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In this work, we have implemented the time-dependent density functional theory approximate auxiliary s function (TDDFT-aas) method, which is an approximate TDDFT method. Instead of calculating the exact two-center electron integrals in the K coupling matrix when solving the Casida equation, we approximate the integrals, thereby reducing the computational cost. In contrast to the related TDDFT plus tight-binding (TDDFT+TB) method, a new type of gamma function is used in the coupling matrix that does not depend on the tight-binding parameters. The calculated absorption spectra of silver and gold nanoparticles using TDDFT-aas show good agreement with TDDFT and TDDFT+TB results. In addition, we have implemented the analytical excited-state gradients for the TDDFT-aas method, which makes it possible to calculate the emission energy of molecular systems. 

Understanding photoluminescent mechanisms has become essential for photocatalytic, biological, and electronic applications. Unfortunately, analyzing excited state potential energy surfaces (PESs) in large systems is computationally expensive, and hence limited with electronic structure methods such as time-dependent density functional theory (TDDFT). Inspired by the sTDDFT and sTDA methods, time-dependent density functional theory plus tight binding (TDDFT + TB) has been shown to reproduce linear response TDDFT results much faster than TDDFT, particularly in large nanoparticles. For photochemical processes, however, methods must go beyond the calculation of excitation energies. Herein, this work outlines an analytical approach to obtain the derivative of the vertical excitation energy in TDDFT + TB for more efficient excited state PES exploration. The gradient derivation is based on the Z vector method, which utilizes an auxiliary Lagrangian to characterize the excitation energy. The gradient is obtained when the derivatives of the Fock matrix, the coupling matrix, and the overlap matrix are all plugged into the auxiliary Lagrangian, and the Lagrange multipliers are solved. This article outlines the derivation of the analytical gradient, discusses the implementation in Amsterdam Modeling Suite, and provides proof of concept by analyzing the emission energy and optimized excited state geometry calculated by TDDFT and TDDFT + TB for small organic molecules and noble metal nanoclusters. 

Developments in nanotechnology have made the creation of functionalized materials with atomic precision possible. Thiolate-protected gold nanoclusters, in particular, have become the focus of study in literature as they possess high stability and have tunable structure–property relationships. In addition to adjustments in properties due to differences in size and shape, heteroatom doping has become an exciting way to tune the properties of these systems by mixing different atomic d characters from transition metal atoms. Au 24 Pt(SR) 18 clusters, notably, have shown incredible catalytic properties, but fall short in the field of photochemistry. The influence of the Pt dopant on the photoluminescence mechanism and excited state dynamics has been investigated by a few experimental groups, but the origin of the differences that arise due to doping has not been clarified thoroughly. In this paper, density functional theory methods are used to analyze the geometry, optical and photoluminescent properties of Au 24 Pt(SR) 18 in comparison with those of [Au 25 (SR) 18 ] 1− . Furthermore, as these clusters have shown slightly different geometric and optical properties for different ligands, the analysis is completed with both hydrogen and propyl ligands in order to ascertain the role of the passivating ligands.