Biomedical Engineering Reference
In-Depth Information
16.1 Roothaan's 1951 Landmark Paper
Our next landmark is the paper by Roothaan (1951). In those days papers did not always
have a synopsis. We can learn a great deal from the first paragraph of the introduction, as
follows.
For dealing with the problems of molecular quantum mechanics, two methods of approx-
imation have been developed which are capable of handling many-electron systems. The
Heitler-London-Pauling-Slater or valence bond (VB) method originated from a chemical
point of view. The atoms are considered as the material from which the molecule is built;
accordingly, the molecular wave function is constructed from the wave functions of the indi-
vidual atoms. The Hund-Mulliken or molecular orbital (MO) method is an extension of the
Bohr theory of electron configurations from atoms to molecules. Each electron is assigned to
a one-electron wave function or molecular orbital, which is the quantum mechanical analog
of an electron orbit...
It is the purpose of this paper to build a rigorous mathematical framework for the MO
method.
Within the Born-Oppenheimer approximation we consider a molecule as a set of
N
point
charges of magnitudes
eZ
1
,
eZ
2
,...,
eZ
N
at fixed position vectors
R
1
,
R
2
,...,
R
N
. Their
mutual potential energy is
N
−
1
N
e
2
4πε
0
Z
i
Z
j
R
ij
U
nuc
=
(16.1)
i
=
1
j
=
i
+
1
Assume for the moment that we have chosen
n
basis functions, which could be the
Slater orbitals from Chapter 14. Basis functions are usually real quantities in the math-
ematical sense, but complex basis functions have to be used in difficult cases such as
when we have to deal with molecular magnetic properties. I will assume that we have
chosen a set of real basis functions written χ
1
(
r
), χ
2
(
r
),...,χ
n
(
r
). The HF-LCAO method
seeks to express each orbital ψ
1
(
r
), ψ
2
(
r
),...,ψ
M
(
r)
as a linear combination of the basis
functions
ψ
i
=
c
i
,1
χ
1
+
c
i
,2
χ
2
+···+
c
i
,
n
χ
n
(16.2)
and the process gives
n
LCAO-MO orbitals in total.
Roothaan's original treatment only applies to molecular electronic states where each
HF-LCAO is doubly occupied (so-called 'closed shell states'). This covers the case of the
vast majority of organic molecules in their electronic ground state, and we think of the
electronic configuration as
(ψ
1
)
2
(ψ
2
)
2
...(ψ
M
)
2
as shown in Figure 16.1. Also, the treatment is restricted to the lowest energy state of each
allowed symmetry type; since most organic molecules have no symmetry, this means that
only the electronic ground state can be treated. The HF-LCAO wavefunction is a single
Slater determinant, and we can assume without any loss of generality that the HF-LCAO
orbitals are normalized and orthogonal. Each orbital is doubly occupied and there are
therefore
m
=
2
M
electrons.
