The following content isprovided under a Creative Commons license. Your support will helpMIT OpenCourseWare continue to offer high-qualityeducational resources for free. To make a donation orview additional materials from hundreds of MIT courses,visit MIT OpenCourseWare at ocw.mit.edu. MICHELE: Welcometo lecture three on quantum mechanical methods. As you may haveguessed, I’m not Jeff, as much as I looklike him, I know. I’m actually a graduatestudent of his. I’m Michele. So feel free to interruptme with any questions you have during this. And this is a really excitingclass today because today, we’re actually going tostart talking about some of the quantum methodsthat we’ve been doing in the background up till now. So first of all, Iguess you guys know that you don’t haveclass on Thursday, but I guess for thoseof you doing projects, you can go meet withProfessor Buehler then.So on our outline, we are here. We’re going to be talking alittle bit about modeling, the beginnings ofmodeling today, and a little bitabout molecules. So last time, we talked abouta single electron system. We talked about hydrogen, andactually, that gave us a lot. We were able to explain howspectral lines came about, just from understanding thebasic structure of hydrogen. But today, we’re going towork on more-than-one electron systems.And you’ll see that this iswhere the computation really becomes necessary. So here’s the outline. So you can seetoday, we’re going to talk a little bit aboutHartree and Hartree-Fock methods, and densityfunctional theory. So I believe youguys are going to be using density functional theoryto your next problem set. So just some reviewfrom last time– here’s the Schrodingerequation that I’m sure you’re all sickof seeing by now. In general, if our systemis time independent, then our Hamiltonian isgoing to be time independent. And we can separatethe variables into the spatial componentand the time component. And then by dividingout the time component, we are left with thisstationary, time-independent Schrodinger equation. And that’s the equationthat we’ve been solving and we will continue to solve. So as I said, last time wetalked about the hydrogen atom. We just solved its Hamiltonian. The Hamiltonian itself is,of course, the kinetic energy and the potential energy. The kinetic energy is this. The potential energy is actuallyjust that Coulomb attraction between the electron and theproton in that hydrogen atom.And this, we saw, waspretty simple to solve. I guess we didn’t actually doit, but the way this is solved is you switch intospherical coordinates. You solve for a radialcomponent of the wave function, and an angular componentof the wave function. So to any of you who havehad a lot of chemistry, these probablylook very familiar. So this is the s, these threeare p, these five are d. And these are basicallyjust the spherical harmonics that you get fromsolving the theta and phi parts of the Hamiltonian.And when we combine these,we get the total spatial wave function for the hydrogen atom. So the spatial wavefunction is described by these quantum numbers. And the principle quantumnumber, l, the angular momentum component, and m l, which isthe angular momentum component, projected along the z axis. So you guys have probablyseen these a million times– that leads to thesespatial wave functions. So this is increasingprincipal quantum number and this is increasingangular momentum. All right, so we’ve allseen this graph now a couple of times. Can somebody tell me whatthis graph represents? AUDIENCE: [INAUDIBLE] MICHELE: Right.And why is it important thatthere are these specific lines that it can go between? I mean, what phenomenon arewe capturing with this idea? AUDIENCE: We can [INAUDIBLE] MICHELE: Right. So this is why peopleobserved spectral lines instead of seeing acontinuous spectrum. and so atoms, instead ofemitting a continuous spectrum, emit specific lines. And that puzzle was solved by– we actually already solved itlast time with the solution to the hydrogen atom, because wesaw that the hydrogen atom had these discrete energy levels. And electrons could onlyexist in those specific energy levels. And so when an electron wentbetween those two energy levels, it would emit theenergy of that difference in energy levels as a photon. And that photon wouldbe a specific wavelength corresponding toa specific color, and that’s exactlywhere this emission line spectrum comes from. So let’s try to addone more electron. So this is our Hamiltoniannow for helium. We still have the kinetic part,the T for electrons 1 and 2.We have the potential partfor electrons 1 and 2. That’s the interaction ofthe electrons with the helium nucleus. But now we have this termthat’s the Coulomb repulsion. It’s the two electronsinteracting with each other. And this makes theproblem suddenly unable to be solvedanalytically, just this additionof one more electron. I guess you could think of it assort of the three-body problem in classical mechanics.So last time, we alsotalked about spin. So we know that electrons areeither spin up or spin down. Does anybody rememberthe experiment that was done to show this? AUDIENCE: Stern-Gerlach. MICHELE: Yeah, good job. So the Stern-Gerlachexperiment, basically it was a stream ofelectrons that were shot through this magneticfield and then were observed on a plate behind it.So electrons being shotthrough this magnetic field is kind of like the spincomponent of the wave function being observed. So now that we’re observingthis wave function, it can only be inits eigenstates. So those eigenstatesare here and here. So you think if youhave an electron that can interact witha magnetic field, its magnetic moment could bepointing sort of any direction. So if it’s pointingperpendicular to the magneticfield, why would you expect there to be anykind of interaction? You just expect it to endup in the middle here.And similarly, it could beanywhere between up or down. But this is quantum mechanics. This magnetic field ismaking a measurement. And so we all know thatonce a measurement is made in quantum mechanics,the wave function collapses to one of itsobservable eigenstates. So it may seem strange to youthat if you know something about where magnetic momentsusually come from, usually we talk about magnetic fieldsbeing produced by electrons physically spinning in space. But this is just someproperty of an electron that creates a magnetic moment,without any kind of movement. So that’s still somethingthat we don’t really understand today, whatexactly that spin means. But we observe it andit seems necessary, so right now, all we can say isthat we know electrons either spin up or they’re spin down. And Jeff showed youlast time this letter to Pauli abouthow even Pauli was a little skeptical ofthis spin quantum number, but it’s actually crucialfor his exclusion principle. So does anybody remember thePauli exclusion principle? Yeah. AUDIENCE: [INAUDIBLE] MICHELE: Exactly. So all the electronshave to have– all four quantumnumbers have to be different between anypair of electrons. So the four quantumnumbers, again, are the threespatial components, and now we’ve added onemore, which is spin. Pauli himself said, “Alreadyin my original paper I stressed thecircumstance that I was unable to give a logicalreason for the exclusion principle or to deduce itfrom more general assumptions. I had always the feeling,and I still have it today, that this is a deficiency.” So Pauli came up with this. He used it. It explained the periodictable, but he had no idea why this had to be true.And doesn’t it bother youguys that there’s just this arbitrary rulethat we’re saying no two electrons can have thesame four quantum numbers? Well, today, we’re actuallygoing to explain this. We’re going to explainwhat Pauli could not. So let’s move on tonew stuff, unless there are any questions so far. OK, so today, we’regoing to be talking about what happenswhen we have more than one electron in a system. So we could betalking about helium with two electrons,iron, 26 electrons. We’ll move up to molecules,solids, things with 100s, 1,000s of electrons. Dirac said in 1929, “Theunderlying physical laws necessary for themathematical theory of a large part of physicsand the whole of chemistry are thus completelyknown, and the difficulty is only that the exactapplication of these laws leads to equations much toocomplicated to be soluble.” So basically, he wasfeeling pretty cocky here. It’s like we understandthe theory of everything.We understand theperiodic table. There’s nothing leftto do in chemistry. We understand now. We can write the equationsfor how all molecules work, how all solids work. And it’s the complication ofthese mathematical formulas that turned outto be the problem. So many years later, he stillwas struggling with these. And he says, “If there’sno complete agreement between the results of one’swork and the experiment, one should not allow himselfto be too discouraged.” So even after allthis time struggling to solve this equation,he still could not agree with experiment.And even today, we still cannotalways agree with experiment, but that’s no reason not to try. So this was ourbasic Hamiltonian that we had for justthe hydrogen system. It was just the kineticand potential energy. It’s as simple as that. Well, now our equation looksa little bit more like this. So I don’t want you to be toointimidated by this equation. It seems to have alot of terms in it, so let’s go throughthem for a simple case. So let’s say we’dhave the simplest molecule you can think of, H2. So let’s say you have anucleus here, nucleus here, and an electron thatoriginally was associated with this one, andanother electron originally associated with this. So what are these termsin the Hamiltonian? So what do we need to consider? So this first term hereis the kinetic energy of the nuclei themselves.So it’s how this nucleus andthis nucleus are moving around. So in this verysimple case, we’d have two terms in this sum. This next is the interaction[? C-C ?] of Z i and a Z j. So it’s the interaction ofthe nuclei with each other. It’s the Coulomb repulsionof nucleus 1 and nucleus 2. So in this case, that’sactually just one term. This next term is the kineticenergy of the electrons. So it’s, in thiscase, two terms. We have electron 1 and electron2 that are zooming around, and that’s their kinetic energy. The next term isthere’s only one Z, so it’s the interaction ofthe nucleus and the electrons. So here, we have fourof those interactions. We have electron 1 interactingwith nucleus 1 and 2, electron 2 interactingwith nucleus 1 and 2.And finally, we have electronsinteracting with themselves, so the Coulombrepulsion of electrons. So in this case, again,there’s just one term because there areonly two electrons. So I think all of theseterms are labeled properly. So our equation now hasbecome rather complicated. Instead of justhaving one coordinate, which was the relativecoordinate between the nucleus and the electron that wehad in the hydrogen case, now we have thisHamiltonian that depends on the positionof all the nuclei and the position ofall the electrons, and our wavefunction that depends on the position of all thenuclei and all the electrons.So this has become a massivelycomplicated equation. So at this point, whatare we going to do? You might think toyourself that you’re stuck, that this equationis too complicated. You’ll never get anywhere. And just as you’redespairing, this man walks in, pulls down theback of his shirt. Might take you a little whileto stare at these equations, because you’ll recognize this asbeing the equation we just had. And he’s giving you theBorn-Oppenheimer approximation. So let’s look atthis approximation, maybe not on somebody’s back. So Born was a prettyfoundational guy in quantum mechanics. He did a lot of workin quantum mechanics. And I’m sure you all recognizethe name Oppenheimer. So these two guys, whatthey decided to do, they looked at this picture,and they thought about the fact that a proton is 2,000 timesthe mass of an electron. And so because thiselectron is so heavy, it’s just moving soslowly with respect to these electrons thatare zipping around. And so basically, whatthey decided to do is just ignore thatmotion of the nucleons.So the nuclei are moving somuch slower than the electrons that basically, you can assumethat the electrons will figure out their ground state, figureout where they need to be, by the time any kindof nucleus has moved even a fraction of an Angstrom. So because we’re nowneglecting the kinetic energy of the nucleons, wecan also just calculate the ion-ion interactionclassically. So that really simplifiesour Hamiltonian.So now, we’re just leftwith these three terms. So it’s the kineticenergy of the electrons only, the interactions of theelectrons with the background positive charge– so where the ions are– and the interaction of theelectrons with themselves. So we’re from nowon, we’re mostly going to call this term justthe “external potential,” so “external” because wedon’t care about the motion of these nuclei anymore. So this leads us toactually starting to talk about how we dothese approximations. So traditionally, therehave been two pathways that have been followed. There’s what a lot ofquantum chemists have taken, and we’ll talk about thatfirst, but there’s also density functional theory, whichis what a lot of physicists have traditionally used. And I’m not going to talk abouteither of these two in detail. But I’ll just mention, theMoller-Plesset perturbation theory is a perturbationtheory.So it’s based on changingthe Hamiltonian, so assuming that the Hamiltonianis pretty much something we can solve, andthen just adding a small correction to it. And the coupledcluster approach is more of a traditional quantumchemistry approach, where instead of modifyingthe Hamiltonian, we just modify thewave functions. And we play aroundwith the wave functions until they become somethingthat’s more close to something that would exist in reality. So let’s talk about the basis ofthe quantum chemistry approach.So this is Hartree. He was working rightafter World War I. I think this wasactually his dissertation, and he got his PhD for doingthis very important work. So what he decided iswe have this system that we want to solve. We have a wavefunction that’s made up of all of these electrons thatare zooming around in space. Well, what if that kindof acted like a bunch of one-electron wave functions? What if we couldjust assume that we could know where one electronwas, and another electron was, and we just could just multiplyall those together to give us the total wave function? So all of a sudden, this reallycomplicated wave function becomes something that’spretty easy to separate. So because we canseparate them like this, we can separate them into a setof– so if we have electrons 1 through n, we’ll havea set of Schrodinger equations 1 through n, foreach electron by itself.So this is great. We’ve separated our Hamiltonian. We now can solvethe system, right? So there is a problem here. This is the densityof electron j. We’re doing a sum over jfrom electron 1 through n, and just skipping theelectron that we’re considering in this case. So all of us is sayingthis electron that we’d like to solve dependson the positions of every single otherelectron in that system. So even in this most simplifiedversion of the Schrodinger equation, we stillhave the one electron depending on all ofthe other electrons. So how do we solvesomething like this? Do you guys know? I think this is even a littlemore complicated than something Mathematica can solve.AUDIENCE: [INAUDIBLE]. MICHELE: Pretty much. Well– what we do is we trya self-consistent approach. We basically just guess whatthese wave functions look like. We say, well inthis case, we know that it’s a sigma bond that’sforming between these two, so probably the electronis somewhere in here. And so if I guess somestructure that looks something like this is my wavefunction, that’s maybe a good starting point. And so we put inputthat into this equation. We take out the firstelectron, and we’re going to now solve theSchrodinger equation for the first electron. So we plug in whatwe’ve guessed as being the wave functions for all theother electrons into this term. And now we solve forthis one electron term, for electron number 1. And then we do thatagain for electron 2, and so on until we’ve gonethrough all of the electrons. And then whenwe’re done, we have a new set of wave functions. So maybe my guess was wrongand it looks something more like this. And so we look atthat, and say, wow, those two really don’t lookanything like each other.I need to try this again. And so we gothrough this process over and over againuntil finally, the solutions to these wavefunctions prior to doing this and after goingthrough and solving all of these eigenfunctionslook pretty much the same. And once we’vedone that, we know that we’ve probably gottensomething at least relatively close to the ground-statesolution, or the solution that we’re looking for.So does that makesense to you guys? Are there any questions? OK, because this is actuallypretty close to how DFT works, too. This is theself-consistent method that is pretty common insolving these problems. So this is great. We have a way of doing this. Given that we can come upwith some reasonable guess for our input wavefunctions, we now can solve the Schrodingerequation, right? Yes and no. This picture is niceand simple and soluble, but it takes out allof the interactions that you get, orall of the effects that you get frominteractions beyond just that coulomb repulsion, ofelectron-electron Coulomb repulsion. Because remember, thisis a quantum system. These electronsdon’t behave like you expect they would in some kindof classical Newtonian system. These electrons are weird. We’ll see later on some of thesymmetries that we can look at. Actually, I think that’s next. But these electrons–you wouldn’t expect to just be ableto just average over all of the effects. Because this truly was amulti-electron function.We’re missing somecritical physics here by just separating it outinto single electron functions and just multiplyingthose back together. So we’re missingtwo important terms. They’re going to becalled the “exchange” and the “correlation” term. And the fix to at leastpart of this problem brings us back to spin. So as I said, inquantum mechanics, symmetry is really,really important. Symmetry tells us a lotabout the way nature works. So even in classicalmechanics, you guys have probably encounteredthe symmetry of real space, the symmetry of time,things like that. And that’s what gives you theconservation laws in mechanics. So just like that’strue in mechanics, it’s also true inquantum mechanics. So we’re going to be lookingat one particular symmetry, and talking about howthat might give us some insight into this problem. So that symmetry is called”exchange symmetry.” So that’s based on thefact that electrons are indistinguishable.So what does it mean to bereally indistinguishable? So if I have, maybe, twopieces of chalk that look– I guess they’re notexactly the same, but they look pretty similar. And let’s say I want to follow– I’m going to call this oneFred and this one George. And I’m going to follow theirmovement throughout space. So if I give theman initial position and some initial velocity,I know throughout all time which one is Fred andwhich one is George. Because I knew theirexact initial position. I knew their exactinitial velocity. I can calculatetheir trajectories. I know which one is which. But that’s nottrue for electrons. So does anybody know theuncertainty principle? Can somebody tell me whatthe uncertainty principle is? Yeah. Did you volunteer? AUDIENCE: No, I didn’t. I [INAUDIBLE]algebraic [INAUDIBLE].. MICHELE: Right. AUDIENCE: [INAUDIBLE]. MICHELE: Right, so the exactmathematical formulation is something like theuncertainty in the momentum times the uncertaintyin the position has to be always greaterthan or equal to basically some constant. So if I have my twostarting electrons, and I have some ideaof where it starts, and with some idea of whatvelocity it starts with, and some other electron– withsimilarly not an exact position but some kind of positionand some kind of knowledge of the velocity– it’s clear that overtime, I’m going to have no idea which one was which. They’ll switch and I won’tbe able to tell that they’ve switched because I can’t followtheir trajectories the same way I can with classical objects. And furthermore, electrons don’thave any identifying marks. They don’t have like a beautymark somewhere that can tell us which one was theelectron that we really fell in love with first.And so electrons actcompletely indistinguishable. And so this is a symmetrywe’re going to exploit. So let’s say we have asystem full of electrons, and I pull a curtainin front of you so you can’t seethe system anymore. I take one electron, andI take another electron, and I swap them exactly. I swap their positions,I swap their momentum, I swap everything about them. And I open thatcurtain again, you won’t be able to tell at allthat anything’s happened, because these electrons actexactly like each other.So it doesn’t matter if Iswapped one with another. So this might sounda little bit dumb. I mean, you might besitting there thinking, OK, this is obvious. Why is she going on so longabout something so obvious? And the reason iswhen we formalize this in terms ofmathematics, suddenly things becomereally interesting. So we’re going to definean exchange operator, which is this chi 1 2.So chi 1 2 actingon a system that has an electron 1 and electron2, all it’s going to do is swap electron1 with electron 2. Just so thisformalism makes sense, this 1 refers to whatparticular wave function it is. So this psi 1 refers to thatset of four quantum numbers, or set of whatever quantumnumbers describes that system. And similarly, psi 2describes a different set of quantum numbers, and r1 isthe position of electron 1, r2 is the position of electron 2. So when we use this exchangeoperator on our system, we get something thatshould be indistinguishable. And you can see thatif we act twice– so if we have electron 1and 2, and we swap them, and then we act againwith the exchange operator and we swap 1 and 2,we’ll switch back. So we know mathematically thatthis exchange operator, acted twice on a system,leaves you exactly with the originalsystem that you had. So we have to assumehere that this wave function is going to be aneigenstate of the exchange operator. And there is a quantummechanical reason for that. But what this tells us,assuming that we’ve now acted with thisexchange operator twice, we know that the value of theeigenvalue of the exchange operator squared isequal to 1, which gives us that theeigenvalue itself has to be plus or minus 1. So just to be clear,what I’ve said is that when I acton this wave function with this exchange operator– so I have my totalwave function, whatever it is, which is a functionof multiple electrons, when I act with thisexchange operator on it, I get something whereI’ve switched r1 and r2, and all the other electronsare left the same. And what I said inthe previous slide is that these two states– this state and this state–have to act exactly the same. But now I’ve alsotold you that this is equal to some eigenfunctiontimes the original wave function. And this eigenfunction, I’vesaid, can be negative 1. So can anybodyreconcile this for me? How can this eigenfunctionbe negative and yet have this originaleigenstate not be changed, or have no measurabledifference between the electrons being in the original positionsor 1 and 2 being switched? So the key to the questionis actually the term “measurable difference.” So remember, Jeff has saidthat in quantum mechanics, we have this wave function.We have no idea what thewave function actually means. It’s the square, the absolutevalue square of the function, that actually is meaningful. So you can see, even ifthis is a negative 1, and we have a negative 1 infront of the wave function, when we square that, thenegative is going to disappear. So we still have this symmetrybecause the thing that’s measurable is the absolute valuesquared of the wave function, and not the wavefunction itself. But this is actually a prettykey piece of quantum mechanics. So we have two possiblevalues for this wave function. So this wave functioncan be positive 1 or it can be negative 1. So any quantummechanical system is going to have a wavefunction that either has the eigenvalue forthis equation positive 1 or negative 1.When it’s positive 1,we call them “bosons.” So this leads to a lotof interesting physics, like the Bose-Einsteincondensate, which I don’t know if youguys have heard about. There’s a lot ofcool experiments that have been doneon helium 4, where at a low enough temperature,all of the helium atoms suddenly find themselves inexactly the same ground state. And there’s some reallybizarre consequences of that. However, whatwe’re interested in are the particles withan eigenvalue of minus 1.Those are called “fermions.” Electrons are fermions. So electrons,therefore, need to have an eigenvalue of negative 1. And this is actually the key tothe Pauli exclusion principle. So this is why the Pauliexclusion principle is true. So can somebodyexplain to me that? It’s kind of a subtle point. Does anybody see that? So let’s look at awave function that doesn’t follow the Pauliexclusion principle and see what happens.So let’s say Ihave two electrons. I’m going to put them bothin the 1s with spin up. So what happens when Ioperate with this exchange operator on this? So these two areexactly the same. So I needed a negative 1here, but I have a positive 1. So because this exchangegives me the wrong sign, I know anything that hassome system like this, where I have the same quantum numberson two different particles, has to be a bosonand not a fermion. So let me just give you anexample of a wave function that does satisfy theproper eigenstate for this. So if we have somethinglike 1 over root 2 1s up for particle 1,1s down for particle 2, minus 1s down for particle 1times 1s up for particle 2. So can everybody see? If I switch 1 and 2, I’m goingto get the negative of this.Because these two are going toswitch and become this term, these two are going toswitch and become this term. Is that clear? I’m seeing a lot of blank faces. I hope that means thatpeople are way past this and have mastered this long ago. Are there any questions on this? To me, this is really exciting. This is great to be able tofinally explain something that Paul himselfcouldn’t explain. We actually can understandquantum mechanically why electrons behavethe way they do, why we have the Pauliexclusion principle, and therefore, why wehave the periodic table.I mean, this is apretty big consequence. [DOOR CLOSING] I guess it was so big,he couldn’t take it. All right, let’s move on towhat the implication of this is in the Hartree method. So a guy named Fockcame along, and he has either a very fortunateor very unfortunate name, depending on howyou feel about him. And he basically just took ananti-symmetrized wave function. So anti-symmetrized justmeans this characteristic, that when you act uponit with this operator, you get a negative 1. So he just put in ananti-symmetrizied wave function into Hartree’s originalequation, which was just, if you remember, just this. It’s the kinetic partand then from the ions, the potential part fromthe other electrons. So this was theoriginal equation. And when he plugged in ananti-symmetrized wave function, suddenly, this term appeared.So this is the term wecall the “exchange term.” If you look veryclosely at it, you’ll see that what it doesis quantum mechanically, two electrons with thesame spin will never be in the same position. And it basically subtractsout that possibility, of two electrons with the samespin being in the same place. So this actually reducesthe energy of the system because suddenly, all electronswith the same spin have to be just a little bitfarther apart from each other. So that minimizes the Coulombinteraction just a little bit. So actually, withthis correction, the results seem to come andmatch up with experiments a lot closer. And this is actuallythe foundation for molecular orbital theory. So just by adding this spinconsideration into the Hartree equation, we suddenlyhave an equation that’s actually pretty functional. So Jeff thinks this isquite an emotional moment. However, as I said, therewere two energy terms that we were missing in thatoriginal Hartree equation.So we found the exchangeterm, but we’re still missing the correlation term. So there are some waysto deal with this, but I think the way we’regoing to deal with this is just by moving on to DFT. So now, we’re going to starttalking about DFT, which I think is what youguys are mostly using, so you might findthis more interesting. So the Schrodinger equationis just really hard to solve with allof these electrons. As we’ve said, ittakes a lot of time. It takes a lot ofcomputational expense. So let’s divide space into theworst grid you could imagine. I guess the worstwould be one point, but the worst grid you canimagine is 2 by 2 by 2. And let’s think aboutthe number of points that we need to keep track ofif we’re calculating something with little n electrons. So if you only have oneelectron, it’s only eight, and it’s not too badto keep track of. If we go up all theway to 1,000 electrons, suddenly we have thishuge number of points that we have to keep track of. And that’s just inconceivable,even with today’s computers. Why try to dosomething this big? That would take forever. So it’d be reallynice if we could think about some otheraspect of the system that we could calculate,instead of keeping track of every single one ofthose electrons separately. And a term like that is density. So if we considerdensity instead, suddenly, it’s prettymuch the same scaling no matter how manyelectrons you have. So the electrondensity definitely seems to be a lot moremanageable than dealing with each electron separately. And so at some point,someone’s, like, well, wouldn’t it be niceif we could just calculate with density insteadof with wave functions? Because think about it–these wave functions don’t mean anything anywayuntil you’ve squared them to get the density.That’s the only thing thatwe can measure anyway. Why don’t we justlook at the density? So what if I could come upwith a function of energy that was dependent on thedensity of electrons? And Walter Kohn actually wonthe Nobel Prize in Chemistry in 1998 with somebody elsefor coming up with this idea. So as I said, one reasonthat we want to go to DFT– it scales so much better thanthe quantum chemistry methods. And we’re going to have to spendso much less time, especially as the systems get larger.So let’s look at an example. So if we have twoatoms of silicon, DFT and the two samplequantum chemistry codes all take aboutthe same amount of time. If we try to calculate100 atoms of silicon, suddenly DFT is 5 hours,which might sound long. But the Moller-Plessetis a year, and the coupled clusterapproach is 2,000 years. I don’t thinkanybody’s tested that, but I wouldn’t recommend it. But suddenly, you can see whyif you want to do anything more than just two atoms,maybe DFT would be a good way to go about it. So this is a graph just showing,again, that DFT scales better than all the other approaches. So again, the basictenet of DFT is we’ve taken thisreally complicated, multi-electron wavefunction, which we already had trouble enough calculatingin the quantum chemistry method.I mean, we alreadyfound problems that if we triedto simplify this by making this a functionof each individual electron, we suddenly lost allthe electron correlation and exchange, which ispretty crucial to getting these calculations right. And we know that the squareof this wave function is the electron density anyway. So let’s just thinkabout the density. So does anybody know what afunctional is, any math people? AUDIENCE: [INAUDIBLE]. MICHELE: Great. Great. So generally a function, youinput a number and output a number. A functional, you input afunction and output a number. So this is called “densityfunctional theory,” because we are writing afunctional of the function density. Density is a function. At every point in space,there is a particular value of the density. And so we would liketo be able to solve this energy as afunction of the density or I guess a functionalof the density. So of course, we’re going tosee all the familiar terms showing up– thekinetic energy, the ion potential, theion-electron interaction, electron-electron interaction. We’re going to see allthe same terms showing up.And we’ll just have to dealwith them slightly differently. So the way we actuallygo about solving this is we write some equation ofall of the different terms, as far as we knowwhat they look like. We write out the density interms of its wave functions, and then do afunctional derivative of one of these wave functions. And so the functionalderivative works– it’s basically just avariational principle. Are you guys familiar withthe variational principle in quantum mechanics? The lowest energy possible fora system is the ground state. And so any time that you canfind any system that gives you a lower energy, you’re gettingcloser to the ground state. So I have my basicSchrodinger equation. If I can change somethingabout this wave function such that when I solve thisequation I get a lower energy, I probably have something that’scloser to the true ground state than I used to have. Because the smallestpossible value for this energy thatI can get out of this equation is goingto be the ground state. So that’s actually used alot in quantum mechanics. So maybe I don’t know exactlywhat this function looks like, but I can kind of guess whatthis function looks like. And I can minimize this energywith respect to what exactly this wave function looks like. And that’s exactlywhat we’re doing here. We’re minimizing thisenergy the best that we can, to try to get something that’sas close to the ground state as we can. So that leads to theKohn-Sham equations. Again, this is prettymuch the same Hamiltonian you’ve been looking at allclass, all last week, too. We have the kinetic term. We have the potential term, andthat acts on a wave function, and gives us some eigenvaluetimes the wave function. So in this case, ourpotential is broken up into a couple ofdifferent terms. So we’ve already made theBorn-Oppenheimer approximation. So we’re ignoring theion-ion potential. We can add that in whenever. That doesn’t change. We have the ion interactionwith the electrons. We have the electronsinteracting with themselves. And now we have this new termthat’s called the “exchange correlation potential.” So this is the energy righthere that we were missing before in the quantumchemistry approach. But there’s aproblem– we actually don’t know what thisexchange correlation function looks like. So people use a bunch ofdifferent approximations. The two most commonly used arethe local density approximation and the generalgradient approximation. And they’re prettyself-explanatory. The local densityapproximation, you just approximate the local densityas being a flat in a local area. And the generalgradient approximation, you assume a first derivative. So if you have some potentialthat looks like this, the local density approximationmight approximate it as looking something like that,whereas the general gradient approximation would getthe slopes for each. So generally, the generalgradient approximation ends up being a littlebit more accurate, but they both have theirstrengths and weaknesses.And depending on whatproblem you want to solve, you really need to payattention to what you should expect from either one. So how does thisprocess actually work? So this is a lot like thesolution to the Hartree equation. We start with our guess at theposition of the ions, which may or may not change. From there, wealso need to start with a couple initial parametersthat we didn’t talk about for the Hartree case. So in this case, we’re goingto pick a cut-off for the plane wave basis. So that’s something I’m goingto talk about in a minute. I haven’t talked about that yet. Then we guess what our electronslook like in the system.From that guessof the electrons, we calculate the terms in theHamiltonian that we’re missing. We solve the Hamiltonian. We just diagonalize it. And that diagonalizationgives us a new density. And then again, we ask,is this density close enough to our input densityor is it pretty far away? And if it’s pretty faraway, we go back up, use that new calculated density,again to calculate the terms in the Hamiltonianthat we were missing, diagonalize the Hamiltonianagain, get a new density. And we just keep doing thisagain and again until we’ve found somethingthat we say is close enough to the originaldensity that we’ve probably found something that’sa good approximation. So you’ll noticethat this is actually different from the Hartree.So this looks very similar tothe solution of the Hartree equation, but you’ll notice thatinstead of solving one electron wave function ata time, every time we have to do theself-consistent cycle. All we have to do here is justdiagonalize this Hamiltonian. So this is why theprocess speeds up so much. There are a bunch ofsoftware that people use for DFT calculations. Some of them are free. Some of them are not. I think you guysare using Siesta, which is not on this table. But all of thesedifferent software have their strengthsand weaknesses. And depending on whatyou want to calculate, you should probablythink about which model is the best for you. But we’re going to be lookingat a PWscf input file, just to give you a generalidea of what these input files look like. and I guess prettysoon you’ll find out anyway, since you’ll have towrite some of them. So I promised you I was goingto talk about basis functions. That was that one parameterin the self-consistent cycle that you have to pick.So a basis function isbasically just whatever function you like, what function you wantto deal with mathematically, that you can write yourwave functions in terms of. So all we do is we take our wavefunctions and we expand them. So we get some linearcombination of whatever basis we like. So usually, people pickGaussians, or plane waves, or things that arereally easy to calculate, because you can writeany wave function in terms of a linearcombination of these, as long as you havea complete basis. So we pick these spaces suchthat it’s orthonormalized. What we can do fromthat is when you multiply one ofthese by another one, you’ll get a delta function.So basically, if Imultiply through by– so I have to integrateover all space because these functionsexist over all space. So I integrate over all space. I pick one of theseparticular basic functions that I multiply by. And that picks out exactlythe j basis function here. So this whole term,when I integrate out, because it’s orthogonal, Ionly get the j term here. Because it’snormalized, I get 1. So I’m just left with thisE and c j on this side. On this side, when I do thatsame integration over all space and multiplication bythis j basis function, I get something morecomplicated because I have this Hamiltonian in between. So this Hamilton,this H i j function, is actually not quiteas simple as it looks.It’s an integration over allspace of the Hamiltonian acting on the basis set i, and thenthe inner product of that with basis set j, thecomplex conjugate. But this is somethingwe can calculate. This actually ends upbeing the Hamiltonian that we diagonalize in the end. So you can see that the choiceof these basis functions, to make it something that’seasy to calculate this term, is going to be really crucialfor getting the shortest possible computation time. One that’s commonlyused is plane waves. This is a plane wave. The exact form of the oneswe use, e to the i G j dot r. You guys know the e tothe i x is equal to cosine x plus i sine x. So you can see why theseare called “plane waves,” because in real space, they’rebasically just some wave. So this r is thereal space vector. What we have to dois pick what j’s we’re going to usein our simulation.So really, if we’re going toexpand any function in terms of this basis, we needthe spaces to be complete, which means basicallywhat it says. It’s “complete,” meaning you canexpand anything in terms of it. But we don’t really wanta complete basis set, because for this tobe complete, this G would have to go to infinity. We’d have to go for every numberfor G between 0 and infinity, and we don’t have thecomputational time to do that, because the size of ourHamiltonian that we diagonalize depends on the number of thesefunctions that we put in. So what we hope to dois pick a maximum G that’s sufficient to captureall of the physics that happens, but not so big that ourcomputational time takes forever. So as I increase G, Iget shorter and shorter wavelengths. And at some point,this wavelength is going to be soshort that it’s going to represent a reallyhigh kinetic energy system.And it’s going tobe just un-physical. I mean, if I have an atom here,an atom here, an atom here, probably the variationin the electron density is not this great rightaround that nucleus. And so at some point,we can just cut off G, and say we have enough. We can probablycapture all the physics we need just going upto the G that we have.So you can see actually how thisplane wave basis works really well for periodic crystals. I just drew nuclei here. So when you have aninfinite crystal, you have a nucleus here, anucleus here, a nucleus here. And because they’reall the same, you don’t expect that thewave function or the density should be differentat this nucleus than it is at this nucleus. So this plane wave is actuallya great representation of that because the density is thesame here as it is here, as it’ll be here. And so these planewaves are actually ideal for calculatingperiodic crystals. So how are we goingto attack molecules? What we’re going to do isput these molecules in a box. We’re going to put them in abox with some amount of vacuum on either side, and then repeatthat box in all directions. In x, in y, in z, we’re goingto infinitely repeat this box.And so now instead of havingan atom here, an atom here, an atom here,we’re going to have a molecule here, and a moleculehere, and a molecule here. So you can see howmaybe with this, these plane waves mightrepresent that system a little bit better. But there are somethings you have to take into consideration with this. Since there’s a molecule hereand some distance away there’s another molecule, they mightinteract with each other. And if you want just theproperties of that molecule by itself in vacuum, youdon’t want them interacting with other molecules. So you need to makesure that you’ve included enough vacuumbetween these molecules, between the molecule andthe edge of its unit cell, that the interactionwill be negligible or will be on theorder of the error that you have from everythingelse in your calculation.So once we’ve donethis, we can then start our self-consistent loop. So what happens? We put in where the ions are. We calculate our potential. We’ve guessed at someoriginal electron density. We’ve calculated all theterms in the Hamiltonian. We’ve diagonalizedthat Hamiltonian. That diagonalizationgives us our energy. And then the program said, well,that new density wasn’t really close to the old density. So let’s do that again. And it does that again,and calculates energy. And you can see how theenergy always goes down.So this is a negative number. The energy always goes down. So that’s thevariational principle– as you approach thecorrect ground state, you get lower in energy. And you can seethat the difference between these successivelygets smaller and smaller. And that’s what you really wantto have happen, because that means that yourenergy is converging, and you’ll end up with apretty good approximation of the ground state energy. You’ve also ended upwith the charge density, because that’s what you’ve beencalculating this whole time to calculate the energy.So just with thosetwo properties, and being kind ofclever, you can actually calculate a huge range ofimportant molecular and solid properties. So you calculate at leastyou can guess at a structure. You can calculate bulkmodulus, shear modulus, elastic constants. You can calculate vibrationalproperties and sound velocity. Binding energy isactually just probably going to be this energythat you’ve calculated. You can kind of guessat reaction pathways. Those are a little bit harderto do with this approach. You can calculateforces, pressure, stress. You can calculate allof these properties just with this simple calculationthat we can now do. But how do we know that thesecalculations are meaningful? How do we know that we’veactually said anything useful? Dirac was complainingthat he found that his calculations neverreally matched experiment, but don’t lose heart.But when should you lose heart? Or when should youmaybe check to make sure that what you’re doing isas accurate as you can get? We need to talkabout convergence. So there are a coupleof different properties that you need to worry about. We talked about isthe basis big enough. So that’s, did Ipick a high enough G that I got enough wigglinessof my wave functions, my input basis wave functions,that I’ve captured all the relevant physics? And so this is the trade-off. It takes more time if youhave a larger basis set, but you also lose someaccuracy by decreasing the size of your basis set. And at some point,when you’ve taken out too many basicfunctions, you suddenly get something that’s notgoing to be meaningful at all because you just don’thave the right structure of the potential. We talked about isthe box big enough. Did you include enoughvacuum so that you don’t get interactions betweenmolecules in neighboring unit cells? Because if you want theproperties of the molecule by itself in vacuum,you don’t want the interaction of those twomolecules to play a part. But here we get, actually, atrade-off with the basis set. So the bigger you makeyour box, the bigger you need your basis set to be. So you can see if this isthe size of my box here, and this is my cut-off– G has a wavelength of this– if I suddenly makemy box much bigger, this wavelengthis going to expand and it’s going to be amuch bigger wavelength.And so maybe nowthis is actually an important wavelengthto my system. And maybe even asmaller wavelength is an importantwavelength to my system. Does that make sense? OK, so making yourbox bigger means that you need a biggerbasis set, which means more computationaltime, but on the other hand, you don’t want to includethe effects of intermolecular interaction. Finally, you can ask if youexited the self-consistent loop at the right time. So I keep saying, well,we compare the old density to the new density andif they’re close enough, then we assume it’s right.But what’s that definitionof “close enough?” And how do you know that thatdefinition of “close enough” is close enough? So that’s anotherproperty that you’re going to have to converge along. So let’s just take a lookat an example input file, and see how some of theseproperties that you need to converge along, and theother properties you might care about, show up in this system.So we might care aboutputting in the right atoms, talk about how theatoms are related to each other spatially,and then these three parameters that we talked aboutthat we need to converge along. So this is just asample of water. And so here, this isthe size of the cell. This is the number of atoms,number of types of atoms. This is ecut, thisis the energy cutoff, which refers tothe size of your G, and then your atoms andthe atomic positions.And these are justthe pseudo potentials. So we haven’t talked at allabout pseudo potentials, but just so you know,we talked about the fact that when we thinkabout atoms we only care about the valence electrons. So all those coreelectrons still exist, but we don’t wantto calculate them. So we use pseudo potentialsto sort of capture those inner electronsa little bit, and tell us how thoseinner electrons are going to affect the valenceelectrons and everything else. Because when wehave an atom that has some core of electrons– so let’s say thishas 27 protons in it. An electron outhere is not going to feel that 27 proton charge. It’s going to feel that27 protons shielded by some number ofcore electrons.And so to get the Coulombinteraction right there, we need to have somekind of modified version of what exactly our atoms’potential looks like. So that’s what the pseudopotentials are about. So that’s it. We’ve covered the basis ofquantum chemistry and density functional theory today. Are there any questions? All right, thank you.
