Some analytical results forABC, ABCD, andXBCD coupled spin 1/2 systems. I

Abstract

By choosing a suitable linear combination of the constants of the motion$H_1$, it is shown that the calculation of the density matrixρ(t) can be simplified by subdividing the Hamiltonian$H$ into$(H_1 + H_2 )$. In particular, this technique can be used to obtain closed form solutions for the eigenfunctions and eigenvalues of spin 1/2ABC andXBCD spin systems, evolving in the presence of Zeeman offsets, scalar coupling and dipolar interactions. In general, the eigenvalues and eigenvalues of$H_1$ are very transparent, while those of$H_2$ require more effort. Nevertheless, simplifications can be made. Firstly, the effective size of the Hamiltonian matrix$H_2$ which needs to be considered, is reduced fromN ×N to at least (N − 2) × (N − 2), while forXBC ... systems it is reduced to (N − 4) × (N − 4). Secondly, the highest rank and highest/lowest order tensor operators available to the spin ensemble are constants of the motion under$H_2$. Finally, by exploiting the fact that$J_z$ is a good quantum number, it is possible to block-diagonalize the$H_2$ matrix into no more than 3 × 3 matrices.