Double-Dimeric Versus Tetrameric Cell for Quantum Cellular Automata

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C: Physical Processes in Nanomaterials and Nanostructures

Double-Dimeric Versus Tetrameric Cell for Quantum Cellular Automata: Semiempirical Approach to Evaluation of Cell-Cell Response Combined with Quantum-Chemical Modeling of Molecular Structures Andrew V. Palii, Shmuel Zilberg, Andrey Rybakov, and Boris Tsukerblat J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.9b05942 • Publication Date (Web): 16 Aug 2019 Downloaded from pubs.acs.org on August 27, 2019

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The Journal of Physical Chemistry

Double-Dimeric Versus Tetrameric Cell for Quantum Cellular Automata: Semiempirical Approach to Evaluation of Cell-Cell Response Combined with Quantum-Chemical Modeling of Molecular Structures Andrew Palii,*a,b Shmuel Zilberg,c Andrey Rybakov,d Boris Tsukerblat*c,e aInstitute

of Problems of Chemical Physics, Chernogolovka, Moscow Region, Russian Federation bInstitute of Applied Physics, Chisinau, Moldova cDepartment

of Chemical Sciences, Ariel University, Ariel, Israel dMoscow Institute of Physics and Technology, Institutskii per. 9, Dolgoprudny, Moscow Region, Russian Federation eDepartment

of Chemistry, Ben-Gurion University of the Negev, Beer-Sheva, Israel

*e-mail: [email protected] (A.P.); [email protected] (B.T.) Abstract Quantum dot cellular automata is a computing paradigm based on transistor-free logic, which in turn relies on the idea to encode binary information in the bistable charge conﬁgurations of quantum dots and process information via Coulomb interaction. In context of molecular implementation of quantum dot cellular automata, we have compared the properties of two possible kinds of molecular square cells, namely, the cell tailored from two one-electron mixed valence dimers (double-dimeric cell) and two-electron mixed valence tetramer. The physical model (based on the Hubbard-type Hamiltonian) of the cells involves the Coulomb interelectronic interaction, electron transfer and vibronic coupling. We have demonstrated that the difference in the transfer pathways in the two types of cells gives rise to a considerable difference in their functional characteristics. Thus, the double-dimeric cell exhibits more abrupt nonlinear cell-cell response which is a prerequisite for the efficient functioning of quantum cellular automata. The difference in the cell-cell responses for the two kinds of cells is shown to be smaller for a weak electron transfer and/or strong vibronic coupling when the mobility of the electronic pair is strongly constrained. The dimeric and tetrameric systems, 1,4-dithia-hexane and crown ether 1,4,7,10-tetrathiocyclododecane were selected as the molecular systems for the implementation of proposed Hubbard type analysis. This choice is prompted by the fact of the positive charge localization on the S-atoms which are not connected covalently. We have performed the quantum-chemical calculations of the 1,4-dithia-compound with two S-atoms connected by saturated carbon bridge CH2CH2 (proposed as a dimeric subunit) and the corresponding tetrameric structures of the crown ethers 1,4,7,10-tetrathio-cyclododecane - parent neutral molecule, cation and dication. The 1 ACS Paragon Plus Environment

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quantum-chemical estimations allowed us to quantitatively unveil the key parameters of the dimeric and tetrameric systems and to conclude that the proposed compounds can serve as the cells with predominantly antipodal charge separation which potentially are able to encode binary information. 1. Introduction Quantum cellular automata (QCA)1-9 became a fascinating multidisciplinary field of nanotechnology which has grown from the idea of Lent et al.1-3 to use the quantum-dot cells coupled via Coulomb interaction to form cellular automata architecture. The QCA have been realized at micron scale as arrays of quantum-dot cells deposited on a silicon substrate. Each cell typically consists of four dots situated in the vertices of a square and two extra electrons (or holes) tunneling between these dots. The binary information in QCA is encoded in the two antipodal localizations of charges within the square cell for which the Coulomb repulsion is minimal. Such charge configurations correspond to binary 1 and 0 associated with polarizations P = 1 and P = 1 of the cell. The binary information is transmitted through the intercell Coulomb interaction which occurs due to electric quadrupole moments of the cells in the two antipodal position of mobile charges. Then, different kinds of logical devices such as wires, fun-out devices, invertors, majority gates, etc. can be composed of the two-electron square cells. As distinguished from the conventional devices based on the field effect transistors, the transistorless QCA devices consume small amount of electrical power, the heat release in QCA is also extremely small, and so these devices can be in principle used to perform the computations at very high switching rate. In course of the development towards further miniaturization of devices, a proposal has emerged to use a single molecule to implement as a QCA cell.10-13 This can be done, for example, by using a suitable mixed valence (MV) molecule containing two mobile electrons or holes.13-25 The role of the quantum dots in such molecule is played by the redox sites, which are linked by the bridging ligands mediating the electron tunneling. It was proposed11 that the molecular square cell can be composed either from the two dimeric units each containing single mobile charge which can be referred to as half-cell 13, 15-17, 20, 25 or represented by a square tetrameric MV molecule with two extra electrons as a ready cell (full-cell).21-24 In this article we report the results of a comparative study of the properties of the two kinds of cells paying attention to such key characteristics of the interacting cells as the dependences of the energy levels, charge densities and polarization of the cell on the polarization of the driver. The last is known as the cell-cell response function which should be essentially non-linear to ensure fast response of the working cell to a small change of the polarization of the driver cell. These characteristics are evaluated using both the electronic model of the cell and within a more 2 ACS Paragon Plus Environment

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comprehensive vibronic model which accounts for the interaction of the mobile electrons with the molecular vibraions. Along with the semiempirical study based on the Hubbard-type Hamiltonian we describe the bottom-up design of the new molecular systems possessing mobile charges which are able to encode binary information and illustrate the transformation of the parameters while passing from the half-cell to full cell. In this respect the quantum-chemical analysis of the of 1,4-dithiacompound with two S-atoms connected by saturated carbon bridge CH2CH2 is performed. This molecule is proposed as a dimeric subunit of the double-dimeric cell.

The corresponding

tetrameric cells are modeled by the structures of the crown ethers 1,4,7,10-tetrathio-cyclododecane as parent neutral molecule, cation and dication radicals. Quantum-chemical evaluations allowed us to quantitatively unveil the key parameters of the two kinds of molecular cells and to find out how the hole transfer parameters change when combining two dimers into a tetramer. Along with the semiempirical study based on the Hubbard-type Hamiltonian we describe the quantum-chemical analysis of the of 1,4-dithia-compound with two S-atoms connected by saturated carbon bridge CH2CH2. This molecule is proposed as a dimeric subunit of the doubledimeric cell. The corresponding tetrameric cells are modeled by the structures of the crown ethers 1,4,7,10-tetrathio-cyclododecane as parent neutral molecule, cation and dication radicals. Quantum-chemical evaluations allowed us to quantitatively unveil the key parameters of the two kinds of molecular cells and to find out how hole transfer parameters change when combining two dimers into a tetramer. We concluded also that in the proposed compounds an efficient antipodal charge separation (resulting in the low-lying antipodal charge configurations) occurs which is a prerequisite for using them as molecular cells for QCA. 2. The model for molecular cells The two kinds of cells are schematically shown in Figure 1 with indication of the two distinct polarizations of the cell (P = 1 and P = 1) and electron transfer (parameter t) pathways.

Figure

1a shows two isolated dimeric units (half-cells) combined into a square cell which we will refer to as double-dimeric cell, while Figure 1b shows a true bi-electronic tetrameric cell. In the first case the electron jumps are restricted to the constituent dimeric units, while in the tetrameric cell all transfer pathways are allowed. At the same time the Coulomb repulsion energies between the charges occupying neighboring (AB, AD, etc.) and antipodal (AC, BD) sites in these two cases are assumed to be identical. To get a rational possibility to compare the functional characteristics of the double-dimeric and tetrameric cells we will introduce the following simplifying assumptions: 1) we will take into account only the most efficient transfer along the sides (Figure

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1) and neglect the diagonal transfer assuming that the molecular structure of the cell does not comprise the diagonal bridges mediating the transfer; 2) we assume that merging double-dimeric

(a)

P = 1

P=1 (b)

Figure1. Two possible kinds of square cells for molecular QCA with indication of the allowed electron transfer pathways: (a) double-dimeric two-electron MV cell; (b) two-electron MV tetramer. cells into the tetrameric one does not affect the geometric parameters and the Coulomb repulsion; 3) the bi-center transfer parameters t are assumed to be the same in both cases as shown in Figure 1; 4) merging cells is assumed not to affect the vibronic parameters (Section 4). It is seen from Figure 1 that even if the diagonal transfer is neglected, the number of the transfer paths in the tetrameric cell proves to be two times larger as compared with that in the double-dimeric cell. Since the transfer paths in the cell should significantly affect its polarizability by the quadrupole Coulomb field induced by the neighboring cell acting as a driver, it is reasonable to expect that the functional characteristics of the molecule-based QCA should be substantially dependent on the type of the cell. 3. Comparison of the double-dimeric and tetrameric cells within the electronic approach 3.1 Isolated cells At the first step we leave out of consideration the interaction of the mobile charges with molecular vibrations (vibronic interaction). To treat the electronic problem, we use for a doubledimeric cell the basis set including the following four states: 4 ACS Paragon Plus Environment

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𝜓𝐴𝐶(𝑆), 𝜓𝐴𝐷(𝑆), 𝜓𝐵𝐶(𝑆), 𝜓𝐵𝐷(𝑆) ,

(1)

where 𝜓𝑖𝑗(𝑆) is the bi-electronic wave-function corresponding to localization of the electrons at the sites i and j and the total spin of the cell S (𝑆 = 0, 1). The electronic basis for the tetrameric cell is larger and comprises the following six states for each spin value: 𝜓𝐴𝐵(𝑆), 𝜓𝐴𝐶(𝑆), 𝜓𝐴𝐷(𝑆), 𝜓𝐵𝐶(𝑆), 𝜓𝐵𝐷(𝑆), 𝜓𝐶𝐷(𝑆).

(2)

The bi-electronic bi-center states 𝜓𝑖𝑘(𝑆,𝑀𝑆) are built from the Slater determinants containing localized non-degenerate orbitals 𝜑𝑖 (available for the extra electron) in a standard way: 𝜓𝑖𝑘(1,1) = |𝜑𝑖 𝜑𝑘| , 𝜓𝑖𝑘(0,0) = (|𝜑𝑖 𝜑𝑘| ― |𝜑𝑖 𝜑𝑘|) 2 ( 𝜑𝑖 ≡ 𝜑𝑖(↑) , 𝜑 ≡ 𝜑𝑖(↓)). The states with two electrons per site are excluded from the bases because they are much higher in energy due to strong on-site interelectronic Coulomb repulsion. The electronic matrix Hamiltonian for the double-dimeric cell does not depend on the spin S and is shown to have the following form: 𝑑𝑖𝑚 ― 𝑑𝑖𝑚

𝐻𝑒

𝑈 (𝑆) = 𝑡[(𝜎0⨂𝜎𝑥) + (𝜎𝑥⨂𝜎0)] + [(𝜎0⨂𝜎0) ― (𝜎𝑧⨂𝜎𝑧)] . 2

(3)

In Eq. (3) 𝜎𝑖 are the Pauli matrices, 𝜎0 is the unit 22-matrix, ⨂ is the symbol of the Kronecker product. In contrast, in the case of tetrameric cell the form of the matrix Hamiltonian does depend on S and looks as follows: 𝐻𝑒 (𝑆) = 𝑡[(𝜎0⨂𝜏𝑥) + (1 ― 𝑆)(𝜎𝑥⨂𝜏𝑥) + 𝑆(𝜎𝑦⨂𝜏𝑦)] +𝑈(𝜎0⨂𝜏2𝑧 ) . 𝑡𝑒𝑡𝑟

(4)

In Eq. (4) the notations 𝜏𝑥 = 2 𝐽𝑥 , 𝜏𝑦 = 2 𝐽𝑦, 𝜏𝑧 = 𝐽𝑧 , 𝐽𝑖 are used for the 33-matrices of the angular momentum operator for J=1. The electronic spin-dependent Hamiltonian in Eq. (4) combines both cases S=0 and S=1. In Eqs. (3) and (4) the value U is the difference between the Coulomb repulsion energy of the electrons occupying neighboring centers in the both kinds of systems (for example, the sites A and D) and Coulomb repulsion energy in antipodal position (sites AC and BD). The parameter U is calculated as:

𝑈=

𝑒2 𝑏

(1 ― ) , 1

(5)

2

where b is the shortest intracell distance (Figure 2). Consideration of the cells can be significantly simplified under the assumption of strong Coulomb repulsion, U >> t. Under this condition the electronic density is mostly concentrated at the antipodal sites (minimizing the Coulomb repulsion) that is just relevant for QCA application. Then, one can apply the second order perturbation procedure, that is to project the matrix Hamiltonians onto the restricted space of the two lowest in energy states 𝜓𝐴𝐶(𝑆) and 𝜓𝐵𝐷(𝑆). We 5 ACS Paragon Plus Environment


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thus pass from the exact Hamiltonian matrices in Eqs. (3) and (4) to the effective 22 matrices 𝑑𝑖𝑚 ― 𝑑𝑖𝑚

𝐻𝑒, 𝑒𝑓𝑓

𝑡𝑒𝑡𝑟

and 𝐻𝑒, 𝑒𝑓𝑓(𝑆). The elements of these effective matrices are calculated using the second-

order perturbation theory: ― 𝑑𝑖𝑚 ⟨𝜓𝐴𝐶(𝑆)|𝐻𝑑𝑖𝑚 |𝜓𝐵𝐷(𝑆)⟩ 𝑒, 𝑒𝑓𝑓 1

𝑑𝑖𝑚 ― 𝑑𝑖𝑚 ― 𝑑𝑖𝑚 |𝜓𝛼(𝑆)⟩⟨𝜓𝛼(𝑆)|𝐻𝑑𝑖𝑚 |𝜓𝐵𝐷(𝑆)⟩ = ― = ― 𝑈∑𝛼 = 𝐴𝐷,𝐵𝐶⟨𝜓𝐴𝐶(𝑆)|𝐻𝑒 𝑒

2𝑡2 𝑈,

(6)

Following this procedure, one obtains for the double-dimeric system the following matrix in the basis 𝜓𝐴𝐶(𝑆) 𝜓𝐵𝐷(𝑆): 𝑑𝑖𝑚 ― 𝑑𝑖𝑚 𝐻𝑒, 𝑒𝑓𝑓

2𝑡2 (𝜎 + 𝜎𝑥) . =― 𝑈 0

(7)

For the tetrameric cell the matrix Hamiltonians for 𝑆 = 0 and 𝑆 = 1 in the same basis sets are obtained in the following forms: 𝑡𝑒𝑡𝑟 𝐻𝑒, 𝑒𝑓𝑓(0)

𝑡𝑒𝑡𝑟

4𝑡2 (𝜎 + 𝜎𝑥) , =― 𝑈 0

𝐻𝑒, 𝑒𝑓𝑓(1) = ―

(8)

4𝑡2 𝜎 . 𝑈 0

(9)

The last equation describes only the stabilization of the spin-triplet level. 3.2. Coupled cells: polarization and cell-cell response Now let us consider the cells subjected to the field of the neighboring cell acting as a driver. As usually in the theory of QCA, we assume that the quadrupole Coulomb field induced by the neighboring polarized driver cell (cell 2), takes a predetermined value, or alternatively, the driver is considered as a source of external acting on the cell device (1). The disposition and the geometrical parameters of the two cells are shown in Figure 2. To describe the cell 1 one has to add to the energy matrices which describe the isolated cells (Eqs. (7), (8), (9)) the matrix of the operator 𝐻𝑐𝑒𝑙𝑙 ― 𝑐𝑒𝑙𝑙 describing the external quadrupole field. The form of this latter matrix is the same for the two kinds of cells. Using the basis 𝜓𝐴𝐶(𝑆) 𝜓𝐵𝐷(𝑆) one can find: 𝐻𝑐𝑒𝑙𝑙 ― 𝑐𝑒𝑙𝑙 = 𝑢𝜎𝑧 .

(10)

The field of the driver creates asymmetry in the electronic distribution in the device cell, so that the Coulomb energies of the electronic pairs occupying the AC and BD diagonals of the cell


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Figure 2. Mutual in-plane disposition of two interacting double-dimeric cells with indication of the geometrical parameters. The same parameters are adopted for the tetrameric cells. 1 become different. The parameter u in Eq. (10), which is proportional to the polarization 𝑃2 of the cell 2 is given by (see also ref. 26):

[

𝑢 = 𝑃2𝑒2

1 𝑏+𝑐

+

1 2 (2𝑏 + 𝑐)2 + 𝑏

2 +

1

2 ―

2 𝑏2 + 𝑐

1 (𝑏 + 𝑐)2 + 𝑏

1

1

]

. 2 ― 2(2𝑏 + 𝑐) ― 2𝑐

(11)

It is seen that the value 2u, which is the measure of the asymmetry, represents the energy gap between the Coulomb energies of the electronic pairs occupying the AC and BD diagonals of the cell 1. 𝑑𝑖𝑚 ― 𝑑𝑖𝑚

The energies of the double-dimeric cell 1 which are the eigenvalues of 𝐻𝑒, 𝑒𝑓𝑓

+ 𝐻𝑐𝑒𝑙𝑙 ― 𝑐𝑒𝑙𝑙 are

given by the expression: 𝐸± = ―

2𝑡2 𝑈

± 𝑢2 +

4𝑡4 𝑈2

(12)

.

Similarly, for the case of the tetrameric cell one finds the following eigen-values of the 𝑡𝑒𝑡𝑟

Hamiltonian 𝐻𝑒, 𝑒𝑓𝑓(𝑆) + 𝐻𝑐𝑒𝑙𝑙 ― 𝑐𝑒𝑙𝑙: 𝐸 ± (1) = ―

4𝑡2 𝑈

± 𝑢,

𝐸 ± (0) = ―

4𝑡2 𝑈

± 𝑢2 +

16𝑡4 𝑈2

.

(13)

It is seen from Eq. (13) that E  0   E 1 , and hence, the ground state of the tetrameric cell is always the spin-singlet. This is true for the both isolated cell and for the cell exposed to the action of the quadrupole Coulomb field of the neighboring cell. We will analyze the cell-cell response in the low-temperature limit when only the ground state is populated. For this reason, in the study of the tetrameric cell we will exclusively focus on the spin-singlet states. The energies, Eqs. (12), (13), and the corresponding wave-functions of the cell 1 depend on the polarization P2 of the driver due to the dependence of the intercell Coulomb energy u of P2. We assume that the driver possesses a given arbitrary polarization P2 falling in the range between 1 and +1. In all subsequent calculations we will use the value b =4 Å for the intracell distance 7 ACS Paragon Plus Environment


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Figure 3. Dependences of the electronic energy levels of the double-dimeric (solid line) and tetrameric with S=0 (dashed) cell 1 on polarization P2 of the driver calculated with b =4 Å, c =9 Å and t  500 cm 1 . for which 𝑈 ≈ 8501 cm ―1 (Eq. (5)) and the value c =9 Å for the intercell distance. The dependences of the energies of cell 1 on the on the polarization P2 of the driver calculated with

t  500 cm 1 for the two considered kinds of cells are shown in Figure 3. This value of t ensures the inequality U >> t required for the employed perturbational approach. At P2 = 0 the two levels of the double-dimeric cell are separated by the gap 4𝑡2 𝑈 ≈ 117.63 cm ―1. These two levels correspond to the wave-functions 𝜓 ± (𝑆) = (1 2)[𝜓𝐴𝐶(𝑆) ± 𝜓𝐵𝐷(𝑆)]. At strong polarization P2 the dependences of the energy levels on the polarization are practically linear (Figure 3, solid lines), and at P2  1 the energy gap between the two levels approximately reaches the value of 2|𝑢(𝑃2 =± 1)| ≈ 503.51 cm ―1. This means that at P2  1 the delocalization of the electronic pair is fully suppressed by the external quadrupole field and so the double-dimeric cell 1 proves to be fully polarized. Such full polarization corresponds to the ground state 𝜓𝐴𝐶(𝑆) for 𝑃2 = +1 and 𝜓𝐵𝐷(𝑆) for 𝑃2 = ― 1, while the excited state for 𝑃2 = +1 is 𝜓𝐵𝐷(𝑆) and for 𝑃2 = ― 1 is 𝜓𝐴𝐶(𝑆). For the tetrameric cell the effective transfer in the S = 0 state is twice larger than that for the double-dimeric cell, and so at P2 =0 the two states 𝜓 ± (0) = (1 2)[𝜓𝐴𝐶(0) ± 𝜓𝐵𝐷(0)] of the tetrameric cell 1 are separated by the second order transfer energy gap 8𝑡2 𝑈 ≈ 235.26 cm ―1. Such comparatively strong electron delocalization precludes from the polarization of the system by the field of the driver. Consequently, even at strong polarization P2 (close to the saturation values ± 1) the dependences of the energies on P2 are not linear (Figure 3, dashed lines), and the energy gap at

P2  1 exceeds the value 2|𝑢(𝑃2 =± 1)|. Thus, as distinguished from the


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double-dimeric cell the tetrameric cell 1 is still not fully polarized at 𝑃2 = ± 1, i.e. the electronic pair is not completely localized along a diagonal of the square cell 1. 𝑑𝑖𝑚 ― 𝑑𝑖𝑚

Using the eigenvectors of the energy matrices 𝐻𝑒, 𝑒𝑓𝑓

𝑡𝑒𝑡𝑟

+ 𝐻𝑐𝑒𝑙𝑙 ― 𝑐𝑒𝑙𝑙 and 𝐻𝑒, 𝑒𝑓𝑓(0) +

𝐻𝑐𝑒𝑙𝑙 ― 𝑐𝑒𝑙𝑙 one can calculate the polarization P1 induced in the cell 1 as a function of the polarization P2 of a driver. This is just the cell-cell response function 𝑃1 = 𝑓(𝑃2) that is a key functional characteristic of QCA. For the efficient action of a QCA device the cell-cell response function should be highly nonlinear to ensure very strong polarization of the working cell as a response to a small polarization of the driver cell. Within the employed perturbative approach, the polarization P1 of the cell 1 is calculated as follows: 𝜌𝐴𝐶 ― 𝜌𝐵𝐷

𝑃1 = 𝜌𝐴𝐶 + 𝜌𝐵𝐷 ,

(14)

where  AC and  BD are the probabilities (normalized electronic densities) of the two diagonal localizations of the electronic pair (  AC   BD  1 ) which can be found from the eigen-vectors for the ground state of the cell 1. Figure 4 presents the results of sample calculations of the cell-cell response functions for the two considered kinds of cells evaluated at the following two illustrative values of the transfer parameters:

t  500 cm 1 and t  200 cm 1 . It is important to note that the steepness (non-

linearity) of the 𝑃1 = 𝑓(𝑃2) dependence is higher for the double-dimeric cell. As distinguished from the double-dimeric cell, polarization P1 of the tetrameric cell calculated at t  500 cm 1 does not reach the saturation values 1 at P2  1 (red dashed line in Fig. 4) in accord with the above

Figure 4. Comparison of cell-cell response functions for the double-dimeric (solid lines) and tetrameric (dashed lines) cells calculated with b =4 Å, c =9 Å and t  500 cm 1 (red lines) and t  200 cm 1 (blue lines). 9 ACS Paragon Plus Environment


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discussion of the energy pattern. This is apparently due to the fact that the electron transfer in the double-dimeric cell is constrained within each constituent dimer, while in the case of the tetrameric cell all four transfer pathways are operative. As a result, the magnitude of the effective second order transfer parameter for the double-dimeric cell is twice smaller as compared with that for tetrameric cell. This gives rise to a lower sensitivity of the tetrameric cell to the quadrupole field induced by the driver. We thus arrive at the conclusion that the double-dimeric cell is more efficient for functioning of QCA as compared with the tetrameric cell. The slopes (effective degree of non-linearity) of cell-cell response curves for the two kinds of cells for small t (blue lines in Fig. 4) are only slightly different because in both cells the field of the driver is able to efficiently suppress the delocalization of the electronic pair. 4. Comparison of the two kinds of cells within the vibronic approach 4.1. Vibronic coupling: the model and Hamiltonian In view of the important role of the vibronic interaction in MV systems, in this Section we will generalize the electronic model so far discussed taking into account the vibronic effects in the description of the functional properties of molecule-based QCA

26-30.

The subsequent

consideration will be based on the Piepho, Krausz and Schatz (PKS) model,31 which involves the interactions of the mobile electrons with the full-symmetric local “breathing” displacements of the redox sites in a MV molecule. The dimensionless vibrational coordinates will be denoted by 𝑞𝐴, 𝑞𝐶 , 𝑞𝐵, 𝑞𝐷 and the frequency of the “breathing” vibration (the same for all sites) is denoted as  . We assume that the Coulomb energy U exceeds not only the transfer integral t but also the vibronic coupling and therefore, the vibronic interaction is assumed to act within the truncated basis of the low-lying electronic states 𝜓𝐴𝐶(𝑆), 𝜓𝐵𝐷(𝑆). A starting point of the PKS model is that the matrix elements of the vibronic coupling Hamiltonian 𝐻𝑣 in the case under consideration can be defines as ⟨𝜓𝐴𝐶(𝑆) │𝐻𝑣│𝜓𝐴𝐶(𝑆)⟩ = 𝜐

(𝑞𝐴 + 𝑞𝐶) , ⟨𝜓𝐵𝐷(𝑆) │𝐻𝑣│𝜓𝐵𝐷(𝑆)⟩ = 𝜐(𝑞𝐵 + 𝑞𝐷) . This form of the vibronic term reflects the key assumption of the PKS model according to which different sites are vibronically independent. Then the vibrationally-dependent part of the Hamiltonian of the cell can be defined as follows: 𝐻𝑜𝑠𝑐 + 𝐻𝑣 =

ℏ𝜔 ∑ 2 𝑖 = 𝐴,𝐵,𝐶,𝐷

(

∂2

)

𝜐

𝑞2𝑖 ― ∂𝑞2 𝜎0 + 2(𝜎0 + 𝜎𝑧)(𝑞𝐴 + 𝑞𝐶), 𝑖

(15)

where the first term (𝐻𝑜𝑠𝑐) represents the Hamiltonian for the free harmonic oscillations and the second term (𝐻𝑣) represents the linear (with respect to the vibrational coordinates) vibronic interaction. Note that the Hamiltonian, Eq. (15) is valid for both considered kinds of cells. Then, one can pass to the symmetry adapted coordinates 𝑞𝛼 = ∑𝑖𝛼𝑐𝑖𝛼𝑞𝑖 (𝑖 = 𝐴, 𝐵, 𝐶, 𝐷) of a cell 10 ACS Paragon Plus Environment

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corresponding to the irreducible representations (irreps) 𝛼 = 𝐴1𝑔, 𝐵1𝑔, 𝐸𝑢 of the point group 𝐷4ℎ as shown in Table 1. Table 1. Symmetry adapted PKS vibrations of a square cell in terms of local displacements. Vib. irreps

𝐴1𝑔

(

𝐵1𝑔

)(

) (

1111 , , , 2222

(𝑐𝑖𝛼)

𝐸𝑢𝑥

11 1 1 , ,― ,― 22 2 2

𝐸𝑢𝑦

)(

1 1 ,0, ― ,0 2 2

0,

)

1 1 ,0, ― 2 2

One can show that contributions of all vibrations except the 𝑞𝐵1𝑔mode are proportional to the unit matrix, and hence, they can be ruled out from the consideration. This conclusion has clear physical sense in terms of the basic issues of PKS model. In fact, as follows from Table 1, the 𝑞𝐵1𝑔coordinate can be represented as the out-of-phase combination 𝑞𝐵1𝑔 = (1 2)( 𝑞𝐴𝐶 ― 𝑞𝐷𝐵) of the two outof-phase vibrations 𝑞𝐴𝐶 = (1 2)(𝑞𝐴 ― 𝑞𝐶) and

𝑞𝐷𝐵 = (1 2)(𝑞𝐷 ― 𝑞𝐵) of the constituent

dimeric units. Each PKS vibration, 𝑞𝐴𝐶, 𝑞𝐷𝐵, is linked to the charge transfer in the corresponding dimeric subunits, so only the 𝑞𝐵1𝑔 vibration is relevant to the concerted bi-electron switching between antipodal localizations 𝐴𝐷↔𝐷𝐵 in the tetramer. One thus arrives at the one-mode vibronic problem with the following Hamiltonian defined in the basis 𝜓𝐴𝐶(𝑆) 𝜓𝐵𝐷(𝑆) : ℏ𝜔 2 ∂2 𝐻𝑜𝑠𝑐 + 𝐻𝑣 = (𝑞 ― 2)𝜎0 + 𝜐𝜎𝑧𝑞 . 2 ∂𝑞

(16)

where the notation 𝑞𝐵1𝑔 ≡ 𝑞 is used in Eq. (16) which is valid for both spin values. Then, the total Hamiltonians for the double-dimeric and tetrameric cells are the following: 𝑑𝑖𝑚 ― 𝑑𝑖𝑚

𝐻𝑑𝑖𝑚 ― 𝑑𝑖𝑚 = 𝐻𝑒, 𝑒𝑓𝑓

+ 𝐻𝑐𝑒𝑙𝑙 ― 𝑐𝑒𝑙𝑙 + 𝐻𝑜𝑠𝑐 + 𝐻𝑣 ,

(17)

𝑡𝑒𝑡𝑟

𝐻𝑡𝑒𝑡𝑟(0) = 𝐻𝑒, 𝑒𝑓𝑓(0) + 𝐻𝑐𝑒𝑙𝑙 ― 𝑐𝑒𝑙𝑙 + 𝐻𝑜𝑠𝑐 + 𝐻𝑣.

(18)

Using these Hamiltonians further on we will consider the adiabatic and quantum-mechanical picture of the energy patterns and cell-cell response functions. 4.2. Adiabatic picture First, we consider the vibronic problem within the adiabatic semiclassical approach neglecting the kinetic energy of the vibrations. The energies of the cells can be associated with the adiabatic potentials which for two considered kinds of cells are found as: ― 𝑑𝑖𝑚 𝑈𝑑𝑖𝑚 (𝑞) = ±

ℏ𝜔 2 2𝑞

―

2𝑡2 𝑈

± (𝑢 + 𝜐𝑞)2 +

4𝑡4 𝑈2

,


(19)


𝑈𝑡𝑒𝑡𝑟 ± (𝑞)

=

ℏ𝜔 2 2𝑞

―

4𝑡2 𝑈

2

± (𝑢 + 𝜐𝑞) +

16𝑡4 𝑈2

Page 12 of 36

.

(20)

Eq. (20) for the adiabatic potentials of tetrameric cell relates to S = 0 (ground electronic term), while Eq. (19) for the double-dimeric cell is valid for both S = 0 and S = 1. ― 𝑑𝑖𝑚 At P2=0 (that is at u = 0) the adiabatic potential 𝑈𝑑𝑖𝑚 (𝑞) possesses the only minimum ―

at qmin =0 for a weak vibronic coupling and two equivalent minima at 𝑞𝑚𝑖𝑛 = ± 𝜐2 (ℏ𝜔)2 ― 4𝑡4 𝑈2𝜐2 for strong enough vibronic coupling. The only minimum in the case of weak vibronic coupling corresponds to the adiabatic state in which the two electrons are fully delocalized over two diagonal positions, while the two minima for strong vibronic coupling correspond to a presumable localization of the electronic pair along diagonals of the cell. Note that the necessary condition for the existence of the double-well potential of the isolated doubledimeric cell is given by the inequality 𝜐2 (ℏ𝜔)2 > 4𝑡4 𝑈2𝜐2, while provided that 𝜐2 (ℏ𝜔)2 ≤ 4𝑡4 𝑈2𝜐2, the curve 𝑈𝑑𝑖𝑚 ― (𝑞) possesses the only fully delocalized minimum at 𝑞𝑚𝑖𝑛 = 0. In a rather general terms, one can say that all other things being equal, in the double-dimeric systems the degree of localization in the antipodal positions is higher as compared with that in the tetrameric ones.

Thus, the necessary condition of localization (i.e. bistability) for the isolated

(b)

(a)

Figure 5. Comparison of the adiabatic potentials of the double-dimeric cell 1 (solid lines) and tetrameric cell 1 with

S=0 (dashed lines) evaluated with b =4Å, c =9Å, 𝑡

= 500 cm ―1, ℏ𝜔 = 200 cm ―1 and 𝜐 = 200 cm ―1 for P2 =0 (a) and P2 =0.1 (b). double-dimeric cell is defined by the inequality 𝜐2 (ℏ𝜔)2 > 16𝑡4 𝑈2𝜐2, which shows that for tetrameric cell the localization occurs at two times larger value of  than for the double-dimeric cell. The two equivalent minima corresponding to the vibronically distorted configuration for the tetrameric cell are situated at 𝑞𝑚𝑖𝑛 = ± 𝜐2 (ℏ𝜔)2 ― 16𝑡4 𝑈2𝜐2 and therefore the deviation from the symmetrical configuration (q = 0) for the double-dimeric cell is stronger.

The adiabatic

potentials for the two considered kinds of isolated cells are shown in Figure 5a for the case when 12 ACS Paragon Plus Environment

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the bistability conditions are fulfilled for both kinds of cells. It is seen that the two minima of the ― 𝑑𝑖𝑚 curve 𝑈𝑑𝑖𝑚 (𝑞) are stronger shifted from q = 0 and separated from each other by a higher ―

energy barrier than the minima of the curve 𝑈𝑡𝑒𝑡𝑟 ― (𝑞). The quadrupole field induced by the driver results in an asymmetry of the adiabatic potential (Figure 5b). This is also seen from Figure 6 showing the position of the lowest minimum as function of P2 calculated at different values of the vibronic coupling. Thus, for the case of strong coupling the two minima at small |P2| become energetically inequivalent and then, at larger |P2 |, the shallow minimum disappears and the double-well adiabatic potential is transformed into the 𝑑𝑖𝑚 ― 𝑑𝑖𝑚 single-well one. From the comparison of 𝑈𝑡𝑒𝑡𝑟 (𝑞) in Figure 5b it is seen that in ― (𝑞) and 𝑈 ―

the case of tetrameric cell smaller value of |P2 | is required to cause the transformation from the double-well adiabatic potential to the single-well one. In the opposite case of a weak vibronic coupling the driver field shifts the position of the only minimum with its simultaneous stabilization. It is seen from Figures 5b and 6 that the coordinate of the global minimum 𝑞𝑚𝑖𝑛 is positive (negative) for negative (positive) values of P2. Providing strong vibronic coupling the coordinate 𝑞𝑚𝑖𝑛 of the ground minimum changes abruptly from positive to negative value when P2 changes from P2 0. For a weak vibronic coupling the coordinate 𝑞𝑚𝑖𝑛 gradually changes from positive to negative values passing through the point 𝑞𝑚𝑖𝑛 = 0 at P2=0. It is seen from Figure 6 that the dependences 𝑞𝑚𝑖𝑛 vs 𝑃2 for the tetrameric cell are more gradual than those for the double-dimeric one provided that the values of  are the same, which evidences in favor of a faster response of the double-dimeric cell to the driver action. Qualitatively, this observation can be explained in terms of the so far discussed degrees of localizations in these two kinds of the discussed systems. In fact, in tetrameric systems the degree of localization in the antipodal positions is higher as compared with that in the double-dimeric ones and therefore the former systems are more resistant against external field. The adiabatic approach allows to approximately evaluate the cell-cell response function with the use of adiabatic wave-functions of the cell 1 evaluated in the ground minimum of the lower branch of the adiabatic potential at different values of P2. At the same time, it should be borne in mind the restriction of applicability of the adiabatic approximation. The adiabatic approach loses its accuracy just in in the crossover region of the potential curves which is especially important in the central (non-linear) part of the cell-cell response function. Thus, the analysis of the two kinds of cells in the framework of the adiabatic approximation gives a benchmark rather than quantitative results. That is why in the next section we apply a quantum mechanical (dynamic) vibronic approach.



Page 14 of 36

― 𝑑𝑖𝑚 Figure 6. Coordinates 𝑞𝑚𝑖𝑛of the minima of the adiabatic potentials 𝑈𝑑𝑖𝑚 (𝑞) (solid line) ―

and 𝑈𝑡𝑒𝑡𝑟 ― (𝑞) with S=0 (dashed lines) as functions of P2 calculated at b = 4Å, c =9Å, 𝑡 = 500 cm ―1, ℏ𝜔 = 200 cm ―1 and the following different values of 𝜐: 𝜐 = 50 cm1 (magenta), 100 cm1 (red), 150 cm1 (green), 200 cm1 (blue), 250 cm1 (cyan). 4.3. Quantum-mechanical consideration Within the quantum-mechanical approach we define the matrices of the full Hamiltonians 𝐻𝑑𝑖𝑚 ― 𝑑𝑖𝑚 and 𝐻𝑡𝑒𝑡𝑟(0) in the basis composed of the products 𝜓𝐴𝐶(𝑆) |𝑛⟩ and 𝜓𝐵𝐷(𝑆) |𝑛⟩ of the electronic wave-functions and harmonic oscillator wave-functions |𝑛⟩ (n = 0, 1, …). Solving numerically this eigen-problem one finds the vibronic energy levels  k of cell 1 and the corresponding vibronic wave-functions 𝑘,𝑆 |𝑘,𝑆⟩ = ∑𝑛[𝑐𝑘,𝑆 𝐴𝐶, 𝑛𝜓𝐴𝐶(𝑆)|𝑛⟩ + 𝑐𝐵𝐷,𝑛𝜓𝐵𝐷(𝑆)] |𝑛⟩,

(21)

where k =1, 2…numerates the vibronic states with a definite S. The dimension of the truncated basis set |𝑛⟩, used in the numerical procedure, is chosen to ensure a good convergence in the evaluation of the vibronic levels with the further increase of the basis set. The vibronic energy levels of the cell 1 as functions of polarization P2 illustrate the effect of the driver (Figure 7) on the working cell. The vibronic levels are somewhat similar to those obtained in the framework of pure electronic approach (Figure 3). However, an essential difference is that the second order transfer gaps (at P2 = 0) are significantly reduced as compared with the pure electronic gaps 4𝑡2 𝑈 and 8𝑡2 𝑈 (see discussion of Figure 3). Consequently, as distinguished from the result of the electronic approach, the low-lying vibronic levels for both kinds of cells are practically linear functions of P2 at large values of P2.

This means that the vibronic coupling tends to suppress


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Figure 7. Comparison of the vibronic energy levels of the double-dimeric (solid lines) and tetrameric cells (1) with S=0 (dashed lines) evaluated as functions of P2 at 𝜐 = 200 cm1 provided that all other parameters are the same as in Figure 6. the electron delocalization making the cells more resistant to the change of the driver field. Using the eigen-vectors of the ground vibronic level (k=1) one can calculate the polarization P1 in the low-temperature limit as follows:

𝑃1 =

2 𝑆 2 1, 𝑆 ∑𝑛[(𝑐1, 𝐴𝐶, 𝑛) ― (𝑐𝐵𝐷, 𝑛) ] 2 𝑆 2 1, 𝑆 ∑𝑛[(𝑐1, 𝐴𝐶, 𝑛) + (𝑐𝐵𝐷, 𝑛) ]

(22)

.

= 1,𝑆 = 1,𝑆 Since the coefficients 𝑐𝑘𝐴𝐶, 𝑛 and 𝑐𝑘𝐵𝐷,𝑛 are functions of the polarization P2 of the driver, Eq. (22)

allows us to calculate the cell-cell response function 𝑃1 = 𝑓(𝑃2). By comparing the cell-cell response functions for the two kinds of cells evaluated at different values of the vibronic parameter (Figure 8), one can see that the above conclusion about more abrupt 𝑃1 = 𝑓(𝑃2) - dependence for the double-dimeric cell derived in the framework of the electronic approach, remains also valid in the framework of a more general dynamic vibronic approach. It is also seen from Figure 8 that the difference in the slopes of the cell-cell response curves for the two kinds of cells is smaller for larger values of the vibronic parameter. Indeed, the vibronic coupling reduces the transfer parameter (which conventionally occurs in MV systems) so that the difference in the effective transfer parameters for the two kinds of systems gets smaller when the vibronic coupling is involved.



Page 16 of 36

Figure 8. Cell-cell response functions for the double-dimeric (solid lines) and tetrameric (dashed lines) cells calculated with b =4 Å, c =9 Å , 𝑡 = 500 cm ―1, ℏ𝜔 = 200 cm ―1 for the following two values of the vibronic coupling parameter  : 𝜐 = 200 cm ―1 (blue), and 𝜐 = 350 cm ―1 (brown). 5. From half-cell to full cell: quantum-chemical modeling In this section we pass from the semiempirical description based on the Hubbard-type model supplemented by the vibronic coupling to the quantum-chemical analysis of molecules as possible candidate to act as cells. In view of the comparative analysis of the two kinds of cells, in this Section we analyze the system that can be imagined as two MV dimers which can be coupled by chemical bonds to get a tetrameric bi-electron cell consisting of the four equivalent (related by C4 rotations) centers accommodating well localized charges (electrons or holes). The dimeric and tetrameric systems, 1,4-dithia-hexane (Figure 9) and crown ether 1,4,7,10-tetrathia-cyclododecane (I, Figures 9, 10) were selected as the molecular systems for the implementation of proposed Hubbard type analysis. 1,4,7,10-tetrathia-cyclododecane (I) also known as a tetrathia-12-crown-4 was synthesized 40 years ago.32 This tetrathia-crown adopts a square conformation with the sulfur atoms at the corners according to X-ray analysis. Sulphur atoms point out of the ring with approximate D4 symmetry.32 Tetrathia-12-crown-4 is an ionophore like other crown-ethers which have extremely rich coordination chemistry.33 This is a physical reason for the choice of tetrathia12-crown-4 for computer design of tetramer. Our choice is prompted by the fact of the positive charge localization on the S-atoms which are not connected covalently and therefore, hopefully, the mobile charges can be well localized. According to our knowledge, no experimental data exist for the oxidized states - cation or dication of tetrathia-12-crown-4. At the same time the quantumchemical evaluation confirms stability of the cationic and dicationic forms.


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DFT calculations of the crown ethers 1,4,7,10-tetrathio-cyclododecane (I) (Figure 10) were done using Gaussian 09 program.34 B3LYP functional 35, 36 and cc-pVDZ basis set was used in all the calculations. Natural charge distributions were evaluated using NBO (Natural Bond Order) procedure as implemented in Gaussian 09.37 CASSCF

38-42

methodology implemented in

PCGAMESS 43, 44 program suite has been used to study the neutral, cation and dication of tetramer – crown ether (I) and corresponding dimer. Calculations of these systems in the different electronic states were performed to compare the quantum-chemical results with the results of Hubbard-type analysis in order to estimate the empirical parameters Hubbard-type model at the B3LYP level of theory. The geometric and electronic ground-state structure of the tetrathio-crown ethers could in the future be used for atomistic and mesoscale simulations of QCA cells and of larger devices. For this purpose, we will start our discussion with the validation of our computational methodology for the well-known case of I and apply the analysis on the other homologues. Molecule I has the highest occupied MOs fully localized on the well-separated S-atoms (S-S distance >2.8Å). The analysis of the electronic properties allows to conclude that the hole is mainly localized on the Satoms and the S∙∙∙S interactions are weak enough. VB resonance schemes (Figure 9) illustrate the charge distribution for the dimer (two S-atoms connected by H2CCH2 bridge, Figure 9a) and the antipodal charge localization for the corresponding tetramer (tetrathio-crown ether, Figure 9b). The hole is predominantly localized at the sulfur atoms because the ionization potential of sulfur is lower than that in the carbon saturated system. B3LYP/cc-pVDZ calculations show that S-atoms carry half of the positive charge of the hole (+0.5) according to NBO analysis. Calculations show also that the hole leads to the shortening of the S-S distance compared to the neutral dimer structure from 3.4 Å to 2.86 Å despite the repulsion between the positive charged Satoms. In terms of MO consideration, we could explain this compression as a result of the removal of electron from S-S antibonding MO (Figure 10d).

(b)

(a)

Figure 9. Schematic presentation for the dimeric and tetrameric molecular forms: (a) resonance structures of 1,4-dithia-compound with two S-atoms connected by saturated carbon 17 ACS Paragon Plus Environment


Page 18 of 36

bridge CH2CH2; (b) resonance structures of the crown ethers 1,4,7,10-tetrathiocyclododecane with indication of the antipodal charge localization in the low-lying states. The dimeric entities can be considered as a prototype system for the tetrameric molecular QCA. The neutral compound 1,4,7,10-tetrathio-cyclododecane is the member of the well-known family of crown ethers. Combination of the two charged dimers produces a tetramer in which each pair of sulfur atoms is connected by the CH2CH2 bridge (Figure 9b). So, the two charges are shared among four sulfur atoms with predominant antipodal localization as shown in Figure 9b. Figure 10 shows the structures of the molecule in different oxidation forms: parent neutral molecule (I), cation (I+) and dication (I+2). The last has two forms according to the full spin of the system, namely spin-singlet and spin-triplet forms S0 and T1 (only spin-singlet form of I+2, corresponding to the ground term is shown in Figure 10). The parent neutral molecule has four-fold symmetry as well as the cation and spin singlet dication, while the spin-triplet dication exhibits a pronounced tetragonal distortion. The last is undoubtedly caused by the Jahn-Teller effect in the excited orbital doublet (Figure 11). Since the symmetry of I+2 in the spin-singlet ground state remains tetragonal, one can conclude that only the coordinate of the full-symmetric vibration is shifted by the presence of the two holes, while the vibronic coupling to the B1g mode is small and hence can be omitted in the following consideration. The energy gap between two lowest electronic states in the cation of the parent dimer is estimated as  900 cm-1 (according to CASSCF). This result is in a good agreement with a calculated gap between two highest MOs of the opposite parity. The lowest bonding orbital (occupied) is odd, while the antibonding is even which allows to conclude that the transfer parameter t

Figure 10. Structures of crown ethers 1,4,7,10-tetrathio-cyclododecane: parent neutral molecule I (a), cation I+1 (b) and dication I+2 (c), HOMO of I (d). Charges (+1 and +2) are indicated by red, bond lengths are shown in Å, S-S distance are shown in bold (data according to B3LYP/cc-pVDZ). 18 ACS Paragon Plus Environment

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is positive. This energy gap corresponds to the value 2|t| in the Hubbard model and therefore the transfer parameter t can be estimated as  450 cm-1. Charged tetramers exhibit the shortening of the S-S distance from 3.539 Å in neutral form to 2.928 Å in dication (Figure 10).

(a)

(b)

(c)

Figure 11. Scheme of the Coulomb configurations and energy pattern of a tetrameric MV unit with two mobile electrons in the case of strong Coulomb repulsion. The energies of the lowlying levels (arising from the d-configurations) are shown out of the scale, the excited levels (d-configurations) are not shown, their labels are indicated in the box. This structural compression in the tetramer is slightly weaker than that in the parent dimer. The singlet-triplet splitting in dication I+2 is estimated as  250 cm-1 (B3LYP results, very structural sensitive) and 75 cm-1 obtained in the framework of CASSCF approach. Let rationalize the results of the ab initio calculations in view of availability of I2+ dioxide to act as a tetrameric cell encoding binary information. To do this, one needs to make sure that the Coulomb repulsion of holes is able to effectively separate them in antipodal distribution. The MO description does not provide the direct access to this information but the results of CASSCF calculations can be match with the energy pattern predicted by the Hubbard Hamiltonian. Actually, the Coulomb repulsion U of holes should be large enough so separate the low-lying levels states corresponding to the two distant charge distribution (d- configurations) from the excited levels arising from the six nearest neighboring n-configurations (Figure 11a, b). The configuration mixing of the group of low-lying levels with the excited ones (they are specified in Figure 10c without indication of their detailed structure) leads to the second order splitting 19 ACS Paragon Plus Environment


Page 20 of 36

(providing large U) in both manifolds and, in particular the three equidistant low lying levels 1B , 3E 1g

, 1A1g separated by the gap ∆ = 4𝑡2𝑛 𝑈 (Figure 11c) . This result is fully compatible with

the results of CASSCF calculations which gives the estimation ∆ ≈ 75 cm ―1. The position of excited group of levels gives the value of U ≈ 5930 cm1 and, therefore, the transfer parameter can be estimated as t ≈ 333.5 cm1. We thus can conclude that the condition of the charge separation required for the candidate for molecular cell is well satisfied. One can see that the value t ≈ 333.5 cm1 obtained for the tetrameric unit I2+ dioxide is comparable with the value t ≈ 450 cm1 estimated (in the framework of CASSCF approach) for the dimeric entity 1,4-dithiacompound. The decrease of t in the tetrameric system is in agreements with fact of the increase of the length of S-S bond from 2.86 Å in the dimer to 2.928 Å in tetramer. This estimation shows that the assumption that the parameter t is the same in the half-cell and full-cell is a reasonable approximation (at least for the modeled compounds) but in a more rigorous calculations the model should be improved. 6. Conclusion The present study combines semiempirical (based on the Hubbard type Hamiltonian) analysis and quantum-chemical study of the two possible compositions of the square molecular cells for QCA: double-dimeric and tetrameric cells which was referred to as half-cell and full cell. The theoretical Hubbard-type model includes electron (hole) transfer, Coulomb repulsion and vibronic coupling. Although both kinds of cells have equivalent networks of Coulomb interactions, the difference in the transfer pathways has been shown to result in considerable difference in their key functional characteristics. Considering strong Coulomb repulsion, we have demonstrated that this difference in the electron transfer pathways leads to the twice larger absolute value of the effective (second order) transfer matrix element linking the antipodal states of tetrameric cell as compared with the double-dimeric cell. As a result, the mobility of the electronic pair is higher in the tetrameric cell. Since the electronic delocalization acts as a factor hindering polarization of the cell by the quadrupole Coulomb field created by the driver cell, the double-dimeric cell is less resistant with respect to the action of the driver. This advantage of the double-dimeric cell is confirmed by the analysis of the dependences of the electronic energy levels of the cell 1 on the polarization P2 of the driver derived in the framework of the electronic approach, and similar dependences of the adiabatic potentials and vibronic energy levels obtained using the dynamic vibronic approach. The comparison of the cellcell response functions evaluated for the two kinds of cells also demonstrates the favorable properties of the double-dimeric cell. Indeed, the evaluated cell-cell response for such cell exhibits more abrupt nonlinear behavior that is one of key requirement for functioning of QCA. In contrast, 20 ACS Paragon Plus Environment

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more gradual P1=f(P2) dependence has been found for the tetrameric cell. It is remarkable, that this difference in the cell-cell responses for the two kinds of cells becomes less pronounced in cases of very weak electron transfer and/or strong vibronic coupling when the mobility of the electronic pair is low enough even in the tetrameric cell. Along with the semiempirical calculations we have performed the

quantum-chemical

analysis of the 1,4-dithia-compound with two S-atoms connected by saturated carbon bridge CH2CH2 (proposed as a dimeric subunit) and the corresponding tetrameric structures of the crown ethers 1,4,7,10-tetrathio-cyclododecane in three valent states: parent neutral molecule, cation and dication. The quantum-chemical calculations allowed us to quantitatively determine the key parameters such as hole transfer integral and Coulomb repulsion energy. It was shown that tetragonal arrangements of the S-atoms remain in parent neutral molecule, cation and dication which allows to conclude that the vibronic coupling with the non-symmetric modes in the systems under consideration is weak. It was demonstrated that combining two dimeric molecules into tetrameric leads to a decrease of the hole hopping parameter. We were able to match the Hubbard type model with the ab initio approach and in this way to conclude that the proposed compounds can serve as the cells with predominantly antipodal charge separation which potentially are able to encode binary information. The ab initio study and implementation of more complicated systems such as molecular logical gates remain a challenging task. Acknowledgements Support from the Ministry of Science and Higher Education of the Russian Federation (Agreement No. 14.W03.31.0001-Institute

of

Problems

of

Chemical

Physics,

Chernogolovka)

is

acknowledged. References 1. Lent, C. S.; Tougaw, P. D.; Porod, W.; Bernstein, G. H. Quantum Cellular Automata. Nanotechnology 1993, 4, 49 −57. 2. Lent, C. S.; Tougaw, P.; Porod, W. Bistable Saturation in Coupled Quantum Dots for Quantum Cellular Automata. Appl. Phys. Lett. 1993, 62, 714-716. 3. Lent, C. S.; Tougaw, P. D. Lines of Interacting Quantum‐Dot Cells: A Binary Wire. J. Appl. Phys. 1993, 74, 6227-6233. 4. Tougaw, P. D.; Lent, C. S. Logical Devices Implemented Using Quantum Cellular Automata. J. Appl. Phys. 1994, 75, 1818-1825. 5. Lent, C. S.; Tougaw, P. D. A Device Architecture for Computing With Quantum Dots. Proc. IEEE 1997, 85, 541 - 557. 21 ACS Paragon Plus Environment


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Double-Dimeric Versus Tetrameric Cell for Quantum Cellular Automata

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