where r_j=S _{(i n.e. j)}r_ij. Written in matrix form, under the presence of homonuclear interactions (for which I_z^j(t)=S_zⁱ(t)) or strong heteronuclear decoupling fields (for which S_zⁱ(t)=0)), the Solomon equations resemble:

Matrix Formalism:
The matrix, R, is composed of elements R_ij = r_j and R_ij = s_ij. Matrix elements may be computed using the coefficients, c_lk, and correlation times, t_lk, shown in Table 1.

Elements, r_j and s_ij, are calculated from the transition rates, W₀, W₁, and W₂, of the dipolar interaction between spins i and j.

The rate matrix may be separated into contributions arising from the respective Hamiltonians of the interactions of interest. For example, one is able to separate contributions to the rate matrix from the dipolar and chemical shift Hamiltonians.

Dipolar Contributions:
The formulation of the dipolar transition rates, W₀^ij, W₁^ij, and W₂^ij, from the elements of Table 1 follows,

where d_l is 1, h_l is 0, and r_ij is the distance between spins i and j.

CSA Contributions:
The CSA contributes to the r_j terms of the relaxation matrix as an R₁ effect. The CSA contribution consists of the anisotropy (d) and asymmetry (h) of the symmetric tensor component, and the antisymmetry (r_X,Y,Z) of the antisymmetric componenet of the chemical shift tensor. The isotropic component of the chemical shift tensor does not contribute to R₁.

Shown below is a review of the contributions to the chemical shift tensor from the symmetric and antisymmetric components.

where,

and,

For a spin, j, the R₁ CSA contributions may be assembled from Table 1 as follows,

Table 1: Shown below are a summary of the spectral density coefficients, c_lk, and correlation times, t_lk, for rank one and two tensor interactions. Symbols, r^l_X,Y,Z, represent antisymmetric tensor properties of the rank one interaction, l. d_l and h_l represent the anisotropy and symmetry, respectively, of the rank two interaction, l.

l	k	tlk	clk
1	+1 -1
	0
2	+2 -2
1	+1 -1
	0

Tumbling Regimes:
Anisotropic tumbling is represented by the angular components (a, b, and g) in the coefficients shown in Table 1. These angles define the projections of spectral density onto the coordinate axes of the diffusion frame. Isotropic tumbling may be represented by averaging the angular components over the unit sphere.

YARM Implimentation: