Chemistry Reference
In-Depth Information
the conditions of the Pauli exclusion principle. The Slater determinant
may be generalized to a system of
N
electrons easily; it is the determinant of
an
N N
matrix of single-electron spin orbitals. By using a Slater deter-
minant, we are ensuring that our method for solving the Schr¨dinger
equation will include exchange. Unfortunately, this is not the only kind of
electron correlation that we need to describe in order to arrive at good compu-
tational accuracy.
The description above may seem a little unhelpful since we know that in any
interesting system the electrons interact with one another. The many different
wave-function-based approaches to solving the Schr¨dinger equation differ in
how these interactions are approximated. To understand the types of approxi-
mations that can be used, it is worth looking at the simplest approach, the
Hartree-Fock method, in some detail. There are also many similarities
between Hartree-Fock calculations and the DFT calculations we have
described in the previous sections, so understanding this method is a useful
way to view these ideas from a slightly different perspective.
In a Hartree-Fock (HF) calculation, we fix the positions of the atomic
nuclei and aim to determine the wave function of
N
-interacting electrons.
The first part of describing an HF calculation is to define what equations are
solved. The Schr¨dinger equation for each electron is written as
x
j
(
x
)
¼ E
j
x
j
(
x
)
h
2
2
m
r
2
þ V
(
r
)
þ V
H
(
r
)
:
:
(1
14)
The third term on the left-hand side is the same Hartree potential we saw in
Eq. (1.5):
V
H
(
r
)
¼ e
2
ð
n
(
r
0
)
jr
d
3
r
0
:
(1
:
15)
r
0
j
In plain language, this means that a single electron “feels” the effect of other
electrons only as an average, rather than feeling the instantaneous repulsive
forces generated as electrons become close in space. If you compare
Eq. (1.14) with the Kohn-Sham equations, Eq. (1.5), you will notice that
the only difference between the two sets of equations is the additional
exchange-correlation potential that appears in the Kohn-Sham equations.
To complete our description of the HF method, we have to define how the
solutions of the single-electron equations above are expressed and how these
solutions are combined to give the
N
-electron wave function. The HF approach
assumes that the complete wave function can be approximated using a single
Slater determinant. This means that the
N
lowest energy spin orbitals of the
