 OK. Shall we start? Yep. So, please. OK, I'll start again. I'll start again. Sorry. OK, sorry. I will continue. I'll start again. We have this mean field. So far we've only spoken about the single, the non-instructing electronic picture, where we basically threw away all of the electronic terms in Hamiltonian, so we can use a single electron separable idea where we can solve the total molecular wave function and its single electron pieces. What we do is we use this mean field. So to take the theory further, we want to introduce the electronic-electronic attraction. So, the true electronic Hamiltonian is the electronic attraction is kind of unsolwable in this single electron picture, but what you can do is you can add in this mean field term. I quite like this figure because it kind of shows you what the mean field is actually doing. It kind of makes a lot of intuitive sense. We have the electron density peaks around the nucleus. We saw that in the previous plots. This is kind of the three-space picture. The mean electron density is always giving greater amount of nuclei because they're attracted due to the electrons, the key-long interactions. You can retrieve its background fixed potential with our Hamiltonian. So the idea is quite simple the time you get this extra mean field term. The mean field, formerly the mean field, is defined as the density. So you integrate the density. This basically gives you a three-dimensional plot of this electronic nuclear cusp. So now we have this term, which essentially is if you notice we've integrated out all the variables here. So this is just the constant. Once we have the density, it's just the constant. So let's assume we have the density for now. Then we can solve this mean field Hamiltonian with the separable methods that we have before. So we can use to solve the global molecular wave function we can use a product of single electronic wave functions and this is known as the famous Hartree product. Again this was determined in about 1929 I think. He was an expert in differential equations from World War I in the ballistics. So then he was at Cambridge and then basically applied all that knowledge to solving differential equations of the Schrodinger equation. So there's a product of single electronic wave functions. And because the mean field operator is still a one-electron operator because it integrated overall space out the distance we can use this. But it's not anti-symmetric and electrons are interesting at the differential. So there's a lot wrong with it but I'm just trying to show you the motivations. So it turns out that although the density, when you have a density it is solvable. Say if we're given some electrons in space and it's been over to it. We can solve this here because this term can be solved and then this is the potential term and this is the kinetic term. But you might notice that when you solve this you get a new solution of the spin orbitals. So this isn't kind of, it's a non-linear equation. So you get this by solving this to input density you get a new density here. For one set of spin orbitals you solve the equation and you get a new set of spin orbitals. So it says non-linear equation. So what you have to do is you have to keep iterating over this and this is called, you might have heard what's called the self-consistent field because the field has to become, this field term has to become consistent and kind of optimised to kind of this constant. Now I must say we can't use the combination of atomic orbitals in this approach because the, I should say, sorry, if, so there are some, and the reason why this, there's no basis here so far, this is all just using differential equations. So this is an integral differential equation and they are solvable in some cases. So the famous one is the, again, the spherically symmetric nuclei. So you can actually solve the Hartree-Flock equations or the Hartree equations for a nuclei. So you can, for example, fluorine with lots of inner core electrons, you can still solve that exactly because of the spherical symmetry. There's some quite nice, if you look online, there's some very good, like step, there's some great resources showing this. But as soon as you go away from spherical symmetry and some molecular picture, then you kind of have to start solving things with a basis. So you need to use a basis in the same way that we use linear combination of atomic orbitals. But due to numerical problems, it's not suitable. OK, same. And I will come on to why that is in a second. But as I said before, this is a very crude picture, I apologise, but when you go to helium, so you start, this is the first solution, this is the first case of differential, the first case of the Schrodinger equation where, sorry, the first case of the, where you have core electrons, two electrons, you start noticing that you get these anti-symmetric solutions, whereas in reality they are only observing these anti-symmetric solutions. So this is kind of the first case where they started to think, it was like experiment, this kind of first proof that that anti-symmetry is needed. OK, so what is anti-symmetry? So it's quite simple. Because the electrons, the physical solutions were anti-symmetric, they realised that electrons had to be anti-symmetric with respect to a particle exchange. I just put, because of the universe. Apparently if you go into the quantum field theory there is actually no regulations for this, but it's kind of an observation for our purposes. So going into the previous method where we have this Hartree product, we now, we take this single electron picture but we have to, we give it the constraints that it has to be anti-symmetric with respect to a particle exchange. So we take the simple two electron Hartree product, two spin-alls of Hartree product and we see that obviously, because there are two terms that have this normalisation factor here, one over root two, and we say that we've moved, that should be, sorry, there's a mistake here, that should be two, so these x's should be swapped. But basically you see you have this swapped-aver term, which is the exchange. And that very neatly can be expressed as a, that can be very neatly expressed as a determinant. So if you take the determinant of the, if you take the determinant of this, you'll then get this, you'll get the equation from the previous slide and then we represent it kind of compactly as a ket. So the ket contains the anti-symmetry here. Now this generalises to n electrons or n spin-all-tools with n electrons. And it has all the same properties. So if you, you know with determinants, if you move columns and rows, you'll flip the signs, things like that. Okay, so what this is showing is that we have this heart of a single electron picture, wave function, this product wave function, but it can actually account for the anti-symmetry. Okay, so you can see here that as you exchange the columns, you get the sign flipping. And exchanging the columns is equivalent to exchanging the electrons. Have this braket notation. Okay, so what does it do to our operator? So I don't have time to derive the Hartree-Fock equations, but you just have to take it from me. If you go back to this equation, this is the mean field equation. So we have this mean field term where we're integrating over all the spin-all-tools, all the positions. Now, in the presence of anti-symmetry, you basically get, you get this result, this falls out. And your mean field term has what's called the Coulon term, which is kind of an electro-nuclear, so the electronic interaction has got a physical understanding. And then you've got the exchange term. Now, this doesn't have a physical interpretation. It's purely a facet of the anti-symmetry, the necessary anti-symmetry in the equation. But you can see the differences. There's still two electron, I'm still integrating over the two electron over the density here, but then on the exchange, here you've exchanged these two particles. So it looks like a two-particle operator, but you're actually integrating out the variable, so it's still a mean field operator, so it's a bit confusing. And this, because this is a one-electron operator, they're integrating out the terms, I think it's quite remarkable really that you have a one-electron operator that still accounts for anti-symmetry, anti-symmetry is the property of two particles. Yeah, so this is a solution for the whole molecule. And we've got the sum over i here, where i is the number of spin orbitals, and then j is the number of spin orbitals as well. So you've got the Coulomb term, which is an average instantaneous point-charge repulsion at that density. And then the exchange term, which exists only to anti-symmetry. And this causes a lot of problems. So one of the exercises will be to kind of try to understand this a bit more. So we can then take the Hartree Fock molecular operator, and then we can, because this is a one-electron operator, we can then use the same trick of using separable differential equations. And we can solve a single-electron picture. And this gives us what's known as the Fock operator. This is really important. So what is the Fock operator? So we have inside the Fock operator, we have our kinetic term, and then we have our term which accounts for the, and the potential that's contained in here as well. So we call this the one-electron operator. So this has got the one-electron kinetic and the one-electron potential. And then we have the mean-feel potential term. The mean-feel potential term, which contains, which is our approach in this picture to have electron-electron interactions, which has this exchange, this Coulomb term, and then this exchange term, where we've split over the exchange there. So it's quite strange, because you're solving, you have the j's, you run over all j's, and then i is the solution that you're trying to solve for. So that's what you get here. So again, you have this kind of feedback loop, where you're solving for i, and i's that go back into the equation. Okay. Then again, we have this iterative self-consistency condition, but now we've got this kind of anti-symmetrized mean-field operator. So you're going to think of Archie Fock as an anti-symmetrized mean-field operator, but it must be solved iteratively, and that's why you might hear self-consistency used. Okay. So this was all solved for the purpose of the previous discussion, we were just treating this as a differential equation, as if it could be solved analytically. But we know that it can't for molecules. So if a multi-nucleosenters, there's no analytical closed-form solution. So we have to solve what's called the Fock, at least it's called the briefly in equations, and essentially it's the secular equation, but with the, it's a secular equation with the Fock operator, and it's a non-linear equation because we have this feedback loop. So we introduced what's known as a finite basis. Now, as I said before, we can't use the linear combination of atomic orbitals due to numerical problems, with orthogonality, but we can still use the ideas in the generalized secular equation. But the problem is we need to find a basis. So what is a basis? This is something which is very subtle, and I think it's where a lot of people kind of get confused in quantum chemistry. So you have what, essentially we have the spin orbital, and we introduced, so let's say that this car analogy is quite nice. The black here is the exact wave function. But we want to introduce a basis, so we have these weightings, these red circles are these eters, and we can weight the eters with this C. And the more eters that you have, the better approximation to your overall wave function will be. So the basis is kind of like you mangle it together to the shape of the orbitals that we showed in the hydrogen picture. So it's quite subtle. You have this molecular bonding picture, but it's made up of these smaller blocks, which is like your LEGO, essentially. So it's really quite subtle. The stuff we show for benzene, things like that, essentially you build that from a smaller set of basis functions, we call them. So if you've got more balls, a larger basis, you'll get a more accurate answer. If you have different shaped balls, typically you have different angular momentum shapes as well, you'll get a better fit to that wave function. So there's a huge amount of research in quantum chemistry into building basis functions for different things. So, for example, you might have S23G, 631G, double Z to basis set, things like that. So typically what you do is, rather than balls, they're formal mathematical functions, where we're trying to fit the exact wave function onto this finite basis here. So typically, a good example of Gaussians, you combine a load of Gaussians. How it works in the 70s and 80s, people spent a long time fitting these basis functions to get the mathematical functions, these etas. The Gaussian parameters, like the tweaking, these were done quite rigorously by the people and his colleagues at the paper basis sets. So now, essentially, we just have this fixed set of LEGO blocks that we apply to our problems. So we can then build these fitted Gaussians, build our density structures, for example, for bonding orbitals from a set of Gaussians, for example, or slater-type orbitals. And what's nice, because of what I said about the variational principle, the secular equation always finds the exact, the minimum energy for a finite basis. So you can obtain your bonding orbitals by directly just solving the fuck secular equation. So it's very powerful and very cool. And this is why we haven't got the quantum computing yet. This is why you need to do all this class, like old-school quantum chemistry first, to build the shape of the vector orbitals, because you have these numerical parameters which show your basis functions, numerically-fitted basis functions, which then build up these orbital shapes like benzene and hydrogen and H2. OK, so the basic examples and the basis functions, the two main ones are, for molecules at least, you have slater-type orbitals, which are exponentiated to the single power of R. Now, we know from the Cato theorem, which basically says that the nuclear cusps are like these finite peaks. This is exactly what a slater-type orbitals, these stater functions, fit, right? The problem is, they require a lot more mathematical work to solve them when you start applying them into their secular equation. So typically, just for numerical purposes, we use Gaussian-type orbitals, because the reason for this is the product of two Gaussians is a Gaussian. So you can then interpret that over these four centre Gaussians, and then you get a single Gaussian. So it makes the numerical stuff much easier. So you can see here in this figure that, so this is the slater-type orbital, and then we've got these Gaussians best approximating it. So if you have lots of Gaussians, you can, in combination with Gaussians, you can quite accurately fit the slater-type functions. So what Popele and the colleagues did was they basically fitted a load of Gaussians to these slater functions. So STOs reduce slater-type orbitals, but it's made of Gaussians. There's slater-type orbitals free Gaussians. Okay. So let's take this, if we now, this is where it gets quite advanced. So we take the spin orbitals that we had before for this, this is the Fock equation, the Fock operator, so for the energy we're integrating. So this is a bra operator cap, but we're just not showing it a direct notation. We're integrating a real-all space for a single spin orbital. Now we expand out the, we expand out the, the operator in terms of the spin orbitals. We're going to get the one electron exchange in a kilo on term. Now when we apply this equation, the basis equation, so we've picked some basis functions eta, we then substitute that into the previous equation and the Fock operator takes this form. So you notice now we've got this extra, so the G's are these orbital integrals and then the C's are the weightings of each, or of each of these basis functions term. So the Fock operator just got a lot more complicated and you can see that because we have, if we go back, because you have these, because we have orbitals inside the operator, when the Fock operator then has these C parameters inside of it. So you can see like, so really importantly, the Fock operator has almost all parameters inside of it in the basis expansion. Now you can see, you can now see why that would be non-linear because you solve F which has C in it to get C. So you still have to start leaving around. Now, and going back to our benzene example, we're not using atomicals now, we're using this arbitrary basis expansion. Okay. Let's say S2, 3G. Now, by solving the Fock operator, the Fock's and the Routhine equation, each eigenvector solution is one of the spin orbitals. Okay. So you can see here, say if we had some rough P orbital like basis functions, but made of Gaussians, then the linear combination of those would add up to be like the delocalised P here. So these are all mathematical objects now. And you can see we end up with the same result. And again, because this is the sector equation which comes in the Rayleigh-Ritz variational principle, this is the minimum energy for each spin orbital. So just to clarify, so inside the Fock operator, we have these basis function overlaps. Now these are just Gaussians here, these eta. So you have to do, you're just doing four centre Gaussian overlaps here. So there's quite a lot of hard-core numerics going on here. Okay. So, summarise. So, Hartree Fock is essentially an anti-symmetrized mean field approach. And it uses, it gets the rough electronic structure correct. So it's kind of, Hartree Fock really does explain a lot of the chemistry quite well. So you can see here, I did this with Gaussian a long time ago. But it gets the chemistry correct. This is what you see in molecules. The qualitative picture is correct for Hartree Fock most of the time. The problem is when you want that 1% of energy to get the transition state correct, that's when you need higher order structures. But Hartree Fock is responsible for all of the kind of chemical properties you see in higher order methods, if that makes sense. The orbital structures, the orbital shapes, they all come from the Hartree Fock calculation that you do. So that's why it's so important to verify. Most importantly, these C eigenvectors are the shape of the spin orbitals. You need to do a Hartree Fock calculation before a higher order quantum computing calculation to get the shape of the spin orbitals that contain the spatially-dependent interactions in the second quantum particle mechanism. So it's passed on to the second... So what I mean by the fact that Hartree Fock has passed on to the second quantised full configuration interaction, a couple cluster or quantum computing calculations, is that now we're coming back to the quantum computing algorithm at the beginning of this lecture. So this is one of the significance of why you have to really respect Hartree Fock to get the interaction parameters. So you can see here... This is your second quantised chemistry Hamiltonian. Now, these ion j's, these are the spin orbitals, which are themselves linear combinations of basis functions. Now, the ion j's... So, obviously, when you multiply out the ion j's in the basis expansion, you get all the linear combinations carefully to come out. So the HIJ terms are these rank 4 tensors. The rank 2 tensor and the GIJKL is the rank 4 tensor of these Gaussian integrals. This double thing is the exchange in the Coulomb term. But you can see that you have these... You might do your quantum computing calculation and you'll get a chemistry Hamiltonian. The terms contain all the information about the shapes of the orbitals. That's the moral of the story. So your interaction parameters in second quantisation contain them at the Hartree Fock and let the orbital to expansion coefficient. That's why you need to do this lower level of theory calculation first and then apply. I'll explain why you need to do a higher order theory in a second. Hartree Fock, almost perfect. It typically gets... 99% of the energy is correct. But at bomb breaking and bomb forming, it breaks down, I'll show you why. Most importantly, strong electron interactions, strong correlation is not treated well by Hartree Fock. The more accurate methods on quantum computing are needed. Here's an example of where Hartree Fock starts to break down. You've got your two hydrogen atoms at equilibrium geometry. You can see here how you stretch your hydrogen, Hartree Fock starts to get way beyond the exact. You have... There's this other method called configuration interaction. The exact here is the infinite basis limit. Very interesting. The... The I is the same basis as Hartree Fock, but it seems to get the dissociation limits correct. What is configuration interaction? If we go back to our nice... We're not thinking about the linear combination of chemical, we're thinking about Hartree Fock, but the solutions of the spin orbitals are the same actually, so before in the benzene example. If we have a single... If we think about the ground state... If we think about the ground state, the single determinant wave function, the ground state of hydrogen, we have this two... It's two spin orbitals, so we have one sigma g solution times another sigma g solution. The sigma g solution, the ground state, as we showed before, has this... We approximate. We're just showing that it's got this s orbital character. It's a positive interaction of two s orbitals. If we just roughly multiply the two things together, we get this... this product, which this is just quadratic... quadratic equation. But what you see when you multiply this out is you get four cross terms, okay? Now the four cross terms actually end up being what we have here is both electrons on one, both electrons shared, both electrons shared, both electrons on the other one. Right? Now, if we think about when we want to go to the really dissociated bonds, you don't want or the electrons want to be on individual atoms. They don't want to be isolated on one. Okay? So you can see that there's really unphysical allowed solutions in this expansion. The electrons, along this, the electrons really want to be separate, basically. So you can see we've got these terms here which shouldn't really be there in the long separation limit. So this single determinant picture is not adequate. So that's why we have to use this method called CI. And CI is really what motivates us to use quantum... It's the same problem that we use for quantum computing as well, and that you start to introduce these extra bases, extra expansions. What do I mean by that? So extra determinants. You have this many-electron multi-determinant wave function now. So what you do is you take the solutions of the Hartree foc. So you've got all these different Hartree focs. So each one of these lines here, one, two, three, four, these are solutions to the foc matrix. So these are eigenvectors of the foc matrix. But I've just put them here as... You can see it. So we've got sigma bonding G, sigma bonding G. And here we've got sigma bonding G multiplied by sigma star U, anti-bonding. Sigma star U. And then sigma G, anti-bonding. So anti-bonding is bonding. Then we've got two anti-bonding. So we introduce these extra determinants. So this kind of seems a bit arbitrary why we're doing this. But if you look now when you combine the two... anti-bonding bonds in a linear combination now this is very important now. So we've gone from the single-electron picture to the many-electron basis. So this is now a wave function of two electron. OK, so we've got... We're out of the single-electron picture now. We're in... So this is what... By introducing... having this many-electron wave function for the linear combination of these two... these two determinants we've got 10 at 1, 10 at 2, 10 at 1, 10 at 2, but they're different. And we've got the weighting parameters. What happens is when you work through the mass basically the dissociated terms you can basically... you have basically the seas cancel basically here. So you have the seas... these are equally contributed but then it causes the very long expansion terms... the separable terms to cancel. You end up with the truly physical solution which is the... is the separators... only the separated SL of course, you've got SL... SA, SP, SA... and so if you were to solve the secular equation for this picture in the main-electron wave function you'll just get this because you want the solution. But you can think about it in an intuitive way so this is what you get. Now the bonding is described correctly at long distances. We only have these separators terms... physical terms. So this is really the motivation for configuration traction. So configuration traction is a complete generalisation of this problem. So now we've left Hartree Fock behind, okay? Hartree Fock gives us each one of these black lines. It gives us the spin orbital shape. And then we have this many determinant wave function which here's a six-electron wave function and then we introduce all possible combinations of excited determinants. Okay, so in... As you saw before in the H2 example we added an extra determinant. But in a general case you just see all combinations of the of the of the of all possible excitations. So you got here we have this is called the reference wave function which is the ground state all the... it's the lowest occupations of the Hartree Fock solutions and then here we have the single excitations. The set of... This is a set. So this is all possible singles. So this is just showing one single so we can have so A is the hole and then R is the particle. So you get this particle hole excitation here. And then we have the same for the doubles where we do all possible double excitations and then the full solution goes to the all possible excitation. And this is kind of... It's equivalent to exact diagonalization in physics but in chemical configurations. Because if you think about it you're introducing all these extra configurations electronic configurations and then the wave function is essentially this. So you have the C0 reference coefficient. This is just the Hartree Fock energy by the way Hartree Fock wave function This is the solution This is the product of all the of the single electron solution and then we have all the weight into the singles and the weight into the doubles some of the occupied virtual etc. So we often call this the occupied space and the virtual space We then get we get our old friend the secular equation because we have a basis and a Hamiltonian we can apply the secular equation and we we still get this optimal solution to this of the optimal for the grand unexcited states. But now this is whereas before the basis scaled with the size of the basis set that you choose those Gaussian functions here you have a combinatorially scaling matrix That's really bad. A lot of quantum chemistry for the past 40 years has been devoted to solving this equation efficiently. It's just a generalized I and my problem and essentially this is scaling with I mean if you were just a naive you solve this by a diagonalisation you can only really do up to like 10 spin over tools The largest known numerical solution to this is an iterative diagonalisation methods which is essentially it's called a Crelog method that basically allows you to rather than solving the matrix exactly you approximately solve it using matrix vector products iterations of that The largest solution of that with the Lanshawth method is 44 spin over tools 22 spin over tools 22 electrons and 44 spin over tools so you can't really do anything any large molecules interested in this with this and you're not going to get over the amount of effort that they put in to get one extra spin over tool there was crazy because you're fighting against a combinatorially scaling problem that really motivates a need for quantum computing because if your basis scales combinatorially it's actually slightly less than two to the end of the combinatorial in this case because of the particle number but you can really see why we need quantum computing methods because quantum chemistry has been stuck here for about 40 years trying to solve this problem we're not going to get past it with classical methods really all the new methods has hacks to solve this slightly more efficiently how can we keep the most of the wave function that treats the interesting part of the wave function but if we really want to solve this problem probably we need quantum computers to go to interesting systems now there's lots of arguments in quantum chemistry that we don't need to treat the whole system we need to treat the sites of interest like the active sites like that all those methods in perturbation theory all in the chromium diner and DMet for example so the argument is true that we need to solve this for everything no but for some systems which have very strong electron correlation this is needed and the nitrogenase example that does the whole process in the soil is a good example of this and very importantly this gives you excited states as well so you get the excited state and give it some images I can write it for them okay I'll briefly talk about this so what is second quantisation I mean I briefly alluded to it just because people are familiar with it the second quantisation essentially takes the distance dependence from the wave function so you don't have any of these functions of R anymore if you're wave function it takes that and then puts it into the operator as you saw in the previous slide then it means that we can treat them the basis in the occupation number for them we're just 1s and 0s and it gives us a rigorous mathematical footing this is what 1, 9 minutes and these these are basically the determinant like properties that we saw in first quantisation where you can exchange the road and column and etc these are all represented in the same way by these fermionic creation you have this vacuum state which is an empty cap and then you excite this and you get a spin able to pee existing same way it happens here you can destroy try pee and get an empty cap and then you're going to supply them in succession and then you can see this is the really important one here so you can see if you destroy pee if you apply them in this order if you exchange pq via the use of the fermionic rather than exchanging the rows of the state determinant if you just do it via the fermionic creation and relation operators you still get the same properties but this is now done rather than at the wave function level by moving around the determinant it's done at the operator level via these operators and as obviously there's all this famous commutation relation here anti-commutation relations I should say, sorry but the main idea is that these operators preserve the symmetry ok, so finally I guess this is kind of what we just spoke about with configuration interaction is known as post-harchery foc but as I said post-harchery foc methods require harchery foc to be done beforehand before you do your quantum computing calculation you run your harchery foc calculation you give it your nuclear coordinates you choose your basis set you then get an output of optimised orbitals and electronic integrals you then give these electronic integrals to your configuration interaction or you could use a couple of clusters or whatever method is more of them but in particular the point here is that quantum chemistry, quantum computing just is exactly the same as configuration interaction basically in terms of what it requires to be run beforehand so you need these optimised integrals in the second quantum performance of quantum computing to work I'll probably stop there I think ok any questions? thanks Nathan, question? no question ok, let's take a break next Nathan talk will next Nathan talk will start at 2020 last year then