The longitudinal and transverse relaxation rates of the various

spin-states of the 15N-ammonium check details AX4 spin-system are calculated using the Bloch-Wangsness-Redfield theory [20], [21], [22] and [23]. We assume here that the geometric structure of the AX4 spin-system is that of a tetrahedron, which for ammonium means that the 15N nucleus is in the centre, with each of the four protons located at the corners of the tetrahedron (see below). Thus, the symmetry-adapted elements of an irreducible basis representation have symmetries that fall within the irreducible representations of the Td point group [24], that is, A1, A2, E, T1, T2. The total spin density operator that completely describes the spin-state of 15NH4+ can be written as a direct product of spin density operators describing Epacadostat the 15N and proton spin-states. The 15N and proton spin density operators can in turn be expressed as linear combinations

of a set of basis operators. Here we derive 15N relaxation rates in terms of two sets of proton spin density basis operators: (1) Proton spin density operators that are the projection operators of the eigenfunctions to the proton Zeeman Hamiltonian. These energy eigenfunctions are denote by |m1m2m3m4〉, where mi (i = 1, 2, 3, 4) is the eigenvalue of the Zeeman Hamiltonian (α ≡ 1/2, β ≡ −1/2). The corresponding projection operator, which is the relevant density operator element, is denoted by |m1m2m3m4〉〈m1m2m3m4|. (2) Proton spin density operators from the basis of Cartesian/shift operator basis, where each basis operator represents a combination HSP90 of longitudinal and zero-quantum magnetisations of the four protons, that is, Hz1, Hz2, … , Hz1Hz2, … , H+1H−2, … , Hz1Hz2Hz3Hz4,

where Hzi is the longitudinal product operator of proton i, and H+i and H−i are the corresponding shift (raising and lowering) operators. As shown below, we use group theory to derive symmetry-adapted proton spin eigenfunctions, thereby simplifying the calculation of the relaxation rates. Symmetry-adapted basis functions for the spin wavefunctions in tetrahedral T  d symmetry can be conveniently constructed with the basic tools of group theory. The major strength of using the symmetry-adapted basis functions, as opposed to non-symmetry adapted functions, is that time-evolutions are simpler since total-symmetric Hamiltonians (A  1 in the T  d point group) cannot mix functions with different symmetry. In the context of NMR spectroscopic investigations of AX4 spin-systems, this means that the time-evolution of the spin-system and the observed relaxation rates are more intuitive. Below we briefly outline how the symmetry-adapted basis functions, which are also eigenfunctions of the proton Zeeman Hamiltonian, H^Z, are constructed.