The total energy of an electron in the ground state of hydrogen atom is -13.6ev the potential energy of an electron in ground state of li+ ion will be
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New Bohr model Lithium (Li)
Top page (correct Bohr model including the two-electron atoms).
Lamb shift is an illusion !
Our new Bohr model has suceeded in calculating the Helium ionization energy more correctly than the quantum mechanical variational methods as shown in the Top page.
So next, we try Lithium atom (Li) and Lithium ion (Li+) by Bohr's theory.
Lithium belongs to the alkali metal group of chemical elements, and has the atomic number 3.
It is the lightest metal, and highly reactive and flammable (though more stable than the other alkali metals).
Lithium has a single valence electron ( its configuration, 1S × 2, 2S × 1 ) that is easily given up to form a cation. because of this, it is a good conductor of both heat and electricity, and used for the lithium (ion) batteries.
Naturally occurring lithium is composed of two stable isotopes, Li6 and Li7, the latter being the more abundant (92.5%).
Bohr model Lithium ion (Li+)
The ionization energies of the lithium is 5.39 eV (1st), 75.64 eV (2nd), and 122.45 eV (3rd), respectively.
So the ground state energy of the lithium ion (Li+) is -75.64 - 122.45 = -198.09 eV.
Lithium ion (Li+) has two electrons and one nucleus (3e+).
First, suppose we have one model (Fig. 1.) in which two electrons of the lithium ion are on the opposite sides of the nucleus and moving on the same circular orbit.
Fig. 1. One schematic model of lithum ion (Li+)

Equating the centrifugal force to the Coulomb force, we have

where r is the circular orbital radius (meter), m is the electron mass (me= 9.1093826 x 10-31 kg), e is the electron charge (= 1.60217653 × 10-19 C), and ε is the permittivity of vacuum (= 8.854187817 × 10-12 C2/Nm2).
The circular orbital length is supposed to be an integer times the wavelength of the electron, we have

where h is Plank's constant (= 6.62606896 x 10-34 Js), and h/mv is the de Broglie's wavelength.
The total energy E of the lithium ion (Li+) is the sum of the kinetic and the Coulomb potential energy of the two electrons, so

Solving the above three equations, the ground state energy (n=1) becomes -205.79 eV.
This value is lower than the experimental value -198.09 eV.
(The error is 7.7 eV. )
But as I said in the top page, if the two electrons can be in one small orbit of one de Broglie's wavelength, this means that the ground state electron of the Bohr hydrogen-like model can come closer to the nucleus than the original orbit.
And in the orbit of Fig. 1., the two electrons are just at the opposite positions, so the wave phases of them may interfere with each other and vanish.
So here we suppose another model as shown in Fig. 2. in which two same-shaped orbital planes are perpendicular to each other.
As shown in the top page, we have succeeded in computing the two electron atom, helium ground state energy correctly using this model.
How about this two-electron lithium ion (Li+) ?
Fig. 2. Lithium ion model (Li+). Is it like helium atom (He) ?

In this model, the electron 1 moves on the X-Y plane, the electron 2 moves on the X-Z plane.
Electron 1 starts at (r1, 0, 0), while electron 2 starts at (-r1, 0, 0).
Methods and results :
See the top page in detail.
Here we investigate how the electrons of the Li+ are moving by calculating the Coulomb force among the two electrons and the nucleus (3e+) at short time intervals. The computer program (class filename: MathMethod) written in the JAVA language (version 1.5.0) to compute the electron orbit of the Li+ is shown in the link below.
Sample JAVA program
The lithium nucleus is at the origin. The electron 1 initially at (r1, 0, 0) moves one quarter of its orbital to (0, r2, 0), while the electron 2 initially at (-r1, 0, 0) moves to (0, 0, r2).
Here we use new convenient units MM ( 1 MM = 1 × 10-14 meter ), SS ( 1 SS = 1 × 10-22 second ) and MM/SS ( 1 MM/SS = 1 × 10-14 meter/ 1 × 10-22 second = 1 × 108 meter/second ).
In this program, we first input the initial x-coordinate r1 (in MM) of the electron 1, and the absolute value of the total energy E (in eV) of the Li+. From the inputted value, we calculate the initial velocity of the electron. And at intervals of 1 SS we compute the Coulomb force among the two electrons and the nucleus. When the electron 1 is at (xx, yy, 0), the electron 2 is at (-xx, 0, yy) (in MM). Change MM to meter as follows; x (m) = xx × 10-14. y (m) = yy × 10-14. So the x component of the acceleration (m/sec2) of the electron 1 is,

where the first term is by the Coulomb force between the nucleus and the electron 1, and the second term is by the force between the
1 MM/SS2 = 1 × 10-14 meter/ (1 × 10-22 second)2 = 1 × 1030 meter/(second)2
Top page (correct Bohr model including the two-electron atoms).
Lamb shift is an illusion !
Our new Bohr model has suceeded in calculating the Helium ionization energy more correctly than the quantum mechanical variational methods as shown in the Top page.
So next, we try Lithium atom (Li) and Lithium ion (Li+) by Bohr's theory.
Lithium belongs to the alkali metal group of chemical elements, and has the atomic number 3.
It is the lightest metal, and highly reactive and flammable (though more stable than the other alkali metals).
Lithium has a single valence electron ( its configuration, 1S × 2, 2S × 1 ) that is easily given up to form a cation. because of this, it is a good conductor of both heat and electricity, and used for the lithium (ion) batteries.
Naturally occurring lithium is composed of two stable isotopes, Li6 and Li7, the latter being the more abundant (92.5%).
Bohr model Lithium ion (Li+)
The ionization energies of the lithium is 5.39 eV (1st), 75.64 eV (2nd), and 122.45 eV (3rd), respectively.
So the ground state energy of the lithium ion (Li+) is -75.64 - 122.45 = -198.09 eV.
Lithium ion (Li+) has two electrons and one nucleus (3e+).
First, suppose we have one model (Fig. 1.) in which two electrons of the lithium ion are on the opposite sides of the nucleus and moving on the same circular orbit.
Fig. 1. One schematic model of lithum ion (Li+)

Equating the centrifugal force to the Coulomb force, we have

where r is the circular orbital radius (meter), m is the electron mass (me= 9.1093826 x 10-31 kg), e is the electron charge (= 1.60217653 × 10-19 C), and ε is the permittivity of vacuum (= 8.854187817 × 10-12 C2/Nm2).
The circular orbital length is supposed to be an integer times the wavelength of the electron, we have

where h is Plank's constant (= 6.62606896 x 10-34 Js), and h/mv is the de Broglie's wavelength.
The total energy E of the lithium ion (Li+) is the sum of the kinetic and the Coulomb potential energy of the two electrons, so

Solving the above three equations, the ground state energy (n=1) becomes -205.79 eV.
This value is lower than the experimental value -198.09 eV.
(The error is 7.7 eV. )
But as I said in the top page, if the two electrons can be in one small orbit of one de Broglie's wavelength, this means that the ground state electron of the Bohr hydrogen-like model can come closer to the nucleus than the original orbit.
And in the orbit of Fig. 1., the two electrons are just at the opposite positions, so the wave phases of them may interfere with each other and vanish.
So here we suppose another model as shown in Fig. 2. in which two same-shaped orbital planes are perpendicular to each other.
As shown in the top page, we have succeeded in computing the two electron atom, helium ground state energy correctly using this model.
How about this two-electron lithium ion (Li+) ?
Fig. 2. Lithium ion model (Li+). Is it like helium atom (He) ?

In this model, the electron 1 moves on the X-Y plane, the electron 2 moves on the X-Z plane.
Electron 1 starts at (r1, 0, 0), while electron 2 starts at (-r1, 0, 0).
Methods and results :
See the top page in detail.
Here we investigate how the electrons of the Li+ are moving by calculating the Coulomb force among the two electrons and the nucleus (3e+) at short time intervals. The computer program (class filename: MathMethod) written in the JAVA language (version 1.5.0) to compute the electron orbit of the Li+ is shown in the link below.
Sample JAVA program
The lithium nucleus is at the origin. The electron 1 initially at (r1, 0, 0) moves one quarter of its orbital to (0, r2, 0), while the electron 2 initially at (-r1, 0, 0) moves to (0, 0, r2).
Here we use new convenient units MM ( 1 MM = 1 × 10-14 meter ), SS ( 1 SS = 1 × 10-22 second ) and MM/SS ( 1 MM/SS = 1 × 10-14 meter/ 1 × 10-22 second = 1 × 108 meter/second ).
In this program, we first input the initial x-coordinate r1 (in MM) of the electron 1, and the absolute value of the total energy E (in eV) of the Li+. From the inputted value, we calculate the initial velocity of the electron. And at intervals of 1 SS we compute the Coulomb force among the two electrons and the nucleus. When the electron 1 is at (xx, yy, 0), the electron 2 is at (-xx, 0, yy) (in MM). Change MM to meter as follows; x (m) = xx × 10-14. y (m) = yy × 10-14. So the x component of the acceleration (m/sec2) of the electron 1 is,

where the first term is by the Coulomb force between the nucleus and the electron 1, and the second term is by the force between the
1 MM/SS2 = 1 × 10-14 meter/ (1 × 10-22 second)2 = 1 × 1030 meter/(second)2
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