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Lecture 22. The Born-Oppenheimer Approximation and H2+

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today in chemistry 131 and we're going to move on From multi electron Adams to the simplest molecule women talk
1st about the born Oppenheimer approximation this was a great idea that was introduced very shortly after the introduction of the shorting her equation itself in fact when you go back to this period in history but you find out is that everybody was really working like crazy on this new field of quantum mechanics much like if he come back to 2 thousand 14 you may find a lot of biologists were working like crazy on stem cells now now that they've found a better way to produce and then after we talk about the born Oppenheimer approximations were going to introduce the simplest molecule age 2 plus and from there will go to H
1st the born Oppenheimer approximations we're going to just look at the simplest molecules H 2 plus and H 2 and we're gonna look at both of them in some detail to try to understand why it is that you can have a chemical bomb In fact it's very important interesting in the framework of classical mechanics an electrostatic to try to figure out if there could be such a thing as for example to positively charged particles in the let's say 1 2 electrons let's say we have to sell the whole thing is neutral in its like age 2 and we can propose that they're doing some classical thing we could forget about the uncertainty principle outlets have them going like this in some pattern and so forth and and have all the electrostatic forces and try to see whether the molecules together and in fact I scientists at the time didn't really understand how molecules could hold together because it doesn't seem to work out very well and we'll see exactly what when we come to the exchange's rules but if we're going to talk about molecules even when we talk about helium we had to use some approximations we had to make a good guess with the variation ulterior driver perturbation theory or something and now that we've got to nucleotide 2 senators even with 1 electron we're going to guess that it's going to be extremely hard to solve things exactly so the 1st idea is that we want to decouple the motion Of the nuclei from the motion of the electron and we know that we can probably do that because the electrons are much less massive than the nuclear and the electrons are therefore moving much more quickly what we're trying to do that is to say let's say we've got an elephant and we've got a bunch of flies around and the elephants moving slowly and so the flies have plenty of time to readjust and if the elephant rolls over slowly the flies have plenty of time to find the best place where they want to alight on the elephant in much the same way as the nuclear coordinates change the electrons almost instantaneously at least from the perspective of the nuclear nuclei readjust and they readjust to the best possible position and so that being the case we can go 1 step further than that and we can just say supposing meant that the we just stop the motion of the nuclear and we solve the electrons then that we can do that and that's essentially what the born Oppenheimer approximations 1st I
want to talk about our notation here are equations are going to get more complicated and that's going to happen quickly and many used begin to develop the masses of the proton or the nucleus and little land to the north the mass of the electrons and are coordinate system is as on this slight 543 where you have to nuclei marked a and B and in this case put in both electrons on which I call 1 and 2 this is a student and this isn't some instant some were trying to set up Hourigan assault for the wave equation for the Schrödinger equations we have all these repulsive and attractive terms between these charged particle the Hambletonian
then I have with all the terms and as a kinetic energy for the 1st nucleus a a kinetic energy for the 2nd nucleus be the interim nuclear repulsion which is a positive terms pushing them apart then the kinetic energy for the 1st electron won the kinetic energy of for the same electron to the the attraction electron 1 the 1st nuclear tests the 2nd nucleus the attraction of electron to after 1st nuclear the 2nd nuclear and then the repulsion 1 are 1 2 which is the distance between the 2 electrons we are not going to be able to solve this exactly with all these terms for the total wave function which includes all the nuclear an electron quarter In fact with the 2 electrons we won't be able to solve but like the HA we can separated out the center of mass motion for sure but we can't rigorously separated the relative nuclear motion we have to make an argument about why it is that we're doing that and
born Oppenheimer wrote a classic paper that I have given you the reference to hear if you'd like to look it up but unfortunately it's in German and so you have to look at the translation of it I had actually learned to read German when I got my degree in chemistry which shows you how old I am but it helps occasionally From time to time if you have to go back and read some of these papers and they added that the paper was on the 1 mechanics of molecules and they have a very elegant their derivation about why it is that you can ignore the nuclear motion but I don't want to go through that because that would take at least this lecture another lecture and it would just take us a little bit too far afield to actually go through that but we can certainly see that's because the nuclear a moving slowly and because the electrons adjust then if I just consider the nuclei here and I saw all the shortening equation for the electrons pretending the nuclei frozen and then I moved in nuclei a little bit and really solve the shorting equation that basically what the electrons are doing continuously as the nuclei moved slowly from here to here is there to solving the shortening equation again and again and again and because they're they're readjusting so quickly it it appears to them as if the nuclei basically frozen and that means to us there basically frozen as well so if we right
that the wave function of the molecule is which depends on all the coordinates is the factorization product of the wave function of the electronic parts and wave function of the nuclear part than the total energy is just as some of the electronic energy and the nuclear energy this is right that particle on a box with the directions and this has been our strategy over and over and over which shows you it's a very good strategy then what we find is that the electronic functions dependence instantaneously so to speak on the corners of the electrons are 1 . 2 and depends parametric on this very slow variable big which is just the distance between the 2 nuclei and we can always use this approximation for molecules that have very high excited electronic states usually to excite an electron from any ground state to some higher state and takes a lot of energy and the slow motion of the nuclei cannot possibly supply that kind of energy where we have to be in trouble or where we have to be extremely careful is if we have radicals are open shall Adams or something like that where we have a lot of electronic states that a nearby than what you can imagine happening is that the motion of the nuclei itself can somehow cause a transition of the electrons to move out of the orbital it was originally and into a new 1 in that case the motions are couple the whole thing's wobbling around and then we're in trouble and we can use the former Oppenheimer approximations of on that case it fails but
the upshot of our factorization is we can forget completely about Connecticut Energy the nuclei which is clamping down freeze them and we solve for the electronic energy and that's really important simplification yeah I've ridden out the Hambletonian in electronic trading in atomic units the electronic Hambletonian still has the kinetic energy of electrons the kinetic energy of the nuclei is gone and I've added to it 1 over are which is just deal the static electrostatic repulsion of the 2 nuclear we have all these instantaneous terms now we have this static terms but we have to add that to the electronic energy because when the nuclei of very close that will be an overall unfavorable position we don't care about the kinetic energy in the nuclei because we we threw that wh and and so what we want to do then it is solved assuring the equations for the electronic coordinator as a function of the anti-nuclear separation based are and then see what that is and in fact if
we do that what we find is that when the nuclei of very close it's very unfavorable the partly it's just the anti-nuclear repulsion but partly it is that the election electrons themselves are pretty much outside and so don't scream the positive charges and as we move them apart the potential comes down and then there's the sweet spot where there is a minimum and then it comes back up again and in fact that's exactly the kind of shape that we argued was like a Morse potential or some potential like that when we're talking about vibrations and the harmonic oscillator and I can see that the nuclei themselves glide around on a potential energy surface that's constantly readjusting we just drive as if it's always there but it's costly readjusting as they move it is there before they get to where they're going so to speak and we talked then about the nuclear accord it's moving around potential energy curve or a potential energy surface which is the total electronic energy plus the interim nuclear propulsion that's what a potential energy surface is how you should think of it and the nuclei will respond itself there to close those gradually pull apart and as they pull apart the electron soldier around and you just the simplest
possible molecules H 2 plus I want to begin with that IRA I wrote down the Hambletonian H too but that's a little bit intimidating for us right now so I wanted but just do it with 1 electron the SEC electrons now not present so I don't even want to I dull and there's only the attraction to the 1st nuclear is the attraction to the 2nd nuclear there's no 1 over our 1 to thank goodness because that was quite a production when we did helium and and then there's 1 over par are which is no big deal which is the interim nuclear propulsion here's our picture than on the bottom of the slide 549 sank coordinate system as before and except the 2nd letter this message
well much like we could solve the hydrogen atoms but once we factored out the center of mass motion and assumed the Proton was essentially fixed by saying Well look this problem has spherical symmetry we should actually write the Schrödinger equation and spherical coordinates and know what happened this we knew we had the right quarantine system because the wave function satirized into 3 parts Dan that made it easy to solve each part this three-dimensional differential equations became 3 one-dimensional differential equation and that is always the strategy if you can get it to work now in this case sphere "quotation mark warnings won't work because we have stretched there charge out in 1 direction well if you take a circle and stretches in 1 dimension so that that you have to senators then you have an ellipse and so not surprisingly there is accorded system called elliptical coordinator ,comma focal elliptical coordinates where you take the side of the distance between them and the electron and the nuclei divided by so there's a dimensional as things landed and then another 1 which is the difference between them are a minor starkly divided but they got a call that you then in the 3rd quarter net is still like it was in the hydrogen atom that fight and the solution for that because that's basically the same that was the particle honoring we can still use that to solve that problems that the couple's sound so we don't have to worry about that the other 2 we have to actually worry about what kind of differential equations that we get you can think of and if you make a triangle between the 2 nuclei and the electron yellow plane 3 points to 5 points and the position of that plane around this axis between the 2 nuclei is the best youthful angle and so this is just like a stretched hydrogen and billing let's
write the wave function again as a product I were really beating a dead horse with this but it's the only horse we've got and we're going to write it as a function of land on which I have called the gland of land as a function of new and unfortunately capital New looks like a man but it's mu of you and then the other side of 5 those suggests that presupposes that we can write write like this and I'm giving you a reference here at the bottom of page 552 slide 551 if you're interested there's quite a long paper and the Philosophical Transactions of the Royal Society where they've gone through exactly how you seperate but these coordinate system it does separated it does separated into a differential equations with respect and you only began and then another differential equations that only has big the problem for us is that these differential equations you know the one-dimensional if you look at their very complicated and they have a lot of moving parts and we can't write down the solution to these kind of equations in any elementary way there may be solutions in terms of repeated fractions where 1 plus 1 over explosive plus and so forth and things like that but that's not really up to illuminating for us at this level and so I don't want to go into writing attempting to write down these very esoteric functions that might solve this because of the stock and move us that far forward he has a
musical solutions or pretty sensible that we're back to the EDI and 5 divided by the square to pipe and entered the 0 plus or minus 1 plus remind us too and the reason why this 1 so simple is that motion around this axis is that's still a good quantum number because we haven't monkey with the symmetry that way when we pull the other thing apart than the other stuff gets thrown out the motion gets thrown off and it gets much more complicated and so it is much more complicated to describe it and so as I as I mentioned we are going to get into Lambda you just to say that they can be figured out their numerical tables and you can compute energies and you can't do everything with them but we're going to actually get an approximate solution that's going to be a lot more insightful than Nederlander New although it won't be so quantitatively but the end equals 0 solutions are called estate's find out the call Excuse me said most states by analogy with their states from and then states with 1 twist around this way are our called state and plus minus 2 adult estates and much like we do with the PX Y and easy orbitals sometimes when we actually plot the solutions we take linear combinations of them to get "quotation mark so that we don't have an imaginary part to the wave function the other really
important consideration for H 2 is that the problem has cemetery if I imagine the origin the middle and a positive charge here and a positive charge here than there is symmetry in the problem and that has to be reflects of swapped 2 nuclei of overriding bird the coordinates than I have nothing to drastic can happen because it's basically the same thing again In particular the this operator I've called P had that inverts the coordinates it just takes all .period xyz and puts some through the origin minus X minus 1 and minus the and this operator if you applied to wave functions and then you apply the Hambletonian and then you apply the Hambletonian the wave function and then this operator you find out is you get the same thing therefore it's not like position and momentum where they didn't commute this operated the inversion operator of the coordinates commutes with the Hambletonian for 2 plots and that means that unlike and function of the Hambletonian also has to be and I can't function of the inversion operator within the inversion operator doesn't have any units of the I can function of the inversion operator is just going to be a number question is what number is and the answer is well supplied inversion operator twice will 1st take xyz and go to minus X minus Y -minus C great now will again we went back to where we started everything has to be the same men because all the coordinates are the same as if we do those 2 steps
we 1st put on inversion operator onto an item functions we get a number C times the Eigen functions that follows because it has to be function Of the operator P can't turn away function into something else and that if we applied again that he squared and we get see square but we get the same way function that and so what we conclude here is that squares once and that means that see Is he the plus 1 or minus 1 there's 2 possibilities well that means that the energy functions whatever they happen to be RE even if it's plus 1 for all know is it's minus 1 under version of all the coordinates and that's important because we can use that to classify the solutions to say well you're and even tan or you're in the yard care and historically we use this the symbol G for even and you 4 all I from the German derided and on the OK
now let's say instead of writing down landed new and writing all these very complicated formulas and perhaps in a certain sense being none the wiser when we finish let's try get a solution 4 the wave function for the electronic part of H 2 plots and then once we get past let's use our our hidden ace in the hole the variation on principle to optimize our against so that it's really good after we've proposed some kind of gets well OK let's explore this there are a lot of kinds of functions I guess I could guess something where I have found something in the exponent or I have something squared or have a lot of things and I could optimize those by the variation of principle but the problem is that might be a lot of work he saw with the parameters stated that we had before but we ended up with very complicated equations and then and then we had to to really stare at them a long time to get what we want out in this and make a contour plot even because we couldn't really see how to find the minimum very easily but in the special case that I the form of the function I said my way function is like a bowling ball plus a Q plus appearance or something and I have percentages of each of these things courses quantum mechanics the percentages could be complex numbers but I don't change the shape of the underlying function that I'm using I just change the amount of the In that case there is a very elegant way to look at how this optimization works and we get a series of equations that lead to something which we call the secular determines which I want to introduce this is not the Slater determinant this is something else less than supposed but we have to to fixed functions which I call 5 1 and 5 2 and we have to pick them cleverly because if we pick them poorly and even when we optimize them we may get terrible answer for the energy so we need to have some physical insight I would say to pick these functions correctly and some people have a lot more physical insight than others but in any case then we have an amount C 1 of 5 1 and amount C 2 of 5 2 and Bulwer were doing now is optimizing the amounts the linear coefficients were adding up the functions we're not multiplying the rear not raising them to powers of each other or anything else where adding them up and we've got these linear coefficients
OK so we have to pick them correctly but obviously that's part of the physical intuition we have to depict functions the goal 2 0 at insanity and and other things like that so that they behave properly otherwise but if we optimize them we may find out that the amount of 0 because we put in a function of the camp but for now let's assume that the functions are fixed and and there real and like the 1 s orbital of hydrogen for example and let's also assume that the C 1 and C 2 are also real so that we don't need the complex conjugate and that's what I've written here there is the expectation values of the energy which is just the integral of C 1 5 1 plus C 2 5 2 there would normally be start let's assume they're real the Hambletonian members the same thing C 1 5 1 policy to fight too and I can then multiply out ideas for terms I guess he wants squared times 5 1 H 5 1 and then I get the cross terms to those C-1 C-2 policy to C C 1 and then Agassi to square and then fly to age 5 2 and when I do the Senate roles the Hambletonian operating on a function and over the other function multiplying out on the left and that's just a number the intervals just a number or whatever it is it could be 0 it could be something and I'm just going to write those numbers as the as the C 1 where I'm going to call that 1 H sub 1 1 and for the 1 to I'm going to call it that number that integral H 1 2 and further to 1 and college to 1 and for the 2 to I'm going to call that that integral H 2 2 and general these these numbers H sub Vijay are called matrix elements and that's because if you're going to see in a 2nd they are positions in matrix and this is a very beautiful systematic way to keep improving our guesses as wide as a linear combination of orbitals is a very very powerful approach well the
Hambletonian is real and the sees a real and the functions real so the matrix elements a symmetric of course the Hambletonian is always for mediation and so if there is no necessary part the the real part symmetrically imaginary artists aren't isometric because H I J stars equaled HAI but we don't have to worry about that well we also because we took these 2 functions we mustn't assume that anything's normalized we should actually make sure we can normalize things is so out of the way you normalize things as you take a square wave function or you can consider the square wave functions a slightly different way rather than putting the Hambletonian in between the 2 sides just going to put the 1 operator 1 operator just takes the wave function and returns there's nothing to it but by writing it that way and then expanding it out I see that I get 4 terms again C 1 1 1 squared I get the angle of fire 1 square and then I get 5 1 5 2 5 2 5 1 and I write those exactly the same way as I did with age c 1 square S 1 1 man I have it's 1 2 2 1 and has 2 2 and best it is called the overlap matrix because it's going to tell us how the hell do different functions we think overlap well if
the the basis functions were than normal then the overlap matrix would just be the identity matrix if I had an functions and what they were like different orbitals on a hydrogen atoms for example on the same nucleus and that were my bases then because those are already off the normal I have nothing to do but if I have a atomic orbitals on different nuclei so that the shifted of these exponential than they certainly are not often normal there there are normalized with respect to themselves if I take this 1 a multiplied by this 1 square go overall space it's 1 and likewise with this 1 but when I take that too multiplying in between it's positive everywhere that can be 0 and therefore they are not orthogonal so a reporter for the 2 basis functions if they were orthogonal this matrix S 1 1 S 1 to S 2 1 which is the same as S 1 2 and as 2 2 would just amount to the unit matrix 1 than 0 the 0 . 1 but but we usually can't if we make a natural choice of functions that we usually can ensure that this happens and we don't want to what we want to do it's just go ahead and compute the matrix S and then use that to figure out what's going on so here is
our formula than for the energy keeping in in mind that the wave function it is not necessarily normalized in fact the exact overlap is going to depend on how far the nuclei Our apart we can write this energy functional the energy as a function of the coefficient C 1 and the efficiency to is the expectation values the Hambletonian on the top of divided by the square of the wave function on the bottom and the top I get those 4 terms but 2 of them were the same so I end up with 2 C 1 and C 2 times age 1 2 1 on the bottom and up with those 4 terms in the U.S. matrix but 2 of them same possessed 2 1 is equaled S 1 2 so I end up with 3 terms there and rather than right that is a ratio the way I have a minimal both sides by the the dominator and writer and as the times this spinach in S is equal to the I'm Hambletonian matrix those matrix elements the numbers that fixed because the functions fixed that's the beauty of it I specified functions once I do the undergo once and then they're done and I don't play around with them anymore if I had to keep doing intervals over and over because I'm playing around with the functions themselves would be very very time-consuming whether I do it on a computer or by hand well the variation of principles says if you've got the energy the quantum systems as a function of a parameter like C 1 and C 2 then if you minimize the energy with respect to the parameter you're going to improve your guests of the wave function and so that's exactly what we're gonna do this now we have a formula here for the energy a lot I've done here
then is I have just taken the derivative of both sides of this equation with respect to see 1 those we have a product so it's a derivative of the 1st as a first-term times a 2nd which is all the S matrix elements plus the 1st which is just the energy plans a derivative of the 2nd half of the term with C-2 goes away because this is a partial derivative with respect to see what an army Hambletonian side I and upward to see 1 H 1 1 plus 2 C 2 H 1 2 and interestingly enough there's a lot to lose in this thing at the minimum or maximum but it's going to be a minimum effect the derivatives should be 0 so therefore we can say that the the times that as part is in fact equal to the other part because that term with DDE DEC 1 is 0 and that's nice because now we have be left over and if we set out write that out we get what I have on the bottom of slides 560 which is that C 1 times H 1 1 minus iii that's 1 1 plus C 2 times stage 1 2 minus the S 1 2 is equal to 0
if we do that derivative with respect to see to you get a similar equations and these at these 2 equations that In 2 unknowns can be written in matrix notation as a matrix that now has elements H 1 1 -minus eh S 1 1 and so forth In each of the entries always page minus CDs times the column vector C-1 C-2 is equal to 0 this is a system of linear equations and 1 solution to this system of linear equations the simplest 1 actually is just say OK no matter what the Matrix is let's let's see 1 and C 2 equals to 0 both of us than anything time 0 plus 0 times anything is 0 that works and the other 1 works and so the whole thing that comes to 0 well that was no good because C 1 and C 2 overall cereal that unfortunately our way function which is seen 1 times the 1st function policy 2 times 2nd is also 0 and that's not going to help us find a good way functions so that trivial solution is not the 1 we want
in some sense than what we want is instead of the column vector being 0 we want the matrix somehow to be 0 but we can't control the matrix to be 0 per cent because it has the 6 numbers and what we can do is we can make the determinant of the Matrix equal to 0 In order to see why this works you really need a course in linear algebra which is why it's very important to take a course in linear algebra so you can get to the bottom of stuff like this once and for all and I'd recommend the straight lines like I did for the Slater determinant the determinant of of this array of numbers is equal to 0 that's the necessary and sufficient condition that we get a solution that we want well this determinant is in fact a call the secular determines the secular determinant then it's just the major Solomon the Hambletonian Take the Function put a chance to take the other functions started the need to have its complex integrated get a number that number is fixed you shove them into a matrix do the same thing with the overlap where there's no Hambletonian shove those numbers and put there and then you will get the equation in eh and following the rules for taking indeterminate members a crisscross rules 2 by to the age minus estate's my C as well H 1 1 H 2 2 are the same In this problem and H 1 2 is the same and so we were going to get a
quite dramatic equation in the energy and that we're going to get to the roots and take the smaller ones to be that our best estimate of the energy as usual when you solve mathematical equations and mathematicians like to give you all possible answers not necessarily the 1 you're looking for and so but because they don't know which 1 you're looking for among other things and so we have to look through and then after we get all the energies of solving the secular determine we have today figure out which ones the best when we got 3 basis functions we end up with a Q because we have put the 3 by 3 determine which means he every term has he times times he's so we've got a cubic equations and to solve which can be very messy it costs got other powers to because there's more than 1 term they and all here we could just take the cue grew and with aquatic it's very messy indeed and so in actual fact we don't ever try to just write down a formula for the energy except if it's a quadratic equations and usually if we're lucky we don't even have to write down the formula for a quadratic equation you that we just look but you can always calculate them numerically and their standard packages to root polynomials once you set up the secular determinants and and bracket out in that with a big polynomial in the energy and you want solve for the values of the better real that make that things equal to 0 and the standard methods to root polynomials packages that you can get and I think you can see why haven't having something like that is like having a hammer or a screwdriver it's incredibly useful if you want pounded a nail or screw in a screw and if you don't have that to all the common stock and there are lots of course mathematical tools like that available to us these days OK let's
go back to H 2 with a secular determined analysts do this 1 then as practice problems 20 8 let's get since we've gotten a lot better idea as to what s orbitals let's state 1 along centered on proton aid and the other 1 centered on proton B the question is what we're going to take C 1 of this 1 and see to it that 1 was performed for the 2 energy from solving the secular determined that's the question well OK we know that the 1 s orbitals I have normalized so we don't have to worry about that we integrate them overall space the answers 1 and unfortunately they're not orthogonal because once the off nuclei de Almeida 1 0 there's something and we have to figure out what the something and I call that something just that's because it's the only thing that's not going to be 1 in overlap matrix is definitely not equal to 0 that much we can see because they're they're both positive exponential says Scott and great to something between 0 and 1 and the
problem has symmetry so luckily we don't have to do all the matrix elements over and over we can just do the 1st the matrix element I'll call H a that's got 1 say the Hambletonian 1 said will work out what it is later but to solve this we don't need to actually work it out we just need to estimate has some value and that's exactly the same as HPV because that we get the same orbital we got the same nuclear charge everything's the same and then there is this 1 it's such a pity which is mixture and that again has cemetery where call this 1 area this 1 billion vice-versa there have to be the same so that sequel to HBO so I want to do 2 of the 4 possible matrix elements and then I can just fill in the other keep in mind here that although a writing too functions 1 S 1 and B the girls are 1 three-dimensional integral because I've got 1 orbital that's the sum of these 2 guys I'm writing the out for terms because that's the way it pans out but this is different than when we took helium where we had a product Of this times that that's different because then you've got all the variables and we had 6 dimensional intervals we had integrated over R 1 and then integrate over 2 as well the Senate just have what I want and I have to figure out however how to express your bills 1 essay and 1 SB in terms of this variable are and maybe some other angular variables like faded and it won't surprise you but the last co-signed this gonna come in again but keep in mind that this is 1 of 3 dimensional integral and nothing like what we were doing with helium even what might look like a it's easy to get mixed up well here's
our secular determined that putting in all the values that are the same we have age 18 -minus and then the next 1 off diagonal H a B minus the Times best were as just that the 1 that isn't 0 and then the same thing and to 1 entry and then same thing on the bottom and so this sciences minus the stances just gives us a while Quality squared minus Page a B-minus yes quantity squared equals to 0 why could expand this out insolvent as a quadratic equations and that would be quite messy to do because rather than just having a square post be exposed C a B and C would be a very messy things and then I'd have to Stratus a simplified and there would be a very slow way to go but
luckily I can factorize this by looking at it because whenever I see the squared minus that's what I remember that that's something passed something times something better something and therefore I can think well if I say squared minus B squared is a plus B times a minor then I can write that out with a and B and my terms the HA A-minus H a B minus the than the other way round equals to 0 at 2 things equal to 0 the only way that can be true as if either this 1 0 or that 1 0 and so that means we get to energy routes the first one i've called the plus you get minus plus -minus yes equals to 0 you solve that for and call the plus and you get H a plus HAD divided by 1 of us the other 1 yes just the other part of an idea H a minus page a B divided by 1 minus and I've called the minus
To qualify the things that what we're going to have to do so that that's the end of the problems but to quantify things we need to know what these numbers are in particular we need to know what these numbers are as a function of a big part the separation between the 2 nuclei if we can work that out and then with them we consider its track he plus and the minus pretending where an electron riding along on this aircraft carrier this molecule we can pretend and we can see where we're going to have the sweet spot of energy or if we have 1 what would be good is to be tried to predict the ionization potential of helium try to to predict whether hydride would be stable or not we saw sometimes it could be tough we want to see if we take something simple like this with just a quadratic equation here and 2 routes and we pick the most stable 1 does it predicts that age to Plus which does in fact exist does exist and if it does what the value of the artists most stable value and how does that compare with the actual value which has been measured of course spectroscopic click and then the next time what will see that it is we we can probably take this and take this orbital that we solved with the and we can probably not another electron and then we can argue where on the electron electron repulsion is an important but it's not all important and so we can just kind of adjust our theory a little bit and then we may be able to predict things about the properties of molecule like H to the next time will will take a bit of time and will actually go through how to calculate with these 1 s orbitals the matrix elements how to do these angles so easy to do their aren't so hot but they are they are trivial leader and it's probably good at 1 time to go through them and see how to do that and then you really appreciate it when the computers doing it for you how much labor-saving idiots if you set up a quantum chemistry calculations sometimes the program will tell you how much per cent overlap you've got on that in in in terms of setting up the calculations and I think you can see basically what is talking about is talking about this matrix ass and the fact that if you don't set up north of normal basis than its own having to do a lot more calculations but sometimes it's very hard to see what kind of off the normal basis I'm going to use when I've got things that different positions in space and I just can't visualize what I'm going pick them and then attend pick atomic orbitals butter centered on the atoms and in that case I certainly can't expect them to all be orthogonal to to each other so next time I will do those integral school work through them systematically and we'll see if we can predict whether age 2 plus is stable or not and if so how stable is that
d-Orbital
Feuer
Methyliodid
Übergangsmetall
Pferdefleisch
Fleischersatz
Morse-Potenzial
Fülle <Speise>
Elektron <Legierung>
Symptomatologie
Operon
Mähdrescher
Germane
Bukett <Wein>
Mannose
Krankheit
Expressionsvektor
Periodate
Chemische Bindung
Zellkern
Erstarrung
Wasserwelle
Klinische Prüfung
Hydride
Süßkraft
Lösung
Werkzeugstahl
Stockfisch
Derivatisierung
CHARGE-Assoziation
Elefantiasis
Aktionspotenzial
Elektron <Legierung>
Chemische Formel
Helium
Cadmiumsulfid
Funktionelle Gruppe
Ionisationsenergie
Lösung
Translationsfaktor
Konjugate
Click-Chemie
Kryosphäre
Physikalische Chemie
Gangart <Erzlagerstätte>
Tellerseparator
Computational chemistry
Mutationszüchtung
Nucleolus
Derivatisierung
Chemische Formel
Spektralanalyse
Radikalfänger
Koordinationszahl
Aktionspotenzial
Sense
Reaktionsmechanismus
Oberflächenchemie
Wildbach
Alkoholgehalt
Molekül
Lactitol
Funktionelle Gruppe
Nucleolus
Bewegung
Krankengeschichte
Sonnenschutzmittel
Reaktionsführung
Base
Ordnungszahl
Agar-Agar
Protonierung
Protonenpumpenhemmer
Bewegung
Thermoformen
Mischen
Elektrostatische Wechselwirkung
Chemische Forschung
Metallmatrix-Verbundwerkstoff
Chemisches Element
Chemische Forschung
Orbital
Explosivität
Konkrement <Innere Medizin>
Stammzelle
Altern
Atomorbital
ACE
Quantenchemie
Nanopartikel
Linker
Operon
Hydroxyethylcellulosen
Systemische Therapie <Pharmakologie>
Atom
Butter
Pipette
Metallmatrix-Verbundwerkstoff
Potenz <Homöopathie>
Querprofil
Trennverfahren
Erdrutsch
Azokupplung
Deformationsverhalten
Replikationsursprung
Herzfrequenzvariabilität
Chemische Eigenschaft
Chemisches Element
Molekül

Metadaten

Formale Metadaten

Titel Lecture 22. The Born-Oppenheimer Approximation and H2+
Alternativer Titel Lecture 22. Quantum Principles: The Born-Oppenheimer Approximation and H2+
Serientitel Chemistry 131A: Quantum Principles
Teil 22
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 10.5446/18901
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: 0:01:03 The Born-Oppenheimer Approximation 0:09:52 The Electronic Hamiltonian 0:11:06 Potential Surfaces 0:12:43 H2+ 0:22:43 Variational Approach for H2+ 0:41:04 Back to H2+

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