 So, today of course I am not going to do anything new, I think I have more or less finished except for a review and just telling qualitatively what are the other theories that are possible. We have mainly described the many electron problems, so in particular Hartree-Fock which is of course the best uncorrelated theory. So this is the reference point from where all the electron correlations start, so then we discuss the electron correlation as the important topic and in particular we developed perturbation, the algebraic as well as diagrammatic of course CI, the configuration interaction as the two major methods of electron correlation. Of course there are several things to do, so our interest has been mostly on the energies, but I must tell you that as you go forward we also need to learn how to get not only energies but also energy derivatives and this is something that we have not covered in this course. How do you get energy derivatives with respect to some external field or perturbation? This is a very important topic and this is often called the response, this is something we have not covered that means how does the energy change if I give an additional field like electric field, magnetic field, this is very important because electric field change gives you dipole moment, if you have second order change it is called polarizability and so on. So these are also very important to calculate analytically, so I must mention the numerical calculation is very easy because you just put an external electric field, your one particle potential changes, you keep doing whatever you want to do and take a numerical derivative. What is of importance is of course analytical derivative, analytic derivative and this is something that is a very important part of quantum chemistry, not only external field equally important or even more important are the nuclear coordinates and this is a very part that we discussed and if I change the nuclear coordinates then of course my Hamiltonian will change because the external potential changes, how does the energy change and these are basically known as, these derivatives will be known as the gradients and the Haitian. Many of these are used for geometry optimization, why I am saying because in quantum chemistry a very important part is to optimize the geometry of a molecule. So what do you mean by optimization of geometry, if you calculate the potential energy surface like a diatomic molecule we have to find out where it is minimum, the simple hydrogen molecule if I do it correctly, so this is the point which is 1.4 atomic unit or whatever for H2 molecule it is a minimum. So for a complex polyatomic molecule we have to find out where the equilibrium geometry is, so what we need to do is to calculate this energy which I call potential energy, I hope all of you remember what is the difference between V and E electronic, what we have done the calculation is E electronic because in the Hamiltonian we have not added the nuclear nuclear repulsion term, so if you add the nuclear nuclear repulsion term which is a constant so it will not change anything you get the potential energy. So for gradient it is very important because this part is also a function of nuclear coordinates. So when I do the gradient I have to calculate del V del R, all R let us say there are n number of R's for nuclear coordinates which are basically degrees of freedom, vibrational degrees of freedom like 3n minus 5, 3n minus 6, then you know that for a minimum each of them has to be 0, so this is a very important equation, so what we mean by analytic derivatives is to calculate these analytically and then set an equation that this is equal to 0 okay and then we have to see whether it is maximum or minimum by looking at the second derivatives and checking the second derivative matrix which is basically called Hessian, so this is gradient and this is Hessian, second derivative matrix I take with respect to one R and another R, so R i, R j this is a matrix as you can see as a function of ij, this is called Hessian matrix and the trick is that if you diagonalize this matrix you get the eigenvalues, if those eigenvalues are all positive then it is called minimum. These eigenvalues are essentially the frequencies, you know many of you might have heard IR frequencies, this is another way to calculate, this is infrared because I am changing potential energy with respect to coordinates, I calculate those frequencies if they are all positive then if they are all positive then it is called the equilibrium geometry and of course there may be negative frequencies, one of them is negative, two of them negative, you will get different states which are actually called transition states, you know many of you will be using the transition states, so we are not done in great details but I thought today I will just tell what are the other things that we have to actually do apart from calculation of energies, so what we have focused in this entire exercise is to calculate the energies okay but the gradient and Hessians are very important part along with other external field response and this in general is called response approach, so this is a very important part how do you calculate this apart from the energy calculator, whichever method you pick up, in the same method you have to do consistently this, we have of course discussed within the energy important problem of size consistency in great details when is a theory size consistent, when is theory not size consistent and in particular we discussed the Hartree-Fock that Hartree-Fock itself may not be size consistent, when a even restricted Hartree-Fock or Hartree-Fock when a closed shell molecule fragments into open plus open it is not size consistent, however if they goes into close plus close it is size consistent but then even if it is size consistent if I do approximate CI the size consistency can go down, so in particular we have taken a system of H4 into H2 plus H2, so with an example we showed that H4 can go to H2 plus H2 correctly in the case of the restricted Hartree-Fock but here this correction breaks down if I do an approximate CI and in particular we discussed DCI, I hope you remember this particular model problem, this model problem was essentially a Hartree-Fock 1, 1 bar note that whenever 1 bar will be used, please remember it is a spin orbital with 1 as the special orbital, beta as the spin orbital, spin, so that is something that you should have no confusion, so this is just called chi 1, chi 2 in our nomenclature where chi 1 is 1 alpha, chi 2 is 1 beta, so that is a spin orbital, so please be careful even whenever we ask be careful of the symbols so if I am saying chi 1, chi 2 then of course this will be chi 1, chi 2, if I am writing in terms of special orbitals this can be written 1, 1 bar and then 2, 2 bar becomes chi 3, chi 4, so a very simple minimal basis problem where this is special orbital 2, this is special orbital 1, we have done in great detail and in particular we discussed the DCI wave function which is 1, 1 bar plus C 2, 2 bar, remember this is a DCI wave function and this is actually full CI for this problem if we restrict ourselves to a symmetry adopted state, so let me just quickly tell you why as you know that this symmetry is sigma G, all of you know for hydrogen molecule when I do this it is a sigma G, one S A and one S B symmetry combination and this is called the sigma U which is anti-symmetric combination, so if I do a one electron excitation from G to U which can be a part of the CI, S D CI, full CI, the one electron excitation will lead to a symmetry which will be G into U, one of the electrons will be in sigma G, one will be in sigma U, so that will become a small U symmetry, remember that this is sigma G square, so sigma G square has a group theoretic symmetry which is singlet sigma G, capital sigma G, I hope all of you are more or less familiar that the capital sigma is for the molecular term symbol, small sigma is for the orbitals, so this you have done in MSA itself, so if I restrict only this wave function to singlet sigma G then you can imagine I cannot have any function which is singly excited because it will involve one G and one U, this will then become U state, I am interested only in G state, further of course I am interested in capital sigma state which will be always generated if I have two sigma so no issue, further I am interested only in a singlet state, so the only state that is possible is 2 2 bar which is doubly excited because if I have put both the electrons in sigma U then sigma U square also is a singlet sigma G state with higher energy normally, if you have look at these two determinants they are higher energy but your actual ground state energy is a combination of these, so this will have a larger dominance, this will have a lower dominance for this kind of states okay but of course as I discussed for the hydrogen molecule if you stretch this apart then both these orbitals come very close and then of course that is the reason RHF breaks down because they become nearly degenerate, so if I go around this part when R tends to infinity for hydrogen molecule these two almost come together and both sigma G square and sigma U square become almost equally important, so that is a failure of the RHF and eventually all the theories have to be geared to take this, so we have discussed the DCI problem, we also have discussed how to solve the DCI equations iteratively, so solution of particularly DCI equations iteratively, of course DCI or any CI can be solved as an eigenvalue equation, so let us not forget but we iteratively essentially mean that we do not want to solve the entire eigenvalue problem, we want to look at only the ground state or the lowest root in an iterative manner and very often we have started with the a correlation expression in terms of e correlation and set e correlation equal to 0 as the first guess and then you continue, so this also we have discussed very elaborately, so I just wanted to mention this, I should now mention few of the other theories for energy calculation which I mentioned, so perturbation I have already told you that we have discussed yesterday also in great detail diagrammatic and connection in the algebraic except that we have not proved, we have not proved the link cluster theorem, we have simply used the link cluster expansion of Goldstone and yesterday we gave you a model problem, I hope you remember practice problem of MP4, so please try to do, I have given a fourth order diagram, so do not say that the fourth order is outside the syllabus, I have given the diagram yesterday, if I give a diagram you should be able to write the algebra with all the Hugen-Hulls rule, there only Hugen-Hulls diagrams is what we have done, so you should be able to write the algebra with this sign and everything correctly, all right, so apart from perturbation and CI we said that we are going to cover at least mention the names of a few more methods, so one of the important methods is what is called the MCSCF, I think the name itself is clear, it is multi-configuration self-consistent field theory, so this is known as the multi-configuration self-consistent field theory, so let me just explain what it is, so this is the full form of this, self-consistent field already you know that is a Hartree-Fock where for a single determinant you try to get the best spin orbitals, okay, you also know what is multi-configuration theories in CI, CI is a multi-configuration, expansion, so this is as the name suggests is a mixture of CI and Hartree-Fock, the spirit, Hartree-Fock is optimizing the orbitals, right, that is self-consistent field, CI is optimizing the coefficients of an expansion, if I have a linear expansion how do I optimize the coefficient, so in MCSCF you do both, so you have a web function which is a combination of determinant, first linear combination of determinants and these determinants have to be chosen carefully, then you optimize the energy with respect to both the linear expansion coefficient and the orbitals involved in the determinants, not just one determinant, there are many determinants, all the orbitals involved in the determinant, you can clearly see that this goes much beyond CI, this of course is beyond Hartree-Fock because I have a multi-configuration function and it can be a very, very accurate method because depending on number of determinants there are many orbitals, all the orbitals I am optimizing, just like in Hartree-Fock you remember I wrote energy as a, optimized with respect to orbitals, right, orbitals essentially could mean LCAO coefficients, orbital expanded in a basis, so then coefficients are optimized but essentially orbitals and in CI it was a reverse problem, orbitals are fixed, if you remember the standard CI that we did, we always went with a basis of orbitals which was Hartree-Fock orbitals, with those Hartree-Fock orbitals we constructed Hartree-Fock determinant singly excited, doubly excited and we only changed the coefficients, so the equation was only to get the coefficient which is an eigenvalue equation, we have again done this, you construct the H effective matrix for the E correlation where the first term is 0 and then all matrix elements of H minus E Hartree-Fock, you with respect to all the determinants but determinants are fixed because orbitals are fixed in CI, only the coefficients you change, in this case we are doing both, so for example, so obviously in such a case we have a restriction that we cannot take such a large CI because there are too many determinants, if I have to do this it's really a no go from the beginning because it's too complex a problem, so what we do is to actually take problems where it is required, so one of the problems is precisely what I have discussed just now, that I have a situation where hydrogen molecule, I mean hydrogen is just an example stretched, so if I stretch it the configuration sigma g square and sigma u square will be nearly same in energy, they may not be degenerate, they may be degenerate only at the separation point but if I stretch they will come closer and closer, so basically this is the equilibrium sigma g and sigma u, as you go forward this will go down, this will go up and something like this, so this kind of system is not degenerate but they have a name in quantum chemistry which is called quasi degenerate, the quasi essentially means that it is nearly degenerate, not exactly degenerate, now in all systems and including degenerate systems I need to construct the wave function not as I discussed the Hartree-Fock does fails but I must need to construct the wave function phi 0 as some C0 sigma g square plus C1 sigma u square, in a small model this is exactly full CI, 1, 1 bar, 2, 2 bar except the C0 is 1, I mean normally we have said C0 is 1 for intermediate normalization but that's unimportant, so if I have only 1, 1 bar and a minimal basis this will be then 2, 2 bar it's of course full CI but if there are many other basis it is a part of the double CI and there may be many other double CI coefficients but knowing the physics here that only these two configurations are important that is two electrons here and two electrons here I can do a much better job than just doing a CI, note again in a minimal basis this is exact, I can repeat for hydrogen molecule but if there are many other orbitals, so there are many other orbitals then it is not exact, so to do exactly even DCI I need to take many, many configurations, correct, I may have to take even singles for full CI because there may be another orbital with g symmetry where I should be able to excite because g into g remains g, right, here it was only g and u minimal basis so I can't do so in principle the problem may be very large even for hydrogen molecule if there are many number of orbitals but I know that only these two configurations are important the rest are not important, in such a case I should be able to write only as a combination of these two determinants and then not only vary these coefficients just like in CI but also vary these orbitals themselves 1 and 2, so I would vary C0 and C1 as well as 1 and 2 remember the sigma g and sigma u normally when I do CI I have taken from a Hartree-Fock calculation and fix it but now what we are suggesting is that anyway this is an approximate calculation let's not just take Hartree-Fock 1 and 2 let's also try to re-optimize 1 and 2, so this the reason it will change is that now that we are optimizing the orbital 1 it is being optimized not just for sigma g square that was the Hartree-Fock but in the presence of another determinant so there will be a coupling term and because of that the orbital 1 will change similarly orbital 2 will change the coefficients will change of course when I vary so it's a very complex problem and this is exactly what is MC SCF multi configurations SCF multi is here 2 that 2 can be 3, 4, 5 but in general you don't do a multi configuration SCF with 1000 configurations because then there are so many orbitals the orbital optimization itself is a non-linear optimization as you know because orbitals are the exponential something quadratic even Gaussian or whatever exponential term so they are extremely difficult to optimize Hartree-Fock we could do because there is only single determinant so obviously if there are 1000 determinants you can't optimize the coefficient and orbital they are very very expensive so MC SCF has to be used judicially those will do MC SCF remember should be done for a particular problem where a few configurations are important so don't do a real CI because CI you are taking everything even in CI of course you do selection I told you by looking at the energy denominator you can select determinants amplitudes but at least here I know that a few configurations are important in such a case you can actually do MC SCF it gives a very very accurate results many times this configurations are chosen by looking at active orbitals and I will tell you what is an active orbital so let's take a problem of more than 2 electrons so where it will be relevant so let us say that these are my occupied orbitals and this is the line these are my virtual orbit this is this is not an orbital they just a line separating the occupied and virtual orbitals so this is my lumo this is the homo and of course I have many other orbitals here so I have let's say 10 electron system okay so up down up down whatever now and then there are lumos so I have to I can generate many configurations by exciting electrons from here to here okay question is I want to do an MC SCF because this gap between homo and lumo has become very very small so it is exactly the similar problem that the gap between homo and lumo has become small so there may be other configurations which are important so you may decide then to choose what are called active orbitals which have to be mixed so if you ask me what are the active orbitals in this problem the active orbitals which were mixed with sigma g and sigma u in this case there are two active orbitals so let us say that I choose similarly one active orbital from here one active orbital from here and then I construct all possible determinants where the electrons active electrons are put within these orbitals within these two special orbital so let's say I choose active orbitals as homo and lumo so two orbitals how many spin orbitals four spin orbitals correct I don't want to change this so all my configurations will have this frozen this eight electron so these are not my active so active electrons are also only two now without looking at the symmetry how many determinants I can form between these active orbitals so that means I will I am free to excite one of the electrons here two electrons here whatever but everything that I do is just between these two special orbitals and involving these two electrons these eight electrons are fixed so how many how many determinants I can make as everybody agree you have four spin orbitals two electrons so you have four c2 without looking at the symmetry right now because you don't know the symmetry of the orbitals they may have the same symmetry so you should be able to excite okay so in general I have six electrons so if I six determinants if I make a wave function which is now a linear combination of these six determinants okay how did I how did I choose these six determinants by first choosing the active orbital remember if I if I say this lumo plus one is also very close to lumo then I can expand my active orbitals as homo lumo lumo plus one so active orbitals are essentially orbitals which are near degenerate or quasi degenerate so look at the degeneracy of within the homo within the lumo and between homo and lumo and choose the active orbitals okay of course if it is only within the homo there is there is nothing to choose because that is already filled so it has to involve lumo of course the unoccupied orbital not only lumo lumo plus one so if I have one more orbital this would become six this this and this and then my number of the six active so it will become six C2 I can also say that homo minus one is also very close then what will happen I will have this this this and maybe one more this is all these four are very very close in energy then what I have to do I have to choose these four orbitals which means eight spin orbitals and I have to choose four electrons as active because this will be also lifted these six will then be frozen so you have to decide the problem looking at the orbital energies how many active orbitals you have to choose but having chosen an active orbital if you make a wave function which is a linear combination of all determinants all determinant that can be constructed out of the active orbitals all then it is complete okay then such a method is called CAS SCF if I make the determinants and then do exactly like MC SCF which means I change the coefficients of this determinant six determinants as well as their orbitals so in this case the orbitals are let's say homo and lumo two active orbitals in this particular example then you change this and these orbitals okay and their coefficients of the six determinant in general and of course this is g and u then you have only one determinant okay g square and a two determinant g square and u square just like this example so in this example it is a CAS SCF actually because I have chosen only sigma u I have two electrons in terms of symmetry this is the only possibility okay so in a way this is a CAS SCF so CAS SCF is like MC SCF but the MC the MC's part is chosen like this that you first look at the active orbitals distribute the active electrons in all the active orbitals possible within the symmetric considerations and then do an MC SCF so that becomes CAS SCF it is called complete active space and then you know self consistent field field so basically just the MC space becomes a CAS complete active space after you do CAS SCF there are many people who do within this multi-reference multi-configuration function another perturbation theory that is very very powerful and that is usually called CAS PT perturbation theory so on so do a perturbation on CAS so these are all methods that are available in fact many of you will see this so you have a CAS and then do a perturbation theory so you get CAS PT so you can have CAS PT 2 PT 3 etc depending on which order you do usually CAS PT 2 itself is extremely time demanding and people do CAS PT 2 and you can see that first of all I have chosen the important determinants I am doing a CAS SCF which means not just CI changing the orbitals and then top of it I am doing second order perturbation where other orbitals will come because right now other orbitals are left out I have taken only homo lumo all these orbitals are left out in CAS okay so when I do CAS SCF remember the dynamic correlation from all these orbitals are left out only these two correlations are taken care and this is usually called the non-dynamic correlation where very few configurations are important and I do not want to introduce these terms late in the course but I just thought I will tell you so MC SCF or CAS SCF does the non-dynamic or static correlation very well but you need these dynamic correlations all the other basis then you have to do a perturbation theory on that and that is where CAS PT 2 gives very good results in fact if you do second order perturbation that is good enough but in many case the dynamic correlation is not important only static correlation is important then you do CAS SCF or MC SCF so these two kinds of correlations are very important static and dynamic so usually static comes from a few configurations few important configurations so in that case you can do just a CAS SCF dynamic comes from a large number of configurations each of which is not very important but their sum is so sum is important or some cannot be neglected so I have to sum them also so this is what we have done so far so today what I am discussing right now MC SCF is really to take the static correlation how do you take small number of correlation and there you have a choice not to optimize the orbitals or to optimize the orbitals but MC SCF does both okay orbitals as well as then coefficients so it is very good for static correlation so MC SCF or CAS SCF MC or CAS whatever SCF is very good for static correlation this you have to do dynamic correlation the perturbation theory or normal CI CI involving all other determinants basically I have to involve the rest of the orbitals not just those few configurations for dynamic correlation so this is something that is today understood that the correlation also has two different parts and different theories can handle each of these very well so I think this is something that I thought I would tell you among the other important theories and there used to be also a theory which is almost now dead today but it is a very important theory which is called the electron pair theory which actually started from Synanoglu's work and then was eventually taken up by Wilfred Meyer as electron pair approximation and there were several approximation independent electron pair approximation and different versions of coupled electron pair approximations from SEPA 0 to SEPA 7 you know there are so many small small modifications this is called independent electron pair approximation coupled electron pair approximation so take electron pairs how do you couple them the way you couple but today the electron pair theory is almost extinct you know most people don't do it