2.4 Hartee–Fock (HF) Method
11
A monoelectronic wavefunction (called orbital) is attributed to each electron.
Molecular orbital (MO) theory assumes that the atomic orbitals of the atoms in the
molecule combine to produce molecular orbitals that are delocalized over the entire
molecule.
Generally, a basis set expansion technique is used. The many-electron wavefunction is written as a product of orthonormal one-electron functions called molecular
orbitals (MOs)
Ψ = ϕ 1 ϕ 2 · · · ϕ n
(2.7)
However, the electrons being Fermions, this function, called Hartree product,
should be antisymmetric upon exchange of two electrons. The simplest antisymmetric
function is a determinant which may be written as follows for a closed-shell system,
introducing the spin functions of the electron: α for spin +1/2 or β for spin −1/2
(r ) =
1
√
n!
ϕ 1 (r 1 )α(1) ϕ 1 (r 1 )β(1) ϕ 2 (r 1 )α(1) ϕ 2 (r 1 )β(1) · · · ϕ n / 2 (r 1 )α(1) ϕ n / 2 (r 1 )β(1)
ϕ 1 (r 2 )α(2) ϕ 1 (r 2 )β(2) ϕ 2 (r 2 )α(2) ϕ 2 (r 2 )β(2) · · · ϕ n / 2 (r 2 )α(2) ϕ n / 2 (r 2 )β(2)
. . .
. . .
. . .
. . .
ϕ 1 (r n )α(n) ϕ 1 (r n )β(n) ϕ 2 (r n )α(n) ϕ 2 (r n )β(n) · · · ϕ n / 2 (r n )α(n) ϕ n / 2 (r n )β(n)
(2.8)
Each of these MOs is expressed as a linear combination of basis functions
ϕ i =
k
c ik χ k
(2.9)
The unknown c ik coefficients are determined by the variational method, i.e., they
are chosen as to minimize the energy (Roothaan 1951)
E = min
Ψ
∗ HΨ dτ
Ψ ∗ Ψ dτ
≥ E exact
(2.10)
It is tempting to use Slater-type atomic orbitals (STOs) as basis functions because
they provide a reasonable representation of atomic orbitals:
χ
STO
abc = N x
a y
b z
c r
n−1 e
−ςr
(2.11)
The integers a, b, c control the angular momentum, n plays the role of principal
quantum number, ς controls the width of the orbital, a large ς gives a tight function
and a small ς a diffuse function.
However, they are now rarely used because it is not convenient to calculate the
multicentric integrals
11
A monoelectronic wavefunction (called orbital) is attributed to each electron.
Molecular orbital (MO) theory assumes that the atomic orbitals of the atoms in the
molecule combine to produce molecular orbitals that are delocalized over the entire
molecule.
Generally, a basis set expansion technique is used. The many-electron wavefunction is written as a product of orthonormal one-electron functions called molecular
orbitals (MOs)
Ψ = ϕ 1 ϕ 2 · · · ϕ n
(2.7)
However, the electrons being Fermions, this function, called Hartree product,
should be antisymmetric upon exchange of two electrons. The simplest antisymmetric
function is a determinant which may be written as follows for a closed-shell system,
introducing the spin functions of the electron: α for spin +1/2 or β for spin −1/2
(r ) =
1
√
n!
ϕ 1 (r 1 )α(1) ϕ 1 (r 1 )β(1) ϕ 2 (r 1 )α(1) ϕ 2 (r 1 )β(1) · · · ϕ n / 2 (r 1 )α(1) ϕ n / 2 (r 1 )β(1)
ϕ 1 (r 2 )α(2) ϕ 1 (r 2 )β(2) ϕ 2 (r 2 )α(2) ϕ 2 (r 2 )β(2) · · · ϕ n / 2 (r 2 )α(2) ϕ n / 2 (r 2 )β(2)
. . .
. . .
. . .
. . .
ϕ 1 (r n )α(n) ϕ 1 (r n )β(n) ϕ 2 (r n )α(n) ϕ 2 (r n )β(n) · · · ϕ n / 2 (r n )α(n) ϕ n / 2 (r n )β(n)
(2.8)
Each of these MOs is expressed as a linear combination of basis functions
ϕ i =
k
c ik χ k
(2.9)
The unknown c ik coefficients are determined by the variational method, i.e., they
are chosen as to minimize the energy (Roothaan 1951)
E = min
Ψ
∗ HΨ dτ
Ψ ∗ Ψ dτ
≥ E exact
(2.10)
It is tempting to use Slater-type atomic orbitals (STOs) as basis functions because
they provide a reasonable representation of atomic orbitals:
χ
STO
abc = N x
a y
b z
c r
n−1 e
−ςr
(2.11)
The integers a, b, c control the angular momentum, n plays the role of principal
quantum number, ς controls the width of the orbital, a large ς gives a tight function
and a small ς a diffuse function.
However, they are now rarely used because it is not convenient to calculate the
multicentric integrals