# A Model for the Evaluation of Monostable Molecule Signal Energy in Molecular Field-Coupled Nanocomputing

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## Abstract

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## 1. Introduction

## 2. Theoretical Methods

#### 2.1. MoSQuiTo Methodology

#### 2.2. Bistable Factor

## 3. Energy Modelling

#### 3.1. Model Definition

#### 3.2. Internal Energy: The Conformation Energy

#### 3.3. Internal Energy: The Polarization Energy

#### 3.4. The Interaction Energy: Intermolecular Energy

#### 3.5. The Interaction Energy: Electric Field Energy

#### 3.6. Final Expression

## 4. Results

#### 4.1. Equilibrium Analysis

#### 4.2. Field-Induced Polarization of the Diallylbutane

#### 4.3. Intermolecular Interaction

#### 4.3.1. The Driver Response

#### 4.4. Bistability Study

#### 4.5. Memory Effect

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

## Abbreviations

CAD | Computer-Aided Design |

CMOS | Complementary Metal-Oxide Semiconductor |

DFT | Density Functiona Theory |

ESP | Electrostatic Potential |

FCN | Field-Coupled Nanocomputing |

MoSQuiTo | Molecular Simulator Quantum-dot cellular automata Torino |

MUT | Molecule Under Test |

SCERPA | Self-Consistent Electrostatic Potential Algorithm |

TSA | Two-State Approximation |

QCA | Quantum-dot Cellular Automata |

VACT | Vin-Aggregated Charge Transcharacteristics |

## Appendix A

**Figure A1.**Expectation value of the Hamiltonian operator (Energy) evaluated in a Two-State Approximation.

## References

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**Figure 1.**Basics of the molecular FCN paradigm and modelling. (

**a**) Information encoding in Molecular FCN cells. The blue and red circles represent the location of the molecule charges; (

**b**) propagation of the information in a molecular FCN wire composed of three cells; (

**c**) derivation of the Aggregated charge of a 1,4-diallylbutane: DFT computed equilibrium geometry; (

**d**) DFT computed atomic charges, colored according to the aggregation groups; (

**e**) aggregated charges ${Q}_{1}$ (blue atomic charges) and ${Q}_{2}$ (red atomic charges).

**Figure 2.**Modelling of the molecular interaction. (

**a**) Molecule under the effect of an electric field ${\mathit{E}}_{in}$ generated by electrodes. The electric field is measured on the molecule through the so-called input voltage ${V}_{in}$, which is obtained by integrating ${\mathit{E}}_{in}$ on a generic path connecting the two carbon atoms highlighted in blue and enlarged in the molecule; (

**b**) molecule under test (MUT) under the effect of an electric field ${\mathit{E}}_{in}$ generated by a driver molecule. The electric field is measured on the MUT through the so-called input voltage ${V}_{in}$. In turn, the MUT creates a second electric field ${\mathit{E}}_{out}$ which impacts other molecules. A fictitious molecule position at the same distance as the driver molecule can be exploited to evaluate the MUT capability to impact other elements by evaluating the so-called output voltage ${V}_{out}$, that is, the input voltage of the fictitious molecule; (

**c**) basic scheme for the evaluation of the centered bistable factor. The MUT is positioned between two N-molecule wires (N = 4).

**Figure 3.**Basic schematic of the energy model: Two diallylbutane molecules contribute to the total energy with their conformation energy ${w}_{0}$. The intermolecular interaction and the electric field $\mathit{E}$ induce a dipole $\mu $, which increases the molecule internal energy. Each molecule is fixed in the space and interact with the other molecules and with the electric field, leading to the Interaction Energy of the system ${w}_{mm}^{(1,2)}+{w}_{E}^{\left(1\right)}+{w}_{E}^{\left(2\right)}$.

**Figure 4.**Modelling of the internal energy. (

**a**) Diallylbutane equipotential surface (3.4 V) evaluated with the molecule at the equilibrium (configuration [A]); (

**b**) diallylbutane equipotential surface (3.4 V) of the evaluated when an electric field induces a dipole moment in the molecule (configuration [B]); (

**c**) internal energy ${W}_{m}$ variation of the molecule when brought from equilibrium (configuration [A]) to a polarized condition (configuration [B]) with an electric field $\mathit{E}$; (

**d**) equivalent mechanical model used for the modelling of the molecule polarizability at the equilibrium (configuration [A]); (

**e**) polarization of the mechanical model under the effect of an electric field $\mathit{E}$ (configuration [B]).

**Figure 5.**Equilibrium configuration of the diallylbutane cation. (

**a**) position of atoms obtained by the geometry optimization procedure; (

**b**) electrostatic iso-potential surface evaluated at 3.4 V.

**Figure 6.**Study of the diallylbutane cation under the effect of external electric fields. (

**a**) Schematic of the procedure for the evaluation of the molecular polarizability; (

**b**) dipole moment obtained with DFT computation (CAM-B3LYP/def2-TZVPP) for an oxidized diallylbutane stimulated under the effect of electric field (E). The polarization $\alpha $ is obtained by linearly interpolating the dipole in the central region (interpolation range).

**Figure 7.**Energy study of the diallylbutane cation under the effect of external electric fields. (

**a**) Total energy ${W}_{TOT}$ obtained with DFT calculation (CAM-B3LYP/def2-TZVPP) and with the proposed energy model for a diallylbutane molecule embedded in an electric field; (

**b**) electric field energy ${w}_{E}$. (

**c**) polarization internal energy ${w}_{\mu}$ obtained with the proposed model, Equation (13), and with the ideal linear assumption, Equation (9); (

**d**) absolute error between the DFT calculated energy and the proposed model result.

**Figure 8.**Energy study of the diallylbutane cation under the effect of surrounding molecules. (

**a**) Interaction energy of two diallylbutane molecules positioned at different distances ‘d’ evaluated with DFT (CAM-B3LYP/def2-TZVPP) and with the proposed model; (

**b**) relative error between the model evaluated energy and the DFT calculation; (

**c**) interaction energy of two diallylbutane molecules evaluated with DFT and with the proposed model by using several charge approximations (ESP, Mulliken and Lowdin) for describing the molecule electrostatic behaviour with the associated relative error between the proposed model and the DFT calculation; (

**d**) relative error between the model evaluated interaction energy obtained with different charge approximations and the DFT calculation.

**Figure 9.**Analysis of the diallylbutane cation under the effect of a driver molecule. (

**a**) Simulation scheme of the driver-molecule system used for the evaluation of the transcharacteristics, two charges aligned with atoms C5 and C8 model the aggregated charge of a possible driver molecule; (

**b**) diallylbutane dipole computed for different input voltages using the driver-molecule system. A linear fitting enables the evaluation of the effective polarizability ${\overline{\alpha}}_{xx}$. The plot also reports the dipole moment obtained with a uniform electric field.

**Figure 10.**The total energy of a driver-molecule system evaluated using the proposed energy model. In this calculation, the value of molecule atomic charges $\left\{{Q}_{i}\right\}$ is fixed to ground state values, whereas the driver charges ${Q}_{D1}$ and ${Q}_{D2}$ are varied. Under this condition, the internal energy of the molecule is constant, and the driver-molecule interaction varies. Points highlight the energy obtained with the proposed model for the ground state configuration (i.e., the driver charges used to obtain the ground state charge distribution).

**Figure 11.**Analysis of the bistable propagation. (

**a**) SCERPA simulation of a wire composed of 21 molecules with intermolecular distance 0.65 nm. The figure reports the voltage generated by the molecule charge distribution evaluated 0.2 nm above the molecules. The zigzag positioned white spots indicate the correctness of the information propagation; (

**b**) SCERPA simulation of a wire composed of 21 molecules with intermolecular distance 0.75 nm. The information fades after a few molecules; (

**c**) Centered Bistable Factor (BF10c) evaluated as a function of the intermolecular distance. The use of the effective polarizability avoid overestimating the bistable factor, correctly describing the propagation of the information in molecular wires.

**Figure 12.**Vin-Aggregated Charge Transcharacteristics (VACT) of the diallylbutane. The Aggregated Charge is calculated by summing the atomic charges calculated with DFT (CAM-B3LYP/def2-TZVPP).

**Figure 13.**Energy study of molecular wires evaluated by varying the polarization of molecules between the two propagating configurations. (

**a**) The energy of a 20 molecule wire with the intermolecular distance 0.65 nm. A barrier between the two states ($\beta =\pm 1$) appears, demonstrating the bistability of the two logic states; (

**b**) the energy of a 20 molecule wire, intermolecular distance 0.75 nm. No barrier between the two states ($\beta =\pm 1$) appears, demonstrating that information propagation is not possible since the logic states are not stable; (

**c**) the energy of a 21 molecule wire, intermolecular distance 0.65 nm. The first molecule polarization is constant and emulates a possible driver. The driver introduces an asymmetry in the energy trend, favoring one of the two stable states; (

**d**) the energy of a 21 molecule wire, intermolecular distance 0.75 nm. The first molecule polarization is constant and emulates a possible driver. The driver makes the energy trend asymmetric, yet information propagation remains not possible.

**Figure 14.**Energy analysis of the bistable propagation. (

**a**) Total energy of a wire composed by 14, 10, and 4 molecules as a function of the $\beta $ parameter, i.e., as a function of the wire molecular charge distribution. The total energy is subtracted by the total energy evaluated with $\beta =1$ for the sake of clarity. (

**b**) SCERPA simulations of a wire composed of 14 molecules (and a driver) with an intermolecular distance of 0.65 nm. The figure reports the voltage generated by the molecule charge distribution evaluated 0.2 nm above the molecules. In timestep [A], the zigzag positioned white spots indicate the correctness of the information propagation. In configuration [B], the driver molecule is switched to propagate the opposite logical information. The new logical information does not propagate correctly as a consequence of a memory effect. (

**c**) SCERPA simulations of a wire composed of 4 molecules (and a driver) with an intermolecular distance of 0.65 nm. The figure reports the voltage generated by the molecule charge distribution evaluated 0.2 nm above the molecules. In timestep [A], the zigzag positioned white spots indicate the correctness of the information propagation. In configuration [B], the driver molecule is switched to propagate the opposite logical information. The molecular wire well propagates the expected logical information in both configurations, with no memory effect.

**Figure 15.**Charge configuration, described in terms of $\beta $, of a molecular wire composed of 3, 4, and 5 molecules. The $\beta $ parameter is plotted as a function of the driver input voltage.

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**MDPI and ACS Style**

Ardesi, Y.; Graziano, M.; Piccinini, G.
A Model for the Evaluation of Monostable Molecule Signal Energy in Molecular Field-Coupled Nanocomputing. *J. Low Power Electron. Appl.* **2022**, *12*, 13.
https://doi.org/10.3390/jlpea12010013

**AMA Style**

Ardesi Y, Graziano M, Piccinini G.
A Model for the Evaluation of Monostable Molecule Signal Energy in Molecular Field-Coupled Nanocomputing. *Journal of Low Power Electronics and Applications*. 2022; 12(1):13.
https://doi.org/10.3390/jlpea12010013

**Chicago/Turabian Style**

Ardesi, Yuri, Mariagrazia Graziano, and Gianluca Piccinini.
2022. "A Model for the Evaluation of Monostable Molecule Signal Energy in Molecular Field-Coupled Nanocomputing" *Journal of Low Power Electronics and Applications* 12, no. 1: 13.
https://doi.org/10.3390/jlpea12010013