# The Bondons: The Quantum Particles of the Chemical Bond

^{1}

^{2}

## Abstract

**:**

_{B̶}), velocity (v

_{B̶}), charge (e

_{B̶}), and life-time (t

_{B̶}). This is quantized either in ground or excited states of the chemical bond in terms of reduced Planck constant ħ, the bond energy E

_{bond}and length X

_{bond}, respectively. The mass-velocity-charge-time quaternion properties of bondons’ particles were used in discussing various paradigmatic types of chemical bond towards assessing their covalent, multiple bonding, metallic and ionic features. The bondonic picture was completed by discussing the relativistic charge and life-time (the actual zitterbewegung) problem, i.e., showing that the bondon equals the benchmark electronic charge through moving with almost light velocity. It carries negligible, although non-zero, mass in special bonding conditions and towards observable femtosecond life-time as the bonding length increases in the nanosystems and bonding energy decreases according with the bonding length-energy relationship ${E}_{\mathit{\text{bond}}}[\mathit{\text{kcal}}/\mathit{\text{mol}}]\times {X}_{\mathit{\text{bond}}}[\stackrel{0}{A}]=182019$, providing this way the predictive framework in which the B̶ particle may be observed. Finally, its role in establishing the virtual states in Raman scattering was also established.

## 1. Introduction

## 2. Method: Identification of Bondons (B̶)

- Considering the de Broglie-Bohm electronic wave-function/spinor Ψ
_{0}formulation of the associated quantum Schrödinger/Dirac equation of motion. - Checking for recovering the charge current conservation law$$\frac{\partial \rho}{\partial t}+\nabla \overrightarrow{j}=0$$
- Recognizing the quantum potential V
_{qua}and its equation, if it eventually appears. - Reloading the electronic wave-function/spinor under the augmented U(1) or SU(2) group form$${\Psi}_{G}(t,x)={\Psi}_{0}(t,x)\mathrm{exp}\left(\frac{i}{\u0127}\frac{e}{c}\aleph (t,x)\right)$$$${\aleph}_{0}=\frac{\u0127c}{e}\sim 137.03599976\left[\frac{\mathit{\text{Joule}}\times \mathit{\text{meter}}}{\mathit{\text{Coulomb}}}\right]$$
^{–19}Joules (electron-volts)! Nevertheless, for establishing the physical significance of such chemical bonding quanta, one can proceed with the chain equivalences$${\aleph}_{\mathit{\text{B\u0336}}}\sim \frac{\mathbf{\text{energy}}\times \mathbf{\text{distance}}}{\mathbf{\text{charge}}}\sim \frac{\left(\begin{array}{l}\mathbf{\text{charge}}\times \mathbf{\text{potentialdifference}}\end{array}\right)\times \mathbf{\text{distance}}}{\mathbf{\text{charge}}}\sim \left(\begin{array}{l}\mathbf{\text{potential}}\\ \mathbf{\text{difference}}\end{array}\right)\times \mathbf{\text{distance}}$$ - Rewriting the quantum wave-function/spinor equation with the group object Ψ
_{G}, while separating the terms containing the real and imaginary ℵ chemical field contributions. - Identifying the chemical field charge current and term within the actual group transformation context.
- Establishing the global/local gauge transformations that resemble the de Broglie-Bohm wave-function/spinor ansatz Ψ
_{0}of steps (i)–(iii). - Imposing invariant conditions for Ψ
_{G}wave function on pattern quantum equation respecting the Ψ_{0}wave-function/spinor action of steps (i)–(iii). - Establishing the chemical field ℵ specific equations.
- Solving the system of chemical field ℵ equations.
- Assessing the stationary chemical field$$\frac{\partial \aleph}{\partial t}\equiv {\partial}_{t}\aleph =0$$
- The manifested bondonic chemical field ℵ
_{bondon}is eventually identified along the bonding distance (or space). - Checking the eventual charge flux condition of Bader within the vanishing chemical bonding field [26]$${\aleph}_{\mathit{\text{B\u0336}}}=0\iff \nabla \rho =0$$
- Employing the Heisenberg time-energy relaxation-saturation relationship through the kinetic energy of electrons in bonding$$v=\sqrt{\frac{2T}{m}}\sim \sqrt{\frac{2}{m}\frac{\u0127}{t}}$$
- Equate the bondonic chemical bond field with the chemical field quanta (6) to get the bondons’ mass$${\aleph}_{\mathit{\text{B\u0336}}}({m}_{\mathit{\text{B\u0336}}})={\aleph}_{0}$$

## 3. Type of Bondons

#### 3.1. Non-Relativistic Bondons

_{G}in the form of (5) is decomposed into imaginary and real parts

_{ℵ}carries specific bonding particles that can be appropriately called bondons, closely related with electrons, in fact with those electrons involved in bonding, either as single, lone pair or delocalized, and having an oriented direction of movement, with an action depending on the chemical field itself ℵ.

^{2}=1). We will apply such writing whenever necessary for avoiding scalar to vector ratios and preserving the physical sense of the whole construction as well. Replacing the gradient of the chemical field (26) into its temporal Equation (25b) one gets the unified chemical field motion description

_{t}ℵ)

^{1/2}; however, they may be considerably simplified when assuming the stationary chemical field condition (8), the step (xi) in the bondons’ algorithm, providing the working equation for the stationary bondonic field

_{A},X

_{B}, to primarily give

#### 3.2. Relativistic Bondons

_{k}≡ ∂/∂x

_{k}and the special operators assuming the Dirac 4D representation

**N**).

## 4. Discussion

_{0}= 0.52917 · 10

^{−10}[m]

_{SI}the corresponding binding time would be given as t → t

_{0}= a

_{0}/v

_{0}= 2.41889 · 10

^{−17}[s]

_{SI}while the involved bondonic mass will be half of the electronic one m

_{0}/2, to assure fast bonding information. Of course, this is not a realistic binding situation; for that, let us check the hypothetical case in which the electronic m

_{0}mass is combined, within the bondonic formulation (38), into the bond distance ${X}_{\mathit{\text{bond}}}=\sqrt{\u0127t/2{m}_{0}}$ resulting in it completing the binding phenomenon in the femtosecond time t

_{bonding}∼ 10

^{−12}[s]

_{SI}for the custom nanometric distance of bonding X

_{bonding}∼ 10

^{−9}[m]

_{SI}. Still, when both the femtosecond and nanometer time-space scale of bonding is assumed in (38), the bondonic mass is provided in the range of electronic mass m

_{B̶}∼ 10

^{−31}[kg]

_{SI}although not necessarily with the exact value for electron mass nor having the same value for each bonding case considered. Further insight into the time existence of the bondons will be reloaded for molecular systems below after discussing related specific properties as the bondonic velocity and charge.

_{bonding}→ ħ/E

_{bond}, thus delivering another working expression for the bondonic mass

_{2}, considerably surpasses the available charge in the system, although this may be eventually explained by the continuous matter-antimatter balance in the Dirac Sea to which the present approach belongs. However, to circumvent such problems, one may further use the result (77) and map it into the Poisson type charge field Equation

_{bond}and X

_{bond}) successively as

_{2}being the most covalent binding considered in Table 1 since it is most closely situated to the electronic pairing at the mass level. The excess in H

_{2}bond mass with respect to the two electrons in isolated H atoms comes from the nuclear motion energy converted (relativistic) and added to the two-sided electronic masses, while the heavier resulted mass of the bondon is responsible for the stabilization of the formed molecule respecting the separated atoms. The H

_{2}bondon seems to be also among the less circulated ones (along the bondon of the F

_{2}molecule) in bonding traveled information due to the low velocity and charge record—offering therefore another criterion of covalency, i.e., associated with better localization of the bonding space.

_{v}<<) or fraction of the light velocity from all C–C types of bonding; in this case also the bondonic highest mass (ς

_{m}>>), smallest charge (ς

_{e}<<), and highest (observed) life-time (t

_{B̶}>>) criteria seem to work well. Other bonds with high covalent character, according with the bondonic velocity criterion only, are present in N≡N and the C=O bonding types and less in the O=O and C–O ones. Instead, one may establish the criteria for multiple (double and triple) bonds as having the series of current bondonic properties as: {ς

_{m}<, ς

_{v}>, ς

_{e}>, t

_{B̶}<}

_{m}<<), is characterized by the highest velocity (ς

_{v}>) and charge (ς

_{e}>) in the CC series (and also among all cases of Table 1). This is an indication that the bond is very much delocalized, thus recognizing the solid state or metallic crystallized structure for this kind of bond in which the electronic pairings (the bondons) are distributed over all atomic centers in the unit cell. It is, therefore, a special case of bonding that widely informs us on the existence of conduction bands in a solid; therefore the metallic character generally associated with the bondonic series of properties {ς

_{m}<<, ς

_{v}>, ς

_{e}>, t

_{B̶}<}, thus having similar trends with the corresponding properties of multiple bonds, with the only particularity in the lower mass behavior displayed—due to the higher delocalization behavior for the associate bondons.

_{m}∼ >, ς

_{v}∼, ς

_{e}∼, t

_{B̶}∼>}; this may explain why these bonds are the most preferred ones in DNA and genomic construction of proteins, being however situated towards the ionic character of chemical bond by the lower bondonic velocities computed; they have also the most close bondonic mass to unity; this feature being due to the manifested polarizability and inter-molecular effects that allows the 3D proteomic and specific interactions taking place.

_{2}, Cl

_{2}, and I

_{2}, only the observed life-time of bondons show high and somehow similar values, while from the point of view of velocity and charge realms only the last two bonding types display compatible properties, both with drastic difference for their bondonic mass respecting the F–F bond—probably due the most negative character of the fluorine atoms. Nevertheless, judging upon the higher life-time with respect to the other types of bonding, the classification may be decided in the favor of covalent behavior. At this point, one notes traces of covalent bonding nature also in the case of the rest of halogen-carbon binding (C–Cl, C–Br, and C–I in Table 1) from the bondonic life-time perspective, while displaying also the ionic manifestation through the velocity and charge criteria {ς

_{v}∼, ς

_{e}∼} and even a bit of metal character by the aid of small bondonic mass (ς

_{m}<). All these mixed features may be because of the joint existence of both inner electronic shells that participate by electronic induction in bonding as well as electronegativity difference potential.

_{2}, O

_{2}, N

_{2}(with impact in environmental chemistry) or in polar compounds like C–F (specific to ecotoxicology) or in the reactions that imply a competition between the exchange in the hydrogen or halogen (e.g., HF). The valence explanation relied on the possibility of higher orders of orbitals’ existing when additional shells of atomic orbitals are involved such as <f> orbitals reaching this way the charge-shift bonding concept [73]; the present bondonic treatment of chemical bonds overcomes the charge shift paradoxes by the relativistic nature of the bondon particles of bonding that have as inherent nature the time-space or the energy-space spanning towards electronic pairing stabilization between centers of bonding or atomic adducts in molecules.

_{bond}, E

_{bond}} combinations.

_{j}, t) acting on the bondons “j”, carrying the kinetic moment p

_{B̶j}= m

_{B̶}v

_{B̶}, charge e

_{B̶}and mass m

_{B̶.}

_{0}, υ

_{0}) and scattered (q⃗, υ) light beams, the interactions driven by H

^{(1)}and H

^{(2)}model the changing in one- and two- occupation numbers of photonic trains, respectively. In this context, the transition probability between the initial |B̶

_{i}〉 and final |B̶

_{f}〉 bondonic states writes by squaring the sum of all scattering quantum probabilities that include absorption (A, with n

_{A}number of photons) and emission (E, with n

_{E}number of photons) of scattered light on bondons, see Figure 1.

_{|B̶v 〉}in terms of bonding energy associated with the initial state

## 5. Conclusion

_{m}and ς

_{e}, and of the bondonic-to-light velocity percent ratio ς

_{v}, along the bondonic observable life-time, t

_{B̶}respectively–here summarized in Table 3.

## Acknowledgments

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**Figure 1.**The Feynman diagrammatical sum of interactions entering the Raman effect by connecting the single and double photonic particles’ events in absorption (incident wave light q⃗

_{0}, υ

_{0}) and emission (scattered wave light q⃗, υ) induced by the quantum first H

^{(1)}and second H

^{(2)}order interaction Hamiltonians of Equations (84) and (85) through the initial |B̶

_{i}〉, final |B̶

_{f}〉, and virtual |B̶

_{v}〉 bondonic states. The first term accounts for absorption (A)-emission (E) at once, the second term sums over the virtual states connecting the absorption followed by emission, while the third terms sums over virtual states connecting the absorption following the emission events.

**Table 1.**Ratios for the bondon-to-electronic mass and charge and for the bondon-to-light velocity, along the associated bondonic life-time for typical chemical bonds in terms of their basic characteristics such as the bond length and energy [71,72] through employing the basic formulas (68), (73), (80) and (81) for the ground states, respectively.

Bond Type | X_{bond} (Å) | E_{bond} (kcal/mol) | ${\zeta}_{m}=\frac{{m}_{\mathit{\text{B\u0336}}}}{{m}_{0}}$ | ${\zeta}_{v}=\frac{{v}_{\mathit{\text{B\u0336}}}}{c}[\%]$ | ${\zeta}_{e}=\frac{{e}_{\mathit{\text{B\u0336}}}}{e}[\times {10}^{3}]$ | t_{B̶}[×10^{15}] (seconds) |
---|---|---|---|---|---|---|

H–H | 0.60 | 104.2 | 2.34219 | 3.451 | 0.3435 | 9.236 |

C–C | 1.54 | 81.2 | 0.45624 | 6.890 | 0.687 | 11.894 |

C–C (in diamond) | 1.54 | 170.9 | 0.21678 | 14.385 | 1.446 | 5.743 |

C=C | 1.34 | 147 | 0.33286 | 10.816 | 1.082 | 6.616 |

C≡C | 1.20 | 194 | 0.31451 | 12.753 | 1.279 | 5.037 |

N≡N | 1.10 | 225 | 0.32272 | 13.544 | 1.36 | 4.352 |

O=O | 1.10 | 118.4 | 0.61327 | 7.175 | 0.716 | 8.160 |

F–F | 1.28 | 37.6 | 1.42621 | 2.657 | 0.264 | 25.582 |

Cl–Cl | 1.98 | 58 | 0.3864 | 6.330 | 0.631 | 16.639 |

I–I | 2.66 | 36.1 | 0.3440 | 5.296 | 0.528 | 26.701 |

C–H | 1.09 | 99.2 | 0.7455 | 5.961 | 0.594 | 9.724 |

N–H | 1.02 | 93.4 | 0.9042 | 5.254 | 0.523 | 10.32 |

O–H | 0.96 | 110.6 | 0.8620 | 5.854 | 0.583 | 8.721 |

C–O | 1.42 | 82 | 0.5314 | 6.418 | 0.64 | 11.771 |

C=O (in CH_{2}O) | 1.21 | 166 | 0.3615 | 11.026 | 1.104 | 5.862 |

C=O (in O=C=O) | 1.15 | 191.6 | 0.3467 | 12.081 | 1.211 | 5.091 |

C–Cl | 1.76 | 78 | 0.3636 | 7.560 | 0.754 | 12.394 |

C–Br | 1.91 | 68 | 0.3542 | 7.155 | 0.714 | 14.208 |

C–I | 2.10 | 51 | 0.3906 | 5.905 | 0.588 | 18.9131 |

**Table 2.**Predicted basic values for bonding energy and length, along the associated bondonic life-time and velocity fraction from the light velocity for a system featuring unity ratios of bondonic mass and charge, respecting the electron values, through employing the basic formulas (81), (73), (68), and (80), respectively.

${X}_{\mathit{\text{bond}}}[\stackrel{0}{A}]$ | E_{bond} [(kcal/mol)] | t_{B̶}[×10^{15}] (seconds) | ${\zeta}_{v}=\frac{{v}_{\mathit{\text{B\u0336}}}}{c}[\%]$ | ${\zeta}_{m}=\frac{{m}_{\mathit{\text{B\u0336}}}}{{m}_{0}}$ | ${\zeta}_{e}=\frac{{e}_{\mathit{\text{B\u0336}}}}{e}$ |
---|---|---|---|---|---|

1 | 87.86 | 10.966 | 4.84691 | 1 | 0.4827 × 10^{−3} |

1 | 182019 | 53.376 | 99.9951 | 4.82699 × 10^{−4} | 1 |

10 | 18201.9 | 533.76 | 99.9951 | 4.82699 × 10^{−5} | 1 |

100 | 1820.19 | 5337.56 | 99.9951 | 4.82699 × 10^{−6} | 1 |

**Table 3.**Phenomenological classification of the chemical bonding types by bondonic (mass, velocity, charge and life-time) properties abstracted from Table 1; the used symbols are: > and ≫ for ‘high’ and ‘very high’ values; < and ≪ for ‘low’ and ‘very low’ values; ∼ and ∼> for ‘moderate’ and ‘moderate high and almost equal’ values in their class of bonding.

Property | ς_{m} | ς_{v} | ς_{e} | t_{B̶} | |
---|---|---|---|---|---|

Chemical bond | |||||

Covalence | >> | << | << | >> | |

Multiple bonds | < | > | > | < | |

Metallic | << | > | > | < | |

Ionic | ∼> | ∼ | ∼ | ∼> |

© 2010 by the authors; licensee Molecular Diversity Preservation International, Basel, Switzerland. This article is an open-access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

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**MDPI and ACS Style**

Putz, M.V. The Bondons: The Quantum Particles of the Chemical Bond. *Int. J. Mol. Sci.* **2010**, *11*, 4227-4256.
https://doi.org/10.3390/ijms11114227

**AMA Style**

Putz MV. The Bondons: The Quantum Particles of the Chemical Bond. *International Journal of Molecular Sciences*. 2010; 11(11):4227-4256.
https://doi.org/10.3390/ijms11114227

**Chicago/Turabian Style**

Putz, Mihai V. 2010. "The Bondons: The Quantum Particles of the Chemical Bond" *International Journal of Molecular Sciences* 11, no. 11: 4227-4256.
https://doi.org/10.3390/ijms11114227