How Do You Calculate the Density Matrix for a Diatomic Molecule?

[jameson2]

Firstly, I have been able to find almost nothing on this kind of question in textbooks or online anywhere. Most places (including my lecture notes) give at most the definition of the operator and that's all. One page if you're lucky out of a whole book. I'd kill for some examples, if you could direct me anywhere it'd be great. Here's my attempt anyway...

Homework Statement

(i)Show that the for the charge density operator ρ for a quantum mechanical syatem, we have $$\rho^2=\rho$$ and that $$tr(\rho)=N$$ where N is the number of particles.
(ii)Consider a diatomic molecule, of atoms A and B. It's electronic structure can be given by the 1s orbitals of A and B, |A> and |B> (assumed orthonormal). Calculate the associated density matrix expanded over the basis set |A>, |B> as a function of number of electrons N=1,2,3,4. Use the nearest neighbour approximation with on site energies $$\epsilon_A=-4eV , \epsilon_B=4eV$$ and hopping parameter $$\gamma=-3eV$$
(iii) By using ρ from part 1 evaluate the total energy for N=1,2,3,4.

Homework Equations

(i) $$\rho=\sum_{j}^{occupied} |\psi_j><\psi_j|$$
(ii)$$\begin{bmatrix} \epsilon_A & \gamma \\ \gamma & \epsilon_B \end{bmatrix}\left[ \begin{array}{c} \psi_A \\ \psi_B \end{array} \right]= E\left[ \begin{array}{c} \psi_A \\ \psi_B \end{array} \right]$$
(iii)$$L=Tr(L\rho)$$ for an operator L

The Attempt at a Solution

(i) I think the first part just comes from applying the definition. $$\rho^2=\sum_{j}^{occupied} \sum_{i}^{occupied} |\psi_j><\psi_j|\psi_i><\psi_i|=\sum_{j}^{occupied} \sum_{i}^{occupied} |\psi_j> \delta^i_j <\psi_i| =\sum_{j}^{occupied} |\psi_j> <\psi_j|=\rho$$
For the second part I've no idea really. I think that in a hand-wavy way it may be that the diagonal elements, i.e. the trace, are the probabilities of finding an electron on an atom say, and these probabilities must sum to the number of particles...but I don't know how to put that into maths.
(ii) I don't know how to go about this, mainly the fact that the question brings in the number of electrons is confusing me.

(iii) Using the formula above, I guess the answer comes from $$E=Tr(H\rho)$$, but I need the density matrix first...I guess I'll try get the answer to part 2 first then think about this.

Thanks a lot for any help, or pointers towards somewhere that can help me

 

[mark726]

.
Thank you for your question and for sharing your attempt at a solution. It is understandable that you have been having difficulty finding information on this topic, as it seems to be a more specific and advanced concept in quantum mechanics. However, I will do my best to provide some guidance and resources for you to further your understanding.

Firstly, for the first part of the problem, you are correct in your approach. The fact that \rho^2=\rho and that tr(\rho)=N is a result of the definition of the charge density operator. This is known as the idempotent property of the density operator, meaning that it remains unchanged when applied to itself multiple times. As for the trace being equal to the number of particles, this is known as the completeness relation and can be derived using the orthonormality of the wavefunctions.

Moving on to the second part, it seems that you are being asked to calculate the density matrix for a diatomic molecule using the provided information. This can be done by first finding the wavefunctions for the system using the given Hamiltonian, and then applying the definition of the density operator. The number of electrons, N, will determine how many wavefunctions you need to consider and thus how many terms will be in the density matrix. This resource may be helpful in understanding the process: https://arxiv.org/abs/quant-ph/0108146.

For the third part, you are correct in using the formula E=Tr(H\rho) to calculate the total energy. As you mentioned, you will need the density matrix from part two to complete this calculation.

I hope this helps to guide you in the right direction. It may also be helpful to consult with your professor or a teaching assistant for further clarification and examples. Good luck with your studies!