 Having discussed dihydrogen cation now it is time for us to put in another electron and talk about dihydrogen molecule and what you see here is an energy diagram which I suspect all of us have seen earlier. But just to remind ourselves this is what we discussed already. We wrote down the Hamiltonian for the H2 plus cation and there are already so many terms when we introduce another electron we have we are going to get more terms here. First of all the second electron is going to be attracted by nucleus A by nucleus B and the two electrons are also going to repel each other. So, these are the terms that come in and one thing that I forgot to say just now is that we will also have a kinetic energy term for this second electron. So, you understand it is impossible to solve this equation directly you cannot solve it analytically. So, how do we go about building a quantum mechanical description of something as simple as neutral dihydrogen molecule which has two electrons. This is how we have already discussed that even for the cation dihydrogen cation when in principle we could have solved Schrodinger equation using electrical polar coordinates we did not do that. Rather we constructed the wave function by taking a linear combination of atomic orbitals. So, the wave functions are already there what we will do and this incidentally is the same strategy that is followed for many electron atoms there the same atomic orbitals that one obtains from solution of Schrodinger equation for hydrogen atom is used with some modification for many electron atoms we will do the same thing for molecules here. So, we are going to use the same molecular orbitals and as we bring in electrons one thing that one needs to change eventually is the effective nuclear charge. So, in a very very simple treatment one needs to account for shielding and one can work out the energy. So, there are techniques of doing that we will not get into that right now here. But what we are saying is that we will use the same LCAO molecular orbitals that we had obtained for H2 plus even in the case of dihydrogen molecule. And we are going to work with the same kind of energy level diagram remember the bonding molecular orbital has an energy that goes through a minimum and the inter nuclear separation where the minimum is obtained that is the equilibrium bond length. In the anti-bonding scenario there is a monotonic increase as the inter nuclear separation keeps on decreasing. So, this here is the energy diagram for H2 plus and another thing that we would better remember is the symmetries of the orbitals. If it is symmetric with respect to inversion we call it j ready we use a subscript g if it is anti-symmetric then we use a u and usually for anti-bonding orbitals we use a superscript star. So, for sigma orbitals the bonding orbital for sigma orbital using s atoms the ones that we have studied already using sigma orbitals using s atomic orbitals that we have studied already we get sigma g anti-bonding combination gives us sigma u star and in case of pi interaction the bonding pi orbital actually has ungerade symmetry and the anti-bonding orbital has gerade symmetry. So, please do not think that bonding orbitals are necessarily gerade and anti-bonding orbitals are necessarily ungerade they are not. And also we can use the subscripts g and u only for homonuclear diatomic molecules like H2 like F2 like N2 if you want to talk about heteronuclear diatomic molecules like say HCl or HF or HI then you cannot use g and u anymore because in any case we are going to have an electron distribution that is not symmetric we will talk about that later right. But coming back to the agenda of this lecture we are trying to build a molecular orbital theoretical description of dihydrogen molecule. So, use the same orbital same molecular orbital that we get for dihydrogen plus H2 plus molecular cation we do have some changes in the forms of the functions here. But for now in this course we are not going to discuss it. So, all we have to do is that we place the second orbit electron in the bonding MO. I think we are all familiar with poly exclusion principle, poly exclusion principle as in the form that we most of us would have studied says that in an atom 2 electrons cannot have all 4 quantum numbers the same right it boils down to the fact that each atomic orbital can accommodate at most 2 electrons something similar holds even though actually it is more complicated. But for now it is enough if you understand that like atomic orbitals each molecular orbital can also accommodate no more than 2 electrons. So, we will place the second electron in the bonding orbital itself and that is how we will obtain the ground state of dihydrogen molecule. So, how do I construct the wave function now? I write like this 1 and 2 are the labels for the electrons what this product means is that electron number 1 is in the bonding orbital electron number 2 is also in the bonding orbital. So, ground state what would happen if I want to draw want to write the wave function for an excited state I would write like this and I am going to write in a little compact form here psi anti-bonding I will just write psi a h 2 excited state or rather I can write psi star h 2 that is going to be I will write I will keep this a for later I will write psi anti-bonding 1 and multiplied by psi bonding 2. But see that is not going to be enough why is that not going to be enough because who has told us that it is electron number 1 and not electron number 2 that is in the anti-bonding orbital electrons are indistinguishable we are going to touch upon this again a little later but electrons are indistinguishable. So, you cannot neglect the other term either which is psi a 2 psi b 1 remember here psi a is the anti-bonding molecular orbital and psi b is the bonding molecular orbital. So, you have to take linear combinations of the two and right away you see that I generate two different wave functions for the excited state two different kinds of excited state what is the implication let us see if we discuss that little later I have actually not prepared slides for it but maybe we can. But now one thing that is related here is when you write like this and this is the psi bonding h 2 well psi bonding h 2 means the wave function the special wave function for the molecule as a whole psi bonding 1 and psi bonding 2 here these psi bonding these mean actually molecular orbitals it would have perhaps been better to write psi plus or something but anyway that is how we have written it what to do. This is your spatial part and you can easily open it out and you are going to get a constant of 1 by 2 plus 2 s and inside the bracket you are going to get 4 terms we will come to that also. But before that it is not enough for a two electron system it is not enough to talk only about the spatial part of the wave function we have to take care of something else once again we have heard about this quantity is a spin of an electron the origin of spin of electron goes back to almost 100 years ago by Stern and Garlash performed this experiment in which they passed a beam of silver atoms through an inhomogeneous magnetic field and then they obtained two lines. So, what was going on there to explain this and that was the formative phase of quantum mechanics right. So, Good-Smith the way Good-Smith puts his story and that story can be actually read here and a more neutral account can be read in the second link here. So, Good-Smith and Ulenbeck they sort of collaborated and they figured that the only way to explain this doublets Good-Smith had this fascination for doublets the only way they found they could explain this kind of a doublet structure is by considering what they called a fourth degree of freedom for the electron and that fourth degree of freedom is the spin quantum number NLM we have studied right from these experiments it was understood that you need a fourth degree of freedom of fourth quantum number and Good-Smith actually thought that it is because of electrons that are spinning about their own axis because this is the time when old quantum theory is at its prime and new quantum theory is just about taking over. So, they thought of extending the classical analog a little further we know the earth rotates about its axis and also revolves around the sun. So, they thought that maybe electrons spins a little bit and immediately this was at the proposed the published paper immediately it met with severe criticism of a lot of people like Pauli and Ulenbeck himself was very very uncomfortable. So, because it was quite obvious by doing a very back of the envelope kind of calculation that the kind of energies we are talking about if it is due to spin of an electron then the electron has to spin with a velocity that is more than the speed of light speed of the electron linear speed of the electron has to be more than the speed of light which is not going to happen. So, only later when Dirac formulated his relativistic quantum mechanics when I say later it I do not mean 20 years 30 years later but later than the first report when Dirac combined this relativity theory with quantum mechanics in a more rigorous fashion see relativity and quantum mechanics were never divorced from each other even Sommerfeld's modification of Bohr theory was a relativistic correction but if a still part of old quantum theory Dirac using this four particular matrices could bring in relativity in quantum mechanics in a much more rigorous fashion and it was established that it is essential in that kind of treatment to consider something that some angular momentum which is beyond NLM. So, this angular momentum is associated with something called spin for historic reasons nothing else but it is important to understand that it is not really arising out of a spinning electron it is a spin with respect to some unknown coordinate S omega does that sound very mysterious enigmatic to the point of being unbelievable if it sounds like that you are not the only one who has felt that way anybody and everybody who is exposed to this for the first time would better feel this way if somebody just accepts this then that means that person is not thinking it is a very disturbing thought and to be honest if you want to understand what exactly spin is you need to study advanced quantum mechanics course not with us chemists but with physicists. So, for now we will take it axiomatically that there is an additional angular momentum associated with something called spin with the understanding that this spin is not really what we can think of the electron spinning like this. However, this angular momentum is such that you can do all the mathematics associated with it. You can write the magnitude of the angular momentum as h cross multiplied by root over s into s plus 1 where s is the spin quantum number does that ring a bell the expression is similar to the expression for the orbital angular momentum right when we did nlm the theta dependent part of Schrodinger equation hydrogen atom we obtain the magnitude of the angular momentum to be h cross multiplied by root over l into l plus 1 very similar expression. The difference is this that l depends on which energy state the electron is in in the hydrogen atom what do I mean l depends on n right n is what determines the energy state. So, if n equal to 1 l can be only 0 if n equal to 5 then l can be 0 1 2 3 4 so on and so forth. So, that is what it is however take any electron it has only one spin quantum number and that is half. So, perhaps this is where some of us should object because we have studied in 11th well that electron can have a spin quantum number of plus half and minus half my submission here is that that thing that we studied in the name of spin perhaps is not really spin quantum number it is ms remember the magnetic quantum number that we studied nlm here this is an analog of m since it is about something to do with spin we call it ms the magnetic quantum number associated with spin. So, and that gives us an idea about the z component of the spin angular momentum maddening maddening is not it we are saying that this spin is about some imaginary coordinate not about xyz not about our known nothing yet in real space in Cartesian space we can draw an arrow that can give us an idea of not only the magnitude of the angular momentum associated with spin but also its orientation why because it is also associated with the magnetic quantum number ms which like the magnetic quantum number we have encountered earlier can go from plus s to minus s. So, 2 s plus 1 values would be there now for all electrons in the universe spin quantum number is half. So, how many different values of ms would be there 2 into half plus 1 that is 2 what are those 2 values plus half and minus half what do they designate the designate the z component of spin angular momentum that would be h cross multiplied by ms. So, for an electron any electron whether it is bound to an atom or whether it is free you have 2 different kinds of z components of angular momentum is h cross by 2 minus h cross by 2 and since the length is specified root over h multiplied by s into s plus 1 and the z component is also specified this angle theta is specified as well what is theta going to be well that is not very difficult to figure out I think cos theta is going to be well I am just working with this theta you can obtain this theta dash by just subtracting the same value of theta from 180 degrees. So, cos theta what is cos theta that will be equal to h cross by 2 divided by root 3 h cross by 2 root over s into s plus 1 remember. So, we are left with 1 by root 3. So, what is theta equal to cos inverse 1 by root 3 you figure out how much it is I will not tell you and when you figure it out think whether you have encountered this quantity somewhere else or not. So, space quantization is there for the angular momentum associated with spin even though spin is not spin is not electron spinning about its own axis very often for the sake of understanding you would see people discussing things in perspective of electron spinning please remember that all that matters is angular momentum we cannot really say that electron is spinning about its axis great. So, this is what we have discussed only one more thing to say the 2 wave functions associated with spin up or spin down that is eigenvalues of h cross by 2 and minus h cross by 2 they are alpha and beta can you write them out in some functional form like you did for hydrogen atom wave functions no you cannot because they are functions of omega and we do not even know what omega is we just call them alpha and beta and that is what we work with spin. Why are we saying all this because we are dealing with the 2 electron system and in 2 electron system we have used this electron labels 1 and 2 right. So, suppose both the electrons have a up spin and this is experimentally determinable then the spin wave function would be alpha 1 alpha 2 no problem with that similarly both electrons have down spin wave function would be beta 1 beta 2. The problem arises when you have a situation where one electron has up spin one electron has down spin what is the wave function? Well alpha 1 beta 2 that can be a wave function who has told us that it is not beta 1 alpha 2. So, this is the same situation that we encountered few minutes ago when we tried to talk about the special part of the excited state of dihydrogen molecule see electrons do not wear jerseys right they do not have number 1 and number 2 branded on them they do not even know that I am number 1 and your number 2 they are indistinguishable you cannot distinguish between the 2. So, while it is experimentally determinable that the 2 electrons have opposite spins it is impossible to tell whether it is electron number 1 or electron number 2 which has up spin or down spin. So, when you have this situation opposed spins then you must take a linear combination you cannot tell which is which. So, you have to take linear combination while taking linear combination plus and minus both are equally valid and of course you need some kind of a normalization constant. So, you get 1 by root 2 alpha 1 beta 2 plus beta 1 alpha 2 1 by root 2 alpha 1 beta 2 minus beta 1 alpha 2. So, you see 4 different wave functions are possible for a 2 electron system now we will introduce something and that is the exchange operator exchange operator means you just interchange the labels instead of 1 you write 2 instead of 2 you write 1. If you see if I interchange the labels what happens well first of all even before going there alpha 1 alpha 2 if I interchange labels will the wave function change no it would not beta 1 beta 2 same alpha 1 beta 2 would actually have changed alpha 1 beta 2 would have become alpha 2 beta 1 but it is not a valid wave function anyway if you take the linear combination the first one the plus combination there if I interchange 1 and 2 then what do I get alpha 2 beta 1 plus beta 2 alpha 1 it is the same thing. So, alpha 1 alpha 2 beta 1 beta 2 and alpha 1 beta 2 plus beta 1 alpha 2 are all symmetric with respect to exchange whereas the last wave function is such that if you interchange 1 and 2 what will happen if you get alpha 2 beta 1 minus beta 2 alpha 1 which is minus of the original wave function. So, the last one the minus combination is actually antisymmetric with respect to invert exchange. Why are you talking about this what is there how does it matter if they are indistinguishable and how does it matter if some wave function is symmetric and some is not it matters because of sixth postulate of quantum mechanics which also goes by the name poly principle not poly exclusion principle poly exclusion principle follows from here poly principle it essentially says that the complete wave function of a system of identical fermions must be antisymmetric what are fermions fermions are fundamental particles with half integral spins. So, electron is a fermion what are bosons bosons are fundamental particles with integral spins well photons for example are actually bosons and I hope you know that bosons are named after an Indian scientist Professor S. N. Bose and Professor Bose is very well known because of his Bose Einstein statistics that he had formulated along with Einstein. I encourage you to read how this Bose Einstein statistics began it began from a simple curiosity Bose just wanted to derive Planck's law in a very different in a different way and this is correct because Bose himself has gone on record several times saying it that is only intention was to derive Planck's law in a different manner he was very good at derivation but that was the beginning of Bose statistics Bose Einstein statistics which led to things like Bose condensation and paved the way of things like superconductivity let that be the story for another day and many engineers as well as basic scientists would be interested in those but for now we just need to know the definition fermions are particles with half integral spins bosons are particles with integral spins electrons are fermions and the postulate or poly principle says that for identical fermions the total wave function complete wave function that means spatial part into spin part must change sign if the labels are exchanged and this is nicely demonstrated if you try to write down the helium atom wave function ground state in helium atom what do we have we have two electrons in the 1s orbital so something like this psi 1s 1 psi 1s 2 very similar to our hydrogen molecule wave function isn't it now that has to well that special part of helium atom wave function as you see is symmetric it does not matter if you exchange 1 and 2 you get the same wave function so in order for the complete wave function to be antisymmetric the only option is to multiply it by the only antisymmetric wave function antisymmetric spin wave function we have at our disposal alpha beta minus beta alpha right so this is what it is what happens if the two electrons in 1s orbital have the same spin I will write alpha 1 alpha 2 special part is well defined this is symmetric right so that is not allowed by sixth postulate of quantum mechanics and that is what leads to poly exclusion principle which says that in the form that we have studied it says that in an orbital the two electrons must have different spins actually poly exclusion principle is more general than that and it says that for a system of two fermions in a quantum mechanical system you are going to get a situation where the complete wave function must be antisymmetric that is what it is okay now knowing that we will come back and plug this into molecular orbital theory look at the ground state wave function of a dihydrogen of course it is symmetric so of all these four spin wave functions that we have the only one that works like in helium atom is the one with the minus combination so this is the complete wave function of dihydrogen molecular dihydrogen molecule okay so this is singlet what would happen if I think of the excited state what would happen if I think of the excited state so as we wrote I will write something like this psi star one that has to be 1 by 2s multiplied by what is the combination we use we use the minus combination isn't it so phi 1s a 1 minus phi 1s b 1 let us see electron number 1 is in the anti-bonding orbital and electron number 2 is still in the ground state again it will interchange actually phi 1s a 2 minus phi 1s b 2 and you can write it in different ways you can expand and you will see that you have access to singlet as well as triplet this state is a singlet because there is only one wave function but if the special part of the wave function is symmetric then you can multiply it by the antisymmetric if the special part is antisymmetric then you can multiply it by the symmetric wave functions and there are 3 such possibilities alpha 1 alpha 2 beta 1 beta 2 alpha 1 beta 2 plus beta 1 alpha 2 that is what takes you to the excited triplet state of dihydrogen the ground state of dihydrogen molecule is definitely singlet okay now one last thing that we wanted to discuss about dihydrogen molecule is just forget the spin part for the moment look at the special part expand it this is what you get and I will write it in a another way instead of writing phi so many times I will just write 1s a 1s b so on and so forth now look at the first two terms the first two terms tell you 1s a has electron number 1 1s a itself has electron number 2 the second term tells you 1s b has electron number 1 1s b itself has electron number 2 third term tells you electron number 1 is in 1s a electron number 2 is in 1s b fourth term tells you electron number 1 is in 1s b electron number 2 is in 1s a so the last two terms are okay they tell you about the neutral dihydrogen molecule what about the first two terms there both the electrons are in the same atomic orbital either 1s a or 1s b so that represents the ionic terms so it takes into account that h plus h minus can also be there but the problem is that it overemphasizes ionic terms of course in dihydrogen molecule you do not have too much of ionic form but here it seems that contribution of ionic and covalent forms are the same which is a mistake so this is the problem with molecular orbital theory the good thing is that it is a general theory the bad thing is that it is too general the theory so sometimes it overdoes things you have to be careful that being said let us now fill in these different electrons let us see what is the situation for different diatomic molecules based on h2 plus so h2 plus we have discussed already h2 we have discussed already what are the bond order and bond order h2 plus has a bond order of half because you have only one electron in the lower energy molecular orbital h2 has a bond order of one and look at the bond length bond length experimentally determined bond length is 106 picometer for h2 plus only 74 picometer for h2 why the second electron actually effectively shields the first one from the nuclear charge okay well rather the electrons shield the nuclei from each other that is why they can come a little closer and you see binding energy is so much more what happens in h2 plus bond order is the same as that of h2 plus but bond length is a little longer binding energy is a little lesser why because now in addition to these two electrons you have an electron in the anti-bonding orbital and destabilization of anti-bonding orbital at equilibrium bond length is actually a little more than the stabilization of the bonding molecular orbital that is why that is the problem and what about h2 plus h2 plus we already discussed h2 what about h2 in h2 you have two electrons here in the bonding orbital two electrons there in the anti-bonding orbital bond order is 0 but energy is not 0 it is still a little less than 1 and actually you get an energy minimum at much longer distance 6000 picometer why why is it that at 6000 picometer we have this so 1 nanometer 1 nanometer is huge for an atom why is it that we have a minimum that is because of inter atomic interaction all right so this is what we learned for the atoms of first row in the next lecture we will talk about homonuclear diatomic molecules once again but of the second row that will be a shortage lecture