2.1 Computational Methods
35
approximates the wavefunction by an antisymmetrised product
1 of N one-electron
wavefunctions, χ i ( x i ), known as spin orbitals, their exact description depends on the
basis set used (Sect. 2.1.3). This product is known as the Slater determinant [4], S D
≈ S D =
1
√
N !
χ 1 ( x 1 ) χ 2 ( x 1 ) . . . χ N ( x 1 )
χ 1 ( x 2 ) χ 2 ( x 2 ) . . . χ N ( x 2 )
. . .
. . .
. . .
. . .
χ 1 ( x N ) χ 2 ( x N ) . . . χ N ( x N )
(2.5)
where the pre-factor ensures the normalization condition that
||
2 dτ = 1. Due to
the use of the antisymmetrised wavefunction in this way, HF theory is said to include
exact exchange. For example, the Slater determinant of a two-electron system can be
formulated according to
( x 1 ,
x 2 ) =
1
√
(2)
(χ 1 ( x 1 )χ 2 ( x 2 ) − χ 2 ( x 1 )χ 1 ( x 2 ))
(2.6)
Hence, the two electrons are indistinguishable, and cannot exist in the same spinorbital.
Derivation of the HF equations are outside the scope of this work, but introductory
texts can be found in Reference [5]. However, it is useful to briefly introduce the
results here, which lead to definitions of electron exchange and electron correlation.
The energy of the one-electron wavefunction can be found using the Fock operator,
ˆ
f ; one electron wavefunctions are found to reduce to,
ˆ
f χ i = i χ i
(2.7)
where i describes the orbital energy of spin-orbital i and
ˆ
f i = −
1
2
∇
2
i −
M
i=1
Z A
r i A
+ V H F (i)
(2.8)
The Fock operator hence represents the kinetic energy and the electrostatic interaction between nuclei and electrons. The third term in Eq. 2.8, V H F (i) is known as
the Hartree-Fock potential. This term describes the average repulsive potential felt by
electron i due to interaction with the other N − 1 electrons. Hence, the two-electron
repulsive operator (term 3 of Eq. 2.3) is reduced to a simple electron operator V H F (i),
in which the electron-electron repulsion is only accounted for in an average way. HF
1 Antisymmetrisation is the result of Pauli’s exclusion principle, which dictates that interchange of any two fermions (e.g. electrons) must result in a change in sign. Hence
x 1 ,
x 2 , . . .
x i ,
x j , . . . ,
x N
= −
x 1 ,
x 2 , . . .
x j ,
x i , . . . ,
x N
.
35
approximates the wavefunction by an antisymmetrised product
1 of N one-electron
wavefunctions, χ i ( x i ), known as spin orbitals, their exact description depends on the
basis set used (Sect. 2.1.3). This product is known as the Slater determinant [4], S D
≈ S D =
1
√
N !
χ 1 ( x 1 ) χ 2 ( x 1 ) . . . χ N ( x 1 )
χ 1 ( x 2 ) χ 2 ( x 2 ) . . . χ N ( x 2 )
. . .
. . .
. . .
. . .
χ 1 ( x N ) χ 2 ( x N ) . . . χ N ( x N )
(2.5)
where the pre-factor ensures the normalization condition that
||
2 dτ = 1. Due to
the use of the antisymmetrised wavefunction in this way, HF theory is said to include
exact exchange. For example, the Slater determinant of a two-electron system can be
formulated according to
( x 1 ,
x 2 ) =
1
√
(2)
(χ 1 ( x 1 )χ 2 ( x 2 ) − χ 2 ( x 1 )χ 1 ( x 2 ))
(2.6)
Hence, the two electrons are indistinguishable, and cannot exist in the same spinorbital.
Derivation of the HF equations are outside the scope of this work, but introductory
texts can be found in Reference [5]. However, it is useful to briefly introduce the
results here, which lead to definitions of electron exchange and electron correlation.
The energy of the one-electron wavefunction can be found using the Fock operator,
ˆ
f ; one electron wavefunctions are found to reduce to,
ˆ
f χ i = i χ i
(2.7)
where i describes the orbital energy of spin-orbital i and
ˆ
f i = −
1
2
∇
2
i −
M
i=1
Z A
r i A
+ V H F (i)
(2.8)
The Fock operator hence represents the kinetic energy and the electrostatic interaction between nuclei and electrons. The third term in Eq. 2.8, V H F (i) is known as
the Hartree-Fock potential. This term describes the average repulsive potential felt by
electron i due to interaction with the other N − 1 electrons. Hence, the two-electron
repulsive operator (term 3 of Eq. 2.3) is reduced to a simple electron operator V H F (i),
in which the electron-electron repulsion is only accounted for in an average way. HF
1 Antisymmetrisation is the result of Pauli’s exclusion principle, which dictates that interchange of any two fermions (e.g. electrons) must result in a change in sign. Hence
x 1 ,
x 2 , . . .
x i ,
x j , . . . ,
x N
= −
x 1 ,
x 2 , . . .
x j ,
x i , . . . ,
x N
.