Starting from a microscopic Hamiltonian we analyze the coupling between tunneling systems (TS's) and phonons in a structural model glass, chosen to describe NiP. We estimate the TS-phonon coupling constants, i.e., the longitudinal and transverse deformation potentials yl and y, . They are proportional to the spatial distance between the two energy minima. This dependence translates into an energy dependence of the deformation potential, .which is relevant for the temperature dependence of the lowtemperature thermal conductivity. On the basis of TS s in NiP found via a computer search in our previous work we obtain the values y1=0.38 eV and yi/y, =2.35. We analyze the inAuence of the structure of typical TS's on the deformation potential. Together with the density of tunneling systems determined in a previous paper the interaction between tunneling systems can be calculated. In contrast to recent proposals, the weak-coupling picture and hence the validity of the standard tunneling model is confirmed. The properties of glasses at very low temperatures ( T ( 1 K) can be nicely explained in the framework of the standard tunneling model (STM), which assumes that glasses contain a large number of low-energy excitations. ' These can be viewed as microscopic tunneling systems (TS's) described as double-well potentials (DWP's) in the high-dimensional configuration space of the glass. The Hamiltonian for dielectric glasses, in which we are mainly interested, can be written generally as &=Ms+'N+&~ . &s describes the TS's, &~ the phonon bath, and 'N the interaction between TS's and phonons. Whereas the contribution to the specific heat due to TS's only depends on &s, contributions to other physical properties like the thermal conductivity, the absorption, or the velocity shift also depend on the interaction between TS's and phonons. Therefore, it is of principal interest to specify 'N as exactly as possible. For low temperatures &s can be projected on the lowest two eigenfunctions of the TS's. In the localized representation we can write &s as a sum of independent TS Hamiltonians hz, where ~0