Lecture 23. LCAOMO Approximation Applied to H2+
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Lecture 23. LCAOMO Approximation Applied to H2+

Alternative Title 
Lecture 23. Quantum Principles: LCAOMO Approximation Applied to H2+

Title of Series  
Part Number 
23

Number of Parts 
28

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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  
Publisher 
University of California Irvine (UCI)

Release Date 
2014

Language 
English

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Subject Area  
Abstract 
UCI Chem 131A Quantum Principles (Winter 2014) Instructor: A.J. Shaka, Ph.D Description: This course provides an introduction to quantum mechanics and principles of quantum chemistry with applications to nuclear motions and the electronic structure of the hydrogen atom. It also examines the Schrödinger equation and study how it describes the behavior of very light particles, the quantum description of rotating and vibrating molecules is compared to the classical description, and the quantum description of the electronic structure of atoms is studied. Index of Topics: 0:01:06 LCAOMO Approximation 0:03:22 Variational Energies 0:04:49 Matrix Elements 0:14:44 The Overlap Integral 0:25:41 The Hab Matrix Element 0:33:50 Is H2+ Stable?

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00:06
well on next episode of can 131 a goes back to the hydrogen molecules ions which we've been looking at and this lecture it is about what's called the linear combination of atomic orbitals molecular orbital approximations that we're going to use here applied to the age 2 plus this is going to involve quite a lot of mathematics I want to lay it all out because sometimes the books gloss over a little bit and by laying it all out in great detail and seeing how we do reach integral and how the answer comes about them there's no mystery as to where these potential energy surfaces that were going to get in the end come from you'll see exactly where they come from exactly how we calculate them and then later on will optimize the manual 6 Shockley how we do that as well although we never gave our trial wave function which was 1 has they and 1 has to be the 1 s or on each hydrogen and we never gave that a proper name it's in fact called the linear combination of atomic orbitals molecular orbital and a molecular orbital is just the generalization of atomic orbital it's 0 1 electron wave function that extends over more than 1 nuclear site in this case there are only 2 Proton so they're stupid in general could be more the beauty of this approach is that we can use linear algebra to optimize the orbital energies and that makes the method really popular and very powerful their huge linear algebra packages that are available there very reliable the very fast and they get you the right answer so if you can get your problem down to a linear algebra problem you can solve it or you can prove that it can be solved if your problem is some sort of nonlinear optimization then you're in trouble usually but not always but you can be in trouble because there may be many solutions you may get caught you you may never be able to find the correct solution and so on so let's now go back to our problems from the secular determined and calculate that the matrix elements here than onsite 570 coordinate system again as a reminder the gods the separation between the 2 protons which we've labeled a and B and then the electron is at radius of a from proton and are said to be from proton B an hour Hambletonian anatomic units which is so clean and nice you can appreciate why we love these units so much is written here it just as 4 terms 1 German kinetic energy than the 2 potential energy terms of attraction and then the Indian nuclear repulsion we don't have want to thank goodness in this system because we don't have a 2nd electron but we had obtained 2 routes of the secular determined and they are Bplus Aminus is equal age AA plus or minus stage in the divided by 1 plus or minus in these terms a and analysts are shorthand for numbers that are in fact but which will depend on big are the numbers that we
03:52
get will depend on the Gaza could consider to be a function of the but there are things that cannot calculate out the coordinates of the electron and just returned the value for the expectation value of of these things up we have 3 intervals to do that we have to do the honorable thing we have the will to get it should be and we have the integral to get past let's have a look at the remember that the Senate roles you know where you have a product of 2 functions Indiana girls because it's an expectation value the Santa Rosa just over 1 3 D space range not like helium where we had a 60 we had R 1 and R 2 where the 2 electrons to integrate the wave function over the suggest 3 D integral so they aren't so bad now let's stop start by calculating the matrix element HA we're going to etc. coordinate system on the proton just for simplicity and that because they are wary function is 1 essay and 1 essay so why not just sent to the quorum system on Monday and then there the 1 over IRA is sending a very easy to calculate on this then has 3 terms the 1st term is just the Connecticut and potential energy of the electrons which looks just like the hydrogen atoms and so that when we done we don't have to do that and then we have another term that's the integral over minus 1 over our but with the way function when I say 1 and then we have the final 1 which is just the integral of 1 over are they got the interim nuclear repulsion that was very easy to do because artists fixed
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member we froze the nuclear fixed so that goal that won over big are just comes out at the interval we we can do that the 1st integral then it is just like for a hydrogen atom we know the answer in atomic units it's minus a hat we get there early on the last integral is simple because fixed it's just pull out the 1 over are outside the integral which I done here on slides 573 and then the 1 essay wave function is normalized and so that it goes 1 Over the and the center integral is similar to the kind of miracle that we had to do for the helium at slightly different but it has retains a lot of the same flavor of what we have to do because we're integrating over DRI we have to express I got 1 over on the In terms of the integration variable parts of the and any other things and to do this we have to come back again to our friend the law of signs in order to do this integral otherwise we can get a formula to express it I want to do that then let's let the z axis B along big are lists that let that be the interim nuclear vector that access will call C and then in our court system which every drawn here again say there will be the angle between big & director from the nucleus aid to the electrons that angle is going to be feted the electron can be anywhere so that angle varies from 0 to party and that's infectious the angle theta in spherical coordinates and then the socalled as a musical angle will be the angle around the interim nuclear axis which which will b The variable 5 and we can write by by the law of cosigns then the ABI is equal to the square root of RA squared plus our square those of the 2 sides of the triangle minus 2 are so very big are cosigned the that's just exactly what we did that gets rid of it R B and it gives us variables that we're going to integrate over the data are a big Are we don't have to worry about the cost and that's going to come through whatever we do in our calculations and as before it's data that's the problem because we've got cosigned data we've got signed data without the square root and it's also but in the denominator therefore let's just concentrate on the fate of integral we 1st let x equals cosigned then Dr is the DX debater is equal to minus sign data or DX features minus sign .period data and that lets us transform the integral on the top of the slide 575 into an integral Over acts but I can do and the limits then instead of being 0 part I like they are data are 1 and minus 1 and we did I've written down the entire derivatives and that works out to be 1 of the big plus 1 over our so there minus the absolute value of our big AAminus hours a day divided by the product are bad that's very similar to what we got doing and the hydride and I and the helium ions so that the helium atom Excuse me and so you get there used to this kind of thing because you've seen it you know that you tend to get an absolute value and you know that you're going to have to break up what you do
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in the following into several cases as you can imagine if you start ending up with a lot of electrons these cases of whose outside who can get pretty complicated pretty quickly and less than integrate over already we have to break it up into 2 cases depending on the absolute value of big AAminus RA if power is bigger than our it's just ominous or if our is less than are it's the other way round and we reflect that by breaking up the undergrowth of RA from 0 to infinity into 2 parts 1 where it's less than big on 1 it's bigger than they are the integral over 5 thank goodness because I appear immune to grand anywhere is just of the 5 from 0 to Part going around the sea access so that 1 is very nice but and that 1 will continue to be very nice until we get to the orbitals or something like that and then boy that wanna be there as well and we can get complicated but in the product of the 1 functions because each 1 has the square root of pie and the denominator gives us a factor of pie In the denominator and that cancels the pie over the integral fire so we're just have to riding along and then this gives us which I last year as homework and for you to do break up the inability to domains as I described get the Yantai derivatives and then check it and this cynical is the culture of our the the Coolum integral and it is equally eager the minus 2 are they got times quality 1 plus 1 over minors 1 overall that's what we get and adding up the street terms that the 1st term was just a hydrogen atom minus a half the last term was 1 over all and the Senate term was J of R and the 1 over our plus 1 over AAminus the 1 over orange debacle was and so this matrix element nature is equal to minus the hat plus the 2 miners to our times quantity 1 plus 1 overall now as calculated this overlap integral but yeah you might say Well why don't we calculate the HA be 1st because going order but when you look at it said it involves tests and so you're going have to calculate anyway and so few foreign calculated that than the other part goes much quicker In the end of course once you've done the Senegal's and you've convinced yourself that you understand how they work then you just
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look them up at that point but if you never actually do any calculations don't expect to have any great understanding about how the terms actually came about the basically come out of Bullwinkle said you will be unwise this scandal seems deceptively easy it's the integral over D R D R of wanna say 1 sp and that's just 1 over par times a over of ETA the miners are something you to the minor source of but final exam on the 2 well surprise I need to express In terms of the variables that I'm integrating over which are all to do with it and unless I can do that in a formula I can't figure out what the integral to do as I have to know what the functional dependence of the underground is on the variables that I'm integrating over In this case once again we're going to have to use the same strategy so it can be a dead horse here with the Allocco cosigned because he how useful the loft cosigned the unfortunately this time it's a little bit different because now we've got 1 over par 5 times you go only to the miners are so bad times e to the miners and then the square root of IRA square closely guards square miles to ah ah "quotation mark data that's different now because now we've got all that spinach up in the exponent of of the integral and that means that that's going to be a separate the little project for us to do as usual what we're going to try to do is get rid of the integral over fatal because that's a sign saying sorghum and do that integral 1st Oregon after looked at the cosigned data probably were going to make the same substitution within the form of maybe we're going to have to do a little bit more work this time once we get rid of the things that we have to decide whether or not are resolved their makes the integral over the other variables and turned out to be 2 or 3 parts where we have to keep track of who's bigger than whom but so let's then focus on the integral over the fate of of this function which has the square root in the exponent and let's not get into a form that we can finally recognize where we can look it up in a book of course it depends on the books you've got if you've got a very extensive table of minerals you'd be surprised what you can look up because that many of them were worked out of course let us then right the capital I here to be the integral from 0 the pride of just eaten the minus what we don't have to worry about of because that has no dependence on the organ integrate that
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later so we just have to do this 1 exponential of the square root as overwritten here on slides 579 signed thank member of the side that comes in from doing spherical coordinates well we make our sense substitution that we did before we let x equals cosigned data txt data is minus Sunday and that means the his minus DX oversight predator so we can substitute debate minus oversight data and the integral becomes the integral from minus 1 to 1 you we swapped the limits of integration as well to get rid of the miners because it would be of course of course the other way round has cosigned 0 oncesolid beauty and go from 1 to minus 1 but we swap because of a minus sign and now we have another integral needed minus the square root of of a squared plus big squared minus 2 hours a day the R X d acts I still don't know how to do that offhand and therefore I'm going to make another substitution I'm going to get rid of the whole thing up there in the exponent now I'm going to let you he was a square of all that I'm going to figure out the DU DX this is what I've written here which simplifies after you're done to minus artists of their times are divided by you and that means that the U.S. is minus you do you over our submitted times are and therefore substitutes that in and substituting new and finally I get an integral I can recognize idea minus 1 over our survey times Degas and I have to be careful with the limits of the lower limit is when you is equal to ah plus they got in the upper limit is when you is equal to the absolute value of our they got minus on Sunday but at least my grand now is just the minus you times you the DUE that's what I can do but I have to do it by parts and then I can get the answer and so that will consider that to be a done deal and as usual the answer breaks up into 2 pieces now the 1st cases they always again 1 so there's less than they are in the 2nd case is 1 back it is bigger than they are we get these 2 terms putting an end the limits here and the reason why we have to put in is of course the absolute value function changes 1 way do that it's what's around because it can ever be negative so at the top and we have these 2 terms With the deminers they are minors but are the miners bigger .period and Alabama where a similar set of terms but with some minor signs instead of pasta and then based on these 2 cases we then have to carry out the immigration over are so that includes the exponential function 1 essay which I took out because remember we just integrated over the data in order to get on what we've got here and so we still have eaten minus are and then we have of course but of the other parts as well as the odds of a square that we get from Chirico polar coordinates fires like usual testimony to parliament will see that the 2nd therefore on slide of 580 to what I've written here it is explicitly for the case where parts of there's less than big when I'm integrated you the minor source of it the of a square and all this stuff and thank goodness that there is a 1 over IRA in the denominator so that cancels 1 of them that means I don't have to integrate by part's over and over and over otherwise and have our severe acute didn't have to do it 3 times but we get to terms here depending on whether the limit 0 over where the limit is big on that those are the limits because our there's less than they got we get eaten minus3 are times 5 forwards plus
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1 over to plus plus either the miners big at times and then there's 4 terms are square over 6 plus or over to minus minus 1 over to ah and all those terms come out naturally from doing you go buy parts that's where they come from whenever they are big or is less than an hour so they we get the integral than from the guard to insanity of the same thing but the immigrant has various things changing science and what we get here years and this often happens is that thing simplified we get minus ito the minus3 are times the 5 was 1 of a twohourplus that's the same as the other terminal with a minus sign and then we get the to the minors they got times threefourths plus 1 over 2 are so there's some terms left over there the total integral over IRA that adding the up the terms in need of a minus3 they got canceled and we ended up with the following things even the Minnesota are times R square over 60 plus our to plus onehalf that's what we end up with well we ended up with a factor of 2 because of the 2 pilots from integrating over fire which we never did we can do that in our head and then the prize goes away because of the square applying in 1 s wave functions and so we just multiplied by a factor of 2 and we get s as a function of the got the number s which is the overlap matrix element as a function of the interim nuclear separation is an exponential while you can see 1 big Oregon it's large there's no overlap when big or small the overlap is very large bets the mines are part of this thing and Times 1 the square plus plus 1 OK not so bad so these matrix elements so far aren't nearly so bad as all those formulas we had in Zadar where things got out of control a little bit with all those terms flying around these things are fairly well behaved we've got 1 more to do however and that let's hope that 1 doesn't sound anything that has won coordinated and then some functions with the other coordinate are called to center and roles in the business and to center in the girls are always a fair bit of work to do when you see that you don't you don't think Well gee I can just write down the answer that usually you're going to have to sit there change coordinate systems play around and look up the end derivatives or do with mathematics but if you don't it changed water systems or you don't do anything don't expect mathematical or or any program like that to necessarily help you it may not be quite bright enough to understand what you're trying to do if you try to do these intervals for example in Cartesian coordinates and just tell you just get on with it in the greatest thing I don't care how you do it you may find that the computer fiddles and travels around for a couple of minutes and then comes back and just as the answer is the unique role of this writing it out In a nice touch that form but not giving you what you want which is the answer so far last matrix element that is the matrix element BJP this 1 we left the last and you'll see why because this 1 is going to require a fair bit of patient here's what we got we got 4 terms h abc is the integral of Dr 1 answer Hambletonian 1 has to be well we 1st of kinetic energy and we have the potential energy 1 over IRA then we have the 3rd term which is minus1 over mixed SA SB and then we have the last 1 in which thank goodness is just 1 over are but which again is a constant and therefore the last term the 4th term is just 1 over times the overlap integral because that's the definition of and we've already calculated that so where does go with that anyway so if we can get the other ones down 2 things where we calculated something in the wind up with the overlap integral again that were done so that's a pretty good
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strategy to do if you know what the overlapping the first one it is just a half again at times the mines have Excuse me that's the overlap integral about one's easy and the last 1 is just 1 overall are 1 of the big times the overlap integral so that when the that leaves us with the 2 center 1 and we could expand an awful lot of effort during 1 of them and doing the other ones and then scratching our head and saying My isn't that something those 2 are the same well let's see why there's going to be the same amendments only calculate 1 of them that's half as much work as well wired is saying here's what the problem is symmetric there's 2 protons it doesn't matter which 1 were calling area which 1 were calling therefore whatever the value that integral if I am a great 1 has times minus 1 over IRA times 1 sp if I just change can be they have to be the same because it's exactly the same integral so that's equal to the angle of 1 S B minus 1 over are be 21 it's a so that means that but but that functions were swamped as well because we we wanna say that 1 has a minus1 over RB 1 sp is the same as what I've got I don't have that idea but that's true because 1 over our B or 1 over our city it is just multiplication bye that number 1 over ourselves 1 over our so there is solid when I have the derivatives where I have to take the if I the derivative I operate on the function with the derivative and then go ahead multiplied by the function and integrate but if I have a number on the border of factors in need Grant doesn't matter at all and therefore I am free to reorder the things anywhere want unlike the case where I might have a derivative where fighters take the function away from the derivatives that I end up with with the legal operation to hear what I've done and this started out with 1 a minus 1 over IRA 1 SPD I've swapped the labels and manages rearranged the order of the 3 things and I can show them that 1 essay a Times minus 1 over our 1 sp is exactly the same as 1 answer minus 1 over our of B 1 sp therefore I am only going to do 1 of the integral and I'm just going to multiplied by 2 at the yet now the the article itself which I have given the symbol here which is the standard symbol for the integral Kayumba is 1 overpowering times the integral Over the the shorthand integrate overall 3 variables EU the miners sorry that's what Over that's 1 minus times minus 1 over hours of the center part of the attractive potential and then I had the other nucleus right away I don't view the miners are B I go ahead and just putting on the lawn of the cosigned and not get the same kind of integral that I encountered when I had to do the overlap integral because I've got the square root of staff in the as usual that we do the order we integrate overstated there's no dependence on fire so we can do that whenever we like we cannot say that we slid up and the cases we integrate by hours of and were very careful because if we make any kind of mistake anywhere along here it's steadily a factor of 2 that's it it's over and you've got a track down this stuff is much like debugging a computer program if you've got something wrong inside post wonder I instead of Jay you've got to track down because you just get something that's completely useless if you make a slider so this kind of thing doesn't go well In a noisy coffee shop usually this goes much further in a quiet room with a big piece of paper and a bright light well ,comma we didn't go over there it's just like before and it doesn't much matter for us whether it's already squared in front when we integrate over our or RA because we've got a minus1 1 over IRA so that actually helps us in this thing rather than hurt because it cancels out 1 of the yard square that means that we don't have to integrate by parts so much but we still have to break into the cases due to a integrated by parts and get the answer I'm going to leave that as another now the problem to do it's exactly the same thing but I'll give you the answer here Kayumba is minus he to the miners are planned 1 plus that's the final answer after a loss a fair amount of work now a final results of the matrix element than HA a is minus onehalf times as of Oct the overlap plus cave are the exchange and grow this is our divided by the grow divided by antinuclear distance the energy there now we have to keep pushing back so we punched down several levels here running the sub programs now I gotta go back to where we were where were we we wanted the energy now we've got all the dependence on what are we going AGA we've got a and we've got 1 of those reminders that we can calculate the plus and the minus and what we want to know is this question he is a student plus stable Is there an energy at which stage to plus can exist remember when we tried hard drive the 1st few times we did it it wasn't stable was only when we played around a bit but we finally got a solution was let's have a look the class Is this giant thing here that I shown slides 591 we put in everything that we've gone and it simplifies down to minus 1 plus j of our plus 1 Iraq's most table look at bar upon are all divided by 1 of the best of all and if you've got these things written down in a little computer code it's best to keep them as the coup along the goal the
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exchanges on the overlap and call their little functions of art and then you can manipulate the but just looking at this may not be at all transparent what it is In set will come to that and set the other route III which I call the miners on these very similar their lease terms and the denominator is 1 minus asked the and that turns out again today possible the simplified into minus ahead the leading term plus all this stuff jail close 1 over on minor scale are minus the overlapping go over all of them divided by 1 minus as the miners onehalf is the energy of a hydrogen atoms In a proton and infinity at rest therefore that's going to be our measure what we want to know is if we bring in a proton 2 hydrogen atoms Will they share the electrons and the more stable as a couple like that then just a single animal and alone I'm not just sitting out and infinity that's the question that being the question let's omitted the minus onehalf because what we're interested in is whether the other part is bigger than 0 or less than 0 that's bigger than 0 which it was for hydride initially that means that the thing is repelling and it's more stable to just be apart if it's less than 0 so that means that it could have bound state in which the nuclei want to stick together and somehow the electrons in this orbital that we've been trying to concoct here holds them together well if you put in all the formulas you end up with these rational functions excuse me they have exponential send them so they're still transcendental functions but unit with these ratios of our functions very fast and Bminus and unless you're extremely talented with formula is still extremely hard to tell at least for me what the shapes look like but I got a simple formula for them and I cannot imagine the days when you could spot that you have and and it would be obtained to calculate all these things too because he didn't have a computer you have this little calculator that was very very under power and yeah he's a graph paper and yet the reason why you learn to take acid totes and figure out when things were 0 and figure out when they were maximum is so that you can draw a few points and then sketched the curve and see what it looks like cost now that's ancient history the minimum is pretty shallow but each 1 of us the lower energy state it is stable and here's the curve and on slide 593 there 2 curves that shown the In the minors and our energies in terms of hot trees and they are in terms of the border radius or not which over here and you can see that there is a minimum In the potential energy surface mark he plus or potential energy occurred in this case because there's only 1 quarter N look a little faster to between 2 and 4 there's a little minimum their about minus 2 . 0 5 oratory and then the mind Is this thing that just goes up and up and up and up and the closer the nuclei again the worst Bminus the first one Amendments :colon attractive potential surface for the obvious reason that if you start with the nuclei apart and their further apart than big are the minimum rent of around 2 .period something they will slide in I intend to interact down there the base and the other services the mind it is cholera pulses potential surface because as you come in no matter how close you get better how rapidly you're coming in on you tend collide together and then just fly apart and if you have a molecule with on bonds but say where and he when you're down there and then you make some kind of an electronic transitions and to where your on service but looks like he's minus here In this In this figure then the 2nd you go up to the miners here now on a different potential surface and you roll apart and so the way that is the the normal way than to think about molecules is to think about the electronic configuration setting up places where the nuclei can be attracted to or not and then the nuclei settling To the most stable position and then maybe rattling around there like the harmonic oscillator around those positions as they go around and then if an electron is promoted what we have to do is stop usually an electron promotion we consider to be instantaneous for the same reason that we said the electrons can adjust to the way the nuclei are all the times of the nuclear effects the electron goes pink and now what we have to do is recalculated the whole potential surface now all the bonds of different or at least some of them a different question is not what happens it could be that the molecule is still stable and what happens is the bond lanterns come up you want to think of coming up in energy and tonight on a different surface and the minimums out here well now on the surface I appear to be compressed and so on I start rattling apart and so I make an electronic transition and I create vibrational energy and if I do that in a condensed face what will happen is the vibrational energy will hit something else and that will cause heat transfer and that's how I can take lying on radiates something and have it just too great to be absorbed like this slack the church and then turn into heat after it's been absorbed that's exactly the mechanisms or it could be but the thing will actually fly apart into Adams and then the question is if I'm in the gas phase the announcement said by later and never recombined again or if I'm in a solid what they may do is they may slide are there on the repulsive curves and now the
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electron has time to get its act together this is that photon excited me but I wasn't happy and that orbital up there anyway sorry guys that I created so much trouble I'll come back down and then the molecule may recombined so if you have molecules in a cold matrix and you photo excite them they format and the Adams may be combined and if they don't that means that maybe there was a whole here that the atom could seek out and get stuck in a different part and that's true of a lot of interest to chemical physicist because they talk about being in a cage and the 2 atoms are an occasional 1 goes to a different Cajun can understand something about the holes between the cages and the structures of the solid and that's the whole field of endeavor well we've got this these curves here that that was pretty good we were able to get that but the question here Is this is interesting but what I want here's how good is our answer N what is our way function we in girls we get the determinant but we never actually go with the wave function once and so on next time but we're going to do you start from where we left off we're going to go ahead and find out the the wave function and we're going to find out how good our energy but we're also going to develop a new technique that we haven't talked about but was part of the actual online wave functions but I showed you it had this cryptic and that's very and then there's something like minus 1 . 9 9 9 kph or we're can I learn what that meant