Then the interaction of the first term and the hyperpolarizability yeilds Eq. ( ), while the second term is from the derivatives of the distance rather than molecular dipole. The rest contributions to the nuclear derivatives of polarizability α x i j in Eq. ( ) includes the expression β MM mnl f QM l , referring to the first-order change in metal's polarizability induced by QM field. As in the QM/DIM method, each metal atom is induced by QM potential and field by atomic capacitance and polarizability and the induced charges and dipoles interact with each other, it wold be difficult to distinguish this mutual polarization from the contributions in Eq. (36) with the use of molecular hyperpolarizability parameter for metal cluster. Therefore, we use Eq. (35) to account the effect of hyperpolarizability of metal in the following. This correction is sufficient for our purpose because the gold cluster has large hyperpolarizability and it has little participation in the vibrational modes of the Raman spectral region we are interested in. The hyperpolarizability of the Au 18 cluster calculated by the finite-difference method with a field strength of 0.001 au is given in Table . Consistent values are obtained with a smaller field strength (0.0002 au). As demonstrated in Table , the calculated hyperpolarizability of the Au 18 cluster has large components along the z axis, comparable with the values reported in the literature. The corrected QM/DIM Raman intensities of selected modes of Py-Au 18 -V (3 Å) complex is shown in Table . There are five variations (a-e) of the hyperpolarizabilityrelated correction for the QM/DIM method since some arbitrariness arises when only the molecular hyperpolarizability parameter is available rather than the atomic ones. The (a), (b), and (c) columns are calculated by taking the distance R AB in Eq. (32) as the distance between the two centers of mass (COM), the COM of pyridine and the location of the nearest gold atom, and the shortest distance of the two fragments, respectively. In other words, the distance R AB reduces gradually when moving from (a) to (c). The (d) and (e) columns collect the Raman intensities corrected using Eq. ( ) without and with QM induced fields, respectively. Even though the QM/DIM Raman intensity of mode 11 is still larger than the full QM one, it is unsurprisingly reduced with the correction of the hyperpolarizability dependent term. On the other hand, the intensity of mode 12 is also slightly increased by the (a), (b), and (c) corrections. Fig. displays the corrected QM/DIM Raman spectrum of the Py-Au 18 -V (3 Å) complex. For approach (c), even though more correct intensity is reproduced for mode 11, several modes are wrongly enhanced such as 5, 16, and 19. It reflects that the distance is too small within this calculation. When the distance is larger, for instance, in the case of approach (b), the green curve moderately approaches the QM one in Fig. . On the other hand, the (e) profile (blue line) improves the intensities of almost all the vibrational modes except that the intensity of mode 16 is overestimated and that of mode 12 is further reduced relative to the uncorrected QM/DIM result (from 236 to 168 Å 4 /AMU, while the full QM reference is 294 Å 4 /AMU). It is acceptable though that these corrections could not produce overall satisfying intensities for all the modes, because only the molecular hyperpolarizability of the gold cluster is employed in these calculations rather than the distributed atomic hyperpolarizabilities while the fields and their derivatives act on individual gold atoms.