文档介绍：Notes on the Density Matrix Theory
Xihua Chen
Department of Chemistry, New York University,
New York, NY 10003, USA
Summer, 2005
Contents
1 Density Operators 2
Electron density . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
The   th-order density matrix . . . . . . . . . . . . . . . . . . . . 3
Density operator . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
Ensemble density operator . . . . . . . . . . . . . . . . . . . . . 6
Reduced density matrix . . . . . . . . . . . . . . . . . . . . . . . 8
The second-order density matrix . . . . . . . . . . . . . . . . . . 9
The first-order density matrix . . . . . . . . . . . . . . . . . . . . 11
  -representability problem . . . . . . . . . . . . . . . . . . . . . 13
2 Spinless Density Matrices 14
Second-order spinless density matrix . . . . . . . . . . . . . . . . 14
First-order spinless density matrix . . . . . . . . . . . . . . . . . 15
Energy with spinless density matrices . . . . . . . . . . . . . . . 17
Exchange-correlation hole . . . . . . . . . . . . . . . . . . . . . 18
1
Some basics of elementary quantum mechanics, in particular, the concepts of
electron density and density matrix that are fundamental to the Density Functional
Theory (DFT) methods are summarized.
1 Density Operators
Electron density
 
Let’s consider a molecular system which posed of ¡ atomic nuclei and
electrons. The geometrical structure of the molecule is defined by the positions of
nuclei which are positively charged with ¢¤£¦¥¨§
©
§§¡ . Within the Born-
Oppenheimer approximation and non-relativistically, the stationary state of the
molecule pletely specified by an antisymmetric electronic wave function
"
! #§
$%
(or state function), ¨ , which belongs to a state space con-
sisting of all possible states of the system and depends on the coordinates of nu-
&
¥'§
©( )
§§¡(
clei, ¢