Dirac spin-exchange identity

The validity of (42) rests upon the well known Dirac spin-exchange identity, Py = [1 + 4S(i) S(j)], which is easily confirmed by testing the effect of each side on the... 

Use the Dirac spin-exchange identity (Problem 4.4) to obtain an alternative proof of the energy formula (7.1.9). Extend your result to the case of N electrons, for any perfect-pairing wavefhnction (p. 219), and express (7.3.7) as the expectation value of a spin Hamiltonian. 

For a symmetrical (D ) diatomic or linear polyatomic molecule with two, or any even number, of identical nuclei having the nuclear spin quantum number (see Equation 1.47) I = n + where n is zero or an integer, exchange of any two which are equidistant from the centre of the molecule results in a change of sign of i/c which is then said to be antisymmetric to nuclear exchange. In addition the nuclei are said to be Fermi particles (or fermions) and obey Fermi Dirac statistics. However, if / = , p is symmetric to nuclear exchange and the nuclei are said to be Bose particles (or bosons) and obey Bose-Einstein statistics. 

For identical hydrons, the symmetry postulate of identical particles has to be fulfilled. For protons and tritons this means that the overall wave function must be antisymmetric under particle exchange and for deuterons it must be symmetric under particle exchange. Due to this correlation of spin and spatial state, the energy difference A between the lowest two spatial eigenstates can be treated as a pure spin Hamiltonian, similar to the Dirac exchange interaction of electronic spins. 

The next term, EX, is positive for all the molecular systems of interest for liquids. The name makes reference to the exchange of electrons between A and B. This contribution to AE is sometimes called repulsion (REP) to emphasize the main effect this contribution describes. It is a true quantum mechanical effect, related to the antisymmetry of the electronic wave function of the dimer, or, if one prefers, to the Pauli exclusion principle. Actually these are two ways of expressing the same concept. Particles with a half integer value of the spin, like electrons, are subjected to the Pauli exclusion principle, which states that two particles of this type cannot be described by the same set of values of the characterizing parameters. Such particles are subjected to a special quantum version of the statistics, the Fermi-Dirac statistics, and they are called fermions. Identical fermions have to be described with an antisymmetric wave function the opposite also holds identical particles described by an... 

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