Adsorption of tungsten clusters W-n (n = 1-4) on the ideal MgO(001) surface has been studied computationally using a scalar relativistic density functional method and a gradient-corrected exchange-correlation functional. Structure and energetic features of the adsorption complexes W-n/MgO(001) have been analyzed. The oxide surface was represented by cluster models embedded in a large array of point charges (PCs). To reduce the artificial polarization of oxygen anions in the immediate vicinity of positive PCs, the cations at the cluster boundaries were treated as Mg2+ ions at either all-electron or pseudopotential (PP) level. Compared to the all-electron + PC embedding, the significantly more economic PP + PC approach is demonstrated to impose cluster model boundary conditions appropriate to the ionic oxide MgO. The cluster size dependence of the adsorption properties is found weak. Like other transition metal clusters considered previously, tungsten species favor adsorption sites in the proximity of oxygen centers of MgO(001). Rather small calculated adsorption-induced deformations of the tungsten clusters manifest notably stronger W-W bonds compared to W-O bonds between metal and substrate. The tetrahedron shape of W-4, most Stable in the gas phase, is calculated to be energetically preferred also in the adsorbed state, in particular over a square-planar adsorbate. This finding is at variance with a model of two-dimensional W-4 clusters on MgO(001) derived from a recent high-resolution electron microscopy investigation (Tanaka, N., et al. Surf: Rev. Lett. 1998, 5, 723). The configuration of W-3 With two W atoms located close to two nearest-neighbor oxygen ions is favored over that where two W atoms are close to next-nearest-neighbor substrate anions. In both cases, the adsorbed W3 cluster tilts considerably from an upright orientation.