 So let's wait for everyone to show up. So I guess the, this was the e-bots. We had to have this like a round table discussion with maybe like some questions that were raised during the meeting. And, you know, kind of get the input from everybody on. Kind of for some of these like longer term planning is like, are there some like things that we want to discuss some like potential directions in which this one ecosystem is going. And we can have some questions. So one was the question of the like symmetry. So the question was why do money functions in the max localized scheme why do they break symmetry. It looks like there are some ways to kind of improve on this. We had some talks. And then you know we had many other questions. So there was a question of, is there anything that could be gained for from non orthonormality of the money functions if they are not orthonormal. Then on Monday and Tuesday in the summer school there was a question of descriptors. I think after Nicholas talk, Raffaele's talk, there was this question of, you know, can we get some meaningful descriptors out of any functions. And how do we deal with this choice of gauge in that case, or is that even a problem. I didn't think Raffaele had this comment that like a lot of the chemical intuition. So once the chemical concepts are not really like, maybe strictly formally mathematically well defined and seems to work just fine. Then we had questions about this idea of when you compute the position matrix element of diagonal and diagonal and then in the diagonal case you use the logarithm formula. And how do you and then you know that for the diagonal you don't so then you know is there some way to fix those. There are certain problems with translation variants that appear. Anyway, so there are a lot of questions so I don't know if anybody wants to start by giving some comments on this or maybe raise some other questions. Please do. Yeah, I think this works now. Yes, I think we should also keep it as an informal conversation. So maybe we go through this topics. And also, David can hear and is a muted. Probably there should be one more microphone around so that people can feel and comment. But maybe we can start with the symmetry David what do you think. Sure. You know there is the possibility that you hear two very different opinions. So I'll give David the chance to have the last word that's usually it is the last. It is the last word. But you know I would even go back for a moment to molecules rather than solids. If you think at molecules, you can have different localization criteria, you can have the foster boys criteria and that is the R square minus R squared, or you can have the admin son Rudenberg criteria and that you maximize the global economic self interaction. And you get not only very different symmetries in the orbitals but also very different chemical intuition that if you want is a negative statement about the fact that there is in my opinion, actually, no symmetry that natural naturally means from localization or no chemical intuition that you know naturally means that say formally comes and then there is a lot of heuristic understanding but I think we made the case of CO2 last week, whereas CO2 localized the foster boys or localized admin son Rudenberg give rise to completely different orbitals. You know something that looks like a triple bond with foster boys that doesn't make any sense in molecules like, you know, ethylene a CH4H2, you get the banana bond so that the chemistry don't look like, don't like. And in molecules, you know, you don't have a, you know, special unique universal physically meaningful representation, and even more so in solids because now you're not only mixing at a K point that would have been gamma, but you are then integrating all over the zone. But I David I don't know what you think on this. Well, I guess I would start at a slightly more elementary level I mean we certainly know lots of cases where the symmetry will get spontaneously broken by the maximum localization I mean the simplest case is just a closed cell Adam with s and p orbitals. So if you start out with s projections and px py and pz projections, you're going to minimize to sp3 hybrids. You know, as we know for finite systems the minimization basically just tries to push the centers as far apart as possible. And so. So certainly that's a case where you break symmetry. I think it might be useful to distinguish two different kinds of broken symmetry there are situations where. You know, suppose you just had a one dimensional s orbital and px orbital, you know, and then they break symmetry to form an x, px hybrid and a p y hybrid. If the two hybrids, if the two one a function still map on to each other under the symmetry operation, then, you know, it will still produce a perfectly symmetric representation when you do one a interpolation and so on. So that's still acceptable, although you have to understand that you're not quite dealing with atomic orbitals, as you put them into the projectors. And then there are cases where it's just going to break symmetry in some kind of random way and scramble things and that's I think what we want to be more careful to avoid. So that's my initial comment. I'm from the audience. Hi Raffaele. I don't know if your camera works but you're super welcome also to. What do people in the audience I think, or, you know, I think, maybe the question is also, when is symmetry very relevant. And I think all the people building low energy amiltonians care a lot about symmetry. As you know I'm obsessed with Cookman's functional and orbital density dependent functional, but again there you know we have actually, if you want an intellectual failure that it's not evident that the spectrum of orbital density dependent. I'm Eltonian as the symmetry properties that the spectrum of a real amiltonian should ever. So in some ways is also related to some of this other development so part of the question actually was even just practical, like why. Okay, so why didn't you implement to begin with my 90 using symmetry that is, instead of summing over all the cave actors in the brilliant zone, just using the irreducible part. So that question that there's a very deep answer. And it's the fact that I knew the code that I was using that was a derivative of cast step, sort of what we call the old bands ensemble the ft code. It was a calculating gradients with respect to everything. And somehow cast step was built, you know, along the goals of doing a large scale simulations. So, you know, maybe in the limit of gamma sampling only but still with k points like the carparinello code that was developed here in the east actually at gamma sampling only. Well, the pwcf code was built with symmetry from the very beginning, but I would say it's just random I mean the code that I was using was was built without symmetry. And I think we discussed at the beginning because David with Dominic King Smith and others have developed, you know, the first ultra soft so the potential code, rightly so but we said that maybe it's better to start not conserving it might be a bit easier. And that was absolutely true actually. So, and so, so that's it just by chance. I think that we had the time reversal just for simplicity but I don't know, David. I think just historically is that, you know, the attitude was that the computer time to do the maximal localization was going to be trivial compared to the computer time to do the DFT calculation. And so we just weren't worried about it from the point of view of computer time of course, maybe we should have been worried about it from the point of view of preserving symmetry but it was sort of two early days for us to worry about that I guess. And just as a side note, you know, some of the interesting thing about breaking symmetry that is actually a numerical broken symmetry is that when we were first doing calculations say with silicon or gallium arsenide. And if you were to do a monk or spark measure that was unshifted containing gamma everything was symmetric but if you would do the shifted measure, you would actually have a numerical breaking of the symmetry with one bunny a function having a center and a spread that was likely different from the other. I think David explained me to me why that was and I keep trying to remind myself why that was, but I remember the phenomenology of this. But, but so, so if I do a calculation in. You know, like, that could plot spread as a function of all of the degrees of freedom in the gauge, which is you know huge number of dimensions if I have 10 by 10 bands with 500 k points that's like 10 by 10 by 500 degrees of freedom in that space. You know, I could have a situation where, you know, so this is my gauge space. And this is my spread. And so maybe there's some point at which, you know, a gauge is symmetric, but there's some choice of matrices where gauge is not symmetric. So, I mean the situation that, you know, we are like this, or like this, you know, so both of these cases, the gauge is symmetric. So, you know, in one case is over here so I've been the situation that when money functions break the symmetry. Then, you know, we actually have two choices for very similar gauges with the same spread but then the code just picks one of them. Or are we kind of guaranteed somehow to be in a situation that the maximum localized gauge is going to be the one, like you know the one that does preserve symmetry. So, this whole gauge space has so many degrees of freedom that we kind of don't think about how does the spread look like as a function of degrees of freedom in the gauge space and you know I remember also this. Until in the 97 paper you guys mentioned magnesium oxide, where apparently we put oxygen orbital on oxygen P orbit on the oxygen that whichever way you're oriented you always got the same spread. So that means that basically that landscape of gauges, your spread was basically like very flat right. So, you know, your spread as a function of gauge, you know, you could have picked, you know, incidentally many points, all have the same. It happens a lot, especially things like manganese oxide and nickel oxide the 3D in an oxide. You said it was like 10 to the minus 12 like you're 10 or something like many, many, many digits so even if you say you're on it. Shouldn't there be like a little variation somehow in there that you know, but the orbitals want to point along the x axis and y and z and not like in some random. So the numerical, not noise but in precision in the finite differences, you know, representation of the position of the later and so. So in there. So, I think there are actually two questions or maybe that yeah we separated. And so if we go back to the first. You can also imagine, you know, we are using our square but we could use our fourth and all of a sudden use which are different physics that could make you swap from one to the other so. But let's pick one, let's pick because but I don't. So there isn't, you know, I mean, so it's accidental that you are in the double well versus the single well situation, depending on the chemistry of the system and the localization of that you have defined. So, again, maybe I'm too generic, but I don't see anything that should say we should always be in the top case with a single well or so. I don't know. Let me say one other thing. I think Cinecia's question was getting at the question of whether sometimes it's exactly flat. And I think sometimes it is exactly flat in some sense so, for example, suppose you had some some Adam with p p x p y and p z orbitals, and it's in a cubic crystal where hydrogen atoms in the plus and minus x plus and minus y and plus and minus z directions, you know, symmetrically disposed around it. And then, you know, you ask, you know, are the maximally localized one a functions going to be the p x p y and p z, or how about p x plus p y over root p x minus p y over root two and p z. And, and so you can rotate the one a functions by some angle theta. And I think you can prove that the spread functional has no higher harmonics than like cosine of two theta or something and in the cubic crystal you can't have anything below, you know, cosine for theta and therefore it has to be exactly flat. So I think there are cases like that where it is exactly flat. And I think that even follows over to the to the crystal environment like, you know, some kind of magnesium oxide or something like that. Exactly. That means that unless you start from a high symmetry point, you're almost guaranteed to end up in a low symmetry state. Yeah, in that case, right. Yeah, but again you swap from square to fourth and is not flat anymore it's almost flat. But, but I think exactly because of this. You know when we did the manganese oxide that we were co diagonalizing our square in order because exactly you would end up in anything that was random basically. And so we decided that, you know, we had the five D when your functions mixed together and we would co diagonalize in that space are square. Okay, yes. So just a comment. If you go to the second, if you go to the second image where we have the more parabolic picture more symmetric minimum and the double minima in the Mexican. So, I also have the feeling that according to the way you minimize you can get either solutions. And so the one thing I've been many something like if I minimize I don't know copper with six funny functions and you obtain something with seven functions, I may need to break the symmetry to get in something that is more localized. Perhaps one interesting question there could be under which conditions one get the symmetric minima if any, or what what is the physics so it's amount that where we have the broken similar to think it looks like a more frustrated situation and I don't know so this is just and waving and feeling. And again I don't know if it's my opinion but you could have an isotropic expansion or contraction of the system so you don't change any symmetry, but our square changes and I would imagine that you can swap between this just by tuning this global handle and that's why I sort of I tend to be a bit pessimistic or whatever not over interpreting. But I think this question of like if you have six and then make seven. You know the thing is that you know like, if you have five the orbit, you know, you know, each, each, each represent, you know each representation of the point group or space group has a certain number of degeneracies right so you know they have to be five these states to have four then you kind of naturally so I think probably as you kind of add more and more you know you're kind of filling up something that exactly has the number of degeneracies in the representation for that group right so. I think if then if you're in the situation that you know you needed a two dimensional representation but you only added one more state, then maybe you naturally will break it but then with another one maybe you want. But I would also separate the, you know, composite separate bands of an insulator as a problem, and the disentangled of the metal work. Yes, everything. Everything goes. Yeah. Something else people want to. I wanted to, to maybe pick your brain on a question that was raised in one of the flash talks, which is that I forget who it was but he was doing supercell calculations on a pristine system with a smaller primitive cell than periodicity. And he found that the maximum localized funny functions broke that primitive cell, the periodicity. And well, so the question is, I guess that has to be numerical. One possibility that came up in the discussions is that this might be related to the lack of size consistency of the discretization of the derivative so there is this work by Stengel and spouting where they solve that problem so I'm just, I guess. So what we are saying is that we take silicon in the primitive cell and we get something with the silicon in a supercell, and we could have a symmetry breaking. Yeah, so the funny functions in the supercell are not exact replicas of one another. Yeah, yeah. So, I think so there are two possibilities that come to mind, one that I find it very physically interesting that is but maybe it's not the one that you want to discuss that is when the system would have a charge transfer and stability so in general in a supercell you could also have an electronic state that breaks the symmetry of the primitive cell and that's one thing. But let's suppose that that is not the case. Again, got feeling is that this would be possible. I mean, in principle, the only thing that comes out all the time is that maybe you know you get the px orbitals and they have swapped signs or things like this or the the orbitals that are mixed that would be mixed in different ways at every site. Now, it's true, but maybe it's good to have David on this. I mean, it's true that in principle, model of the constraint that you need to get a periodic charge density out of this that probably is the hardest constraint in terms of, you know, breaking things again in the principle, you have actually a larger degrees of freedom, you know, you have a larger set of degrees of freedom or you have a different set of degrees of freedom, if you're not imposing the symmetries. I don't know David what do you think on this. I don't know it sounds. It sounds unlikely to me but so I mean I guess. Well, one comment. I think if you if you do the two calculations in a completely parallel way so that for example the K point sampling that you choose in the supercell maps on to the, you know, in the usual mapping sense maps on to the same K point sampling that you're using for the supercell calculation. Then, in principle, and let's say I'm we're just talking about maximum localization of a of the occupied states in an insulator for the simpler case. It seems to me in principle the algorithm ought to end up doing exactly the same thing. I mean actually in the supercell calculations some of the matrices will be blocked diagonal, although there's some instability and so on. So, now it could be that the, again that you're actually at a saddle point not at a minimum and that there's some instability in which the system can lower its spread by breaking the translational symmetry. There is such a case to see it, you know, to see it, you know somebody find one, I mean, it sounds unlikely to me but I certainly can't think of any a priori reason why it couldn't happen so it would be interesting if someone can find a clear example where that where that does happen. But I think if I could, I don't know if I done the calculations right you might not have the same number of degrees of freedom because you have let's say in one dimension a primitive cell with two bands and two k points you have two two by two matrices, but if you double the supercell and keep the consistency on the k points you have one four by four matrix rather than having a block diagonal. You know, you have a four by four, but everything is in principle non zero rather than being to zero zero so is it correct I think you have more degrees of freedom. So you do have the degrees of freedom to break the translational symmetry. I'm just saying that, you know, I think it's analogous to the other kinds of symmetry suppose you have some mirror symmetry and you choose projection functions that, you know, make your positions be mirror Eigen states, but that's actually a saddle point and as you run the minimization you eventually wander off the saddle point and get into some other minimum. So, you know, if the situation is like that. That's one plausible scenario, although, you know, I, as I said I, I find it a priori surprising so if there is such a situation I'd like to know. Somebody want to comment on. Yeah, I may say something. Oh, yeah. Now what they say what be interesting in those cases since the maximum localized you are maximizing the localization of all together that is minimizing the spread of the sum of the spreads. It may happen you have an instability this in the sense to to when if I should should be equivalent one is a has a larger spread and another as a smaller spread and with that you gain in the total spread. Could be a let's say that instability coming from the fact that you want to minimize the sum, and you can minimize the sum not taking both equal but taking, possibly very much different. But you still need to have a periodic charge density that I think is a. No, I mean, so what you say, sorry, you still need to have a periodic charge density that is a hard constraint. Yes, the system as a true. It may happen that you may have a periodic charge density with two to financial which are no longer equivalent, just because one has a much larger spread and the other one has a much smaller price, and you send it to spread. But it seems very accidental. This actually, and sorry, I let you talk reminds me of a question that I'm very fond of and that is, is the dynamics of the centers of charge. Continuous in a molecular dynamic simulation by my wild guess would be that is not but I'm not, I'm not aware actually of never really monitor that. So I have a short comment about the super. I have short comment about the Supercell thing. So, if you have if you consider a premium standard Supercell, then due to the definition the final difference approximation for the spread functional that that the definition in the Supercell 1997 paper breaks super the size consistency so even if you use the exact same one your function and put that in that equation, it will get different results so that might explain some. Yeah, yeah, very good point. Let me make one other point which is that in one dimension this could not happen. Because in one dimension, you know you have a non iterative construction of maximum localized one a functions, and, and you're guaranteed I think to get one a functions that are periodic images of one another. So, if it cannot happen in one dimension but it can happen in two dimensions or three dimensions that also seems to me a little odd. I mentioned it can happen if you, you know, diagonalize the, you know, if you use the procedure from one day but if you follow the procedure from the paper and apply to one day. It might be that you know for a finite came as you somehow run this problem. Because I think the claim is that in the finite came came as the way you kind of compute these imagine part of logarithm causes trouble. I mean, I guess I'm thinking of it in terms of a more abstract question which is that, you know, to suppose you just asked the question what are the maximally localized one a functions is there ever a situation where the most localized maximally localized one a function to break the periodicity. Now, I would be very surprised if the answer is yes, so I suspected some kind of miracle thing like that. Yeah, what like was just discussed. Yes, I think the question was mostly about the miracle. So just as I comment, unless I mistaken at the impression that the symmetry the rotational symmetry problem we were discussing before it's just the alternative view of this discussion here in terms of translation or symmetry isn't it. I mean, we should have the same answer to the two aspects. There is a follow up. Maybe in the case in which there is there is some degeneracy. Maybe this can happen, maybe try to draw because it's a bit easier maybe if you take a linear carbon chain. If you take p orbitals, right, it's a bit similar to what we were saying before about having some flat region. In principle, you could rotate if you're say pie and pie orbitals on the, on the direction of the chain, but you could have the p orbit as a p x and p y here, you could rotate them. And then you could, in the end, they are still the same spread and just rotating them. And so in principle, you know, in a super in a unit cell, you have the publicity but in a supercell. I guess, I know this is a special case because it's flat in a sense but in a specific unit cell you could have now here, 090 here 45 degrees. Some other random angle, the Hamiltonia would have maybe is what Nicole was saying I would have some half diagonal components. It's not any more unblock repeated, at least in this facial case if you want, which is some degeneracy. You can have a lot of very equivalent cases in which probably numerically if you start with or with projections which are p x p y everywhere the system will not numerically will not try to go somewhere else but yeah, maybe I'm wrong. I think it's absolutely correct if they don't talk with each other. And if they start talking with each other. It might actually not be the case because they might want to format this banana. Yes, but still you have, still you have, I will turn it will be fully invariant for rotations. So in principle, you could rotate each of them. I don't know maybe it's true. Yeah, I mean, in this case, if you're, if you're just doing a one dimensional maximal localization and in the, in the long dimension. You'll get, you know, one a functions that come in pairs with the degenerate center. If you have degenerate centers then yes then you can, you can rotate those one a functions on each individual unit cell independently and I don't think there's any cross talk. So that would be, I mean, that would be a case where we're breaking the periodicity doesn't raise the energy but it doesn't raise the spread doesn't lower the spread but it also doesn't raise the spread. Another question is, is there really a case where you're really at a saddle point and that by breaking the symmetry you can lower the spread that I doubt. At least in one day. So should we move to another topic. So, in the case you talk a little bit about this chemical intuition thing. That was, I think also topic in the summer school. Paper from 1984, which has a very nice quote in it. It says that the money functions are still one of the most useful but underutilized methodologies of solid state physics. And in particular it is in the language of money functions that I feel the chemical implications of Ben theory are most effectively expressed. And this paper from 1984. So kind of the question of like you know how do we build these. This, you know, in the light of this question or maybe some other problems like how do we build like, you know, also what Nicole said about you know you don't have to minimize our square maybe you have to minimize something else. Like, is there some way to kind of agree on some kind of descriptors that would be useful or maybe there's a way to find kind of descriptors of this chemical intuition which are gauging variant. Or, you know, we just have to like pick whatever, you know, at a certain situation makes no sense. Or, like, what are kind of common descriptors, do we just look at the bunny centers, or is there something that the game from, you know, looking at some other operators in the bunny representation like Tony or maybe some other operator. All the questions I don't know. Yeah, no, I mean David, do you want to go first. I don't think I have much to say about that one why don't you take a shot. I, you know, I think the chemist agonized on this for a long time. So, I mean, I mentioned already foster boys Edmondson Rudberg, but also people may say, so there is actually a very intense literature from the late 60s and the early 70s, where exactly chemist think what is the chemical intuition that is attached to this chemical bonds and to this localized orbitals. And at the end, it's not that there is a consensus, I mean there is a, if anything, you know the consensus at the time was that the admin son Rudenberg was the one that was more, you know, similar to what the chemical intuition that people had built was. If you can't do it for molecules in a sort of isolated system. I don't see any other reason why you know the solids should be sort of better or easier or different. So this alternative minimization. The principle is, did anybody ever do it for a solid or the difficult to. So, that's the one way where you mean myself interaction. So you maximize this, you maximize the self interaction. No. Okay, so, did anybody try that with like a points for soul. I, I'm not super sure. I mean, you know, when we do stuff with the Purdue Zunger functional. So what the harness young son has been doing with the Purdue Zunger functional and what we do with the cook months okay IPC functional that gives rise to localized orbitals that are more in the spirit of maximizing the self interaction that is then subtracted. So in some ways that's another, you know, localization criteria with the caveat that I think the Purdue Zunger. So if I remember correctly if you apply Purdue Zunger to a Gaussian there is a sort of switch where at certain point you make it less localized things rather than more localized. I forgotten who who did this work I think is a and read you remember the guy. He works with Andy Gore link course door for Thomas course or for I think so I mean that there is some stuff but I wouldn't over emphasize it if anything we you know I've seen a lot of that. You know, the key IPC localized orbitals that is driven by the Purdue Zunger term in solids look a lot like maximally localized when you're function so so I think the general statement is that all these localization schemes look very similar. Unless they look different and they look different more in some you know small high symmetry molecules, but this is super realistic guide. And you know in terms of chemical intuition I think, you know, the something that I really liked, I mean, David and I are also co authors but it was really driven by the first effort of Michele Parinello and Perino Sylvester Ali to apply. So, you know, they, they, they have functions for chemical intuition so there is this 1998 solid state communication paper on amorphous silicone in which you know they took the point of view that I find it very interesting to have, you know, exactly as you describe amorphous or different solid with per correlation functions, rather than having ion ion per correlation function, they had the vanier function center ion per correlation function. And that was actually very insightful to understand that because of, you know, one of the conclusions that is that you could have geometrical environment that actually look very similar. And one doesn't compare and one doesn't. And so this, you know, electronic awareness was actually very interesting. And there have been of course works using many a function centers in this respect, but also the symmetrization of the hydrogen bond in high pressure ice and the like, but I always find it quite interesting and maybe underexploited in a more way that some liquids and this kind of, yeah, systems. Some other comments. No need to be shy. Okay, so, for example, these. Let's maybe try this SCDM so this automated thing that Linlin talked about the people have understanding for. So basically works by the vanier is automatically by you know you want five functions and it finds a five five points in space. Where you basically put delta function, the way I understand it and then you could use that as initial projections. Is there some kind of understanding for like, how are those points selected this is some kind of chemical picture of like what's going on there like, it's a very mathematical algorithm the way to represent this QR activity. Yeah, there's some. Yeah, so Giovanni understands it. And that's why we did the project ability disentanglement of which I will comment in a moment but Giovanni go ahead. I don't know if I understand that I understand the first part of the algorithm and then now it continues. It's harder to explain intuitively but in the end, you can imagine. So you have to construct this density matrix and say the simplest way I have to picture in my head as that you imagine to be in a grid real space grid. And so a column would be the so this is a density matrix is a projector on the balance. So we have a two column, not a single column, a two column. That's the same in the Hilbert space. If you're only looking at the violence states. And so every column can be fought as the multiplication of this matrix we have a two zero zero zero one zero zero zero so as you say it's a delta function in a specific position in space. And you can do a lot of projections and if you do very near by points you get very similar. near sight. Essentially they are all localized. And so the question how you do get not a million because you have a very dense grid but you get only a few and so you do this QRCP which essentially is an algorithm to find the few which are most orthogonal. So you have two which are very close you don't want to have this. Of course you pick n of them so n being the number of any functions it will most probably anyway expand the space but it would be very close together in medical evidence table. So you want to get the most orthogonal and then in practice you also do a load in your calculation at the end. The way the algorithm works in practice is that the first one it picks is the one with the largest charge density I think the maximum. Yeah so it finds the point with the largest charge density and that's how it picks the first one. And then from there on it's more like finding the most orthogonal one in the rest of what you have. And then maybe it's less artistic means that's easy to picture in a sense but you know you find one and you find one with highest density and then you find the next one which is the most orthogonal to it and then you continue like this. And since they are localized you get a basic set non-orthogonal yet but as orthogonal as possible if you want. And then so in this way the idea is that once you do the load in orthogonalization you remain as close as possible without destroying the localization. I don't know if this was clear but there's a way I understood it. And of course this was a gamma molecule then if you have k points. I think David said that he has to leave it's four foot five in our time zone so. I can stay for another five or ten minutes. I have another meeting I have to run to so. So like maybe so okay so there were questions about um like is there something I don't know who posed this question is there something that could be gained by having running functions which are not orthonormal and also do we really need exponential localization so I don't know who posted that question yeah but but like for some of the kind of population purposes um actually I think that this paper that I cited I think the from Phil Andersen's you call them it's ultra localized I don't think I think his are important let's start with the north or maybe again David do you um well I don't know uh I mean there's a lot to be gained and also much to be lost I mean obviously the uh non-orthogonal 1a functions can be more localized and there's a huge chemistry literature on using non-orthogonal wave functions as a basis set but then of course you lose uh you know all the fact that you can relate the 1a functions to each other by unitary rotations they just become generic rotations and and then you have to keep things balanced in some way um I guess you make sure that all your 1a functions have unit norm but are not necessarily orthogonal to each other um but then you know for example the representation of electric polarization in terms of 1a centers is broken um you know the change in spread having to do with 1a centers moving apart from each other is broken a lot of things get broken uh so uh that's my reaction um uh you know in terms of actual practical uh experimenting with this idea um you know I don't I don't have much experience so I'm speaking somewhat from ignorance but but that's that's my my perspective no yeah absolutely and I think there is also an ill conditioning if you think of it because just think sp3 hybrids if we were uh you know not forced to be orthogonal you could have basically an s orbital with infinitesimal amounts of p and you know that there's you know exactly any combination yeah yeah so so yeah okay uh so I don't know who posed that question so I don't know if somebody wants to comment maybe further um yes yes you know I yeah I raised the point but I tend to totally agree with the with the answers in a sense I also had the feeling that there's not much to gain yeah the question was exactly is there anything to gain uh following that path maybe before David signs off uh it would be interesting to to hear his thoughts on Jemos question which I think was uh if we are in a situation where there is for example a topological obstruction to makes me to exponential localized vani functions and they are just polynomially localized is that still good enough to use them for numerical work did he have in mind like interpolation or okay he's nodding so I think he's kind of agreeing but maybe you can elaborate uh is that a question to me well let me comment so uh I don't think so isn't my short answer I mean first of all I mean when you have a topological obstruction it means that the gauge uh has to have a there's a point in case space where there's basically a vortex present and and first of all you know that will give you parallel tails that makes it I think impossible to cut to to define position matrix elements and then secondly the location of that vortex in case space is completely arbitrary by changing the gauge you can move it around and so it just sounds to me like a rather unpleasant way to proceed that's my reaction but I you know I've never tried it I had a comment on that that is if even if the localization is not exponential when you write the inverse transformation you get a periodic gauge so so I think this is by absurdum you you can prove that that is suppose I have topological material if they can find or turn over when you're functional no matter how localized they are I write the inverse transformation and I find that the block states uh fulfill a periodic gauge which cannot be it can be if there's a if there's a singularity in the gauge right in other words uh you know like I say a vortex basically there's a point in the gauge where an infinitesimal circle around the gauge gives you two pi um uh you know you can you can soak up the uh the very uh you know you can soak up the churn number by putting the vortex in instead of by having the gauge be not be uh non-periodic but but it seems to me that if you write the inverse transformation that is you write from the when you're functional to the block then the block is an is a periodic is a periodic gauge for any k yeah but if you start with 1a functions that have these horrible power law tails and then do the 1a you know the inverse transformation because they're so poorly localized in real space to get a singularity in case yeah yeah I can be non-converged at this I'm serious yeah yeah yeah I see I would say it's also important to distinguish between uh the topological side and the metal side because when we disentangle we have actually something that is basically exponentially localized uh exactly because it's smooth I mean in some ways that you know forgets of the discontinuity so but as anyone try that to localize that topological material because uh you know being an engineer also you know algebraic and exponential are more or less the same so I think actually when scdm was started yeah I had conversation with Lulein he there is a paper by David and uh tonauser on the aldane model where it is very clear that the overlap matrix you cannot invert it anymore okay oh that's a plastic reference the 2010 is David's yeah yeah yeah that's basically the top lot that's yeah that's uh that's related to this topological obstruction that we've just been discussing there were papers um I'm I'm not going to remember who but for metals there were papers where people constructed a a kind of a scaling um renormalization group type of structure where you have most of your 1a functions are very localized and then you have a few that have a larger distance and then a few that are very delocalized and as you get close to the Fermi level you have a you know a very small fraction of 1a functions that are extremely delocalized it might have been a rowy buyer I think yeah that sounds right that sounds right I mean I think in metals the issue is that you want to localize a broader manifold so that's the issue I mean of course if you want just to represent the occupied states you have exactly what you described but the interesting thing to do is just localize the disentangled manifold that includes also the conduction yeah I think I remember when when I talked to Lin Lin that he took a finite Haldane model by the SCDM and then most of them were localized and some of them looked at like this long tails but there is I mean you have you know the representation of the Fermi operator on the disentangled so there is no need to just transform the occupied exact exact exact okay I do have to leave any final question for me maybe maybe a slightly related question is what about this fragile topology why you can't bannerize those bands I have these fragile topology but that you know the overall chair number is still zero but yeah well I mean again it I think that is always embedded in questions of symmetry I think it's always questions about whether you can I mean if you're willing to break symmetry in the in the generation of 1a functions I think those problems you know I think the fragile fragile topology is defined in the context of certain symmetry constraints and because you need to reproduce the symmetry representations at high symmetry points in the Breland zone in a certain way I don't think I better say anymore right now because I'll just I'm not up on this enough to have a clear a clear picture in my head at the moment but that's my first reaction thank you David it's very time thank you this was fun any questions to wrap up or I didn't get a chance to ask David if he had any regrets for example like okay so this when people there was a lot to talk about the electron phonon right so you know use vanier functions you take this operator and then you can compute whatever you want with however many k and q points you want it's very useful if you kind of think more broadly are there some other solid state calculations where you would gain from vanier ising for example like beta cell peter gw are kind of expensive calculations okay in gw you can get away usually just with shifting bands because gw doesn't change orbitals too much but let's say you want to you do beta cell peter for example something else on like a coarse mesh and then you vanierize it somehow like would it in in principle be possible to kind of vanierize everything or maybe there are some things that you can't or maybe you can think of something else or I mean there's some other operator you could you know also vanierize you know I think anything that depends you know completely on the Fermi surface on states of the Fermi surface really benefits from vanierization when you have to do this you know integrals to capture the response from a line of states or anything actually we you know very early on we had a project with myilda zack and francesco maury on nmr in metals and we needed to calculate the spin susceptibility the orbital susceptibility and the night shifts and those were horrendous sort of integrals to converge because exactly they they they require that response so I mean in general I think anything that requires exactly a low dimensional manifold with respect to your entire brain zone I will will need a lot of k-points to be calculated precisely and then when realization is good so for example there's a prl from louis group on betasalpeter and I think molybdenum disulfide I think where they to get the spectrum of excitons correct they need to I think effectively had to do like a 700 by 700 k-point grid and you know that's like a and they had some tricks to do it but you know there's a way to kind of speed up those or maybe you know there's some other code you know something else that people compute that could be sped up I don't know if people have some yes yeah I think that concerning betasalpeter there were some early attempts of using vanille functions by claudia tacalite and a few other people but 10-15 years ago and I think that is still something that could be further explored in a sense because you typically need a few bands so it's okay you also involve empty states but typically those are a few just for the lowest line parts of the spectrum but you have a very incredible number of key points or somehow the shape of the betasalpeter metric should look pretty differently once on the k basis other than the vanille space so I think there's room for improvement but paulo maria has been doing a lot of you know the gwl code the gw the betasalpeter all in a vanille basis or a combination of vanille functions yeah I mean others I suppose but yeah whenever you need empty states so let's say gw that's where playing where it's shine because all this localized business and let's say let's put aside vanille functions because like siesta and everything the the basicity is not complete so you cannot go you cannot converge actually so maybe what the right thing to do is do a rewrite things in terms of the occupy states only so people like star night doing star americ equations there that that makes sense because you have the occupy states only and and you can work in a localized basis but yeah so we have with the empty states I don't think it's a good idea to go near factions yeah so basically in johann stonk there was a you kind of basically vanille is turnheimer equation in the fbt in some sense sorry but I I haven't thought deep about that aspects but normally the bottleneck for solving but the superior occasion is to diagonalize the case space that they all couple to each other so even if you obtain somehow those huge matrix in case space like several thousand by thousand it's not it was difficult to diagonalize them so there is that problem oh I see so normally that's the the difficult part for when it comes to the exit on physics I don't know whether there is a way that one can do entirely localized moment sorry real space thing but that's because exit on is made of all the transitions together and then somehow I see so I have two comments about this so first for a bsc thing I have very I have little experience I have no experience with bsc but maybe iterative dilation methods might be my soul the problem should have mentioned if you can represent the Hamiltonian in some localized basis so in a sparse manner and regarding the previous topic about the empty states so as sinisha mentioned our work on linear function perturbation theory is a way to solve the problem of empty states in in case of perturbative calculations so you can represent the wave function perturbation which includes a lot of empty bands but in a localized manner so it's not applicable to all problems but in my soul some problems for example the self-energy real part of the self-energy which we calculated in our paper okay good uh so when are we doing this meeting again next year I would the developers meeting from what I've seen that is very little since I've been very successful in bringing together so it's not for me to say but I thought there was a lot of enthusiasm of making this a more regular event because I think it really brings people you know together to think and do problems and and it's very easy to get bits of funding from everywhere so actually funding is not a problem and it's actually maybe you know this is a general comment that in some ways I mean I think Xavier Gons said it very early around I mean Xavier both said that you know biodiversity of codes when referring to electronics such a cause is a very positive thing so but very early around actually use the you know case of the 90s a very interesting effort that was actually sort of you know working with all these different different communities and you know I've always been convinced that you know software and simulation software is really you know one of the powerful pillars of 21st century science and I think this is one of probably the nicest examples so where we exactly bring together very different communities pieces people with different interests and different codes because again you know biodiversity is important but you know we have the long view that I think is good to develop you know a powerful environment so so I see this as a very interesting very successful and also very pleasant example and you know probably we should do our best as we are as you guys are doing I don't do anything these days apart from writing grants basically but I think you should really continue because it's such a nice example of how to do science and computational science this century without somehow the stress that some other communities have I think maybe you know I think it was good to have kind of people who kind of go in depth to understand the code very well and are discussing a lot of like technical things that are you know useful for us who kind of use this more than develop but then also kind of have the breath with people who kind of came from wide range of things where areas where they kind of use some of these funny or late ideas and also in terms of diversity I think also the idea that you know you know maybe there are people who are kind of more in a like coding side or more in like development side I think that's good to have a kind of depth that that makes okay until next time I guess okay thank you guys so thank you thanks everybody we still need tomorrow so I think it's nine we've got summaries of all the kind of things that were achieved during the meeting at nine