The Quantum Condition That Should Have Been Assumed by Bohr When Deriving the Energy Levels of a Hydrogen Atom

Bohr assumed a quantum condition when deriving the energy levels of a hydrogen atom. This famous quantum condition was not derived logically, but it beautifully explained the energy levels of the hydrogen atom. Therefore, Bohr’s quantum condition was accepted by physicists. However, the energy levels predicted by the eventually completed quantum mechanics do not match perfectly with the predictions of Bohr. For this reason, it cannot be said that Bohr’s quantum condition is a perfectly correct assumption. Since the mass of an electron which moves inside a hydrogen atom varies, Bohr’s quantum condition must be revised. However, the newly derived relativistic quantum condition is too complex to be assumed at the beginning. The velocity of an electron in a hydrogen atom is known as the Bohr velocity. This velocity can be derived from the formula for energy levels derived by Bohr. The velocity v of an electron including the principal quantum number n is given by αc/n. This paper elucidates the fact that this formula is built into Bohr’s quantum condition. It is also concluded in this paper that it is precisely this velocity formula that is the quantum condition that should have been assumed in the first place by Bohr. From Bohr’s quantum condition, it is impossible to derive the relativistic energy levels of a hydrogen atom, but they can be derived from the new quantum condition. This paper proposes raising the status of the previously-known Bohr velocity formula.

Catalytic activity of V_(2)CO_(2) MXene supported transition metal single atoms for oxygen reduction and hydrogen oxidation reactions:A density functional theory calculation study

Two-dimensional(2D)MXene and single-atom(SA)catalysts are two frontier research fields in catalysis.2D materials with unique geometric and electronic structures can modulate the catalytic performance of supported SAs,which,in turn,affect the intrinsic activity of 2D materials.Density functional theory calculations were used to systematically explore the potential of O-terminated V2C MXene(V_(2)CO_(2))-supported transition metal(TM)SAs,including a series of 3d,4d,and 5d metals,as oxygen reduction reaction(ORR)and hydrogen oxidation reaction(HOR)catalysts.The combination of TM SAs and V_(2)CO_(2)changes their electronic structure and enriches the active sites,and consequently regulates the intermediate adsorption energy and catalytic activity for ORR and HOR.Among the investigated TM-V_(2)CO_(2)models,Sc-,Mn-,Rh-,and PtMCCh showed high ORR activity,while Sc-,Ti-,V-,Cr-,and Mn-V_(2)CO_(2)exhibited high HOR activity.Specifically,Mn-and Sc-V_(2)CO_(2)are expected to serve as highly efficient and cost-effective bifunctional catalysts for fuel cells because of their high catalytic activity and stability.This work provides theoretical guidance for the rational design of efficient ORR and HOR bifunctional catalysts.

Relativistic Reduction of the Electron-Nucleus Force in Bohr’s Hydrogen Atom and the Time of Electron Transition between the Neighbouring Quantum Energy Levels

The aim of the paper is to get an insight into the time interval of electron emission done between two neighbouring energy levels of the hydrogen atom. To this purpose, in the first step, the formulae of the special relativity are applied to demonstrate the conditions which can annihilate the electrostatic force acting between the nucleus and electron in the atom. This result is obtained when a suitable electron speed entering the Lorentz transformation is combined with the strength of the magnetic field acting normally to the electron orbit in the atom. In the next step, the Maxwell equation characterizing the electromotive force is applied to calculate the time interval connected with the change of the magnetic field necessary to produce the force. It is shown that the time interval obtained from the Maxwell equation, multiplied by the energy change of two neighbouring energy levels considered in the atom, does satisfy the Joule-Lenz formula associated with the quantum electron energy emission rate between the levels.

Zero Strength of the Lorentz Force and Lack of a Classical Energy Radiation in the Bohr's Hydrogen Atom?

It is demonstrated that the Lorentz force acting on the electron particle in the Bohr's hydrogen atom, as well as the classical radiation energy of that atom, tends to be zero on condition both the electric and magnetic fields in the atom are considered in a definite quantum state. The ratio of the mentioned fields becomes of importance for discussion of the occurence of the electron spin.

Phase-Space Areas of the Body Motion in the Solar System Deduced from the Bohr-Sommerfeld Atomic Theory and Approximate Invariance of Their Ratios for the Pairs of Planets and Satellites

Energy-time and momentum-position phase spaces defined by the electron orbits in the hydrogen-like atom exhibit special properties of equivalence. It is demonstrated that equivalence of the same kind can be obtained for the phase-space areas defined by the orbit pairs of planets, or satellites, which compose the solar system. In the choice of the examined areas it is useful to be guided by the Bohr-Sommerfeld atomic theory.

Experimental and Theoretical Study of Hydrogen Atom Abstraction from C2H6 and C4H10 by Zirconium Oxide Clusters Anions

The reactions of anionic zirconium oxide clusters ZrxOy- with C2H6 and C4H10 are investi-gated by a time of flight mass spectrometer coupled with a laser vaporization cluster source.Hydrogen containing products Zr2O5H- and Zr3O7H- are observed after the reaction. Den-sity functional theory calculations indicate that the hydrogen abstraction is favorable in the reaction of Zr2O5- with C2H6, which supports that the observed Zr2O5H- and Zr3O7H- are due to hydrogen atom abstraction from the alkane molecules. This work shows a newpossible pathway in the reaction of zirconium oxide cluster anions with alkane molecules.

Bohr’s Spectrum of Quantum States in the Atomic Hydrogen Deduced from the Uncertainty Principle for Energy and Time

A modified uncertainty principle coupling the intervals of energy and time can lead to the shortest distance attained in course of the excitation process, as well as the shortest possible time interval for that process. These lower bounds are much similar to the interval limits deduced on both the experimental and theoretical footing in the era when the Heisenberg uncertainty principle has been developed. In effect of the bounds existence, a maximal nuclear charge Ze acceptable for the Bohr atomic ion could be calculated. In the next step the velocity of electron transitions between the Bohr orbits is found to be close to the speed of light. This result provides us with the energy spectrum of transitions similar to that obtained in the Bohr’s model. A momentary force acting on the electrons in course of their transitions is estimated to be by many orders larger than a steady electrostatic force existent between the atomic electron and the nucleus.

Emission Intensity in the Hydrogen Atom Calculated from a Non-Probabilistic Approach to the Electron Transitions

Quantum aspects of the Joule-Lenz law for the transmission of energy allowed us to calculate the time rate of energy transitions between the quantum states of the hydrogen atom in a fully non-probabilistic way. The calculation has been extended to all transitions between p and s states having main quantum numbers not exceeding 6. An evident similarity between the intensity pattern obtained from the Joule-Lenz law and the corresponding quantum-mechanical transition pro-babilities has been shown.

Two Problems of Time Entering Respectively the Relativistic Mechanics and Electron Transport in Quantum Theory

In the relativistic mechanics, we calculate a minimal distance between the time scale of a one-dimensional motion having a larger velocity and the time scale of a similar motion with a lower velocity. Concerning the quantum theory, we demonstrate that mechanical parameters entering the electron motion in the Bohr hydrogen atom can provide us with a correct size of the time interval entering the Joule-Lenz law for the emission energy between two neighbouring quantum levels of the atom.

Model of the Pulsing Atom

In this work, we reanalyzed the movement of an electron in the electrostatic field of nucleus. The trajectory of the electron’s motion is an ellipse with a minor semiaxis, tending towards zero. From a mathematical point of view the movement of an electron in such an orbit will be equivalent to the oscillation of an electron. The action produced by electrons in movement between stationary points is discrete and proportional to a Planck constant. This condition sets the allowable values of the electron energy and the radius of their orbit. Electrons on the same shell perform symmetric synchronous oscillations. Their frequency is of the order of 1016 Hz. Most of the time the electrons are located on the periphery of the atom, periodically they simultaneously rush to the nucleus, the atom rapidly compresses and immediately decompresses, i.e. pulsates. The model gives Bohr formula for the energy of single-electron atom and suitable values of ionization potentials of the atoms of the second period of the Periodic Table.

Relativistic Correction of the Rydberg Formula

The relationship E = −K holds between the energy E and kinetic energy K of the electron constituting a hydrogen atom. If the kinetic energy of the electron is determined based on that relationship, then the energy levels of the hydrogen atom are also determined. In classical quantum theory, there is a formula called the Rydberg formula for calculating the wavelength of a photon emitted by an electron. In this paper, in contrast, the formula for the wavelength of a photon is derived from the relativistic energy levels of a hydrogen atom derived by the author. The results show that, although the Rydberg constant is classically a physical constant, it cannot be regarded as a fundamental physical constant if the theory of relativity is taken into account.

Hydrogen Bonding Character Between the Glycine and BF4^-

The hydrogen bonding character between the BF4- and glycine was theoretically studied at the level of B3LYP/6-31＋G^＊, single point energies were performed at the level of B3LYP/6-311＋＋G^＊＊. The relevant geometrical characteristics, energy properties, as well as the characters of the intramolecular hydrogen bonds have been studied. Atoms in molecule theory topological analysis indicated the （3,-1） critical points for hydrogen bonds. In addition, the electron density and Laplacian were in the range suggested for the hydrogen bonds. Especially, the changes of atomic charge, hydrogen upon hydrogen bonds formation dipole moment, enegry as well as volume of the were systemitically discussed.

Relativized Quantum Physics Generating <i>N</i>-Valued Coulomb Force and Atomic Hydrogen Energy Spectrum

Though not well-known, Einstein endeavored much of his life to general-relativize quantum mechanics, (rather than quantizing gravity). Albeit he did not succeed, his legacy lives on. In this paper, we begin with the general relativistic field equations describing flat spacetime, but stimulated by vacuum energy fluctuations. In our precursor paper, after straightforward general relativistic calculations, the resulting covariant and contravariant energy-momentum tensors were identified as n-valued operators describing graviton excitation. From these two operators, we were able to generate all three boson masses (including the Higgs mass) in precise agreement as reported in the 2010 CODATA (NIST);moreover local, as-well-as large-scale, accelerated spacetimes were shown to naturally occur from this general relativized quantum physics approach (RQP). In this paper, applying the same approach, we produce an n-valued Coulombs Force Law leading to the energy spectrum for atomic hydrogen, without assuming quantized atomic radii, velocity and momentum, as Bohr did.

Investigation of the Antioxidant and UV Absorption Properties of 2-(2’-hydroxy-5’-methylphenyl)-benzotriazole and Its Ortho-Substituted Derivatives via DFT/TD-DFT

The demand and pursuit of chemical entities with UV filtration and antioxidant properties for enhanced photoprotection have been driven in recent times by acute exposure of humans to solar ultraviolet radiations. The structural, electronic, antioxidant and UV absorption properties of drometrizole (PBT) and designed ortho-substituted derivatives are reported via DFT and TD-DFT in the gas and aqueous phases. DFT and TD-DFT computations were performed at the M062x-D3Zero/6-311++G(d,p)//B97-3c and PBE0-D3(BJ)/def2-TZVP levels of theory respectively. Reaction enthalpies related to hydrogen atom transfer (HAT), single-electron transfer followed by proton transfer (SET-PT), and sequential proton loss electron transfer (SPLET) mechanisms were computed and compared with those of phenol. Results show that the presence of -NH2 substituent reduces the O-H bond dissociation enthalpy and ionization potential, while that of -CN increases the proton affinity. The HAT and SPLET mechanisms are the most plausible in the gas and aqueous phases respectively. The molecule with the -NH2 substituent (PBT1) was identified to be the compound with the highest antioxidant activity. The UV spectra of the studied compounds are characterized by two bands in the 280 - 400 nm regions. Results from this study provide a better comprehension antioxidant mechanism of drometrizole and present a new perspective for the design of electron-donor antioxidant molecules with enhanced antioxidant-photoprotective efficiencies for applications in commercial sunscreens.

The dynamical properties of Rydberg hydrogen atom near a metal surface

The dynamical properties of Rydberg hydrogen atom near a metal surface are presented by using the methods of phase space analysis and closed orbit theory. Transforming the coordinates of the Hamiltonian, we find that the phase space of the system is divided into vibrational and rotational region. Both the Poincaré surface of section and the closed orbit theory verify the same conclusion clearly. In this paper we choose the atomic principal quantum number as n = 20. The dynamical character of the exited hydrogen atom depends sensitively on the atom-surface distance d. When d is sufficiently large, the atom-surface potential can be expressed by the traditional van der Waals force and the system is integrable. When d becomes smaller, there exists a critical value dc. For d > dc, the system is near-integrable and the motion is regular. While chaotic motion appears for d < dc, and the system tends to be non-integrable. The trajectories become unstable and the electron might be captured onto the metal surface.

A Mechanistic Switch in C-H Bond Activation by Elusive Fe^(V)(O)(TAML)Reaction Intermediate:A Theoretical Study

The divergent behavior of C-H bond oxidations of aliphatic substrates compared to those of aromatic substrates shown in Gupta’s experiment was mechanistically studied herein by means of density functional theory calculations.Our calculations reveal that such difference is caused by different reaction mechanisms between two kinds of substrates(the aliphatic cyclohexane,2,3-dimethylbutane and the aromatic toluene,ethylbenzene and cumene).For the aliphatic substrates,C-H oxidation by the oxidant Fe^(V)(O)(TAML)is a hydrogen atom transfer process;whereas for the aromatic substrates,C-H oxidation is a proton-coupled electron transfer(PCET)process with a proton transfer character on the transition state,that is,a proton-coupled electron transfer process holding a proton transfer-like transition state(PCET(PT)).This difference is caused by the strongπ-πinteractions between the tetra-anionic TAML ring and the phenyl ring of the aromatic substrates,which has a“pull”effect to make the electron transfer from substrates to the Fe=O moiety inefficient.

Quantum Mechanics of Moving Bound States

The quantum mechanics of bound states with discrete energy levels is well understood. The quantum mechanics of scattering processes is also well understood. However, the quantum mechanics of moving bound states is still debatable. When it is at rest, the space-like separation between the constituent particles is the primary variable. When the bound state moves, this space-like separation picks up the time-like separation. The time-separation is not a measurable variable in the present form of quantum mechanics. The only way to deal with this un-observable variable is to treat it statistically. This leads to rise of the statistical variables such entropy and temperature. Paul A. M. Dirac made efforts to construct bound-state wave functions in Einstein’s Lorentz-covariant world. In 1927, he noted that the c-number time-energy relation should be incorporated in the relativistic world. In 1945, he constructed four-dimensional oscillator wave functions with one time coordinate in addition to the three-dimensional space. In 1949, Dirac introduced the light-cone coordinate system for Lorentz transformations. It is then possible to integrate these contributions made by Dirac to construct the Lorentz-covariant harmonic oscillator wave functions. This oscillator system can explain the proton as a bound state of the quarks when it is at rest, and explain the Feynman’s parton picture when it moves with a speed close to that of light. While the un-measurable time-like separation becomes equal to the space-like separation at this speed, the statistical variables become prominent. The entropy and the temperature of this covariant harmonic oscillator are calculated. It is shown that they rise rapidly as the proton speed approaches that of light.

Theoretical Investigation on Interaction between Guanine and Luteolin

The interacting patterns of the luteolin and guanine have been investigated by using the density functional theory B3LYP method with 6-31＋G＊ basis set. Eighteen stable structures for the luteolin-guanine complexes have been found respectively. The results indicate that the complexes are mainly stabilized by the hydrogen bonding interactions. Meanwhile, both the number and strength of hydrogen bond play important roles in determining the stability of the complexes which can form two or more hydrogen bonds. Theories of atoms in molecules and natural bond orbital have also been utilized to investigate the hydrogen bonds involved in all the systems. The interaction energies of all the complexes which were corrected by basis set superposition error are 6.04-56.94 kJ/mol. The calculation results indicate that there are strong hydrogen bonding interactions in the luteolin-guanine complexes. We compared the interaction between luteolin and four bases of DNA, and found luteolin-thymine was the strongest and luteolin-adenine was the weakest. The interaction between luteolin and DNA bases are all stronger than luteolin-water.

Theoretical Study on the Hydrogen Bonding Interactions in Complexes of 5-Hydroxytryptamine with Water

The energies, geometries and harmonic vibrational frequencies of 1 ： 1 5-hydroxytryptamine-water （5-HT-H20） complexes are studied at the MP2/6-311 ＋ ＋ G（d,p） level. Natural bond orbital （NBO）, quantum theory of atoms in molecules （QTAIM） analyses and the localized molecular orbital energy decomposition analysis （LMO-EDA） were performed to explore the nature of the hydrogen-bonding interactions in these complexes. Various types of hydro- gen bonds （H-bonds） are formed in these 5-HT-H20 complexes. The intermolecular C4H55HT＇＂Ow H-bond in HTW3 is strengthened due to the cooperativity, whereas no such cooperativity is found in the other 5-HT-H20 complexes. H-bond in which nitrogen atom of amino in 5-HT acted as proton donors was stronger than other H-bonds. Our researches show that the hydrogen bonding interaction plays a vital role on the relative stabilities of 5-HT-H20 complexes.

氢原子低能级的Stark效应

用微扰法计算了氢原子低能级的Stark效应 ,基态精确到二级修正 ,能级产生移动 ;第一激发态、第二激发态精确到一级修正 ,能级产生分裂 ,简并部分解除 .