So, yesterday we did this psi 0 theta 1 psi 0. I want to make sure that everybody understands how to do it, not you, but let me see others. So, let us see how we do it. We are not going to use whole particle creation, normal creation analysis operator. So, how do you write this? You write this as ij psi 0, ai dagger aj psi 0 and some h ij. So, your theta 1 is sum over h of i. This is in first quantization. Again, note the difference. This is in first quantization, this is in second quantization, where the sum over ij are now no longer electron coordinates but above spin orbitals. So, this is the first thing. So, this is the second quantized notation. So, now how do you show that this is equal to sum over h a? That is what we have to show. So, what is the next thing that we do? So, what should be aj, first of all? So, j should be occupied. What should be i? Should i also be occupied? Does everybody understand it? If j is not occupied, then you cannot destroy. So, in our later language, this becomes a whole creation operator. It will become xj dagger in later language. Ai dagger also has to be a occupied orbital i because it is acting on the left side to psi 0. So, if you look at this part, it is the adjoint of ai psi 0. So, if only i is whole, this will survive. The adjoint will survive. Otherwise, the adjoint will also become 0. So, this can be now written as sum over ab psi 0, a a dagger ab psi 0, h a b. So, we have put j as b. Now, note that this a and b by our notation are occupied orbitals. Later, we can change it to whole operator, whole creation, whole annihilation. So, that is the standard notation but right now we are doing. Is it clear? Otherwise, there will be 0. If any other ij, it will become 0 anyway. Then, how do I manipulate this? Because ab can now act on psi 0. Psi 0 is not my vacuum now. Psi 0 is actually, my vacuum is a physical vacuum because this is a normal creation annihilation operator. So, ab can act on psi 0. If I would have made this vacuum, then I have to write this as a whole creation operator. So, of course, it can create. It is basically creating a whole. So, it is possible. So, what you do is actually to write a whole annihilation operator on this side. So, to do that, you do an interchange. So, do the commutation. That is the next thing. So, write a a dagger ab as delta ab minus ab a dagger psi 0. Is it okay? So, I am now using the anti-commutator that this plus this equal to delta ab. So, I am writing this as delta ab minus this. So, then the first term is very easy. Delta ab comes out with h ab. Psi 0, psi 0 is 1. Psi 0, psi 0 is 1. So, basically this is multiplied by psi 0, psi 0. Then, I have a second term which is minus psi 0, ab a dagger psi 0 h ab. That is the second term. Psi 0, ab a dagger psi 0 h ab. Now, you can see this is actually a whole annihilation operator. If this is a whole particle vacuum, it will become 0 or you can argue that I cannot create an occupied orbital in psi 0. Hence, this part is anyway 0. So, this goes off. So, you are left with only this quantity. This is already 1 because of delta ab. The result is sum over, sorry, there is a sum over h a. So, there is a sum. I forgot to write the sum. So, there is a sum everywhere. Sum over ab is there. So, sum over ab is there. So, this becomes sum over a h a, which is my first quantized result. See, any one particle operator can create one spin orbital and annihilate one. So, that is the idea of the one electron operator. So, that is the reason this has come. There has to be, it is a matrix. So, it is like a linear combination. My string of operators, so basically when I am writing theta 1, my string of operators are ai dagger aj with some combination coefficient like a matrix superposition. So, I am writing one operator as a product of a set of operators. The set of operators, the only idea is the set of operators cannot be just creation or annihilation, but one annihilation, one creation because it is eventually a one particle operator which acts on a wave function not to change its number of electrons. So, when theta 1 acts on a psi which is an electron, the result should be an n electron function. Operator acting on a, so my number of electron must be conserved. So, I have to have one pair of annihilation and creation. Two electron, I have two pairs, but always number conserving. So, the question that is asking is that why ai dagger aj? The rest is just a linear combination. So, there has to be coefficients. We are calling it Hij. So, it is a general expansion. Like a function we expand in a complete set, we are expanding operator in another set of annihilation and creation operator. And it is self-consistent because of the anti-commutation rules that everything gets clear. You can actually do the same with theta 2. I will not do it, but please understand that you will exactly have the same thing, ai dagger aj dagger al ak. And again you can argue that al ak must be occupied orbital, ij must also be occupied orbital. So, it will become a dagger, a b dagger, a d ac. All four are occupied orbital just as here. And then show by anti-commutation that you will have two sets of integral.  So, I think this is, I just wanted to tell you that you should be able to do this algebra. It is very simple algebra, but you should be able to do this. Later we said that because of this idea that we have to show that a dagger acting on size 0 is 0, which is little artificial because creation operator acting on something is 0 is artificial, we like to make it annihilation operator. So, that is the reason the whole and particle annihilation operators were created. And we called this as a vacuum instead of expanding this. Here I do not have to expand because physically I can understand what is there. But in a larger expression, it is convenient to work with the Hartree-Fock as a vacuum because all our correlation expressions are based on Hartree-Fock and using whole creation and particle creation or whole annihilation and particle annihilation. In which case anyway this should have become an annihilation operator x a and this should have become x b dagger. So, now if this is annihilation operator, if this is a vacuum, then automatically x a vacuum is 0. So, the idea would have come more mathematically. So, only for that reason we have switched to whole particle creation annihilation operator. But otherwise you can do exactly the same thing in normal creation annihilation operator, you have to be just little bit more careful. And in a longer algebra, it is much more difficult to do because the keeping track is very difficult. Keeping track becomes much more difficult. So, I think we will not go into much more details of second quantization, but we will only restrict to the form of the wave function, form of the operators in second quantization and all anti-commutation rules in the second quantization. That is very important. The rules of anti-commutation in second quantization, I think this may be enough as a background because we are not going to really implement second quantization in our correlation theory. That will take longer time to understand and do it, but eventually of course those who are interested have to do that. So, this is only a background that we will do. With that we said that we will come back to diagrammatic interpretation of perturbation theory. So, that is what we will do and that diagrammatic interpretation when you write the diagram has a throwback on the second quantization, that you write the operators in second quantization, expand everything in second quantization or MP2 energy expression, then convert this into diagram. Again there is a lot of rigor, lot of procedures to do that. What we can only do is to tell you the rules, how to do that. So, before I do, let me again recast your second order energy. Remember again, Psi 0 0 is my Hartree-Fock. I hope you remember that Psi 0 0 is Hartree-Fock. V Psi 0 1, this was the original expression of E naught 2 and then we wrote Psi 0 1 as a linear combination K not equal to 0. Remember it was intermediately normalized, so K not equal to 0, CK 1 Psi K 0. So, we find out CK 1 as overlap of Psi K 0 with Psi 0 1 or any specific L, I whatever you call it. This K is here a dummy index however. So, if I can calculate this integral, then I am in the business. I can calculate Psi 0 1, I can eventually calculate E naught 2. So, how did I calculate this integral? I went to the first order perturbed equation projected with Psi K 0. So, can somebody tell me this value, this we have already done. What is CK 0? So, it is Psi K 0 V Psi 0 0 divided by E 0 0 minus E K 0. That is a general form. So, that Psi K Psi 0 1 could be written as sum over K not equal to 0 Psi K 0 Psi K 0 V Psi 0 0 divided by E 0 0 minus E K 0. Note that for our MP perturbation, Mila plus MP perturbation, our V is sum over 1 by Rij. Again, I want to make sure that you understand sum over 1 by Rij minus the Hartree-Fock potential. That is your V. So, when I put this here, note that this quantity has to be only W excited determinant because of Brillouin's theorem. So, why is the Brillouin's theorem applied? Can you now tell me, because V has also one particle part. So, what I am saying is that when I calculate Psi K 0, V Psi 0 0. So, not K for an MP perturbation, we have Hartree-Fock, singly excited, doubly excited, all these perturbations are there. We are saying singly excited cannot come here. What is the argument? For 1 by Rij anyway, it is more than why it cannot come for 1 by Rij? It can come. Both of them will come, but there is a cancellation. There is a cancellation and that is a statement of the Brillouin's theorem that they will cancel because Brillouin's theorem essentially says that any Psi singly excited, H Psi 0 0 or Psi Hartree-Fock is 0. I can write H as H0 plus V. H0 is sum of the Fock operator. So, sum of the Fock operator, they are eigenfunctions of the sum of the Fock operator. So, this is 0. So, with V itself, it should be 0 and we have actually shown by elaborate calculation by a letter rule this and this that they cancel each other. Each of them survives, but they will cancel each other actually. So, that is the reason this is only doubly excited determinant. So, that is the first important thing to understand. I am going from Hartree-Fock to a doubly excited determinant and in my E0 2, which just comes as a cover for Psi 0 0 V Psi 0 1, I can now write this as sum over k not equal to 0 Psi 0 0 V and then push this Psi 0 1. So, which means this Psi k 0 come here and Psi k 0 V Psi 0 0. For the same reason, now k must contain only doubly excited determinant. So, you have to understand because later on these quantities will come everywhere.  So, that was the final formula when we expanded this Psi k 0 in terms of this and we also said that if you construct this as a resolvent, then you can write it more nicely with including sum over k, then you can write this as a Psi 0 0 V resolvent V Psi 0 0, where resolvent is sum over all the states Psi k 0 Psi k 0 divide by the energy. Now, what we want to do is of course we can write everything in terms of second quantization and then follow this as a diagram. So, what we will do? We will directly write a diagrammatic form of the E naught 2. There are various diagrammatic conventions. So, I must first tell you there are many many diagrammatic conventions. I have already told some name like Feynman diagram, Huygen-Holtz diagram, Brando diagram, Goldstone diagram, several. What we are going to do is one of the simplest diagrammatic conventions and for this course we will just stick to that and that is called the Huygen-Holtz diagrams. So, I first want to show you that this expression, how it can be written in terms of diagrams. So, there has to be some rules so that from the diagram I can get back this expression and then we will see how to proceed further. So, the first rule is that for every one particle operator, in this case there is no one particle operator because you have only one by Rij. So, one particle operator like Hf whatever is there is represented by a vertex which I can just call some dot vertex with one incoming and one outgoing line. You will see that how the second quantization really comes in. So, for example, if I have to, I have a matrix element which is Hij let us say or Fij whatever depends on what is the vertex then this will be constructed as a vertex. The vertex will be either H or F depending on what I am going to do with one incoming and one outgoing line. So, what is the second quantized form of this Ai dagger Aj? Whenever I have Hij I know that the operator is Ai dagger Aj. So, that means J is annihilated, I is created. So, but J can be occupied, J can be virtual, I can be occupied, I can be virtual. So, the first second rule is that whatever is annihilated must be incoming. That means incoming is it goes into the vertex. The one which is created must be outgoing, it must go away from the vertex, away from the vertex. So, let us assume that J is occupied, I is also occupied. So, both of them are occupied. So, I and J are occupied, occupied. So, remember my one particle operator theta 1 is sum over Ij whatever Hij, Fij does not matter, Ai dagger Aj. Of course, for the Hartree-Fock perturbation it will become F of Ij. So, let me write this as F of Ij for the time being. Do not have to assume that this canonical, non-canonical nothing, it has everything possible. So, let us say my I and J are occupied orbital. So, this is now A and B. So, how do I represent this operator in terms of diagram? So, this operator wherever it is there will be a vertex and now this is actually Ai dagger A B, both of them are occupied. Let us just take an example. So, A B is annihilated. So, the line B must go into the vertex, the line A must come out of the vertex. Then there is a rule 3 before we do that, that all holes, now if this is a hole or occupied orbital lines are lines with arrow pointing downwards and the particles which are virtual orbitals should point upwards, arrow direction should be upwards.  So, my A B is first an occupied orbital, but it is an annihilation operator. So, first thing is that it must be incoming, it must have an arrow pointing downwards. So, a line pointing downwards comes into the vertex. So, it must come only from the top and B like this. So, this becomes my B, it is occupied line. I am actually writing in terms of normal creation annihilation operator, because in terms of this it is actually whole creation operator. A line but it is being going down because it is a whole line, it is coming into the vertex. So, it should actually end in the vertex because this is the line which is annihilated, electron annihilates. So, this will be actually a whole creation. Then I have a A dagger, this A is also occupied. So, the line again should go down and it should go away from the vertex. So, then obviously it should be like this. And this should have a matrix element attached. So, every time I have an operator like this, I have a matrix element attached. Matrix element will connect these two lines, which is whatever is the value of whatever is the operator. So, if it is H, it is HAB, if it is Fock operator, Fock AB and so on, then of course there will be simplification for the canonical Fock operator that we will see later. For canonical operator of course it is very clear that A must be equal to B, otherwise it is 0, because both A and B are Eigen function of the Fock operator. So, FAB is delta AB times the orbital energy. So, that is a different matter but for a general operator vertex, this should be the operator. Now, let us take i and j something else so that you learn how to draw the diagrams. So, let us say now j is particle, i continue to be occupied. So, now what I am doing? I am annihilating a particle orbital. So, this is basically now AA dagger AR. So, I am annihilating a particle orbital into the vertex, but what is particle orbital? They are upwards, but it must go inside into the vertex. How will you do this? So, it will be like this. All particle lines will be like this, all occupied lines will be like this, but it must come in. So, it must come from the down and go into the vertex. On the other hand, AA dagger is also an occupied orbital. So, its arrow must point downwards and it is now actually going away from the vertex. So, then what is the other option? It must go like this. See, every operator has one incoming and one outgoing line, one particle operator. So, one of them is and this is why second quantization comes. Every operator has one creation, one annihilation, because I told you any theta 1 is AA dagger aj. So, let us take that i as occupied, j as virtual. I had taken here both occupied. So, now I am changing j as occupied, j as unoccupied, i is occupied. So, then AR has to be annihilated. A particle orbital has to be annihilated. So, it will go like this. An occupied orbital is being created. So, actually hole is being destroyed. So, this must go like this. So, for one particle theta 2, we have given i dagger aj dagger kkl. So, in terms of normal quantization, they will be all identical, not in terms of whole particle, because then it is all creation. Yeah, so that is a different matter, because there are four subsets. If you average over the all subsets, again it is same. So, this is a occupied orbital. So, a sense of the line must be downwards and it must go out. So, the only way to make is like this. And AR, this is the only way to make it. Now, I can take the other one, AR dagger AA, just opposite, that is particle creation, whole orbital destroyed. So, what will be the line? This will be exactly opposite. So, this will be now AA, this will be now R, R is being created upwards, A is being annihilated going in and line should be downwards. And all of you should be able to write this very easily, because remember, this is a Hermitian conjugate of this. AR dagger AR, AR dagger AR, Hermitian conjugates. So, diagram wise also, it is a conjugate, you just flip it. Diagram wise, it is just flip it. So, diagrammatically you can see it very easily, that it is a conjugate. This is, this you can already see, it is self conjugate. This is Hermitian, nothing else to do, because both of the lines are going this. This can be flipped. The final one is AP dagger AQ, both are particle. So, very clearly this line will be both up going. This is P, this is Q, correct. And note that again, this and this are very similar, except that the lines are just reversed. Both are unoccupied here. So, once you get into that mood, you can very easily do this. Both are occupied, both are occupied, it will go down. Both are unoccupied, it will go like this. One of them will be whole particle, another will be creation, another will be annihilation, whole particle annihilation. So, these are the two operators.  So, in my nomenclature, this will be actually whole particle creation. This will be whole particle destruction. Actually, there is a one pair of creation and annihilation operators in terms of A. But in terms of whole and particle, this becomes both whole particle creation, this becomes whole annihilation as well as particle annihilation. In fact later on you will realize for any operator, one or two particle operator, if I have all lines up, if I have all lines up, it is a whole particle creation operator. It is very easy to understand. Because if I have all lines up, it must have two lines coming from the top incoming and they must be occupied orbitals. And the ones which are going up are also occupied orbitals. So, that is basically a whole particle creation operator. It is very easy to see. If you have turned it around, it must be whole particle destruction operators. So, from the figure, we should be able to say whether it is whole creation, particle creation, whole destruction, particle destruction, from the figure itself. And you will get used to it once you start drawing. This is for the one particle operator. Of course, you have to go to the two particle operator, which is the essential point here because mp2 has two particle operators. So, let me write further rules for two particle operators just as here, they will be represented by a vertex with two incoming and two outgoing lines. Now, they can be occupied, virtual, anything, but the two incoming and outgoing lines. And you attach a matrix element. Just like you attach a matrix element here for every operator, here also I should write for every operator that I draw, there is a matrix element. So, this is let us say rule 4. So, for every operator, there is a matrix element attached. So, for example, for one particle operator, the matrix element is Hij, where the creation annihilation operators are in this sequence, a i dagger aj. So, if i is created, j is annihilated, the matrix element is just Hij, which is basically iHj in the first quantized notation, just to make sure that what is Hij. Similarly, for two particle operators, remember the whole and particle convention remains the same that arrow down is whole, arrow up is particle. That is a general rule. For two particle operator in Hugen-Hull's convention, remember now we are talking of Hugen-Hull's convention and here the convention to convention it differs. The attached matrix element is an anti-symmetrized two electron integral, ijkl for a sequence of operators which is ai dagger aj dagger al ak. So, basically k is going to i, j is going to l, l is going to j and the attached sequence is ij anti-symmetrized k. I will write the two particle operator right now. So, let me write a vortex of a two particle operator by taking a specific indices ijkl because otherwise I cannot write because the arrow line direction must depend on occupied or virtual orbit.  So, I am trying to write this vortex. Of course, the attached matrix element is very clear. It will be pq anti-symmetrized. So, that will be the attached matrix element for this and I am writing this two particle operator now. p and q are created. So, p and q will go away from the vortex and they are up going line. They will go away from the vortex. There will be two pairs of up going line. Now, the two particle operators with two incoming and two outgoing lines. So, two outgoing lines are already defined. My two ingoing lines are a and b. So, let me write this as p and q. This is a and b. So, a and b now must be arrow direction going downwards and incoming. So, they must be like this a b. Both of them are coming. So, if I write little bit nicely, all four lines are upwards. Two incoming, two outgoing. So, this is a, this is b, this is b, this is q. All four lines are upwards to the top of the vortex. So, if you look at the table, there is nothing below but two of them are incoming, two of them are outgoing. So, that defines this operator a p dagger a q dagger a b a. Actually, it is very easy to read from the diagram. From diagram, it is very easy. Just see the arrow. I hope you can read this. p is going out. You can easily see it. Start to think in terms of figures. A p dagger a q dagger a b a.  That will resolve later. The anti-symmetric matrix element will only change sign depending on what is the sign. There will be sign convention which will take care of that. So, do not have to worry right now. But the sequence of operators is very easy to write that there will be a p dagger a q dagger. There is a a a b or a b a a. Which sequence they are destroying annihilating? That is the only issue now. Actually, diagrams are much easy to understand. It is very easy to understand the diagram. Let me do the other diagram.  So, two lines are now annihilated which are particle. So, what will be the diagram? So, this will become p and q. They are up going lines. Up going lines annihilated. a and b are down going lines created.  So, exactly opposite to this. This is again not a surprise. If you look at the string of operators, this is exactly conjugate to this. I have two particle creation, two whole annihilation. Here I have a whole annihilation particle creation. Exactly opposite. So, diagrammatically you just flip it. It is very easy. Diagrammatically thinking is very easy. Of course, there are many other possibilities. All four can be particles. All four can be whole. There are so many possibilities. You can have so many lines. For example, this this can be another way of writing. So, this can be like this. So, all four are occupied. I can see from the beginning all four are occupied because lines are all going down. Two occupied are created, two occupied are destroyed on the top. You can have a situation like this, even this asymmetry. Two incoming, one outgoing, one outgoing, two outgoing. I can make incoming outgoing any way I want. Now, you can see these two are occupied. This is particle line which is created. This is another occupied line which is created. Actually, this is a vertex which has three of them occupied, one of them particle. You can call it three whole one particle kind of line. Where i, j, k, l are such that three of them are occupied, one of them are virtual. I am not drawing all diagrams because you can clearly see there are many, many possibilities. All holes. So, i can be whole particle, j can be whole. There are sixteen possibilities basically. So, all sixteen possibilities will transfer into sixteen shapes of diagrams. Then, I start writing the MP2. I have some more rules. Order, I am not going to suggest right now. The order will be taken care of by the sign convention because it does not matter. If it is PQAB, it is just a negative sign. The sign convention, when I write the entire diagram, there will be sign convention which will take care of the order, which will fix this order. I have to sum. This is only a vertex. I mean, there will be this plus this plus this. It is an operator, but then only one will be used. Finally, only one like an MP2 will show. Only one of them will be used. If all of them are used, then another will be used. Another diagram will be there. Let me write the actual energy expression now. Let me go back to MP2 diagram and write again the E correlation at second order. 1 by 4 sum over ABRS, AB anti-symmetrize RS, RS anti-symmetrize AB divided by epsilon A plus epsilon B minus epsilon R minus epsilon S. Now, what I am doing with these basic rules, there will be many more rules which will come up. I am translating this into the correlation energy diagram. First is to write this vertex. This vertex is ABRS. You can clearly see this is like this, ABQP. RS is unoccupied, AB is occupied. This vertex will translate into this. Is it clear? All four lines to the bottom just like here because it is A dagger, AB dagger, AS, AR. It is exactly like that. Two of them will be AB, two of them will be RS. Of course, the lines will be like this. This will be outgoing, these two will be incoming. You can see that they are being annihilated, R and S and the arrow direction is pointing upwards. These are created, A dagger, AB dagger, arrow direction is pointing downwards. These are occupied, these are virtual. Is it clear? I have also got the basic rule done, two incoming, two outgoing. You draw this vertex, RS AB. The rule is that for every vertex that I have, when I multiply, order them on the board one after another. How will this look like? I have already drawn this, exactly conjugate of this because that is a conjugate of this. Again two of them will be here. You can see that this is my RS, this is my AB. It is AR dagger, AS dagger. These two are going out. These two are coming in. They are occupied, they are virtual. Is it okay? Now I have a sum. I must need to know how to sum in the rule. The sum is performed in a very simple manner by what is called the contraction. This is something that I will just loosely define. The contraction comes because of the second quantization. The contraction is now the product of the second quantized operators. I have a second quantized operator here. I have another second quantized operator here. Only the matrix elements are what is seen in the actual expression. The summation is done by summing over lines, joining lines which are equal in sense. You can see this is occupied, this is occupied. I can join this line. This is occupied, this is occupied. I can join this line. I can join this line. Let me write this little bit better. This is a joining of two what is very simple. These two are AB, these two are RS. You can see automatically one represents this, one represents this. With the same sense you can join. This means I have summed over all AB, RS and I have an anti-symmetrized matrix element attached here. I have an anti-symmetrized matrix element attached here. Everything else would now vanish. The actual sense of creation and annihilation operators will no longer be there. What will be there is only the matrix elements when I write the energy expression. A further rule between the two what is says draw a line. Draw a line between the two what is says. The line will intersect some holes in particle's line. What is it intersected here? This, this, this, this. These four lines. This is just a line between the two what is says which intersects these four lines. Then attach a denominator where you put sum of the whole lines minus sum of the particle lines. This is now A and B. These are whole lines.  That is a rule. Then multiply the matrix elements. If I give this rule you can automatically see that I am actually getting to this MP2 result and sum over all lines where whatever is joint sum that automatically comes. Of course I have to still explain one by four and all that. I will come to that. There are some more rules are there. So every pair of vertices, so this is very important. Between two vertices means every pair of vertices. So this is important when I go to MP3. There will be three V terms. So there will be another pair of vertices. So I have to draw another line. So later on it will be useful. So now let us go to the other rules. The next rule is that count the number of holes. Please write down in the rule numbers and all that I have lost it but count the number of holes and number of particles which are summed up. So these are called internal lines. Internal lines are lines which are summed up completely. So how many holes are there? After contracted holes, this is one line now. So you can count the number of contracted holes and contracted particles basically. So do not think of four, two here and two here. They are now contracted. So here are actually two hole lines, two particle lines. Then attach a sign which is minus 1 to the power h which is the number of, so h is the number of hole lines, number of internal holes. This is called internal holes. Particle does not come. It is only minus 1 to the power h. The next is count the number of loops, L, number of loops L. So let me show how to count. First of all I must write here when you are drawing this energy diagram. These energy diagrams must be closed and that is understood here. There are no open lines. Closed means all lines are contracted. As you can see here, nothing is dangling. All are contracted to each other. That means basically there are four lines here. Four lines here, they have a handshake. So all completely handshaken. So all the MP2 diagrams or MP3 diagrams must be a closed diagram. That is very important. With the closed diagram now you have to count the number of loops. What is a loop? Loop is the following. You start from here A, go to R and come back. If you start from here and come back to the same point like A to R, then it is one loop. Then again you can come back with another line. B, come back here. B to S. So that is another loop. How many such loops are present? Two. You go from here and come back here. So that is one. That is over. I had another chalk. I could see the color. Then you go from here and come back here. I have traversed everything. I have traversed all the four lines. There is no other loop. You cannot start from here and come back again. It is the same loop. You cannot traverse the same line twice. If A R is there, B S is there, then A S cannot be there. So it is a question of how you write your element. So you can see my element is A to R, R to A, S to B, B to S. So you follow the lexical order here and that is your loop. Number of electrons? Yes. It depends on how I write. That is also important. I could have one loop also and I will tell you how. That will answer your question, the previous question. I can go from A to R, A to R but R comes back to B. B goes back to S. The loop is completed. When I go from A to R and then R comes back to A. So A goes to R but R may come back to B. Then it is a different loop. B goes to S. So that would essentially mean it is A B anti-symmetrized R S, R S anti-symmetrized B A. Then you see the loop. A just follow this. A goes to R, R goes to B, then B goes to S, A goes to A. How many loops are there? One. I have started from A and coming back to A, I have traversed all the lines. So this actually answers your question. How do I write the second quantized rule? It does not matter how you write. Had you written this R S anti-symmetrized B A, my loop will become one less and then the sign is minus 1 to the power L. So there will be a negative sign because your loop is changing depending on how you are writing this. So at the end you need not even see. You write the anti-symmetrized matrix elements any way you want. Keep the outgoing here, keep the incoming here, any way you want. The actual sign of loops will take care of the value. So that is how it is quite sophisticated rule. So let us assume that I am writing it in the manner I wrote. In the manner I wrote, you do not even have to see this. You just look at the algebra. Start from A, go to R, R back to A. I have finished one rule. Then you start from B, go to S or S, go to B, whatever. B go to S and then S comes back to B. So another loop. So two loops. See remember, it is a direct notation. A1, R1. So that is how you should see. So A is going to R, R is going to A. If on the other hand you flip it, then A goes to R, R goes to B, B goes to S, S goes to A. So that actually does the change and that brings the anti, the minus, because you have a minus 1 to the power L, the sign will be automatically taken care. So if I have internal holes H and internal loops L, my actual sign becomes minus 1 to the power H plus L. I mean I can do it together instead of saying minus 1 to the power H and minus 1 to the power L. So that takes care of the sign. In this case, I have two internal holes. I have two loops. So there is no further sign plus. Minus 1 to the power 4 is plus. So it remains plus here. The way I have written this. This is very important. The way I wrote this, because if I write it opposite, then loops will change. That will automatically take care of the anti-symmetry that we are talking of. So I don't have to be specifically careful when I write the two electron integral. What is RSAB or RSBA? I don't have to bother. The loops will automatically take care. Is it clear? Now the final one, 1 by 4. You still have 1 by 4. 1 by 4 comes for the following reason. How many pairs of equivalent lines? So that is the last rule. Count the number of equivalent lines. Count the pair of equivalent lines. What is an equivalent lines? Equivalent lines must be either holes, both of them must be holes or particles. They are outgoing and incoming on the same vertices. Now tell me how many equivalent lines are there? If you look at A and B, that is an equivalent pair of lines. Two lines which are both holes which are starting from this vertex, ending at this vertex. So that is one pair. There is a second pair which is R and S. Both of them are particles starting from the same vertex, ending the same vertex. So in this case, I have two pairs of lines. So let us say, count the pairs of equivalent line and I call these pairs that is a P, then attach a factor which is 1 by 2 to the power P. So in this case, of course, it is very clear that there will be 1 by 4, 1 by 2 whole square and sum is over all lines. So I don't have to specifically bother about A less than B or less than S, etc. Is it clear? So you can see you have so many facts, so many rules to of course eventually write this. But once you understand, the writing becomes very easy.


Income tax 2022-2023. Ira deduction questionnaire. Let's do some wealth preservation with some tax preparation. Support accounting instruction by clicking the link below giving you a free month membership to all of the content on our website broken out by category further broken out by course. Each course then organized in a logical reasonable fashion making it much more easy to find what you need than can be done on a YouTube page. We also include added resources such as Excel practice problems PDF files and more like QuickBooks backup files when applicable. So once again click the link below for a free month membership to our website and all the content on it. In a prior presentation we look at the Ira deduction in general now we're going to look at a worksheet which you can find on the form 1040 instructions at the IRS website irs.gov irs.gov helping us to calculate the actual deduction note that in practice this would often be done with the software. The software can help you to calculate this but just taking a quick look at this worksheet can help us to look at what the limitations are in terms of the max amount of the deductions possible in scenarios that are simple scenarios and scenarios that are more complex scenarios scenarios where we might also have access to say a 401k plan or a 403 B plan or we're married to someone who has access to a 401k or 403 B plan or something like that. Now remember the general scenario I think the government concept the idea the government has would be they would like everybody to be basically employees of some kind and have access to like a 401k type of plan and if you have access to the 401k plan plan what plan then you need to max out whatever you can put into that 401k plan in the taxable year that being 2022 that we are talking about here and then if you don't have access to a 401k plan that's when kind of like the IRA kicks in so you could still get some of the advantages of being able to save for retirement and have some tax benefits but it gets confusing when there's overlap in that you put money into a 401k plan can you still put money into say an IRA or if your spouse has access to a 401k plan can you put money into an IRA and you've got the income thresholds and so on and so forth. Fortunately, fortunately, you could put money into an IRA generally up until the point that you file the tax return not including extensions and therefore you can do a bit of the last minute tax planning so you want to be able to express that to the clients and obviously you would typically do a tax planning situation by using actual software not usually an actual form like this but like I say looking at it in terms of a form can get us an idea of what's going to happen so that we can communicate what's going to happen to clients when we're speaking with them so let's take a look at it.  My lifestyle is my retirement plan. So notice if they were a single filer for example we might have W2 income and then the question is did you have access to say a 401k or a 403b or something like that which would be shown on the W2 and marked off in the software from the W2 and that can complicate whether or not we would be able to access and get more benefit but putting money into an IRA. B, if married filing jointly was your spouse covered by a retirement plan. So now when we're married that's going to add a level of complexity because even if just our spouse had access to the 401k plan depending on the income thresholds it could have an impact on how much we can put into the retirement plan or an IRA I should say we're focused on the IRA. Next if you check no online 1A and no online 1B if married filing jointly skip lines 2 through 6 enter the applicable amount below online 7A and line 7B if applicable and go to line 8. So we've got the 6000 if under age 50 so that's going to be the maximum contribution typically 6000 for an individual unless they are over age 50 where it can be boosted up to 7000. So this is one of those types of things that's kind of applied to person per person in general. So if you're married then you're still could have some some overlap in terms of the rules meaning like if your spouse has access to a 401k plan maybe it still has an impact on how much you can put in but it's also kind of separate at the same time because obviously we have two individuals which we would think could max out their their IRA contributions each. So we'd have 6000 each or 7000 each depending on the age if under 50 or older 50. So enter the amount shown below that applies to you and then we have these are kind of our thresholds now that are going to be starting to apply the income thresholds. So single head of household or married filing separate and you live apart from your spouse for all of 2022 you enter the 78000. So married filing jointly 129000 in both columns but if you checked no on either line one A or one B enter 124000 to the person who wasn't covered married filing separately and you lived with your spouse at any time during 2022 10000. So you're going to be severely limited as is often the case if filing married filing separately because the IRS is going to be kind of suspicious of those scenarios. Now I'm not going to go through each of these items more one by one so you kind of read it over if you would like to but we're going to be then kind of taking these general ideas and then applying them to tax software in a future presentation. Let's just I'll try to just kind of recap the general scenarios that might come up. So let's say you're married your spouse has a 401k and you're modified adjusted gross income. This is the adjusted gross income you know after the above the line deductions but it's kind of modified. So it's still a general idea of your adjusted gross income. In 2022 if you are married filing jointly and your spouse has a 401k you can take the full deduction for the IRA contribution as long as your modified adjusted gross income is below 204000 I believe if your modified adjusted gross income is between 204000 and 214000 you can take a partial deduction. Meaning it starts to phase out at that point. So let's say that you are married you have a 401k and your modified adjusted gross income is below the threshold. So now you're saying you have the 401k plan. So if you are married filing jointly and you have a 401k you can take the full deduction for your contribution to a traditional IRA as long as your modified adjusted gross income is 109000 or less in this situation if you're modified adjusted gross income is between 109000 and 129000 you can take a partial deduction.  So if you have a 401k or other retirement plan at work you can take the full deduction if you're modified adjusted gross income is 68000 or less. If you're modified adjusted gross income is between 68000 and 78000 you can take a partial deduction. Note that in all of these scenarios the workplace retirement plan doesn't have to be a 401k plan. It could be a 403b or whatever kind of qualified retirement plan. That's just the kind that most people have. So that's the general concept. So then we have if either your or your spouse is covered by a retirement plan at work and your combined income exceeds the threshold set by the IRS you may not take the full tax deduction for your IRA contribution for married taxpayers filing jointly. Jointly the limit is 129000. If you have a 401k and 214000 if your spouse has a 401k for single taxpayers the limit is 78000. And of course the general IRA contribution limit is going to be the 6000 per person generally and 7000 if you are 50 or older as we can see in our worksheets here. Now again that whole scenario of course is quite complex. So the general rules would be the IRA contribution 6000 if you're under 50 7000 if you're older 50 if you're have access to a 401k plan either you or your spouse it's going to get more complicated in those scenarios. And as your income goes up you may have limited limitations and phase outs to be able to access the IRA. So you will be able to do tax planning and the software can kind of help us out with that last minute tax planning would be the general scenario that you might want to kind of have in mind when these questions come up as they do because the IRA is always going to be that last minute type of thing that you could take a look at.


Hey everybody, welcome back to magic orthodoxy. My name is David and this is a magic review Hey today, we're gonna look at the Networker deck from Paul Brooke. Alright, so what is the Networker deck? Well, basically it's a well it's a deck of cards that you can do card tricks with but it's very unique Every single card is a unique business card And you know what you're gonna be able to do is take this deck and do a ton of really killer Mentalism effects. You want to see the trailer? This is it For quite some time, one of the most powerful things I've been using in close-up is a Svengali deck made out of business cards I was home making them by collecting business cards Trimming some short and using them to force information on people The problem is that business cards vary a great deal in quality So the poorer quality ones would bend or become out of shape and would essentially destroy the Svengali nature of the decks Equally home trimming them didn't always guarantee 100% that it was a straight line again when things are slightly askew with the different sizes and shapes of the business cards This would render the Svengali principle useless In order to get over these hurdles. I've had my own Svengali business card decks created This has meant that I can keep the quality high have them precisionally cut and All of the information on the front of the business cards has been designed by me Which has meant that I can hide certain things certain mentalism effects within that information As well as keeping every single business card completely unique These are the most congruent use of a Svengali deck in the corporate environment And they will allow you to springboard into one of four effects One of the effects is its own mini routine, which is pretty cool in itself Welcome to the Networker deck All right, so that was the Networker deck from Paul Brook couple questions We always ask about all of our magic reviews and the first of which is What's in the box? What's in the box? You're gonna get the Networker deck and then with that You're gonna get a URL that'll get you to a download for a 54 page PDF Is it what I thought it is what I thought but only because I had read a lot about it and just seen Some different reviews, so I knew exactly what I was getting into how are the angles the angles on this It's gonna work exactly like a long and short deck sometimes called a Svengali deck So if you already have some familiarity with that as a means to force the card You're gonna be very familiar with the Networker deck. Can it be inspected? The individual cards can be inspected, of course It is made as a deck where you would hand the various business cards out and the specter would hold them in Examine them. However, I would not hand the entire deck out because that would be bad What's the overall quality and production value of the PDF? The PDF has 23 color photographs and Images and it makes every step of the process very easy to understand. Is it well made? The business cards are printed on a premium quality stock with a protective matte finish And it's gonna provide you with long lasting durability for years and years and years Each one of these cards has been machine cut to precision and it's delivered in a beautiful plastic box that makes carrying the deck Very simple and it keeps everything organized. How much practice does this require?  Okay, so just a random Image that they pick from the deck and they're gonna write it down or have it in their mind And you'll be able to do a drawing duplication from that They're also gonna teach you how to have your Participant choose a website and then a random word from that site And then you're gonna be able to determine what that word is and yes The person can use their own cell phone and go to the website themselves and never show you the website And you're able to determine the word that they're thinking of and aside from all the different Formal effects that are taught on the PDF Paul Brooks is going to share with you a number of techniques and tips It'll help you to get even more from this wonderful deck of cards and you may even generate even more work and Develop clients and create new contacts with the Networker deck. How much setup and reset is there? It is a deck stack. So you're already familiar with the longest short deck or a spengali deck. You understand that But the mentalism aspect of it will require you to learn some details Memorize some details that are on the individual business cards Positives one of the positives to the Networker deck for one. It's very creative It allows you to do a spengali style trick but with a deck of business cards, which looks seemingly Innocent, you know, and then when you throw that on top of all the mentalism tricks You know the star sign the word divination the drawing duplication you put all those together You really have a solid Routine of tricks that you can do with this negatives one of the negatives to the Networker deck You know like we say right here. There's negatives to everything. These are just the negatives I see They don't have to be the negatives that you see Again with Paul Brooke, I was just a little disappointed that there wasn't a DVD with this There was no video learning obviously he made a commercial that has you know a video sales pitch So I would have liked Coupled with that at least a downloadable video that just had some sort of aspect of Picture, you know or showed some performances or showed his routine. I thought something like that would have greatly enhanced the buy value on That end for me is it worth your money. It's 40 bucks $40 is super super super affordable for what you're getting I mean you're there's a lot of decks of cards out there that are $40 that are magic decks of cards And they only do one trick this will do multiple tricks This is a mentalism routine that you can carry in your pocket for $40 and it looks completely innocent So highly recommended well worth your money if that sounds like something you would like I would say who would like this you would like this Definitely mentalists that are trying to add something new to their routine or who are just looking for something fresh I think should definitely check out the Networker deck from Paul Brooke All right, so that is the Networker deck from Paul Brooke I purchased my copy from Penguin magic calm and if you'd like to purchase one for yourself I would suggest you go there as well. Thanks. Bye


So the last chapter began to add those rules for analytical reading, and a lot of it covered skills that you use, skills where you use inspectional reading, remember that then. Not skimming, or not speed reading, but maybe skimming or, you know, special reading. To refine the structure of the book, try to find the problem that the philosopher, excuse me, the author is trying to solve what question the author is trying to answer, what the author believes answers that question. The structure and how the steps that the author is going to take to get there, to go from the question to the answer. This chapter begins kind of the next stage for analytical reading, and it's coming to terms with the author. And what Adler is dealing with here is to find the key words, the key concepts, the important words, the important concepts that the author uses to get from that question to that answer. So to start, Adler gives us a distinction between words and terms. Words are collections of, you know, if they're written down, collections of letters, which together, you know, follow rules of spelling to be a word. If it's spoken, it's a collection of sounds. Again, following certain rules, phonetics, sign language has words. They're signs, hand signs. I'm not going to try to insult people by, you know, imitating them right now because I just don't know enough about sign language to do that. But sure, those are words too. Those are words. Terms are different from words. Terms are a word used unambiguously, which is kind of a clumsy phrase there. The medieval said used univocally, right? And that means that they're used with one and only one meaning. There's a difference between words and terms. There can be many different meanings for a single word. I know I've looked up the word innocent before. Innocent has something like 16 different definitions depending upon which dictionary you look at, ranging from anything to not guilty, to not guilty because of ignorance, to, you know, moral purity, something like that. And those are three very different things. So a word can have many different meanings. A meaning can have several different words, right? These are synonyms. What liquid, water, agua, right? Well, that's, I guess that's Spanish at that point. But these are rough synonyms for all the same meaning, right? Okay. So there's a difference between words and terms. And what you have to do as a reader is come to terms with the author. You have to find those important words and figure out what term the author intends with that word. This brings us to rule five of analytical reading. You have to identify the key words and come to terms with the author. Now notice, and Adler takes pains to point this out. This is a two-step process. You have to first identify the key words, right? You don't necessarily understand fully what they mean yet, or maybe not at all, right? But you have to identify the key words. Once you identify the key words, then you try to understand what the author means by those words. What are the terms? So you have words on the one hand, terms on the other. And Adler recommends, by the way, Adler recommends in your notes, you know, creating a two-column list, right? One column with the words, one column with the meanings and the terms, okay? Now this process isn't going to be straightforward. It's not like we have a finite number of steps to determine what the key words are. I know maybe you've done assignments, right? Find five key words in this text. It's not going to be like that.  So, you know, that's perhaps another discussion. But identifying the key words is not going to be straightforward. You've got to hunt for these words. And Adler has several pieces of advice on how to do so. And then once you identify those words, you have to find the term. You have to find the meaning. Finding the key words is not going to be straightforward. You've got to hunt for them. One of the initial pieces of advice that Adler gives as far as that's concerned is, you know, first find those words you don't already know, right? Find these words with which you're unfamiliar. And you don't know the meanings of the terms. Or even maybe, I don't know, you kind of have an idea, but it's not really clear. That's just as good practice, so you can understand what the author has to say. I mean, the whole purpose behind identifying the key words and coming to terms with the author is to understand what the author has to say. Okay, so you first start with those words you don't already know. Secondly, pay attention to those words that the author takes pains to identify and to define. And sometimes this happens. Sometimes this happens, especially if the author is, you know, maybe using a word in a very, very specific way and not necessarily using it just in a common way of speaking about it. There's lots of words that we use that in common speak just really don't mean something very specific. It can mean a variety of different things, right? I'm going to take a walk, okay? I'm walking, right? Those are two, you know, walk is kind of a common word between those two. But it probably means something different, right?  There's lots of ways that we use these words that, you know, in common speak, we just, it could be in any one of them and it doesn't really matter. But the author really wants to pay attention to a particular word and spends time defining it, right? You know, pointing it out, giving a definition, then yes, right? You should write that one down and also write down whatever meaning the philosopher believes is appropriate for them. You should pay attention to the section titles, right? If you have an article or a book or a chapter or, you know, sections within a chapter, those are probably going to be important words. You should write those down and make sure you understand what they mean. Often or, you know, a common practice is to, you know, title a section with something very, very important, you know, for the main point. It's not just going to be, you know, here's a Sunday afternoon, right, unless somehow Sunday afternoons are important to the author's work. Now I don't necessarily give this advice, but I think it's a good piece of advice. Remember in the last chapter, Arthur wanted us to pay, you know, to identify the problem and to, you know, try to figure out what the, you know, the book is about. All right. This is a good idea. Now once you identify that question and that answer, you should write down the subjects and the predicates, right, the important, these key concepts in that question, in that answer. Right. Chances are this is going to be important to the, to the author's works, right. And you know, if you've identified that structure, chances are there's going to be some kind of link from one concept to another, right, where, you know, if we're talking about subjects and predicates, that a subject is, is described by the predicate or excluded by, from the predicate, these important links, you want to identify those key terms all along the way from that question to that answer, that'll be helpful in identifying the, these key words. All right. So that's the first step, find the key words. The next step, come to terms with the author. So once you identify those, the key words. Now you have to come to terms with the author. You have to figure out what the author means by those words.  I mean, there's no guarantee that the author means the same thing that the dictionary means. And frankly, there's, you know, there's going to be lots of problems with this trying to just pull out meanings from the dictionary. First of all, you're not thinking anymore. Right. That's a start. You're letting somebody else think for you.  And you know, there's the problem of synonyms. Right. Actually, not even, I'm sorry, not synonyms at this point. This is the problem of ambiguity, where there's multiple meanings for any given word in the dictionary. Yeah. I mean, there's rare exceptions, but, you know, there can be multiple meanings for any given word in the dictionary. So you have to come to terms with the author and like finding the key, the key words, this is going to be difficult. So earlier I mentioned, you know, paying attention to those words that the author, you know, tries really hard to define. So that's kind of, that's kind of an easy step.   But don't think your job's done, right? Because the author can provide multiple definitions for the same word. And they may not necessarily be identical. So what you'll have to do is, you know, kind of catalog all those different definitions and then write them all down, right? Have them in your, you know, two columns, right? You got your word and your definitions, right? So that's, that's kind of a, you know, an upfront way of identifying or coming to terms with the author is looking for those technical definitions or the ones that the author specifies, like, I mean this and only this, right? That sort of thing. And that's, that's kind of a good way forward. The other way to do this is pay attention to examples that the author gives. You'll, you know, want to write those down too. Now, I either doesn't necessarily state this, but I think it's a good idea. If the author provides examples of the word, that's part of defining that term, right? So write down those examples too. Try to identify what those examples have in common. What makes those things what they are and what distinguishes those things from others of its kind, right? If you're paying attention, you just notice that I gave genus a differential. Adler is unapologetically Aristotelian, so it's not a surprise that he's going to help himself to Aristotle here. Now, you know, the more that you work with this, right? You're going to struggle because, you know, this comes through experience. You're going to struggle with finding the term, the meaning that the author is using. Now, you know, you might wonder, well, when do I really understand what the author is doing? Well, you can start coming up with your own examples and see if they fit with the examples that the author gives. This is part of defining these terms, is the things to which the meaning refers, right? That's really, really great. They helps. And when you start coming up with your own test cases that match up with all the claims that the author makes about those terms, yeah, you've probably successfully identified at least the objects to which the term refers. The other part of it is actually providing your own definition, right? The author is going to provide one definition and that's fine. I'm not saying you, you know, you almost do better, but you can do better. Or at least you can provide a definition that's going to be pretty much the same, right? It is possible to give multiple definitions, right, without using, that they all mean the same thing without using the same set of words. It can be done. Now, but this is, you know, this is this project that you have with coming to terms with the author, with understanding what the author says.  And this is a really great way of doing this, right? Unfortunately, when we, you know, what happens a lot when teachers they put it in your own words, you know, students will hit the the source and try to find synonyms for these words and, you know, have at it and say, OK, well, that doesn't really help. But now this coming to terms with the author is a practice in defining at this art of defining, defining does not mean going to the dictionary and finding the meaning in the dictionary, right? Merriam Webster is not the end all be all of your thought, right? You've got to learn to think and by the way, those dictionaries aren't even like, aren't even necessarily that great. I mean, really good dictionaries and these are only multi volume works will sit there and painstakingly define a single term and it'll take, you know, a few pages, right? Sometimes we think of these as encyclopedias, but it isn't necessarily the case. So learning to define is an important skill and it is a skill. This is something you can acquire and perfect and home. So what do you do and broad strokes, right? You, you know, give examples, these clear cut cases of what you're talking about. You give clear cut cases of it like the exact opposite, right? What's really not this, maybe even give cases where, you know, you're trying to say, OK, I'm talking about this, but not that, right? Maybe some close cases, right? And you give, you know, maybe. Famous cases, if it applies, right? If that applies point to examples, this is going to help you to find the term so that the reader, right? This is when you're writing and somebody else is reading your work. The reader will be able to identify what you're talking about. Provide, look for synonyms. Synonyms are great. Now, it can be helpful to say, look, I'm kind of talking about this sort of thing. But you might also want to distinguish the synonym, right? You know, to distinguish some cases in the synonym. That's not what you're dealing with.   But being able to provide a single sentence definition is a real skill, right? And that means providing the genus and the differentiate, right? You identify the kind to which the thing belongs. And then you spell out what distinguishes that thing from other members of its kind. Now, shameless plug moment. I have a whole video set of videos, really, on defining terms, right? I'm going to put a link in the description. You want to develop your skill in this, start there, that will help.   I've yet to see a paper in the humanities that didn't require defining at least three terms.  If you are, you know, if you're not nearly so precise, if you're just speaking casually, or you're just not trying to shave off words, you can probably do it in about 300. And it's about a page. And all of a sudden that three to five page paper, it's a lot easier to fill the pages, right? And this isn't, you know, this isn't bold work. This is just, you know, making stuff up. This is defining terms. It's defining terms. Okay. So that's rule three, that's rule five, identifying the key words and coming to terms with the author. You know, and Adler even says that, you know, the more you work on this, the more you can become interested in defining the difference between a literal definition and a metaphorical one. Concrete terms versus abstract ones. Emotive terms versus cognitive. And the more you can define, the more you learn how to think. It's very literally, understanding what you're saying. I think we're all familiar with more than a few people who speak without understanding what they mean.