where H AB IJ are the matrix elements for a double excitation from I, J to A, B. If the picked hole indices A, B are already occupied in D i the excitation is discarded. The nominator of equation 15 involves no approximation. However, compared to the on the fly heat bath method, the denominator contains some non-zero elements which would vanish in the non-approximated heat-bath algorithm if X or Y were already occupied in D i . Note that p(AB; IJ) is not yet the complete p gen value needed for equation 9. It has to be multiplied with the probability to perform a double excitation and to draw the particles I and J.