K-theoretic Defect in Chern Class Identity for a Free Divisor

Let X be a nonsingular variety defined over an algebraically closed field of characteristic 0, and D be a free divisor. We study the motivic Chern class of D in the Grothendieck group of coherent sheaves G(0)(X), and another class defined by the sheaf of logarithmic differentials along D. We give explicit calculations of the difference of these two classes in two particular cases: D is a divisor on a nonsingular surface; D is a hyperplane arrangement whose affine cone is free.