A simplified method of calculation of the surface plasmon energy states of the Ag colloidal aggregates characterized by varying inter-particle (inter-sphere) distance has been developed. Ag colloidal aggregate is approximated by a linear (one-dimensional) assembly of N silver spheres (of identical radii r and identical inter-sphere distances D) mutually interacting by a dipole-dipole interaction. The calculations use the following parameters: N from 1 to 25, r = 2, 5 and 10 nm, D = 0, 0.5, 1 and 2 nm, water and/or vacuum embedding media. The perturbation energies Vmin (stabilization energy) and Vmax (destabilization energy) of the excited plasmon state of a linear aggregate of N spheres interacting by the dipole-dipole interaction were calculated as the eigenvalues of perturbation matrix using the above-mentioned parameters. The stabilization energy Vmin increases with increasing number of spheres in the aggregate and with increasing sphere radius, while it decreases with increasing inter-particle (inter-sphere) distance. Calculations of the square values of the eigenvector coefficients show that the contribution of a particular single sphere to the total energy of the aggregate is the highest for the central sphere in the odd-N aggregates and for the two central spheres in the even-N aggregates. The results of the model calculations support the hypothesis that the differences between the surface plasmon absorption curves of the Ag colloid/monomeric adsorbate and of the Ag colloid/polymeric (oligomeric) adsorbate systems have their origin in the difference in the inter-particle distance distributions.

Šesták, O., Matějka, P., & Vlčková, B. (1996). The Model of Linear Aggregate of Ag Colloidal Particles with Variable Inter-Particle Distances. Collection of Czechoslovak Chemical Communications, 61(1), 59–69.