Abstract
Molecular quantumdot cellular automata (mQCA) has received considerable attention in nanoscience. Unlike the currentbased molecular switches, where the digital data is represented by the on/off states of the switches, in mQCA devices, binary information is encoded in charge configuration within molecular redox centers. The mQCA paradigm allows high device density and ultralow power consumption. Digital mQCA gates are the building blocks of circuits in this paradigm. Design and analysis of these gates require quantum chemical calculations, which are demanding in computer time and memory. Therefore, developing simple models to probe mQCA gates is of paramount importance. We derive a semiclassical model to study the steadystate output polarization of mQCA multidriver gates, directly from the twostate approximation in electron transfer theory. The accuracy and validity of this model are analyzed using full quantum chemistry calculations. A complete set of logic gates, including inverters and minority voters, are implemented to provide an appropriate test bench in the twodot mQCA regime. We also briefly discuss how the QCADesigner tool could find its application in simulation of mQCA devices.
Keywords:
Electron transfer reactions; Molecular electronics; Molecular gates; Molecular quantumdots; Quantum cellular automataBackground
Recent advances in molecular electronics on one hand and the limitations of conventional semiconductor devices, on the other hand, have driven a surge of activities towards the realization of molecular devices, circuits, and systems. Achieving the ultimate diminution in size, power consumption, and delay of electronic devices and systems has always been a challenging endeavor of scientists and designers in this field. Due to the prospect of size reduction in electronics offered by molecularlevel control of properties, molecular electronics provides means to extend the Moore's law beyond the foreseen limits of smallscale conventional silicon integrated circuits. The small size of molecules allows high device density in the range of 10^{11} to 10^{12} devices/cm^{2}[1]. Besides, the chemical selfassembly capacity in manufacturing molecular devices provides many advantages to conventional semiconductor manufacturing technology, including lower manufacturing cost and uniform device reproducibility. Molecular electronics endeavors to use the nonlinear current–voltage characteristics of individual molecules or molecular assemblies as active devices (diodes, transistors, etc.) in electronic circuits. However, the power consumption of molecular current switches at very high frequencies is still a drawback [2]. The πσπ mixedvalence type molecules, which provide doublewell potentials for electrons, have been proposed and studied by Aviram towards the synthesis of memory, logic, and amplification [3]. Lent proposed using molecules in representing binary information within the molecular quantumdot cellular automata (mQCA) paradigm [1,4]. Molecular QCA provides an alternative approach to represent and process data, where binary representation lies in the charge configuration within molecules rather than in the on/off states of current switches. A cell in the mQCA model consists of a number of molecular quantum dots (or redox centers) and a few electrons. The electrons tend to occupy antipodal sites as a result of Coulomb repulsion. The Columbic interactions cause electrons to tunnel from one redox center to another in a cell, but not between cells. Thus, it is likely that no current flows, since the neighboring cells are coupled by electrostatic field. Figure 1 depicts twodot and fourdot mQCA cells and how binary “1” or “0” is represented. The first QCA device was implemented and tested using metal dots at near 0 K [5]. Semiconductor implementation of QCA using GaAs/AlGaAs heterostructure materials has been reported in [6,7] as well. Molecules are good containers for keeping electric charge and mQCA cells have a more promising future to work at room temperature [8]. Nonbonding π or d orbitals of a single molecule (or multiple molecules) can function as quantum dots, where the electric charge is localized in each cell. Synthesis of twodot and fourdot mixedvalence candidate molecules for mQCA has been reported in [914]. Many of these molecules are mixedvalence type and include transition metals to enable fast electron transfer reactions [15]. Molecular QCA gates are the building blocks of circuits in this paradigm. Calculation of the electronic structure of mQCA gates composed of these molecules is challenging, since the number of basis set functions grows exponentially as the number of molecules and atoms are increased. Besides, many of the ab initio methods fail in describing charge distribution in mixedvalence complexes. Therefore, developing semiclassical models to study mQCA gates is of high importance. Currently QCADesigner [16] utilizes nonlinear and twostate approximations to solve metallicbased QCA circuits. Several QCA circuits including combinational as well as sequential circuits have been studied using QCADesigner. Examples are adders, shift registers, RAM, digital data storage, and simple microprocessors [1726]. In this paper, we center on twodot mQCA and analyze the validity and accuracy of the twostate model approximation for studying multidriver mQCA gates. This study provides an approach to enhance the QCADesigner tool for simulation of mQCA devices in the future.
Figure 1. Binary representation in the mQCAparadigm. (a) Schematic structure of a twodot mQCA cell. (b) A twodot molecule. (c) In twodot cells, depending on which of the upper (A) or lower (B) quantumdot is occupied, binary “1” or “0” is represented. (d) In fourdot cells, binary “1” and “0” is represented within the occupation of AB’ or A’B dots correspondingly.
Methods
Twodot molecular QCA test bench
The majority voter (MV) and the inverter (INV) gates [25] are the fundamental building blocks of any circuit in the fourdot QCA architecture. These gates have been schematically shown in Figure 2a, b. Particularly, the MV gate is referred to as the universal QCA gate, since the AND and OR logical operations can be done by this gate, as evident from the truth table shown in Figure 2c.
Figure 2. Fourdot QCA gates. (a) The universal MV gate. The majority of the three fixed inputs, which is “0” in this figure, appears at the output as a result of Coulombic interactions and minimum energetics. (b) The inverter gate. (c) Truth table of the MV gate. When d_{3} = ”0”, the MV gate performs AND logical operations on d_{1} and d_{2,} and when d_{3} = ”1” the MV gate functions as a twoinput OR gate [25].
Our multidriver minority voter (MinV) gate is composed of m drivers, where m is an odd number, as inputs and one output. Figure 3a schematically illustrates the threedriver MinV model gate in the twodot mQCA regime. When only one driver (e.g., the d_{1}) is present, the model gate serves as an INV gate (Figure 3b). In the multidriver MinV model gate, all the distances between the centers of the molecules are l, which is equal to the distance between the middle of upper and lower πbonds as shown in Figure 3c. The inputs of the gates are kept fixed, while the output cells switch to their stable states. To this end, two point charges q and 1q separated by distance l are used to mimic each input as depicted in Figure 3c.
Figure 3. Twodot QCA gates. (a) Structure of the threeinput MinV gate. This gate is composed of three fixed inputs (d_{1}, d_{2}, and d_{3}) and a twodot molecule (AB) as output. The minority of the three fixed inputs, which is “0” in this figure, appears at the output. (b) When there is only one input, the MinV gate functions as an inverter gate. (c) Two point charges q and 1q separated by distance l, which is the distance between the middle of upper and lower πbonds, are used to mimic a fixed input. (d) Truth table of the MinV gate. When d_{3} = ”0”, the MinV gate performs NAND logical operation on d_{1} and d_{2,} and when d_{3} = ”1” the MinV gate functions as a twoinput NOR gate.
The MinV gate is an alternative to the MV gate in the fourdot QCA, where the output is inverted. Compared to MV and INV gates in the fourdot architecture, which require 16 and 28 quantumdots correspondingly, the MinV and INV gates require only 8 and 4 quantumdots in the twodot mQCA regime. Consequently, these gates provide a small twodot mQCA test bench, which make high level quantum chemical calculations feasible. The MinV gate can perform NAND and NOR logical operations, as shown in Figure 3d, and provides a functionally complete logic set to implement any logic function in the twodot mQCA framework. Additionally, it is possible to implement multiinput (or multidriver) MinV gates, which in turn decrease the total number of gates required to implement a logic circuit. It is important to note that since the MinV gate is not a planar gate, circuits implemented in the twodot mQCA regime are not planar circuits. We highlight that the practical QCA circuits require clockedcontrol cells and clocking schemes [21,2729], which are not addressed in this paper.
Twostate model for molecular QCA gates
The charge configuration in a QCA cell is quantified by the so called ‘polarization’, and is defined as [30]
where q_{A}q_{B}q_{A′} and q_{B′} are the charges localized at four quantum dots labeled in Figure 1. For twodot mQCA cells, the polarization is given by the charges q_{A} and q_{B} at the corresponding redox centers in Equation 1. The polarization of a QCA cell varies between −1 and 1, while negative and positive polarizations represent binary “0” and “1”, respectively. In twodot mQCA cells, the normalized dipole moment of the used twodot molecule is also identical to the polarization, which is given by
where μ_{α} denotes the component of the molecular dipole moment that is parallel to the bridge direction, and the origin is in the middle of the bridge. The dipole moment of an mQCA cell can be obtained through full quantum chemical calculations. An important parameter of a QCA device is the Kink energy (E_{k}), which is the required energy to excite the system from the ground state to the first excited state. To distinguish a bit value from the thermal environment, E_{k} must be greater than k_{B}T[31], where T is the operation temperature in Kelvin, and k_{B} is the Boltzmann's constant. The E_{k} represents the energy cost of cells i and j having opposite polarizations. That is, the electrostatic interaction between all the charges in cells i and j is calculated by [16]
where ϵ_{0} is the permittivity of free space, and ϵ_{r} is the relative permittivity of the material system. The Kink energy is then given by [16]
where E′_{i,j} and E_{i,j} denote the electrostatic energy of cells i and j having opposite and same polarizations correspondingly.
Tougaw and Lent have used a simple Hamiltonian of the extendedHubbard type to describe the dynamic behavior of fourdot metallicbased QCA nanodevices [32]. Although this Hamiltonian describes the dynamics of the coherent system composed of arrays of fourdot QCA cells elegantly in theory, it is only possible to model the small systems employing this scheme, since the total required number of directproduct basis sets grows exponentially with the number of cells. In other words, an array with N number of fourdot cells and B number of basis sets in each cell requires the total number of directproduct basis sets as [32]
By reducing the number of basis sets for each cell and picking up the two orthogonal ones, the Hamiltonian of a fourdot QCA wire can be mapped to Ising model as [32,33]
where E_{k} is the kink energy of fourdot cells, γ is the tunneling energy, and σ_{x} and σ_{z} are Pauli spin matrices. In semiconductor and metaldot QCA, the tunneling barriers of the cells are connected to electrodes, and their heights are controlled externally by voltage sources [33]. The steadystate polarization of any cell, j in a block of cells, can be obtained as a solution to the HartreeFock intercellular approximation. This approximation decouples the line of N cells into N single cell subsystems and assumes that the cells are only coupled through expectation values of polarizations. The consequent solution is [33],
where P^{—}_{j} is the sum of the polarizations of the neighboring fourdot QCA cells. Equation 7 is currently used in the nonlinear and twostate simulation engine of QCADesigner to solve the metallicbased QCA circuits. It is important to note that mQCA utilizes nonabrupt clocking to reduce the probability of Kink, the property that is not currently present in the QCADesigner as it is based on metallic QCA. In mQCA, the tunneling barriers can be controlled by external electric field [27]. It is demanding to enhance the tool to be able to simulate mQCA circuits. As a primary step towards this end, we present how a similar equation to (7) can be derived directly from the twostate approximation in electron transfer theory [34,35] for twodot mQCA. We then discuss how these approximations affect the results compared to those obtained from full quantum chemistry calculations.
Equation 8 describes the electron transfer (ET) process in a twodot mQCA cell, where the two redox centers, A and B, are linked through an intervening bridge, I.
The electronic coupling between the redox centers, which is time independent, is an important factor in the ET process. Within the twostate approximation, the LandauZener model [36,37] for avoided crossing of energy surfaces may be applied, where the two diabatic states “1” and “0” denoted by ψ_{a} and ψ_{b}, and with energies H_{aa} and H_{bb} interact. The ground state ψ_{1} and the first excited state of a QCA cell, ψ_{2}, can be related to the diabatic states ψ_{a} and ψ_{b} by a unitary transformation [34]
In Equations 9 and 10ψ_{1} and ψ_{2} are orthonormal, whereas ψ_{a} and ψ_{b} are not orthogonal in general. The correspondence between diabatic and adiabatic twostate models arises from the secular determinant (S_{ab}= <ψ_{a}ψ_{b}> is neglected) [38]
where E is the adiabatic energy eigenvalue. The η in Equations 9 and 10 satisfies [38]
The energy difference between the two diabatic states in the output cell of the MinV gate can be approximated by calculating the difference between the electrostatic energies of the gate for the two output configurations, where the unit charge is localized at sites A and B correspondingly. Using Equation 3, for the “1” and “0” output states (Figure 3a), we obtain
Inserting Equation 13 into Equation 12 we have
The Kink energy of twodot cells can be calculated from Equation 3 and 4 for two neighboring cells as
and for each driver, the polarization is defined using Equation 1
Thus, Equation 14 can be rewritten in terms of the Kink energy and the input polarizations
And finally, using Equation 1 and the transformation coefficients in Equations 9 and 10, the output polarization of the MinV gate is obtained
Inserting Equation 17 into Equation 18, we can find the polarization of the output cell of the MinV gate in terms of the polarizations of the inputs straightforwardly as
Equation 19 in twodot mQCA is analogous to Equation 7 in fourdot metalbased QCA, where the tunneling energy γ appears as electronic coupling of redox centers (H_{ab}) in Equation 19. They also imply
The additivity relation in Equation 20 originates from the additivity of electrostatic potential energy in Equation 13 for diabatic states.
Multidriver MinV gates help reduce the number of needed gates for implementation of a logic circuit. Similarly, for an minput MinV gate, we obtain
We refer to Equaiton 21 as the twostate model (TSM) for mQCA gates along this paper. The E_{k} and the H_{ab} are the only parameters of the TSM. Once the geometrical parameter l is determined, experimentally or from theoretical calculations, the Kink energy can be calculated using Equation 15. The electronic coupling matrix element, H_{ab}, can be calculated using various quantum chemistry techniques [34,3841] or obtained via spectroscopic experiments, including absorption [42,43], EPR [44], and ultraviolet photoelectron spectroscopy [45]. As we will present, the parameter μ = E_{k}/2H_{ab} plays an important role in the accuracy of the TSM. It is also the slope of the switching response function at the origin i.e.,
Results and discussion
The chemistry of mixedvalence complexes has received considerable attention recently in mQCA device implementation, where the intramolecular electron transfer and charge localization at redox sites are the important key factors. Mixedvalence compounds contain more than one redox site in the same molecule or molecular unit. Simple model molecules for twodot cells are the πσπ mixedvalence types, which were proposed by Aviram and studied later by Hush [3,46]. In the Aviram's model molecule (1, 4diallyl butane cation), the two πallyl groups form two redox centers and are connected by a σbutyl bridge. One of the allyl groups is a neutral radical, while the other one is anionic (or cationic). The possibility of charge localization in some mixedvalence mQCA candidate molecules has been examined theoretically as well as experimentally [15,47,48]. Advances in quantum chemistry in the past half century provide reliable methods to explore the electronic structure of molecules; however, many of the ab initio techniques fail in describing charge distribution in mixedvalence complexes. The unrestricted HartreeFock method overestimates the charge localization due to the neglect of electron correlation effects [49]. In the density functional theory (DFT) method, the exchange potential defined in hybrid functional leads to underestimation of charge localization [47,49]. The complete active space selfconsistent field (CASSCF) method [50,51] is believed to be the most reliable for describing charge distribution in mixedvalence complexes [50]. However, the multideterminant CASSCF calculations scale with the system size, which makes this method highly demanding in computer time and memory. The number of Slater determinants has factorial dependence on both the number of active electrons and particularly on the number of active orbitals generating manyelectron configurations (full configuration interaction (CI) within the active space). This is much more significant than any dependence on the number of oneelectron basis functions. The number of Slater determinants in a full CI calculation is given by:
where M is the number of active orbitals, N_{α} and N_{β} are the numbers of active electrons with α and βspins, respectively, and the quantities in parentheses are binomial coefficients:
We present full quantum chemistry calculations of the steadystate output polarization of the universal MinV model gate serving as INV and threeinput MinV gates. The results based on full quantum chemical calculations are compared to the results obtained from the TSM. The πσπ mixedvalence type molecules, descended from Aviram's original idea are analyzed. These molecules include 1, 6heptadiene, 1, 8nonadiene, and 1, 4diallyl butane radical cations, which will be referred to as molecule 1, molecule 2, and molecule 3 in this paper, respectively (Figure 4). We optimized the geometry of these monocations using the DFT/B3LYP method. The dotdot distance, l, in these molecules is between 0.5 to 0.8 Å. Bistability and electron localizability of these molecules have been studied in [3,46,49].
Figure 4. Geometry of the molecules we used in our calculations. 1, 6heptadiene, 1, 8nonadiene, and 1, 4diallyl butane radical monocations from left to right. In the first two molecules, coordinates of the three highlighted atoms have been set to xy plane, while the coordinates of the central carbon have been set to origin. For 1, 4diallyl butane, the origin has been set in the middle of the bond between the two central carbon atoms.
Koopmans' theorem [52] has found extensive application in calculation of the ET matrix element, H_{ab} for symmetric molecules. Under the twostate approximation, H_{ab} is related to adiabatic energies of the ground and first excited state (E_{1} and E_{2}) as [34]
When no driver is present or the sum of the input drivers is zero, P_{o} = 0; and from Equation 18 it is clear that Cos2η = 0, thus
According to the one electron Koopmans' theorem, the ionization potential of the highest occupied molecular orbital (HOMO) and HOMO1 can be expressed in terms of the molecular orbital (MO) energies, i.e., [3841]
Inserting Equations 27 and 28 into Equation 26, the electronic coupling element is obtained in terms of the MO energies as [3841]
The stateaveraged CASSCF (SA/CASSCF) method [53] can be used to calculate the electronic coupling element of asymmetric molecules. One can obtain H_{ab} by calculating the energies of the ground and first excited states and use them in Equation 26 within the twostate approximation. We have calculated the electronic coupling elements using both methods. The calculations for molecule 1 and 2 were based on SA/CASSCF(34), and the calculations for molecule 3 were based on SA/CASSCF(56). In 1, 4diallyl butane cation, the allyl πbonds are aromatic, and the active space is extended to five electrons in six orbitals. The calculated ET matrix elements have been compiled in Table 1. The Kink energies of the molecules have also been calculated using Equation 15. Table 1 includes all required parameters of the TSM. All calculations reported here were performed in Gaussian 09 [54], using 631 G(d) basis set. Various basis sets have been extensively tested to examine the basis set dependency of the results. Application of larger basis sets did not significantly influence the energy difference between the ground state and the first excited state. The results from the two methods are in good agreement. The ET matrix element, H_{ab} decays exponentially with dotdot distance, l[55]. The dotdot distance for molecule 3 is less than that of molecule 2; however, the H_{ab} has been remarkably decreased. This is due to the symmetry of this molecule and the aromatic bonds of the radical allyls. The geometrical parameter, l, and the type of head groups, play an important role in determining the ET matrix element. To obtain a more accurate electronic coupling element, the overlap integral, S_{ab}, should be taken into account as described in the work of Farazdel et al [56]. Aviram [3] has obtained a negligible overlap integral for molecule 3 with dotdot distance of 7 Å. In mQCA, the electron transfer drama should have a little effect on the geometric parameters [4]. Consequently, candidate molecules should possess fast electron transfer reactions, and the relaxation of nuclear degrees of freedom can be ignored. Table 1 also lists the changes in the head groups' πbonds (Δζ) as a consequence of ET reactions. It is seen that ET reactions in molecule 3 should be faster compared to the other two molecules.
Table 1. Twostate model parameters for the used molecules
INV gates
The INV gate in twodot mQCA is the nucleus of all other gates. Once its operation and switching properties are clearly understood, the properties of more intricate structures such as multidriver MinV gates can be derived from extrapolating the results obtained from the inverters, based on the additivity relation (Equation 20). The analysis of inverters can be extended to explain the behavior of more complex gates, which in turn form the building blocks for modules such as adders, multipliers, and processors. Table 2 compares the output polarizations of the INV gates, obtained from full quantum chemistry calculations and the TSM. The normalized dipole moments (Equation 2) of the monocations adjacent to fixed inputs (point charges as fixed drivers) have been calculated based on SA/CASSCF(3,4) for molecule 1 and 2, and SA/CASSCF(5,6) for molecule 3. The root mean square errors (RMSE) of the results obtained from the two methods have been calculated. The RMSE decreases with the increase of the μ parameter (or decrease of the ET matrix element, H_{ab}), determining the degree of agreement between the results. The saturation polarization of the output is also dependant on the μ parameter. For INV gates, it is obtained by setting the sum of the input drivers to one in Equation 21 as
Table 2. INV gates
Equation 30 shows that the saturation polarization of the output increases with the increase of μ. This is also evident from the results in Table 2.
Twodriver devices
In the model MinV gate, the number of input drivers (m) should be odd. No logic operation is performed when m is even. However, neglecting the logic, the twodriver device is an appropriate small model system for studying the additivity relation, and how the accuracy of the TSM is influenced by the number of drivers. Here, the MinV gate is probed when only the two input drivers d_{1} and d_{3} are present (Figure 3a). We have calculated the normalized dipole moments of the gates' outputs based on the SA/CASSCF calculations. The output polarizations have also been calculated by the TSM. The results obtained from the two methods are compiled in Table 3, which are in good agreement. The conclusions from analysis of the INV gates can be extrapolated to twodriver devices as well. Compared to INV gates, the increase in the RMSE of the twodriver devices, composed of molecule 1 or 2, is mainly attributed to the asymmetric head groups. In other words, the effect of d_{1} on the head groups is different from that of d_{3}, where P_{d1}= P_{d3}. In molecule 3, the allyl head groups are symmetric, and the TSM error mainly arises from the classical approximation of the intercellular interactions. It is important to note that the output polarization of the twodriver devices can be calculated by employing the additivity relation on the output polarizations of the INV gates. The additivity relation has been validated for the SA/CASSCF method as well. Through full quantum chemistry calculations, the output polarizations of the twodriver devices were obtained. We also used the results of the INV gates, P(P_{d1}+ P_{d2}) from Table 2, to examine the additivity relation for SA/CASSCF method. As expected, RMSE is highly dependent on the symmetry of the head groups. Unlike molecule 1 and 2, for the case of molecule 3, each driver has an exactly same effect on the head allyl groups, which leads to smaller RMSE. We also highlight that employing additivity relation decreases the computational cost of SA/CASSCF calculations. Table 2 and Table 3 also show how the accuracy of the TSM is affected by the number of input drivers. It is seen that RMSE is approximately doubled when the number of input drivers is scaled up by a factor of two.
Table 3. Twodriver devices
Threeinput MinV gates
The multidriver MinV gate is a universal gate in the twodot mQCA. The output polarizations of these gates with three fixed input drivers are shown in Table 4. This table also shows that the output polarizations of the MinV gates can be obtained from extrapolating the INV output polarizations using the additivity relation. For the model molecules, considering the spatial location of the d_{2,} the effect of d_{2} on the head groups is different from the same effect of d_{1} and d_{3}, while in the TSM, drivers with equal polarizations have same effects on the head groups and are treated the same. Quantum chemical calculations show that despite the equal sum of the input polarizations, the output polarizations are not equal particularly when the sum of the drivers is zero. Table 4 also displays that the SA/CASSCF method returns different output polarizations, while the sum of the input polarizations is zero. This is the main reason of the decrease in the accuracy of the results obtained from the TSM for MinV gates. Ignoring these points by avoiding the null state logic occurrence, the twostate approximation results are fairly in good agreement with the quantum chemical calculations. Table 2 and Table 4 show how the accuracy of the twostate model is decreased with the number of drivers. It is seen that RMSE is tripled when the number of input drivers is scaled up by a factor of three.
Table 4. ThreedriverMinVgates
Conclusions
Molecular QCA gates are the building blocks of more complex modules. Probing molecular devices requires quantum chemical calculations, which are challenging as the molecular system grows in size. A semiclassical model was derived directly from the twostate approximation in the ET theory, serving as a device for studying mQCA gates. This model is very similar to the twostate model which is currently the core of the QCADesigner simulation engine for solving circuits based on metallic QCA. The range of applications and limitations of this model for mQCA gates was investigated carefully. The parametric TSM can be used to study more complex mQCA gates composed of practical candidate mixedvalence molecules, where exploiting the SA/CASSCF method is of high computational cost. A complete set of logic gates were implemented within the twodot mQCA framework. These gates include INV and MinV gates, which provide a small molecular test bench, making further analysis by quantum chemistry methods, particularly SA/CASSCF, practical. The INV gate was studied as a nucleus of all other gates. It was also presented that output polarizations of all other gates can be derived from extrapolating the results obtained from inverters based on the additivity relation. We compared the results obtained from the TSM to those obtained from SA/CASSCF calculations for INV and MinV gates. The degree of agreement between the TSM and quantum chemical calculations is highly dependent on the μ parameter and the symmetry of the head groups. Additionally, application of the additivity relation for CASSCF method can in turn reduce the computational cost. It is important to note that we did not address questions of surface attachment, input/output, clocked control, layout, and patterning, which are the requirements of a practical QCA system. Moreover, we did not consider the relaxation of nuclear degrees of freedom associated with electron transfer. It is presented that for mQCA, the electron localization and Coulombic interactions play the key roles, and nuclear positions can be considered frozen (nuclear relaxation even assists charge localization) [4]. Although we limited our focus on the twodot mQCA, it merits highlighting that the model can also be used for fourdot cells, since they can be considered as double twodot cells. Our focus was on the mQCA gates as building blocks of circuits. The twostate model may be applied to simulate mQCA circuits as well, as it is currently used iteratively for simulation of metallic QCA circuits in the QCADesigner. However, to determine the additive error resulting from exploiting the twostate model for solving mQCA circuits, further quantum chemical calculations on the mQCA clocked circuits composed of several molecules are required, which are extremely challenging at the time, and have not been addressed in this paper.
Competing interests
The authors declare that they have no competing interests.
Authors' contributions
ER developed the theories and carried out the quantum chemical calculations. SMN supervised the project. All authors read and approved the final manuscript.
Acknowledgment
ER was affiliated with Norwegian University of Science and Technology. Calculations presented in this work have been carried out on Stallo. ER also thanks Professor Sven Larsson for many enlightening discussions on electron transfer theory at Chalmers University of Technology.
References

Lent CS: Bypassing the transistor paradigm.
Science 2000, 288:15971599. Publisher Full Text

Niemier M, Kogge P, Murphy R, Rodrigues A, Dysart T, Frost S: Data flow in molecular QCA: logic can “sprint”, but the memory wall can still be a "hurdle". University of Notre Dame: Technical report; 2005.

Aviram A: Molecules for memory, logic, and amplification.
J Am Chem Soc 1988, 110:56875692. Publisher Full Text

Lent CS, Isaksen B, Lieberman M: Molecular quantumdot cellular automata.
J Am Chem Soc 2003, 125:10561063. PubMed Abstract  Publisher Full Text

Orlov AO, Amlani I, Kummamuru RK, Ramasubramaniam R, Toth G, Lent CS, Bernstein GH, Snider GL: Experimental demonstration of clocked singleelectron switching in quantumdot cellular automata.
Appl Phys Lett 2000, 77:295297. Publisher Full Text

Smith C: Realization of quantumdot cellular automata using semiconductor quantum dots.
Superlatt Microst 2003, 34:195203. Publisher Full Text

Gardelis S, Smith C, Cooper J, Ritchie D, Linfield E, Jin Y: Evidence for transfer of polarization in a quantumdot cellular automata cell consisting of semiconductor quantum dots.

Lent CS, Isaksen B: Clocked molecular quantumdot cellular automata.
IEEE T Electron Dev 2003, 50:18901896. Publisher Full Text

Qi H, Sharma S, Li Z, Snider GL, Orlov AO, Lent CS, Fehlner TP: Molecular quantum cellular automata cells. electric field driven switching of a silicon surface bound array of vertically oriented twodot molecular quantum cellular automata.
J Am Chem Soc 2003, 125:1525015259. PubMed Abstract  Publisher Full Text

Jiao J, Long GJ, Grandjean F, Beatty AM, Fehlner TP: Building blocks for the molecular expression of quantum cellular automata. Isolation and characterization of a covalently bonded square array of two ferrocenium and two ferrocene complexes.
J Am Chem Soc 2003, 125:75227523. PubMed Abstract  Publisher Full Text

Jiao J, Long GJ, Rebbouh L, Grandjean F, Beatty AM, Fehlner TP: Properties of a mixedvalence (feII)2(feIII)2 square cell for utilization in the quantum cellular automata paradigm for molecular electronics.
J Am Chem Soc 2005, 127:1781917831. PubMed Abstract  Publisher Full Text

Qi H, Gupta A, Noll BC, Snider GL, Lu Y, Lent C, Fehlner TP: Dependence of field switched ordered arrays of dinuclear mixedvalence complexes on the distance between the redox centers and the size of the counterions.
J Am Chem Soc 2005, 127:1521815227. PubMed Abstract  Publisher Full Text

Li Z, Fehlner TP: Molecular QCA cells. 2. Characterization of an unsymmetrical dinuclear mixedvalence complex bound to a Au surface by an organic linker.
Inorg Chem 2003, 42:57155721. PubMed Abstract  Publisher Full Text

Lu Y, Lent CS: Theoretical study of molecular quantumdot cellular automata.
J Comput Elec 2005, 4:115118. Publisher Full Text

BraunSand SB, Wiest O: Theoretical studies of mixedvalence transition metal complexes for molecular computing.

Walus K, Dysart TJ, Jullien GA, Budiman RA: QCADesigner: a rapid design and simulation tool for quantumdot cellular automata.
IEEE T Nanotechnol 2004, 3:2631. Publisher Full Text

Vetteth A, Walus K, Dimitrov SV, Jullien GA: Quantumdot cellular automata carrylookahead adder and barrel shifter. Richardson, TX: IEEE Conf on Emerging Telecom Tech; 2002.

Frost SE, Rodrigues AF, Janiszewski AW, Rausch RT, Kogge PM: Memory in Motion: a Study of Storage Structures in QCA.

Niemier MT, Kogge PM: Logic in wire: using quantum dots to implement a microprocessor.
ICECS '99 6th IEEE Int Conf Circuits and Systems 1999, 3:12111215. PubMed Abstract  Publisher Full Text

Walus K, Vetteth A, Jullien GA, Dimitrov VS: RAM design using quantumdot cellular automata.

Lent CS, Tougaw PD: A device architecture for computing with quantum dots.
Proc of the IEEE 1997, 85:541557. Publisher Full Text

Rahimi E, Nejad SM: Secure clocked QCA logic for implementation of quantum cryptographic processors.

Rahimi E, Nejad SM: QuantumDot Cellular ROM: A nanoscale level approach to digital data storage.
6th IEEE Int Conf on Communication Systems, Networks and Digital Signal Processing 2008, 618621.

Rahimi E, Nejad SM: A novel architecture for quantumdot cellular ROM.
7th IEEE Int Conf on Communication Systems, Networks and Digital Signal Processing 2010, 347350.

Tougaw PD, Lent CS: Logical devices implemented using quantum cellular automata.
J Appl Phys 1994, 75:18181825. Publisher Full Text

Mohammad Nejad S, Rahimi E: QCA: The prospective technology for digital telecommunication systems. In Nanotechnology for Telecommunications. CRC Press; 2010:275307.

Hennessy K, Lent CS: Clocking of molecular quantumdot cellular automata.

Orlov AO, Kummamuru R, Ramasubramaniam R, Lent CS, Bernstein GH, Snider GL: Clocked quantumdot cellular automata shift register.

Karim F, Walus K, Ivanov A: Analysis of fielddriven clocking for molecular quantumdot cellular automata based circuits.
J Comput Elec 2010, 9:1630. Publisher Full Text

Lent CS, Tougaw PD, Porod W, Bernstein GH: Quantum cellular automata.
Nanotechnology 1993, 4:49. Publisher Full Text

Timler J, Lent CS: Power gain and dissipation in quantumdot cellular automata.
J Appl Phys 2002, 91:823831. Publisher Full Text

Tougaw PD, Lent CS: Dynamic behavior of quantum cellular automata.
J Appl Phys 1996, 80:4722. Publisher Full Text

Géza T: Correlation and coherence in quantumdot cellular automata. PhD Thesis. University of Notre Dame; 2000.

Newton MD: Quantum chemical probes of electrontransfer kinetics: the nature of donoracceptor interactions.
Chem Rev 1991, 91:767792. Publisher Full Text

Newton MD: Control of electron transfer kinetics: models for medium reorganization and donor–acceptor coupling.

Zener C: Nonadiabatic crossing of energy levels.
Proc R Soc A 1932, 137:696702. Publisher Full Text

Zener C: Discussion of excited diatomic molecules by external perturbations.
Proc R Soc A 1933, 140:660668. Publisher Full Text

Larsson S, Braga M: Transfer of mobile electrons in organic molecules.
Chem Phys 1993, 176:367375. Publisher Full Text

Li XY, Tang XS, Xiao SQ, He FC: Application of Koopmans’ theorem in evaluating electron transfer matrix element of longrange electron transfer.
J Mol Struct (THEOCHEM) 2000, 530:4958. Publisher Full Text

Li XY, Tang XS, He FC: Electron transfer in poly (pphenylene) oligomers: effect of external electric field and application of Koopmans theorem.
Chem Phys 1999, 248:137146. Publisher Full Text

RodriguezMonge L, Larsson S: Conductivity in polyacetylene.
IV. Ab initio calculations for a twosite model for electron transfer between allyl anion and allyl. Int J Quantum Chem 1997, 61:847857.

Hush NS: Intervalencetransfer absorption. Part 2. Theoretical considerations and spectroscopic data.

Hush NS: Homogeneous and heterogeneous optical and thermal electron transfer.
Electrochim Acta 1968, 13:10051023. Publisher Full Text

Bailey SE, Zink JI, Nelsen SF: Contributions of symmetric and asymmetric normal coordinates to the intervalence electronic absorption and resonance raman spectra of a strongly coupled pphenylenediamine radical cation.
J Am Chem Soc 2003, 125:59395947. PubMed Abstract  Publisher Full Text

Coropceanu V, Gruhn NE, Barlow S, Lambert C, Durivage JC, Bill TG, Nöll G, Marder SR, Brédas JL: Electronic couplings in organic mixedvalence compounds: the contribution of photoelectron spectroscopy.
J Am Chem Soc 2004, 126:27272731. PubMed Abstract  Publisher Full Text

Hush NS, Wong AT, Bacskay GB, Reimers JR: Electron and energy transfer through bridged systems. 6. Molecular switches: the critical field in electric field activated bistable molecules.
J Am Chem Soc 1990, 112:41924197. Publisher Full Text

Cabrero J, Calzado CJ, Maynau D, Caballol R, Malrieu JP: Metal−ligand delocalization in magnetic orbitals of binuclear complexes.
J Phys Chem A 2002, 106:81468155. Publisher Full Text

Lu Y, Quardokus R, Lent CS, Justaud F, Lapinte C, Kandel SA: Charge localization in isolated mixedvalence complexes: an STM and theoretical study.
J Am Chem Soc 2010, 132:1351913524. PubMed Abstract  Publisher Full Text

Lu Y, Lent CS: A metric for characterizing the bistability of molecular quantumdot cellular automata.
Nanotechnology 2008, 19:155703. PubMed Abstract  Publisher Full Text

Malmqvist PÅ, Roos BO: The CASSCF state interaction method.
Chem Phys Lett 1989, 155:189194. Publisher Full Text

Yamamoto N, Vreven T, Robb MA, Frisch MJ, Bernhard Schlegel H: A direct derivative MCSCF procedure.
Chem Phys Lett 1996, 250:373378. Publisher Full Text

Koopmans T: Über die zuordnung von wellenfunktionen und eigenwerten zu den einzelnen elektronen eines atoms.
Physica 1934, 1:104113. Publisher Full Text

Ruedenberg K, Cheung LM, Elbert ST: MCSCF optimization through combined use of natural orbitals and the brillouin–levy–berthier theorem.
Int J Quantum Chem 1979, 16:10691101. Publisher Full Text

Frisch MJ, Trucks GW, Schlegel HB, Scuseria GE, Robb MA, Cheeseman JR, Scalmani G, Barone V, Mennucci B, Petersson GA, et al.:

Newton MD, Cave RJ: Molecular Control of Electron and Hole Transfer Processes: Theory and Applications.

Farazdel A, Dupuis M, Clementi E, Aviram A: Electricfield induced intramolecular electron transfer in spiro.pi.electron systems and their suitability as molecular electronic devices. A theoretical study.
J Am Chem Soc 1990, 112:42064214. Publisher Full Text