# Functional Properties of Tetrameric Molecular Cells for Quantum Cellular Automata: A Quantum-Mechanical Treatment Extended to the Range of Arbitrary Coulomb Repulsion

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

**0**and

**1**) as shown in Figure 1. The QCA devices composed of quantum dots have several important potential advantages over standard silicon ones, such as current-free signal propagation and switching, strongly reduced power dissipation in the circuits, and also much smaller size of the logic gates, which allow to achieve much higher density of the devices.

_{4}Ru

^{II}

_{2}Ru

^{III}

_{2}](pz)

_{4}]

^{10+}and [Fe

^{II}

_{2}Fe

^{III}

_{2}(L)

_{4}]

^{2+}(cyclen = 1,4,7,10-tetraazacyclododecane, pz = pyrazine a nd H

_{2}L = bis[phenyl (2-pyridyl) methanone] thiocarbohydrazone) [16,17,18]. Additionally, the grid-type architectures comprising two-dimensional arrays of molecular cells have been obtained [21], which has interrelated the synthesis and characterization of the molecular cell prototypes with nanotechnological applications of the controlled 2D monolayers of such cells.

## 2. Cell-Cell Response: Why Do We Need Quantum-Mechanical Approach

_{A}, ρ

_{B}, ρ

_{C}and ρ

_{D}, (numeration of the sites is shown in Figure 1) which are, in general, different. One can define the induced polarization of the working cell as follows:

_{dc}≠ 0 the adiabatic potential is asymmetric due to the action of the quadrupole field of the driver-cell. At the beginning of the switching circle (P

_{dc}= −1) the working cell is localized in a left deep minimum and its induced polarization acquires the value P

_{wc}= −1. The left global minimum becomes shallower with decreasing of |P

_{dc}|, and the second exited (right) minimum appears. The |P

_{wc}| defined in the global minimum gradually decreases with decreasing |P

_{dc}| until the latter reaches the narrow critical range of values near P

_{dc}= 0. In this range, polarization of the working cell abruptly changes from P

_{wc}= −1 to P

_{wc}= +1 which is accompanied by interchanging of the global and the excited minima. Finally, at the end of the cycle we arrive at the same energy of the working cell but the cell proves to be localized in the right minimum with polarization P

_{wc}= +1.

_{dc}= −1 and P

_{dc}= +1. However, when |P

_{dc}| approaches the critical area (P

_{dc}| ≈ 0) the minima are becoming shallower and the left and the right minima become almost energetically equivalent. As a result, the quantum tunneling effects are expected to play important role in this area.

_{dc}| values playing crucially important roles in switching the cell between

**0**and

**1**states. Meanwhile, a more exact quantum-mechanical vibronic approach is free from this shortcoming. This is true for the case of strong Coulomb repulsion [28] for which the quantum-mechanical approach is shown to result in a more smooth ${P}_{wc}\left({P}_{dc}\right)$ dependence as compared with semiclassical one due to presence of the tunneling processes facilitating reorientation of the cell polarization [29]. The difference between semiclassical and quantum-mechanical description of cell-cell response is expected to be even more significant for general cases when the intracell Coulomb repulsion is comparable with the ET. Just such a general case will be discussed below.

## 3. Electronic and Vibronic Interactions in a Two-Electron Mixed-Valence Molecular Square

_{1}, E, B

_{1}of the point group D

_{4h}. The corresponding unitary transformation is the following:

_{u}and B

_{1g}vibrations illustrating their different physical role. Indeed, ${q}_{{B}_{1g}}$ type displacements are interrelated with the ground Coulomb manifold comprising antipodal (diagonal type) charge configurations, while the double degenerate vibration ${q}_{{E}_{u}x},{q}_{{E}_{u}y}$ operates within the excited Coulomb manifold.

_{u}and B

_{1g}- modes is described by the terms $\upsilon {q}_{i}\left(i=2,3,4\right)$ in which $\upsilon $ is the vibronic coupling parameter that is the same for all vibrations in the PKS model dealing with equivalent redox sites. The coupling with the full-symmetric mode does not depend on the charge configuration and so it has been eliminated from the matrix in Equation (4) by means of proper redetermination of the reference vibrational configuration.

## 4. Electronic Densities in a Free Cell: Quantum-Mechanical Results

_{2}, n

_{3}and n

_{4}are the vibrational quantum numbers related to the three active vibrational modes. To obtain a numerical solution of the dynamic problem the infinite matrix has to be truncated. The dimension of the truncated matrix should be large enough to ensure a good convergence that provides a satisfactory accuracy in the evaluation of the low-lying vibronic levels. This dimension depends on the set of parameters involved in the Hamiltonian.

^{−1}for the vibrational PKS quantum and the ET parameter.

^{−1}) ${\rho}_{d}$ and ${\rho}_{n}$ gradually depend on υ. At this range of the values of υ the lowest adiabatic potential was shown to possess the only minimum or the two shallow minima separated by a barrier whose top is lower in energy than the energy of the ground vibronic level found from quantum-mechanical treatment. At moderate vibronic coupling, the functions ${\rho}_{d}\left(\upsilon \right)$ and ${\rho}_{n}\left(\upsilon \right)$ show more rapid changes, which become again more gradual in the limit of strong vibronic coupling when the tunneling is almost fully suppressed. Therefore, the obtained dependences confirm the conclusion derived based on the semiclassical consideration [28] according to which the strong vibronic coupling is able to effectively restore the dominance of the diagonal-type electronic configurations typical of the strong limit of strong Coulomb repulsion, which is broken down when the inequality t << U fails.

## 5. Effect of Quantum Tunneling on Cell-Cell Response

_{ik}. Figure 6a shows a family of cell-cell response functions calculated with the aid of quantum-mechanical vibronic approach at four different values of υ. It is seen that for weak vibronic coupling, the cell-cell response is rather weak and almost linear (curves 3 and 4 in Figure 6a). Increase in the coupling leads to a pronounced non-linearity in the response with the slope of ${P}_{wc}\left({P}_{dc}\right)$ curve being increased with the increase in υ (curves 1 and 2 in Figure 6a). Finally, providing strong vibronic coupling the cell-cell response function shows abrupt stepwise behavior, so that even weak driver-cell polarization causes almost full polarization of the working cell (curve 1 in Figure 6a).

^{−1}curve 3 in Figure 6b), the quantum-mechanical cell-cell response calculated with the same υ proves to be rather weak and practically linear (curve 3 in Figure 6a). The difference between semiclassical and quantum-mechanical curves diminishes with the increase in the vibronic coupling, so that at strong coupling (case of υ = 600 cm

^{−1}) the cell-cell responses evaluated with these two approaches prove to be quite similar in the sense that both curves show abrupt non-linear behavior, with full polarization of the working cell being reached at a weak driver-cell polarization (curves 1 in Figure 6a,b). Still, even in this case the quantum-mechanically evaluated ${P}_{wc}\left({P}_{dc}\right)$-dependences proves to be more gradual as compared with the semiclassical one.

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Scheme of a square-planar cell encoding binary information

**0**and

**1**in the two diagonal positions of the electronic pair denoted by red balls. The labels of the dots A, B, C, D used through the article are also shown.

**Figure 2.**Illustration for the strongly non-linear cell-cell response (upper part) accompanied by corresponding semiclassical representation of the switching cycle showing the evolution of the lower branch U

_{−}(q) of the adiabatic potential of the working cell caused by the change of the driver cell polarization (lower part). Cell polarizations range from −1 (binary

**0**) to +1 (binary

**1**). For the sake of utmost clarity, the case of one-dimensional adiabatic potential is shown.

**Figure 3.**Pictorial representation of the coordinates of symmetry adapted PKS vibrations of the square-planar unit. The balls mimic expanded (large) and compressed (small) redox sites.

**Figure 4.**Electronic energy levels in the form E/U vs. t/U. The ratio t/U corresponding to the sample set of t and U values is shown by dashed red section. Numbers in the parentheses indicate multiplicities of the levels.

**Figure 5.**Overall probabilities of the diagonal-type and the side-type two-electron populations in a free cell calculated as functions of the vibronic coupling parameter υ.

**Figure 6.**Cell-cell response functions calculated in the framework of quantum-mechanical (

**a**) and semiclassical (

**b**) vibronic approaches for the following four values of the vibronic coupling parameter: 1—υ = 600 cm

^{−1}, 2—υ = 500 cm

^{−1}, 3—υ = 300 cm

^{−1}, 4—υ = 265 cm

^{−1}.

**Figure 7.**Dependences of the ground and the first excited vibronic levels of the working cell on the driver-cell polarization P

_{dc}evaluated for two values of the vibronic coupling parameter υ: (

**a**) $\upsilon =500{\mathrm{cm}}^{-1}$; (

**b**) $\upsilon =600{\mathrm{cm}}^{-1}$.

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Palii, A.; Aldoshin, S.; Tsukerblat, B.
Functional Properties of Tetrameric Molecular Cells for Quantum Cellular Automata: A Quantum-Mechanical Treatment Extended to the Range of Arbitrary Coulomb Repulsion. *Magnetochemistry* **2022**, *8*, 92.
https://doi.org/10.3390/magnetochemistry8080092

**AMA Style**

Palii A, Aldoshin S, Tsukerblat B.
Functional Properties of Tetrameric Molecular Cells for Quantum Cellular Automata: A Quantum-Mechanical Treatment Extended to the Range of Arbitrary Coulomb Repulsion. *Magnetochemistry*. 2022; 8(8):92.
https://doi.org/10.3390/magnetochemistry8080092

**Chicago/Turabian Style**

Palii, Andrew, Sergey Aldoshin, and Boris Tsukerblat.
2022. "Functional Properties of Tetrameric Molecular Cells for Quantum Cellular Automata: A Quantum-Mechanical Treatment Extended to the Range of Arbitrary Coulomb Repulsion" *Magnetochemistry* 8, no. 8: 92.
https://doi.org/10.3390/magnetochemistry8080092