Quantum Mechanical Calculation of the Optical Absorption of Silver and Gold Nanoparticles by Density Functional Theory

Abstract

Problem statement: Metal nanoparticles confine the motion of conduction electrons and exhibit a strong optical absorption of electromagnetic radiation in the UV-vis-NIR region. The absorption is classically derived from the collective oscillations of free electrons in a metallic nanostructure as a consequence of incident electromagnetic radiation polarizing the particle optically embedded in a dielectric matrix. These oscillations, known as the localized surface Plasmon resonance has been modelled by Gustav Mie in 1908 using the Maxwell's equations. Nevertheless, the electrodynamics approach cannot account for the electronic transitions often displayed in experiment as a broad UV-vis optical absorption spectrum originated from the conduction electrons of metal nanoparticles. A quantum mechanical approach is required to address the optical absorption spectra of metal nanoparticles systemically. Approach: In this study, an attempt was made to calculate the optical absorption spectra of conduction electrons of metal nanoparticle quantum mechanically using the density functional theory. The particle was an isolated spherical metal nanoparticle containing N atoms confined in a face-centered cubic lattice structure. When light strikes the particle, the occupied ground-state conduction electrons absorbed the energy and excite to the unoccupied higher energy-state of the conduction band. In this development, we used time-independent Schrodinger equation for the ground-state energy of ThomasFermi-Dirac-Weizsacker atomic model for the total energy functional and the density function in the Euler-Lagrange equation is algebraically substituted with the absorption function. The total energy functional was computed numerically for silver and gold nanoparticles at various diameters. Results: The results showed broad absorption spectra derived from the occupied ground-state conduction electrons at the orbital {n = 5 and l = 0 or 5s} for silver and {n = 6 and l = 0 or 6s} for gold, which excite to the unoccupied higher energy of conduction band at the orbital {n≥6 and l = 0 or 1} for silver and {n≥7 and l = 0 or 1} for gold. A nonlinear red-shift of the absorption peak λ max , appearing at 404.79, 408.36, 412.55, 415.73, 418.42 and 420.96 nm for silver and at 510.28, 520.91, 533.11, 542.35, 549.74 and 556.04 nm for gold when the particle diameter varies at 4, 5, 7, 10, 15 and 25 nm respectively. The quantum confinement effect of the conduction bands is stronger for silver and gold nanoparticles of less than about 20 nm in diameter. Conclusion: The optical absorption spectra of silver and gold nanoparticles have been successfully calculated using a quantum treatment and this calculation could be extended to other transition metal nanoparticles of interest in nanoscience and nanotechnology.

Size-Controlled and Optical Properties of Monodispersed Silver Nanoparticles Synthesized by the Radiolytic Reduction Method

“…For a technical understanding, however, perhaps it should be left to the experts to scrutinize the complex physics and mathematics for obtaining and interpreting the theoretical data. In quantum physics, the absorption spectrum of metal nanoparticles may be described as a result of the intra-band excitations of conduction electrons from the lowest energy state to higher energy states within the conduction band of metal nanoparticles [56,57]. …”

“…The final term is the non-classical exchange-correlation energy containing all the remaining quantum effects not captured by the kinetic energy; the classical Coulomb potential with C e is the Thomas–Fermi–Dirac non-classical exchange-correlation energy constant [63]. Details of the quantum mechanical treatment can be found in our earlier publications [56,57]. It can be shown that the relation between the density and absorption may be written as ρ ≈ ( Z /σ) 3/2 , where Z is the atomic number.…”

“…The transformation of density energy functional E [ρ( r )] into absorption energy functional E [σ( r )] can be calculated with a relatively simple mathematics involving only the integration and differentiation. The second order differential equation of the final absorption energy functional may be solved numerically [56,57]. All possible intra-band electronic transitions allowed would produce the same absorption maximum for a given diameter since the energy states of quantum numbers n ≥ 6 near the Fermi level are very close to or overlap each other [56,57].…”

Structural, Optical and Electrical Properties of PVA/PANI/Nickel Nanocomposites Synthesized by Gamma Radiolytic Method

“…Recently, optical absorption of metal nanoparticles based on quantum mechanics has been proposed [43,44] and confirmed experimentally [38,45]. In this theory of metal nanoparticles, the absorption of light can be described as a result of non-radiative intra-band excitations of conduction electrons from the lowest energy state assigned by the quantum numbers to higher energy states near the Fermi level of the conduction band.…”

Effect of nano-silver particles on postharvest life of Lilium orientalis cv. 'Shocking'

“…Moreover, decreasing bacterial population in vase solution by SNPs might be decrease bacterial contamination in vessels. When metallic particles grown and became big, its peak at spectrophotometers graph will shift to long wave in red area (Gharibshahi and Saion, 2010). However, it can be concluded that plant extracts or bacteria extracts did not affect on nano-particles stability and aggregations (Ashrafi et al, 2010).…”