Operator electron spin

To consider the question in more detail, you need to consider spin eigenfunctions. If you have a Hamiltonian X and a many-electron spin operator A, then the wave function T for the system is ideally an eigenfunction of both operators ... 

In the limit of infinite atom separations, or if we switch off the Coulomb repui. sion between two electrons, all four wavefunctions have the same energy. But they correspond to different eigenvalues of the electron spin operator the first combination describes the singlet electronic ground state, and the other three combinations give an approximate description of the components of the first triplet excited state. 

For every electronic wavefunction that is an eigenfunction of the electron spin operator S, the one-electron density function always comprises an spin part... 

Having stated the limitations (non-relativistic Hamilton operator and the Bom-Oppenheimer approximation), we are ready to consider the electronic Schrodinger equation. It can only be solved exactly for the Hj molecule, and similar one-electron systems. In the general case we have to rely on approximate (numerical) methods. By neglecting relativistic effects, we also have to introduce electron spin as an ad hoc quantum effect. Each electron has a spin quantum number of 1 /2. In the presence of an... 

Coulomb-Breit potential gives the following set of operators, where the QED correction to the electronic spin has been introduced by means of the ge pa factor. 

The equivalent of the spin-other-orbit operator in eq. (8.30) splits into two contributions, one involving the interaction of the electron spin with the magnetic field generated by the movement of the nuclei, and one describing the interaction of the nuclear spin with the magnetic field generated by the movement of the electrons. Only the latter survives in the Born-Oppenheimer approximation, and is normally called the Paramagnetic Spin-Orbit (PSO) operator. The operator is the one-electron part of... 

New (magnetic) interactions in the Hamilton operator due to electron spin. This destroys the picture of an orbital having a definite spin. 

The paramagnetism of the triplet state can be observed by electron spin resonance spectroscopy. This is perhaps the most reliable means of determining the existence of a triplet state since the ESR signals can be predicted using the following Hamiltonian operator ... 

The Hamiltonian energy operator for an electron spin is given by Equation (5a)1... 

In Equation (6) ge is the electronic g tensor, yn is the nuclear g factor (dimensionless), fln is the nuclear magneton in erg/G (or J/T), In is the nuclear spin angular momentum operator, An is the electron-nuclear hyperfine tensor in Hz, and Qn (non-zero for fn > 1) is the quadrupole interaction tensor in Hz. The first two terms in the Hamiltonian are the electron and nuclear Zeeman interactions, respectively the third term is the electron-nuclear hyperfine interaction and the last term is the nuclear quadrupole interaction. For the usual systems with an odd number of unpaired electrons, the transition moment is finite only for a magnetic dipole moment operator oriented perpendicular to the static magnetic field direction. In an ESR resonator in which the sample is placed, the microwave magnetic field must be therefore perpendicular to the external static magnetic field. The selection rules for the electron spin transitions are given in Equation (7)... 

A comprehensive discussion and illustrations of the effects of spectrometer operating parameters on ESR spectra are given by the author in Electron spin resonance, in Physical Methods of Chemistry, ed. A. Weissberger and B. W. Rossiter, John Wiley and Sons, Inc., New York, 1972, part IIIA, ch. VI,. 

The singlet function corresponds to zero total electron spin angular momentum, S = 0 the triplet functions correspond to S = 1. Operating on these functions with the spin Hamiltonian, we get ... 

Our next goal is to transform this expression into one based on the total electron spin operator, S = si + s2. The first three terms can be simplified by making use of the identity (derived using raising and lowering operators) ... 

In the above derivation, we have made no explicit assumption about the total electron spin quantum number S so that the results should be correct for S = 1 /2 as well as higher values. However, the fine structure term is not usually included in spin Hamiltonians for 5=1/2 systems. The fine structure term can be ignored since in that case the results of operating on a spin-1/2 wave function is always zero ... 

Here, /3 and / are constants known as the Bohr magneton and nuclear magneton, respectively g and gn are the electron and nuclear g factors a is the hyperfine coupling constant H is the external magnetic field while I and S are the nuclear and electron spin operators. The electronic g factor and the hyperfine constant are actually tensors, but for the hydrogen atom they may be treated, to a good approximation, as scalar quantities. 

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