Cross-Validation and Cosine Similarity-based Deep Correlation Analysisof Nonlinear Properties in Transition Metal Clusters

Abstract A new approach for the rapid and accurate correlation study of the nonlinear properties in the Transition Metal (TM) clusters based on the Deep Leave-One-Out Cross-Validation (LOO-CV) method is presented. This study shows that the DNN-based approach proposes a more efficient method for predicting several properties of the fourth-row TM nanoclusters in comparison with the conventional methods based on Density Functional Theory (DFT), which are computationally expensive and significantly time-consuming. The feature space or equivalently called descriptors are defined based on a wide range of electronic and physical properties. Considering the similarities between these clusters, the DNN-based model is employed to investigate the correlation between the TM cluster properties. The method together with the cosine similarity delivers significant accuracy in the order of at most 10 − 9 for the prediction of total energy, lowest vibrational mode, binding energy and HOMO-LUMO energy gap of TM 2 , TM 3 , and TM 4 nanoclusters. Based on the correlation errors, the most coupling TM clusters are obtained. In this regard, Mn and Ni clusters has the maximum and minimum amount of energy couplings with other transition metals, respectively. In general, energy prediction errors of TM 2 , TM 3 , and TM 4 demonstrate comparable patterns while an even-odd behavior is observed for vibrational modes and binding energies. In addition, Ti, V and Co demonstrate maximum binding energy coupling to the TM 2 , TM 3 and TM 4 sets, respectively. For the case of the energy gap, Ni shows the maximum correlation in the smallest TM 2 clusters while Cr dependence is highest for TM 3 and TM 4 sets. Finally, Zn has the highest error for HOMO-LUMO energy gap in all sets and consequently the maximum independent energy gap characteristics.