36
2 Experimental and Computational Methods
theory thus introduces so-called average electron correlation.
2 To see this better, it
is worth considering the Hartree-Fock potential in further detail,
V H F ( x 1 ) =
N
j
(J
j ( x 1 ) − K
j ( x 1 ))
(2.9)
The term J
, the Coulomb operator, is defined as
J
j ( x 1 ) =
χ j ( x 2 )
2 1
r 12
d
x 2
(2.10)
This operator describes the repulsive potential experienced by an electron at position
x 1 due to the average charge distribution of another electron in a second spin
orbital χ j .
3
The second term in Eq. 2.9, K
j describes the exchange contribution of the
HF potential. This term has no classical interpretation, and is due entirely to the
interaction of spin orbitals,
ˆ
K j ( x 1 )χ i ( x 1 ) =
χ
∗
j ( x 2 )
1
r 12
χ i ( x 2 )d
x 2 χ j ( x 1 )
(2.11)
This term only exists for electrons of like spin, and results from the antisymmetry
of the Slater determinant. Hence, this term is computed without approximation in
HF theory.
HF theory therefore offers an approach for approximating the solution of the
Schrödinger equation for an N-electron system, by assuming N non-interacting
particles that move in an effective potential, V H F ,
ˆ
H H F S D = E
0
H F S D =
N
i
ˆ
f i S D =
N
i
ε i S D
(2.12)
Despite its simplifications, HF is able to reproduce overall system energies to
within ca. 10% of the most accurate computational approaches (i.e. couple cluster
methods). However, the approximations made by neglect of correlation can lead to
issues surrounding calculation of system properties. Numerous post-HF methods
2 Electron correlation within HF is taken as the difference in energy between the real system and the
HF-derived energy, E H F
c
= E 0 − E H F . Under normal bonding conditions, this difference is mainly
due to the short-range instantaneous repulsion that occur between electrons (dynamic correlation).
In HF, this potential is treated only as an average, and hence is underestimated. Typically, correlation
energies are quite small (ca. 0.04 E h in H 2 ).
3 Note that the Borne interpretation of the wavefunction states that
χ j ( x 2 )
2 d
x 2 describes the
probability of finding the electron within volume d
x 2 .
2 Experimental and Computational Methods
theory thus introduces so-called average electron correlation.
2 To see this better, it
is worth considering the Hartree-Fock potential in further detail,
V H F ( x 1 ) =
N
j
(J
j ( x 1 ) − K
j ( x 1 ))
(2.9)
The term J
, the Coulomb operator, is defined as
J
j ( x 1 ) =
χ j ( x 2 )
2 1
r 12
d
x 2
(2.10)
This operator describes the repulsive potential experienced by an electron at position
x 1 due to the average charge distribution of another electron in a second spin
orbital χ j .
3
The second term in Eq. 2.9, K
j describes the exchange contribution of the
HF potential. This term has no classical interpretation, and is due entirely to the
interaction of spin orbitals,
ˆ
K j ( x 1 )χ i ( x 1 ) =
χ
∗
j ( x 2 )
1
r 12
χ i ( x 2 )d
x 2 χ j ( x 1 )
(2.11)
This term only exists for electrons of like spin, and results from the antisymmetry
of the Slater determinant. Hence, this term is computed without approximation in
HF theory.
HF theory therefore offers an approach for approximating the solution of the
Schrödinger equation for an N-electron system, by assuming N non-interacting
particles that move in an effective potential, V H F ,
ˆ
H H F S D = E
0
H F S D =
N
i
ˆ
f i S D =
N
i
ε i S D
(2.12)
Despite its simplifications, HF is able to reproduce overall system energies to
within ca. 10% of the most accurate computational approaches (i.e. couple cluster
methods). However, the approximations made by neglect of correlation can lead to
issues surrounding calculation of system properties. Numerous post-HF methods
2 Electron correlation within HF is taken as the difference in energy between the real system and the
HF-derived energy, E H F
c
= E 0 − E H F . Under normal bonding conditions, this difference is mainly
due to the short-range instantaneous repulsion that occur between electrons (dynamic correlation).
In HF, this potential is treated only as an average, and hence is underestimated. Typically, correlation
energies are quite small (ca. 0.04 E h in H 2 ).
3 Note that the Borne interpretation of the wavefunction states that
χ j ( x 2 )
2 d
x 2 describes the
probability of finding the electron within volume d
x 2 .