Using lattice QCD, we calculate the twist-2 contribution a(2) to the third Mellin moment of the spin structure functions g(1) and g(2) in the nucleon. In addition we evaluate the twist-3 contribution d(2). Our computations make use of N-f = 2 + 1 gauge field ensembles generated by the Coordinated Lattice Simulations effort. Neglecting quark-line disconnected contributions we obtain as our best estimates a(2)((p)) = 0.069(17), d(2)((p)) = 0.0105(68) and a(2)((n)) = 0.0068(88), d(2)((n)) = -0.0009(70) for the proton and the neutron, respectively, where we use the normalizations given in Eqs. (58) and (59). While the a(2) results have been converted to the (MS) over bar scheme using three-loop perturbation theory, the numbers for d(2) are given in the regularization independent momentum subtraction scheme, i.e., the conversion has been performed only in tree-level perturbation theory. The d(2) results can be interpreted as corresponding to a transverse color Lorentz force on a quark in a transversely polarized proton of size F-(u) = 116(61) and F-(d) = -38(66) MeV/fm for u and d quarks, respectively. The error estimates quoted include statistical and systematic uncertainties added in quadrature.