#
In Quest of Molecular Materials for Quantum Cellular Automata: Exploration of the Double Exchange in the Two-Mode Vibronic Model of a Dimeric Mixed Valence Cell^{ †}

^{1}

^{2}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

**0**and

**1**) in the two antipodal (diagonal) distributions of the charges. To illustrate encoding and operating with binary information underlying the actions of electronic devices, a dimeric system can be used, as illustrated in Figure 1. The dimeric unit can be considered as a “half-cell” from which the “full-cell” (tetrameric unit) can be constructed. Figure 1 illustrates a dimeric cell in which the delocalized pair the mobile electron is evenly distributed between two sites and the two predominantly localized configurations corresponding to the binary

**0**and

**1**.

**1**in Figure 2), while the second one is unpolarized. Let us assume that the polarized state of the cell 2 can be induced and controlled so that this cell can be termed as the “driver cell”. The electrostatic effect of the driver cell 2 with a given polarization affects the neighboring cell 1 forcing this cell to acquire polarization

**0**. The polarization of the cell 1 obeys the action driver cell and in this way the driver cell can transmit the binary information to the surrounding cells. Thus the cell 1 can be referred to as the “working cell”.

## 2. Magnetic Interactions in a ${\mathit{d}}^{2}-{\mathit{d}}^{1}$—Type Cell

_{0}(S

_{0}= 1/2 in the present case). It is assumed that we are dealing with the high-spin metal ions so that the spin of ${d}^{2}$ ion is S

_{0}+ 1/2 = 1. These two spins are combined to give the total spin S of the dimer, which takes the values 1/2 and 3/2.

_{0}and S

_{0}+ 1/2 (within each localized configuration of the dimer). In order to take into account the effect of the driver cell, we consider also the contribution describing the Coulomb interaction between the cells (terms (terms $\pm u{P}_{2}$), where ${P}_{2}$ is the polarization of the driver- cell $\mathrm{A}\prime -\mathrm{B}\prime $ (cell 2). The value ${P}_{2}=+1$ corresponds to localization of the mobile charge on site A, while providing ${P}_{2}=-1$ the charges localized on site B, as shown in Figure 3. Usually, in the theory of QCA, it is assumed that the driver-cell is a source of a Coulomb field acting on the working cell (cell 1), which causes its polarization. Finally, u is the characteristic energy of the Coulomb interaction between the cells. The physical meaning of this parameter is clear from Figure 3 showing the relative disposition of the working cell and the driver-cell, and the two possible electronic distributions in a pair of interacting cells. As observable, the energy 2u is the difference between the Coulomb repulsion energies of the excess electrons occupying neighboring and distant (energetically favorable) positions in the two interacting dimers.

## 3. Polarization of a Cell

_{gr}= 2S

_{0}+ 1/2 = 3/2). When ${P}_{2}\ne 0$ the working cell is subjected to the action of an electrostatic field created by the driver-cell, which tends to localize the excess electron in the working cell. Since this field restricts the mobility of the excess electron it leads to a partial suppression of the ferromagnetic DE. As a result of such suppression, the antiferromagnetic HDVV exchange (that acts within localized configurations and hence is not affected by the field) can become the dominant interaction, which can lead to a stabilization of the spin-state with lower total spin value or, in other words, to cause a spin switching effect.

## 4. Two-Mode Vibronic Model

_{1g}symmetry). The vibronic interaction in MV compounds (particularly, in molecular cells) can be described in the framework of generally accepted Piepho-Kraus-Schatz (PKS) vibronic model [28,29,30]. Although this model is rather simplified, it successfully describes the key features of MV systems, such as the occurrence of a potential barrier between localized configurations. In particular, this model underlies the Robin and Day classification of MV compounds according to the degree of localization.

## 5. Dynamic Vibronic Problem

## 6. Spin-Vbronic Levels and Cell-Cell Response Function

## 7. Conclusions

- (1)
- in the case of the dominating DE, the character of the ground spin-state of a free and polarized cells has been shown to be strongly influenced by the interactions of the electronic subsystem with both types of vibrational modes included in the model. Therefore, in the case of vanishingly weak vibronic interaction, as well as in the case of strong coupling with the intercenter vibration, the ferromagnetic effect produced by the DE proves to be the dominating interaction;
- (2)
- in this case of strong DE the ${P}_{1}\left({P}_{2}\right)$ dependence (cell-cell response function) demonstrates weak and almost linear cell-cell response that is an unfavorable case for the QCA function. Moreover, the spin-switching is not possible in this case;
- (3)
- in contrast, at strong vibronic PKS coupling, the ferromagnetic DE is largely suppressed, which leads to the stabilization of the state with a minimum total spin, along with the appearance of a strong nonlinear cell-cell response. This case is definitely favorable for the design of QCA-based devices.
- (4)
- finally, when the contributions of both types of vibrations are comparable, the magnetic cell has been shown to exhibit properties of a spin-switcher, i.e., the electrostatic field of the driver change spin of the working cell along with the charge distribution. In relation to the QCA application, this feature manifests itself in the specific shape of the cell-cell response function exhibiting sharp steps. More precisely, the polarizability of the working cell is efficiently increased that is favorable for the QCA action.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

QCA | Quantum Cellular Automata |

CMOS | Complimentary Metal-Oxide-Semiconductor Structure |

MV | Mixed Valence |

DE | Double Exchange |

HDVV | Heisenberg-Dirac-Van Vleck (exchange, model) |

PKS | Piepho-Kraus-Schatz (coupling, model) |

## References

- Day, P. Molecular information processing: Will it happen? Proc. R. Inst. Great Br.
**1998**, 69, 85–106. [Google Scholar] - Robin, M.B.; Day, P. Mixed valence chemistry—A survey and classification. Adv. Inorg. Chem. Radiochem.
**1967**, 10, 247–422. [Google Scholar] - Lent, C.S.; Tougaw, P.D.; Porod, W.; Bernstein, G.H. Quantum Cellular Automata. Nanotechnology
**1993**, 4, 49–57. [Google Scholar] [CrossRef] - Lent, C.S.; Tougaw, P.; Porod, W. Bistable saturation in coupled quantum dots for quantum cellular automata. Appl. Phys. Lett.
**1993**, 62, 714–716. [Google Scholar] [CrossRef][Green Version] - Lent, C.S.; Tougaw, P.D. Lines of interacting quantum-dot cells: A binary wire. J. Appl. Phys.
**1993**, 74, 6227–6233. [Google Scholar] [CrossRef][Green Version] - Lent, C.S.; Isaksen, B.; Lieberman, M. Molecular Quantum-Dot Cellular Automata. J. Am. Chem. Soc.
**2003**, 125, 1056–1063. [Google Scholar] [CrossRef] - Tsukerblat, B.; Palii, A.; Aldoshin, S. Molecule Based Materials for Quantum Cellular Automata: A Short Overview and Challenging Problems. Israel J. Chem.
**2020**, 60, 527–543. [Google Scholar] [CrossRef] - Jiao, J.; Long, G.J.; Grandjean, F.; Beatty, A.M.; Fehlner, T.P. Building Blocks for the Molecular Expression of Quantum Cellular Automata. Isolation and Characterization of a Covalently Bonded Square Array of Two Ferrocenium and Two Ferrocene Complexes. J. Am. Chem. Soc.
**2003**, 125, 7522–7523. [Google Scholar] [CrossRef] - Li, Z.; Beatty, A.M.; Fehlner, T.P. Molecular QCA Cells. 1. Structure and Functionalization of an Unsymmetrical Dinuclear Mixed-Valence Complex for Surface Binding. Inorg. Chem.
**2003**, 42, 5707–5714. [Google Scholar] [CrossRef] [PubMed] - Li, Z.; Fehlner, T.P. Molecular QCA Cells. 2. Characterization of an Unsymmetrical Dinuclear Mixed-Valence Complex Bound to a Au Surface by an Organic Linker. Inorg. Chem.
**2003**, 42, 5715–5721. [Google Scholar] [CrossRef] - Qi, H.; Sharma, S.; Li, Z.; Snider, G.L.; Orlov, A.O.; Lent, C.S.; Fehlner, T.P. Molecular Quantum Cellular Automata Cells. Electric Field Driven Switching of a Silicon Surface Bound Array of Vertically Oriented Two-Dot Molecular Quantum Cellular Automata. J. Am. Chem. Soc.
**2003**, 125, 15250–15259. [Google Scholar] [CrossRef] [PubMed] - Braun-Sand, S.B.; Wiest, O. Theoretical Studies of Mixed Valence Transition Metal Complexes for Molecular Computing. J. Phys. Chem. A
**2003**, 107, 285–291. [Google Scholar] [CrossRef] - Braun-Sand, S.B.; Wiest, O. Biasing Mixed-Valence Transition Metal Complexes in Search of Bistable Complexes for Molecular Computing. J. Phys. Chem. B
**2003**, 107, 9624–9628. [Google Scholar] [CrossRef] - Qi, H.; Gupta, A.; Noll, B.C.; Snider, G.L.; Lu, Y.; Lent, C.; Fehlner, T.P. Dependence of Field Switched Ordered Arrays of Dinuclear Mixed-Valence Complexes on the Distance between the Redox Centers and the Size of the Counterions. J. Am. Chem. Soc.
**2005**, 127, 15218–15227. [Google Scholar] [CrossRef] [PubMed] - Jiao, J.; Long, G.J.; Rebbouh, L.; Grandjean, F.; Beatty, A.M.; Fehlner, T.P. Properties of a Mixed-Valence (Fe
^{II})_{2}(Fe^{III})_{2}Square Cell for Utilization in the Quantum Cellular Automata Paradigm for Molecular Electronics. J. Am. Chem. Soc.**2005**, 127, 17819–17831. [Google Scholar] [CrossRef] - Lu, Y.; Lent, C.S. Theoretical Study of Molecular Quantum Dot Cellular Automata. J. Comput. Electron.
**2005**, 4, 115–118. [Google Scholar] [CrossRef] - Zhao, Y.; Guo, D.; Liu, Y.; He, C.; Duan, C. A Mixed-Valence (Fe
^{II})_{2}(Fe^{III})_{2}Square for Molecular Expression of Quantum Cellular Automata. Chem. Commun.**2008**, 5725–5727. [Google Scholar] [CrossRef] - Nemykin, V.N.; Rohde, G.T.; Barrett, C.D.; Hadt, R.G.; Bizzarri, C.; Galloni, P.; Floris, B.; Nowik, I.; Herber, R.H.; Marrani, A.G.; et al. Electron-Transfer Processes in Metal-Free Tetraferrocenylporphyrin. Understanding Internal Interactions to Access Mixed Valence States Potentially Useful for Quantum Cellular Automata. J. Am. Chem. Soc.
**2009**, 131, 14969–14978. [Google Scholar] [CrossRef][Green Version] - Wang, X.; Yu, L.; Inakollu, V.S.S.; Pan, X.; Ma, J.; Yu, H. Molecular Quantum-Dot Cellular Automata Based on Diboryl Radical Anions. J. Phys. Chem. C
**2018**, 122, 2454–2460. [Google Scholar] [CrossRef][Green Version] - Burgun, A.; Gendron, F.; Schauer, P.A.; Skelton, B.W.; Low, P.J.; Costuas, K.; Halet, J.-F.; Bruce, M.I.; Lapinte, C. Straightforward Access to Tetrametallic Complexes with a Square Array by Oxidative Dimerization of Organometallic Wires. Organometallics
**2013**, 32, 5015–5025. [Google Scholar] [CrossRef][Green Version] - Schneider, B.; Demeshko, S.; Neudeck, S.; Dechert, S.; Meyer, F. Mixed-Spin [2 × 2] Fe
_{4}Grid Complex Optimized for Quantum Cellular Automata. Inorg. Chem.**2013**, 52, 13230–13237. [Google Scholar] [CrossRef] [PubMed] - Christie, J.A.; Forrest, R.P.; Corcelli, S.A.; Wasio, N.A.; Quardokus, R.C.; Brown, R.; Kandel, S.A.; Lu, Y.; Lent, C.S.; Henderson, K.W. Synthesis of a Neutral Mixed-Valence Diferrocenyl Carborane for Molecular Quantum-Dot Cellular Automata Applications. Angew. Chem. Int. Ed.
**2015**, 54, 15448–15671. [Google Scholar] [CrossRef] [PubMed] - Makhoul, R.; Hamon, P.; Roisnel, T.; Hamon, J.-R.; Lapinte, C. A Tetrairon Dication Featuring Tetraethynylbenzene Bridging Ligand: A Molecular Prototype of Quantum Dot Cellular Automata. Chem. Eur. J.
**2020**, 26, 8368–8371. [Google Scholar] [CrossRef] - Rahimia, E.; Reimers, J.R. Molecular quantum cellular automata cell design trade-offs: Latching vs. power dissipation. Phys. Chem. Chem. Phys.
**2018**, 20, 17881–17888. [Google Scholar] [CrossRef] [PubMed] - Ardesi, Y.; Pulimeno, A.; Graziano, M.; Riente, F.; Piccinini, G. Effectiveness of Molecules for Quantum Cellular Automata as Computing Devices. J. Low Power Electron. Appl.
**2018**, 8, 24. [Google Scholar] [CrossRef][Green Version] - Palii, A.; Clemente-Juan, J.M.; Rybakov, A.; Aldoshin, S.; Tsukerblat, B. Exploration of the double exchange in quantum cellular automata: Proposal for a new class of cells. Chem. Comm.
**2020**, 56, 10682–10685. [Google Scholar] [CrossRef] [PubMed] - Anderson, P.W.; Hasegawa, H. Consideration of Double Exchange. Phys. Rev.
**1955**, 100, 675–681. [Google Scholar] [CrossRef] - Palii, A.; Aldoshin, S.; Zilberg, S.; Tsukerblat, B. A parametric two-mode vibronic model of a dimeric mixed-valence cell for molecular quantum cellular automata and computational ab initio verification. Phys. Chem. Chem. Phys.
**2020**, 22, 25982–25999. [Google Scholar] [CrossRef] [PubMed] - Piepho, S.B.; Krausz, E.R.; Schatz, P.N. Vibronic coupling model for calculation of mixed valence absorption profiles. J. Am. Chem. Soc.
**1978**, 100, 2996–3005. [Google Scholar] [CrossRef] - Piepho, S.B. Vibronic coupling model for the calculation of mixed-valence line shapes: The interdependence of vibronic and MO effects. J. Am. Chem. Soc.
**1988**, 110, 6319–6326. [Google Scholar] [CrossRef]

**Figure 1.**Two charge distributions in a two-dot cell or in a dimeric (mixed-valence) MV molecule with one mobile electron corresponding to the delocalized (unpolarized) configuration and localized configurations corresponding to binary and

**1**. The red balls indicate the populated sites and their sizes symbolize the degree of localization of the mobile electrons.

**Figure 2.**Scheme of the elementary process of the control of the binary information through the action of the dimeric driver cell 2 to the working cell 1. Left part shows the initial step of the information processing: unpolarized driver and unpolarized working cell. Then at the next step the driver gets polarization corresponding to the binary

**1**. Finally, the polarized driver cell acts on the unpolarized working cell and gives rise to its polarization corresponding to binary

**0**.

**Figure 3.**Mutual disposition of the driver-cell and the working cell, and the two possible electronic distributions in a pair of interacting dimeric cells shown to explain the physical meaning of the intercell Coulomb energy u. The sites belonging to the driver-cell and the working cell are primed and unprimed correspondingly, the site comprising (in a definite electronic distribution) the excess electron is shown as a red ball.

**Figure 4.**Spin-vibronic energy levels of the working cell d

^{2}− d

^{1}calculated as a function of polarization of the driver cell P

_{2}at $u=600{\mathrm{cm}}^{-1}$, $\hslash \omega =\hslash \Omega =200{\mathrm{cm}}^{-1}$, $t=1000{\mathrm{cm}}^{-1}$, $J=-125{\mathrm{cm}}^{-1}$ and the following sets of the vibronic coupling parameters: $\upsilon =0,\zeta =0\left(a\right);$ $\upsilon =300{\mathrm{cm}}^{-1},\zeta =0\left(b\right);$ $\upsilon =400{\mathrm{cm}}^{-1},\zeta =0\left(c\right);$ $\upsilon =500{\mathrm{cm}}^{-1},\zeta =0\left(d\right);$ $\upsilon =500{\mathrm{cm}}^{-1},\zeta =200{\mathrm{cm}}^{-1}\left(e\right);$ $\upsilon =500{\mathrm{cm}}^{-1},\zeta =300{\mathrm{cm}}^{-1}\left(f\right);$ $\upsilon =500{\mathrm{cm}}^{-1},\zeta =400{\mathrm{cm}}^{-1}\left(g\right);$ $\upsilon =500{\mathrm{cm}}^{-1},\zeta =450{\mathrm{cm}}^{-1}\left(h\right).$ The ground spin-vibronic level is chosen as a reference point for the energy.

**Figure 5.**Cell-cell response functions evaluated for the ${d}^{2}-{d}^{1}$–type cells at $u=600{\mathrm{cm}}^{-1}$, $\hslash \omega =\hslash \Omega =200{\mathrm{cm}}^{-1}$, $t=1000{\mathrm{cm}}^{-1}$, $J=-125{\mathrm{cm}}^{-1}$ and following sets of the vibronic coupling parameters: $\upsilon =0,\zeta =0\left(a\right);\upsilon =300{\mathrm{cm}}^{-1},\zeta =0\left(b\right);$ $\upsilon =400{\mathrm{cm}}^{-1},\zeta =0\left(c\right);$ $\upsilon =500{\mathrm{cm}}^{-1},\zeta =0\left(d\right);\upsilon =500{\mathrm{cm}}^{-1},\zeta =200{\mathrm{cm}}^{-1}\left(e\right);\upsilon =500{\mathrm{cm}}^{-1},\zeta =300{\mathrm{cm}}^{-1}\left(f\right);\upsilon =500{\mathrm{cm}}^{-1},\zeta =400{\mathrm{cm}}^{-1}\left(g\right)\mathrm{and}\upsilon =500{\mathrm{cm}}^{-1},\zeta =450{\mathrm{cm}}^{-1}\left(h\right).$

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Tsukerblat, B.; Palii, A.; Aldoshin, S. In Quest of Molecular Materials for Quantum Cellular Automata: Exploration of the Double Exchange in the Two-Mode Vibronic Model of a Dimeric Mixed Valence Cell. *Magnetochemistry* **2021**, *7*, 66.
https://doi.org/10.3390/magnetochemistry7050066

**AMA Style**

Tsukerblat B, Palii A, Aldoshin S. In Quest of Molecular Materials for Quantum Cellular Automata: Exploration of the Double Exchange in the Two-Mode Vibronic Model of a Dimeric Mixed Valence Cell. *Magnetochemistry*. 2021; 7(5):66.
https://doi.org/10.3390/magnetochemistry7050066

**Chicago/Turabian Style**

Tsukerblat, Boris, Andrew Palii, and Sergey Aldoshin. 2021. "In Quest of Molecular Materials for Quantum Cellular Automata: Exploration of the Double Exchange in the Two-Mode Vibronic Model of a Dimeric Mixed Valence Cell" *Magnetochemistry* 7, no. 5: 66.
https://doi.org/10.3390/magnetochemistry7050066