Chemistry Reference
In-Depth Information
If we consider only the nuclei of a molecule in fixed positions and just one
electron in the field of the nuclei, the eigenfunctions of (3) are called molecular
orbitals (MOs). In quantum chemistry an MO is often expanded in a sum of atomic
orbitals, for example, the familiar hydrogenic atomic orbitals,
f
A
0
1
s
,2
s
,2
p
,3
s
,3
p
,
3
d
, etc.:
X
N
c
el;k
¼
a
i;k
f
A
0
:
(5)
i¼
1
The Hartree-Fock (HF) method is widely used in electronic structure calcula-
tions, which is based on the following assumptions: (1) Born-Oppenheimer approx-
imation, (2) the many-electron Hamiltonian is replaced with an effective one-electron
Hamiltonian which acts on orbitals (one-electron wave functions), (3) the Coulomb
repulsion between electrons is represented in an averaged way.
In practice the HF method is applied in the form of so-called Roothaan equations
(see [
4
]):
FC
k
¼ e
k
SC
k
;
(6)
where F is the Fock matrix and S contains the overlap elements between the basis
set functions. The term C is a square matrix of the basis set expansion coefficients,
and
represents a diagonal matrix of the orbital energies.
The HF iteration scheme is given in Fig.
3a
. The HF wave function |HF
e
is thus
the variationally best one - determinant approximation to the electronic ground
i
Choose basis set (s)
Choose a basis set
Choose a molecular geometry
Choose a molecular geometry
Compute and store all overlap,
one-electron, and two-electron integrals
Compute and store all overlap
and one-electron integrals
(0)
Guess initial density matrix
P
(0)
Guess initial density matrix
P
Construct and solve Hartree-Fock
secular equation
Construct and solve Kohn-Sham
secular equation
Construct density matrix
from occupied MO's
Construct density matrix from
occupied Kohn-Sham MO's
(
n
)
sufficiently similar to old density
matrix
(
n
)
sufficiently similar to old density
matrix
P
Is new density matrix
P
Is new density matrix
P
(
n
-1)
?
(
n
-1)
?
P
yes
no
no
yes
(
n
-1)
with
(
n
)
Replace
P
P
Output data
Output data
(
n
-1)
with
P
(
n
)
Replace
P
Fig. 3 Hartree-Fock and Kohn-Sham SCF scheme
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