4. Consider a system of two electrons in a molecular environment that can have either paired or unpaired spins (e.g. a biradical). The energy of the system depends on the relative orientation of the spins. (a) Show that the singlet state and triplet states are not degenerate with respect to the coupling operator (J/h?)51 - 52. (b) The system is now exposed to a magnetic field in the z-direction. Because the two electrons are in different environments (e.g. at different ends of a molecule), they experience different local magnetic fields and their interaction energy can be written as (kb/H)B(91912 +92822) with 91 +92. The full Hamiltonian for the spin pair is a combination of the local magnetic interactions and the spin-spin coupling: H = (J/2)51 82 + (MB/)B(91812 + 92822), where pb = le/2mis the Bohr magneton and g; is the electron spin g-factor. Write out the matrix representations of the Hamiltonian in both the uncoupled basis (mg1, m.2) and the coupled basis S, Ms), where S corresponds to the total spin si + $2. The spin-, Clebsch-Gordon coefficients are given in Table 1. (c) Show that when Jug the off-diagonal matrix elements of the Hamiltonian are smaller in the coupled representation, and find the eigenvalues of the matrix when these elements are neglected (set to zero). (d) Show that when I