Density Matrix in that DFT bible book

[wolich22]

-- i know there were threads about reduced density matrix in this forum, but I am reading "Density-functional theory of atoms and molecules" by Parr R., Yang W., their notation is quite confusing to me... their notation is the same as shown in this page:
http://www-theor.ch.cam.ac.uk/people/ross/thesis/node34.html"
-- for a N-electron system (without the number of electrons changing), the matrix element of density operator is:
$$\gamma$$(x'₁x'₂...x'_N, x₁x₂...x_N)=psi(x'₁x'₂...x'_N)psi*(x₁x₂...x_N) but in my mind, in the place of x'₁x'₂...x'_N and x₁x₂...x_N, there should be a basis vector from a complete set of basis of the hilbert space for that the system, like
psi(<C_i|)psi*(|C_j>) = <C_i||psi><psi||C_j> (for a pure state).
-- it becomes more comfusing for the mix state, when the ensemble density operator is:
$$\Gamma$$ = sum( p_i|psi_i|><|psi_i|)

questions:
1. for the pure states in ensemble operator $$\Gamma$$, are they in the same hilbert space or not OR do they share the same set of basis? if not, then we can't to sandwich density operator $$\Gamma$$ with the same basis...
2. what dose x₁x₂...x_N actually mean? each electron x_i is a system? and the basis set for them is ALL SPACE plus spins?
3. the reduced density matrix is written as
(n choose p)$$\int$$psi(x'₁x'₂...x'_p, x_p+1...x_N)psi*(x₁x₂...x_p, x_p+1...x_N) dx_p+1...dx_N
(this is for order p, you can refer to order 2 reduced density matrix in the above webpage). The ' is gone after x_p in psi(...). Is it because all the cross terms actually vanished during integration so the author of the book just wrote down what is left?

soooo many questions thanks!

 

[azntoon]

1. For the pure states in the ensemble operator \Gamma, they do not necessarily have to be in the same Hilbert space. However, it is important to note that all of the basis vectors must be compatible with each other, meaning that they can all be expressed in terms of a common set of basis states. 2. The x1x2...xN actually represents the coordinates of each electron in the system. The basis set for them is usually the complete set of atomic orbitals, which can be thought of as the space plus spins.3. Yes, the ' is gone after xp in psi(...) because all of the cross terms vanish during integration, so the author of the book just wrote down what is left. This is because the density matrix elements are only non-zero when the two sets of coordinates (x'1x'2... and x1x2...) correspond to the same state.

 

[Entropia]

1. The pure states in the ensemble operator \Gamma are in the same Hilbert space, as they represent different possible states that the system could be in. However, they may not necessarily share the same set of basis. The density operator \Gamma can be sandwiched with the same basis if the basis is complete and orthonormal, as shown in the equation you mentioned.
2. The notation x1x2...xN represents the coordinates of all N electrons in the system, including their positions and spins. The basis set for them is typically the complete set of atomic orbitals or plane wave basis functions, depending on the system being studied.
3. The reduced density matrix is obtained by integrating out the coordinates of some electrons in the system, leaving only the coordinates of the remaining p electrons. The ' is gone after xp in psi(...) because the integration has been performed, and as you mentioned, the cross terms have vanished. The remaining terms represent the probability of finding the remaining p electrons in the given state.