 So one of the first kind of silly and stupid things I did in Excel and spreadsheet was to plot the atomic orbitals This should really be done with proper software and you get Patterns like this is basically kind of a probability distribution of where you will find an electron around an atom They look a little bit like this as well in 3d and for the pedants We're not plotting the original wave functions were plotting kind of the real component of a Superposition of two of them, but the physics is not really worth getting into It's more of a teaching aid in a demonstration to show that the maths work. This is not some kind of quantum mechanical magic It is just plotting things. So they do kind of have some horrible looking equations But what you can do the analysis a little bit you realize most of them are just constants What really comes into one is in this one, this is an angular component It tells us how it varies as we go around a circle It's really just next component just divided by the radius and here we mostly just an exponential function It's an exponential decay row here being the radius, but it's kind of scaled Not really worth going into it all so this is just to quickly visualize You can modify all these equations to include every single variable Correctly if you want They're all written down either in the Atkins textbook or on Mark Winter's Orbitron site If you go to the D orbitals for instance and click equations. He's got the list of them here So we can take them from there now. How do you put this together in a spreadsheet and plot them? Well the turn that it's actually quite easy now the first time I did this it would involve a lot of array functions and dragging and dropping but now it's it's kind of really easy, so I'm going to start with a sequence function and Let's make this big. Let's do 51 rows One column. I'm going to start from minus 25 Take a single step. We go from minus 25 to plus 25 except that's a bit big so I'm going to divide that by 10 there we go now we go from minus 2.5 to 2.5 I'm gonna copy that and paste it into here and swap these two numbers around so I'm actually going to go for 51 columns of one row Same thing we're going from minus 2.5 to 2.5 and it always passes through zero as well. That's why I've used an odd number of rows and columns I am Let's resize that so now we're looking at it's about 40 to 38 pixels on this screen and Let's zoom out so it's Quite a big one that we're going to plot here. Usually when I'm testing this out. I'll just do 10 by 10 I think you can do it up to a hundred, two hundreds if you want, depending on how many cells Do you want to include in it? It's like each of these is going to be a pixel So What I'm going to do is now start giving these names because what I need to do is make these things look a little bit more Like equations and makes it a bit easier. So I'm going to create a new name. I'm going to call this X and it's B1 I'm going to put a hash on the end because it is a dynamic array and I want this Named range to reflect that so B1 hash. I've got X. I'm going to do Y Which is actually going to be a Two hash there So if I okay that one basically got Two sequences, so let me just check that's worked. I take Y. There you go. There's all the numbers associated with Y axis And I type in here in the top left corner. I can do X times Y Here we go. I've now spilled across here Automatically, it's understood that this is 51 Across and this is 51 down and it spilled everything in and you can kind of see there's going to be zero in the middle It's negative here because we've got one positive number and a negative number It's positive here because it's negative times a negative all works really nicely now if I wanted to work out See the radius from this origin. I'm going to type in square root of X to the power of two plus Y to the power of two So now if I return that I've got a radius instead It's zero in the middle and it gets bigger and bigger and bigger as we get up to the corner and Numbers are fine. You can kind of see a little circular pattern to them But if I press control shift right control shift down to select all of that I can do it with conditional formatting. So let's come to home conditional formatting color scales so basic blue to red color scale is kind of kind of a very usual color to highlight these in But this is not the end of the story for the conditional formatting because this if I edit the rule and look at it It's going from the lowest value, which in this case will be zero to the highest value, which is blue We don't want that for plotting Orbitals because they are positive and negative so I want one color for the positive value of the function and Another color for the negative value. So I'm going to go stretch it from minus one to Plus one just by selecting number from all of these and I'll make the color a little bit more Intense just to customize it a little bit So now What I can see is there's a white dot in the middle for zero It looks kind of like it's gluing but blue as it gets more intense And that's not the kind of the end of the story because we don't want these numbers on either so I'm going to select all of those again and Pick this option to format the number and I don't want numbers or anything. I'm going to put custom and you just put three semi-colons in The semi-colons kind of separate Different types the positives negatives and zeros and you don't put anything in there So it just doesn't format the number and what happens if you okay that it's rid of it So there are numbers associated here, but the software is not rendering those numbers. It's just rendering it as a color Great Final thing. I'm going to copy that square root Function here. That's Pythagoras's theorem go back to my formulas Create a new name I'm gonna call this radius you could call it R if you wanted to but it will reject it and turn it into R underscore So it may as well. Just call it radius Now if I type in radius here Same thing it still tells me that it's gonna be radius now How do we start plotting orbitals? Well, I start with the simplest of a go to the Orbitron 1s these are perfectly circular The functions will be simple it's just well kind of e to the power of minus row over two and row is a function of The radius so I'm just going to simplify it and just say that this is going to be exp Minus radius, and I'm going to keep the over to just to scale it correctly if I return that Well, I've got the peak of the wave function in the middle And then it exponentially decays down so you can kind of see that circular Pattern that's kind of circular symmetric won't you just taking a slice through the middle of them you can in theory render This in three dimensions, but you'd have to like then to take a slice through it and just set your value of z You couldn't quite plot this in 3d This is not the software to do this. Okay, come on. This is this is just to show that it can be done And it's just there's nothing magic about these things Now if I wanted to turn that into a p orbital, I just need to take X and times that So if I come back to the Orbitron again Come to 2p and look at the equations. There's a little bit of a More complicated bit of normalization I need to use I'm just kind of ignore that for now And it's just x divided by r but also it kind of multiplies by r there So we just do it by x and what you can see is that's the The shape it is negative on one side positive on the other. Maybe we could scale that up and down if we wanted to Render it in a particular way right and what about a d orbital well in that case it is something like x times y So what we can see here is we've got a negative positive negative positive That's one of these things here And you can see this is actually called xz. There's another one called xy And we've also got x squared minus y squared. So let's try what that is. Let's do x squared minus y squared we turn that it's a Completely different thing. It's kind of rotated it 45 degrees. That's quite nice To see so this is kind of what we expect from this function and if I change these Sequences just a little bit remove that by 10 you can see it's scaled in a slightly different way we can now see it Really good so far when we can play with the scaling kind of manually if we want Right, so that is the x squared minus y squared that makes sense, right? So where x squared is zero straight down here It's always going to be equal to y squared, but the negative of it. So this is all negative down this area right here and Where y squared is zero this is always going to be positive So it's going to be equal to the x squared have a Maths works there Another couple of things we can do. Well, let's go back to it being x times that I'm going to copy all of that and put plus Y squared there now there is usually a normalization constant here to scale it down But I'm not going to bother with it for now If I add an x px orbital to a p y but all it kind of rotates at 45 degrees. So what I could do is stick something like Let's say a sign of the number or converted to radians 45 degrees times that Copy all of that and put it before here change that to cosine Then go 45 degrees. So what if I change this 45 degrees to maybe 20 degrees? You can see the rotation is slightly less intense. I could probably multiply the whole thing up just to make it a bit more visible That's about four. There we go a bit more intense there So if I change that to maybe minus 20 that spins it in a different direction So we can do super positions with this and rotate things away And we could also get rid of maybe the y component of This Gonna get rid of the angle as well. So we've got what's basically That p orbital and s orbital we have those together you get hybrid orbital I need to multiply that one up to visualize it a bit more It kind of becomes a bit of a Oops Make sure my brackets are on my place here. Yeah, well, we're getting here is Kind of big on one side that is what happens with a We'll put that one bit a hybrid orbital kind of bulges off to one side. So we do a lot of Interesting things with it just plotting the equations out Like that and we can kind of prove that this maths does form these shapes