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Lecture 17. Approximation Methods: Variational Principle, Atomic Units, and Preparation for Two-Electron Systems

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In welcome back to chemistry 131 in today what we're going to do it is pick up where we left off last time Morgan expand our vision we're going to talk about approximation methods including the variation or principle were going introduce atomic units because when we do calculations in quantum chemistry of carrying around big and small numbers turns into a big problem and we're going to prepare the ground to study to electron system we can hope to get the exact solution for these 2 electron systems but we can get as close as solution as we want to do and that's usually good enough for chemical accuracy other we saw from the last lecture than that 2 electrons in the same spatial orbital would have to have their spins pared the another words the overall wave function has to be antisymmetric so that if the spatial part of symmetric spent part has to be antisymmetric single state 50 50 down minus downtown was the state now when we look at 2 electron systems hydride might be the simplest 1 will start with that and we'll see that although conceptually it's very simple and fact computationally it's extremely difficult and helium is very similar to hydride except there's no negative charge but there's 2 electrons announced that of a plus 1 charge in the nucleus there's a plus to charges that will make all the difference in terms of how easy it is to calculate properties of helium we can just assume anyway that the electrons in the systems are the single and cannot forget
that for the time being and just focus on the spatial part we want to figure out what's the energy of the Saturn can we figure out its ionization energy can we figure out its properties and if so how accurately can we get our results to compare with the experiment and so that's the idea behind what I'm going to introduce now which is called the variation of principal harks back to they are observations earlier on that but I can't functions for different eigenvalues of a linear her mediation operator are orthogonal we can think of them like vectors in space they're all at right angles to each other and that means that we can take any function and decomposes into Eigen functions of some linear information operator and what we will be trying to do is to decomposes and functions of an energy operator so I've written here what orthogonal the means for 2 functions fight and fight and that means if we take the integral which is the continuous analog of the dark problem if we integrate every point in space up against each other we get 0 and that literally means these 2 functions have nothing to do with each other they're in different directions and of course we can always normalize any function we did that early on we just take the raw function square integrated and see what we get and then we divided by the square root of whatever we get to make sure that it comes out to be 1 that means that when we do that what we're insuring is that we have these factors in different directions but we wouldn't want to be measuring the x-direction in meters and the wise direction and miles we want to make sure that the units of our measurement the amounts of each functional we're going to make to this recipe to get our final functions are all in the same units and normalizing make sure that that happens then the analogy just like we can free of a two-dimensional surface we can specify any forwarded on the surface as ordered pair X ,comma Y X tells us how on the x-axis and wine on the Y axis we can specify any function any unknown function as a linear combinations of these basis functions or these I can function and in fact you drawn on slide for 17 a picture which is kind of the analog of accorded picture of 4 points but here I am a function f I visualized as it as just an arrow pointing somewhere nothing more than that and then I can visualize my basis functions as arrows pointing along the corridor next season this funny functions based in this 1 there's a red car 5 1 2 s and there is a blue part fight to end in general and there could be an infinite number of parts but usually we won't have to go that far to get a decent approximation to the function that we're trying to get How can we calculated the amounts well we calculated the amounts just taking the integral sold by the same token that we calculated the amount of and infighting that was once it's 100 per cent and the amount of fight and fight and that's 0 per cent if we have a function deaths here I've written and we integrated with each basis function and verify 1 5 to so forth and so on we get a series of numbers the numbers are the result of the integral and the number that we get by doing an integral is the amount of that function that is present in the unknown function that's all we have to be able to do to figure out how much of our basis functions in an unknown function is to an integral and while intervals can be intimidating to beginners intervals are considered to be easy to do 1 where another numerically or analytically and so that that's a very good that we have a close solutions to calculate these coefficients we don't have to try to guess the or something crazy and then see how close we get to actually have a way to systematically chip away and get our unknown functions as accurately as we want and sometimes we have to do the intervals numerically and in that case we use a very powerful computer and we set small step sizes so we do the integral but the idea then that some unknown function that we're trying to find a can be represented
as a linear combination of Eigen functions leads us on to this very powerful and general method to find approximate solutions to difficult problems and that method is called a variation on principle the variation occasional principle is just going to be an inequality but an inequality is very important when you're trying to figure out which way to go even as a kid when you play a game where you're blindfolded people say you're getting warmer you're getting colder if they never said anything about where you were going you could never find whatever it is that you were looking for and the computers not too smart and we are to Smarty there and we need a criterion we need some way to figure out if this new thing is better or worse than the last 1 and if so How much better and it's this very variation principle is going to give us this machinery to do that we can suppose always even for an unknown problem that we haven't solved we can suppose that there does exist a set but I can state energy Oregon State's in particular for the problem we are trying to solve we haven't yet solved it but we suppose that if we could solve it that the states would exist and we know that the states are orthogonal because we proved that generally for linear Hermitian operator which the Hambletonian the energy operator it's a linear mission operate then based on our orthogonal set of states we can write some way function side as a linear combinations of these unknowns states now this might seem to be the vaguest equation never proposed I an unknown thing that I got here outside of our and then I've got some unknown coefficients and then I've got some unknown solutions to a problem that I'm trying to solve but nevertheless this is an exact relationship between this and the method to calculate the coefficients is to do the integral of these functions so we don't know the coefficients yet but we can assume that they exist and they are complex numbers this being quantum mechanics and they can be calculated by doing integral so we're trying to find the let's say the the ground state wave function the lowest energy the most stable state of an atom or some other system and we don't know what it is but we make a guess we make a guess based on a similar system or by analogy or we just find a function that looks like it might be a good way functions and we take that as a guest and a very good guests to make it would be a function that would be easy to integrate because when we won actually figure out what's going on in real calculation we may have to do integral so picking a function that's very very hard to integrate takes a long time or tricky is not usually a good choice the condition and that are unknown gas wave function be normalized I've written here on the bottom of slides for 20 in this rather long equation basically we take the integral of star so and then we expand each 1 4 so I star we put CNN star finance star because they could both the complex numbers and then for the last the other side not started we put the sum over and above C M fire fight on the important thing here to do is to always when you introduce a song and you're breaking up an unknown vector into its components used a different letter don't use then and then use and again if you do that and that happens when your 1st starting out and you run into a terrible mess because then you're inadvertently and coupling the coefficients together and they have nothing to do with each other so we just want people mall separate but now so we've got to have some unknown let's say we've got 16 functions in the inside C 1 to C 16 and then another seeds 1 through C 16 but they all go away though because the I knew functions are orthogonal so whenever the 2 functions on the same you know we don't know what they are but the term vanishes so we don't have anything except the terms work on a sequel to and we end up down in the 2nd line with the summer Over end of CNN squared five-star but then fight and that because the Pfizer normalized we always assumed that the I can't states are normalized if not we normalize them just ends up to be that the sum of the square of the coefficients is equal to 1 so that the coefficients themselves we take each 1 and you take its length and the animals it
adds up to 1 that's the condition we don't know the explicit form of these energy functions fight and we don't know the energy needed but we can order the were trying to get the lowest energy was called the lowest energy E 0 0 and let's suppose that there are other energies he won he too and so forth that a higher we don't know where they are we don't know what they are but that's unimportant for this discussion we guess at trial wave function side we mentally decomposes into a linear combination of these and functions with certain coefficients the complex numbers and then using because we can get I'm very very useful inequality to do with the energy which will help us to systematically improve our initial guests here's what we do we follow through the same principle that we did it to show that the sum of the coefficients has to be 1 but now we put the energy operator Indiana crawl so were calculating the expectation value of the energy in the state so so I star H so we integrate that all we don't know how to do that cause we know the Hambletonian for unknown system that's for sure otherwise we would be nowhere we have to know what forces in energy source play knowing those doesn't give us the answer unfortunately and that we take are unknown function and again we know that we can always decomposes into this linear combination just like any point on the XY plane has 7 x coordinate and OY coordinate and any point in 3 D space xyz course who we can decompose and what I've written that out here again 1 summer's over and then age than another summer is over then and now we can use the fact that we know these are energy Oregon State's these functions fight to put H on each of the the ones that is marked fight am and we get the and then we can use the fact that all the functions are orthogonal to realize that when we have the the other son with all those other things that only the term where n is equal to end is going to come through and in that case we have the
square of the coefficient which is a real positive and then we have the wave function fight the N 5 which integrates to the song over and over he had time CNN square Ian is always greater than or equal to the not because we ordered the energies and so if if if there's only 1 term not then it's equally not but if there is any other amount if there's a 50 50 of 0 energy and 1 then that's higher than 100 per cent of 0 energy and because these numbers CNN square are a real positive numbers there can't be any consolation and that means that the sum Of the end cnn's square which is the expectation value of the energy for our guests is always bigger than are equal to the sum over and of you not instead of putting in and I put in the not breached terms and then I can't believe you not out of the sun because it doesn't have an index that depends on them and then before I showed you that the sum of and squares 1 so that means that the expectation value of the energy for or against always has to be greater than or equal to the ground state energy he not we get something we calculate the energy and then we guess again or we're just something or we minimize something by calculus because we can find the minimum of many things by finding where the derivative is 0 and we go downhill and we know automatically that if our energy loss that we're going to be improving our estimate of the ground state of whatever quantum system it is that we're looking at that's the variation of principle is really 1 of the fundamental and most powerful tools that we have to find approximate solutions for these complex systems because we can minimize things in many ways we can use a computer Wikinews calculus and so on there many tricks minimizing functions and so forth is a well-studied area so if we can take this problem in quantum mechanics of figuring out this way functions and cast into a minimization problem that mathematicians have studied for ages and we made a lot of progress so this was a very very important result now it may seem like this is a bit empty because we broke our function up into the design and functions but we can actually write down the Eigen functions because if we did right down the Eigen functions we wouldn't be doing any of this we just write down the ground state I didn't function and we would know by quantum mechanics and everything that is possible to know about the ground state of the art and so the problem would already be solved but that's not the the main thing because we don't need to know what they are all we need to do is calculated the energy and we don't need to know them and to calculate the the energy because we know the energy operators it's got some derivatives Senate and it's got 1 over on some charges and other things in it and we can calculate for function that we guess exactly what the energy is by just doing integral we don't need to actually try to break it apart into these unknown functions that's just a mental exercise to show that when the energy gets lower it gets better we calculate the energy we tweak our way function if the energy goes lower is better automatically and that means that we're closer 2 the correct solution it is obviously we have to make a very good gas and whatever we add that were twisting Awad adding things and some things we hadn't may not be very important so we we work like crazy and do all these articles and the energy goes down a little bit that's disappointing but if we find the right thing to add to make the energy really drop down very sharply down into very close to the minimum then were pretty sure that we've got a very good description of what's going on and what we see what it took to make the system better we start to get some physical insight into what's going on here how we should think of where the electrons are and how they may be interacting with each other at some point we're just getting a lower limit or we're going to get tired of doing so many integral so we might give up and we might have to do a lot more computational effort much more than we want to bother with to get a more accurate result it all depends on the problem and whether you're trying to set a benchmark for accuracy or really deeply look into the theory or whether you're trying to figure out whether sister trans isomer of something might be more stable or whether this confirmation may be more stable than another 1 it just all depends on on the problem but whatever it is once the energy matches the ground state energy which we can usually measured experimentally than we know the whatever wave function we have is a pretty good approximation to the ground state waste functions and then we can look at the way functions and it'll tell us the expectation value of many of the things that don't have to anything to do with the energy but I have to do with other things we might be interested in measuring and then of course we can always compare with experiment ionization energies of Adams and such systems can be measured quite well and have dementia and so we can compare the ionization energy of helium for example to make the helium plus 1 ion and electron but with what we calculate the ionization energy should be which we can calculate by calculating the energy the helium atoms and then the energy of the helium ions which is like a hydrogen atom because now there's 1 electron so we can just use a formula for that and then an electron of insanity at rest which has not 0 energy because there is no potential and no kinetic energy let's try some examples but 1st before we undertake any of these calculations we're going to need atomic units if we carry around and MKS a SI units in these calculations but with all constants and so forth it'll get extremely tedious to keep track of everything the let's see so let's have a look at them and how we could minimize this effort so to streamline the equations atomic physicists adopted units of measure such that that although constants in front like East squared m so be it h bar and all those things that appear in all these equations where did not involve larger small numbers if you take H bar which is tender minus 34 and Hubert on a digital computer it under flows it makes a number so small in most other languages that I need to get 0 and then later if divided by the divide by 0 and you have a big problem and therefore we don't want to have larger small numbers like the electron charge
raised to the 4th power these can quickly get out of hand and you're adding and subtracting small and multiplying and dividing small and large numbers so for accuracy as well it's much better to keep everything near 1 that way you have lots of headroom lots of floor on bigger and smaller numbers and your calculation usually stays accurate and these units are called Hot atomic units and their named after the British physicist an applied mathematician Douglas Hogg tree who was instrumental in proposing many of these methods including ways to calculate the electronic structure of here's what we do then on slides for 26 we agreed to measure charge energy and so forth in units such that the mass of the electron as 1 unit the charges the electron is 1 unit h far is 1 unit and 1 over for clients not which comes in all that time whenever you have charges interacting is 1 unit and if we do that then all the units disappear and we've got a much simpler equations of course although they disappeared the units are still there and they're riding along and we have to put the man at the end if we want to go back to MKS units the thing is sometimes we just don't want to go back and we'd rather express everything in these units in Hartford and the unit of energy is called the hot tree said here I've written is the mass of the electrons times the charges the electron to the 4th power divided by for pirates along not age by quantity square that's the heart that's 1 unit of energy we can convert as I said once we're done we can convert to more conventional units and in chemistry we might want to convert to electron volts and chemistry or atomic physics and 1 hot is 27 . 2 1 1 3 8 5 electron volts or about twice the ionization energy of a hydrogen atom and that's equivalent to 219 thousand 470 wave numbers if we're doing spectroscopy so it's quite a very big lot of unit of energy or if we're talking to organic chemist and they're talking in terms of killer jewels per mole which is common with thermal chemistry and Bonn stability than 1 hot trees 2 thousand 625 and have killer jewels from all so again it's a very big unit that is the strongest chemical bond I believe carbon-monoxide and that's about a thousand killer jewels from also 1 hot tree is much stronger than typical chemical bond in terms of energy there is another more subtle reasons to use atomic units and that's 1 that you probably don't ever think of unless you're doing very very very accurate calculations and you're trying to compare your calculations with benchmarks of people said in the past for various numbers and the problem that you run into is that if you're making a calculation about 8 or 9 digits of accuracy and you're introducing small things in Europe arguing about small energy terms in this Hambletonian is this important and not how exactly how well can we calculate these things so you pushing the frontier the problem is if you can't be achieved "quotation mark your energy in electron volts or some other form then the value depends on what the value of the fundamental constants was when you work writing your paper because you insert that age the speed of light the charge of the electron and normally we don't question of us because we look them up in this database or something like that and we take the most accurate values but all those constants are subject to change in other words the there's slight variables quiet because somebody comes along with a more clever experiment to actually get the accuracy better than you could do before and that would be very bad it's a constant changed and then I went back to a paper in 1965 and in addition to looking at the energy had deft also decide well what was evaluated far back in 1965 because they had determined it quite as accurately as today and so forth but if I work in hockey I don't have to do the calculation over because we are using any of those they're all 1 so whatever they are are you quote the value and hard trees only today when you want to convert to use the best possible values of all the constant and that makes comparison with previous calculations much much easier to do that's a hidden benefit that of using these atomic units so let's go on this OK let's try practice problem here excuse me on slide for 20 not less do you practice problem 23 that's right down the Hambletonian for the hydride and for the helium atoms In conventional units and also in atomic units this will be our lead and actually using these Hambletonian to do some calculations to figure out the way function for the state so here's the answer we have what the kinetic energy of 2 electrons the potential energy of attraction to the nucleus they're both attracted to the positive nucleus and then we have the electrons electron repulsion try a remember we've always factored out the center of mass and then we pretend that the nucleus is fixed in space and so the Peace in its are just to the electrons themselves 4 a hydride is conventional units we have minus a over 2 ends of the Dell 1 squared -minus age spots where over 2 ends of the same massively electron Dell to square wider 102 well each electron has some caught sex 1 1 1 0 1 st 2 white Uzi too and the wave function will depend on these coordinates the first one means look when you see Annex 1 Y 1 0 0 1 take the 2nd derivatives and that's going to figure out what the kinetic energy of electron 1 is doing when you get to number 2 look at only X to Y Tu and see to treat others as constant that's can isolate the kinetic energy of the 2nd electron so looks a bit funny that the subscript 1 and 2 on the but it's perfectly natural and then we have the attraction minus eastward over 4 plants land 1 hour 1 is the distance electron 1 To the nucleus the same thing for our 2 nowhere plus because these repel each other eastward over 4 pirates along are 1 2 are 1 2 is the
distance between the 2 electrons and we'll get to that in a minute In atomic units it's much cleaner because getting rid of the ends of the Nth Power and so forth now my Hambletonian is this minus one-half Dell 1 square -minus 1 have dealt to square -minus 1 over 1 -minus 1 over are too plus 1 over on 1 . 2 that is a nice the equation that is easy to deal with both by paper and buy a digital computer 1st he added In conventional units with the same thing exactly as hydride except 1 well the attraction to the nucleus of the nucleus has charged him so we have to times so we have minus 2 weeks were were for clients 1 hour 1 same thing for a R 2 and atomic units the Hambletonian is simply again minus one-half Dell 1 square minus 1 have still to square foot minus 2 over 1 -minus 2 over are too plus 1 again the 2 electronic over are 1 to those just much much much cleaner and easier now it's as I said it's you have to be clear about the notation and hear what I've written is our coordinate system on the bottom of slide 431 we think of these things as factors are 1 1 electron to
to another and then how 1 who is the factor between the tip of R to and the tip of R 1 and it's a director with the direction between the 2 of electrons and With respect to the figure little 1 without boldface Sofres boldface I'm talking about a vector of a magnitude with direction and quantity and if I'm using just a regular italic typeface I'm talking about the length of the factory just the distance between the nucleus and the electron questions so are once the length of the vector bold
1 hour to select the vector bald 2 and are 1 2 is the length of the of the vector 1 bold are 1 2 which is just the vector are 1 minus the vector are too I know it has to be that because I know when I had the actors I put them in detail and when I too and with the
factory and detail are what what I called 1 2 I got our once and that means that are 1 2 plus too should be equaled are ones and that means that are 1 2 should R 1 minus are too so when I got to get our 1 and that's how you go about it don't try the memorize which way round things go you'll get you'll always get lost just say I'm directors I put him had tail was that dry start with what I had to it was the better and that was set up an equation and just solve that even in classical mechanics I believe it was planned showed that the and problem where n is bigger than 2 can be solved because you can't be written down as a simple formula because it shows chaotic behavior it can do all kinds of things and to think that you can write a function for that is very naive so it's just a little bit beyond the scope of functions to be able to encapsulate the behavior it's too complicated and therefore any kind of symbol of way of solving this problem in quantum mechanics is completely out of the question were quite have to guess a very good solution and then we're going to have to work very hard to get a better solution and the harder we were the better it gets the booklets new that's life often works so we're going to have to approximate the solutions let's have a look at this I'm calling hydride and I am trying number 1 I'm not quite sure at this point how many tries were going to have but it might be quite a few before we actually get a hydride and I am that we actually like the 1st try here's what we're gonna do we're going to take time independent perturbation theory which are few during 2nd because I was a while ago and we're going to use it to try to figure out what the ground state energy of the hydride and 9 work on it and use it as to compute the correction to our naive deaths both the Hydro so recall now from lecture that what we did is In perturbation theory we had total Hambletonian which we broke apart hopefully into a big part in a small park but usually what it is it's unknown part but we know already with the energies are in an unknown part and we hope that big and small applies but it doesn't offer doesn't always applied so we have to be a bit careful and then for the Hambletonian we put it on the wave function we get an energy on the wave function and we expand everything out the Hambletonian is age not plus Lander parameter which 1 landed is 0 it's the solve problems when Landis 1 it's a problem was solved where we've turned on the perturbation full force and then we expand the wave function in terms of land that we expand the energy in terms of land he not plus landed the wonderful slander Square D 2 etc. and then we write everything out and we say Look if this is going to be true certainly true when land is equal to 0 because we solved that problem and we know the ground state energy of the known system but said particle in the box of hydrogen atoms he 0 we do know that that's not as not open to question if it's going to solve it for all possible values of land then what's going to have to happen is that the various powers have to match and that's an organizing principle than to get a set of equations that we can solve and as I mentioned lecture 8 we just set the same powers of land on both sides we had to do some algebra right out all these terms and landed there we have to collect them together was a 0 power was the 1st power was the 2nd power and so on and the 1st 2 anyway which is always going to have to use thank goodness for that is the 0 power was just age not Sinai is equally not sign up that's the solve problems so good at that came back if there is no provision of all then that's what we did and then the 1st power we got this and more complex equation based not on side was the correction to the wave functions plus H 1 the perturbation on Sinai which is giving a different answer there is equally easy not on sigh 1 plus the 1 on sign and then we solved that knowing the 1st equation we solve that and we got a very important part of the equation for the correction To the energy he was and that was an integral side not star H 1 the perturbation signed off but the wave function we know now because we've got so we don't do perturbation theory unless we've got a solution for some related problem if we don't we can profitably users so we can assume that we have some functional form for Sinai on and we certainly know what each 1 is we know what the perturbation is for our 2 electron system it'll be the electron electrons repulsion and so calculating the energy astounded doing an integral but unfortunately the animals have to be a lot harder than it seems especially when we have 2 electrons and that is because we're going to have to do with 6 dimensional integral we're going to have to integrate over the coordinates of the 1st electrons which has are a faded and a fly in spherical coordinates call them are 1 data 1 want 5 1 and then we have the 2nd electron are to fight to think that too and in order to calculate this
thing which is the number we have to get rid of all these variables in other words we have to integrate them out get rid of them and that's means 6 integral sitting there and that means we have to know the entire derivative of those or figure out a way to do it numerically 1 or the other and then we have to have a way if we do it numerically of guaranteeing the accuracy because if the roles of especially if they go out to infinity like they do with are they can be tricky to figure out what the functions getting smaller but it is isn't 0 how much areas left under the curve if you stop if you say Look I'm tired of integrating out here is taking too much time but it may be small but because it goes out along way it may be a slightly bigger than you thought and that may influence the accuracy luckily we can do the Senegal's by hand unluckily they're going to be difficult and so I'm gonna spent some considerable time going through this like a tutorial rather than leaving all of it this as a practice problems 2 now without the repulsion turn we could write our solution as a product of hydrogen orbitals just hydrogen 1 s orbitals M other the here I've written that I've written size 0 Our unperturbed function is a product of the 1 wave function for electron 1 which I've written sigh when FAA 1 and then so I went as far too and the product of the way functions means that the energies are added and I've read that in the next equation here he 0 is equal to the 1 s for the 1st electrons plus the 1 for the 2nd electrons and that's equal to minors the the H that the because of equal to 2 times the ionization energy the energies are additive because I the Hambletonian 2 not interacting terms I can break the Hambletonian into a 1st term that only has to do with electron 1 and 2nd term that only has to do with electron too and I can prove that the 2 terms you and therefore the energies add up and that's just what we found with the particle in a box for example now what we have to do however is we have had in our are messy turned the electron electron repulsion are 1 2 in there and unfortunately that Rex everything here's our correction and that we have to calculate and I've written a shorthand here I've written the integral of vector are 1 because at this point that these equations are going to get very long and I will do it systematically but I don't want to write a 3 D integral over defied an odd squared signed dated the Dr and so forth for all these equations because they won't even fit on the slide if I start doing that we have today take our way function which has the corners of 2 and 1 we have to salvage perturbation between and now a very thankful that we've got atomic units because our perturbation is just 1 over are 1 to assist the distance between the 2 electron that's that's operation and we know but we've got formulas for the solutions for the hydrogen atom for the 1 as orbital we wrote those down before and we have to do these 2 3 intervals rolls of 1 after another we have to integrate overall the Hornets Dr 2 and D R 1 and if we figure that out there we can add that to the energy that we got from the 2 to 1 as terms and we should get the energy of the hydride and I that corrected and we know this energy is going to be positive because the 2 electrons repel each other and so that's what we expected and as I said the shorthand D factor are just means txt wide easy don't worry about it at this point I will get to it when we actually introduce all the spherical coordinates and will do the Ingalls properly now however we've got a problem we've got this thing are 1 2 but we're integrating Over either are ones or are too and that means there is unknown thing 1 over are 1 to the distance between them and we have to reformulate that in terms of something that depends on R 1 and R 2 and not anything else that we can't figure out so otherwise while could depend on data fight too but it has to depend on the variables that were integrating over and we can't just leave it as 1 over a R 1 2 we don't know what the onto derivative of status let's fix them and this is a trick that's quite important so let's fix the 1st electrons along the c axes wherever it is we're going rotated back so it's along the sea that's quite important in that wall as long as you rotate everything that won't change the energy at all because the energy doesn't depend on what's north and south reason West it just depends on the distance between the 2 electrons and so always always going put the 1st electron along sea and that way but the angle between the directors for the 1st electron and the 2nd electron is going to be the angle faded in the coordinate system of our integration for the 2nd electron without trick this problem gets extremely messy very quickly and so you have to convince yourself that it's legitimate to do that because when you make an argument like this if in fact it's illegitimate he did something wrong mathematically it changes the energy they don't have a big mess on your hands but you have to be extremely careful cautious at especially when you start our problem if you make an assumption that you have to verify that it's OK before you start doing
the calculations otherwise you waste the rest of the afternoon calculating something that turns out to be nonsense later on and sometimes it's not so easy the see because it's not so easy as saying well 3 is greater than 2 I know that it can be a lot deeper than that to figure out whether something is OK to do or not and it takes some experience some time so what I'm going to do now it is stop at this point because that's enough material for us to digest in 1 go and next time what I'm going to do is I'm going to introduce the coordinate system for the 2 electrons show you how to calculate the distance between the 2 electrons in terms of our 1 and R no matter what they are that could end the functions and then show you some tricks for how to do the integral because even when we get it in terms of our 1 . 2 1 and integrating things with the square root of something in the denominator of something it can be tricky to figure out what the at derivative this and if we didn't do it and a nice coordinate system that we be totally dead we'd never be able to figure out if we did Cartesian coordinates so this is around problem we definitely want EU spherical polar coordinates to solve this problem and not some foreign like box which is not what this problem so leave it there and pick it up next time when we talk about the loss of Kosovo
Chemische Forschung
Elektron <Legierung>
Zellkern
Reaktionsführung
Chemische Forschung
Orbital
Konkrement <Innere Medizin>
Lösung
CHARGE-Assoziation
Chemische Eigenschaft
Quantenchemie
Elektron <Legierung>
Helium
Orbital
Systemische Therapie <Pharmakologie>
Permakultur
Single electron transfer
Verrottung
Lösung
Konkrement <Innere Medizin>
Atom
Oberflächenchemie
Elektron <Legierung>
Pommes frites
Operon
Funktionelle Gruppe
Ionisationsenergie
Weibliche Tote
Systemische Therapie <Pharmakologie>
Atom
Sonnenschutzmittel
Fleischersatz
Operon
Mähdrescher
Gangart <Erzlagerstätte>
Tellerseparator
Verrottung
Blauschimmelkäse
Erdrutsch
Azokupplung
Biologisches Material
Löschwirkung
Krankheit
Orbital
Zimt
Expressionsvektor
Altern
Thermoformen
Quellgebiet
Krankheit
Klinische Prüfung
Operon
Mähdrescher
Funktionelle Gruppe
Systemische Therapie <Pharmakologie>
Atom
Klinisches Experiment
Chemische Forschung
Zellkern
Wasserscheide
Ordnungszahl
Klinische Prüfung
Konvertierung
Zusatzstoff
Hydride
Graphiteinlagerungsverbindungen
Bleitetraethyl
Lösung
Konkrement <Innere Medizin>
Edelstein
Altern
Härteprüfung
Derivatisierung
Aktionspotenzial
Elektron <Legierung>
Chemische Bindung
Helium
Paste
Operon
Gletscherzunge
Ionisationsenergie
Funktionelle Gruppe
Systemische Therapie <Pharmakologie>
Atom
Entfestigung
Physikalische Chemie
Elektron <Legierung>
Symptomatologie
Hydride
Potenz <Homöopathie>
Querprofil
Helium
Graphiteinlagerungsverbindungen
Ordnungszahl
Erdrutsch
Isomer
Blei-208
Herzfrequenzvariabilität
CHARGE-Assoziation
Bukett <Wein>
Thermoformen
Chemische Formel
Mannose
Chemieanlage
Golgi-Apparat
Sonnenschutzmittel
Expressionsvektor
Zellkern
Elektron <Legierung>
Elektron <Legierung>
Hydride
Helium
Ordnungszahl
Hydride
Expressionsvektor
Erdrutsch
Single electron transfer
Elektron <Legierung>
Symptomatologie
Hydride
Potenz <Homöopathie>
Seltenerdverbindungen
Hydride
Lösung
Käse
Altern
Expressionsvektor
Bukett <Wein>
Thermoformen
Chemische Formel
Nanopartikel
Funktionelle Gruppe
Expressionsvektor
Systemische Therapie <Pharmakologie>
Weibliche Tote
Sonnenschutzmittel
d-Orbital
Elektron <Legierung>
Hydride
Querprofil
Orbital
Hydride
Lösung
Erdrutsch
Wasserstofferzeugung
Derivatisierung
Herzfrequenzvariabilität
Chemische Formel
Nanopartikel
Operon
Ionisationsenergie
Funktionelle Gruppe
Expressionsvektor
Elektron <Legierung>
Hydride
Funktionelle Gruppe
Systemische Therapie <Pharmakologie>
Terminations-Codon
Konkrement <Innere Medizin>

Metadaten

Formale Metadaten

Titel Lecture 17. Approximation Methods: Variational Principle, Atomic Units, and Preparation for Two-Electron Systems
Alternativer Titel Lecture 17. Quantum Principles: Approximation Methods: Variational Principle, Atomic Units, and Preparation for Two-Electron Systems
Serientitel Chemistry 131A: Quantum Principles
Teil 17
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 10.5446/18895
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: 0:00:55 Symmetry 0:02:23 Orthogonality 0:05:53 The Variational Principle 0:23:17 Atomic Units 0:34:25 Notation 0:37:29 Hydride Anion, Try #1

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