 We are learning about many electron atoms. Today we are going to learn how to write their wave functions using a particular form. We learn how to write their wave functions as what are called slater determinants which looks sort of like this. But before that let us recapitulate very quickly what we have studied in the earlier modules. We have learned that the way you write the Hamiltonian for many electron atom is that first of all you separate out the term for the kinetic energy of the nucleus or the center of mass what you are left with is sum of n number of kinetic energy terms one for each electron sum of n number of nucleus electron proton electron you can say attraction terms minus qn e square sum over i to n 1 by ri and you have this problematic plus qe square it is a double summation 1 by ri j terms. So these turn out to be the one electron Hamiltonians these ones that is not a problem but what you are left with is this electron electron repulsion term which we have established that we cannot ignore. So what we do is that we incorporate this electron electron repulsion term into the shielding constant and we work with not the actual nuclear charge but the effective nuclear charge and that is how we write these one electron wave functions by replacing the actual nuclear charge by effective nuclear charge and for helium atom for example the Hamiltonian is going to be a sum of four terms and I have shown the terms due to each electron in a different color and the wave function would also be a product of two wave functions one in electron 1 one in electron 2 and to start with we work within the ambit of orbital approximation which says that the orbitals here are essentially the well the wave functions here are essentially atomic orbitals in case of helium these are 1s orbitals we are now familiar with the form of 1s orbitals only radius dependent part no theta phi dependence and here once again the only change from the hydrogen atom system hydrogen atom scenario is that you write z effective instead of actual set. So this is how we have formulated the Hamiltonian this is how we have written the wave function and then we did a quick fact check and we saw that for helium atom when we work with z effective actually we have not told you how we get this sigma but there are ways I think most of you would know of the empirical rules by which one can determine the value of sigma for different kinds of electrons in an atom and using this we showed you that you get a theoretical value of energy which is close to the actual experimental value that is there and the point to note is that the theoretical value is more positive than the experimental value which is the real value we are going to have more to say about this in the future modules. Next we came to this very very important concept of spin I sort of rush through this part and once again I am going to rush through this part today but I hope this is something familiar to us we know very well that there is something called a spin quantum number the only confusion that is usually there is that in many books it is written for electrons s equal to plus minus half no please remember for electrons the spin quantum number s is half and m s m s is the spin magnetic quantum number that can take up values of plus half and minus half so z component of angular momentum can be plus minus h cross by 2 and the length of this arrow or the magnitude of the spin angular momentum as such is equal to h cross multiplied by square root of s into s plus 1. So we have actually learnt about angular momentum and we know very well why is it that the magnetic quantum number cannot take up sorry about this is not mom recapitulation spin angular momentum unfortunately some kind of reformatting has taken place please do not worry about it. Okay so we have studied angular momentum and we know that the magnetic angular momentum has an upper cap let us do a small calculation now which we sort of started but we did not complete in one of the earlier modules what I want to say is this that this length here as we said is root over 3 by 2 is not it why because s is equal to half half into 3 by 2 that is 3 by 4 under root sign root 3 divided by well 4 comes out of the root sign and you get 2 in the denominator. Of course if you want I can write root 3 h cross by 2 reason why I did not write it is there in quantum mechanics often you write quantities in terms of h cross h cross is like the unit okay and here it is shown here that the z component is equal to h cross by 2 right. So if we take say this angle theta what is cos theta cos theta is equal to h cross by 2 divided by root 3 h cross by 2 you are left with 1 by root 3 is that right 1 by root 3 do you remember where we encountered this 1 by root 3 earlier what this essentially means is if you take square on the both sides you get 3 cos square theta minus 1 equal to 0. So you see this cos theta well theta is again the same magic angle that we get okay so this is the value of theta what is the value of phi we do not know cannot say okay that is why usually it is written like this this tip of the arrow is usually shown to define the edge of a cone and in the classical picture interestingly for spin one can actually get away using a classical nice classical vector picture and it works okay so sometimes people get confused that is a problem the name spin itself is very confusing because you might get the idea that the electron is spinning about its own axis pretty much like the spinning motion of the earth which is there in addition to its revolution around the sun now the thing is this spin quantum number or spin angular momentum they do not arise from actual spinning motion of electron if you work out the energy is associated you find out that the electron has to actually spin at velocities greater than light if this angular momentum has to arise from a really circular motion of some sort okay so that does that is not realistic so let me just tell you that spin arises out of relativistic treatment of electron okay spin is a product of relativistic quantum mechanics which you are not going to study in this course okay interestingly the spinning is in terms of some kind of an imaginary coordinate omega but you can have its component in real space so it is a rather intriguing quantity it is highly possible that we do not exactly understand what is going on here but the mathematics is not impossible to understand and also it is manifested in experiments as I told you earlier experimental results are the only truth as Max Planck had said so this spin what we will do is we are going to just deal with this spin angular quantum number in the way that we are talking about right now okay so what we have said is spin is associated with some kind of an angular momentum we know the magnitude we know the direction also and we have also understood that phi is undefined phi is completely undefined so this tip of the arrow could be actually anywhere if you are talking about one spin so since for electron m can take up m s can take up values of plus half and minus half your z component can be h cross by 2 or minus h cross by 2 that determines whether the angular momentum spin angular momentum is pointing up or pointing down these are called spin up or spin down states spin up state is assigned a wave function alpha spin down is assigned a wave function beta right now let us not worry about what the actual functional form of alpha and beta is it suffices our purpose if we just think that the wave function for up spin is alpha wave function for down spin is beta but remember these are wave functions spin wave functions that is all right now let us go ahead so what we said is now if you want to talk about multi electron atoms it is not sufficient to talk about n l and m you also have to talk about m s not s remember m s for all electrons s is half anyway so this total wave function psi which is a function of r theta phi and omega the spin coordinate is given by psi r theta phi the wave function is spatial coordinates multiplied by the wave function in spin coordinates alpha or beta so if you just take one electron then it can reside in the same orbital in two ways one in which it has up spin one in which it has down spins so two wave functions are now possible when you incorporate spin now this state is called a doublet state why doublet because there are actually two wave functions when you consider the spin part as far okay now let us make things a little interesting let us talk about two electron systems remember that is what we want to do anyway not only two we want to talk about maybe 5 electron 6 electron 10 12 electron systems to start with let us understand how to go about it by considering a two electron system so here we are using electron levels of 1 and 2 so 1 and 2 are like the names of the electron electron number 1 electron number 2 so as you understand we can construct 4 different spin wave functions using electron numbers 1 and 2 and at this stage in case you do not have a pen and notebook please pause the video go and fetch them and start writing right as I said earlier we have to keep writing so that we understand okay so what are the 4 spin functions that we can think of first of all both electron numbers 1 and 2 can I have up spin so that is written as alpha 1 alpha 2 alpha then 1 in brackets then alpha 2 in brackets this means electron number 1 has alpha spin electron number 2 also has alpha spin alpha 1 alpha 2 similarly both can have beta spins that wave function will be beta 1 beta 2 no problem so far things get interesting when we think of mixed spin systems what happens when one spin is alpha one spin is beta you might think of writing like this alpha 1 beta 2 okay or you might think of writing like this beta 1 alpha 2 now if you look at these terms what does this first wave function say the first wave function says electron number 1 has up spin electron number 2 has down spin this one says electron number 1 has down spin electron number 2 has up spin do you get the point do you get the point if not let us just say it once again what it says here this alpha 1 beta 2 is that it is electron number 1 and not electron number 2 that has alpha spin it also says it is electron number 2 and not electron number 1 that has beta spin similarly in this function we say that it is electron number 1 and not electron number 2 that has beta spin and electron number 2 and not electron number 1 that has alpha spin now we encounter a problem what is the problem if you say alpha 1 beta 2 and imply it is 1 and not 2 electron I mean electron number 1 and not electron number 2 that has alpha spin then what we are saying essentially is that we can differentiate between electron numbers 1 and 2 can we really do that yeah actually we cannot so before going there alpha 1 alpha 2 is fine beta 1 beta 2 is fine problem is 1 and 2 are indistinguishable electrons do not wear jerseys with their names or numbers written on them electrons do not have the numbers imprinted on them we are designating them 1 and 2 to formulate the problem but actually the 2 electrons are absolutely indistinguishable you cannot say that it is electron number 1 and electron number 2 or electron number 2 and electron number 1 that has a particular spin state what you can say is like one of the electrons has alpha spin and one of the electrons has beta spin that much is okay but you cannot say that is this and that or this and not that or that and not this that has alpha spin or beta spin I hope I am able to make the point yeah so I have say 2 2 boys in a room and there is an apple and then after a while I come back and I see that the apple is not there somebody has eaten it without asking them shall I be able to say that well this is a bad example because they could have shared the apple let us say they cannot let us say that it is an apple that cannot be shared then I cannot say that it is boy a and not boy b who has eaten the apple I cannot say the other way round what I can say is that one of them has eaten it provided the apple has not been shared that is another problematic situation fortunately which is not relevant to our discussion so what we do is we write the wave function like this since we cannot figure out which which is which we take a linear combination when we take a linear combination we end up getting two functions one with a plus sign between these two and one with minus sign and we have to normalize so we multiply by 1 by root 2 now if I look at this what does it say the first term says that it is electron number 1 and not number 2 that has alpha spin and electron number 2 and not number 1 that has beta spin the second one says exactly the opposite it says electron number 1 has beta spin electron number 2 has alpha spin so if I combine the two remember probability what is the probability of simultaneous occurrence product what is the probability of either this or that happening some yeah I can probability of raining is half probability of not raining is half so it cannot rain and not rain if it could then the probability of that would have been one fourth but the total probability that it might rain might not rain that is half plus half equal to 1 once again I am thinking of very bad examples today anyway so what is the significance of this plus and minus at this point nothing significance is there profound significance is there we will come to that but at this point we are just combining the two terms using plus or minus the two terms which say that one of the electrons has alpha spin and the other one has beta spin I can say that using plus or minus there is no reason at least now to discard one in favor of the other so we write both so what have we learned we have learned that if both the spins are up or both the spins are down they can happily write alpha 1 alpha 2 or beta 1 beta 2 if one is of spin one is down spin then we must necessarily write two terms in the first one electron number one should have alpha spin electron number two should have beta spin in the second one it should be other way round and then we connect them using either plus or minus sign so we take these two linear combinations now to develop the discussion of why these linear combinations are important at all we will bring in the concept of symmetry what is the meaning of symmetry let us say I have an exchange operator exchange operator means an operator by which the labels are interchanged when you apply exchange operator to psi 1 2 then it can become either plus psi 2 1 or minus psi 2 1 something like that not all wave functions will be like that but we are going to work only with wave functions that are either symmetric symmetric means eigen function of 1 the sign is plus or anti symmetric anti symmetric means eigen function of minus 1 we have minus sign here so we are going to work only with wave functions that are either symmetric or anti symmetric with respect to exchange exchange means exchanging the labels whatever is in 1 becomes in terms of 2 whatever is in terms of 2 becomes in terms of 1 that is all what about the first one is it symmetric or is it anti symmetric with respect to exchange and here how I wish there was a chat window where you people could respond unfortunately we do not but please think first the first one the first linear combination that we have shown upon applying exchange operator does it change sign does it not change sign obviously it does not change sign so it is symmetric with respect to exchange what about the second one if you exchange the labels 1 and 2 then what do you get simply exchange the labels you get of course you get 1 by root 2 let us not worry about that then you get alpha 2 beta 1 minus beta 2 alpha 1 minus this is beta 2 alpha 1 whatever I done I have just interchanged 1 and 2 so what do I get here I get 1 by root 2 2 let us take this second term first and write in the same sequence as far as possible beta 2 alpha 1 is the same as alpha 1 beta 2 minus but wait there is a minus sign in front of this so I better take this minus sign out minus 1 by root 2 alpha 1 beta 2 now I can write another minus because this is plus sign minus and minus is plus minus I will write it as beta 1 alpha 2 so what is that you can recognize it if this is psi then this is minus psi so after erasing so what we see is that upon applying the exchange operator the second wave function changes sign so the second wave function must be anti-symmetric with respect to exchange so what we see here is that the wave functions that we generate by taking linear combinations are either symmetric or anti-symmetric with respect to exchange why would we bother about that we bother about that because there is something called sixth postulate of quantum mechanics and I will just read it for you because it is important to understand this it says the complete wave function of a system of identical fermions fermion is what is fermion spin half integral multiple of half that is what the spin number is so fermions like electrons must be anti-symmetric with respect to interchange of all their coordinates spatial and spin of any two particles I read that again for identical fermions the complete wave function must be anti-symmetric with respect to interchange of all their coordinates of any two particles which means that if I just interchange one and two then psi should become minus that is all it means right that this is true of fermions for bosons it is not the case for bosons they are actually symmetric the wave functions are symmetric with respect to interchange so now let us try to write the helium atom wave function knowing very well that the complete wave function has to be anti-symmetric with respect to exchange of the coordinates first of all let us start with this psi 1 s 1 psi 1 s 2 we encountered this in the previous slide I hope you remember yeah this means both the electrons are in the 1 s orbital then what will the corresponding spin part be to say that first tell me well tell yourself is this psi s 1 psi s 2 symmetric or anti-symmetric with respect to exchange that is a very easy question because just take one here take two here psi 1 s 1 psi 1 s 2 becomes psi 1 s 2 psi 1 s 1 and these are functions these are not operators so the sequence of multiplication does not matter right they are commutative so what we get then is helium atom wave function given by psi s 1 multiplied by psi s 2 is symmetric with respect to exchange now let us go back to the 6 postulate this is also called by the way poly principle ring a bell remember something well it does not say poly exclusion principle I did not say poly exclusion principle I said poly principle poly principle is that for fermions the total wave function including the spin part has to be anti-symmetric with respect to exchange as we will see our familiar poly exclusion principle is going to follow from here but until now what we have stated the 6 postulate it is really poly principle okay so now so this is symmetric so if the total wave function has to be anti-symmetric then I must multiply this by an anti-symmetric spin part remember there are two spin parts that were there well two kinds of spin parts alpha 1 alpha 2 symmetric or anti-symmetric symmetric so if I just write if I write say alpha 1 alpha 2 then what is the problem the problem is this is symmetric with respect to exchange this here is symmetric with respect to exchange so the product is also symmetric but then we have just learned that for identical fermions the complete wave function must be anti-symmetric with respect to exchange so this is not an acceptable wave function the moment you bring in the spin part some of the special wave functions are not going to be allowed anymore okay sorry about that let us go ahead so now what we have then is we multiplied by the anti-symmetric combination remember there are 4 wave spin wave functions alpha alpha which is symmetric beta beta which is symmetric alpha beta plus beta alpha which is symmetric the only anti-symmetric spin wave function in our hand is 1 by root 2 multiplied by alpha 1 beta 2 minus beta 1 alpha 2 so what you are saying is that only this anti-symmetric spin part can be multiplied with this symmetric spatial part to give us an acceptable total wave function okay so this is this only spin part that is going to work all other spin wave functions are not compatible you can say with psi 1 s 1 multiplied by psi 1 s 2 because we need an anti-symmetric total wave function all right shall we go ahead let us now see what happens if the 2 electrons in 1 s orbital has same spin so we as what we imply by saying this is that when the moment we say that alpha 1 alpha 2 beta 1 beta 2 these are not going to exist it automatically says that 2 electrons in 1 s orbital must not have the same MS numbers not the same spin if I say loosely okay and that essentially is your poly exclusion principle that in an atom for an electron all 4 quantum numbers cannot be the same okay if NLM are the same at least MS should be different okay so now we are asking the question what happens if 2 electrons in 1 s orbital has the same spin so it is like this alpha 1 alpha 2 multiplied by psi he is equal to psi 1 s 1 multiplied by psi 1 s 2 so see this is symmetric with respect to exchange isn't it so since it is symmetric it does not stand the test of your 6 postulate of quantum mechanics so it cannot be taken it turns out that upon applying the exchange operator you get back the same psi 1 2 psi 2 1 wave function right no change in sign so this is not allowed by the 6 postulate of quantum mechanics the only thing that is allowed I hope you it is not very difficult for you to see is 1 by root 2 alpha 1 beta 2 minus beta 1 alpha 2 right so expand this is what you get what I have done here is I have labeled all the functions in a particular coordinate using the same color and all functions in another particular coordinate in a different color now we are already familiar with matrices because we talked about them while discussing your angular momentum now see this not sorry not matrices determinants now when you look at this do you see that it is very simple to collect the terms in the same atom sorry same electron and write this whole thing in the form of a determinant let me do it for you first of all in this 1 1 location we are going to write psi s 1 alpha 1 what is that psi s 1 alpha 1 that is a spin orbital then psi s 1 alpha 1 is multiplied by psi 1 s 2 multiplied by beta by 2 so the only way you can get it is by incorporating psi 1 s 1 sorry psi 1 s 2 beta 2 in the diagonal block of the determinant we are getting and the other 2 terms come here okay simple let us get ahead all right so this is what we have we have this determinant and now this is called a stator determinant let us have a closer look at the determinant what has happened look at row number 1 when you go from left to right what has remained the same what has changed but let us do something let us break here we will come back and continue from here in the next module