is, if you think about a lab chemist, they're always thinking about the shape, these shapes and how they can fit together, things like that. So this, this is really, really where the chemical intuition comes from. Okay. And this was actually experimentally verified in 2013. You think you're using extra diffraction, I think. And you can see that 90 years after the Huxer Schrodinger derived the equations only about nine years ago, they actually were able to verify this. So these are the excited hydrogen, extra diffraction, I think, which is really cool. So essentially in the presence of no extra electron, so the single electron picture of the hydrogen atom is essentially correct with that, because there's no relativistic effect, sorry. Okay. So what's very, very interesting is that these, the solutions of the hydrogen atom, they predict the structure of these orbitals. It turns out that the periodic table structure, which is determined before the solutions of the hydrogen atom is grouped into essentially SP, D and F orbitals, which are essentially the N, L, M solutions of the Schrodinger equation. So Schrodinger actually essentially verified the structure of the periodic table, termed by Mendevez and Lehev. So you can see here, you have, we have going down in principle quantum number, you have N1, 2, 3, 4. This block here is called the S block, these are all the spherically symmetric, these, these are all the, these are all the outer electrons in the spherically symmetric. So yeah, the periodic table is structured on the valence electrons, which is the outer most electrons. And then the electron, the electrons in this block, I don't know, is it, so they're almost in this block, this block in particular is known as the S block, because the outer electrons are all in S orbitals. And then this block here is known as the P block. And the outer electrons are all in these P orbitals, which are these dumbbell-like shape ones. Shall I get back? Yeah, so the S orbitals, the S orbitals are these one, and these are the P orbitals here, so that's one P orbitals, he made this figure. So this is, this is one P orbitals dumbbell, this is another P orbitals dumbbell, another P orbitals dumbbell. So yeah, so and then in the middle you get these what are called the D orbitals. And essentially these are grouped into kind of these, these are all, all the, all the blocks have the same principle, or the same outer electrons, that have outer electrons in the same, in a spin or in an orbital which has the same quantum number solutions in the hydrogen atom, which is very cool. And then you got the, the lanthanize, and that's nice down here, which is the F orbitals. Okay. So you, and you can kind of see this with, so you kind of see the, what this is showing is, this now this is, rather than taking a qualitative approach, this is kind of the quantitative analysis of this. So this is the, this is the ionization energy, experimental ionization energy. So what this is showing is the energy it takes to take your most outer electron and, and remove it. So that's, that's kind of telling you the energy of your, your orbital, your, your last filled orbital. And you can, you can see that that, like, it's, it's quite good. Like, so you, you can see here this is H to H E, this is the first principle quantum number. And that just goes up monotonically. And then L to, L to neon, this is the second principle quantum number. And then we have the third principle quantum number. And then things start to get a bit confusing when we have the D orbitals. Then we have the F orbitals. But you can see the blocks do monotonically increase which is kind of what is, what is expected from, from the solution to the hydrogen atom. But the problem is when we start to introduce, and electron, electron, when we start to include the electron and contraction of the, of the inner electrons, then the, the theory kind of breaks down. But it's still like a quantitative level. But they're roughly qualitatively correct. And it's basically due to, as you increase the nuclear potential as you go up the group for the same quantum number. Because obviously as you go along the group you're adding more new, more protons than nucleus. So your, the attraction of electrons to the new, the protons increases. So it requires more energy to ionize them. But obviously this, this whole approximation that we've been talking about so far, neglects electronic, electronic interactions. And of course it, the relativistic effects for the lanthanides. So like, for example gold would not be gold if we didn't include the relativistic interactions. Because they, the, the core orbitals travel at near the speed of light. So this is kind of what you, what, this is what is really seen. You might have seen this in your chemistry classes at A level. So if the, the solutions that we just derived before, these would have, for all the things with two, these would be the same, same level. And then all the things with three would be the same level. But obviously what happens is that because the, because the, the peel orbitals are in these dumbbell shapes and they're more diffuse. When you start to include the core electrons, the peel orbitals, and yeah this is like confusing. So ionization is up here. So this is more energy to ionize here. So yeah, so this is larger energy gap. So obviously the peel orbitals are less strongly bound to the nucleus because they, they interact less strongly with the inner electrons. And this is what it's called shielding. So the peel orbitals are shielded less, sorry, shielded more. But yes, the peel orbitals feel the effects of the, they're repelled by the nucleus. Sorry, they're repelled by the inner electrons more. And that's why the ionization energy is lower. But so, okay, so to solve this we need, we need to include these electronic, electronic interactions. Okay. And this, this is where Hartree-Fock theory and all the motivation for Hartree-Fock comes in. So obviously molecular energies, we have all of these, we have, there's lots of terms in the Hamiltonian which are neglected. So you have vibrational, rotational, and nuclear spins. These all decrease in value. And we essentially neglect, and this translation as well. We basically neglect all of these and just focus on the electronic, electronic energy. Okay, so now we're starting to think about molecules. Okay. So we have, so now we have, this is, this is, these are all the terms from the molecular, for the molecular electronic Hamilton, just for electronic energy. So we have here, I hate you like my drawing, it's color coded. So we have the kinetic energy operator of the electrons. So we're assumed, what would, I'm going to say, we're using the Born-Oppenheimer approximation here. So we're assuming that the nuclei are fixed in position, and then the electrons are just whizzing around these fixed nuclear positions. And the, so you can see the kinetic energy is represented by the yellow, and then we have the, the nuclear, the nuclear electron interaction is represented by green. You can see this is a coulombic term. There's one over R. That should be, yeah, that should be swear to us. The, as you can see here, we have, this is the mass of the electron times by the electron mass times the number of protons in the nucleus. Which is that. And then, obviously, RIA is the distance between, is the distance between, so is the, is the distance interrupt, is the distance between the nuclei on the electron. And then we have this red term, which is this kind of incident, this is a two-body electronic interaction term. And these, because these are moving electrons, this is not, you can't solve this term like very easily. It's, it's, there's actually not, this is two, but it's a three-body problem. So it becomes very difficult. And then you have this, this fourth term, which is the, which is the electron, the nuclear, nuclear interactions. But we tend to remove that because the nuclei fix, so they're not going to change over that. So you can, so you can actually just calculate that and add it in at the end. So the problem that we're trying to solve with electronic Hamiltonian is actually this one. It should be squared, I apologize. These are, here, this Coulomb interaction, sorry, to the charge-charge interaction. So, so now we have these three terms in electronic Hamiltonian, molecular electronic Hamiltonian. We have the, we have a Coulomb interaction of the positive nucleus and the negative electron. We have the kinetic energy of the moving electrons. And then we have the electron-electron interaction. So these are what are really difficult to solve. And actually most of modern quantum chemistry is actually trying to solve this term, rather than this term. So it turns out, if you notice the, the nuclei, yeah, because the nuclear are fixed in position, they're not variable. So it turns out that we have a single electron term here and a single electron term here. And what that means is we basically have a non-interacting set of terms in the first part of the equation. Okay, so so this, so this can be solved with this, this product type approach where you have, well, you, when you have a separable differential equation, you could obviously use separation of variables and solve the individual term each time. And then this term is not separate, obviously, because you have these electron-electron cross terms. And it's essentially because the instantaneous positions are affected by each other. So this is, this is incredibly, yeah, so what we do is we just get rid of it as an initial, initial qualitative approximation. So, and then this is called the linear combination of Thomas Hortons and Hamiltonian. Okay, we literally just have the kinetic energy of the electron and then the kind of the electronic interaction, the, we have the kilometric interaction with the electron and the nuclear. So you can, you can see the sums in the simple picture, yeah, so you have these four sums here, which is represented by the arrows. Then we have two terms for kinetic energy. Okay. All right, so now we've got the separable differential equation, or separable linear operator. So it means that we can use separation of variables and we can solve the global molecular picture with a single electronic product wave function. Okay, but so the way we go about this is we use, we suggest we use the first ideas to try and solve this molecular problem with this very simplified Hamiltonian that we've picked. As we use, we use the solutions of the hydrogen atom, hydrogen like atom to these shapes, these orbital shapes. We use those as a basis for our, for this Hamiltonian. Okay, so and I must say that there's something called the power exclusion principle, which essentially means that the solutions of the, the Schrodinger, the hydrogen atom that I showed you before, they have space for two electrons, and that's because you can have, basically have no repeating quantum number, but there is a quantum number that I ignored, which is called spin. Spin's a fundamental property of the universe. It was discovered by Jordan, and it has this S2 symmetry. But what we basically have here is we have what is called a spin orbital. So this is really important, you might have heard this, you might have heard this from, you might have heard this like from your lecture or something like that. So what spin orbital is essentially is you have this psi here, the psi is the spatial orbital wave function. The spatial orbital is the, is that hydrogen atom thing, the shape, but then that's got space with two electrons, and then the individual electronic wave function is called the spin orbital. It has two variables, we have r, which is a distance, and then we have sigma, which is a discrete spin variable, and it typically represents a spin half, or minus spin half, and we often greet them together into this thing called the position spin variable, called x. Okay, so that's what it means. Okay, so we then, so we take the Hamiltonian and then we solve it in the single electron picture, and that's the single spin orbital picture. Okay, so we take one term from that, one, basically one of those Hamiltonians for each electron, and then that's totally valid to then just solve it for each one, and then form a total molecular wave function and add them together, and then we use what we use, the combination of atomic orbitals, as you can tell by the name, we form a molecular orbital from a linear combination of atomic orbitals. Okay, so you can see here that we have a sum over atomic orbitals, weighted, which is the spin orbital like things that I just showed you before, and then we have this linear weighting for each spatial, for each spin orbital. It's a bad quantitative picture, but this is actually an adequate way of getting kind of quantitative results in the lab. It motivates all the arrow pushing. So let's consider the simplest molecular system now. Okay, so we're doing chemistry finally. So we've got two hydrogens interacting, and then we say we've got these, we take the simplest single electron solution, and due to symmetry arguments, you can just isolate all the S orbitals. Don't ask me about that, but it's to do a group theory if we're interested. Well, please do ask me about that, actually. That's my favourite topic. So basically you can isolate the S orbitals due to symmetry arguments, and then you can solve the interaction, you can basically say the two S orbitals on each hydrogene, you let them interact, and these are spin orbitals, so there's a single electron. You have the spatial orbital, which is the S orbital, and only one spin occupation on each side. And you solve it. Now I've interested basis, when we apply this basis to the Hamiltonian that we showed, you get what's called the secular equation. Now I don't have time to derive this in this equation in this lecture, but essentially when you solve, when you have a Hamiltonian, you introduce a basis, you will always form a not a convex optimization problem, and that's due to what's known as the Rayleigh Ritz Variational Principle, and there's an exercise in the things that solve this. What that gives you, basically, is that whenever you solve this, you get from Rayleigh Ritz Variational Principle, you solve it for a minimum, this equation drops out, and what this equation shows is that the solution, this Eigen value equation, will always be the global minimum for this basis, which is extremely powerful. So you see this in all the quantum chemistry and all the kind of exact ganglization methods, things like that, and this is kind of really fundamental. It says that the secular equation and the Rayleigh Ritz Variational Principle are two of the founding principles of quantum chemistry. Okay, so what does this look like from a mathematical setting? So we have the, so we have, sorry I should get back, so this H, this H here is called the Hamiltonian matrix, and now we've got this non-orthogonal basis. Now obviously the atomic orbitals are orthogonal in space, but bear in mind, we're not in this, they're not on the same, they're not in the same point in space, so they're not actually orthogonal wave functions. And then you have this overlap matrix which counts for orthogonality, so this is a generalized Eigenwave problem now. So what do these elements look like? So a lot of quantum chemistry really comes down to solving these Hamiltonian elements. Now this is one of the, because we're working in that kind of a first quantization picture here, you notice that the, but their functions are distance away functions, which is not actually the case in now, second quantization, but it gives you an idea of how you build up these elements. So there's a whole field of first quantization quantum computing, but basically what you see here is you have the wave function on each side, so you got this, you see your wave function, you operate it, and you get your wave function on the right side, integrated over all the space. Now the operator, you can see here, this is the, if we just take the s-obletoes, we've shown that these come from the radial solutions, really symmetric solutions of the Schrodinger equation on both sides, and then we have that kind of sandwiched by this, you have this operator which is sandwiched by both of them, and you can see the operator now has, we've got the kinetic energy term, remember this is, remember we're in a single electron picture here, so we have the kinetic energy of one electron, and then we have the kinetic energy of one electron interacting with the nuclei on the first hydrogen and the second hydrogen. Okay, so this is a pretty massive equation, but you can actually solve this with, kind of, side-by-dot integrated or something like that, and I think it would be really cool to actually write, if someone wanted to write a, this solver for this, for the s-obletoes, so just, you can see that you can just plug this in, these are all functions that you can integrate, you just need to discretize them, you may have grid over the radius, the radial function, and then we have the overlap matrix, and I think the overlap matrix have, kind of, an intuitive picture, because it's literally in this first quantizational picture, where you have, where the wave function at a distance dependent, you get, that's probably the wrong, the wave function at always distance dependent, but in first quantizational, the range at the distance dependent lives on the wave function rather than the operator, like in second quantizational, but anyway, you get this, kind of, physical overlap picture, where you have these, kind of, exponentially decaying functions, when the nuclear cusp, and you can see that the overlap will increase with how much they physically overlap, okay, so I'll just quickly, I did a few more slides then I'll stop, so, okay, so, so this is kind of what we, you can use this, so if you notice that equation was an eigenvalue equation, it turns, and we had two basis functions, so that means there will be two solutions with the S orbitals and hydrogen, we have, so it, we solve for the one electron picture, and then we fill it up with two electrons, because we get one solution for one electron and one solution for the other electron, but they're the same due to the spatial orbitals, being the same, so, but basically you have two interacting hydrogen atoms, and then you get what's called a bonding orbital, bonding molecular orbital, this is super important now, and then you have an anti-bonding molecular orbital, so we take the two spin orbitals, and we form a molecular orbital, now it's quite confusing because you get us two molecular orbitals down here, one for the spin up and one for the spin down, and two molecular orbitals up here for this, where these are vacant, but they're, what we call the sigma G, this is the bonding orbital, and this is kind of a positive linear combination to us orbitals, and then you get the sigma U star, stands for Gerardo and Ungerardo, I think which is German for spherically symmetric, symmetry across the century, I think, an anti-symmetry, and you get a negative combination here, so this is really kind of what you get, this bonding anti-bonding picture, and it's not normally this simple when you have lots more electron, lots more spin orbitals, but in the kind of the two spin orbital system it's quite nice to see that you get bonding anti-bonding, so what this is showing just to summarize, you're joining up these two, and then you form these two here, and then you can kind of look at the Born rule now, the densities for this new molecular anti-bonding orbital picture, okay, so if you look at the bonding orbital here, you can multiply this out into the two spin orbitals if you wanted to, what you see is the Born rule, you get these kind of this peaky, the two orbitals are really these peak nuclear cuts, and you get the bonding interaction here, which is like this favorable density in the center of the bonding, and that's why you get this joined electron cloud, these all just mathematical solutions, and then you have the anti-bonding, which is obviously negative, you have a positive and negative interaction here of the same function, and then that's where you get this nasty function here of the wave function, but obviously when you take the Born rule, you're squaring it, so then you get this unfavorable like area in the center where the electrons are like, you get away from me, and that's why you get this, that's why you see this anti-bonding, and that's why you have the positive and the negative colors here, these are called the phasors, okay, so the summary is you can build up a molecular bonding, a very simple molecular bonding picture with the hydrogen like atomic orbital, so you get the simplest idea, you don't even have to think about the mass really, it basically boils down to if they have the same shape and overlapping of the same color, there will be a favorable interaction, so this is where the chemists get so much of that chemical intuition from, that they're kind of implicitly solving the atomic orbitals in their head by kind of like overlaps with like favorably, and that's why you get this bonding picture, so here you have the two S orbitals, favorably interacting to form this sigma bond, we call the sigma bond because it has totally, it's totally spherically symmetric, circle symmetric around the bonding axis, and when you have two favorably interacting P orbitals, you get this kind of delocalized, you get the positive phase delocalizing at the top favorably, and the positive phase is the favorably at the bottom, so you get this two sausages almost above and below that, but below the bonding axis, and if you, this has kind of 180 degree symmetry, okay, and this all comes from, if you're interested in group three, these labels all come from point group symmetry for the irreducible representations over the finite groups, and then yeah, so you can see that now if you start wanting to go to, you can obviously add all the there's an infinite number of solutions for the hydrogen atom, right, so you can start to form all these bonding pictures of two hydrogen atoms, so you can kind of see, it's quite intuitive you just have favorable interaction, non favorable, favorable, non favorable for the same shape, then the P orbitals they're all the same, but they're on different axes, so they can actually spatially interact slightly differently, so you get sigma bonding P's here, pi bonding P's, anti-bonding pi P's, anti-bonding sigma P, sigma P bonds, sigma P orbitals, you can kind of see the P orbitals interact less strongly than this, so the pi bonds are less strong than the sigma bonds because they have less physical overlap, and that actually comes from the overlap term in the secular equation having less magnitude for this two combinations, so yeah, so this is the chemical intuition that people sort of have, and it's basically just also overlap, that's all you think about, so benzene is these six carbon atoms, you know carbon is in the P block, it's got these these Pz, we call them, so you have Px, Py and Pz, now all the Pz's from the carbons, you have six of them, so this would be a six-dimensional basis, right, one Pz for each carbon atom, and you get these favorable interactions here, and this is the pi-bonding interactions of benzene, you get this delocalized cloud, and then you can kind of, this is really cool because if you start to think about like graphene, you can see why you have all this delocalization property of graphene, because the electron clouds are all delo... it's basically extrapolation of this to like tessellations of these columns. Okay, I'm going to stop there for now. Any questions? Let me know. Okay, do we have a question, or yeah, okay. Maybe it's a bit early, but I need help building my chemical intuition, so if we look at the diagrams of the atomic orbitals, where you have light and dark patches, if I want to make them fit together, should I be trying to put light on top of light, or light on top of dark? You want them to overlap with the same colors, so if I go back, you can see it here actually. So if you look at this, this is a very strong, so you have the, this is the most strongly bonding, because we have, vertically we have four, six of the same color interacting, and then you have the negative phase, six of the same color interacting on the bottom. Then you can see the, now the energy of the bond is less strong here, you start to have, there's more mixed, you see up here you've got one positive, one negative, two positive, two negative, etc. And then you can see up here is the most weakly bonding, the strongest anti-bonding, this is because you have totally out of phase things. So if you have more, more things of the different colors touching, that's bad basically. And then, so I noticed when you were doing the ionization energies of various elements, indium and gallium have really low ionization energies. Does this have something to do with why they're used in three, five semiconductors? I'm not an expert, that's my, but, so I believe that's to do the fact that they form 2D materials like graphene, gallium, arsenic, things like that, that's why it's a fabrication. It's easy. But the gallium is there, I mean, so let's go to the, I don't have a period table in my head. It's been a long time since I've actually done chemistry properly. So where's gallium? Someone help me. So the right one column and there's, so it's the first P orbital. So you've changed your quantum number and then the P orbitals, it's the first one. So that's why it's at the bottom. Why it's lower. Okay, so one electron has just been pushed up into an orbital and it's easy to peel that electron off. Yes, exactly. Yeah. Okay, one quick question. You can see here, actually Ben, you can see it's the first P block, because you can see the start of the P block ionization. So the six P orbitals you got, so gallium is the first one and you got the rest of it. And when you have more nuclei, you've got obviously strongest potential interaction. So it's like the nuclei hold onto electrons much more. Okay, sorry. Thank you for the wonderful lecture. I'm here. Okay, so in the linear combination of atomic orbitals, there was a section equation, right? Yeah, so you have four parameters for variable C, right? Can you just add up all possible linear atomic orbitals and make a linear combination of them? And I don't know, by brute force or machine learning, just go up to as much as you can and learn everything. You could just have all of them if you wanted to. And would there be any significance to it? What's the implications? You'll see the symmetry, if you just were naively to throw all the spatial orbitals, you will see, and if you had to machine learn it, you will see them group into sweet symmetry objects. So you'll see this Hamiltonian will get blocked into symmetry groups. So irreducing representations. Yeah, so I imagine the machine learning, if you were to throw it all of the map, all the orbitals at it, it will just find the physical symmetries present in your problem. And that will be represented by the blocks of this Hamiltonian, which will then result in kind of, you'll have eigenvectors which have a lot of zeros, which allow for interaction. Okay, good. So let's have a quick break. Thank our speaker for the first hour. We'll be back soon. Okay.