-- 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:(adsbygoogle = window.adsbygoogle || []).push({});

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:

[tex]\gamma[/tex](x'_{1}x'_{2}...x'_{N}, x_{1}x_{2}...x_{N})=psi(x'_{1}x'_{2}...x'_{N})psi*(x_{1}x_{2}...x_{N}) but in my mind, in the place of x'_{1}x'_{2}...x'_{N}and x_{1}x_{2}...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:

[tex]\Gamma[/tex] = sum( p_{i}|psi_{i}|><|psi_{i}|)

questions:

1. for the pure states in ensemble operator [tex]\Gamma[/tex], 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 [tex]\Gamma[/tex] with the same basis....

2. what dose x_{1}x_{2}...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 writen as

(n choose p)[tex]\int[/tex]psi(x'_{1}x'_{2}...x'_{p}, x_{p+1}...x_{N})psi*(x_{1}x_{2}...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!!!

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# Density Matrix in that DFT bible book

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