Algebraic Symmetry and Quantum Secular Equations

The algebraic symmetry of a Hamiltonian of “spin–Hamiltonian type” having the form $\mathfrak{H}\mathrm{\hspace{0.17em}=\hspace{0.17em}}{A·J}\mathrm{\hspace{0.17em}+\hspace{0.17em}}{J·B·J}$ is examined. The form of the coefficients in the characteristic equation is deduced, and symmetry‐adapted parameters and coupling coefficients are introduced. Any polynomial function of the matrix elements of the Hamiltonian which is invariant under all transformations of the frame of reference can be written as a polynomial in the symmetrized parameters. By use of coupling coefficients direct calculation of all but a limited number of matrix elements can often be avoided. The study of nuclear hyperfine interactions in solids by spectroscopic techniques (NMR, NQR, Mössbauer) is taken as an illustrative example.