Electronic Dynamics of a Molecular System Coupled to a Plasmonic Nanoparticle Combining the Polarizable Continuum Model and Many-Body Perturbation Theory

The efficiency of plasmonic metallic nanoparticles in harvesting and concentrating light energy in their proximity triggers a wealth of important and intriguing phenomena. For example, spectroscopies are able to reach single-molecule and intramolecule sensitivities, and important chemical reactions can be effectively photocatalyzed. For the real-time description of the coupled dynamics of a molecule’s electronic system and of a plasmonic nanoparticle, a methodology has been recently proposed (J. Phys. Chem. C. 120, 2016, 28774−28781) which combines the classical description of the nanoparticle as a polarizable continuum medium with a quantum-mechanical description of the molecule treated at the time-dependent configuration interaction (TDCI) level. In this work, we extend this methodology by describing the molecule using many-body perturbation theory: the molecule’s excitation energies, transition dipoles, and potentials computed at the GW/Bethe–Salpeter equation (BSE) level. This allows us to overcome current limitations of TDCI in terms of achievable accuracy without compromising on the accessible molecular sizes. We illustrate the developed scheme by characterizing the coupled nanoparticle/molecule dynamics of two prototype molecules, LiCN and p-nitroaniline.


LiCN NP mesh convergence
For the determination of the optimal mesh the LiCN molecule is aligned along the x axis, and the center of the NP is located along the z axis as shown in Fig. 2. The presence of the NP breaks the rotational symmetry along the x-axis. Figure S2: LiCN/NP geometry adopted for the determination of the optimal mesh.
The convergence of the NP tessellation was tested at 3Ådistance, where the mutual NPmolecule interaction is most sensitive to the details of the tessellation close to the molecule.
Five tessellations were compared: a uniform mesh with mesh element size (mes) fixed to 1.2 nm, and four non uniform meshes with decreasing mes upon reaching the surface point S3 closest to the molecule. In these cases the mesh was determined by fixing the mes couple t-b where t stands for the mes of the furthest top point of the NP and b for the closest bottom point.
As already mentioned in the text, also without any external field, the presence of a metallic NP in proximity of the molecule affects the molecule's spectrum and ground state polarization; in turn the NP acquires a finite polarization. For the study of the optimal tessellation we included only the ground and the first three excited states. In Tab. the modified energies of the second and third excited state E 2 and E 3 , as well as degree of degeneracy breaking∆ E = E 3 − E2, the sum of the square modulus of the z-component of the transition dipole moments of the two states d 2 z = | 0|d z |2 > | 2 + 0|d z |3 > | 2 , and the NP induced dipole moment µ are reported. of Kohn-Sham orbitals and of their occupations. 5 We thus studied how Rabi oscillations are described in our formalism and what is the impact on them of a plasmonic NP. Before going into the details of the calculation, however, it is important to stress that in this case we should not look at the results as the predicted behaviour of a real LiCN molecule (for instance we are not including the possibility for the molecule to ionize), nor as the quantum behavior of the NP, but more as the behaviour of a model system made of a given number of energy levels differently coupled to the external field, to the NP, and among themselves. τ 0 which we define as the first time in which the ground state population is 0 for ideal Rabi oscillations of a two level system whose transition dipole is the same as that of LiCN resonant excited state.  Fig. 4. Following the common notation, 6 we split the total field (in magenta) in three contributions: incident (in black), image (red) and scattered (green) field. Scattered fields are those fields produced by the polarization that is induced within the NP by the external field, whereas image fields are due to the polarization that the NP develops responding to the fluctuation of the molecule charge density. The total field amplitude decreases by two order of magnitude increasing the distance from 3Å to 100Å, scattered fields give the strongest contribution in all cases. It is worth noting that for the 3Å distance the image field is comparable to the incident field but still much smaller than the scattered field.

LiCN high fields dynamics in vacuum
From Rabi frequencies, obtained by Fourier transforming the population dynamics, the field amplification for each distance is computed by inverting the relation ω R = E · µ. At the frequency ω R of the external pulse, an analytic estimation of the field amplification can moreover be obtained considering a classical dielectric sphere in an uniform external field. 7 The results are listed in Tab. 3, compared with the field amplification obtained from the values of the amplitude of the total field at the molecule center. Table S3: Amplification of the electric field experienced by the molecule compared to the value extracted by Rabi frequencies and to the analytic model of a dielectric sphere placed in a uniform external field. The dielectric constant of the sphere is given by the value of the Drude-Lorentz dielectric function at the equilibrated molecule resonance energy, specific for each distance. In the = (ω d=3 ) case the sphere dielectric constant is kept fixed at the 3Å distance evalue for all distances. The 3Å case is qualitatively different from the rest. As seen from the left panel of Fig. 4 and, for a longer simulation, in The inclusion of excitations up to 10 eV (20 states) has strong effects on the population dynamics, as shown in Fig. 6. For the 6Å and 10Å distances, when the field experienced by the molecule is maximally enhanced, the population of the resonant excited states is almost entirely quenched and the population is transferred to higher energy excitations. Slowly, as the distance increases and the field decreases, a two-states Rabi-like oscillatory behaviour is recovered.

LiCN active space size convergence
The convergence of the population dynamics with respect to the active space size, i.e. the number of excited states included in the dynamics, depends on the system under investigation, due to system-dependent spacing of the excitation energies and strengths of the transition dipoles, on the NP-molecule distance and on the magnitude of the external field.
In Fig. 7 we report the population dynamics of the dipole switch states for different num-  Experimental visible-UV absorption spectra 11 report a first excitation with finite oscillator strength at 4.24 eV in rough agreement with our first excitation at 4.77 eV.

PNA Determination of an optimal tesselation
An optimal tessellation of the NP surface is critical for an accurate and computationally affordable calculation. We generated non uniform tesselations with decreasing tesserae dimension when approaching the cone tip using the open source 3D finite element mesh generator Gmsh 12 by fixing the desired mesh element size (mes) at specific points on the surface. In Fig. 8 the points were the mes has been fixed are shown. As the cone is obtained by a 360 degree rotation of half of its section, all the equivalent points in terms of such rotation have S13 the same mes. In particular we fixed the mes of the top surface points (D and equivalent Figure S8: Section of the tip NP, with the points were the mes has been fixed.  Figure S10: Convergence of the excited state population dynamics with respect to the mes at point B.

PNA 18 vs 4 levels dynamics
In the case of LiCN a simplification of the population dynamics and a Rabi-like behaviour (with the exception of the 3Ådistance case) could be obtained once the number of the excitations involved in the dynamics was reduced. A different behaviour is found for the PNA case where including excited states up to 8 eV (first 17 excitations) does not lead to a much different dynamics with respect to including only the first 3 excitations as shown in Fig. 11 for the case of geometry A. S16 Figure S11: Ground state and excited state population dynamics for calculations involving 18 and 4 levels. The PNA molecule is located at geometry A with respect to the NP. S17