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This work advances the modeling of bondonic effects on graphenic and honeycomb structures, with an original twofold generalization: (i) by employing the fourth order path integral bondonic formalism in considering the high order derivatives of the Wiener topological potential of those 1D systems; and (ii) by modeling a class of honeycomb defective structures starting from graphene, the carbonbased reference case, and then generalizing the treatment to Si (silicene), Ge (germanene), Sn (stannene) by using the fermionic twodegenerate statistical states function in terms of electronegativity. The honeycomb nanostructures present
With the irresistible rise of graphene, great attention has been paid by the scientific community to the spectacular properties of this carbon monolayer, the “Nobel prized” new carbon allotrope which ‒ a decade after its discovery in 2004 [
From a general perspective, ten years of investigations on graphenic honeycomb lattices point out the scientific relevance of
The natural candidate for such a class of material is
In this context therefore a comprehensive treatment of the electron behavior in nanostructures made of GroupIV elements is highly necessary and represents in fact the main scope of this work. Our original approach is based on the properties of the
In this section the main characteristics of 1D nanoribbons made of GroupIV elements are briefly presented. Like graphene, these systems exhibit the genuine honeycomb structure given in
(
Such extended Si nanoribbons are built by the action of the twocomponents
The chemical stability of buckled honeycomb structures is substantially granted by the “puckering induced” dehybridizationeffect which allows
Remarkably, silicene and SiNR exhibit, like graphene, massless relativistic Dirac fermions arising, for the nanoribbons case from the 1D projection of π and π* Dirac cones [
The electronic properties of 1D nanoribbons are discussed here by considering the recent concept of bondon, the new bosonic quasiparticle arising from the Bohmian quantum picture applied to the quantization of the chemical bond [
In Equation (1) the energy and the proper length of action obey to the Heisenberg analogous relationship [
This description of the chemical bonding has been recently applied to extended nanostructures (
One considers a particle (the bondon) with mass
In terms of the classical path dependence connecting the endpoints
However, Equation (4) can be further simplified by using the Gauss theorem (see
At this point one implements the bondonic information regarding the mass quantification in the valence state (the first or “ground” state of the bonding spectra) in terms of bonding energy and length of Equation (1),
Now, facing the superior potential first, second, and fourth order derivatives, they can be systematically treated by replacing them with associate topological invariants and higher orders over the concerned bonds, networks or lattices,
Nevertheless, attention should be paid at this passage from physical to topological quantities since it actually replaces
Next, one should fix the energylength realm of the bondon in the 0^{th} order of the partition function which renders the classical observability by the involved thermal length, here mapped into the topological space and energy so defining the bondonic unitary cell of action [
Then, Equation (8) is used for internal energy computing of the bondon as the average energy condensed in the network responsible for bonding at periodicalrange action:
It immediately fixes the longrange length of periodic action of bondon by recalling Equation (2):
Remarkably, when the asymptotic limits are considered for both periodic energy and length of bondon, one sees that they naturally appear associated with the topological potential and with the Coulombian interaction for the lowtemperature case, while rising and localizing the bonding information (like the deltaDirac signal) for the hightemperature range, respectively, being the last case an observational manifestation of bondonic chemistry. This feature will be used in a moment below.
Returning to the full partition function now the bondonic periodicity information on length and energy action maybe included to rewrite Equation (6) to the actual form:
However, for workable measures of macroscopic observables, one employs the partition function of Equation (11) to compute the canonical associated partition function according with the custom statistical rule assuming the
with the help of Equation (12) one is provided with the canonical (macroscopic) internal energy contributed by
Finally; by continuing the inverse thermal energy derivatives; the internal energy of bonding of Equation (13) may be employed also for estimating the allied caloric capacity:
The treatment of pristine (“0”)–to–defect (“D”) networks goes now by equating the respective formed caloric capacities from Equation (14) towards searching for the
Now one may use the above mentioned high temperature regime, (
In the next section, this model will be applied to the study of the bondonic properties of graphenic (Cbased), silicenic (Sibased), germanenic (Gebased), and stannenic (Snbased) nanoribbons with StoneWales defects.
Here we will progress on the investigations of SW defects in graphene and related layers, as silicene germanene, and stannene, by analyzing the
Propagation of the StoneWales wavelike defect along the zigzag direction caused by the insertion of pairs of hexagons at
The topological skeletons of the systems considered in the present article are basically represented by a mesh of fused hexagons entirely paving the nanoribbons (
From the topological perspective, the key concepts applied in comparing graphene with silicone and related honeycomb networks’ properties are basically two:
First, the two 57 pentagonheptagon units SW constituting the SW defect, also called 5775 dipole, are
Such a structural modification reflects
The second conceptual instrument used in the present analysis regards the physicaltotopological passage introduced by Equation (7):
The evolution of the nanoribbon defective structure is controlled by a pure
Equally important,
It is worth remembering here the “basal properties of distancebased topological potentials“ making those mathematical object exceptionally suitable for determining delocalized bondonic properties:
Actually, a practical introduction to lattice topological descriptors is provided by looking to the nanoribbon structure in
The invariant Equation (17) provides a powerful rank of isomeric chemical graphs, privileging the most compact structures [
Systems like graphene and, to some extent, the related ones, including silicene, that are rich in
In case of large structures only the first terms
Interested readers may find the formal derivation of Equation (19) contributions and related asymptotic properties in the original work [
Numerical values abstracted from topological potentials of Equations (21)–(24) then used to generate the interpolations polynomials of Equations (27)–(34) as a function of the
η  W^{[}°^{]}  W^{[1]}  W^{[2]}  W^{[4]} 

0  4,467,960  40,453,200  267,186,000  1,410,130,000 
0.2  4,466,600  40,424,600  266,868,000  1,407,660,000 
0.4  4,465,060  40,392,500  266,508,000  1,404,880,000 
0.6  4,463,470  40,359,500  266,134,000  1,402,000,000 
0.8  4,461,900  40,329,100  265,764,000  1,399,170,000 
1  4,460,420  40,305,200  265,416,000  1,396,500,000 
2  4,455,370  40,542,500  264,226,000  1,387,400,000 
3  4,453,620  42,849,300  263,807,000  1,384,160,000 
4  4,452,140  51,493,100  263,439,000  1,381,240,000 
5  4,450,930  74,805,700  263,132,000  1,378,770,000 
The supercell in
Synopsis of the toporeactive parameters for the defects instances (starting from pristine net at the step
Defect Step  Instant Structure  Electronic Energy (eV)  Total Energy (eV)  Binding Energy (eV)  Parabolic Energy (eV) 

2858.69979  2595.306  7308.17  13,063.1207  
2425.90314  2409.47  7494.0069  102,344.109  
2641.35644  2410.023  7493.4534  100,091.3384  
90063.404  10331.43  −427.9522  6770.427006  
2428.23769  2408.129  7495.3472  102,353.338  
2484.84517  2468.133  7676.8912  107,394.1399  

0.00384593  0.000846 

0.01614977  

0.00029961  7.01 × 10^{−5} 

0.001488221  

6.4681 × 10^{−5}  1.42 × 10^{−5}  0.000271766  

1.2299 × 10^{−5}  2.71 × 10^{−6}  5.16926 × 10^{−5}  

0.21643315  0.622413 

0.727612957  

0.16519269  0.538169 

0.777077187  

0.21568411  0.621567 

0.725935217  

0.21522938  0.621048 

0.724864925 
The appropriate
Equivalently, within the frozen core approximation or by Koopmans’ theorem [
Accordingly,
Also interesting, the parabolic based chemical reactivity analysis furnishes the second best results after the pure binding energy correlations; this behavior justifies both the pro and contra regarding its use in modern chemical reactivity theory, namely:
the proargument, largely advocated by Parr works in last decades of conceptual chemistry research with application in inorganic and organic reactivity alike [
the contraargument, defended by late Szentpaly works on various inorganic systems [
Therefore, although valuable, the parabolic reactivity calibration is also by this approach taken over by the cute binding energy for the correlation coefficients with topological potentials in
the energetically calibrated topological potentials for the forming SW defect instance (still corresponding to the “0” structure) within [01] range of the
The polynomials for the topological potentials describing the SW waves still corresponding to the defective “D” structures) within [05] range of the
It is worth evidencing the advantage of this procedure which effectively allows an easy energetic calibration and a separate description of the 0forming and Dpropagating steps of the SWw defect by providing the associate polynomials that are: (i) energetically realistic and (ii) with “equal importance” despite the different information contained: see for instance the numeric form of the fourth order topological potential Equation (24) respecting those provided by Equations (30) and (34). This computationallyconvenient method assures that higher order topological potentials will contribute in providing the bondonic related quantities of Equations (13), (14) and (16).
Nevertheless, all the present computational algorithms were implemented for graphenic structures, having the carbon atom as the basic motive; however they can be for further used in predicting similar properties also for similar atomic group like Si, Ge, Sn, through appropriate topological potential factorization depending on the displayed reactivity differences; since such differences are usually reflected in gap band or bonding distance differences, one may recall again the electronegativity as the atomic measure marking the passage from an atomic motive to another keeping the honeycomb structure. The influence of the lattice will be implemented by considering the (electronegativity dependant) function of the fermionic statistical type with 2degeneracy of states spread over the graphenic type lattice–taken as a reference. Such a function accounts for the electronic pairing in chemical bonding is analytically taken as:
Numerically, Equation (35) features the factorization with unity for CC bonding, while departing to fractions from it when the Group AIV of elements are considered as motives for honeycomb lattices with graphenic reference: SiSi honeycomb bonding will carry statistically the Si atomic electronegativity χ(Si) = 4.68 [eV] in Equation (35) with X = Y = Si, and successively for GeGe with χ(Ge) = 4.59 [eV], and SnSn with χ(Sn) = 4.26 [eV] for the corresponding silicone as well as for similarly designed germanene and stannene nanoribbon structures. Note that atomic electronegativity were considered within the Mulliken type formulation of ionization potential and electronic affinity as in the first term of Equation (26); furthermore, their geometric mean was “measured” against the referential graphenic CC chemical bonding, while their difference was normalized under exponential of Equation (35) to the socalled “universal” geometrical averaged form of Parr and Bartolotti,
Going to have the final and most important part of discussion of the obtained bondonic observable properties through the present fourth order topologicalpotential formalism, they will be displayed through jointly implementing the short and medium (
One starts with the topoenergetic Wiener potentials of Equations (27)–(34) with representations in
The topological potentials modeling the forming (“0to1”) step of SWw are all monotonically descending, meaning their eventual release into defective structures;
The topological defective potentials have quite constant behavior over the entire computational
The Dto0 difference shows some short range fluctuations for all potentials unless the first order one, yet ending into the Dpotential definite rising barrier on the long range behavior; the difference for the first order potential displays such energetic barrier rise just from the short range, paralleling the defective “D” shape;
Concerning the C > Si > Ge > Sn potential (paralleling the electronegativity) hierarchy, one sees that the CtoSi large energetic gap for pristine “0” structure is considerably attenuated for the “D” propagation of the SWw defect, manifested especially for second and 4th order, while the SitoGe energetic curves almost coincides for these orders; even more, all Si, Ge, and Sn shapes are practically united under first order defective potential “D1”.
The Wiener based topological potentials of Equations (27)–(34): from top to bottom in successive orders and from left to right for the forming (“0”) SW and for propagating (“D”) of the SW defects on the medium (
The critical (left column), forming (middle column) and transforming (right column) of SWw in
These topological potential features stay at the foreground for further undertaking of the remaining observable properties in a comparative analysis framework. As such, when analyzing the critical “temperature” through the inverse of the thermal energy of Equation (16) one actually gets information on the phase transition
The general feature for the
The situation changes on the long range “echo” when the critical signal may be recorded closer to the defective than to the pristine structures, when one should record also a shrink signal pulses gap between the CtoSitoGetoSn;
The differences between the criticaltodefectivetopristine structures’ pulses shapes follows on a short range the generally recorded topological potential difference in that range, while noticing definite cupolas for the long range behavior–especially on the critical regime, meaning that indeed the SWw echo is disappearing after about 50 isomeric topological transformation of the considered honeycomb nanosystem (see the right bottom line picture of
It is worth noting that the
The bondonic “length” for SWw in forming (left column), defective propagation (middle column) along their differences (right column) behavior: on short (upper row) and long (lower row) ranges, upon considering critical information of Equation (16) into Equation (10).
Even more, these times are in fact the bondonic times for concerned lattices whose periodic radii of action is determined upon considering the
The identical boning length for pristine and defective structures on the short range transformations (up to seven topological rearrangements paralleling the SW wave propagation into extended lattice); notably, the bondonic lengths are correctly shorter than the detected or previously estimated bonding length for CC bond (in graphene) and elongated in SiSi bond (in silicene), see the Introduction, since the bondonic agent nature, in assuring the bonding action for the basic atomic pairing in honeycomb nanoribbons.
The decrease of boning length and of the consequent action on the long range dynamics, paralleling the decreasing of the interelongation difference in bonding for CtoSitoGetoSn;
The prediction on the bondonic longest “echo” in an extended lattice, limited to 50 transformations, or dipole extensions steps continuing the
The Dto0 bondonic length differences parallels those recorded for the criticaltoD one found for the
Sidebyside canonical internal energies of bondons in honeycomb supercells of
Passing to the canonical measures one has in
No shape differences other than the overall scales along a quasi invariant energygap between CtoSitoGetoSn related lattices are recorded between the [II] and [IV] order pathintegral bondonic formalisms, with natural higher energetic records for the later approach since more interaction/interconnection effects are included, for the internal energies of honeycomb lattice without and with topological defects as triggered on the short range as in
The previous situation changes for the long range SW dipole transformations, noticing the same type of energetic increase as for the forth order topological potential/barrier in
The Dto0 differences are nevertheless replicating the defective behavior for both the [II] and [IV] order analysis on the long range, while showcasing some different types of fluctuations and inversions along the CtoSitoGetoSn honeycomb nanosystems for the shortrange of SW dipole evolution;
with special reference to [IV] short range behavior one notes the the pristine “0” state is still present as an “echo” over the defective “D” state, due to its energetic dominance, that nevertheless has the contribution in replicating “the learning” mechanism of generating SW defects in between each short range steps as was the case in between η = 0 and η = 1; remarkably, this may have future exciting consequences in better understanding the cellular morphogenesis by “replicating the learning” machinery of the StoneWales transformation, found to be present also at the celllifecycle phenomenology, see [
Going to the last but the most “observable” quantity which is the caloric capacity of Equation (14) within the present [IV] order path integral–bondonic approach, one has the results, and comparison with the previous [II] order formalism of reference [
from the scale values, one obtains actual quite impressive accordance with the previously calculated or predicted values for the graphene and silicone networks: take for instance just the pristine “0” output, in [IV] other environment; for it one notes the constant results about
As previously noted the “D” effect is to shrink the energetic gap between CtoSitoGetoSn lattice structural behavior, respecting the “0” pristine or defect forming transition state;
The short range DtoO differences closely follow the previous internal energy shapes of
For the long range behavior, instead, what was previously a parabolic increase in internal energy acquires now a plateau behavior in all [IV] and [II] order representations: “0”, “D”, and their “Dto0” differences; nevertheless it seems that the “echo”/signal about η = 10 is particularly strong in [IV] order modeling of Dto0 differences in caloric capacities, while we notice for the “D” state the graphenic apex curvature about η = 7 followed by that of silicone at η = 10, in full consistency with above bondonic energetic analysis (
The same type of representations as in
The present results fully validate bondonic analysis as a viable tool for producing reliable observable characters, while modeling and predicting the complex, and subtle, chemical phenomenology of bonding in isomers and topological transformations in the space of chemical resonances. Further works are therefore called in applying the present algorithm and bondonic treatment for other nanosystems as well as in deep treatment for the symmetrybreaking in chemical bonding formation of atomsencountering in molecules and in large nanosystems.
The current article aims to contribute in advancing the fascinating
Highly intriguing novel properties are theoretically derived here, namely: the topological potentials up to the fourth order, the so called betasignal accounting for the time scale of bondonic pulses in a lattice supercell of graphenic type, the associate length of action, along the total internal energy of topological isomers and the remarkable behavior of the caloric capacity of the nanosystems. These quantities were evaluated and discussed for a critical regime by modeling the phase transition from pristine to defective nanoribbons with StoneWales dislocation dipoles and the creation of isomeric bondons; antibonding particle creation have been also described.
As an overall comment, the ability to indicate peculiar scalethreshold (like the
Nevertheless, future applications triggered by the present study of graphenic systems are extendible to the design of reactive supports for pharmaceutical and cosmetic compounds, due to their unique electronic, magnetic and chemical saturation properties (recognized by the Nobel Prize in Physics in 2010), while the technological passage downward GroupIV elements is expected to enrich the
Finally, the bondonic quantum condensate distribution picture allows the computation of the energetic (observable) energies involved in the isomeric nanostructures with the exciting perspective of simulating the creation and dissipation of SWw defects in the grapheniclike regions characterizing the surface of
This work was supported by the Romanian National Council of Scientific Research (CNCSUEFISCDI) through project TE16/20102013 within the PN IIRUTE20101 framework.
M. V. Putz set up the bondonic phasetransition algorithm, and performed the input, parabolic energetic and critical regime calculations for pristine to defective nanolattices; O. Ori performed the topological calculation and supervised their further implementation. Putz and Ori jointly created and corrected the text to the final manuscript.
The authors declare no conflict of interest.
Path integrals quantum formalism represents the viable integral alternative to the differential orbital approaches of manyelectronic systems at nanoscale; accordingly, by employing path integral formalism one actually avoids the cumbersome computation and modeling of the orbital wavefunction. Current method is therefore best suited for the path integral approach to the extended systems in which topological and bondonic chemistry appropriately describe longrange structures and the longrange interactions, respectively.
Accordingly, one looks for evaluating the spacetime quantum amplitude/propagator (
Note that we have maintained here the physical constants appearances so that the semiclassical expansion procedure is clearly understood in orders of Planck’s orders, see below. As such, the path integral propagator representation can be summarized as:
All in all, the semiclassical form of path integral representation of evolution amplitude looks in the fourth order of Planck constant’s expansion:
Now, we are faced with expressing the averaged values of the fluctuation paths in single or multiple time connection,
However, by involving the pairwise (Wick) decomposition of the
One can easily obtain the higher orders of correlations, however observing that all connected orders of events are reduced to the combinations of
With
With the Heaviside stepfunction:
With expressions (A14) and (A15) back in Equations (A8), (A11)–(A13) some imaginarytime integrals vanish, namely:
while for the nonvanishing imaginarytime integrals appearing in Equation (A7), one obtains:
With these, the earlier form Equation (A7) takes the particular expression for the propagator as in Equation (3), for
One may apply the Gauss theorem successively for integrals of gradient of a given long range defined quantity (