  We study the equivalence of four different approaches to calculate the
excitonic linear and nonlinear optical response of multiband semiconductors.
These four methods derive from two choices of gauge, i.e. length and velocity
gauges, and two ways of computing the current density, i.e. direct evaluation
and evaluation via the time-derivative of the polarization density. The linear
and quadratic response functions are obtained for all methods by employing a
perturbative density matrix approach within the mean-field approximation. The
equivalence of all four methods is shown rigorously, when a correct interaction
Hamiltonian is employed for the velocity gauge approaches. The correct
interaction is written as a series of commutators containing the unperturbed
Hamiltonian and position operators, which becomes equivalent to the
conventional velocity gauge interaction in the limit of infinite Coulomb
screening and infinitely many bands. As a case study, the theory is applied to
hexagonal boron nitride monolayers, and the linear and nonlinear optical
response found in different approaches are compared.
