1.1 Slater Determinants and Slater–Condon Rules
5
Similar Coulomb integrals J are obtained for other combinations of electron coordinates. The next step is to see what integrals are obtained for γ = 1 interchanging
electrons 1 and 2:
−−φ a (1)φ b (2)φ c (3)...φ ω (N )|
1
r 12
|φ b (1)φ a (2)φ c (3)...φ ω (N )
=−−φ a (1)φ b (2)|
1
r 12
|φ b (1)φ a (2)φ c (3)...φ ω (N )|φ c (3)...φ ω (N )
=−−φ a φ b |
1
r 12
|φ b φ a =−K ab
(1.12)
This integral is known as the exchange integral and usually written as K ab . Other
combinations of permutations and electron coordinates lead to similar K ′ s but higherorder permutations will always result in zero contributions due to the orthogonality.
Hence, the terms can be collected and the expression given in the top row of Table 1.1
emerges.
The evaluation of the interaction matrix elements between Slater determinants
with different occupations follows the same mechanics and can be derived as a
useful exercise by the reader.
1.2 Generation of Many Electron Spin Functions
In a non-relativistic setting the N -electron wave function Ψ can be chosen to be also
an eigenfunction of ˆ
S 2 and one of its components, we denote this component ˆ
S z .
ˆ
S
2 Ψ S,M S = S(S + 1)Ψ S,M S
(1.13a)
ˆ
S z Ψ S,M S = M S Ψ S,M S
(1.13b)
with S the total spin quantum number and the magnetic spin quantum number M S
running from −S to S in steps of 1.
1.3 Give the degeneracy of Ψ S,M S in terms of S assuming that spin-orbit
coupling (see Sect. 2.1) can be neglected.
Before looking in more detail to the N -electron wave functions, we will first shortly
summarize the most important aspects of the spin part of a one-electron wave
function. We will follow the common practice to use lower case symbols when
dealing with one-particle wave functions and uppercase for many-particle systems.
5
Similar Coulomb integrals J are obtained for other combinations of electron coordinates. The next step is to see what integrals are obtained for γ = 1 interchanging
electrons 1 and 2:
−−φ a (1)φ b (2)φ c (3)...φ ω (N )|
1
r 12
|φ b (1)φ a (2)φ c (3)...φ ω (N )
=−−φ a (1)φ b (2)|
1
r 12
|φ b (1)φ a (2)φ c (3)...φ ω (N )|φ c (3)...φ ω (N )
=−−φ a φ b |
1
r 12
|φ b φ a =−K ab
(1.12)
This integral is known as the exchange integral and usually written as K ab . Other
combinations of permutations and electron coordinates lead to similar K ′ s but higherorder permutations will always result in zero contributions due to the orthogonality.
Hence, the terms can be collected and the expression given in the top row of Table 1.1
emerges.
The evaluation of the interaction matrix elements between Slater determinants
with different occupations follows the same mechanics and can be derived as a
useful exercise by the reader.
1.2 Generation of Many Electron Spin Functions
In a non-relativistic setting the N -electron wave function Ψ can be chosen to be also
an eigenfunction of ˆ
S 2 and one of its components, we denote this component ˆ
S z .
ˆ
S
2 Ψ S,M S = S(S + 1)Ψ S,M S
(1.13a)
ˆ
S z Ψ S,M S = M S Ψ S,M S
(1.13b)
with S the total spin quantum number and the magnetic spin quantum number M S
running from −S to S in steps of 1.
1.3 Give the degeneracy of Ψ S,M S in terms of S assuming that spin-orbit
coupling (see Sect. 2.1) can be neglected.
Before looking in more detail to the N -electron wave functions, we will first shortly
summarize the most important aspects of the spin part of a one-electron wave
function. We will follow the common practice to use lower case symbols when
dealing with one-particle wave functions and uppercase for many-particle systems.