The calculation was conducted mainly by using the open source DDSCAT 7.3 code. It is based on DDA and can be used to calculate scatter and absorbance to electromagnetic wave of any body of arbitrary geometry [12-13].

In the DDA method, geometry is replaced by a matrix of dipoles, each of which has a microscopic polarizability determined by the optical constants of metal or insulator. The scattering and absorption of electromagnetic field caused by the interaction between incident light and dipole matrix can be solved accurately by discretization of the geometry into dipole matrix. Theoretically, the DDA method can calculate arbitrary scattering and absorption if the dipole matrix representing the geometry can be correctly generated. In order to ensure the convergence of DDA calculation, the inter-dipole spacing used is much smaller than the incident light wavelength.

The optical cross sections of geometry in DDA method are calculated in the following formula [12],

where k is the wavenumber, Cext, Cabs, Csca are the extinction, absorption, scattering cross section, N is the number of dipoles, E0 is the electric field of the incident plane wave, Einc,j is the electric field at position j due to the incident plane wave, Pj is the instantaneous dipole moment of dipole j, and αj is the polarizability tensor of dipole j.

In this paper, the two spherical nanoparticles are of same diameter (Scheme 1a). The adjustable parameters include D and L. The incident light is a monochromatic plane wave. The polarization of the electric field is parallel or perpendicular to the line connecting the centers of the two nanospheres. The model is submerged in a non-absorbent ambient medium; therefore, the light intensity does not change with the distance from the light source to the target. The material of the nanoparticles is copper, and the refractive index data comes from the Ref 14.

In order to investigate the effect of oxidation on the LSPR, core-shell structure was used to replace CuNPs, as shown in Scheme 1b. The core is copper, and the shell is Cu2O or CuO. The core-shell two-sphere model uses an extra parameter to label materials to distinguish the core and the shell. t/R was used to characterize the oxidation degree. It is assumed that the volume of the nanoparticles remains constant during oxidation. The refractive index data of Cu2O and CuO are from Ref 15.

We emphasize that this calculation can reflect the coupling effect of two nanoparticles. This is because coupling between two nanoparticles is caused by the specific geometric structure, and is related to the setting of boundary conditions, rather than the expression of Maxwell's equations. The boundary conditions have been defined by the geometric shape formed via dipole matrix, therefore, coupling between CuNPs is naturally reflected in the calculated results via the interaction between or among the CuNPs [16].

The refractive index n was set as 1, 1.5, 2, and D was set as 5, 10, 15, 50 nm. The inter-particle spacing was set as L = 1.01D, 1.1D, 1.5D, 2D. When L/D = 1.01, quantum tunneling might take place [17-18], which is out of the consideration of this model, therefore under this case, the calculated extinction spectrum can be considered as an extrapolation of classic model and be used as a reference.