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

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#### Automatisierte Medienanalyse

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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
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
Mineralbildung
Koordinationszahl
Chemische Forschung
Hydride
Vitalismus
Konkrement <Innere Medizin>
Eisenherstellung
Glykosaminoglykane
Elektron <Legierung>
Alkoholgehalt
Helium
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