There's many different density matrices that can correspond to a given Hamiltonian.

For the 8x8 matrix in your question, there's 8 different "eigenstate" density matrices that can be obtained, one for each of the 8 eigenvectors. The density matrices are constructed by doing the outer product of the eigenvectors. For the $i^{\rm{th}}$ eigenstate of the Hamiltonian, the density matrix $\rho_i$ is:

$ \rho_i = |\psi_i\rangle_ \langle \psi_i| $.

A system can also be in a "pure" superposition of eigenstates, for example:

$|\psi \rangle = \frac{1}{\sqrt{2}}|\psi_1\rangle + \frac{1}{\sqrt{2}}|\psi_2\rangle $.

Then the density matrix is once again made by doing the outer product of he pure wave function $|\psi\rangle$.

A system can also be in a "mixed" state, which means it's a linear combination of "pure" states.

In this case you would construct the density matrix like this (for example):

$\rho = 0.5 \rho_1 + 0.5\rho_2$,

which descrbes a state which is a 50% mixture of $\rho_1$ and a 50% mixture of $\rho_2$.

There's many different density matrices that can correspond to a given Hamiltonian.

For the 8x8 matrix in your question, there's 8 different "eigenstate" density matrices that can be obtained, one for each of the 8 eigenvectors. The density matrices are constructed by doing the outer product of the eigenvectors. For the $i^{\rm{th}}$ eigenstate of the Hamiltonian, the density matrix $\rho_i$ is:

$ \rho_i = |\psi_i\rangle_ \langle \psi_i| $.

A system can also be in a "pure" superposition of eigenstates, for example:

$|\psi \rangle = \frac{1}{\sqrt{2}}|\psi_1\rangle + \frac{1}{\sqrt{2}}|\psi_2\rangle $.

Then the density matrix is once again made by doing the outer product of the pure wave function $|\psi\rangle$ with itself.

A system can also be in a "mixed" state, which means it's a linear combination of "pure" states.

In this case you would construct the density matrix like this (for example):

$\rho = 0.5 \rho_1 + 0.5\rho_2$,

which descrbes a state which is a 50% mixture of $\rho_1$ and a 50% mixture of $\rho_2$.