The Science of Everything Podcast - Episode 119: Computational Chemistry Part 1

Episode Date: June 30, 2021

An overview of techniques of determining molecular structure, including an introduction to valance bond theory covering bond formation and orbital hybridisation, and a discussion of the basics of molecular orbital theory, covering the basics of the Hartree-Fock method for solving the Schrödinger equation and finding molecular orbitals. In the process I also discuss the Pauli exclusion principle, the effect of electron spin, and the indistinguishability of electrons. If you enjoyed the podcast please consider supporting the show by making a PayPal donation or becoming a Patreon supporter. https://www.patreon.com/jamesfodor https://www.paypal.me/ScienceofEverything

you're listening to The Science of Everything podcast episode 119, Computational Chemistry, Part 1. I'm your host, James Fodor. In this episode, we're going to look at methods for determining chemical structure. So this falls into under the rubric of computational chemistry. And as part of that, we're going to discuss how chemists basically solve Schrodinger's equation, from quantum mechanics, to determine the structure and other chemical properties of molecules. And this is a really interesting part of chemistry that doesn't necessarily get discussed very much in introductory courses or in popular science outlets, at least as far as I've seen. But I think it's extremely interesting.

So basically we're looking at sort of real world quantum mechanics. A lot of the time when you read up on quantum mechanics, it's or it's discussed in the popular literature. It's very simple systems, you know, electron in a box and a single electron atom and things like that, which are obviously important for understanding the concepts, but they're not really very useful for practical applications to, you know, real materials. And so computational chemistry and molecular structure is where sort of the rubber hits the road, if you like, and we actually apply some of these methods to understanding and predicting the structure and other properties of molecules. Now, this is going to be a fairly technical episode, or actually two-part episode, this is part one. And so you're strongly encouraged to listen to the prerequisite, which is episode 14, principles of quantum mechanics. And generally, somewhat of a background in chemistry and quantum mechanics will be useful.

So check out some of the other episodes I've done on that, those topics as well. But 14 would be a good place to start. In particular, I'm going to be talking about a sort of assuming basic familiarity with terms like, you know, molecular orbitals, Hamiltonian, Schrodinger's equation, electron spin, power exclusion principle, and other such concepts. So if you've never even heard of many of those things, probably a good idea to go back and listen to one of the previous episodes, such as the intro to quantum mechanics or similar ones that I've covered. because otherwise some of this stuff is not going to make a lot of sense. That big said, I will give a refresher for these things when I mention them,

but it will be a bit brief if you've never heard of them before. So again, you have been warned this is going to be a more technical episode. But I do know that there are those out there who really enjoy the more technical, sort of more advanced stuff, so hopefully you find this series interesting. Now, in this first part, we're going to start by talking about Valence Bond Theory, which is sort of an older way of describing molecular structure. And then we're going to move on to molecular orbital theory, which is what we're going to spend most of our time on.

And in particular, I'm going to go through in some detail the Hartree-Fock method and how that's sort of derived and the assumptions made and the reasoning behind it and so forth and talk about how those equations are solved and how that gives rise to our molecular orbitals which determine the structure of a molecule. That probably will take us to the end of this first part. In the next part, I'll talk about how the equations,

the Hartree-Fok equations are actually solved in practice, and so the use of the linear combination of atomic orbitals, and the basis functions and how they're sort of computed to solve for the actual, you know, real solutions to give us some numbers. And I'll also talk about how we include effects of electron correlation and the techniques involved in that. I'll also in the next part talk a bit about some other methods in a bit less detail, including semi-empirical methods, molecular mechanics slash molecular dynamics, and also another approach which is called density functional theory. So I'll give you sort of an introduction to some other ways of working out some of these problems.

But at the heart of a lot of this is the Hartree-Fock formalism, and that's what I'm going to spend most of my time talking about. And a lot of this is quite mathematical. Obviously, I can't show you the mathematics because this is an audio podcast, but that's part of, you know, what we do here is to take ideas that are usually represented using diagrams or equations or whatever else, and to try to explain them verbally. And I think that there's an advantage of doing that. Now, I have got a fairly detailed working out of a lot of this stuff, which I'm going to post a link to on the Facebook page. and hopefully in the podcast description as well, so you can click through and have a look at these notes for yourself if you're interested, but that's not required for following the episode.

All right, so that's enough preamble. Let's get started and talk about Valence Bond Theory. So the basic question that we're trying to address here is what shape do molecules have and what are their chemical properties, such as how much energy is released when the molecules are formed, or maybe they're spectroscopic qualities in terms of what frequency. of light they emit when they're excited, or magnetic properties, optical properties. So there's lots of things that we might want to know about molecules, but sort of shape and energy are kind of the two

most basic ones that we can think about. And molecular structure is the broad sort of area that we might use to describe this, the sort of set of questions. And the techniques to solve them are computational in nature. So valence bond theory is sort of the oldest theory that attempts to explain chemical bonding in quantum mechanical terms. There are other methods that are not not really incorporating quantum mechanics, but here we're looking at a quantum mechanical framework. How valence bond theory works is that it takes the atomic orbitals of single atoms and then combines them, basically adds them together to generate chemical bonds for the molecule as a whole. So hopefully you recall, again, this is where I'm appealing to sort of assumed knowledge of

basic quantum mechanics, that when you solve the, when you solve Schrodinger's equation for the hydrogen atom, so electron going about to. single proton, you get a series of orbitals, which are basically like positions slash energies that the electron can exist in. So there's your S orbitals, your P orbitals, your D orbitals, and so forth. And within each of those sort of sets, there's different numbers depending on the energy level. So, you know, there's the 1s orbital, the 2s orbital, 3s orbital, and then there's the 2P, the 3P, and so forth. So different orbitals, different shells with different energy levels. And these orbitals are different shapes. So the S orbitals basically a,

sphere. The P orbital is kind of like a dumbbell or an hourglass shape in different directions. The d-orbitales are more complicated still. So when I say orbital, if that's a bit abstract, just always think electron cloud shape. It's like the shape of the electron clouds surrounding the atom, representing the probability of an electron being in a particular position. And each orbital, or any type of orbital really, has a particular corresponding shape, and also energy. Because the shape of the electron cloud around the nucleus is directly related to how far away the electron is on average from the nucleus. And the distance of the electron on average from the nucleus is going to determine its energy. So obviously the electrons are negatively charged,

the nucleus is positive a charge. The electron tends to be attracted to the nucleus, other things being equal. And therefore, the distance it achieves with respect to the nucleus determines its energy level. So the closer it can get on average, then the lower its energy level is. So that's why, you know, in the S orbital, for example, the 1S orbital, it's as close as you can get to the nucleus and therefore is the lowest energy state. And then the two S orbital is the next closest and so on. So one orbital means one shape and one energy level. A different orbital will have a different shape and a different energy level. Okay, that's for a hydrogen atom. We have these standard atomic orbitals. Atoms with more protons will have a different set of orbitals. Unfortunately,

we can't solve for those exactly, because now as soon as you introduce an extra electron, the situation gets a lot more complicated. I'll talk a bit more about that a bit later. But the point is, you've got your atomic orbitals for hydrogen, and then other atoms, I mean, going to be similar-ish to those, at least the smaller atoms. Once you get transition metals, it becomes even more complicated. But at least small atoms, you know, things like carbon, you know, fluorine and sulfur and stuff like that, especially sort of molecules, atoms fan and organic molecules. You know, they're going to be roughly approximated by the hydrogenic orbitals, as they're called.

But when we come to molecules, it's a whole different ballgame. Because molecules, you know, a bunch of atoms covalently bonded together, they don't have atomic orbitals. They have a different set of orbitals for the molecule as a whole. And Valens Bond theory is an attempt to predict those molecular orbitals on the basis of your atomic orbitals. So basically what we do is we say, like take H2O, for example, very simple molecule, one oxygen, two hydrogen atoms. So I'm going to give my oxygen atom a set of hydrogenic atomic orbitals, you know, the SPD and so forth. And then I'm going to give my hydrogen atoms, each of them, a set of hydrogenic atomic orbitals.

In that case, it'll be exact because they're actually hydrogen. In the oxygen case, it's going to be an approximation, but still good enough. Because again, the real oxygen animal will have slightly different orbitals, but using the hydrogen ones will be close enough, at least for Valence Bond theory, right? So I stick a set of those atomic orbitals on each of the atoms, and then basically see how they overlap. For example, if I have two S orbitals close enough to each other, they overlap. And since they're basically spheres, what you get is a sort of an oblong shape,

because the sphere sort of partially overlap when you can sort of draw an oblong around it. that overlap of the orbitals will represent a bond, and that bond has a name, it's called a sigma bond. That's a Greek letter that's named after. So whenever you have 2s orbital sort of overlapping it to form an oblong, it's called a sigma bond. You can actually form a sigma bond in a different way. You can have two P orbitals.

A P orbital is kind of in like an hourglass shape. If you imagine kind of rotating the hourglasses so that the tops were pointing to each other and then sort of merging them together at the top so that the tops were overlapping of two different hourglasses, now you've got kind of, well, it's kind of like an infinity symbol, except there's three. I don't quite know how to explain that. Sort of three ovals next to each other. So, you know, the infinity symbol with an extra one, or the figure, you know, figure eight on its side,

but with another circle there. That's what you've got, right, when you overlap the ends of the hourglass on each other. Hopefully you can visualize what I'm saying there. That is also a sigma bond, basically because it's sort of two blobs overlapping with each other, you know, face on, if you like. So there's a couple of different ways of forming these sigma bonds. But the point about the Valence Bond theory is that you start with the atomic orbital, you overlap them, and then you form a new type of, well, you form a bond, which is basically a combination of these individual orbitals superimposed on each other. So a sigma bond is a single bond, which forms when either S-orbital overlap or two P-orbital overlap with each other. You can also have an S-orbital overlap

with a P-orbital. That's also a sigma bond, because it's basically two blobs, you know, sort of face-on hitting each other. You might ask, well, what other type of bonds are there, according to valence bond theory. Well, you can also have a pi bond, again, after the Greek letter pi. Now, pie bond is quite different because in a pie bond, you have two P orbitals, but instead of meeting sort of face on, remember, you've got the hourglasses, you rotate them towards each other, and then you sort of push them in towards each other, and they overlap, and you get that sort of triplet blob structure. That was a sigma bond. In a pi bond, you've got your two hourglasses, but instead of rotating them towards each other and sort of pushing their heads together, you keep them both upright, and you push them towards each other, So you've got two upright hourglasses kind of next to each other. And now they're not meeting head-on, they're meeting side to side.

That is a pie bond, because you've got the sort of two top bits touching on the sides, and the two bono bits touching on the side and kind of a gap in the middle, if you can visualize that hourglass structure. That is a pie bond. It's when two P orbitals come side to side next to each other, as opposed to face-on. That type of bond is particularly formed, for example, between carbon atom. You can also combine sigma and pi bonds, because, you remember, carbons can form double bonds between each other, and you can.

bonds between each other and other atoms as well. So in that case, you might have a sigma bond and a pi bond. And so basically, one of the double bonds is sigma and the other one is a pi bond. Obviously, you know, we've only given a few examples of these types of bonds. But this is an example of how you do valence bond theory. You start with the atomic orbitals. You overlap them, depending on obviously, you know, where the atoms are with respect to each other. And then you see what sort of bonds are going to be formed. And you can then estimate how strong the bond's going to be and its energies and so forth, right? So is it a sigma bond? Is it a pi bond? Is it both? because it's a double bond and so on and so forth.

So that's a really useful technique. It's a very simple technique, right? Because you don't really have to calculate anything too difficult. It's just take your atomic orbitals, smash them together, work out the geometries, work out whether it's sigma or a pie or, you know, another type of bond. Quite straightforward. This is why it was one of the earliest of these techniques to be developed, because it's quite simple. And it's a good qualitative model for these sorts of bonds and still widely taught and used

as a sort of a first simple level of analysis. Now, it's important to understand this is not quite right. And the reason it's not quite right is because a molecule is not the same as a bunch of atoms smashed together. It's a whole, a molecule is its own thing. And as it turns out, we're going to have to solve Schrodinger's equation separately for the whole molecule. You can't just solve it for one atom and then another atom and then kind of just atom together. It doesn't actually work like that. But Valence Bond theory says, okay, maybe it doesn't really work like that,

but we can approximate it like that. And it turns out it works quite well in a lot of cases. Now there's another thing that I wanted to talk about with respect to Valence Bond theory, and that's orbital hybridization. If you studied chemistry, even at a first year level at uni, you've probably heard about this, your SP3, SP2, and SP orbitals. So what do these mean?

Well, basically the idea here is that in addition to orbitals overlapping to form bonds, orbitals can also mix together and hybridize to form what are called hybrid orbitals. It's called a hybrid because it's basically two different types of atomic orbitals mixed together to form something different. An orbital that you wouldn't find in a single atom, or at least not a single hydrogen atom, but you will find in a molecule with multiple different atoms. So let's take the so-called SP3 hybrid orbital.

What does that mean? Well, it's called SP3 because it's got 1S orbital, remember a sphere, and then three P orbitals, which are your figure rates or your hourglasses. And those are three in different orientations. So if you like, that's left and right, up and down, back and forward. You can imagine your hourglass in three different orientations.

You take those three plus your S orbital and you mix them together. What do you get? Well, you get this tetrahedral arrangement where it's kind of like four party balloons. You know, those balloons that you blow up and they have the little nozzle at the end that you tie up when you've blown the air in. And so they form that kind of a balloon-like shape. But if I were to get four of those and point them sort of in different directions out from each other and tie up all of the tied nozzles, you know, with sticker tape or whatever, I'd have this weird contraption where there were four of these balloons pointing outwards,

but they're all pointing in different directions, and they form sort of a tetrahedral, which means it's just four bits to it. If I actually did that with balloons, it would probably look a bit different to this, but hopefully that gives you the idea. The reason why this is important is because the tetrahedral structure is symmetrical. So there's four of these orbitals now,

or four hybrid orbitals within the tetrahedral hybrid orbital. So there's room for eight electrons, remember two electrons per orbital, because you've got your spin up and you spin down. And this sort of hybrid orbital is what we see in certain molecules that have dissimetry. So there's sort of four different directions that the orbitals can point in. But there's also other ways that you can hybridize orbitals. So you can have SP2 orbitals, hybrid orbitals, which is 1S and 2P orbitals,

or SPP orbitals, which is 1S and 1P orbital. And they'll look different. So SP3 is tetrahedral, as I said, SP2 is trigonal planar. So this is basically like a fan blade. Now I've got three blobs pointing in different directions. but only in like a two-dimensional plane, not in three dimensions as the tetrahedral is. And then in the SP hybrid orbital, it's just a single hourglass shape, except it's kind of elongated relative to the P orbital one.

So I know this is a bit hard to sort of visualize, but the point is, depending on how many atomic orbitals you mix together and what type of orbitals, you can get different hybrid orbitals. And I've been talking about the SP orbitals, but there's others as well. You can mix S with D or you can mix S with P and D orbitals. There's probably hybrids with F as well, although I've not heard of those. So these hybrid orbitals are a really useful way of thinking about molecular structure because they enable us to predict things like bond strengths and geometries and so forth

with really very simple heuristics about, okay, I've got this orbital and I've got these other orbitals they hybridize together, what structures are that going to give, how strong is that bond going to be, and what geometry will a molecule probably have given that. So this is a really useful approach, and this is all under the rubric of what's called valence bond theory. Again, the idea is you take the valence electrons, remember those are in the outer shell, and you see what orbitals they're in, and then you look at also the sort of geometry of the atoms with respect to each other, like how close they are and so forth, and then you see what type of

bonds will form and what sort of hybrid orbitals will form, just by sort of adding them together in a simple way. Very simple technique, very powerful. However, as I did say before, this technique is only a crude approximation to what's actually happening. And the basic problem is that when atoms come together to form a molecule and they form bonds between each other, the atomic orbitals don't actually exist anymore. And this is the sort of thing that Valence Bond theory gets wrong. The Valence Bond theory assumes that atomic orbitals still exist. They just kind of merge together or add up or something like that. Or maybe they hybridize, but you know, it's still formed from the same basic ingredients. But that's not actually true. And you might have even learned this

in chemistry right, because this is typically taught for at least, I think, the first two years from memory, that atomic orbitals always exist, and then they just kind of add together or maybe hybridize in molecules, but it's not actually true, not literally true. Again, it's a good approximation for a lot of things, but it's not really what's happening. What really is happening when a molecule forms

is that atomic orbitals like go away, they disappear. They don't exist anymore. Instead, you get a new set of orbitals, completely different set, which may look nothing at all like the original atomic orbitals. And these are called molecular orbitals. They're orbitals for the whole molecule. The electron-neutral.

doesn't care which atom it's part of. You can't even say that this electron is part of this atom or that atom anymore. That's purely conventional. Instead, you can just say, this whole molecule has how of many electrons, and they sit in these different orbitals. So the point is you have to consider the whole molecule as the new quantum system with its own set of orbitals and, you know, orbital energies and so forth. That is not how chemistry students, at least in the earlier years, typically think about molecules, because, you know, they get you to draw these Lewis dot structures and so forth. And for those you, for those who know what I'm talking about, You put the electrons surrounding specific atoms and then you see, oh, there's this, they're forming a single or a double bond here or whatever, and then you can talk about orbital hybridization and sigma and pi bonds and so forth.

But again, that's only an approximation. The electrons don't actually belong to this or that atom. Instead, there's a whole new set of molecular orbitals, and the electrons belong to those orbitals. And those orbitals are in general delocalized across the whole molecule. I mean, they don't have to be delocalized. they may be more in one side or more in another, but the point is in general, they belong to the molecule as a whole, not individual atoms within the molecule. So that poses a problem, because instead of taking out individual atomic orbitals and kind of adding them together, what we're going to have

to do is find a whole new set of molecular orbitals for each new molecule that we want to look at. Every different molecule has its own set of molecular orbitals, and we're going to have to solve for those individually for each molecule we're interested in. And there are millions of molecules, at least when we're looking at organic chemistry, right? So this is a big task. And also, there are lots of electrons here. You know, a carbon atom has 12 electrons, right? But if I have even a simple compound that has, say, you know, three carbon atoms, that's 36 electrons right there. And then a bunch of hydrogens, you know, bring in a few more, even a very small molecule. You might have, you know, 40, 50, 60 electrons. Bigger molecules could have hundreds. And that might not sound like a lot. But remember I said that

electrons interact with each other? They repel each other. So it's not just each electron interacting with the nucleus. It's each electron interacting with the nucleus and with all of the other electrons. So it's electron one interacting with the other 39 and then electron two interacting with the other 38 and then electron three interacting with the other 37 and so forth. So you can see that there's a huge amount of complexity in the system now as soon as we introduce extra electrons and with more atoms. So the point is this analysis becomes quite complicated and that's why molecular orbital theory, although it's more precise and accurate compared to Valance Bond theory, a more recent development. The theory was developed, I think, back in the 20s or 30s, but it wasn't

really able to be used until late 50s, early 60s, and even then it was fairly simple systems. It's really taken off since I'd say the 80s, with widely accessible, fairly easy to use computational software programs, accessibility of digital computers and so forth, because you really need these to actually compute these molecular orbitals. Once we get our molecular orbitals, then we can compute, we can calculate whatever properties we want about the target molecule, because we're got the orbitals, and that fully specifies the energy structure of the electrons and the energy levels of the electrons, as well as the physical geometrical structure of the molecule. So what it all comes down to then, molecular orbital theory and particularly the Hartree-Fock

method, which I'm about to talk about, is solving Schrodinger's equation for a molecule instead of for a single atom, and in particular instead of a hydrogen atom, because the hydrogen atom is the only atom, you know, with one proton, one electron, that we have a closed form analytic solution for the Schroedges equation. That means that you can write out an equation that exactly solves, that exactly represents the Schrodinger equation for that system as a series of functions. For all other systems, we only have approximations, although those approximations can be made arbitrarily accurate. So remember, Schroedger's equation, that's the quantum mechanics equation, it describes the energy of a quantum system. And of course, it depends on your quantum

system, but you can define an atom, for example, as a bunch of electrons orbiting a nucleus, and they have a certain set of interactions that give rise to different forms of energy, and you solve Schrodinger's equation, and that gives you the wave function, which fully describes all of the quantum mechanical behavior of that system. So if you want to know about a system, a quantum system, you need to know the wave function, or at least if you know the wayfunction, you're able to tell anything you like about the system. You better calculate. You'll be able to predict it theoretically from the way function,

if you know what the wave function is. And the way function is an abstract mathematical description of the system, essentially. So the wave function is just basically sort of defined as being everything you need to know about the system or something like that. Again, I'm not going to go through precisely mathematically what it is. See the advanced quantum mechanics series for that. That's not really the point here. I'm just giving a bit of sort of background to understand the project here.

The project is solve Schroedges equation for an arbitrary molecule. So any molecule I give you, at least if it's small enough, if it gets too large, it's going to be impossible to solve. But if at least for small molecules, you give me the molecule, I solve Schrodinger's equation and give you the wave function for that molecule. Because then I can calculate whatever I like. I can calculate, again, the orbitals, the energies, geometric structure, excitation states, magnetic properties, whatever you'd like. So that would be great. But how do we do that? I've said you can only do that exactly for the hydrogen atom or hydrogen atoms.

Well, that's what molecular orbal theory is about, essentially, but there's a lot of complexities, as I said, because now we're dealing with multiple electron systems, and that's really where you get the problem coming in. So what I'm going to be talking about is how the Hartree-Fock method works. And the Hartree-Fock method is an attempt to get around all of the complexity that is involved in solving Schroedges equation for a molecule. So before I explain how the Hartree-Fock method works, it's named after its originators, basically, Hartree and Fock. It's not really a specific method so much as a process for getting to, for making this work or for making it tractable, I suppose that's a better way to put it. It's a process for solving Schroedges equation for molecules tractable.

So how does this work? Well, let's start by talking about the form of Schroedges equation for a molecule. What form does it take? Shrowne's equation, it basically is an equation that tells you the energy of the system. So it says that if I take an Hamiltonian operator, which is just an energy operator, and apply that operator to my wave function of the molecule, and that gives me back the energy of that molecule in that state, multiplied by the original wave function. The technical way to say this is that the wave function is an ligand function of the Hamiltonian operator, but don't worry about that if that's meaningless to you. The important point is the Hamiltonian is basically an total energy operator, which acts on the wave function for the molecule.

and each molecule will have a different wave function. So if I can solve Schroedges equation for a molecule, then I've got its wave function and I can calculate all the stuff that I want to, which would be great, because then you can make all these predictions and understand all of the chemical properties of the molecule, but how do I do it? Well, I need to know the form of the Hamiltonian for that molecule. Now, that's not too hard.

I can write out the form of the Hamiltonian, because it's just the sum of all the different types of energy. Well, what are the different types of energy? Well, for a start, there's the kinetic energy in the nucleus, because you've got protons and neutrons and stuff that jiggle and vibrate around. So you need to add that on. You need the kinetic energy of the electrons as they move around the nucleus. You need the potential energy of the electrons because of their attraction to the nucleus.

So that's electromagnetic potential energy effectively. And so you need to add that on. Then there's the potential energy of the electrons with their repulsive attractions to repulsion to each other. So not with the nucleus, but between electrons. They repel each other, so that gives rise to potential energy, so you need to add that on. And then finally, there's the potential energy of the nucleus. So basically, I've got two types of kinetic energy, nucleus and electrons, and three types of potential energy. Nuclear electron reaction, electron electron reaction, and just nuclear reaction.

Now, when I tell you that, already I've made some assumptions, and we're going to keep track of the assumptions that we've made here along the way, because Hartree Fock is about making assumptions, right? But it's about making the right ones, or useful ones. useful ones. So when I tell you that Hamiltonian operator, implicitly, I've ignored relativistic effects, because there are other types of energy interactions in addition to those that I've mentioned. And basically, these result from the effects of relativity. I mean, it's a little more complicated because, for example, you may know that the form of kinetic energy changes when you go from classical mechanics to special relativity, and so it depends on the form that I'm using.

But throughout, I'm just going to be assuming that special relativity is not a thing. We're just going to be assuming classical, and by that I just mean non-relativistic. In giving the Hamiltonian operator as I have, and in choosing the particular form of the kinetic energy and so forth, I'm just ignoring relativistic effects. Turns out this is fine for most molecules, but we'll talk more about that later. So already we've assumed no relativistic effects. Now, here's the next assumption that we're going to make. These nuclear terms, the nuclear kinetic energy and the nuclear potential energy, they make things complicated, well, particularly the kinetic energy of the nucleus, because we have to keep track of all of the sort of vibrational modes and stuff of the nucleus. And we can't be bothered doing that.

That's too difficult to do. So to avoid this, we use what's called the Born Oppenheimer approximation. Born Oppenheimer approximation. Very, very important concept here. This is the assumption essentially that we can treat the way function of the nucleus and the electrons separately. And the justification for this is that the nuclei are much heavier than the electrons. A proton is about 2,000 times heavier than an electron. And there is also neutrons in the nucleus, so that adds to the mass as well. So I mean it's a reasonable assumption. They're much, much heavy than any of the electrons. So basically a way to think about this is we're just going to assume that the nucleus is fixed in place and doesn't move. Or when I say the nucleus,

I should actually say nuclei, because remember it's a molecule, so there's going to be different nuclei one for each atom. So we're just going to assume that they're all fixed in place and none of them move, which is not true, they do move and they can move and they are affected by the electrons that surround them, but it's a good enough approximation for many purposes. So this is the Born Oberheimer approximation, and when we introduce it, we can basically just ignore all of the nuclear terms. So the kinetic energy of the nucleus, gone, basically because we assume it's static. The potential energy of the nucleus, well, it turns out that that can just be treated as a constant if it's not affected by the electrons, and so basically

you can ignore that as well, because it's just a constant that's added on at the end. So we don't need to worry about it in our calculations. So already, by ignoring relativistic effects and introducing the Born Oberheim approximation, basically the static nucleus approximation, we've simplified the which is nice because instead of five terms, now there's only three. There's kinetic energy of the electrons, potential energy of the nuclear and electron interactions, you know, attraction, and then there's the repulsive potential energy of the electrons with each other. So those are the three energy terms that I have to worry about here. I can write that out because I know the form of those interactions. So, you know, the form for kinetic energy is basically half mass times

velocity squared. And the form of the force between, the two charged particles is just the, well, it's proportional to the product of the charges divided by the distance between them squared. And so I can write that out for the nuclear interaction and also the electron-electron-election force, and put that all into my Hamiltonian. So this is good. We've made some progress. We've simplified the Hamiltonian by introducing the no relativistic effects and Bournematheim approximation, and I've written out the functional form of these kinetic and potential energy operators using just, you know,

standard knowledge from mechanics and from electromagnetism. So, that's all good. Now I know what my Hamiltonian looks like. But I still need to actually solve the equation, right? I still need to find the wave function that satisfies this equation, and I need to find the corresponding energies. So how am I going to do that? Well, here is where the Hartree Fock method really comes into its own.

We really start applying this method specifically, because there's a very clever idea here. Here's what the Hartree Fock method says to do. Let us write the total molecular wave function, the wave function for the whole molecule, as the product of individual molecular orbitals. Remember, an orbital is basically just like a region of space with probabilities associated with the likelihood of finding an electron in that space

or in that region, and there are going to be two electrons, or up to two electrons in each orbital, a spin-up and a spin-down. Although, if that's confusing, it may be easier to just think one electron, one orbital, because although it's not, like, there's two electrons in each orbital, it's sort of helpful as a way of conceptualizing it, that, like, okay, each electron has its own orbital and all the orbitals have different shapes. It's just at the end you have to kind of times it by two because there's two electrons in each orbital. But turns out that that's actually an okay way to do it because the electrons are indistinguishable.

But anyway, I'm going to write my molecular wave function as the product of individual molecular orbitals. Now remember, these are not our atomic orbitals, the SPs and Ds and so forth that we know and love. These are unknown molecular orbitals. We don't know what they are and they're going to be different for each molecule that I look at. You might ask, well, how's that helpful? I've just said that a thing that I don't know is equal to the product of a bunch of other things. things that I don't know. Well, it turns out that it's a step forward, but we'll have to sort of wait and see how that's a step forward. But unfortunately, physics makes things more complicated for us,

because this isn't going to do as it stands, because I've been ignoring one very important aspect of electrons. That is electron spin. I didn't mention this a couple of times. Electrons can be spin up or spin down, so there's two spin states that they can have. Each electron has to be one or the other. But at any given time, you know, you don't know which ones it's going to be and only determined after measurement, but that's standard quantum mechanics. The point, though, is I need to represent that in my wave function. And so, in addition to the spatial part of the wave function, I need also a spin part of the wave function to form what's called a spin orbital, which is one word spin orbital, and basically just multiply them together, the way function as a whole, but also

each molecular orbital consists of a spatial orbital part, which says how likely is it that the electrons going to be found in this region of space, times the spin part, which says, how likely is it that when I measure it, the electron's going to be spin up or spin down. That's called a spin orbital. It's important that I mention this, although I'm not going to talk about spin orbitals too much, because you can factor the two out, because the spin is separable from the spatial component. So why do I mention? Well, because it's very important for what's about to happen next, because there's one more thing that's kind of complicated about electrons. And it is that electrons are indistinguishable from each other, which means that there's no such thing as this

electron or that electron. There's just electrons. They're all identical to each other. Now, they may have different properties, like the electrons may be in different places, for example, or they may have one bump you spin up and one might be spin down. But the point is, suppose I have two electrons, electron A and electron B, and then I measure them, and I see one, there's one that spin up and there's one that spin down. Here's what you can't say. You can't say that electron A is spin up and electron B is spin down. Why not? Well, because electrons are indistinguishable. That doesn't mean anything. There's no electron one being spin up and electron two being spin down. All you have is there's two and one is up and one is down.

You don't know which is which. And it's not even just that you don't know. There's no fact of the matter about which is which. This is kind of hard to understand because classically things don't work that way. You might as well just look and see that this one is up and this one is down. You just can't do that. It doesn't make sense because you don't know. It's not like you can write a label on this electron and say, ah, this is electron A.

I don't know when I measure an electron which electron it is. And in fact, there's no fact of the matter. it's just an electron. They're all identical to each other. We can put labels on them, but it's meaningless, right? It's just our way of keeping track of them. There's no actual difference between them. So that means we have to treat the maths differently. And the way to do this, I don't think it's going to be helpful to try to explain mathematically why it works this way. You really just have to look at it. But the way that you incorporate this indistinguishable element to the mathematics is to use something called a Slater determinant. For those who know about this, the determinant is

an operation that you apply to matrices that you sort of perform on the matrix elements. So for a simple two by two matrix, the determinant is equal to the product of the diagonal terms minus the product of the cross diagonal terms. For those who don't know anything about matrices, doesn't matter, just forget about that. A determinant is a thing that you do to a matrix, which is just a list of, a matrix is just an order list of numbers, essentially, in two dimensions, like a grid and you put the numbers across it. Turns out that you can represent the molecular wave function, the total molecular wave function, as a slater determinant. And doing this is really useful because it allows you to incorporate the fact that electrons

are indistinguishable. The intuition for that, if that seems a bit weird, is just that I can't write it down and just say electron A is an orbital 1 and electron 2 is an orbital 2 because that doesn't really make sense. What I have to say is electron A is an orbital 1 and electron B is an orbital 2, minus electron B is an orbital 1 and electron A is an orbital 2. I have to write both of the possibility. and I have to subtract them from each other. But obviously I just gave two orbitals as a simple example, but you have to do that for all orbital combinations. So what I have is a big long list of different combinations

of A and 1 and B and 3 and 4 in D and whatever else, different combinations of electrons in different orbitals. And this is because they're all indistinguishable, so I have to sort of consider all of the possibilities when I'm writing out my wave function, because otherwise I've sort of missed out some of what's there. And a slated determinant, it turns out, it's just a really convenient way of doing that.

Now, the name is important because we're going to be talking about slated determinants a lot more in the next episode when we talk about considering electron correlation. But we'll worry about that later. So just bear in mind, the way you represent a molecular wave function is with a slater determinant, which is a mathematical structure, which is basically a bunch of combinations of my molecular orbitals. Many molecular orbitals times together gives me back my molecular wave function.

Don't get confused between a wave function and an orbital. A wave function describes the whole molecule. and all of the electrons in it. An orbital is just for two electrons, one spin up and one spin down, and it only describes part of the molecule. So it's like many orbitals make up one molecule, and a slated determinant describes how to combine the orbitals together

to get the wave function, which describes a molecule. All right, so we're already making a lot of progress. We've written the wave function in terms of a complex combination, as defined by a slated determinant, of the individual molecular orbitals. The problem is we still don't know what the molecular orbitals are. At this point, if I told you what the molecular orbitals are, then you could use those to, you could use a slated determinant to work out the molecular wave function, and then stick that with

the Hamiltonian operator that we've worked out and then solve your shoronics equation, and solve your shoronidious equation, find your energies. But we don't know what our molecular orbitals are. But as I said, already we've made a lot of progress because we've gone from having no idea how to solve shroner's equation to now the specific question is, how do I find my molecular orbitals. Again, a reminder, these are not our good old hydrogenic atomic orbitals, SPD, and so forth. So we can't just use those. They're different molecular orbitals, and they'll be different for each molecule that I want to solve this for. So I need to find a general method to solve for these. And this is quite difficult, but let me explain how the Hartree-Fock method does this. It's quite,

it's quite ingenious, I think. So the way it does this is to use a principle called the variational principle. Now the variational principle and its use here is really, really nifty, I think. So let me explain how it works. We want to find our molecular orbitals so that we can find our molecular wave function and solve for energy and so forth. But we don't know what these molecular orbitals are. And in general, there could be very, very complicated functions with very complicated shapes. And so, you know, how would you even guess those? But there is a way to do this. And the way we can do it is to utilize the fact that we know that the molecular wave function, the whole wave function, has to be the lowest energy state of the molecule, at least when it's in its ground state.

You know, electrons kind of excited states, but we're talking about the ground state. When it's in its ground state, it must give rise to the lowest possible energy of that molecule. What does that mean? Well, what it means, and this is the really cool bit, suppose that I guessed a different wave function that's not the correct wave function, and then I stick that into Schrodinger's equation and solve for the energy, because I can still do that, right, even if it's the wrong wave function, like I can still stick it in and see what I get. What you will get, though, you'll get an energy, right, and it will be the wrong energy, but here's the important part. It will always be higher than the true energy, the energy that you would get if you use the correct wave function.

So if I give you a wave function, a molecular wave function, and you stick it into Stroudanages equation, and calculate an energy, the energy of that ground system, it's always going to be too high. If it is correct, it will be the lowest possible energy. Why is that? Well, because remember, systems always want to sort of reach the lowest energy. state. So that's what the true, the ground system of the true wave function is going to be. It's going to be the lowest energy state. It's not going to be something that's not the lowest. Because then, well, it could reach a lower energy state, right? That's how it works. So this key principle, the true wave function will have the lowest possible energy allows us to actually solve for the molecular wave functions. Well,

kind of, but you'll see what I mean in a minute. It at least allows us to get an equation for them, which we then have to solve. Basically, we can use calculus to find the minimum or maximum of a particular function as long as we can differentiate it. For those who've done calculus, this will be very familiar. You differentiate a function, equate the derivative to zero, and solve, and that will tell you where the minimum or maximum is. In this case, it's going to be a minimum. This is finding the extrema of that function. For those who haven't done calculus, the principle is fairly common-sensical, if you think about it. Imagine that I go to the top of a hill. Imagine it's a sort of a simple hill that's sort of curved. When I go to the top of the hill, think about what the ground looks like.

at the top of the hill, the ground is sloping downwards in all directions. But at the very top of the hill, like at the very, very peak, the ground has to be flat, right? Because the top of the hill is where it goes from sloping downwards on one side to sloping downwards on the other side. So there has to be a part in between it, assuming it's smooth, where it's exactly flat. And as it turns out, this actually is true for all possible geometries, as long as it's sort of smooth. There's always going to be a point where the slope is zero, and that will be the maximum point or the minimum point. It works in a valley as well. If you imagine going down in a valley, it slopes downwards on all the sides, but there must be a point right in the middle there at the very lowest point where it's exactly flat, because it's at that point that it sort of stops being sloping down on one side to start sloping up on the other side, if you see what I mean. It's where the marble will roll down to if you let it roll. That very point at the bottom there, it will be flat. And that is how the variational principle works. It says, look, where is it flat? That is where is the energy surface flat? of the function, it's going to be flat at the minimum energy.

And therefore, all I have to do is find the molecular orbitals that give that minimum energy. And those will be the true correct molecular orbitals for my true correct wave function. So that's awesome. I've found a method to find those molecular orbitals. However, I need to first actually calculate this derivative. And doing that is a bit tricky because I have to write an equation for the energy of the whole molecule. Now, doing that takes quite a bit of maths. The basic idea is fairly simple enough.

We know how to write the energy of a quantum system. It's basically the integral of the wave function times the Hamiltonian operator, which gives you the energy times the wave function again over all of space and also spin space. Don't worry if you don't really know what that means. But it's basically applying the Hamiltonian operator and integrating over that. So you solve an integral. And to do that, we need a Hamiltonian operator. Well, I've said what that is.

You know, we grab the potential energies and kinetic energies of the electrons, and we know what that is. So we stick that in, and we write that as an equation, and then we do a bunch of maths. Luckily, we're able to do this because we know what the form of the wave function looks like. Remember, we know the Hamiltonian, that's the energy operator, and we know that the wave function is a slated determinant operating over molecular orbitals. So we can stick that slated determinant into the equation for the energy,

chug the sausage machine, and what we get at the end of this is an equation for the energy of the molecule, the whole molecule, not just one orbital, but the whole energy of the molecule. And it turns out that there's a fairly ready interpretation to give of this equation here. You can write it in fairly simple terms. The way it's typically written is in terms of H, J, and K, if you're sort of following this along. And these HG and K terms are then themselves complicated integrals. But here's what it all comes down to.

When I find the energy of the whole molecule, it turns out that there's only three. separate bits of the energy that I need to worry about. And each of these bits is a complicated integral that contains the molecular orbitals. But still, it's a big simplification. So here are the three different terms, the bits of the energy that need to worry about. The first is the kinetic energy of a single electron in a particular orbital, as well as its potential energy with the nucleus. So these are the bits, this is the H part, and these are the bits that refer to basically a single electron orbiting in an orbital and reacting, interacting with the nucleus. Technically it's all the nuclei, but I'm just going to say one nucleus just to simplify things.

Remember this is a molecule, so there's multiple nuclei. So that's the simple bit, the kinetic energy and the potential energy with the nucleus. Then there's the J term, which is the interaction potential energy. And that arises from the electrostatic repulsion between electrons in different orbitals. So if I put two electrons near each other, they're going to repel. And that gives rise to a potential energy of interaction. So that's another source of energy here. Now there's also a third aspect or component of the energy, and this is kind of a complicated one.

This is denoted as a K for those following along. This is a correction to the interaction energy, which arises from the Pali Exclusion Principle. The Pali Exclusion Principle, if you recall, says that you can only have two electrons in any single orbital, because two electrons cannot have the same quantum state in the same quantum system. Basically, the way to think about this is you can't have two electrons with the same spin in the same space, or they can't get too close to each other. And so electrons that have the same spin repel each other effectively.

They push further apart from each other than they otherwise would. And this gives rise to a correction to the electrostatic repulsion, the J term, between individual electrons. So this correction arising from the Palli Exclusion principle is an entirely quantum phenomenon.

There's no classical analog. It arises from spin effects, which again we don't really see in the classical world. So if it seems a bit weird, Don't worry, because it is. It's a quantum effect. But it's very important, and you have to put it in. Remember when I mentioned electron spin a ways back, and I said, well, actually, you need spin orbitals, and I need to consider the different spin combinations and slated determinants and all that. That was critical, because if we didn't do all that, we wouldn't get this correction term there in the energy.

And if we don't have it here, then we'd get the wrong answer. So it's very important that we included that spin stuff, even though it's quite complicated. So, to summarize, we've just now written the total energy of the whole molecule in terms of the kinetic and potential energy of a single electron in a particular orbital interacting with the nuclei, the interaction between electrons as they repel each other and they're in different orbitals, and also a correction to that interaction arising from the power exclusion principle and different spins relating to each other. So there's three terms here in our energy function. Now, I just apply the variational principle, basically differentiate the energy, equate the derivative to zero, solve, and I get an equation. I get an equation that, if I can solve it, will tell me, will show me what the molecular orbital should be.

There's different ways to write this equation. The simplest way to write the equation, though, is just as, oh, I can value your equation, which is just like the Schroeneges equation. Remember, Schroeneg's equation, is the energy operator, H, times your mind, you molecular wave function gives the total energy times the wave function back again. Turns out that when I go through the variational principle and solve it all and tidy it all up, what I get is a similar thing back. But with a twist, instead of the Hamiltonian, I get a new operator called the FOC operator, denoted with an F. And it's multiplied not by the total

molecular wave function, but by the set of molecular orbitals in my wave function. And then that gives me the molecular orbital energies times by the set of molecular orbitals back again. So Fock operator times molecular orbitals gives me energies of those molecular orbitals times the molecular orbitals again. So it's an eigenvalue equation. And these equations, it's not really one equation, it's a bunch of equations, are the Hartree Fok equations. They're really, really useful because if I tell you what my operator F is, the Fok operator, then I can just solve these equations using matrix methods to solve a bunch of simultaneous equations, to get the molecular orbitals. And then with the molecular orbitals, I can just stick those in my slated determinant,

and that's a complicated combination of molecular orbitals, and get my total molecular wave function, which is the thing I'm looking for. So the longer and the short of it is, applying the variational principle allows us to find an equation for the molecular orbitals, which we didn't have before. Previously, we just had no idea what they were, but now we have an equation for them. So that's pretty cool. And it all uses the property that the true wave function must have the lowest energy of all possible wave functions. So we can just write an energy equation, minimize that, and then find the molecular orbitals, then minimize it. And then boom, those are our molecular orbitals. So we stick them into the slay determinant. You get your total

molecular wave function. But there's a problem. You knew this was too good to be true, right? It can't be that simple. Well, unfortunately, it's not that simple. Because that fuck operator, the one that I mentioned that if I just told you that, you can solve the equation and find your molecular orbitals. Well, it turns out, it's a complicated equation. I won't try to explain it, but it's got the different energy terms in it, including the HJ and the K terms, and I mentioned the interaction terms, the spin correction term, the kinetic and potential energy terms. It's got all those terms in it, right? But just for each orbital instead of for the whole molecule as a whole, as the H operator did. But the problem is that in the FOC operator, buried in there in all the maths, are the very

molecular orbitals that I'm trying to find. So remember I said if you give me the Fock operator, I can solve the Harsary Fok equations and get the molecular orbitals that I need. But the problem is in order for you to give me the Fok operator, I need to give you the molecular orbitals. But those are the things that I'm trying to solve for in the first place. So it's a chicken and egg problem. I need to have the molecular orbitals to find the thing that I need to solve for the molecular orbitals. So how do we do this? It seems impossible, right? And so is all our work wasted? Are we stuck? Well, no, because problems like this in mathematics are not impossible to solve, they just take extra effort. Sometimes you can solve them analytically, but not in this case, because the

Fok operator is too complicated. It's got all these different energy terms and integrals in it. Instead, the way we do this is we solve it iteratively. What does that mean? Well, it basically means we take a guess at what the molecular orbitals are, then we work out what the Fock operator is for those orbitals, stick that into the Hartree-Fock equations to solve for the orbitals, and then see how close they are to our guess. So you basically make a guess and see that if the result that you get is close enough to your guess. In general, of course, the first guess that you make is going to be way off. And so the molecular orbitals that you get are going to be, after your first round of calculation, are going to be very different to the ones you started with. So what do you do? Well,

you take those new ones, and you use them to update the FOC operator using the new molecular orbital. So you've got a new FOC operator, version 2, if you like. Use that, stick that into your heart tree fog equations to calculate version 2, well, actually version 3 now of your molecular orbitals. And then you use version 3 of your molecular orbitals to calculate version 3 of your FOC operator, which gives you version 4 of your molecular orbitals, which gives you version 5 of your molecular orbitals, and you go round and round on round, until you basically breach a situation where version N of your molecular orbitals is basically the same as version N plus one of your molecular orbitals. So they converge to each other. And then you basically say, well, I found it. This is it.

It's self-consistent, and this is why this technique is called a self-consistent field method. Because you basically start with a guess, and then see if that guess is consistent with the equation that you're trying to solve. And if it's not, well, then you need to try again. But if it is, well, then that's your solution. So this takes a lot of computing power because you have to iterate many, many times before you're going to get the solution. And every time I want to go around this sort of circle of iteration, I have to calculate lots and lots of integrals. Because in the FOC operator, it turns out that there's basically not just interactions between each electron and the nuclei, but also there's interactions between each electron and all the other electrons in all of the different orbitals.

It turns out therefore that these Hartree-Fock methods, the difficulty of solving them grows to the fourth power of the number of electrons. And this means that it's very, very difficult to apply these Hartree-Fock methods to anything but fairly simple systems, because there's just too many integrals to solve. Even with small and a few dozen electrons, you get millions and millions of integrals that you have to solve each time you go around the iterative process. This can be done with modern computers, but you can see it's expensive because there's just lots and lots of computations to do. There's one wrinkle that I haven't really explained yet. Well, I mean, there's a few wrinkles, but there's one that I'm going to mention here before finishing out, and then as a lead up to the next episode, or part two of this. And the wrinkle is this.

I've said that in order to solve these Hantry Falk equations, we need to take a guess at what the initial molecular orbitals are, and then we find the Fock operators, stick that into the Hantry Fock equation, solve for the new molecular orbitals and then update. But the problem is, how do I even know what to guess, and how do I know how to update that guess? Because this is just, the molecule the orbitals is just some function. Like, how do I know what the function looks like? There's arbitrarily many different functions.

There's an infinitely many different functions that it could be. How do I even know where to start? I've got to have somewhere to start and some way to sort of update the guess. And that's absolutely correct, right? So you do need some way to represent these functions, and you need a language for how to do that. And it turns out that there is a way to do this. And that is you need to select a particular basis set.

And it's in the next episode that we're going to talk more about that basis set. and how that gives rise to the ruthan equations, which are the actual equations that you solve when you're doing this. You don't actually directly solve the Hartree Fok equation. You're actually solved the Ruthan equations, which is kind of like version 2.0 of the Hartree Fok equations, if you like. And this involves selecting a basis set, and then going from there to actually solve the equations

and get your molecular orbitals. And so in the next episode, I'm going to explain how that works, how you choose your basis set, what that means, how you put all the ingredients together to solve for ultimately your molecular orbitals, and then your molecular wave function. But before we get to that, or before closing out this episode, I'm just going to briefly summarize where we're up to so that we can put the pieces together. So remember, what we're trying to do is we're trying to solve Schrodinger's equation

for an arbitrary molecule that contains multiple electrons. Unfortunately, there's no closed-form mathematical functional solution for this, because it's too complicated. So we're going to have to have approximate solutions. Some of those approximations include ignoring relativistic effects, and the born Oppenheimer approximation, which basically involves ignoring the kinetic energy of the nucleus, and assuming it's basically just stuck there. There are other approximations, which I'll mention later. So those approximations and knowledge of the nature of kinetic energy and electrostatic interactions allows me to write down

the H, which is the Hamiltonian function, the energy function essentially for that molecule. So I can write that out, and then I can put that in Schroeneges equation, but how do I solve for the wave function of the molecule? Well, the Hartree-Fock method says that you can write the wave function as a product of molecular orbitals, where a molecular orbital is basically just a function over the molecule space that says where an electron is at a given time, or the probability that is in a particular space at a given time. That's complicated because then I need to incorporate electron spin into that, so it's not just a spatial orbital, it's a spatial and a spin orbital, or a spin orbital, as it's called. And also I need to factor in electrons being indistinguishable. So I can put all these pieces

together by writing the total molecular wave function as a slater determinant, which is basically just a big complicated combination of molecular orbitals. Each orbital will have two different electrons in it, or up to two electrons. And that's good, because now I've got a form for my total molecular wave function, and I've got a form for the Hamiltonian. So I can stick those together, but I still need a way of solving the actual equation. How do I know what form the molecular wave orbital take. The answer is I use the variational principle. I write an equation for the total molecular energy, which includes the kinetic energy of the electrons, the potential energy of the electrons interacting with the nucleus, the interaction potential of the electrons repelling each other,

and the Pally exclusion principle correction to that repulsion interaction. So all of those terms will go into the total energy function for the whole molecule. So I write that out, and then I use calculus to minimize that energy with respect to the molecular orbitals. That's the variational principle, which then allows me to write out what I call the Hartree-Fock equations, which is kind of like the Trondon's equations, except they apply for the molecular orbitals instead of the whole wave function. So I've greatly simplified the problem. The Hartree-Fock equation say that if I take my basically single electron operator, energy operator, which is called the Fok operator, multiply it by a list of all my molecular orbitals, then that gives me back a list of the

energies of each of those molecular orbitals times back the list of those molecular orbitals. So it's an eigenfunction equation. If I can solve this equation, I can solve for the molecular orbitals, and with those, then I can stick them into the Slater Determinate, and I've got the total molecular wave function. The problem is, I can't solve the hydrogen-fok equations directly, because I need to actually know the solution in order to know what the fog operator is. And so there's a chicken and egg problem. I get around this by solving them iteratively. So you start with the guess, then you update the guess using the equation, and then you use that as the new guess, which then you update using the equation again, and you go around and around until you meet,

sort of meet in the middle with a consistent set of guesses and answers, and that is your solution for the molecular orbitals. So that's where we're up to. But I haven't quite explained how you get a guess in the first place and how you kind of update that guess. Like, how do you even know what form to start with for the molecular orbitals? And that's what I'm going to talk about next time I'm going to talk about the basis functions that you use and how you put those together to form your molecular orbitals and use that to solve the Hartree-Focke equation. Next time I'm also going to talk about a couple of the other approximations that have been made. In particular, we have ignored electron correlation, and I'm going to talk about how we resolve that and get more accurate solutions.

Also, I'm going to talk about some of the other methods that appear in computational chemistry. So I've just been talking about what I call ab initio techniques with the Hartree-Fock method. But there are other techniques as well. There are semi-emperical methods, and there's density functional theory, which is a different take on there. So I'm going to not go in those as much detail, but we'll talk about them briefly. And also I'll talk a little bit about molecular mechanics or molecular dynamics, which is something I've done work on. So stay tuned for those next time. I hope you found this interesting. I know it was a more technical episode, but again, I did warn at the start. And so if it was difficult, feel free to go back and listen

to some previous episodes that might be good for a background and then have a listen again. as I said there will be a link to the notes that are relevant on the hopefully the episode description and also the Facebook page so you can refer to that if you're interested speaking of which jump on the Facebook page and give a like just type in the Science of Everything podcast and helps to promote the show you can also send me an email if you'd like to ask a question or make any suggestions my email address is fods12 at gml.com that's fODs1.gmol. If you'd like to support the show financially you can do so by making a one-off donation. via PayPal or you can join the Patreon page and make a monthly pledge there.

You can find the details to that in the show description. Thank you very much for listening and I'll talk to you next time.