Personal preface:
The work included below I tried to publish in three important journals of the area of theoretical and computational chemistry. My attempts were completely unsuccessful (detailed information on the request). Censors of these journals rejected this work at the first stage of publication process. They gave different reasons. The first answer and the second one were typical: theoretical and computational chemists are not interested in this theme. Okay – disregard. The third one was more important: A study of the hydrogen atom, which is an exactly solvable problem, is not sufficient for the aims and objectives of [here was a journal name].
And then: In the event you develop a paper that goes further and that reports results on more than the hydrogen atom, it may be that such a paper would be suitable for consideration.
These opinions cited above are very important. It seemed to me that hydrogen atom is the obligatory object for new research attempts in the area of physics. It means that theoretical chemistry have to examine this atom initially. What do mean that "hydrogen atom is exactly solvable problem"? It means, we have Schrödinger equation to calculate hydrogen atom properties. My work is about something else. Please, try to understand this. Actually, I found the new description of the density function for hydrogen atom ground state only. It turned out that it is not enough. But the central point of my work is elsewhere. I showed a completely new ability to describe quantum processes in the hydrogen atom. Maybe the work listed below is written in fatal English, maybe arguments are weak but these reasons could not stopped publication. A this moment, I feel exempted from publication this work in an official journal.
Wojciech Szczepankiewicz
(Employment data were removed because the work is unofficial)
E-mail: Wojciech.Szczepankiewicz@polsl.pl
Abstract: In this paper is presented a new theoretical method for determination of a hydrogen atom electron density at the ground state. A set of chaotic points created by the iteration of a logarithmic function was used to generate an electron density function characterized by the discrete property and limited spatial reach.
Keywords: Hydrogen atom, electron density, Lyapunov exponent, deterministic chaos, logarithmic function, return map
1. Introduction
2. Determination of chaotic electron density
The non-normalized density function of the hydrogen atom ground state is given by the equation (in atomic units)
where ρ – electron density, r – electron radius vector, c – variational constant.
The Lyapunov exponents λ [13], calculated for r (at c <0.001, 4.0>), are less than zero. This means that the iterated equation
The work included below I tried to publish in three important journals of the area of theoretical and computational chemistry. My attempts were completely unsuccessful (detailed information on the request). Censors of these journals rejected this work at the first stage of publication process. They gave different reasons. The first answer and the second one were typical: theoretical and computational chemists are not interested in this theme. Okay – disregard. The third one was more important: A study of the hydrogen atom, which is an exactly solvable problem, is not sufficient for the aims and objectives of [here was a journal name].
And then: In the event you develop a paper that goes further and that reports results on more than the hydrogen atom, it may be that such a paper would be suitable for consideration.
These opinions cited above are very important. It seemed to me that hydrogen atom is the obligatory object for new research attempts in the area of physics. It means that theoretical chemistry have to examine this atom initially. What do mean that "hydrogen atom is exactly solvable problem"? It means, we have Schrödinger equation to calculate hydrogen atom properties. My work is about something else. Please, try to understand this. Actually, I found the new description of the density function for hydrogen atom ground state only. It turned out that it is not enough. But the central point of my work is elsewhere. I showed a completely new ability to describe quantum processes in the hydrogen atom. Maybe the work listed below is written in fatal English, maybe arguments are weak but these reasons could not stopped publication. A this moment, I feel exempted from publication this work in an official journal.
Wojciech Szczepankiewicz
(Employment data were removed because the work is unofficial)
E-mail: Wojciech.Szczepankiewicz@polsl.pl
Abstract: In this paper is presented a new theoretical method for determination of a hydrogen atom electron density at the ground state. A set of chaotic points created by the iteration of a logarithmic function was used to generate an electron density function characterized by the discrete property and limited spatial reach.
Keywords: Hydrogen atom, electron density, Lyapunov exponent, deterministic chaos, logarithmic function, return map
1. Introduction
Deterministic chaos is mainly used in the research of nonlinear dynamical processes for description of classical chaotic processes in the category of quantum mechanics [1-3], Brown motions in a solution [4], dynamics of time dependent hamiltonians [5] or chaotic processes in molecular vibrations [6,7]. These processes were also examined at the excited atoms in external magnetic fields [8]. According to our knowledge, the procedures for generation of deterministic chaos has not been used in the context of the creation of electron density in an atom.
Because the procedures for generation of deterministic chaos return limited number of points in the limited space, remind results of experimental measurements of electron density. For instance, structure determination of chemical compounds by the X-Ray spectroscopy [9] or the discrete electron diffraction picture created at a double slit Young experiment [10-12]. These results can be theoretically generalized in the term of continuous exponential type functions vanishing at infinity by using the wave theory, Schrödinger equation (wave functions) or Density Functional Theories. In other words, continuous and unlimited (with the exception of nodal planes) wave or density functions are mathematical canvas for describing of limited and discrete experimental results. In this work we will show that logarithmic function, associated with the expotential function by the inverse procedure, can simulate electron density of a hydrogen atom ground state. This simulation returns discrete and spatially limited density results. The central point of the following discussion is the chaotic behaviour of iterated logarithmic function characterized by positive values of the Lyapunov exponents [13]. These functions can be plotted on the so called return maps [14].
2. Determination of chaotic electron density
The non-normalized density function of the hydrogen atom ground state is given by the equation (in atomic units)
where ρ – electron density, r – electron radius vector, c – variational constant.
The Lyapunov exponents λ [13], calculated for r (at c <0.001, 4.0>), are less than zero. This means that the iterated equation
(where rn+1 corresponds to the rn value on the return map) shows one or more attractors and does not add any significant information concerning the density function. From another side, the logarithmic function given by Eq. (1) is an inverse function of r.
One can construct chaotic functions by the iterative processes starting from an initial r variable value. Unfortunately, the values of the right side of Eq. (1) are negative for ρ > 1. Thus, it is impossible to calculate r values for these ρ ones. However, the introduction of the module to the right side of Eq. (1) resolves this problem. The new Eq. (2) is infinitely iterable and can be used for an identification of the chaotic processes.
Lyapunov exponents λ for Eq. (2), expressed in Fig. 1, are positive for the interval c<0.19,1.35>. It means additionally that Eq. 2 shows the chaotic behavior in the special case for the non-normalized density function (c = 1) and for the normalized one ( c = 1/π).
Figure 1. Lyapunov exponents for logarithmic Eq. 2. Values λ > 0 indicate the chaotic domain of the function.
The return map of Eq. 2 (c = 1) plotted for the 30000 iteration steps (number of choice), shows the discrete set points concentrated near the origin (Fig. 2).
Figure 2. Return map of the logarithmic Eq. 2. Value rn correspond to a ρ density.
The first step for the electron density function to be recovered, needs the transposition of the axes shown in Fig. 2. We call the new curve expressed in Fig. 3, a pseudo density function ρ’’, because the values ρ’’ > 1 are situated on the positive side of the r axis (Fig. 3). Those values have no physical significance and have to be removed. It is a computational cost that arises because of the application of a module to the iteration procedure of the logarithmic Eq. 2. For example, 5655 points (about 19.8%) are lost for 30000 iteration steps (see supplementary content).
Fig. 3. Return map of the so called pseudo density function ρ’’. Values greater than 1 are nonphysical.
Removing values greater than one from Fig. 3 can characterize the non-normalized density function of the hydrogen atom ground state in atomic units i.e.
(Fig. 4.)
Fig. 4. The chaotic electron density function of the hydrogen atom ground state.
Since the function on Fig. 4 was created by a chaotic process, we suggest to call it a Chaotic Electron DEnsity function (CEDE for short). This function is a discrete one, its set point being limited by a number of iterations. The domain range r is limited by the rmin > 0 and rmax < INF. Both rmin and rmax depend on the number of the iteration steps. These values change abruptly at the different number of iterations and these changes present an opposite direction i.e. an increase of the number of iterations results in an increase of the rmax value and a decrease of the rmin. The rmin stabilizes over 1.8×107 iterations at the value of 1.38×10-08 and rmax over 2.1×108 at the value of 9.72 respectively. These values did not change up to 1×109 iterations. However, it is possible that a greater number of iterations will expand these values. (Calculation details and procedures are available from the author upon request)
3. Summary
CEDE of the ground state of the hydrogen atom is discrete and is limited by the minimum and maximum radius vector of the electron. This fact implies a diffuse but a finite size of the atom. For billion iterations maximum radius CEDE exceeds the Bohr radius of the hydrogen atom 9.72 times (see supplementary content). This means that CEDE can interact with the electron densities of adjacent atoms with a non-zero probability.
Assuming that the iteration that creates CEDE is an image of a physical process, the question should be put about the number of iterations since the creation of the atom until a point on the time. The answer to this question is not yet known, but it suggests that the stationary point of an atom is rather a process that occurs with a certain, unknown frequency.
If the stationary state of an atom is treated as a chaotic process of creation of electron density, it is consistent with the Heisenberg uncertainty principle. Chaotic "appearing and disappearing" of the electron around the nucleus at certain points, preventing the definition of the physical path of the electron in three-dimensional space.
Due to the fact that the electron density of the ground state of the hydrogen atom has a spherical symmetry, the values of the angles in a spherical coordinate system can be selected randomly.
The disadvantage of this description is the fact that it is dependent on the scale, which means that, only for a certain range of electron density equation coefficients this equation does behave chaotically. In the example discussed above, this value domain is well expressed by the scale of atomic units.
4. Reference
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