# A Quantum Chemistry Approach Based on the Analogy with π-System in Polymers for a Rapid Estimation of the Resonance Wavelength of Nanoparticle Systems

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## Abstract

**:**

## 1. Introduction

## 2. The Variational Method Applied to NPs Systems

_{ij}= β when $\left|i-j\right|=1$ and is equal to 0 when $\left|i-j\right|>1$;

_{0}) while the rest are dark modes (ω > ω

_{0}).

_{0}and Δω

_{dim}. As is well known, dielectric polarization plays an important role in the plasmon resonant energy. For this reason, the input data should be selected properly, taking into account the dielectric medium where the NPs system is embedded.

## 3. NPs Chain Systems

_{0}and the dimer shift Δω

_{dim}are ω

_{0}= 2.375 eV, β = −0.28 eV and ω

_{0}= 3.10 eV and β = −0.52 eV, respectively, for AuNPs of 40 nm and AgNPs of 50 nm [22]. Results of the VM are reported in Figure 2, which shows an increase of the wavelength as a function of the number of NPs in the chain, which, in turn, results in the red-shift of the resonance wavelength of the lowest energy mode, obtained with the solution of the secular determinant of Equation (5).

_{0,}and the corresponding wavefunction represents a system with a net total dipole. On the contrary, when m < N, then ω > ω

_{0}, and the wavefunction describes a system where no effective dipole exists, which is a dark mode [33]. In the case of the dark mode, the total wavefunction presents more nodes than coherent interactions and, therefore, in agreement with the general knowledge, they cannot be excited.

_{0}and β used for comparisons of Figure 4 are ω

_{0}= 2.34 eV, β = −0.57 eV for the comparison with experimental results (EM), ω

_{0}= 2.36 eV, β = −0.33 eV for the comparison with Electrodynamics simulation (MS) and ω

_{0}= 2.37 eV, β = −0.27 eV for the comparison with the Finit Integration Technique (MFIT). The agreement of the proposed method with literature data shows the similarity of the plasmonic characteristics of NPs system with a conjugated dienic system, where this theoretical approach is considered one of the most accurate ways to determine the energy of π-bonding.

## 4. NPs Cluster Systems

_{0}= 2.375 eV and β = −0.28 eV. We can note that the linear chain and ring chain had a similar energy shift, 1.86 eV and 1.88 eV, respectively. However, while in the linear chain, there was no degeneracy of the modes, the ring chain presented a degeneracy equal to 2 of all the modes except the one with the higher energy with k = 7. This effect was due to the geometrical symmetry of the ring system. When comparing the chain systems with the heptamer cluster, a further redshift was observed as a result of the strong coupling with the NP placed in the center in the heptamer cluster. It interacts with all the surrounding NPs, which increases the number of coherent interactions. The latter observation introduces a general feature of 2D systems, which is an increase of the number of bright modes with respect to the dark modes. In this frame, the heptamer shows a general condition where the NPs packing (i.e., the ratio between NP diameter and interparticle mean distances) is optimized, which allows the central NP to have a full interaction with several other units, since it will be discussed below in the section about the 2D lattice of NPs.

_{0}, the energy stabilization and, in turn, the wavelength increasing is dependent on the number of interactions that every NP is able to perform.

_{0}= 2.375 eV, β = −0.28 eV.

## 5. Effect of the Surface

_{k}is the resonance energy of the mode k of the chain, ω

_{S}is the plasma energy of the metal, and the interaction integral α is the shift in energy due to the interaction of the chain with the surface. A sketch of the level formation is shown in Figure 8 for the chain system and for the chain on the metallic surface.

_{s}= 2n − k for the positive interaction with the surface while the number of nodes remains the same as the insulated chain. On the contrary, for the destructive interaction of the chain with the surface, the number of nodes becomes N

_{s}= n + k − 1, while the number of coherent dipole interactions remains the same as the insulated chain. This effect is shown in Figure 9 in the case of a four NPs chain.

_{s}= 8.9 eV, α = −2.6 eV, with ω

_{k}calculated using the VM, as described in the previous paragraph with ω

_{0}= 2.38 and β = −0.28 eV. By a formal point of view, it would be more correct to use the energy of the metal surface plasmon (6.4 eV) and the corresponding interaction integral, but, as from the practical point of view, the energy of the metal is only the baseline for the calculation. We decided to use the plasma frequency (8.9 eV) and the corresponding interaction integral, which is more easy to estimate from literature data.

## 6. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**Lowest energy mode of the chain as a function of the number of NPs, for gold and silver NPs. Input data for AuNPs and AgNPs is, respectively,ω

_{0}= 2.375 eV, β = −0.28 eV, ω

_{0}= 3.10 eV, and β = −0.52 eV.

**Figure 3.**The image of the normalized number density of modes for chains with five different lengths. Black vertical lines represent single modes as obtained with the Variational Method, blue curves represent the density of states, calculated as a convolution of gauss curves modeled for each mode using a broadening of 80 nm for each state.

**Figure 4.**Comparison of results obtained by the variational method with data presented in literature. E (Experimental), S (electrodynamic Simulation), and FIT (Finit Integration Technique) have been taken from References [22,24,26], respectively. ME (ω

_{0}= 2.34 eV, β = −0.57 eV), MS (ω

_{0}= 2.36 eV, β = −0.33 eV), and MFIT (ω

_{0}= 2.37 eV, β = −0.27 eV) are the results calculated by the Variational Method for the appropriate systems.

**Figure 5.**Energy diagram of the linear combination obtained in the case of heptamer geometries: linear, ring, and cluster. Input data are ω

_{0}= 2.375 eV and β = −0.28 eV.

**Figure 6.**Energy diagram of the linear combination obtained in the case of 2D geometries: (

**A**) effect of increasing the number of NPs in the cluster on mode formation, (

**B**) effect of packing on mode energy. Input data are ω

_{0}= 2.375 eV and β = −0.28 eV.

**Figure 8.**Sketch of the linear combination in the Variational Method for a 20 NPS chain on a gold surface. Black lines symbolize modes of three different systems (from left to right): 20 NPs chain, 20 NPs chain on the surface, and the surface with the plasmon energy ω

_{p}= 8.87 eV. The green region illustrates the position of the bright mode ω < ω

_{0}with ω

_{0}= 2.38 eV.

**Figure 9.**Dipole orientation during four NPs chain on a metallic surface in the case of (

**left**) bright and (

**right**) dark modes. Potential hot spots are indicated with red spots and nodes are indicated with black spots.

**Figure 10.**The lowest and highest energy bright mode of the chain on the surface (white square) and the chain (black square) as a function of the number of NPs in the chain. Values of ω

_{0}and β taken from Reference [22] and the value of α = −2.6 eV.

**Figure 11.**Sketch of the linear combination in the Variational Method for the heptamer on a gold surface. The black lines symbolize modes of three different systems (from left to right): heptamer cluster, heptamer cluster on the surface, and the surface with plasmon energy ω

_{p}= 8.87 eV. The green region illustrates the position of bright mode ω < ω

_{0}with ω

_{0}= 2.38 eV.

**Table 1.**Dimer resonance energy from different published works for Ag and Au NPs, to be used in VM for the investigation of more complex system. In the table ω

_{0}, w

_{dimer}, Δω

_{dim}correspond to single nanoparticle resonant energy, dimer resonant energy, and the energy shift, respectively.

NPs | Size (nm) | Id (nm) | Medium | ω_{0} | ω_{dimer_longitudinal mode} | |Δω| | Reference |
---|---|---|---|---|---|---|---|

AuNPs | 20 | 0.5 | Water | 2.36 | 1.99 | 0.37 | [21] |

AuNPs | 40 | 1 | Air | 2.37 | 2.09 | 0.28 | [22] |

AuNPs | 18 | 3.8 | water | 2.36 | 2.32 | 0.04 | [23] |

AuNPs | 18 | 0.2 | Water | 2.36 | 1.97 | 0.39 | [23] |

AuNPs | 40 | 10 | Water | 2.34 | 2.34 | 0 | [23] |

AuNPs | 10 | 0.8 | Water | 2.34 | 1.97 | 0.37 | [23] |

AuNPs | 64 | 1 | air | 2.21 | 1.77 | 0.44 | [24] |

AuNPs | 60 | 1 | n = 1 | 2.05 | 1.88 | 0.17 | [25] |

AuNPs | 60 | 1.5 | n = 1.5 | 2.21 | 1.65 | 0.56 | [25] |

AuNPs | 40 | 1.5 | water | 2.36 | 2.03 | 0.33 | [26] |

AuNPs | 80 | 1 | water | 2.17 | 1.59 | 0.58 | [27] |

AuNPs | 35 | 0.34 | Vacuum (STEM) | 2.34 | 2.09 | 0.25 | [28] |

AgNPs | 60 | 3 | Air | 3.31 | 2.67 | 0.64 | [29] |

AgNPs | 50 | 1 | Air | 3.10 | 2.58 | 0.52 | [22] |

AgNPs | 36 | 2 | Air | 2.95 | 2.38 | 0.57 | [30] |

AgNPs | 30 | 0.3 | Vacuum (STEM) | 2.91 | 2.45 | 0.46 | [28] |

**Table 2.**Energy and wavelength of the peak of the resonance band for different cluster geometries considering ω

_{0}= 2.375 eV (522 nm) for single AuNP and β = −0.28 eV.

CLUSTER | ω (eV) | λ (nm) |
---|---|---|

2.38 | 522 | |

1.98 | 625 | |

2.10 | 570 | |

1.94 | 638 | |

2.02 | 615 | |

1.86 | 668 | |

1.88 | 661 | |

1.82 | 683 | |

1.84 | 674 | |

1.94 | 641 | |

1.88 | 661 | |

1.82 | 683 |

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**MDPI and ACS Style**

De Giacomo, A.; Salajkova, Z.; Dell’Aglio, M.
A Quantum Chemistry Approach Based on the Analogy with π-System in Polymers for a Rapid Estimation of the Resonance Wavelength of Nanoparticle Systems. *Nanomaterials* **2019**, *9*, 929.
https://doi.org/10.3390/nano9070929

**AMA Style**

De Giacomo A, Salajkova Z, Dell’Aglio M.
A Quantum Chemistry Approach Based on the Analogy with π-System in Polymers for a Rapid Estimation of the Resonance Wavelength of Nanoparticle Systems. *Nanomaterials*. 2019; 9(7):929.
https://doi.org/10.3390/nano9070929

**Chicago/Turabian Style**

De Giacomo, Alessandro, Zita Salajkova, and Marcella Dell’Aglio.
2019. "A Quantum Chemistry Approach Based on the Analogy with π-System in Polymers for a Rapid Estimation of the Resonance Wavelength of Nanoparticle Systems" *Nanomaterials* 9, no. 7: 929.
https://doi.org/10.3390/nano9070929