Lec 1 MIT 5.74 Introductory Quantum Mechanics II

so I'm gonna tell you about the spectroscopic Antonian okay we call that the effective Hamiltonian it's a matrix it's a fit model it represents everything that you can observe first diatomic molecule it is not the exact cameltoe a quantum chemist would not calculate that they could derive it if they chose to from what they can calculate but it's different in several important ways okay fundamental to the effective Hamiltonian is a separation of variables to nuclear variables and electronic Y that's called the born-oppenheimer approximation so we get electronic vibrational and rotational these are nuclear this is electronic [Music] No and now we're all out okay and so the born-oppenheimer approximation that allows us to make this separation nothing really allows us to make this separation what we do it what is it what is this separation good form well it enables us to approach the problem of a mini electron atom how many electrons molecules without having to deal with all of the degrees of freedom that would be associated with many electrons plus several nuclei it enables us to think about things called electronic states and rotation vibration levels of electronic states and that's an enormous simplification now allies looked under the right these things that are slipped under the rug are going to be critical to understanding dynamics but first you need to have a picture of what would the molecules behave like if they were randomly simple if Nature has been more calming but then there'd be no dynamics there'd be no life there you know that'd be specific and so we're going to create a world which we understand which is free of all these difficult terms and then we're going to bring these difficult terms in use this simple picture that we started with in order to be able to calculate all of the complexities that we would we have suppressed when we define the simple world okay why diatomic molecules well they're not atoms and chemists are generally interested in molecules this is the simplest thing that has all of the complexity of chemistry diatomic molecules are also special because of several things you can go from the spectrum to the effective Hamiltonian and there is an exact this is not true for any other kind of moment so one can approach the problem of diatomic molecules without having to make any assumptions of models one can then take that approach and a lot of the approximations that work and use them for more complicated systems with confidence that you know the limits of these assumptions and approaches and models you can also have a complete sample of all relevant states now you're not going to look at all states of the molecule but for the diatomic molecule you can essentially measure everything that is relevant to whatever questions you want to ask because the vibrational density of states and the electronic density of states increases very very slowly and so one can resolve all of the stuff that you need to create your model and to answer your questions and this is not generally cute molecules in fact it's almost never true so we need somehow to develop ways of extrapolating into Kali atomic molecules to deal with this rapid increase of density of states and complexity and we use diatomic pictures as a way of doing okay I said there is an effective Hamiltonian which is our model for representing everything in a diatomic molecule and how is it different from the exact cameltoe well you might start with the exact Hamiltonian and you would partition the exact Hamiltonian into zero order picture plus everything else okay the zero art picture is based on models that you understand and can calculate Li this is the simple world it might be rigid rotor harmonic oscillator hydrogen atom particle in a box the particle in a box is a good representation for valence electronic states the hydrogen atom is a good representation for rid burg electronic states the vibration of the harmonic oscillators vibration interpreter for rotation and so we might put in things for which we have exact solutions and we don't have to work and we have eigenfunctions of this which we can use to evaluate matrix elements of whatever is in feet now there's a lot of stuff that I'm hiding here but that's the formal approach and this is an extremely good time to ask questions so this enables us to define a complete basis in other words we have a complete set of wave functions for bras and ket's that enables us to evaluate the effects of the terms that we've decided to exclude for make 0 we can choose which terms we put in a 0 depending on which interactions we think are important and ought to be dealt with in the zero order model and which ones are only small perturbations that we won't have to that will we will have to deal with but will not be necessary for being able to recognize in either the ab initio calculations respective places and we'll see how that works okay so we divide the divide the Hamiltonian into a zero order part and everything else then we have essentially an infinite pages which is no use and we have to approximately block diagonal is't and that's called the van black or contact translation this is basically second order permit it's a procedure which is described well exactly and it enables us to take this infinite matrix which is the exact Hamiltonian and cut it down to small pieces blocks that we can diagonalize and we think of that as our fit model as the effective Hamiltonian this is an essential part about which I will talk very little but it is the difference between what a quantum chemist would calculate and what a spectroscopy is perpendicular so that gives us the effective Hamiltonian which is a matrix that has a bunch of blocks along the diagonal and some small stuff off the diagonal that we're allowed to neglect except whenever they're cut occurs accidental degeneracy it's possible that we have a bunch of states associated with this block and they have their own energies and bunch of states associated with this block and there's a tiny term here that couples as it states in these two blocks and normally the energies are very different the difference in 0ur energies or block energies are large compared to the coupling term and so we can ignore them we do but when my accident levels get close together we can't and we have to do something special that's called dynamics it's called a perturbation it's called the interesting stuff it's the stuff that most textbooks slip under the rug so we have a procedure for simplifying the exact dental phobia and there's complicated dynamics because that these blocks also have off diagonal elements but those are sort of part of our zero order bottom and then there are these exceptional things which lead to unexpected and useful energy flow between different parts of a systems that are normally not in communication yes Pat two questions so is the vent like the transformation the purposes is to minimize those off diagonal elements between the blocks exactly exactly is it designed such that it's an optimization routine to minimize that or is it a set routine which which does that which is known to do that it's a set routine isn't it so you'd approve the procedure is to is to correct to add terms on the diagonal and off the diagonal in each of these blocks did compensate for these out of block terms and so so let's call this the M and this the M block okay so we have a matrix element between one member of the end block and one member of and one member of the end okay so that's an optimal matrix element it's an auto block matrix M okay and so second order perturbation theory would would say well we can consider we can correct the energies okay so we have an off title matrix element between the eye of the idol events of the in lock and the empty out of eject elements at the in life and so then we come back via the same from the same place but to a different member of the the end block so this is going to be a direction to the element of the end block so that's an off title or diagonal element of that one block so this is second our vision for matrix element squared and this is an energy denominator we're taking the average of these two guys that are near each other and now this remote and that's the correction that you automatically calculate to the elements in these various parts and you don't have to diagonalize a matrix or anything it's just you look at what the hamiltonian tells you and you automatically make these Corrections and that leads to all sorts of horrible ugly second-order constants and confusion about what is in the effective Hamiltonian because we call the things in the effect of healthily by the same names that the quantum chemist call those elements in the exact penalty and so there is always contamination diggsy are called it confusion or mixing of the mechanical and electronic significance parameters and he wrote a very beautiful paper in the early seventies where he talked about how you separate these mechanical and electronic meanings of parameters but mostly spectroscopy don't care about that because you need to fit your spectrum and what you get when you fit the spectrum is not just energy a formula that fits the energy levels but you get quasi eigenvectors which enable you to calculate whatever you want and you do make some small errors but you because you have to worry about what was that transformation you did but it's completely mechanical and you can undo it in a sense to make a connection between the exact that will tell you then this papers cover you said you have two questions still just practically helping others block diagonals like if you had a probably got an actual problem you were working on they really did but again that's those blocks are more or less self defining from your zero order model so you try to make it so the block diagonals are as big as that computer can handle unless it's not necessary right okay yes those typically it's like each block represent practically like a vibration and so within the blog your diagnostic implications do I mean what it's a represent well have you heard of say something like that triplet I see a triplet Pisces is a collection of six independent electronic states that are somehow represented as one state okay so it's it's a thing six things that belong together and there's our file elements and there's stuff in there or suppose you have a a triplet PI state a triplet sync with state that are near each other and there are so the little arc is an electronic each block might be one electronic staying or two or sometimes when there's true complex as some serious interstate coupling you have all of the vibrational levels of two states in one mid block so it's whatever you need whenever the interaction matrix element is large compared to the energy differences get this ticket into something that you diagonalize rather than something you use a formula from second order perturbation okay so are you okay defend yourself okay I am NOT determined to finish 15 pages of notes it makes my life easier are we all to me try to formulate a question I know for the people who only had 561 this is a reach but I'm trying to talk about the ideas rather than the equations that's why I didn't want to write this I have to think about this for a second yes [Music] vibration [Music] this thing will reproduce what you would observe in the experiment okay so the this effective Hamiltonian and make sure that I answer your question rather than when I take your answer this is a model which is expressed in terms of molecular constants like maybe spin orbit constant rotational constant you know some perturbation term it's a bunch of numbers that when you adjust them so that the eigenvalues of this matrix match the energy levels you say I've got the optimum values of these parameters but this is a month expressed in terms of a bunch of adjustable constants that we think have physical significance and we have a least squares fitting procedure which adjusts these parameters to match the experiment or if we haven't have an issue of calculation if we calculated the wave functions and we want to calculate the molecular properties we convert the quantities in the initial calculations to these molecular constants we put them into a matrix then we come did that help anybody okay so now we're ready to actually start to talk about the critical thing that enabled us to do all this and that's the born-oppenheimer process and there's a lot of confusion about the born-oppenheimer approximation there's a lot of people worrying about is this approximation valid or not we don't care it's a self fixing approximation the it's a method for defining a basis of a set of functions which is complete and which enables us to calculate all of the born-oppenheimer breakdown phenomena all of the stuff is outside of the point of time and in fact as you'll see in the next few minutes spectroscopy don't even use the born-oppenheimer functions they use something else of the diabetic okay the born-oppenheimer approximation is the basis for separating electronic motions from nuclear motions and the basic justification for that which is I can we Bureau is the electrons move fast relative to nuclei so because of this we have the idea of electronic states and potential energy curves potential energy services what is the born-oppenheimer approximation so there's no nuclear motion no nuclear kinetic energy no rotation okay so that's a perfectly sensible procedure because what it does is it says we saw the electronic problem and there is no nuclear it's nothing no rotation that causes trouble no vibration that causes trouble and we get a bunch of so what we do is we calculate for a diatomic molecule I can draw this for a polyatomic molecule it's a little harder but we thought get a bunch of energy levels and each hitter in clear distance so we we might have a series of energy levels like this or we might have a series of energy levels so we say ok here is a electronics thing here is a repulsive electronic state and well now we're at the beginning for the bound electronic state we can solve for the vibrational levels of that problem simply by saying okay here's a potential energy curve let us use numerically or some numerical integration procedure to calculate the vibrational wave functions of vibrational eigenstates of this isolated potential we can find the continuum wave functions again by using the same idea so you solve a differential equation and you get the nuclear wave functions associated with these electronic wave functions so now we've allowed the nuclei to vibrate we've included the nuclear kinetic energy but only when it operates on the nuclei and not on the electrons the electronic wave functions are explicit functions the electron nuclear coordinate but implicit functions of the inter nuclear chord so the nuclear kinetic energy which is basically that's that operator that has that acts on this it gives and so we can calculate the first derivative outside and that's going to give some constant I have some other labels so this might be a small constant this operator when operating on the product of electronic and vibrational wave function can you have the second derivative this type of the vibrational way yeah first rid of this time the first derivative the vibrational wave function you can know derivative this is a second derivative so there's a whole bunch of stuff that we're not yet bringing it and that slipped under the right that's the part that's one of the born-oppenheimer break down terms but we can calculate them there's nothing mysterious about it we have when we solve the shorter quit electronic shutter equation we get these curves we can find the way vibrational wave functions of these curves we can find the rotational wave functions of these curves so we can let the nuclei start to move and we know what to do but we know when we do that we're T there's something we have to go back and worry about but we normally don't worry okay we expect attention girls to look like one or the other of these but sometimes we get a potential curve it looks like this another one that looks sort of like that or sometimes we get potential curves that have to minimize like that and this is the truth this is what you get by clamping the nuclei in solving the electronic trigger question and these things are called avoided crossings because States of the same symmetry cannot cross crossing this is bad because these curves have shapes that are offensive to us to our to our sense of simplicity and it's also a warning that the vibrate of the electronic leg function is going to undergo a very dramatic change of electronic character as you go through this and a slower change of electronic character as you go through that well because why would this curve do that that's you know it can't do that unless the electronic character change so we we say well there must be another kind of approximation that does not allow the electronic wave function to change rapidly so these kinds of curves are the born-oppenheimer baby of editors and what we want are called diabetic the adiabatic is the sense of moving slowly a diabetic is a sense of movement fast this gets to the last lecture we talked about landau-zener but the diabetic curves are the curves that would be obtained by allowing these things to extrapolate and there the the curves that don't change electronic character remember if the electronic character changes with our rapidly matrix elements D by D our electronic matrix M is the D by V are are going to be huge and if they're big then we're going to add they're going to be large compared to the separations between the zero order states and we're going to have to diagonalize a big matrix which means trouble it means you either have to know the answer before you start or you have to do a lot of work in order to get from what you can observe and what you can calculate now what you need to do is clear the diabetic representation when you have this almost singularity or customer in the wave function in the potential curve when that occurs you know this is the wrong representation when this occurs it might be the right representation right wrong is based on convenience not based on something gives you a good or bad fit to the energy levels either representation all representations which are complete give you the same quality of representation of fitting the observed energy levels and calculating all of the dynamical quantities it's just what their call is different and if you chose up and a physically appropriate model your Labor's will be less and the the resemblance between the zero order model and what you could observe will calculate will be much greater than if you chose an inappropriate model and this is good for experimentalist because your model before you include a lot of off hang on all of us needs to resemble when you can observe otherwise you're lost okay the distance of closest approach between these potential curves is twice the coupling matrix of ice the electronic coupling matrix we know from perturbation theory and as you try to make two levels get degenerate they can never get closer together than twice the basics on the zero-order levels those of you who don't know that new Miller but so this are the energies as a function of R and so these curves tell you can't get any closer the list so that means that there is some electronic coupling term which is half that distance of closest approach now in the adiabatic picture we diagonalize the electronic Hamiltonian and we ignore the nuclear kinetic energy the other approach is we're going to diagonalize the nuclear kinetic energy simply by making the electronic wave functions not that strongly R and we're not going to diagonalize the electronic cameltoe this is a simple idea but it's computationally ill-defined because we know how to collect the nuclei we not to say the nuclear kinetic energy terms and the rotation they're going to be ignored the inter electronic repulsion that's 1 over R IJ a lot of terms and you don't know which one to make do to ignore there's no way computationally from scratch to suppress a term in electronic Hamilton so the Hamiltonian electronic spin-orbit these are the important terms in the Hamiltonian and these are the two terms that we would want to play fast and loose with if we wanted to define a diabetic basis a rotation well that's just ignored I'm sorry so we can turn off the spin orbit but we can't turn off specific terms you know the inter electronic repulsion Hamiltonian so we have two different adiabatic pictures adiabatic relativistic that's the full sphere that's that's the born-oppenheimer every time everything is clamped everywhere solving the electronic plus spin orbit and then we have relative adiabatic non relativistic where we turn this thing off and so a weekend when spin orbit is important we can die appetize by turning off this term but there is no way computationally we can diabeties by turning off something in here instead we have to use a simple model okay so let me describe we do so we start with the adiabatic potential curves so they might look like this okay and so immediately from these curves we know two electronic pulses the coupling matrix then we have to make two approximations one is that there are only two curves that are interactive that the effects of other curves can be ignored this is usually a good approximation but it's not always a good approximation so two states the other approximation is that the diabetic curves are linear in other words we have a diabetic okay and so what we want to determine is the interview for crossing the eater different distance of the crossing and a coupling matrix over okay and there's a couple of pages of what I call not lecture because I don't want to go through the formulas that the crossing these two are equal and so you get an equation for the value of the crossing you also have the type a today which is what you're given of an issue you don't have to make any assumptions about the form of these but at the crossing radius the difference between these two is a minimum so the derivative of the difference is zero and that gives you the value of the r-value in which that difference is smallest that's the cross ingredients the energy difference at this R value is is twice the matrix element the average energy of the crossing value is the energy of the crossing so basically you have everything you need so given these things you can calculate these critical quantities and now we have a 2x2 matrix which describes everything that happens involving these two spins yes it's it's the included I'm sorry I should have said the crossing inter nuclear distance okay so at some inter nuclear distance the diabetic curves cross and the adiabatic curves have reached their minimum vertical energy separation and that's what we want to determine so we want to know our CDC and and this and they are all obtainable by exploring the of additio curve so then you splice you say okay the diabetic curves just go through that crossing they're linear at the crossing and then they sort of join me the adiabatic curves or we can so we have any two-by-two matrix that describes [Music] so [Music] okay whenever you have a 2x2 problem it's always useful to separate out so suppose you have a 2-bike you probably won me it's always useful to separate out the average okay so whenever you're facing a 2x2 problem it's useful to write it this way because this you know the solutions they're just V squared minus plus so we have whenever we have a 2 by Q problem we take out the average energy and we get the symmetric 2 by 2 matrix we know the solutions so what I wrote on page six of the notes is the symmetrize the average energy in this diabetic representation and the what remains and so we have one define diabetic curves this way this is the matrix that we need to solve it's a two by two matrix it's diagonalized by unitary transformation the unitary transformation has the form cosine theta sine minus sine cosine but this transformation is a function of our internet resistance and so we have the transformation that diagonalize this matrix which is an explicit function of our now this theta is complicated combination of all of the control parameters so there's some algebra but basically you know what this is and so one can write the eigen functions of this two-by-two matrix what are that those are the a teabag stings we can write the eigenfunctions as an explicit linear combination of the diabetic states we can calculate matrix elements of g by dr because now we have an explicit function of our and so all the necessary matrix elements are now soluble and so the this transformation which is a model between adiabatic and diabetic enables us conveniently to calculate all of the off diagonal matrix elements in terms of parameters that describe either the diabetic or adiabatic potential factors things that we can get from either experiment or from initial calculation so it is also very common when you have a two-state interaction so here is here's diabetic state number one and here's a diabetic state number two and here is your ground state and if we excite this system with the short pulse it is very common that the only thing only one of these two states can be excited by an allowed transition from the ground state and so what one makes is some kind of super position of the so we have the diabetic states and we have a thief some of these is this is the higher energy eigenstate the lower energy eigenstate so so you have a short pulse short enough to cover both of these so we make a superposition of these two eigenstates which is at T equals zero purely the basis city that has the allowed transition and now we've got quantities we have the system going back and forth between these two eigenstates I'm sorry we have this the system going back and forth between the two zero order stings so at the time of the excitation it looks like the allowed zero order state and then moves towards the other disallowance your order state so there's a kind of dynamics the system is taking its diabetic character with time now we can also record a spectrum in the frequency committee and measure these eigenstates and measure characteristics out there wave functions and include the same thing about the potential dynamics so alright that wasn't what I had planned the same on this yes so just to sum this up the only reason that you're [Music] you're just mixing the two different limits that's right some clever ways you and that's all you're getting out of it reality is neither look right and it's often true that the energy levels that you observe in the spectrum or that you calculate I'm an issue by doing everything with the exact Hamiltonian are closer to one limit or the other and there's a tape with on the last page on the next-to-last patient what I have you to where there's a to electronic stick interaction and silicon monoxide and what you see is that the vibrational energy levels calculated from the adiabatic potential curve or from the diabetic without considering the interactions between the two potential groups in what the dietbet the adiabatic potential curve in this case gives a better representation we observe vibrational levels but not perfect this is the result of an exact fit so when one includes either the nuclear kinetic energy coupling between the levels of the adiabatic curve or the electronic coupling between levels of the diabetic curves you get an exactly to the energy and you get the eigenvectors and you can have the eigenvectors in either representation so if it's not this table is not saying that neither representation is very good both are perfect but one is closer to reality than the other and usually you want to use the representation that's closer to reality in this case there's only a small advantage to one picture over the other usually there is a huge advantage and one can know in advance whether you want the idiomatic or diabetic by comparing certain quantities which is not what I want to talk about that okay so now go to electronic vibrations from patient [Music] so we get a potential clue somehow it could be adiabatic retirement it's for the eye electronic sleep I we just count up from the line we and we can find the all of the vibrational levels with this potential curve by solving by doing a numerical integration by solving the nuclear kinetic energy to the nuclear ternary question and so we have the vibrational wave functions all of them for all vibrational levels and to continue associated with this potential so we get the rate function we get the energy levels we can fit the energy levels to its top the Donovans fashion [Music] these are done in constants like 1 0 is Omega 0 Omega E and y 0 the vibrational frequency rotational constant this lots more so these are this is a no insight no knowledge empirical representation of the eigen energies but calculated from the potential truth and these are the things that spectroscopy is published so one can do a least-squares fit to the eigenvalues or one can do a least-squares fit but what they write v of our side power side it's our my discography re is the internet equally every minute including distance and this is a convenient parameter and that's the Dunham's expansion of the potential and these are a bunch of empirical parameter is determined by fitting this to the potential and there is a ton of inversion that enable you to calculate these things to calculate these things from these things there's lots of ways of going between the potential and the constants or the constants in the potential and they mostly involve perturbation theory okay now I slipped something under the rug or at least since I've not looking at my notes I'm doing it out of order we have the effective potential which has gotten from the clap dutifully high-potential in other words J equals zero plus we explicitly know since rhythmical correction the potential okay this is the error of the distance this is the reduced mass this is the rotational angular momentum so this is the bear potential no rotation now we fix claps nuclei approximation by allowing the nuclei to rotate that's one of the terms will be neglected that enables us to calculate to derive the JT dependent effective potential and the eigenvalues of that they give us V of the engine okay so the OP initial potential curve enabled us to calculate all of these things all of these things allow us to compute the molecular constants by fitting or by fitting occur and doing the dominant version of reusing formation theory there lots of connections but so everything is okay we've included the rotation this is the very difficult god we call this the rotational operator it becomes a constant we take the vibrational matrix element of it a very slight rotation value of it so we do this [Music] okay remember up these nine so the born-oppenheimer approximation enable us to calculate electronic energy levels for the ID state but they're normally called T City and it enabled us to get a potential curve so then by allowing the molecules of roughly have this and we still haven't included all of the couplings between electronics needs either the nuclear kinetic energy couplings or we've gone to a diabetic representation the electronic but this provides the separation that we need between electronic of nuclear motions and then we can say all right are we going to have a love that describes the different parts of the problem the electronic part of the problem are we going to try to explain it as something related to something that we understand well like the hydrogen atom or like the particle in a box and valence states are reasonably well described by orbitals which correspond to different eigen states of the particle in a box energy levels get farther and farther apart as you go up and as you go up and interview have more and more notes or they could be the hydrogen atom for reverse States so you can sort of have a model for this and then you can have a model for this you can say it really wanted to be a harmonic oscillator but there so some terms that involve cubic and quartic powers of R or we can make it a more Sasa later and we can represent a lot of the molecular constants by using the harmonic oscillator plus and our minor Corrections with a Morse oscillator and we get relationships between molecular constants that help us in leveraging our finite number of observation so we're ready to begin so in order to really begin we need to say well what do we really become I'm going to summarize this for the electronic problem if we take a diabetic picture and most spectroscopy discs are most comfortable in the diabetic representation because it is the one where the electronic wave function doesn't do gymnastics as a fast molecular geometry so we're more comfortable thinking about the electronic problem in a diabetic picture and that's no speaking so and in that case this is or a better by Brian vibrational we could use it harmonic oscillator in powers of our cubed 4 etc well we could use a more sphincter and so treating cubic and quartic terms or the power series expansion of the Morse potential in cubic and quartic terms we can treat them as perturbations so there's lots of things you can do for the vibration and the rotation now I lied before I said the rotational angular momentum that goes with the centrifugal potential is J squares it's not it's the nuclear rotation it's Jenny minus L minus s square which is equal to R squared where this is now not the inner the distance 51 of the operator associated with the nuclear rotation and so we've got some something some work to do here to correct for the fact that you could have nonzero electron orbital and electron spin angular momentum so we wanted to use the JK plus 1 because this is the rotational levels are given by a constant times an integer times an integer plus 1 we know Jake could also be a half integer but we we have a pattern of rotational energy levels they're given us basically by one constant and some magic expression in the quota numbers well it's going to be more complicated here and we're going to discover what spaces so the Hamiltonian is written as the sum of the electronic Hamiltonian the vibrational cameltoe nyan which is really sort of part of this plus and now I need to tell you about the various terms in each of the calculators and classify them according to their type and so we can know something about the selection rules for non zero who want it in a simple way of doing this and this is the way I do it for the perturbation term associated with the electronic Hamiltonian okay this guy hates as well well what is it our I J is the square root of our ones this is a scalar operator with respect to electron I this is a scalar operator with respect to electron J this is a vector operator with respect to either electron I or J this is a dot product of two different vector operators so this guy destroys all of the one-electron it as matrix elements Delta L plus or minus 1 or 0 l de l j plus or minus 1 or 0 but this is our vector with respect to any many electron negative momentum so the total J for the total hell this is a vector this is a vector dot product of two vectors is a scalar so this is a scalar operator with respect to all of the many electrons and get momentum so that means Delta JL and there are independent of Omega which is the eigen value of JC lambda which is the eigen value of LZ and Sigma that's the property of a scalar operator it's diagonal and it's independent projection corner so this says well this thing is only one hopper make between electronic states of the same symmetry hence the nine crossing okay now for the rotational Pendleton it could have to stop pretty soon but I got pretty far [Music] this becomes V of our times bunch of stuff chase where - jz s where exist those are all of the diagonal elements in space but you see these things that you say I'm pretty happy and then we have minus J plus L plus these are raising and lowering operators - that's it so these have matrix elements off diagonal and Omega and lambda behind Omega and sigma alpha angle and Lambeth and signal this is called L uncoupling this is called spin a coupler and this is small but it's called rotation electronic or spin electron so this is what we have to consider so a basis set of this form enables us to deal with all of the terms here now I put L in parentheses the reason I put L in parentheses is the atom and molecule is not round an orbit in order for the orbital angular momentum to be defined it needs to be round you can think about the effect of the two nuclei on a wave functions around each there's a start it causes L to mix but it preserves the projections of L along this axis so L is totally my god but its projections are preserved so I'm sorry doesn't laugh of it objection now lambda is the projection of elf but because of the cylindrical symmetry lambda gets preserved but L is destroyed okay when we lose this quantum number we lose the ability to evaluate these matrix elements explicitly and these ladies implement this matrix element explicitly and we lose the ability to use clebsch-gordon coefficients to perform transformations of angular momentum from one coupling case to another so it's kind of a tragedy but there are ways around the reason it's not such a bad thing is that states with different values of lambda are usually relatively far apart states with different values of Sigma are usually close together and so we talk about things called multiple states which we go net denote by 2's plus 1 lambda and these multiple states have several values of Omega where Omega can go from lambda plus s down to lambda minus s there's a little bit of complexity your lambda might expect in steps of 1 and so depending on whether lambda is larger or smaller than s we get a certain number of components to s plus 1 and these states are essentially replications of each other we don't say for each of the Omega is a state like this we have a separate potential curve so we get a bunch of states that belong together in spectroscopy there are naturally groupings of states the most common of which is called a multiplet saint-like triplet pi or triplet Sigma or double it Sigma whatever the only time the textbook is going to give you what you would like for a rotational levels is for a state which has no spin or over the labor movement ok then J is equal to R and everything was correct but whenever you have or nonzero lambda or non 0s or both you get these multiple states and this is a grouping of states that belong together it's a natural block diagonal ization of the hamiltonian and the optical elements within that block given hind this this is between different electronic States so that's why we aren't really so harmed by the inability to know so this brings me to the last thing that I'm going to talk briefly about and that's on the bottom of page 13 there is a very beautiful paper by Nikita and zero which says okay we've got several terms of the Hamiltonian we have the electronic term and we're going to choose a zero-order Hamiltonian which is a good representation of the rotational and v structure that is different for each of the relative orderings of magnitudes of these terms so the usual ordering is the electronic is large compared to the spin-orbit compared to the rotation so we have multiplet states the rotation is small to spin order this big we have multiple states and case a is what happens in this limit if we make spit up at small compared to rotation then then we get case the rotational pattern form and quantum number is J here the energy levels are given by the j j plus 1 the rotational pattern from quantum number is n which is defined as J minus s the total angular momentum exclusive of spin and the rotational pattern for model numbers N and there are additional levels are different by this this is often half integer this is always integer they're different pattern of energy levels is different it's nice to know what which is going to be so if you have three objects you can order them in six different ways and they correspond to hoods case ABCD and nothing de nothing