In compliance with the Moore’s law the number of computing units on a chip doubled every 18 months. This tendency was observed during more than two decades, and presently a natural physical limit of practical implementation restricted by the heat release and quantum effects is not too far from being achieved. It became clear that the devices based on the traditional electronic elements such as semiconductor transistors, wires or diodes could not more ensure further progress in the informational technologies. A new revolutionary computing paradigm was proposed by Lent et al. in 1993  as a physical implementation of an automaton at nano-scale using quantum-dot cellular automata (QCA) . This discovery gave a strong impact to the development of the new multidisciplinary field of nanotechnology that combines physics, chemistry, biosciences, material science, computer science, and electrical engineering [2–15]. The metal-based and semiconductor-based QCA have been realized as micron scale as metal dots composing wires and majority gates on a silicon substrate (see, for review ref. ). The electronic QCA devices consuming extremely small amounts of electrical power and very small heat release are capable of performing computation at very high switching speeds.
The central idea of a four-dot QCA cell is schematized in Fig. 1 [6, 12]. The cell consists of the four dots occupied by the two extra electrons (or holes) which can tunnel between the sites. The information is encoded in the two antipodal charge configurations of the cell which are energy beneficial and transferred via Coulomb interaction between the neighboring cells. Just these two quasi-stable charge distributions that conventionally are referred to as cell polarizations P = –1 and P = +1 encode the binary information 1 and 0 (Fig. 1). The arrays of the cells compose logical devices such as wires, majority gates, invertors, etc. from which more complex circuits can be designed. The architectures of these circuits had been proposed in underlying papers [1–5] and then discussed thoroughly [6, 10–17].
Recently the idea of using molecular systems as QCA units has been proposed and discussed [18–25]. This proposal marked a new step towards further miniaturization of microelectronic devices while receiving a substantial increase of their advantages compared to the conventional ones. In fact, molecular QCA (mQCA) offers promise of nanometer-scale devices with accompanying ultra-high device densities as well as room-temperature operation providing at the same time options to control the key properties of the molecule by chemical means. A molecule acting as structured charge containers acceptable as a mQCA cell should have two or more positions in which the electrons occupying the nonbonding orbitals can be localized mimicking the dots and tunnel among these positions. As a single molecule implementation of QCA Lent and Tougaw  proposed to use 1,4-dialyl butane radical cation consisting of butyl bridge linking two allyl groups accommodating electron. Then the problem of mQCA and other attractive candidates for molecular cells have been proposed and discussed in detail in Ref. [18–31].
The proposal  to realize QCA cells by MV molecules seems to be especially promising because they naturally have two or more metal sites with different oxidation degrees forming specific charge configurations while the electron-lattice interaction (vibronic coupling) can lead to the self-trapping and consequently to the barrier between these localized configurations. This refers to the MV systems well studied in molecular magnetism such as dimeric clusters and mainly to the large scale MV systems, like biologically important metal clusters, iron-sulfur proteins [32–36], polynuclear reduced polyoxometalates (e.g., Keggin and Wells-Dawson systems) accommodating two or several electrons delocalized over a metal network [37–44] (see review article ) and tetranuclear metal clusters [46–48].
In this article we consider the energy levels of the delocalized electronic pair and vibronic interaction in tetranuclear MV systems that can be exemplified by the MV tetra–ruthenium complexes 2Ru(II)+ 2Ru(III) (assembled as two coupled Creutz–Taube dimers) in which two holes are delocalized over four metal sites. These systems have been proposed as possible candidates for implementations in mQCA  and they are attractive as examples of MV systems in which all physically important interactions, such as electron transfer, electron correlations and electron-lattice interaction, are involved in a competitive manner. The mQCA implementations are closely related to the localization/delocalization features of the named compounds and therefore the vibronic coupling proves to be one of the key factor defining the stability of the charge configurations, response to the varying electric field of the driver and switching period in QCA. In a more wide context the vibronic coupling includes along with the molecular degrees of freedom also the environmental decoherence which can influence the system dynamics, heat release and QCA operation. As it was mentioned in Ref.  the MV ruthenium complexes are representatives of the Robin and Day classes I, II, and III [49–51] and therefore the vibronic coupling is needed for the adequate description of the magnetic characteristics and profiles of the intervalence optical bands. The square planar moiety can be a part of a more complex system, like polyoxovanadate [V12As8O40(HCO2)]n– (n = 3,5) [41, 43] whose magnetic properties are closely related to the interplay between the exchange interactions in localized and delocalized tetranuclear subunits [41, 43].
The main interactions
To model the molecular systems suitable for use as four dot mQCA we will consider the structures exemplified by the tetra–ruthenium MV complexes suggested in Ref.  to encode the binary information. These MV species based on the two Creutz-Taube complexes assembled in the center-bridged and side-bridged square-planar tetramers within which two holes are shared among the four sites as schematically shown in Fig. 2. The four sites A, B,C and D will be enumerated as shown in Fig. 3 along with the molecular coordinate frame. The following consideration is applicable to the both cases. The DFT calculation of these systems is presented in Ref. .
The following consideration is applicable to delocalized holes or electrons and for the sake of definiteness we will refer to the electrons. The Hamiltonian of the system is assumed to consist of three terms:
Here He is the electronic (Hubbard type) Hamiltonian, including all interactions between electrons and ions provided that the reference configuration of the ions is fixed to the full-symmetric one (D4h). This reference configuration will be specified below in more detail. The next two terms (they will be defined later) are the Hamiltonians of the free vibrations and the electron-vibrational interaction V. The vibronic interaction [49, 50] is known to produce bistability in MV system which leads to a barrier between the two charge distribution and therefore acts as one of the key factors in the mQCA action.
The main factors controlling the electronic spectrum (eigen-system of the Hamiltonian He) are the following:
Coulomb repulsion between the two itinerant electrons which can be instantly localized either at the adjacent sites (along the sides of the square) or at the remote sites (antipodal positions) as shown in Fig. 4a. The Coulomb repelling forces of electrons tend to localize them as far as possible in order to minimize the energy, i.e., on the sites located at opposite vertices. These two localized configurations which will be referred to as distant pairs form the ground manifold. The difference in the energies of the Coulomb repulsion between the distant pairs and the neighboring ones (d- and n-pairs, the last have four possible configurations) will be denoted as U as illustrated in Fig. 4a, b. The charge configurations having the lowest energies are assumed to encode the binary information in nQCA.
The transfer of the two electrons among the four sites. Only the one-electron transfer processes will be taken into consideration. The electron transfer parameters tn and td correspond to the jumps between the distant and adjacent sites respectively (Fig. 3). The parameters of the isotropic exchange interaction acting in the localized configurations are normally smaller than the transfer integrals in MV compounds and therefore the magnetic exchange will be neglected. There are some additional interactions mentioned in Ref.  but they are expected to be relatively small (like bi-electron transfer, etc.) and therefore will be ruled out from the present consideration.
The energy levels are labeled by the irreducible representations (irreps) of the point group D4h of the molecule and by the quantum number of the full spin of the system (S = 0 and S = 1 for two electrons). Group-theoretical classification (see ) enables to classify the allowed terms (spin and irrep) of MV clusters with due account of the Pauli principle and point symmetry prior explicit evaluation of the wave-function and energy levels. Assuming that each center has an orbitally non-degenerate ground state one can find the following spin-singlet and spin-triplet terms for the d- configurations 1B1g(d), 1A1g(d), 3Eu(d) (the low-lying group of levels) and n-configurations (excited Coulomb states) 1A1g(n), 1B2g(n), 1Eu(n), 3A2g(n), 3B1g(n), Eu(n) as indicated in Fig. 4c by symbols d and n. The Hamiltonian He is diagonal in the symmetry adapted electronic basis except the two 2×2-blocks corresponding to 1A1g(d), 1A1g(n) and 3Eu(d), 3Eu(n) pairs originating from the different Coulomb configurations (Fig. 4). Using the well developed technique  the wave-functions corresponding to the allowed spin multiplets have been expressed in terms of the bi-electronic determinants. The energy pattern evaluated within the model involving the main interactions so far indicated is given in Table 1 in which the energy of low-lying group is chosen as the reference point.
The first general conclusion one can draw from this consideration is that the electron transfer processes in a bi-electron system separate spin triplets and spin singlets resulting thus in an effective magnetic interaction between the electrons. This kind of the effective magnetic exchange caused by electron delocalization has been recognized in our earlier consideration of MV double reduced polyoxometales with Keggin and Wells–Dawson structures which allowed to expound the riddle of the unusual antiferromagnetic properties of these compounds [37–39]. The correlation diagrams (Fig. 5a and b) give the pictures of the levels (in dimensionless units E/Uvs t/U) in the two limiting cases: td = 0, tn ≠ 0 and td ≠ 0, tn = 0. The case of a strong transfer between the distant sites can probably be relevant to the center-bridged (Fig. 2a) MV compound although in this case the transfer between neighboring site can not be excluded as the bridge can mediate also the electron transfer of this type. On the contrary, for the side-bridged mixed-valence (Fig. 2b) the transfer between neighboring sites is undoubtedly dominant.
In the case of strong transfer between neighboring sites (td=0, tn ≠ 0) the ground state is accidentally degenerate and is represented by a paramagnetic mixture of spin triplet and spin singlet terms. In both cases 5a and b the energy patterns are symmetric with respect to the signs of td and tn respectively. In the general case (tn ≠ 0 and td ≠ 0) the energy pattern (Fig. 5c) proves to be more complicated and it is important to emphasize that it becomes asymmetric. Providing td/|tn| < 1.6 (at U/|tn| = 2) the ground level is spin singlet 1A1g while at td/|tn| > 1.6 the ground state involves spin singlet and spin-triplet and exhibits accidental degeneracy (1B2g,3A2g) as in the case in Fig. 5a. Therefore, in general case (tn ≠ 0, td ≠ 0) the magnetic properties are essentially dependent on both, the values and the signs of the transfer parameters. It is remarkable that the ground state of the system can be either diamagnetic or represented by a paramagnetic mixture of spin-singlet and spin-triplet but can never be ferroramagnetic.
Low lying levels in the limit of strong coulomb repulsion
Let us consider a region of strong inter-site Coulomb repulsion U which is most likely the realistic case just actual for utilization of the system as mQCA cell. Assuming thus that U>> |tn|, |td| the energy levels (Table 1) belonging to d-configurations can be approximately expressed as:
These low lying group of levels is well separated from the group of excited ones (by a large gap ∝U) and represented as a series in terms of small parameters |tn|/U, |td|/U. One can see that the mixing of d- and n-configurations through the transfer processes stabilizes and at the same time splits the d-levels of interest as shown in Fig. 6 in which the absolute values of the gaps [defined in Eq. (2) are denoted as Δ and Δ1]. The main contribution to the gaps is produced by the tn- transfer which is active in the two virtual steps (corresponding to the 2nd order perturbation theory) of switching between the two d-localizations of the pair. For this reason the corresponding splitting is independent of the sign of tn in compliance with the general level diagram (Fig. 5b). This is schematically shown in Fig. 7a illustrating a two-step processes through the intermediate excited Coulomb levels which are virtually achieved via the tn-jump at the step 1 and then the system is converted to the final d-configuration at the step 2 through the tn-processes. On the contrary, the td-transfer switches only between d- and n-configurations and therefore appears only in the next term of expansion in Eq. (4), corresponding to the combined three-step processes (involving two tn- and one td- jumps) as is clear from Fig. 7b. This first non-vanishing term containing td parameter is also independent of the sign of tn but is linear (i.e. odd) with respect to td. Therefore the td transfer processes stabilize (td < 0) or destabilize (td > 0) the level 1A1g giving rise to either antiferro- or ferromagnetic contributions (at non-zero temperature) correspondingly.
Notably, both kinds of the electron transfer are not operative within the d-configurations as was observed in the case of the bi-electron reduced Keggin and Wells–Dawson anions [37–39]. In particular, the td- transfer transforms d-configurations into n-ones and therefore is not operative within the d-manifold. Therefore, in the strong Coulomb repulsion limit the contribution of the td transfer is much smaller than that of tn as follows from Eq. (2). If the correction ∝td is neglected the levels of the d-configurations are equidistant (that means Δ = 2Δ1).
Vibronic model for mixed-valence square-planar species: adiabatic picture in strong coulomb repulsion limit
We have seen that the Coulomb interaction effectively reduces the switching rate between d-configurations (at least by a small |tn|/U factor) but the switching processes remain barrierless because the states of the excited n-configurations are virtual (they participate in the second order processes as specified in Fig. 7). In other words the Coulomb energy U can not be associated with the height of the barrier between the two charge configurations of interest. In general, the quantum electronic states can be referred to as “full symmetry” states and therefore the problem of localization is conceptually out of the scope of this approach. This means that the probabilities of populations Pi of the sites are equal in the stationary quantum electronic states (PA = PB = PC = PD = 1/4) and as a result the preference in a certain charge configuration can be achieved only by applying an external electric field. On the contrary, the vibronic coupling which will be considered in this Section is known to produce the localization effect resulting (under due conditions) in the long living quasi-stationary states (“broken symmetry” states) separated by a barrier. If the barrier is high enough the quantum tunneling is considerably suppressed and therefore the localized states can be employed to encode information in mQCA and/or drastically influence the time of switching in mQCA operation. To rationalize this concept and to visualize the physical picture we will employ here the adiabatic approximation although the efficient symmetry assisted quantum-mechanical vibronic approach has been recently developed and applied to treat the magnetic and optical properties of complex MV systems [55–57].
In the full Hamiltonian, Eq. (1), the second term describes the free harmonic vibrations of the system, Qi are the dimensionless vibrational coordinates, ωi are the vibrational frequencies. We shall employ the conventional vibronic model for MV systems formulated by Piepho, Krausz and Schatz [49, 50] (referred to as PKS model) which deals with the independent “breathing” displacements of the ligands around the sites of the electron localization. Using the projection operator  one can obtain the following explicit expressions for the PKS coordinates for a square-planar tetrameric system belonging to the point group D4h:
In Eq. (3) Qα (α = A1g, B1g, Eux, Euy) are the symmetry adapted PKS coordinates (actually, the normal coordinates) composed of the four “breathing” vibrations Qi of the sites (i=A, B, C, D). The PKS coordinates (or so called reaction coordinates) are schematically shown in Fig. 8 for the phase Qα > 0 [see Fig. 8 and Eqs. (3) in which the local displacements are interrelated with the collective coordinates] and later on we shall use the notation The vibronic coupling [the term V in Eq. (1)] is linear with respect to the vibrational coordinates as adopted in the PKS model. The interaction of the electrons with these vibrations results in the mixing of the electronic states with the same spin which is specific for the Jahn–Teller (JT) or pseudo JT effect in MV systems. This mixing can be represented in the matrix form as following:
In Eq. (4) υi is the coupling parameter (having dimension of energy) of the “extra” electrons with the vibrational mode i. Taking into account the assignment of the electronic states (Section 2) one can formulate the independent pseudo JT vibronic problems for spin-singlets and spin-triplets terms which can be symbolized as follows: (21A1g + 1B1g + 1B2g + 1Eu) ⊗ (b1g + eu) and (3A2g + 3B1g + 23Eu) ⊗ (b1g + eu) where we use small letters for the vibrational irreps. The matrices Oi are defined in the basis of the electronic wave-functions expressed in terms of the bi-electronic Slater’s determinants (we omit here the details and intermediate results) belonging to the definite irreps for spin triplet and spin singlet terms so far indicated. Within the PKS model the frequencies and the vibronic parameters prove to be equal for all active vibrations (because they originate from the couplings of the same strength with the breathing modes at the equivalent centers) and will be denoted as υ and ω correspondingly. The coupling with the full-symmetric mode (A1g) is proportional to the unit matrix and therefore is eliminated from the further consideration by a due shift of the equilibrium position along this coordinate which does not affect magnetic and optical properties. More precisely, this shift means that the presence of the “extra” electrons changes the sizes of the sites (increase them due to additional repulsion between the electrons and negatively charged ligands) so that the real reference configuration (left image in Fig. 8) for the non-symmetric coordinates (B1g, Eu) is defined with the due account for this change.
Under the condition of strong Coulomb repulsion (approximate estimate could be υ2/ℏω << U) the vibronic problem can also be restricted to the ground manifold and therefore is reduced to the (1B1g + 1A1g) ⊗ b1g and 3Eu ⊗ b1g sub-problems. This means that a pair of the S = 0 levels (1B1gand 1A1g) are mixed with the only vibration b1g and the same vibration is active in the S = 1 term 3Eu. One can see that the e- modes do not act within the ground manifold so that the vibronic problem in the limit of strong U proves to be one-dimensional and therefore it is essentially simplified. It is to be noted that the vibronic mixing between the d- and n-terms is neglected which is expected to ensure a good enough accuracy providing large U and at the same time allows to judge about the localization of the electronic pair. To build up the vibronic matrices Oi one can use the zeroth-order wave-functions, i.e., the wave-functions for 1A1g(d) and 3Eu(d) belonging to the definite (d) charge configuration while 1B1g(d) is not mixed with the excited states. Actually, this means that a small (providing U >> |tn|, |td|) partial delocalization of the electronic pair arising from the admixture of n-states is assumed to have a negligible influence on the vibronic parameter in the ground manifold.
Using the vibronic matrices (that have been explicitly calculated) one can find the following expressions for the adiabatic potentials:
In Eq. (5) the functions ε±(Q) are the two branches of the adiabatic potential of the (1B1g+1A1g) ⊗ b1g problem which is mathematically equivalent to that of the one-electron dimer  (two levels mixed by the only vibration) while the 3Ee ⊗ b1g coupling leads to the two intersecting parabolas εx,y(Q). Figure 9 shows the composition of the adiabatic potentials ε±(Q) and εx,y(Q) for some qualitatively different cases. Depending of the interrelation between the gap Δ and the strength of the vibronic coupling υ the lower sheet ε–(Q) can have either the only minimum at Q± = 0 (providing weak coupling, υ2 < Δℏω/2, Fig. 9a–c) or two minima in the points (strong coupling, υ2 > Δℏω/2, Fig. 9d–f). The energy of the only minimum of ε–(Q) in which the electronic pair is fully delocalized is just the electronic energy (–Δ) while the energies of the two equivalent minima at Q± are –υ2/2ℏω–Δ/2–Δ2ℏω/8υ2. As one can see from Fig. 10 the electronic pair produces a molecular distortions, namely, expansion of the B, D sites with simultaneous compression of A,C sites at Q = Q– and compression of the B, D sites with simultaneous expansion of A,C sites at Q = Q+. This is an evidence of the vibronic self-trapping which produces also a barrier between two localized states in antipodal positions. The qualitative estimation of the degree of localization of the electronic pair in the spin-singlet states of a terameric unit is quite similar to that conventionally adopted for the MV dimers in which one extra electron is shared between the two sites. The last is based on the Robin and Day classification scheme [49, 50] which is applicable to the spin-singlet states of a terameric unit.
The potential curves εx,y(Q) represent two intersecting parabolas and exhibit two minima irrespectively of the value of the vibronic coupling (static JT effect) which are disposed at Qx = –Qy = υ/ℏω and have the energies –Δ1 – υ2/2ℏω. These two minima correspond to the localization of the electronic pair in the antipodal positions along the molecular X and Y axes (“broken symmetry” states). In this case of the static JT effect the tunneling between the minima is strictly forbidden so that the spin-triplet electronic pair is fully localized in each of the two minima.
It is convenient to introduce a dimensionless parameter κ = (2Δ1 – Δ)/Δ1 which is expressed through td as κ = 4td/U and serves as a measure of the position of the spin-triplet 3Ee within the gap between two spin-singlets 1B1g and 1A1g (providing κ = 0 the spin-triplet is located in middle of this gap). In the framework of the adiabatic approach the minima of the potential curves can be roughly associated with the energy levels of the system. Fig. 9a–c show that weak vibronic coupling (υ/ℏω = 1.0) effectively reduces the singlet-triplet gap (1A1g, 3Eu) from its value εx,y(Q = 0)–ε_(Q = 0) = Δ–Δ1 in the electronic spectrum to εx,y(Q = Q±)–ε_(Q = 0) = Δ – Δ1 – υ2/2ℏω (gap between the minima) giving rise to a ferromagnetic contribution. Additionally, the vibronic gap depends on the value and sign of κ (Fig. 9b, c) which reflects the role of the td processes in the disposition of the electronic levels.
The most interesting case of strong vibronic coupling is illustrated in Fig. 9d and e. In the limit of strong coupling (υ/ℏω = 3.0 in Fig. 9d–f) providing κ = 0 (Fig. 9d) the minima related to the spin-singlet and spin-triplet states become degenerate. The minima corresponds to the localization of the charges along the diagonals of the square. This is a result of the full quenching of the tn processes by a strong vibronic interaction which leads to the localization of the electronic pair and consequently to the paramagnetic ground state. On the contrary, when the td processes are also involved (κ ≠ 0) the quenching of the transfer is incomplete that lifts the degeneracy of the minima even in the limit of strong vibronic coupling. Providing td < 0 the minima of spin-singlet states are deeper than those for the spin-triplet states (Fig. 9e, κ = –0.86) although the singlet-triplet gap in the electronic spectrum (at Q = 0) is reduced. In the case of td > 0 (Fig. 9c, κ = 0.52) the strong enough vibronic coupling leads to the spin-crossover of the levels so that the spin-triplet state becomes the ground one. In the last case the system is locked in the minima. One can see that the magnetic properties of the system are closely related to the interrelation between the two relevant transfer parameters and qualitatively and quantitatively are affected by the vibronic coupling.
We have considered the mechanisms of the vibronic self-trapping of charge polarized states in a square-planar MV systems proposed as a charge container in the four-dot mQCA. These systems are exemplified by the two kinds tetra–ruthenium [2Ru(II)+ 2Ru(III)] clusters (assembled as two coupled Creutz–Taube complexes). We employ the model which takes into account two relevant kinds of the electron transfer processes as well as the inter-site Coulomb repulsive energies in the different instant positions of localization. The electron transfer processes in a bi-electron system separate spin triplets and spin singlets resulting thus in an effective exchange interaction between the electrons. It is shown that depending on interrelation between the electron transfer parameters in the absence of the vibronic coupling the system can be either diamagnetic or paramagnetic. The vibronic self-trapping is considered within the conventional vibronic PKS model adapted to the bi-electronic MV species with the square topology. The adiabatic potentials are evaluated for the low lying Coulomb levels in which the antipodal sites are occupied, the case just actual for utilization in mQCA. It is essential that the conditions for the vibronic localization are spin-dependent. These conditions for spin-singlet and spin-triplet states are revealed in terms of the two actual transfer pathways parameters and strength of the vibronic coupling. It is demonstrated that the spin-singlet states can be either localized or delocalized while the spin-triplet state always localized and the tunneling between the minima is forbidden.
We could demonstrate that the magnetic properties of the system are closely related to the interrelation between the two relevant transfer parameters being at the same time qualitatively and quantitatively affected by the vibronic coupling. In particular, the vibronic coupling leads to the spin-crossover of the adiabatic levels. The interrelation between the magnetic nature of the ground vibronic state and the localization/delocalization features can be used for the control of the cell by an external magnetic field.
The results so far mentioned are based on a simplified model of strong Coulomb repulsion and adiabatic approximation which allowed us to simplify and visualize the results. In future study we plan to extend the adiabatic model employed in this article beyond the limit of strong Coulomb repulsion. In fact, the vibronic coupling in MV compounds can be strong enough as compare to the Coulomb gap as indicated by the fact of incomplete reduction of the electron transfer in the employed model. Although the adiabatic approximation is an appropriate tool to get insight into the effects of electron correlation and localization vs delocalization this approximation is inapplicable (due to fast non-adiabatic Landau–Zener transitions in the area of avoiding crossing of the adiabatic potentials) to the adequate description of the intervalence optical bands. Moreover, the discrete structure of the lines which peculiar to the quantum-mechanical description is missed in the semiclassical approach. Therefore, we intend to solve the dynamic JT problem for these systems with the aid of the recently developed approach [55–57]. Finally, the developed model could be a good background for the description of the dissipative dynamics in MV systems  closely related the heat release in the switching processes. We hope to discuss this in the forthcoming paper.
B.T. acknowledges financial support of the Israel Science Foundation (ISF, grant no. 168/09). B.T. and A.P. are grateful to COST Action CM1203 “Polyoxometalate Chemistry for molecular Nanoscience (PoCheMon)” for supporting this work. A.P. acknowledges the STCU (project N 5929) and the Supreme Council for Science and Technological Development of the Republic of Moldova (Project 11.817.05.03A) for financial support. J.M.C-J thanks EU (Project ELFOS and ERC Advanced Grant SPINMOL), the Spanish MINECO (grants MAT2011-22785), and the Generalidad Valenciana (Prometeo and ISIC Programmes of excellence) for the financial support of this work.
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