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Lecture 18. The Hydride Ion (Continued): Two-Electron Systems

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I welcome back today what we're going to discuss is the hydride iron and we're going to continue our calculation on that remain recall we had just gotten to the point where we were looking at the electron electron repulsion term as a perturbation and we're going to look at some other 2 electron systems and how we're going to be forced to work pretty hard today in order to try and reproduce any kind of properties that are known about the
systems and this is a very good tested the theory of course because you should be able to reproduce things like the ionization energy of helium and so on your theories correct and you can take the approximations far I recall here's what we have we have an expression from first-order time independent perturbation theory and we were going to use it to compute the correction to the the sum of the 2 hydrogen atom energies from each electrons interacting independently with the positive please so what we have is we have this sandwiched integral with star on the left and then won overall are 1 2 and then sign on the right and the thing is the way functions themselves are functions of the coordinates R 1 and R 2 and therefore what I have to do is I have to be able to express one-over over are 1 to the distance between them in terms of only R 1 R 2 and maybe some other coordinate that I'm integrating over because I can integrate a function if I don't know the functional dependence on on top of the integration and I can just have some variable in an integral that's why and I don't know how wide depends on the facts or if it does then I can't do the integral with respect facts so there without a mind let's take a look at this figure what I'm John here is an obtuse triangle with R 1 oriented along the sea remember we were going to do that we were going to reorient the electrons every time we do the integral over the other variables so that the first one is a long sea and because the spherically symmetric that doesn't change the answer at all In terms of the energy and what I've done is I added a little lengths to once so that by the time he gets started to it's a right triangle and the little length I've added I've called a and the distance on the other side of the right triangle I've called B and the angle between R 1 and R 2 which is bigger than 90 degrees and this figure I've called figure and that means that the other angle to the other side of the line is pipelines there are 180 degrees minus in order to get an expression than for the distances are 1 2 which is between that and the 2 electrons there too right there is 1 that involves the small figure which is a square closed the square is our 2 square and there's another 1 which is a big triangles which is ah 1 plus a quantity square always 1st and menswear plus the square is equal to R 1 2 square that's that's the big triangle expanding the 2nd equation that and just riding it out we get 1 2 squared is equal a square close to a R 1 police are 1 square was peaceful in squared plus the squared I can gather together and by the other triangle that's hard to square so are 1 2 square is equal to are 2 square good that's a variable to plus R R 1 square good that's variable are ones plus 2 times they funds are what no good what I have to know what areas to be able integrate over the thing but luckily a by that triangle it is equal to our 1 times co-signers the angle that's nearest today and that angle is minus the and therefore I could go back to Oilers identity and put me in the eye hadn't figured out but I know fact just changes signs are to "quotation mark data and that's so I substitute the value of a and I get what's called a lot of the cosigns which you could go back to the Pythagorean some Euclid and find that they were smart enough to figure this out as well are 1 2 square is equal to our 2 square close are 1 squared plus 2 R 1 hour to co-sign data there is OK because remember in spherical coordinates data of being the angle between the 2 vectors that that's perfect because of Quantum's along Cedar data was the variable that I was integrating over for the other 1 so now were set to go and we can do the integral but in that case that data is less than 90 so it's not enough to strangle but it's an acute triangle I'll let you draw the triangles you draw them slightly differently but we come to exactly the same conclusion namely that this formula is always valid and of course this status equal to 90 degrees so that it's just a right triangle than the co-signed data 0 and that goes away very conveniently and then we just have the Pythagorean theorem so the Allocco it's just a generalization of the Pythagorean theorem for the case where it's not exactly a writer trying so now what we've done here in the bottom of slides for 440 His we put in 1 over 1 2 and because the 1st wave function that depends only on our wine and has no dependence on art to the independent variables I've factory about factored out and I have a shorthand notation here that I'm using and that's just to try to fit the equations onto the slide basically but I do a single integral D vector are 2 what that really means is under integrated overfly a integrated overstated and I'm going to integrate over the scalar from 0 to infinity and gonna do all those things when it comes right down to it but just to keep track as a placeholder I've got that integral it's going to be a triple and goal but I just right and this was to make the equation a little bit easier to to see but don't let that notation throw you off will will get to them now this still is not so easy because How do I have to I'll do this integral here I've got the square root of all the spinach and the denominator and it doesn't necessarily suggest the answer right away well let's put on the atomic waste functions that's 1 over Roop won over a nod to the 3 have stumps III to the miners are over a knot in assessing whether it's are 1 A R 2 is just the coordinator of the electron but the wave function is the same just as a different variables and atomic units it's 1 0 Group I and to the miners are and therefore leave the 1st again and you see what I can I can barely 50 equation on the slide I'm going to leave the 1st integration with respect or 1 as just a symbolic thing and then the 2nd integration with respect to our too I have a the integral over fate of scientific data because remember that was part of the volume element
that I needed and then in the bottom I have this to our 1 hour to co-sign Satan that I didn't go over a fire that doesn't bother me and go over the fight to outstanding dependence on file that I didn't go over are and I have to remember To put in the arts squared I'm OK so if I make a substitution which you tend to do when you have trigonometric functions and if you have an algebraic functions can't do you intend to make a trigonometric substitution and if you have a trigonometric functions can do to make an algebraic substitutions here what I'm gonna do is I'm gonna let access the variable XP co-signed a faded to the DX is equal to minus sign today that the failure to and therefore the integral over to have signed Sunday 2 over the radical is equal to minus the Annagrove from 1 of which is "quotation mark there 1 day this equals 0 "quotation mark 1 2 2 minus 1 of DX over the square of 1 square closer to square foot minus 2 our ones are too and then I can change the limits make it from minus 1 get rid of them negative side at that 1 I can look up the ante derivatives because that 1 is the standard very easy to do and I'll let you verify it by taking this actual entered derivative I've given you hear on the bottom of slide 442 and please differentiated with respect to X and verify that you get and grant that we started we get this minus the square root of are 1 square .period to squared minus 2 R 1 hour to access divided by R 1 2 I think it's pretty easy to see where that came from once you start doing derivatives if we put it the limits then and do the subtraction we get the following Our 1 plus are too minus the absolute value those are 1 minus 2 divided the are 1 of and recalled the square X squared is the absolute value of next to the square root of positive therefore are integral over the data is equal to that so already kind of interesting because she may not have encountered this before the integral over the of that size data over the square root is equal to 2 over par 1 if 1 is bigger than are too the bigger than are to it's too over our 1 but it's equally to all over are too if I wanted less than arteries so it's equal to 2 over the bigger of the 2 and that
means when I integrate over the other variable are 1 I have to be very careful here that I picked the right answer 1 and integrating the right up there at all if I fix are too and I had a great 1 but are too I should use 1 formula and that when I degrade are 1 the rest of the way to insanity I should use the other formula and if I'm not careful about how I do that then I did the wrong answer they can imagine if you have a lot of electrons and you have a lot of integral state keep doing this kind of thing that they depend on who's where it can get mighty tricky to figure out how to keep track of the right way to do things that so this might be your 1st introduction to this kind of straightforward problem that seems to give a conditional enter derivatives that depends on on what's going on the Honourable over 5 to gives to teapot big deal nothing to do their the control over data 1 and there's no they don't want last status gone there it was stated to there's no they 1 last and final and just gives for prior like it always does because the fourth-highest is the angular the extent of of the spherical coordinates and so there's nothing else there to do except the radio parts and we have to break the radio parts into 2 integral we have to 1st break it into 1 over are 1 and the inability 2 R 2 squared meter minus 2 or 2 From 0 are 1 and then the rest of the way from 1 to insanity it's just E R 2 artillery because it was 1 of too so 1 1 away you minus to our both of both of these we have 1 that's are too he didn't minus 2 are too weak and the other 1 hour to square you much to our 2 boys you get good at doing In fact you you get so good you just know them by heart if you start doing this kind of thing often because you don't want to waste time to flipped through pages despite knowing somebody's phone number if you column of a lot just know it and you will know this stuff if you work through these things and you will suddenly seemed very smart people who don't do these kinds of calculations or very nearly hour any way you can do these by parts and here's what we get the integral part of our security demands to or 2 is equal to minus the demise to over 4 times a polynomial not to 1 us to our 2 plus 2 our to square and the other 1 has 1 last term and it is in the middle of slide for 45 but again if you worry about whether these correct or you just want to reassure yourself take the derivatives and whenever you're by yourself and you don't have software new proposing an entire derivative it's the actor derivative is always a bit harder but it's it's like dividing and away you have to kind of see it it's gonna work the derivative is more like a multiplication you can just do it so you can always go backwards and figure out now as putting the limits and we can finish off the gold Over are too and we put in these limits 0 are 1 of the 1st integral we get the expression of the top of slide 446 and are 1 to insanity on the 2nd integral 1 infinity the exponential vanishes soldiers to 0 and then we have turned the shot and if you tidy everything up and you get the following you get that this radial integral is equal to 1 4 times 1 over our 1 minus he to the minors to over 1 time 1 over a 1 plus 1 little bit messy but not too bad the AI integral over 1 can also be done now but parts so you've got the interval over overarching which was conditional died down at the end over they ones I wanted said was for pride in go over are 1 nite that this function of our alliance you've got to do it by parts again and that illegal turns out to just 5 over 120 and that's the way these things sometimes work and that usually means you've done it right if it works like this but what well we get a factor of 16 because we get the 2 4 pies but the price where it goes away because of the 1 of over the square Pienaar each 1 has wave function I got 2 of them but I got them on both sides price square that goes away conveniently and so we did that the energy correction by first-order perturbation theory he won is equal to 5 8 because of the factor of 16 the 1st reaction when you do this calculation it 2 curves answer 5 eights also what 5 eights well it's
Aloha tree because we were working on atomic units and we know that that's the unit of energy anatomically so 1 way is to close your eyes and just say Look I didn't do anything wrong I said these units I know this is energy it's got to be a hot treaty if that doesn't reassure you you can go back and put in all the constants that you've done everything let them ride along very messy messy stuff and you can see that it is firing of a hockey all constitutes right along not too electrons speech 1 interacting with the nucleus is minus half a heart because that's what hockey twice the ionization hydrogen at approximately so that mind-set at minus the have "quotation mark dates and therefore are added the the total energy of the hydride and now is the energy of the 1st electron a half plus the energy SEC electron minus a half plus 5 eights minus 3 perfect you might think it's negative so that means that H minus is stable compared to a proton and electron In another electrons at rest at insanity well unfortunately that is a pretty low both To have to meet because that's not the question the question really is hydride stable compared to a hydrogen atoms which we know is stable an electron and incentive and that's a real sour ending because the energy of the hydride is minus 3 eights of archery and the energy of a hydrogen atoms plus electron it is my user and therefore what we're predicting is if we have a hydrogen atoms and we have electron and we bring it up that the energy becomes more unstable in other words it just takes it back out it ionizes it back out takes it out and the repulsion force wins and that would mean that hydride wouldn't exist and if I I didn't exist we wouldn't have a name for what proper maybe I should say that we have names for plenty of things that don't exist except in our heads but hydride is a real thing that we can see it has a very big radius the radius of the hydride ion is bigger than the fluoride and so it's very comfortable at but it does exist and unfortunately what this means is that could lead to thinks it could mean that quantum mechanics is a crock and it doesn't work and this is the proof or it could mean that perturbation theory to 1st order is not good enough To give us the correct answer and in fact in this case is the 2nd that that means that we've got to somehow work harder in order to figure out a better way function or a better way to calculate the energy but we know it's more accurate than what we've
done here after all that work then we can reproduce the simple fact that age minus exists as a well-known radius that has a positive ionization energy what we should have anticipated that In retrospect and here's why ah perturbation has exactly the same size as the 2 attractive part recall that the idea behind perturbation theory was that you had a large problem and that was simple but you it's all and then you an income small complicated part of that was you couldn't solve exactly but because it was small you could expand it and you could close in on answer is a power series and you can decide when to stop but if we look at this then suppose we proposes a parameter acts that indicates the size of the various energies and the ratio of X is sort of the ratio of the repulsion energy to the attraction energy for any particular electrons than if trying to get a solution acts and axes near to be 1 then I X doesn't get small very quickly and so justice assuming that edX for example is 1 plus tax effects is not small Is it is a very bad approximation especially Texas near 1 of year comparing you know to well this way off and it backs in the world perturbation serious happens to be bigger than you then what that means is that you calculate successive corrections they might get more and more violent we might say Well 1st the energy should go down like this manager go up like that and you go down like this and you might have found some of the terms that just doesn't add up to anything just diverges and furthermore becomes very inaccurate after you do all this of work you still get garbage out the end so we shouldn't be so surprised that this is not so it's not so easy what to do and whenever you whenever you look at a problem like this it's a good idea to try to estimate how big these energies are likely to be and get an idea of whether perturbation theory there may not work here in retrospect we didn't expect it to work well but of course if you go through the calculations do all those integral to do all that work and it doesn't work then you really scratches pattern and you remember that a long time how can we improve that what's wrong well using the hydrogen 1 s orbitals is not a good idea because the hydrogen 1 s orbital set the maximum probability of finding the electron Michel at the border radius but we know experimentally the hydride ion as of March they radius the hydrogen atom and is therefore we would be much better because electron electoral repulsion among other things possible we would expect on that it might be too tight a squeeze they just sit there with the hydrogen wave functions and just calculate the correction that we could go further in perturbation theory and calculate the corrections to the hydrogen wave function based on perturbation theory and what we would find is that it would get bigger when we corrected but that would be a lot of the integral today not a few a lot and will take a long long time even if we had software helping us out along the way and still have to keep track of things very accurately and add them all up but I want to try on a slightly different approach we would use perturbation theory to correct the wave function but why don't we correct the wave function with physical intuition why don't we introduce an artificial parameter into the orbitals also the controls the radius before we have the the race the we function is either the miners are 1 for example now let's put someone else in there who was put in the the minus Z R 1 and now say there is something we can control it's a dialog we can dial in and out and we can compute the energy of the hydride on iron as a function of this parameter Zeta and then we can find the optimal the it won't necessarily guarantee success but you learn a lot by failing in this kind of endeavor it's very important never to look at the answer before you've really tried like crazy because if you do you miss everything about site somebody getting the answer to crosswords or anything else telling you a message of hope for the 3 there is it completely ruins in a way and you never really learned all right so let's keep in mind that our evaluation of the energy although we didn't buy perturbation theory there it's insect exact because we have the hydrogen atoms energy that's exact minus one-half oratory and we did the integral over are 1 2 exactly we didn't make any approximations there so that's the exact energy 4 that way function in order to get a better estimate we have to correct the wave functions and we're just going to correct the wave function now and hopefully if we just tweak the wave function slightly but don't change it in any essential way the math won't get too difficult we can still use all our other results and do our intervals and so forth and then see what we did so will recycle most of our work I'll refer back to those in if you want to look back at how do them and just slightly adjust the most probable radius and what we expect physically is that it should get more stable if we make it a little bit bigger now if we make too big it will be very bad because of its very far apart than the energy 0 because it's like Proton into electrons at rest and Finland and Finley so as I said rather than doing perturbation theory to the next order which would be interesting but will take a long time which is gonna us away to adjust the way function and then we're going to try this variation of approach we have energy that we've calculated that depends on a parameter and then we know if the energy go slower but it's a closer approximation to the truth so we minimize the energy with respect to the press so
we introduce a parameter that I'm calling Zeta which looks like agreed squiggle a lot of people know what the letter is but now you're part of the club there's 2 of the squiggles there's a and there's sign and don't get a mixed up say has less squiggles and so let's put on our new normalized gets here at the bottom of slight 454 the wave function which is a function of R 1 and R 2 sigh of R 1 R 2 is equal to say to divided by height and speed of the miners say they are 1 I can't see the miners are too on Wednesday this was we get exactly the same thing but we had before and the question is is it equal to 1 of the best guess what we can do worse by introducing it may not improve but we can't do worse because we can always be examined is equal to 1 and we got we got to what we got before now we have to calculate the expectation value of the energy with this wave function and that means we have to do the 2 hydrogen like terms each electron interacting with the protons and then we had to do .period repulsion integral again here we have our 3 Hambletonian H 1 it's true In atomic units and 1 to alcohol which is the interaction term the mark set up prevents the energy from just being the sum and we have 3 and also but thank goodness 2 of them are identical just with electron 1 electron to there's no point in doing that and here's what we get from the energy which is equal to the double and again use the shorthand here of In deep director are 1 of vector "quotation mark h Excuse me so I stoppage size but in this case the way functions real so it does not matter whether we have the conflict commentary is not now and that rates in the future terms the integral with H 1 sandwiched in between the to Saladin between wherever they are the same no difference between 1 and 2 and then the repulsive integral With age 1 2 which is just 1 over are 1 2 sandwiched in between which we saw how to do and therefore it's not going to be too bad because the only real differences this thing Zadar in the exponent and that doesn't change in any fundamental way whether we can do the integral or not it's not like we suddenly switched to our square in in the on something that might make it quite difficult or change or not you of make it difficult to answer to do the integral the 1st 2 terms are the same and that the angular parts integrated for in each case because there's no angular dependence in the wave function and therefore we have to do to do either of the 1st 2 terms is the angular excuse me the radial and what I've written down in full here for the variable are 1 of the bottom of slide 457 we have are 1 square dance with comes in from the volume element but we've got Zaid accused over pride the 1st wave function that we've got the Hambletonian than we've got the 2nd wave functions and the four-part comes out on the angular variables the lets them take a closer look and how to do this well we got the kinetic energy part now you might say Well why don't you just kind divine this is so similar to to the idea that 1 answerable hydrogen 1 divine were put a Zadar in into the answer and the answer is I don't trust myself to be able to get that right so I'm going to go back and putting in what the kinetic energy is an atomic United's minus 1 hostile square and I know how to write that out In spherical polar coordinates and the wave function has no dependence on fire and therefore the part that I end up with is the 2nd derivative with respect to our 1 square -minus 1 over R 1 times the 1st through the expect or want and I have to keep in mind about the factor of minus 1 which I concluded that I therefore here's what we've got now this is pretty messy because finally integrating with respect to our 1 there's the wave function and then there's this thing that's taking a lot of derivatives with respect ones and then there is the potential energy which is minus-1 over are ones that I've put it so we've got all those 3 things In that and then there are operating on the way function softcover go step by step and better take the derivatives write down everything put them there right in there and then I'm going have to integrate by part anything that doesn't depend on our 1 can be pulled out front so I pulled out 2 from from the 2 exponential said depend on our and out of this mess to do with our 1 but the derivatives are pretty easy all they do is bring down as Zeta each time and change the sign depending on how many derivatives sector so I end up with this little thing here to integrate our 1 square feet of the miners say they are 1 times minus status square over to plus 1 over 1 times said of minus 1 and then there's another you the minuses are 1 and he undergoes a standard by which I mean while you just look among Nevada and the miners are and if you go ahead and do it and go on and do without you get a result that looks pretty nice it's 1 over a it was a minus 1 over for sadist square and that's that resolved if you integrate over are too then the leading Constance out front including the leading Constance upfront Excuse me and you do the whole thing then you get the following result for the single electron energy but the expectation value just the Hambletonian 1 is saved over to times said a minus 2 and when we could say equals 1 we get minus ahead and so I wouldn't at I wouldn't have been able to figure out that it had this functional form without a very quiet room without actually doing the Senate rules and that's why don't you just put in something like Well it's minus 8 over 2 because that's not correct it might be that that also gives a half say this 1 but it might not be an infected it is isn't itself and there's good reason for us to have a linear quadratic terms the electron repulsion integral is the same thing and I'm going to let you do that because that once we did in great detail with sign that the conditional to over R 1 and if you want to do that and you go through and you do it with a big piece of paper the quiet room you will get 5 Zeta divided by 8 as the answer for the electron electron propulsion and that would be quite a lot of paper The Hendon when you do it as
1 of our problems no well what whenever you get something you should check whether it makes sense sometimes things don't seem to make sense like the electron going through both slits that in that case you keep doing the experiment over and over in this case you should expect that calculation to tell you that the thing that when we look at this formula 5 Zeta overrate its status reduced that means since it said reduced the thing gets larger but then the repulsion is slower and that makes sense that's exactly what we expect to happen the total expectation value of the energy than as a function of Zeta is the sum of these 3 terms finally we get this formula when everything comes out in the wash we had everything up and we do the algebra very carefully and don't make any errors we get serious squared minus 11 saves over it that is the Zeta dependent energy on Wednesday it is equal to what we get our prior value of minus three-eighths of a hot treat now however we have this energy as a function of Zeta we have variation of principle that says when the energy goes lower you did better and I this kind of function with a quadratic in a linear terms with different signs clearly as a minimum and so we can optimize this by the variation of principle and get a much better estimate of the some of the energy of the hydride and less to the standards of practice problem this is practice problem 24 let's optimize the value of Zadar to minimize the energy Is the hydride on on predicted to be stable here's the answer the is simple problem and calculus what do we do if there is a minimum of that means that's the lowest it was the slope is 0 and so while saying the function has a minimum is kind of a statement and away saying that the derivative is equal to 0 it's something you can actually work with it's an actual equation gives you a solution and so that's the course when you translate it to me and therefore we take the derivative of that functions and we get absolutely Zeta squared minus 11 aides say that if we take the derivative again Tuesday at minus 11 aces and if we said that equals to 0 then Zeta should be 11 16 now again we say that makes sense and the answer is yes because 616 scenes was 1 and what we did is we have to go out we let it go out quite a bit more and that lowered the energy because of repulsion terms going away faster than the attraction state and now the question is yes the hydride iron stable by which I mean is the energy of the Don hydride ion less than half a heart tree which is the energy of hydrogen atoms and an electron just hanging out window well it's putting the optimum value of Zeta the energy minimum them as a function of Zeta is 11 over 16 square minus 11 over 8 pounds 11 over 60 and then becomes minus 121 Over 256 so there's no joy in Montreal here because to be stable we would have to get energy lower then minus 120 8 Over 256 wherever minus 121 or much better than the 3 before we and that was pretty bad this is better for sure and that shows that workers were closing in on the correct way of looking at the problem but unfortunately it's not good enough it looks like hydride and it is a tough nut to crack and indeed it is we're gonna crack it but we're going to have to go into the kitchen and we're going to have to get some tools out and you know some of Brazil nuts and other not subpoena peanuts you open with your fingers this is definitely a macadamia or Brazil now it's going to be tough to crack it's not we didn't do anything wrong we've taken a perfectly sensible approach it's just that this is a tricky system and it's interesting that it's so simple and yet so far at the same time to get the right answer but for some reassurance but it's not the quantum mechanics that that but it's not completely wrong or something crazy let's try helium why would helium that well helium as opposed to charge for the nucleus so that's going attract the electrons much more strongly there and they would be single charge of the protons and that might mean that it's much easier to get to be stable and actually get reasonable and it also teaches you why you love atomic units so here over the Hambletonian in atomic units for helium the kinetic energy is exactly the same line a after square and then the only thing that changes is we've got minus 2 over are 1 minus 2 over are too and 1 over a 1 2 so that propulsion and the same that's done and the only other thing is the other things to and so that's easy today so that we can scale the kinetic and potential energies are going to be both double for each electron and the repulsion it's going be increased as well because the orbitals are gonna get pulled in 2 the nucleus and what we did when we when we work it out is the and instead of getting -minus that have minus a half we get minus 2 square for the 0 lower energy and instead of just getting 5 days we get 5 times too so therefore the total energy of the helium matter if you follow through the exact same Mathis what we did hydride and iron and each sentenced to instead of 1 everywhere through you could do it very quickly if you just keep track of you get minus 11 forks but that means that they see the ad is stable already by perturbation theory compared to a helium ions and electrons at infinity at rest and it's the experimental energy that's listed in the literature and hard cheese is minus 2 . 9 0 3 3 and recall I said a few coatings and hot region ever after reading "quotation mark them because you don't don't depend on what do units or just depends on what the calculation it's doesn't that on the fundamental constants and are calculated value minus 11 forwards is minus 2 . 7 5 hot trees here's the real
yeah good helium experimentally down here here is the best we can do before we just by perturbation theory so at least we got that it was stable was an unstable but it's quite a bit of difference between the correct that the correct answer 5 per cent error in these kinds of things is no way where way way off unfortunately that that's like you didn't even belong the right City Union bomb right country you're on a different planet in terms of both your Jakarta and therefore we can't do that but what what and why don't we just I do the same thing is what we did with high with helium since it's already better at least is predicted to be stable letters the orbitals up a little bit was Seda and take the minimum and follow through said Madigan with 2 so 1 and we get a similar expression To be what we had before but now said is Zeta squared minus 27 over a and the same technique obviously get 27 over 16 for sake being the optimum value and then if you put that on the to the minimum what you get is you get minus 27 over 16 square that's what exactly what you get and what that approximately becomes is minus 2 . 8 4 7 6 6 Hartford that is looking pretty good because the the real 1 minus 2 . 9 but the real questions well how good our results in terms of something that just might be interested in could we calculated based with this kind of approach could we calculate something accurately enough that somebody would really invest money in some scheme based on the calculations and the answer is no the energy of the helium ion is minus 2 3 because it's just like a hydrogen atoms of it with different charge that is not an hour ionization energy if minus 2 . 4 6 or ionization energy energy who 1 electron off is . 8 4 7 6 6 the true value as . 9 0 3 7 to boot an electron off helium when we convert them from archery to kill jewels from all which are more familiar units to attend lest we find that were in error by about a hundred 47 killer jewels per mole that is similar to the value of many chemical bonds many chemical bonds value that could be a weak bond but that's a lot of energy as far as a catalyst is concerned the not tiny air that's a huge share of and so we're nowhere close even doing this very occasional calculations of and inserting this parameter I say they're doing all this work doing all these girls taken during setting at 0 optimizing and putting it on hold your breath calculated and we're still nowhere close to what we would call chemical accuracy I think you can see why this get this kind of field in atomic physics and quantum chemistry gets call computational chemistry very quickly because you may have to introduce functions that are not so very easy to integrate that happened to be very close to the true wave functions and may be part of our trouble is where introducing exponential speakers we know how to integrate them we have a closed form for the entire derivative but maybe once we start getting more than 1 electron in there they aren't so close to the correct result next time and what we're going to look at is can we With a little bit more physical inside so we said what's it puffed out but could we do better the that could we somehow change the way function in such a way that it's not so bad to integrate but that it's much much better in terms of results and that will be what I call hydride try number 3 where you are going to strike out or we're going to please get a single and hopefully we can figure out the hydride is in fact stable so pick it up the next
Mineralbildung
Koordinationszahl
Chemische Forschung
Hydride
Vitalismus
Konkrement <Innere Medizin>
Eisenherstellung
Glykosaminoglykane
Elektron <Legierung>
Alkoholgehalt
Helium
Linker
Ionisationsenergie
Funktionelle Gruppe
Systemische Therapie <Pharmakologie>
Atom
Elektron <Legierung>
Hydride
Zellkern
Querprofil
Vitalismus
Ordnungszahl
Genexpression
Knoten <Chemie>
Erdrutsch
Herzfrequenzvariabilität
Chemische Eigenschaft
Bukett <Wein>
Chemische Formel
Feinkost
Expressionsvektor
Chemisches Element
Reglersubstanz
Sonnenschutzmittel
Elektron <Legierung>
Fülle <Speise>
Reaktionsführung
Feuer
Hydride
Zellkern
Sonnenschutzmittel
Vitalismus
Gold
Genexpression
Vitalismus
Alben
Konkrement <Innere Medizin>
Erdrutsch
Substitutionsreaktion
Derivatisierung
Chemische Reaktion
Bukett <Wein>
Derivatisierung
Chemische Formel
Funktionelle Gruppe
Singulettzustand
Periodate
Mineralbildung
d-Orbital
Zetapotenzial
Zellkern
Feuer
Hydride
Orbital
Vitalismus
Lösung
Konkrement <Innere Medizin>
VSEPR-Modell
Altern
Eisenherstellung
Elektron <Legierung>
Gezeitenstrom
Ionisationsenergie
Funktionelle Gruppe
Atom
Aktives Zentrum
Reglersubstanz
Physikalische Chemie
Fülle <Speise>
Elektron <Legierung>
Hydride
Zellkern
Quellgebiet
Vitalismus
Trennverfahren
Fruchtmark
Protonierung
Chemische Reaktion
Komplikation
Magnetisierbarkeit
Hope <Diamant>
Bohrium
Fluoride
Zetapotenzial
d-Orbital
Potenz <Homöopathie>
Alkohol
Feuer
Oktanzahl
Härteprüfung
Sense
Gletscherzunge
Elektron <Legierung>
Hydride
Vitalismus
Ordnungszahl
Protonierung
Reaktivität
Zähigkeit
Bukett <Wein>
Thermoformen
Zetapotenzial
Expressionsvektor
Mineralbildung
Zuchtziel
Zellkern
Chemische Forschung
Labkäse
Hydride
Orbital
Konkrement <Innere Medizin>
Lösung
Vitalismus
Altern
Derivatisierung
Eisenherstellung
Elektron <Legierung>
Helium
Funktionelle Gruppe
Systemische Therapie <Pharmakologie>
Atom
Potenz <Homöopathie>
Helium
Zigarre
Gangart <Erzlagerstätte>
Zuchtziel
Windsichten
Primärer Sektor
Erdrutsch
Herzfrequenzvariabilität
CHARGE-Assoziation
Chemische Formel
Derivatisierung
Chemisches Element
Chemische Forschung
Zetapotenzial
Zitronensaft
Chemische Forschung
Orbital
Hydride
Konkrement <Innere Medizin>
Vitalismus
Computeranimation
Edelstein
Quantenchemie
Chemische Bindung
Elektron <Legierung>
Helium
Funktionelle Gruppe
Ionisationsenergie
Atom
Physikalische Chemie
Elektron <Legierung>
Reaktionsführung
Helium
Vitalismus
Genexpression
Ordnungszahl
CHARGE-Assoziation
Reaktivität
Thermoformen

Metadaten

Formale Metadaten

Titel Lecture 18. The Hydride Ion (Continued): Two-Electron Systems
Alternativer Titel Lecture 18. Quantum Principles: The Hydride Ion (Continued): Two-Electron Systems
Serientitel Chemistry 131A: Quantum Principles
Teil 18
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 10.5446/18896
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: 0:00:53 Where We Left Off 0:02:06 Figuring r12 0:03:55 Law of Cosines 0:07:58 First-Order Correction 0:09:11 The Theta Integral 0:17:42 What's the Energy? 0:19:56 Stability of Hydride 0:24:37 Improving our Estimate 0:30:19 Evaluating the Energy 0:37:09 The Repulsion Term 0:43:59 The Helium Atom

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