This work is the first in a series devoted to applying mode coupling diffusion theory to the derivation of local dynamics properties of proteins in solution. The first-order mode-coupling approximation, or optimized Rouse-Zimm local dynamics (ORZLD), is applied here to derive the rotational dynamics of the bonds and compare the calculated with the experimental nmr 15N spin-lattice relaxation time behavior of the vnd/NK-2 homeodomain from Drosophila melanogaster. The starting point for the calculations is the experimental three-dimensional structure of the homeodomain determined by multidimensional nmr spectroscopy. The results of the computations are compared with experimentally measured 15N spin-lattice relaxation times T1, at 34.5 and 60.8 MHz, to check the first-order approximation. To estimate the relative importance of internal and overall rotation, both rigid and fluctuating dynamic models are examined, with fluctuations evaluated using molecular dynamics (MD) simulations. The correlation times for the fundamental bond vector time correlation function and for the second-order bond orientational TCF are obtained as a function of the residue number for vnd/NK-2. The stability of the corresponding local dynamics pattern for the fluctuating structure as a function of the length of the MD trajectory is presented. Diffusive dynamics, which is essentially free of model parameters even at first order in the mode-coupling diffusion approach, confirm that local dynamics of proteins can be described in terms of rotational diffusion of a fluctuating quasi-rigid structure. The comparison with the nmr data shows that the first-order mode coupling diffusion approximation accounts for the correct order of magnitude of the results and of important qualitative aspects of the data sensitive to conformational changes. Indications are obtained from this study to efficiently extend the theory to higher order in the mode-coupling expansion. These results demonstrate the promise of the mode-coupling approach, where the local dynamics of proteins is described in terms of rotational diffusion of a fluctuating quasi-rigid structure, to analyze nmr spin-lattice relaxation behavior.