 Actually I thought all this material will be complete in one module but as usual I my estimate was not right. So we will just start from here it is okay. So see we are talking about helium atom wave function, many electron atom wave function and we have learned this elegant way of writing the wave function in the form of a slater determinant. Let us have a look at the slater determinant. Look at the first row. What is the constant and what is varying? Electron number is constant is not it? Psi 1 S 1 alpha 1 Psi 1 S 1 beta 1. What has remained same is the electron number and what has changed? Even Psi 1 S is same what has changed is from alpha we have gone to beta. So if you look at the whole thing what has really changed is your spin orbital. Similarly in the second row go from first column to the second one electron number remains 2 but from Psi S 1 sorry Psi S 2 alpha 2 Psi 1 S 2 alpha 2 we go to Psi 1 S 2 beta 2. Okay so on going from left to right in slater determinant these spin orbital changes and the electron label remains the same and upon going from top to bottom then what happens Psi S let us look at the first column now Psi 1 S 1 alpha 1 Psi 1 S 2 alpha 2. So Psi 1 S alpha that is your spin orbital that has remained same when you went from top to bottom what has changed is the electron label has changed from 1 to 2 and same here. So for the record let me write that in many textbooks slater determinant is written in a slightly different way it means the same so you should not get confused but very often the same thing would be written like this. So you write Psi 1 S alpha and then write 1 Psi 1 S alpha then write sorry Psi 1 S alpha 2 and this one is Psi 1 S beta 1 this is Psi 1 S beta 2. This is another way of writing the same thing matter of convention nothing else please do not get confused if you read a textbook that uses this kind of notation instead of this as you can see they mean exactly the same but in one way the second way of writing is better because here it is very clear Psi 1 S alpha Psi 1 S alpha same spin orbital you have changed from 1 to 2 here also Psi 1 S beta Psi 1 S beta you only changed from 1 to 2 and go from left to right 1 is same Psi 1 S alpha has become Psi 1 S beta. So maybe this is actually easier to read but we are going to follow this convention for now. But what is the advantage of writing the wave function all of a sudden in terms of a determinant first of all as we are going to say very soon eventually when you talk about complex systems atoms or molecules you have to use computers computational chemistry is a very very big field nowadays not nowadays and it has been a very very big field for several decades now because you cannot do all this calculation by hand you want to find energy right now we are working within the ambit of orbital approximation as you will see there are ways to actually generate these wave functions without resorting to 1 S 2 S also then you might want to relate them that is a different issue how do you do all that all that is done using computer and computer can work only when you give it the input in a very nice manner or if you let it keep provide the outputs in nice organized manner that is why matrices determinants these things come very handy because they are essentially data arranged in array forms right. So that is one thing second thing very elegantly something nice comes out if you go back to the properties of determinants in mathematics those of you ask who has studied determinants in mathematics would know that if you exchange two rows or two columns the determinant changes sign okay not very difficult to understand because determinant means if you write a b c d it is a d minus b c just interchange it is going to change sign obviously there is a talking about this determinant a b c d this is well forgive me for writing d a I am feeling lazy and do not want to erase d a d are the same minus b c now suppose I just interchange the two rows then what happens you write c d first row a b in the second row what do you get b c minus since I have written d a there I will write d a you see what happened if you call this d 1 and you call this d 2 then obviously d 2 is equal to minus d 1 satisfy yourselves by interchanging two columns that the same thing happens once again so exchange of two rows or two columns leads to change in sign of determinant so what which means that they are anti-symmetric now we are talking remember our total wave function has to be anti-symmetric okay so in any case in whichever form we write it would we have we are working with wave functions but now by property of determinants you are always going to get anti-symmetric functions so you cannot make a mistake while writing it by mistake I write plus 1 here the determinant will not be the same right you cannot write the determinant that way so that is where we will understand that we have made a mistake right so or since the determinants change sign upon exchange of two rows and or two columns when you write the wave function in the determinant form you are assured that you have written an anti-symmetric wave function you do not have to worry okay another property is if any two rows are the same or any two columns are the same then the determinant becomes 0 once again since we have promised some of students to do things from scratch by and large we will do it here I have a matrix I call them d 1 d 2 so determinants so not matrix already determinant I have this bad habit of referring to determinants as matrices please do not get confused as far as this part is concerned we are only dealing with determinants so if I say matrices please correct yourself and understand that I am talking about determinant I do sometimes I am sufficiently careless to use this terms interchangeably but you do not do it so I will call this determinant d 3 what are we saying we want to prove that if any two rows are the same or any two columns are the same then the determinant is 0 so let us write two rows to be the same what do I get a b minus a b obviously that is equal to 0 so if two rows are the same or two columns are the same you do the column you would write a determinant with two identical columns and satisfy yourselves that this holds the determinant becomes 0 that leads to something very very interesting and something that we are familiar with something that we have been using axiomatically all the time if we are going there let me remind you where we started from we started from poly principle or this sixth rule of quantum mechanics sixth postulate of quantum mechanics which said essentially that for fermions like electrons the total wave function has to be antisymmetric since it is antisymmetric you can write in determinant form now since in a determinant if two rows are the same or two columns are the same that determinant becomes 0 it is essential that no two electrons can occupy the same spin orbital what am I saying here no two electrons can occupy the same spin orbital so what is the meaning of two electrons occupying the same spin orbital so it will be psi 1s alpha 1 and psi 1s alpha 2 at the same time right psi 1s alpha 1 psi 1s alpha 2 and the other one will be psi 1s beta 1 psi 1s beta 2 simultaneously so you try and do that you will get a 0 determinant okay that is what leads to our very familiar poly exclusion principle poly exclusion principle what we might have learnt earlier excludes the possibility that in a particular atom all the two electrons can have all four quantum numbers are the same if n l m are same at least m s has to be different this is what we studied qualitatively what we are saying now in a little more quantitative manner is that no two electrons can occupy the same spin orbital if electron number 1 has occupies psi 1s alpha spin orbital then electron number 2 must necessarily occupy psi 1s beta otherwise you get a 0 determinant which means the wave function is 0 if wave function is 0 then probability density and therefore probability of finding the electron anywhere is 0 and then what are we talking about yeah so very beautifully using this concept of stator determinants we arrive at poly exclusion principle okay so that is one great thing but I made things I kept things very very simple so far we have only worked with helium we have only worked with the two electron system what happens if I want to include more electrons for many electron atoms you can write similar determinant wave functions which means you can write the wave functions as slater determinant and here I have shown you one for an n electron atom okay let us see what the determinant would look like look at the first row the first row we have kept the label the same as usual and spin orbitals have changed from phi 1 alpha phi 1 beta next one will be phi 2 alpha next one will be phi 2 beta so on and so forth okay we are going to say something what were they going to say first of all two things yes now I remember first of all why phi 1 alpha first why not phi 1 beta the answer is convention okay we have to speak the same language we do not want to create a tower of Babel where if you do not know what the tower of Babel is just do a Google search it is a biblical beautiful biblical story that you can learn a tower of Babel essentially means where one does not understand the other person's language so we must formulate things in such a way that we understand each other's language so just convention that is the need of convention that is point number one that is why we write phi 1 alpha first and not phi 1 beta first point number 2 why is it that we keep the label constant while going from left to right in a row we might as well have kept label constant while going from top to bottom in a column it is a determinant determinant will remain unchanged if we just transpose it yeah so why do not we do it the answer once again is convention see Slater wrote it this way and Slater is a famous scientist after whom these determinants have been named so until we become more famous than Slater and until the scientific community is agreeable to accept our convention over Slaters we better follow Slater's convention that being said there are textbooks in which the opposite convention transpose convention is used actually they are one and the same but since we do not want to create a tower of Babel for the purpose of this course let us write Slater determinants in this way upon going from left to right the electron level remains the same in a row and the spin orbitals change going from lower energy to higher energy and upon going from top to bottom the spin orbital remains the same level change from 1 to 2 to 3 and so on and so forth up to n all right what is the last one here this is Phi m why not Phi n because remember each spin orbital for any given orbital I can generate two spin orbitals right one with alpha one with beta so m has to be less than n so is m equal to n by 2 yes provided n is even understand what I am saying each so Phi 1 gives you two spin orbitals Phi 2 gives you two spin orbitals Phi 10 gives you two spin orbitals so up to Phi 10 how many spin orbitals do we have 20 right and what will the value of m be if n equal to 10 5 well 10 all right now what I am saying is that let me say that once again I think I made a mistake so let us say I am working with a 10 electron system so what 10 electron system what will happen you will have Phi 1 alpha 1 sorry Phi 1 alpha 1 yes and then next one will be Phi 1 beta 1 next we will have Phi 2 alpha 1 Phi 1 alpha 1 next we will have sorry Phi 2 alpha 1 next we will have Phi 2 beta 1 then we will have Phi 3 alpha 1 next we will have Phi 3 beta 1 and so on and so forth how far will we go we have already accounted for 6 electrons when we went up to Phi 3 so to account for 10 electrons we have to go up to Phi m Phi 5 so in this case m is equal to n by 2 what happens if n equal to 11 where will I stop Phi 1 alpha 1 Phi 1 beta 1 Phi 2 alpha 1 Phi 2 beta 1 Phi 3 alpha 1 Phi 3 beta 1 so on and so forth up to Phi 5 alpha 1 Phi 5 beta 1 okay so 10 electrons are accounted for one is left next one will be Phi 6 1 alpha 1 Phi 6 1 alpha 1 okay so this m is going to be either n by 2 or n by 2 plus 1 depending on whether n is even or odd in the preceding couple of minutes I might have said things that might have confused you a little bit but I mean you just do it yourself it should not be difficult okay it will not be difficult so this is how we write later determinants let us show you one example for lithium what will happen how many electrons are there 3 so you write like this 1s 1 alpha 1 1s 1 beta 1 2s 1 alpha 1 then 1s 2 alpha 2s 1 and so forth finally the third row is for electron number 3 is this the complete wave function this is something that I want you to take as homework what I am asking is this wave function I have written first of all is it normalizable secondly is it complete or do I need some other term okay I said that as a matter of convention I use alpha but is beta not equally probable how do I incorporate that is there a need to incorporate that these are things that I would like you to work out by yourself but to conclude this part of the discussion this is what we have learned that you can write it in determinant form any 2 rows of columns same then the determinant would become 0 once again I would like you to take this as homework set one of these 2 I mean what I am saying is put electron number 3 in 1s orbital then what happens then this becomes electron number 3 in 1s orbital so this 2s will be replaced by 1s everywhere then what happens the first row and the third row they became one and the same what I am saying is the third electron actually when I see electron number 3 that can be a little confusing for you because you might think that I am talking about the level I am not talking about the level I am saying the third electron okay in whichever way we are filled in 1 2 3 3 2 1 2 1 3 whatever what I am saying is let us say that the third electron also goes to 1s then this last row is going to become sorry last column is going to become 1s 1 alpha 1 1s 2 alpha 2 1s 3 alpha 3 okay now what has happened this 1s 1 alpha 1 1s 2 alpha 2 1s 3 alpha 3 is exactly the same as the first column so the 2 columns are same therefore the determinant is 0 so here we have shown that even for lithium poly exclusion principle follows very nicely from this 6th postulate of quantum mechanics so we have learned Slater determinant and we have learned how to express the wave function of the ground state of helium and lithium using Slater determinants in the next module we want to see whether we can extend this discussion to excited states of these atoms