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Lecture 15. Hydrogen Wavefunctions, Quantum Numbers, Term Symbols and Transitions


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Title Lecture 15. Hydrogen Wavefunctions, Quantum Numbers, Term Symbols and Transitions
Alternative Title Lecture 15. Quantum Principles: Hydrogen Wavefunctions, Quantum Numbers, Term Symbols and Transitions
Title of Series Chemistry 131A: Quantum Principles
Part Number 15
Number of Parts 28
Author Shaka, Athan J.
License CC Attribution - ShareAlike 4.0 International:
You are free to use, adapt and copy, distribute and transmit the work or content in adapted or unchanged form for any legal purpose as long as the work is attributed to the author in the manner specified by the author or licensor and the work or content is shared also in adapted form only under the conditions of this license.
DOI 10.5446/18893
Publisher University of California Irvine (UCI)
Release Date 2014
Language English

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Subject Area Chemistry
Abstract UCI Chem 131A Quantum Principles (Winter 2014) Instructor: A.J. Shaka, Ph.D Description: UCI Chem 131A covers principles of quantum chemistry with applications to nuclear motions and the electronic structure of the hydrogen atom. Index of Topics: 0:00:21 Quantum Numbers 0:02:40 Configurations 0:04:32 Term SYmbols 0:12:02 Holes and Electrons 0:13:47 Shapes of H Wavefunctions 0:17:39 Hydrogen Orbitals 0:18:41 Nodes 0:23:05 Spin-Orbit Coupling 0:24:09 Internal Magnetic Field 0:32:23 Other Alkali Metals 0:37:22 Cesium
higher today and chemistry 131 they were going to talk about hydrogen wave functions quantum numbers term symbols and transitions in atomic spectroscopy remember that the solution for the hydrogen atom gave us 3 quantum numbers and that had to do with the energy at all that had to do with the square of the total angular momentum and so Bell which was the magnetic quantum number which was the projection of the angular momentum on the sea access and then there was another number ends up best that came from the intrinsic magnetic moment of the electron itself and so With we have these for quantum numbers than we know everything that we can about an electron in a hydrogen atom however once we have more than 1 electron then there's the well-known Pauli exclusion principle which you've probably learned from freshman chemistry and that is no 2 electrons can have all for quantum numbers the same so if an L and some all of the same for example the electrons in 2 PX orbital then ends up there so as to be different for the 2 electrons 1 has to be so called up and the other has to be done in other words they have to be paired will see a little bit later what that actually means and so on as I said electrons in the same spatial orbital have to be paired up magnetically with respect to this day let's look at the possibilities for a neutral carbon atoms with 6 electron the configuration from the from the periodic table for a car and that is 1 has to 2 hours to To P-2 and that means that there's 2 electrons in the PD shell they can occupy any of the 3 P orbitals the other orbitals art-filled and the L equals 1 state for P M so that all can be 1 0 minus 1 and we can figure out than the limits on Dell and which depend on each other because of the exclusion principle
let's have a look at this the 1st thing to note is that only the P electrons have any flexibility the 1 hand shell is filled that's out of commission the 2 show is filled that's the commission and it's only in the frontier orbitals
that that we're putting electrons into that have any flexibility as to how the electrons Comorian whatever it's already happened to 1 s and to as has happened and that's water under the bridge for example in the Quran slide 389 we have all these possible but configurations of the electrons in these 3 Oracle's relied on the left we've got 2 electrons in the hands of Bell equals plus 1 orbital and they have to be paired because of the exclusion principle or we could have won in the 1 orbital and 1 the 0 and they could be also appeared a 1 up 1 down we could I wanted 1 and 1 in the minus 1 and we can have them parallel as well as 1 and the 1 and 1 in the 0 all those possible configurations of the electrons in the few workable shells are OK they're all allowed but on the bottom of the slide here none of these are allowed in the 1st 1 we have Isabel equals plus 1 there 2 electrons and they're both that means they have all 4 quantum numbers the same because analysts to Ellis 1 and was 1 amends of base's plus a hat for both of them are not allowed to say anything with the others whenever we have 2 electrons in the same orbital they can have the same remember that we had a shorthand way to keep track of these various atomic levels which was the term symbol which gave us it's sort of a summary of how and as with magnetically coupled to give the total angular momentum J and would allow us to keep track of which transitions were allowed to remember that the term symbol is to ask plus 1 on the left hand superscript capital L which was SPD and so forth for the Central letter and then of value of J which is the way Arsenault couple and they can couple in various ways the various allowed values according to the clutch scored series that we'd arrive maybe up to us plus 1 is called the multiplicity but keep in mind that it is only really the number of states there yes in fact is greater than or equal to ask because if I less than or equal to ask then there could be less states allowed so it s a 0 is called a single this is a habit Dublin investors 1 in Triplett however if we have to ask 1 half where Allen 0 then it's not really a Dublin there's only 1 state because there's only 1 level that for for the electron to occupy so it's not really a Dublin in that case let's try another example and have a look at the Eastman and see which values are allowed so this is practice problems 20 what terms can arise from a neutral carbon atoms With the electron configuration 1 has to do to us to 2 Peter well remember the way we approach these problems we have to figure out possible values as well based on me values of little else each electrons we couple together then we figure out possible values of biggest based on spin 1 half for the electron we couple those together and that we use the platform series together about possible values of J which naively go from El plus process down to the absolute value of L -minus s so if we can figure out these possible values of bigger and bigger as we should be able to come up with the term symbols but what will see in this case is that it's a little bit more complicated than it might 1st appear in the reason why is that some of the values that we'd like to pick up Polly excludes from possibility and the possible values of Big with to the electrons are too 1 and 0 those are the 3 you can get from 2 values of Little equals plus 1 and the possible values they guessed the total spin angular momentum are 1 and 0 that's which you can get from to spend one-half electrons therefore if we just took the clutch Gordon series it would seem that could be well l processes 3 down to the absolute value of the online access is 0 so it would seem that Jake could be 3 2 1 or 0 but we can't have J. equals 3 when we have both electrons in the same shell and the reasons why is that if we had 3 we have to have it all equals 2 so let's put the 2 electrons in and be plus 1 orbital and then to get 3 we have the best equals 1 so we have to
have a parallel that means that we have to have both in the same orbital there and then we have to have them with the same standard that's not allowed because if they're in the same orbital they have their offices spent so we are allowed to take the maximum value in that case if we had an excited carbon atoms and that's why some books give you these weird problems worsen excited where 1 of the electrons isn't to be and once in a key then you just told them all up because they're different shelves and so quality exclusion principle does not imply they are on the same orbital in this case which is actually more important work out because usually want to work out the terms for the ground state of the art and to figure out what kind of spectroscopy you can see you have to be quite a bit more careful how you do it I therefore if we have well equals 2 we have to have appeared so we have as equals 0 the electrons have opposite spin and the term that arises from that is single D so if it's England then that means that J has to be too because there s a 0 2 1 1 answer 0 DE so that the series starts with J-PAL's too and so the term that arises is single D 2 for L equals 1 where OK because then we can have 1 electron here and the other Over here somewhere and in that case we get a triplet key terms and there are 3 values because now addresses
1 and Dallas once a get Triplett key 2 Triple P 1 and Triplett key 0 and then the last term that we can have is allocable was 0 S equals 0 and therefore the only possible value of J 0 and this would be called singlet as what you have to do that is you have to go back and say what were the possibilities of the state's I started with recall that we did that when we did this before we just coupling to electrons we made sure that we accounted for all the possibilities that were there and then make sure that our terms cover all those and no extra once unlike anything I like doing crosswords a pseudo who anything but this takes some practice but it can be quite rewarding puzzle to work out how to make sure that you've got it right that your including any non allowed state in in the turn if you've got an open and shut with multiple electrons on opened the show with multiple electrons working out the allowed terms can really require a quiet room because it can get quite complicated to keep track of which ones are allowed and which ones are excluded because you can't have 2 electrons in the same orbital with the same spent suppose on the other hand we have to fight the say flourish I if we took all 5 electrons and then we said boy this is a mess because all these orbitals filled and so there's these millions ,comma a combinations that are no good because we can't have the electrons that way it would be very very difficult to do and it would take a long time and so the trick we can do is we can't feed the missing electrons the hole as if it were an electron and we can move it around and we can just treat it the same way we would if we had 1 electron and that's because once it's more than half full it's really the holes that are dictating which terms can arise the electrons filling all the orbitals and it turns out that it's exactly the same and so you don't have to do it over there for the worst case is when you've got 3 electrons certified the electrons when she got more than that it just reduces to a previous case where you have less electron so it's no heart and soul sir fluorine the I exceeding 4 oxygen which has to be for for electrons you get the same terms as carbon had to 2 that we tested and likewise flaring gives rise to double P 3 have and Dublin one-half terms now I want to talk about the shapes of the hydrogen wave functions we can plot solutions we had the equations but lost just look at the shapes of the way functions themselves not the radial distribution function but the angular part as well and it's common practice to plot believes that as a contour plots in which we pick some arbitrary contraints space so we say there is a certain likelihood that the electron is inside this contour and then we can pick a color to cold weather at the phase of the wave function is plus or minus remember it's a waiver so it has a face and has a face and 3 D just like the sine wave has a face and 1 day the sometimes on before people use color the markup plus on 1 side of the wave function as an A-minus on the other the problem with using plus and minus is that if you look at it and you aren't quite sure what it refers to you might think it has something to do with the charge but it does anything to do with charge because electrons are always negative and it's just whether the phase of the wave function as a positive or negative and in between there's a node where the where the wave function has to be 0 I the the thing is we tend to avoid using the true I states for the hydrogen atom and the reason why I which I referred to in a previous lecture is that if we include and we've got heated the iron fire going around and that's difficult to figure out what you're gonna plot because it's much easier to say Well I'm gonna plot if it's if if this number is equal to this thing I'm gonna plot that surface where this number is equal to 1 . 9 or something like that but it as you go around the face is changing with respect I then you'd have to figure out some way to cope that that and you might be able to do that by coding in a rainbow but on the other hand that might be confusing when you include the face so as dodge what tends to be done is instead of following the true Oregon State's we linear combinations of plus and minus and so that all that he give the co-signed part or the science park and recall from boilers identity that EDI data is cos they plus I signed it so we can pick linear combinations plot and that's usually what stance we picked then the real part we throw away the eye and we think the real part and we just plot the things however keep in mind that the plots depends a lot on the particular value of the contour but you can do I pick but at the electron has 99 per cent probability of being inside this thing off 50 per cent probability of being inside the thing or whatever usually 90 per cent is a typical value but as we saw the hydrogen atoms if we go out just to the most likely radius of where the electron is the border radius there's only a 32 per cent probability that the electrons inside there so perhaps you could argue that maybe you should take a more conservative but percentage to get a more realistic idea of the size of the spatial extent of the way functions so here are some ways functions then and red is positive face in these plots and blew is negative phase and the 1 as is the little guy
on the left that's a little red things and it just as spherical as we would expect and the 2 West is the biggest fear there's a little tiny red dot the the center and then there's a bigger blue area and that because as we saw the radio function for to us as a node at some point where there's 0 probability of finding the electron and then we get these big fluffy orbitals the 2 PC To the acts and to PY the look really quite big and standard and that has to do course with the particular controversy were chosen to plot the delegates recalled that I know this is a place where an electron were the wave function changes sign is strictly speaking it should just go to 0 but it should be passed on 1 side and minus on the other side and node in a 3 D wave is a surface when the 3 D wave has a value of 0 the radio loads they are are in fact spheres we saw that Friesz states there nothing but radio modems and then the angular nose are angles which turned out 2 planes where the wave function is 0 a 1 has function has 0 nodes at to access 1 radial load and free access to radio and in general the number of nodes is and minus 1 if we let them know that the 2 P C orbital which I have here on slide 398 we see that there is an angular knows there's a plane in which the wave function vanishes it's positive on 1 side rather one-sided blue on the other side everywhere in between the wave function vanishes on the plane z is equal 2 0 the deed functions look like these here's 3 D X Y looking like a big cloverleaf and you can clearly see that there's I I a disturbance building up in the angular variations of the electron is it assumes more quantity of angular momentum that makes perfect sense and so there there 5 3 d orbitals and they are we saw what they were we wrote them all out but we didn't see exactly what they looked like so here's 3 DXY here with these 4 notes and unlike the blue or across from each other and the rhetoric across from each other P changes signed when you rotated 180 degrees this deep changes sign if you rotated 90 degrees and that's because it's too quantum of angular momentum and not 1 here's a different view Of the d-wave functions and how they may add up to give us a shell that has spherical symmetry they 4 of them are pretty much like clover release therefore things red and blue here they're just grace and here there also plotted at a different level of the Contras so they look a lot thinner than in any other plot and then there's this funny 1 3 cents squared which almost looks like P orbital and unlike the others where I can clearly see these planes where this thing is vanishing on this 1 especially without color is a little bit harder to see but what we have with the 3 has said square is we have a doughnut around the equator and then we have this thing that looks a little bit like a orbital but the difference is this thing is all the same color it doesn't change sign and this John it's all the same color and you still look at it and you say Well where are the angles where the notes and the answer is there at the magic angle if you look back the formula for this particular orbital you'll find that there an angle tilted where the wave function damages and advantages all along that angle all the way out and so that's a that's how that 1 works so looks a little bit different but in fact it's exactly the same in terms of vanishing values of an angle just like the other ones too now let's talk about spin orbit coupling we saw that we had these 2 lines for the sodium the spectrum and they were closely separated and that the 2 transitions were "quotation mark P 3 have goes to W S 1 half and W P 1 half goes to Dublin has won him recall Delta J. can be 0 as long as J is not equal to 0 the energy levels and break down like this as I've shown on this figure on slide for old ones there are 2 levels too excited states above the ground state please peace states and there's led by an amount of coal Delta in this slide called the spin orbit couplings and what we'd like to do is explore this a little bit and figure out exactly where this comes from the only difference between the 2 states is the orientation of the span relative to the orbital angular momentum because they have the same value of arson and all but jails difference so it mail can add up like a triangle and make J the 3rd leg of the triangle than Jake can be different because there can be several different values that the work and it must be then the energy difference is just whether this magnetic intrinsic bar magnet of the electron is aligned or against the magnetic field that it sees if from the perspective of the electron the nucleus is going around the other way and it creates a magnetic field that acts on the electron if we know the measured optical frequency difference between the 2 levels than we can estimate the magnetic field that the 3 key electron and sodium appears to be
experiencing from from going around the nucleus of the sodium the magnetic take excuse me the magnetic energy difference of an electron spin is Delta III is is equal to achieve the so-called electron chief factor which has a value of about equal to 2 and then a conversion the format the time that converts magnetic field to energy and then be the value of the magnetic field that the spinners experience that's the magnetic energy that this bar magnet but the electron has is going to feel the let's do a practice problem and figure out what kind of feel it experiences after we figure out exactly what the spin orbit coupling what's the spin or coupling in weight numbers In electron volts and what is the apparent magnetic field but the electron and sodium atoms in the in India to the orbital experiences I'm OK what about the wavelength of the 2 transitions are 589 animators and 589 . 6 centimeters and we know that energy differences Delta using each new ones -minus HUD to and that's HC times 1 Overland 1 -minus 1 too and we can convert that to I always numbers as just times new bar 1 minus new bar to those of the wave numbers and centimeters inverse centimeters and that we can just write that as Delta Nu Bop so we can sulfur Delta Nu bar and we just have to take the inverse of the the wavelength which is in animators to the minus 1 time instead of the 7 animators per centimeter and have to be careful if you're doing an exam you may want to convert from animators 2 meters and meters 2 centimeters it seems easy but it's easy to get also factories 10 thousand offered few go too quickly and happened to go the wrong way with 1 of the conversions if you do that the spin-off splitting and sodium is 17 . 2 8 wave numbers if we convert that Evie which we do but by converting to jewels and then we just simply convert from jewels by multiplying but but by dividing rather by 1 . 6 cents and minus 19 jewels briefly we get a very small value of 2 . 1 4 2 times 10 to minus 3 electron volts keep in mind that the ionization energy of hydrogen to kick the electron completely out was about 13 . 6 electron volts and here were talking to a million electron volts so these magnetic interactions in in these systems are very very much smaller energy differences than the main electrostatic potential that the electron fields from the charge and that that's a common theme basically magnetic energies are always quite a bit smaller than electric energy and in the hydrogen atoms spin orbit splitting is very very tiny compared to sodium and so it took a lot of detailed work even work out that there was something there and then to explain it and it's very interesting historically it's too bad we don't have time to go through all the thought processes that went through finding the fine structure and a hybrid fine structure of these systems but it is very interesting detective work from science standpoint finally then OK we know this spin orbit splitting let's use our formula for this the energy difference of electron the magnetic field we know the energy difference we know that you value of electron we know or magnet on that's a constant we look that up let's figure out what the apparent magnetic field is this select electron and while we get if we take Delta be divided by G 8 times the poor management time is we get 18 . 5 has 18 . 5 Tesla it is an absolutely huge magnetic field if we tried to make a magnetic field like that and in the laboratory and we would be very hard pressed even if we took tons and tons of lions 1 of the gigantic electromagnet we would not get such a big magnetic field in fact the very biggest NMR spectrometer at UCI has a magnetic field of 18 . 6 Tesla but the Mariners 12 feet tall and it has several miles of superconducting wire round into a gigantic solenoid To make this magnetic field which is enormous vacancy that because the particles were so close and things there are moving so rapidly but there are very large magnetic interactions on the electron very large compared to what we can do in the laboratory and so when we take Adams and we put them in a even a pretty strong magnetic field the energy levels do Split and move and do things but they split only a little bit compared to how big their already Split but due to these internal magnetic fields from the nucleus and that's good because that means that if we have to levels like this and then we turn on a magnetic field and they split they go like that and that makes it easier to keep track of what's going on if they went like this and all over the place it might be really hard to figure out but the spectrum was doing if we look at other alkali metals I like lithium potassium rubidium cesium then because they have a single electron outside a closed shop they also give rise to exactly the same terms as sodium and so the spectra are
very similar and they have the same allowed transitions assault on except for the absolute energy differences which can be different because of the differences in the energy of the various orbitals S P and so forth why are they different well different numbers of electrons and their indifferent shelves so of course they're going to be different and it turns out for these systems there's a slightly jaded formula for the energy but that works for the single electron systems when there's a single electron outside a closed shell sorta like hydrogen and you could think of a hydrogen as a single electron outside an empty Shell and in this case we can write e of and the energy of the state with respect to 0 energy just like with hydrogen and which it depends on an and not just as being miners are the river constant divided by quantity and mine Delta square and Delta has to do with the other electron spinning around changing the value the parent value of the energy Delta depends only on not on em and that's important because since it doesn't depend on and if we get a couple of em values we can sell for Delta and then we can figure out for example With an estimated the ionization energy of manner that some non-hydrogen atoms if it's an alkali metal by the this means unfortunately for 4 other atoms carbon or something else no simple theory like this is even going to be closer to the truth and so there too many other problems but at least for a single electron outside a closed shop there's a slightly a different formula in this so-called 1 defect as it has been called historically this has a value that is bigger for s orbitals they know they have a bigger defect than he then and you go out it gets to be more and more ideal because it is you increase the annual momentum you're just out there and so basically what you see is much more like a hydrogen atom because while the other electrons cancel out with all the charge except the extra charge that makes the neutral with this electron here way out there but for S even if you're in a big gas shell you can penetrate again and so then you see that the the energy is not nearly so ideal so I've said that on the slide the US electrons can penetrate and hear what I drawn here is an attempt to try to rationalize why this is so why we have to have this term the potential when you get from the inside looks like a multiple charge because now you have other electrons outside you and so you use actually see this gigantic ubiquitous and so you start dialing down in energy and mass this curve I've marked multiple charge that's with some value that's not equal to 1 and then as you go way out and all the electrons are inside then what you would expect is that it would look like a single charge and on the interior part I've drawn the same potential but with a single charge now not with the multiple charge and then the true potential has to look like the single potential far out but it has a little like the multiple charged 1 as she buried in inside all the other electrons so it has to sort of interpolate between the 2 curves that we've got and in fact that's exactly what that formula basically does for let's look at the emission spectrum now Of the neutral cesium as a practice problems and we're gonna look at it in some detail in fact we're gonna look at it probably in enough detail that we will be at the finish the whole thing in this lecture that will be good because what would have time to digest what we did and then come back and have another look at practice problem 20 to them is there the strongest lines in the emission spectrum of the neutral cesium atoms are at 11 thousand 178 and 11 thousand 732 wave numbers and these lines are also seen an absorption where numbers for related lines in the emission spectra are 73 57 68 0 3 then there's the son Michael that's important 33 21 28 65 27 67 another son Michael 11 thousand 411 10 thousand 900 and the consummate 57 what construct an energy level diagram for the cesium atoms and a sign as many quantum numbers as you can 2 could the ionization potential of cesium the estimated from the state the 1st time we get a problem like this especially if it's on an exam is 1 horrible moment because you start to wonder if you ever had any experience with doing a calculation like this on a multi electronic system at all and you might also wonder if I've got more than 1 electron how do I know that it's only the outer electron that's that's monkeying about what if I start to wonder suddenly may be 1 of the other ones is doing something to and how can I figure out what's going on the answer is it's always the other 1 that's that's going and you never can too excited because if he had to excited the outer 1 would have lionized off
before that and so you'd be doing spectroscopy on the cesium on then you don't have to worry about that is just the outer success in this case electrons that's doing on the various transitions here but we've got all these various lines and we've got to figure out what's going on so we know but the ground state is success 1 but that electrons seasons on the 6th of roads in the 1st column is basically what the periodic table is telling us about the electron configurations but we have those 2 strong line the strongest lines recalled the resonance line the strongest lines and sodium are those 2 yellow lines that's why the sodium lamps is yellow because those they're doing all the emissions there are other transitions going on but there among the less intense the fact that those lines the scene in absorption In absorption you have to start on the ground state you don't start with the electron up here because if you have the electron up here and it's an absorption the temperature would have to be extremely hot and the temperatures not extremely high and the temperatures just whatever it is and so we know that those 2 lines it terminates In the ground state of the ad and that's an important clue To help was why are there 2 of them while we go back to our turn symbols the term symbols for cesium are the same as sodium so although this seems hard is actually not that far hard because we know the ground state doubtless 1 there is a single electron outside a closed shop and we know that the 2 excited states but in the 1st 2 there R. double P 3 has and Dublin P 1 half and we know but there's splendor and the splintered by the spin-off that coupling and we know the Wolf we've got a very big positive charge on the cesium that that that's been coupling should be better and if we happen to remember oftentimes you do if you're under pressure but the Soviets splitting was 17 we expected some splitting that's going to be quite a bit bigger than 17 when we had this enormous cesium charge moving around that my goodness I can make an enormous magnetic field at the site of the the electron and therefore these residents lines give us the following picture which I shall neurons line for 10 we have that too "quotation mark P-3 have stumbled 1 have states and they are 6 p the other mistake you can make is you might Markham as 7 p there no the 6 p that
they this experience success in cesium have a huge energy difference because the success can penetrate the other levels and the 6 he hasn't node the nucleus of the 6 p electrons hardly ever sees the cesium nucleus because it has a 0 there and when you look at the probability of 2 square so it's far less likely the vast has a finite probability of being right at the nucleus and in fact in some cases as electrons actually get gobbled up by the nucleus and that turns out to be a mechanism for radioactive decay cold electron capture which is just like what it so we can drive to transitions 1 of them is 11 1 7 8 for the law was the other 1 is 11 7 3 to wave numbers and the difference between his 554 wave numbers which is a very very huge value but maybe not surprising because cesium has a lot more charge than the so in that OK I'm going to leave it there and we're going to digest this problem and come back to it in the next lecture meanwhile we have to think why do we have the groups of 2 and groups of 3 I think you can understand where the groups of 2 are going to be coming from and then how can we figure out but these random collection of lines that we get and where numbers actually means in terms of the energy level so we have to have some way to kind of sort of and what we're going to use to sort them as you'll see in the next lectures were going to use this value of 554 554 is going to be our flashlight in the dark were gonna keep looking for things that differ by 554 if they do that means 2 things that means that these 2 states these 2 two-piece states are involved in the offense and that whatever level came down with these peace state was allowed to make a transition to both of them so we have a common level and it's going to these pieces so leave it there and when we come back next time I will try to do is fill in all the other energy levels for this that and make all the assignments as we asked to do and then will also try to see if we can estimate the ionization potentials of course if you get a problem like this on an exam which I actually didn't you don't say Well No you can't estimate the ionization potential use and you can and then you figure out how to do it To will come back to that next time
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