Atomic excited states and the related energy levels

This article is about generalization and extension of the Bohr atomic model as well as the Rydberg formula to make them applicable to all atomic/ionic excited states and their energy levels. Bohr and Rydberg’s original works were deemed only for hydrogen and the hydrogen-like ions but in time many mistakenly have come to the conclusion that those original forms of the theory are applicable to all species. This article clarifies the subject and helps with the misunderstandings. The article reviews first the original theory of atoms, the related excited states, and the related energy levels. It then shows the shortcoming of the original formulations and makes changes to generalize the theory and extend their applications to all atoms and their related ions. The theory of atomic excited states is re-formulated using a newly defined parameter called “characteristic exponent k” and the corresponding ionization energy. Numerical calculations and detailed works for several elements are presented to establish a better understanding of excited states. The article seeks also for a connection between the atomic energy levels and the internal structures and inner electrons of atoms. Furthermore, a small data bank is generated using the calculated “characteristic exponents k” for elements to be utilized for future simulations, studies, and research activities.


Introduction
The primary objective of this article is to introduce an updated generalized version of the classical atomic model in order to extend its application to all atoms, their excited states, and energy levels. The article is a continuation of a few earlier works [1][2][3][4] as well as the previous research activities [5,6] to establish a better understanding of atomic and molecular structures, characteristics, and the related energies. To achieve the goal of this article, a quick review of earlier concepts and theories is necessary. Thus, a review over the original classical theory of motion, atomic energy, and the concept of excited states is made before presenting the new generalized form of the model.

Classical energy model and concepts [7-16]
In 1924, a French physicist Louis de Broglie postulated a new concept that all moving objects have a wave-like motion with an associated wavelength. The concept is known as wave-particle duality which forms the foundation of quantum mechanics. According to his postulate, the wavelength of any moving object is inversely proportional to the linear momentum of the object (Eq. (1)).
where λ is the wavelength of object, h is Plank's constant, m is the mass, and v is the velocity of object. Figure 1 presents De Broglie's perception for motion of different objects. According to his concept, an electron with a mass of 9.109534 × 10 −31 kg moving with 1/10 of speed of light around nucleus would have a wavelength of 2.42 × 10 −11 meter. A baseball with 5 ounce mass thrown by the baseball pitcher, Nolan Ryan in 1974 with a speed of 100.9 mph would have a wavelength of 1.036 × 10 −34 m. And finally, the planet earth with a mass of 5.972 × 10 24 kg and a speed of 30 km/s around the sun would have a wavelength of 3.698 This article is submitted in the honor of University of New Orleans (La), Departments of Chemistry, Mathematics and Physics where the author received parts of his education.
× 10 −63 m. Hence in brief, smaller objects with common speeds would have larger wavelengths compared to larger objects.

Original classical Bohr atomic model
In 1913, Niels Bohr and Ernest Rutherford presented their atomic model as a model for one-electron atom (i.e., hydrogen). The concept assumed circular orbit for the electron in equilibrium under two forces. One force was the centripetal force acting on the electron due to its rotational speed which was assumed to be equal to the attracting coulombic force acting on electron by the nucleus (Eq. (2)).
where m e is the mass of electron, v is the velocity of electron, r is the orbital radius, Z is the atomic number, and e is the electron's charge, all in atomic units. To assure that the electron would stay in a certain orbits consistent with common observations and the notion of ground and excited state (2) m e v 2 r = Ze 2 r 2 energy levels, Bohr needed to make an additional assumption. He postulated that the circumference of the electronic path should be able to accommodate a whole number of electronic wavelength to assure separate electronic orbits as well as different vibrational modes for both ground (n = 1) and excited states (n = 2, 3, 4, …). This way, he tied the atomic excitation states to vibrational modes and standing waves of electron.
where r is the electronic radius, n is the number of excited state (or vibrational mode), and λ the wavelength of moving electron. Figure 2 presents Bohr's view of electronic orbits in ground and excited states of hydrogen. Hence, the concept of excited state could be viewed as electronic vibrational mode. The theory needed one more component which was Planck's formula (or Plank's energyfrequency relationship) which is presented by Eq. (4).
where ΔE is the energy change, h is Plank's coefficient, and presents the frequency. Combining Eq. (1) and Eq. (3), one can derive Eq. (5) which relates the rotational momentum (mvr) to the excitation state or vibrational mode n as: Then, the Eq. (6) can be derived by the rearrangement of Eq. (2) as: Furthermore, combining Eqs. (2), (5), and (6), one can obtain an equation for the ground state and excited states radii (Eq. (7)) as: where R 0 is the ground state electronic radius for one-electron atom, R n the electronic radius in excited states, and n is the number of excited state or the vibrational mode. n can take only whole numbers representing different excited states. Using the formal classical mechanics, then one can write Eq. (8) for the kinetic energy of the electron and Eq. (9) for the potential energy of interaction between electron and nucleus as: But, the magnitude of potential energy (E p ) is twice of the kinetic energy (E k ). And hence, the total energy of the oneelectron system can be written as summation of those two energies as presented by Eq. (10) in atomic units: Combining Eqs. (2) and (5), one can easily conclude that velocity of the electron in excited states would be inversely proportional to the n which is the related number for excited state and vibrational mode. This would provide a new equation (Eq. (11)) for the total energy of the one-electron model in ground and excited state modes (E n ) as: But, we know that the total energy of one-electron atom in ground state is equal to its related ionization energy (Eq. (12)) which can be found experimentally. Therefore,

Generalization of Bohr atomic model
Now we are ready to expand Bohr's theory to multi-electron atoms by making minor changes to our classical atomic energy theory. What we know that having inner electrons in the model would only impact the Eq. (2). Figure 3 presents the case of multi-electron case.
As one can observe in Fig. 3, the outer electron in multielectron atoms would be subjected to two forces and would be interacting not only with the nucleus at R o radius but also would be interacting with inner electrons which are at R i radii. The existence of inner electrons will transform Eq.  is the number of excited state in our classical model would now be changing to a different number between 0 and 2 such as 0.92 for rubidium. We will be calling this new parameter "characteristic exponent k" for the rest of this article. Therefore, our governing equations would look more like Eq. (14) and Eq. (15) in which the exponent 2 are replaced by k which would be a number less than 2 depending on the inner electronic structure of atom. The k factor will be equal to 2 only for hydrogen and would have different values for the rest of atoms which would be needed to be determined.
Characteristic exponent k is the exponent of n in our updated model and is a parameter which would provide us information about the inner structure of atoms. Also, the characteristic exponent k of an atom would help us to determine all excited state energy levels of the each atom or ion. The characteristic exponent k of several elements are found and presented at the end of this article.
Hence, the updated generalized form of Bohr atomic model can be presented as: To better understand the case, let's summarize the author's conclusion from his observations. Readers would check all of the points for themselves in upcoming figures and discussions.
During the excitation of atoms only one electron which is the one furthest away from nucleus would move away into excited states. 2. Analogy of a closed chest with one opening for only one electron at the time to move out applies to all of the inner electrons of the atoms. All inner electrons are kept in place as the outer electron goes through the excitation. 3. If an outer electron leaves away from the atom, then a new electron will be able to move out and fill the excited states. But now, the atom would turn to a positively charged ion with a new ionization energy (or E G.S. ). 4. In all possible excitation cases, there would be only one electron at the time going through the excitation and only one ionization energy figure would govern the excited energy levels. 5. The two factors governing the excited energy levels would be the ionization energy of the specie and the related excited state or vibrational mode of the electron. 6. Hence the equation of E n = (I.E.)/n k can be used to find all excited energy levels as long as k and I.E. are known.

Review of the original Rydberg's formula [17, 18]
The Rydberg formula is a correlation which predicts the wavelength of emitted light resulting from an electron moving from one energy level of an atom to another for hydrogen and hydrogen-like ions only. The formula was suggested by the Swedish physicist Johannes Rydberg in1888. The formula can be presented as: where λ is the wavelength of the emitted photon, R H Rydberg's constant, Z the atomic number of the atom, n 1 and n 2 integers corresponding to the energy levels. This form of Rydberg equation is very well known in spectroscopy.
With the first look at the Rydberg formula and the existence of Z which is the atomic number, most of people mistakenly assume that the formula should be applicable to all elements in the periodic table which is not true. Upcoming graphs and calculated data obtained from the Rydberg formula shows a large discrepancy with the experimental data. One can use the Eq. (4) as ΔE = h = h C λ and combine it with Eq. (16) to get an easier form of formula in terms of energy as: where C is the speed of light.

Fig. 3 Schematic diagram of multi-electron atoms
Comparing Eq. (17) with Eq. (11), one can easily then see that the ground state energy of atom as E G.S. = R H .h.C.Z 2 , and hence, Eq. (17) can be rewritten as: Comparing Eq. (11) of Bohr and Eq. (18) of Rydberg, one can see that both equations are actually the same and cannot generate different results.

Updated generalized form of the Rydberg formula
With simple analogy and comparison of the Rydberg formula to the updated version of Bohr atomic formula (Eq. (15)), one can come to the conclusion that an updated version of the Rydberg's formula would look like Eq. (19). The Bohr model defines the energy levels and Rydberg's formula finds the difference between those energy levels.
The original form of Rydberg's formula uses the R H coefficient specifically calculated for hydrogen. And, most of people are familiar with that related Rydberg constant for hydrogen and the hydrogen-like species as R H = 1.097373177 × 10 7 m −1 where the atomic number Z is taken equal to 1. Equation (20) utilizes h the Plank's constant (6.62607015 × 10 −34 m 2 kg/s), C the speed of light (2.99792458 × 10 8 m/s), and E G.S. the first experimental ionization energy of atom/ion. Now by the generalization of the Rydberg's formula, one needs to re-adjust the R H coefficient for the targeted element. Hence, a general equation for Rydberg's constant can be suggested as: Without a correct Rydberg's coefficient consistent with the atom/ion under the study, one cannot expect to obtain correct results using Rydberg's formula. Also, readers should keep in mind that the ground state energies of atoms/ ions mentioned in this article correspond to the first experimental ionization energy figure of the specie. Hence, by knowing the experimental ionization energy of atoms/ions, one can easily calculate the Rydberg constant of elements. Therefore one should not forget that the Rydberg constant is different for different elements and Table 1 dramatic off-limit values for the calculated excited state energy values.
Although Rydberg formula and Bohr's atomic model are widely accepted but one should not forget that they were made for hydrogen and hydrogen-like ions and their application should not be stretched to cover all atoms and ions.

Methods
To find the characteristic exponents k of atoms, the author needed all experimental ionization energies as well as the experimental energy levels of atoms and ions. These data have been experimentally measured throughout years by different researchers and the collected measured values are available through different sources [19][20][21][22]. But, one of the best reliable sources of experimental data for our purpose is the online NIST Database website [19] which provides reliable "atomic spectra energy level and ionization energy" values. NIST stands for the National Institute of Standards and Technology and is a governmental institute which collects and maintains reliable experimental data for research and industrial use. All available experimental energy values of atoms and ions were downloaded and collected from NIST websites for being used in this article.
But, it needs to be told that the downloaded energy level data from NIST websites were loaded with all sorts of notations, configurations, splitting, and maybe duplications and degeneracies which made it real hard for the author to identify and separate the actual individual atomic energy levels of the excited states. Hence, the author had to resort to the concept that actual reported energy values for excited states should be consistent in their magnitudes within the arbitrary figure of 0.01 eV for each energy level, and therefore, all reported energy values within 0.01 eV should be actually originated from the same root and should not be considered as separate energy levels. This remedy generally condensed the huge number of energy levels to a manageable level below 100. In some occasions, the reported energy values within 0.05 eV were considered having a common root and were considered to be one energy level.

Results and discussion
This article is not just a correlation but it is an extension to the classical atomic theory. It generalizes the classical model and makes it applicable to all elements within the periodic table. Looking at the huge number of reported atomic excitation data, many might have a false common belief that the Rydberg formula and Bohr equation should be sufficiently capable to sort and order them. But, they will learn in this article that the notion of the adequacy of Rydberg formula and Bohr equation to predict atomic excitation energy levels for all elements is absolutely false. This article updates the classical model using a newly defined "characteristic exponent" together with the related ionization energy to predict the excitation energy levels of all atoms/ions within a reasonable accuracy. The original Rydberg formula and Bohr's equation are not capable to predict atomic energy levels accurately beyond hydrogen and the hydrogen-like ions. The capability of the suggested methodology, having a rigorous theory behind it and its applicability to all elements in the periodic table tells us that the approach suggested in this article is more of a "law" rather than a correlation.
Results and graphs are presented for a few selected atoms in the article while all of the collected data, results, and graphs are shown in the "Supplementary Data" for the readers to review. It should be noted that the original Rydberg and original Bohr calculated data are both shown in the figures and tables of the article. The data from both original sources are basically identical using the correct Rydberg constant and are shown under the name of "classical energy model." We will start with hydrogen atom.

Hydrogen (H)
The one-electron hydrogen is the basis for our classical theory. The atom does not have any inner electron, and therefore, it is completely consistent with Bohr's atomic model and Rydberg formula. All theoretical and experimental data overlap on each other for this atom. Figure 4 presents both tabulated and graphical displays of the energy levels for the atom. As one can observe, the experimental figures in this case are completely consistent and in line with our classical theoretical model data of Bohr and Rydberg which gives credibility to both approaches for hydrogen and hydrogenlike ions. Errors are zero and the experimental data fall all on the theoretical graph. This atom belongs to group 1 (IA) of s block.

Helium (He I)
Helium atom has two electrons with one to be considered an inner electron for the other outer electron. So the atom is not completely in line with our Bohr atomic model and Rydberg formula due to the inner electron. Figure 5 presents both tabulated and graphical displays of the energy levels for this atom. As one can observe, the experimental data are not in line with the classical theoretical models but in better agreement with the method suggested in this article. Due to the inner electron, a departure of characteristic exponent (k from 2 to 1.35 happens for this atom. An average of 0.28 eV absolute error was observed for this atom. This atom belongs to group 18 (O) of s-block.

Lithium (Li I)
Lithium atom has three electrons with two to be considered inner electrons for the outer electron. So the atom is not in line with our Bohr and Rydberg classical models due to the inner electron. Figure 6 presents both tabulated and graphical displays of the energy levels for this atom. As one can observe, the experimental figures are not in line with the classical theoretical models but in better agreement with the method suggested in this article. Due to the inner electron, a departure of characteristic exponent (k from 2 to 0.96) happens to this atom. An average absolute error value of 0.14 eV was observed for this atom. This atom belongs to group 1 (IA) of s-block.

Beryllium (Be I, II, III, IV)
Beryllium atom has four electrons with three to be considered inner electrons for the outer electron. So the atom is not in line with our Bohr and Rydberg classical models due to the inner electron. Figure 7 presents the graphical displays of the energy levels for this atom and its ions. As one can observe, none of the experimental data is in line with our classical model. This is due to the inner electrons and a departure of 2 to 1.75 for Be IV, a departure of 2 to 1.13 for Be III, a departure of 2 to 0.93 for Be II, and a departure of 2 to 0.79 for Be I happens to the characteristic exponent k of this atom and its ions to adjust the simulated graphs closer to the experimental data. Average absolute error values of 0.16 eV for Be I, 0.57 eV for Be II, 3.28 eV for Be III, and 2.29 eV for Be IV were observed for this atom and its ions. This atom belongs to group 2 (IIA) of s-block. Now it is important to make some conclusions about the excited states using our observations of this atom and its ions. First, we can see that each ion/specie uses its own outer ionization energy to establish its excited energy levels which means Be I uses E n = (9.323)/n 0.79 , Be II uses E n = (18.211)/ n 0.93 , Be III uses E n = (153.896)/n 1.13 , and Be IV uses E n = (217.718)/n 1.75 to establish the energy levels. This proves some sort of independency among the ions which would indicate that the inner electrons are not affected by an outer electron in excitation mode. As a reminder, the equation of E n = (I.E.)/n k is always used to simulate the excited energy levels of each specie.   4 of Be IV should have also played a role and must have affected the energy levels of Be I due to lack of independency which is not observed.
So during the excitation of atoms/ions only one electron which is the one furthest away from nucleus would move away into excited states and the analogy of a closed chest with one opening for only one electron at the time to move out should apply to inner electrons of atoms. Hence, all inner electrons must been have kept in place as the outer electron goes through the excitation. And if an outer electron leaves the atom, then a new electron will be able to move out and fill the excited states. But in all possible ions of an atom, there would be only one electron at the time going through the excitation and only one ionization energy figure would govern the excited energy levels of that specie.

Boron (B I)
Boron atom has five electrons with four to be considered inner electrons for the outer electron. So the atom is not in line with our Bohr and Rydberg classical models due to the inner electron. Figure 8 presents the graphical display of energy levels for this atom. As one can observe, the experimental figures are not in line with the classical theoretical models but in better agreement with the method suggested in this article. This is due to the inner electrons and a departure of 2 to 1.10 happens to the characteristic exponent k of this atom to bring the simulated graph closer to the experimental data. An average absolute error value of 0.21 eV was observed for this atom. This atom belongs to group 13 (IIIA) of p-block.

Nitrogen (N I)
Nitrogen atom has seven electrons with six to be considered inner electrons for the outer electron. So the atom is not in line with our Bohr and Rydberg classical models due to the inner electron. Figure 9 presents the graphical display of energy levels for this atom. As one can observe, the experimental figures are not in line with the classical theoretical models but in better agreement with the method suggested in this article. Due to the inner electrons, a departure of 2 to 0.57 happens to the characteristic exponent k of this atom to bring the simulated graph closer to the experimental data. An average absolute error value of 0.26 eV was observed for this atom. This atom belongs to group 15 (VA) of p-block.

Magnesium (Mg I)
Magnesium atom has 12 electrons with 11 to be considered inner electrons for the outer electron. So the atom is not in line with our Bohr and Rydberg classical models due to the inner electron. Figure 10 presents the graphical display of energy levels for this atom. As one can observe, the experimental figures are not in line with the classical theoretical  Graphical presentation of energy levels of magnesium models but in better agreement with the method suggested in this article. Due to the inner electrons, a departure of 2 to 0.9 happens to the characteristic exponent k of this atom to bring the simulated graph closer to the experimental data. An average absolute error value of 0.18 eV was observed for this atom. This atom belongs to group 2 (IIA) of s-block.

Aluminum (Al I)
Aluminum atom has 13 electrons with 12 to be considered inner electrons for the outer electron. So the atom is not in line with our Bohr and Rydberg classical models due to the inner electron. Figure 11 presents the graphical display of energy levels for this atom. As one can observe, the experimental figures are not in line with the classical theoretical models but in better agreement with the method suggested in this article. Due to the inner electrons, a departure of 2 to 0.85 happens to the characteristic exponent k of this atom to bring the simulated graph closer to the experimental data. An average absolute error value of 0.05 eV was observed for this atom. This atom belongs to group 13 (IIIA) of p-block.

Potassium (K I)
Potassium atom has 19 electrons with 18 to be considered inner electrons for the outer electron. So the atom is not in line with our Bohr and Rydberg classical models due to the inner electron. Figure 12 presents the graphical display of energy levels for this atom. As one can observe, the experimental figures are not in line with the classical theoretical models but in better agreement with the method suggested in this article. Due to the inner electrons, a departure of 2 to 0.95 happens to the characteristic exponent k of this atom to bring the simulated graph closer to the experimental data. An average absolute error value of 0.12 eV was observed for this atom. This atom belongs to group 1 (IA) of s-block.

Zinc (Zn I)
Zinc atom has 30 electrons with 29 to be considered inner electrons for the outer electron. So the atom is not in line with our Bohr and Rydberg classical models due to the inner electron. Figure 13 presents the graphical display of energy levels for this atom. As one can observe, the experimental figures are not in line with the classical theoretical models but in better agreement with the method suggested in this article. Due to the inner electrons, a departure of 2 to 0.91 happens to the characteristic exponent k of this atom to bring the simulated graph closer to the experimental data. An average absolute error value of 0.37 eV was observed for this atom. This atom belongs to group 12 (IIB) of d-block.

Bromine (Br I)
Bromine atom has 35 electrons with 34 to be considered inner electrons for the outer electron. So the atom is not in line with our Bohr and Rydberg classical models due to the inner electron. Figure 14 presents the graphical display of energy levels for this atom. As one can observe, the experimental figures are not in line with the classical theoretical models but in better agreement with the method suggested in this article. Due to the inner electrons, a departure of 2 to 0.72 happens to the characteristic exponent k of this atom to bring the simulated graph closer to the experimental data. An average absolute error value of 0.35 eV was observed for this atom. This atom belongs to group 17 (VIIA) of p-block.

Rubidium (Rb I)
Rubidium atom has 37 electrons with 36 to be considered inner electrons for the outer electron. So the atom is not in line with our Bohr and Rydberg classical models due to the inner electron. Figure 15 presents the graphical display of energy levels for this atom. As one can observe, the experimental figures are not in line with the classical theoretical models but in better agreement with the method suggested in this article. Due to the inner electrons, a departure of 2 to 0.92 happens to the characteristic exponent k of this atom to bring the simulated graph

Molybdenum (Mo I)
Molybdenum atom has 42 electrons with 41 to be considered inner electrons for the outer electron. So the atom is not in line with our Bohr and Rydberg classical models due to the inner electron. Figure 16 presents the graphical display of energy levels for this atom. As one can observe, the experimental figures are not in line with the classical theoretical models but in better agreement with the method suggested in this article. Due to the inner electrons, a departure of 2 to 0.17 happens to the characteristic exponent k of this atom to bring the simulated graph closer to the experimental data. An average absolute error value of 0.12 eV was observed for this atom. This atom belongs to group 6 (VIB) of d-block.

Mercury (Hg I)
Mercury atom has 80 electrons with 79 to be considered inner electrons for the outer electron. So the atom is not in line with our Bohr and Rydberg classical models due to the inner electron. Figure 17 presents the graphical display of energy levels for this atom. As one can observe, the experimental figures are not in line with the classical theoretical models but in better agreement with the method suggested in this article. Due to the inner electrons, a departure of 2 to 0.83 happens to the characteristic exponent k of this atom to bring the simulated graph closer to the experimental data. An average absolute error value of 0.23 eV was observed for this atom. This atom belongs to group 12 (IIB) of d-block.

Thallium (Tl I)
Thallium atom has 81 electrons with 80 to be considered inner electrons for the outer electron. So the atom is not in line with our Bohr and Rydberg classical models due to the inner electron. Figure 18 presents the graphical display of  16 Graphical presentation of energy levels of molybdenum energy levels for this atom. As one can observe, the experimental figures are not in line with the classical theoretical models but in better agreement with the method suggested in this article. Due to the inner electrons, a departure of 2 to 0.86 happens to the characteristic exponent k of this atom to bring the simulated graph closer to the experimental data. An average absolute error value of 0.16 eV was observed for this atom. This atom belongs to group 13 (IIIA) of p-block. Figure 19 and 20, and Table 2 present variations of characteristic exponent k among the elements of periodic table. Figure 19 shows the variation of k value in segmented rows of periodic table. Each segment/row has been shown with a different color for clarity. Figure 20 presents the variation of characteristic exponent k for different groups     The drop of k value below 0.5 for the d-block (G3-G10) indicated of much larger and more disorganized inner electronic systems. This group was the hardest to fit a simulating curve on their experimental data and their data sometimes showed a linear behavior. The rest of elements were similar and had their k values between 0.5 and 1.0 in general. The author does not understand much of the essence and sources of the differences. Table 2 presents the tabulated values of the characteristic exponent (k) for elements of periodic table with the corresponding average absolute error in eV. Most of the data available in the data bank was processed for this study.

Conclusions
Most of the experimental energy levels of different elements available in NIST data bank were utilized for this study. The author did his best to separate the energy levels correctly and come up with a descent list of experimental data for the calculations and the study. It seems that the methodology of predicting the excited energy levels for different elements suggested in this article works sufficiently well. The mechanism of atomic excitation with the inner electrons in place and intact as suggested in this article is a new concept which may need further investigations. Also, the introduction of "characteristic exponent k" together with the related ionization energies as a tool to predict the excited energy levels of elements seemed effective and successful. Meanwhile, the characteristic exponent k showed potential to provide us with clues about the status of inner electrons of atoms as well as better understanding of electronic structures. This article is considered a ground work for future developments.
Author contribution Author contributed everything to the study from conception, design, material preparation, data collection, and analysis all were performed by the author. There is only one author who contributed all of the research work.
Data availability All data and material were submitted with full transparency.
Code availability There was no software or code prepared for this article.

Declarations
Ethics approval This article does not involve research involving human or animal subjects and therefore no ethical approval is required.

Consent to participate
This article does not involve human subjects and therefore does require consent to participate.