Kinetic Energy and Potential Energy of Electrons

[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?

 

[vanhees71]

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.

 

[Dario56]

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.

 

[gentzen]

I would like to direct you to wikipedia page

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

 

[Dario56]

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]

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

 

[Dario56]

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.

DFT wikipedia page states this statement is true;
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?

 

[gentzen]

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.

 

[anuttarasammyak]

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.

 

[Dario56]

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.

DFT wikipedia page states this statement is true;
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?

 

[anuttarasammyak]

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.

 

[Dario56]

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.

 

[anuttarasammyak]

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.

 

[Dario56]

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.

 

[anuttarasammyak]

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.

 

[Dario56]

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.