24
1 Basic Concepts
Taking into account the orthogonality of the zeroth-order eigenvectors, the only
non-zero terms in the summation are those when i = k and the equation simplifies to
a k E
(0)
k ++ψ
(0)
k | ˆ
V |ψ
(0)
0 =a k E
(0)
0 + 0
(1.69)
from which a k and ψ (1) follow immediately
a k =
ψ
(0)
k | ˆ
V |ψ
(0)
0
E
(0)
0 − E
(0)
k
ψ
(1) =
i =0
ψ
(0)
i | ˆ
V |ψ
(0)
0
E
(0)
0 − E
(0)
i
ψ
(0)
i
(1.70)
The second-order correction to the energy is obtained from the quadratic equation
in λ Eq. 1.62c in a similar fashion as the first-order correction. First, we multiply the
equation by ψ
(0)
0
∗ and then we integrate over the electron coordinates
ψ
(0)
0 | ˆ
H
(0) |ψ
(2)
0 ++ψ
(0)
0 | ˆ
V |ψ
(1)
0
= E
(0)
0 ψ
(0)
0 |ψ
(2)
0 +E
(1)
0 ψ
(0)
0 |ψ
(1)
0 +E
(2)
0 ψ
(0)
0 |ψ
(0)
0
(1.71)
Orthogonality causes the first and second term on the right-hand-side to be zero and
the substitution of ψ
(2)
0 by a linear combination of zeroth-order eigenfunctions leads
to the following equation
j =0
b j ψ
(0)
0 | ˆ
H
(0) |ψ
(0)
j ++ψ
(0)
0 | ˆ
V |ψ
(1)
0 =E
(2)
0
(1.72)
The first left-hand-side term is zero and after substituting Eq. 1.70, the second order
correction to the energy is obtained
E
(2)
0 =
i =0
ψ
(0)
0 | ˆ
V |ψ
(0)
i ψ
(0)
i | ˆ
V |ψ
(0)
0
E
(0)
0 − E
(0)
i
(1.73)
Higher order corrections can be derived in a similar way, but the expressions get
more complicated rapidly.
When excited state energies of the model system (E
(0)
i ) are close to E
(0)
0 ,t h e
corresponding terms in the summations of ψ (1) and E (2) diverge, unless the matrix
elements of these terms are zero. In case of a degenerate ground state of the model
system, say E
(0)
0 = E
(0)
i = ··· = E
(0)
k , we can solve this problem by first diagonalizing the full ˆ
H in the basis of ψ
(0)
0 ,ψ
(0)
i ,...,ψ
(0)
k . This yields linear combinations
of ψ
(0)
0 ,ψ
(0)
i ,...,ψ
(0)
k that are equally valid as zeroth-order wave functions while
the divergence problem is avoided since the diagonalization process made all nondiagonal matrix elements equal to zero. Note that in case of near, but not strict
1 Basic Concepts
Taking into account the orthogonality of the zeroth-order eigenvectors, the only
non-zero terms in the summation are those when i = k and the equation simplifies to
a k E
(0)
k ++ψ
(0)
k | ˆ
V |ψ
(0)
0 =a k E
(0)
0 + 0
(1.69)
from which a k and ψ (1) follow immediately
a k =
ψ
(0)
k | ˆ
V |ψ
(0)
0
E
(0)
0 − E
(0)
k
ψ
(1) =
i =0
ψ
(0)
i | ˆ
V |ψ
(0)
0
E
(0)
0 − E
(0)
i
ψ
(0)
i
(1.70)
The second-order correction to the energy is obtained from the quadratic equation
in λ Eq. 1.62c in a similar fashion as the first-order correction. First, we multiply the
equation by ψ
(0)
0
∗ and then we integrate over the electron coordinates
ψ
(0)
0 | ˆ
H
(0) |ψ
(2)
0 ++ψ
(0)
0 | ˆ
V |ψ
(1)
0
= E
(0)
0 ψ
(0)
0 |ψ
(2)
0 +E
(1)
0 ψ
(0)
0 |ψ
(1)
0 +E
(2)
0 ψ
(0)
0 |ψ
(0)
0
(1.71)
Orthogonality causes the first and second term on the right-hand-side to be zero and
the substitution of ψ
(2)
0 by a linear combination of zeroth-order eigenfunctions leads
to the following equation
j =0
b j ψ
(0)
0 | ˆ
H
(0) |ψ
(0)
j ++ψ
(0)
0 | ˆ
V |ψ
(1)
0 =E
(2)
0
(1.72)
The first left-hand-side term is zero and after substituting Eq. 1.70, the second order
correction to the energy is obtained
E
(2)
0 =
i =0
ψ
(0)
0 | ˆ
V |ψ
(0)
i ψ
(0)
i | ˆ
V |ψ
(0)
0
E
(0)
0 − E
(0)
i
(1.73)
Higher order corrections can be derived in a similar way, but the expressions get
more complicated rapidly.
When excited state energies of the model system (E
(0)
i ) are close to E
(0)
0 ,t h e
corresponding terms in the summations of ψ (1) and E (2) diverge, unless the matrix
elements of these terms are zero. In case of a degenerate ground state of the model
system, say E
(0)
0 = E
(0)
i = ··· = E
(0)
k , we can solve this problem by first diagonalizing the full ˆ
H in the basis of ψ
(0)
0 ,ψ
(0)
i ,...,ψ
(0)
k . This yields linear combinations
of ψ
(0)
0 ,ψ
(0)
i ,...,ψ
(0)
k that are equally valid as zeroth-order wave functions while
the divergence problem is avoided since the diagonalization process made all nondiagonal matrix elements equal to zero. Note that in case of near, but not strict