 We have discussed in detail the acceptable wave functions of hydrogen atom and we said that these are called orbitals for the umpteenth time in this course orbitals are acceptable solutions for one electron acceptable one electron wave functions acceptable solutions for Schrodinger equation for a one electron or hydrogenic systems. So we have learned how to plot them we have learned how to draw 3D and contour diagrams like these and we also said what happens when you draw with pen and paper qualitative sketches that also we learned that you draw the nodes first and then every time you change a node sign must change that is how so what you see here I am sure you remember is a 3D depiction of 3 pz orbital along with the contours. And now the question that arises is we have put in so much of time and effort to learn about atomic orbitals of what use are they I mean it is not as if we want to only study hydrogen. Hydrogen is the simplest atom we want to talk about atoms with many electrons we want to talk about molecules so of what use are the simple one electron wave functions are they of any use as we will discover they are of some use a lot of use at least at a starting point. Eventually of course when things get more complicated we move on and we move on to more complex systems more complex solutions but to start with atomic orbitals actually have an important role to play and for a lot of chemists orbitals actually make this very complicated quantum mechanical business very simple because they can be presented in the form of pictures and you can develop qualitative theories just based on the symmetry of orbitals what kind of distribution is there in space and so on and so forth so orbitals are actually very very important in chemistry. So we start our discussion now about multi-electron atoms and there the first approach we will discuss is orbital approximation. So what you see here is a very popular depiction of multi-electron atom with three electrons even nucleus of course this very good looking picture is from Bohr-Sommerfeld model so right now it is not really state of the art but it finds use in depiction of science in many many different platforms I leave it to you to find out which major government agency has this picture or at least its adaptation in its logo. So but to continue our discussion we have learned what sense to make of the surface plots and one thing that I should say before moving out from this topic is that very often you see pictures like this and they are useful as I said because they present a qualitative picture but please remember that what you see here most likely is the region of space in which finding the probability is maximum so you decide how much of probability you want to include as you have seen discussed earlier it is not only radius theta also matters phi does not matter because you get rid of phi when you generate real orbitals. So theta also matters and from theta and r you can figure out at what are and what values of theta what kind of intense of probabilities are there and you can construct the surfaces which contain the maybe 90 percent 95 percent 99 percent of probability and additionally what you do is you show the charge of sorry the sign of the wave functions on the appropriate lobes but please remember these are not really orbitals even though they are called so they are qualitatively similar looking but these are regions of space orbitals as you know are actually one electron wave functions and one danger of this is that very often you see popular figures like this where earlier one was orbital outer structure this one is inner structure the purpose is to show the nodes and all in fact you can buy structures made of styrofoam using this there is a chemical general chemical education paper in which it says how you can cut open a styrofoam structure to actually see the nodes the problem is you debit like this l equal to 1 m equal to 1 one might think that m is equal to 1 here that is not what the meaning is the meaning is these real pictures that are there for n equal to 2 3 and so on and so forth they are all generated by taking appropriate linear combinations of m equal to 1 and m equal to minus 1 orbitals because as you understand e to the power i m phi that is the phi part so you cannot really draw an imaginary function in real space same is true for say l equal to 2 m equal to 1 l equal to 2 m equal to 2 if you just take m equal to 1 or m equal to minus 1 or m equal to 2 m equal to minus 2 they are imaginary functions these are actually linear combinations that you get right and that is how you generate them so please remember that I hope we will have no confusion about what orbitals really are and what these pictures really are from this point on but now we come back to our disturbing question hydrogen atom has 1 electron fine first of all why will we even bother to spend 3 modules on so many orbitals of what use are they and the answer is that hydrogen has 1 electron true it occupies only 1 s orbital in the ground state true but do not forget one of the origins of quantum mechanics is hydrogen spectrum the moment you talk about spectroscopy you get involved with excited states which means somehow you have to promote that electron from 1 s to maybe well whichever orbital the energy takes it to and then if it is emission spectroscopy it has to come down so if you want to talk about excited states you better know about the wave functions that are involved in the excited states and without them you cannot really talk about spectra and what we are going to discuss now is these orbitals with a little bit of adaptation can actually be used for many electron atoms as well to some extent and you already know that without me having to tell you so what is the configuration of helium you will say 1 s 2 how 1 s is an orbital 1 electron wave function so how is it that for helium you are saying 1 s 2 lithium beryllium nitrogen oxygen if I just tell you the element you will be able to rattle off the electron configuration like 1 s 2 2 s 2 2 p 3 is 1 and so forth how is it that this 1 s 2 s 2 p 3 s 3 p 3 d 4 s how is it that these 1 electron wave functions are being used happily in talking about electron configuration of multi electron atoms how is it that we talk about sp hybrid orbitals when we talk about bonding there are many many electrons in molecules anyway that is what we will learn slowly to start with let us keep things as simple as possible let us start with helium helium is the simplest many electron atom that one can think of and if I draw a very rough model of helium we always start from there we draw simple pictures here we have the nucleus this is electron number 1 this is electron number 2 do the electrons know do the electrons know that they are number 1 or number 2 no they do not electrons do not wear jerseys electrons do not have numbers written on them we are putting the numbers for our convenience so that we can formulate the problem but electrons are actually indistinguishable we are going to come back to this point several times later in our discussion but right for now this is what you have you have this position vector of electron number 1 r 1 position vector of electron number 2 r 2 or you can just think separation and separation between the two is the vector sum of r 1 and r 2 here we have written r 1 minus r 2 not really vector sum we subtract one vector from the other so how do we write the Hamiltonian for a system like this for any quantum mechanical problem what you need to do is you need to write the Hamiltonian then you need to think if you can write the wave function in some way then you can think of how you can solve it okay so this is what the Hamiltonian is going to be I have shown you the entire thing at one go but every term is actually labeled with a different color let us go one by one the first term we have written is minus h cross square by 2 m n del capital N square capital N for nucleus this is the term for the kinetic energy of the nucleus and as you can well imagine understand that we can do separation of variables so that we can express things in terms of a center of mass coordinates and relative coordinates this kinetic energy of nucleus will be expressed completely in center of mass coordinates and we will not worry about it the second term of course see I have written it this way you can write it in any order does matter okay second one that I have written is minus h cross square by 2 m e m e means mass of electron del e square okay that is your I think sorry this is del one square so this is the kinetic energy of electron number one next one will be exactly the same thing except for the fact that instead of one we have written 2 what about m e does m e change no they do not the electrons were the same mass is the same for both the electrons we do not write m e 1 and m e 2 makes no sense but you have to write del 1 and del 2 because x 1 y 1 z 1 x 2 y 2 z 2 these are distinct r 1 r 2 these are distinct great so so far we have kinetic energy of nucleus kinetic energy of electrons kinetic energy electron number 1 electron number 2 then you have to think of the attraction between nucleus and the electrons first one minus 1 by 4 pi epsilon 0 z n nuclear charge e square divided by r 1 this stands for attraction between nucleus and electron 1 next term is exactly the same once again 1 replaced by 2 so attraction between nucleus and electron 2 there is a type who here is not electron 1 electron 2 please correct it yourself so far so good this is just an extension of what we did for hydrogen now comes the additional term the term is e square divided by r 1 2 this stands for repulsion between electron number 1 and electron number 2 as we will see this becomes a major player and major headache in the subsequent discussion and a lot of our efforts going to how to account for this great so what you do is first of all you separate the nuclear and electronic coordinates and while doing that you write the terms in electron number 1 together write the terms in electron number 2 together so write instead of 1 by 4 pi epsilon 0 like what we did in hydrogen atom we write q and this is what we get out of this the first term is basically the nuclear Hamiltonian do not worry about it we only worry about the electronic part of the Hamiltonian and that is because the Hamiltonian for the nuclear part operates on the nuclear part of the wave function you take the wave function to be a product of a nuclear part and electronic part like in hydrogen atom and you just do the separation exactly same treatment as hydrogen atom so we will not discuss it again what we now start worrying about is this electronic part of the Hamiltonian for helium atom okay how many terms do you have 1 2 3 4 5 first term kinetic energy of electron 1 second term potential energy of electron 1 due to the nucleus third term kinetic energy of electron 2 fourth term potential energy of electron 2 due to the nucleus the last term is what you would not get in a modern electron system the electron electron repulsion great so I have color I have labeled the terms in 1 and those in 2 I cannot put label here I mean I can but it has to be a different color because this r 1 2 involves coordinates of r 1 as well as 2 so cannot be separated so easily so what you see is that the first one this one is essentially a 1 electron Hamiltonian for electron number 1 is not it minus h cross square by 2 me del 1 square minus q z n e square divided by r 1 exactly what we had encountered in hydrogen atom and the second one is also exactly the same as what we encountered in the hydrogen atom but for electron number 2 1 electron Hamiltonians which is sort of a relief but then what we need to also understand is that see these Hamiltonians are all operating on wave functions that are more complex than what they were for hydrogen so each wave function here say let us take the first one psi e this is a function of not only r 1 theta 1 phi 1 it is also a function of r 2 theta 2 phi 2 because that electron electron repulsion is there right 2 electrons are there same charge they will repel each other so we have more number of coordinates in these systems right to simplify what we do is to start with we invoke what is called orbital approximation see we have learnt about atomic orbitals with so much of great effort so it makes sense for us to try and retain them to the maximum extent possible so what we do is we write the wave function of any of these as a product of 2 1 electron wave functions electron number 1 electron number 2 and if you had n number of electrons here we would have written it as product of phi 1 phi 2 so on and so forth phi n we can always write products as we have discussed while talking about separation of variables earlier these are in different dimensions you cannot add them but you can take products and the good thing is since Hamiltonian is a derivative Hamiltonian contains a d 2 d q 2 kind of term you take derivative then derivative of products as you know is a sum so the eigenvalue energy nicely separates into different parts also but will it separate here we will see okay so this is where we are now what I have done is I have collected everything in 1 collected everything in 2 okay so what we have is h 1 operating on psi 1 e multiplied by psi 2 e plus h 2 operating on psi 1 e multiplied by psi 2 e plus q e square multiplied by the product is what you have okay now what is h 1 psi 1 e r 1 psi 1 e which is function of r 1 theta 1 phi 1 h psi we know Schrodinger equation h psi equal to e psi so this will be epsilon 1 where epsilon 1 is the energy of this is associated with this wave function okay and we are talking about helium remember so this wave function is essentially a 1 s wave function so we write epsilon 1 what about here and what about the second term psi 2 e function of r 2 theta 2 phi 2 it is going to be a constant as far as this h 1 is concerned so it will come out once we have got this epsilon 1 multiplied by this we can rewrite it in this order you are quite simple things we have done many times second one also h 2 for h 2 this psi 1 e is constant goes out but h 2 does operate on psi 2 e to give you once again we can write epsilon 2 psi 2 e but then epsilon 1 and epsilon 2 are actually they have the same value minus 13.6 z square by n square e v same energy as that of 1 s electron of hydrogen yeah why do you get it because we are operating a 1 electron Hamiltonian on a 1 electron wave function whether we write 1 or 2 how does it matter okay so this is what we have got left hand side h psi right hand side e psi so this is your e this is the energy of the system and we write the product as a product of 2 1 s orbitals 1 in terms of electron number 1 1 in terms of electron number 2 okay everything looks uncutory so far then that is because we have so far not thought how the situation gets modified because of the presence of q e square by r 1 2 okay that is the problematic part that we have conveniently kept aside until now but we cannot do that any longer we have to address it to address it the first thing we can do is we can pretend that it is not there okay we have a problem I mean I do not you might know the 7 stages of acceptance of some bad news or 7 stages of grief the first stage is always denial so right now we are in denial we pretend as if that q e square by r 1 2 term is not even there knowing fully well that this is not going to hold water but come on you have 2 electrons they will repel each other strongly they are confined in a small place still we have to start somewhere even if it is to prove that this approach is wrong we have to start so we do that let us set it to 0 for now so this is the wave function this is a equation you get very nice h e psi e equal to epsilon 1 plus epsilon 2 multiplied by psi e very nice so this here is your eigen function eigen value epsilon 1 plus epsilon 2 we know the values for hydrogen atoms so we expect minus 108.8 electron volt this is the theoretical value that we get using orbital approximation and neglecting this electron-electron repulsion term the problem is if you do an experiment experiment that is pertinent here is photoelectron spectroscopy for now we take a rain check on discussing photoelectron spectroscopy when we talk about molecules we will have the scope to at least tell you how it works for now just believe me when I say that this is an experimental way of determining energy the energy turns out to be minus 78.99 electron volt and the theoretical value that we get is minus 108.8 electron volt now again we are going to talk about it in a little more detail later on but there is something called variation theorem which says that your theoretically observed value can never be less this is negative value stabilization can never go beyond the experimentally observed value because experimental value is the truth this is what Max Planck has said and I always end up quoting this it has become a cliche in my courses Max Planck the great Max Planck had said that experimental results are the only truth everything else is poetry and imagination not to undermine poetry and imagination they set human beings apart from other animals but then science is a pursuit of truth to do that you must do experiments and you must do experiments properly you must not make mistakes and you must have confidence in your experimental results because they reflect what the situation actually is theory is a way of understanding it reaching it explaining it so obviously the theory that we have used is inadequate we cannot wish this q e square by r 1 2 term away we have to find a way of accommodating it somehow within the ambit of our theory how do we do that before saying that let me write this same problem in a little more general terms for many electron atoms what is the Hamiltonian minus h cross square by 2 m n this is common next we write it as a summation minus h cross square by 2 m e sum over i equal to 1 to n we are assuming an n electron atom del i square so this is the kinetic energy of each atom summed over sorry kinetic energy of each electron summed over all the electrons next we have minus q z n e square sum over i equal to 1 to n 1 by r i what is this this is the sum of potential energy of all the electrons due to their presence near the nucleus so this term accounts for electron nucleus attraction finally we have our problematic term and it it is it has to be a double summation because if you take i equal to 1 j can be 2 3 4 whatever if you take i equal to 500 and if there are 500 electrons j can be 1 2 3 4 so on and so forth up to 499 then 500 once onwards so i equal to 1 to n j i would prefer to write j is not equal to i okay this is the problematic part this one as usual we do not worry about there is a nuclear part you worry about the electronic part of the Hamiltonian I hope you see like we had discussed in the simple helium case what we have in the first two terms well the first two summations rather is a summation of n number of 1 electron Hamiltonians and the terms that are left over the summations that are left over q square again double summation 1 by r i j these are the electron electron repulsion terms see when you have more than 2 electron situation is more complex right suppose you have 3 then you have 1 3 repulsion 3 2 repulsion 2 3 repulsion but then do not go back and say you also have 3 1 repulsion 3 2 repulsion 1 3 repulsion so on so forth so it is you have to take combinations not permutations okay this is what it is so this is the simple form in which I can write it cannot be ignored as we know these are the inter-electron repulsion terms so the only way to go ahead so hydrogen atom your free electron particle in a box these were all nice systems where we could exactly solve Schrodinger equation right analytically now you cannot you have to use numerical methods and you are going to spend considerable effort in understanding some of the numerical methods but for now let us study the simplest one we are working under the ambit of orbital approximation and this here is our Hamiltonian what we do is this we think in this way and well this is nothing new for anybody because this thought process is there from the very early days that you study atomic structure okay the thought process is this is 1 by r i j this is basically repulsive interaction between the electrons okay now what it does is that it offsets part of the attraction each electron fills with the nucleus okay so what we are saying is that the actual nuclear attraction fed by the electron is a little less net nuclear attraction is actual nuclear attraction minus repulsion by other electrons so this phenomenon is called shielding okay shielding everybody knows what a shield is isn't it so well when we were kids star trek and all used to be very popular and there you would hear these people going out into outer space screaming that the shield is down to 50 percent shield is down to 20 percent protective layer so what we are saying here is that shield 1 each electron partially screens the nuclear charge for the other electron so 2 electrons due to presence of 1 electron electron number 1 electron number 2 does not feel suppose a nuclear charge is plus 3 it does not feel plus 3 it feels a nuclear charge of say plus 2 or less or more whatever but each electron acts as sort of an electrostatic shield or screen for the other one please remember for the other one and electron does not shield itself from the nucleus it shields the other ones okay so that brings us to the concept of effective nuclear charge I just said it without mentioning the name I said that due to the presence of 1 electron the other one feels instead of 3 maybe electric nuclear charge of 2.5 or 2 or 1.5 or more or less so we invoke the concept of effective nuclear charge which is z minus sigma per sigma is called the shielding constant what are we doing here this here is our Hamiltonian first term no problem second term no problem third term is a problematic part which we are not being able to handle so we have conveniently shifted the effect of the third term into the second term itself into z itself because then we can we do not have to write the third term anymore it is accounted for in this shielding constant this is how you write it he equal to minus h cross square by 2 me sum over i equal to 1 to n del i square minus q into z nuclear charge effective nuclear charge e square sum over i equal to 1 to n 1 by r i okay so this r i j problematic r i j vanishes have we neglected it we have not actually we have accounted for it in sigma but the good thing is that allows us to write our Hamiltonian in terms of 2 in some terms of n number of 1 electron Hamiltonians with the modification that screening is accounted for okay so for helium atom this is going to be the Hamiltonian and your wave function is going to be something like this the only change in the way so remember wave function will also change the change is that see these are 1s wave functions right for helium 1s wave functions will be used but look at this e to the power minus z effective r look at the constant z effective by a 0 what does that mean if z effective is less than z then this constant becomes smaller so for any given value of r psi becomes a little smaller also the fall off is not as much as it would have been for full z z effective is a smaller number so the shape and size of orbitals if you want to put it that way is going to change because of effective nuclear charge now let we will just demonstrate using a simple calculation so let us say that we talk about helium atom as usual z effective is 1.69 n equal to 1 so energy of helium accounting for your shielding turns out to be minus 77.68 whereas the experimental value is minus 78.99 so is that a good agreement or is that a good agreement it is a good agreement there is no scope of saying is not good given the kind of approximation we have invoked it is as good as it gets okay and the effect of this effective nucleus charge is such that one thing that we need to remember you see for one electron systems hydrogen atom the 2s 2p everything had the same energy the moment you put in more electrons the moment the system has more electrons and one 2s and 2p are no longer degenerate they have different energies okay and shielding of different and that is because shielding of different uh well electrons in different orbitals extent of shielding is actually different and that is why you have this you know this very familiar with periodic properties and that is why ionization energy exhibits this kind of a sawtooth variation because it is not exactly like hydrogen uh your different electrons in different orbitals even if n equal to n is in is same actually have different penetrations different shielding that is one part of the story electron electron repulsion the second part of the story which at the moment I am in a little bit of dilemma about to what extent we will develop in this course is that of spin spin is something that you have heard and many of us might have the idea that the spin spin quantum number arises out of the electron actually spinning on the axis that is what people thought in the first place that is why it was given the name spin however it is not origin of spin one of the origins is Tarn Gerlach experiment of 1922 by this beam of silver atoms which passed through an inhomogeneous magnetic field showed two lines which means that the ensemble splits into there are two kinds of atoms in the ensemble and the explanation was the presence of two angular momentum states remember angular momentum we have talked about angular momentum at some length but these angular momentum are different in the sense that they are intrinsic to the electron if you take an electron inside an atom then you get nlm and you get s you do not I mean you just take a free electron there is no n no l no m s is still there but one thing we will have to develop is what is the meaning of s and what is the meaning of m s I think our earlier discussion of your angular momentum rigid rotor and hydrogen atom is going to help us big time so very briefly we can say that it is not a result of actual rotation it is written in an unknown coordinate but interestingly you can talk about it is z component it really arises out of relativistic quantum treatment that did act had performed and many people still continue with it so it is associated with the spin angular momentum s capital S and its magnitude as usual is h cross multiplied by root over s into s plus 1 but s is a spin quantum number you see the analogy between this and the azimuthal quantum number or that rotational quantum number j of rigid rotor capital J in rigid rotor l in hydrogen atom s in spin quantum number all have identical behavior and s z is m s h cross m s as you know denotes the z component of angular momentum it has 2s plus 1 values as we have demonstrated earlier same derivation as what we have done for j and m. Now for electrons spin is half this is another common source of error please remember for electrons spin is not plus half and minus half that those are m s m s is plus half and minus half spin is half what does spin designate the length of the arrow what do this plus minus half designate whether the arrow is pointing up or down okay we will talk about spin at a greater length in the next module but for now let us conclude here that you have to incorporate spin and you cannot just write these orbitals in terms of spatial coordinates when you include spin along with spatial coordinates then you get modified orbitals that are called spin orbitals. So the total wave function is a product of spatial and spin parts each atomic orbitals now becomes doubly degenerate m s can be plus half or minus half that is why later on we come to this rule where no more than 2 electrons can occupy each orbital right poly exclusion principle that comes because you have only 2 values of m s and then please remember spin orbitals are orthogonal and normalized. So total quantum numbers that we now have are n l m and m s please remember m s and not s we went through this last portion a little hurriedly it was meant to be only an introduction we are going to develop the concept of spin in little more detail slowly then we will continue with our discussion of atoms with more than one electron.