Chapter 1
Systems of Identical Particles
In the first section, we take a look at the basic ingredients in the quantum-theoretical
formulation of many-electron systems: wave functions (states) and operators. For
an in-depth discussion of physical aspects relevant here, the reader is referred to
Chap. XIV in the textbook by A. Messiah [1]. In dealing with many-particle systems,
the handling of matrix elements involving Slater determinants is required. Here, the
essential tool is a set of simple rules, referred to as Slater–Condon rules, which will
be considered in the second section of this chapter.
1.1 Many-Electron Wave Functions
In the following, we will consider a system of N identical (more strictly: indistinguishable) particles, specifically, electrons in an atom or molecule. Each particle is
associated with a set of three spatial coordinates, x, and a spin variable, σ. Accordingly, an N -particle wave function,
= (x 1 σ 1 , . . . , x N σ N )
(1.1)
is a function of the N sets of variables, x k σ k , k = 1, . . . , N . The wave function is a
representation of an underlying abstract state |,
(x 1 σ 1 , . . . , x N σ N ) = =x N σ N | . . . x 1 σ 1 |
(1.2)
Here, |xσ denotes a (formal) one-particle eigenstate of the position and spin operators. For notational brevity, we shall occasionally combine the spatial and spin
variables,
ξ i ≡ x i σ i
(1.3)
© Springer Nature Switzerland AG 2018
J. Schirmer, Many-Body Methods for Atoms, Molecules and Clusters, Lecture
Notes in Chemistry 94, https://doi.org/10.1007/978-3-319-93602-4_1
3
Systems of Identical Particles
In the first section, we take a look at the basic ingredients in the quantum-theoretical
formulation of many-electron systems: wave functions (states) and operators. For
an in-depth discussion of physical aspects relevant here, the reader is referred to
Chap. XIV in the textbook by A. Messiah [1]. In dealing with many-particle systems,
the handling of matrix elements involving Slater determinants is required. Here, the
essential tool is a set of simple rules, referred to as Slater–Condon rules, which will
be considered in the second section of this chapter.
1.1 Many-Electron Wave Functions
In the following, we will consider a system of N identical (more strictly: indistinguishable) particles, specifically, electrons in an atom or molecule. Each particle is
associated with a set of three spatial coordinates, x, and a spin variable, σ. Accordingly, an N -particle wave function,
= (x 1 σ 1 , . . . , x N σ N )
(1.1)
is a function of the N sets of variables, x k σ k , k = 1, . . . , N . The wave function is a
representation of an underlying abstract state |,
(x 1 σ 1 , . . . , x N σ N ) = =x N σ N | . . . x 1 σ 1 |
(1.2)
Here, |xσ denotes a (formal) one-particle eigenstate of the position and spin operators. For notational brevity, we shall occasionally combine the spatial and spin
variables,
ξ i ≡ x i σ i
(1.3)
© Springer Nature Switzerland AG 2018
J. Schirmer, Many-Body Methods for Atoms, Molecules and Clusters, Lecture
Notes in Chemistry 94, https://doi.org/10.1007/978-3-319-93602-4_1
3