Charge Density Matrix: Calculating & Evaluating for N Electrons

[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)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 $$\gamma=-3eV$$
(ii) 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 don't know how to go about this, mainly the fact that the question brings in the number of electrons is confusing me. Here's what I've reasoned so far: Since it's a molecule, all the electrons are in the bonding state. Solving the matrix equation to find the bonding state, I get bonding energy of -5eV which leads to a bonding state of $$\frac{1}{\sqrt{10}} (3,1)$$. Applying the formula for the density matrix for N electrons in the bonding state, I get $$\rho= N\frac{1}{10}\begin{bmatrix} 9 & 3 \\ 3 & 1 \end{bmatrix}$$

(ii) Using the formula above, I guess the answer comes from $$E=Tr(H\rho)$$, but I need the density matrix first. Assuming I have the right density matrix, plugging in the Hamiltonian $$\begin{bmatrix} -4 & -3 \\ -3 & 4 \end{bmatrix}$$ I get the energy to be -5N for N=1,2,3,4.

Thanks a lot for any help, or pointers towards somewhere that can help me. As I said, I'm having trouble finding info on this topic, you're my last chance!

 

[mark726]

Thank you for your question. It is understandable that you are having trouble finding information on this topic, as it is a more specific and advanced concept in quantum mechanics. However, I will do my best to provide some guidance and resources for you.

Firstly, let's break down the problem into smaller parts. In part (i), you are asked to calculate the density matrix for a diatomic molecule with a given electronic structure and energy parameters. The density matrix is a mathematical tool used to describe the quantum state of a system. In this case, it will give us information about the electronic structure of the molecule and how many electrons are in each orbital.

To calculate the density matrix, you will need to use the formula you provided: \rho=\sum_{j}^{occupied} |\psi_j><\psi_j| , where |\psi_j> represents the wavefunction for each occupied orbital. In this case, you have two orbitals, |A> and |B>, and you will need to find the wavefunction for each one. You can do this by solving the matrix equation you provided: \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] , where \epsilon_A and \epsilon_B are the on-site energies and \gamma is the hopping parameter. The energy, E , will depend on the number of electrons in the system, which brings us to the next part of the problem.

In part (ii), you are asked to use the density matrix you calculated in part (i) to evaluate the total energy of the system for different numbers of electrons. To do this, you will use the formula: E=Tr(H\rho) , where H is the Hamiltonian for the system. This will give you the total energy for a given number of electrons. You can then repeat this calculation for different numbers of electrons to get the energy for N=1,2,3,4.

As for finding examples of similar problems, I suggest looking into textbooks or online resources on quantum chemistry or molecular physics. These topics often cover the electronic structure of molecules and may