5 Basics of Molecular Modeling and Molecular Simulation
203
−
1
2
∇
2
φ i (r) + V ion (r)φ i (r) + U (r)φ i (r) = ε i φ i (r)
(5.2.11)
We can obtain a quasi-eigen equation, which is called the HF equation:
ε i |φ i =
−
1
2
∇ 2 + V ion (r)
φ i (r) +
j
d r
φ
j (r )
2
r − r
φ i (r) −
j
δ σ i σ j
d r
φ ∗
j (r )φ i (r )
r − r
φ j (r)
= ˆ
F i |φ i
(5.2.12)
where
ˆ
F i = ˆ
h i +
N
j=1
j =i
ˆ
J j − ˆ
K j
(5.2.13)
The Fock operator ˆ
F i in the above equations consists of three terms. Because
the Fock operator depends on the solution of wave functions, the HF equation is a
quasi-eigen equation which should be solved self-consistently. The term ˆ
h i consists
of the kinetic energy contribution and the electron-ion potential. The term
N
j=1
j =i
ˆ
J j , or
Hartree term, is simply the electrostatic potential arising from the charge distribution
of N electrons. The exchange term
N
j=1
j =i
ˆ
K j results from our inclusion of the Pauli
principle and the assumed determinant form of the wave function, which has no
classic counterpart.
The SCF technique is used to solve the HF equation self-consistently, which
expands the wave functions by an assigned basis set,
φ
i
=
M
α=1
c αi |χ α
(5.2.14)
which leads to the Roothan-Hall equation in the matrix form:
FC = SCε
(5.2.15)
where F is the Fock matrix, C is the coefficient matrix, S is the overlap matrix of
the basis functions, and ε is the matrix of orbital energies. Accordingly, it should be
solved iteratively since F and C depend on each other.
Assuming a single-determinant form for the wave function, the HF theory neglects
correlation between electrons. As a result, the electrons are subject to an average nonlocal potential arising from the other electrons, which leads to about 2% deviation
of the total energy. Although the deviation sounds small, most of it comes from
the outmost electrons which determines the physical and chemical properties of the
molecular system. Thus, it is usually insufficient to compute at the HF level and the
correlation effect should be included in some ways.
203
−
1
2
∇
2
φ i (r) + V ion (r)φ i (r) + U (r)φ i (r) = ε i φ i (r)
(5.2.11)
We can obtain a quasi-eigen equation, which is called the HF equation:
ε i |φ i =
−
1
2
∇ 2 + V ion (r)
φ i (r) +
j
d r
φ
j (r )
2
r − r
φ i (r) −
j
δ σ i σ j
d r
φ ∗
j (r )φ i (r )
r − r
φ j (r)
= ˆ
F i |φ i
(5.2.12)
where
ˆ
F i = ˆ
h i +
N
j=1
j =i
ˆ
J j − ˆ
K j
(5.2.13)
The Fock operator ˆ
F i in the above equations consists of three terms. Because
the Fock operator depends on the solution of wave functions, the HF equation is a
quasi-eigen equation which should be solved self-consistently. The term ˆ
h i consists
of the kinetic energy contribution and the electron-ion potential. The term
N
j=1
j =i
ˆ
J j , or
Hartree term, is simply the electrostatic potential arising from the charge distribution
of N electrons. The exchange term
N
j=1
j =i
ˆ
K j results from our inclusion of the Pauli
principle and the assumed determinant form of the wave function, which has no
classic counterpart.
The SCF technique is used to solve the HF equation self-consistently, which
expands the wave functions by an assigned basis set,
φ
i
=
M
α=1
c αi |χ α
(5.2.14)
which leads to the Roothan-Hall equation in the matrix form:
FC = SCε
(5.2.15)
where F is the Fock matrix, C is the coefficient matrix, S is the overlap matrix of
the basis functions, and ε is the matrix of orbital energies. Accordingly, it should be
solved iteratively since F and C depend on each other.
Assuming a single-determinant form for the wave function, the HF theory neglects
correlation between electrons. As a result, the electrons are subject to an average nonlocal potential arising from the other electrons, which leads to about 2% deviation
of the total energy. Although the deviation sounds small, most of it comes from
the outmost electrons which determines the physical and chemical properties of the
molecular system. Thus, it is usually insufficient to compute at the HF level and the
correlation effect should be included in some ways.