Theoretical Hints to Optimize Energy Dissipation and Cell–Cell Response in Quantum Cellular Automata Based on Tetrameric and Bidimeric Cells ^†

Abstract

This article is largely oriented towards the theoretical foundations of the rational design of molecular cells for quantum cellular automata (QCA) devices with optimized properties. We apply the vibronic approach to the analysis of the two key properties of such molecular cells, namely the cell–cell response and energy dissipation in the course of the non-adiabatic switching of the electric field acting on the cell. We consider two kinds of square planar cells, namely cells represented by a two-electron tetrameric mixed valence (MV) cluster and bidimeric cells composed of two one-electron MV dimeric half-cells. The model includes vibronic coupling of the excess electrons with the breathing modes of the redox sites, electron transfer, intracell interelectronic Coulomb repulsion, and also the interaction of the cell with the electric field of polarized neighboring cells. For both kinds of cells, the heat release is shown to be minimal in the case of strong delocalization of excess electrons (weak vibronic coupling and/or strong electron transfer) exposed to a weak electric field. On the other hand, such a parametric regime proves to be incompatible with a strong nonlinear cell–cell response. To reach a compromise between low energy dissipation and a strong cell–cell response, we suggest using weakly interacting MV molecules with weak electron delocalization as cells. From this point of view, bidimeric cells are advantageous over tetrameric ones due to their smaller number of electron transfer pathways, resulting in a lower extent of electron delocalization. The distinct features of bidimeric cells, such as their two possible mutual arrangements (“side-by-side” and “head-to-tail”), are discussed as well. Finally, we briefly discuss some relevant results from a recent ab initio study on electron transfer and vibronic coupling from the perspective of the possibility of controlling the key parameters of molecular QCA cells.

1. Introduction

Molecular quantum cellular automata (QCA) represent a new promising paradigm of transistorless electronics and computing in which digital (binary) information is encoded in charge configurations of mixed-valence (MV) molecules [1,2,3]. QCA technology holds promise as a new revolutionary technology in nanoelectronics and computing based on molecular units, having essential advantages over traditional electronic devices. The idea of molecular QCA has given rise to a new area of research combining chemistry, physics, molecular magnetism, and materials science, with an emphasis on 2D systems. Since the first proposal to use MV molecules as QCA cells or half-cells [3], numerous MV systems have been suggested for the molecular implementation of QCA (see examples in Refs. [4,5,6,7,8,9,10,11,12,13,14,15,16,17,18] and a recent exhaustive review [19]). Figure 1 shows two selected examples of square-planar molecular cells [3,8,20], which are tetraruthenium Ru(III)₂Ru(II)₂ complexes composed of a pair of famous Creutz–Taube ions [21] (Nobel Prize in Chemistry for discoveries on electron transfer). Each of these molecules comprises two excess holes, delocalized among the four sites occurring either via the bridges located on the edges of the square (Figure 1a) or through the central bridge (Figure 1b).

These two topologies of the electron transfer pathways are closely related to the two kinds of spin arrangements (as will be discussed below), which in turn demonstrate the interrelation between the areas of the molecular magnetism and the QCA.

Theoretical work aimed at identifying and understanding the specific properties (sometimes conflicting with each other) of active molecules, which allows us to consider their suitability for use as QCA cells, has acquired a special role. Among the requirements that the MV molecule should meet in order to act as a cell or half-cell, two should be specifically mentioned. First, the molecule should exhibit low energy dissipation in the course of the switching cycle [22,23,24,25,26,27], and second, the molecule should be able to easily switch between the two charge configurations encoding binary information under the action of the electric field created by the polarized neighboring molecule, which manifests itself in the strongly nonlinear (stepwise) cell–cell response function. The latter requirement, first formulated for QCA based on quantum dots [28,29,30,31,32], is general for all kinds of QCA devices and all types of cells.

Concerning the first requirement, it has recently been shown that the higher the extent of delocalization of the excess charges in an MV dimer, the lower the specific heat release (heat release per one cell during one switching cycle) [26,27]. It follows from the vibronic theory of molecular QCA based on MV compounds [33,34,35,36,37] that the extent of delocalization is determined by the interplay of two competing interactions, namely electron transfer promoting delocalization and vibronic coupling with the breathing vibrations of the redox sites producing a self-trapping effect. Then, a high degree of delocalization means that electron transfer dominates over the vibronic coupling. In addition, heat release increases with an increase in the Coulomb interaction strength.

Regarding the second requirement, it has been demonstrated [33] that the extent of the electron delocalization should be low in order to obtain a strong nonlinear cell–cell response, and so the vibronic coupling should dominate over the electron transfer. This means that the two requirements are in some sense contradictory to each other. Indeed, if one uses weakly interacting MV dimers with strong electron delocalization to minimize the energy dissipation (first requirement), one faces an impossibility of fulfilling the second requirement, since in this case the cell–cell response proves to be weak and almost linear. By considering the simplest dimeric cells, we have demonstrated that the best way to reconcile low heat release and strong nonlinear cell–cell response is to use weakly delocalized MV dimers separated from each other by a long distance as cells in order to prevent strong interdimer Coulomb interaction [26,27]. This follows from the fact that such dimers can easily be polarized even by a rather weak Coulomb field, which, in turn, can significantly reduce the heat release in spite of the high extent of the electron localization.

In this work, based on the vibronic theory developed in Refs. [33,34,35,36,37], we generalize the approach proposed in Refs. [26,27] to the cases of more practically applicable molecular cells, which are more complex than the dimeric ones. These the square-planar cells composed of two one-electron MV dimers playing the role of half-cells (“bidimeric cells”) and cells representing two-electron tetrameric square-planar MV molecules (“tetrameric cell”). A number of candidates for the molecular implementation of tetrameric and bidimeric cells have been considered [4,5,6,7,8,9,10,11,12,13,14,15,16,17,18], and also the attempt to compare the functional properties of these two kinds of cells has been addressed [34]. Still, this attempt has been restricted to a special limiting case of strong intracell Coulomb interaction, and, even more importantly, only the cell–cell response has been analyzed, while the energy dissipation remained outside of the scope of consideration in Ref. [34]. In this paper, based on the quantum-mechanical solution of the vibronic problems for such cells [36,37], we analyze both the dependences of the cell–cell response functions and the specific heat release in the nonadiabatic switching cycle on the electronic and vibronic parameters as well as on the strength of the cell–cell interaction. Then, we establish parametric regimes at which competing requirements of low heat release and strong nonlinear cell–cell response can be simultaneously fulfilled. Along the way, a complete comparative analysis of bidimeric and tetrameric cells is carried out in terms of their advantages for creating QCA devices. Finally, we analyze some important ab initio results for a number of specifically selected MV compounds, with a particular focus on the ability to control and optimize the key parameters responsible for functioning of the cells.

2. Description of the Molecular Cells within the Vibronic Approach

2.1. Hamiltonians of the Bidimeric and Tetrameric Working Cells

We consider two interacting cells of the same type, with one of them playing the role of the driver-cell that acts as a source of the electric field (it is either bidimeric or tetrameric square cell A′B′C′D′, Figure 2) and the other one represents the working cell that can be polarized by this field (bidimeric or tetrameric cell ABCD). The square planar driver-cell is assumed to be polarized in such a way that the densities on the sites situated along one diagonal of the square are both equal to $\rho$, while the densities on the sites belonging to another diagonal are $1-\rho$ (Figure 2). Such a cell acts as an electric quadrupole.

Polarization of the square-planar driver-cell $A^{\prime }{B}^{\prime }{C}^{\prime }{D}^{\prime }$ (bidimeric and tetrameric) is defined as a normalized difference in populations (${\rho }_A^{\prime },\mathrm{e}\mathrm{t}\mathrm{c}.$) of the two diagonals of the square, that is

$$P_{dc}={\displaystyle \frac{{\rho }_A^{\prime }+{\rho }_C^{\prime }-{\rho }_B^{\prime }-{\rho }_D^{\prime }}{{\rho }_A^{\prime }+{\rho }_C^{\prime }+{\rho }_B^{\prime }+{\rho }_D^{\prime }}}=2\rho -1.$$

(1)

The polarized driver-cell creates a quadrupolar electric field inducing polarization of the neighboring working cell. The polarization $P_{wc}$ of the working cell is defined by similar expression. It is evident that the polarization of the working cell should depend on the polarization of the driver-cell. This dependence $P_{wc}\left(P_{dc}\right)$ is the so-called “cell–cell response function” that is one of the key functional characteristics of QCA.

To calculate the cell–cell response function and the heat release, one needs a proper Hamiltonian of the working cell subjected to the action of the electric field created by the polarized driver-cell. The specific form of the Hamiltonian depends on the type of the cell under consideration, the employed approximations and also on the adopted vibronic model. In this work, we consider two types of square-planar ABCD-cells, namely bidimeric cells consisting of two dimeric one-electron half-cells AB and CD and tetrameric cells represented by a two-electron MV tetramer. As the vibronic model, we use the conventional Piepho–Krausz–Schatz (PKS) vibronic model, which takes into account the interaction of the excess electrons with the fully symmetric (“breathing”) vibrations of the redox sites [38]. As to the approximations employed, for both considered types of cells, we consider the limiting case of strong Coulomb interaction inside the cell, which is defined by the inequality t << U, where U is the difference between the intracell Coulomb energies of the two close (edges of the square cell) and two distant (diagonals of the cell) excess electrons, while t is the one-electron transfer parameter. In the framework of this approximation, we also consider the general case of the arbitrary relationship between the parameters t and U.

Let us start with the general case discussed previously in view of the problem of the evaluation of cell–cell response (see Refs. [35,36,37]). We briefly discuss the key issues related to this case with short comments. For the description of tetrameric and bidimeric cells, we use the following Hamiltonians:

$$\begin{array}{l}{\widehat{H}}_{bidim}^{ta}=t{\displaystyle \sum _{\sigma }}\left(c_{A\sigma }^{+}{c}_{B\sigma }+c_{C\sigma }^{+}{c}_{D\sigma }+h.c.\right)\\ +u_{AC}{\widehat{n}}_A{\widehat{n}}_C+u_{BD}{\widehat{n}}_B{\widehat{n}}_D+\left(U+u_{AD}^{ta}\right){\widehat{n}}_A{\widehat{n}}_D+\left(U+u_{BC}^{ta}\right){\widehat{n}}_B{\widehat{n}}_C\\ +\hslash \omega \left({\widehat{b}}_1^{+}{\widehat{b}}_1+{\widehat{b}}_2^{+}{\widehat{b}}_2+1\right)\left({\widehat{n}}_A{\widehat{n}}_C+{\widehat{n}}_B{\widehat{n}}_D+{\widehat{n}}_A{\widehat{n}}_D+{\widehat{n}}_B{\widehat{n}}_C\right)\\ +\upsilon \hspace{0.33em}\left[\left({\widehat{b}}_1^{+}+{\widehat{b}}_1\right)({\widehat{n}}_A{\widehat{n}}_C-{\widehat{n}}_B{\widehat{n}}_D)+\left({\widehat{b}}_2^{+}+{\widehat{b}}_2\right)({\widehat{n}}_A{\widehat{n}}_D-{\widehat{n}}_B{\widehat{n}}_C)\right],\end{array}$$

(2)

$$\begin{array}{l}{\widehat{H}}_{tetr}=t{\displaystyle \sum _{\sigma }}\left(c_{A\sigma }^{+}{c}_{B\sigma }+c_{B\sigma }^{+}{c}_{C\sigma }+c_{C\sigma }^{+}{c}_{D\sigma }+c_{A\sigma }^{+}{c}_{D\sigma }+h.c.\right)\\ +u_{AC}{\widehat{n}}_A{\widehat{n}}_C+u_{BD}{\widehat{n}}_B{\widehat{n}}_D+\left(U+u_{AD}\right){\widehat{n}}_A{\widehat{n}}_D+\left(U+u_{BC}\right){\widehat{n}}_B{\widehat{n}}_C+\left(U+u_{AB}\right){\widehat{n}}_A{\widehat{n}}_B +\left(U+u_{CD}\right){\widehat{n}}_C{\widehat{n}}_D\\ +\hslash \omega \left({\widehat{b}}_1^{+}{\widehat{b}}_1+{\widehat{b}}_2^{+}{\widehat{b}}_2+{\widehat{b}}_3^{+}{\widehat{b}}_3+{\displaystyle \frac{3}{2}}\right)\left({\widehat{n}}_A{\widehat{n}}_C+{\widehat{n}}_B{\widehat{n}}_D+{\widehat{n}}_A{\widehat{n}}_D+{\widehat{n}}_B{\widehat{n}}_C+{\widehat{n}}_A{\widehat{n}}_B+{\widehat{n}}_C{\widehat{n}}_D\right)\\ +\upsilon \hspace{0.33em}\left[\left({\widehat{b}}_1^{+}+{\widehat{b}}_1\right)({\widehat{n}}_A{\widehat{n}}_C-{\widehat{n}}_B{\widehat{n}}_D)+\left({\widehat{b}}_2^{+}+{\widehat{b}}_2\right)({\widehat{n}}_A{\widehat{n}}_D-{\widehat{n}}_B{\widehat{n}}_C)+\left({\widehat{b}}_3^{+}+{\widehat{b}}_3\right)({\widehat{n}}_A{\widehat{n}}_B-{\widehat{n}}_C{\widehat{n}}_D)\right].\end{array}$$

(3)

The first terms in Equations (2) and (3) describe the electron transfer contributions, where t is the electron transfer parameter, $c_{i\sigma }^{+}$ and $c_{i\sigma }$ are the fermionic creation and annihilation operators, and σ is the projection of the spin of the electron. The form of the electron transfer Hamiltonian in Equation (2) assumes that the transfer in the bidimeric cell is allowed within each dimeric half-cell, and no transfer is possible between different half-cells. In contrast, in tetrameric cells, the transfer is allowed between each of the two nearest neighboring sites, while the transfer between the antipodal sites of the square (along the diagonal) is neglected. For this reason, the Hamiltonian in Equation (3) contains twice as many electron transfer connections than that in Equation (2). In this context, the tetramer shown in Figure 1a represents an example of a compound with predominant transfer between neighboring sites, while in the tetramer shown in Figure 1b, both the transfer processes occurring along the edges and the diagonal transfer processes can produce comparable contributions.

The second terms in Equations (2) and (3) describe the Stark interaction of bidimeric and tetrameric working cells with the quadrupolar electric field created by the polarized driver-cell, where ${\widehat{n}}_i$ is the operator of the number of electrons on the site i, whose eigenvalues can either be 1 (site occupied by the electron) or 0 (empty site). For the description of the Stark interaction in the case of bidimeric cells, one needs four energies of the Coulomb interactions between the polarized driver-cell and the working cell with four possible different configurations of the excess electrons, which are denoted as $u_{AC}$, $u_{BD}$, $u_{AD}^{ta},$and $u_{BC}^{ta}$ in Equation (2). These energies depend on the size of the cell, the distance between the cells, the driver-cell polarization $P_{dc}$ and the type of mutual arrangement of the interacting cells, which can be either “head-to-tail” or “side-by-side”, as shown in Figure 3a,b, respectively. The index ta (type of arrangement) in Equation (2) is used to indicate the type of mutual arrangement of bidimeric cells. It should be noted that for diagonal-type electronic configurations AC and BD, the Coulomb energies are independent of the type of cells arrangement, and so the index ta is omitted in the notations of the corresponding Coulomb energies. As distinguished from the bidimeric cells, the Stark interaction in the case of tetrameric cells is described by six intercell Coulomb energies $u_{AC},$ $u_{BD}$, $u_{AD}$, $u_{BC}$, $u_{AB}$, and $u_{CD}$, as can be seen from Equation (3). The smaller number of intercell Coulomb energies for bidimeric cells is because for such cells, there are fewer electronic configurations as compared with the tetrameric cells, since configurations with two excess electrons per dimeric half-cell are forbidden. The dependences of the intercell Coulomb energies for tetrameric and bidimeric cells on the dimensions of the cell and the intercell distance are given in Refs. [35,37], respectively.

Besides the intercell interaction, the Hamiltonians, Equations (2) and (3), also include the contributions of the intracell Coulomb interaction. This interaction is described by the Coulomb energy gap U between the diagonal-type electronic configurations AC and BD in which the two electrons are distributed far from each other, forming the ground Coulomb manifold of the cell and the excited configurations (AD and BC in the case of bidimeric cell and AD, BC, AB and CD in the case of tetrameric cell) with closely spaced electrons.

Finally, the last two terms in Equations (3) and (4) describe the contribution of active molecular vibrations. We apply the conventional vibronic Piepho–Krausz–Schatz (PKS) model [38], which duly takes into account all the key features of the localization–delocalization phenomenon responsible for the functional properties of the molecular cells. Within this model, the molecular vibrations are built as symmetry-adapted combinations of the full-symmetric (“breathing”) vibrations of the constituent redox sites. It has been shown [37] that in the case of bidimeric cells, one should take into account two molecular vibrational out-of-phase modes. The first such mode is that describing compression of the two sites situated along one diagonal accompanied by expanding of the two sites on another diagonal of the square. The second mode describes compression (expansion) of the two closely spaced sites A and D belonging to different half-cells accompanied by expansion (compressions) of the other two sites B and C. For tetrameric cells, in addition to these two active modes, one more active out-of-phase mode appears, which describes compression (expansion) of the two closely spaced sites A and B accompanied by expansion (compressions) of the other two sites C and D [35]. The contributions of the out-of-phase vibrations are expressed through the bosonic creation and annihilation operators ${\widehat{b}}_j^{+}$ and ${\widehat{b}}_j$, where the subindex j enumerates the active PKS vibrations (j = 1, 2 in the case of bidimeric cell and j = 1, 2, 3 for tetrameric cell). The terms, containing the products ${\widehat{b}}_j^{+}{\widehat{b}}_j$ in Equations (2) and (3), are the Hamiltonians of the free vibrations occurring with the frequency ω, while the linear terms with respect to the operators ${\widehat{b}}_j^{+}$ and ${\widehat{b}}_j$ describe the interaction of the excess electron with the out-of-phase modes, i. e., the vibronic coupling, where υ is the vibronic coupling parameter. Note that all out-of-phase PKS vibrations occur with the same frequency ω, which is equal to the frequency of the local breathing mode (it is assumed that the frequency of the breathing vibration is independent of the oxidation degree of the site), and also the vibronic coupling with all such vibrations is described by the only vibronic parameter υ, since these modes are composed of the breathing vibrations of the equivalent redox sites.

Along with the general case, we also analyze the limit of strong intracell Coulomb interaction. This limit has been considered for bidimeric and tetrameric cells in [34] in the context of the analysis of the cell–cell response, while the problem of heat release has not been discussed. Considering the intracell Coulomb interaction as a zero-order Hamiltonian, and applying the perturbation theory to all other interactions, we reduce the initial Hamiltonians, Equations (2) and (3), to the following simpler ones:

$$\begin{array}{l}{\widehat{H}}_{tc}=k_{tc}{t}_{eff}{\displaystyle \sum _{\sigma {\sigma }^{\prime }}}\left(c_{D\sigma }^{+}{c}_{B{\sigma }^{\prime }}^{+}{c}_{C\sigma }{c}_{A{\sigma }^{\prime }}+h.c.\right)+u{P}_{dc}\left({\widehat{n}}_A{\widehat{n}}_C-{\widehat{n}}_B{\widehat{n}}_D\right)\\ +\hslash \omega \left({\widehat{b}}_1^{+}{\widehat{b}}_1+{\displaystyle \frac{1}{2}}\right)\left({\widehat{n}}_A{\widehat{n}}_C+{\widehat{n}}_B{\widehat{n}}_D\right)+\upsilon \hspace{0.33em}\left({\widehat{b}}_1^{+}+{\widehat{b}}_1\right)({\widehat{n}}_A{\widehat{n}}_C-{\widehat{n}}_B{\widehat{n}}_D),\end{array}$$

(4)

where $t_{eff}=t^2/U$ is the effective second-order electron transfer parameter, the index $tc$ specifies the kind of cell ($tc=tetr, bidim$), and $k_{tc}$ is a factor that is different for tetrameric and bidimeric cells, namely $k_{tetr}=4 \mathrm{and} k_{bidim}=2$; finally, u is the parameter that is equal to the half of the so-called kink energy, defined as the difference in Coulomb energies of pairs of cells with different and identical polarizations.

As distinguished from the general Hamiltonians, Equations (2) and (3), whose electronic contributions act within the full electronic basis containing six electronic configurations for tetrameric cells and four configurations for bidimeric cells, the electronic parts of the effective Hamiltonians, Equation (4), operate within the truncated electronic basis comprising only two ground electronic configurations, namely the diagonal-type configurations AC and BD possessing the minimal intracell Coulomb energy. The first term in Equation (4) describes the effective two-electron transfer (occurring via virtual one-electron transfers mixing the ground and excited electronic configurations), which transforms one diagonal configuration to another one. The second term in Equation (4) describes the intercell Stark interaction of the working cell with the polarized driver-cell with polarization $P_{dc}$. As distinguished from the general case, only one intercell Coulomb energy $u{P}_{dc}$ is required to describe such an interaction.

The last two terms in Equation (4) are the free harmonic oscillator Hamiltonian and the Hamiltonian of the linear vibronic coupling. Only the PKS out-of-phase vibration involving the two diagonals of the square is active within the truncated basis of diagonal configurations and so instead of two-mode and three-mode vibrations problems arising in the general case, we face a one-mode vibronic problem for both types of cells, provided that the limit of strong Coulomb intracell interaction is fulfilled.

2.2. Quantum-Mechanical Vibronic Approach: Evaluation of Cell–Cell Response

To calculate the eigenvalues and the eigenvectors of the Hamiltonians, Equations (2)–(4), one has to present these Hamiltonians as a matrix defined in the full electron-vibrational space. To achieve this, we use as the basis the product $\left|D,n_1,...n_N\right.⟩\equiv \left|D\right.⟩\hspace{0.33em}\left|n_1,...n_N\right.⟩$,. where $\left|D\right.⟩$ represents the electronic wave functions, each characterized by a certain electronic distribution $D$, and $\left|n_1,...n_N\right.⟩$ represent the wave functions of free N-dimensional harmonic oscillator (N = 1 for bidimeric and tetrameric cells, for which the limiting case of strong Coulomb interaction is satisfied, and N = 2 and 3 for bidimeric and tetrameric cells with a violated limit of strong Coulomb interaction).

Diagonalization of the obtained matrices provides the sets of the vibronic eigenvalues and eigenvectors, which depend on the type of cell, its size, the distance between the cells, and the values of the parameters describing the intracell and intercell interactions. The knowledge of the vibronic wave functions evaluated with different values of the driver-cell polarization $P_{dc}$ allows us to calculate the dependence of the induced polarization $P_{wc}$ of the working cell on $P_{dc}$, that is, the cell–cell response function. Also, based on the solution of the vibronic problem, one can evaluate the specific heat release assuming fast nonadiabatic switching of the electric field acting on the working cell.

2.3. Quantum–Mechanical Vibronic Approach: Evaluation of Specific Heat Release

At this point, we introduce a simplifying assumption that restricts the generality of the results to some extent, but at the same time considerably facilitates the calculations and makes them more transparent. We assume that the process of repolarization of the working cell is caused by a sudden (nonadiabatic) switching of the field of the driver-cell, as schematically shown in Figure 4 for the case of bidimeric cells considered within the limit of strong intracell Coulomb interaction. The left side of Figure 4 shows a pair of cells, each of which is in the Boolean state 0, which corresponds to the minimum energy of the Coulomb interaction between the cells. The middle part of Figure 4 shows an intermediate transition state arising upon sudden switching of the driver-cell from the state 0 to the state 1, while the working cell still remains in the state 0. This corresponds to the Coulomb excitation of the pair of cells (circled area in Figure 4), with the Coulomb excitation energy being just equal to the kink energy 2u. At the last stage of the cycle, the excitation energy is converted into the heat through vibrational relaxation and transferred to the phonon bath, after which the pair of cells acquires a new stable charge configuration with both cells being in the state 1 (right part of Figure 4). It is seen that the kink energy can be considered as a measure of the specific heat release, or, more precisely, as will be shown below, as the maximum possible value of the specific heat release in the course of nonadiabatic switching.

Note that in this consideration, switching is assumed to be so fast that there is no time for energy exchange between the cell and the phonon bath to occur in the course of the transition from the initial thermodynamically equilibrium state (left part of Figure 4) to the nonequilibrium state (middle part of Figure 4). Under such conditions, the bath does not influence the switching event and the energy exchange occurs only in the course of subsequent vibrational relaxation, resulting in the final equilibrium state (right part of Figure 4).

To calculate the heat release for all types of cells, one needs the eigenvalues and the eigenfunctions of the Hamiltonians, Equations (2)–(4), evaluated at two polarizations of the driver-cell, namely at$P_{dc}=-1$ (beginning of the switching cycle) and $P_{dc}=1$ (end of the cycle). The energies and the wave functions of the working cell calculated at $P_{dc}=-1$ are denoted by ${\tilde{E}}_{\tilde{\nu }}$ and $\left|\tilde{\nu }\right.⟩$, while $E_{\nu }$ and |ν⟩ are the energies and the wave functions calculated at $P_{dc}=1$. The wave functions obtained for these values of $P_{dc}$ are the linear combinations of the basis vectors

$$\begin{array}{l}\left|\tilde{\nu }\right.⟩={\displaystyle \sum _D}\left|D\right.⟩{\displaystyle \sum _{n_1\dots n_N}}{c}^D(n_1\dots n_N,\tilde{\nu })\left|n_1\dots n_N\right.⟩,\\ \left|\nu \right.⟩={\displaystyle \sum _D}\left|D\right.⟩{\displaystyle \sum _{n_1\dots n_N}}{c}^D(n_1\dots n_N,\nu )\left|n_1\dots n_N\right.⟩,\end{array}$$

(5)

in which the coefficients $c^D(n_1\dots n_N,\tilde{\nu })$ and $c^D\left(n_1\dots n_N,\nu \right)$depend on the set of parameters describing the working cell, the size of the cell, the intercell distance, and the polarization of the driver-cell.

Upon the action of sudden (stepwise) perturbation of the form

$$\widehat{V}\left(\tau \right)=\left[{\widehat{H}}_{wc}\left(P_{dc}=1\right)-{\widehat{H}}_{wc}\left(P_{dc}=-1\right)\right]\theta \left(\tau \right)=2u\theta \left(\tau \right),$$

(6)

expressed in terms of the Heaviside step function of time $\theta \left(\tau \right)$, the working cell being initially (at time $\tau <0$) in a stationary state $\left|\tilde{\nu }\right.⟩$ will occur after sudden switching (i.e., at $\tau >0$) with a certain probability in a new stationary state $\left|\nu \right.〉$, and, as follows from the nonstationary perturbation theory for the case of instant perturbation [39], this probability is defined as a square of the overlap integral

$$a_{\nu \tilde{\nu }}\hspace{0.33em}\hspace{0.33em}=\hspace{0.33em}\hspace{0.33em}⟨\nu |\tilde{\nu }⟩$$

(7)

of the vibronic wave functions calculated at $P_{dc}=-1$ and $P_{dc}=1$.

Using the calculated probability amplitudes $a_{\nu \tilde{\nu }}$ and the calculated eigenvalues ${\tilde{E}}_{\tilde{\nu }}$ ${\mathrm{a}\mathrm{n}\mathrm{d} E}_{\nu }$, one can derive the following expression for the specific heat release W occurring in course of the non-adiabatic switching:

$$W={\displaystyle \frac{\sum _{\nu }{E}_{\nu }\sum _{\tilde{\nu }}{\left(a_{\nu \tilde{\nu }}\right)}^2 exp\left(-{\beta }_T\hspace{0.33em}{\tilde{E}}_{\tilde{\nu }}\right)}{\sum _{\mu }\sum _{\tilde{\nu }}{a_{\mu \tilde{\nu }}}^{2}exp\left(-{\beta }_T\hspace{0.33em}{\tilde{E}}_{\tilde{\nu }}\right)}}-{\displaystyle \frac{{\sum }_{\nu }{E}_{\nu }\mathit{exp}\left(-\hspace{0.33em}{\beta }_T\hspace{0.33em}{E}_{\nu }\right) }{{\sum }_{\mu }\mathit{exp}\left(-\hspace{0.33em}{\beta }_T\hspace{0.33em}{E}_{\mu }\right) }},$$

(8)

where${\beta }_T=1/k_{B}T.$ Although the derivation of Equation (8) in [27] has been carried out for the simplest case of a dimeric cell, this expression is valid for bidimeric and tetrameric cells as well, while all specifical features (both electronic and vibronic) of the cells and the approximations made so far are taken into account by using a specific form of the cell Hamiltonian.

3. Results and Discussion

3.1. Consideration of Bidimeric and Tetrameric Cells: Limit of Strong Coulomb Repulsion

Based on the general formalism described above, we study the energy dissipation for the two types of square-planar two-electron molecular QCA cells, namely for bidimeric cells composed of one-electron MV dimers, such as the d¹-d⁰-type transition metal clusters or similar organic MV molecules, and for two-electron tetrameric cells represented by the d¹-d¹-d⁰-d⁰ type 3d-metal clusters and similar organic complexes.

Let us first consider the limiting case of strong Coulomb interaction when$U\gg t.$ Before analyzing the results of calculation, it is appropriate to make some preliminary remarks concerning the spin-states of the system. Since we are dealing with the bi-electronic systems, their full spin ($S=0 \mathrm{a}\mathrm{n}\mathrm{d} S=1)$ is a good quantum number. At $U\gg t$, the energy pattern of a two-electron square-planar MV tetramer consists of two groups of levels separated by a large Coulomb gap $U$ [40]. The low-lying group corresponds to the localization of the mobile electrons at the antipodal sites lying on the diagonal of the square, while the electronic population of the adjacent sites lying on the sides of the square gives rise to the highly excited (having energies $\approx U$) group of levels. Figure 5a shows the pattern of the low-lying spin-singlets and spin-triplets evaluated for a square-planar MV tetramer of the d¹-d¹-d⁰-d⁰-type within the adopted model of the electron transfer pathways in which only the transfer occurring along the edge of the molecular square is assumed to be nonzero. One can see that mixing of different Coulomb configurations through the transfer splits the lower Coulomb manifold in such a way that the spin-singlet proves to be the ground state. The pattern of the potential curves evaluated within the PKS model (Figure 5b) shows that the vibronic ground state associated with the potential minima also remains the spin-singlet. Moreover, one can show that the spin-singlet proves to be the ground state even when the polarizing electric field created by a neighboring cell (driver-cell) is applied.

Here, we focus on the analysis of the cell–cell response and heat release in the low-temperature limit. In view of the above discussion and mentioned approximation, this means that in the case under consideration, the excited levels with S = 1 do not contribute to the low-temperature cell–cell response, meaning that the following consideration will be confined to the low-lying S = 0 states. Considering the power dissipation, one should note that the instant switching of the polarizing field does not induce the transitions with a spin change, so that in the low-temperature limit, the transitions from the ground spin-singlet occur only to the excited levels with S = 0. This means that in the model of the transfer pathways adopted here, the spin-vibronic levels of the tetrameric cells with S = 1 do not contribute to power dissipation in the low temperature limit. In the bidimeric cells, the two half-cells are not connected through the transfer processes and hence all their states are degenerated with respect to the total spin S of the cell. Therefore, while considering the bidimeric cells, one can also select only the S = 0 pattern for the calculation of the cell’s characteristics.

Based on the numerical solution of the one-mode vibronic problem, one can evaluate the vibronic energy levels and the corresponding wave functions of the cells, and also calculate the specific heat release and cell–cell response functions for different sets of cell parameters. These results should allow us to conduct a comparative analysis of the two types of cells in order to establish the best “architecture” of the molecular cells and the arrays of cells which would ensure the coexistence of a low energy dissipation with a strong substantially nonlinear cell–cell response.

Figure 6 shows the dependences of the specific heat release on the vibronic PKS coupling parameter, calculated for tetrameric cells at $\hslash \omega =1605\hspace{0.33em}\mathrm{c}{\mathrm{m}}^{-1},$ two values of u shown in the plots and the following two values of $t_{eff}$for each u: $t_{eff}=100\hspace{0.33em}\mathrm{c}{\mathrm{m}}^{-1}$ (dashed lines) and $t_{eff}=50\hspace{0.33em}\mathrm{c}{\mathrm{m}}^{-1}$ (solid lines). It is seen from Figure 6 that the heat release W increases with increasing υ, provided that u and $t_{eff}$ are fixed, and also W increases with decreasing $t_{eff}$ for fixed u and υ values. In all cases, W tends to the maximum value $W_{max}=2u$, which is equal to the kink energy. It also follows from Figure 4 that for fixed $t_{eff}$ and υ, the heat release increases with increasing u.

It should be noted that exactly the same dependencies as in Figure 6 are obtained for bidimeric cells with two times larger $t_{eff}$ values, namely for $t_{eff}=100\hspace{0.33em}\mathrm{c}{\mathrm{m}}^{-1}$ (solid lines) and $t_{eff}=200\hspace{0.33em}\mathrm{c}{\mathrm{m}}^{-1}$ (dashed lines). This result is a consequence of the fact that in the limit of strong Coulomb interaction, the only difference between the Hamiltonians describing the two types of molecular cells is the factor in the matrix describing the effective second-order electron transfer, which is equal to 2 for a bidimeric cell and 4 for a tetrameric cell. Therefore, for a bidimeric cell, as in the case of a tetrameric cell, the specific heat release is a decreasing function of the parameters $t_{eff}$ and υ and an increasing function of the intercell Coulomb energy u.

Finally, Figure 6 allows us to compare the dependences of W vs. υ evaluated in the low-temperature limit for the bidimeric and tetrameric cells with the same values $\hslash \omega =1605\hspace{0.33em}\mathrm{c}{\mathrm{m}}^{-1} \mathrm{a}\mathrm{n}\mathrm{d}{t}_{eff}=100\hspace{0.33em}\mathrm{c}{\mathrm{m}}^{-1}$. One can see that for all the same parameters, the heat release is lower for the tetrameric cell. This is explained by the fact that in a tetrameric cell, there are four electron transfer pathways (along the four sides of the square), while in a bidimeric cell, there are only two such pathways (along the two dimeric half-cells). For this reason, the contribution of effective transfer in the tetrameric cell turns out to be twice as large as in the dimer ($4t_{eff}$ against $2t_{eff}$), and, consequently, the tetrameric cell is characterized by a higher degree of electronic delocalization. It is seen that in the limit of strong vibronic coupling, the difference in the heat releases for the two kinds of cells disappears, since the effective transfer in both types of cells in this limit is almost completely suppressed. In contrast, in the case of moderate vibronic coupling, the heat release for a tetrameric cell can be significantly lower than that for a bidimeric cell.

The results obtained would seem to suggest that in order to reduce energy dissipation, one should use weakly interacting tetrameric or bidimeric cells characterized by weak vibronic coupling and or/strong electron transfer, and also that a tetrameric cell should exhibit better functional properties than a bidimeric cell. Below, we show that this statement is erroneous because it is based exclusively on the requirement of low energy dissipation and does not take into account the other key requirement for the molecular cell, that is the strong nonlinear response of the working cell.

When both of these requirements are considered simultaneously, one immediately arrives at the conclusion that the use of MV molecules with weak vibronic interaction and/or strong electron transfer as cells or half-cells is counterproductive. In fact, the MV molecules exhibiting such properties are strongly delocalized (systems belonging to the Robin and Day class III), and therefore they do not have their own electric dipole (or quadrupole) moment in a minimum of a single-well adiabatic potential. Therefore, a very high value of energy u is required to polarize such a cell, which, in turn, leads to a dramatic increase in heat release. In contrast, using weakly delocalized (strong vibronic coupling and/or weak electron transfer) In this case, MV molecules acting as cells or half-cells ensures a strongly nonlinear cell–cell response, even providing for a weak intercell Coulomb interaction (small u). Therefore, one can expect that the heat release could be relatively low even if it is close to the upper limit $W=2u$.

Also, a bidimeric cell should have an advantage over a tetrameric cell, since for a bi-dimeric cell, a strong nonlinear response can be achieved at lower values of $u$, which allows the heat release to be significant reduced.

The reasoning presented so far is essentially qualitative in nature and indicates only the range of parameters in which the required functional properties of the cells could be expected. The quantitative results confirming the advantages of weakly delocalized molecules as cells are shown in Figure 7a,b. These figures present the low-temperature cell–cell response functions $P_{wc} \mathrm{v}\mathrm{s}. P_{dc}$calculated for the tetrameric cells characterized by different sets of parameters and the corresponding specific heat release values W evaluated with these sets of parameters.

Figure 7a shows the cell–cell response functions calculated for $\hslash \omega =1605\hspace{0.33em}\mathrm{c}{\mathrm{m}}^{-1}\mathrm{a}\mathrm{n}\mathrm{d}\upsilon =3500\hspace{0.33em}\mathrm{c}{\mathrm{m}}^{-1}$ and three different sets of u and $t_{eff}$. Comparison of the three cases in Figure 6 shows that although the lowest heat release is achieved at $u=10\hspace{0.33em}\mathrm{c}{\mathrm{m}}^{-1}$ and $t_{eff}=200\hspace{0.33em}\mathrm{c}{\mathrm{m}}^{-1}$ ($W\approx 15.9\hspace{0.33em}\mathrm{c}{\mathrm{m}}^{-1}$), that is, with a minimum value of u and a maximum value of $t_{eff}$ (within the parameter sets used in Figure 7a), the cell–cell response in this case is rather weak and the function $P_{wc}\left(P_{dc}\right)$ has a slight slope in a wide range of $P_{dc}$. This is definitely an unfavorable factor from the point of view of the functioning of QCA devices. To ensure a strong nonlinear cell–cell response at such large values of the effective transfer parameter, it is necessary to significantly enhance (by approximately five times) the intercell Coulomb interaction (in Figure 7a, this is the case when $u=50\hspace{0.33em}\mathrm{c}{\mathrm{m}}^{-1}$ and $t_{eff}=200\hspace{0.33em}\mathrm{c}{\mathrm{m}}^{-1}$), but such a strong increase in u leads to a sharp (up to $W\approx 98.1\hspace{0.33em}\mathrm{c}{\mathrm{m}}^{-1}$) increase in heat release. In contrast, for $u=10\hspace{0.33em}\mathrm{c}{\mathrm{m}}^{-1}$ and $t_{eff}=50\hspace{0.33em}\mathrm{c}{\mathrm{m}}^{-1}$ (weak intercell interaction and weak effective transfer), the cell–cell response function has a highly nonlinear stepwise shape (red curve in Figure 7a), and at the same time, the corresponding heat release value $W\approx 19.7\hspace{0.33em}\mathrm{c}{\mathrm{m}}^{-1}$ turns out to be only slightly higher than that found at $u=10\hspace{0.33em}\mathrm{c}{\mathrm{m}}^{-1}$ and $t_{eff}=200\hspace{0.33em}\mathrm{c}{\mathrm{m}}^{-1}$. One can thus conclude that the optimal rational strategy to design highly efficient molecular QCA cells is using weakly interacting MV tetramers characterized by weak electron transfer.

Figure 7b shows the cell–cell response functions calculated at $\hslash \omega =1605\hspace{0.33em}\mathrm{c}{\mathrm{m}}^{-1} \mathrm{a}\mathrm{n}\mathrm{d}{t}_{eff}=50\hspace{0.33em}\mathrm{c}{\mathrm{m}}^{-1}$ and three different sets of $u$ and $\upsilon$. Although the lowest heat release occurs when the intercell interaction and the vibronic interaction are both relatively weak (when $u=10\hspace{0.33em}\mathrm{c}{\mathrm{m}}^{-1}$ and $\upsilon =3000\hspace{0.33em}\mathrm{c}{\mathrm{m}}^{-1}$, for which we obtain $W\approx 17\hspace{0.33em}\mathrm{c}{\mathrm{m}}^{-1}$), the cell–cell response is weak and exhibits an almost linear slope, which makes such a parametric regime unsuitable for the proper functioning of QCA devices. To ensure a strong cell–cell response at such a weak vibronic coupling, it is necessary, as in the case considered above, to significantly enhance the intercell Coulomb interaction (case of $u=40 \mathrm{c}{\mathrm{m}}^{-1}$ and $\upsilon =3000 \mathrm{c}{\mathrm{m}}^{-1}$ in Figure 7b), but this leads to a sharp (up to $W\approx 79\hspace{0.33em}\mathrm{c}{\mathrm{m}}^{-1}$) increase in the heat release. The favorable regime is that of weak intercell interaction and strong vibronic coupling (case of $u=10\hspace{0.33em}\mathrm{c}{\mathrm{m}}^{-1}$ and $\upsilon =3500\hspace{0.33em}\mathrm{c}{\mathrm{m}}^{-1}$ in Figure 7b), because in this case both the nonlinearity of the cell–cell response function and the relatively low heat release of $W\approx 19.7\hspace{0.33em}\mathrm{c}{\mathrm{m}}^{-1}$ prove to be attainable (the red curve in Figure 7b, which coincides with red curve in Figure 7a, since both these plots are obtained with the same set of parameters). Therefore, it follows from Figure 7b that weakly interacting tetramers characterized by strong vibronic coupling are promising for use as cells.

Based on the observations following from Figure 7a,b, one can conclude that the high efficiency of tetrameric molecular QCA cells in which the limit of strong Coulomb interaction holds can be achieved for weakly delocalized MV molecules which are allowed to only weakly interact with each other. The first requirement can be satisfied provided that the transfer of the excess charge is weak and/or the vibronic coupling is strong, while in order to meet the second requirement, the tetramers should be located at long distances from each other and/or separated by groups of atoms capable to strongly shield the electric field.

The same is true for the bidimeric cells in the limiting case of strong Coulomb interaction, since they differ from the tetrameric cells only in the factor appearing in the effective transfer matrix. Moreover, the results obtained allow us to conclude that (ceteris paribus) the bidimeric molecular cells can have functional advantages over the tetrameric ones. This is explained by the weaker electron delocalization in the bidimeric cell, which means that strong nonlinear response for such a cell is achieved at lower values of u, which, in turn, helps to reduce the heat release.

3.2. Consideration of Bidimeric and Tetrameric Cells: Common Case of Arbitrary Coulomb Interaction

In addition to the limiting case of strong Coulomb interaction, an actual issue is the general case of an arbitrary interrelation between the Coulomb energy gap U and the electron transfer parameter t in tetrameric and bidimeric cells. The calculation of heat release and cell–cell response in this case is based on solving more complex multimode vibronic problems such as a three-mode problem for a tetrameric cell and a two-mode problem for a bidimeric cell. Along with the complications in the vibronic problem, an expansion of the electronic basis is also required. Thus, for a tetrameric cell, the electronic basis includes six states corresponding to six possible distributions of the two excess electrons in an MV tetramer. In the case of a bidimeric cell, the electronic distributions with two excess electrons per dimeric half-cell should be excluded, and so the electric basis is reduced to only four states. Note that in this general case, the intercell Coulomb interaction cannot be described by a single parameter u; instead, to describe such an interaction, we used the distance c between the cells. Here, we use the parameters evaluated with the aid of ab initio quantum chemical calculations as representatives of the series of organic MV compounds, which includes oxidized norbornadiene and its polycyclic derivatives [41].

Figure 6 shows the dependence of the low-temperature specific heat release of a tetrameric cell on the vibronic interaction parameter evaluated at $t=1573\hspace{0.33em}\mathrm{c}{\mathrm{m}}^{-1},\mathrm{\hslash }\omega =1605\hspace{0.33em}\mathrm{c}{\mathrm{m}}^{-1},\mathrm{and} b=6.973 \AA$ and two different intercell distances c. It is seen that the heat release increases with the increase in $\upsilon$ and saturates in the limit of strong vibronic coupling. It also follows from Figure 8 that the higher the heat release, the shorter the distance between the cells, and the stronger the Coulomb interaction between them. One can also conclude (the corresponding plots are not shown) that the heat release decreases with increasing t. Qualitatively, these results coincide with those described above for the limiting case of strong intracell Coulomb interaction.

By comparing the cell–cell response functions calculated for a tetrameric cell with three sets of c and υ values (Figure 9), one can conclude that the most favorable case for the proper functioning of a cell is the case of strong vibronic interaction and a large intercell distance. This is the red curve in Figure 9, which corresponds to $c=30 \AA \mathrm{a}\mathrm{n}\mathrm{d} \upsilon =4932 \mathrm{c}{\mathrm{m}}^{-1}.$ Only in this case does the cell demonstrate a strong nonlinear response in combination with a relatively low heat release ($W\approx 22 \mathrm{c}{\mathrm{m}}^{-1}$). Indeed, although in the case of weaker vibronic coupling $\upsilon =4000 \mathrm{c}{\mathrm{m}}^{-1}$ and the same c value, even lower heat release is obtained, the cell–cell response in this case turns out to be weak and has a slight slope (almost linear behavior). To improve the cell–cell response function while retaining this υ value, it is necessary to significantly bring the cells closer together, but such an approach of the cells leads to a significant (more than an order of magnitude) increase in the heat release. Calculations also show that cells separated from each other by long distances and characterized by small values of the parameter t should demonstrate a strong nonlinear response in combination with low heat release. Qualitatively, these conclusions coincide with those made for the case of strong intracell Coulomb interaction. Note that the conclusions made for tetrameric cells based on the analysis of the dependencies presented in Figure 8 and Figure 9 are also valid for the bidimeric cells with a violated limit of strong Coulomb interaction.

All of the regularities discussed so far are common to all types of cells and valid not only in the limiting case of strong Coulomb interaction but also in a general case of an arbitrary interrelation between the parameters t and U. Besides these common features, bidimeric cells with a violated limit of strong Coulomb interaction also have important specific characteristics, arising from the fact that the intercell Coulomb interaction in this case proves to be dependent on the mutual arrangement of the interacting cells, which can be of either “side-by-side” or “head-to-tail” type, as discussed earlier (see Figure 3). The influence of the relative arrangement of bidimeric cells on their polarization characteristics was studied in detail in Ref. [34]. In this regard, it seems useful to analyze the influence of the relative arrangement of such cells on the energy dissipation under nonadiabatic switching conditions.

The results of calculations of the specific heat release for the two types of mutual arrangement of bidimeric cells are presented in Figure 10. One can see that the heat dissipation increases when increasing the vibronic coupling parameter, which is a common feature of all types of cells discussed so far. It also follows from Figure 10 that, all other things being equal, the heat release is lower for the head-to-tail-type arrangement of cells. This observation correlates rather well with the shapes of calculated cell–cell response functions shown in Figure 11, from which it follows that the saturation cell–cell response value is lower for the head-to-tail-type arrangement. The difference in specific heat release for the two arrangements decreases with the increase in the strength of the vibronic coupling, and fully disappears in the limit of strong vibronic coupling or, in other words, in the saturation region. Note that in this limit, the difference in response functions also disappears (see Ref. [37]), and at the same time, the cell–cell response function, which is the same for two mutual arrangements of cells, takes on a highly nonlinear stepwise shape. Thus, under the condition of strong vibronic coupling, both important functional characteristics of the cells (heat release and cell–cell response) become independent of their mutual arrangement.

The latter result seems to be quite appealing from the point of view of the use of the bidimeric cells for building up QCA gates and circuits. In fact, almost all QCA devices are usually built in such a way that their architecture includes both types of mutual arrangements of the constituent cells, which are desired to be equivalent in terms of their functional characteristics (cell–cell response functions, heat release, etc.). Therefore, the presence of a strong vibronic interaction in a bidimeric cell makes it possible to simultaneously ensure the fulfillment of the two key conditions required for the proper functioning of the QCA devices, namely to obtain a strong nonlinear cell–cell response even for a weak intercell interaction, ensuring low energy dissipation, and also to eliminate undesired effects of the non-equivalence of the two types of cell arrangement.

4. Some Relevant Ab Initio Results and Possibility of Controlling the Key Parameters

The above theoretical consideration and conclusions are formulated within the framework of a semi-empirical approach dealing with adjustable parameters which are to be evaluated separately. In this section, we go beyond the parametric theory and briefly discuss the results of ab initio calculations of the key parameters such as electron transfer parameter t and the vibronic coupling parameter υ, while the intercell distance remains variable. The evaluation of these parameters and identification of their microscopic origin is necessary not only to identify the real areas of their changes that critically affect the heat release and cell–cell response, but also to reveal the possibilities for their rational control. The discussion below is inspired by the regularities recently found for the series of organic MV molecules in which either t or υ has been shown to vary in a systematical way within each such series [41,42].

The first such series includes MV oxidized norbornadiene [C₇H₈]⁺ and four of its polycyclic derivatives, [C₁₂H₁₂]⁺, [C₁₇H₁₆]⁺, [C₂₇H₂₄]⁺, and [C₃₂H₂₈]⁺ (compounds I–V in Ref. [41]). Each MV molecule in this series comprises two terminal C=C chromophores (redox sites) connected by a bridge, which mediates the intramolecular charge transfer [41]. The accuracy of the description of the vibronic and electron transfer parameters and consequently the degree of localization (and even possibility of trapping) in MV molecules crucially depend upon the level of the quantum-chemical approach employed. In particular, the single-configuration diabatic wave function could not provide a satisfactory description of the system. In particular, single-configuration methods are inadequate for the calculation of radical cations, because they lead to an adequate condition for charge-delocalized structures (see Ref. [41] references therein). The quantum-chemical study of norbornylogous compounds I–V in [41] included evaluation of the tunneling energy gap and determination of the fully optimized molecular structures. In this way, the barrier between the two with localized structures has been found along with Mulliken charges in symmetric delocalized and localized configurations. The CASSCF methodology [42,43] (GAMESS program suite [44,45]) has been used to study the localization the charge in I–V norbornylogous cation radicals. Numerical vibrational frequency calculations have been performed for the systems I, II and III using the GAMESS program. The cc-pVDZ basis set was used for calculations of I–III and the 6-311G* basis set was used for calculations of IV and V. High-level ab initio calculations combined with the parametric description within the vibronic model have revealed a quite weak dependence of the vibronic parameter and frequency of the vibrations associated with the redox sites on the length of the bridge, which justifies the applicability of PKS model for treating of these compounds. Unfortunately, we do not have the results of ab initio calculations for the tetrameric square-planar molecular cell for QCA representing the tetraruthenium Creutz–Taube derivatives mentioned so far. Due to the presence of a bridge connecting the Creutz–Taube dimers, the electronic properties and vibronic parameters of the tetraruthenium units are expected to differ significantly from those of the dimeric systems. That is why in this article, we limit ourselves to considering norbornylogous compounds, leaving the important case of tetraruthenium systems for detailed consideration in the future.

One can combine two identical MV dimers (e.g., two [C₁₂H₁₂]⁺ (I) and [C₁₇H₁₆]⁺ (II) molecules, as shown in Figure 12a,b) to obtain bidimeric square cells. One can adopt a reasonable assumption that the values of t and υ parameters are the same in the isolated dimeric entities and in the bidimeric cell combined of the dimeric units. In view of the results of the present study, one can conclude that the cells composed of dimers (in series I–V [41]) with longer bridges have greater polarizability than those consisting of dimers with shorter bridges even with smaller kink energy u, which, in turn, allows the lower power dissipation of such cells. This example shows the possibility of controlling the transfer parameter by modifying the bridge connecting the redox sites without changing the vibronic coupling.

Another series of organic MV dimers includes the radical cation forms of the parent 1,4-diallyl-butane (the system initially proposed by Lent et al. for molecular implementation of quantum cellular automata [1]) and its derivatives such as 1,4-diallyl-butene-2 (II), 1,4-diallyl-hexane (III), 1,4-diallyl-hexene-3(IV), 1,4-diallyl-octane (V) and 1,4-diallyl-octene-4 (VI) (numbered from I to V according to Ref. [46] containing a detailed quantum-chemical study of these compounds). In these MV compounds, the bridge of variable length connecting allyl groups playing the role of redox sites can be either saturated or unsaturated and modified by the π-spacer [46]. As distinguished from the series of oxidized norbornadiene derivatives in which one can control the electron transfer parameter t through variation of the bridge length, in the series I–V based on 1,4-diallyl-butane, one can modify the vibronic coupling υ [46]. In fact, the parameter υ strongly depends on the electronic state of the bridge being essentially different in compounds with saturated and unsaturated bridges. This dependence has been explained by polarization of the π-chromophore on the bridge by the excess charge at the redox sites, which has been referred to as the polaronic mechanism of localization [46]. In view of the present study, this means that bidimeric cells (Figure 12a,b) composed of the dimers with saturated bridges (Figure 12b) are preferable in the design of a bidimeric cell, since they require lower kink energy 2u for their efficient polarization and so one can expect lower heat release for such cells.

Summarizing the results emanating from the ab initio consideration of the two families of MV organic compounds, one can see that the key parameters of MV systems used for the design of the QCA cells can be varied in a controllable manner, providing options to optimize the heat release and the shape of the cell–cell response. By changing either the length of the bridge or its electronic structure in constituent dimeric half-cells, one can tune polarization and dissipative properties of bidimeric cells.

5. Conclusions

In this work, we have generalized the vibronic approach to calculating the specific heat release during nonadiabatic switching, previously proposed for the simplest case of dimeric molecular cells, to the case of more complex and practically promising square-planar bidimeric and tetrameric cells. Using the general expression, Equation (8), we have evaluated the specific heat release based on previously found quantum-mechanical solutions of the vibronic problems, which are specific to different types of cells and also dependent on the approximations adopted in the course of their description. This expression is applicable to estimating temperature-variable specific heat release in bidimeric and tetrameric cells (or, more precisely, its upper limit) as a function of the electronic and vibronic parameters of the cell, as well as on the energy of the intercell Coulomb interaction.

First, based on the developed general approach, specific heat release and cell–cell response functions have been evaluated for molecular QCA based on bidimeric (composed of $\mathrm{d}¹-\mathrm{d}⁰$-type MV dimers or organic MV compounds playing the role of half-cells) and tetrameric (represented by $\mathrm{d}¹-\mathrm{d}¹-\mathrm{d}⁰-\mathrm{d}⁰$-type clusters or organic MV compounds) two-electron square-planar cells, for which an important limiting case occurs, when the energy of the intracell interelectronic Coulomb repulsion significantly exceeds the electron transfer energy. The description of such cells has been performed with the aid of a quantum-mechanical solution of the vibronic problem arising in the framework of the vibronic PKS model, with this problem being significantly simplified in the limit of strong Coulomb interaction, reducing to a single-mode problem. Based on solving this problem, it has been shown that specific heat release in bidimeric and tetrameric cells increases with increasing vibronic coupling and/or weakening of electron transfer, reaching in the limit of strong vibronic coupling and/or weak transfer its maximum value equal to the Coulomb excitation energy of the pair of interacting cells. It has also been shown that, provided that the values of all parameters are the same, the heat release for a tetrameric cell is lower than for a bidimeric one. We have explained this difference by the fact that in a tetrameric cell, there are four electron transfer pathways (along the sides of the square), while in a bidimeric cell, there are only two such pathways (one within each dimeric half-cell). As a result, the tetrameric cell is characterized by a higher extent of the electronic delocalization, which leads to lower energy dissipation in the tetrameric cell. It follows from the results obtained that this difference disappears in the limit of strong vibronic coupling. This can be explained by the fact that in this limit, the effective electron transfer in both types of cells is almost fully suppressed.

Also, specific heat release in the course of nonadiabatic switching has been calculated for tetrameric and bidimeric cells in the general case of an arbitrary relationship between the energies of intracell Coulomb interaction and the electron transfer. These calculations are based on the numerical solutions of the three-mode vibronic problem for a tetrameric cell and two-mode problem for a bidimeric cell. Since, unlike the strong Coulomb interaction limit, in the general case the intercell Coulomb interaction cannot be described by using a single parameter, the intercell distance has been used to describe such interaction. It has been shown that as the vibronic coupling parameter increases, the heat release increases and reaches saturation in the limit of strong vibronic coupling, and the higher the heat release, the shorter the distance between the cells, i.e., the stronger the Coulomb interaction between them. Qualitatively, these results coincide with the results obtained in the limit of strong Coulomb interaction. In addition to these regularities, which are common to all types of cells, bidimeric cells with a violated limit of strong Coulomb interaction have been shown to possess some specific behavioral features arising from the dependence of the intercell Coulomb interaction on the mutual arrangement of the interacting cells, which can be either head-to-tail or side-by-side. We have demonstrated that, all other things being equal, heat release is lower in the case of a head-to-tail arrangement of cells, and the difference in specific heat release for the two arrangements decreases with increasing vibronic coupling, completely disappearing in the strong vibronic coupling limit. Therefore, bidimeric cells with a violated strong Coulomb interaction limit and strong vibronic coupling are equivalent to cells for which the strong Coulomb interaction limit is satisfied, since in both these cases, the mutual arrangement of the cells does not affect the heat release. This result seems to be important from the point of view of the rational design of QCA devices, as far as different types of cells’ arrangement in such devices should be equivalent in terms of their functional characteristics, including energy dissipation.

Based on the developed vibronic approach aimed at the evaluation of the specific heat release in the course of nonadiabatic switching as well as the previously developed approach to calculating vibronic cell–cell response functions, a theoretical methodology for optimizing functional properties of cells of various types has been proposed. This methodology involves the procedure for establishing the parametric regimes, which provide low heat release in combination with a strong nonlinear response of the working cell to the electric field created by a polarized driver-cell. During the search for such optimal regimes, both the electronic and the vibronic parameters of the cell and also the intercell Coulomb interaction energies dependent on the distances between the cells have been varied. The use of the proposed methodology allows us to formulate criteria for the rational design of QCA based on the molecular cells. Thus, it has been shown that the optimal design strategy is to use weakly interacting molecular cells (cells located at long distances from each other or separated by the groups of atoms which can strongly screen the electric field) exhibiting weak delocalization of the excess charges. Suitable systems to satisfy the condition of weak delocalization are MV molecules characterized by weak electron transfer and/or strong vibronic coupling.

A comparative analysis of the efficiency of the two types of square-planar cells has been carried out from the point of view of the possibility of minimizing heat release, while maintaining strong nonlinear cell–cell response. We have thus concluded that bidimeric cells have an advantage over tetrameric cells. This advantage follows from the fact that, due to the weaker delocalization of the excess electrons in the bidimeric cells, a strong nonlinear response for such cells can be achieved with weaker intercell interactions as compared with tetrameric cells, which makes it possible to significantly reduce the heat release. Although the bidimeric cells are advantageous over tetrameric cells provided that the two kinds of mutual arrangements of the bidimeric cells (head-to-tail and side-by-side) are equivalent in terms of their functional properties (cell–cell response and heat release), such equivalence does not contradict the main criterion formulated above. Indeed, its fulfillment also requires the presence of weak electronic delocalization in the molecule.

Finally, we have discussed some relevant ab initio results of the study of electron transfer and vibronic coupling in view of the possibility of controlling the key parameters responsible for the functioning of molecular QCA cells. We elaborate upon the computer-aided methodology to control the electron transfer and the vibronic coupling, which are the interactions governing the heat release and the cell–cell response in molecular QCA.

The above-proposed methodology for optimizing the properties of molecular cells is based on the two key requirements that such cells should meet, namely low energy dissipation and a strong nonlinear cell–cell response. Along with these, there are other important conditions for the applicability of molecules as QCA cells, such as their ability to perform clocking [3,47,48,49,50,51,52,53], a special disposition of the counterion preventing violation of the high symmetry of the cell [15,16,17], and finally, the ability of the molecule to be attached to a substrate without breaking down its required properties (see, e.g., [18]). Elaboration of a more general approach to optimizing the properties of molecular cells, which takes into account all aforenamed criteria, as well as the synthesis of suitable molecular candidates, is still a challenge in the design of high-performance molecular QCA.