# Density Matrix in that DFT bible book

1. May 2, 2009

### 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...xN)=psi(x'1x'2...x'N)psi*(x1x2...xN) but in my mind, in the place of x'1x'2...x'N and x1x2...xN, there should be a basis vector from a complete set of basis of the hilbert space for that the system, like
psi(<Ci|)psi*(|Cj>) = <Ci||psi><psi||Cj> (for a pure state).
-- it becomes more comfusing for the mix state, when the ensemble density operator is:
$$\Gamma$$ = sum( pi|psii|><|psii|)

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...xN actually mean? each electron xi is a system? and the basis set for them is ALL SPACE plus spins???
3. the reduced density matrix is writen as
(n choose p)$$\int$$psi(x'1x'2...x'p, xp+1...xN)psi*(x1x2...xp, xp+1...xN) dxp+1...dxN
(this is for order p, you can refer to order 2 reduced density matrix in the above webpage). The ' is gone after xp 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!!!

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