Kinetic Energy and Potential Energy of Electrons

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• Dario56
In summary, the time independent Schrödinger equation for a system (atom or molecule) consisting of N electrons can be written as a Hamiltonian operator consisting of the sum of kinetic energy of electrons, potential energy of electron-nuclei interaction, and potential energy of electron-electron interaction. The terms in the Hamiltonian are universal for all systems with the same number of electrons, except for the potential energy of electron-nuclei interaction which is system dependent. This is due to the fact that the excluded electrons in the detailed analysis can affect the potential of the nuclei, making it non-universal. This is supported by DFT and Hohenberg-Kohn theorems, where the Hamiltonian is written as a sum of a
Dario56
Time indepedendent Schrödinger equation for a system (atom or molecule) consisting of N electrons can be written as (with applying Born - Oppenheimer approximation): $$[(\sum_{i=1}^N - \frac {h^2} {2m} \nabla _i ^2) + \sum_{i=1}^N V(r_i) + \sum_{i < j}^N U(r_i,r_j)] \Psi = E \Psi$$

Terms in Hamiltonian are as follows:
1) Kinetic energy of electrons
2) Potential energy of electron - nuclei interaction
3) Potential energy of electron - electron interaction

It is said that for N electron system, kinetic and potential energy of electron - electron interaction are system independent which means that their value depends only on number of electrons ##N##and nothing else. Potential energy of electron - nuclei interaction depends on specific system and isn't determined only by ##N##.

Why is this?

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I don't understand the question, but in your formula there's a factor 1/2 missing in the last term, because you should have only one interaction potential per electron pair and not two.

Lord Jestocost and Dario56
vanhees71 said:
I don't understand the question, but in your formula there's a factor 1/2 missing in the last term, because you should have only one interaction potential per electron pair and not two.
I would like to direct you to wikipedia page: Density Functional Theory, section: Derivation and Formalism. 2nd paragraph states what I do here.

https://en.m.wikipedia.org/wiki/Density_functional_theory#:~:text=Density-functional theory (DFT),molecules, and the condensed phases.

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Dario56 said:
vanhees71 is right, you wrote ##i \neq j## below the sum, but wikipedia wrote ##i < j##. Therefore you are missing a factor 1/2.

PeroK, Dario56 and vanhees71
gentzen said:
vanhees71 is right, you wrote ##i \neq j## below the sum, but wikipedia wrote ##i < j##. Therefore you are missing a factor 1/2.
Yes I've made the correction. However, this doesn't really answer my question.

vanhees71 said:
As I said, I don't understand your question. The correct Hamiltonian can be found in the Wikipedia article on the BO approximation:

https://en.wikipedia.org/wiki/Born–Oppenheimer_approximation
I think we don't understand each other. I made a correction about Hamiltonian, but that is not the point of this question. Question is why are kinetic and electron - electron potential energy operators universal while electron - nuclei potential energy operator is not.

Section: Derivation and Formalism, 2nd paragraph. I would like you to read the 2nd paragraph, I am not reffering to the Schrödinger equation.

https://en.m.wikipedia.org/wiki/Density_functional_theory

Other example is this page where Hohenberg - Kohn theorems are proved:

http://cmt.dur.ac.uk/sjc/thesis_ppr/node12.html

Hamiltonian is written as:
$$H = F + V_{ext}$$

Where ##F## is called universal operator since it is the same for all systems with the same number of electrons and ##V_{ext} ##is system dependent. ##F## is defined as sum of kinetic and electron - electron potential energy operators.

I don't understand why is ##F## universal and why ##V_{ext}## isn't?

Dario56 said:
I don't understand why is ##F## universal and why ##V_ext## isn't?
One reason is that you often decide for yourself which electrons you want to study in detail, and which electrons you want to just include in the external potential. For example, including inner shell electrons in the detailed (DFT or other) analysis for heavy elements like gold will just make the analysis computationally infeasible, without adding significant value.

Since ##F## only depends on the number of electrons, it is universal in this context. But ##V_ext## depends on details of how the electrons excluded from the detailed analysis shield the potential of the nuclei. Therefore it is not universal.

Dario56 said:
It is said that for N electron system, kinetic and potential energy of electron - electron interaction are system independent which means that their value depends only on number of electrons N and nothing else.
Determining energy level structure of N electron gas not free but with mutual Coulomb repulsive forces requires boundary condition e.g. contained in a box of volume V. I do not think it depends only on N.

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anuttarasammyak said:
Determining energy level structure of N electron gas not free but with mutual Coulomb repulsive forces requires boundary condition e.g. contained in a box of volume V. I do not think it depends only on N.
Section: Derivation and Formalism, 2nd paragraph.

https://en.m.wikipedia.org/wiki/Density_functional_theory

Other example is this page where Hohenberg - Kohn theorems are proved:

http://cmt.dur.ac.uk/sjc/thesis_ppr/node12.html

Hamiltonian is written as: $$H = F + V_{ext}$$
Where ##F## is called universal operator since it is the same for all systems with the same number of electrons and ##V_{ext}## is system dependent. ##F## is defined as sum of kinetic and electron - electron potential energy operators.

I don't understand why is ##F## universal and why ##V_{ext}## isn't?

vanhees71
Dario56 said:
I don't understand why is F universal and why Vext isn't?
Operators F and V_ext are both functions or functionals of coordinate operators of electrons. In addition V_ext is also function of coordinate of positive ions or nuclei R_i with distance to electrons, ##\mathbf{R_i}-\mathbf{r_j}##. I think it is the difference.

vanhees71
anuttarasammyak said:
F and V_ext are both functions of coordinates of electrons. In addition V_ext is function of coordinate of positive ions or nuclei with distance ##\mathbf{R_i}-\mathbf{r_j}##. I think it is the difference.
Electron - electron interaction energy also depends on their distance not only electron - nuclei interaction energy.

vanhees71
Electron positions are operators but nuclei position R_i s are fixed value which should specify the system we are considering. If we take R_i variables or operators also, it is not external potential. The system would not be crystal but plasma.

vanhees71
anuttarasammyak said:
Electron positions are operators but nuclei position R_i s are fixed value which should specify the system we are considering. If we take R_i variables or operators also, it is not external potential. The system would not be crystal but plasma.
Yes, electrons move and their position isn't fixed while nuclei have fixed positions in Born - Oppenheimer approximation.

I am still not sure what does that have to do with kinetic and electron - electron interaction energy being universal, though.

Dario56 said:
I am still not sure what does that have to do with kinetic and electron - electron interaction energy being universal, though.
I take it the form or formula of operators ,i.e.
$$-\hbar^2\frac{d^2}{dx_i^2}$$ and
$$\frac{e^2}{|r_i-r_j|}$$
are universal in the sense that we can write it down with no knowledge on positions of positive ions which specifies the system. But
$$-\frac{Ne^2}{|R_i-r_j|}$$
depends on R_i so it is not universal.

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gentzen and Dario56
anuttarasammyak said:
I take it the form or formula of operators ,i.e.
$$-\hbar^2\frac{d^2}{dx_i^2}$$ and
$$\frac{e^2}{|r_i-r_j|}$$
are universal in the sense that we can write it down with no knowledge on positions of positive ions which specifies the system. But
$$-\frac{Ne^2}{|R_i-r_j|}$$
depends on R_i so it is not universal.
Everything is clear now. Thank you.

1. What is the difference between kinetic energy and potential energy?

Kinetic energy is the energy an object possesses due to its motion, while potential energy is the energy an object possesses due to its position or state. In the case of electrons, kinetic energy refers to the energy they possess due to their movement around an atom's nucleus, while potential energy refers to the energy they possess due to their position in relation to the nucleus.

2. How are kinetic energy and potential energy related to each other for electrons?

According to the Bohr model of the atom, electrons exist in specific energy levels or shells around the nucleus. The further away an electron is from the nucleus, the higher its potential energy. As an electron moves closer to the nucleus, its potential energy decreases, but its kinetic energy increases. This is because the electron's speed increases as it moves closer to the nucleus.

3. How does temperature affect the kinetic energy of electrons?

Temperature is a measure of the average kinetic energy of particles in a substance. As temperature increases, the kinetic energy of electrons also increases. This is because higher temperatures cause atoms and molecules to vibrate faster, which in turn increases the speed and kinetic energy of the electrons within them.

4. What is the role of kinetic and potential energy in chemical reactions?

In chemical reactions, electrons are transferred or shared between atoms. This transfer or sharing of electrons involves changes in both kinetic and potential energy. For example, in an exothermic reaction, potential energy is converted into kinetic energy as the reacting particles gain energy and move faster. In an endothermic reaction, kinetic energy is converted into potential energy as the reacting particles absorb energy and move to higher energy levels.

5. How is the concept of kinetic and potential energy of electrons applied in technology?

The understanding of kinetic and potential energy of electrons is essential in fields such as electronics and semiconductors. For example, in electronic devices, electrons are manipulated to move from one energy level to another, which allows for the flow of electricity. In semiconductors, the movement of electrons between energy levels is controlled to create specific properties and functions, such as in computer chips and solar cells.

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