Chemistry Reference
In-Depth Information
The correlation energy amounts to about 1 eV per electron pair and
is quite difficult to account for, as we shall briefly outline in Chapter 8.
We shall introduce, first, the essential lines of the HF theory for closed
shells, with its practical implementation by Hall (1951) and Roothaan
(1951b) yielding the so-called self-consistent-field (SCF)method inside the
Ritz linear-combination-of-atomic-orbitals (LCAO) approach, and, next,
the LCAOmethod devised by H
€
uckel (1931) to deal in a topological way
€
with the
uckel
theory has recently been used by the author to introduce an elementary
model of the chemical bond (Magnasco, 2002, 2003, 2004a, 2005).
p
electrons of conjugated and aromatic hydrocarbons. H
7.1 ELEMENTS OF HARTREE-FOCK THEORY
FOR CLOSED SHELLS
Let
Y ¼jjc
1
c
2
...c
N
jj
hc
i
jc
j
i¼d
ij
S
¼
M
S
¼
0
ð
7
:
2
Þ
be a normalized single determinant wavefunction of doubly occupied
orthonormal spin-orbitals {
c
i
}. We shall introduce consistently the fol-
lowing notation:
N
¼
number of electrons
¼
number of spin-orbitals {
c
i
(x)}, x
¼
rs
;
i
¼
1
;
2
; ...;
N
n
¼
N
=
2 number of doubly occupied spatialMOs {
f
i
(r)}, i
¼
1
;
2
; ...;
n
m
¼
number of basic spatial AOs {
x
m
(r)},
m ¼
1
;
2
; ...;
m
m
nm
¼
n means minimal basis
m
>
n usually includes polarization functions (e.g. 3d, 4f, 5g,
...
on C,
N, O, F atoms and 2p, 3d, 4f,
...
on H atoms, as explained in
Section 7.3).
We then see that the many-electron single determinant wavefunction
Y
has the following properties.
7.1.1 The Fock-Dirac Density Matrix
All physical properties of the system are determined by the fundamental
physical invariant
r
, a one-electron bilinear function in the coordinates x
