Derivation of two-electron density operator

In summary, the conversation discussed the derivation of the second term in 2.11a of the two-electron density. The expert explained that the term eliminates the i=j products through the use of certain identities and integration methods. They also clarified that the two-electron density is not commonly written in this form.
  • #1
Mart1234
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Derivation of two electron density operator using single electron density operator
Hello, I am going over the derivation for two-electron density. I am having a hard time understanding how the second term in 2.11a seen below is derived. I know this term must eliminate the i=j products but can't seem to understand how. Thanks for the help.
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  • #2
Where is this coming from? The 2nd equality of (2.11a) seems to indicate that you consider a special state of uncorrelated/free particles, but we need more context to make sense of it.
 
  • #3
I've never seen the two-electron density written like that. Here are my thoughts but I can't say for sure.

Mart1234 said:
I know this term must eliminate the i=j products but can't seem to understand how.
Considering the terms where ##i=j## $$\sum_{i}\delta(\mathbf r-\mathbf r_i)\delta(\mathbf r' - \mathbf r_i)$$We should be able to use the identities ##\delta(x - y) = \delta(y - x)## and ##\int dy \delta(x-y)\delta(y-x') = \delta(x-x')## So we rewrite the above as $$\sum_{i}\delta(\mathbf r-\mathbf r_i)\delta(\mathbf r_i - \mathbf r')$$ and when we are computing the electron density function we will be integrating over ##\mathbf r_i## so the latter identity suggests $$\sum_i\int d\mathbf r_1\dots d\mathbf r_N \delta(\mathbf r - \mathbf r_i)\delta(\mathbf r_i - \mathbf r')|\Psi(\mathbf r_1,\dots, \mathbf r_N)|^2 = N\int d\mathbf r_2\dots d\mathbf r_N \delta(\mathbf r - \mathbf r')|\Psi(\mathbf r,\dots, \mathbf r_N)|^2$$For the last line. I am assuming $$\int dx\int dy \delta(x-y)\delta(y-x')f(y) = \int dx \delta(x-x')f(x)$$
 
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Morbert said:
I've never seen the two-electron density written like that. Here are my thoughts but I can't say for sure.Considering the terms where ##i=j## $$\sum_{i}\delta(\mathbf r-\mathbf r_i)\delta(\mathbf r' - \mathbf r_i)$$We should be able to use the identities ##\delta(x - y) = \delta(y - x)## and ##\int dy \delta(x-y)\delta(y-x') = \delta(x-x')## So we rewrite the above as $$\sum_{i}\delta(\mathbf r-\mathbf r_i)\delta(\mathbf r_i - \mathbf r')$$ and when we are computing the electron density function we will be integrating over ##\mathbf r_i## so the latter identity suggests $$\sum_i\int d\mathbf r_1\dots d\mathbf r_N \delta(\mathbf r - \mathbf r_i)\delta(\mathbf r_i - \mathbf r')|\Psi(\mathbf r_1,\dots, \mathbf r_N)|^2 = N\int d\mathbf r_2\dots d\mathbf r_N \delta(\mathbf r - \mathbf r')|\Psi(\mathbf r,\dots, \mathbf r_N)|^2$$For the last line. I am assuming $$\int dx\int dy \delta(x-y)\delta(y-x')f(y) = \int dx \delta(x-x')f(x)$$

Morbert said:
I've never seen the two-electron density written like that. Here are my thoughts but I can't say for sure.Considering the terms where ##i=j## $$\sum_{i}\delta(\mathbf r-\mathbf r_i)\delta(\mathbf r' - \mathbf r_i)$$We should be able to use the identities ##\delta(x - y) = \delta(y - x)## and ##\int dy \delta(x-y)\delta(y-x') = \delta(x-x')## So we rewrite the above as $$\sum_{i}\delta(\mathbf r-\mathbf r_i)\delta(\mathbf r_i - \mathbf r')$$ and when we are computing the electron density function we will be integrating over ##\mathbf r_i## so the latter identity suggests $$\sum_i\int d\mathbf r_1\dots d\mathbf r_N \delta(\mathbf r - \mathbf r_i)\delta(\mathbf r_i - \mathbf r')|\Psi(\mathbf r_1,\dots, \mathbf r_N)|^2 = N\int d\mathbf r_2\dots d\mathbf r_N \delta(\mathbf r - \mathbf r')|\Psi(\mathbf r,\dots, \mathbf r_N)|^2$$For the last line. I am assuming $$\int dx\int dy \delta(x-y)\delta(y-x')f(y) = \int dx \delta(x-x')f(x)$$
Got it. I appreciate the help.
 

FAQ: Derivation of two-electron density operator

What is a two-electron density operator?

A two-electron density operator is a mathematical representation used in quantum mechanics to describe the probability distribution of two electrons in a given quantum state. It provides information about the spatial and spin correlations between the two electrons, allowing us to understand their joint behavior within a quantum system.

How is the two-electron density operator derived?

The two-electron density operator is derived from the many-body wave function of a system containing two electrons. By taking the appropriate partial trace over the degrees of freedom of the other particles in the system, one can obtain the reduced density operator that describes the two-electron subsystem. This involves integrating the full wave function with respect to the coordinates of the other particles while keeping the coordinates of the two electrons fixed.

What is the significance of the two-electron density operator in quantum chemistry?

The two-electron density operator plays a crucial role in quantum chemistry as it allows for the calculation of various properties of electronic systems, such as electron-electron interactions, correlation effects, and the determination of molecular orbitals. It serves as a foundation for methods like Hartree-Fock and Density Functional Theory (DFT), which are essential for predicting the behavior of electrons in molecules.

What are the limitations of using the two-electron density operator?

One limitation of the two-electron density operator is that it does not account for the full many-body correlations present in systems with more than two electrons. As the number of electrons increases, the complexity of the interactions grows, making it challenging to accurately describe the system using only the two-electron density operator. Additionally, approximations are often needed, which may lead to inaccuracies in the predicted properties.

How does the two-electron density operator relate to the electron correlation problem?

The two-electron density operator is directly related to the electron correlation problem, which refers to the inability of single-particle approaches to fully capture the correlated motion of electrons in a system. By analyzing the two-electron density operator, one can gain insights into the correlation effects between pairs of electrons, which are essential for understanding the overall electron correlation in many-electron systems. This understanding is critical for improving computational methods in quantum chemistry.

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