106
3 Fundamentals of the Analysis Tools
H
core Core Hamiltonian operator
H
= −
1
2
∇
2
−
A
Z A
r A
(3.12)
J
i Coulomb repulsion operator
J
i =
ψ i
r
2
|r − r |
dr
(3.13)
K
i Exchange repulsion operator
K
i = ψ i (r)
ψ
∗
i
r
P
rr
|r − r |
dr
(3.14)
P
rr Permutation operator to change the variable r to r
.
In order to build the above operators J
i and K
i , one needs the very MO’s to
enumerate the averaged potential, which means that one first has to assume a set of
certain tentative MO’s to start to solve the HF equation and then the resulting set of
MO’s obtained as the solution is used to build up the new averaged potential. The HF
equation is thus iteratively solved until one eventually gets the stable set of MO’s,
which is called self-consistent field (SCF) solution or simply SCF-MO. The solution
of the HF equation consists of the MO’s ψ i and their energies ε i . Information of the
solved MO’s is explicitly represented by the actual values of coefficients of the basis
set for each MO. Note there can also be obtained the MO’s unoccupied with electrons
which is called unoccupied or virtual MO’s in addition to the occupied MO’s in the
ground state.
In order to express the total electron configuration of a molecule, one has to
construct an antisymmetrized combination of the product of the MO’s occupied with
electrons to satisfy Pauli’s exclusion principle. This combination eventually leads to
a determinant referred to as a Slater determinant or single determinant in Eq. (3.5).
A single determinant thus constructed with the occupied MO’s gives a relatively
simple wavefunction of the molecule in the ground state Ψ 0 describing a single
configuration as shown in Fig. 3.1. Figure 3.1a shows an example of the ground state
of a molecule with an even number of electrons and is often called the closed-shell
structure, whereas Fig. 3.1b with odd numbers of electrons and is called the openshell structure. There are two kinds of representations for the open-shell structure as
shown in Figs. 3.1b, c. In the former, two electrons with α and β spins occupy the
same spatial part of MO and, in the latter, those occupy the different spatial parts.
In this sense, the former is called restricted open HF (ROHF) scheme (Roothaan
1960) and the latter is unrestricted HF (UHF) scheme (Pople and Nesbet 1954). The
unrestricted model is also expressed as different orbitals for different spins (DODS).
Although UHF method inevitably includes the spin contamination from the quantum
3 Fundamentals of the Analysis Tools
H
core Core Hamiltonian operator
H
= −
1
2
∇
2
−
A
Z A
r A
(3.12)
J
i Coulomb repulsion operator
J
i =
ψ i
r
2
|r − r |
dr
(3.13)
K
i Exchange repulsion operator
K
i = ψ i (r)
ψ
∗
i
r
P
rr
|r − r |
dr
(3.14)
P
rr Permutation operator to change the variable r to r
.
In order to build the above operators J
i and K
i , one needs the very MO’s to
enumerate the averaged potential, which means that one first has to assume a set of
certain tentative MO’s to start to solve the HF equation and then the resulting set of
MO’s obtained as the solution is used to build up the new averaged potential. The HF
equation is thus iteratively solved until one eventually gets the stable set of MO’s,
which is called self-consistent field (SCF) solution or simply SCF-MO. The solution
of the HF equation consists of the MO’s ψ i and their energies ε i . Information of the
solved MO’s is explicitly represented by the actual values of coefficients of the basis
set for each MO. Note there can also be obtained the MO’s unoccupied with electrons
which is called unoccupied or virtual MO’s in addition to the occupied MO’s in the
ground state.
In order to express the total electron configuration of a molecule, one has to
construct an antisymmetrized combination of the product of the MO’s occupied with
electrons to satisfy Pauli’s exclusion principle. This combination eventually leads to
a determinant referred to as a Slater determinant or single determinant in Eq. (3.5).
A single determinant thus constructed with the occupied MO’s gives a relatively
simple wavefunction of the molecule in the ground state Ψ 0 describing a single
configuration as shown in Fig. 3.1. Figure 3.1a shows an example of the ground state
of a molecule with an even number of electrons and is often called the closed-shell
structure, whereas Fig. 3.1b with odd numbers of electrons and is called the openshell structure. There are two kinds of representations for the open-shell structure as
shown in Figs. 3.1b, c. In the former, two electrons with α and β spins occupy the
same spatial part of MO and, in the latter, those occupy the different spatial parts.
In this sense, the former is called restricted open HF (ROHF) scheme (Roothaan
1960) and the latter is unrestricted HF (UHF) scheme (Pople and Nesbet 1954). The
unrestricted model is also expressed as different orbitals for different spins (DODS).
Although UHF method inevitably includes the spin contamination from the quantum