 Today, we are going to discuss the quantum mechanical treatment of various interactions of an atom and the atom is the simplest possible atom we can think of that is the hydrogen atom. Why you discuss this because the simplest system which is paramagnetic. So, to discuss any quantum mechanical properties we must start with Hamiltonian. So, what is the Hamiltonian of the hydrogen atom? As you all know Hamiltonian operator is a sum of this kinetic energy operator and the potential energy operator. What a hydrogen atom is T and B can be written as in the kinetic energy operator which is kinetic energy of the electron and the kinetic energy of the proton. So, this will be this is the kinetic energy operator for the electron and the kinetic energy operator of the proton. These are the Laplacian operator and the potential energy B is the electron the Coulomb attraction between the two. So, this is minus z e square by r where r is the distance between the electron and proton. So, in a particular coordinate system if I have x y and z let us say here the electron is sitting this proton is sitting then this distance r and for each of this coordinates here and here I have got the corresponding operators. Now, in magnetic resonance we really do not worry at all about these type of interactions. These are treated to be a constant energy that the system has. What we are interested in is over and above this interaction what are the magnetic interactions which are responsible for the magnetic resonance spectrum in general and EPR spectrum in particular. So, these terms are not used at all they are just giving rise to a constant energy of the system. So, what are the magnetic interactions that this atom has? So, if we put this hydrogen atom in a magnetic field then the interactions that we can think of is the Zeeman interaction here. So, this is the magnetic moment interacting with the magnetic field give rise to the corresponding operate energy. So, for electron the magnetic moment comes from the spin angle momentum and this is the relationship. So, its operator for Zeeman interaction is G e beta e s dot b. Similarly, for the nucleus it is G n beta n i is the magnetic moment of the nucleus. So, that gives the Hamiltonian for the nuclear Zeeman interaction which is G n beta n i dot b. In addition there is another interaction that is possible which comes from the magnetic dipole interaction. I have one let us say magnetic dipole coming from the proton another magnetic dipole coming from the electron. So, these two magnetic dipoles can interact and that is shown here this slide this is the let us say direction of the electric dipole moment this is the direction of the magnetic dipole moment in sudden x y z coordinate system and this is the distance between the two. So, this energy of these two magnetic dipoles depends on the orientation of these two dipoles. In fact, it is the relative orientation of them that matters the energy is given as this. So, it not only depends on the magnitude of the dipole, but also distance between them and the orientation of mu with respect to r mu with respect to r. So, therefore, this interaction is directional dependent and you call it anisotropic interactions. Now, what will be the corresponding operator for this we simply replace the magnetic moment of each of them by their corresponding operators in terms of the angular momentum. So, mu e is replaced by G e beta e s mu n is replaced by G e n beta n i in a similar fashion this is the expression for the dipole dipole Hamiltonian operator for electron nuclear interaction. Now, this dipole dipole interaction of is such that its average value turns out to be 0. So, when the species is tumbling rapidly let us say liquid or gas in their collisions of atoms then the average value of this energy becomes 0. How do you visualize that this energy of interaction becomes 0 it is not very obvious from this type of operator or this type of energy expression for that of course, one can calculate and turn and find out the indeed that is 0 if we take care of all possible orientations, but let us try to understand at least qualitatively whether this energy can be 0. So, for that this is the schematic representation of two magnets let us say mu 1 and mu 2 they are kept at a distance r and they are aligned such a way that mu 1 forms an angle phi with respect to r and mu 2 forms an angle theta. Then what we have here is mu 1 mu 2 and this is r 1 mu 2 mu 2 mu 2 mu vector. So, if I keep this in a magnetic field then both of them will get aligned along the magnetic field. So, you put it here then what will be the corresponding energy of interaction if their orientation changes and also go from let us say one alignment to other alignment. This is shown here this is the direction of the extra magnetic field and let us say one magnetic moment is fixed here and other one is moved from this arrangement to this arrangement. Here these two magnets are parallel to each other. So, north pole is point in north pole south pole is point in south pole. So, this energy will be repulsive. So, we call the energy is positive same is true if this magnet is brought here. On the other hand when second magnet is pointing in this direction and at the top of this first magnetic moment mu 1 then you see here the north and south here north south are pointing in such a way that there will be attractive interaction. So, this north is pointing to south. So, the energy of interaction will be negative here same is true here. So, approximately in this region the energy of interaction is negative and same is true here and in this region is positive and this energy is positive. So, it turns out that if we really look at the interaction operator in detail the angle of approximately 54 degree is the defining angle within which this rather within this cone of this interaction region where the energy is negative here also energy is negative and exactly in this along this region the energy of interaction becomes 0. So, the two regions are divided in such a way that somewhere energy is positive and somewhere energy of interaction is negative. So, here now if the system is tumbling very rapidly that all possible regions are covered equally then one could considerably assume that this total energy actually becomes 0 that is indeed the case if we do the calculation exactly for all possible orientations. So, in solution if your spectrum for example when the species is tumbling very rapidly then one can ignore this interaction but it cannot be ignored if let us say solution is viscous and tumbling is not very rapid or if the species is held in a solid state or in a frozen solution then the tumbling is restricted very severely. So, we cannot neglect this interaction but for our purpose right now let us neglect this. So, what other type of interaction this electron nuclear can have that is called electron nuclear hyperfine interaction it is a special type of interaction which does not depend on the direction it is called an isotropic interaction and also a special requirement is that the electron sort of seats on the magnetic nucleus. So, unlike here the dipole level interaction there here if the let us say this is the nucleus which is proton for hydrogen atom of course electron is moving all over this is the decided by the orbital of the electron. Now if there is some finite probability that electron actually occupies this place this is what I call sitting on the proton then that interaction is a special type of interaction that is called the electron nuclear hyperfine interaction it was proposed by Fermi the expression is given here. See just like dipole level interaction this also depends on the strength of the magnetic moment of electron and the magnetic moment nucleus but it also depends on this probability of finding the electron at the nucleus. So if the psi is the wave function of this electron which is describing its motion around the nucleus the psi 0 and the square of that gives the probability of finding the electron at the nucleus. Now for those orbitals which satisfies this requirement that this has to be non-zero then this expression shows that this isotropic electron nuclear is hyperfine interaction will be non-zero. So not all orbitals have this property we know from the property of hydrogen orbitals it is that s type of orbital for example 1 s, 2 s etcetera. So here if you plot the wave function square as a function of the distance from this nucleus r then this is psi r square. So this for 1 s orbital it goes down exponentially. So at r equal to 0 this is value is non-zero similarly for 2 s orbital it looks like something like this this could be 2 s orbital. So for all the s orbitals there is finite probability of finding the electron at the nucleus. In contrast if you take other type orbital let us say p orbital this is the 2 p orbital similarly d and other orbitals there here the at the nucleus the probability is exactly 0. So for these sort of orbitals there will not be any such electron nuclear hyperfine interaction but for these orbitals it will be. So for hydrogen atom this is of course the exact form but for other system let us say free radical species this will be the corresponding wave function of the electron which is producing the interaction and probability of finding the electron at the particular nuclear site has to be used here. So this whole of this term here is replaced by a constant called A is a parameter in fact. So this constant is called hyperfine coupling constant or hyperfine splitting constant. Now these 2 terms though they are used often synonymously they may not mean exactly the same thing. Let us right now let us understand this just a subtle difference what I mean by that in this Hamiltonian isotropic Hamiltonian I write A s dot i. So here this is an energy of interaction. So here A gives the strength of the interaction the A gives the strength of interaction. So here I will be calling it as hyperfine coupling constant with coupling interaction between the electron dipole and the nuclear dipole. So when it is measured in the units of energy this is the coupling constant. Now we will see that this will of course show its presence by splitting the EPR spectrum. So EPR spectrum is measured as a function of magnetic field generally. So we measure something from the experiment so which is a measure of the splitting in terms of the magnetic field. So this interaction causes splitting in the spectrum. So whatever property we measure from here the measure of the splitting that is of course related to this one but that splitting that we measure from the experiment would be called the hyperfine splitting constant and we will see how they are related later. So I have already said that psi 0 square gives the probability of finding the electron at the nucleus and this is non-zero only for electrons in S type of orbitals. So this Hamiltonian where we are dealing only with the magnetic interactions such as this electron Z1 interaction, nuclear Z1 interaction and the hyperfine interaction. So this type of Hamiltonian are called spin Hamiltonian where we see that we do not include the kinetic energy of the various constituent particles or potentiality that they have. Only this interaction energies are included in the spin Hamiltonian. Now if the magnetic field B is along the Z direction then this Hamilton offered a curvature as B0 SZ minus GE G electric nuclear Z1 SZ IZ. This can be written as the magnetic sum of two components plus this is one component plus another component. We will see why you do that in a moment. This term which involves the X and Y this mistake SX IX plus SY IY. So this can be written in terms of the raising and lowering of order S plus minus equal to AX plus minus on SY. So SX gives S plus plus I minus 2 and SY is equal to SY is equal to SY minus I minus Y 2I. Similarly we can write for the IX and IY. Then SX plus IX for example this becomes S plus plus S minus by 2 into S which gives SX plus IX plus. Similarly if we take the other one by putting this one then this term will cancel out. So what we see here that the simplification of this will turn out to be this way that the electron Z1 nuclear Z1 and this SZ as a plus this is the term that comes out to be S plus I minus plus S minus I plus by 2. So we have the total Hamiltonian is tan B0 SZ IZ plus A SZ IZ plus A by 2 and this S plus I minus plus S minus I plus. Here now let us see the relative magnitude of the various interactions. This is the electron Z1 term this is nuclear Z1. So we know that magnetic moment of electron is about 2000 times higher than this. So this will be the most dominant interaction. This will be very very small compared to that one and this magnitude of the hyperbolic constant is also not very large. So this will still be the dominant interaction. This is usually very small. So what we do now is that we treat this as the unperturbed Hamiltonian or main Hamiltonian plus this is the perturbation. So A0 is equal to this is the main Hamiltonian and perturbation Hamiltonian is A by 2 minus plus S minus I plus this one. With this approximation we will see how the energy level that appears for hydrogen atom in the next discussion. We stop here now.