104
3 Fundamentals of the Analysis Tools
electron designated as r. This feature is completely different from that of the formal
wavefunction Φ obtained from the Schrödinger equation in Eq. (3.1), since Φ is a
function of the coordinates of all the electrons in the molecule.
In case of a molecule, the orbitals are called MO’s as is well known. We set a
definite function called basis set to represent each MO in terms of their linear combinations. The basis set functions originally ought to consist of atomic orbitals (AO’s)
for general multielectron atoms named by Slater-type orbitals (STO’s) having the
exponential-function type of exp(-ζ r) inherent to each atom in the molecule, where
r stands for the nucleus-electron distance and ζ is the orbital exponent representing
the degree of expansion of the orbital. However, since there is some difficulty to use
STO’s in the actual calculation, they are usually further decomposed into a linear
combination of Gaussian-type orbitals (GTO’s) having the exponential-function type
of exp(−ζ r
2 ) with which the calculation becomes far more accessible. The MO is
thus represented with the linear combination of the STO’s as follows:
ψ i (r) =
p
c ip χ
STO
p
(r)
(3.3)
where each χ
STO
p
is further expanded with χ
GTO
q
as follows:
χ
STO
p
(r) =
q
d pq χ
GTO
q
(r)
(3.4)
the coefficients d pq being pre-determined.
There are several ways of selection of the basis set to represent MO in accordance
with the characteristics and expansion numbers of GTO’s, so that one should be
careful to this selection depending on one’s problem. The details of the basis set will
be summarized in Sect. 3.6. In any case, the coefficients of the basis set, e.g., c ip in
Eq. (3.3), to build up each MO are to be decided by the HF calculation. There is even
a tendency to directly call the selected GTO’s basis set.
3.1.1.2 Ground State
Within the framework of the HF method, the ground-state wavefunction Ψ HF and the
electronic energy E HF of a molecule having N electrons are represented as follows:
Φ(r 1 , r 2 , · · · · · · , r N ) Ψ HF =
1
√
N !
det[ψ 1 (r 1 )ψ 2 (r 2 ) · · · · · · ψ N (r N )] (3.5)
E HF =
N
i=1
H i +
1
2
N
i,j=1
J ij − K ij
(3.6)
3 Fundamentals of the Analysis Tools
electron designated as r. This feature is completely different from that of the formal
wavefunction Φ obtained from the Schrödinger equation in Eq. (3.1), since Φ is a
function of the coordinates of all the electrons in the molecule.
In case of a molecule, the orbitals are called MO’s as is well known. We set a
definite function called basis set to represent each MO in terms of their linear combinations. The basis set functions originally ought to consist of atomic orbitals (AO’s)
for general multielectron atoms named by Slater-type orbitals (STO’s) having the
exponential-function type of exp(-ζ r) inherent to each atom in the molecule, where
r stands for the nucleus-electron distance and ζ is the orbital exponent representing
the degree of expansion of the orbital. However, since there is some difficulty to use
STO’s in the actual calculation, they are usually further decomposed into a linear
combination of Gaussian-type orbitals (GTO’s) having the exponential-function type
of exp(−ζ r
2 ) with which the calculation becomes far more accessible. The MO is
thus represented with the linear combination of the STO’s as follows:
ψ i (r) =
p
c ip χ
STO
p
(r)
(3.3)
where each χ
STO
p
is further expanded with χ
GTO
q
as follows:
χ
STO
p
(r) =
q
d pq χ
GTO
q
(r)
(3.4)
the coefficients d pq being pre-determined.
There are several ways of selection of the basis set to represent MO in accordance
with the characteristics and expansion numbers of GTO’s, so that one should be
careful to this selection depending on one’s problem. The details of the basis set will
be summarized in Sect. 3.6. In any case, the coefficients of the basis set, e.g., c ip in
Eq. (3.3), to build up each MO are to be decided by the HF calculation. There is even
a tendency to directly call the selected GTO’s basis set.
3.1.1.2 Ground State
Within the framework of the HF method, the ground-state wavefunction Ψ HF and the
electronic energy E HF of a molecule having N electrons are represented as follows:
Φ(r 1 , r 2 , · · · · · · , r N ) Ψ HF =
1
√
N !
det[ψ 1 (r 1 )ψ 2 (r 2 ) · · · · · · ψ N (r N )] (3.5)
E HF =
N
i=1
H i +
1
2
N
i,j=1
J ij − K ij
(3.6)