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Lecture 20. Hartree-Fock Calculations, Spin, and Slater Determinants

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we're gonna continue and chemistry 131 at the and
today we're going to talk about heart she calculations spin and Slater determinants we left
off by the way in introduced Douglas Hogg tree who devised a method to solve the Schrödinger equation for an Adam the starting point for the helium atom is to write the wave function for the 2 electrons system as a product of 2 1 electron orbitals that's the essential simplification and as we've seen earlier when we went through this calculation before looking at more generally and this is a somewhat flawed approach will see exactly why it's flawed in in a minute but it's much better than nothing and according to our interpretation of probability density the probability density of electron to then is just five-star are 2 five-star or are too Dr to that's the probability density that you're going to be in that region of space and rather
than attempting to solve the equations in 1 go we instead converge this the 2nd electron In 2 just a classical charge distribution that's the essence of the simplification when we have the 2 electrons were really they're both buzzing around we have to keep track of both of them are in some sense and that means a faraway function has both of those coordinates in there it's 6 dimensional and it's very hard for us to understand exactly what it is but if we say well they it will will average the 2nd electron out into a blur and will assume we can kept calculates some time average property like the stable energy of the atom by 1st averaging about To a charge distribution and then solving for the energy of the 1st electron in this charge distribution to do that then what we need to do is we need to figure out what the energy is the average energy of interaction between the 1st electrons and the charge distribution but we can do that because we can calculate the effective potential as a function of R 1 the position of the 1st electron by doing this integral can average five-star are too 1 over are 1 2 sigh are too and we can integrate that and that's a function and that only depends on R 1 because in the differences are to minus R 1 R 2 is integrated out so we get them an effective potential that only depends on where the 1st electron years you can imagine the 2nd electron assembler it gets last you go farther away from the Adam and if the 1st electron is out on the fringe of the blur it doesn't have very much repulsive but potential energy pushing away if on the other hand it's that at a position where the blurred as I got a heavy density than not but it feels a lot of repulsive and that the energy goes up and it may tend to try to avoid of that particular region With this trick them of integrating out the 2nd electron we get a one-dimensional problem which is just like all the other one-dimensional problems that we've done the hydrogen atom and the particle in a box the main thing is that we've got something that depends just on the coordinates of 1 electron that's the main simplification
let's look at how we would implement this Stanford helium we solved helium and hydride guest various wave functions tried a variation all approach perturbation theory so on and so forth but say we wanted to take a systematic approach this is going to be very good then for bigger Adams where were never going to define such a good gasses we may have been able to do for a to electron system we can write in atomic units for helium and but the effective Hambletonian which depends only on the corner bar 1 here is the kinetic energy electron 1 as minus have dealt 1 squared minus 2 over R 1 because he is too there's cause to charge for helium plus this the effective for electron 1 of our 1 which has this interaction with the cloud of electrons to that that that is it and this looks very much like any other kind of one-dimensional 1 coordinate problem it's a three-dimensional problem of course but in actual fact sense is going to have some sphere it's only going to depend on R 1 is the main thing so it's not going to be multidimensional the key is that the the 1st electron sees only the average and not where the instantaneous positions of the 2nd electron and that that's a key difference here well what we get
then is we get an equation that's very much like that for the hydrogen atoms we then have this effective Hambletonian which has the kinetic energy the attraction to the nucleus and the average repulsion with the 2nd electron that is our Hambletonian we put them in some way function we wanna get energy which called Epsilon 1 times the wave functions so we wanna find the Eigen function for that but there's a the 1st catch is that we've got to know the orbital flight to for the 2nd electrons before we can know the effective potential for the 1st electrons and set out the effective Hambletonian for the 1st electrons and then saw Florida and we don't know the wave function for the state of the orbital from the 2nd electron the 2nd catch is that the 2 orbitals are the same for the spatial para and therefore what it amounts to is that we have to know the 1st orbital To Know the 1st that same circular and it seems like we're going to be completely stopped because we need to know something to get what we need to know but in
fact in a lot of things like that you can solve things like that iteratively you can get something and then go round and round and that's essentially the essence of the self-consistent field approach we get something reasonable we calculate an effective potential and then we iterate Salesi how how how we would do this where demanded get something we're going to calculate an effective potential and we're going to put an index on our guests we're going to get some function it could be just an exponential function parameters Ada could be anything we want we're gonna put that in calculating effective potential within index that as our 0 guests doesn't matter what we call it but we want to keep track of that we have a function we put it in we then use this to calculate the effect of potential then we got the Hambletonian and then we calculate this energy Epsilon 1 for the Hambletonian and we then also get an updated orbital unimproved orbital which solves this equation based on the potential for the old orbital this new orbital were gonna call 1 superscript 1 that's the 1st interim and then using
but so this is an interesting approach but the question is this How do we know but the orbital that we get at the end let's say for helium is any good and the question is is it any good will it turns out that the variation of principle helps tell us that's what we get with the self-consistent field approach it is extremely good and if we calculate the expectation value of the energy In the orbital approximation we get this equation furry that I've shown on slide 500 and Our Hambletonian for helium I've broken up into 2 parts per 3 parts Excuse me the Connecticut Energy of and potential energy you 1st electron kinetic energy and potential energy the SEC electrons and then the intellect electron repulsion
and that breaks up into 3 terms because when we do the integral if the variable only depends on electron 1 then nothing happens to the wave function to so we get the arrival of start to fly but we assume it's normalize the that's what just out of our hair we get 5 1 which is the integral the 1st electron coordinator and we get I to which is the interval 2nd and now we get this thing I've called J 1 1 which is the integral 0 5 1 start a to stop 1 over 1 to and that's a repulsive term that we've seen before when we were setting up helium and hydride and in fact this integral is called the Coolum integral because it looks exactly like to to charge distributions repelling each other if
we write the energy them as equal I 1 plus side too plus J 1 1 then the exact hot tree flock equation results if we minimized the energy with respect to 5 and what that means is that the very occasional approach is telling us that if we want to find the best orbital this product of orbitals approximation than what we should solve is exactly the equations that Douglas archery suggested that we should look at and that means that in the orbital approximation this is going to be the best we have to be careful go with the hot tree frog energies and that's because these energies when we solve we solve for H 1 sigh 1 is equal to Epsilon One fly 1 and St. for fight to In the helium they're the same another Adams they would they would be separate things depending on which electron we're looking at if we don't add up those energies those energy eigenvalues Epsilon One and epsilon too we don't get the total energy of the helium atoms because what we get is we get 5 1 plus J 1 1 that's for Epsilon once and that we get right to plus J 1 1 again for that and 1 too and that's not equal to the total energy because the total energy as I want plus I two-plus J 1 1 In
fact what you can show them is Epsilon 1 is equal to the total energy the Adams -minus I too but I too but you can go back to slide 5 0 2 is nothing more than the energy of a helium ions only calculated with the orbital solved for from the helium atoms the that's what that means that Is that Epsilon 1 isn't in fact a direct approximation for the ionizing ionization energy of the helium atoms mother words minus epsilon 1 is equal to the ionization energy Of the helium atoms that's an approximation because it assumes that you can use this orbital but you solved for for the helium atoms to figure that out to figure out the energy of the helium ions just by by taking the other electron away yeah what this is called is Krugman's approximations and it all it does is it assumes that we can use the same orbitals in the ion Indiana which is not quite true but the fact that it's even close at all indicates that the orbital approximation and the self-consistent field approach is reasonably good for a lot of Adams but now I
want to talk about something called the price that we pay by dividing these things up in treating them 1 at a time is that if you treat things 1 at a time then you can never have both of them interacting together because you're treating them 1 at a time when every treating this like the other ones of aboard when you're treating this 1 this was a blur and that's a flaw because in fact the real thing they are avoiding each other at all times that means that they tend to do stuff like this amount stuff like that and that's impossible to take into account because we don't let the other 1 exist if we had a lot of electrons all at once In the and they're all doing stuff I think you can see it gets extremely complicated the wave function is depending on all these variables we don't know what it looks like these electrons avoiding each other all the time I was never going to guess what the wave function In effect it would be very difficult to even have a systematic method of organizing our thoughts of even the fact that when we build up the periodic table we talk about electron configurations 1 S 1 1 as to were talking about orbitals there but if we don't actually have any orbitals if the atom is actually far more complicated than that than were actually letting on it is just a giant wave function then we've got problems even understanding how we're going to organize our thoughts so the orbital approximation is incredibly important but but we what we have to be able to do if we want to get the energy better it is we have to take into account that there is some correlation the electron correlation I'm going on in the system In the Heart tree equations the motion of the electrons is called uncorrelated because that's what it is we just bought a 1 we never asked what the other 1 was doing when we did that because it doesn't matter for treating them independently so if they are correlated were getting rid of them and therefore the correlation energy this is what we left out and that is just the difference here this equation the correlation is equal to the exact energy whatever that is could be measured by experiments or by some other kind a calculation that gets a better energy -minus the energy In the orbital approximation by doing the hotly fought treatment for helium something simple like that we know the exact energy and we know the hearty soccer energy
and the coral at what we find out is the correlation energy for helium is the exact which is minus 2 . 9 0 3 7 and in hot trees minors what we calculate From the Heart follow-up limit minus 2 . 8 6 1 7 are trees or we get minors .period for 2 cartridge and unfortunately but energy and hard trees atomic units is a huge unit of energy and so on we converters Indian units that are more unfamiliar to chemists we get minus 110 killer jewels from more that means that although the poetry Fok energy accounts for almost 99 per cent of the exact energy the air that we've got and in doing this calculation is way too large because it's roughly on the energy of a chemical bond what we're saying is if we I have an approach like this we can't tell if we've got a long doorway or we don't have a bond we don't have a fine enough magnifying glass to be sure the we've got the energy down to where we can tell that something is there or not there that's very bad and that's because a deck it's worse to with bigger Adams and the reason why it gets worse is that we have a very very big energies for the 1 s and 2 Western so for orbitals but but they are playing much role in the chemistry that chemistry has to do with the electrons that are in the outer orbitals that can make bonds and so forth and so on so we have this huge saying with a lot of energy and we've got this subtle thing that we're trying to look at and we if we want take a submarine and then a submarine plus me and tell the difference between that we have to have a scale that has a lot of difference in that a lot of digits because the submarine Saddam had that we need a measure way way out there to tell whether I'm on the submarine or not in percentage terms we've gotta get the mass of the submarine to 99 . 9 9 9 9 9 percent accuracy and that's the same problem that we've got here calculating these kinds of energy users that they get big and so the differences can be big even if we think our calculation is pretty accurate this to summarize this is a common problem and so-called chemical accuracy In doing any kinds of these computational studies is very hard to obtain usually what you try to do it is I have 2 systems that are very similar in a lot of ways and then you have 1 thing that's different and you're hoping to calculate the different things and what you want you pray is that the things that are similar if you make mistakes or approximations or of some kind there are about the same for both sides and so they just go out whether you got them right or wrong doesn't so much matter because you're making comparisons so rather than making some absolute things saying the energy of this molecule is right here what you may be actually more interested in anyhow is how how does it compare To this isomer that for that configuration and you may be able to do a lot better on that because a lot of the things will be the same and therefore when you calculate them accurately or not as long as you make the same kind of mistakes on both of the sides it goes out
now let's take spin into account because we haven't talked about that we've been we've been really glossing over the Polly principle namely that the wave function be antisymmetric under the exchange of the 2 electrons How are we going to take that into account well we could like our book does we could use the superscript Alpha I probably won't stick with that because I I don't care for that all the time or beta did indicate a wave function for an electron that's either spent or spend down and you can also right that is just a spatial part kind of spent part taken think of the spatial part as the probability amplitude you can square to give the probability density and you could think of spent part is just a bar magnet riding along and can atop down and that's we really quite simple picture of what's going on and they have nothing to do with each other per saying other words the span of coordinates this up-and-down orientation of the magnitude and there is no function of space for that that's something that originates from a point from the Electron itself and in the office state if we measure of the spin orientation led by convention Lizzie component is a over to and from the Bay stay busy component is minus H bar over to otherwise identical because
there's only 2 values of the spin we can represent the spin rather than having a function like a wave function where we we have this density all over space we can just write the span as column vectors and we can write here everyone else it said it can correspond to a column vector with the 1 in the town and a 0 in the bar and data it can be a column vector would 0 in the top and 1 in the body these 2 by 1 column vectors can then represent the state of the the spin in these problems when we want to integrate the spin we have no integrated over rather what we do it is we rearrange the column so it's like a role here I've written Alpha's Star then is equal to 1 star which is 1 because there's no imaginary part 0 as a role data star is equal to 0 1 star which is just 0 1 as a role then when I want to take the product and figure out that I'm normalized I just take the role times the column like matrix multiplications so I follow the entries the role 1st entry in the road times the 1st entry in the column plus a 2nd entry in the row times a 2nd entry in the column and if it's normalized that should be 1 and if it's not normalized it's something else and if the orthogonal it should be 0 this can be
written them in this notation which is very common which was introduced by called Iraq and is extremely useful especially for spin in that we instead of from having an integral both sides star sign DX or something like that we have this angle bracket with alignment between so-called inner product broadcast and I've written out then what it would be for alpha alpha it's 1 0 times 1 0 which is 1 times 1 plus 0 times 0 which is 1 beta is just the other way that comes out to be 1 as well that means that our spin states often data written this way are normalized and then if I take the inner product with on data I have 1 0 0 times 0 1 1 time 0 that's 0 0 times 1 that's 0 so that because they're in different positions in this BUY 1 :colon there automatically orthogonal therefore the spent part if I write like this is automatically normalized and orthogonal we can use
an index then to keep track of which electron were actually talking about if we have alpha and then we have a parenthesis ones what that is is the shorthand notation to say Look Spanish the electron calling 1 it is then on the bar magnets on electron 1 therefore in the orbital approximation we can write the shorthand way function so I 1 ,comma too which just means the 1st electron the SEC electron this is quite abbreviated could we we don't even have that are 1 we get tired of even writing or are we just understand what we mean 1 s also won 1 has offered to the beta to for example what is that me that means the spatial part of electron 1 is 0 1 as type orbital heeded the miners on R & B spent part is up for electron 1 and for the 2nd electron is also on a 1 as spatial orbital same and the Spanish down that's all it the electrons however are indistinguishable all electrons are identical and so we could equally equally well right side a 2 1 is equal to 1 has offered to 1 as beta 1 the area 1 of these would be In a cost 1 we've got both
combinations what you learned quickly in this game is that when you've got 2 choices in quantum mechanics you always use both of them some you don't choose 1 because there is no reason to choose 1 over the other so you have to use both of them and the question is that what combination do you choose well there's a symmetric combination so I 1 2 plus site 2 1 and that's 1 Celtel 1 1 has been added to plus 1 offered to 1 has been 1 and there is an antisymmetric combination but sigh 1 2 minus side 2 1 1 SL 1 1 spirited to minus-1 itself 2 1 especially 1 but the 1st combination is symmetric the 2nd is isometric and neither 1 is normalized
let's then as practice problem 25 normalized by just following through this little spent algebra exercise let's normalize the antisymmetric orbital which are called site sub 84 antisymmetric so here's the answer we can assume the spatial parts are normalized the 1 as part is normalized and all we have to do is figure out the spin parts therefore what we have to do is figure out the inner products of the spin parts and then make sure after we include the spent parts that the integral of sigh star and they cite a is equal to 1 1 we integrate the whole thing up using the
shorthand notation here which gets a little bit longer the integral of size start size is equal why get a ride out those 2 terms the antisymmetric part started while the 1 s is real the other part then this factor rises in this case it does not always factorize but in this case it does into 1 dated too on Alpha 1 dated 2 times a double integral of the space part which we can assume is 1 -minus well-versed in terms so we're going to get you know 1st inside-outside last minus Alpha 1 dated to beta 1 altitude 2 -minus Beta 1 al-Sadr to offer 1 bit 2 Plaza they 1 off the tube anyone else to and since the space pods just integrate 1 I don't have to bother with that and I know that the 1 answers are already normalized and that an up with these 4 spent parts and you can either write them out as 1 by 2 vectors to buy 1 common rule or you can just say Look if its Alpha 1 on Alpha 1 if those arrested sandwich that's 1 if it's powerful 1 on bail was then that means that it's 0 because alpha and beta are orthogonal and of course the sand which applies to the EU the electron because yeah yeah you don't take on things that have different electrons and compare the spent parts because those are irrelevant what you wanna do when you normalizing as compared to the same part of the functions and if we do that the 1st terms 1 minus the second-term 0 the 3rd term minus is 0 the 4th term not surprisingly is plus 1 and so the whole thing is plus to with that and we did this before but it's nice to do it again just to review the enables 2 rather than 1 so the normalization constant it is just the square root of 2 Over 2 you could write 1 over the square root is too but it's nicer to so-called rationalize the denominator that means if you have a fraction you don't have imaginary numbers and the denominator and you don't have square roots and things like that in the denominator if you can avoid it you keep them all in the numerator instead where it's easier to see so you want rational number In the denominator not all books and stick to the principle however the only I wave function with the correct path to satisfy the public principle is the antisymmetric spent part of the symmetric once no good and therefore Our wave functions that should be route to over 2 1 4 1 1 as 2 -minus 1 as too 1 they don't want great bats pretty easy to do for 2 electrons the question is what we do if we've got 3 or 4 5 the whole electrons and if we swallow any 2 of them not just the 2 and the 1 s orbital or something like that but if we swapped any too number 1 with 1 28 we swap what we're calling them an our equations we swapped the labels the Pollack tells us that the wave function should change sign the overall wave function on that gets tricky them to think about what combinations you're going to use and at that point you know you really enjoy having a mathematician down the hall to help you figure out something that could be but useful and in fact John Slater devised a very ingenious and very compact notation to solve this problem which is now called the Slater determinant to show you how important it is he got his name on the thing let's have a look recall
but if you if you haven't had a course in linear algebra you may need to review but what's new throughout this course sentence saying there various techniques and mathematics that you need to be able to use like you need be able use wrenches screwdrivers wouldn't take something apart and here we've got 2 by 2 determinant it's written like a matrix but a matrix has brackets around a determined has straight lines and you have to look carefully sometimes it's something to see what object it is because a determinant is just a number if it as numbers and the entries and matrix is not just a number a matrix is a matrix with a whole bunch of dimensions to it we fully right then had to buy to determine how one-by-one determinant is just a number nothing new there I'm if we're to bite determined with a and B. and the end the topper and C and In the bottom row then you do a crisscross should take a deep -minus he said Of course it doesn't matter whether you right at sepia BC I'll keep the alphabet in order that's that that's the determinant of a two-by-two matrix and if we swapped the Rose so we put CD at the time and baby at the bottom what we get is we get CB well into the crisscrossed minus a D and that is minus baby financed BC and therefore the determinant as this interesting property at least for the 2 by 2 but if we swapped 2 rows it automatically changes sign this sounds very promising but if the 2 rows are identical yeah they're both the same entry then we get a B-minus being a we at 0 and that could be very useful to because that could mean that if we could set this up as electron orbitals but if they were all the same the the wave function wouldn't exist we would allow that wave functions and we label them as the Rose than if we swap into rows it changes sign automatically force and arrested is and down to figuring out how to do it
OK on slide 515 I've written kind of an intimidating formula for a 3 by 3 determine the top ABC the middle rowers DVDs bomb rose GH these are just all numbers so they could be functions of the could be anything but the key is I go through them you start on the top left and you take a and then you block of the row and column the ayes and you block out the column under a which is D & G and you block out the role which is B and C then you've got to buy 2 left which is easier for a child and you write the determinant is 8 times the 2 by 2 determined he asked then you go to the 2nd round entry on the top row which is B and because of its the 2nd 1 rather than the first one you put a minus sign in front of me and then block out the column under B which is he in nature and you block out the role that the and which is a and C. and you write to buy to determine DEI she asked and then you go to the next 1 over and takes C and because you the last 1 was minus you put a plus and then you block out the wrong columns and you have the 2 by 2 determinant DH on and the GH which is just DH minus Cheney you can I take any determined that no matter how big and you can grab the 1st entry and block out that call me and say this determination of this whole thing is this thing times the determinant of the smaller things which is now if this was followed by 5 this is now 4 wife or and then you go across the road so now you have 5 4 by 4 and then for the four-by-fours you go across the Rio and they collapse and the 3 by 3 days and then you go to Dubai to use strikers crisscrossed formula and you're done and boy do you get a mass of terms which you but right out this huge thing of all these things with miners and plus minus and plus and so on I the determined itself just writing them in this array is such a brilliant compact waited to do it you don't have to ride out all this stuff you've got I'm all right there it's it's really very very slick let's see how we can use that as well I we can prove with any determinant of any size that if we swapped to Rosie changes sign I won't do that here but that's a fairly straightforward thing to do and as I said any size determination can be decomposed into on size and minus 1 until you get down to the 2 by 2 they just not then on this
landslide 516 I'm sorry it's come a little bit too small to read if we've got a closed shell added With 2 electrons to an electron after the mayor spent about half of them were down and there and spatial orbital and opens shell electron has unpaired electrons in different orbital those kinds of systems are are harder to talk about the let's now do something like helium or beryllium on the on or something like that that's not a not an open shell and there's still quite a few Adams that weakened that can be written as a single Slater determinant but as you can probably guess the other ones that are open shell I have to be written as a bunch of Slater determinants and that just gets more complicated and what we do then it is we take the 1st spatial orbital and we put it across I would we take the 1st quarter and we put across all the possible spatial orbitals so we write 5 1 L 4 1 5 1 beta 1 fight too L 4 1 5 2 they don't want and so forth and all the 1st row refers to coordinate electron 1 and then in the 2nd row we have everything the same except not refers to electron too In the 3rd row here refers to electron 3 and as we go down the columns in the in in the column it always has the same function coming down and spend state so column 1 always has functioned 1 and spin state up and then you just referring to the different electrons as you go down and because you have to and electrons it's to end by 2 and determinants and the normalization but which again is an interesting problem to work out which I won't do here is 1 divided by the square root of To and factorial that because of the way you end up with a factorial number of terms in the Slater determinant the beauty of this thing is that if we swap into roles the changes and you can think of swapping roles in 2 ways you can think of swapping roses grabbing wrote to and putting it in 1 and grabbing role 1 and putting it into or you can think of it just as he broke 1 there but change the 1 2 at 2 all the way across and keep wrote to their and change the two-door what and the latter is just changing the label so were using to label the electrons and police's says when you change the labels like that the wave function has best change signed and of course the probability and the probability density has to remain the same practice
problem 26 now let's write down the slate Slater determinant for the helium well there are only 2 electrons so I determinant is a 2 by 2 the following our formula on the previous slide so I want to it's route to over 2 straight lines 1 herself 1 1 as Spadea 1 2nd Round 1 has offered to 1 aspect to and then we just work out the determination and we get the exact same thing we had before 1 S 1 1 has 2 times alpha-beta minus Beta Alpha the Tennessee is a
shorthand because attended not want to ride out so many things so if you see alpha-beta something like that what it means is the 1st electron a spin-off of and the 2nd electron spin Bader the person's got so sick of writing Alpha bracket 1 that they've got carpal tunnel from doing and they just write Alpha and they expect you to understand that if the office and the 1st position then it means the 1st electron so the ordering is what it's all about and argues that common shorthand here as well and 1 s in this context for helium and means an optimized 1 has orbital but with the best value of Zeta and not just a simple hydrogen 1 of orbit OK it will stop their next time what we'll do this will actually set up but not solve or I'll show you that there are resources online where you can look at these things out 4 while the electron items that have a lot of electrons in fact all the way up to see even and you can go look at the orbitals that have been calculated boy has a lot of work put into them and a course that's a great way to start to build molecules is to have animals that have optimized orbitals so we'll leave it there and pick it up next time on the speaker
Chemische Forschung
Atom
d-Orbital
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Atom
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Zellkern
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Aktionspotenzial
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d-Orbital
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Helium
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Konkrement <Innere Medizin>
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f-Element
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Funktionelle Gruppe
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Atom
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Chemische Forschung
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Tube
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Sand
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Metallmatrix-Verbundwerkstoff
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Sepia <Pigment>
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Erdrutsch
Konkrement <Innere Medizin>
Chemische Eigenschaft
Bukett <Wein>
Chemische Formel
f-Element
Gletscherzunge
Funktionelle Gruppe
Mühle
d-Orbital
Elektron <Legierung>
Helium
Isotopenmarkierung
Orbital
Alphaspektroskopie
Erdrutsch
Beryllium
Bukett <Wein>
Elektron <Legierung>
Chemische Formel
Chemische Formel
Helium
Tonschiefer
Karat
Funktionelle Gruppe
Beta-Faltblatt
Systemische Therapie <Pharmakologie>
Atom
Erdrutsch
Zetapotenzial
Elektron <Legierung>
Elektron <Legierung>
Helium
Molekül
Alphaspektroskopie
Orbital
Computeranimation