1.1 Many-Electron Wave Functions
5
It is useful to expand general N -electron states in terms of products of orthonormal
one-particle states
|qγ = |q|γ
(1.11)
Here, q and γ = ±
1
2
are spatial and spin quantum numbers, respectively. We shall
also use the notation γ = α, β, established in quantum chemistry. Note that the oneparticle states are themselves products of spatial states, |q, and spin states, |γ,
fulfilling the orthonormal conditions
q|q
= δ qq ,
γ|γ
= δ γγ
(1.12)
The corresponding wave functions are
ϕ q (x) = =x|q
χ γ (σ) = =σ|γ
The spin functions may also be written in spinor form,
χ α =
1
0
,
χ β =
0
1
A common choice of spatial orbitals is the set of molecular orbitals (MOs) generated
by a Hartree–Fock (HF) or self-consistent field (SCF) computation of the N -electron
ground state.
For a more abstract representation, it is helpful to replace the pair of a spatial and
a spin quantum number by single comprehensive spin-orbital quantum number
qγ ≡ q
(1.13)
which, for notational economy, may be labeled by the same latin letter. (Whether
q labels a spin-orbital or merely a spatial orbital will be clear from the respective
context.) Accordingly, spin-orbital wave functions may be written as
ψ q (ξ) = =ξ|q
(1.14)
A simple (not yet antisymmetric) product state of N electrons, in which the ith
electron “occupies” the spin-orbital q i , may be written as
| = |q 1 |q 2 . . . |q N
(1.15)
The corresponding wave function takes on the form (Hartree product)
(ξ 1 , . . . , ξ N ) = ψ q 1 (ξ 1 ) . . . ψ q N (ξ N ) = =ξ N | . . . ξ 1 |q 1 . . . |q N
(1.16)
5
It is useful to expand general N -electron states in terms of products of orthonormal
one-particle states
|qγ = |q|γ
(1.11)
Here, q and γ = ±
1
2
are spatial and spin quantum numbers, respectively. We shall
also use the notation γ = α, β, established in quantum chemistry. Note that the oneparticle states are themselves products of spatial states, |q, and spin states, |γ,
fulfilling the orthonormal conditions
q|q
= δ qq ,
γ|γ
= δ γγ
(1.12)
The corresponding wave functions are
ϕ q (x) = =x|q
χ γ (σ) = =σ|γ
The spin functions may also be written in spinor form,
χ α =
1
0
,
χ β =
0
1
A common choice of spatial orbitals is the set of molecular orbitals (MOs) generated
by a Hartree–Fock (HF) or self-consistent field (SCF) computation of the N -electron
ground state.
For a more abstract representation, it is helpful to replace the pair of a spatial and
a spin quantum number by single comprehensive spin-orbital quantum number
qγ ≡ q
(1.13)
which, for notational economy, may be labeled by the same latin letter. (Whether
q labels a spin-orbital or merely a spatial orbital will be clear from the respective
context.) Accordingly, spin-orbital wave functions may be written as
ψ q (ξ) = =ξ|q
(1.14)
A simple (not yet antisymmetric) product state of N electrons, in which the ith
electron “occupies” the spin-orbital q i , may be written as
| = |q 1 |q 2 . . . |q N
(1.15)
The corresponding wave function takes on the form (Hartree product)
(ξ 1 , . . . , ξ N ) = ψ q 1 (ξ 1 ) . . . ψ q N (ξ N ) = =ξ N | . . . ξ 1 |q 1 . . . |q N
(1.16)