Understanding the second and third order nonlinear optical responses of M@b66/64Al12N12: a comprehensive DFT and TD-DFT study

Abstract

Materials with significant first hyperpolarizability values are essential for application in second harmonic generation to achieve frequency doubling. Therefore, ideal NLO materials must not only exhibit a substantial NLO response but also maintain transparency when exposed to laser light. In this study, we investigate two series of nanoparticles, namely M@b₆₄Al₁₂N₁₂ and M@b₆₆Al₁₂N₁₂ (M ranges from Sc to Zn). The aim is to evaluate the second and third NLO responses through DFT and TD-DFT calculations. These evaluations are performed using the CAM-B3LYP/6-311+G(d) level of theory and the sum-over-states method in the static and dynamic regime (λ = ∞, 1906, 1341, and 1064 nm). These properties are further explained by considering factors such as molecular topology, delocalization indices, Waber–Cromer radius, excitation energy, oscillator strengths, variations of dipole moment of the excited state, and one/two-photon resonance effects. The results indicate that incorporating transition metals into Al₁₂N₁₂ substantially increases both the first and second hyperpolarizability. The delocalization index values reveal a higher degree of electron delocalization between the transition metal and nitrogen compared to that between the transition metal and aluminum. The QTAIM analysis displays that the presence of a closed quasi-ring structure between the metal and the nanocage, combined with electron delocalization, significantly enhances the first hyperpolarizability. TD-DFT calculations suggest potential application of these compounds in deep ultraviolet laser devices due to their transparency below 200 nm. The SOS approach reveals that the most critical excited states are local excitations, characterized by high S_r, small D, and negative t values. On the other hand, in the dynamic regime, our results indicated that the values of β_HRS, β_SHG(−2ω; ω, ω) and γ_ESHG(−2ω; ω, ω, 0) are larger than their static counterparts. Additionally, one/two photon resonance energy, along with substantial oscillator strength, plays a pivotal role in enhancing the dynamic hyperpolarizability of the investigated nanoparticles. Our findings suggest that the increase in β^λ is primarily linked to two-photon resonance rather than one-photon resonance. Based on our current understanding, this study provides novel evidence that, at λ = ∞, the first hyperpolarizability of M@b_64/66Al₁₂N₁₂ is correlated with the Waber–Cromer radius of the transition metal. Additionally, in the dynamic regime, the first hyperpolarizability is correlated with the second hyperpolarizability.