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Lecture 24. A closer look at our Molecular Orbital: The Virial Theorem in Action

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let's continue on where we left off we had the potential energy curve each plus and the minus 4 the molecular ion H 2 plus in this lecture I want to take a closer look at the molecular orbital we got In the 1st I want to figure out what it actually is we started out with these 2 basic functions we still need to figure out the coefficients you can probably guess by the name C plus and the minus what the coefficients are going to be and the symmetry of the problem but let's play down for the time being and pretend we don't know and then if we don't know we will have an algorithm to figure it out and then we're going to look at some measure of the quality it was the kind of orbital that we got and that's a little bit of a deeper investigation that's going lead us into something called the virial theorem and which I believe was developed by Klaus years of the classes "quotation mark Pienaar questions well we have some experimental values but we will always have experimental values for everything but in this case we do less them and then also we have very extensive calculations of the variation on principle tells us that no matter how big we make our basis said we're going to be above the ground state energy unless we have very bad round off here so we have to do an accurate calculation is to make things too big and you too inaccurate you can end up below by mistake because of numerical round of problems we end up with our in newness of a not being 2 . 4 9 that's what I said last time was about to or between 2 and 4 which is about 132 peak meters and the minimum energy it is getting rid of the minus ahead which is just the hydrogen at this minus . 0 6 4 8 3 1 archery which is minus 1 . 7 5 8 electron volts which is minus 169 . 6 killer jewels per mole in units that can have some worries the true values of course are better the true values are that big art is just to a on which is about 160 commuters instead of 132 and the energy is better Of course the true energy is minus . 1 0 0 2 6 4 archery minus 2 . 7 9 electron volts or minus 269 . 5 killer jewels from all this leads us to believe that you know we did OK but after all week we took a very simple approach so it's amazing that it even works at all and you can go back and ask yourself why it doesn't work and what you will conclude is that it's only on account of that integral the exchange integral that it's become possibly the stable you can test that out on your own as little practice problems really and see how it goes and you will see that it's only this non-classical thing where we have to wave functions interacting with this operator in between that we end up with on something that can be stable and track now it's kind of ironic that after fiddling around with the secular determined and so forth which after all we we want to do that in the 1st place To solve for the coefficients the seeds that could tell us how much of what I say and how much of 1 sp it is in the answer when and this linear algebra problem we said 1 answer to be 0 policies a 0 that wasn't done pretty uninteresting the other answer is the same as the determinant then we had those matrix elements that we had to work out what those were now we know what those are so now we go back to our problems to figure out what these coefficients which I'm going to call CA NCB R series for 1 SAC 1 sp let's go back to the linear equations which began with a and substituting in each of the the energy solutions in terms because they will have different coefficients because they're different energies and their different function the 1st of equation we get is C times HAA -minus plus CG Times HAD might CDs remember those were our equation equals to 0 but now we can put the it is a clustering minors if I put it the last 1 I get is what I've got on slides 597 this is big equations CHA it's so forth and so on was CBD another thing equals to 0 and if I expanded out which have done in the 2nd line and then simplify what I find is that I have CA times staff times might suggest plus CBD HAD -minus as stages which is the opposite equals to 0 so that means that CA and CDs are the same because the other things are like 1 minus 1 that means serious people the
and that means that for the class energy I have 1 said plus 1 indeed and so that's not surprisingly why we called plus so now here I'm writing I plus is equal to some constant CA UCB times 1 as there was 1 sp now I still need to normalize because the unit growth side plus square overall station B 1 and I've done that here the heroes easy because I remember 1 essay plus 1 sp times 1 as a plus 1 B 1 Estates where his wife that's life 1 has be square is 1 BA and a R S and we know where therefore will we get is on the handle is equal to CA square times 2 plus 2 at which factors 2001 was therefore CA is is equal to 1 over the square root of 2 times quantity 1 process and so I can write my final result sigh Plus is 1 over the square of 2 times 1 was the overlap integral plants 1 as it was 1 and the and I tidied up and I know exactly what the energy as I know exactly what the minimum of future nuclear distances and what the potential surface looks like Ballack scraped what exactly the same sequence of steps but instead of substituting and positive energy I substituted negative energy I get 1 over the squared off 2 times 1 -minus rather than 1 person times quantity 1 answer -minus 1 sp and I have to choose which ones to be plus and minus the doesn't matter because remember the faces the wave function doesn't matter we usually chooses to be real we can because we're very pious about that but whether there is a minus-1 downtrodden on it doesn't change the overall probability density and sewers 1 as a 1 experience were around it's the same thing it doesn't change anything about the problem the orbital also here I've I've drawn and I wouldn't swear that these are 100 per cent accurate but they're probably pretty accurate when the other 2 nuclei are far apart 30 cases that i've got here and red red and blue come angry-looking rather than of 2 1 1 sp the bold red all are at 1 of on is the opposite face and negative for about wave function member there's nothing to do with charge density that's just the face the weather's minus plots and that's in blue and that if we bring the 2 red once together we bring these exponential and they have a cost that the nucleus of look like that like a little tense would bring sense together as they get closer and in between where the both positive they can make a can catenary like a bridge to warrant movie studio where they keep you out the movie theater that they're willing to you out by hanging out of the lower things and then if square and where it's been in it adds up when you when he squared gets bigger and that means in some sense that the electron is spending a lot of time in between the 2 positive charges which is perfect when you want to think of it in the electron gluing the nuclei together that's where you would expect it to be and you get this kind of sausage shape then the exact shape of the sausage depends on how close you allow the nuclei together and then in the 2nd line was 1 of them's blue and it's the I would face combinations now I've got to think about 1 of hot 1 of them's cold and in the middle it's as you bring them together they just cancel out more and more in effective you brought them right on top of each other they disappear that's very bad and what happens is as they come together In between the 2 nuclear it's always 0 In other words there is no legal claim and 5 colors that atop a red and the bottom part blue in the in the figure on the 2nd line and then when you square that Of course when you swear they should both be be ready because when you square it's a positive number for what I've done is that left 1 of them blew and I squared anyway to show that in between where they cancel then when he square there be canceled in between and that really explains that 1 is repulsive because it's canceling and as you get closer and closer it cancels more and more and more than all that happens the protons see each other right closed and they each each other because they got a huge positive repulsion there both positive charges and that explains completely wide the miners curve just goes up In the end and only when they're very far apart and there is essentially no cancelations today than at the same energy is 0 of a hydrogen atom the lowest
energy solutions with the atomic orbitals interface is :colon bonding all because it makes the matter at the nuclei stick together and still resembles a month to indicate that it has a symmetry it's allegedly symmetrical it's the most like as it can be and not being added and the other solutions with the atomic orbitals of opposite face is called time bonding or all and it's given the symbol 6 months start usually means excited or unfavorable or something like that and that's exactly what this means if you put an electron and there is unfavorable for the future of the molecule the bombing Oracle has even cemetery and the entire bonding orbital has ordered symmetry so the notation with RG new might be segment G for the 1 with the event's symmetry and so it must stop are you for the 1 with on tree and oftentimes molecular orbitals labeled this way to help you understand what they look like the surface span of the entire bonding orbital is purely repulsive there's no between the 2 nuclei and you cannot ever get the molecule to be stable with that group now let's have a look at how good our solution is we know it has amassed experiment but will find is that it's slot in another way that we haven't even talked about there is a powerful general observation from classical mechanics the relates the expectation value of the kinetic energy and the potential energy to whatever the Forest Law is between the particles in the Forest Law is several of them and simple to figure out and you may not have the a proper course classical mechanics maybe that's later on in the series and never got to it and so I'm just going to tell you what this virial theorem says without arriving at if we had a potential the alarm that is something quite a constant let's say a times already this year some power the material appearances the 2 times the expectation value Of the kinetic energy which I call T here for sure and the case is equal to end times the expectation value of the potential energy and that's equal to minus 1 times the expectation value potential energy because are potential with charged particles always has won over are that's the potential to force laws 1 over a square the initial over part so now it if we've got a different kind of force law then we end up with a different ratio of tea and the the question is does our solutions are way functions does that satisfy the virial theorem or not and that's let's so I have a lot because if it doesn't satisfy the material here and then it's not very good and there could be another reason why it's not very good and that means if we can just as if we can tweak it so that it doesn't
satisfy the virial theorem it's liable to be much much better this is kind of an independent check on our system because nowhere when we did this calculation did we say all by the way 2 times the expectation value of the kinetic energy should be minus 1 times the expectation value of the potential energy this is coming in and like an auditor to accompany and just saying how good the books and we couldn't sell the books because we didn't have anything to do with less than a practice problems applied virial theorem 1st to classical harmonic oscillator because it came from classical mechanics there was no quantum mechanics when Rudolph classes was proposing and then let apply it to the ground state of the simple harmonic oscillator while while we already have that way function and we'll see if the quantum of slavery in the classical oscillator both satisfied virial theorem and then after we do that and we're confident that we know what we're doing will apply it to our system and see how we do it OK practice problem 29 consider a classical harmonic oscillator it has reduced mass and force constant K doesn't conform to the real part How about a later with the same values of k and OK here's our answer part the total energy of the classical oscillator is the is equality policy which is one-half in the square was one-half K X square and of course conserved over time in the ocelot it just changes for if we take this news Newton's equations which is what we have to do cost to solve problems of classical mechanics or if we get more advanced we get into the oil and Cronje equations but we won't do that here forces Eagle a mass times acceleration that is equal to MA acceleration is the derivative of the velocity DVD today and he is the director of the position so that the sequel and the square square yeah but forest is also equal to the minus derivative of the potential so that also equal to minus DVDs but DVDs is the derivative of 1 have square which is to scale so what we end up with is this equations on the middle of slight 604 the square STT Square is equal skate over at times well this equation are you really really familiar because look it's exactly the same kind of thing languages just some different variables as what we ran into with particle of box With the 2nd derivative and this is equal to itself so it resembles a part Boston's with change of variables with time here rather than X and X being sought the wave function rather than so but isn't exactly so let's suppose the time 0 but the oscillator is extended to whatever the maximum that can be let's call it next match at time people 0 or the
compressed blood stated that the Max and let's assume that at the maximum that it's stationary 1 has to be stationary otherwise you go farther and if it were moving the other way ,comma got there in the 1st place so it's stationary none of this matters to them problem 1 idea but it makes it easier to calculate that the solution is exiting evil acts Max co-signer making and the of is equal to miners Omega x Max Cyanamid because BAT is the director of Texas and I'll make the angular velocity it is equal to the square root of K upon yeah next we have to say what we interpret the average kinetic energy and average potential energy to to be the interpretation classical mechanics and quantum mechanics is different in classical mechanics we interpreted to be a time-averaged because the energy of this particle that has definite position momentum at all times is changing between Connecticut and potential we want take the time averaged a 1 take the time averaged over 1 side goes you go out and come back to go outside that's the correct damage we don't include anything else so we want to integrate over to high the variable at 360 degrees around in circles thing if we do that than the average of the kinetic energy as recall if you take the means that you have something is divided by the land so it's 1 over 2 5 times in a row from is equal to 0 0 2 0 making it easier for the 2 types of dt on 1 hand and the square and if I put in what happened the square I'm with the and Omega's squared X squared Max squirrel make tea I've got do that individual lawmakers "quotation mark squared by parts look it up and if I do that and go through the whole thing and with the sequence of steps that the highest cancel out as they always do and I end up with a K X Max square over 4 that's happened the available energy because the most that can be when it's stationary out X Max's one-half kayaks Maxwell this is half of that not surprisingly than the other park the expectation value the potential is going to be the other half and you could just assume that the right because energy is conserved but I'm not going to assume that I'm going to slug it out in calculated so I can have more confidence in that I know what I'm doing the expectation value potential energy is again 1 of a two-part transient role from all major 2 years equal to 0 2 0 making is equal to 2 pints of deep-sea one-half kayak square X depends on the I put in my ex square co-signed square the I do the integral again and outcomes KS squares Mac over 4 that's the other the average potential energy is thus exactly equal to the average kinetic energy for the classical harmonic oscillator but what the virial theorems the virial theorem says that 2 times the average of the kinetic energy is equal to an times the average of the the potential energy but it is too because our potential was one-half K X square so that to is equal and the becomes duties to the and that's just equals the that's what we got therefore that satisfies the virial theorem part of the for the "quotation mark Massa later recalling lectures 8 we solve that boy that seems like a long time ago we've covered a lot of ground since then but here's the here's the ground state wave functions for the simple harmonic oscillator I have and homemaker over each bar all divided by party race to the one-fourth of power funny thing to to normalize and then Gallician functions each the miners and will make X squared over to which part I've just written exactly the same thing in slightly different terms to keep in the enemy of maker so that it it's more comparable to the classical oscillator but there's nothing different OK now however we calculate the average of the kinetic energy of the potential energy here as well we know how to we calculate the expectation value that that the angular brackets means in quantum mechanics in classical mechanics they may mean time which like we saw with quantum-mechanical we know exactly what to do and so the expectation value tea then is the integral part of the ground state wave functions and I have my message part square over to at times a 2nd room with respect to act again on calcium I take the the derivative downtowns access and then I take a derivative of exceeded the estimated 2 terms and I and other with this term out in front of this state terms with the squared along
with two-time troop I'm nominated and I end up with 2 terms 1 H the others minus animal may get at square at all times the Garcia I have to do that in a go buy parts twice and then I get the answer and the answer comes out to be H Omega over 4 however meet because we know that the 0 . energy is a made over to Palestinian energy that had to be there by the uncertainty principle the and Soviets again it's happy the available 0 . energy is assigned to the kinetic energy the grounds to harm on costly once again we can assume while the other has potential but actually is you get good at doing is the kind of fun and so why not do it so here's the calculation of expectation value potential I put in 1 hand and will make a square X square and the integration by parts twice but I'm getting awfully good that fact get so good that I can just do my head almost so that turns out again without too much trouble to be H-bomb over 4 so for the Hamas leader the expectation value the kinetic energy is in Parliament over 4 the expectation value potential energy in Statesboro makeover for they are equal it satisfies the cereal there are once again good now let us check our wave function for H 2 plus the virial theorem says with the minus 1 and forced rather the potential function part minus 1 of the 2 times the kinetic energy should equal to minus 1 tons of potential bomb we've got our grounds wave function plus let's go ahead and do it so this is now an independent check and now you can go back to slide 526 to that online thing and you can see that there's this cryptic then off to the side very old equals and then at some number that is very close to minus 2 and that's kind of the quality control measures 1 . 9 9 9 9 who knows that's just round sometimes get something like that at the bottom here than a slide 611 I had the expectation values of the kinetic energy to is equal to the unit already on site plus atomic years minus one-half square cyclists lost in that it is equal to if I do Everything everything because I have all the integral so that I have to do and just write it down that's equal to one-half minus as Safarova too -minus Carol all divided by 1 plus a cigar but that's not any big deal because the the derivative of each of the past Arias is just in the yard so that and the other ones in the definitions of K for the potential we have to integrate over sigh plus of minus 1 over on side plots plus the cyclists of minus 1 B the truck plants and that when we end up with minus 1 plus charitable plus TKO divided by 1 busses plus 1 over I know what all these functions are and both of them depend on the parameters Degas because all these articles depending on how close the nuclei are to each other but so obviously it can be minus 2 for all the values that they got that that's ridiculous because of what we want to know it is at the most stable .period does it satisfy because that's what we're going around so when article 2 Part the equilibrium position the minimum of the well how close to the country's satisfied the virial all we know the minimum is 2 2 . 4 9 3 times of 4 areas and if we just put him on a sequel to that value and because where an atomic units we just put in horror sequel to 2 . 4 9 3 we ended up with the expectation value of of the kinetic energy being . 3 8 2 7 Hopkins and the expectation values of the potential energy being minus 0 . 9 4 7 5 so that's sad because the conditions of the best conditions article already Nunavut with this ratio which is minus 2 . 4 8 which is way off minus 2 so the auditors have come in and say you are missing a lot of money In fact we have about 25 per cent error With this function that we worked like crazy to get and it's still no good now the question is how we can say well 2 the 1 thing that we didn't let up Adams do is we don't like the 1 orbital expand and you can guess that in order to I have both of them and the sleeping bag there that discovered the little that and therefore that's what I will try unfortunately that means that we would have to go back home and calculate everything with our friend say that so although this horrible is not very good but we have to take a pretty deep Brad but before we do that now that you know about the virial theorem you can go back to the other problems we did on because it holds for adults as well and you can go back to our attempts on high and dry and helium and even the 1 has stayed at the heart of the matter the line and you can figure out the the expectation value tea and the it's a very good exercise ABC now you've got an independent check out those other way functions that because we're just playing around there looking at the energy trying to get to the stable we never went to this this level well we're going to have to introduce our friends say that again to let the orbitals adjust to the new environment and that means we're going to have to go back and do all our matrix elements over the HA HA B S I can hear groaning even value on here and I feel your pain so here's the new orbital we've got 1 out of the square prior to the minor was a simple 1 has to someone 1 has now goes to say you go over at all To the one-half per hour you might say that are so-called Slater won a sort where is it is a parameter and we would have to go through them with this new that all our orbital all are matrix elements over and
calculate the luckily we don't have to do that at home because somebody's already done in Slater orbitals are already tabulated and so all we have to do is set them up so we recognize whether it's as chair and then what about the answers and we can figure it out pretty quickly and therefore rather than working them out blow-by-blow like I have done for some of the others were just gonna look them up and where as a function of Zadar then there were going to cut to the chase and minimize the thing as a function of sale and even that will be so very easy but let's have a look at most of the terms are pretty predictable so here on-site 616 what I've read his team that the expectation value of the kinetic energy as a function of Zadar and they are and basically a very similar to the other things there is a sadist square over to miners in a square essence k now the S & K are functions of state times are because that's what's in the export of their there together but then there's some Zetas floating around on itself and for the potential energy it's again very similar there's the 1 German lost 1 over which is the repulsion of the nuclear and that 1 those depend on which electronic warble stupid because that's just depends on the 2 positive charges pushing the party could guess that once I'm going to have to say that and now we've got these 2 things and we can add them up and then we can plot the energy E as a function of both Zeta which are plotted on the Y axis on the slide from Haydn always said it was but I guess that they would have to be bigger than 1 because I didn't see any mileage and having the orbital shrink but that give me like I'd write probably bigger than 1 so I got it from 1 to 1 and a half and I didn't know what are over a knot should be either but I because the true value was around 2 and the value of before was like 2 . 4 9 and I thought if it improves it should get smaller so that the orbital should expand but the nuclei should be held together better I thought of that from about 1 and a half to enhance herself and bingo I got the minimum that Little Red Spot which is almost in the Senate by coincidence that I had applied again because I couldn't believe I would say that laughter but I was and I found that the lowest energy it was our Over a knot is equal to 2 . 0 0 3 which is fantastic football just about right on and that was
106 speaker commuters that and Zeta is bigger than 1 it's 1 . 2 3 8 that's the numerical answer if you minimize yeah In the middle of managing that tell us getting rid of the months ahead because the mother His miners . 0 8 6 5 1 archery minus 2 . 3 4 5 electron volts or minus 226 . 3 killer jewels from all weight madam than before but now after we introduce this say let's see what are auditors said let's see if the operator says we made a good move while now I can calculate the expectation value tea with this wave function no big deal .period 5 8 6 5 oratory for the potential minus 1 . 1 7 3 and guess what it's minus 2 . 0 0 0 0 right on Monday look we improved every aspect of our solution what letting the town 1 as that's kind of interesting because we might have taken another strategy we might just said Look we've got to find some say that such that the virial theorem satisfied of of course that we can use it as a check because we use it as the end of but we didn't do that always said is let's minimize the energy with this extra flexibility and the very old the order came back and said the books right now but that doesn't mean that that's the correct energy that just means it's not wrong and so on lots of things like that it's it's it's not obviously wrong but that doesn't mean it's absolutely correctly the somewhere in between there's lots of solutions that have the feral criteria met and they have different qualities How could we interpret this well with just the 1 who elect wave functions is too restrictive they have to be allowed to stretch to incorporate the under the presence of the of the nucleus and that's how I would interpret it anyway the energy is much better it's still not perfect but at least the virial theorem satisfied now if we want to do better we know the prescription we have to include other functions in the mix and if you're doing this by hand you better think long and hard about which functions you're going to include because whichever ones you include the beginning have amassed integrated and when you start including more of you have many many many more integral to do as they do a ton of work and then it doesn't come out and better and I'm going to show you then and in the last part of this this -minus sob story from going to pick something the the 1 that isn't very smart pick it turns out the next time we're gonna learn why it isn't very smart pig and how to make a much smarter well you might have guessed the I'm sort of like hydride remember that we could included 2 1 has to ask wants to add a we didn't do that we include the values of said but that'll work and so I could include here is I can't say I'm going to take some local efficiency 1 of 1 as plus 1 speed and the later on but also they already can explain never the worst I have plus a 2nd coefficient of USA 2 SBA which economists later into 7 are and I really like that because if I have also later orbitals they're all tabulated the and I don't have to do anything in the world before the advent of software that would do the angles for you that it's extremely important because there's no way she had to do them all by hand could make a profit interestingly enough years a table there's the wave function here is e-mail and here's are the 1st entry is the very 1st thing we didn't wanna say plus 1 be which I called 1 of their equals 1 the minimum energy including the minus half now this minus 2 . 5 6 4 8 8 3 and RE is 2 . 4 9 and if I expanded will say that is equal to 1 . 2 3 8 the lineup with minus . 5 8 6 5 1 and are used 2 and now I do time work and I get this funky wave function . 7 0 7 1 1 analysts with say it is equal to 1 . 2 4 clubs and all 1 2 Westwood said it is equal to 1 . 2 and I have the same language -minus . 5 8 6 5 1 and the same equilibrium bond distance 2 . 0 0 that's really disappointing if we had actually done the calculations but if you come up if you say let's include smidge of something in the wave functions and this marriage is really like a spider recently and liked a pinch of salt .period 1 well that's telling you right away is that whatever you selected is not very important but it's not going to be on the top of the list of ingredients what you want to do is you want to figure out what you can and that will end up with a reasonable coefficient online .period 1 something higher we get a clue that it's not going to be very good just by doing that next time then what we're going to do is figure out 1st by thinking about the problem a little bit more what kind of function we could add without necessarily going through all the calculations because it would be quite difficult but what kind of function we can that would improve it even more compared to the Slater won its orbital slot of expanded and that will lead us to the ideas including polarization but it's really comes down to the fact that is still sphere all and the problem is elliptical but so we win expected to improve too much by just adding s because we already led around an expanded in order to make the optimized orbital that satisfied the period there and we need something that shaped more like a sausage which will see it's going to be a PC orbital and will do that next time
Chemische Bindung
Elektron <Legierung>
Bukett <Wein>
Monomolekulare Reaktion
Konkrement <Innere Medizin>
Essenz <Lebensmittel>
Funktionelle Gruppe
Systemische Therapie <Pharmakologie>
Aktives Zentrum
Endokrin wirksamer Stoff
Potenz <Homöopathie>
Setzen <Verfahrenstechnik>
Gangart <Erzlagerstätte>
Knoten <Chemie>
Braunes Fettgewebe


Formale Metadaten

Titel Lecture 24. A closer look at our Molecular Orbital: The Virial Theorem in Action
Alternativer Titel Lecture 24. Quantum Principles: a closer look at our Molecular Orbital: The Virial Theorem in Action
Serientitel Chemistry 131A: Quantum Principles
Teil 24
Anzahl der Teile 28
Autor Shaka, Athan J.
Lizenz CC-Namensnennung - Weitergabe unter gleichen Bedingungen 4.0 International:
Sie dürfen das Werk bzw. den Inhalt zu jedem legalen Zweck nutzen, verändern und in unveränderter oder veränderter Form vervielfältigen, verbreiten und öffentlich zugänglich machen, sofern Sie den Namen des Autors/Rechteinhabers in der von ihm festgelegten Weise nennen und das Werk bzw. diesen Inhalt auch in veränderter Form nur unter den Bedingungen dieser Lizenz weitergeben.
DOI 10.5446/18903
Herausgeber University of California Irvine (UCI)
Erscheinungsjahr 2014
Sprache Englisch

Inhaltliche Metadaten

Fachgebiet Chemie
Abstract UCI Chem 131A Quantum Principles (Winter 2014) Instructor: A.J. Shaka, Ph.D Description: This course provides an introduction to quantum mechanics and principles of quantum chemistry with applications to nuclear motions and the electronic structure of the hydrogen atom. It also examines the Schrödinger equation and study how it describes the behavior of very light particles, the quantum description of rotating and vibrating molecules is compared to the classical description, and the quantum description of the electronic structure of atoms is studied. Index of Topics: 0:01:11 Comparing the Results 0:04:04 H2+ Molecular Orbitals 0:05:32 The Coefficients 0:06:58 Normalization 0:09:00 The Orbitals are Different 0:14:22 The Virial Theorem 0:28:27 Checking our MO 0:37:01 Optimizing the Energy

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