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Through introducing the size (Nt) and the shape factor (λ), the size- and shape-dependent bond number Ba of quantum dots, respectively, with icosahedral, truc-decohedral, cuboctahedral, octahedral, decohedral and tetrahedral structures is established in this work. It is found that Nt and λ have reverse contribution to Ba, that is, Ba increases with increase in Nt, while it decreases with increase in λ. As the basic parameter, the size- and shape-dependent Ba function is extended to predict the cohesive energy Ec(Nt) of quantum dots. Similar to Ba, Ec(Nt) shows strong dependence on both the size and shape. Larger Nt leads to higher Ec(Nt), whereas larger λ results in a smaller Ec(Nt) value. There is a sequence: Ec(IH) > Ec(CO) > Ec(truc-DH) > Ec(OT) > Ec(DH) > Ec(TH) if Nt is certain, which is similar to Ba since Ba(IH) > Ba(CO) > Ba(truc-DH) > Ba(OT) > Ba(DH) > Ba(TH) is tested in the whole size range. To some extent, this is due to λ(IH) = λ(truc-DH) < λ(CO) < λ(OT) < λ(DH) < λ(TH), however, Ba(IH) > Ba(truc-DH) despite λ(IH) = λ(truc-DH). In addition, λ is no longer constant and increases with increase in Nt when the shape is given. The fact that whatever the shape is, Ba or Ec(Nt) increases upon increasing Nt, meaning that the shape is a secondary factor if compared with the size. The validity of the size- and shape-related model for the Ec(Nt) function is also confirmed by the simulation results of the size- and shape-dependent thermodynamic stability of Au, Ag, Cu, Ca, Sr, and Si quantum dots with different atomic structures.

Significant variations in Raman shifts with decreasing material size, D, have been detected in Raman spectroscopy. In this study, we propose a simple and unified model to determine and explain the size-dependent Raman shift, ω(D), of low-dimensional semiconductor nanomaterials. ω(D) was found to be a function of bond number in a system, with an obvious decline in Raman shift observed when size dropped to the nanoscale. This arose from a decrease in coordination number, Z(D), and increase in single bond strength, ε(D). The predicted results show good agreement with experimental data for a series of semiconductor nanomaterials, showing that bond number can be used to calculate Raman shifts of nanomaterials. Moreover, this theoretical model was successfully applied to both single crystals and some binary semiconductor nanomaterials. Furthermore, bond number, which is directly related to the nanomaterial shape and size, becomes the only parameter required to determine ω(D) in this model, as both Z(D) and ε(D) can be determined from the bond number. This indicates that the established model has the potential to determine Raman shifts of nanomaterials with different shapes and sizes.

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